Sample records for bohr approximation

  1. The Bohr paradox

    NASA Astrophysics Data System (ADS)

    Crease, Robert P.

    2008-05-01

    In his book Niels Bohr's Times, the physicist Abraham Pais captures a paradox in his subject's legacy by quoting three conflicting assessments. Pais cites Max Born, of the first generation of quantum physics, and Werner Heisenberg, of the second, as saying that Bohr had a greater influence on physics and physicists than any other scientist. Yet Pais also reports a distinguished younger colleague asking with puzzlement and scepticism "What did Bohr really do?".

  2. Niels Bohr as philosopher of experiment: Does decoherence theory challenge Bohr's doctrine of classical concepts?

    NASA Astrophysics Data System (ADS)

    Camilleri, Kristian; Schlosshauer, Maximilian

    2015-02-01

    Niels Bohr's doctrine of the primacy of "classical concepts" is arguably his most criticized and misunderstood view. We present a new, careful historical analysis that makes clear that Bohr's doctrine was primarily an epistemological thesis, derived from his understanding of the functional role of experiment. A hitherto largely overlooked disagreement between Bohr and Heisenberg about the movability of the "cut" between measuring apparatus and observed quantum system supports the view that, for Bohr, such a cut did not originate in dynamical (ontological) considerations, but rather in functional (epistemological) considerations. As such, both the motivation and the target of Bohr's doctrine of classical concepts are of a fundamentally different nature than what is understood as the dynamical problem of the quantum-to-classical transition. Our analysis suggests that, contrary to claims often found in the literature, Bohr's doctrine is not, and cannot be, at odds with proposed solutions to the dynamical problem of the quantum-classical transition that were pursued by several of Bohr's followers and culminated in the development of decoherence theory.

  3. "Bohr's Atomic Model."

    ERIC Educational Resources Information Center

    Willden, Jeff

    2001-01-01

    "Bohr's Atomic Model" is a small interactive multimedia program that introduces the viewer to a simplified model of the atom. This interactive simulation lets students build an atom using an atomic construction set. The underlying design methodology for "Bohr's Atomic Model" is model-centered instruction, which means the central model of the…

  4. Timing and Impact of Bohr's Trilogy

    NASA Astrophysics Data System (ADS)

    Jeong, Yeuncheol; Wang, Lei; Yin, Ming; Datta, Timir

    2014-03-01

    In their article- Genesis of the Bohr Atom Heilbron and Kuhn asked - what suddenly turned his [Bohr's] attention, to atom models during June 1912- they were absolutely right; during the short period in question Bohr had made an unexpected change in his research activity, he has found a new interest ``atom'' and would soon produce a spectacularly successful theory about it in his now famous trilogy papers in the Phil Mag (1913). We researched the trilogy papers, Bohr`s memorandum, his own correspondence from that time in question and activities by Moseley (Manchester), Henry and Lawrence Bragg. Our work suggests that Bohr, also at Manchester that summer, was likely to have been inspired by Laue's sensational discovery in April 1912, of X-ray interference from atoms in crystals. The three trilogy papers include sixty five distinct (numbered) references from thirty one authors. The publication dates of the cited works range from 1896 to 1913. Bohr showed an extraordinary skill in navigating thru the most important and up-to date works. Eleven of the cited authors (Bohr included, but not John Nicholson) were recognized by ten Noble prizes, six in physics and four in chemistry.

  5. From correspondence to complementarity: The emergence of Bohr's Copenhagen interpretation of quantum mechanics

    NASA Astrophysics Data System (ADS)

    Tanona, Scott Daniel

    I develop a new analysis of Niels Bohr's Copenhagen interpretation of quantum mechanics by examining the development of his views from his earlier use of the correspondence principle in the so-called 'old quantum theory' to his articulation of the idea of complementarity in the context of the novel mathematical formalism of quantum mechanics. I argue that Bohr was motivated not by controversial and perhaps dispensable epistemological ideas---positivism or neo-Kantianism, for example---but by his own unique perspective on the difficulties of creating a new working physics of the internal structure of the atom. Bohr's use of the correspondence principle in the old quantum theory was associated with an empirical methodology that used this principle as an epistemological bridge to connect empirical phenomena with quantum models. The application of the correspondence principle required that one determine the validity of the idealizations and approximations necessary for the judicious use of classical physics within quantum theory. Bohr's interpretation of the new quantum mechanics then focused on the largely unexamined ways in which the developing abstract mathematical formalism is given empirical content by precisely this process of approximation. Significant consistency between his later interpretive framework and his forms of argument with the correspondence principle indicate that complementarity is best understood as a relationship among the various approximations and idealizations that must be made when one connects otherwise meaningless quantum mechanical symbols to empirical situations or 'experimental arrangements' described using concepts from classical physics. We discover that this relationship is unavoidable not through any sort of a priori analysis of the priority of classical concepts, but because quantum mechanics incorporates the correspondence approach in the way in which it represents quantum properties with matrices of transition probabilities, the

  6. Rutherford-Bohr atom

    NASA Astrophysics Data System (ADS)

    Heilbron, J. L.

    1981-03-01

    Bohr used to introduce his attempts to explain clearly the principles of the quantum theory of the atom with an historical sketch, beginning invariably with the nuclear model proposed by Rutherford. That was sound pedagogy but bad history. The Rutherford-Bohr atom stands in the middle of a line of work initiated by J.J. Thomson and concluded by the invention of quantum mechanics. Thompson's program derived its inspiration from the peculiar emphasis on models characteristic of British physics of the 19th century. Rutherford's atom was a late product of the goals and conceptions of Victorian science. Bohr's modifications, although ultimately fatal to Thomson's program, initially gave further impetus to it. In the early 1920s the most promising approach to an adequate theory of the atom appeared to be the literal and detailed elaboration of the classical mechanics of multiply periodic orbits. The approach succeeded, demonstrating in an unexpected way the force of an argument often advanced by Thomson: because a mechanical model is richer in implications than the considerations for which it was advanced, it can suggest new directions of research that may lead to important discoveries.

  7. Resisting the Bohr Atom: The Early British Opposition

    NASA Astrophysics Data System (ADS)

    Kragh, Helge

    2011-03-01

    When Niels Bohr's theory of atomic structure appeared in the summer and fall of 1913, it quickly attracted attention among British physicists. While some of the attention was supportive, others was critical. I consider the opposition to Bohr's theory from 1913 to about 1915, including attempts to construct atomic theories on a classical basis as alternatives to Bohr's. I give particular attention to the astrophysicist John W. Nicholson, who was Bohr's most formidable and persistent opponent in the early years. Although in the long run Nicholson's objections were inconsequential, for a short period of time his atomic theory was considered to be a serious rival to Bohr's. Moreover, Nicholson's theory is of interest in its own right.

  8. Special Bohr-Sommerfeld Lagrangian submanifolds

    NASA Astrophysics Data System (ADS)

    Tyurin, N. A.

    2016-12-01

    We introduce a new notion in symplectic geometry, that of speciality for Lagrangian submanifolds satisfying the Bohr- Sommerfeld condition. We show that it enables one to construct finite-dimensional moduli spaces of special Bohr- Sommerfeld Lagrangian submanifolds with respect to any ample line bundle on an algebraic variety with a Hodge metric regarded as the symplectic form. This construction can be used to study mirror symmetry.

  9. Bohr effect of hemoglobins: Accounting for differences in magnitude.

    PubMed

    Okonjo, Kehinde O

    2015-09-07

    The basis of the difference in the Bohr effect of various hemoglobins has remained enigmatic for decades. Fourteen amino acid residues, identical in pairs and located at specific 'Bohr group positions' in human hemoglobin, are implicated in the Bohr effect. All 14 are present in mouse, 11 in dog, eight in pigeon and 13 in guinea pig hemoglobin. The Bohr data for human and mouse hemoglobin are identical: the 14 Bohr groups appear at identical positions in both molecules. The dog data are different from the human because three Bohr group positions are occupied by non-ionizable groups in dog hemoglobin; the pigeon data are vastly different from the human because six Bohr group positions are occupied by non-ionizable groups in pigeon hemoglobin. The guinea pig data are quite complex. Quantitative analyses showed that only the pigeon data could be fitted with the Wyman equation for the Bohr effect. We demonstrate that, apart from guinea pig hemoglobin, the difference between the Bohr effect of each of the other hemoglobins and of pigeon hemoglobin can be accounted for quantitatively on the basis of the occupation of some of their Bohr group positions by non-ionizable groups in pigeon hemoglobin. We attribute the anomalous guinea pig result to a new salt-bridge formed in its R2 quaternary structure between the terminal NH3(+) group of one β-chain and the COO(-) terminal group of the partner β-chain in the same molecule. The pKas of this NH3(+) group are 6.33 in the R2 and 4.59 in the T state. Copyright © 2015 Elsevier Ltd. All rights reserved.

  10. Einstein, Bohr, and Bell

    NASA Astrophysics Data System (ADS)

    Bellac, Michel Le

    2014-11-01

    The final form of quantum physics, in the particular case of wave mechanics, was established in the years 1925-1927 by Heisenberg, Schrödinger, Born and others, but the synthesis was the work of Bohr who gave an epistemological interpretation of all the technicalities built up over those years; this interpretation will be examined briefly in Chapter 10. Although Einstein acknowledged the success of quantum mechanics in atomic, molecular and solid state physics, he disagreed deeply with Bohr's interpretation. For many years, he tried to find flaws in the formulation of quantum theory as it had been more or less accepted by a large majority of physicists, but his objections were brushed away by Bohr. However, in an article published in 1935 with Podolsky and Rosen, universally known under the acronym EPR, Einstein thought he had identified a difficulty in the by then standard interpretation. Bohr's obscure, and in part beyond the point, answer showed that Einstein had hit a sensitive target. Nevertheless, until 1964, the so-called Bohr-Einstein debate stayed uniquely on a philosophical level, and it was actually forgotten by most physicists, as the few of them aware of it thought it had no practical implication. In 1964, the Northern Irish physicist John Bell realized that the assumptions contained in the EPR article could be tested experimentally. These assumptions led to inequalities, the Bell inequalities, which were in contradiction with quantum mechanical predictions: as we shall see later on, it is extremely likely that the assumptions of the EPR article are not consistent with experiment, which, on the contrary, vindicates the predictions of quantum physics. In Section 3.2, the origin of Bell's inequalities will be explained with an intuitive example, then they will be compared with the predictions of quantum theory in Section 3.3, and finally their experimental status will be reviewed in Section 3.4. The debate between Bohr and Einstein goes much beyond a

  11. Paul Ehrenfest, Niels Bohr, and Albert Einstein: Colleagues and Friends

    NASA Astrophysics Data System (ADS)

    Klein, Martin J.

    2010-09-01

    In May 1918 Paul Ehrenfest received a monograph from Niels Bohr in which Bohr had used Ehrenfest's adiabatic principle as an essential assumption for understanding atomic structure. Ehrenfest responded by inviting Bohr, whom he had never met, to give a talk at a meeting in Leiden in late April 1919, which Bohr accepted; he lived with Ehrenfest, his mathematician wife Tatyana, and their young family for two weeks. Albert Einstein was unable to attend this meeting, but in October 1919 he visited his old friend Ehrenfest and his family in Leiden, where Ehrenfest told him how much he had enjoyed and profited from Bohr's visit. Einstein first met Bohr when Bohr gave a lecture in Berlin at the end of April 1920, and the two immediately proclaimed unbounded admiration for each other as physicists and as human beings. Ehrenfest hoped that he and they would meet at the Third Solvay Conference in Brussels in early April 1921, but his hope was unfulfilled. Einstein, the only physicist from Germany who was invited to it in this bitter postwar atmosphere, decided instead to accompany Chaim Weizmann on a trip to the United States to help raise money for the new Hebrew University in Jerusalem. Bohr became so overworked with the planning and construction of his new Institute for Theoretical Physics in Copenhagen that he could only draft the first part of his Solvay report and ask Ehrenfest to present it, which Ehrenfest agreed to do following the presentation of his own report. After recovering his strength, Bohr invited Ehrenfest to give a lecture in Copenhagen that fall, and Ehrenfest, battling his deep-seated self-doubts, spent three weeks in Copenhagen in December 1921 accompanied by his daughter Tanya and her future husband, the two Ehrenfests staying with the Bohrs in their apartment in Bohr's new Institute for Theoretical Physics. Immediately after leaving Copenhagen, Ehrenfest wrote to Einstein, telling him once again that Bohr was a prodigious physicist, and again

  12. Uncertainty in Bohr's response to the Heisenberg microscope

    NASA Astrophysics Data System (ADS)

    Tanona, Scott

    2004-09-01

    In this paper, I analyze Bohr's account of the uncertainty relations in Heisenberg's gamma-ray microscope thought experiment and address the question of whether Bohr thought uncertainty was epistemological or ontological. Bohr's account seems to allow that the electron being investigated has definite properties which we cannot measure, but other parts of his Como lecture seem to indicate that he thought that electrons are wave-packets which do not have well-defined properties. I argue that his account merges the ontological and epistemological aspects of uncertainty. However, Bohr reached this conclusion not from positivism, as perhaps Heisenberg did, but because he was led to that conclusion by his understanding of the physics in terms of nonseparability and the correspondence principle. Bohr argued that the wave theory from which he derived the uncertainty relations was not to be taken literally, but rather symbolically, as an expression of the limited applicability of classical concepts to parts of entangled quantum systems. Complementarity and uncertainty are consequences of the formalism, properly interpreted, and not something brought to the physics from external philosophical views.

  13. Revisiting the Bohr Atom 100 Years Later

    NASA Astrophysics Data System (ADS)

    Wall, Ernst

    2013-03-01

    We use a novel electron model wherein the electron is modeled as a point charge behaving as a trapped photon revolving in a Compton wavelength orbit at light speed. The revolving point charge gives rise to spiraling Compton wavelets around the electron, which give rise to de Broglie waves. When applied to the Bohr model, the orbital radius of the electron scales to the first Bohr orbit's radius via the fine structure constant. The orbiting electron's orbital velocity, Vb, scales to that of the electron's charge's internal velocity (the velocity of light, c) via the fine structure constant. The Compton wavelets, if they reflect off the nucleus, have a round trip time just long enough to allow the electron to move one of its diameters in distance in the first Bohr orbit. The ratio of the electron's rotational frequency, fe, to its rotational frequency in the Bohr orbit fb, is fe/fb = 1/α2, which is also the number of electron rotations in single orbit. If we scale the electron's rotational energy (h*fe) to that of the orbit using this, the orbital energy value (h*fb) would be 27.2114 eV. However, the virial theorem reduces it to 13.6057, the ground state energy of the first Bohr orbit. Ref: www.tachyonmodel.com.

  14. When champions meet: Rethinking the Bohr-Einstein debate

    NASA Astrophysics Data System (ADS)

    Landsman, N. P.

    Einstein's philosophy of physics (as clarified by Fine, Howard, and Held) was predicated on his Trennungsprinzip, a combination of separability and locality, without which he believed objectification, and thereby "physical thought" and "physical laws', to be impossible. Bohr's philosophy (as elucidated by Hooker, Scheibe, Folse, Howard, Held, and others), on the other hand, was grounded in a seemingly different doctrine about the possibility of objective knowledge, namely the necessity of classical concepts. In fact, it follows from Raggio's Theorem in algebraic quantum theory that-within an appropriate class of physical theories-suitable mathematical translations of the doctrines of Bohr and Einstein are equivalent. Thus-upon our specific formalization-quantum mechanics accommodates Einstein's Trennungsprinzip if and only if it is interpreted à la Bohr through classical physics. Unfortunately, the protagonists themselves failed to discuss their differences in this constructive way, since their debate was dominated by Einstein's ingenious but ultimately flawed attempts to establish the "incompleteness" of quantum mechanics. This aspect of their debate may still be understood and appreciated, however, as reflecting a much deeper and insurmountable disagreement between Bohr and Einstein about the knowability of Nature. Using the theological controversy on the knowability of God as a analogy, we can say that Einstein was a Spinozist, whereas Bohr could be said to be on the side of Maimonides. Thus Einstein's off-the-cuff characterization of Bohr as a 'Talmudic philosopher' was spot-on.

  15. Bohr, Heisenberg and the divergent views of complementarity

    NASA Astrophysics Data System (ADS)

    Camilleri, Kristian

    The fractious discussions between Bohr and Heisenberg in Copenhagen in 1927 have been the subject of much historical scholarship. However, little attention has been given to Heisenberg's understanding of the notion of complementary space-time and causal descriptions, which was presented for the first time in Bohr's lecture at the 1927 Como conference. In this paper, I argue that Heisenberg's own interpretation of this notion differed substantially from Bohr's. Whereas Bohr had intended this form of complementarity to entail a choice between a space-time description of the electron in an atom, and defining the energy of a stationary state, Heisenberg interpreted the 'causal' description in terms of ψ -function in configuration space. In disentangling the two views of complementarity, this paper sheds new light on the hidden philosophical disagreements between the proponents of these two founders of the so-called 'Copenhagen interpretation' of quantum mechanics.

  16. Teaching Bohr Theory.

    ERIC Educational Resources Information Center

    Latimer, Colin J.

    1983-01-01

    Discusses some lesser known examples of atomic phenomena to illustrate to students that the old quantum theory in its simplest (Bohr) form is not an antiquity but can still make an important contribution to understanding such phenomena. Topics include hydrogenic/non-hydrogenic spectra and atoms in strong electric and magnetic fields. (Author/JN)

  17. Niels Bohr on the wave function and the classical/quantum divide

    NASA Astrophysics Data System (ADS)

    Zinkernagel, Henrik

    2016-02-01

    It is well known that Niels Bohr insisted on the necessity of classical concepts in the account of quantum phenomena. But there is little consensus concerning his reasons, and what he exactly meant by this. In this paper, I re-examine Bohr's interpretation of quantum mechanics, and argue that the necessity of the classical can be seen as part of his response to the measurement problem. More generally, I attempt to clarify Bohr's view on the classical/quantum divide, arguing that the relation between the two theories is that of mutual dependence. An important element in this clarification consists in distinguishing Bohr's idea of the wave function as symbolic from both a purely epistemic and an ontological interpretation. Together with new evidence concerning Bohr's conception of the wave function collapse, this sets his interpretation apart from both standard versions of the Copenhagen interpretation, and from some of the reconstructions of his view found in the literature. I conclude with a few remarks on how Bohr's ideas make much sense also when modern developments in quantum gravity and early universe cosmology are taken into account.

  18. Correction of the exciton Bohr radius in monolayer transition metal dichalcogenides

    NASA Astrophysics Data System (ADS)

    Li, Run-Ze; Dong, Xi-Ying; Li, Zhi-Qing; Wang, Zi-Wu

    2018-07-01

    We theoretically investigate the correction of exciton Bohr radius in monolayer transition metal dichalcogenides (TMDCs) on different polar substrates arising from the exciton-optical phonon coupling, in which both the intrinsic longitudinal optical phonon and surface optical phonon modes couple with the exciton are taken into account. We find that the exciton Bohr radius is enlarged markedly due to these coupling. Moreover, it can be changed on a large scale by modulating the polarizability of polar substrate and the internal distance between the monolayer TMDCs and polar substrate. Theoretical result provides a potential explanation for the variation of the exciton Bohr radius in experimental measurement.

  19. Bohr effect of human hemoglobin: Separation of tertiary and quaternary contributions based on the Wyman equation.

    PubMed

    Okonjo, Kehinde Onwochei

    2017-09-01

    As a prelude to separating tertiary from quaternary structure contributions to the Bohr effect, we employed the Wyman equation to analyze Bohr data for human hemoglobin to which 2,3-bisphosphoglycerate, 2,3-BPG, is bound. Changes in the pK a s of the histidine Bohr groups result in a net reduction of their contributions to the Bohr effect at pH 7.4 compared to their contributions in stripped hemoglobin. The non-histidine 2,3-BPG binding groups - the β-chain terminal amino group and Lys82β - make negative and positive contributions, respectively, to the Bohr effect. The final result is that the Bohr effect at physiological pH is higher for 2,3-BPG bound compared to stripped hemoglobin. Contributions linked to His2β, His77β and His143β enable us to separate tertiary from quaternary Bohr contributions in stripped and in 2,3-BPG bound hemoglobin. Both contributions serve to make the Bohr effect for 2,3-BPG bound hemoglobin higher than for stripped hemoglobin at physiological pH. Copyright © 2017 Elsevier B.V. All rights reserved.

  20. A Simple Relativistic Bohr Atom

    ERIC Educational Resources Information Center

    Terzis, Andreas F.

    2008-01-01

    A simple concise relativistic modification of the standard Bohr model for hydrogen-like atoms with circular orbits is presented. As the derivation requires basic knowledge of classical and relativistic mechanics, it can be taught in standard courses in modern physics and introductory quantum mechanics. In addition, it can be shown in a class that…

  1. The BOHR Effect before Perutz

    ERIC Educational Resources Information Center

    Brunori, Maurizio

    2012-01-01

    Before the outbreak of World War II, Jeffries Wyman postulated that the "Bohr effect" in hemoglobin demanded the oxygen linked dissociation of the imidazole of two histidines of the polypeptide. This proposal emerged from a rigorous analysis of the acid-base titration curves of oxy- and deoxy-hemoglobin, at a time when the information on the…

  2. Bohr's Electron was Problematic for Einstein: String Theory Solved the Problem

    NASA Astrophysics Data System (ADS)

    Webb, William

    2013-04-01

    Neils Bohr's 1913 model of the hydrogen electron was problematic for Albert Einstein. Bohr's electron rotates with positive kinetic energies +K but has addition negative potential energies - 2K. The total net energy is thus always negative with value - K. Einstein's special relativity requires energies to be positive. There's a Bohr negative energy conflict with Einstein's positive energy requirement. The two men debated the problem. Both would have preferred a different electron model having only positive energies. Bohr and Einstein couldn't find such a model. But Murray Gell-Mann did! In the 1960's, Gell-Mann introduced his loop-shaped string-like electron. Now, analysis with string theory shows that the hydrogen electron is a loop of string-like material with a length equal to the circumference of the circular orbit it occupies. It rotates like a lariat around its centered proton. This loop-shape has no negative potential energies: only positive +K relativistic kinetic energies. Waves induced on loop-shaped electrons propagate their energy at a speed matching the tangential speed of rotation. With matching wave speed and only positive kinetic energies, this loop-shaped electron model is uniquely suited to be governed by the Einstein relativistic equation for total mass-energy. Its calculated photon emissions are all in excellent agreement with experimental data and, of course, in agreement with those -K calculations by Neils Bohr 100 years ago. Problem solved!

  3. The Bohr effect before Perutz.

    PubMed

    Brunori, Maurizio

    2012-01-01

    Before the outbreak of World War II, Jeffries Wyman postulated that the Bohr effect in hemoglobin demanded the oxygen linked dissociation of the imidazole of two histidines of the polypeptide. This proposal emerged from a rigorous analysis of the acid-base titration curves of oxy- and deoxy-hemoglobin, at a time when the information on the chemistry and structure of the protein was essentially nil. The magnetochemical properties of hemoglobin led Linus Pauling to hypothesize that the (so called) Bohr histidines were coordinated to the heme iron in the fifth and sixth positions; and Wyman shared this opinion. However, this structural hypothesis was abandoned in 1951 when J. Wyman and D. W. Allen proposed the pK shift of the oxygen linked histidines to be the result of "...a change of configuration of the hemoglobin molecule as a whole accompanying oxygenation." This shift in paradigm, that was published well before the 3D structure of hemoglobin was solved by M.F. Perutz, paved the way to the concept of allostery. After 1960 the availability of the crystallographic structure opened new horizons to the interpretation of the allosteric properties of hemoglobin. Copyright © 2012 Wiley Periodicals, Inc.

  4. Bohr effect of avian hemoglobins: Quantitative analyses based on the Wyman equation.

    PubMed

    Okonjo, Kehinde O

    2016-12-07

    The Bohr effect data for bar-headed goose, greylag goose and pheasant hemoglobins can be fitted with the Wyman equation for the Bohr effect, but under one proviso: that the pK a of His146β does not change following the T→R quaternary transition. This assumption is based on the x-ray structure of bar-headed goose hemoglobin, which shows that the salt-bridge formed between His146β and Asp94β in human deoxyhemoglobin is not formed in goose deoxyhemoglobin. When the Bohr data for chicken hemoglobin were fitted by making the same assumption, the pK a of the NH 3 + terminal group of Val1α decreased from 7.76 to 6.48 following the T→R transition. When the data were fitted without making any assumption, the pK a of the NH 3 + terminal group increased from 7.57 to 7.77 following the T→R transition. We demonstrate that avian hemoglobin Bohr data are readily fitted with the Wyman equation because avian hemoglobins lack His77β. From curve-fitting to Bohr data we estimate the pK a s of the NH 3 + terminal group of Val1α in the R and T states to be 6.33±0.1 and 7.22±0.1, respectively. We provide evidence indicating that these pK a s are more accurate than estimates from kinetic studies. Copyright © 2016 Elsevier Ltd. All rights reserved.

  5. Steering Quantum States Towards Classical Bohr-Like Orbits

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Dunning, F. B.; Reinhold, Carlos O; Yoshida, S.

    2010-01-01

    This article furnishes an introduction to the properties of time-dependent electronic wavefunctions in atoms and to physics at the interface between the quantum and classical worlds. We describe how, almost 100 years after the introduction of the Bohr model of the atom, it is now possible using pulsed electric fields to create in the laboratory localized wavepackets in high-n (n ~ 300) Rydberg atoms that travel in near-circular Bohr-like orbits mimicking the behavior of a classical electron. The control protocols employed are explained with the aid of quantum and classical dynamics. Remarkably, while many aspects of the underlying behavior canmore » be described using classical arguments, even at n ~ 300 purely quantum effects such as revivals can be seen.« less

  6. Emergence of complementarity and the Baconian roots of Niels Bohr's method

    NASA Astrophysics Data System (ADS)

    Perovic, Slobodan

    2013-08-01

    I argue that instead of a rather narrow focus on N. Bohr's account of complementarity as a particular and perhaps obscure metaphysical or epistemological concept (or as being motivated by such a concept), we should consider it to result from pursuing a particular method of studying physical phenomena. More precisely, I identify a strong undercurrent of Baconian method of induction in Bohr's work that likely emerged during his experimental training and practice. When its development is analyzed in light of Baconian induction, complementarity emerges as a levelheaded rather than a controversial account, carefully elicited from a comprehensive grasp of the available experimental basis, shunning hasty metaphysically motivated generalizations based on partial experimental evidence. In fact, Bohr's insistence on the "classical" nature of observations in experiments, as well as the counterintuitive synthesis of wave and particle concepts that have puzzled scholars, seem a natural outcome (an updated instance) of the inductive method. Such analysis clarifies the intricacies of early Schrödinger's critique of the account as well as Bohr's response, which have been misinterpreted in the literature. If adequate, the analysis may lend considerable support to the view that Bacon explicated the general terms of an experimentally minded strand of the scientific method, developed and refined by scientists in the following three centuries.

  7. Bohr's Creation of his Quantum Atom

    NASA Astrophysics Data System (ADS)

    Heilbron, John

    2013-04-01

    Fresh letters throw new light on the content and state of Bohr's mind before and during his creation of the quantum atom. His mental furniture then included the atomic models of the English school, the quantum puzzles of Continental theorists, and the results of his own studies of the electron theory of metals. It also included the poetry of Goethe, plays of Ibsen and Shakespeare, novels of Dickens, and rhapsodies of Kierkegaard and Carlyle. The mind that held these diverse ingredients together oscillated between enthusiasm and dejection during the year in which Bohr took up the problem of atomic structure. He spent most of that year in England, which separated him for extended periods from his close-knit family and friends. Correspondence with his fianc'ee, Margrethe Nørlund, soon to be published, reports his ups and downs as he adjusted to J.J. Thomson, Ernest Rutherford, the English language, and the uneven course of his work. In helping to smooth out his moods, Margrethe played an important and perhaps an enabling role in his creative process.

  8. Essential Ambiguity and Essential Influence: Rereading Bohr's Reply to EPR

    NASA Astrophysics Data System (ADS)

    Plotnitsky, Arkady

    2006-01-01

    The article offers a rereading of Bohr's reply to the argument of A. Einstein, B. Podolsky, and N. Rosen (EPR) concerning the incompleteness, or else nonlocality, of quantum mechanics. Bohr shows EPR's argument to be deficient on their own terms, by virtue of an ambiguity found in their application of the criterion of reality they propose to the phenomena in question. He also offers an alternative interpretation of the EPR experiment itself, the thought experiment proposed by EPR, as part of his overall interpretation of quantum mechanics as complementarity.

  9. How Sommerfeld extended Bohr's model of the atom (1913-1916)

    NASA Astrophysics Data System (ADS)

    Eckert, Michael

    2014-04-01

    Sommerfeld's extension of Bohr's atomic model was motivated by the quest for a theory of the Zeeman and Stark effects. The crucial idea was that a spectral line is made up of coinciding frequencies which are decomposed in an applied field. In October 1914 Johannes Stark had published the results of his experimental investigation on the splitting of spectral lines in hydrogen (Balmer lines) in electric fields, which showed that the frequency of each Balmer line becomes decomposed into a multiplet of frequencies. The number of lines in such a decomposition grows with the index of the line in the Balmer series. Sommerfeld concluded from this observation that the quantization in Bohr's model had to be altered in order to allow for such decompositions. He outlined this idea in a lecture in winter 1914/15, but did not publish it. The First World War further delayed its elaboration. When Bohr published new results in autumn 1915, Sommerfeld finally developed his theory in a provisional form in two memoirs which he presented in December 1915 and January 1916 to the Bavarian Academy of Science. In July 1916 he published the refined version in the Annalen der Physik. The focus here is on the preliminary Academy memoirs whose rudimentary form is better suited for a historical approach to Sommerfeld's atomic theory than the finished Annalen-paper. This introductory essay reconstructs the historical context (mainly based on Sommerfeld's correspondence). It will become clear that the extension of Bohr's model did not emerge in a singular stroke of genius but resulted from an evolving process.

  10. Noninvasive measurement of mean alveolar carbon dioxide tension and Bohr's dead space during tidal breathing.

    PubMed

    Koulouris, N G; Latsi, P; Dimitroulis, J; Jordanoglou, B; Gaga, M; Jordanoglou, J

    2001-06-01

    The lack of methodology for measuring the alveolar carbon dioxide tension (PA,CO2) has forced investigators to make several assumptions, such as that PA,CO2 is equal to end-tidal (PET,CO2) and arterial CO2 tension (Pa,CO2). The present study measured the mean PA,CO2 and Bohr's dead space ratio (Bohr's dead space/tidal volume (VD,Bohr/VT)) during tidal breathing. The method used is a new, simple and noninvasive technique, based on the analysis of the expired CO2 volume per breath (VCO2) versus the exhaled VT. This curve was analysed in 21 normal, healthy subjects and 35 chronic obstructive pulmonary disease (COPD) patients breathing tidally through a mouthpiece apparatus in the sitting position. It is shown that: 1) PA,CO2 is similar to Pa,CO2 in normal subjects, whilst it is significantly lower than Pa,CO2 in COPD patients; 2) PA,CO2 is significantly higher than PET,CO2 in all subjects, especially in COPD patients; 3) VD,Bohr/VT is increased in COPD patients as compared to normal subjects; and 4) VD,Bohr/VT is lower than the "physiological" dead space ratio (VD,phys/VT) in COPD patients. It is concluded that the expired carbon dioxide versus tidal volume curve is a useful tool for research and clinical work, because it permits the noninvasive and accurate measurement of Bohr's dead space and mean alveolar carbon dioxide tension accurately during spontaneous breathing.

  11. The Modified Hartmann Potential Effects on γ-rigid Bohr Hamiltonian

    NASA Astrophysics Data System (ADS)

    Suparmi, A.; Cari, C.; Nur Pratiwi, Beta

    2018-04-01

    In this paper, we present the solution of Bohr Hamiltonian in the case of γ-rigid for the modified Hartmann potential. The modified Hartmann potential was formed from the original Hartmann potential, consists of β function and θ function. By using the separation method, the three-dimensional Bohr Hamiltonian equation was reduced into three one-dimensional Schrodinger-like equation which was solved analytically. The results for the wavefunction were shown in mathematically, while for the binding energy was solved numerically. The numerical binding energy for the presence of the modified Hartmann potential is lower than the binding energy value in the absence of modified Hartmann potential effect.

  12. Why has the bohr-sommerfeld model of the atom been ignoredby general chemistry textbooks?

    PubMed

    Niaz, Mansoor; Cardellini, Liberato

    2011-12-01

    Bohr's model of the atom is considered to be important by general chemistry textbooks. A major shortcoming of this model was that it could not explain the spectra of atoms containing more than one electron. In order to increase the explanatory power of the model, Sommerfeld hypothesized the existence of elliptical orbits. This study has the following objectives: 1) Formulation of criteria based on a history and philosophy of science framework; and 2) Evaluation of university-level general chemistry textbooks based on the criteria, published in Italy and U.S.A. Presentation of a textbook was considered to be "satisfactory" if it included a description of the Bohr-Sommerfeld model along with diagrams of the elliptical orbits. Of the 28 textbooks published in Italy that were analyzed, only five were classified as "satisfactory". Of the 46 textbooks published in U.S.A., only three were classified as "satisfactory". This study has the following educational implications: a) Sommerfeld's innovation (auxiliary hypothesis) by introducing elliptical orbits, helped to restore the viability of Bohr's model; b) Bohr-Sommerfeld's model went no further than the alkali metals, which led scientists to look for other models; c) This clearly shows that scientific models are tentative in nature; d) Textbook authors and chemistry teachers do not consider the tentative nature of scientific knowledge to be important; e) Inclusion of the Bohr-Sommerfeld model in textbooks can help our students to understand how science progresses.

  13. 'Let the stars shine in peace!' Niels Bohr and stellar energy, 1929-1934.

    PubMed

    Kragh, Helge

    2017-04-01

    Faced with various anomalies related to nuclear physics in particular, in 1929 Niels Bohr suggested that energy might not be conserved in the atomic nucleus and the processes involving it. By this radical proposal he hoped not only to get rid of the anomalies but also saw a possibility to explain a puzzle in astrophysics, namely the energy generated by stars. Bohr repeated his suggestion of stellar energy arising ex nihilo on several occasions but without ever going into detail. In fact, it is not very clear what he meant or how seriously he took the stellar energy hypothesis. This paper relates Bohr's comments to the period's attempts to find a mechanism for stellar energy and also to the role played by astrophysics at the Copenhagen institute. Moreover, it looks at how Bohr's hypothesis was received not only by physicists but also by astronomers. In this regard the disciplinary status of astrophysics and its contemporary relation to the new quantum mechanics is of relevance. It turns out that, with very few exceptions, the hypothesis was met with silence by astronomers and astrophysicists concerned with the problem of stellar energy production. And yet, for a brief period of time it did have an impact on how physicists thought about the interior of the stars.

  14. Relativistic Corrections to the Bohr Model of the Atom

    ERIC Educational Resources Information Center

    Kraft, David W.

    1974-01-01

    Presents a simple means for extending the Bohr model to include relativistic corrections using a derivation similar to that for the non-relativistic case, except that the relativistic expressions for mass and kinetic energy are employed. (Author/GS)

  15. Why we should teach the Bohr model and how to teach it effectively

    NASA Astrophysics Data System (ADS)

    McKagan, S. B.; Perkins, K. K.; Wieman, C. E.

    2008-06-01

    Some education researchers have claimed that we should not teach the Bohr model of the atom because it inhibits students’ ability to learn the true quantum nature of electrons in atoms. Although the evidence for this claim is weak, many have accepted it. This claim has implications for how to present atoms in classes ranging from elementary school to graduate school. We present results from a study designed to test this claim by developing a curriculum on models of the atom, including the Bohr and Schrödinger models. We examine student descriptions of atoms on final exams in transformed modern physics classes using various versions of this curriculum. We find that if the curriculum does not include sufficient connections between different models, many students still have a Bohr-like view of atoms rather than a more accurate Schrödinger model. However, with an improved curriculum designed to develop model-building skills and with better integration between different models, it is possible to get most students to describe atoms using the Schrödinger model. In comparing our results with previous research, we find that comparing and contrasting different models is a key feature of a curriculum that helps students move beyond the Bohr model and adopt Schrödinger’s view of the atom. We find that understanding the reasons for the development of models is much more difficult for students than understanding the features of the models. We also present interactive computer simulations designed to help students build models of the atom more effectively.

  16. Realization of localized Bohr-like wave packets.

    PubMed

    Mestayer, J J; Wyker, B; Lancaster, J C; Dunning, F B; Reinhold, C O; Yoshida, S; Burgdörfer, J

    2008-06-20

    We demonstrate a protocol to create localized wave packets in very-high-n Rydberg states which travel in nearly circular orbits around the nucleus. Although these wave packets slowly dephase and eventually lose their localization, their motion can be monitored over several orbital periods. These wave packets represent the closest analog yet achieved to the original Bohr model of the hydrogen atom, i.e., an electron in a circular classical orbit around the nucleus. The possible extension of the approach to create "planetary atoms" in highly correlated stable multiply excited states is discussed.

  17. EPR before EPR: A 1930 Einstein-Bohr thought Experiment Revisited

    ERIC Educational Resources Information Center

    Nikolic, Hrvoje

    2012-01-01

    In 1930, Einstein argued against the consistency of the time-energy uncertainty relation by discussing a thought experiment involving a measurement of the mass of the box which emitted a photon. Bohr seemingly prevailed over Einstein by arguing that Einstein's own general theory of relativity saves the consistency of quantum mechanics. We revisit…

  18. Experimental Observation of Bohr's Nonlinear Fluidic Surface Oscillation.

    PubMed

    Moon, Songky; Shin, Younghoon; Kwak, Hojeong; Yang, Juhee; Lee, Sang-Bum; Kim, Soyun; An, Kyungwon

    2016-01-25

    Niels Bohr in the early stage of his career developed a nonlinear theory of fluidic surface oscillation in order to study surface tension of liquids. His theory includes the nonlinear interaction between multipolar surface oscillation modes, surpassing the linear theory of Rayleigh and Lamb. It predicts a specific normalized magnitude of 0.416η(2) for an octapolar component, nonlinearly induced by a quadrupolar one with a magnitude of η much less than unity. No experimental confirmation on this prediction has been reported. Nonetheless, accurate determination of multipolar components is important as in optical fiber spinning, film blowing and recently in optofluidic microcavities for ray and wave chaos studies and photonics applications. Here, we report experimental verification of his theory. By using optical forward diffraction, we measured the cross-sectional boundary profiles at extreme positions of a surface-oscillating liquid column ejected from a deformed microscopic orifice. We obtained a coefficient of 0.42 ± 0.08 consistently under various experimental conditions. We also measured the resonance mode spectrum of a two-dimensional cavity formed by the cross-sectional segment of the liquid jet. The observed spectra agree well with wave calculations assuming a coefficient of 0.414 ± 0.011. Our measurements establish the first experimental observation of Bohr's hydrodynamic theory.

  19. Darwinism in disguise? A comparison between Bohr's view on quantum mechanics and QBism.

    PubMed

    Faye, Jan

    2016-05-28

    The Copenhagen interpretation is first and foremost associated with Niels Bohr's philosophy of quantum mechanics. In this paper, I attempt to lay out what I see as Bohr's pragmatic approach to science in general and to quantum physics in particular. A part of this approach is his claim that the classical concepts are indispensable for our understanding of all physical phenomena, and it seems as if the claim is grounded in his reflection upon how the evolution of language is adapted to experience. Another, recent interpretation, QBism, has also found support in Darwin's theory. It may therefore not be surprising that sometimes QBism is said to be of the same breed as the Copenhagen interpretation. By comparing the two interpretations, I conclude, nevertheless, that there are important differences. © 2016 The Author(s).

  20. The cognitive nexus between Bohr's analogy for the atom and Pauli's exclusion schema.

    PubMed

    Ulazia, Alain

    2016-03-01

    The correspondence principle is the primary tool Bohr used to guide his contributions to quantum theory. By examining the cognitive features of the correspondence principle and comparing it with those of Pauli's exclusion principle, I will show that it did more than simply 'save the phenomena'. The correspondence principle in fact rested on powerful analogies and mental schemas. Pauli's rejection of model-based methods in favor of a phenomenological, rule-based approach was therefore not as disruptive as some historians have indicated. Even at a stage that seems purely phenomenological, historical studies of theoretical development should take into account non-formal, model-based approaches in the form of mental schemas, analogies and images. In fact, Bohr's images and analogies had non-classical components which were able to evoke the idea of exclusion as a prohibition law and as a preliminary mental schema. Copyright © 2016 Elsevier Ltd. All rights reserved.

  1. "It's best not to think about it at all-like the new taxes": Reality, observer, and complementarity in Bohr and Pauli

    NASA Astrophysics Data System (ADS)

    Plotnitsky, Arkady

    2012-12-01

    This article considers the concepts of reality, observer, and complementarity in Pauli and Bohr, and the similarities and, especially, differences in their understanding of these concepts, differences defined most essentially by their respective views of the role of the human observer in quantum measurement. These differences are significant even in the case of their respective interpretations of quantum phenomena and quantum mechanics, where the influence of Bohr's ideas on Pauli's understanding of quantum physics is particularly strong. They become especially strong and even radical in the case of their overall philosophical visions, where the impact of Jungean psychology, coupled to that of the earlier archetypal thinking of such figures as Kepler and Fludd, drives Pauli's thinking ever further away from that of Bohr.

  2. Molecular Basis of the Bohr Effect in Arthropod Hemocyanin

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Hirota, S.; Kawahara, T; Beltramini, M

    2008-01-01

    Flash photolysis and K-edge x-ray absorption spectroscopy (XAS) were used to investigate the functional and structural effects of pH on the oxygen affinity of three homologous arthropod hemocyanins (Hcs). Flash photolysis measurements showed that the well-characterized pH dependence of oxygen affinity (Bohr effect) is attributable to changes in the oxygen binding rate constant, kon, rather than changes in koff. In parallel, coordination geometry of copper in Hc was evaluated as a function of pH by XAS. It was found that the geometry of copper in the oxygenated protein is unchanged at all pH values investigated, while significant changes were observedmore » for the deoxygenated protein as a function of pH. The interpretation of these changes was based on previously described correlations between spectral lineshape and coordination geometry obtained for model compounds of known structure A pH-dependent change in the geometry of cuprous copper in the active site of deoxyHc, from pseudotetrahedral toward trigonal was assigned from the observed intensity dependence of the 1s ? 4pz transition in x-ray absorption near edge structure (XANES) spectra. The structural alteration correlated well with increase in oxygen affinity at alkaline pH determined in flash photolysis experiments. These results suggest that the oxygen binding rate in deoxyHc depends on the coordination geometry of Cu(I) and suggest a structural origin for the Bohr effect in arthropod Hcs.« less

  3. Quantum Humor: The Playful Side of Physics at Bohr's Institute for Theoretical Physics

    NASA Astrophysics Data System (ADS)

    Halpern, Paul

    2012-09-01

    From the 1930s to the 1950s, a period of pivotal developments in quantum, nuclear, and particle physics, physicists at Niels Bohr's Institute for Theoretical Physics in Copenhagen took time off from their research to write humorous articles, letters, and other works. Best known is the Blegdamsvej Faust, performed in April 1932 at the close of one of the Institute's annual conferences. I also focus on the Journal of Jocular Physics, a humorous tribute to Bohr published on the occasions of his 50th, 60th, and 70th birthdays in 1935, 1945, and 1955. Contributors included Léon Rosenfeld, Victor Weisskopf, George Gamow, Oskar Klein, and Hendrik Casimir. I examine their contributions along with letters and other writings to show that they offer a window into some issues in physics at the time, such as the interpretation of complementarity and the nature of the neutrino, as well as the politics of the period.

  4. Clinical Management of Patients with ASXL1 Mutations and Bohring-Opitz Syndrome, Emphasizing the Need for Wilms Tumor Surveillance

    PubMed Central

    Russell, Bianca; Johnston, Jennifer J; Biesecker, Leslie G.; Kramer, Nancy; Pickart, Angela; Rhead, William; Tan, Wen-Hann; Brownstein, Catherine A; Clarkson, L Kate; Dobson, Amy; Rosenberg, Avi Z; Schrier Vergano, Samantha A.; Helm, Benjamin M.; Harrison, Rachel E; Graham, John M

    2016-01-01

    Bohring-Opitz syndrome is a rare genetic condition characterized by distinctive facial features, variable microcephaly, hypertrichosis, nevus flammeus, severe myopia, unusual posture (flexion at the elbows with ulnar deviation, and flexion of the wrists and metacarpophalangeal joints), severe intellectual disability, and feeding issues. Nine patients with Bohring-Opitz syndrome have been identified as having a mutation in ASXL1. We report on eight previously unpublished patients with Bohring-Opitz syndrome caused by an apparent or confirmed de novo mutation in ASXL1. Of note, two patients developed bilateral Wilms tumors. Somatic mutations in ASXL1 are associated with myeloid malignancies, and these reports emphasize the need for Wilms tumor screening in patients with ASXL1 mutations. We discuss clinical management with a focus on their feeding issues, cyclic vomiting, respiratory infections, insomnia, and tumor predisposition. Many patients are noted to have distinctive personalities (interactive, happy, and curious) and rapid hair growth; features not previously reported. PMID:25921057

  5. The unitary convolution approximation for heavy ions

    NASA Astrophysics Data System (ADS)

    Grande, P. L.; Schiwietz, G.

    2002-10-01

    The convolution approximation for the impact-parameter dependent energy loss is reviewed with emphasis on the determination of the stopping force for heavy projectiles. In this method, the energy loss in different impact-parameter regions is well determined and interpolated smoothly. The physical inputs of the model are the projectile-screening function (in the case of dressed ions), the electron density and oscillators strengths of the target atoms. Moreover, the convolution approximation, in the perturbative mode (called PCA), yields remarkable agreement with full semi-classical-approximation (SCA) results for bare as well as for screened ions at all impact parameters. In the unitary mode (called UCA), the method contains some higher-order effects (yielding in some cases rather good agreement with full coupled-channel calculations) and approaches the classical regime similar as the Bohr model for large perturbations ( Z/ v≫1). The results are then used to compare with experimental values of the non-equilibrium stopping force as a function of the projectile charge as well as with the equilibrium energy loss under non-aligned and channeling conditions.

  6. Bohr effect and temperature sensitivity of hemoglobins from highland and lowland deer mice.

    PubMed

    Jensen, Birgitte; Storz, Jay F; Fago, Angela

    2016-05-01

    An important means of physiological adaptation to environmental hypoxia is an increased oxygen (O2) affinity of the hemoglobin (Hb) that can help secure high O2 saturation of arterial blood. However, the trade-off associated with a high Hb-O2 affinity is that it can compromise O2 unloading in the systemic capillaries. High-altitude deer mice (Peromyscus maniculatus) have evolved an increased Hb-O2 affinity relative to lowland conspecifics, but it is not known whether they have also evolved compensatory mechanisms to facilitate O2 unloading to respiring tissues. Here we investigate the effects of pH (Bohr effect) and temperature on the O2-affinity of high- and low-altitude deer mouse Hb variants, as these properties can potentially facilitate O2 unloading to metabolizing tissues. Our experiments revealed that Bohr factors for the high- and low-altitude Hb variants are very similar in spite of the differences in O2-affinity. The Bohr factors of deer mouse Hbs are also comparable to those of other mammalian Hbs. In contrast, the high- and low-altitude variants of deer mouse Hb exhibited similarly low temperature sensitivities that were independent of red blood cell anionic cofactors, suggesting an appreciable endothermic allosteric transition upon oxygenation. In conclusion, high-altitude deer mice have evolved an adaptive increase in Hb-O2 affinity, but this is not associated with compensatory changes in sensitivity to changes in pH or temperature. Instead, it appears that the elevated Hb-O2 affinity in high-altitude deer mice is compensated by an associated increase in the tissue diffusion capacity of O2 (via increased muscle capillarization), which promotes O2 unloading. Copyright © 2016 Elsevier Inc. All rights reserved.

  7. Why We Should Teach the Bohr Model and How to Teach it Effectively

    ERIC Educational Resources Information Center

    McKagan, S. B.; Perkins, K. K.; Wieman, C. E.

    2008-01-01

    Some education researchers have claimed that we should not teach the Bohr model of the atom because it inhibits students' ability to learn the true quantum nature of electrons in atoms. Although the evidence for this claim is weak, many have accepted it. This claim has implications for how to present atoms in classes ranging from elementary school…

  8. H2+, HeH and H2: Approximating potential curves, calculating rovibrational states

    NASA Astrophysics Data System (ADS)

    Olivares-Pilón, Horacio; Turbiner, Alexander V.

    2018-06-01

    Analytic consideration of the Bohr-Oppenheimer (BO) potential curves for diatomic molecules is proposed: accurate analytic interpolation for a potential curve consistent with its rovibrational spectra is found. It is shown that in the BO approximation for four lowest electronic states 1 sσg and 2 pσu, 2 pπu and 3 dπg of H2+, the ground state X2Σ+ of HeH and the two lowest states 1 Σg+ and 3 Σu+ of H2, the potential curves can be analytically interpolated in full range of internuclear distances R with not less than 4-5-6 s.d. Approximation based on matching the Laurant-type expansion at small R and a combination of the multipole expansion with one-instanton type contribution at large distances R is given by two-point Padé approximant. The position of minimum, when exists, is predicted within 1% or better. For the molecular ion H2+ in the Lagrange mesh method, the spectra of vibrational, rotational and rovibrational states (ν , L) associated with 1 sσg and 2 pσu, 2 pπu and 3 dπg potential curves are calculated. In general, it coincides with spectra found via numerical solution of the Schrödinger equation (when available) within six s.d. It is shown that 1 sσg curve contains 19 vibrational states (ν , 0) , while 2 pσu curve contains a single one (0 , 0) and 2 pπu state contains 12 vibrational states (ν , 0) . In general, 1 sσg electronic curve contains 420 rovibrational states, which increases up to 423 when we are beyond BO approximation. For the state 2 pσu the total number of rovibrational states (all with ν = 0) is equal to 3, within or beyond Bohr-Oppenheimer approximation. As for the state 2 pπu within the Bohr-Oppenheimer approximation the total number of the rovibrational bound states is equal to 284. The state 3 dπg is repulsive, no rovibrational state is found. It is confirmed in Lagrange mesh formalism the statement that the ground state potential curve of the heteronuclear molecule HeH does not support rovibrational states. Accurate

  9. Bohr-Sommerfeld quantization condition for Dirac states derived from an Ermakov-type invariant

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Thylwe, Karl-Erik; McCabe, Patrick

    2013-05-15

    It is shown that solutions of the second-order decoupled radial Dirac equations satisfy Ermakov-type invariants. These invariants lead to amplitude-phase-type representations of the radial spinor solutions, with exact relations between their amplitudes and phases. Implications leading to a Bohr-Sommerfeld quantization condition for bound states, and a few particular atomic/ionic and nuclear/hadronic bound-state situations are discussed.

  10. Niels Bohr's discussions with Albert Einstein, Werner Heisenberg, and Erwin Schroedinger: the origins of the principles of uncertainty and complementarity

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Mehra, J.

    1987-05-01

    In this paper, the main outlines of the discussions between Niels Bohr with Albert Einstein, Werner Heisenberg, and Erwin Schroedinger during 1920-1927 are treated. From the formulation of quantum mechanics in 1925-1926 and wave mechanics in 1926, there emerged Born's statistical interpretation of the wave function in summer 1926, and on the basis of the quantum mechanical transformation theory - formulated in fall 1926 by Dirac, London, and Jordan - Heisenberg formulated the uncertainty principle in early 1927. At the Volta Conference in Como in September 1927 and at the fifth Solvay Conference in Brussels the following month, Bohr publiclymore » enunciated his complementarity principle, which had been developing in his mind for several years. The Bohr-Einstein discussions about the consistency and completeness of quantum mechanics and of physical theory as such - formally begun in October 1927 at the fifth Solvay Conference and carried on at the sixth Solvay Conference in October 1930 - were continued during the next decades. All these aspects are briefly summarized.« less

  11. Bohr Hamiltonian for γ = 30° with Davidson potential

    NASA Astrophysics Data System (ADS)

    Yigitoglu, Ibrahim; Gokbulut, Melek

    2018-03-01

    A γ-rigid solution of the Bohr Hamiltonian for γ = 30° is constructed with the Davidson potential in the β part. This solution is going to be called Z(4)-D. The energy eigenvalues and wave functions are obtained by using the analytic method developed by Nikiforov and Uvarov. The calculated intraband and interband B(E2) transitions rates are presented and compared with the Z(4) model predictions. The staggering behavior in γ-bands is considered to search Z(4) -D candidate nuclei. A variational procedure is applied to demonstrate that the Z(4) model is a solution of the critical point at the shape phase transition from spherical to rigid triaxial rotor.

  12. Oxygen affinity and Bohr effect responses to 2,3-diphosphoglycerate in equine and human blood.

    PubMed

    diBella, G; Scandariato, G; Suriano, O; Rizzo, A

    1996-05-01

    The dependence of blood oxygen affinity and the Bohr effect on the concentration of 2,3-diphosphoglycerate (DPG) in erythrocytes was investigated in 24 trotter horses and 24 healthy men. The oxygen tension at half saturation and standard conditions (P50st at pH 7.4, PCO2(40) mmHg and 37 degrees C) and the carbon dioxide or fixed-acid-induced Bohr effect (dlogP50/dpH) were determined. Samples of fresh blood and blood depleted of or enriched with DPG were studied. In the absence of measurable DPG, the equine and human blood had similar mean (SD) values of P50st (16.6 [0.6] and 16.2 [0.7] mmHg, respectively). In both species these values increased with increasing DPG, but the response of equine blood was significantly lower, at least up to physiological values (P50st = 24.6 [0.6] and 26.2 [0.7]) mmHg; DPG = 14([1.8] and 12.8 [1.2] mumol gHb-1, respectively, in fresh blood). For concentrations above 20 to 25 mumol gHb-1 of DPG the difference between the values of P50st in the two species tended to decrease because the response in human blood reached a plateau. The interactions between the Bohr effect and the concentration of DPG showed that in the horses, as in the men, the level of DPG played an important role in governing the relative magnitude of carbon dioxide and fixed acid factors. The difference between them, which is associated with the oxylabile carbamino binding, was greatest in DPG-depleted blood, but whereas in the men the difference was suppressed by an above normal DPG concentration, in the horses it was still measurable.

  13. A New Contribution for WYP 2005: The Golden Ratio, Bohr Radius, Planck's Constant, Fine-Structure Constant and g-Factors

    NASA Astrophysics Data System (ADS)

    Heyrovska, R.; Narayan, S.

    2005-10-01

    Recently, the ground state Bohr radius (aB) of hydrogen was shown to be divided into two Golden sections, aB,p = aB/ø2 and aB,e = aB/ø at the point of electrical neutrality, where ø = 1.618 is the Golden ratio (R. Heyrovska, Molecular Physics 103: 877-882, and the literature cited therein). The origin of the difference of two energy terms in the Rydberg equation was thus shown to be in the ground state energy itself, as shown below: EH = (1/2)e2/(κaB) = (1/2)(e2/κ) [(1/aB,p - (1/aB,e)] (1). This work brings some new results that 1) a unit charge in vacuum has a magnetic moment, 2) (e2/2κ) in eq. (1) is an electromagnetic condenser constant, 3) the de Broglie wavelengths of the proton and electron correspond to the Golden arcs of a circle with the Bohr radius, 4) the fine structure constant (α) is the ratio of the Planck's constants without and with the interaction of light with matter, 5) the g-factors of the electron and proton, ge/2 and gp/2 divide the Bohr radius at the magnetic center and 6) the ``mysterious'' value (137.036) of α -1 = (360/ø2) - (2/ø3), where (2/ø3) arises from the difference, (gp - ge).

  14. Semiclassical quantization of Bohr orbits in the helium atom

    NASA Astrophysics Data System (ADS)

    Belov, V. V.; Maksimov, V. A.

    2007-05-01

    We use the complex WKB-Maslov method to construct the semiclassical spectral series corresponding to the resonance Bohr orbits in the helium atom. The semiclassical energy levels represented as the Rydberg tetra series correspond to the doubly symmetrically excited states of helium-like atoms. This level series contains the Rydberg triple series reported by Richter and Wintgen in 1991, which corresponds to the Z2+e-e- configuration of electrons observed by Eichmann and his collaborators in experiments on the laser excitation of the barium atom in 1992. The lower-level extrapolation of the formula obtained for the semiclassical spectrum gives the value of the ground state energy, which differs by 6% from the experimental value obtained by Bergeson and his collaborators in 1998. We also calculate the fine structure of the semiclassical spectrum due to the spin-orbit and spin-spin interactions of electrons.

  15. Effectiveness of an Asynchronous Online Module on University Students' Understanding of the Bohr Model of the Hydrogen Atom

    ERIC Educational Resources Information Center

    Farina, William J., Jr.; Bodzin, Alec M.

    2018-01-01

    Web-based learning is a growing field in education, yet empirical research into the design of high quality Web-based university science instruction is scarce. A one-week asynchronous online module on the Bohr Model of the atom was developed and implemented guided by the knowledge integration framework. The unit design aligned with three identified…

  16. Diffusive Insights: On the Disagreement of Christian Bohr and August Krogh at the Centennial of the Seven Little Devils

    ERIC Educational Resources Information Center

    Gjedde, Albert

    2010-01-01

    The year 2010 is the centennial of the publication of the "Seven Little Devils" in the predecessor of "Acta Physiologica". In these seven papers, August and Marie Krogh sought to refute Christian Bohr's theory that oxygen diffusion from the lungs to the circulation is not entirely passive but rather facilitated by a specific cellular activity…

  17. The divine clockwork: Bohr's correspondence principle and Nelson's stochastic mechanics for the atomic elliptic state

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Durran, Richard; Neate, Andrew; Truman, Aubrey

    2008-03-15

    We consider the Bohr correspondence limit of the Schroedinger wave function for an atomic elliptic state. We analyze this limit in the context of Nelson's stochastic mechanics, exposing an underlying deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This solves the long standing problem of obtaining Kepler's laws of planetary motion in a quantum mechanical setting. In this quantum mechanical setting, local mild instabilities occur in the Keplerian orbit for eccentricities greater than (1/{radical}(2)) which do not occur classically.

  18. Investigation of Bohr Hamiltonian in presence of Killingbeck potential using bi-confluent Heun functions

    NASA Astrophysics Data System (ADS)

    Sobhani, Hadi; Hassanabadi, Hassan; Chung, Won Sang

    2018-05-01

    In this study, Bohr Hamiltonian is studied for the triaxial and rotational cases. In both cases, Killingbeck potential is used as interaction. The wave function and energy of these cases are found using bi-confluent Heun functions. The results are examined by reproducing experimental data of some isotopes for each case. Energy levels of the isotopes are shown graphically as well as theoretical results for staggering in γ bands of the isotopes is discussed. In the next step, we argue about B (E 2) transition rates of the isotopes for each case. The results have a good agreement with experimental data.

  19. Can we close the Bohr-Einstein quantum debate?

    NASA Astrophysics Data System (ADS)

    Kupczynski, Marian

    2017-10-01

    Recent experiments allow one to conclude that Bell-type inequalities are indeed violated; thus, it is important to understand what this means and how we can explain the existence of strong correlations between outcomes of distant measurements. Do we have to announce that Einstein was wrong, Nature is non-local and non-local correlations are produced due to quantum magic and emerge, somehow, from outside space-time? Fortunately, such conclusions are unfounded because, if supplementary parameters describing measuring instruments are correctly incorporated in a theoretical model, then Bell-type inequalities may not be proved. We construct a simple probabilistic model allowing these correlations to be explained in a locally causal way. In our model, measurement outcomes are neither predetermined nor produced in an irreducibly random way. We explain why, contrary to the general belief, the introduction of setting-dependent parameters does not restrict experimenters' freedom of choice. Since the violation of Bell-type inequalities does not allow the conclusion that Nature is non-local and that quantum theory is complete, the Bohr-Einstein quantum debate may not be closed. The continuation of this debate is important not only for a better understanding of Nature but also for various practical applications of quantum phenomena. This article is part of the themed issue `Second quantum revolution: foundational questions'.

  20. Can we close the Bohr-Einstein quantum debate?

    PubMed

    Kupczynski, Marian

    2017-11-13

    Recent experiments allow one to conclude that Bell-type inequalities are indeed violated; thus, it is important to understand what this means and how we can explain the existence of strong correlations between outcomes of distant measurements. Do we have to announce that Einstein was wrong, Nature is non-local and non-local correlations are produced due to quantum magic and emerge, somehow, from outside space-time? Fortunately, such conclusions are unfounded because, if supplementary parameters describing measuring instruments are correctly incorporated in a theoretical model, then Bell-type inequalities may not be proved. We construct a simple probabilistic model allowing these correlations to be explained in a locally causal way. In our model, measurement outcomes are neither predetermined nor produced in an irreducibly random way. We explain why, contrary to the general belief, the introduction of setting-dependent parameters does not restrict experimenters' freedom of choice. Since the violation of Bell-type inequalities does not allow the conclusion that Nature is non-local and that quantum theory is complete, the Bohr-Einstein quantum debate may not be closed. The continuation of this debate is important not only for a better understanding of Nature but also for various practical applications of quantum phenomena.This article is part of the themed issue 'Second quantum revolution: foundational questions'. © 2017 The Author(s).

  1. De novo truncating mutations in ASXL3 are associated with a novel clinical phenotype with similarities to Bohring-Opitz syndrome

    PubMed Central

    2013-01-01

    Background Molecular diagnostics can resolve locus heterogeneity underlying clinical phenotypes that may otherwise be co-assigned as a specific syndrome based on shared clinical features, and can associate phenotypically diverse diseases to a single locus through allelic affinity. Here we describe an apparently novel syndrome, likely caused by de novo truncating mutations in ASXL3, which shares characteristics with Bohring-Opitz syndrome, a disease associated with de novo truncating mutations in ASXL1. Methods We used whole-genome and whole-exome sequencing to interrogate the genomes of four subjects with an undiagnosed syndrome. Results Using genome-wide sequencing, we identified heterozygous, de novo truncating mutations in ASXL3, a transcriptional repressor related to ASXL1, in four unrelated probands. We found that these probands shared similar phenotypes, including severe feeding difficulties, failure to thrive, and neurologic abnormalities with significant developmental delay. Further, they showed less phenotypic overlap with patients who had de novo truncating mutations in ASXL1. Conclusion We have identified truncating mutations in ASXL3 as the likely cause of a novel syndrome with phenotypic overlap with Bohring-Opitz syndrome. PMID:23383720

  2. Memories of Crisis: Bohr, Kuhn, and the Quantum Mechanical ``Revolution''

    NASA Astrophysics Data System (ADS)

    Seth, Suman

    2013-04-01

    ``The history of science, to my knowledge,'' wrote Thomas Kuhn, describing the years just prior to the development of matrix and wave mechanics, ``offers no equally clear, detailed, and cogent example of the creative functions of normal science and crisis.'' By 1924, most quantum theorists shared a sense that there was much wrong with all extant atomic models. Yet not all shared equally in the sense that the failure was either terribly surprising or particularly demoralizing. Not all agreed, that is, that a crisis for Bohr-like models was a crisis for quantum theory. This paper attempts to answer four questions: two about history, two about memory. First, which sub-groups of the quantum theoretical community saw themselves and their field in a state of crisis in the early 1920s? Second, why did they do so, and how was a sense of crisis related to their theoretical practices in physics? Third, do we regard the years before 1925 as a crisis because they were followed by the quantum mechanical revolution? And fourth, to reverse the last question, were we to call into the question the existence of a crisis (for some at least) does that make a subsequent revolution less revolutionary?

  3. What is complementarity?: Niels Bohr and the architecture of quantum theory

    NASA Astrophysics Data System (ADS)

    Plotnitsky, Arkady

    2014-12-01

    This article explores Bohr’s argument, advanced under the heading of ‘complementarity,’ concerning quantum phenomena and quantum mechanics, and its physical and philosophical implications. In Bohr, the term complementarity designates both a particular concept and an overall interpretation of quantum phenomena and quantum mechanics, in part grounded in this concept. While the argument of this article is primarily philosophical, it will also address, historically, the development and transformations of Bohr’s thinking, under the impact of the development of quantum theory and Bohr’s confrontation with Einstein, especially their exchange concerning the EPR experiment, proposed by Einstein, Podolsky and Rosen in 1935. Bohr’s interpretation was progressively characterized by a more radical epistemology, in its ultimate form, which was developed in the 1930s and with which I shall be especially concerned here, defined by his new concepts of phenomenon and atomicity. According to this epistemology, quantum objects are seen as indescribable and possibly even as inconceivable, and as manifesting their existence only in the effects of their interactions with measuring instruments upon those instruments, effects that define phenomena in Bohr’s sense. The absence of causality is an automatic consequence of this epistemology. I shall also consider how probability and statistics work under these epistemological conditions.

  4. Density functional theory of electron transfer beyond the Born-Oppenheimer approximation: Case study of LiF.

    PubMed

    Li, Chen; Requist, Ryan; Gross, E K U

    2018-02-28

    We perform model calculations for a stretched LiF molecule, demonstrating that nonadiabatic charge transfer effects can be accurately and seamlessly described within a density functional framework. In alkali halides like LiF, there is an abrupt change in the ground state electronic distribution due to an electron transfer at a critical bond length R = R c , where an avoided crossing of the lowest adiabatic potential energy surfaces calls the validity of the Born-Oppenheimer approximation into doubt. Modeling the R-dependent electronic structure of LiF within a two-site Hubbard model, we find that nonadiabatic electron-nuclear coupling produces a sizable elongation of the critical R c by 0.5 bohr. This effect is very accurately captured by a simple and rigorously derived correction, with an M -1 prefactor, to the exchange-correlation potential in density functional theory, M = reduced nuclear mass. Since this nonadiabatic term depends on gradients of the nuclear wave function and conditional electronic density, ∇ R χ(R) and ∇ R n(r, R), it couples the Kohn-Sham equations at neighboring R points. Motivated by an observed localization of nonadiabatic effects in nuclear configuration space, we propose a local conditional density approximation-an approximation that reduces the search for nonadiabatic density functionals to the search for a single function y(n).

  5. Interrelationship among Fe-His Bond Strengths, Oxygen Affinities, and Intersubunit Hydrogen Bonding Changes upon Ligand Binding in the β Subunit of Human Hemoglobin: The Alkaline Bohr Effect.

    PubMed

    Nagatomo, Shigenori; Okumura, Miki; Saito, Kazuya; Ogura, Takashi; Kitagawa, Teizo; Nagai, Masako

    2017-03-07

    Regulation of the oxygen affinity of human adult hemoglobin (Hb A) at high pH, known as the alkaline Bohr effect, is essential for its physiological function. In this study, structural mechanisms of the alkaline Bohr effect and pH-dependent O 2 affinity changes were investigated via 1 H nuclear magnetic resonance and visible and UV resonance Raman spectra of mutant Hbs, Hb M Iwate (αH87Y) and Hb M Boston (αH58Y). It was found that even though the binding of O 2 to the α subunits is forbidden in the mutant Hbs, the O 2 affinity was higher at alkaline pH than at neutral pH, and concomitantly, the Fe-His stretching frequency of the β subunits was shifted to higher values. Thus, it was confirmed for the β subunits that the stronger the Fe-His bond, the higher the O 2 affinity. It was found in this study that the quaternary structure of α(Fe 3+ )β(Fe 2+ -CO) of the mutant Hb is closer to T than to the ordinary R at neutral pH. The retained Aspβ94-Hisβ146 hydrogen bond makes the extent of proton release smaller upon ligand binding from Hisβ146, known as one of residues contributing to the alkaline Bohr effect. For these T structures, the Aspα94-Trpβ37 hydrogen bond in the hinge region and the Tyrα42-Aspβ99 hydrogen bond in the switch region of the α 1 -β 2 interface are maintained but elongated at alkaline pH. Thus, a decrease in tension in the Fe-His bond of the β subunits at alkaline pH causes a substantial increase in the change in global structure upon binding of CO to the β subunit.

  6. Target Z dependence of Xe L x-ray emission in heavy ion-atom collision near the Bohr velocity: influence of level matching

    NASA Astrophysics Data System (ADS)

    Ren, Jieru; Zhao, Yongtao; Zhou, Xianming; Cheng, Rui; Lei, Yu; Sun, Yuanbo; Wang, Xing; Xu, Ge; Wang, Yuyu; Liu, Shidong; Yu, Yang; Li, Yongfeng; Zhang, Xiaoan; Xu, Zhongfeng; Xiao, Guoqing

    2013-09-01

    X-ray yields for the projectile L-shell have been measured for collisions between Xe20+ and thick solid targets throughout the periodic table with incident energies near the Bohr velocity. The yields show a very pronounced cyclic dependence on the target atomic number. This result indicates that Xe L x-ray emission intensity is greatly enhanced either in near-symmetric collisions or if the binding energy of the Xe M-shell matches the L- or N-shell binding energy of the target.

  7. Density functional theory of electron transfer beyond the Born-Oppenheimer approximation: Case study of LiF

    NASA Astrophysics Data System (ADS)

    Li, Chen; Requist, Ryan; Gross, E. K. U.

    2018-02-01

    We perform model calculations for a stretched LiF molecule, demonstrating that nonadiabatic charge transfer effects can be accurately and seamlessly described within a density functional framework. In alkali halides like LiF, there is an abrupt change in the ground state electronic distribution due to an electron transfer at a critical bond length R = Rc, where an avoided crossing of the lowest adiabatic potential energy surfaces calls the validity of the Born-Oppenheimer approximation into doubt. Modeling the R-dependent electronic structure of LiF within a two-site Hubbard model, we find that nonadiabatic electron-nuclear coupling produces a sizable elongation of the critical Rc by 0.5 bohr. This effect is very accurately captured by a simple and rigorously derived correction, with an M-1 prefactor, to the exchange-correlation potential in density functional theory, M = reduced nuclear mass. Since this nonadiabatic term depends on gradients of the nuclear wave function and conditional electronic density, ∇Rχ(R) and ∇Rn(r, R), it couples the Kohn-Sham equations at neighboring R points. Motivated by an observed localization of nonadiabatic effects in nuclear configuration space, we propose a local conditional density approximation—an approximation that reduces the search for nonadiabatic density functionals to the search for a single function y(n).

  8. Effectiveness of an Asynchronous Online Module on University Students' Understanding of the Bohr Model of the Hydrogen Atom

    NASA Astrophysics Data System (ADS)

    Farina, William J.; Bodzin, Alec M.

    2017-12-01

    Web-based learning is a growing field in education, yet empirical research into the design of high quality Web-based university science instruction is scarce. A one-week asynchronous online module on the Bohr Model of the atom was developed and implemented guided by the knowledge integration framework. The unit design aligned with three identified metaprinciples of science learning: making science accessible, making thinking visible, and promoting autonomy. Students in an introductory chemistry course at a large east coast university completed either an online module or traditional classroom instruction. Data from 99 students were analyzed and results showed significant knowledge growth in both online and traditional formats. For the online learning group, findings revealed positive student perceptions of their learning experiences, highly positive feedback for online science learning, and an interest amongst students to learn chemistry within an online environment.

  9. Self-consistent frequencies of the electron-photon system

    NASA Astrophysics Data System (ADS)

    Hawton, Margaret

    1993-09-01

    The Heisenberg equations describing the dynamics of coupled Fermion photon operators are solved self-consistently. Photon modes, for which ω~=kc, and particlelike Bohr modes with frequencies ωnI~=(En-EI)/ħ are both approximate solutions to the system of equations that results if the current density is the source in the operator Maxwell equations. Current fluctuations associated with the Bohr modes and required by a fluctuation-dissipation theorem are attributed to the point nature of the particle. The interaction energy is given by the Casimir-force-like expression ΔE=1/2ħtsum(ΔωnI+Δωkc) or by the expectation value of 1/2(qcphi-qp^.A^/mc+q2A2/mc2). It is verified that the equal-time momentum-density and vector-potential operators commute if the contributions of both the Bohr modes and vacuum fluctuations are included. Both electromagnetic and Bohr or radiation-reaction modes are found to contribute equally to spontaneous emission and to the Lamb shift.

  10. How to understand the tunneling in attosecond experiment?. Bohr-Einstein photon box Gedanken experiment, tunneling time and the wave particle duality

    NASA Astrophysics Data System (ADS)

    Kullie, Ossama

    2018-02-01

    The measurement of the tunneling time (T-time) in today's attosecond and strong field (low-frequency) experiments, despite its controversial discussion, offers a fruitful opportunity to understand time measurement and the time in quantum mechanics. In addition, as we will see in this work, a related controversial issue is the particulate nature of the radiation. The T-time in attosecond experiment and its different aspects and models, is discussed in this work, especially in relation to my model of real T-time (Kullie, 2015), where a good agreement with the experiment and an intriguing similarity to the Bohr-Einstein photon box Gedanken experiment was found. The tunneling process itself is still not well understood, but I am arguing that a scattering mechanism (by the laser wave packet) offers a possibility to understand the tunneling process in the tunneling region. This is related to the question about the corpuscular nature of light which is widely discussed in modern quantum optics experiments.

  11. Bringing physics to bear on the phenomenon of life: the divergent positions of Bohr, Delbrück, and Schrödinger.

    PubMed

    Domondon, Andrew T

    2006-09-01

    The received view on the contributions of the physics community to the birth of molecular biology tends to present the physics community as sharing a basic level consensus on how physics should be brought to bear on biology. I argue, however, that a close examination of the views of three leading physicists involved in the birth of molecular biology, Bohr, Delbrück, and Schrödinger, suggests that there existed fundamental disagreements on how physics should be employed to solve problems in biology even within the physics community. In particular, I focus on how these three figures differed sharply in their assessment of the relevance of complementarity, the potential of chemical methods, and the relative importance of classical physics. In addition, I assess and develop Roll-Hansen's attempt to conceptualize this history in terms of models of scientific change advanced by Kuhn and Lakatos. Though neither model is fully successful in explaining the divergence of views among these three physicists, I argue that the extent and quality of difference in their views help elucidate and extend some themes that are left opaque in Kuhn's model.

  12. Bell Nonlocality, Signal Locality and Unpredictability (or What Bohr Could Have Told Einstein at Solvay Had He Known About Bell Experiments)

    NASA Astrophysics Data System (ADS)

    Cavalcanti, Eric G.; Wiseman, Howard M.

    2012-10-01

    The 1964 theorem of John Bell shows that no model that reproduces the predictions of quantum mechanics can simultaneously satisfy the assumptions of locality and determinism. On the other hand, the assumptions of signal locality plus predictability are also sufficient to derive Bell inequalities. This simple theorem, previously noted but published only relatively recently by Masanes, Acin and Gisin, has fundamental implications not entirely appreciated. Firstly, nothing can be concluded about the ontological assumptions of locality or determinism independently of each other—it is possible to reproduce quantum mechanics with deterministic models that violate locality as well as indeterministic models that satisfy locality. On the other hand, the operational assumption of signal locality is an empirically testable (and well-tested) consequence of relativity. Thus Bell inequality violations imply that we can trust that some events are fundamentally unpredictable, even if we cannot trust that they are indeterministic. This result grounds the quantum-mechanical prohibition of arbitrarily accurate predictions on the assumption of no superluminal signalling, regardless of any postulates of quantum mechanics. It also sheds a new light on an early stage of the historical debate between Einstein and Bohr.

  13. Approximate Dynamic Programming: Combining Regional and Local State Following Approximations.

    PubMed

    Deptula, Patryk; Rosenfeld, Joel A; Kamalapurkar, Rushikesh; Dixon, Warren E

    2018-06-01

    An infinite-horizon optimal regulation problem for a control-affine deterministic system is solved online using a local state following (StaF) kernel and a regional model-based reinforcement learning (R-MBRL) method to approximate the value function. Unlike traditional methods such as R-MBRL that aim to approximate the value function over a large compact set, the StaF kernel approach aims to approximate the value function in a local neighborhood of the state that travels within a compact set. In this paper, the value function is approximated using a state-dependent convex combination of the StaF-based and the R-MBRL-based approximations. As the state enters a neighborhood containing the origin, the value function transitions from being approximated by the StaF approach to the R-MBRL approach. Semiglobal uniformly ultimately bounded (SGUUB) convergence of the system states to the origin is established using a Lyapunov-based analysis. Simulation results are provided for two, three, six, and ten-state dynamical systems to demonstrate the scalability and performance of the developed method.

  14. Countably QC-Approximating Posets

    PubMed Central

    Mao, Xuxin; Xu, Luoshan

    2014-01-01

    As a generalization of countably C-approximating posets, the concept of countably QC-approximating posets is introduced. With the countably QC-approximating property, some characterizations of generalized completely distributive lattices and generalized countably approximating posets are given. The main results are as follows: (1) a complete lattice is generalized completely distributive if and only if it is countably QC-approximating and weakly generalized countably approximating; (2) a poset L having countably directed joins is generalized countably approximating if and only if the lattice σ c(L)op of all σ-Scott-closed subsets of L is weakly generalized countably approximating. PMID:25165730

  15. Approximate symmetries of Hamiltonians

    NASA Astrophysics Data System (ADS)

    Chubb, Christopher T.; Flammia, Steven T.

    2017-08-01

    We explore the relationship between approximate symmetries of a gapped Hamiltonian and the structure of its ground space. We start by considering approximate symmetry operators, defined as unitary operators whose commutators with the Hamiltonian have norms that are sufficiently small. We show that when approximate symmetry operators can be restricted to the ground space while approximately preserving certain mutual commutation relations. We generalize the Stone-von Neumann theorem to matrices that approximately satisfy the canonical (Heisenberg-Weyl-type) commutation relations and use this to show that approximate symmetry operators can certify the degeneracy of the ground space even though they only approximately form a group. Importantly, the notions of "approximate" and "small" are all independent of the dimension of the ambient Hilbert space and depend only on the degeneracy in the ground space. Our analysis additionally holds for any gapped band of sufficiently small width in the excited spectrum of the Hamiltonian, and we discuss applications of these ideas to topological quantum phases of matter and topological quantum error correcting codes. Finally, in our analysis, we also provide an exponential improvement upon bounds concerning the existence of shared approximate eigenvectors of approximately commuting operators under an added normality constraint, which may be of independent interest.

  16. Padé Approximant and Minimax Rational Approximation in Standard Cosmology

    NASA Astrophysics Data System (ADS)

    Zaninetti, Lorenzo

    2016-02-01

    The luminosity distance in the standard cosmology as given by $\\Lambda$CDM and consequently the distance modulus for supernovae can be defined by the Pad\\'e approximant. A comparison with a known analytical solution shows that the Pad\\'e approximant for the luminosity distance has an error of $4\\%$ at redshift $= 10$. A similar procedure for the Taylor expansion of the luminosity distance gives an error of $4\\%$ at redshift $=0.7 $; this means that for the luminosity distance, the Pad\\'e approximation is superior to the Taylor series. The availability of an analytical expression for the distance modulus allows applying the Levenberg--Marquardt method to derive the fundamental parameters from the available compilations for supernovae. A new luminosity function for galaxies derived from the truncated gamma probability density function models the observed luminosity function for galaxies when the observed range in absolute magnitude is modeled by the Pad\\'e approximant. A comparison of $\\Lambda$CDM with other cosmologies is done adopting a statistical point of view.

  17. Approximate spatial reasoning

    NASA Technical Reports Server (NTRS)

    Dutta, Soumitra

    1988-01-01

    Much of human reasoning is approximate in nature. Formal models of reasoning traditionally try to be precise and reject the fuzziness of concepts in natural use and replace them with non-fuzzy scientific explicata by a process of precisiation. As an alternate to this approach, it has been suggested that rather than regard human reasoning processes as themselves approximating to some more refined and exact logical process that can be carried out with mathematical precision, the essence and power of human reasoning is in its capability to grasp and use inexact concepts directly. This view is supported by the widespread fuzziness of simple everyday terms (e.g., near tall) and the complexity of ordinary tasks (e.g., cleaning a room). Spatial reasoning is an area where humans consistently reason approximately with demonstrably good results. Consider the case of crossing a traffic intersection. We have only an approximate idea of the locations and speeds of various obstacles (e.g., persons and vehicles), but we nevertheless manage to cross such traffic intersections without any harm. The details of our mental processes which enable us to carry out such intricate tasks in such apparently simple manner are not well understood. However, it is that we try to incorporate such approximate reasoning techniques in our computer systems. Approximate spatial reasoning is very important for intelligent mobile agents (e.g., robots), specially for those operating in uncertain or unknown or dynamic domains.

  18. Two Theories Are Better Than One

    NASA Astrophysics Data System (ADS)

    Jones, Robert

    2008-03-01

    All knowledge is of an approximate character (B. Russell, Human Knowledge, 1948, pg 497 and 507). Our formalisms abstract, idealize, and simplify (R. L. Epstein, Propositional Logics, 2001, Ch XI and E. Bender, An Intro. to Math. Modeling, 1978, pg v and 2). Each formalism is an idealization, often times approximating in its own DIFFERENT ways, each offering somewhat different coverage of the domain. Having MULTIPLE overlaping theories of a knowledge domain is then better than having just one theory (R. Jones, APS general meeting, April 2004). Theories are not unique (T. M. Mitchell, Machine Learning, 1997, pg 65-66 and Cooper, Machine Learning, vol. 9, 1992, pg 319). In the future every field will possess multiple theories of its domain and scientific work and engineering will be performed based on the ensemble predictions of ALL of these. In some cases the theories may be quite divergent, differing greatly one from the other. This idea can be considered an extension of Bohr's notion of complementarity, ``...different experimental arrangements...described by different physical concepts...together and only together exhaust the definable information we can obtain about the object.'' (H. J. Folse, The Philosophy of Neils Bohr, 1985, pg 238)

  19. Legendre-Tau approximation for functional differential equations. Part 3: Eigenvalue approximations and uniform stability

    NASA Technical Reports Server (NTRS)

    Ito, K.

    1984-01-01

    The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. A charactristic equation is derived for the eigenvalues of the resulting approximate system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.

  20. News Almost dry but never dull: ASE 2014 EuroPhysicsFun shows physics to Europe Institute of Physics for Africa (IOPfA) South Sudan Report October 2013 Celebrating the centenary of x-ray diffraction The Niels Bohr Institute—an EPS Historical Site Nordic Research Symposium on Science Education (NFSUN) 2014: inquiry-based science education in technology-rich environments Physics World Cup 2013

    NASA Astrophysics Data System (ADS)

    2014-03-01

    Almost dry but never dull: ASE 2014 EuroPhysicsFun shows physics to Europe Institute of Physics for Africa (IOPfA) South Sudan Report October 2013 Celebrating the centenary of x-ray diffraction The Niels Bohr Institute—an EPS Historical Site Nordic Research Symposium on Science Education (NFSUN) 2014: inquiry-based science education in technology-rich environments Physics World Cup 2013

  1. Approximate spatial reasoning

    NASA Technical Reports Server (NTRS)

    Dutta, Soumitra

    1988-01-01

    A model for approximate spatial reasoning using fuzzy logic to represent the uncertainty in the environment is presented. Algorithms are developed which can be used to reason about spatial information expressed in the form of approximate linguistic descriptions similar to the kind of spatial information processed by humans. Particular attention is given to static spatial reasoning.

  2. CONTRIBUTIONS TO RATIONAL APPROXIMATION,

    DTIC Science & Technology

    Some of the key results of linear Chebyshev approximation theory are extended to generalized rational functions. Prominent among these is Haar’s...linear theorem which yields necessary and sufficient conditions for uniqueness. Some new results in the classic field of rational function Chebyshev...Furthermore a Weierstrass type theorem is proven for rational Chebyshev approximation. A characterization theorem for rational trigonometric Chebyshev approximation in terms of sign alternation is developed. (Author)

  3. Approximate Genealogies Under Genetic Hitchhiking

    PubMed Central

    Pfaffelhuber, P.; Haubold, B.; Wakolbinger, A.

    2006-01-01

    The rapid fixation of an advantageous allele leads to a reduction in linked neutral variation around the target of selection. The genealogy at a neutral locus in such a selective sweep can be simulated by first generating a random path of the advantageous allele's frequency and then a structured coalescent in this background. Usually the frequency path is approximated by a logistic growth curve. We discuss an alternative method that approximates the genealogy by a random binary splitting tree, a so-called Yule tree that does not require first constructing a frequency path. Compared to the coalescent in a logistic background, this method gives a slightly better approximation for identity by descent during the selective phase and a much better approximation for the number of lineages that stem from the founder of the selective sweep. In applications such as the approximation of the distribution of Tajima's D, the two approximation methods perform equally well. For relevant parameter ranges, the Yule approximation is faster. PMID:17182733

  4. Approximate kernel competitive learning.

    PubMed

    Wu, Jian-Sheng; Zheng, Wei-Shi; Lai, Jian-Huang

    2015-03-01

    Kernel competitive learning has been successfully used to achieve robust clustering. However, kernel competitive learning (KCL) is not scalable for large scale data processing, because (1) it has to calculate and store the full kernel matrix that is too large to be calculated and kept in the memory and (2) it cannot be computed in parallel. In this paper we develop a framework of approximate kernel competitive learning for processing large scale dataset. The proposed framework consists of two parts. First, it derives an approximate kernel competitive learning (AKCL), which learns kernel competitive learning in a subspace via sampling. We provide solid theoretical analysis on why the proposed approximation modelling would work for kernel competitive learning, and furthermore, we show that the computational complexity of AKCL is largely reduced. Second, we propose a pseudo-parallelled approximate kernel competitive learning (PAKCL) based on a set-based kernel competitive learning strategy, which overcomes the obstacle of using parallel programming in kernel competitive learning and significantly accelerates the approximate kernel competitive learning for large scale clustering. The empirical evaluation on publicly available datasets shows that the proposed AKCL and PAKCL can perform comparably as KCL, with a large reduction on computational cost. Also, the proposed methods achieve more effective clustering performance in terms of clustering precision against related approximate clustering approaches. Copyright © 2014 Elsevier Ltd. All rights reserved.

  5. Non-invasive measurement of the mean alveolar O(2) tension from the oxygen uptake versus tidal volume curve.

    PubMed

    Jordanoglou, J; Latsi, P; Chroneou, A; Koulouris, N G

    2007-10-01

    The classical equations for measuring the mean and the ideal alveolar O(2) tension are based on assumptions, which are shown to be invalid. So we thought to develop a new, non-invasive method for measuring the mean alveolar P,O(2) within the volume domain (PA,O(2(Bohr))). This method is based on the oxygen uptake vs. tidal volume curve (VO(2) vs. VT) obtained during tidal breathing of room air and/or air enriched with oxygen. PA,O(2(Bohr)) and the ideal alveolar PO(2) (PA,O(2(ideal))) were simultaneously measured in 10 healthy subjects and 34 patients suffering from chronic obstructive pulmonary disease (COPD) breathing tidally room air at rest. Additionally, 10 subjects (three healthy subjects and seven COPD patients) were studied while breathing initially room air and subsequently air enriched with oxygen. According to the results, PA,O(2(Bohr)) considerably differed from PA,O(2(ideal)) (P = 0.004). The cause of the difference, at the individual's R, is: (1) the difference between the arterial and Bohr's alveolar CO(2) tension, mainly in COPD patients, and (2) the inequality between Bohr's alveolar part of the tidal volume for CO(2) and O(2). Furthermore, end-tidal gas tension (PET,CO(2) and PET,O(2)) differed from Pa,CO(2) and PA,O(2(Bohr)) respectively. The deviation of PA,O(2(Bohr)) from PA,O(2(ideal)) has a definite impact on Bohr's dead space ratio for O(2) and CO(2), and on the alveolar-arterial O(2) difference. The difference (PA,O(2(Bohr)) - PA,O(2(ideal))) is not related to the pathology of the disease. So, gas exchange within the lungs should be assessed at the subject's R from PA,O(2(Bohr)) and PA,CO(2(Bohr)) but not from PA,O(2(ideal)) nor Pa,CO(2).

  6. Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix

    DOE PAGES

    Miller, William H.; Cotton, Stephen J.

    2016-08-28

    It is pointed out that the classical phase space distribution in action-angle (a-a) variables obtained from a Wigner function depends on how the calculation is carried out: if one computes the standard Wigner function in Cartesian variables (p, x), and then replaces p and x by their expressions in terms of a-a variables, one obtains a different result than if the Wigner function is computed directly in terms of the a-a variables. Furthermore, the latter procedure gives a result more consistent with classical and semiclassical theory - e.g., by incorporating the Bohr-Sommerfeld quantization condition (quantum states defined by integer valuesmore » of the action variable) as well as the Heisenberg correspondence principle for matrix elements of an operator between such states - and has also been shown to be more accurate when applied to electronically non-adiabatic applications as implemented within the recently developed symmetrical quasi-classical (SQC) Meyer-Miller (MM) approach. Moreover, use of the Wigner function (obtained directly) in a-a variables shows how our standard SQC/MM approach can be used to obtain off-diagonal elements of the electronic density matrix by processing in a different way the same set of trajectories already used (in the SQC/MM methodology) to obtain the diagonal elements.« less

  7. Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix.

    PubMed

    Miller, William H; Cotton, Stephen J

    2016-08-28

    It is pointed out that the classical phase space distribution in action-angle (a-a) variables obtained from a Wigner function depends on how the calculation is carried out: if one computes the standard Wigner function in Cartesian variables (p, x), and then replaces p and x by their expressions in terms of a-a variables, one obtains a different result than if the Wigner function is computed directly in terms of the a-a variables. Furthermore, the latter procedure gives a result more consistent with classical and semiclassical theory-e.g., by incorporating the Bohr-Sommerfeld quantization condition (quantum states defined by integer values of the action variable) as well as the Heisenberg correspondence principle for matrix elements of an operator between such states-and has also been shown to be more accurate when applied to electronically non-adiabatic applications as implemented within the recently developed symmetrical quasi-classical (SQC) Meyer-Miller (MM) approach. Moreover, use of the Wigner function (obtained directly) in a-a variables shows how our standard SQC/MM approach can be used to obtain off-diagonal elements of the electronic density matrix by processing in a different way the same set of trajectories already used (in the SQC/MM methodology) to obtain the diagonal elements.

  8. Cosmological applications of Padé approximant

    NASA Astrophysics Data System (ADS)

    Wei, Hao; Yan, Xiao-Peng; Zhou, Ya-Nan

    2014-01-01

    As is well known, in mathematics, any function could be approximated by the Padé approximant. The Padé approximant is the best approximation of a function by a rational function of given order. In fact, the Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. In the present work, we consider the Padé approximant in two issues. First, we obtain the analytical approximation of the luminosity distance for the flat XCDM model, and find that the relative error is fairly small. Second, we propose several parameterizations for the equation-of-state parameter (EoS) of dark energy based on the Padé approximant. They are well motivated from the mathematical and physical points of view. We confront these EoS parameterizations with the latest observational data, and find that they can work well. In these practices, we show that the Padé approximant could be an useful tool in cosmology, and it deserves further investigation.

  9. Approximate number and approximate time discrimination each correlate with school math abilities in young children.

    PubMed

    Odic, Darko; Lisboa, Juan Valle; Eisinger, Robert; Olivera, Magdalena Gonzalez; Maiche, Alejandro; Halberda, Justin

    2016-01-01

    What is the relationship between our intuitive sense of number (e.g., when estimating how many marbles are in a jar), and our intuitive sense of other quantities, including time (e.g., when estimating how long it has been since we last ate breakfast)? Recent work in cognitive, developmental, comparative psychology, and computational neuroscience has suggested that our representations of approximate number, time, and spatial extent are fundamentally linked and constitute a "generalized magnitude system". But, the shared behavioral and neural signatures between number, time, and space may alternatively be due to similar encoding and decision-making processes, rather than due to shared domain-general representations. In this study, we investigate the relationship between approximate number and time in a large sample of 6-8 year-old children in Uruguay by examining how individual differences in the precision of number and time estimation correlate with school mathematics performance. Over four testing days, each child completed an approximate number discrimination task, an approximate time discrimination task, a digit span task, and a large battery of symbolic math tests. We replicate previous reports showing that symbolic math abilities correlate with approximate number precision and extend those findings by showing that math abilities also correlate with approximate time precision. But, contrary to approximate number and time sharing common representations, we find that each of these dimensions uniquely correlates with formal math: approximate number correlates more strongly with formal math compared to time and continues to correlate with math even when precision in time and individual differences in working memory are controlled for. These results suggest that there are important differences in the mental representations of approximate number and approximate time and further clarify the relationship between quantity representations and mathematics. Copyright

  10. Producing approximate answers to database queries

    NASA Technical Reports Server (NTRS)

    Vrbsky, Susan V.; Liu, Jane W. S.

    1993-01-01

    We have designed and implemented a query processor, called APPROXIMATE, that makes approximate answers available if part of the database is unavailable or if there is not enough time to produce an exact answer. The accuracy of the approximate answers produced improves monotonically with the amount of data retrieved to produce the result. The exact answer is produced if all of the needed data are available and query processing is allowed to continue until completion. The monotone query processing algorithm of APPROXIMATE works within the standard relational algebra framework and can be implemented on a relational database system with little change to the relational architecture. We describe here the approximation semantics of APPROXIMATE that serves as the basis for meaningful approximations of both set-valued and single-valued queries. We show how APPROXIMATE is implemented to make effective use of semantic information, provided by an object-oriented view of the database, and describe the additional overhead required by APPROXIMATE.

  11. Structural optimization with approximate sensitivities

    NASA Technical Reports Server (NTRS)

    Patnaik, S. N.; Hopkins, D. A.; Coroneos, R.

    1994-01-01

    Computational efficiency in structural optimization can be enhanced if the intensive computations associated with the calculation of the sensitivities, that is, gradients of the behavior constraints, are reduced. Approximation to gradients of the behavior constraints that can be generated with small amount of numerical calculations is proposed. Structural optimization with these approximate sensitivities produced correct optimum solution. Approximate gradients performed well for different nonlinear programming methods, such as the sequence of unconstrained minimization technique, method of feasible directions, sequence of quadratic programming, and sequence of linear programming. Structural optimization with approximate gradients can reduce by one third the CPU time that would otherwise be required to solve the problem with explicit closed-form gradients. The proposed gradient approximation shows potential to reduce intensive computation that has been associated with traditional structural optimization.

  12. Bounded-Degree Approximations of Stochastic Networks

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Quinn, Christopher J.; Pinar, Ali; Kiyavash, Negar

    2017-06-01

    We propose algorithms to approximate directed information graphs. Directed information graphs are probabilistic graphical models that depict causal dependencies between stochastic processes in a network. The proposed algorithms identify optimal and near-optimal approximations in terms of Kullback-Leibler divergence. The user-chosen sparsity trades off the quality of the approximation against visual conciseness and computational tractability. One class of approximations contains graphs with speci ed in-degrees. Another class additionally requires that the graph is connected. For both classes, we propose algorithms to identify the optimal approximations and also near-optimal approximations, using a novel relaxation of submodularity. We also propose algorithms to identifymore » the r-best approximations among these classes, enabling robust decision making.« less

  13. Limitations of shallow nets approximation.

    PubMed

    Lin, Shao-Bo

    2017-10-01

    In this paper, we aim at analyzing the approximation abilities of shallow networks in reproducing kernel Hilbert spaces (RKHSs). We prove that there is a probability measure such that the achievable lower bound for approximating by shallow nets can be realized for all functions in balls of reproducing kernel Hilbert space with high probability, which is different with the classical minimax approximation error estimates. This result together with the existing approximation results for deep nets shows the limitations for shallow nets and provides a theoretical explanation on why deep nets perform better than shallow nets. Copyright © 2017 Elsevier Ltd. All rights reserved.

  14. More on approximations of Poisson probabilities

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Kao, C

    1980-05-01

    Calculation of Poisson probabilities frequently involves calculating high factorials, which becomes tedious and time-consuming with regular calculators. The usual way to overcome this difficulty has been to find approximations by making use of the table of the standard normal distribution. A new transformation proposed by Kao in 1978 appears to perform better for this purpose than traditional transformations. In the present paper several approximation methods are stated and compared numerically, including an approximation method that utilizes a modified version of Kao's transformation. An approximation based on a power transformation was found to outperform those based on the square-root type transformationsmore » as proposed in literature. The traditional Wilson-Hilferty approximation and Makabe-Morimura approximation are extremely poor compared with this approximation. 4 tables. (RWR)« less

  15. Approximate circuits for increased reliability

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Hamlet, Jason R.; Mayo, Jackson R.

    2015-08-18

    Embodiments of the invention describe a Boolean circuit having a voter circuit and a plurality of approximate circuits each based, at least in part, on a reference circuit. The approximate circuits are each to generate one or more output signals based on values of received input signals. The voter circuit is to receive the one or more output signals generated by each of the approximate circuits, and is to output one or more signals corresponding to a majority value of the received signals. At least some of the approximate circuits are to generate an output value different than the referencemore » circuit for one or more input signal values; however, for each possible input signal value, the majority values of the one or more output signals generated by the approximate circuits and received by the voter circuit correspond to output signal result values of the reference circuit.« less

  16. Risk approximation in decision making: approximative numeric abilities predict advantageous decisions under objective risk.

    PubMed

    Mueller, Silke M; Schiebener, Johannes; Delazer, Margarete; Brand, Matthias

    2018-01-22

    Many decision situations in everyday life involve mathematical considerations. In decisions under objective risk, i.e., when explicit numeric information is available, executive functions and abilities to handle exact numbers and ratios are predictors of objectively advantageous choices. Although still debated, exact numeric abilities, e.g., normative calculation skills, are assumed to be related to approximate number processing skills. The current study investigates the effects of approximative numeric abilities on decision making under objective risk. Participants (N = 153) performed a paradigm measuring number-comparison, quantity-estimation, risk-estimation, and decision-making skills on the basis of rapid dot comparisons. Additionally, a risky decision-making task with exact numeric information was administered, as well as tasks measuring executive functions and exact numeric abilities, e.g., mental calculation and ratio processing skills, were conducted. Approximative numeric abilities significantly predicted advantageous decision making, even beyond the effects of executive functions and exact numeric skills. Especially being able to make accurate risk estimations seemed to contribute to superior choices. We recommend approximation skills and approximate number processing to be subject of future investigations on decision making under risk.

  17. Combining global and local approximations

    NASA Technical Reports Server (NTRS)

    Haftka, Raphael T.

    1991-01-01

    A method based on a linear approximation to a scaling factor, designated the 'global-local approximation' (GLA) method, is presented and shown capable of extending the range of usefulness of derivative-based approximations to a more refined model. The GLA approach refines the conventional scaling factor by means of a linearly varying, rather than constant, scaling factor. The capabilities of the method are demonstrated for a simple beam example with a crude and more refined FEM model.

  18. Spline approximation, Part 1: Basic methodology

    NASA Astrophysics Data System (ADS)

    Ezhov, Nikolaj; Neitzel, Frank; Petrovic, Svetozar

    2018-04-01

    In engineering geodesy point clouds derived from terrestrial laser scanning or from photogrammetric approaches are almost never used as final results. For further processing and analysis a curve or surface approximation with a continuous mathematical function is required. In this paper the approximation of 2D curves by means of splines is treated. Splines offer quite flexible and elegant solutions for interpolation or approximation of "irregularly" distributed data. Depending on the problem they can be expressed as a function or as a set of equations that depend on some parameter. Many different types of splines can be used for spline approximation and all of them have certain advantages and disadvantages depending on the approximation problem. In a series of three articles spline approximation is presented from a geodetic point of view. In this paper (Part 1) the basic methodology of spline approximation is demonstrated using splines constructed from ordinary polynomials and splines constructed from truncated polynomials. In the forthcoming Part 2 the notion of B-spline will be explained in a unique way, namely by using the concept of convex combinations. The numerical stability of all spline approximation approaches as well as the utilization of splines for deformation detection will be investigated on numerical examples in Part 3.

  19. Monotone Boolean approximation

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Hulme, B.L.

    1982-12-01

    This report presents a theory of approximation of arbitrary Boolean functions by simpler, monotone functions. Monotone increasing functions can be expressed without the use of complements. Nonconstant monotone increasing functions are important in their own right since they model a special class of systems known as coherent systems. It is shown here that when Boolean expressions for noncoherent systems become too large to treat exactly, then monotone approximations are easily defined. The algorithms proposed here not only provide simpler formulas but also produce best possible upper and lower monotone bounds for any Boolean function. This theory has practical application formore » the analysis of noncoherent fault trees and event tree sequences.« less

  20. Testing approximations for non-linear gravitational clustering

    NASA Technical Reports Server (NTRS)

    Coles, Peter; Melott, Adrian L.; Shandarin, Sergei F.

    1993-01-01

    The accuracy of various analytic approximations for following the evolution of cosmological density fluctuations into the nonlinear regime is investigated. The Zel'dovich approximation is found to be consistently the best approximation scheme. It is extremely accurate for power spectra characterized by n = -1 or less; when the approximation is 'enhanced' by truncating highly nonlinear Fourier modes the approximation is excellent even for n = +1. The performance of linear theory is less spectrum-dependent, but this approximation is less accurate than the Zel'dovich one for all cases because of the failure to treat dynamics. The lognormal approximation generally provides a very poor fit to the spatial pattern.

  1. Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation

    NASA Astrophysics Data System (ADS)

    Bervillier, C.; Boisseau, B.; Giacomini, H.

    2008-02-01

    The relation between the Wilson-Polchinski and the Litim optimized ERGEs in the local potential approximation is studied with high accuracy using two different analytical approaches based on a field expansion: a recently proposed genuine analytical approximation scheme to two-point boundary value problems of ordinary differential equations, and a new one based on approximating the solution by generalized hypergeometric functions. A comparison with the numerical results obtained with the shooting method is made. A similar accuracy is reached in each case. Both two methods appear to be more efficient than the usual field expansions frequently used in the current studies of ERGEs (in particular for the Wilson-Polchinski case in the study of which they fail).

  2. An Approximate Approach to Automatic Kernel Selection.

    PubMed

    Ding, Lizhong; Liao, Shizhong

    2016-02-02

    Kernel selection is a fundamental problem of kernel-based learning algorithms. In this paper, we propose an approximate approach to automatic kernel selection for regression from the perspective of kernel matrix approximation. We first introduce multilevel circulant matrices into automatic kernel selection, and develop two approximate kernel selection algorithms by exploiting the computational virtues of multilevel circulant matrices. The complexity of the proposed algorithms is quasi-linear in the number of data points. Then, we prove an approximation error bound to measure the effect of the approximation in kernel matrices by multilevel circulant matrices on the hypothesis and further show that the approximate hypothesis produced with multilevel circulant matrices converges to the accurate hypothesis produced with kernel matrices. Experimental evaluations on benchmark datasets demonstrate the effectiveness of approximate kernel selection.

  3. Approximate error conjugation gradient minimization methods

    DOEpatents

    Kallman, Jeffrey S

    2013-05-21

    In one embodiment, a method includes selecting a subset of rays from a set of all rays to use in an error calculation for a constrained conjugate gradient minimization problem, calculating an approximate error using the subset of rays, and calculating a minimum in a conjugate gradient direction based on the approximate error. In another embodiment, a system includes a processor for executing logic, logic for selecting a subset of rays from a set of all rays to use in an error calculation for a constrained conjugate gradient minimization problem, logic for calculating an approximate error using the subset of rays, and logic for calculating a minimum in a conjugate gradient direction based on the approximate error. In other embodiments, computer program products, methods, and systems are described capable of using approximate error in constrained conjugate gradient minimization problems.

  4. Applied Routh approximation

    NASA Technical Reports Server (NTRS)

    Merrill, W. C.

    1978-01-01

    The Routh approximation technique for reducing the complexity of system models was applied in the frequency domain to a 16th order, state variable model of the F100 engine and to a 43d order, transfer function model of a launch vehicle boost pump pressure regulator. The results motivate extending the frequency domain formulation of the Routh method to the time domain in order to handle the state variable formulation directly. The time domain formulation was derived and a characterization that specifies all possible Routh similarity transformations was given. The characterization was computed by solving two eigenvalue-eigenvector problems. The application of the time domain Routh technique to the state variable engine model is described, and some results are given. Additional computational problems are discussed, including an optimization procedure that can improve the approximation accuracy by taking advantage of the transformation characterization.

  5. Analysis of corrections to the eikonal approximation

    NASA Astrophysics Data System (ADS)

    Hebborn, C.; Capel, P.

    2017-11-01

    Various corrections to the eikonal approximations are studied for two- and three-body nuclear collisions with the goal to extend the range of validity of this approximation to beam energies of 10 MeV/nucleon. Wallace's correction does not improve much the elastic-scattering cross sections obtained at the usual eikonal approximation. On the contrary, a semiclassical approximation that substitutes the impact parameter by a complex distance of closest approach computed with the projectile-target optical potential efficiently corrects the eikonal approximation. This opens the possibility to analyze data measured down to 10 MeV/nucleon within eikonal-like reaction models.

  6. Approximations of Two-Attribute Utility Functions

    DTIC Science & Technology

    1976-09-01

    preferred to") be a bina-zy relation on the set • of simple probability measures or ’gambles’ defined on a set T of consequences. Throughout this study it...simplifying independence assumptions. Although there are several approaches to this problem, the21 present study will focus on approximations of u... study will elicit additional interest in the topic. 2. REMARKS ON APPROXIMATION THEORY This section outlines a few basic ideas of approximation theory

  7. Born approximation in linear-time invariant system

    NASA Astrophysics Data System (ADS)

    Gumjudpai, Burin

    2017-09-01

    An alternative way of finding the LTI’s solution with the Born approximation, is investigated. We use Born approximation in the LTI and in the transformed LTI in form of Helmholtz equation. General solution are considered as infinite series or Feynman graph. Slow-roll approximation are explored. Transforming the LTI system into Helmholtz equation, approximated general solution can be found for any given forms of force with its initial value.

  8. Approximate Model of Zone Sedimentation

    NASA Astrophysics Data System (ADS)

    Dzianik, František

    2011-12-01

    The process of zone sedimentation is affected by many factors that are not possible to express analytically. For this reason, the zone settling is evaluated in practice experimentally or by application of an empirical mathematical description of the process. The paper presents the development of approximate model of zone settling, i.e. the general function which should properly approximate the behaviour of the settling process within its entire range and at the various conditions. Furthermore, the specification of the model parameters by the regression analysis of settling test results is shown. The suitability of the model is reviewed by graphical dependencies and by statistical coefficients of correlation. The approximate model could by also useful on the simplification of process design of continual settling tanks and thickeners.

  9. Energy conservation - A test for scattering approximations

    NASA Technical Reports Server (NTRS)

    Acquista, C.; Holland, A. C.

    1980-01-01

    The roles of the extinction theorem and energy conservation in obtaining the scattering and absorption cross sections for several light scattering approximations are explored. It is shown that the Rayleigh, Rayleigh-Gans, anomalous diffraction, geometrical optics, and Shifrin approximations all lead to reasonable values of the cross sections, while the modified Mie approximation does not. Further examination of the modified Mie approximation for the ensembles of nonspherical particles reveals additional problems with that method.

  10. Recognition of computerized facial approximations by familiar assessors.

    PubMed

    Richard, Adam H; Monson, Keith L

    2017-11-01

    Studies testing the effectiveness of facial approximations typically involve groups of participants who are unfamiliar with the approximated individual(s). This limitation requires the use of photograph arrays including a picture of the subject for comparison to the facial approximation. While this practice is often necessary due to the difficulty in obtaining a group of assessors who are familiar with the approximated subject, it may not accurately simulate the thought process of the target audience (friends and family members) in comparing a mental image of the approximated subject to the facial approximation. As part of a larger process to evaluate the effectiveness and best implementation of the ReFace facial approximation software program, the rare opportunity arose to conduct a recognition study using assessors who were personally acquainted with the subjects of the approximations. ReFace facial approximations were generated based on preexisting medical scans, and co-workers of the scan donors were tested on whether they could accurately pick out the approximation of their colleague from arrays of facial approximations. Results from the study demonstrated an overall poor recognition performance (i.e., where a single choice within a pool is not enforced) for individuals who were familiar with the approximated subjects. Out of 220 recognition tests only 10.5% resulted in the assessor selecting the correct approximation (or correctly choosing not to make a selection when the array consisted only of foils), an outcome that was not significantly different from the 9% random chance rate. When allowed to select multiple approximations the assessors felt resembled the target individual, the overall sensitivity for ReFace approximations was 16.0% and the overall specificity was 81.8%. These results differ markedly from the results of a previous study using assessors who were unfamiliar with the approximated subjects. Some possible explanations for this disparity in

  11. Consistent Yokoya-Chen Approximation to Beamstrahlung(LCC-0010)

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Peskin, M

    2004-04-22

    I reconsider the Yokoya-Chen approximate evolution equation for beamstrahlung and modify it slightly to generate simple, consistent analytical approximations for the electron and photon energy spectra. I compare these approximations to previous ones, and to simulation data.I reconsider the Yokoya-Chen approximate evolution equation for beamstrahlung and modify it slightly to generate simple, consistent analytical approximations for the electron and photon energy spectra. I compare these approximations to previous ones, and to simulation data.

  12. Exponential approximations in optimal design

    NASA Technical Reports Server (NTRS)

    Belegundu, A. D.; Rajan, S. D.; Rajgopal, J.

    1990-01-01

    One-point and two-point exponential functions have been developed and proved to be very effective approximations of structural response. The exponential has been compared to the linear, reciprocal and quadratic fit methods. Four test problems in structural analysis have been selected. The use of such approximations is attractive in structural optimization to reduce the numbers of exact analyses which involve computationally expensive finite element analysis.

  13. Combination of the pair density approximation and the Takahashi–Imada approximation for path integral Monte Carlo simulations

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Zillich, Robert E., E-mail: robert.zillich@jku.at

    2015-11-15

    We construct an accurate imaginary time propagator for path integral Monte Carlo simulations for heterogeneous systems consisting of a mixture of atoms and molecules. We combine the pair density approximation, which is highly accurate but feasible only for the isotropic interactions between atoms, with the Takahashi–Imada approximation for general interactions. We present finite temperature simulations results for energy and structure of molecules–helium clusters X{sup 4}He{sub 20} (X=HCCH and LiH) which show a marked improvement over the Trotter approximation which has a 2nd-order time step bias. We show that the 4th-order corrections of the Takahashi–Imada approximation can also be applied perturbativelymore » to a 2nd-order simulation.« less

  14. Better approximation guarantees for job-shop scheduling

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Goldberg, L.A.; Paterson, M.; Srinivasan, A.

    1997-06-01

    Job-shop scheduling is a classical NP-hard problem. Shmoys, Stein & Wein presented the first polynomial-time approximation algorithm for this problem that has a good (polylogarithmic) approximation guarantee. We improve the approximation guarantee of their work, and present further improvements for some important NP-hard special cases of this problem (e.g., in the preemptive case where machines can suspend work on operations and later resume). We also present NC algorithms with improved approximation guarantees for some NP-hard special cases.

  15. Linear Approximation SAR Azimuth Processing Study

    NASA Technical Reports Server (NTRS)

    Lindquist, R. B.; Masnaghetti, R. K.; Belland, E.; Hance, H. V.; Weis, W. G.

    1979-01-01

    A segmented linear approximation of the quadratic phase function that is used to focus the synthetic antenna of a SAR was studied. Ideal focusing, using a quadratic varying phase focusing function during the time radar target histories are gathered, requires a large number of complex multiplications. These can be largely eliminated by using linear approximation techniques. The result is a reduced processor size and chip count relative to ideally focussed processing and a correspondingly increased feasibility for spaceworthy implementation. A preliminary design and sizing for a spaceworthy linear approximation SAR azimuth processor meeting requirements similar to those of the SEASAT-A SAR was developed. The study resulted in a design with approximately 1500 IC's, 1.2 cubic feet of volume, and 350 watts of power for a single look, 4000 range cell azimuth processor with 25 meters resolution.

  16. Chemical Laws, Idealization and Approximation

    NASA Astrophysics Data System (ADS)

    Tobin, Emma

    2013-07-01

    This paper examines the notion of laws in chemistry. Vihalemm ( Found Chem 5(1):7-22, 2003) argues that the laws of chemistry are fundamentally the same as the laws of physics they are all ceteris paribus laws which are true "in ideal conditions". In contrast, Scerri (2000) contends that the laws of chemistry are fundamentally different to the laws of physics, because they involve approximations. Christie ( Stud Hist Philos Sci 25:613-629, 1994) and Christie and Christie ( Of minds and molecules. Oxford University Press, New York, pp. 34-50, 2000) agree that the laws of chemistry are operationally different to the laws of physics, but claim that the distinction between exact and approximate laws is too simplistic to taxonomise them. Approximations in chemistry involve diverse kinds of activity and often what counts as a scientific law in chemistry is dictated by the context of its use in scientific practice. This paper addresses the question of what makes chemical laws distinctive independently of the separate question as to how they are related to the laws of physics. From an analysis of some candidate ceteris paribus laws in chemistry, this paper argues that there are two distinct kinds of ceteris paribus laws in chemistry; idealized and approximate chemical laws. Thus, while Christie ( Stud Hist Philos Sci 25:613-629, 1994) and Christie and Christie ( Of minds and molecules. Oxford University Press, New York, pp. 34--50, 2000) are correct to point out that the candidate generalisations in chemistry are diverse and heterogeneous, a distinction between idealizations and approximations can nevertheless be used to successfully taxonomise them.

  17. On Nash-Equilibria of Approximation-Stable Games

    NASA Astrophysics Data System (ADS)

    Awasthi, Pranjal; Balcan, Maria-Florina; Blum, Avrim; Sheffet, Or; Vempala, Santosh

    One reason for wanting to compute an (approximate) Nash equilibrium of a game is to predict how players will play. However, if the game has multiple equilibria that are far apart, or ɛ-equilibria that are far in variation distance from the true Nash equilibrium strategies, then this prediction may not be possible even in principle. Motivated by this consideration, in this paper we define the notion of games that are approximation stable, meaning that all ɛ-approximate equilibria are contained inside a small ball of radius Δ around a true equilibrium, and investigate a number of their properties. Many natural small games such as matching pennies and rock-paper-scissors are indeed approximation stable. We show furthermore there exist 2-player n-by-n approximation-stable games in which the Nash equilibrium and all approximate equilibria have support Ω(log n). On the other hand, we show all (ɛ,Δ) approximation-stable games must have an ɛ-equilibrium of support O(Δ^{2-o(1)}/ɛ2{log n}), yielding an immediate n^{O(Δ^{2-o(1)}/ɛ^2log n)}-time algorithm, improving over the bound of [11] for games satisfying this condition. We in addition give a polynomial-time algorithm for the case that Δ and ɛ are sufficiently close together. We also consider an inverse property, namely that all non-approximate equilibria are far from some true equilibrium, and give an efficient algorithm for games satisfying that condition.

  18. Legendre-tau approximations for functional differential equations

    NASA Technical Reports Server (NTRS)

    Ito, K.; Teglas, R.

    1986-01-01

    The numerical approximation of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.

  19. Legendre-Tau approximations for functional differential equations

    NASA Technical Reports Server (NTRS)

    Ito, K.; Teglas, R.

    1983-01-01

    The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.

  20. Modulators of haemocyanin oxygen affinity in the hypoxia- and sulphide-tolerant Baltic isopod Saduria entomon (L.).

    PubMed

    Hagerman, L; Vismann, B

    2001-11-01

    Dialysed haemocyanin from the isopod Saduria entomon had a considerably increased oxygen affinity (lower P50) and Bohr factor (-1.71) compared to native haemocyanin (Bohr factor -1.36) indicating that dialysis removes a small molecule size modulating factor decreasing the affinity of native haemolymph. Dialysed haemocyanin had a slightly lower co-operativity (2.42 +/- 0.3) than native haemocyanin (2.9 +/- 0.2). L-Lactate (10 mmol l(-1)) improved oxygen affinity by 1-1.5 torr while urate had no effect. Mg2+ affected affinity in a pH-dependent manner (Bohr-factor increased to -1.67) while Ca2+ had no effect on the Bohr factor but increased affinity with ca 1 torr. Thiosulphate changed the Bohr factor to -1.75 to -1.82, similar to dialysed blood. Co-operativity was in neither case affected. The haemocyanin characteristics of S. entomon are similar to those of crustaceans from hydrothermal vents. These characteristics are probably general for crustaceans that are more or less permanently exposed to sulphide.

  1. Piecewise linear approximation for hereditary control problems

    NASA Technical Reports Server (NTRS)

    Propst, Georg

    1987-01-01

    Finite dimensional approximations are presented for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The approximation scheme is applied to optimal control problems when a quadratic cost integral has to be minimized subject to the controlled retarded system. It is shown that the approximate optimal feedback operators converge to the true ones both in case the cost integral ranges over a finite time interval as well as in the case it ranges over an infinite time interval. The arguments in the latter case rely on the fact that the piecewise linear approximations to stable systems are stable in a uniform sense. This feature is established using a vector-component stability criterion in the state space R(n) x L(2) and the favorable eigenvalue behavior of the piecewise linear approximations.

  2. Approximation concepts for efficient structural synthesis

    NASA Technical Reports Server (NTRS)

    Schmit, L. A., Jr.; Miura, H.

    1976-01-01

    It is shown that efficient structural synthesis capabilities can be created by using approximation concepts to mesh finite element structural analysis methods with nonlinear mathematical programming techniques. The history of the application of mathematical programming techniques to structural design optimization problems is reviewed. Several rather general approximation concepts are described along with the technical foundations of the ACCESS 1 computer program, which implements several approximation concepts. A substantial collection of structural design problems involving truss and idealized wing structures is presented. It is concluded that since the basic ideas employed in creating the ACCESS 1 program are rather general, its successful development supports the contention that the introduction of approximation concepts will lead to the emergence of a new generation of practical and efficient, large scale, structural synthesis capabilities in which finite element analysis methods and mathematical programming algorithms will play a central role.

  3. Approximations of e and ?: An Exploration

    ERIC Educational Resources Information Center

    Brown, Philip R.

    2017-01-01

    Fractional approximations of e and p are discovered by searching for repetitions or partial repetitions of digit strings in their expansions in different number bases. The discovery of such fractional approximations is suggested for students and teachers as an entry point into mathematics research.

  4. Topics in Multivariate Approximation Theory.

    DTIC Science & Technology

    1982-05-01

    once that a continuous function f can be approximated from Sa :o span (N3 )B63 to within *(f, 131 ), with 13 t- sup3 e3 dian PS The simple approximation...N(C) 3- U P s P3AC 0 0 ) . Then, as in Lebesgue’s inequality, we could conclude that f - Qf - f-p - Q(f-p) , for all p e k k therefore I(f-0f) JCI 4 I

  5. Approximation methods for combined thermal/structural design

    NASA Technical Reports Server (NTRS)

    Haftka, R. T.; Shore, C. P.

    1979-01-01

    Two approximation concepts for combined thermal/structural design are evaluated. The first concept is an approximate thermal analysis based on the first derivatives of structural temperatures with respect to design variables. Two commonly used first-order Taylor series expansions are examined. The direct and reciprocal expansions are special members of a general family of approximations, and for some conditions other members of that family of approximations are more accurate. Several examples are used to compare the accuracy of the different expansions. The second approximation concept is the use of critical time points for combined thermal and stress analyses of structures with transient loading conditions. Significant time savings are realized by identifying critical time points and performing the stress analysis for those points only. The design of an insulated panel which is exposed to transient heating conditions is discussed.

  6. A result about scale transformation families in approximation

    NASA Astrophysics Data System (ADS)

    Apprato, Dominique; Gout, Christian

    2000-06-01

    Scale transformations are common in approximation. In surface approximation from rapidly varying data, one wants to suppress, or at least dampen the oscillations of the approximation near steep gradients implied by the data. In that case, scale transformations can be used to give some control over overshoot when the surface has large variations of its gradient. Conversely, in image analysis, scale transformations are used in preprocessing to enhance some features present on the image or to increase jumps of grey levels before segmentation of the image. In this paper, we establish the convergence of an approximation method which allows some control over the behavior of the approximation. More precisely, we study the convergence of an approximation from a data set of , while using scale transformations on the values before and after classical approximation. In addition, the construction of scale transformations is also given. The algorithm is presented with some numerical examples.

  7. Quirks of Stirling's Approximation

    ERIC Educational Resources Information Center

    Macrae, Roderick M.; Allgeier, Benjamin M.

    2013-01-01

    Stirling's approximation to ln "n"! is typically introduced to physical chemistry students as a step in the derivation of the statistical expression for the entropy. However, naive application of this approximation leads to incorrect conclusions. In this article, the problem is first illustrated using a familiar "toy…

  8. Approximations to camera sensor noise

    NASA Astrophysics Data System (ADS)

    Jin, Xiaodan; Hirakawa, Keigo

    2013-02-01

    Noise is present in all image sensor data. Poisson distribution is said to model the stochastic nature of the photon arrival process, while it is common to approximate readout/thermal noise by additive white Gaussian noise (AWGN). Other sources of signal-dependent noise such as Fano and quantization also contribute to the overall noise profile. Question remains, however, about how best to model the combined sensor noise. Though additive Gaussian noise with signal-dependent noise variance (SD-AWGN) and Poisson corruption are two widely used models to approximate the actual sensor noise distribution, the justification given to these types of models are based on limited evidence. The goal of this paper is to provide a more comprehensive characterization of random noise. We concluded by presenting concrete evidence that Poisson model is a better approximation to real camera model than SD-AWGN. We suggest further modification to Poisson that may improve the noise model.

  9. Approximate techniques of structural reanalysis

    NASA Technical Reports Server (NTRS)

    Noor, A. K.; Lowder, H. E.

    1974-01-01

    A study is made of two approximate techniques for structural reanalysis. These include Taylor series expansions for response variables in terms of design variables and the reduced-basis method. In addition, modifications to these techniques are proposed to overcome some of their major drawbacks. The modifications include a rational approach to the selection of the reduced-basis vectors and the use of Taylor series approximation in an iterative process. For the reduced basis a normalized set of vectors is chosen which consists of the original analyzed design and the first-order sensitivity analysis vectors. The use of the Taylor series approximation as a first (initial) estimate in an iterative process, can lead to significant improvements in accuracy, even with one iteration cycle. Therefore, the range of applicability of the reanalysis technique can be extended. Numerical examples are presented which demonstrate the gain in accuracy obtained by using the proposed modification techniques, for a wide range of variations in the design variables.

  10. Approximate Bayesian evaluations of measurement uncertainty

    NASA Astrophysics Data System (ADS)

    Possolo, Antonio; Bodnar, Olha

    2018-04-01

    The Guide to the Expression of Uncertainty in Measurement (GUM) includes formulas that produce an estimate of a scalar output quantity that is a function of several input quantities, and an approximate evaluation of the associated standard uncertainty. This contribution presents approximate, Bayesian counterparts of those formulas for the case where the output quantity is a parameter of the joint probability distribution of the input quantities, also taking into account any information about the value of the output quantity available prior to measurement expressed in the form of a probability distribution on the set of possible values for the measurand. The approximate Bayesian estimates and uncertainty evaluations that we present have a long history and illustrious pedigree, and provide sufficiently accurate approximations in many applications, yet are very easy to implement in practice. Differently from exact Bayesian estimates, which involve either (analytical or numerical) integrations, or Markov Chain Monte Carlo sampling, the approximations that we describe involve only numerical optimization and simple algebra. Therefore, they make Bayesian methods widely accessible to metrologists. We illustrate the application of the proposed techniques in several instances of measurement: isotopic ratio of silver in a commercial silver nitrate; odds of cryptosporidiosis in AIDS patients; height of a manometer column; mass fraction of chromium in a reference material; and potential-difference in a Zener voltage standard.

  11. Leaky, Pulsating Laser-Plasma Filaments in the Nonlinear Schrödinger Equation Approximation

    NASA Astrophysics Data System (ADS)

    Johnston, Tudor Wyatt

    1996-11-01

    For studying self-focusing and filamention of electromagnetic beams in plasmas (and other media) (including laser beams in ICF plasmas) with power well over the critical value for self-focusing, considerable success has recently been achieved using the well-known nonlinear Schrodinger equation with "saturating" nonlinearities (i.e. those for which the nonlinearity is not infinite even if the field becomes indefinitely large). High-power beams with noticeable structure can indeed break up into numerous filaments, but these filaments emerge from the phase of filament creation in states rather close to the known filament equilibria. This is seen to be due to the loss of excess power from the filaments as they are forming and pulsating, so that each filament thus tends to condense to an equilibrium state. Analysis of filaments near equilibrium is thus seen to be more relevant than anticipated, but radiation from the filament(s) can often play an important role. Concepts have been borrowed from normal quantum mechanics (e.g. Bohr oscillations at the appropriate Rabi frequency) to explain the oscillation spectrum and whether significant radiation damping is present. Apparent momentum transfer during multiple filament creation seems common. Some discussion is given of the application of this kind of analysis to the radial confinement of sub-picosecond laser pulses over distances of tens of meters. (François Vidal (INRS) is the major collaborator in this work.)

  12. Difference equation state approximations for nonlinear hereditary control problems

    NASA Technical Reports Server (NTRS)

    Rosen, I. G.

    1982-01-01

    Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.

  13. Approximation algorithms for planning and control

    NASA Technical Reports Server (NTRS)

    Boddy, Mark; Dean, Thomas

    1989-01-01

    A control system operating in a complex environment will encounter a variety of different situations, with varying amounts of time available to respond to critical events. Ideally, such a control system will do the best possible with the time available. In other words, its responses should approximate those that would result from having unlimited time for computation, where the degree of the approximation depends on the amount of time it actually has. There exist approximation algorithms for a wide variety of problems. Unfortunately, the solution to any reasonably complex control problem will require solving several computationally intensive problems. Algorithms for successive approximation are a subclass of the class of anytime algorithms, algorithms that return answers for any amount of computation time, where the answers improve as more time is allotted. An architecture is described for allocating computation time to a set of anytime algorithms, based on expectations regarding the value of the answers they return. The architecture described is quite general, producing optimal schedules for a set of algorithms under widely varying conditions.

  14. Approximation methods in gravitational-radiation theory

    NASA Technical Reports Server (NTRS)

    Will, C. M.

    1986-01-01

    The observation of gravitational-radiation damping in the binary pulsar PSR 1913 + 16 and the ongoing experimental search for gravitational waves of extraterrestrial origin have made the theory of gravitational radiation an active branch of classical general relativity. In calculations of gravitational radiation, approximation methods play a crucial role. Recent developments are summarized in two areas in which approximations are important: (a) the quadrupole approxiamtion, which determines the energy flux and the radiation reaction forces in weak-field, slow-motion, source-within-the-near-zone systems such as the binary pulsar; and (b) the normal modes of oscillation of black holes, where the Wentzel-Kramers-Brillouin approximation gives accurate estimates of the complex frequencies of the modes.

  15. Difference equation state approximations for nonlinear hereditary control problems

    NASA Technical Reports Server (NTRS)

    Rosen, I. G.

    1984-01-01

    Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems. Previously announced in STAR as N83-33589

  16. Approximate number word knowledge before the cardinal principle.

    PubMed

    Gunderson, Elizabeth A; Spaepen, Elizabet; Levine, Susan C

    2015-02-01

    Approximate number word knowledge-understanding the relation between the count words and the approximate magnitudes of sets-is a critical piece of knowledge that predicts later math achievement. However, researchers disagree about when children first show evidence of approximate number word knowledge-before, or only after, they have learned the cardinal principle. In two studies, children who had not yet learned the cardinal principle (subset-knowers) produced sets in response to number words (verbal comprehension task) and produced number words in response to set sizes (verbal production task). As evidence of approximate number word knowledge, we examined whether children's numerical responses increased with increasing numerosity of the stimulus. In Study 1, subset-knowers (ages 3.0-4.2 years) showed approximate number word knowledge above their knower-level on both tasks, but this effect did not extend to numbers above 4. In Study 2, we collected data from a broader age range of subset-knowers (ages 3.1-5.6 years). In this sample, children showed approximate number word knowledge on the verbal production task even when only examining set sizes above 4. Across studies, children's age predicted approximate number word knowledge (above 4) on the verbal production task when controlling for their knower-level, study (1 or 2), and parents' education, none of which predicted approximation ability. Thus, children can develop approximate knowledge of number words up to 10 before learning the cardinal principle. Furthermore, approximate number word knowledge increases with age and might not be closely related to the development of exact number word knowledge. Copyright © 2014 Elsevier Inc. All rights reserved.

  17. Computing Functions by Approximating the Input

    ERIC Educational Resources Information Center

    Goldberg, Mayer

    2012-01-01

    In computing real-valued functions, it is ordinarily assumed that the input to the function is known, and it is the output that we need to approximate. In this work, we take the opposite approach: we show how to compute the values of some transcendental functions by approximating the input to these functions, and obtaining exact answers for their…

  18. Approximating lens power.

    PubMed

    Kaye, Stephen B

    2009-04-01

    To provide a scalar measure of refractive error, based on geometric lens power through principal, orthogonal and oblique meridians, that is not limited to the paraxial and sag height approximations. A function is derived to model sections through the principal meridian of a lens, followed by rotation of the section through orthogonal and oblique meridians. Average focal length is determined using the definition for the average of a function. Average univariate power in the principal meridian (including spherical aberration), can be computed from the average of a function over the angle of incidence as determined by the parameters of the given lens, or adequately computed from an integrated series function. Average power through orthogonal and oblique meridians, can be similarly determined using the derived formulae. The widely used computation for measuring refractive error, the spherical equivalent, introduces non-constant approximations, leading to a systematic bias. The equations proposed provide a good univariate representation of average lens power and are not subject to a systematic bias. They are particularly useful for the analysis of aggregate data, correlating with biological treatment variables and for developing analyses, which require a scalar equivalent representation of refractive power.

  19. Explicit approximations to estimate the perturbative diffusivity in the presence of convectivity and damping. I. Semi-infinite slab approximations

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Berkel, M. van; Fellow of the Japan Society for the Promotion of Science; FOM Institute DIFFER-Dutch Institute for Fundamental Energy Research, Association EURATOM- FOM, Trilateral Euregio Cluster, PO Box 1207, 3430 BE Nieuwegein

    2014-11-15

    In this paper, a number of new approximations are introduced to estimate the perturbative diffusivity (χ), convectivity (V), and damping (τ) in cylindrical geometry. For this purpose, the harmonic components of heat waves induced by localized deposition of modulated power are used. The approximations are based on semi-infinite slab approximations of the heat equation. The main result is the approximation of χ under the influence of V and τ based on the phase of two harmonics making the estimate less sensitive to calibration errors. To understand why the slab approximations can estimate χ well in cylindrical geometry, the relationships betweenmore » heat transport models in slab and cylindrical geometry are studied. In addition, the relationship between amplitude and phase with respect to their derivatives, used to estimate χ, is discussed. The results are presented in terms of the relative error for the different derived approximations for different values of frequency, transport coefficients, and dimensionless radius. The approximations show a significant region in which χ, V, and τ can be estimated well, but also regions in which the error is large. Also, it is shown that some compensation is necessary to estimate V and τ in a cylindrical geometry. On the other hand, errors resulting from the simplified assumptions are also discussed showing that estimating realistic values for V and τ based on infinite domains will be difficult in practice. This paper is the first part (Part I) of a series of three papers. In Part II and Part III, cylindrical approximations based directly on semi-infinite cylindrical domain (outward propagating heat pulses) and inward propagating heat pulses in a cylindrical domain, respectively, will be treated.« less

  20. Pawlak Algebra and Approximate Structure on Fuzzy Lattice

    PubMed Central

    Zhuang, Ying; Liu, Wenqi; Wu, Chin-Chia; Li, Jinhai

    2014-01-01

    The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties. PMID:25152922

  1. Pawlak algebra and approximate structure on fuzzy lattice.

    PubMed

    Zhuang, Ying; Liu, Wenqi; Wu, Chin-Chia; Li, Jinhai

    2014-01-01

    The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties.

  2. A Generalization of the Karush-Kuhn-Tucker Theorem for Approximate Solutions of Mathematical Programming Problems Based on Quadratic Approximation

    NASA Astrophysics Data System (ADS)

    Voloshinov, V. V.

    2018-03-01

    In computations related to mathematical programming problems, one often has to consider approximate, rather than exact, solutions satisfying the constraints of the problem and the optimality criterion with a certain error. For determining stopping rules for iterative procedures, in the stability analysis of solutions with respect to errors in the initial data, etc., a justified characteristic of such solutions that is independent of the numerical method used to obtain them is needed. A necessary δ-optimality condition in the smooth mathematical programming problem that generalizes the Karush-Kuhn-Tucker theorem for the case of approximate solutions is obtained. The Lagrange multipliers corresponding to the approximate solution are determined by solving an approximating quadratic programming problem.

  3. Dynamical cluster approximation plus semiclassical approximation study for a Mott insulator and d-wave pairing

    NASA Astrophysics Data System (ADS)

    Kim, SungKun; Lee, Hunpyo

    2017-06-01

    Via a dynamical cluster approximation with N c = 4 in combination with a semiclassical approximation (DCA+SCA), we study the doped two-dimensional Hubbard model. We obtain a plaquette antiferromagnetic (AF) Mott insulator, a plaquette AF ordered metal, a pseudogap (or d-wave superconductor) and a paramagnetic metal by tuning the doping concentration. These features are similar to the behaviors observed in copper-oxide superconductors and are in qualitative agreement with the results calculated by the cluster dynamical mean field theory with the continuous-time quantum Monte Carlo (CDMFT+CTQMC) approach. The results of our DCA+SCA differ from those of the CDMFT+CTQMC approach in that the d-wave superconducting order parameters are shown even in the high doped region, unlike the results of the CDMFT+CTQMC approach. We think that the strong plaquette AF orderings in the dynamical cluster approximation (DCA) with N c = 4 suppress superconducting states with increasing doping up to strongly doped region, because frozen dynamical fluctuations in a semiclassical approximation (SCA) approach are unable to destroy those orderings. Our calculation with short-range spatial fluctuations is initial research, because the SCA can manage long-range spatial fluctuations in feasible computational times beyond the CDMFT+CTQMC tool. We believe that our future DCA+SCA calculations should supply information on the fully momentum-resolved physical properties, which could be compared with the results measured by angle-resolved photoemission spectroscopy experiments.

  4. Differential equation based method for accurate approximations in optimization

    NASA Technical Reports Server (NTRS)

    Pritchard, Jocelyn I.; Adelman, Howard M.

    1990-01-01

    This paper describes a method to efficiently and accurately approximate the effect of design changes on structural response. The key to this new method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in msot cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacement are used to approximate bending stresses.

  5. Differential equation based method for accurate approximations in optimization

    NASA Technical Reports Server (NTRS)

    Pritchard, Jocelyn I.; Adelman, Howard M.

    1990-01-01

    A method to efficiently and accurately approximate the effect of design changes on structural response is described. The key to this method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in most cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacements are used to approximate bending stresses.

  6. Variationally consistent approximation scheme for charge transfer

    NASA Technical Reports Server (NTRS)

    Halpern, A. M.

    1978-01-01

    The author has developed a technique for testing various charge-transfer approximation schemes for consistency with the requirements of the Kohn variational principle for the amplitude to guarantee that the amplitude is correct to second order in the scattering wave functions. Applied to Born-type approximations for charge transfer it allows the selection of particular groups of first-, second-, and higher-Born-type terms that obey the consistency requirement, and hence yield more reliable approximation to the amplitude.

  7. Uniform analytic approximation of Wigner rotation matrices

    NASA Astrophysics Data System (ADS)

    Hoffmann, Scott E.

    2018-02-01

    We derive the leading asymptotic approximation, for low angle θ, of the Wigner rotation matrix elements, dm1m2 j(θ ) , uniform in j, m1, and m2. The result is in terms of a Bessel function of integer order. We numerically investigate the error for a variety of cases and find that the approximation can be useful over a significant range of angles. This approximation has application in the partial wave analysis of wavepacket scattering.

  8. Inversion and approximation of Laplace transforms

    NASA Technical Reports Server (NTRS)

    Lear, W. M.

    1980-01-01

    A method of inverting Laplace transforms by using a set of orthonormal functions is reported. As a byproduct of the inversion, approximation of complicated Laplace transforms by a transform with a series of simple poles along the left half plane real axis is shown. The inversion and approximation process is simple enough to be put on a programmable hand calculator.

  9. Einstein and the Quantum: The Secret Life of EPR

    NASA Astrophysics Data System (ADS)

    Fine, Arthur

    2006-05-01

    Locality, separation and entanglement -- 1930s style. Starting with Solvay 1927, we'll explore the background to the 1935 paper by Einstein, Podolsky and Rosen: how it was composed, the actual argument and principles used, and how the paper was received by Schroedinger, and others. We'll also look at Bohr's response: the extent to which Bohr connects with what Einstein was after in EPR and the extent to which EPR marks a shift in Bohr's thinking about the quantum theory. Time permitting, we will contrast EPR with Bell's theorem.

  10. Minimal entropy approximation for cellular automata

    NASA Astrophysics Data System (ADS)

    Fukś, Henryk

    2014-02-01

    We present a method for the construction of approximate orbits of measures under the action of cellular automata which is complementary to the local structure theory. The local structure theory is based on the idea of Bayesian extension, that is, construction of a probability measure consistent with given block probabilities and maximizing entropy. If instead of maximizing entropy one minimizes it, one can develop another method for the construction of approximate orbits, at the heart of which is the iteration of finite-dimensional maps, called minimal entropy maps. We present numerical evidence that the minimal entropy approximation sometimes outperforms the local structure theory in characterizing the properties of cellular automata. The density response curve for elementary CA rule 26 is used to illustrate this claim.

  11. Laplace approximation for Bessel functions of matrix argument

    NASA Astrophysics Data System (ADS)

    Butler, Ronald W.; Wood, Andrew T. A.

    2003-06-01

    We derive Laplace approximations to three functions of matrix argument which arise in statistics and elsewhere: matrix Bessel A[nu]; matrix Bessel B[nu]; and the type II confluent hypergeometric function of matrix argument, [Psi]. We examine the theoretical and numerical properties of the approximations. On the theoretical side, it is shown that the Laplace approximations to A[nu], B[nu] and [Psi] given here, together with the Laplace approximations to the matrix argument functions 1F1 and 2F1 presented in Butler and Wood (Laplace approximations to hyper-geometric functions with matrix argument, Ann. Statist. (2002)), satisfy all the important confluence relations and symmetry relations enjoyed by the original functions.

  12. Approximation Set of the Interval Set in Pawlak's Space

    PubMed Central

    Wang, Jin; Wang, Guoyin

    2014-01-01

    The interval set is a special set, which describes uncertainty of an uncertain concept or set Z with its two crisp boundaries named upper-bound set and lower-bound set. In this paper, the concept of similarity degree between two interval sets is defined at first, and then the similarity degrees between an interval set and its two approximations (i.e., upper approximation set R¯(Z) and lower approximation set R_(Z)) are presented, respectively. The disadvantages of using upper-approximation set R¯(Z) or lower-approximation set R_(Z) as approximation sets of the uncertain set (uncertain concept) Z are analyzed, and a new method for looking for a better approximation set of the interval set Z is proposed. The conclusion that the approximation set R 0.5(Z) is an optimal approximation set of interval set Z is drawn and proved successfully. The change rules of R 0.5(Z) with different binary relations are analyzed in detail. Finally, a kind of crisp approximation set of the interval set Z is constructed. We hope this research work will promote the development of both the interval set model and granular computing theory. PMID:25177721

  13. Function approximation using combined unsupervised and supervised learning.

    PubMed

    Andras, Peter

    2014-03-01

    Function approximation is one of the core tasks that are solved using neural networks in the context of many engineering problems. However, good approximation results need good sampling of the data space, which usually requires exponentially increasing volume of data as the dimensionality of the data increases. At the same time, often the high-dimensional data is arranged around a much lower dimensional manifold. Here we propose the breaking of the function approximation task for high-dimensional data into two steps: (1) the mapping of the high-dimensional data onto a lower dimensional space corresponding to the manifold on which the data resides and (2) the approximation of the function using the mapped lower dimensional data. We use over-complete self-organizing maps (SOMs) for the mapping through unsupervised learning, and single hidden layer neural networks for the function approximation through supervised learning. We also extend the two-step procedure by considering support vector machines and Bayesian SOMs for the determination of the best parameters for the nonlinear neurons in the hidden layer of the neural networks used for the function approximation. We compare the approximation performance of the proposed neural networks using a set of functions and show that indeed the neural networks using combined unsupervised and supervised learning outperform in most cases the neural networks that learn the function approximation using the original high-dimensional data.

  14. A Survey of Techniques for Approximate Computing

    DOE PAGES

    Mittal, Sparsh

    2016-03-18

    Approximate computing trades off computation quality with the effort expended and as rising performance demands confront with plateauing resource budgets, approximate computing has become, not merely attractive, but even imperative. Here, we present a survey of techniques for approximate computing (AC). We discuss strategies for finding approximable program portions and monitoring output quality, techniques for using AC in different processing units (e.g., CPU, GPU and FPGA), processor components, memory technologies etc., and programming frameworks for AC. Moreover, we classify these techniques based on several key characteristics to emphasize their similarities and differences. Finally, the aim of this paper is tomore » provide insights to researchers into working of AC techniques and inspire more efforts in this area to make AC the mainstream computing approach in future systems.« less

  15. Meta-Regression Approximations to Reduce Publication Selection Bias

    ERIC Educational Resources Information Center

    Stanley, T. D.; Doucouliagos, Hristos

    2014-01-01

    Publication selection bias is a serious challenge to the integrity of all empirical sciences. We derive meta-regression approximations to reduce this bias. Our approach employs Taylor polynomial approximations to the conditional mean of a truncated distribution. A quadratic approximation without a linear term, precision-effect estimate with…

  16. Thin-wall approximation in vacuum decay: A lemma

    NASA Astrophysics Data System (ADS)

    Brown, Adam R.

    2018-05-01

    The "thin-wall approximation" gives a simple estimate of the decay rate of an unstable quantum field. Unfortunately, the approximation is uncontrolled. In this paper I show that there are actually two different thin-wall approximations and that they bracket the true decay rate: I prove that one is an upper bound and the other a lower bound. In the thin-wall limit, the two approximations converge. In the presence of gravity, a generalization of this lemma provides a simple sufficient condition for nonperturbative vacuum instability.

  17. Smooth function approximation using neural networks.

    PubMed

    Ferrari, Silvia; Stengel, Robert F

    2005-01-01

    An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented. In this paper, the approach is implemented for the approximation of smooth batch data containing the function's input, output, and possibly, gradient information. The training set is associated to the network adjustable parameters by nonlinear weight equations. The cascade structure of these equations reveals that they can be treated as sets of linear systems. Hence, the training process and the network approximation properties can be investigated via linear algebra. Four algorithms are developed to achieve exact or approximate matching of input-output and/or gradient-based training sets. Their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques.

  18. Random-Phase Approximation Methods

    NASA Astrophysics Data System (ADS)

    Chen, Guo P.; Voora, Vamsee K.; Agee, Matthew M.; Balasubramani, Sree Ganesh; Furche, Filipp

    2017-05-01

    Random-phase approximation (RPA) methods are rapidly emerging as cost-effective validation tools for semilocal density functional computations. We present the theoretical background of RPA in an intuitive rather than formal fashion, focusing on the physical picture of screening and simple diagrammatic analysis. A new decomposition of the RPA correlation energy into plasmonic modes leads to an appealing visualization of electron correlation in terms of charge density fluctuations. Recent developments in the areas of beyond-RPA methods, RPA correlation potentials, and efficient algorithms for RPA energy and property calculations are reviewed. The ability of RPA to approximately capture static correlation in molecules is quantified by an analysis of RPA natural occupation numbers. We illustrate the use of RPA methods in applications to small-gap systems such as open-shell d- and f-element compounds, radicals, and weakly bound complexes, where semilocal density functional results exhibit strong functional dependence.

  19. Monotonically improving approximate answers to relational algebra queries

    NASA Technical Reports Server (NTRS)

    Smith, Kenneth P.; Liu, J. W. S.

    1989-01-01

    We present here a query processing method that produces approximate answers to queries posed in standard relational algebra. This method is monotone in the sense that the accuracy of the approximate result improves with the amount of time spent producing the result. This strategy enables us to trade the time to produce the result for the accuracy of the result. An approximate relational model that characterizes appromimate relations and a partial order for comparing them is developed. Relational operators which operate on and return approximate relations are defined.

  20. Approximate isotropic cloak for the Maxwell equations

    NASA Astrophysics Data System (ADS)

    Ghosh, Tuhin; Tarikere, Ashwin

    2018-05-01

    We construct a regular isotropic approximate cloak for the Maxwell system of equations. The method of transformation optics has enabled the design of electromagnetic parameters that cloak a region from external observation. However, these constructions are singular and anisotropic, making practical implementation difficult. Thus, regular approximations to these cloaks have been constructed that cloak a given region to any desired degree of accuracy. In this paper, we show how to construct isotropic approximations to these regularized cloaks using homogenization techniques so that one obtains cloaking of arbitrary accuracy with regular and isotropic parameters.

  1. Explicit approximations to estimate the perturbative diffusivity in the presence of convectivity and damping. III. Cylindrical approximations for heat waves traveling inwards

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Berkel, M. van; Fellow of the Japan Society for the Promotion of Science; FOM Institute DIFFER-Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM, Trilateral Euregio Cluster, P.O. Box 1207, 3430 BE Nieuwegein

    In this paper, a number of new explicit approximations are introduced to estimate the perturbative diffusivity (χ), convectivity (V), and damping (τ) in cylindrical geometry. For this purpose, the harmonic components of heat waves induced by localized deposition of modulated power are used. The approximations are based on the heat equation in cylindrical geometry using the symmetry (Neumann) boundary condition at the plasma center. This means that the approximations derived here should be used only to estimate transport coefficients between the plasma center and the off-axis perturbative source. If the effect of cylindrical geometry is small, it is also possiblemore » to use semi-infinite domain approximations presented in Part I and Part II of this series. A number of new approximations are derived in this part, Part III, based upon continued fractions of the modified Bessel function of the first kind and the confluent hypergeometric function of the first kind. These approximations together with the approximations based on semi-infinite domains are compared for heat waves traveling towards the center. The relative error for the different derived approximations is presented for different values of the frequency, transport coefficients, and dimensionless radius. Moreover, it is shown how combinations of different explicit formulas can be used to estimate the transport coefficients over a large parameter range for cases without convection and damping, cases with damping only, and cases with convection and damping. The relative error between the approximation and its underlying model is below 2% for the case, where only diffusivity and damping are considered. If also convectivity is considered, the diffusivity can be estimated well in a large region, but there is also a large region in which no suitable approximation is found. This paper is the third part (Part III) of a series of three papers. In Part I, the semi-infinite slab approximations have been treated. In

  2. Structural Reliability Analysis and Optimization: Use of Approximations

    NASA Technical Reports Server (NTRS)

    Grandhi, Ramana V.; Wang, Liping

    1999-01-01

    This report is intended for the demonstration of function approximation concepts and their applicability in reliability analysis and design. Particularly, approximations in the calculation of the safety index, failure probability and structural optimization (modification of design variables) are developed. With this scope in mind, extensive details on probability theory are avoided. Definitions relevant to the stated objectives have been taken from standard text books. The idea of function approximations is to minimize the repetitive use of computationally intensive calculations by replacing them with simpler closed-form equations, which could be nonlinear. Typically, the approximations provide good accuracy around the points where they are constructed, and they need to be periodically updated to extend their utility. There are approximations in calculating the failure probability of a limit state function. The first one, which is most commonly discussed, is how the limit state is approximated at the design point. Most of the time this could be a first-order Taylor series expansion, also known as the First Order Reliability Method (FORM), or a second-order Taylor series expansion (paraboloid), also known as the Second Order Reliability Method (SORM). From the computational procedure point of view, this step comes after the design point identification; however, the order of approximation for the probability of failure calculation is discussed first, and it is denoted by either FORM or SORM. The other approximation of interest is how the design point, or the most probable failure point (MPP), is identified. For iteratively finding this point, again the limit state is approximated. The accuracy and efficiency of the approximations make the search process quite practical for analysis intensive approaches such as the finite element methods; therefore, the crux of this research is to develop excellent approximations for MPP identification and also different

  3. Variational Gaussian approximation for Poisson data

    NASA Astrophysics Data System (ADS)

    Arridge, Simon R.; Ito, Kazufumi; Jin, Bangti; Zhang, Chen

    2018-02-01

    The Poisson model is frequently employed to describe count data, but in a Bayesian context it leads to an analytically intractable posterior probability distribution. In this work, we analyze a variational Gaussian approximation to the posterior distribution arising from the Poisson model with a Gaussian prior. This is achieved by seeking an optimal Gaussian distribution minimizing the Kullback-Leibler divergence from the posterior distribution to the approximation, or equivalently maximizing the lower bound for the model evidence. We derive an explicit expression for the lower bound, and show the existence and uniqueness of the optimal Gaussian approximation. The lower bound functional can be viewed as a variant of classical Tikhonov regularization that penalizes also the covariance. Then we develop an efficient alternating direction maximization algorithm for solving the optimization problem, and analyze its convergence. We discuss strategies for reducing the computational complexity via low rank structure of the forward operator and the sparsity of the covariance. Further, as an application of the lower bound, we discuss hierarchical Bayesian modeling for selecting the hyperparameter in the prior distribution, and propose a monotonically convergent algorithm for determining the hyperparameter. We present extensive numerical experiments to illustrate the Gaussian approximation and the algorithms.

  4. Approximate Computing Techniques for Iterative Graph Algorithms

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Panyala, Ajay R.; Subasi, Omer; Halappanavar, Mahantesh

    Approximate computing enables processing of large-scale graphs by trading off quality for performance. Approximate computing techniques have become critical not only due to the emergence of parallel architectures but also the availability of large scale datasets enabling data-driven discovery. Using two prototypical graph algorithms, PageRank and community detection, we present several approximate computing heuristics to scale the performance with minimal loss of accuracy. We present several heuristics including loop perforation, data caching, incomplete graph coloring and synchronization, and evaluate their efficiency. We demonstrate performance improvements of up to 83% for PageRank and up to 450x for community detection, with lowmore » impact of accuracy for both the algorithms. We expect the proposed approximate techniques will enable scalable graph analytics on data of importance to several applications in science and their subsequent adoption to scale similar graph algorithms.« less

  5. Pumping approximately integrable systems

    PubMed Central

    Lange, Florian; Lenarčič, Zala; Rosch, Achim

    2017-01-01

    Weak perturbations can drive an interacting many-particle system far from its initial equilibrium state if one is able to pump into degrees of freedom approximately protected by conservation laws. This concept has for example been used to realize Bose–Einstein condensates of photons, magnons and excitons. Integrable quantum systems, like the one-dimensional Heisenberg model, are characterized by an infinite set of conservation laws. Here, we develop a theory of weakly driven integrable systems and show that pumping can induce large spin or heat currents even in the presence of integrability breaking perturbations, since it activates local and quasi-local approximate conserved quantities. The resulting steady state is qualitatively captured by a truncated generalized Gibbs ensemble with Lagrange parameters that depend on the structure but not on the overall amplitude of perturbations nor the initial state. We suggest to use spin-chain materials driven by terahertz radiation to realize integrability-based spin and heat pumps. PMID:28598444

  6. Sensitivity analysis and approximation methods for general eigenvalue problems

    NASA Technical Reports Server (NTRS)

    Murthy, D. V.; Haftka, R. T.

    1986-01-01

    Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive. Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and accuracy. Proper eigenvector normalization is discussed. An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity. Important numerical aspects of this method are also discussed. To alleviate the problem of reanalysis, various approximation methods for eigenvalues are proposed and evaluated. Linear and quadratic approximations are based directly on the Taylor series. Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. Approximation methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the approximations are given. General recommendations are made for the selection of appropriate approximation technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which approximation is sought.

  7. Convergence Rates of Finite Difference Stochastic Approximation Algorithms

    DTIC Science & Technology

    2016-06-01

    dfferences as gradient approximations. It is shown that the convergence of these algorithms can be accelerated by controlling the implementation of the...descent algorithm, under various updating schemes using finite dfferences as gradient approximations. It is shown that the convergence of these...the Kiefer-Wolfowitz algorithm and the mirror descent algorithm, under various updating schemes using finite differences as gradient approximations. It

  8. Capacitor-Chain Successive-Approximation ADC

    NASA Technical Reports Server (NTRS)

    Cunningham, Thomas

    2003-01-01

    A proposed successive-approximation analog-to-digital converter (ADC) would contain a capacitively terminated chain of identical capacitor cells. Like a conventional successive-approximation ADC containing a bank of binary-scaled capacitors, the proposed ADC would store an input voltage on a sample-and-hold capacitor and would digitize the stored input voltage by finding the closest match between this voltage and a capacitively generated sum of binary fractions of a reference voltage (Vref). However, the proposed capacitor-chain ADC would offer two major advantages over a conventional binary-scaled-capacitor ADC: (1) In a conventional ADC that digitizes to n bits, the largest capacitor (representing the most significant bit) must have 2(exp n-1) times as much capacitance, and hence, approximately 2(exp n-1) times as much area as does the smallest capacitor (representing the least significant bit), so that the total capacitor area must be 2(exp n) times that of the smallest capacitor. In the proposed capacitor-chain ADC, there would be three capacitors per cell, each approximately equal to the smallest capacitor in the conventional ADC, and there would be one cell per bit. Therefore, the total capacitor area would be only about 3(exp n) times that of the smallest capacitor. The net result would be that the proposed ADC could be considerably smaller than the conventional ADC. (2) Because of edge effects, parasitic capacitances, and manufacturing tolerances, it is difficult to make capacitor banks in which the values of capacitance are scaled by powers of 2 to the required precision. In contrast, because all the capacitors in the proposed ADC would be identical, the problem of precise binary scaling would not arise.

  9. The Kantian framework of complementarity

    NASA Astrophysics Data System (ADS)

    Cuffaro, Michael

    A growing number of commentators have, in recent years, noted the important affinities in the views of Immanuel Kant and Niels Bohr. While these commentators are correct, the picture they present of the connections between Bohr and Kant is painted in broad strokes; it is open to the criticism that these affinities are merely superficial. In this essay, I provide a closer, structural, analysis of both Bohr's and Kant's views that makes these connections more explicit. In particular, I demonstrate the similarities between Bohr's argument, on the one hand, that neither the wave nor the particle description of atomic phenomena pick out an object in the ordinary sense of the word, and Kant's requirement, on the other hand, that both 'mathematical' (having to do with magnitude) and 'dynamical' (having to do with an object's interaction with other objects) principles must be applicable to appearances in order for us to determine them as objects of experience. I argue that Bohr's 'complementarity interpretation' of quantum mechanics, which views atomic objects as idealizations, and which licenses the repeal of the principle of causality for the domain of atomic physics, is perfectly compatible with, and indeed follows naturally from a broadly Kantian epistemological framework.

  10. Rational Approximations with Hankel-Norm Criterion

    DTIC Science & Technology

    1980-01-01

    REPORT TYPE ANDu DATES COVERED It) L. TITLE AND SLWUIlL Fi901 ia FUNDING NUMOIRS, RATIONAL APPROXIMATIONS WITH HANKEL-NORM CRITERION PE61102F i...problem is proved to be reducible to obtain a two-variable all- pass ration 1 function, interpolating a set of parametric values at specified points inside...PAGES WHICH DO NOT REPRODUCE LEGIBLY. V" C - w RATIONAL APPROXIMATIONS WITH HANKEL-NORM CRITERION* Y. Genin* Philips Research Lab. 2, avenue van

  11. Approximate reasoning using terminological models

    NASA Technical Reports Server (NTRS)

    Yen, John; Vaidya, Nitin

    1992-01-01

    Term Subsumption Systems (TSS) form a knowledge-representation scheme in AI that can express the defining characteristics of concepts through a formal language that has a well-defined semantics and incorporates a reasoning mechanism that can deduce whether one concept subsumes another. However, TSS's have very limited ability to deal with the issue of uncertainty in knowledge bases. The objective of this research is to address issues in combining approximate reasoning with term subsumption systems. To do this, we have extended an existing AI architecture (CLASP) that is built on the top of a term subsumption system (LOOM). First, the assertional component of LOOM has been extended for asserting and representing uncertain propositions. Second, we have extended the pattern matcher of CLASP for plausible rule-based inferences. Third, an approximate reasoning model has been added to facilitate various kinds of approximate reasoning. And finally, the issue of inconsistency in truth values due to inheritance is addressed using justification of those values. This architecture enhances the reasoning capabilities of expert systems by providing support for reasoning under uncertainty using knowledge captured in TSS. Also, as definitional knowledge is explicit and separate from heuristic knowledge for plausible inferences, the maintainability of expert systems could be improved.

  12. Comparison of dynamical approximation schemes for nonlinear gravitaional clustering

    NASA Technical Reports Server (NTRS)

    Melott, Adrian L.

    1994-01-01

    We have recently conducted a controlled comparison of a number of approximations for gravitational clustering against the same n-body simulations. These include ordinary linear perturbation theory (Eulerian), the lognormal approximation, the adhesion approximation, the frozen-flow approximation, the Zel'dovich approximation (describable as first-order Lagrangian perturbation theory), and its second-order generalization. In the last two cases we also created new versions of the approximation by truncation, i.e., by smoothing the initial conditions with various smoothing window shapes and varying their sizes. The primary tool for comparing simulations to approximation schemes was cross-correlation of the evolved mass density fields, testing the extent to which mass was moved to the right place. The Zel'dovich approximation, with initial convolution with a Gaussian e(exp -k(exp 2)/k(sub G(exp 2)), where k(sub G) is adjusted to be just into the nonlinear regime of the evolved model (details in text) worked extremely well. Its second-order generalization worked slightly better. We recommend either n-body simulations or our modified versions of the Zel'dovich approximation, depending upon the purpose. The theoretical implication is that pancaking is implicit in all cosmological gravitational clustering, at least from Gaussian initial conditions, even when subcondensations are present. This in turn provides a natural explanation for the presence of sheets and filaments in the observed galaxy distribution. Use of the approximation scheme can permit extremely rapid generation of large numbers of realizations of model universes with good accuracy down to galaxy group mass scales.

  13. Embedding impedance approximations in the analysis of SIS mixers

    NASA Technical Reports Server (NTRS)

    Kerr, A. R.; Pan, S.-K.; Withington, S.

    1992-01-01

    Future millimeter-wave radio astronomy instruments will use arrays of many SIS receivers, either as focal plane arrays on individual radio telescopes, or as individual receivers on the many antennas of radio interferometers. Such applications will require broadband integrated mixers without mechanical tuners. To produce such mixers, it will be necessary to improve present mixer design techniques, most of which use the three-frequency approximation to Tucker's quantum mixer theory. This paper examines the adequacy of three approximations to Tucker's theory: (1) the usual three-frequency approximation which assumes a sinusoidal LO voltage at the junction, and a short-circuit at all frequencies above the upper sideband; (2) a five-frequency approximation which allows two LO voltage harmonics and five small-signal sidebands; and (3) a quasi five-frequency approximation in which five small-signal sidebands are allowed, but the LO voltage is assumed sinusoidal. These are compared with a full harmonic-Newton solution of Tucker's equations, including eight LO harmonics and their corresponding sidebands, for realistic SIS mixer circuits. It is shown that the accuracy of the three approximations depends strongly on the value of omega R(sub N)C for the SIS junctions used. For large omega R(sub N)C, all three approximations approach the eight-harmonic solution. For omega R(sub N)C values in the range 0.5 to 10, the range of most practical interest, the quasi five-frequency approximation is a considerable improvement over the three-frequency approximation, and should be suitable for much design work. For the realistic SIS mixers considered here, the five-frequency approximation gives results very close to those of the eight-harmonic solution. Use of these approximations, where appropriate, considerably reduces the computational effort needed to analyze an SIS mixer, and allows the design and optimization of mixers using a personal computer.

  14. Exponential Approximations Using Fourier Series Partial Sums

    NASA Technical Reports Server (NTRS)

    Banerjee, Nana S.; Geer, James F.

    1997-01-01

    The problem of accurately reconstructing a piece-wise smooth, 2(pi)-periodic function f and its first few derivatives, given only a truncated Fourier series representation of f, is studied and solved. The reconstruction process is divided into two steps. In the first step, the first 2N + 1 Fourier coefficients of f are used to approximate the locations and magnitudes of the discontinuities in f and its first M derivatives. This is accomplished by first finding initial estimates of these quantities based on certain properties of Gibbs phenomenon, and then refining these estimates by fitting the asymptotic form of the Fourier coefficients to the given coefficients using a least-squares approach. It is conjectured that the locations of the singularities are approximated to within O(N(sup -M-2), and the associated jump of the k(sup th) derivative of f is approximated to within O(N(sup -M-l+k), as N approaches infinity, and the method is robust. These estimates are then used with a class of singular basis functions, which have certain 'built-in' singularities, to construct a new sequence of approximations to f. Each of these new approximations is the sum of a piecewise smooth function and a new Fourier series partial sum. When N is proportional to M, it is shown that these new approximations, and their derivatives, converge exponentially in the maximum norm to f, and its corresponding derivatives, except in the union of a finite number of small open intervals containing the points of singularity of f. The total measure of these intervals decreases exponentially to zero as M approaches infinity. The technique is illustrated with several examples.

  15. Ventilatory inhomogeneity determined from multiple-breath washouts during sustained microgravity on Spacelab SLS-1.

    PubMed

    Prisk, G K; Guy, H J; Elliott, A R; Paiva, M; West, J B

    1995-02-01

    We used multiple-breath N2 washouts (MBNW) to study the inhomogeneity of ventilation in four normal humans (mean age 42.5 yr) before, during, and after 9 days of exposure to microgravity on Spacelab Life Sciences-1. Subjects performed 20-breath MBNW at tidal volumes of approximately 700 ml and 12-breath MBNW at tidal volumes of approximately 1,250 ml. Six indexes of ventilatory inhomogeneity were derived from data from 1) distribution of specific ventilation (SV) from mixed-expired and 2) end-tidal N2, 3) change of slope of N2 washout (semilog plot) with time, 4) change of slope of normalized phase III of successive breaths, 5) anatomic dead space, and 6) Bohr dead space. Significant ventilatory inhomogeneity was seen in the standing position at normal gravity (1 G). When we compared standing 1 G with microgravity, the distributions of SV became slightly narrower, but the difference was not significant. Also, there were no significant changes in the change of slope of the N2 washout, change of normalized phase III slopes, or the anatomic and Bohr dead spaces. By contrast, transition from the standing to supine position in 1 G resulted in significantly broader distributions of SV (P < 0.05) and significantly greater changes in the changes in slope of the N2 washouts (P < 0.001), indicating more ventilatory inhomogeneity in that posture. Thus these techniques can detect relatively small changes in ventilatory inhomogeneity. We conclude that the primary determinants of ventilatory inhomogeneity during tidal breathing in the upright posture are not gravitational in origin.

  16. Ventilatory inhomogeneity determined from multiple-breath washouts during sustained microgravity on Spacelab SLS-1

    NASA Technical Reports Server (NTRS)

    Prisk, G. Kim; Guy, Harold J. B.; Elliott, Ann R.; Paiva, Manuel; West, John B.

    1995-01-01

    We used multiple-breath N2 washouts (MBNW) to study the homogeneity of ventilation in four normal humans (mean age 42.5 yr) before, during, and after 9 days of exposure to microgravity on Spacelab Life Sciences-1. Subjects performed 20-breath MBNW at tidal volumes of approximately 700 ml and 12-breath MBNW at tidal volumes of approximately 1,250 ml. Six indexes of ventilatory inhomogeneity were derived from data from (1) distribution of specific ventilation (SV) from mixed-expired and (2) end-tidal N2, (3) change of slope of N2 washout (semilog plot) with time, (4) change of slope of normalized phase III of successive breaths, (5) anatomic lead dead space, and (6) Bohr dead space. Significant ventilatory inhomogeneity was seen in the standing position at normal gravity (1 G). When we compared standing 1 G with microgravity, the distributions of SV became slightly narrower, but the difference was not significant. Also, there were no significant changes in the change of slope of the N2 washout, change of normalized phase III slopes, or the anatomic and Bohr dead spaces. By contrast, transition from the standing to supine position in 1 G resulted in significantly broader distributions of SV and significantly greater changes in the changes in slope of the N2 washouts, indicating more ventilatory inhomogeneity in that posture. Thus these techniques can detect relatively small changes in ventilatory inhomogeneity. We conclude that the primary determinants of ventilatory inhomogeneity during tidal breathing in the upright posture are not gravitational in origin.

  17. A Gaussian-based rank approximation for subspace clustering

    NASA Astrophysics Data System (ADS)

    Xu, Fei; Peng, Chong; Hu, Yunhong; He, Guoping

    2018-04-01

    Low-rank representation (LRR) has been shown successful in seeking low-rank structures of data relationships in a union of subspaces. Generally, LRR and LRR-based variants need to solve the nuclear norm-based minimization problems. Beyond the success of such methods, it has been widely noted that the nuclear norm may not be a good rank approximation because it simply adds all singular values of a matrix together and thus large singular values may dominant the weight. This results in far from satisfactory rank approximation and may degrade the performance of lowrank models based on the nuclear norm. In this paper, we propose a novel nonconvex rank approximation based on the Gaussian distribution function, which has demanding properties to be a better rank approximation than the nuclear norm. Then a low-rank model is proposed based on the new rank approximation with application to motion segmentation. Experimental results have shown significant improvements and verified the effectiveness of our method.

  18. Fundamentals and Recent Developments in Approximate Bayesian Computation

    PubMed Central

    Lintusaari, Jarno; Gutmann, Michael U.; Dutta, Ritabrata; Kaski, Samuel; Corander, Jukka

    2017-01-01

    Abstract Bayesian inference plays an important role in phylogenetics, evolutionary biology, and in many other branches of science. It provides a principled framework for dealing with uncertainty and quantifying how it changes in the light of new evidence. For many complex models and inference problems, however, only approximate quantitative answers are obtainable. Approximate Bayesian computation (ABC) refers to a family of algorithms for approximate inference that makes a minimal set of assumptions by only requiring that sampling from a model is possible. We explain here the fundamentals of ABC, review the classical algorithms, and highlight recent developments. [ABC; approximate Bayesian computation; Bayesian inference; likelihood-free inference; phylogenetics; simulator-based models; stochastic simulation models; tree-based models.] PMID:28175922

  19. Dead space and slope indices from the expiratory carbon dioxide tension-volume curve.

    PubMed

    Kars, A H; Bogaard, J M; Stijnen, T; de Vries, J; Verbraak, A F; Hilvering, C

    1997-08-01

    The slope of phase 3 and three noninvasively determined dead space estimates derived from the expiratory carbon dioxide tension (PCO2) versus volume curve, including the Bohr dead space (VD,Bohr), the Fowler dead space (VD,Fowler) and pre-interface expirate (PIE), were investigated in 28 healthy control subjects, 12 asthma and 29 emphysema patients (20 severely obstructed and nine moderately obstructed) with the aim to establish diagnostic value. Because breath volume and frequency are closely related to CO2 elimination, the recording procedures included varying breath volumes in all subjects during self-chosen/natural breathing frequency, and fixed frequencies of 10, 15 and 20 breaths x min(-1) with varying breath volumes only in the healthy controls. From the relationships of the variables with tidal volume (VT), the values at 1 L were estimated to compare the groups. The slopes of phase 3 and VD,Bohr at 1 L VT showed the most significant difference between controls and patients with asthma or emphysema, compared to the other two dead space estimates, and were related to the degree of airways obstruction. Discrimination between no-emphysema (asthma and controls) and emphysema patients was possible on the basis of a plot of intercept and slope of the relationship between VD,Bohr and VT. A combination of both the slope of phase 3 and VD,Bohr of a breath of 1 L was equally discriminating. The influence of fixed frequencies in the controls did not change the results. The conclusion is that Bohr dead space in relation to tidal volume seems to have diagnostic properties separating patients with asthma from patients with emphysema with the same degree of airways obstruction. Equally discriminating was a combination of both phase 3 and Bohr dead space of a breath of 1 L. The different pathophysiological mechanisms in asthma and emphysema leading to airways obstruction are probably responsible for these results.

  20. A test of the adhesion approximation for gravitational clustering

    NASA Technical Reports Server (NTRS)

    Melott, Adrian L.; Shandarin, Sergei; Weinberg, David H.

    1993-01-01

    We quantitatively compare a particle implementation of the adhesion approximation to fully non-linear, numerical 'N-body' simulations. Our primary tool, cross-correlation of N-body simulations with the adhesion approximation, indicates good agreement, better than that found by the same test performed with the Zel-dovich approximation (hereafter ZA). However, the cross-correlation is not as good as that of the truncated Zel-dovich approximation (TZA), obtained by applying the Zel'dovich approximation after smoothing the initial density field with a Gaussian filter. We confirm that the adhesion approximation produces an excessively filamentary distribution. Relative to the N-body results, we also find that: (a) the power spectrum obtained from the adhesion approximation is more accurate than that from ZA or TZA, (b) the error in the phase angle of Fourier components is worse than that from TZA, and (c) the mass distribution function is more accurate than that from ZA or TZA. It appears that adhesion performs well statistically, but that TZA is more accurate dynamically, in the sense of moving mass to the right place.

  1. A test of the adhesion approximation for gravitational clustering

    NASA Technical Reports Server (NTRS)

    Melott, Adrian L.; Shandarin, Sergei F.; Weinberg, David H.

    1994-01-01

    We quantitatively compare a particle implementation of the adhesion approximation to fully nonlinear, numerical 'N-body' simulations. Our primary tool, cross-correlation of N-body simulations with the adhesion approximation, indicates good agreement, better than that found by the same test performed with the Zel'dovich approximation (hereafter ZA). However, the cross-correlation is not as good as that of the truncated Zel'dovich approximation (TZA), obtained by applying the Zel'dovich approximation after smoothing the initial density field with a Gaussian filter. We confirm that the adhesion approximation produces an excessively filamentary distribution. Relative to the N-body results, we also find that: (a) the power spectrum obtained from the adhesion approximation is more accurate that that from ZA to TZA, (b) the error in the phase angle of Fourier components is worse that that from TZA, and (c) the mass distribution function is more accurate than that from ZA or TZA. It appears that adhesion performs well statistically, but that TZA is more accurate dynamically, in the sense of moving mass to the right place.

  2. On the origins of approximations for stochastic chemical kinetics.

    PubMed

    Haseltine, Eric L; Rawlings, James B

    2005-10-22

    This paper considers the derivation of approximations for stochastic chemical kinetics governed by the discrete master equation. Here, the concepts of (1) partitioning on the basis of fast and slow reactions as opposed to fast and slow species and (2) conditional probability densities are used to derive approximate, partitioned master equations, which are Markovian in nature, from the original master equation. Under different conditions dictated by relaxation time arguments, such approximations give rise to both the equilibrium and hybrid (deterministic or Langevin equations coupled with discrete stochastic simulation) approximations previously reported. In addition, the derivation points out several weaknesses in previous justifications of both the hybrid and equilibrium systems and demonstrates the connection between the original and approximate master equations. Two simple examples illustrate situations in which these two approximate methods are applicable and demonstrate the two methods' efficiencies.

  3. Alternative approximation concepts for space frame synthesis

    NASA Technical Reports Server (NTRS)

    Lust, R. V.; Schmit, L. A.

    1985-01-01

    A structural synthesis methodology for the minimum mass design of 3-dimensionall frame-truss structures under multiple static loading conditions and subject to limits on displacements, rotations, stresses, local buckling, and element cross-sectional dimensions is presented. A variety of approximation concept options are employed to yield near optimum designs after no more than 10 structural analyses. Available options include: (A) formulation of the nonlinear mathematcal programming problem in either reciprocal section property (RSP) or cross-sectional dimension (CSD) space; (B) two alternative approximate problem structures in each design space; and (C) three distinct assumptions about element end-force variations. Fixed element, design element linking, and temporary constraint deletion features are also included. The solution of each approximate problem, in either its primal or dual form, is obtained using CONMIN, a feasible directions program. The frame-truss synthesis methodology is implemented in the COMPASS computer program and is used to solve a variety of problems. These problems were chosen so that, in addition to exercising the various approximation concepts options, the results could be compared with previously published work.

  4. Approximating Integrals Using Probability

    ERIC Educational Resources Information Center

    Maruszewski, Richard F., Jr.; Caudle, Kyle A.

    2005-01-01

    As part of a discussion on Monte Carlo methods, which outlines how to use probability expectations to approximate the value of a definite integral. The purpose of this paper is to elaborate on this technique and then to show several examples using visual basic as a programming tool. It is an interesting method because it combines two branches of…

  5. Spline approximations for nonlinear hereditary control systems

    NASA Technical Reports Server (NTRS)

    Daniel, P. L.

    1982-01-01

    A sline-based approximation scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The approximating framework reduces the original control problem to a sequence of optimization problems governed by ordinary differential equations. Convergence proofs, which appeal directly to dissipative-type estimates for the underlying nonlinear operator, are given and numerical findings are summarized.

  6. Approximate supernova remnant dynamics with cosmic ray production

    NASA Technical Reports Server (NTRS)

    Voelk, H. J.; Drury, L. O.; Dorfi, E. A.

    1985-01-01

    Supernova explosions are the most violent and energetic events in the galaxy and have long been considered probably sources of Cosmic Rays. Recent shock acceleration models treating the Cosmic Rays (CR's) as test particles nb a prescribed Supernova Remnant (SNR) evolution, indeed indicate an approximate power law momentum distribution f sub source (p) approximation p(-a) for the particles ultimately injected into the Interstellar Medium (ISM). This spectrum extends almost to the momentum p = 1 million GeV/c, where the break in the observed spectrum occurs. The calculated power law index approximately less than 4.2 agrees with that inferred for the galactic CR sources. The absolute CR intensity can however not be well determined in such a test particle approximation.

  7. Properties of the Boltzmann equation in the classical approximation

    DOE PAGES

    Epelbaum, Thomas; Gelis, François; Tanji, Naoto; ...

    2014-12-30

    We examine the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical approximation. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical approximation of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since onemore » has also access to the non-approximated result for comparison.« less

  8. Recent advances in approximation concepts for optimum structural design

    NASA Technical Reports Server (NTRS)

    Barthelemy, Jean-Francois M.; Haftka, Raphael T.

    1991-01-01

    The basic approximation concepts used in structural optimization are reviewed. Some of the most recent developments in that area since the introduction of the concept in the mid-seventies are discussed. The paper distinguishes between local, medium-range, and global approximations; it covers functions approximations and problem approximations. It shows that, although the lack of comparative data established on reference test cases prevents an accurate assessment, there have been significant improvements. The largest number of developments have been in the areas of local function approximations and use of intermediate variable and response quantities. It also appears that some new methodologies are emerging which could greatly benefit from the introduction of new computer architecture.

  9. Is approximated de-epithelized glanuloplasty beneficial for hypospadiologist?

    PubMed

    ZakiEldahshoury, M; Gamal, W; Salem, E; Rashed, E; Mamdouh, A

    2016-05-01

    Further evaluation of the cosmetic and functional results of approximated de-epithelized glanuloplasty in different degree of hypospadias. This study included 96 male patients (DPH=68 & MPH=28). Patients selected for repair with glans approximation should have wide urethral plate & grooved glans. All cases were repaired with the classic TIP and glans approximation technique. Follow up was for one year by clinical examination of the meatal shape, size & site, glans shape, skin covering, suture line, urethral catheter, edema & fistula in addition to parent satisfaction. Mean operative time was 49±9minutes. As regards the functional and cosmetic outcomes, success was reported in 95.8%, while failure was in 4.16% in the form of glanular disruption in two patients and subcoronal urethrocutaneous fistula in another two patients. Glans approximation has many advantages, good cosmetic and functional results, short operative time, less blood loss, no need for tourniquet. Study of a large number of cases and comparing glans approximation with the classic TIP technique. Copyright © 2015 AEU. Publicado por Elsevier España, S.L.U. All rights reserved.

  10. Adaptive control using neural networks and approximate models.

    PubMed

    Narendra, K S; Mukhopadhyay, S

    1997-01-01

    The NARMA model is an exact representation of the input-output behavior of finite-dimensional nonlinear discrete-time dynamical systems in a neighborhood of the equilibrium state. However, it is not convenient for purposes of adaptive control using neural networks due to its nonlinear dependence on the control input. Hence, quite often, approximate methods are used for realizing the neural controllers to overcome computational complexity. In this paper, we introduce two classes of models which are approximations to the NARMA model, and which are linear in the control input. The latter fact substantially simplifies both the theoretical analysis as well as the practical implementation of the controller. Extensive simulation studies have shown that the neural controllers designed using the proposed approximate models perform very well, and in many cases even better than an approximate controller designed using the exact NARMA model. In view of their mathematical tractability as well as their success in simulation studies, a case is made in this paper that such approximate input-output models warrant a detailed study in their own right.

  11. Rational trigonometric approximations using Fourier series partial sums

    NASA Technical Reports Server (NTRS)

    Geer, James F.

    1993-01-01

    A class of approximations (S(sub N,M)) to a periodic function f which uses the ideas of Pade, or rational function, approximations based on the Fourier series representation of f, rather than on the Taylor series representation of f, is introduced and studied. Each approximation S(sub N,M) is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of S(sub N,M) agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these 'Fourier-Pade' approximations converge point-wise to (f(x(exp +))+f(x(exp -)))/2 more rapidly (in some cases by a factor of 1/k(exp 2M)) than the Fourier series partial sums on which they are based. The approximations are illustrated by several examples and an application to the solution of an initial, boundary value problem for the simple heat equation is presented.

  12. Self-consistent approximation beyond the CPA: Part II

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Kaplan, T.; Gray, L.J.

    1981-08-01

    In Part I, Professor Leath has described the substantial efforts to generalize the CPA. In this second part, a particular self-consistent approximation for random alloys developed by Kaplan, Leath, Gray, and Diehl is described. This approximation is applicable to diagonal, off-diagonal and environmental disorder, includes cluster scattering, and yields a translationally invariant and analytic (Herglotz) average Green's function. Furthermore Gray and Kaplan have shown that an approximation for alloys with short-range order can be constructed from this theory.

  13. The "Hard Problem" and the Quantum Physicists. Part 1: The First Generation

    ERIC Educational Resources Information Center

    Smith, C. U. M.

    2006-01-01

    All four of the most important figures in the early twentieth-century development of quantum physics--Niels Bohr, Erwin Schroedinger, Werner Heisenberg and Wolfgang Pauli--had strong interests in the traditional mind--brain, or "hard," problem. This paper reviews their approach to this problem, showing the influence of Bohr's complementarity…

  14. A Gaussian Approximation Potential for Silicon

    NASA Astrophysics Data System (ADS)

    Bernstein, Noam; Bartók, Albert; Kermode, James; Csányi, Gábor

    We present an interatomic potential for silicon using the Gaussian Approximation Potential (GAP) approach, which uses the Gaussian process regression method to approximate the reference potential energy surface as a sum of atomic energies. Each atomic energy is approximated as a function of the local environment around the atom, which is described with the smooth overlap of atomic environments (SOAP) descriptor. The potential is fit to a database of energies, forces, and stresses calculated using density functional theory (DFT) on a wide range of configurations from zero and finite temperature simulations. These include crystalline phases, liquid, amorphous, and low coordination structures, and diamond-structure point defects, dislocations, surfaces, and cracks. We compare the results of the potential to DFT calculations, as well as to previously published models including Stillinger-Weber, Tersoff, modified embedded atom method (MEAM), and ReaxFF. We show that it is very accurate as compared to the DFT reference results for a wide range of properties, including low energy bulk phases, liquid structure, as well as point, line, and plane defects in the diamond structure.

  15. CMB-lensing beyond the Born approximation

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Marozzi, Giovanni; Fanizza, Giuseppe; Durrer, Ruth

    2016-09-01

    We investigate the weak lensing corrections to the cosmic microwave background temperature anisotropies considering effects beyond the Born approximation. To this aim, we use the small deflection angle approximation, to connect the lensed and unlensed power spectra, via expressions for the deflection angles up to third order in the gravitational potential. While the small deflection angle approximation has the drawback to be reliable only for multipoles ℓ ∼< 2500, it allows us to consistently take into account the non-Gaussian nature of cosmological perturbation theory beyond the linear level. The contribution to the lensed temperature power spectrum coming from the non-Gaussianmore » nature of the deflection angle at higher order is a new effect which has not been taken into account in the literature so far. It turns out to be the leading contribution among the post-Born lensing corrections. On the other hand, the effect is smaller than corrections coming from non-linearities in the matter power spectrum, and its imprint on CMB lensing is too small to be seen in present experiments.« less

  16. New Hardness Results for Diophantine Approximation

    NASA Astrophysics Data System (ADS)

    Eisenbrand, Friedrich; Rothvoß, Thomas

    We revisit simultaneous Diophantine approximation, a classical problem from the geometry of numbers which has many applications in algorithms and complexity. The input to the decision version of this problem consists of a rational vector α ∈ ℚ n , an error bound ɛ and a denominator bound N ∈ ℕ + . One has to decide whether there exists an integer, called the denominator Q with 1 ≤ Q ≤ N such that the distance of each number Q ·α i to its nearest integer is bounded by ɛ. Lagarias has shown that this problem is NP-complete and optimization versions have been shown to be hard to approximate within a factor n c/ loglogn for some constant c > 0. We strengthen the existing hardness results and show that the optimization problem of finding the smallest denominator Q ∈ ℕ + such that the distances of Q·α i to the nearest integer are bounded by ɛ is hard to approximate within a factor 2 n unless {textrm{P}} = NP.

  17. Local approximation of a metapopulation's equilibrium.

    PubMed

    Barbour, A D; McVinish, R; Pollett, P K

    2018-04-18

    We consider the approximation of the equilibrium of a metapopulation model, in which a finite number of patches are randomly distributed over a bounded subset [Formula: see text] of Euclidean space. The approximation is good when a large number of patches contribute to the colonization pressure on any given unoccupied patch, and when the quality of the patches varies little over the length scale determined by the colonization radius. If this is the case, the equilibrium probability of a patch at z being occupied is shown to be close to [Formula: see text], the equilibrium occupation probability in Levins's model, at any point [Formula: see text] not too close to the boundary, if the local colonization pressure and extinction rates appropriate to z are assumed. The approximation is justified by giving explicit upper and lower bounds for the occupation probabilities, expressed in terms of the model parameters. Since the patches are distributed randomly, the occupation probabilities are also random, and we complement our bounds with explicit bounds on the probability that they are satisfied at all patches simultaneously.

  18. Fostering Formal Commutativity Knowledge with Approximate Arithmetic

    PubMed Central

    Hansen, Sonja Maria; Haider, Hilde; Eichler, Alexandra; Godau, Claudia; Frensch, Peter A.; Gaschler, Robert

    2015-01-01

    How can we enhance the understanding of abstract mathematical principles in elementary school? Different studies found out that nonsymbolic estimation could foster subsequent exact number processing and simple arithmetic. Taking the commutativity principle as a test case, we investigated if the approximate calculation of symbolic commutative quantities can also alter the access to procedural and conceptual knowledge of a more abstract arithmetic principle. Experiment 1 tested first graders who had not been instructed about commutativity in school yet. Approximate calculation with symbolic quantities positively influenced the use of commutativity-based shortcuts in formal arithmetic. We replicated this finding with older first graders (Experiment 2) and third graders (Experiment 3). Despite the positive effect of approximation on the spontaneous application of commutativity-based shortcuts in arithmetic problems, we found no comparable impact on the application of conceptual knowledge of the commutativity principle. Overall, our results show that the usage of a specific arithmetic principle can benefit from approximation. However, the findings also suggest that the correct use of certain procedures does not always imply conceptual understanding. Rather, the conceptual understanding of commutativity seems to lag behind procedural proficiency during elementary school. PMID:26560311

  19. A Discrete Approximation Framework for Hereditary Systems.

    DTIC Science & Technology

    1980-05-01

    schemes which are included in the general framework and which may be implemented directly on high-speed computing machines are developed. A numerical...an appropriately chosen Hilbert space. We then proceed to develop general approximation schemes for the solutions to the homogeneous AEE which in turn...rich classes of these schemes . In addition, two particular families of approximation schemes included in the general framework are developed and

  20. Best uniform approximation to a class of rational functions

    NASA Astrophysics Data System (ADS)

    Zheng, Zhitong; Yong, Jun-Hai

    2007-10-01

    We explicitly determine the best uniform polynomial approximation to a class of rational functions of the form 1/(x-c)2+K(a,b,c,n)/(x-c) on [a,b] represented by their Chebyshev expansion, where a, b, and c are real numbers, n-1 denotes the degree of the best approximating polynomial, and K is a constant determined by a, b, c, and n. Our result is based on the explicit determination of a phase angle [eta] in the representation of the approximation error by a trigonometric function. Moreover, we formulate an ansatz which offers a heuristic strategies to determine the best approximating polynomial to a function represented by its Chebyshev expansion. Combined with the phase angle method, this ansatz can be used to find the best uniform approximation to some more functions.

  1. Approximating the Helium Wavefunction in Positronium-Helium Scattering

    NASA Technical Reports Server (NTRS)

    DiRienzi, Joseph; Drachman, Richard J.

    2003-01-01

    In the Kohn variational treatment of the positronium- hydrogen scattering problem the scattering wave function is approximated by an expansion in some appropriate basis set, but the target and projectile wave functions are known exactly. In the positronium-helium case, however, a difficulty immediately arises in that the wave function of the helium target atom is not known exactly, and there are several ways to deal with the associated eigenvalue in formulating the variational scattering equations to be solved. In this work we will use the Kohn variational principle in the static exchange approximation to d e t e e the zero-energy scattering length for the Ps-He system, using a suite of approximate target functions. The results we obtain will be compared with each other and with corresponding values found by other approximation techniques.

  2. Minimax rational approximation of the Fermi-Dirac distribution.

    PubMed

    Moussa, Jonathan E

    2016-10-28

    Accurate rational approximations of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known approximations use O(log(βΔ)log(ϵ -1 )) poles to achieve an error tolerance ϵ at temperature β -1 over an energy interval Δ. We apply minimax approximation to reduce the number of poles by a factor of four and replace Δ with Δ occ , the occupied energy interval. This is particularly beneficial when Δ ≫ Δ occ , such as in electronic structure calculations that use a large basis set.

  3. Minimax rational approximation of the Fermi-Dirac distribution

    NASA Astrophysics Data System (ADS)

    Moussa, Jonathan E.

    2016-10-01

    Accurate rational approximations of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known approximations use O(log(βΔ)log(ɛ-1)) poles to achieve an error tolerance ɛ at temperature β-1 over an energy interval Δ. We apply minimax approximation to reduce the number of poles by a factor of four and replace Δ with Δocc, the occupied energy interval. This is particularly beneficial when Δ ≫ Δocc, such as in electronic structure calculations that use a large basis set.

  4. On direct theorems for best polynomial approximation

    NASA Astrophysics Data System (ADS)

    Auad, A. A.; AbdulJabbar, R. S.

    2018-05-01

    This paper is to obtain similarity for the best approximation degree of functions, which are unbounded in L p,α (A = [0,1]), which called weighted space by algebraic polynomials. {E}nH{(f)}p,α and the best approximation degree in the same space on the interval [0,2π] by trigonometric polynomials {E}nT{(f)}p,α of direct wellknown theorems in forms the average modules.

  5. Background magnetic spectra - Approximately 10 to the -5th to approximately 10 to the 5th Hz

    NASA Astrophysics Data System (ADS)

    Lanzerotti, L. J.; Maclennan, C. G.; Fraser-Smith, A. C.

    1990-09-01

    The determination of the amplitude and functional form of the geomagnetic fluctuations measured at the Arrival Heights area of the Hut Point Peninsula on Ross Island in June 1986 is presented. The frequency range covered is from approximately 10 to the -5th to approximately 10 to the 5th Hz, with a gap between 0.1 and 10 Hz due to instrumentation limitations. In spite of this gap, it is thought that these magnetic fluctuation spectra, obtained from data acquired simultaneously with two instruments, cover the broadest frequency range to date. Schematic spectra derived from the data obtained are provided.

  6. Unveiling the Structural Basis That Regulates the Energy Transduction Properties within a Family of Triheme Cytochromes from Geobacter sulfurreducens

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Dantas, Joana M.; Simões, Telma; Morgado, Leonor

    A family of triheme cytochromes from Geobacter sulfurreducens plays an important role in extracellular electron transfer. In addition to their role in electron transfer pathways, two members of this family (PpcA and PpcD) were also found to be able to couple e(-)/H+ transfer through the redox Bohr effect observed in the physiological pH range, a feature not observed for cytochromes PpcB and PpcE. In attempting to understand the molecular control of the redox Bohr effect in this family of cytochromes, which is highly homologous both in amino acid sequence and structures, it was observed that residue 6 is a conservedmore » leucine in PpcA and PpcD, whereas in the other two characterized members (PpcB and PpcE) the equivalent residue is a phenylalanine. To determine the role of this residue located close to the redox Bohr center, we replaced Leu(6) in PpcA with Phe and determined the redox properties of the mutant, as well as its solution structure in the fully reduced state. In contrast with the native form, the mutant PpcAL6F is not able to couple the e(-)/H+ pathway. We carried out the reverse mutation in PpcB and PpcE (i.e., replacing Phe(6) in these two proteins by leucine) and the mutated proteins showed an increased redox Bohr effect. The results clearly establish the role of residue 6 in the control of the redox Bohr effect in this family of cytochromes, a feature that could enable the rational design of G. sulfurreducens strains that carry mutant cytochromes with an optimal redox Bohr effect that would be suitable for various biotechnological applications.« less

  7. Modeling and Reality in Early Twentieth-Century Physics

    NASA Astrophysics Data System (ADS)

    Seth, Suman

    2011-04-01

    Towards the end of 1913, Arnold Sommerfeld, Professor of theoretical physics at Munich University, sent a letter of congratulations to a young Niels Bohr. The Dane's now-classic trilogy of papers, which coupled Rutherford's conception of the atom with a ``planetary'' configuration of electrons, had just appeared. Sommerfeld saw the calculation of the Rydberg constant as a singular triumph and immediately spotted an opportunity to try to explain the Zeeman effect. Yet he also sounded a note of caution, confessing that he remained ``somewhat skeptical'' of atomic models in general. In this, of course, he was hardly alone. Bohr's atom was a particularly egregious example of a peculiar model, one requiring what even its creator considered ``horrid assumptions.'' Nonetheless, success bred conviction. Expanding upon Bohr's original ideas, Sommerfeld soon produced the so-called ``Bohr-Sommerfeld quantization conditions,'' using them to calculate a myriad of results. Experimental evidence, Sommerfeld argued in 1915, showed that quantised electron-paths ``correspond exactly to reality'' and possess ``real existence.'' This kind of realism would not, of course, last long. In 1925, Werner Heisenberg (earlier a student of Sommerfeld's) made scepticism about the details of the Bohr model into a methodological dictum, one later enshrined in the ``Copenhagen interpretation'' of quantum mechanics. This paper uses Sommerfeld's work from the turn of the twentieth century to the mid-1920s as a window onto a landscape involving multiple contestations over the legitimacy of atomic modelling. The surprise that greeted Heisenberg's and others' phenomenological insistences, we will see, can only be understood with reference to what should be considered a ``realist interlude'' in the history of twentieth century atomic physics, one inspired by the astonishing successes of Rutherford's and Bohr's imaginings.

  8. Making It Visual: Creating a Model of the Atom

    ERIC Educational Resources Information Center

    Pringle, Rose M.

    2004-01-01

    This article describes a lesson in which students construct Bohr's planetary model of the atom. Niels Bohr's atomic model provides a framework for discussing with middle and high school students the historical development of our understanding of the structure of the atom. The model constructed in this activity will enable students to visualize the…

  9. Compression of strings with approximate repeats.

    PubMed

    Allison, L; Edgoose, T; Dix, T I

    1998-01-01

    We describe a model for strings of characters that is loosely based on the Lempel Ziv model with the addition that a repeated substring can be an approximate match to the original substring; this is close to the situation of DNA, for example. Typically there are many explanations for a given string under the model, some optimal and many suboptimal. Rather than commit to one optimal explanation, we sum the probabilities over all explanations under the model because this gives the probability of the data under the model. The model has a small number of parameters and these can be estimated from the given string by an expectation-maximization (EM) algorithm. Each iteration of the EM algorithm takes O(n2) time and a few iterations are typically sufficient. O(n2) complexity is impractical for strings of more than a few tens of thousands of characters and a faster approximation algorithm is also given. The model is further extended to include approximate reverse complementary repeats when analyzing DNA strings. Tests include the recovery of parameter estimates from known sources and applications to real DNA strings.

  10. Approximating smooth functions using algebraic-trigonometric polynomials

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Sharapudinov, Idris I

    2011-01-14

    The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form p{sub n}(t)+{tau}{sub m}(t), where p{sub n}(t) is an algebraic polynomial of degree n and {tau}{sub m}(t)=a{sub 0}+{Sigma}{sub k=1}{sup m}a{sub k} cos k{pi}t + b{sub k} sin k{pi}t is a trigonometric polynomial of order m. The precise order of approximation by such polynomials in the classes W{sup r}{sub {infinity}(}M) and an upper bound for similar approximations in the class W{sup r}{sub p}(M) with 4/3

  11. Minimax rational approximation of the Fermi-Dirac distribution

    DOE PAGES

    Moussa, Jonathan E.

    2016-10-27

    Accurate rational approximations of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known approximations use O(log(βΔ)log(ϵ –1)) poles to achieve an error tolerance ϵ at temperature β –1 over an energy interval Δ. We apply minimax approximation to reduce the number of poles by a factor of four and replace Δ with Δ occ, the occupied energy interval. Furthermore, this is particularly beneficial when Δ >> Δ occ, such as in electronic structure calculations that use a large basis set.

  12. Incorporating approximation error in surrogate based Bayesian inversion

    NASA Astrophysics Data System (ADS)

    Zhang, J.; Zeng, L.; Li, W.; Wu, L.

    2015-12-01

    There are increasing interests in applying surrogates for inverse Bayesian modeling to reduce repetitive evaluations of original model. In this way, the computational cost is expected to be saved. However, the approximation error of surrogate model is usually overlooked. This is partly because that it is difficult to evaluate the approximation error for many surrogates. Previous studies have shown that, the direct combination of surrogates and Bayesian methods (e.g., Markov Chain Monte Carlo, MCMC) may lead to biased estimations when the surrogate cannot emulate the highly nonlinear original system. This problem can be alleviated by implementing MCMC in a two-stage manner. However, the computational cost is still high since a relatively large number of original model simulations are required. In this study, we illustrate the importance of incorporating approximation error in inverse Bayesian modeling. Gaussian process (GP) is chosen to construct the surrogate for its convenience in approximation error evaluation. Numerical cases of Bayesian experimental design and parameter estimation for contaminant source identification are used to illustrate this idea. It is shown that, once the surrogate approximation error is well incorporated into Bayesian framework, promising results can be obtained even when the surrogate is directly used, and no further original model simulations are required.

  13. The closure approximation in the hierarchy equations.

    NASA Technical Reports Server (NTRS)

    Adomian, G.

    1971-01-01

    The expectation of the solution process in a stochastic operator equation can be obtained from averaged equations only under very special circumstances. Conditions for validity are given and the significance and validity of the approximation in widely used hierarchy methods and the ?self-consistent field' approximation in nonequilibrium statistical mechanics are clarified. The error at any level of the hierarchy can be given and can be avoided by the use of the iterative method.

  14. Book Review

    NASA Astrophysics Data System (ADS)

    Tanona, Scott

    There is more to this book than readers of this journal might expect from an author who has associated Bohr and Heisenberg with Derrida and Lacan (Plotnitsky, 1994, 2002). Plotnitsky has written in the past on trying to bridge the "two cultures" of the "science wars" by defending Derrida and others against what he perceives to have been too-quickly dismissive critiques (Plotnitsky, 1997), but he has not provided an extended philosophical analysis of Bohr to support the connections he has made. This book seems to be an attempt to engage in a careful analysis of Bohr in a way meant to satisfy critical readers and also support what he here calls "nonclassical epistemology".

  15. Adequacy of selected evapotranspiration approximations for hydrologic simulation

    USGS Publications Warehouse

    Sumner, D.M.

    2006-01-01

    Evapotranspiration (ET) approximations, usually based on computed potential ET (PET) and diverse PET-to-ET conceptualizations, are routinely used in hydrologic analyses. This study presents an approach to incorporate measured (actual) ET data, increasingly available using micrometeorological methods, to define the adequacy of ET approximations for hydrologic simulation. The approach is demonstrated at a site where eddy correlation-measured ET values were available. A baseline hydrologic model incorporating measured ET values was used to evaluate the sensitivity of simulated water levels, subsurface recharge, and surface runoff to error in four ET approximations. An annually invariant pattern of mean monthly vegetation coefficients was shown to be most effective, despite the substantial year-to-year variation in measured vegetation coefficients. The temporal variability of available water (precipitation minus ET) at the humid, subtropical site was largely controlled by the relatively high temporal variability of precipitation, benefiting the effectiveness of coarse ET approximations, a result that is likely to prevail at other humid sites.

  16. The Effects of the Pauli Exclusion Principle in Determining the Ionization Energies of the Helium Atom and Helium-Like Ions

    ERIC Educational Resources Information Center

    Deeney, F. A.; O'Leary, J. P.

    2012-01-01

    For helium and helium-like ions, we have examined the differences between the values of the ionization energies as calculated from the Bohr theory and those measured in experiments. We find that these differences vary linearly with the atomic number of the system. Using this result, we show how the Bohr model for single-electron systems may be…

  17. Approximate analytic expression for the Skyrmions crystal

    NASA Astrophysics Data System (ADS)

    Grandi, Nicolás; Sturla, Mauricio

    2018-01-01

    We find approximate solutions for the two-dimensional nonlinear Σ-model with Dzyalioshinkii-Moriya term, representing magnetic Skyrmions. They are built in an analytic form, by pasting different approximate solutions found in different regions of space. We verify that our construction reproduces the phenomenology known from numerical solutions and Monte Carlo simulations, giving rise to a Skyrmion lattice at an intermediate range of magnetic field, flanked by spiral and spin-polarized phases for low and high magnetic fields, respectively.

  18. On Born approximation in black hole scattering

    NASA Astrophysics Data System (ADS)

    Batic, D.; Kelkar, N. G.; Nowakowski, M.

    2011-12-01

    A massless field propagating on spherically symmetric black hole metrics such as the Schwarzschild, Reissner-Nordström and Reissner-Nordström-de Sitter backgrounds is considered. In particular, explicit formulae in terms of transcendental functions for the scattering of massless scalar particles off black holes are derived within a Born approximation. It is shown that the conditions on the existence of the Born integral forbid a straightforward extraction of the quasi normal modes using the Born approximation for the scattering amplitude. Such a method has been used in literature. We suggest a novel, well defined method, to extract the large imaginary part of quasinormal modes via the Coulomb-like phase shift. Furthermore, we compare the numerically evaluated exact scattering amplitude with the Born one to find that the approximation is not very useful for the scattering of massless scalar, electromagnetic as well as gravitational waves from black holes.

  19. Efficient solution of parabolic equations by Krylov approximation methods

    NASA Technical Reports Server (NTRS)

    Gallopoulos, E.; Saad, Y.

    1990-01-01

    Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms.

  20. Precise analytic approximations for the Bessel function J1 (x)

    NASA Astrophysics Data System (ADS)

    Maass, Fernando; Martin, Pablo

    2018-03-01

    Precise and straightforward analytic approximations for the Bessel function J1 (x) have been found. Power series and asymptotic expansions have been used to determine the parameters of the approximation, which is as a bridge between both expansions, and it is a combination of rational and trigonometric functions multiplied with fractional powers of x. Here, several improvements with respect to the so called Multipoint Quasirational Approximation technique have been performed. Two procedures have been used to determine the parameters of the approximations. The maximum absolute errors are in both cases smaller than 0.01. The zeros of the approximation are also very precise with less than 0.04 per cent for the first one. A second approximation has been also determined using two more parameters, and in this way the accuracy has been increased to less than 0.001.

  1. An approximation method for configuration optimization of trusses

    NASA Technical Reports Server (NTRS)

    Hansen, Scott R.; Vanderplaats, Garret N.

    1988-01-01

    Two- and three-dimensional elastic trusses are designed for minimum weight by varying the areas of the members and the location of the joints. Constraints on member stresses and Euler buckling are imposed and multiple static loading conditions are considered. The method presented here utilizes an approximate structural analysis based on first order Taylor series expansions of the member forces. A numerical optimizer minimizes the weight of the truss using information from the approximate structural analysis. Comparisons with results from other methods are made. It is shown that the method of forming an approximate structural analysis based on linearized member forces leads to a highly efficient method of truss configuration optimization.

  2. Geometrical-optics approximation of forward scattering by coated particles.

    PubMed

    Xu, Feng; Cai, Xiaoshu; Ren, Kuanfang

    2004-03-20

    By means of geometrical optics we present an approximation algorithm with which to accelerate the computation of scattering intensity distribution within a forward angular range (0 degrees-60 degrees) for coated particles illuminated by a collimated incident beam. Phases of emerging rays are exactly calculated to improve the approximation precision. This method proves effective for transparent and tiny absorbent particles with size parameters larger than 75 but fails to give good approximation results at scattering angles at which refractive rays are absent. When the absorption coefficient of a particle is greater than 0.01, the geometrical optics approximation is effective only for forward small angles, typically less than 10 degrees or so.

  3. Revised Thomas-Fermi approximation for singular potentials

    NASA Astrophysics Data System (ADS)

    Dufty, James W.; Trickey, S. B.

    2016-08-01

    Approximations for the many-fermion free-energy density functional that include the Thomas-Fermi (TF) form for the noninteracting part lead to singular densities for singular external potentials (e.g., attractive Coulomb). This limitation of the TF approximation is addressed here by a formal map of the exact Euler equation for the density onto an equivalent TF form characterized by a modified Kohn-Sham potential. It is shown to be a "regularized" version of the Kohn-Sham potential, tempered by convolution with a finite-temperature response function. The resulting density is nonsingular, with the equilibrium properties obtained from the total free-energy functional evaluated at this density. This new representation is formally exact. Approximate expressions for the regularized potential are given to leading order in a nonlocality parameter, and the limiting behavior at high and low temperatures is described. The noninteracting part of the free energy in this approximation is the usual Thomas-Fermi functional. These results generalize and extend to finite temperatures the ground-state regularization by R. G. Parr and S. Ghosh [Proc. Natl. Acad. Sci. U.S.A. 83, 3577 (1986), 10.1073/pnas.83.11.3577] and by L. R. Pratt, G. G. Hoffman, and R. A. Harris [J. Chem. Phys. 88, 1818 (1988), 10.1063/1.454105] and formally systematize the finite-temperature regularization given by the latter authors.

  4. Double power series method for approximating cosmological perturbations

    NASA Astrophysics Data System (ADS)

    Wren, Andrew J.; Malik, Karim A.

    2017-04-01

    We introduce a double power series method for finding approximate analytical solutions for systems of differential equations commonly found in cosmological perturbation theory. The method was set out, in a noncosmological context, by Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases where perturbations are on subhorizon scales. The FSN method is essentially an extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding approximate analytical solutions for ordinary differential equations. The FSN method we use is applicable well beyond perturbation theory to solve systems of ordinary differential equations, linear in the derivatives, that also depend on a small parameter, which here we take to be related to the inverse wave-number. We use the FSN method to find new approximate oscillating solutions in linear order cosmological perturbation theory for a flat radiation-matter universe. Together with this model's well-known growing and decaying Mészáros solutions, these oscillating modes provide a complete set of subhorizon approximations for the metric potential, radiation and matter perturbations. Comparison with numerical solutions of the perturbation equations shows that our approximations can be made accurate to within a typical error of 1%, or better. We also set out a heuristic method for error estimation. A Mathematica notebook which implements the double power series method is made available online.

  5. Approximation abilities of neuro-fuzzy networks

    NASA Astrophysics Data System (ADS)

    Mrówczyńska, Maria

    2010-01-01

    The paper presents the operation of two neuro-fuzzy systems of an adaptive type, intended for solving problems of the approximation of multi-variable functions in the domain of real numbers. Neuro-fuzzy systems being a combination of the methodology of artificial neural networks and fuzzy sets operate on the basis of a set of fuzzy rules "if-then", generated by means of the self-organization of data grouping and the estimation of relations between fuzzy experiment results. The article includes a description of neuro-fuzzy systems by Takaga-Sugeno-Kang (TSK) and Wang-Mendel (WM), and in order to complement the problem in question, a hierarchical structural self-organizing method of teaching a fuzzy network. A multi-layer structure of the systems is a structure analogous to the structure of "classic" neural networks. In its final part the article presents selected areas of application of neuro-fuzzy systems in the field of geodesy and surveying engineering. Numerical examples showing how the systems work concerned: the approximation of functions of several variables to be used as algorithms in the Geographic Information Systems (the approximation of a terrain model), the transformation of coordinates, and the prediction of a time series. The accuracy characteristics of the results obtained have been taken into consideration.

  6. Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation

    ERIC Educational Resources Information Center

    Gordon, Sheldon P.; Yang, Yajun

    2017-01-01

    This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…

  7. Finding the Best Quadratic Approximation of a Function

    ERIC Educational Resources Information Center

    Yang, Yajun; Gordon, Sheldon P.

    2011-01-01

    This article examines the question of finding the best quadratic function to approximate a given function on an interval. The prototypical function considered is f(x) = e[superscript x]. Two approaches are considered, one based on Taylor polynomial approximations at various points in the interval under consideration, the other based on the fact…

  8. Dynamic Analyses of Result Quality in Energy-Aware Approximate Programs

    NASA Astrophysics Data System (ADS)

    RIngenburg, Michael F.

    Energy efficiency is a key concern in the design of modern computer systems. One promising approach to energy-efficient computation, approximate computing, trades off output precision for energy efficiency. However, this tradeoff can have unexpected effects on computation quality. This thesis presents dynamic analysis tools to study, debug, and monitor the quality and energy efficiency of approximate computations. We propose three styles of tools: prototyping tools that allow developers to experiment with approximation in their applications, online tools that instrument code to determine the key sources of error, and online tools that monitor the quality of deployed applications in real time. Our prototyping tool is based on an extension to the functional language OCaml. We add approximation constructs to the language, an approximation simulator to the runtime, and profiling and auto-tuning tools for studying and experimenting with energy-quality tradeoffs. We also present two online debugging tools and three online monitoring tools. The first online tool identifies correlations between output quality and the total number of executions of, and errors in, individual approximate operations. The second tracks the number of approximate operations that flow into a particular value. Our online tools comprise three low-cost approaches to dynamic quality monitoring. They are designed to monitor quality in deployed applications without spending more energy than is saved by approximation. Online monitors can be used to perform real time adjustments to energy usage in order to meet specific quality goals. We present prototype implementations of all of these tools and describe their usage with several applications. Our prototyping, profiling, and autotuning tools allow us to experiment with approximation strategies and identify new strategies, our online tools succeed in providing new insights into the effects of approximation on output quality, and our monitors succeed in

  9. Comparison of dynamical approximation schemes for non-linear gravitational clustering

    NASA Technical Reports Server (NTRS)

    Melott, Adrian L.

    1994-01-01

    We have recently conducted a controlled comparison of a number of approximations for gravitational clustering against the same n-body simulations. These include ordinary linear perturbation theory (Eulerian), the adhesion approximation, the frozen-flow approximation, the Zel'dovich approximation (describable as first-order Lagrangian perturbation theory), and its second-order generalization. In the last two cases we also created new versions of approximation by truncation, i.e., smoothing the initial conditions by various smoothing window shapes and varying their sizes. The primary tool for comparing simulations to approximation schemes was crosscorrelation of the evolved mass density fields, testing the extent to which mass was moved to the right place. The Zel'dovich approximation, with initial convolution with a Gaussian e(exp -k(exp 2)/k(exp 2, sub G)) where k(sub G) is adjusted to be just into the nonlinear regime of the evolved model (details in text) worked extremely well. Its second-order generalization worked slightly better. All other schemes, including those proposed as generalizations of the Zel'dovich approximation created by adding forces, were in fact generally worse by this measure. By explicitly checking, we verified that the success of our best-choice was a result of the best treatment of the phases of nonlinear Fourier components. Of all schemes tested, the adhesion approximation produced the most accurate nonlinear power spectrum and density distribution, but its phase errors suggest mass condensations were moved to slightly the wrong location. Due to its better reproduction of the mass density distribution function and power spectrum, it might be preferred for some uses. We recommend either n-body simulations or our modified versions of the Zel'dovich approximation, depending upon the purpose. The theoretical implication is that pancaking is implicit in all cosmological gravitational clustering, at least from Gaussian initial conditions, even

  10. Application of Approximate Unsteady Aerodynamics for Flutter Analysis

    NASA Technical Reports Server (NTRS)

    Pak, Chan-gi; Li, Wesley W.

    2010-01-01

    A technique for approximating the modal aerodynamic influence coefficient (AIC) matrices by using basis functions has been developed. A process for using the resulting approximated modal AIC matrix in aeroelastic analysis has also been developed. The method requires the unsteady aerodynamics in frequency domain, and this methodology can be applied to the unsteady subsonic, transonic, and supersonic aerodynamics. The flutter solution can be found by the classic methods, such as rational function approximation, k, p-k, p, root locus et cetera. The unsteady aeroelastic analysis using unsteady subsonic aerodynamic approximation is demonstrated herein. The technique presented is shown to offer consistent flutter speed prediction on an aerostructures test wing (ATW) 2 and a hybrid wing body (HWB) type of vehicle configuration with negligible loss in precision. This method computes AICs that are functions of the changing parameters being studied and are generated within minutes of CPU time instead of hours. These results may have practical application in parametric flutter analyses as well as more efficient multidisciplinary design and optimization studies.

  11. Singularly Perturbed Lie Bracket Approximation

    DOE PAGES

    Durr, Hans-Bernd; Krstic, Miroslav; Scheinker, Alexander; ...

    2015-03-27

    Here, we consider the interconnection of two dynamical systems where one has an input-affine vector field. We show that by employing a singular perturbation analysis and the Lie bracket approximation technique, the stability of the overall system can be analyzed by regarding the stability properties of two reduced, uncoupled systems.

  12. Computational aspects of pseudospectral Laguerre approximations

    NASA Technical Reports Server (NTRS)

    Funaro, Daniele

    1989-01-01

    Pseudospectral approximations in unbounded domains by Laguerre polynomials lead to ill-conditioned algorithms. Introduced are a scaling function and appropriate numerical procedures in order to limit these unpleasant phenomena.

  13. Explicitly solvable complex Chebyshev approximation problems related to sine polynomials

    NASA Technical Reports Server (NTRS)

    Freund, Roland

    1989-01-01

    Explicitly solvable real Chebyshev approximation problems on the unit interval are typically characterized by simple error curves. A similar principle is presented for complex approximation problems with error curves induced by sine polynomials. As an application, some new explicit formulae for complex best approximations are derived.

  14. Approximate convective heating equations for hypersonic flows

    NASA Technical Reports Server (NTRS)

    Zoby, E. V.; Moss, J. N.; Sutton, K.

    1979-01-01

    Laminar and turbulent heating-rate equations appropriate for engineering predictions of the convective heating rates about blunt reentry spacecraft at hypersonic conditions are developed. The approximate methods are applicable to both nonreacting and reacting gas mixtures for either constant or variable-entropy edge conditions. A procedure which accounts for variable-entropy effects and is not based on mass balancing is presented. Results of the approximate heating methods are in good agreement with existing experimental results as well as boundary-layer and viscous-shock-layer solutions.

  15. Kernel K-Means Sampling for Nyström Approximation.

    PubMed

    He, Li; Zhang, Hong

    2018-05-01

    A fundamental problem in Nyström-based kernel matrix approximation is the sampling method by which training set is built. In this paper, we suggest to use kernel -means sampling, which is shown in our works to minimize the upper bound of a matrix approximation error. We first propose a unified kernel matrix approximation framework, which is able to describe most existing Nyström approximations under many popular kernels, including Gaussian kernel and polynomial kernel. We then show that, the matrix approximation error upper bound, in terms of the Frobenius norm, is equal to the -means error of data points in kernel space plus a constant. Thus, the -means centers of data in kernel space, or the kernel -means centers, are the optimal representative points with respect to the Frobenius norm error upper bound. Experimental results, with both Gaussian kernel and polynomial kernel, on real-world data sets and image segmentation tasks show the superiority of the proposed method over the state-of-the-art methods.

  16. Dipole Approximation to Predict the Resonances of Dimers Composed of Dielectric Resonators for Directional Emission: Dielectric Dimers Dipole Approximation

    DOE PAGES

    Campione, Salvatore; Warne, Larry K.; Basilio, Lorena I.

    2017-09-29

    In this paper we develop a fully-retarded, dipole approximation model to estimate the effective polarizabilities of a dimer made of dielectric resonators. They are computed from the polarizabilities of the two resonators composing the dimer. We analyze the situation of full-cubes as well as split-cubes, which have been shown to exhibit overlapping electric and magnetic resonances. We compare the effective dimer polarizabilities to ones retrieved via full-wave simulations as well as ones computed via a quasi-static, dipole approximation. We observe good agreement between the fully-retarded solution and the full-wave results, whereas the quasi-static approximation is less accurate for the problemmore » at hand. The developed model can be used to predict the electric and magnetic resonances of a dimer under parallel or orthogonal (to the dimer axis) excitation. This is particularly helpful when interested in locating frequencies at which the dimer will emit directional radiation.« less

  17. Best packing of identical helices

    NASA Astrophysics Data System (ADS)

    Huh, Youngsik; Hong, Kyungpyo; Kim, Hyoungjun; No, Sungjong; Oh, Seungsang

    2016-10-01

    In this paper we prove the unique existence of a ropelength-minimizing conformation of the θ-spun double helix in a mathematically rigorous way, and find the minimal ropelength {{{Rop}}}* (θ )=-\\tfrac{8π }{t} where t is the unique solution in [-θ ,0] of the equation 2-2\\cos (t+θ )={t}2. Using this result, the pitch angles of the standard, triple and quadruple helices are around 39.3771^\\circ , 42.8354^\\circ and 43.8351^\\circ , respectively, which are almost identical with the approximated pitch angles of the zero-twist structures previously known by Olsen and Bohr. We also find the ropelength of the standard N-helix.

  18. Resumming the large-N approximation for time evolving quantum systems

    NASA Astrophysics Data System (ADS)

    Mihaila, Bogdan; Dawson, John F.; Cooper, Fred

    2001-05-01

    In this paper we discuss two methods of resumming the leading and next to leading order in 1/N diagrams for the quartic O(N) model. These two approaches have the property that they preserve both boundedness and positivity for expectation values of operators in our numerical simulations. These approximations can be understood either in terms of a truncation to the infinitely coupled Schwinger-Dyson hierarchy of equations, or by choosing a particular two-particle irreducible vacuum energy graph in the effective action of the Cornwall-Jackiw-Tomboulis formalism. We confine our discussion to the case of quantum mechanics where the Lagrangian is L(x,ẋ)=(12)∑Ni=1x˙2i-(g/8N)[∑Ni=1x2i- r20]2. The key to these approximations is to treat both the x propagator and the x2 propagator on similar footing which leads to a theory whose graphs have the same topology as QED with the x2 propagator playing the role of the photon. The bare vertex approximation is obtained by replacing the exact vertex function by the bare one in the exact Schwinger-Dyson equations for the one and two point functions. The second approximation, which we call the dynamic Debye screening approximation, makes the further approximation of replacing the exact x2 propagator by its value at leading order in the 1/N expansion. These two approximations are compared with exact numerical simulations for the quantum roll problem. The bare vertex approximation captures the physics at large and modest N better than the dynamic Debye screening approximation.

  19. An approximate classical unimolecular reaction rate theory

    NASA Astrophysics Data System (ADS)

    Zhao, Meishan; Rice, Stuart A.

    1992-05-01

    We describe a classical theory of unimolecular reaction rate which is derived from the analysis of Davis and Gray by use of simplifying approximations. These approximations concern the calculation of the locations of, and the fluxes of phase points across, the bottlenecks to fragmentation and to intramolecular energy transfer. The bottleneck to fragment separation is represented as a vibration-rotation state dependent separatrix, which approximation is similar to but extends and improves the approximations for the separatrix introduced by Gray, Rice, and Davis and by Zhao and Rice. The novel feature in our analysis is the representation of the bottlenecks to intramolecular energy transfer as dividing surfaces in phase space; the locations of these dividing surfaces are determined by the same conditions as locate the remnants of robust tori with frequency ratios related to the golden mean (in a two degree of freedom system these are the cantori). The flux of phase points across each dividing surface is calculated with an analytic representation instead of a stroboscopic mapping. The rate of unimolecular reaction is identified with the net rate at which phase points escape from the region of quasiperiodic bounded motion to the region of free fragment motion by consecutively crossing the dividing surfaces for intramolecular energy exchange and the separatrix. This new theory generates predictions of the rates of predissociation of the van der Waals molecules HeI2, NeI2 and ArI2 which are in very good agreement with available experimental data.

  20. Representation of Ice Geometry by Parametric Functions: Construction of Approximating NURBS Curves and Quantification of Ice Roughness--Year 1: Approximating NURBS Curves

    NASA Technical Reports Server (NTRS)

    Dill, Loren H.; Choo, Yung K. (Technical Monitor)

    2004-01-01

    Software was developed to construct approximating NURBS curves for iced airfoil geometries. Users specify a tolerance that determines the extent to which the approximating curve follows the rough ice. The user can therefore smooth the ice geometry in a controlled manner, thereby enabling the generation of grids suitable for numerical aerodynamic simulations. Ultimately, this ability to smooth the ice geometry will permit studies of the effects of smoothing upon the aerodynamics of iced airfoils. The software was applied to several different types of iced airfoil data collected in the Icing Research Tunnel at NASA Glenn Research Center, and in all cases was found to efficiently generate suitable approximating NURBS curves. This method is an improvement over the current "control point formulation" of Smaggice (v.1.2). In this report, we present the relevant theory of approximating NURBS curves and discuss typical results of the software.

  1. Approximations to the exact exchange potential: KLI versus semilocal

    NASA Astrophysics Data System (ADS)

    Tran, Fabien; Blaha, Peter; Betzinger, Markus; Blügel, Stefan

    2016-10-01

    In the search for an accurate and computationally efficient approximation to the exact exchange potential of Kohn-Sham density functional theory, we recently compared various semilocal exchange potentials to the exact one [F. Tran et al., Phys. Rev. B 91, 165121 (2015), 10.1103/PhysRevB.91.165121]. It was concluded that the Becke-Johnson (BJ) potential is a very good starting point, but requires the use of empirical parameters to obtain good agreement with the exact exchange potential. In this work, we extend the comparison by considering the Krieger-Li-Iafrate (KLI) approximation, which is a beyond-semilocal approximation. It is shown that overall the KLI- and BJ-based potentials are the most reliable approximations to the exact exchange potential, however, sizable differences, especially for the antiferromagnetic transition-metal oxides, can be obtained.

  2. On the dipole approximation with error estimates

    NASA Astrophysics Data System (ADS)

    Boßmann, Lea; Grummt, Robert; Kolb, Martin

    2018-01-01

    The dipole approximation is employed to describe interactions between atoms and radiation. It essentially consists of neglecting the spatial variation of the external field over the atom. Heuristically, this is justified by arguing that the wavelength is considerably larger than the atomic length scale, which holds under usual experimental conditions. We prove the dipole approximation in the limit of infinite wavelengths compared to the atomic length scale and estimate the rate of convergence. Our results include N-body Coulomb potentials and experimentally relevant electromagnetic fields such as plane waves and laser pulses.

  3. Rational-spline approximation with automatic tension adjustment

    NASA Technical Reports Server (NTRS)

    Schiess, J. R.; Kerr, P. A.

    1984-01-01

    An algorithm for weighted least-squares approximation with rational splines is presented. A rational spline is a cubic function containing a distinct tension parameter for each interval defined by two consecutive knots. For zero tension, the rational spline is identical to a cubic spline; for very large tension, the rational spline is a linear function. The approximation algorithm incorporates an algorithm which automatically adjusts the tension on each interval to fulfill a user-specified criterion. Finally, an example is presented comparing results of the rational spline with those of the cubic spline.

  4. Short-Path Statistics and the Diffusion Approximation

    NASA Astrophysics Data System (ADS)

    Blanco, Stéphane; Fournier, Richard

    2006-12-01

    In the field of first return time statistics in bounded domains, short paths may be defined as those paths for which the diffusion approximation is inappropriate. This is at the origin of numerous open questions concerning the characterization of residence time distributions. We show here how general integral constraints can be derived that make it possible to address short-path statistics indirectly by application of the diffusion approximation to long paths. Application to the moments of the distribution at the low-Knudsen limit leads to simple practical results and novel physical pictures.

  5. Cosmological collapse and the improved Zel'dovich approximation.

    NASA Astrophysics Data System (ADS)

    Salopek, D. S.; Stewart, J. M.; Croudace, K. M.; Parry, J.

    Using a general relativistic formulation, the authors show how to compute the higher order terms in the Zel'dovich approximation which describes cosmological collapse. They evolve the 3-metric in a spatial gradient expansion. Their method is an advance over earlier work because it is local at each order. Using the improved Zel'dovich approximation, they compute the epoch of collapse.

  6. Robustness of controllers designed using Galerkin type approximations

    NASA Technical Reports Server (NTRS)

    Morris, K. A.

    1990-01-01

    One of the difficulties in designing controllers for infinite-dimensional systems arises from attempting to calculate a state for the system. It is shown that Galerkin type approximations can be used to design controllers which will perform as designed when implemented on the original infinite-dimensional system. No assumptions, other than those typically employed in numerical analysis, are made on the approximating scheme.

  7. Analytic Interatomic Forces in the Random Phase Approximation

    NASA Astrophysics Data System (ADS)

    Ramberger, Benjamin; Schäfer, Tobias; Kresse, Georg

    2017-03-01

    We discuss that in the random phase approximation (RPA) the first derivative of the energy with respect to the Green's function is the self-energy in the G W approximation. This relationship allows us to derive compact equations for the RPA interatomic forces. We also show that position dependent overlap operators are elegantly incorporated in the present framework. The RPA force equations have been implemented in the projector augmented wave formalism, and we present illustrative applications, including ab initio molecular dynamics simulations, the calculation of phonon dispersion relations for diamond and graphite, as well as structural relaxations for water on boron nitride. The present derivation establishes a concise framework for forces within perturbative approaches and is also applicable to more involved approximations for the correlation energy.

  8. Polynomial approximation of the Lense-Thirring rigid precession frequency

    NASA Astrophysics Data System (ADS)

    De Falco, Vittorio; Motta, Sara

    2018-05-01

    We propose a polynomial approximation of the global Lense-Thirring rigid precession frequency to study low-frequency quasi-periodic oscillations around spinning black holes. This high-performing approximation allows to determine the expected frequencies of a precessing thick accretion disc with fixed inner radius and variable outer radius around a black hole with given mass and spin. We discuss the accuracy and the applicability regions of our polynomial approximation, showing that the computational times are reduced by a factor of ≈70 in the range of minutes.

  9. Flexible Approximation Model Approach for Bi-Level Integrated System Synthesis

    NASA Technical Reports Server (NTRS)

    Sobieszczanski-Sobieski, Jaroslaw; Kim, Hongman; Ragon, Scott; Soremekun, Grant; Malone, Brett

    2004-01-01

    Bi-Level Integrated System Synthesis (BLISS) is an approach that allows design problems to be naturally decomposed into a set of subsystem optimizations and a single system optimization. In the BLISS approach, approximate mathematical models are used to transfer information from the subsystem optimizations to the system optimization. Accurate approximation models are therefore critical to the success of the BLISS procedure. In this paper, new capabilities that are being developed to generate accurate approximation models for BLISS procedure will be described. The benefits of using flexible approximation models such as Kriging will be demonstrated in terms of convergence characteristics and computational cost. An approach of dealing with cases where subsystem optimization cannot find a feasible design will be investigated by using the new flexible approximation models for the violated local constraints.

  10. The complex variable boundary element method: Applications in determining approximative boundaries

    USGS Publications Warehouse

    Hromadka, T.V.

    1984-01-01

    The complex variable boundary element method (CVBEM) is used to determine approximation functions for boundary value problems of the Laplace equation such as occurs in potential theory. By determining an approximative boundary upon which the CVBEM approximator matches the desired constant (level curves) boundary conditions, the CVBEM is found to provide the exact solution throughout the interior of the transformed problem domain. Thus, the acceptability of the CVBEM approximation is determined by the closeness-of-fit of the approximative boundary to the study problem boundary. ?? 1984.

  11. Approximation methods for stochastic petri nets

    NASA Technical Reports Server (NTRS)

    Jungnitz, Hauke Joerg

    1992-01-01

    Stochastic Marked Graphs are a concurrent decision free formalism provided with a powerful synchronization mechanism generalizing conventional Fork Join Queueing Networks. In some particular cases the analysis of the throughput can be done analytically. Otherwise the analysis suffers from the classical state explosion problem. Embedded in the divide and conquer paradigm, approximation techniques are introduced for the analysis of stochastic marked graphs and Macroplace/Macrotransition-nets (MPMT-nets), a new subclass introduced herein. MPMT-nets are a subclass of Petri nets that allow limited choice, concurrency and sharing of resources. The modeling power of MPMT is much larger than that of marked graphs, e.g., MPMT-nets can model manufacturing flow lines with unreliable machines and dataflow graphs where choice and synchronization occur. The basic idea leads to the notion of a cut to split the original net system into two subnets. The cuts lead to two aggregated net systems where one of the subnets is reduced to a single transition. A further reduction leads to a basic skeleton. The generalization of the idea leads to multiple cuts, where single cuts can be applied recursively leading to a hierarchical decomposition. Based on the decomposition, a response time approximation technique for the performance analysis is introduced. Also, delay equivalence, which has previously been introduced in the context of marked graphs by Woodside et al., Marie's method and flow equivalent aggregation are applied to the aggregated net systems. The experimental results show that response time approximation converges quickly and shows reasonable accuracy in most cases. The convergence of Marie's method and flow equivalent aggregation are applied to the aggregated net systems. The experimental results show that response time approximation converges quickly and shows reasonable accuracy in most cases. The convergence of Marie's is slower, but the accuracy is generally better. Delay

  12. A Subsonic Aircraft Design Optimization With Neural Network and Regression Approximators

    NASA Technical Reports Server (NTRS)

    Patnaik, Surya N.; Coroneos, Rula M.; Guptill, James D.; Hopkins, Dale A.; Haller, William J.

    2004-01-01

    The Flight-Optimization-System (FLOPS) code encountered difficulty in analyzing a subsonic aircraft. The limitation made the design optimization problematic. The deficiencies have been alleviated through use of neural network and regression approximations. The insight gained from using the approximators is discussed in this paper. The FLOPS code is reviewed. Analysis models are developed and validated for each approximator. The regression method appears to hug the data points, while the neural network approximation follows a mean path. For an analysis cycle, the approximate model required milliseconds of central processing unit (CPU) time versus seconds by the FLOPS code. Performance of the approximators was satisfactory for aircraft analysis. A design optimization capability has been created by coupling the derived analyzers to the optimization test bed CometBoards. The approximators were efficient reanalysis tools in the aircraft design optimization. Instability encountered in the FLOPS analyzer was eliminated. The convergence characteristics were improved for the design optimization. The CPU time required to calculate the optimum solution, measured in hours with the FLOPS code was reduced to minutes with the neural network approximation and to seconds with the regression method. Generation of the approximators required the manipulation of a very large quantity of data. Design sensitivity with respect to the bounds of aircraft constraints is easily generated.

  13. An approximation function for frequency constrained structural optimization

    NASA Technical Reports Server (NTRS)

    Canfield, R. A.

    1989-01-01

    The purpose is to examine a function for approximating natural frequency constraints during structural optimization. The nonlinearity of frequencies has posed a barrier to constructing approximations for frequency constraints of high enough quality to facilitate efficient solutions. A new function to represent frequency constraints, called the Rayleigh Quotient Approximation (RQA), is presented. Its ability to represent the actual frequency constraint results in stable convergence with effectively no move limits. The objective of the optimization problem is to minimize structural weight subject to some minimum (or maximum) allowable frequency and perhaps subject to other constraints such as stress, displacement, and gage size, as well. A reason for constraining natural frequencies during design might be to avoid potential resonant frequencies due to machinery or actuators on the structure. Another reason might be to satisy requirements of an aircraft or spacecraft's control law. Whatever the structure supports may be sensitive to a frequency band that must be avoided. Any of these situations or others may require the designer to insure the satisfaction of frequency constraints. A further motivation for considering accurate approximations of natural frequencies is that they are fundamental to dynamic response constraints.

  14. Meta-regression approximations to reduce publication selection bias.

    PubMed

    Stanley, T D; Doucouliagos, Hristos

    2014-03-01

    Publication selection bias is a serious challenge to the integrity of all empirical sciences. We derive meta-regression approximations to reduce this bias. Our approach employs Taylor polynomial approximations to the conditional mean of a truncated distribution. A quadratic approximation without a linear term, precision-effect estimate with standard error (PEESE), is shown to have the smallest bias and mean squared error in most cases and to outperform conventional meta-analysis estimators, often by a great deal. Monte Carlo simulations also demonstrate how a new hybrid estimator that conditionally combines PEESE and the Egger regression intercept can provide a practical solution to publication selection bias. PEESE is easily expanded to accommodate systematic heterogeneity along with complex and differential publication selection bias that is related to moderator variables. By providing an intuitive reason for these approximations, we can also explain why the Egger regression works so well and when it does not. These meta-regression methods are applied to several policy-relevant areas of research including antidepressant effectiveness, the value of a statistical life, the minimum wage, and nicotine replacement therapy. Copyright © 2013 John Wiley & Sons, Ltd.

  15. Low rank approximation in G 0W 0 calculations

    DOE PAGES

    Shao, MeiYue; Lin, Lin; Yang, Chao; ...

    2016-06-04

    The single particle energies obtained in a Kohn-Sham density functional theory (DFT) calculation are generally known to be poor approximations to electron excitation energies that are measured in tr ansport, tunneling and spectroscopic experiments such as photo-emission spectroscopy. The correction to these energies can be obtained from the poles of a single particle Green’s function derived from a many-body perturbation theory. From a computational perspective, the accuracy and efficiency of such an approach depends on how a self energy term that properly accounts for dynamic screening of electrons is approximated. The G 0W 0 approximation is a widely used techniquemore » in which the self energy is expressed as the convolution of a noninteracting Green’s function (G 0) and a screened Coulomb interaction (W 0) in the frequency domain. The computational cost associated with such a convolution is high due to the high complexity of evaluating W 0 at multiple frequencies. In this paper, we discuss how the cost of G 0W 0 calculation can be reduced by constructing a low rank approximation to the frequency dependent part of W 0 . In particular, we examine the effect of such a low rank approximation on the accuracy of the G 0W 0 approximation. We also discuss how the numerical convolution of G 0 and W 0 can be evaluated efficiently and accurately by using a contour deformation technique with an appropriate choice of the contour.« less

  16. Pythagorean Approximations and Continued Fractions

    ERIC Educational Resources Information Center

    Peralta, Javier

    2008-01-01

    In this article, we will show that the Pythagorean approximations of [the square root of] 2 coincide with those achieved in the 16th century by means of continued fractions. Assuming this fact and the known relation that connects the Fibonacci sequence with the golden section, we shall establish a procedure to obtain sequences of rational numbers…

  17. Approximate strip exchanging.

    PubMed

    Roy, Swapnoneel; Thakur, Ashok Kumar

    2008-01-01

    Genome rearrangements have been modelled by a variety of primitives such as reversals, transpositions, block moves and block interchanges. We consider such a genome rearrangement primitive Strip Exchanges. Given a permutation, the challenge is to sort it by using minimum number of strip exchanges. A strip exchanging move interchanges the positions of two chosen strips so that they merge with other strips. The strip exchange problem is to sort a permutation using minimum number of strip exchanges. We present here the first non-trivial 2-approximation algorithm to this problem. We also observe that sorting by strip-exchanges is fixed-parameter-tractable. Lastly we discuss the application of strip exchanges in a different area Optical Character Recognition (OCR) with an example.

  18. An Origami Approximation to the Cosmic Web

    NASA Astrophysics Data System (ADS)

    Neyrinck, Mark C.

    2016-10-01

    The powerful Lagrangian view of structure formation was essentially introduced to cosmology by Zel'dovich. In the current cosmological paradigm, a dark-matter-sheet 3D manifold, inhabiting 6D position-velocity phase space, was flat (with vanishing velocity) at the big bang. Afterward, gravity stretched and bunched the sheet together in different places, forming a cosmic web when projected to the position coordinates. Here, I explain some properties of an origami approximation, in which the sheet does not stretch or contract (an assumption that is false in general), but is allowed to fold. Even without stretching, the sheet can form an idealized cosmic web, with convex polyhedral voids separated by straight walls and filaments, joined by convex polyhedral nodes. The nodes form in `polygonal' or `polyhedral' collapse, somewhat like spherical/ellipsoidal collapse, except incorporating simultaneous filament and wall formation. The origami approximation allows phase-space geometries of nodes, filaments, and walls to be more easily understood, and may aid in understanding spin correlations between nearby galaxies. This contribution explores kinematic origami-approximation models giving velocity fields for the first time.

  19. Comparison of universal approximators incorporating partial monotonicity by structure.

    PubMed

    Minin, Alexey; Velikova, Marina; Lang, Bernhard; Daniels, Hennie

    2010-05-01

    Neural networks applied in control loops and safety-critical domains have to meet more requirements than just the overall best function approximation. On the one hand, a small approximation error is required; on the other hand, the smoothness and the monotonicity of selected input-output relations have to be guaranteed. Otherwise, the stability of most of the control laws is lost. In this article we compare two neural network-based approaches incorporating partial monotonicity by structure, namely the Monotonic Multi-Layer Perceptron (MONMLP) network and the Monotonic MIN-MAX (MONMM) network. We show the universal approximation capabilities of both types of network for partially monotone functions. On a number of datasets, we investigate the advantages and disadvantages of these approaches related to approximation performance, training of the model and convergence. 2009 Elsevier Ltd. All rights reserved.

  20. Integral approximations to classical diffusion and smoothed particle hydrodynamics

    DOE PAGES

    Du, Qiang; Lehoucq, R. B.; Tartakovsky, A. M.

    2014-12-31

    The contribution of the paper is the approximation of a classical diffusion operator by an integral equation with a volume constraint. A particular focus is on classical diffusion problems associated with Neumann boundary conditions. By exploiting this approximation, we can also approximate other quantities such as the flux out of a domain. Our analysis of the model equation on the continuum level is closely related to the recent work on nonlocal diffusion and peridynamic mechanics. In particular, we elucidate the role of a volumetric constraint as an approximation to a classical Neumann boundary condition in the presence of physical boundary.more » The volume-constrained integral equation then provides the basis for accurate and robust discretization methods. As a result, an immediate application is to the understanding and improvement of the Smoothed Particle Hydrodynamics (SPH) method.« less

  1. A Varifold Approach to Surface Approximation

    NASA Astrophysics Data System (ADS)

    Buet, Blanche; Leonardi, Gian Paolo; Masnou, Simon

    2017-11-01

    We show that the theory of varifolds can be suitably enriched to open the way to applications in the field of discrete and computational geometry. Using appropriate regularizations of the mass and of the first variation of a varifold we introduce the notion of approximate mean curvature and show various convergence results that hold, in particular, for sequences of discrete varifolds associated with point clouds or pixel/voxel-type discretizations of d-surfaces in the Euclidean n-space, without restrictions on dimension and codimension. The variational nature of the approach also allows us to consider surfaces with singularities, and in that case the approximate mean curvature is consistent with the generalized mean curvature of the limit surface. A series of numerical tests are provided in order to illustrate the effectiveness and generality of the method.

  2. Approximating exponential and logarithmic functions using polynomial interpolation

    NASA Astrophysics Data System (ADS)

    Gordon, Sheldon P.; Yang, Yajun

    2017-04-01

    This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is analysed. The results of interpolating polynomials are compared with those of Taylor polynomials.

  3. Solving the infeasible trust-region problem using approximations.

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Renaud, John E.; Perez, Victor M.; Eldred, Michael Scott

    2004-07-01

    The use of optimization in engineering design has fueled the development of algorithms for specific engineering needs. When the simulations are expensive to evaluate or the outputs present some noise, the direct use of nonlinear optimizers is not advisable, since the optimization process will be expensive and may result in premature convergence. The use of approximations for both cases is an alternative investigated by many researchers including the authors. When approximations are present, a model management is required for proper convergence of the algorithm. In nonlinear programming, the use of trust-regions for globalization of a local algorithm has been provenmore » effective. The same approach has been used to manage the local move limits in sequential approximate optimization frameworks as in Alexandrov et al., Giunta and Eldred, Perez et al. , Rodriguez et al., etc. The experience in the mathematical community has shown that more effective algorithms can be obtained by the specific inclusion of the constraints (SQP type of algorithms) rather than by using a penalty function as in the augmented Lagrangian formulation. The presence of explicit constraints in the local problem bounded by the trust region, however, may have no feasible solution. In order to remedy this problem the mathematical community has developed different versions of a composite steps approach. This approach consists of a normal step to reduce the amount of constraint violation and a tangential step to minimize the objective function maintaining the level of constraint violation attained at the normal step. Two of the authors have developed a different approach for a sequential approximate optimization framework using homotopy ideas to relax the constraints. This algorithm called interior-point trust-region sequential approximate optimization (IPTRSAO) presents some similarities to the two normal-tangential steps algorithms. In this paper, a description of the similarities is presented

  4. Lognormal Approximations of Fault Tree Uncertainty Distributions.

    PubMed

    El-Shanawany, Ashraf Ben; Ardron, Keith H; Walker, Simon P

    2018-01-26

    Fault trees are used in reliability modeling to create logical models of fault combinations that can lead to undesirable events. The output of a fault tree analysis (the top event probability) is expressed in terms of the failure probabilities of basic events that are input to the model. Typically, the basic event probabilities are not known exactly, but are modeled as probability distributions: therefore, the top event probability is also represented as an uncertainty distribution. Monte Carlo methods are generally used for evaluating the uncertainty distribution, but such calculations are computationally intensive and do not readily reveal the dominant contributors to the uncertainty. In this article, a closed-form approximation for the fault tree top event uncertainty distribution is developed, which is applicable when the uncertainties in the basic events of the model are lognormally distributed. The results of the approximate method are compared with results from two sampling-based methods: namely, the Monte Carlo method and the Wilks method based on order statistics. It is shown that the closed-form expression can provide a reasonable approximation to results obtained by Monte Carlo sampling, without incurring the computational expense. The Wilks method is found to be a useful means of providing an upper bound for the percentiles of the uncertainty distribution while being computationally inexpensive compared with full Monte Carlo sampling. The lognormal approximation method and Wilks's method appear attractive, practical alternatives for the evaluation of uncertainty in the output of fault trees and similar multilinear models. © 2018 Society for Risk Analysis.

  5. Direct application of Padé approximant for solving nonlinear differential equations.

    PubMed

    Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Garcia-Gervacio, Jose Luis; Huerta-Chua, Jesus; Morales-Mendoza, Luis Javier; Gonzalez-Lee, Mario

    2014-01-01

    This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant. 34L30.

  6. Dual methods and approximation concepts in structural synthesis

    NASA Technical Reports Server (NTRS)

    Fleury, C.; Schmit, L. A., Jr.

    1980-01-01

    Approximation concepts and dual method algorithms are combined to create a method for minimum weight design of structural systems. Approximation concepts convert the basic mathematical programming statement of the structural synthesis problem into a sequence of explicit primal problems of separable form. These problems are solved by constructing explicit dual functions, which are maximized subject to nonnegativity constraints on the dual variables. It is shown that the joining together of approximation concepts and dual methods can be viewed as a generalized optimality criteria approach. The dual method is successfully extended to deal with pure discrete and mixed continuous-discrete design variable problems. The power of the method presented is illustrated with numerical results for example problems, including a metallic swept wing and a thin delta wing with fiber composite skins.

  7. Comparison of approximate solutions to the phonon Boltzmann transport equation with the relaxation time approximation: Spherical harmonics expansions and the discrete ordinates method

    NASA Astrophysics Data System (ADS)

    Christenson, J. G.; Austin, R. A.; Phillips, R. J.

    2018-05-01

    The phonon Boltzmann transport equation is used to analyze model problems in one and two spatial dimensions, under transient and steady-state conditions. New, explicit solutions are obtained by using the P1 and P3 approximations, based on expansions in spherical harmonics, and are compared with solutions from the discrete ordinates method. For steady-state energy transfer, it is shown that analytic expressions derived using the P1 and P3 approximations agree quantitatively with the discrete ordinates method, in some cases for large Knudsen numbers, and always for Knudsen numbers less than unity. However, for time-dependent energy transfer, the PN solutions differ qualitatively from converged solutions obtained by the discrete ordinates method. Although they correctly capture the wave-like behavior of energy transfer at short times, the P1 and P3 approximations rely on one or two wave velocities, respectively, yielding abrupt, step-changes in temperature profiles that are absent when the angular dependence of the phonon velocities is captured more completely. It is shown that, with the gray approximation, the P1 approximation is formally equivalent to the so-called "hyperbolic heat equation." Overall, these results support the use of the PN approximation to find solutions to the phonon Boltzmann transport equation for steady-state conditions. Such solutions can be useful in the design and analysis of devices that involve heat transfer at nanometer length scales, where continuum-scale approaches become inaccurate.

  8. Test particle propagation in magnetostatic turbulence. 2: The local approximation method

    NASA Technical Reports Server (NTRS)

    Klimas, A. J.; Sandri, G.; Scudder, J. D.; Howell, D. R.

    1976-01-01

    An approximation method for statistical mechanics is presented and applied to a class of problems which contains a test particle propagation problem. All of the available basic equations used in statistical mechanics are cast in the form of a single equation which is integrodifferential in time and which is then used as the starting point for the construction of the local approximation method. Simplification of the integrodifferential equation is achieved through approximation to the Laplace transform of its kernel. The approximation is valid near the origin in the Laplace space and is based on the assumption of small Laplace variable. No other small parameter is necessary for the construction of this approximation method. The n'th level of approximation is constructed formally, and the first five levels of approximation are calculated explicitly. It is shown that each level of approximation is governed by an inhomogeneous partial differential equation in time with time independent operator coefficients. The order in time of these partial differential equations is found to increase as n does. At n = 0 the most local first order partial differential equation which governs the Markovian limit is regained.

  9. Blocking performance approximation in flexi-grid networks

    NASA Astrophysics Data System (ADS)

    Ge, Fei; Tan, Liansheng

    2016-12-01

    The blocking probability to the path requests is an important issue in flexible bandwidth optical communications. In this paper, we propose a blocking probability approximation method of path requests in flexi-grid networks. It models the bundled neighboring carrier allocation with a group of birth-death processes and provides a theoretical analysis to the blocking probability under variable bandwidth traffic. The numerical results show the effect of traffic parameters to the blocking probability of path requests. We use the first fit algorithm in network nodes to allocate neighboring carriers to path requests in simulations, and verify approximation results.

  10. Congruence Approximations for Entrophy Endowed Hyperbolic Systems

    NASA Technical Reports Server (NTRS)

    Barth, Timothy J.; Saini, Subhash (Technical Monitor)

    1998-01-01

    Building upon the standard symmetrization theory for hyperbolic systems of conservation laws, congruence properties of the symmetrized system are explored. These congruence properties suggest variants of several stabilized numerical discretization procedures for hyperbolic equations (upwind finite-volume, Galerkin least-squares, discontinuous Galerkin) that benefit computationally from congruence approximation. Specifically, it becomes straightforward to construct the spatial discretization and Jacobian linearization for these schemes (given a small amount of derivative information) for possible use in Newton's method, discrete optimization, homotopy algorithms, etc. Some examples will be given for the compressible Euler equations and the nonrelativistic MHD equations using linear and quadratic spatial approximation.

  11. Theory and applications of a deterministic approximation to the coalescent model

    PubMed Central

    Jewett, Ethan M.; Rosenberg, Noah A.

    2014-01-01

    Under the coalescent model, the random number nt of lineages ancestral to a sample is nearly deterministic as a function of time when nt is moderate to large in value, and it is well approximated by its expectation E[nt]. In turn, this expectation is well approximated by simple deterministic functions that are easy to compute. Such deterministic functions have been applied to estimate allele age, effective population size, and genetic diversity, and they have been used to study properties of models of infectious disease dynamics. Although a number of simple approximations of E[nt] have been derived and applied to problems of population-genetic inference, the theoretical accuracy of the formulas and the inferences obtained using these approximations is not known, and the range of problems to which they can be applied is not well understood. Here, we demonstrate general procedures by which the approximation nt ≈ E[nt] can be used to reduce the computational complexity of coalescent formulas, and we show that the resulting approximations converge to their true values under simple assumptions. Such approximations provide alternatives to exact formulas that are computationally intractable or numerically unstable when the number of sampled lineages is moderate or large. We also extend an existing class of approximations of E[nt] to the case of multiple populations of time-varying size with migration among them. Our results facilitate the use of the deterministic approximation nt ≈ E[nt] for deriving functionally simple, computationally efficient, and numerically stable approximations of coalescent formulas under complicated demographic scenarios. PMID:24412419

  12. Discussion of CoSA: Clustering of Sparse Approximations

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Armstrong, Derek Elswick

    2017-03-07

    The purpose of this talk is to discuss the possible applications of CoSA (Clustering of Sparse Approximations) to the exploitation of HSI (HyperSpectral Imagery) data. CoSA is presented by Moody et al. in the Journal of Applied Remote Sensing (“Land cover classification in multispectral imagery using clustering of sparse approximations over learned feature dictionaries”, Vol. 8, 2014) and is based on machine learning techniques.

  13. Green-Ampt approximations: A comprehensive analysis

    NASA Astrophysics Data System (ADS)

    Ali, Shakir; Islam, Adlul; Mishra, P. K.; Sikka, Alok K.

    2016-04-01

    Green-Ampt (GA) model and its modifications are widely used for simulating infiltration process. Several explicit approximate solutions to the implicit GA model have been developed with varying degree of accuracy. In this study, performance of nine explicit approximations to the GA model is compared with the implicit GA model using the published data for broad range of soil classes and infiltration time. The explicit GA models considered are Li et al. (1976) (LI), Stone et al. (1994) (ST), Salvucci and Entekhabi (1994) (SE), Parlange et al. (2002) (PA), Barry et al. (2005) (BA), Swamee et al. (2012) (SW), Ali et al. (2013) (AL), Almedeij and Esen (2014) (AE), and Vatankhah (2015) (VA). Six statistical indicators (e.g., percent relative error, maximum absolute percent relative error, average absolute percent relative errors, percent bias, index of agreement, and Nash-Sutcliffe efficiency) and relative computer computation time are used for assessing the model performance. Models are ranked based on the overall performance index (OPI). The BA model is found to be the most accurate followed by the PA and VA models for variety of soil classes and infiltration periods. The AE, SW, SE, and LI model also performed comparatively better. Based on the overall performance index, the explicit models are ranked as BA > PA > VA > LI > AE > SE > SW > ST > AL. Results of this study will be helpful in selection of accurate and simple explicit approximate GA models for solving variety of hydrological problems.

  14. Strong washout approximation to resonant leptogenesis

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Garbrecht, Björn; Gautier, Florian; Klaric, Juraj, E-mail: garbrecht@tum.de, E-mail: florian.gautier@tum.de, E-mail: juraj.klaric@tum.de

    We show that the effective decay asymmetry for resonant Leptogenesis in the strong washout regime with two sterile neutrinos and a single active flavour can in wide regions of parameter space be approximated by its late-time limit ε=Xsin(2φ)/(X{sup 2}+sin{sup 2}φ), where X=8πΔ/(|Y{sub 1}|{sup 2}+|Y{sub 2}|{sup 2}), Δ=4(M{sub 1}-M{sub 2})/(M{sub 1}+M{sub 2}), φ=arg(Y{sub 2}/Y{sub 1}), and M{sub 1,2}, Y{sub 1,2} are the masses and Yukawa couplings of the sterile neutrinos. This approximation in particular extends to parametric regions where |Y{sub 1,2}|{sup 2}>> Δ, i.e. where the width dominates the mass splitting. We generalise the formula for the effective decay asymmetry to themore » case of several flavours of active leptons and demonstrate how this quantity can be used to calculate the lepton asymmetry for phenomenological scenarios that are in agreement with the observed neutrino oscillations. We establish analytic criteria for the validity of the late-time approximation for the decay asymmetry and compare these with numerical results that are obtained by solving for the mixing and the oscillations of the sterile neutrinos. For phenomenologically viable models with two sterile neutrinos, we find that the flavoured effective late-time decay asymmetry can be applied throughout parameter space.« less

  15. An approximation theory for the identification of linear thermoelastic systems

    NASA Technical Reports Server (NTRS)

    Rosen, I. G.; Su, Chien-Hua Frank

    1990-01-01

    An abstract approximation framework and convergence theory for the identification of thermoelastic systems is developed. Starting from an abstract operator formulation consisting of a coupled second order hyperbolic equation of elasticity and first order parabolic equation for heat conduction, well-posedness is established using linear semigroup theory in Hilbert space, and a class of parameter estimation problems is then defined involving mild solutions. The approximation framework is based upon generic Galerkin approximation of the mild solutions, and convergence of solutions of the resulting sequence of approximating finite dimensional parameter identification problems to a solution of the original infinite dimensional inverse problem is established using approximation results for operator semigroups. An example involving the basic equations of one dimensional linear thermoelasticity and a linear spline based scheme are discussed. Numerical results indicate how the approach might be used in a study of damping mechanisms in flexible structures.

  16. Reply to Steele & Ferrer: Modeling Oscillation, Approximately or Exactly?

    ERIC Educational Resources Information Center

    Oud, Johan H. L.; Folmer, Henk

    2011-01-01

    This article addresses modeling oscillation in continuous time. It criticizes Steele and Ferrer's article "Latent Differential Equation Modeling of Self-Regulatory and Coregulatory Affective Processes" (2011), particularly the approximate estimation procedure applied. This procedure is the latent version of the local linear approximation procedure…

  17. Boson expansions based on the random phase approximation representation

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Pedrocchi, V.G.; Tamura, T.

    1984-04-01

    A new boson expansion theory based on the random phase approximation is presented. The boson expansions are derived here directly in the random phase approximation representation with the help of a technique that combines the use of the Usui operator with that of a new bosonization procedure, called the term-by-term bosonization method. The present boson expansion theory is constructed by retaining a single collective quadrupole random phase approximation component, a truncation that allows for a perturbative treatment of the whole problem. Both Hermitian, as well as non-Hermitian boson expansions, valid for even nuclei, are obtained.

  18. Dissociation between exact and approximate addition in developmental dyslexia.

    PubMed

    Yang, Xiujie; Meng, Xiangzhi

    2016-09-01

    Previous research has suggested that number sense and language are involved in number representation and calculation, in which number sense supports approximate arithmetic, and language permits exact enumeration and calculation. Meanwhile, individuals with dyslexia have a core deficit in phonological processing. Based on these findings, we thus hypothesized that children with dyslexia may exhibit exact calculation impairment while doing mental arithmetic. The reaction time and accuracy while doing exact and approximate addition with symbolic Arabic digits and non-symbolic visual arrays of dots were compared between typically developing children and children with dyslexia. Reaction time analyses did not reveal any differences across two groups of children, the accuracies, interestingly, revealed a distinction of approximation and exact addition across two groups of children. Specifically, two groups of children had no differences in approximation. Children with dyslexia, however, had significantly lower accuracy in exact addition in both symbolic and non-symbolic tasks than that of typically developing children. Moreover, linguistic performances were selectively associated with exact calculation across individuals. These results suggested that children with dyslexia have a mental arithmetic deficit specifically in the realm of exact calculation, while their approximation ability is relatively intact. Copyright © 2016 Elsevier Ltd. All rights reserved.

  19. A quantum relaxation-time approximation for finite fermion systems

    NASA Astrophysics Data System (ADS)

    Reinhard, P.-G.; Suraud, E.

    2015-03-01

    We propose a relaxation time approximation for the description of the dynamics of strongly excited fermion systems. Our approach is based on time-dependent density functional theory at the level of the local density approximation. This mean-field picture is augmented by collisional correlations handled in relaxation time approximation which is inspired from the corresponding semi-classical picture. The method involves the estimate of microscopic relaxation rates/times which is presently taken from the well established semi-classical experience. The relaxation time approximation implies evaluation of the instantaneous equilibrium state towards which the dynamical state is progressively driven at the pace of the microscopic relaxation time. As test case, we consider Na clusters of various sizes excited either by a swift ion projectile or by a short and intense laser pulse, driven in various dynamical regimes ranging from linear to strongly non-linear reactions. We observe a strong effect of dissipation on sensitive observables such as net ionization and angular distributions of emitted electrons. The effect is especially large for moderate excitations where typical relaxation/dissipation time scales efficiently compete with ionization for dissipating the available excitation energy. Technical details on the actual procedure to implement a working recipe of such a quantum relaxation approximation are given in appendices for completeness.

  20. RATIONAL APPROXIMATIONS TO GENERALIZED HYPERGEOMETRIC FUNCTIONS.

    DTIC Science & Technology

    Under weak restrictions on the various free parameters, general theorems for rational representations of the generalized hypergeometric functions...and certain Meijer G-functions are developed. Upon specialization, these theorems yield a sequency of rational approximations which converge to the

  1. Plasma Physics Approximations in Ares

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Managan, R. A.

    Lee & More derived analytic forms for the transport properties of a plasma. Many hydro-codes use their formulae for electrical and thermal conductivity. The coefficients are complex functions of Fermi-Dirac integrals, F n( μ/θ ), the chemical potential, μ or ζ = ln(1+e μ/θ ), and the temperature, θ = kT. Since these formulae are expensive to compute, rational function approximations were fit to them. Approximations are also used to find the chemical potential, either μ or ζ . The fits use ζ as the independent variable instead of μ/θ . New fits are provided for A α (ζmore » ),A β (ζ ), ζ, f(ζ ) = (1 + e -μ/θ)F 1/2(μ/θ), F 1/2'/F 1/2, F c α, and F c β. In each case the relative error of the fit is minimized since the functions can vary by many orders of magnitude. The new fits are designed to exactly preserve the limiting values in the non-degenerate and highly degenerate limits or as ζ→ 0 or ∞. The original fits due to Lee & More and George Zimmerman are presented for comparison.« less

  2. 36 CFR 254.11 - Exchanges at approximately equal value.

    Code of Federal Regulations, 2010 CFR

    2010-07-01

    ... equal value. 254.11 Section 254.11 Parks, Forests, and Public Property FOREST SERVICE, DEPARTMENT OF AGRICULTURE LANDOWNERSHIP ADJUSTMENTS Land Exchanges § 254.11 Exchanges at approximately equal value. (a) The authorized officer may exchange lands which are of approximately equal value upon a determination that: (1...

  3. Basis Function Approximation of Transonic Aerodynamic Influence Coefficient Matrix

    NASA Technical Reports Server (NTRS)

    Li, Wesley Waisang; Pak, Chan-Gi

    2010-01-01

    A technique for approximating the modal aerodynamic influence coefficients [AIC] matrices by using basis functions has been developed and validated. An application of the resulting approximated modal AIC matrix for a flutter analysis in transonic speed regime has been demonstrated. This methodology can be applied to the unsteady subsonic, transonic and supersonic aerodynamics. The method requires the unsteady aerodynamics in frequency-domain. The flutter solution can be found by the classic methods, such as rational function approximation, k, p-k, p, root-locus et cetera. The unsteady aeroelastic analysis for design optimization using unsteady transonic aerodynamic approximation is being demonstrated using the ZAERO(TradeMark) flutter solver (ZONA Technology Incorporated, Scottsdale, Arizona). The technique presented has been shown to offer consistent flutter speed prediction on an aerostructures test wing [ATW] 2 configuration with negligible loss in precision in transonic speed regime. These results may have practical significance in the analysis of aircraft aeroelastic calculation and could lead to a more efficient design optimization cycle

  4. Basis Function Approximation of Transonic Aerodynamic Influence Coefficient Matrix

    NASA Technical Reports Server (NTRS)

    Li, Wesley W.; Pak, Chan-gi

    2011-01-01

    A technique for approximating the modal aerodynamic influence coefficients matrices by using basis functions has been developed and validated. An application of the resulting approximated modal aerodynamic influence coefficients matrix for a flutter analysis in transonic speed regime has been demonstrated. This methodology can be applied to the unsteady subsonic, transonic, and supersonic aerodynamics. The method requires the unsteady aerodynamics in frequency-domain. The flutter solution can be found by the classic methods, such as rational function approximation, k, p-k, p, root-locus et cetera. The unsteady aeroelastic analysis for design optimization using unsteady transonic aerodynamic approximation is being demonstrated using the ZAERO flutter solver (ZONA Technology Incorporated, Scottsdale, Arizona). The technique presented has been shown to offer consistent flutter speed prediction on an aerostructures test wing 2 configuration with negligible loss in precision in transonic speed regime. These results may have practical significance in the analysis of aircraft aeroelastic calculation and could lead to a more efficient design optimization cycle.

  5. Approximation methods in relativistic eigenvalue perturbation theory

    NASA Astrophysics Data System (ADS)

    Noble, Jonathan Howard

    In this dissertation, three questions, concerning approximation methods for the eigenvalues of quantum mechanical systems, are investigated: (i) What is a pseudo--Hermitian Hamiltonian, and how can its eigenvalues be approximated via numerical calculations? This is a fairly broad topic, and the scope of the investigation is narrowed by focusing on a subgroup of pseudo--Hermitian operators, namely, PT--symmetric operators. Within a numerical approach, one projects a PT--symmetric Hamiltonian onto an appropriate basis, and uses a straightforward two--step algorithm to diagonalize the resulting matrix, leading to numerically approximated eigenvalues. (ii) Within an analytic ansatz, how can a relativistic Dirac Hamiltonian be decoupled into particle and antiparticle degrees of freedom, in appropriate kinematic limits? One possible answer is the Foldy--Wouthuysen transform; however, there are alter- native methods which seem to have some advantages over the time--tested approach. One such method is investigated by applying both the traditional Foldy--Wouthuysen transform and the "chiral" Foldy--Wouthuysen transform to a number of Dirac Hamiltonians, including the central-field Hamiltonian for a gravitationally bound system; namely, the Dirac-(Einstein-)Schwarzschild Hamiltonian, which requires the formal- ism of general relativity. (iii) Are there are pseudo--Hermitian variants of Dirac Hamiltonians that can be approximated using a decoupling transformation? The tachyonic Dirac Hamiltonian, which describes faster-than-light spin-1/2 particles, is gamma5--Hermitian, i.e., pseudo-Hermitian. Superluminal particles remain faster than light upon a Lorentz transformation, and hence, the Foldy--Wouthuysen program is unsuited for this case. Thus, inspired by the Foldy--Wouthuysen program, a decoupling transform in the ultrarelativistic limit is proposed, which is applicable to both sub- and superluminal particles.

  6. Comparison of the Radiative Two-Flux and Diffusion Approximations

    NASA Technical Reports Server (NTRS)

    Spuckler, Charles M.

    2006-01-01

    Approximate solutions are sometimes used to determine the heat transfer and temperatures in a semitransparent material in which conduction and thermal radiation are acting. A comparison of the Milne-Eddington two-flux approximation and the diffusion approximation for combined conduction and radiation heat transfer in a ceramic material was preformed to determine the accuracy of the diffusion solution. A plane gray semitransparent layer without a substrate and a non-gray semitransparent plane layer on an opaque substrate were considered. For the plane gray layer the material is semitransparent for all wavelengths and the scattering and absorption coefficients do not vary with wavelength. For the non-gray plane layer the material is semitransparent with constant absorption and scattering coefficients up to a specified wavelength. At higher wavelengths the non-gray plane layer is assumed to be opaque. The layers are heated on one side and cooled on the other by diffuse radiation and convection. The scattering and absorption coefficients were varied. The error in the diffusion approximation compared to the Milne-Eddington two flux approximation was obtained as a function of scattering coefficient and absorption coefficient. The percent difference in interface temperatures and heat flux through the layer obtained using the Milne-Eddington two-flux and diffusion approximations are presented as a function of scattering coefficient and absorption coefficient. The largest errors occur for high scattering and low absorption except for the back surface temperature of the plane gray layer where the error is also larger at low scattering and low absorption. It is shown that the accuracy of the diffusion approximation can be improved for some scattering and absorption conditions if a reflectance obtained from a Kubelka-Munk type two flux theory is used instead of a reflection obtained from the Fresnel equation. The Kubelka-Munk reflectance accounts for surface reflection and

  7. Ranking Support Vector Machine with Kernel Approximation

    PubMed Central

    Dou, Yong

    2017-01-01

    Learning to rank algorithm has become important in recent years due to its successful application in information retrieval, recommender system, and computational biology, and so forth. Ranking support vector machine (RankSVM) is one of the state-of-art ranking models and has been favorably used. Nonlinear RankSVM (RankSVM with nonlinear kernels) can give higher accuracy than linear RankSVM (RankSVM with a linear kernel) for complex nonlinear ranking problem. However, the learning methods for nonlinear RankSVM are still time-consuming because of the calculation of kernel matrix. In this paper, we propose a fast ranking algorithm based on kernel approximation to avoid computing the kernel matrix. We explore two types of kernel approximation methods, namely, the Nyström method and random Fourier features. Primal truncated Newton method is used to optimize the pairwise L2-loss (squared Hinge-loss) objective function of the ranking model after the nonlinear kernel approximation. Experimental results demonstrate that our proposed method gets a much faster training speed than kernel RankSVM and achieves comparable or better performance over state-of-the-art ranking algorithms. PMID:28293256

  8. Ranking Support Vector Machine with Kernel Approximation.

    PubMed

    Chen, Kai; Li, Rongchun; Dou, Yong; Liang, Zhengfa; Lv, Qi

    2017-01-01

    Learning to rank algorithm has become important in recent years due to its successful application in information retrieval, recommender system, and computational biology, and so forth. Ranking support vector machine (RankSVM) is one of the state-of-art ranking models and has been favorably used. Nonlinear RankSVM (RankSVM with nonlinear kernels) can give higher accuracy than linear RankSVM (RankSVM with a linear kernel) for complex nonlinear ranking problem. However, the learning methods for nonlinear RankSVM are still time-consuming because of the calculation of kernel matrix. In this paper, we propose a fast ranking algorithm based on kernel approximation to avoid computing the kernel matrix. We explore two types of kernel approximation methods, namely, the Nyström method and random Fourier features. Primal truncated Newton method is used to optimize the pairwise L2-loss (squared Hinge-loss) objective function of the ranking model after the nonlinear kernel approximation. Experimental results demonstrate that our proposed method gets a much faster training speed than kernel RankSVM and achieves comparable or better performance over state-of-the-art ranking algorithms.

  9. Traveling-cluster approximation for uncorrelated amorphous systems

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Sen, A.K.; Mills, R.; Kaplan, T.

    1984-11-15

    We have developed a formalism for including cluster effects in the one-electron Green's function for a positionally disordered (liquid or amorphous) system without any correlation among the scattering sites. This method is an extension of the technique known as the traveling-cluster approximation (TCA) originally obtained and applied to a substitutional alloy by Mills and Ratanavararaksa. We have also proved the appropriate fixed-point theorem, which guarantees, for a bounded local potential, that the self-consistent equations always converge upon iteration to a unique, Herglotz solution. To our knowledge, this is the only analytic theory for considering cluster effects. Furthermore, we have performedmore » some computer calculations in the pair TCA, for the model case of delta-function potentials on a one-dimensional random chain. These results have been compared with ''exact calculations'' (which, in principle, take into account all cluster effects) and with the coherent-potential approximation (CPA), which is the single-site TCA. The density of states for the pair TCA clearly shows some improvement over the CPA and yet, apparently, the pair approximation distorts some of the features of the exact results.« less

  10. Approximate Locality for Quantum Systems on Graphs

    NASA Astrophysics Data System (ADS)

    Osborne, Tobias J.

    2008-10-01

    In this Letter we make progress on a long-standing open problem of Aaronson and Ambainis [Theory Comput. 1, 47 (2005)1557-2862]: we show that if U is a sparse unitary operator with a gap Δ in its spectrum, then there exists an approximate logarithm H of U which is also sparse. The sparsity pattern of H gets more dense as 1/Δ increases. This result can be interpreted as a way to convert between local continuous-time and local discrete-time quantum processes. As an example we show that the discrete-time coined quantum walk can be realized stroboscopically from an approximately local continuous-time quantum walk.

  11. Mean-field approximation for spacing distribution functions in classical systems

    NASA Astrophysics Data System (ADS)

    González, Diego Luis; Pimpinelli, Alberto; Einstein, T. L.

    2012-01-01

    We propose a mean-field method to calculate approximately the spacing distribution functions p(n)(s) in one-dimensional classical many-particle systems. We compare our method with two other commonly used methods, the independent interval approximation and the extended Wigner surmise. In our mean-field approach, p(n)(s) is calculated from a set of Langevin equations, which are decoupled by using a mean-field approximation. We find that in spite of its simplicity, the mean-field approximation provides good results in several systems. We offer many examples illustrating that the three previously mentioned methods give a reasonable description of the statistical behavior of the system. The physical interpretation of each method is also discussed.

  12. Facial approximation-from facial reconstruction synonym to face prediction paradigm.

    PubMed

    Stephan, Carl N

    2015-05-01

    Facial approximation was first proposed as a synonym for facial reconstruction in 1987 due to dissatisfaction with the connotations the latter label held. Since its debut, facial approximation's identity has morphed as anomalies in face prediction have accumulated. Now underpinned by differences in what problems are thought to count as legitimate, facial approximation can no longer be considered a synonym for, or subclass of, facial reconstruction. Instead, two competing paradigms of face prediction have emerged, namely: facial approximation and facial reconstruction. This paper shines a Kuhnian lens across the discipline of face prediction to comprehensively review these developments and outlines the distinguishing features between the two paradigms. © 2015 American Academy of Forensic Sciences.

  13. Discrete-Time Stable Generalized Self-Learning Optimal Control With Approximation Errors.

    PubMed

    Wei, Qinglai; Li, Benkai; Song, Ruizhuo

    2018-04-01

    In this paper, a generalized policy iteration (GPI) algorithm with approximation errors is developed for solving infinite horizon optimal control problems for nonlinear systems. The developed stable GPI algorithm provides a general structure of discrete-time iterative adaptive dynamic programming algorithms, by which most of the discrete-time reinforcement learning algorithms can be described using the GPI structure. It is for the first time that approximation errors are explicitly considered in the GPI algorithm. The properties of the stable GPI algorithm with approximation errors are analyzed. The admissibility of the approximate iterative control law can be guaranteed if the approximation errors satisfy the admissibility criteria. The convergence of the developed algorithm is established, which shows that the iterative value function is convergent to a finite neighborhood of the optimal performance index function, if the approximate errors satisfy the convergence criterion. Finally, numerical examples and comparisons are presented.

  14. Approximate Arithmetic Training Improves Informal Math Performance in Low Achieving Preschoolers

    PubMed Central

    Szkudlarek, Emily; Brannon, Elizabeth M.

    2018-01-01

    Recent studies suggest that practice with approximate and non-symbolic arithmetic problems improves the math performance of adults, school aged children, and preschoolers. However, the relative effectiveness of approximate arithmetic training compared to available educational games, and the type of math skills that approximate arithmetic targets are unknown. The present study was designed to (1) compare the effectiveness of approximate arithmetic training to two commercially available numeral and letter identification tablet applications and (2) to examine the specific type of math skills that benefit from approximate arithmetic training. Preschool children (n = 158) were pseudo-randomly assigned to one of three conditions: approximate arithmetic, letter identification, or numeral identification. All children were trained for 10 short sessions and given pre and post tests of informal and formal math, executive function, short term memory, vocabulary, alphabet knowledge, and number word knowledge. We found a significant interaction between initial math performance and training condition, such that children with low pretest math performance benefited from approximate arithmetic training, and children with high pretest math performance benefited from symbol identification training. This effect was restricted to informal, and not formal, math problems. There were also effects of gender, socio-economic status, and age on post-test informal math score after intervention. A median split on pretest math ability indicated that children in the low half of math scores in the approximate arithmetic training condition performed significantly better than children in the letter identification training condition on post-test informal math problems when controlling for pretest, age, gender, and socio-economic status. Our results support the conclusion that approximate arithmetic training may be especially effective for children with low math skills, and that approximate arithmetic

  15. Approximate Arithmetic Training Improves Informal Math Performance in Low Achieving Preschoolers.

    PubMed

    Szkudlarek, Emily; Brannon, Elizabeth M

    2018-01-01

    Recent studies suggest that practice with approximate and non-symbolic arithmetic problems improves the math performance of adults, school aged children, and preschoolers. However, the relative effectiveness of approximate arithmetic training compared to available educational games, and the type of math skills that approximate arithmetic targets are unknown. The present study was designed to (1) compare the effectiveness of approximate arithmetic training to two commercially available numeral and letter identification tablet applications and (2) to examine the specific type of math skills that benefit from approximate arithmetic training. Preschool children ( n = 158) were pseudo-randomly assigned to one of three conditions: approximate arithmetic, letter identification, or numeral identification. All children were trained for 10 short sessions and given pre and post tests of informal and formal math, executive function, short term memory, vocabulary, alphabet knowledge, and number word knowledge. We found a significant interaction between initial math performance and training condition, such that children with low pretest math performance benefited from approximate arithmetic training, and children with high pretest math performance benefited from symbol identification training. This effect was restricted to informal, and not formal, math problems. There were also effects of gender, socio-economic status, and age on post-test informal math score after intervention. A median split on pretest math ability indicated that children in the low half of math scores in the approximate arithmetic training condition performed significantly better than children in the letter identification training condition on post-test informal math problems when controlling for pretest, age, gender, and socio-economic status. Our results support the conclusion that approximate arithmetic training may be especially effective for children with low math skills, and that approximate arithmetic

  16. Analytic approximations to the modon dispersion relation. [in oceanography

    NASA Technical Reports Server (NTRS)

    Boyd, J. P.

    1981-01-01

    Three explicit analytic approximations are given to the modon dispersion relation developed by Flierl et al. (1980) to describe Gulf Stream rings and related phenomena in the oceans and atmosphere. The solutions are in the form of k(q), and are developed in the form of a power series in q for small q, an inverse power series in 1/q for large q, and a two-point Pade approximant. The low order Pade approximant is shown to yield a solution for the dispersion relation with a maximum relative error for the lowest branch of the function equal to one in 700 in the q interval zero to infinity.

  17. Approximate Dispersion Relations for Waves on Arbitrary Shear Flows

    NASA Astrophysics Data System (ADS)

    Ellingsen, S. À.; Li, Y.

    2017-12-01

    An approximate dispersion relation is derived and presented for linear surface waves atop a shear current whose magnitude and direction can vary arbitrarily with depth. The approximation, derived to first order of deviation from potential flow, is shown to produce good approximations at all wavelengths for a wide range of naturally occuring shear flows as well as widely used model flows. The relation reduces in many cases to a 3-D generalization of the much used approximation by Skop (1987), developed further by Kirby and Chen (1989), but is shown to be more robust, succeeding in situations where the Kirby and Chen model fails. The two approximations incur the same numerical cost and difficulty. While the Kirby and Chen approximation is excellent for a wide range of currents, the exact criteria for its applicability have not been known. We explain the apparently serendipitous success of the latter and derive proper conditions of applicability for both approximate dispersion relations. Our new model has a greater range of applicability. A second order approximation is also derived. It greatly improves accuracy, which is shown to be important in difficult cases. It has an advantage over the corresponding second-order expression proposed by Kirby and Chen that its criterion of accuracy is explicitly known, which is not currently the case for the latter to our knowledge. Our second-order term is also arguably significantly simpler to implement, and more physically transparent, than its sibling due to Kirby and Chen.Plain Language SummaryIn order to answer key questions such as how the ocean surface affects the climate, erodes the coastline and transports nutrients, we must understand how waves move. This is not so easy when depth varying currents are present, as they often are in coastal waters. We have developed a modeling tool for accurately predicting wave properties in such situations, ready for use, for example, in the</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950012483','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950012483"><span>On the convergence of local <span class="hlt">approximations</span> to pseudodifferential operators with applications</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Hagstrom, Thomas</p> <p>1994-01-01</p> <p>We consider the <span class="hlt">approximation</span> of a class pseudodifferential operators by sequences of operators which can be expressed as compositions of differential operators and their inverses. We show that the error in such <span class="hlt">approximations</span> can be bounded in terms of L(1) error in <span class="hlt">approximating</span> a convolution kernel, and use this fact to develop convergence results. Our main result is a finite time convergence analysis of the Engquist-Majda Pade <span class="hlt">approximants</span> to the square root of the d'Alembertian. We also show that no spatially local <span class="hlt">approximation</span> to this operator can be convergent uniformly in time. We propose some temporally local but spatially nonlocal operators with better long time behavior. These are based on Laguerre and exponential series.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28675757','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28675757"><span>Designing quantum information processing via structural physical <span class="hlt">approximation</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Bae, Joonwoo</p> <p>2017-10-01</p> <p>In quantum information processing it may be possible to have efficient computation and secure communication beyond the limitations of classical systems. In a fundamental point of view, however, evolution of quantum systems by the laws of quantum mechanics is more restrictive than classical systems, identified to a specific form of dynamics, that is, unitary transformations and, consequently, positive and completely positive maps to subsystems. This also characterizes classes of disallowed transformations on quantum systems, among which positive but not completely maps are of particular interest as they characterize entangled states, a general resource in quantum information processing. Structural physical <span class="hlt">approximation</span> offers a systematic way of <span class="hlt">approximating</span> those non-physical maps, positive but not completely positive maps, with quantum channels. Since it has been proposed as a method of detecting entangled states, it has stimulated fundamental problems on classifications of positive maps and the structure of Hermitian operators and quantum states, as well as on quantum measurement such as quantum design in quantum information theory. It has developed efficient and feasible methods of directly detecting entangled states in practice, for which proof-of-principle experimental demonstrations have also been performed with photonic qubit states. Here, we present a comprehensive review on quantum information processing with structural physical <span class="hlt">approximations</span> and the related progress. The review mainly focuses on properties of structural physical <span class="hlt">approximations</span> and their applications toward practical information applications.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017RPPh...80j4001B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017RPPh...80j4001B"><span>Designing quantum information processing via structural physical <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Bae, Joonwoo</p> <p>2017-10-01</p> <p>In quantum information processing it may be possible to have efficient computation and secure communication beyond the limitations of classical systems. In a fundamental point of view, however, evolution of quantum systems by the laws of quantum mechanics is more restrictive than classical systems, identified to a specific form of dynamics, that is, unitary transformations and, consequently, positive and completely positive maps to subsystems. This also characterizes classes of disallowed transformations on quantum systems, among which positive but not completely maps are of particular interest as they characterize entangled states, a general resource in quantum information processing. Structural physical <span class="hlt">approximation</span> offers a systematic way of <span class="hlt">approximating</span> those non-physical maps, positive but not completely positive maps, with quantum channels. Since it has been proposed as a method of detecting entangled states, it has stimulated fundamental problems on classifications of positive maps and the structure of Hermitian operators and quantum states, as well as on quantum measurement such as quantum design in quantum information theory. It has developed efficient and feasible methods of directly detecting entangled states in practice, for which proof-of-principle experimental demonstrations have also been performed with photonic qubit states. Here, we present a comprehensive review on quantum information processing with structural physical <span class="hlt">approximations</span> and the related progress. The review mainly focuses on properties of structural physical <span class="hlt">approximations</span> and their applications toward practical information applications.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_13");'>13</a></li> <li><a href="#" onclick='return showDiv("page_14");'>14</a></li> <li class="active"><span>15</span></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_15 --> <div id="page_16" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_14");'>14</a></li> <li><a href="#" onclick='return showDiv("page_15");'>15</a></li> <li class="active"><span>16</span></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="301"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/22400556','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/22400556"><span>Mean-field <span class="hlt">approximation</span> for spacing distribution functions in classical systems.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>González, Diego Luis; Pimpinelli, Alberto; Einstein, T L</p> <p>2012-01-01</p> <p>We propose a mean-field method to calculate <span class="hlt">approximately</span> the spacing distribution functions p((n))(s) in one-dimensional classical many-particle systems. We compare our method with two other commonly used methods, the independent interval <span class="hlt">approximation</span> and the extended Wigner surmise. In our mean-field approach, p((n))(s) is calculated from a set of Langevin equations, which are decoupled by using a mean-field <span class="hlt">approximation</span>. We find that in spite of its simplicity, the mean-field <span class="hlt">approximation</span> provides good results in several systems. We offer many examples illustrating that the three previously mentioned methods give a reasonable description of the statistical behavior of the system. The physical interpretation of each method is also discussed. © 2012 American Physical Society</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD1013972','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD1013972"><span>Reliable Function <span class="hlt">Approximation</span> and Estimation</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>2016-08-16</p> <p>AUSTIN , TX 78712 08/16/2016 Final Report DISTRIBUTION A: Distribution approved for public release. Air Force Research Laboratory AF Office Of Scientific...UNIVERSITY OF TEXAS AT AUSTIN 101 EAST 27TH STREET STE 4308 AUSTIN , TX 78712 DISTRIBUTION A: Distribution approved for public release. INSTRUCTIONS...AFRL-AFOSR-VA-TR-2016-0293 Reliable Function <span class="hlt">Approximation</span> and Estimation Rachel Ward UNIVERSITY OF TEXAS AT AUSTIN 101 EAST 27TH STREET STE 4308</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4865172','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4865172"><span>Solving Math Problems <span class="hlt">Approximately</span>: A Developmental Perspective</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Ganor-Stern, Dana</p> <p>2016-01-01</p> <p>Although solving arithmetic problems <span class="hlt">approximately</span> is an important skill in everyday life, little is known about the development of this skill. Past research has shown that when children are asked to solve multi-digit multiplication problems <span class="hlt">approximately</span>, they provide estimates that are often very far from the exact answer. This is unfortunate as computation estimation is needed in many circumstances in daily life. The present study examined 4th graders, 6th graders and adults’ ability to estimate the results of arithmetic problems relative to a reference number. A developmental pattern was observed in accuracy, speed and strategy use. With age there was a general increase in speed, and an increase in accuracy mainly for trials in which the reference number was close to the exact answer. The children tended to use the sense of magnitude strategy, which does not involve any calculation but relies mainly on an intuitive coarse sense of magnitude, while the adults used the <span class="hlt">approximated</span> calculation strategy which involves rounding and multiplication procedures, and relies to a greater extent on calculation skills and working memory resources. Importantly, the children were less accurate than the adults, but were well above chance level. In all age groups performance was enhanced when the reference number was smaller (vs. larger) than the exact answer and when it was far (vs. close) from it, suggesting the involvement of an <span class="hlt">approximate</span> number system. The results suggest the existence of an intuitive sense of magnitude for the results of arithmetic problems that might help children and even adults with difficulties in math. The present findings are discussed in the context of past research reporting poor estimation skills among children, and the conditions that might allow using children estimation skills in an effective manner. PMID:27171224</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006CoPhC.174..447W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006CoPhC.174..447W"><span>A Mathematica program for the <span class="hlt">approximate</span> analytical solution to a nonlinear undamped Duffing equation by a new <span class="hlt">approximate</span> approach</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Wu, Dongmei; Wang, Zhongcheng</p> <p>2006-03-01</p> <p>According to Mickens [R.E. Mickens, Comments on a Generalized Galerkin's method for non-linear oscillators, J. Sound Vib. 118 (1987) 563], the general HB (harmonic balance) method is an <span class="hlt">approximation</span> to the convergent Fourier series representation of the periodic solution of a nonlinear oscillator and not an <span class="hlt">approximation</span> to an expansion in terms of a small parameter. Consequently, for a nonlinear undamped Duffing equation with a driving force Bcos(ωx), to find a periodic solution when the fundamental frequency is identical to ω, the corresponding Fourier series can be written as y˜(x)=∑n=1m acos[(2n-1)ωx]. How to calculate the coefficients of the Fourier series efficiently with a computer program is still an open problem. For HB method, by substituting <span class="hlt">approximation</span> y˜(x) into force equation, expanding the resulting expression into a trigonometric series, then letting the coefficients of the resulting lowest-order harmonic be zero, one can obtain <span class="hlt">approximate</span> coefficients of <span class="hlt">approximation</span> y˜(x) [R.E. Mickens, Comments on a Generalized Galerkin's method for non-linear oscillators, J. Sound Vib. 118 (1987) 563]. But for nonlinear differential equations such as Duffing equation, it is very difficult to construct higher-order analytical <span class="hlt">approximations</span>, because the HB method requires solving a set of algebraic equations for a large number of unknowns with very complex nonlinearities. To overcome the difficulty, forty years ago, Urabe derived a computational method for Duffing equation based on Galerkin procedure [M. Urabe, A. Reiter, Numerical computation of nonlinear forced oscillations by Galerkin's procedure, J. Math. Anal. Appl. 14 (1966) 107-140]. Dooren obtained an <span class="hlt">approximate</span> solution of the Duffing oscillator with a special set of parameters by using Urabe's method [R. van Dooren, Stabilization of Cowell's classic finite difference method for numerical integration, J. Comput. Phys. 16 (1974) 186-192]. In this paper, in the frame of the general HB method</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28129193','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28129193"><span>High-Dimensional Function <span class="hlt">Approximation</span> With Neural Networks for Large Volumes of Data.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Andras, Peter</p> <p>2018-02-01</p> <p><span class="hlt">Approximation</span> of high-dimensional functions is a challenge for neural networks due to the curse of dimensionality. Often the data for which the <span class="hlt">approximated</span> function is defined resides on a low-dimensional manifold and in principle the <span class="hlt">approximation</span> of the function over this manifold should improve the <span class="hlt">approximation</span> performance. It has been show that projecting the data manifold into a lower dimensional space, followed by the neural network <span class="hlt">approximation</span> of the function over this space, provides a more precise <span class="hlt">approximation</span> of the function than the <span class="hlt">approximation</span> of the function with neural networks in the original data space. However, if the data volume is very large, the projection into the low-dimensional space has to be based on a limited sample of the data. Here, we investigate the nature of the <span class="hlt">approximation</span> error of neural networks trained over the projection space. We show that such neural networks should have better <span class="hlt">approximation</span> performance than neural networks trained on high-dimensional data even if the projection is based on a relatively sparse sample of the data manifold. We also find that it is preferable to use a uniformly distributed sparse sample of the data for the purpose of the generation of the low-dimensional projection. We illustrate these results considering the practical neural network <span class="hlt">approximation</span> of a set of functions defined on high-dimensional data including real world data as well.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1327571-hyperspherical-sparse-approximation-techniques-high-dimensional-discontinuity-detection','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1327571-hyperspherical-sparse-approximation-techniques-high-dimensional-discontinuity-detection"><span>Hyperspherical Sparse <span class="hlt">Approximation</span> Techniques for High-Dimensional Discontinuity Detection</span></a></p> <p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p> <p>Zhang, Guannan; Webster, Clayton G.; Gunzburger, Max; ...</p> <p>2016-08-04</p> <p>This work proposes a hyperspherical sparse <span class="hlt">approximation</span> framework for detecting jump discontinuities in functions in high-dimensional spaces. The need for a novel approach results from the theoretical and computational inefficiencies of well-known approaches, such as adaptive sparse grids, for discontinuity detection. Our approach constructs the hyperspherical coordinate representation of the discontinuity surface of a function. Then sparse <span class="hlt">approximations</span> of the transformed function are built in the hyperspherical coordinate system, with values at each point estimated by solving a one-dimensional discontinuity detection problem. Due to the smoothness of the hypersurface, the new technique can identify jump discontinuities with significantly reduced computationalmore » cost, compared to existing methods. Several approaches are used to <span class="hlt">approximate</span> the transformed discontinuity surface in the hyperspherical system, including adaptive sparse grid and radial basis function interpolation, discrete least squares projection, and compressed sensing <span class="hlt">approximation</span>. Moreover, hierarchical acceleration techniques are also incorporated to further reduce the overall complexity. In conclusion, rigorous complexity analyses of the new methods are provided, as are several numerical examples that illustrate the effectiveness of our approach.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2137171','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2137171"><span>Two-Term Asymptotic <span class="hlt">Approximation</span> of a Cardiac Restitution Curve*</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Cain, John W.; Schaeffer, David G.</p> <p>2007-01-01</p> <p>If spatial extent is neglected, ionic models of cardiac cells consist of systems of ordinary differential equations (ODEs) which have the property of excitability, i.e., a brief stimulus produces a prolonged evolution (called an action potential in the cardiac context) before the eventual return to equilibrium. Under repeated stimulation, or pacing, cardiac tissue exhibits electrical restitution: the steady-state action potential duration (APD) at a given pacing period B shortens as B is decreased. Independent of ionic models, restitution is often modeled phenomenologically by a one-dimensional mapping of the form APDnext = f(B – APDprevious). Under some circumstances, a restitution function f can be derived as an asymptotic <span class="hlt">approximation</span> to the behavior of an ionic model. In this paper, extending previous work, we derive the next term in such an asymptotic <span class="hlt">approximation</span> for a particular ionic model consisting of two ODEs. The two-term <span class="hlt">approximation</span> exhibits excellent quantitative agreement with the actual restitution curve, whereas the leading-order <span class="hlt">approximation</span> significantly underestimates actual APD values. PMID:18080006</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22250629-mathematical-treatment-born-oppenheimer-approximation','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22250629-mathematical-treatment-born-oppenheimer-approximation"><span>On the mathematical treatment of the Born-Oppenheimer <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Jecko, Thierry, E-mail: thierry.jecko@u-cergy.fr</p> <p>2014-05-15</p> <p>Motivated by the paper by Sutcliffe and Woolley [“On the quantum theory of molecules,” J. Chem. Phys. 137, 22A544 (2012)], we present the main ideas used by mathematicians to show the accuracy of the Born-Oppenheimer <span class="hlt">approximation</span> for molecules. Based on mathematical works on this <span class="hlt">approximation</span> for molecular bound states, in scattering theory, in resonance theory, and for short time evolution, we give an overview of some rigorous results obtained up to now. We also point out the main difficulties mathematicians are trying to overcome and speculate on further developments. The mathematical approach does not fit exactly to the common usemore » of the <span class="hlt">approximation</span> in Physics and Chemistry. We criticize the latter and comment on the differences, contributing in this way to the discussion on the Born-Oppenheimer <span class="hlt">approximation</span> initiated by Sutcliffe and Woolley. The paper neither contains mathematical statements nor proofs. Instead, we try to make accessible mathematically rigourous results on the subject to researchers in Quantum Chemistry or Physics.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19850022868&hterms=functional+structure&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D80%26Ntt%3Dfunctional%2Bstructure','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19850022868&hterms=functional+structure&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D80%26Ntt%3Dfunctional%2Bstructure"><span><span class="hlt">Approximation</span> of Optimal Infinite Dimensional Compensators for Flexible Structures</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Gibson, J. S.; Mingori, D. L.; Adamian, A.; Jabbari, F.</p> <p>1985-01-01</p> <p>The infinite dimensional compensator for a large class of flexible structures, modeled as distributed systems are discussed, as well as an <span class="hlt">approximation</span> scheme for designing finite dimensional compensators to <span class="hlt">approximate</span> the infinite dimensional compensator. The <span class="hlt">approximation</span> scheme is applied to develop a compensator for a space antenna model based on wrap-rib antennas being built currently. While the present model has been simplified, it retains the salient features of rigid body modes and several distributed components of different characteristics. The control and estimator gains are represented by functional gains, which provide graphical representations of the control and estimator laws. These functional gains also indicate the convergence of the finite dimensional compensators and show which modes the optimal compensator ignores.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5375850','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5375850"><span><span class="hlt">Approximate</span> Sensory Data Collection: A Survey</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Cheng, Siyao; Cai, Zhipeng; Li, Jianzhong</p> <p>2017-01-01</p> <p>With the rapid development of the Internet of Things (IoTs), wireless sensor networks (WSNs) and related techniques, the amount of sensory data manifests an explosive growth. In some applications of IoTs and WSNs, the size of sensory data has already exceeded several petabytes annually, which brings too many troubles and challenges for the data collection, which is a primary operation in IoTs and WSNs. Since the exact data collection is not affordable for many WSN and IoT systems due to the limitations on bandwidth and energy, many <span class="hlt">approximate</span> data collection algorithms have been proposed in the last decade. This survey reviews the state of the art of <span class="hlt">approximate</span> data collection algorithms. We classify them into three categories: the model-based ones, the compressive sensing based ones, and the query-driven ones. For each category of algorithms, the advantages and disadvantages are elaborated, some challenges and unsolved problems are pointed out, and the research prospects are forecasted. PMID:28287440</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19980232929','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19980232929"><span>Using <span class="hlt">Approximations</span> to Accelerate Engineering Design Optimization</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Torczon, Virginia; Trosset, Michael W.</p> <p>1998-01-01</p> <p>Optimization problems that arise in engineering design are often characterized by several features that hinder the use of standard nonlinear optimization techniques. Foremost among these features is that the functions used to define the engineering optimization problem often are computationally intensive. Within a standard nonlinear optimization algorithm, the computational expense of evaluating the functions that define the problem would necessarily be incurred for each iteration of the optimization algorithm. Faced with such prohibitive computational costs, an attractive alternative is to make use of surrogates within an optimization context since surrogates can be chosen or constructed so that they are typically much less expensive to compute. For the purposes of this paper, we will focus on the use of algebraic <span class="hlt">approximations</span> as surrogates for the objective. In this paper we introduce the use of so-called merit functions that explicitly recognize the desirability of improving the current <span class="hlt">approximation</span> to the objective during the course of the optimization. We define and experiment with the use of merit functions chosen to simultaneously improve both the solution to the optimization problem (the objective) and the quality of the <span class="hlt">approximation</span>. Our goal is to further improve the effectiveness of our general approach without sacrificing any of its rigor.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19910047246&hterms=1089&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3D%2526%25231089','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19910047246&hterms=1089&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3D%2526%25231089"><span>Eigenvalue and eigenvector sensitivity and <span class="hlt">approximate</span> analysis for repeated eigenvalue problems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Hou, Gene J. W.; Kenny, Sean P.</p> <p>1991-01-01</p> <p>A set of computationally efficient equations for eigenvalue and eigenvector sensitivity analysis are derived, and a method for eigenvalue and eigenvector <span class="hlt">approximate</span> analysis in the presence of repeated eigenvalues is presented. The method developed for <span class="hlt">approximate</span> analysis involves a reparamaterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order <span class="hlt">approximations</span> of changes in both the eigenvalues and eigenvectors associated with the repeated eigenvalue problem. Examples are given to demonstrate the application of such equations for sensitivity and <span class="hlt">approximate</span> analysis.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20090026532','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20090026532"><span>Validity of the Aluminum Equivalent <span class="hlt">Approximation</span> in Space Radiation Shielding</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Badavi, Francis F.; Adams, Daniel O.; Wilson, John W.</p> <p>2009-01-01</p> <p>The origin of the aluminum equivalent shield <span class="hlt">approximation</span> in space radiation analysis can be traced back to its roots in the early years of the NASA space programs (Mercury, Gemini and Apollo) wherein the primary radiobiological concern was the intense sources of ionizing radiation causing short term effects which was thought to jeopardize the safety of the crew and hence the mission. Herein, it is shown that the aluminum equivalent shield <span class="hlt">approximation</span>, although reasonably well suited for that time period and to the application for which it was developed, is of questionable usefulness to the radiobiological concerns of routine space operations of the 21 st century which will include long stays onboard the International Space Station (ISS) and perhaps the moon. This is especially true for a risk based protection system, as appears imminent for deep space exploration where the long-term effects of Galactic Cosmic Ray (GCR) exposure is of primary concern. The present analysis demonstrates that sufficiently large errors in the interior particle environment of a spacecraft result from the use of the aluminum equivalent <span class="hlt">approximation</span>, and such <span class="hlt">approximations</span> should be avoided in future astronaut risk estimates. In this study, the aluminum equivalent <span class="hlt">approximation</span> is evaluated as a means for estimating the particle environment within a spacecraft structure induced by the GCR radiation field. For comparison, the two extremes of the GCR environment, the 1977 solar minimum and the 2001 solar maximum, are considered. These environments are coupled to the Langley Research Center (LaRC) deterministic ionized particle transport code High charge (Z) and Energy TRaNsport (HZETRN), which propagates the GCR spectra for elements with charges (Z) in the range I <= Z <= 28 (H -- Ni) and secondary neutrons through selected target materials. The coupling of the GCR extremes to HZETRN allows for the examination of the induced environment within the interior' of an idealized spacecraft</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19920051906&hterms=Rule+thumb&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D60%26Ntt%3DRule%2Bthumb','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19920051906&hterms=Rule+thumb&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D60%26Ntt%3DRule%2Bthumb"><span>A comparison of polynomial <span class="hlt">approximations</span> and artificial neural nets as response surfaces</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Carpenter, William C.; Barthelemy, Jean-Francois M.</p> <p>1992-01-01</p> <p>Artificial neural nets and polynomial <span class="hlt">approximations</span> were used to develop response surfaces for several test problems. Based on the number of functional evaluations required to build the <span class="hlt">approximations</span> and the number of undetermined parameters associated with the <span class="hlt">approximations</span>, the performance of the two types of <span class="hlt">approximations</span> was found to be comparable. A rule of thumb is developed for determining the number of nodes to be used on a hidden layer of an artificial neural net, and the number of designs needed to train an <span class="hlt">approximation</span> is discussed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27832739','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27832739"><span>libFLASM: a software library for fixed-length <span class="hlt">approximate</span> string matching.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Ayad, Lorraine A K; Pissis, Solon P P; Retha, Ahmad</p> <p>2016-11-10</p> <p><span class="hlt">Approximate</span> string matching is the problem of finding all factors of a given text that are at a distance at most k from a given pattern. Fixed-length <span class="hlt">approximate</span> string matching is the problem of finding all factors of a text of length n that are at a distance at most k from any factor of length ℓ of a pattern of length m. There exist bit-vector techniques to solve the fixed-length <span class="hlt">approximate</span> string matching problem in time [Formula: see text] and space [Formula: see text] under the edit and Hamming distance models, where w is the size of the computer word; as such these techniques are independent of the distance threshold k or the alphabet size. Fixed-length <span class="hlt">approximate</span> string matching is a generalisation of <span class="hlt">approximate</span> string matching and, hence, has numerous direct applications in computational molecular biology and elsewhere. We present and make available libFLASM, a free open-source C++ software library for solving fixed-length <span class="hlt">approximate</span> string matching under both the edit and the Hamming distance models. Moreover we describe how fixed-length <span class="hlt">approximate</span> string matching is applied to solve real problems by incorporating libFLASM into established applications for multiple circular sequence alignment as well as single and structured motif extraction. Specifically, we describe how it can be used to improve the accuracy of multiple circular sequence alignment in terms of the inferred likelihood-based phylogenies; and we also describe how it is used to efficiently find motifs in molecular sequences representing regulatory or functional regions. The comparison of the performance of the library to other algorithms show how it is competitive, especially with increasing distance thresholds. Fixed-length <span class="hlt">approximate</span> string matching is a generalisation of the classic <span class="hlt">approximate</span> string matching problem. We present libFLASM, a free open-source C++ software library for solving fixed-length <span class="hlt">approximate</span> string matching. The extensive experimental results presented here</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1985SPIE..473....2M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1985SPIE..473....2M"><span>Current Research In The Institut D'optique - Orsay</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Marechal, A.</p> <p>1985-01-01</p> <p>I - FUNDAMENTAL PROBLEMS A. ASPECT : Non locality in wave mechanics. The Einstein-<span class="hlt">Bohr</span> dispute and the optical test of Bell's inequalities. A. Einstein and N. <span class="hlt">Bohr</span> had a discussion about the role of probabilities in microphysics; as is well known the location and momentum of particles cannot be known precisely but only through probabilistic expressions. Einstein thought that complementary variables (hidden variables) are unknown at the present time and that the knowledge of those variables would suppress the actual uncertainty "God does not play random games". On the contrary, <span class="hlt">Bohr</span> advocated the fundamental role of probabilities. During the sixties J. Bell established inequalities supporting the Einstein point of view. These inequalities are in contradiction with the predictions of quantum mechanics in some very rare situations. A.Aspect designed and realised an optical experiment where the correlation of polarisation of photons emitted by a single atom was measured with a very good signal to noise ratio. Moreover the orientation of the polarisation analyser was changed during the time of flight of photons. The experimental result is in full agreement with the point of view of N. <span class="hlt">Bohr</span> : the uncertainty on the movement of particles is a fundamental component of the laws of nature and Bell's inequalities do not hold.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19890051477&hterms=Theory+constraints&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3DTheory%2Bconstraints','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19890051477&hterms=Theory+constraints&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3DTheory%2Bconstraints"><span>Parameter estimation in nonlinear distributed systems - <span class="hlt">Approximation</span> theory and convergence results</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Banks, H. T.; Reich, Simeon; Rosen, I. G.</p> <p>1988-01-01</p> <p>An abstract <span class="hlt">approximation</span> framework and convergence theory is described for Galerkin <span class="hlt">approximations</span> applied to inverse problems involving nonlinear distributed parameter systems. Parameter estimation problems are considered and formulated as the minimization of a least-squares-like performance index over a compact admissible parameter set subject to state constraints given by an inhomogeneous nonlinear distributed system. The theory applies to systems whose dynamics can be described by either time-independent or nonstationary strongly maximal monotonic operators defined on a reflexive Banach space which is densely and continuously embedded in a Hilbert space. It is demonstrated that if readily verifiable conditions on the system's dependence on the unknown parameters are satisfied, and the usual Galerkin <span class="hlt">approximation</span> assumption holds, then solutions to the <span class="hlt">approximating</span> problems exist and <span class="hlt">approximate</span> a solution to the original infinite-dimensional identification problem.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016AdSpR..57..137K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016AdSpR..57..137K"><span>Spacecraft attitude control using neuro-fuzzy <span class="hlt">approximation</span> of the optimal controllers</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Kim, Sung-Woo; Park, Sang-Young; Park, Chandeok</p> <p>2016-01-01</p> <p>In this study, a neuro-fuzzy controller (NFC) was developed for spacecraft attitude control to mitigate large computational load of the state-dependent Riccati equation (SDRE) controller. The NFC was developed by training a neuro-fuzzy network to <span class="hlt">approximate</span> the SDRE controller. The stability of the NFC was numerically verified using a Lyapunov-based method, and the performance of the controller was analyzed in terms of <span class="hlt">approximation</span> ability, steady-state error, cost, and execution time. The simulations and test results indicate that the developed NFC efficiently <span class="hlt">approximates</span> the SDRE controller, with asymptotic stability in a bounded region of angular velocity encompassing the operational range of rapid-attitude maneuvers. In addition, it was shown that an <span class="hlt">approximated</span> optimal feedback controller can be designed successfully through neuro-fuzzy <span class="hlt">approximation</span> of the optimal open-loop controller.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2005AN....326..781.','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2005AN....326..781."><span>Editor's note</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p></p> <p>2005-11-01</p> <p>Nordita, the Nordic Institute for Theoretical Physics, was founded in 1957 by Niels <span class="hlt">Bohr</span> and Torsten Gustafsson at Blegdamsvej in Copenhagen, joint to <span class="hlt">Bohr</span>'s legendary Institute. Today, memories of <span class="hlt">Bohr</span> and his famous visitors -- Albert Einstein, Werner Heisenberg, Lev Landau and many others -- strongly contribute to Nordita's genius loci and inspire next generations of her visitors. Nordita awards ``Nordic Project'' grants to individual Nordic physicists to help conduct a world-class research in Nordic countries (Denmark, Finland, Island, Norway, and Sweden). Research reported here was generously supported by the Nordic Project "Quasi Periodic Oscillations in Black Hole and Neutron Star sources" awarded in 2005 to Marek Abramowicz. The Project supported the ``Nordita Workdays on QPO" (March 25 -- April 1, 2005) organized by Marek Abramowicz, Axel Brandenburg and Juri Poutanen with help of Hanne Bergen, Helle http://www.nordita.dk/positions/norproject.html</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2001AnP...513..151E','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2001AnP...513..151E"><span>The emergence of Quantum Schools: Munich, Göttingen and Copenhagen as new centers of atomic theory</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Eckert, M.</p> <p>2001-01-01</p> <p>The institutes of Arnold Sommerfeld in Munich, Niels <span class="hlt">Bohr</span> in Copenhagen, and Max Born in Göttingen became the leading centers for the study of quantum theory in the first decades of the twentieth century. Although founded for a broader range of theoretical physics, the quantum became the major topic of research in Munich after the <span class="hlt">Bohr</span>-Sommerfeld-model of the atom (1913-16). The heyday came in the 1920s, when <span class="hlt">Bohr</span>'s and Born's institutes started operation and became further attractive centers for ambitious theorists all over the world. The discovery of quantum mechanics (1925) should be regarded not only as the achievement of a few young geniuses (in particular Werner Heisenberg and Wolfgang Pauli) but also as the result of a collaborative effort emerging in the new social and intellectual environment of their teachers' schools in Munich, Göttingen and Copenhagen.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_14");'>14</a></li> <li><a href="#" onclick='return showDiv("page_15");'>15</a></li> <li class="active"><span>16</span></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_16 --> <div id="page_17" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_15");'>15</a></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li class="active"><span>17</span></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="321"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20050199427','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20050199427"><span>Engine With Regression and Neural Network <span class="hlt">Approximators</span> Designed</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Patnaik, Surya N.; Hopkins, Dale A.</p> <p>2001-01-01</p> <p>At the NASA Glenn Research Center, the NASA engine performance program (NEPP, ref. 1) and the design optimization testbed COMETBOARDS (ref. 2) with regression and neural network analysis-<span class="hlt">approximators</span> have been coupled to obtain a preliminary engine design methodology. The solution to a high-bypass-ratio subsonic waverotor-topped turbofan engine, which is shown in the preceding figure, was obtained by the simulation depicted in the following figure. This engine is made of 16 components mounted on two shafts with 21 flow stations. The engine is designed for a flight envelope with 47 operating points. The design optimization utilized both neural network and regression <span class="hlt">approximations</span>, along with the cascade strategy (ref. 3). The cascade used three algorithms in sequence: the method of feasible directions, the sequence of unconstrained minimizations technique, and sequential quadratic programming. The normalized optimum thrusts obtained by the three methods are shown in the following figure: the cascade algorithm with regression <span class="hlt">approximation</span> is represented by a triangle, a circle is shown for the neural network solution, and a solid line indicates original NEPP results. The solutions obtained from both <span class="hlt">approximate</span> methods lie within one standard deviation of the benchmark solution for each operating point. The simulation improved the maximum thrust by 5 percent. The performance of the linear regression and neural network methods as alternate engine analyzers was found to be satisfactory for the analysis and operation optimization of air-breathing propulsion engines (ref. 4).</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/EJ1128175.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/EJ1128175.pdf"><span>Tension and <span class="hlt">Approximation</span> in Poetic Translation</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Al-Shabab, Omar A. S.; Baka, Farida H.</p> <p>2015-01-01</p> <p>Simple observation reveals that each language and each culture enjoys specific linguistic features and rhetorical traditions. In poetry translation difference and the resultant linguistic tension create a gap between Source Language and Target language, a gap that needs to be bridged by creating an <span class="hlt">approximation</span> processed through the translator's…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19940019695','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19940019695"><span>On the dynamics of <span class="hlt">approximating</span> schemes for dissipative nonlinear equations</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Jones, Donald A.</p> <p>1993-01-01</p> <p>Since one can rarely write down the analytical solutions to nonlinear dissipative partial differential equations (PDE's), it is important to understand whether, and in what sense, the behavior of <span class="hlt">approximating</span> schemes to these equations reflects the true dynamics of the original equations. Further, because standard error estimates between <span class="hlt">approximations</span> of the true solutions coming from spectral methods - finite difference or finite element schemes, for example - and the exact solutions grow exponentially in time, this analysis provides little value in understanding the infinite time behavior of a given <span class="hlt">approximating</span> scheme. The notion of the global attractor has been useful in quantifying the infinite time behavior of dissipative PDEs, such as the Navier-Stokes equations. Loosely speaking, the global attractor is all that remains of a sufficiently large bounded set in phase space mapped infinitely forward in time under the evolution of the PDE. Though the attractor has been shown to have some nice properties - it is compact, connected, and finite dimensional, for example - it is in general quite complicated. Nevertheless, the global attractor gives a way to understand how the infinite time behavior of <span class="hlt">approximating</span> schemes such as the ones coming from a finite difference, finite element, or spectral method relates to that of the original PDE. Indeed, one can often show that such <span class="hlt">approximations</span> also have a global attractor. We therefore only need to understand how the structure of the attractor for the PDE behaves under <span class="hlt">approximation</span>. This is by no means a trivial task. Several interesting results have been obtained in this direction. However, we will not go into the details. We mention here that <span class="hlt">approximations</span> generally lose information about the system no matter how accurate they are. There are examples that show certain parts of the attractor may be lost by arbitrary small perturbations of the original equations.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19930083541','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19930083541"><span>Short-bearing <span class="hlt">approximation</span> for full journal bearings</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Ocvirk, F W</p> <p>1952-01-01</p> <p>A short-bearing <span class="hlt">approximation</span> of pressure distribution in the oil film is presented which is an extension of the pressure-distribution function of Michell and Cardullo and includes end-leakage effects. Equations giving applied load, attitude angle, location and magnitude of peak film pressure, friction, and required oil flow rate as functions of the eccentricity ratio are also given. The capacity number, a basic non dimensional quantity resulting from this analysis is the product of the Sommerfeld number and the square of the length-diameter ratio. Curves determined by this analysis are compared with previously published experimental data and theoretical curves of Sommerfeld and Cameron and Wood. Conclusions reached indicate that this <span class="hlt">approximation</span> is of practical value for analysis of short bearings.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1407456-incomplete-sparse-approximate-inverses-parallel-preconditioning','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1407456-incomplete-sparse-approximate-inverses-parallel-preconditioning"><span>Incomplete Sparse <span class="hlt">Approximate</span> Inverses for Parallel Preconditioning</span></a></p> <p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p> <p>Anzt, Hartwig; Huckle, Thomas K.; Bräckle, Jürgen; ...</p> <p>2017-10-28</p> <p>In this study, we propose a new preconditioning method that can be seen as a generalization of block-Jacobi methods, or as a simplification of the sparse <span class="hlt">approximate</span> inverse (SAI) preconditioners. The “Incomplete Sparse <span class="hlt">Approximate</span> Inverses” (ISAI) is in particular efficient in the solution of sparse triangular linear systems of equations. Those arise, for example, in the context of incomplete factorization preconditioning. ISAI preconditioners can be generated via an algorithm providing fine-grained parallelism, which makes them attractive for hardware with a high concurrency level. Finally, in a study covering a large number of matrices, we identify the ISAI preconditioner as anmore » attractive alternative to exact triangular solves in the context of incomplete factorization preconditioning.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19950031803&hterms=blue+light&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D80%26Ntt%3Dblue%2Blight','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19950031803&hterms=blue+light&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D80%26Ntt%3Dblue%2Blight"><span>HST images of very compact blue galaxies at z <span class="hlt">approximately</span> 0.2</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Koo, David C.; Bershady, Matthew A.; Wirth, Gregory D.; Stanford, S. Adam; Majewski, Steven R.</p> <p>1994-01-01</p> <p>We present the results of Hubble Space Telescope (HST) Wide-Field Camera (WFC) imaging of seven very compact, very blue galaxies with B less than or equal to 21 and redshifts z <span class="hlt">approximately</span> 0.1 to 0.35. Based on deconvolved images, we estimate typical half-light diameters of <span class="hlt">approximately</span> 0.65 sec, corresponding to <span class="hlt">approximately</span> 1.4 h(exp -1) kpc at redshifts z <span class="hlt">approximately</span> 0.2. The average rest frame surface brightness within this diameter is mu(sub v) <span class="hlt">approximately</span> 20.5 mag arcsec(exp -2), <span class="hlt">approximately</span> 1 mag brighter than that of typical late-type blue galaxies. Ground-based spectra show strong, narrow emission lines indicating high ionization; their very blue colors suggest recent bursts of star-formation; their typical luminosities are <span class="hlt">approximately</span> 4 times fainter than that of field galaxies. These characteristics suggest H II galaxies as likely local counterparts of our sample, though our most luminous targets appear to be unusually compact for their luminosities.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1414139-analytic-saddlepoint-approximation-ionization-energy-loss-distributions','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1414139-analytic-saddlepoint-approximation-ionization-energy-loss-distributions"><span>Analytic saddlepoint <span class="hlt">approximation</span> for ionization energy loss distributions</span></a></p> <p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p> <p>Sjue, Sky K. L.; George, Jr., Richard Neal; Mathews, David Gregory</p> <p>2017-07-27</p> <p>Here, we present a saddlepoint <span class="hlt">approximation</span> for ionization energy loss distributions, valid for arbitrary relativistic velocities of the incident particle 0 < v/c < 1, provided that ionizing collisions are still the dominant energy loss mechanism. We derive a closed form solution closely related to Moyal’s distribution. This distribution is intended for use in simulations with relatively low computational overhead. The <span class="hlt">approximation</span> generally reproduces the Vavilov most probable energy loss and full width at half maximum to better than 1% and 10%, respectively, with significantly better agreement as Vavilov’s κ approaches 1.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012PhRvE..85d6705L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012PhRvE..85d6705L"><span><span class="hlt">Approximated</span> maximum likelihood estimation in multifractal random walks</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Løvsletten, O.; Rypdal, M.</p> <p>2012-04-01</p> <p>We present an <span class="hlt">approximated</span> maximum likelihood method for the multifractal random walk processes of [E. Bacry , Phys. Rev. EPLEEE81539-375510.1103/PhysRevE.64.026103 64, 026103 (2001)]. The likelihood is computed using a Laplace <span class="hlt">approximation</span> and a truncation in the dependency structure for the latent volatility. The procedure is implemented as a package in the r computer language. Its performance is tested on synthetic data and compared to an inference approach based on the generalized method of moments. The method is applied to estimate parameters for various financial stock indices.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017NIMPB.407..270S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017NIMPB.407..270S"><span>Analytic saddlepoint <span class="hlt">approximation</span> for ionization energy loss distributions</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Sjue, S. K. L.; George, R. N.; Mathews, D. G.</p> <p>2017-09-01</p> <p>We present a saddlepoint <span class="hlt">approximation</span> for ionization energy loss distributions, valid for arbitrary relativistic velocities of the incident particle 0 < v / c < 1 , provided that ionizing collisions are still the dominant energy loss mechanism. We derive a closed form solution closely related to Moyal's distribution. This distribution is intended for use in simulations with relatively low computational overhead. The <span class="hlt">approximation</span> generally reproduces the Vavilov most probable energy loss and full width at half maximum to better than 1% and 10%, respectively, with significantly better agreement as Vavilov's κ approaches 1.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JPhCS.966a2050G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JPhCS.966a2050G"><span>Subtraction method in the Second Random Phase <span class="hlt">Approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Gambacurta, Danilo</p> <p>2018-02-01</p> <p>We discuss the subtraction method applied to the Second Random Phase <span class="hlt">Approximation</span> (SRPA). This method has been proposed to overcome double counting and stability issues appearing in beyond mean-field calculations. We show that the subtraction procedure leads to a considerable reduction of the SRPA downwards shift with respect to the random phase <span class="hlt">approximation</span> (RPA) spectra and to results that are weakly cutoff dependent. Applications to the isoscalar monopole and quadrupole response in 16O and to the low-lying dipole response in 48Ca are shown and discussed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/1414139-analytic-saddlepoint-approximation-ionization-energy-loss-distributions','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/1414139-analytic-saddlepoint-approximation-ionization-energy-loss-distributions"><span>Analytic saddlepoint <span class="hlt">approximation</span> for ionization energy loss distributions</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Sjue, Sky K. L.; George, Jr., Richard Neal; Mathews, David Gregory</p> <p></p> <p>Here, we present a saddlepoint <span class="hlt">approximation</span> for ionization energy loss distributions, valid for arbitrary relativistic velocities of the incident particle 0 < v/c < 1, provided that ionizing collisions are still the dominant energy loss mechanism. We derive a closed form solution closely related to Moyal’s distribution. This distribution is intended for use in simulations with relatively low computational overhead. The <span class="hlt">approximation</span> generally reproduces the Vavilov most probable energy loss and full width at half maximum to better than 1% and 10%, respectively, with significantly better agreement as Vavilov’s κ approaches 1.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JChPh.148m4110M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JChPh.148m4110M"><span>A walk through the <span class="hlt">approximations</span> of ab initio multiple spawning</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Mignolet, Benoit; Curchod, Basile F. E.</p> <p>2018-04-01</p> <p>Full multiple spawning offers an in principle exact framework for excited-state dynamics, where nuclear wavefunctions in different electronic states are represented by a set of coupled trajectory basis functions that follow classical trajectories. The couplings between trajectory basis functions can be <span class="hlt">approximated</span> to treat molecular systems, leading to the ab initio multiple spawning method which has been successfully employed to study the photochemistry and photophysics of several molecules. However, a detailed investigation of its <span class="hlt">approximations</span> and their consequences is currently missing in the literature. In this work, we simulate the explicit photoexcitation and subsequent excited-state dynamics of a simple system, LiH, and we analyze (i) the effect of the ab initio multiple spawning <span class="hlt">approximations</span> on different observables and (ii) the convergence of the ab initio multiple spawning results towards numerically exact quantum dynamics upon a progressive relaxation of these <span class="hlt">approximations</span>. We show that, despite the crude character of the <span class="hlt">approximations</span> underlying ab initio multiple spawning for this low-dimensional system, the qualitative excited-state dynamics is adequately captured, and affordable corrections can further be applied to ameliorate the coupling between trajectory basis functions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29892139','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29892139"><span>Functional Data <span class="hlt">Approximation</span> on Bounded Domains using Polygonal Finite Elements.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Cao, Juan; Xiao, Yanyang; Chen, Zhonggui; Wang, Wenping; Bajaj, Chandrajit</p> <p>2018-07-01</p> <p>We construct and analyze piecewise <span class="hlt">approximations</span> of functional data on arbitrary 2D bounded domains using generalized barycentric finite elements, and particularly quadratic serendipity elements for planar polygons. We compare <span class="hlt">approximation</span> qualities (precision/convergence) of these partition-of-unity finite elements through numerical experiments, using Wachspress coordinates, natural neighbor coordinates, Poisson coordinates, mean value coordinates, and quadratic serendipity bases over polygonal meshes on the domain. For a convex n -sided polygon, the quadratic serendipity elements have 2 n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, rather than the usual n ( n + 1)/2 basis functions to achieve quadratic convergence. Two greedy algorithms are proposed to generate Voronoi meshes for adaptive functional/scattered data <span class="hlt">approximations</span>. Experimental results show space/accuracy advantages for these quadratic serendipity finite elements on polygonal domains versus traditional finite elements over simplicial meshes. Polygonal meshes and parameter coefficients of the quadratic serendipity finite elements obtained by our greedy algorithms can be further refined using an L 2 -optimization to improve the piecewise functional <span class="hlt">approximation</span>. We conduct several experiments to demonstrate the efficacy of our algorithm for modeling features/discontinuities in functional data/image <span class="hlt">approximation</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/11325187','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/11325187"><span><span class="hlt">Approximation</span> of reliabilities for multiple-trait model with maternal effects.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Strabel, T; Misztal, I; Bertrand, J K</p> <p>2001-04-01</p> <p>Reliabilities for a multiple-trait maternal model were obtained by combining reliabilities obtained from single-trait models. Single-trait reliabilities were obtained using an <span class="hlt">approximation</span> that supported models with additive and permanent environmental effects. For the direct effect, the maternal and permanent environmental variances were assigned to the residual. For the maternal effect, variance of the direct effect was assigned to the residual. Data included 10,550 birth weight, 11,819 weaning weight, and 3,617 postweaning gain records of Senepol cattle. Reliabilities were obtained by generalized inversion and by using single-trait and multiple-trait <span class="hlt">approximation</span> methods. Some reliabilities obtained by inversion were negative because inbreeding was ignored in calculating the inverse of the relationship matrix. The multiple-trait <span class="hlt">approximation</span> method reduced the bias of <span class="hlt">approximation</span> when compared with the single-trait method. The correlations between reliabilities obtained by inversion and by multiple-trait procedures for the direct effect were 0.85 for birth weight, 0.94 for weaning weight, and 0.96 for postweaning gain. Correlations for maternal effects for birth weight and weaning weight were 0.96 to 0.98 for both <span class="hlt">approximations</span>. Further improvements can be achieved by refining the single-trait procedures.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29626896','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29626896"><span>A walk through the <span class="hlt">approximations</span> of ab initio multiple spawning.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Mignolet, Benoit; Curchod, Basile F E</p> <p>2018-04-07</p> <p>Full multiple spawning offers an in principle exact framework for excited-state dynamics, where nuclear wavefunctions in different electronic states are represented by a set of coupled trajectory basis functions that follow classical trajectories. The couplings between trajectory basis functions can be <span class="hlt">approximated</span> to treat molecular systems, leading to the ab initio multiple spawning method which has been successfully employed to study the photochemistry and photophysics of several molecules. However, a detailed investigation of its <span class="hlt">approximations</span> and their consequences is currently missing in the literature. In this work, we simulate the explicit photoexcitation and subsequent excited-state dynamics of a simple system, LiH, and we analyze (i) the effect of the ab initio multiple spawning <span class="hlt">approximations</span> on different observables and (ii) the convergence of the ab initio multiple spawning results towards numerically exact quantum dynamics upon a progressive relaxation of these <span class="hlt">approximations</span>. We show that, despite the crude character of the <span class="hlt">approximations</span> underlying ab initio multiple spawning for this low-dimensional system, the qualitative excited-state dynamics is adequately captured, and affordable corrections can further be applied to ameliorate the coupling between trajectory basis functions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JChPh.146n4106L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JChPh.146n4106L"><span>Mean-trajectory <span class="hlt">approximation</span> for electronic and vibrational-electronic nonlinear spectroscopy</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Loring, Roger F.</p> <p>2017-04-01</p> <p>Mean-trajectory <span class="hlt">approximations</span> permit the calculation of nonlinear vibrational spectra from semiclassically quantized trajectories on a single electronically adiabatic potential surface. By describing electronic degrees of freedom with classical phase-space variables and subjecting these to semiclassical quantization, mean-trajectory <span class="hlt">approximations</span> may be extended to compute both nonlinear electronic spectra and vibrational-electronic spectra. A general mean-trajectory <span class="hlt">approximation</span> for both electronic and nuclear degrees of freedom is presented, and the results for purely electronic and for vibrational-electronic four-wave mixing experiments are quantitatively assessed for harmonic surfaces with linear electronic-nuclear coupling.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017EntIS..11..434Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017EntIS..11..434Z"><span>Identification of <span class="hlt">approximately</span> duplicate material records in ERP systems</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Zong, Wei; Wu, Feng; Chu, Lap-Keung; Sculli, Domenic</p> <p>2017-03-01</p> <p>The quality of master data is crucial for the accurate functioning of the various modules of an enterprise resource planning (ERP) system. This study addresses specific data problems arising from the generation of <span class="hlt">approximately</span> duplicate material records in ERP databases. Such problems are mainly due to the firm's lack of unique and global identifiers for the material records, and to the arbitrary assignment of alternative names for the same material by various users. Traditional duplicate detection methods are ineffective in identifying such <span class="hlt">approximately</span> duplicate material records because these methods typically rely on string comparisons of each field. To address this problem, a machine learning-based framework is developed to recognise semantic similarity between strings and to further identify and reunify <span class="hlt">approximately</span> duplicate material records - a process referred to as de-duplication in this article. First, the keywords of the material records are extracted to form vectors of discriminating words. Second, a machine learning method using a probabilistic neural network is applied to determine the semantic similarity between these material records. The approach was evaluated using data from a real case study. The test results indicate that the proposed method outperforms traditional algorithms in identifying <span class="hlt">approximately</span> duplicate material records.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24875786','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24875786"><span>Optimal sparse <span class="hlt">approximation</span> with integrate and fire neurons.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Shapero, Samuel; Zhu, Mengchen; Hasler, Jennifer; Rozell, Christopher</p> <p>2014-08-01</p> <p>Sparse <span class="hlt">approximation</span> is a hypothesized coding strategy where a population of sensory neurons (e.g. V1) encodes a stimulus using as few active neurons as possible. We present the Spiking LCA (locally competitive algorithm), a rate encoded Spiking Neural Network (SNN) of integrate and fire neurons that calculate sparse <span class="hlt">approximations</span>. The Spiking LCA is designed to be equivalent to the nonspiking LCA, an analog dynamical system that converges on a ℓ(1)-norm sparse <span class="hlt">approximations</span> exponentially. We show that the firing rate of the Spiking LCA converges on the same solution as the analog LCA, with an error inversely proportional to the sampling time. We simulate in NEURON a network of 128 neuron pairs that encode 8 × 8 pixel image patches, demonstrating that the network converges to nearly optimal encodings within 20 ms of biological time. We also show that when using more biophysically realistic parameters in the neurons, the gain function encourages additional ℓ(0)-norm sparsity in the encoding, relative both to ideal neurons and digital solvers.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/1030363-combinatorial-approximation-algorithms-maxcut-using-random-walks','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/1030363-combinatorial-approximation-algorithms-maxcut-using-random-walks"><span>Combinatorial <span class="hlt">approximation</span> algorithms for MAXCUT using random walks.</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Seshadhri, Comandur; Kale, Satyen</p> <p></p> <p>We give the first combinatorial <span class="hlt">approximation</span> algorithm for MaxCut that beats the trivial 0.5 factor by a constant. The main partitioning procedure is very intuitive, natural, and easily described. It essentially performs a number of random walks and aggregates the information to provide the partition. We can control the running time to get an <span class="hlt">approximation</span> factor-running time tradeoff. We show that for any constant b > 1.5, there is an {tilde O}(n{sup b}) algorithm that outputs a (0.5 + {delta})-<span class="hlt">approximation</span> for MaxCut, where {delta} = {delta}(b) is some positive constant. One of the components of our algorithm is a weakmore » local graph partitioning procedure that may be of independent interest. Given a starting vertex i and a conductance parameter {phi}, unless a random walk of length {ell} = O(log n) starting from i mixes rapidly (in terms of {phi} and {ell}), we can find a cut of conductance at most {phi} close to the vertex. The work done per vertex found in the cut is sublinear in n.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012AIPC.1497..184Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012AIPC.1497..184Z"><span><span class="hlt">Approximation</span> by the iterates of Bernstein operator</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Zapryanova, Teodora; Tachev, Gancho</p> <p>2012-11-01</p> <p>We study the degree of pointwise <span class="hlt">approximation</span> of the iterated Bernstein operators to its limiting operator. We obtain a quantitative estimates related to the conjecture of Gonska and Raşa from 2006.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_15");'>15</a></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li class="active"><span>17</span></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_17 --> <div id="page_18" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li class="active"><span>18</span></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="341"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26731788','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26731788"><span>Minimal-<span class="hlt">Approximation</span>-Based Decentralized Backstepping Control of Interconnected Time-Delay Systems.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Choi, Yun Ho; Yoo, Sung Jin</p> <p>2016-12-01</p> <p>A decentralized adaptive backstepping control design using minimal function <span class="hlt">approximators</span> is proposed for nonlinear large-scale systems with unknown unmatched time-varying delayed interactions and unknown backlash-like hysteresis nonlinearities. Compared with existing decentralized backstepping methods, the contribution of this paper is to design a simple local control law for each subsystem, consisting of an actual control with one adaptive function <span class="hlt">approximator</span>, without requiring the use of multiple function <span class="hlt">approximators</span> and regardless of the order of each subsystem. The virtual controllers for each subsystem are used as intermediate signals for designing a local actual control at the last step. For each subsystem, a lumped unknown function including the unknown nonlinear terms and the hysteresis nonlinearities is derived at the last step and is estimated by one function <span class="hlt">approximator</span>. Thus, the proposed approach only uses one function <span class="hlt">approximator</span> to implement each local controller, while existing decentralized backstepping control methods require the number of function <span class="hlt">approximators</span> equal to the order of each subsystem and a calculation of virtual controllers to implement each local actual controller. The stability of the total controlled closed-loop system is analyzed using the Lyapunov stability theorem.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19920016496','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19920016496"><span>The blind leading the blind: Mutual refinement of <span class="hlt">approximate</span> theories</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Kedar, Smadar T.; Bresina, John L.; Dent, C. Lisa</p> <p>1991-01-01</p> <p>The mutual refinement theory, a method for refining world models in a reactive system, is described. The method detects failures, explains their causes, and repairs the <span class="hlt">approximate</span> models which cause the failures. The approach focuses on using one <span class="hlt">approximate</span> model to refine another.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/22779638','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/22779638"><span>Orbitally invariant internally contracted multireference unitary coupled cluster theory and its perturbative <span class="hlt">approximation</span>: theory and test calculations of second order <span class="hlt">approximation</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Chen, Zhenhua; Hoffmann, Mark R</p> <p>2012-07-07</p> <p>A unitary wave operator, exp (G), G(+) = -G, is considered to transform a multiconfigurational reference wave function Φ to the potentially exact, within basis set limit, wave function Ψ = exp (G)Φ. To obtain a useful <span class="hlt">approximation</span>, the Hausdorff expansion of the similarity transformed effective Hamiltonian, exp (-G)Hexp (G), is truncated at second order and the excitation manifold is limited; an additional separate perturbation <span class="hlt">approximation</span> can also be made. In the perturbation <span class="hlt">approximation</span>, which we refer to as multireference unitary second-order perturbation theory (MRUPT2), the Hamiltonian operator in the highest order commutator is <span class="hlt">approximated</span> by a Mo̸ller-Plesset-type one-body zero-order Hamiltonian. If a complete active space self-consistent field wave function is used as reference, then the energy is invariant under orbital rotations within the inactive, active, and virtual orbital subspaces for both the second-order unitary coupled cluster method and its perturbative <span class="hlt">approximation</span>. Furthermore, the redundancies of the excitation operators are addressed in a novel way, which is potentially more efficient compared to the usual full diagonalization of the metric of the excited configurations. Despite the loss of rigorous size-extensivity possibly due to the use of a variational approach rather than a projective one in the solution of the amplitudes, test calculations show that the size-extensivity errors are very small. Compared to other internally contracted multireference perturbation theories, MRUPT2 only needs reduced density matrices up to three-body even with a non-complete active space reference wave function when two-body excitations within the active orbital subspace are involved in the wave operator, exp (G). Both the coupled cluster and perturbation theory variants are amenable to large, incomplete model spaces. Applications to some widely studied model systems that can be problematic because of geometry dependent quasidegeneracy, H4, P4</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2009IEITI..92..158S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2009IEITI..92..158S"><span>A Space-Saving <span class="hlt">Approximation</span> Algorithm for Grammar-Based Compression</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Sakamoto, Hiroshi; Maruyama, Shirou; Kida, Takuya; Shimozono, Shinichi</p> <p></p> <p>A space-efficient <span class="hlt">approximation</span> algorithm for the grammar-based compression problem, which requests for a given string to find a smallest context-free grammar deriving the string, is presented. For the input length n and an optimum CFG size g, the algorithm consumes only O(g log g) space and O(n log*n) time to achieve O((log*n)log n) <span class="hlt">approximation</span> ratio to the optimum compression, where log*n is the maximum number of logarithms satisfying log log…log n > 1. This ratio is thus regarded to almost O(log n), which is the currently best <span class="hlt">approximation</span> ratio. While g depends on the string, it is known that g =Ω(log n) and g=\\\\Omega(\\\\log n) and g=O\\\\left(\\\\frac{n}{log_kn}\\\\right) for strings from k-letter alphabet[12].</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26383040','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26383040"><span>A faster 1.375-<span class="hlt">approximation</span> algorithm for sorting by transpositions.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Cunha, Luís Felipe I; Kowada, Luis Antonio B; Hausen, Rodrigo de A; de Figueiredo, Celina M H</p> <p>2015-11-01</p> <p>Sorting by Transpositions is an NP-hard problem for which several polynomial-time <span class="hlt">approximation</span> algorithms have been developed. Hartman and Shamir (2006) developed a 1.5-<span class="hlt">approximation</span> [Formula: see text] algorithm, whose running time was improved to O(nlogn) by Feng and Zhu (2007) with a data structure they defined, the permutation tree. Elias and Hartman (2006) developed a 1.375-<span class="hlt">approximation</span> O(n(2)) algorithm, and Firoz et al. (2011) claimed an improvement to the running time, from O(n(2)) to O(nlogn), by using the permutation tree. We provide counter-examples to the correctness of Firoz et al.'s strategy, showing that it is not possible to reach a component by sufficient extensions using the method proposed by them. In addition, we propose a 1.375-<span class="hlt">approximation</span> algorithm, modifying Elias and Hartman's approach with the use of permutation trees and achieving O(nlogn) time.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016CMaPh.342...47B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016CMaPh.342...47B"><span>Product-State <span class="hlt">Approximations</span> to Quantum States</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Brandão, Fernando G. S. L.; Harrow, Aram W.</p> <p>2016-02-01</p> <p>We show that for any many-body quantum state there exists an unentangled quantum state such that most of the two-body reduced density matrices are close to those of the original state. This is a statement about the monogamy of entanglement, which cannot be shared without limit in the same way as classical correlation. Our main application is to Hamiltonians that are sums of two-body terms. For such Hamiltonians we show that there exist product states with energy that is close to the ground-state energy whenever the interaction graph of the Hamiltonian has high degree. This proves the validity of mean-field theory and gives an explicitly bounded <span class="hlt">approximation</span> error. If we allow states that are entangled within small clusters of systems but product across clusters then good <span class="hlt">approximations</span> exist when the Hamiltonian satisfies one or more of the following properties: (1) high degree, (2) small expansion, or (3) a ground state where the blocks in the partition have sublinear entanglement. Previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero. Our <span class="hlt">approximations</span> allow an extensive error in energy, which is the scale considered by the quantum PCP (probabilistically checkable proof) and NLTS (no low-energy trivial-state) conjectures. Thus our results put restrictions on the possible Hamiltonians that could be used for a possible proof of the qPCP or NLTS conjectures. By contrast the classical PCP constructions are often based on constraint graphs with high degree. Likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum Hamiltonians, unless qPCP is false. The main technical tool behind our results is a collection of new classical and quantum de Finetti theorems which do not make any symmetry assumptions on the underlying states.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19890002382','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19890002382"><span><span class="hlt">Approximation</span> theory for LQG (Linear-Quadratic-Gaussian) optimal control of flexible structures</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Gibson, J. S.; Adamian, A.</p> <p>1988-01-01</p> <p>An <span class="hlt">approximation</span> theory is presented for the LQG (Linear-Quadratic-Gaussian) optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite dimensional compensators that <span class="hlt">approximate</span> closely the optimal compensator. The optimal LQG problem separates into an optimal linear-quadratic regulator problem and an optimal state estimation problem. The solution of the former problem lies in the solution to an infinite dimensional Riccati operator equation. The <span class="hlt">approximation</span> scheme <span class="hlt">approximates</span> the infinite dimensional LQG problem with a sequence of finite dimensional LQG problems defined for a sequence of finite dimensional, usually finite element or modal, <span class="hlt">approximations</span> of the distributed model of the structure. Two Riccati matrix equations determine the solution to each <span class="hlt">approximating</span> problem. The finite dimensional equations for numerical <span class="hlt">approximation</span> are developed, including formulas for converting matrix control and estimator gains to their functional representation to allow comparison of gains based on different orders of <span class="hlt">approximation</span>. Convergence of the <span class="hlt">approximating</span> control and estimator gains and of the corresponding finite dimensional compensators is studied. Also, convergence and stability of the closed-loop systems produced with the finite dimensional compensators are discussed. The convergence theory is based on the convergence of the solutions of the finite dimensional Riccati equations to the solutions of the infinite dimensional Riccati equations. A numerical example with a flexible beam, a rotating rigid body, and a lumped mass is given.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22416066-second-derivatives-approximate-spin-projection-methods','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22416066-second-derivatives-approximate-spin-projection-methods"><span>Second derivatives for <span class="hlt">approximate</span> spin projection methods</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Thompson, Lee M.; Hratchian, Hrant P., E-mail: hhratchian@ucmerced.edu</p> <p>2015-02-07</p> <p>The use of broken-symmetry electronic structure methods is required in order to obtain correct behavior of electronically strained open-shell systems, such as transition states, biradicals, and transition metals. This approach often has issues with spin contamination, which can lead to significant errors in predicted energies, geometries, and properties. <span class="hlt">Approximate</span> projection schemes are able to correct for spin contamination and can often yield improved results. To fully make use of these methods and to carry out exploration of the potential energy surface, it is desirable to develop an efficient second energy derivative theory. In this paper, we formulate the analytical secondmore » derivatives for the Yamaguchi <span class="hlt">approximate</span> projection scheme, building on recent work that has yielded an efficient implementation of the analytical first derivatives.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/6234737-analytic-approximation-random-muffin-tin-alloys','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/6234737-analytic-approximation-random-muffin-tin-alloys"><span>Analytic <span class="hlt">approximation</span> for random muffin-tin alloys</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Mills, R.; Gray, L.J.; Kaplan, T.</p> <p>1983-03-15</p> <p>The methods introduced in a previous paper under the name of ''traveling-cluster <span class="hlt">approximation</span>'' (TCA) are applied, in a multiple-scattering approach, to the case of a random muffin-tin substitutional alloy. This permits the iterative part of a self-consistent calculation to be carried out entirely in terms of on-the-energy-shell scattering amplitudes. Off-shell components of the mean resolvent, needed for the calculation of spectral functions, are obtained by standard methods involving single-site scattering wave functions. The single-site TCA is just the usual coherent-potential <span class="hlt">approximation</span>, expressed in a form particularly suited for iteration. A fixed-point theorem is proved for the general t-matrix TCA, ensuringmore » convergence upon iteration to a unique self-consistent solution with the physically essential Herglotz properties.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA182358','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA182358"><span><span class="hlt">Approximate</span> Evaluation of Reliability and Availability via Perturbation Analysis.</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>1986-12-01</p> <p>generating the dxact answers to which the results of the <span class="hlt">approximation</span> will be compared are discussed:." - 87 7" 2 04 2LOISTRIGUTIONIAVAILASILITYI OF...appropriately, constructing the Markov process that <span class="hlt">approximately</span> governs interclass behavior from the result above (this is called the enlarged... compared to a numerical or analytical computation of the aame quantities. This work and its continuation represents our progress so far on Goal 3. 2.4I</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/1237574','SCIGOV-STC'); return false;" href="https://www.osti.gov/servlets/purl/1237574"><span>NONLINEAR MULTIGRID SOLVER EXPLOITING AMGe COARSE SPACES WITH <span class="hlt">APPROXIMATION</span> PROPERTIES</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Christensen, Max La Cour; Villa, Umberto E.; Engsig-Karup, Allan P.</p> <p></p> <p>The paper introduces a nonlinear multigrid solver for mixed nite element discretizations based on the Full <span class="hlt">Approximation</span> Scheme (FAS) and element-based Algebraic Multigrid (AMGe). The main motivation to use FAS for unstruc- tured problems is the guaranteed <span class="hlt">approximation</span> property of the AMGe coarse spaces that were developed recently at Lawrence Livermore National Laboratory. These give the ability to derive stable and accurate coarse nonlinear discretization problems. The previous attempts (including ones with the original AMGe method, [5, 11]), were less successful due to lack of such good <span class="hlt">approximation</span> properties of the coarse spaces. With coarse spaces with <span class="hlt">approximation</span> properties, ourmore » FAS approach on un- structured meshes should be as powerful/successful as FAS on geometrically re ned meshes. For comparison, Newton's method and Picard iterations with an inner state-of-the-art linear solver is compared to FAS on a nonlinear saddle point problem with applications to porous media ow. It is demonstrated that FAS is faster than Newton's method and Picard iterations for the experiments considered here. Due to the guaranteed <span class="hlt">approximation</span> properties of our AMGe, the coarse spaces are very accurate, providing a solver with the potential for mesh-independent convergence on general unstructured meshes.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=20090014845&hterms=highly+charged+heavy+ions&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dhighly%2Bcharged%2Bheavy%2Bions','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=20090014845&hterms=highly+charged+heavy+ions&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dhighly%2Bcharged%2Bheavy%2Bions"><span>Estimates of Sputter Yields of Solar-Wind Heavy Ions of Lunar Regolith Materials</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Barghouty, Abdulmasser F.; Adams, James H., Jr.</p> <p>2008-01-01</p> <p>At energies of <span class="hlt">approximately</span> 1 keV/amu, solar-wind protons and heavy ions interact with the lunar surface materials via a number of microscopic interactions that include sputtering. Solar-wind induced sputtering is a main mechanism by which the composition of the topmost layers of the lunar surface can change, dynamically and preferentially. This work concentrates on sputtering induced by solar-wind heavy ions. Sputtering associated with slow (speeds the electrons speed in its first <span class="hlt">Bohr</span> orbit) and highly charged ions are known to include both kinetic and potential sputtering. Potential sputtering enjoys some unique characteristics that makes it of special interest to lunar science and exploration. Unlike the yield from kinetic sputtering where simulation and <span class="hlt">approximation</span> schemes exist, the yield from potential sputtering is not as easy to estimate. This work will present a preliminary numerical scheme designed to estimate potential sputtering yields from reactions relevant to this aspect of solar-wind lunar-surface coupling.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JETAI..29..823S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JETAI..29..823S"><span>Metaheuristic optimisation methods for <span class="hlt">approximate</span> solving of singular boundary value problems</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Sadollah, Ali; Yadav, Neha; Gao, Kaizhou; Su, Rong</p> <p>2017-07-01</p> <p>This paper presents a novel <span class="hlt">approximation</span> technique based on metaheuristics and weighted residual function (WRF) for tackling singular boundary value problems (BVPs) arising in engineering and science. With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic optimisation algorithms, singular BVPs can be <span class="hlt">approximated</span> as an optimisation problem with boundary conditions as constraints. The target is to minimise the WRF (i.e. error function) constructed in <span class="hlt">approximation</span> of BVPs. The scheme involves generational distance metric for quality evaluation of the <span class="hlt">approximate</span> solutions against exact solutions (i.e. error evaluator metric). Four test problems including two linear and two non-linear singular BVPs are considered in this paper to check the efficiency and accuracy of the proposed algorithm. The optimisation task is performed using three different optimisers including the particle swarm optimisation, the water cycle algorithm, and the harmony search algorithm. Optimisation results obtained show that the suggested technique can be successfully applied for <span class="hlt">approximate</span> solving of singular BVPs.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22471898-use-neural-networks-approximation-nuclear-data','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22471898-use-neural-networks-approximation-nuclear-data"><span>The use of neural networks for <span class="hlt">approximation</span> of nuclear data</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Korovin, Yu. A.; Maksimushkina, A. V., E-mail: AVMaksimushkina@mephi.ru</p> <p>2015-12-15</p> <p>The article discusses the possibility of using neural networks for <span class="hlt">approximation</span> or reconstruction of data such as the reaction cross sections. The quality of the <span class="hlt">approximation</span> using fitting criteria is also evaluated. The activity of materials under irradiation is calculated from data obtained using neural networks.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/17676137','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/17676137"><span>Geometrical-optics <span class="hlt">approximation</span> of forward scattering by gradient-index spheres.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Li, Xiangzhen; Han, Xiang'e; Li, Renxian; Jiang, Huifen</p> <p>2007-08-01</p> <p>By means of geometrical optics we present an <span class="hlt">approximation</span> method for acceleration of the computation of the scattering intensity distribution within a forward angular range (0-60 degrees ) for gradient-index spheres illuminated by a plane wave. The incident angle of reflected light is determined by the scattering angle, thus improving the <span class="hlt">approximation</span> accuracy. The scattering angle and the optical path length are numerically integrated by a general-purpose integrator. With some special index models, the scattering angle and the optical path length can be expressed by a unique function and the calculation is faster. This method is proved effective for transparent particles with size parameters greater than 50. It fails to give good <span class="hlt">approximation</span> results at scattering angles whose refractive rays are in the backward direction. For different index models, the geometrical-optics <span class="hlt">approximation</span> is effective only for forward angles, typically those less than 60 degrees or when the refractive-index difference of a particle is less than a certain value.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19930029110&hterms=berenji&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3Dberenji','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19930029110&hterms=berenji&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3Dberenji"><span>On the integration of reinforcement learning and <span class="hlt">approximate</span> reasoning for control</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Berenji, Hamid R.</p> <p>1991-01-01</p> <p>The author discusses the importance of strengthening the knowledge representation characteristic of reinforcement learning techniques using methods such as <span class="hlt">approximate</span> reasoning. The ARIC (<span class="hlt">approximate</span> reasoning-based intelligent control) architecture is an example of such a hybrid approach in which the fuzzy control rules are modified (fine-tuned) using reinforcement learning. ARIC also demonstrates that it is possible to start with an <span class="hlt">approximately</span> correct control knowledge base and learn to refine this knowledge through further experience. On the other hand, techniques such as the TD (temporal difference) algorithm and Q-learning establish stronger theoretical foundations for their use in adaptive control and also in stability analysis of hybrid reinforcement learning and <span class="hlt">approximate</span> reasoning-based controllers.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhRvL.118p3203H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhRvL.118p3203H"><span>Strong Field Theories beyond Dipole <span class="hlt">Approximations</span> in Nonrelativistic Regimes</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>He, Pei-Lun; Lao, Di; He, Feng</p> <p>2017-04-01</p> <p>The exact nondipole Volkov solutions to the Schrödinger equation and Pauli equation are found, based on which a strong field theory beyond the dipole <span class="hlt">approximation</span> is built for describing the nondipole effects in nonrelativistic laser driven electron dynamics. This theory is applied to investigate momentum partition laws for multiphoton and tunneling ionization and explicitly shows that the complex interplay of a laser field and Coulomb action may reverse the expected photoelectron momentum along the laser propagation direction. The magnetic-spin coupling does not bring observable effects on the photoelectron momentum distribution and can be neglected. Compared to the strong field <span class="hlt">approximation</span> within the dipole <span class="hlt">approximation</span>, this theory works in a much wider range of laser parameters and lays a solid foundation for describing nonrelativistic electron dynamics in both short wavelength and midinfrared regimes where nondipole effects are unavoidable.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014pmqm.book..112P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014pmqm.book..112P"><span>Bohrian Complementarity in the Light of Kantian Teleology</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Pringe, Hernán</p> <p>2014-03-01</p> <p>The Kantian influences on <span class="hlt">Bohr</span>'s thought and the relationship between the perspective of complementarity in physics and in biology seem at first sight completely unrelated issues. However, the goal of this work is to show their intimate connection. We shall see that <span class="hlt">Bohr</span>'s views on biology shed light on Kantian elements of his thought, which enables a better understanding of his complementary interpretation of quantum theory. For this purpose, we shall begin by discussing <span class="hlt">Bohr</span>'s views on the analogies concerning the epistemological situation in biology and in physics. Later, we shall compare the Bohrian and the Kantian approaches to the science of life in order to show their close connection. On this basis, we shall finally turn to the issue of complementarity in quantum theory in order to assess what we can learn about the epistemological problems in the quantum realm from a consideration of Kant's views on teleology.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29882849','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29882849"><span>Time and Memory Efficient Online Piecewise Linear <span class="hlt">Approximation</span> of Sensor Signals.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Grützmacher, Florian; Beichler, Benjamin; Hein, Albert; Kirste, Thomas; Haubelt, Christian</p> <p>2018-05-23</p> <p>Piecewise linear <span class="hlt">approximation</span> of sensor signals is a well-known technique in the fields of Data Mining and Activity Recognition. In this context, several algorithms have been developed, some of them with the purpose to be performed on resource constrained microcontroller architectures of wireless sensor nodes. While microcontrollers are usually constrained in computational power and memory resources, all state-of-the-art piecewise linear <span class="hlt">approximation</span> techniques either need to buffer sensor data or have an execution time depending on the segment’s length. In the paper at hand, we propose a novel piecewise linear <span class="hlt">approximation</span> algorithm, with a constant computational complexity as well as a constant memory complexity. Our proposed algorithm’s worst-case execution time is one to three orders of magnitude smaller and its average execution time is three to seventy times smaller compared to the state-of-the-art Piecewise Linear <span class="hlt">Approximation</span> (PLA) algorithms in our experiments. In our evaluations, we show that our algorithm is time and memory efficient without sacrificing the <span class="hlt">approximation</span> quality compared to other state-of-the-art piecewise linear <span class="hlt">approximation</span> techniques, while providing a maximum error guarantee per segment, a small parameter space of only one parameter, and a maximum latency of one sample period plus its worst-case execution time.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA256799','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA256799"><span><span class="hlt">Approximation</span> Algorithms for Multicommodity Flow and Shop Scheduling Problems</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>1992-09-01</p> <p>DARPA N00014-89-J-1988 11. SUPPLEMENTARY NOTES Ŗa. oIs7RIBU ric.c / AVAILAaILITY STATEMENT, 1.2. 3ISTRIBUT;CN C:. E In this thesis , we give efficient...University Thesis Supervisor Accepted by Campbell L. Searle Chairman, Departmental Committee on Graduate Students <span class="hlt">Approximation</span> Algorithms for Multicommodity...partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis , we give efficient <span class="hlt">approximation</span> algorithms for</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_16");'>16</a></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li class="active"><span>18</span></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_18 --> <div id="page_19" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li class="active"><span>19</span></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="361"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1989rtfs.rept.....K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1989rtfs.rept.....K"><span>Radiative transfer in falling snow: A two-stream <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Koh, Gary</p> <p>1989-04-01</p> <p>Light transmission measurements through falling snow have produced results unexplainable by single scattering arguments. A two-stream <span class="hlt">approximation</span> to radiative transfer is used to derive an analytical expression that describes the effects of multiple scattering as a function of the snow optical depth and the snow asymmetry parameter. The <span class="hlt">approximate</span> solution is simple and it may be as accurate as the exact solution for describing the transmission measurements within the limits of experimental uncertainties.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22611442-multi-level-methods-approximating-distribution-functions','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22611442-multi-level-methods-approximating-distribution-functions"><span>Multi-level methods and <span class="hlt">approximating</span> distribution functions</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Wilson, D., E-mail: daniel.wilson@dtc.ox.ac.uk; Baker, R. E.</p> <p>2016-07-15</p> <p>Biochemical reaction networks are often modelled using discrete-state, continuous-time Markov chains. System statistics of these Markov chains usually cannot be calculated analytically and therefore estimates must be generated via simulation techniques. There is a well documented class of simulation techniques known as exact stochastic simulation algorithms, an example of which is Gillespie’s direct method. These algorithms often come with high computational costs, therefore <span class="hlt">approximate</span> stochastic simulation algorithms such as the tau-leap method are used. However, in order to minimise the bias in the estimates generated using them, a relatively small value of tau is needed, rendering the computational costs comparablemore » to Gillespie’s direct method. The multi-level Monte Carlo method (Anderson and Higham, Multiscale Model. Simul. 10:146–179, 2012) provides a reduction in computational costs whilst minimising or even eliminating the bias in the estimates of system statistics. This is achieved by first crudely <span class="hlt">approximating</span> required statistics with many sample paths of low accuracy. Then correction terms are added until a required level of accuracy is reached. Recent literature has primarily focussed on implementing the multi-level method efficiently to estimate a single system statistic. However, it is clearly also of interest to be able to <span class="hlt">approximate</span> entire probability distributions of species counts. We present two novel methods that combine known techniques for distribution reconstruction with the multi-level method. We demonstrate the potential of our methods using a number of examples.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JHyd..556..674B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JHyd..556..674B"><span>The validity of flow <span class="hlt">approximations</span> when simulating catchment-integrated flash floods</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Bout, B.; Jetten, V. G.</p> <p>2018-01-01</p> <p>Within hydrological models, flow <span class="hlt">approximations</span> are commonly used to reduce computation time. The validity of these <span class="hlt">approximations</span> is strongly determined by flow height, flow velocity and the spatial resolution of the model. In this presentation, the validity and performance of the kinematic, diffusive and dynamic flow <span class="hlt">approximations</span> are investigated for use in a catchment-based flood model. Particularly, the validity during flood events and for varying spatial resolutions is investigated. The OpenLISEM hydrological model is extended to implement both these flow <span class="hlt">approximations</span> and channel flooding based on dynamic flow. The flow <span class="hlt">approximations</span> are used to recreate measured discharge in three catchments, among which is the hydrograph of the 2003 flood event in the Fella river basin. Furthermore, spatial resolutions are varied for the flood simulation in order to investigate the influence of spatial resolution on these flow <span class="hlt">approximations</span>. Results show that the kinematic, diffusive and dynamic flow <span class="hlt">approximation</span> provide least to highest accuracy, respectively, in recreating measured discharge. Kinematic flow, which is commonly used in hydrological modelling, substantially over-estimates hydrological connectivity in the simulations with a spatial resolution of below 30 m. Since spatial resolutions of models have strongly increased over the past decades, usage of routed kinematic flow should be reconsidered. The combination of diffusive or dynamic overland flow and dynamic channel flooding provides high accuracy in recreating the 2003 Fella river flood event. Finally, in the case of flood events, spatial modelling of kinematic flow substantially over-estimates hydrological connectivity and flow concentration since pressure forces are removed, leading to significant errors.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22680072-slow-roll-approximation-loop-quantum-cosmology','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22680072-slow-roll-approximation-loop-quantum-cosmology"><span>Slow-roll <span class="hlt">approximation</span> in loop quantum cosmology</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Luc, Joanna; Mielczarek, Jakub, E-mail: joanna.luc@uj.edu.pl, E-mail: jakub.mielczarek@uj.edu.pl</p> <p></p> <p>The slow-roll <span class="hlt">approximation</span> is an analytical approach to study dynamical properties of the inflationary universe. In this article, systematic construction of the slow-roll expansion for effective loop quantum cosmology is presented. The analysis is performed up to the fourth order in both slow-roll parameters and the parameter controlling the strength of deviation from the classical case. The expansion is performed for three types of the slow-roll parameters: Hubble slow-roll parameters, Hubble flow parameters and potential slow-roll parameters. An accuracy of the <span class="hlt">approximation</span> is verified by comparison with the numerical phase space trajectories for the case with a massive potential term.more » The results obtained in this article may be helpful in the search for the subtle quantum gravitational effects with use of the cosmological data.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19930091318','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19930091318"><span><span class="hlt">Approximations</span> for column effect in airplane wing spars</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Warner, Edward P; Short, Mac</p> <p>1927-01-01</p> <p>The significance attaching to "column effect" in airplane wing spars has been increasingly realized with the passage of time, but exact computations of the corrections to bending moment curves resulting from the existence of end loads are frequently omitted because of the additional labor involved in an analysis by rigorously correct methods. The present report represents an attempt to provide for <span class="hlt">approximate</span> column effect corrections that can be graphically or otherwise expressed so as to be applied with a minimum of labor. Curves are plotted giving <span class="hlt">approximate</span> values of the correction factors for single and two bay trusses of varying proportions and with various relationships between axial and lateral loads. It is further shown from an analysis of those curves that rough but useful <span class="hlt">approximations</span> can be obtained from Perry's formula for corrected bending moment, with the assumed distance between points of inflection arbitrarily modified in accordance with rules given in the report. The discussion of general rules of variation of bending stress with axial load is accompanied by a study of the best distribution of the points of support along a spar for various conditions of loading.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD0704567','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD0704567"><span>FINITE-STATE <span class="hlt">APPROXIMATIONS</span> TO DENUMERABLE-STATE DYNAMIC PROGRAMS,</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p></p> <p>AIR FORCE OPERATIONS, LOGISTICS), (*INVENTORY CONTROL, DYNAMIC PROGRAMMING), (*DYNAMIC PROGRAMMING, <span class="hlt">APPROXIMATION</span>(MATHEMATICS)), INVENTORY CONTROL, DECISION MAKING, STOCHASTIC PROCESSES, GAME THEORY, ALGORITHMS, CONVERGENCE</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2001SPIE.4320..339C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2001SPIE.4320..339C"><span>Analytical <span class="hlt">approximations</span> to the Hotelling trace for digital x-ray detectors</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Clarkson, Eric; Pineda, Angel R.; Barrett, Harrison H.</p> <p>2001-06-01</p> <p>The Hotelling trace is the signal-to-noise ratio for the ideal linear observer in a detection task. We provide an analytical <span class="hlt">approximation</span> for this figure of merit when the signal is known exactly and the background is generated by a stationary random process, and the imaging system is an ideal digital x-ray detector. This <span class="hlt">approximation</span> is based on assuming that the detector is infinite in extent. We test this <span class="hlt">approximation</span> for finite-size detectors by comparing it to exact calculations using matrix inversion of the data covariance matrix. After verifying the validity of the <span class="hlt">approximation</span> under a variety of circumstances, we use it to generate plots of the Hotelling trace as a function of pairs of parameters of the system, the signal and the background.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.ars.usda.gov/research/publications/publication/?seqNo115=348350','TEKTRAN'); return false;" href="http://www.ars.usda.gov/research/publications/publication/?seqNo115=348350"><span><span class="hlt">Approximating</span> genomic reliabilities for national genomic evaluation</span></a></p> <p><a target="_blank" href="https://www.ars.usda.gov/research/publications/find-a-publication/">USDA-ARS?s Scientific Manuscript database</a></p> <p></p> <p></p> <p>With the introduction of standard methods for <span class="hlt">approximating</span> effective daughter/data contribution by Interbull in 2001, conventional EDC or reliabilities contributed by daughter phenotypes are directly comparable across countries and used in routine conventional evaluations. In order to make publishe...</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19820021683','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19820021683"><span>Burgers <span class="hlt">approximation</span> for two-dimensional flow past an ellipse</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Dorrepaal, J. M.</p> <p>1982-01-01</p> <p>A linearization of the Navier-Stokes equation due to Burgers in which vorticity is transported by the velocity field corresponding to continuous potential flow is examined. The governing equations are solved exactly for the two dimensional steady flow past an ellipse of arbitrary aspect ratio. The requirement of no slip along the surface of the ellipse results in an infinite algebraic system of linear equations for coefficients appearing in the solution. The system is truncated at a point which gives reliable results for Reynolds numbers R in the range 0 R 5. Predictions of the Burgers <span class="hlt">approximation</span> regarding separation, drag and boundary layer behavior are investigated. In particular, Burgers linearization gives drag coefficients which are closer to observed experimental values than those obtained from Oseen's <span class="hlt">approximation</span>. In the special case of flow past a circular cylinder, Burgers <span class="hlt">approximation</span> predicts a boundary layer whose thickness is roughly proportional to R-1/2.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26580039','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26580039"><span>Correlation Energies from the Two-Component Random Phase <span class="hlt">Approximation</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Kühn, Michael</p> <p>2014-02-11</p> <p>The correlation energy within the two-component random phase <span class="hlt">approximation</span> accounting for spin-orbit effects is derived. The resulting plasmon equation is rewritten-analogously to the scalar relativistic case-in terms of the trace of two Hermitian matrices for (Kramers-restricted) closed-shell systems and then represented as an integral over imaginary frequency using the resolution of the identity <span class="hlt">approximation</span>. The final expression is implemented in the TURBOMOLE program suite. The code is applied to the computation of equilibrium distances and vibrational frequencies of heavy diatomic molecules. The efficiency is demonstrated by calculation of the relative energies of the Oh-, D4h-, and C5v-symmetric isomers of Pb6. Results within the random phase <span class="hlt">approximation</span> are obtained based on two-component Kohn-Sham reference-state calculations, using effective-core potentials. These values are finally compared to other two-component and scalar relativistic methods, as well as experimental data.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=electroquimica+AND+en+AND+baterias+AND+libros%7d&id=EJ1060255','ERIC'); return false;" href="https://eric.ed.gov/?q=electroquimica+AND+en+AND+baterias+AND+libros%7d&id=EJ1060255"><span>The Role of Intuitive <span class="hlt">Approximation</span> Skills for School Math Abilities</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Libertus, Melissa E.</p> <p>2015-01-01</p> <p>Research has shown that educated children and adults have access to two ways of representing numerical information: an <span class="hlt">approximate</span> number system (ANS) that is present from birth and allows for quick <span class="hlt">approximations</span> of numbers of objects encountered in one's environment, and an exact number system (ENS) that is acquired through experience and…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018SPIE10648E..0CC','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018SPIE10648E..0CC"><span>Series <span class="hlt">approximation</span> to probability densities</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Cohen, L.</p> <p>2018-04-01</p> <p>One of the historical and fundamental uses of the Edgeworth and Gram-Charlier series is to "correct" a Gaussian density when it is determined that the probability density under consideration has moments that do not correspond to the Gaussian [5, 6]. There is a fundamental difficulty with these methods in that if the series are truncated, then the resulting <span class="hlt">approximate</span> density is not manifestly positive. The aim of this paper is to attempt to expand a probability density so that if it is truncated it will still be manifestly positive.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018PhyE..100...24W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018PhyE..100...24W"><span><span class="hlt">Approximate</span> solutions to Mathieu's equation</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Wilkinson, Samuel A.; Vogt, Nicolas; Golubev, Dmitry S.; Cole, Jared H.</p> <p>2018-06-01</p> <p>Mathieu's equation has many applications throughout theoretical physics. It is especially important to the theory of Josephson junctions, where it is equivalent to Schrödinger's equation. Mathieu's equation can be easily solved numerically, however there exists no closed-form analytic solution. Here we collect various <span class="hlt">approximations</span> which appear throughout the physics and mathematics literature and examine their accuracy and regimes of applicability. Particular attention is paid to quantities relevant to the physics of Josephson junctions, but the arguments and notation are kept general so as to be of use to the broader physics community.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/677169','SCIGOV-STC'); return false;" href="https://www.osti.gov/servlets/purl/677169"><span>Towards syntactic characterizations of <span class="hlt">approximation</span> schemes via predicate and graph decompositions</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Hunt, H.B. III; Stearns, R.E.; Jacob, R.</p> <p>1998-12-01</p> <p>The authors present a simple extensible theoretical framework for devising polynomial time <span class="hlt">approximation</span> schemes for problems represented using natural syntactic (algebraic) specifications endowed with natural graph theoretic restrictions on input instances. Direct application of the technique yields polynomial time <span class="hlt">approximation</span> schemes for all the problems studied in [LT80, NC88, KM96, Ba83, DTS93, HM+94a, HM+94] as well as the first known <span class="hlt">approximation</span> schemes for a number of additional combinatorial problems. One notable aspect of the work is that it provides insights into the structure of the syntactic specifications and the corresponding algorithms considered in [KM96, HM+94]. The understanding allows them tomore » extend the class of syntactic specifications for which generic <span class="hlt">approximation</span> schemes can be developed. The results can be shown to be tight in many cases, i.e. natural extensions of the specifications can be shown to yield non-<span class="hlt">approximable</span> problems. The results provide a non-trivial characterization of a class of problems having a PTAS and extend the earlier work on this topic by [KM96, HM+94].« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008PhyD..237.3109A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008PhyD..237.3109A"><span>ɛ-connectedness, finite <span class="hlt">approximations</span>, shape theory and coarse graining in hyperspaces</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Alonso-Morón, Manuel; Cuchillo-Ibanez, Eduardo; Luzón, Ana</p> <p>2008-12-01</p> <p>We use upper semifinite hyperspaces of compacta to describe ε-connectedness and to compute homology from finite <span class="hlt">approximations</span>. We find a new connection between ε-connectedness and the so-called Shape Theory. We construct a geodesically complete R-tree, by means of ε-components at different resolutions, whose behavior at infinite captures the topological structure of the space of components of a given compact metric space. We also construct inverse sequences of finite spaces using internal finite <span class="hlt">approximations</span> of compact metric spaces. These sequences can be converted into inverse sequences of polyhedra and simplicial maps by means of what we call the Alexandroff-McCord correspondence. This correspondence allows us to relate upper semifinite hyperspaces of finite <span class="hlt">approximation</span> with the Vietoris-Rips complexes of such <span class="hlt">approximations</span> at different resolutions. Two motivating examples are included in the introduction. We propose this procedure as a different mathematical foundation for problems on data analysis. This process is intrinsically related to the methodology of shape theory. This paper reinforces Robins’s idea of using methods from shape theory to compute homology from finite <span class="hlt">approximations</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/20774799-testing-ginzburg-landau-approximation-three-flavor-crystalline-color-superconductivity','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/20774799-testing-ginzburg-landau-approximation-three-flavor-crystalline-color-superconductivity"><span>Testing the Ginzburg-Landau <span class="hlt">approximation</span> for three-flavor crystalline color superconductivity</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Mannarelli, Massimo; Sharma, Rishi; Rajagopal, Krishna</p> <p>2006-06-01</p> <p>It is an open challenge to analyze the crystalline color superconducting phases that may arise in cold dense, but not asymptotically dense, three-flavor quark matter. At present the only <span class="hlt">approximation</span> within which it seems possible to compare the free energies of the myriad possible crystal structures is the Ginzburg-Landau <span class="hlt">approximation</span>. Here, we test this <span class="hlt">approximation</span> on a particularly simple 'crystal' structure in which there are only two condensates <us>{approx}{delta}exp(iq{sub 2}{center_dot}r) and <ud>{approx}{delta}exp(iq{sub 3}{center_dot}r) whose position-space dependence is that of two plane waves with wave vectors q{sub 2} and q{sub 3} at arbitrary angles. For this case, we are able tomore » solve the mean-field gap equation without making a Ginzburg-Landau <span class="hlt">approximation</span>. We find that the Ginzburg-Landau <span class="hlt">approximation</span> works in the {delta}{yields}0 limit as expected, find that it correctly predicts that {delta} decreases with increasing angle between q{sub 2} and q{sub 3} meaning that the phase with q{sub 2} parallel q{sub 3} has the lowest free energy, and find that the Ginzburg-Landau <span class="hlt">approximation</span> is conservative in the sense that it underestimates {delta} at all values of the angle between q{sub 2} and q{sub 3}.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20040090534','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20040090534"><span>A Comparison of <span class="hlt">Approximation</span> Modeling Techniques: Polynomial Versus Interpolating Models</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Giunta, Anthony A.; Watson, Layne T.</p> <p>1998-01-01</p> <p>Two methods of creating <span class="hlt">approximation</span> models are compared through the calculation of the modeling accuracy on test problems involving one, five, and ten independent variables. Here, the test problems are representative of the modeling challenges typically encountered in realistic engineering optimization problems. The first <span class="hlt">approximation</span> model is a quadratic polynomial created using the method of least squares. This type of polynomial model has seen considerable use in recent engineering optimization studies due to its computational simplicity and ease of use. However, quadratic polynomial models may be of limited accuracy when the response data to be modeled have multiple local extrema. The second <span class="hlt">approximation</span> model employs an interpolation scheme known as kriging developed in the fields of spatial statistics and geostatistics. This class of interpolating model has the flexibility to model response data with multiple local extrema. However, this flexibility is obtained at an increase in computational expense and a decrease in ease of use. The intent of this study is to provide an initial exploration of the accuracy and modeling capabilities of these two <span class="hlt">approximation</span> methods.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19970019007&hterms=noncoherent&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Dnoncoherent','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19970019007&hterms=noncoherent&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Dnoncoherent"><span>Partially Coherent Scattering in Stellar Chromospheres. Part 4; Analytic Wing <span class="hlt">Approximations</span></span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Gayley, K. G.</p> <p>1993-01-01</p> <p>Simple analytic expressions are derived to understand resonance-line wings in stellar chromospheres and similar astrophysical plasmas. The results are <span class="hlt">approximate</span>, but compare well with accurate numerical simulations. The redistribution is modeled using an extension of the partially coherent scattering <span class="hlt">approximation</span> (PCS) which we term the comoving-frame partially coherent scattering <span class="hlt">approximation</span> (CPCS). The distinction is made here because Doppler diffusion is included in the coherent/noncoherent decomposition, in a form slightly improved from the earlier papers in this series.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017MSSP...84....2V','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017MSSP...84....2V"><span><span class="hlt">Approximate</span> Bayesian Computation by Subset Simulation using hierarchical state-space models</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Vakilzadeh, Majid K.; Huang, Yong; Beck, James L.; Abrahamsson, Thomas</p> <p>2017-02-01</p> <p>A new multi-level Markov Chain Monte Carlo algorithm for <span class="hlt">Approximate</span> Bayesian Computation, ABC-SubSim, has recently appeared that exploits the Subset Simulation method for efficient rare-event simulation. ABC-SubSim adaptively creates a nested decreasing sequence of data-<span class="hlt">approximating</span> regions in the output space that correspond to increasingly closer <span class="hlt">approximations</span> of the observed output vector in this output space. At each level, multiple samples of the model parameter vector are generated by a component-wise Metropolis algorithm so that the predicted output corresponding to each parameter value falls in the current data-<span class="hlt">approximating</span> region. Theoretically, if continued to the limit, the sequence of data-<span class="hlt">approximating</span> regions would converge on to the observed output vector and the <span class="hlt">approximate</span> posterior distributions, which are conditional on the data-<span class="hlt">approximation</span> region, would become exact, but this is not practically feasible. In this paper we study the performance of the ABC-SubSim algorithm for Bayesian updating of the parameters of dynamical systems using a general hierarchical state-space model. We note that the ABC methodology gives an <span class="hlt">approximate</span> posterior distribution that actually corresponds to an exact posterior where a uniformly distributed combined measurement and modeling error is added. We also note that ABC algorithms have a problem with learning the uncertain error variances in a stochastic state-space model and so we treat them as nuisance parameters and analytically integrate them out of the posterior distribution. In addition, the statistical efficiency of the original ABC-SubSim algorithm is improved by developing a novel strategy to regulate the proposal variance for the component-wise Metropolis algorithm at each level. We demonstrate that Self-regulated ABC-SubSim is well suited for Bayesian system identification by first applying it successfully to model updating of a two degree-of-freedom linear structure for three cases: globally</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19880015862','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19880015862"><span>Galerkin <span class="hlt">approximation</span> for inverse problems for nonautonomous nonlinear distributed systems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Banks, H. T.; Reich, Simeon; Rosen, I. G.</p> <p>1988-01-01</p> <p>An abstract framework and convergence theory is developed for Galerkin <span class="hlt">approximation</span> for inverse problems involving the identification of nonautonomous nonlinear distributed parameter systems. A set of relatively easily verified conditions is provided which are sufficient to guarantee the existence of optimal solutions and their <span class="hlt">approximation</span> by a sequence of solutions to a sequence of <span class="hlt">approximating</span> finite dimensional identification problems. The approach is based on the theory of monotone operators in Banach spaces and is applicable to a reasonably broad class of nonlinear distributed systems. Operator theoretic and variational techniques are used to establish a fundamental convergence result. An example involving evolution systems with dynamics described by nonstationary quasilinear elliptic operators along with some applications are presented and discussed.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li class="active"><span>19</span></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_19 --> <div id="page_20" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li class="active"><span>20</span></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="381"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017CG....109...51M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017CG....109...51M"><span>Big geo data surface <span class="hlt">approximation</span> using radial basis functions: A comparative study</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Majdisova, Zuzana; Skala, Vaclav</p> <p>2017-12-01</p> <p><span class="hlt">Approximation</span> of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) <span class="hlt">approximation</span> is appropriate for big scattered datasets in n-dimensional space. It is a non-separable <span class="hlt">approximation</span>, as it is based on the distance between two points. This method leads to the solution of an overdetermined linear system of equations. In this paper the RBF <span class="hlt">approximation</span> methods are briefly described, a new approach to the RBF <span class="hlt">approximation</span> of big datasets is presented, and a comparison for different Compactly Supported RBFs (CS-RBFs) is made with respect to the accuracy of the computation. The proposed approach uses symmetry of a matrix, partitioning the matrix into blocks and data structures for storage of the sparse matrix. The experiments are performed for synthetic and real datasets.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19870020664','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19870020664"><span>UNAERO: A package of FORTRAN subroutines for <span class="hlt">approximating</span> unsteady aerodynamics in the time domain</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Dunn, H. J.</p> <p>1985-01-01</p> <p>This report serves as an instruction and maintenance manual for a collection of CDC CYBER FORTRAN IV subroutines for <span class="hlt">approximating</span> the unsteady aerodynamic forces in the time domain. The result is a set of constant-coefficient first-order differential equations that <span class="hlt">approximate</span> the dynamics of the vehicle. Provisions are included for adjusting the number of modes used for calculating the <span class="hlt">approximations</span> so that an accurate <span class="hlt">approximation</span> is generated. The number of data points at different values of reduced frequency can also be varied to adjust the accuracy of the <span class="hlt">approximation</span> over the reduced-frequency range. The denominator coefficients of the <span class="hlt">approximation</span> may be calculated by means of a gradient method or a least-squares <span class="hlt">approximation</span> technique. Both the <span class="hlt">approximation</span> methods use weights on the residual error. A new set of system equations, at a different dynamic pressure, can be generated without the <span class="hlt">approximations</span> being recalculated.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/1441457-spectral-risk-measures-risk-quadrangle-optimal-approximation','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/1441457-spectral-risk-measures-risk-quadrangle-optimal-approximation"><span>Spectral risk measures: the risk quadrangle and optimal <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Kouri, Drew P.</p> <p></p> <p>We develop a general risk quadrangle that gives rise to a large class of spectral risk measures. The statistic of this new risk quadrangle is the average value-at-risk at a specific confidence level. As such, this risk quadrangle generates a continuum of error measures that can be used for superquantile regression. For risk-averse optimization, we introduce an optimal <span class="hlt">approximation</span> of spectral risk measures using quadrature. Lastly, we prove the consistency of this <span class="hlt">approximation</span> and demonstrate our results through numerical examples.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1441457-spectral-risk-measures-risk-quadrangle-optimal-approximation','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1441457-spectral-risk-measures-risk-quadrangle-optimal-approximation"><span>Spectral risk measures: the risk quadrangle and optimal <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p> <p>Kouri, Drew P.</p> <p>2018-05-24</p> <p>We develop a general risk quadrangle that gives rise to a large class of spectral risk measures. The statistic of this new risk quadrangle is the average value-at-risk at a specific confidence level. As such, this risk quadrangle generates a continuum of error measures that can be used for superquantile regression. For risk-averse optimization, we introduce an optimal <span class="hlt">approximation</span> of spectral risk measures using quadrature. Lastly, we prove the consistency of this <span class="hlt">approximation</span> and demonstrate our results through numerical examples.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19810002914','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19810002914"><span>An analytical technique for <span class="hlt">approximating</span> unsteady aerodynamics in the time domain</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Dunn, H. J.</p> <p>1980-01-01</p> <p>An analytical technique is presented for <span class="hlt">approximating</span> unsteady aerodynamic forces in the time domain. The order of elements of a matrix Pade <span class="hlt">approximation</span> was postulated, and the resulting polynomial coefficients were determined through a combination of least squares estimates for the numerator coefficients and a constrained gradient search for the denominator coefficients which insures stable <span class="hlt">approximating</span> functions. The number of differential equations required to represent the aerodynamic forces to a given accuracy tends to be smaller than that employed in certain existing techniques where the denominator coefficients are chosen a priori. Results are shown for an aeroelastic, cantilevered, semispan wing which indicate a good fit to the aerodynamic forces for oscillatory motion can be achieved with a matrix Pade <span class="hlt">approximation</span> having fourth order numerator and second order denominator polynomials.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/ED107144.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/ED107144.pdf"><span>Cloze, Discourse, and <span class="hlt">Approximations</span> to English.</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Oller, John W., Jr.</p> <p></p> <p>Five orders of <span class="hlt">approximation</span> to normal English prose were constructed; 5th, 10th, 25th, 50th, and 100th plus. Five cloze tests were then constructed by inserting blanks for deleted words in 5 word segments (5th order), 10 word segments (10th), 25 word segments (25th), 50 word segments (50th), and 100 word segments of five different passages of…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018IzMat..82..186P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018IzMat..82..186P"><span>Certain <span class="hlt">approximation</span> problems for functions on the infinite-dimensional torus: Lipschitz spaces</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Platonov, S. S.</p> <p>2018-02-01</p> <p>We consider some questions about the <span class="hlt">approximation</span> of functions on the infinite-dimensional torus by trigonometric polynomials. Our main results are analogues of the direct and inverse theorems in the classical theory of <span class="hlt">approximation</span> of periodic functions and a description of the Lipschitz spaces on the infinite-dimensional torus in terms of the best <span class="hlt">approximation</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2011hiqm.book.....C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2011hiqm.book.....C"><span>Heisenberg and the Interpretation of Quantum Mechanics</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Camilleri, Kristian</p> <p>2011-09-01</p> <p>Preface; 1. Introduction; Part I. The Emergence of Quantum Mechanics: 2. Quantum mechanics and the principle of observability; 3. The problem of interpretation; Part II. The Heisenberg-<span class="hlt">Bohr</span> Dialogue: 4. The wave-particle duality; 5. Indeterminacy and the limits of classical concepts: the turning point in Heisenberg's thought; 6. Heisenberg and <span class="hlt">Bohr</span>: divergent viewpoints of complementarity; Part III. Heisenberg's Epistemology and Ontology of Quantum Mechanics: 7. The transformation of Kantian philosophy; 8. The linguistic turn in Heisenberg's thought; Conclusion; References; Index.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2009hiqm.book.....C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2009hiqm.book.....C"><span>Heisenberg and the Interpretation of Quantum Mechanics</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Camilleri, Kristian</p> <p>2009-02-01</p> <p>Preface; 1. Introduction; Part I. The Emergence of Quantum Mechanics: 2. Quantum mechanics and the principle of observability; 3. The problem of interpretation; Part II. The Heisenberg-<span class="hlt">Bohr</span> Dialogue: 4. The wave-particle duality; 5. Indeterminacy and the limits of classical concepts: the turning point in Heisenberg's thought; 6. Heisenberg and <span class="hlt">Bohr</span>: divergent viewpoints of complementarity; Part III. Heisenberg's Epistemology and Ontology of Quantum Mechanics: 7. The transformation of Kantian philosophy; 8. The linguistic turn in Heisenberg's thought; Conclusion; References; Index.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017elsp.book..207F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017elsp.book..207F"><span>Condensed Matter Physics: Does Quantum Mechanics Matter?</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Fisher, Michael E.</p> <p></p> <p>Herman Feshbach, the organizer of this Symposium in honor of Niels <span class="hlt">Bohr</span>, asked me, in his original invitation, for a review of the present state of condensed matter physics, with emphasis on major unsolved problems and comments on any overlap with <span class="hlt">Bohr</span>'s ideas regarding the fundamentals of quantum mechanics. That is surely a difficult assignment and, indeed, goes well beyond what is attempted here; nevertheless, I will take the liberty of raising one issue of a philosophical or metaphysical flavor.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19860048060&hterms=formula+one&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D40%26Ntt%3Dformula%2Bone','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19860048060&hterms=formula+one&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D40%26Ntt%3Dformula%2Bone"><span>An <span class="hlt">approximation</span> formula for a class of fault-tolerant computers</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>White, A. L.</p> <p>1986-01-01</p> <p>An <span class="hlt">approximation</span> formula is derived for the probability of failure for fault-tolerant process-control computers. These computers use redundancy and reconfiguration to achieve high reliability. Finite-state Markov models capture the dynamic behavior of component failure and system recovery, and the <span class="hlt">approximation</span> formula permits an estimation of system reliability by an easy examination of the model.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017MRE.....4h5017W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017MRE.....4h5017W"><span>Excitonic effects and related properties in semiconductor nanostructures: roles of size and dimensionality</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Wu, Shudong; Cheng, Liwen; Wang, Qiang</p> <p>2017-08-01</p> <p>The size- and dimensionality-dependence of excitonic effects and related properties in semiconductor nanostructures are theoretically studied in detail within the effective-mass <span class="hlt">approximation</span>. When nanostructure sizes become smaller than the bulk exciton <span class="hlt">Bohr</span> radius, excitonic effects are significantly enhanced with reducing size or dimensionality. This is as a result of quantum confinement in more directions leading to larger exciton binding energies and normalized exciton oscillator strengths. These excitonic effects originate from electron-hole Coulombic interactions, which strongly enhance the oscillator strength between the electron and hole. It is also established that the universal scaling of exciton binding energy versus the inverse of the exciton <span class="hlt">Bohr</span> radius follows a linear scaling law. Herein, we propose a stretched exponential law for the size scaling of optical gap, which is in good agreement with the calculated data. Due to differences in the confinement dimensionality, the radiative lifetime of low-dimensional excitons becomes shorter than that of bulk excitons. The size dependence of the exciton radiative lifetimes is in good agreement with available experimental data. This strongly enhanced electron-hole exchange interaction is expected in low-dimensional structures due to enriched excitonic effects. The main difference in nanostructures compared to the bulk can be interpreted in terms of the enhanced excitonic effects induced by exciton localization. The enhanced excitonic effects are expected to be of importance in developing stable and high-efficiency nanoscale excitonic optoelectronic devices.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19920024698','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19920024698"><span>Rational <span class="hlt">approximations</span> from power series of vector-valued meromorphic functions</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Sidi, Avram</p> <p>1992-01-01</p> <p>Let F(z) be a vector-valued function, F: C yields C(sup N), which is analytic at z = 0 and meromorphic in a neighborhood of z = 0, and let its Maclaurin series be given. In this work we developed vector-valued rational <span class="hlt">approximation</span> procedures for F(z) by applying vector extrapolation methods to the sequence of partial sums of its Maclaurin series. We analyzed some of the algebraic and analytic properties of the rational <span class="hlt">approximations</span> thus obtained, and showed that they were akin to Pade <span class="hlt">approximations</span>. In particular, we proved a Koenig type theorem concerning their poles and a de Montessus type theorem concerning their uniform convergence. We showed how optical <span class="hlt">approximations</span> to multiple poles and to Laurent expansions about these poles can be constructed. Extensions of the procedures above and the accompanying theoretical results to functions defined in arbitrary linear spaces was also considered. One of the most interesting and immediate applications of the results of this work is to the matrix eigenvalue problem. In a forthcoming paper we exploited the developments of the present work to devise bona fide generalizations of the classical power method that are especially suitable for very large and sparse matrices. These generalizations can be used to <span class="hlt">approximate</span> simultaneously several of the largest distinct eigenvalues and corresponding eigenvectors and invariant subspaces of arbitrary matrices which may or may not be diagonalizable, and are very closely related with known Krylov subspace methods.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19910027253&hterms=ito&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D80%26Ntt%3Dito','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19910027253&hterms=ito&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D80%26Ntt%3Dito"><span>Finite-dimensional compensators for infinite-dimensional systems via Galerkin-type <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Ito, Kazufumi</p> <p>1990-01-01</p> <p>In this paper existence and construction of stabilizing compensators for linear time-invariant systems defined on Hilbert spaces are discussed. An existence result is established using Galkerin-type <span class="hlt">approximations</span> in which independent basis elements are used instead of the complete set of eigenvectors. A design procedure based on <span class="hlt">approximate</span> solutions of the optimal regulator and optimal observer via Galerkin-type <span class="hlt">approximation</span> is given and the Schumacher approach is used to reduce the dimension of compensators. A detailed discussion for parabolic and hereditary differential systems is included.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20090041751','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20090041751"><span>Quickly <span class="hlt">Approximating</span> the Distance Between Two Objects</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Hammen, David</p> <p>2009-01-01</p> <p>A method of quickly <span class="hlt">approximating</span> the distance between two objects (one smaller, regarded as a point; the other larger and complexly shaped) has been devised for use in computationally simulating motions of the objects for the purpose of planning the motions to prevent collisions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/5296447-uniform-semiclassical-sudden-approximation-rotationally-inelastic-scattering','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/5296447-uniform-semiclassical-sudden-approximation-rotationally-inelastic-scattering"><span>Uniform semiclassical sudden <span class="hlt">approximation</span> for rotationally inelastic scattering</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Korsch, H.J.; Schinke, R.</p> <p>1980-08-01</p> <p>The infinite-order-sudden (IOS) <span class="hlt">approximation</span> is investigated in the semiclassical limit. A simplified IOS formula for rotationally inelastic differential cross sections is derived involving a uniform stationary phase <span class="hlt">approximation</span> for two-dimensional oscillatory integrals with two stationary points. The semiclassical analysis provides a quantitative description of the rotational rainbow structure in the differential cross section. The numerical calculation of semiclassical IOS cross sections is extremely fast compared to numerically exact IOS methods, especially if high ..delta..j transitions are involved. Rigid rotor results for He--Na/sub 2/ collisions with ..delta..j< or approx. =26 and for K--CO collisions with ..delta..j< or approx. =70 show satisfactorymore » agreement with quantal IOS calculations.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27785559','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27785559"><span>Estimation of under-reporting in epidemics using <span class="hlt">approximations</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Gamado, Kokouvi; Streftaris, George; Zachary, Stan</p> <p>2017-06-01</p> <p>Under-reporting in epidemics, when it is ignored, leads to under-estimation of the infection rate and therefore of the reproduction number. In the case of stochastic models with temporal data, a usual approach for dealing with such issues is to apply data augmentation techniques through Bayesian methodology. Departing from earlier literature approaches implemented using reversible jump Markov chain Monte Carlo (RJMCMC) techniques, we make use of <span class="hlt">approximations</span> to obtain faster estimation with simple MCMC. Comparisons among the methods developed here, and with the RJMCMC approach, are carried out and highlight that <span class="hlt">approximation</span>-based methodology offers useful alternative inference tools for large epidemics, with a good trade-off between time cost and accuracy.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012PhDT........16N','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012PhDT........16N"><span>Visualizing, <span class="hlt">Approximating</span>, and Understanding Black-Hole Binaries</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Nichols, David A.</p> <p></p> <p>Numerical-relativity simulations of black-hole binaries and advancements in gravitational-wave detectors now make it possible to learn more about the collisions of compact astrophysical bodies. To be able to infer more about the dynamical behavior of these objects requires a fuller analysis of the connection between the dynamics of pairs of black holes and their emitted gravitational waves. The chapters of this thesis describe three approaches to learn more about the relationship between the dynamics of black-hole binaries and their gravitational waves: modeling momentum flow in binaries with the Landau-Lifshitz formalism, <span class="hlt">approximating</span> binary dynamics near the time of merger with post-Newtonian and black-hole-perturbation theories, and visualizing spacetime curvature with tidal tendexes and frame-drag vortexes. In Chapters 2--4, my collaborators and I present a method to quantify the flow of momentum in black-hole binaries using the Landau-Lifshitz formalism. Chapter 2 reviews an intuitive version of the formalism in the first-post-Newtonian <span class="hlt">approximation</span> that bears a strong resemblance to Maxwell's theory of electromagnetism. Chapter 3 applies this <span class="hlt">approximation</span> to relate the simultaneous bobbing motion of rotating black holes in the superkick configuration---equal-mass black holes with their spins anti-aligned and in the orbital plane---to the flow of momentum in the spacetime, prior to the black holes' merger. Chapter 4 then uses the Landau-Lifshitz formalism to explain the dynamics of a head-on merger of spinning black holes, whose spins are anti-aligned and transverse to the infalling motion. Before they merge, the black holes move with a large, transverse, velocity, which we can explain using the post-Newtonian <span class="hlt">approximation</span>; as the holes merge and form a single black hole, we can use the Landau-Lifshitz formalism without any <span class="hlt">approximations</span> to connect the slowing of the final black hole to its absorbing momentum density during the merger. In Chapters 5</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ghr.nlm.nih.gov/condition/bohring-opitz-syndrome','NIH-MEDLINEPLUS'); return false;" href="https://ghr.nlm.nih.gov/condition/bohring-opitz-syndrome"><span>Genetics Home Reference: <span class="hlt">Bohring</span>-Opitz syndrome</span></a></p> <p><a target="_blank" href="http://medlineplus.gov/">MedlinePlus</a></p> <p></p> <p></p> <p>... Most cases of the condition result from new (de novo) mutations in the gene that occur during ... BW, Rodríguez-Santiago B, Gilissen C, Vissers LE, de Vries P, Janssen I, van Lier B, Hastings ...</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017AIPC.1880f0006I','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017AIPC.1880f0006I"><span><span class="hlt">Approximate</span> formulas for elasticity of the Tornquist functions and some their advantages</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Issin, Meyram</p> <p>2017-09-01</p> <p>In this article functions of demand for prime necessity, second necessity and luxury goods depending on the income are considered. These functions are called Tornquist functions. By means of the return model the demand for prime necessity goods and second necessity goods are <span class="hlt">approximately</span> described. Then on the basis of a method of the smallest squares <span class="hlt">approximate</span> formulas for elasticity of these Tornquist functions are received. To receive an <span class="hlt">approximate</span> formula for elasticity of function of demand for luxury goods, the linear asymptotic formula is constructed for this function. Some benefits of <span class="hlt">approximate</span> formulas for elasticity of Tornquist functions are specified.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li class="active"><span>20</span></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_20 --> <div id="page_21" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li class="active"><span>21</span></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="401"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4490033','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4490033"><span><span class="hlt">Approximating</span> a DSM-5 Diagnosis of PTSD Using DSM-IV Criteria</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Rosellini, Anthony J.; Stein, Murray B.; Colpe, Lisa J.; Heeringa, Steven G.; Petukhova, Maria V.; Sampson, Nancy A.; Schoenbaum, Michael; Ursano, Robert J.; Kessler, Ronald C.</p> <p>2015-01-01</p> <p>Background Diagnostic criteria for DSM-5 posttraumatic stress disorder (PTSD) are in many ways similar to DSM-IV criteria, raising the possibility that it might be possible to closely <span class="hlt">approximate</span> DSM-5 diagnoses using DSM-IV symptoms. If so, the resulting transformation rules could be used to pool research data based on the two criteria sets. Methods The Pre-Post Deployment Study (PPDS) of the Army Study to Assess Risk and Resilience in Servicemembers (Army STARRS) administered a blended 30-day DSM-IV and DSM-5 PTSD symptom assessment based on the civilian PTSD Checklist for DSM-IV (PCL-C) and the PTSD Checklist for DSM-5 (PCL-5). This assessment was completed by 9,193 soldiers from three US Army Brigade Combat Teams <span class="hlt">approximately</span> three months after returning from Afghanistan. PCL-C items were used to operationalize conservative and broad <span class="hlt">approximations</span> of DSM-5 PTSD diagnoses. The operating characteristics of these <span class="hlt">approximations</span> were examined compared to diagnoses based on actual DSM-5 criteria. Results The estimated 30-day prevalence of DSM-5 PTSD based on conservative (4.3%) and broad (4.7%) <span class="hlt">approximations</span> of DSM-5 criteria using DSM-IV symptom assessments were similar to estimates based on actual DSM-5 criteria (4.6%). Both <span class="hlt">approximations</span> had excellent sensitivity (92.6-95.5%), specificity (99.6-99.9%), total classification accuracy (99.4-99.6%), and area under the receiver operating characteristic curve (0.96-0.98). Conclusions DSM-IV symptoms can be used to <span class="hlt">approximate</span> DSM-5 diagnoses of PTSD among recently-deployed soldiers, making it possible to recode symptom-level data from earlier DSM-IV studies to draw inferences about DSM-5 PTSD. However, replication is needed in broader trauma-exposed samples to evaluate the external validity of this finding. PMID:25845710</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27704592','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27704592"><span>Guidelines for Use of the <span class="hlt">Approximate</span> Beta-Poisson Dose-Response Model.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Xie, Gang; Roiko, Anne; Stratton, Helen; Lemckert, Charles; Dunn, Peter K; Mengersen, Kerrie</p> <p>2017-07-01</p> <p>For dose-response analysis in quantitative microbial risk assessment (QMRA), the exact beta-Poisson model is a two-parameter mechanistic dose-response model with parameters α>0 and β>0, which involves the Kummer confluent hypergeometric function. Evaluation of a hypergeometric function is a computational challenge. Denoting PI(d) as the probability of infection at a given mean dose d, the widely used dose-response model PI(d)=1-(1+dβ)-α is an <span class="hlt">approximate</span> formula for the exact beta-Poisson model. Notwithstanding the required conditions α<β and β>1, issues related to the validity and <span class="hlt">approximation</span> accuracy of this <span class="hlt">approximate</span> formula have remained largely ignored in practice, partly because these conditions are too general to provide clear guidance. Consequently, this study proposes a probability measure Pr(0 < r < 1 | α̂, β̂) as a validity measure (r is a random variable that follows a gamma distribution; α̂ and β̂ are the maximum likelihood estimates of α and β in the <span class="hlt">approximate</span> model); and the constraint conditions β̂>(22α̂)0.50 for 0.02<α̂<2 as a rule of thumb to ensure an accurate <span class="hlt">approximation</span> (e.g., Pr(0 < r < 1 | α̂, β̂) >0.99) . This validity measure and rule of thumb were validated by application to all the completed beta-Poisson models (related to 85 data sets) from the QMRA community portal (QMRA Wiki). The results showed that the higher the probability Pr(0 < r < 1 | α̂, β̂), the better the <span class="hlt">approximation</span>. The results further showed that, among the total 85 models examined, 68 models were identified as valid <span class="hlt">approximate</span> model applications, which all had a near perfect match to the corresponding exact beta-Poisson model dose-response curve. © 2016 Society for Risk Analysis.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=20050207392&hterms=ab-initio&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3Dab-initio','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=20050207392&hterms=ab-initio&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3Dab-initio"><span>Surface Segregation Energies of BCC Binaries from Ab Initio and Quantum <span class="hlt">Approximate</span> Calculations</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Good, Brian S.</p> <p>2003-01-01</p> <p>We compare dilute-limit segregation energies for selected BCC transition metal binaries computed using ab initio and quantum <span class="hlt">approximate</span> energy method. Ab initio calculations are carried out using the CASTEP plane-wave pseudopotential computer code, while quantum <span class="hlt">approximate</span> results are computed using the Bozzolo-Ferrante-Smith (BFS) method with the most recent parameterization. Quantum <span class="hlt">approximate</span> segregation energies are computed with and without atomistic relaxation. The ab initio calculations are performed without relaxation for the most part, but predicted relaxations from quantum <span class="hlt">approximate</span> calculations are used in selected cases to compute <span class="hlt">approximate</span> relaxed ab initio segregation energies. Results are discussed within the context of segregation models driven by strain and bond-breaking effects. We compare our results with other quantum <span class="hlt">approximate</span> and ab initio theoretical work, and available experimental results.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19770021819&hterms=rational+better&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Drational%2Bbetter','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19770021819&hterms=rational+better&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Drational%2Bbetter"><span>Rational <span class="hlt">approximation</span> to e to the -x power with negative poles</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Cuthill, E.</p> <p>1977-01-01</p> <p>MACSYMA was applied to the generation of an expansion in terms of Laguerre polynomials to obtain <span class="hlt">approximations</span> to e to the -x power on 0, infinity. These <span class="hlt">approximations</span> are compared with those developed by Saff, Schonhage, and Varga.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25450112','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25450112"><span>Formal properties of the probability of fixation: identities, inequalities and <span class="hlt">approximations</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>McCandlish, David M; Epstein, Charles L; Plotkin, Joshua B</p> <p>2015-02-01</p> <p>The formula for the probability of fixation of a new mutation is widely used in theoretical population genetics and molecular evolution. Here we derive a series of identities, inequalities and <span class="hlt">approximations</span> for the exact probability of fixation of a new mutation under the Moran process (equivalent results hold for the <span class="hlt">approximate</span> probability of fixation under the Wright-Fisher process, after an appropriate change of variables). We show that the logarithm of the fixation probability has particularly simple behavior when the selection coefficient is measured as a difference of Malthusian fitnesses, and we exploit this simplicity to derive inequalities and <span class="hlt">approximations</span>. We also present a comprehensive comparison of both existing and new <span class="hlt">approximations</span> for the fixation probability, highlighting those <span class="hlt">approximations</span> that induce a reversible Markov chain when used to describe the dynamics of evolution under weak mutation. To demonstrate the power of these results, we consider the classical problem of determining the total substitution rate across an ensemble of biallelic loci and prove that, at equilibrium, a strict majority of substitutions are due to drift rather than selection. Copyright © 2014 Elsevier Inc. All rights reserved.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017ZNatA..72..673W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017ZNatA..72..673W"><span>Highly Accurate Analytical <span class="hlt">Approximate</span> Solution to a Nonlinear Pseudo-Oscillator</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Wu, Baisheng; Liu, Weijia; Lim, C. W.</p> <p>2017-07-01</p> <p>A second-order Newton method is presented to construct analytical <span class="hlt">approximate</span> solutions to a nonlinear pseudo-oscillator in which the restoring force is inversely proportional to the dependent variable. The nonlinear equation is first expressed in a specific form, and it is then solved in two steps, a predictor and a corrector step. In each step, the harmonic balance method is used in an appropriate manner to obtain a set of linear algebraic equations. With only one simple second-order Newton iteration step, a short, explicit, and highly accurate analytical <span class="hlt">approximate</span> solution can be derived. The <span class="hlt">approximate</span> solutions are valid for all amplitudes of the pseudo-oscillator. Furthermore, the method incorporates second-order Taylor expansion in a natural way, and it is of significant faster convergence rate.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016ascl.soft05016E','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016ascl.soft05016E"><span>zeldovich-PLT: Zel'dovich <span class="hlt">approximation</span> initial conditions generator</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Eisenstein, Daniel; Garrison, Lehman</p> <p>2016-05-01</p> <p>zeldovich-PLT generates Zel'dovich <span class="hlt">approximation</span> (ZA) initial conditions (i.e. first-order Lagrangian perturbation theory) for cosmological N-body simulations, optionally applying particle linear theory (PLT) corrections.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19760012070','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19760012070"><span>The Investigation of Optimal Discrete <span class="hlt">Approximations</span> for Real Time Flight Simulations</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Parrish, E. A.; Mcvey, E. S.; Cook, G.; Henderson, K. C.</p> <p>1976-01-01</p> <p>The results are presented of an investigation of discrete <span class="hlt">approximations</span> for real time flight simulation. Major topics discussed include: (1) consideration of the particular problem of <span class="hlt">approximation</span> of continuous autopilots by digital autopilots; (2) use of Bode plots and synthesis of transfer functions by asymptotic fits in a warped frequency domain; (3) an investigation of the various substitution formulas, including the effects of nonlinearities; (4) use of pade <span class="hlt">approximation</span> to the solution of the matrix exponential arising from the discrete state equations; and (5) an analytical integration of the state equation using interpolated input.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006JCoAM.189..494S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006JCoAM.189..494S"><span>Capturing planar shapes by <span class="hlt">approximating</span> their outlines</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Sarfraz, M.; Riyazuddin, M.; Baig, M. H.</p> <p>2006-05-01</p> <p>A non-deterministic evolutionary approach for <span class="hlt">approximating</span> the outlines of planar shapes has been developed. Non-uniform Rational B-splines (NURBS) have been utilized as an underlying <span class="hlt">approximation</span> curve scheme. Simulated Annealing heuristic is used as an evolutionary methodology. In addition to independent studies of the optimization of weight and knot parameters of the NURBS, a separate scheme has also been developed for the optimization of weights and knots simultaneously. The optimized NURBS models have been fitted over the contour data of the planar shapes for the ultimate and automatic output. The output results are visually pleasing with respect to the threshold provided by the user. A web-based system has also been developed for the effective and worldwide utilization. The objective of this system is to provide the facility to visualize the output to the whole world through internet by providing the freedom to the user for various desired input parameters setting in the algorithm designed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2005JCrGr.281..188S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2005JCrGr.281..188S"><span>Depth-resolved cathodoluminescence of a homoepitaxial AlN thin film</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Silveira, E.; Freitas, J. A.; Slack, G. A.; Schowalter, L. J.; Kneissl, M.; Treat, D. W.; Johnson, N. M.</p> <p>2005-07-01</p> <p>In the present work we will report on the optical properties of an AlN film homoepitaxially grown on a high-quality large bulk AlN single crystal. The latter was grown by a sublimation-recondensation technique, while the film was grown by organometallic vapor-phase epitaxy. Cathodoluminescence measurements were performed using electron beam energies between 2 and 10 keV in order to excite the sample and so to probe different sample depths, making it possible to differentiate between different features which originate in the AlN homoepitaxial film. The penetration depth has been determined through the calculation of the <span class="hlt">Bohr</span>-Bethe maximum range of excitation using the <span class="hlt">approximation</span> to the Everhart-Hoff expression for the energy loss within a solid.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JPhCS.699a2017O','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JPhCS.699a2017O"><span>Sparse <span class="hlt">approximation</span> problem: how rapid simulated annealing succeeds and fails</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Obuchi, Tomoyuki; Kabashima, Yoshiyuki</p> <p>2016-03-01</p> <p>Information processing techniques based on sparseness have been actively studied in several disciplines. Among them, a mathematical framework to <span class="hlt">approximately</span> express a given dataset by a combination of a small number of basis vectors of an overcomplete basis is termed the sparse <span class="hlt">approximation</span>. In this paper, we apply simulated annealing, a metaheuristic algorithm for general optimization problems, to sparse <span class="hlt">approximation</span> in the situation where the given data have a planted sparse representation and noise is present. The result in the noiseless case shows that our simulated annealing works well in a reasonable parameter region: the planted solution is found fairly rapidly. This is true even in the case where a common relaxation of the sparse <span class="hlt">approximation</span> problem, the G-relaxation, is ineffective. On the other hand, when the dimensionality of the data is close to the number of non-zero components, another metastable state emerges, and our algorithm fails to find the planted solution. This phenomenon is associated with a first-order phase transition. In the case of very strong noise, it is no longer meaningful to search for the planted solution. In this situation, our algorithm determines a solution with close-to-minimum distortion fairly quickly.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015JHyd..523..278B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015JHyd..523..278B"><span><span class="hlt">Approximation</span> of the exponential integral (well function) using sampling methods</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Baalousha, Husam Musa</p> <p>2015-04-01</p> <p>Exponential integral (also known as well function) is often used in hydrogeology to solve Theis and Hantush equations. Many methods have been developed to <span class="hlt">approximate</span> the exponential integral. Most of these methods are based on numerical <span class="hlt">approximations</span> and are valid for a certain range of the argument value. This paper presents a new approach to <span class="hlt">approximate</span> the exponential integral. The new approach is based on sampling methods. Three different sampling methods; Latin Hypercube Sampling (LHS), Orthogonal Array (OA), and Orthogonal Array-based Latin Hypercube (OA-LH) have been used to <span class="hlt">approximate</span> the function. Different argument values, covering a wide range, have been used. The results of sampling methods were compared with results obtained by Mathematica software, which was used as a benchmark. All three sampling methods converge to the result obtained by Mathematica, at different rates. It was found that the orthogonal array (OA) method has the fastest convergence rate compared with LHS and OA-LH. The root mean square error RMSE of OA was in the order of 1E-08. This method can be used with any argument value, and can be used to solve other integrals in hydrogeology such as the leaky aquifer integral.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/15004305','SCIGOV-STC'); return false;" href="https://www.osti.gov/servlets/purl/15004305"><span>Compressibility Corrections to Closure <span class="hlt">Approximations</span> for Turbulent Flow Simulations</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Cloutman, L D</p> <p>2003-02-01</p> <p>We summarize some modifications to the usual closure <span class="hlt">approximations</span> for statistical models of turbulence that are necessary for use with compressible fluids at all Mach numbers. We concentrate here on the gradient-flu <span class="hlt">approximation</span> for the turbulent heat flux, on the buoyancy production of turbulence kinetic energy, and on a modification of the Smagorinsky model to include buoyancy. In all cases, there are pressure gradient terms that do not appear in the incompressible models and are usually omitted in compressible-flow models. Omission of these terms allows unphysical rates of entropy change.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950018436','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950018436"><span>Exponentially accurate <span class="hlt">approximations</span> to piece-wise smooth periodic functions</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Greer, James; Banerjee, Saheb</p> <p>1995-01-01</p> <p>A family of simple, periodic basis functions with 'built-in' discontinuities are introduced, and their properties are analyzed and discussed. Some of their potential usefulness is illustrated in conjunction with the Fourier series representations of functions with discontinuities. In particular, it is demonstrated how they can be used to construct a sequence of <span class="hlt">approximations</span> which converges exponentially in the maximum norm to a piece-wise smooth function. The theory is illustrated with several examples and the results are discussed in the context of other sequences of functions which can be used to <span class="hlt">approximate</span> discontinuous functions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19880031324&hterms=equations+quadratics&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dequations%2Bquadratics','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19880031324&hterms=equations+quadratics&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dequations%2Bquadratics"><span>Legendre-tau <span class="hlt">approximation</span> for functional differential equations. II - The linear quadratic optimal control problem</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Ito, Kazufumi; Teglas, Russell</p> <p>1987-01-01</p> <p>The numerical scheme based on the Legendre-tau <span class="hlt">approximation</span> is proposed to <span class="hlt">approximate</span> the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good <span class="hlt">approximations</span> at low orders and provides an <span class="hlt">approximation</span> technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline <span class="hlt">approximations</span>) is made.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19760013825','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19760013825"><span>Multidimensional stochastic <span class="hlt">approximation</span> using locally contractive functions</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Lawton, W. M.</p> <p>1975-01-01</p> <p>A Robbins-Monro type multidimensional stochastic <span class="hlt">approximation</span> algorithm which converges in mean square and with probability one to the fixed point of a locally contractive regression function is developed. The algorithm is applied to obtain maximum likelihood estimates of the parameters for a mixture of multivariate normal distributions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.fs.usda.gov/treesearch/pubs/32874','TREESEARCH'); return false;" href="https://www.fs.usda.gov/treesearch/pubs/32874"><span>Variance <span class="hlt">approximations</span> for assessments of classification accuracy</span></a></p> <p><a target="_blank" href="http://www.fs.usda.gov/treesearch/">Treesearch</a></p> <p>R. L. Czaplewski</p> <p>1994-01-01</p> <p>Variance <span class="hlt">approximations</span> are derived for the weighted and unweighted kappa statistics, the conditional kappa statistic, and conditional probabilities. These statistics are useful to assess classification accuracy, such as accuracy of remotely sensed classifications in thematic maps when compared to a sample of reference classifications made in the field. Published...</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19810004009','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19810004009"><span>Contextual classification of multispectral image data: <span class="hlt">Approximate</span> algorithm</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Tilton, J. C. (Principal Investigator)</p> <p>1980-01-01</p> <p>An <span class="hlt">approximation</span> to a classification algorithm incorporating spatial context information in a general, statistical manner is presented which is computationally less intensive. Classifications that are nearly as accurate are produced.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.loc.gov/pictures/collection/hh/item/ca2805.photos.377499p/','SCIGOV-HHH'); return false;" href="https://www.loc.gov/pictures/collection/hh/item/ca2805.photos.377499p/"><span>Interior, building 810, view to west from <span class="hlt">approximately</span> midhangar. Area ...</span></a></p> <p><a target="_blank" href="http://www.loc.gov/pictures/collection/hh/">Library of Congress Historic Buildings Survey, Historic Engineering Record, Historic Landscapes Survey</a></p> <p></p> <p></p> <p>Interior, building 810, view to west from <span class="hlt">approximately</span> mid-hangar. Area of photo encompasses <span class="hlt">approximately</span> 1/4 of the interior space, with the KC-10 tanker aircraft and the figures beneath it giving an idea of scale, 90mm lens plus electronic flash fill lightening. - Travis Air Force Base, B-36 Hangar, Between Woodskill Avenue & Ellis, adjacent to Taxiway V & W, Fairfield, Solano County, CA</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017AGUFMMR34B..08B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017AGUFMMR34B..08B"><span>The Linear Mixing <span class="hlt">Approximation</span> for Planetary Ices</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Bethkenhagen, M.; Meyer, E. R.; Hamel, S.; Nettelmann, N.; French, M.; Scheibe, L.; Ticknor, C.; Collins, L. A.; Kress, J. D.; Fortney, J. J.; Redmer, R.</p> <p>2017-12-01</p> <p>We investigate the validity of the widely used linear mixing <span class="hlt">approximation</span> for the equations of state (EOS) of planetary ices, which are thought to dominate the interior of the ice giant planets Uranus and Neptune. For that purpose we perform density functional theory molecular dynamics simulations using the VASP code.[1] In particular, we compute 1:1 binary mixtures of water, ammonia, and methane, as well as their 2:1:4 ternary mixture at pressure-temperature conditions typical for the interior of Uranus and Neptune.[2,3] In addition, a new ab initio EOS for methane is presented. The linear mixing <span class="hlt">approximation</span> is verified for the conditions present inside Uranus ranging up to 10 Mbar based on the comprehensive EOS data set. We also calculate the diffusion coefficients for the ternary mixture along different Uranus interior profiles and compare them to the values of the pure compounds. We find that deviations of the linear mixing <span class="hlt">approximation</span> from the real mixture are generally small; for the EOS they fall within about 4% uncertainty while the diffusion coefficients deviate up to 20% . The EOS of planetary ices are applied to adiabatic models of Uranus. It turns out that a deep interior of almost pure ices is consistent with the gravity field data, in which case the planet becomes rather cold (T core ˜ 4000 K). [1] G. Kresse and J. Hafner, Physical Review B 47, 558 (1993). [2] R. Redmer, T.R. Mattsson, N. Nettelmann and M. French, Icarus 211, 798 (2011). [3] N. Nettelmann, K. Wang, J. J. Fortney, S. Hamel, S. Yellamilli, M. Bethkenhagen and R. Redmer, Icarus 275, 107 (2016).</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li class="active"><span>21</span></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_21 --> <div id="page_22" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li class="active"><span>22</span></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="421"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20120003924','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20120003924"><span>On the Accuracy of Double Scattering <span class="hlt">Approximation</span> for Atmospheric Polarization Computations</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Korkin, Sergey V.; Lyapustin, Alexei I.; Marshak, Alexander L.</p> <p>2011-01-01</p> <p>Interpretation of multi-angle spectro-polarimetric data in remote sensing of atmospheric aerosols require fast and accurate methods of solving the vector radiative transfer equation (VRTE). The single and double scattering <span class="hlt">approximations</span> could provide an analytical framework for the inversion algorithms and are relatively fast, however accuracy assessments of these <span class="hlt">approximations</span> for the aerosol atmospheres in the atmospheric window channels have been missing. This paper provides such analysis for a vertically homogeneous aerosol atmosphere with weak and strong asymmetry of scattering. In both cases, the double scattering <span class="hlt">approximation</span> gives a high accuracy result (relative error <span class="hlt">approximately</span> 0.2%) only for the low optical path - 10(sup -2) As the error rapidly grows with optical thickness, a full VRTE solution is required for the practical remote sensing analysis. It is shown that the scattering anisotropy is not important at low optical thicknesses neither for reflected nor for transmitted polarization components of radiation.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20030067936','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20030067936"><span>An <span class="hlt">Approximate</span> Dissipation Function for Large Strain Rubber Thermo-Mechanical Analyses</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Johnson, Arthur R.; Chen, Tzi-Kang</p> <p>2003-01-01</p> <p>Mechanically induced viscoelastic dissipation is difficult to compute. When the constitutive model is defined by history integrals, the formula for dissipation is a double convolution integral. Since double convolution integrals are difficult to <span class="hlt">approximate</span>, coupled thermo-mechanical analyses of highly viscous rubber-like materials cannot be made with most commercial finite element software. In this study, we present a method to <span class="hlt">approximate</span> the dissipation for history integral constitutive models that represent Maxwell-like materials without <span class="hlt">approximating</span> the double convolution integral. The method requires that the total stress can be separated into elastic and viscous components, and that the relaxation form of the constitutive law is defined with a Prony series. Numerical data is provided to demonstrate the limitations of this <span class="hlt">approximate</span> method for determining dissipation. Rubber cylinders with imbedded steel disks and with an imbedded steel ball are dynamically loaded, and the nonuniform heating within the cylinders is computed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JCAP...10..022M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JCAP...10..022M"><span><span class="hlt">Approximating</span> tunneling rates in multi-dimensional field spaces</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Masoumi, Ali; Olum, Ken D.; Wachter, Jeremy M.</p> <p>2017-10-01</p> <p>Quantum mechanics makes the otherwise stable vacua of a theory metastable through the nucleation of bubbles of the new vacuum. This in turn causes a first order phase transition. These cosmological phase transitions may have played an important role in settling our universe into its current vacuum, and they may also happen in future. The most important frameworks where vacuum decay happens contain a large number of fields. Unfortunately, calculating the tunneling rates in these models is very time-consuming. In this paper we present a simple <span class="hlt">approximation</span> for the tunneling rate by reducing it to a one-field problem which is easy to calculate. We demonstrate the validity of this <span class="hlt">approximation</span> using our recent code "Anybubble" for several classes of potentials.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22667643-approximating-tunneling-rates-multi-dimensional-field-spaces','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22667643-approximating-tunneling-rates-multi-dimensional-field-spaces"><span><span class="hlt">Approximating</span> tunneling rates in multi-dimensional field spaces</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Masoumi, Ali; Olum, Ken D.; Wachter, Jeremy M., E-mail: ali@cosmos.phy.tufts.edu, E-mail: kdo@cosmos.phy.tufts.edu, E-mail: Jeremy.Wachter@tufts.edu</p> <p></p> <p>Quantum mechanics makes the otherwise stable vacua of a theory metastable through the nucleation of bubbles of the new vacuum. This in turn causes a first order phase transition. These cosmological phase transitions may have played an important role in settling our universe into its current vacuum, and they may also happen in future. The most important frameworks where vacuum decay happens contain a large number of fields. Unfortunately, calculating the tunneling rates in these models is very time-consuming. In this paper we present a simple <span class="hlt">approximation</span> for the tunneling rate by reducing it to a one-field problem which ismore » easy to calculate. We demonstrate the validity of this <span class="hlt">approximation</span> using our recent code 'Anybubble' for several classes of potentials.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2010LNCS.6508..309C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2010LNCS.6508..309C"><span><span class="hlt">Approximating</span> Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Chen, Zhixiang; Fu, Bin</p> <p></p> <p>This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum multilinear monomial when the input is a ΠΣΠ polynomial. We first prove that the first problem is #P-hard and then devise a O *(3 n s(n)) upper bound for this problem for any polynomial represented by an arithmetic circuit of size s(n). Later, this upper bound is improved to O *(2 n ) for ΠΣΠ polynomials. We then design fully polynomial-time randomized <span class="hlt">approximation</span> schemes for this problem for ΠΣ polynomials. On the negative side, we prove that, even for ΠΣΠ polynomials with terms of degree ≤ 2, the first problem cannot be <span class="hlt">approximated</span> at all for any <span class="hlt">approximation</span> factor ≥ 1, nor "weakly <span class="hlt">approximated</span>" in a much relaxed setting, unless P=NP. For the second problem, we first give a polynomial time λ-<span class="hlt">approximation</span> algorithm for ΠΣΠ polynomials with terms of degrees no more a constant λ ≥ 2. On the inapproximability side, we give a n (1 - ɛ)/2 lower bound, for any ɛ> 0, on the <span class="hlt">approximation</span> factor for ΠΣΠ polynomials. When the degrees of the terms in these polynomials are constrained as ≤ 2, we prove a 1.0476 lower bound, assuming Pnot=NP; and a higher 1.0604 lower bound, assuming the Unique Games Conjecture.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018PhRvD..97k6001A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018PhRvD..97k6001A"><span>Dynamically assisted Schwinger effect beyond the spatially-uniform-field <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Aleksandrov, I. A.; Plunien, G.; Shabaev, V. M.</p> <p>2018-06-01</p> <p>We investigate the phenomenon of electron-positron pair production from vacuum in the presence of a strong electric field superimposed by a weak but fast varying pulse which substantially increases the total particle yield. We employ a nonperturbative numerical technique and perform the calculations beyond the spatially-uniform-field <span class="hlt">approximation</span>, i.e., dipole <span class="hlt">approximation</span>, taking into account the coordinate dependence of the fast component. The analysis of the main characteristics of the pair-production process (momentum spectra of particles and total amount of pairs) reveals a number of important features which are absent within the previously used <span class="hlt">approximation</span>. In particular, the structure of the momentum distribution is modified both qualitatively and quantitatively, and the total number of pairs created as well as the enhancement factor due to dynamical assistance become significantly smaller.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014CEJPh..12..111D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014CEJPh..12..111D"><span>A Jacobi collocation <span class="hlt">approximation</span> for nonlinear coupled viscous Burgers' equation</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Doha, Eid H.; Bhrawy, Ali H.; Abdelkawy, Mohamed A.; Hafez, Ramy M.</p> <p>2014-02-01</p> <p>This article presents a numerical <span class="hlt">approximation</span> of the initial-boundary nonlinear coupled viscous Burgers' equation based on spectral methods. A Jacobi-Gauss-Lobatto collocation (J-GL-C) scheme in combination with the implicit Runge-Kutta-Nyström (IRKN) scheme are employed to obtain highly accurate <span class="hlt">approximations</span> to the mentioned problem. This J-GL-C method, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled viscous Burgers' equation to a system of nonlinear ordinary differential equation which is far easier to solve. The given examples show, by selecting relatively few J-GL-C points, the accuracy of the <span class="hlt">approximations</span> and the utility of the approach over other analytical or numerical methods. The illustrative examples demonstrate the accuracy, efficiency, and versatility of the proposed algorithm.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3296542','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3296542"><span>Modeling boundary measurements of scattered light using the corrected diffusion <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Lehtikangas, Ossi; Tarvainen, Tanja; Kim, Arnold D.</p> <p>2012-01-01</p> <p>We study the modeling and simulation of steady-state measurements of light scattered by a turbid medium taken at the boundary. In particular, we implement the recently introduced corrected diffusion <span class="hlt">approximation</span> in two spatial dimensions to model these boundary measurements. This implementation uses expansions in plane wave solutions to compute boundary conditions and the additive boundary layer correction, and a finite element method to solve the diffusion equation. We show that this corrected diffusion <span class="hlt">approximation</span> models boundary measurements substantially better than the standard diffusion <span class="hlt">approximation</span> in comparison to numerical solutions of the radiative transport equation. PMID:22435102</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016AIPC.1776i0020G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016AIPC.1776i0020G"><span>The TSP-approach to <span class="hlt">approximate</span> solving the m-Cycles Cover Problem</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Gimadi, Edward Kh.; Rykov, Ivan; Tsidulko, Oxana</p> <p>2016-10-01</p> <p>In the m-Cycles Cover problem it is required to find a collection of m vertex-disjoint cycles that covers all vertices of the graph and the total weight of edges in the cover is minimum (or maximum). The problem is a generalization of the Traveling salesmen problem. It is strongly NP-hard. We discuss a TSP-approach that gives polynomial <span class="hlt">approximate</span> solutions for this problem. It transforms an <span class="hlt">approximation</span> TSP algorithm into an <span class="hlt">approximation</span> m-CCP algorithm. In this paper we present a number of successful transformations with proven performance guarantees for the obtained solutions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/21954203','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/21954203"><span><span class="hlt">Approximate</span> dynamic programming for optimal stationary control with control-dependent noise.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Jiang, Yu; Jiang, Zhong-Ping</p> <p>2011-12-01</p> <p>This brief studies the stochastic optimal control problem via reinforcement learning and <span class="hlt">approximate</span>/adaptive dynamic programming (ADP). A policy iteration algorithm is derived in the presence of both additive and multiplicative noise using Itô calculus. The expectation of the <span class="hlt">approximated</span> cost matrix is guaranteed to converge to the solution of some algebraic Riccati equation that gives rise to the optimal cost value. Moreover, the covariance of the <span class="hlt">approximated</span> cost matrix can be reduced by increasing the length of time interval between two consecutive iterations. Finally, a numerical example is given to illustrate the efficiency of the proposed ADP methodology.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.fs.usda.gov/treesearch/pubs/11557','TREESEARCH'); return false;" href="https://www.fs.usda.gov/treesearch/pubs/11557"><span>A first-<span class="hlt">approximation</span> urban-air-quality indicator</span></a></p> <p><a target="_blank" href="http://www.fs.usda.gov/treesearch/">Treesearch</a></p> <p>David M. Paproski; Julian R. Walker</p> <p>1977-01-01</p> <p>Development of the first-<span class="hlt">approximation</span>-urban-air-quality indicator was reported by the Economic Council of Canada. The indicator takes account of ambient concentrations of five pollutants: sulfur dioxide, particulate matter, oxides of nitrogen, carbon monoxide, and total oxidants. Epidemiological evidence indicating the potential impact of these pollutants on human...</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4989465','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4989465"><span><span class="hlt">Approximate</span> Uncertainty Modeling in Risk Analysis with Vine Copulas</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Bedford, Tim; Daneshkhah, Alireza</p> <p>2015-01-01</p> <p>Many applications of risk analysis require us to jointly model multiple uncertain quantities. Bayesian networks and copulas are two common approaches to modeling joint uncertainties with probability distributions. This article focuses on new methodologies for copulas by developing work of Cooke, Bedford, Kurowica, and others on vines as a way of constructing higher dimensional distributions that do not suffer from some of the restrictions of alternatives such as the multivariate Gaussian copula. The article provides a fundamental <span class="hlt">approximation</span> result, demonstrating that we can <span class="hlt">approximate</span> any density as closely as we like using vines. It further operationalizes this result by showing how minimum information copulas can be used to provide parametric classes of copulas that have such good levels of <span class="hlt">approximation</span>. We extend previous approaches using vines by considering nonconstant conditional dependencies, which are particularly relevant in financial risk modeling. We discuss how such models may be quantified, in terms of expert judgment or by fitting data, and illustrate the approach by modeling two financial data sets. PMID:26332240</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25152920','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25152920"><span>Analytical <span class="hlt">approximate</span> solutions for a general class of nonlinear delay differential equations.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Căruntu, Bogdan; Bota, Constantin</p> <p>2014-01-01</p> <p>We use the polynomial least squares method (PLSM), which allows us to compute analytical <span class="hlt">approximate</span> polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing <span class="hlt">approximate</span> solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with <span class="hlt">approximate</span> solutions previously computed using other methods.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhDT........87C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhDT........87C"><span>Nonlinear Schroedinger <span class="hlt">Approximations</span> for Partial Differential Equations with Quadratic and Quasilinear Terms</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Cummings, Patrick</p> <p></p> <p>We consider the <span class="hlt">approximation</span> of solutions of two complicated, physical systems via the nonlinear Schrodinger equation (NLS). In particular, we discuss the evolution of wave packets and long waves in two physical models. Due to the complicated nature of the equations governing many physical systems and the in-depth knowledge we have for solutions of the nonlinear Schrodinger equation, it is advantageous to use <span class="hlt">approximation</span> results of this kind to model these physical systems. The <span class="hlt">approximations</span> are simple enough that we can use them to understand the qualitative and quantitative behavior of the solutions, and by justifying them we can show that the behavior of the <span class="hlt">approximation</span> captures the behavior of solutions to the original equation, at least for long, but finite time. We first consider a model of the water wave equations which can be <span class="hlt">approximated</span> by wave packets using the NLS equation. We discuss a new proof that both simplifies and strengthens previous justification results of Schneider and Wayne. Rather than using analytic norms, as was done by Schneider and Wayne, we construct a modified energy functional so that the <span class="hlt">approximation</span> holds for the full interval of existence of the <span class="hlt">approximate</span> NLS solution as opposed to a subinterval (as is seen in the analytic case). Furthermore, the proof avoids problems associated with inverting the normal form transform by working with a modified energy functional motivated by Craig and Hunter et al. We then consider the Klein-Gordon-Zakharov system and prove a long wave <span class="hlt">approximation</span> result. In this case there is a non-trivial resonance that cannot be eliminated via a normal form transform. By combining the normal form transform for small Fourier modes and using analytic norms elsewhere, we can get a justification result on the order 1 over epsilon squared time scale.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018PhFl...30d0910P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018PhFl...30d0910P"><span>Padé <span class="hlt">approximant</span> for normal stress differences in large-amplitude oscillatory shear flow</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Poungthong, P.; Saengow, C.; Giacomin, A. J.; Kolitawong, C.; Merger, D.; Wilhelm, M.</p> <p>2018-04-01</p> <p>Analytical solutions for the normal stress differences in large-amplitude oscillatory shear flow (LAOS), for continuum or molecular models, normally take the inexact form of the first few terms of a series expansion in the shear rate amplitude. Here, we improve the accuracy of these truncated expansions by replacing them with rational functions called Padé <span class="hlt">approximants</span>. The recent advent of exact solutions in LAOS presents an opportunity to identify accurate and useful Padé <span class="hlt">approximants</span>. For this identification, we replace the truncated expansion for the corotational Jeffreys fluid with its Padé <span class="hlt">approximants</span> for the normal stress differences. We uncover the most accurate and useful <span class="hlt">approximant</span>, the [3,4] <span class="hlt">approximant</span>, and then test its accuracy against the exact solution [C. Saengow and A. J. Giacomin, "Normal stress differences from Oldroyd 8-constant framework: Exact analytical solution for large-amplitude oscillatory shear flow," Phys. Fluids 29, 121601 (2017)]. We use Ewoldt grids to show the stunning accuracy of our [3,4] <span class="hlt">approximant</span> in LAOS. We quantify this accuracy with an objective function and then map it onto the Pipkin space. Our two applications illustrate how to use our new <span class="hlt">approximant</span> reliably. For this, we use the Spriggs relations to generalize our best <span class="hlt">approximant</span> to multimode, and then, we compare with measurements on molten high-density polyethylene and on dissolved polyisobutylene in isobutylene oligomer.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016AIPC.1708g0012T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016AIPC.1708g0012T"><span>Weber's gravitational force as static weak field <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Tiandho, Yuant</p> <p>2016-02-01</p> <p>Weber's gravitational force (WGF) is one of gravitational model that can accommodate a non-static system because it depends not only on the distance but also on the velocity and the acceleration. Unlike Newton's law of gravitation, WGF can predict the anomalous of Mercury and gravitational bending of light near massive object very well. Then, some researchers use WGF as an alternative model of gravitation and propose a new mechanics theory namely the relational mechanics theory. However, currently we have known that the theory of general relativity which proposed by Einstein can explain gravity with very accurate. Through the static weak field <span class="hlt">approximation</span> for the non-relativistic object, we also have known that the theory of general relativity will reduce to Newton's law of gravity. In this work, we expand the static weak field <span class="hlt">approximation</span> that compatible with relativistic object and we obtain a force equation which correspond to WGF. Therefore, WGF is more precise than Newton's gravitational law. The static-weak gravitational field that we used is a solution of the Einstein's equation in the vacuum that satisfy the linear field <span class="hlt">approximation</span>. The expression of WGF with ξ = 1 and satisfy the requirement of energy conservation are obtained after resolving the geodesic equation. By this result, we can conclude that WGF can be derived from the general relativity.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19890011266','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19890011266"><span>Optimal feedback control infinite dimensional parabolic evolution systems: <span class="hlt">Approximation</span> techniques</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Banks, H. T.; Wang, C.</p> <p>1989-01-01</p> <p>A general <span class="hlt">approximation</span> framework is discussed for computation of optimal feedback controls in linear quadratic regular problems for nonautonomous parabolic distributed parameter systems. This is done in the context of a theoretical framework using general evolution systems in infinite dimensional Hilbert spaces. Conditions are discussed for preservation under <span class="hlt">approximation</span> of stabilizability and detectability hypotheses on the infinite dimensional system. The special case of periodic systems is also treated.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=combining+AND+chemicals&pg=2&id=EJ717172','ERIC'); return false;" href="https://eric.ed.gov/?q=combining+AND+chemicals&pg=2&id=EJ717172"><span>Unified <span class="hlt">Approximations</span>: A New Approach for Monoprotic Weak Acid-Base Equilibria</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Pardue, Harry; Odeh, Ihab N.; Tesfai, Teweldemedhin M.</p> <p>2004-01-01</p> <p>The unified <span class="hlt">approximations</span> reduce the conceptual complexity by combining solutions for a relatively large number of different situations into just two similar sets of processes. Processes used to solve problems by either the unified or classical <span class="hlt">approximations</span> require similar degrees of understanding of the underlying chemical processes.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018EPJWC.17508006B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018EPJWC.17508006B"><span>Real-time dynamics of matrix quantum mechanics beyond the classical <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Buividovich, Pavel; Hanada, Masanori; Schäfer, Andreas</p> <p>2018-03-01</p> <p>We describe a numerical method which allows to go beyond the classical <span class="hlt">approximation</span> for the real-time dynamics of many-body systems by <span class="hlt">approximating</span> the many-body Wigner function by the most general Gaussian function with time-dependent mean and dispersion. On a simple example of a classically chaotic system with two degrees of freedom we demonstrate that this Gaussian state <span class="hlt">approximation</span> is accurate for significantly smaller field strengths and longer times than the classical one. Applying this <span class="hlt">approximation</span> to matrix quantum mechanics, we demonstrate that the quantum Lyapunov exponents are in general smaller than their classical counterparts, and even seem to vanish below some temperature. This behavior resembles the finite-temperature phase transition which was found for this system in Monte-Carlo simulations, and ensures that the system does not violate the Maldacena-Shenker-Stanford bound λL < 2πT, which inevitably happens for classical dynamics at sufficiently small temperatures.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19880014126','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19880014126"><span>An <span class="hlt">approximation</span> theory for the identification of nonlinear distributed parameter systems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Banks, H. T.; Reich, Simeon; Rosen, I. G.</p> <p>1988-01-01</p> <p>An abstract <span class="hlt">approximation</span> framework for the identification of nonlinear distributed parameter systems is developed. Inverse problems for nonlinear systems governed by strongly maximal monotone operators (satisfying a mild continuous dependence condition with respect to the unknown parameters to be identified) are treated. Convergence of Galerkin <span class="hlt">approximations</span> and the corresponding solutions of finite dimensional <span class="hlt">approximating</span> identification problems to a solution of the original finite dimensional identification problem is demonstrated using the theory of nonlinear evolution systems and a nonlinear analog of the Trotter-Kato <span class="hlt">approximation</span> result for semigroups of bounded linear operators. The nonlinear theory developed here is shown to subsume an existing linear theory as a special case. It is also shown to be applicable to a broad class of nonlinear elliptic operators and the corresponding nonlinear parabolic partial differential equations to which they lead. An application of the theory to a quasilinear model for heat conduction or mass transfer is discussed.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li class="active"><span>22</span></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_22 --> <div id="page_23" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li class="active"><span>23</span></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li><a href="#" onclick='return showDiv("page_25");'>25</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="441"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19860035396&hterms=rational+better&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3Drational%2Bbetter','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19860035396&hterms=rational+better&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3Drational%2Bbetter"><span>Monte Carlo turbulence simulation using rational <span class="hlt">approximations</span> to von Karman spectra</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Campbell, C. W.</p> <p>1986-01-01</p> <p>Turbulence simulation is computationally much simpler using rational spectra, but turbulence falls off as f exp -5/3 in frequency ranges of interest to aircraft response and as predicted by von Karman's model. Rational <span class="hlt">approximations</span> to von Karman spectra should satisfy three requirements: (1) the rational spectra should provide a good <span class="hlt">approximation</span> to the von Karman spectra in the frequency range of interest; (2) for stability, the resulting rational transfer function should have all its poles in the left half-plane; and (3) at high frequencies, the rational spectra must fall off as an integer power of frequency, and since the -2 power is closest to the -5/3 power, the rational <span class="hlt">approximation</span> should roll off as the -2 power at high frequencies. Rational <span class="hlt">approximations</span> to von Karman spectra that satisfy these three criteria are presented, along with spectra from simulated turbulence. Agreement between the spectra of the simulated turbulence and von Karman spectra is excellent.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015JCoPh.297..530B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015JCoPh.297..530B"><span>Stable finite element <span class="hlt">approximations</span> of two-phase flow with soluble surfactant</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Barrett, John W.; Garcke, Harald; Nürnberg, Robert</p> <p>2015-09-01</p> <p>A parametric finite element <span class="hlt">approximation</span> of incompressible two-phase flow with soluble surfactants is presented. The Navier-Stokes equations are coupled to bulk and surfaces PDEs for the surfactant concentrations. At the interface adsorption, desorption and stress balances involving curvature effects and Marangoni forces have to be considered. A parametric finite element <span class="hlt">approximation</span> for the advection of the interface, which maintains good mesh properties, is coupled to the evolving surface finite element method, which is used to discretize the surface PDE for the interface surfactant concentration. The resulting system is solved together with standard finite element <span class="hlt">approximations</span> of the Navier-Stokes equations and of the bulk parabolic PDE for the surfactant concentration. Semidiscrete and fully discrete <span class="hlt">approximations</span> are analyzed with respect to stability, conservation and existence/uniqueness issues. The approach is validated for simple test cases and for complex scenarios, including colliding drops in a shear flow, which are computed in two and three space dimensions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/22697250','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/22697250"><span>Normal and compound poisson <span class="hlt">approximations</span> for pattern occurrences in NGS reads.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Zhai, Zhiyuan; Reinert, Gesine; Song, Kai; Waterman, Michael S; Luan, Yihui; Sun, Fengzhu</p> <p>2012-06-01</p> <p>Next generation sequencing (NGS) technologies are now widely used in many biological studies. In NGS, sequence reads are randomly sampled from the genome sequence of interest. Most computational approaches for NGS data first map the reads to the genome and then analyze the data based on the mapped reads. Since many organisms have unknown genome sequences and many reads cannot be uniquely mapped to the genomes even if the genome sequences are known, alternative analytical methods are needed for the study of NGS data. Here we suggest using word patterns to analyze NGS data. Word pattern counting (the study of the probabilistic distribution of the number of occurrences of word patterns in one or multiple long sequences) has played an important role in molecular sequence analysis. However, no studies are available on the distribution of the number of occurrences of word patterns in NGS reads. In this article, we build probabilistic models for the background sequence and the sampling process of the sequence reads from the genome. Based on the models, we provide normal and compound Poisson <span class="hlt">approximations</span> for the number of occurrences of word patterns from the sequence reads, with bounds on the <span class="hlt">approximation</span> error. The main challenge is to consider the randomness in generating the long background sequence, as well as in the sampling of the reads using NGS. We show the accuracy of these <span class="hlt">approximations</span> under a variety of conditions for different patterns with various characteristics. Under realistic assumptions, the compound Poisson <span class="hlt">approximation</span> seems to outperform the normal <span class="hlt">approximation</span> in most situations. These <span class="hlt">approximate</span> distributions can be used to evaluate the statistical significance of the occurrence of patterns from NGS data. The theory and the computational algorithm for calculating the <span class="hlt">approximate</span> distributions are then used to analyze ChIP-Seq data using transcription factor GABP. Software is available online (www</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1983PhyBC.115..271F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1983PhyBC.115..271F"><span>Strong shock implosion, <span class="hlt">approximate</span> solution</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Fujimoto, Y.; Mishkin, E. A.; Alejaldre, C.</p> <p>1983-01-01</p> <p>The self-similar, center-bound motion of a strong spherical, or cylindrical, shock wave moving through an ideal gas with a constant, γ= cp/ cv, is considered and a linearized, <span class="hlt">approximate</span> solution is derived. An X, Y phase plane of the self-similar solution is defined and the representative curved of the system behind the shock front is replaced by a straight line connecting the mappings of the shock front with that of its tail. The reduced pressure P(ξ), density R(ξ) and velocity U1(ξ) are found in closed, quite accurate, form. Comparison with numerically obtained results, for γ= {5}/{3} and γ= {7}/{5}, is shown.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/957785','SCIGOV-STC'); return false;" href="https://www.osti.gov/servlets/purl/957785"><span>Optimal <span class="hlt">approximation</span> of harmonic growth clusters by orthogonal polynomials</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Teodorescu, Razvan</p> <p>2008-01-01</p> <p>Interface dynamics in two-dimensional systems with a maximal number of conservation laws gives an accurate theoreticaI model for many physical processes, from the hydrodynamics of immiscible, viscous flows (zero surface-tension limit of Hele-Shaw flows), to the granular dynamics of hard spheres, and even diffusion-limited aggregation. Although a complete solution for the continuum case exists, efficient <span class="hlt">approximations</span> of the boundary evolution are very useful due to their practical applications. In this article, the <span class="hlt">approximation</span> scheme based on orthogonal polynomials with a deformed Gaussian kernel is discussed, as well as relations to potential theory.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008LNP...728..283R','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008LNP...728..283R"><span>An Integrable <span class="hlt">Approximation</span> for the Fermi Pasta Ulam Lattice</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Rink, Bob</p> <p></p> <p>This contribution presents a review of results obtained from computations of <span class="hlt">approximate</span> equations of motion for the Fermi-Pasta-Ulam lattice. These <span class="hlt">approximate</span> equations are obtained as a finite-dimensional Birkhoff normal form. It turns out that in many cases, the Birkhoff normal form is suitable for application of the KAM theorem. In particular, this proves Nishida's 1971 conjecture stating that almost all low-energetic motions of the anharmonic Fermi-Pasta-Ulam lattice with fixed endpoints are quasi-periodic. The proof is based on the formal Birkhoff normal form computations of Nishida, the KAM theorem and discrete symmetry considerations.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/21310946','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/21310946"><span><span class="hlt">Approximate</span> likelihood calculation on a phylogeny for Bayesian estimation of divergence times.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>dos Reis, Mario; Yang, Ziheng</p> <p>2011-07-01</p> <p>The molecular clock provides a powerful way to estimate species divergence times. If information on some species divergence times is available from the fossil or geological record, it can be used to calibrate a phylogeny and estimate divergence times for all nodes in the tree. The Bayesian method provides a natural framework to incorporate different sources of information concerning divergence times, such as information in the fossil and molecular data. Current models of sequence evolution are intractable in a Bayesian setting, and Markov chain Monte Carlo (MCMC) is used to generate the posterior distribution of divergence times and evolutionary rates. This method is computationally expensive, as it involves the repeated calculation of the likelihood function. Here, we explore the use of Taylor expansion to <span class="hlt">approximate</span> the likelihood during MCMC iteration. The <span class="hlt">approximation</span> is much faster than conventional likelihood calculation. However, the <span class="hlt">approximation</span> is expected to be poor when the proposed parameters are far from the likelihood peak. We explore the use of parameter transforms (square root, logarithm, and arcsine) to improve the <span class="hlt">approximation</span> to the likelihood curve. We found that the new methods, particularly the arcsine-based transform, provided very good <span class="hlt">approximations</span> under relaxed clock models and also under the global clock model when the global clock is not seriously violated. The <span class="hlt">approximation</span> is poorer for analysis under the global clock when the global clock is seriously wrong and should thus not be used. The results suggest that the <span class="hlt">approximate</span> method may be useful for Bayesian dating analysis using large data sets.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/ED583731.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/ED583731.pdf"><span>Communicating Professional Noticing through Animations as a Transformational <span class="hlt">Approximation</span> of Practice</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Amador, Julie M.; Estapa, Anna; Weston, Tracy; Kosko, Karl</p> <p>2016-01-01</p> <p>This paper explores the use of animations as an <span class="hlt">approximation</span> of practice to provide a transformational technology experience for elementary mathematics preservice teachers. Preservice teachers in mathematics methods courses at six universities (n = 126) engaged in a practice of decomposing and <span class="hlt">approximating</span> components of a fraction lesson. Data…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2003CRMec.331..575Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003CRMec.331..575Z"><span>Joseph Boussinesq et son <span class="hlt">approximation</span> : un aperçu actuel</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Zeytounian, Radyadour Kh.</p> <p>2003-08-01</p> <p>A hundred years ago, in his 1903 volume II of the monograph devoted to 'Théorie Analytique de la Chaleur', Joseph Valentin Boussinesq observes that: "The v ariations of density can be ignored except were they are multiplied by the acceleration of gravity in equation of motion for the vertical component of the velocity vector." A spectacular consequence of this Boussinesq observation (called, in 1916, by Rayleigh, the 'Boussinesq <span class="hlt">approximation</span>') is the possibility to work with a quasi-incompressible system of coupled dynamic, (Navier) and thermal (Fourier) equations where buoyancy is the main driving force. After a few words on the life of Boussinesq and on his observation, the applicability of this <span class="hlt">approximation</span> is briefly discussed for various thermal, geophysical, astrophysical and magnetohydrodynamic problems in the framework of 'Boussinesquian fluid dynamics'. An important part of our contemporary view is devoted to a logical (100 years later) justification of this Boussinesq <span class="hlt">approximation</span> for a perfect gas and an ideal liquid in the framework of an asymptotic modelling of the full fluid dynamics (Euler and Navier-Stokes-Fourier) equations with especially careful attention given to the validity of this <span class="hlt">approximation</span>. To cite this article: R.Kh. Zeytounian, C. R. Mecanique 331 (2003).</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27596808','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27596808"><span>Non-symbolic <span class="hlt">approximate</span> arithmetic training improves math performance in preschoolers.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Park, Joonkoo; Bermudez, Vanessa; Roberts, Rachel C; Brannon, Elizabeth M</p> <p>2016-12-01</p> <p>Math proficiency at early school age is an important predictor of later academic achievement. Thus, an important goal for society should be to improve math readiness in preschool-age children, especially in low-income children who typically arrive in kindergarten with less mathematical competency than their higher income peers. The majority of existing research-based math intervention programs target symbolic verbal number concepts in young children. However, very little attention has been paid to the preverbal intuitive ability to <span class="hlt">approximately</span> represent numerical quantity, which is hypothesized to be an important foundation for full-fledged mathematical thinking. Here, we tested the hypothesis that repeated engagement of non-symbolic <span class="hlt">approximate</span> addition and subtraction of large arrays of items results in improved math skills in very young children, an idea that stems from our previous studies in adults. In the current study, 3- to 5-year-olds showed selective improvements in math skills after multiple days of playing a tablet-based non-symbolic <span class="hlt">approximate</span> arithmetic game compared with children who played a memory game. These findings, collectively with our previous reports, suggest that mental manipulation of <span class="hlt">approximate</span> numerosities provides an important tool for improving math readiness, even in preschoolers who have yet to master the meaning of number words. Copyright © 2016 Elsevier Inc. All rights reserved.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28384496','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28384496"><span><span class="hlt">Approximate</span> number sense correlates with math performance in gifted adolescents.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Wang, Jinjing Jenny; Halberda, Justin; Feigenson, Lisa</p> <p>2017-05-01</p> <p>Nonhuman animals, human infants, and human adults all share an <span class="hlt">Approximate</span> Number System (ANS) that allows them to imprecisely represent number without counting. Among humans, people differ in the precision of their ANS representations, and these individual differences have been shown to correlate with symbolic mathematics performance in both children and adults. For example, children with specific math impairment (dyscalculia) have notably poor ANS precision. However, it remains unknown whether ANS precision contributes to individual differences only in populations of people with lower or average mathematical abilities, or whether this link also is present in people who excel in math. Here we tested non-symbolic numerical <span class="hlt">approximation</span> in 13- to 16-year old gifted children enrolled in a program for talented adolescents (the Center for Talented Youth). We found that in this high achieving population, ANS precision significantly correlated with performance on the symbolic math portion of two common standardized tests (SAT and ACT) that typically are administered to much older students. This relationship was robust even when controlling for age, verbal performance, and reaction times in the <span class="hlt">approximate</span> number task. These results suggest that the <span class="hlt">Approximate</span> Number System is linked to symbolic math performance even at the top levels of math performance. Copyright © 2017 Elsevier B.V. All rights reserved.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5053875','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5053875"><span>Non-symbolic <span class="hlt">approximate</span> arithmetic training improves math performance in preschoolers</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Park, Joonkoo; Bermudez, Vanessa; Roberts, Rachel C.; Brannon, Elizabeth M.</p> <p>2016-01-01</p> <p>Math proficiency in early school age is an important predictor of later academic achievement. Thus, an important goal for society should be to improve math readiness in pre-school age children, especially in low-income children who typically arrive in kindergarten with less mathematical competency than their higher-income peers. The majority of existing research-based math intervention programs target symbolic, verbal number concepts in young children. However, very little attention has been paid to the preverbal, intuitive ability to <span class="hlt">approximately</span> represent numerical quantity, which is hypothesized to be an important foundation for full-fledged mathematical thinking. Here, we test the hypothesis that repeated engagement of non-symbolic <span class="hlt">approximate</span> addition and subtraction of large array of items results in improved math skills in very young children, an idea that stems from our previous studies in adults. Three to five year-old children showed selective improvements in math skills after multiple days of playing a tablet-based non-symbolic <span class="hlt">approximate</span> arithmetic game compared to children who played a memory game. These findings, collectively with our previous reports, suggest that mental manipulation of <span class="hlt">approximate</span> numerosities provides an important tool for improving math readiness, even in preschoolers who have yet to master the meaning of number words. PMID:27596808</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1982MolPh..46..449M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1982MolPh..46..449M"><span>SCF-Xα-SW electron densities with the overlapping sphere <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>McMaster, Blair N.; Smith, Vedene H., Jr.; Salahub, Dennis R.</p> <p></p> <p>Self consistent field-Xα-scattered wave (SCF-Xα-SW) calculations have been performed for a series of eight first and second row homonuclear diatomic molecules using both the touching (TS) and 25 per cent overlapping sphere (OS) versions. The OS deformation density maps exhibit much better quantitative agreement with those from other Xα methods, which do not employ the spherical muffin-tin (MT) potential <span class="hlt">approximation</span>, than do the TS maps. The OS version thus compensates very effectively for the errors involved in the MT <span class="hlt">approximation</span> in computing electron densities. A detailed comparison between the TS- and OS-Xα-SW orbitals reveals that the reasons for this improvement are surprisingly specific. The dominant effect of the OS <span class="hlt">approximation</span> is to increase substantially the electron density near the midpoint of bonding σ orbitals, with a consequent reduction of the density behind the atoms. A similar effect occurs for the bonding π orbitals but is less pronounced. These effects are due to a change in hybridization of the orbitals, with the OS <span class="hlt">approximation</span> increasing the proportion of the subdominant partial waves and hence changing the shapes of the orbitals. It is this increased orbital polarization which so effectively compensates for the lack of (non-spherically symmetric) polarization components in the MT potential, when overlapping spheres are used.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20050215274','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20050215274"><span>Subsonic Aircraft With Regression and Neural-Network <span class="hlt">Approximators</span> Designed</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Patnaik, Surya N.; Hopkins, Dale A.</p> <p>2004-01-01</p> <p>At the NASA Glenn Research Center, NASA Langley Research Center's Flight Optimization System (FLOPS) and the design optimization testbed COMETBOARDS with regression and neural-network-analysis <span class="hlt">approximators</span> have been coupled to obtain a preliminary aircraft design methodology. For a subsonic aircraft, the optimal design, that is the airframe-engine combination, is obtained by the simulation. The aircraft is powered by two high-bypass-ratio engines with a nominal thrust of about 35,000 lbf. It is to carry 150 passengers at a cruise speed of Mach 0.8 over a range of 3000 n mi and to operate on a 6000-ft runway. The aircraft design utilized a neural network and a regression-<span class="hlt">approximations</span>-based analysis tool, along with a multioptimizer cascade algorithm that uses sequential linear programming, sequential quadratic programming, the method of feasible directions, and then sequential quadratic programming again. Optimal aircraft weight versus the number of design iterations is shown. The central processing unit (CPU) time to solution is given. It is shown that the regression-method-based analyzer exhibited a smoother convergence pattern than the FLOPS code. The optimum weight obtained by the <span class="hlt">approximation</span> technique and the FLOPS code differed by 1.3 percent. Prediction by the <span class="hlt">approximation</span> technique exhibited no error for the aircraft wing area and turbine entry temperature, whereas it was within 2 percent for most other parameters. Cascade strategy was required by FLOPS as well as the <span class="hlt">approximators</span>. The regression method had a tendency to hug the data points, whereas the neural network exhibited a propensity to follow a mean path. The performance of the neural network and regression methods was considered adequate. It was at about the same level for small, standard, and large models with redundancy ratios (defined as the number of input-output pairs to the number of unknown coefficients) of 14, 28, and 57, respectively. In an SGI octane workstation (Silicon Graphics</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/12093194','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/12093194"><span>Relationship between clinical signs and pulmonary function estimated by the single breath diagram for CO(2) (SBD-CO(2)) in horses with chronic obstructive pulmonary disease.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Herholz, C; Straub, R; Gerber, V; Wampfler, B; Lüthi, S; Imhof, A; Moens, Y; Busato, A</p> <p>2002-03-01</p> <p>The pulmonary health of 66 horses was assessed by a clinical examination and simple supplementary diagnostic methods. Single breath diagrams for CO(2) (SBD-CO(2)) and derived lung function indices were used to determine pulmonary function. The clinical signs in different groups were related to the results of the lung function indices derived from the SBD-CO(2). In horses with moderate to severe chronic obstructive pulmonary disease (COPD), a significant relationship was found between the respiratory frequency and the ratio of <span class="hlt">Bohr</span>'s dead space to the tidal volume (VD(<span class="hlt">Bohr</span>)/VT), and between the physiological dead space/tidal volume ratio (VD(phys)/VT) and the ratio of the alveolar dead space to the alveolar tidal volume (VD(alv)/VT(alv)), but no significant associations were found between the arterial oxygen tension (P(a)O(2)) and lung function indices derived from the SBD-CO(2). The occurrence of cough, the viscosity of tracheobronchial mucus and the amount of polynuclear neutrophils in tracheobronchial aspirates were significantly related to the expiratory tidal volume (VT), the total expired volume of CO(2) (VCO(2)), VD(<span class="hlt">Bohr</span>)/VT, VD(phys)/VT and VD(alv)/VT(alv). We conclude that abnormal findings in these clinical parameters indicate a measurable ventilation and perfusion (V(A)/Q) mismatch which is reflected by increases in dead space, VD(<span class="hlt">Bohr</span>)/VT and VD(phys)/VT as well as VD(alv)/VT(alv). Copyright 2002 Elsevier Science Ltd. All rights reserved.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22390649-data-approximation-using-blending-type-spline-construction','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22390649-data-approximation-using-blending-type-spline-construction"><span>Data <span class="hlt">approximation</span> using a blending type spline construction</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Dalmo, Rune; Bratlie, Jostein</p> <p>2014-11-18</p> <p>Generalized expo-rational B-splines (GERBS) is a blending type spline construction where local functions at each knot are blended together by C{sup k}-smooth basis functions. One way of <span class="hlt">approximating</span> discrete regular data using GERBS is by partitioning the data set into subsets and fit a local function to each subset. Partitioning and fitting strategies can be devised such that important or interesting data points are interpolated in order to preserve certain features. We present a method for fitting discrete data using a tensor product GERBS construction. The method is based on detection of feature points using differential geometry. Derivatives, which aremore » necessary for feature point detection and used to construct local surface patches, are <span class="hlt">approximated</span> from the discrete data using finite differences.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014JCAP...06..054F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014JCAP...06..054F"><span>Swiss-cheese models and the Dyer-Roeder <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Fleury, Pierre</p> <p>2014-06-01</p> <p>In view of interpreting the cosmological observations precisely, especially when they involve narrow light beams, it is crucial to understand how light propagates in our statistically homogeneous, clumpy, Universe. Among the various approaches to tackle this issue, Swiss-cheese models propose an inhomogeneous spacetime geometry which is an exact solution of Einstein's equation, while the Dyer-Roeder <span class="hlt">approximation</span> deals with inhomogeneity in an effective way. In this article, we demonstrate that the distance-redshift relation of a certain class of Swiss-cheese models is the same as the one predicted by the Dyer-Roeder approach, at a well-controlled level of <span class="hlt">approximation</span>. Both methods are therefore equivalent when applied to the interpretation of, e.g., supernova obervations. The proof relies on completely analytical arguments, and is illustrated by numerical results.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014JGRD..11910471L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014JGRD..11910471L"><span>Estimating ice particle scattering properties using a modified Rayleigh-Gans <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Lu, Yinghui; Clothiaux, Eugene E.; Aydin, Kültegin; Verlinde, Johannes</p> <p>2014-09-01</p> <p>A modification to the Rayleigh-Gans <span class="hlt">approximation</span> is made that includes self-interactions between different parts of an ice crystal, which both improves the accuracy of the Rayleigh-Gans <span class="hlt">approximation</span> and extends its applicability to polarization-dependent parameters. This modified Rayleigh-Gans <span class="hlt">approximation</span> is both efficient and reasonably accurate for particles with at least one dimension much smaller than the wavelength (e.g., dendrites at millimeter or longer wavelengths) or particles with sparse structures (e.g., low-density aggregates). Relative to the Generalized Multiparticle Mie method, backscattering reflectivities at horizontal transmit and receive polarization (HH) (ZHH) computed with this modified Rayleigh-Gans approach are about 3 dB more accurate than with the traditional Rayleigh-Gans <span class="hlt">approximation</span>. For realistic particle size distributions and pristine ice crystals the modified Rayleigh-Gans approach agrees with the Generalized Multiparticle Mie method to within 0.5 dB for ZHH whereas for the polarimetric radar observables differential reflectivity (ZDR) and specific differential phase (KDP) agreement is generally within 0.7 dB and 13%, respectively. Compared to the A-DDA code, the modified Rayleigh-Gans <span class="hlt">approximation</span> is several to tens of times faster if scattering properties for different incident angles and particle orientations are calculated. These accuracies and computational efficiencies are sufficient to make this modified Rayleigh-Gans approach a viable alternative to the Rayleigh-Gans <span class="hlt">approximation</span> in some applications such as millimeter to centimeter wavelength radars and to other methods that assume simpler, less accurate shapes for ice crystals. This method should not be used on materials with dielectric properties much different from ice and on compact particles much larger than the wavelength.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JChPh.148j2316R','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JChPh.148j2316R"><span>Harmonic-phase path-integral <span class="hlt">approximation</span> of thermal quantum correlation functions</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Robertson, Christopher; Habershon, Scott</p> <p>2018-03-01</p> <p>We present an <span class="hlt">approximation</span> to the thermal symmetric form of the quantum time-correlation function in the standard position path-integral representation. By transforming to a sum-and-difference position representation and then Taylor-expanding the potential energy surface of the system to second order, the resulting expression provides a harmonic weighting function that <span class="hlt">approximately</span> recovers the contribution of the phase to the time-correlation function. This method is readily implemented in a Monte Carlo sampling scheme and provides exact results for harmonic potentials (for both linear and non-linear operators) and near-quantitative results for anharmonic systems for low temperatures and times that are likely to be relevant to condensed phase experiments. This article focuses on one-dimensional examples to provide insights into convergence and sampling properties, and we also discuss how this <span class="hlt">approximation</span> method may be extended to many-dimensional systems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/5546968-new-approximate-orientation-averaging-water-molecule-interacting-thermal-neutron','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/5546968-new-approximate-orientation-averaging-water-molecule-interacting-thermal-neutron"><span>New <span class="hlt">approximate</span> orientation averaging of the water molecule interacting with the thermal neutron</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Markovic, M.I.; Minic, D.M.; Rakic, A.D.</p> <p>1992-02-01</p> <p>This paper reports that exactly describing the time of thermal neutron collisions with water molecules, orientation averaging is performed by an exact method (EOA{sub k}) and four <span class="hlt">approximate</span> methods (two well known and two less known). Expressions for the microscopic scattering kernel are developed. The two well-known <span class="hlt">approximate</span> orientation averaging methods are Krieger-Nelkin (K-N) and Koppel-Young (K-Y). The results obtained by one of the two proposed <span class="hlt">approximate</span> orientation averaging methods agree best with the corresponding results obtained by EOA{sub k}. The largest discrepancies between the EOA{sub k} results and the results of the <span class="hlt">approximate</span> methods are obtained using the well-knowmore » K-N <span class="hlt">approximate</span> orientation averaging method.« less</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li class="active"><span>23</span></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li><a href="#" onclick='return showDiv("page_25");'>25</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_23 --> <div id="page_24" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li class="active"><span>24</span></li> <li><a href="#" onclick='return showDiv("page_25");'>25</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="461"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4260907','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4260907"><span>Dynamic <span class="hlt">Approximate</span> Entropy Electroanatomic Maps Detect Rotors in a Simulated Atrial Fibrillation Model</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Ugarte, Juan P.; Orozco-Duque, Andrés; Tobón, Catalina; Kremen, Vaclav; Novak, Daniel; Saiz, Javier; Oesterlein, Tobias; Schmitt, Clauss; Luik, Armin; Bustamante, John</p> <p>2014-01-01</p> <p>There is evidence that rotors could be drivers that maintain atrial fibrillation. Complex fractionated atrial electrograms have been located in rotor tip areas. However, the concept of electrogram fractionation, defined using time intervals, is still controversial as a tool for locating target sites for ablation. We hypothesize that the fractionation phenomenon is better described using non-linear dynamic measures, such as <span class="hlt">approximate</span> entropy, and that this tool could be used for locating the rotor tip. The aim of this work has been to determine the relationship between <span class="hlt">approximate</span> entropy and fractionated electrograms, and to develop a new tool for rotor mapping based on fractionation levels. Two episodes of chronic atrial fibrillation were simulated in a 3D human atrial model, in which rotors were observed. Dynamic <span class="hlt">approximate</span> entropy maps were calculated using unipolar electrogram signals generated over the whole surface of the 3D atrial model. In addition, we optimized the <span class="hlt">approximate</span> entropy calculation using two real multi-center databases of fractionated electrogram signals, labeled in 4 levels of fractionation. We found that the values of <span class="hlt">approximate</span> entropy and the levels of fractionation are positively correlated. This allows the dynamic <span class="hlt">approximate</span> entropy maps to localize the tips from stable and meandering rotors. Furthermore, we assessed the optimized <span class="hlt">approximate</span> entropy using bipolar electrograms generated over a vicinity enclosing a rotor, achieving rotor detection. Our results suggest that high <span class="hlt">approximate</span> entropy values are able to detect a high level of fractionation and to locate rotor tips in simulated atrial fibrillation episodes. We suggest that dynamic <span class="hlt">approximate</span> entropy maps could become a tool for atrial fibrillation rotor mapping. PMID:25489858</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25489858','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25489858"><span>Dynamic <span class="hlt">approximate</span> entropy electroanatomic maps detect rotors in a simulated atrial fibrillation model.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Ugarte, Juan P; Orozco-Duque, Andrés; Tobón, Catalina; Kremen, Vaclav; Novak, Daniel; Saiz, Javier; Oesterlein, Tobias; Schmitt, Clauss; Luik, Armin; Bustamante, John</p> <p>2014-01-01</p> <p>There is evidence that rotors could be drivers that maintain atrial fibrillation. Complex fractionated atrial electrograms have been located in rotor tip areas. However, the concept of electrogram fractionation, defined using time intervals, is still controversial as a tool for locating target sites for ablation. We hypothesize that the fractionation phenomenon is better described using non-linear dynamic measures, such as <span class="hlt">approximate</span> entropy, and that this tool could be used for locating the rotor tip. The aim of this work has been to determine the relationship between <span class="hlt">approximate</span> entropy and fractionated electrograms, and to develop a new tool for rotor mapping based on fractionation levels. Two episodes of chronic atrial fibrillation were simulated in a 3D human atrial model, in which rotors were observed. Dynamic <span class="hlt">approximate</span> entropy maps were calculated using unipolar electrogram signals generated over the whole surface of the 3D atrial model. In addition, we optimized the <span class="hlt">approximate</span> entropy calculation using two real multi-center databases of fractionated electrogram signals, labeled in 4 levels of fractionation. We found that the values of <span class="hlt">approximate</span> entropy and the levels of fractionation are positively correlated. This allows the dynamic <span class="hlt">approximate</span> entropy maps to localize the tips from stable and meandering rotors. Furthermore, we assessed the optimized <span class="hlt">approximate</span> entropy using bipolar electrograms generated over a vicinity enclosing a rotor, achieving rotor detection. Our results suggest that high <span class="hlt">approximate</span> entropy values are able to detect a high level of fractionation and to locate rotor tips in simulated atrial fibrillation episodes. We suggest that dynamic <span class="hlt">approximate</span> entropy maps could become a tool for atrial fibrillation rotor mapping.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3835605','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3835605"><span><span class="hlt">Approximate</span> Single-Diode Photovoltaic Model for Efficient I-V Characteristics Estimation</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Ting, T. O.; Zhang, Nan; Guan, Sheng-Uei; Wong, Prudence W. H.</p> <p>2013-01-01</p> <p>Precise photovoltaic (PV) behavior models are normally described by nonlinear analytical equations. To solve such equations, it is necessary to use iterative procedures. Aiming to make the computation easier, this paper proposes an <span class="hlt">approximate</span> single-diode PV model that enables high-speed predictions for the electrical characteristics of commercial PV modules. Based on the experimental data, statistical analysis is conducted to validate the <span class="hlt">approximate</span> model. Simulation results show that the calculated current-voltage (I-V) characteristics fit the measured data with high accuracy. Furthermore, compared with the existing modeling methods, the proposed model reduces the simulation time by <span class="hlt">approximately</span> 30% in this work. PMID:24298205</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=20000032346&hterms=Clustering&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3DClustering','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=20000032346&hterms=Clustering&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3DClustering"><span><span class="hlt">Approximation</span> Of Multi-Valued Inverse Functions Using Clustering And Sugeno Fuzzy Inference</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Walden, Maria A.; Bikdash, Marwan; Homaifar, Abdollah</p> <p>1998-01-01</p> <p>Finding the inverse of a continuous function can be challenging and computationally expensive when the inverse function is multi-valued. Difficulties may be compounded when the function itself is difficult to evaluate. We show that we can use fuzzy-logic <span class="hlt">approximators</span> such as Sugeno inference systems to compute the inverse on-line. To do so, a fuzzy clustering algorithm can be used in conjunction with a discriminating function to split the function data into branches for the different values of the forward function. These data sets are then fed into a recursive least-squares learning algorithm that finds the proper coefficients of the Sugeno <span class="hlt">approximators</span>; each Sugeno <span class="hlt">approximator</span> finds one value of the inverse function. Discussions about the accuracy of the <span class="hlt">approximation</span> will be included.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/21502460-analytic-derivation-approximate-su-symmetry-inside-symmetry-triangle-interacting-boson-approximation-model','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/21502460-analytic-derivation-approximate-su-symmetry-inside-symmetry-triangle-interacting-boson-approximation-model"><span>Analytic derivation of an <span class="hlt">approximate</span> SU(3) symmetry inside the symmetry triangle of the interacting boson <span class="hlt">approximation</span> model</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Bonatsos, Dennis; Karampagia, S.; Casten, R. F.</p> <p>2011-05-15</p> <p>Using a contraction of the SU(3) algebra to the algebra of the rigid rotator in the large-boson-number limit of the interacting boson <span class="hlt">approximation</span> (IBA) model, a line is found inside the symmetry triangle of the IBA, along which the SU(3) symmetry is preserved. The line extends from the SU(3) vertex to near the critical line of the first-order shape/phase transition separating the spherical and prolate deformed phases, and it lies within the Alhassid-Whelan arc of regularity, the unique valley of regularity connecting the SU(3) and U(5) vertices in the midst of chaotic regions. In addition to providing an explanation formore » the existence of the arc of regularity, the present line represents an example of an analytically determined <span class="hlt">approximate</span> symmetry in the interior of the symmetry triangle of the IBA. The method is applicable to algebraic models possessing subalgebras amenable to contraction. This condition is equivalent to algebras in which the equilibrium ground state and its rotational band become energetically isolated from intrinsic excitations, as typified by deformed solutions to the IBA for large numbers of valence nucleons.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20080040183','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20080040183"><span><span class="hlt">Approximation</span> of Failure Probability Using Conditional Sampling</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Giesy. Daniel P.; Crespo, Luis G.; Kenney, Sean P.</p> <p>2008-01-01</p> <p>In analyzing systems which depend on uncertain parameters, one technique is to partition the uncertain parameter domain into a failure set and its complement, and judge the quality of the system by estimating the probability of failure. If this is done by a sampling technique such as Monte Carlo and the probability of failure is small, accurate <span class="hlt">approximation</span> can require so many sample points that the computational expense is prohibitive. Previous work of the authors has shown how to bound the failure event by sets of such simple geometry that their probabilities can be calculated analytically. In this paper, it is shown how to make use of these failure bounding sets and conditional sampling within them to substantially reduce the computational burden of <span class="hlt">approximating</span> failure probability. It is also shown how the use of these sampling techniques improves the confidence intervals for the failure probability estimate for a given number of sample points and how they reduce the number of sample point analyses needed to achieve a given level of confidence.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018PhRvA..97c2121T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018PhRvA..97c2121T"><span>Accuracy of the adiabatic-impulse <span class="hlt">approximation</span> for closed and open quantum systems</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Tomka, Michael; Campos Venuti, Lorenzo; Zanardi, Paolo</p> <p>2018-03-01</p> <p>We study the adiabatic-impulse <span class="hlt">approximation</span> (AIA) as a tool to <span class="hlt">approximate</span> the time evolution of quantum states when driven through a region of small gap. Such small-gap regions are a common situation in adiabatic quantum computing and having reliable <span class="hlt">approximations</span> is important in this context. The AIA originates from the Kibble-Zurek theory applied to continuous quantum phase transitions. The Kibble-Zurek mechanism was developed to predict the power-law scaling of the defect density across a continuous quantum phase transition. Instead, here we quantify the accuracy of the AIA via the trace norm distance with respect to the exact evolved state. As expected, we find that for short times or fast protocols, the AIA outperforms the simple adiabatic <span class="hlt">approximation</span>. However, for large times or slow protocols, the situation is actually reversed and the AIA provides a worse <span class="hlt">approximation</span>. Nevertheless, we found a variation of the AIA that can perform better than the adiabatic one. This counterintuitive modification consists in crossing the region of small gap twice. Our findings are illustrated by several examples of driven closed and open quantum systems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19910050732&hterms=family+beliefs&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dfamily%2Bbeliefs','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19910050732&hterms=family+beliefs&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dfamily%2Bbeliefs"><span>The CFL condition for spectral <span class="hlt">approximations</span> to hyperbolic initial-boundary value problems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Gottlieb, David; Tadmor, Eitan</p> <p>1991-01-01</p> <p>The stability of spectral <span class="hlt">approximations</span> to scalar hyperbolic initial-boundary value problems with variable coefficients are studied. Time is discretized by explicit multi-level or Runge-Kutta methods of order less than or equal to 3 (forward Euler time differencing is included), and spatial discretizations are studied by spectral and pseudospectral <span class="hlt">approximations</span> associated with the general family of Jacobi polynomials. It is proved that these fully explicit spectral <span class="hlt">approximations</span> are stable provided their time-step, delta t, is restricted by the CFL-like condition, delta t less than Const. N(exp-2), where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of <span class="hlt">approximate</span> L(exp 2)-weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. The result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral <span class="hlt">approximations</span> in the nonperiodic case.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19900015517','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19900015517"><span>The CFL condition for spectral <span class="hlt">approximations</span> to hyperbolic initial-boundary value problems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Gottlieb, David; Tadmor, Eitan</p> <p>1990-01-01</p> <p>The stability of spectral <span class="hlt">approximations</span> to scalar hyperbolic initial-boundary value problems with variable coefficients are studied. Time is discretized by explicit multi-level or Runge-Kutta methods of order less than or equal to 3 (forward Euler time differencing is included), and spatial discretizations are studied by spectral and pseudospectral <span class="hlt">approximations</span> associated with the general family of Jacobi polynomials. It is proved that these fully explicit spectral <span class="hlt">approximations</span> are stable provided their time-step, delta t, is restricted by the CFL-like condition, delta t less than Const. N(exp-2), where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of <span class="hlt">approximate</span> L(exp 2)-weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. The result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral <span class="hlt">approximations</span> in the nonperiodic case.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/16446022','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/16446022"><span>The 'hard problem' and the quantum physicists. Part 1: the first generation.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Smith, C U M</p> <p>2006-07-01</p> <p>All four of the most important figures in the early twentieth-century development of quantum physics-Niels <span class="hlt">Bohr</span>, Erwin Schroedinger, Werner Heisenberg and Wolfgang Pauli-had strong interests in the traditional mind-brain, or 'hard,' problem. This paper reviews their approach to this problem, showing the influence of <span class="hlt">Bohr</span>'s complementarity thesis, the significance of Schroedinger's small book, 'What is life?,' the updated Platonism of Heisenberg and, perhaps most interesting of all, the interaction of Carl Jung and Wolfgang Pauli in the latter's search for a unification of mind and matter.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017EGUGA..19.9666V','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017EGUGA..19.9666V"><span>Integrating flood modelling in a hydrological catchment model: flow <span class="hlt">approximations</span> and spatial resolution.</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>van den Bout, Bastian; Jetten, Victor</p> <p>2017-04-01</p> <p>Within hydrological models, flow <span class="hlt">approximations</span> are commonly used to reduce computation time. The validity of these <span class="hlt">approximations</span> is strongly determined by flow height, flow velocity, the spatial resolution of the model, and by the manner in which flow routing is implemented. The assumptions of these <span class="hlt">approximations</span> can furthermore limit emergent behavior, and influence flow behavior under space-time scaling. In this presentation, the validity and performance of the kinematic, diffusive and dynamic flow <span class="hlt">approximations</span> are investigated for use in a catchment-based flood model. Particularly, the validity during flood events and for varying spatial resolutions is investigated. The OpenLISEM hydrological model is extended to implement these flow <span class="hlt">approximations</span> and channel flooding based on dynamic flow. The kinematic routing uses a predefined converging flow network, the diffusive and dynamic routing uses a 2D flow solution over a DEM. The channel flow in all cases is a 1D kinematic wave <span class="hlt">approximation</span>. The flow <span class="hlt">approximations</span> are used to recreate measured discharge in three catchments of different size in China, Spain and Italy, among which is the hydrograph of the 2003 flood event in the Fella river basin (Italy). Furthermore, spatial resolutions are varied for the flood simulation in order to investigate the influence of spatial resolution on these flow <span class="hlt">approximations</span>. Results show that the kinematic, diffusive and dynamic flow <span class="hlt">approximation</span> provide least to highest accuracy, respectively, in recreating measured temporal variation of the discharge. Kinematic flow, which is commonly used in hydrological modelling, substantially over-estimates hydrological connectivity in the simulations with a spatial resolution of below 30 meters. Since spatial resolutions of models have strongly increased over the past decades, usage of routed kinematic flow should be reconsidered. In the case of flood events, spatial modelling of kinematic flow substantially over</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22080326-symmetric-rotating-wave-approximation-generalized-single-mode-spin-boson-system','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22080326-symmetric-rotating-wave-approximation-generalized-single-mode-spin-boson-system"><span>Symmetric rotating-wave <span class="hlt">approximation</span> for the generalized single-mode spin-boson system</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Albert, Victor V.; Scholes, Gregory D.; Brumer, Paul</p> <p>2011-10-15</p> <p>The single-mode spin-boson model exhibits behavior not included in the rotating-wave <span class="hlt">approximation</span> (RWA) in the ultra and deep-strong coupling regimes, where counter-rotating contributions become important. We introduce a symmetric rotating-wave <span class="hlt">approximation</span> that treats rotating and counter-rotating terms equally, preserves the invariances of the Hamiltonian with respect to its parameters, and reproduces several qualitative features of the spin-boson spectrum not present in the original rotating-wave <span class="hlt">approximation</span> both off-resonance and at deep-strong coupling. The symmetric rotating-wave <span class="hlt">approximation</span> allows for the treatment of certain ultra- and deep-strong coupling regimes with similar accuracy and mathematical simplicity as does the RWA in the weak-coupling regime.more » Additionally, we symmetrize the generalized form of the rotating-wave <span class="hlt">approximation</span> to obtain the same qualitative correspondence with the addition of improved quantitative agreement with the exact numerical results. The method is readily extended to higher accuracy if needed. Finally, we introduce the two-photon parity operator for the two-photon Rabi Hamiltonian and obtain its generalized symmetric rotating-wave <span class="hlt">approximation</span>. The existence of this operator reveals a parity symmetry similar to that in the Rabi Hamiltonian as well as another symmetry that is unique to the two-photon case, providing insight into the mathematical structure of the two-photon spectrum, significantly simplifying the numerics, and revealing some interesting dynamical properties.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018InvPr..34e5002T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018InvPr..34e5002T"><span>Sparsest representations and <span class="hlt">approximations</span> of an underdetermined linear system</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Tardivel, Patrick J. C.; Servien, Rémi; Concordet, Didier</p> <p>2018-05-01</p> <p>In an underdetermined linear system of equations, constrained l 1 minimization methods such as the basis pursuit or the lasso are often used to recover one of the sparsest representations or <span class="hlt">approximations</span> of the system. The null space property is a sufficient and ‘almost’ necessary condition to recover a sparsest representation with the basis pursuit. Unfortunately, this property cannot be easily checked. On the other hand, the mutual coherence is an easily checkable sufficient condition insuring the basis pursuit to recover one of the sparsest representations. Because the mutual coherence condition is too strong, it is hardly met in practice. Even if one of these conditions holds, to our knowledge, there is no theoretical result insuring that the lasso solution is one of the sparsest <span class="hlt">approximations</span>. In this article, we study a novel constrained problem that gives, without any condition, one of the sparsest representations or <span class="hlt">approximations</span>. To solve this problem, we provide a numerical method and we prove its convergence. Numerical experiments show that this approach gives better results than both the basis pursuit problem and the reweighted l 1 minimization problem.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28538985','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28538985"><span>Numerical <span class="hlt">Approximation</span> of Elasticity Tensor Associated With Green-Naghdi Rate.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Liu, Haofei; Sun, Wei</p> <p>2017-08-01</p> <p>Objective stress rates are often used in commercial finite element (FE) programs. However, deriving a consistent tangent modulus tensor (also known as elasticity tensor or material Jacobian) associated with the objective stress rates is challenging when complex material models are utilized. In this paper, an <span class="hlt">approximation</span> method for the tangent modulus tensor associated with the Green-Naghdi rate of the Kirchhoff stress is employed to simplify the evaluation process. The effectiveness of the approach is demonstrated through the implementation of two user-defined fiber-reinforced hyperelastic material models. Comparisons between the <span class="hlt">approximation</span> method and the closed-form analytical method demonstrate that the former can simplify the material Jacobian evaluation with satisfactory accuracy while retaining its computational efficiency. Moreover, since the <span class="hlt">approximation</span> method is independent of material models, it can facilitate the implementation of complex material models in FE analysis using shell/membrane elements in abaqus.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3533448','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3533448"><span><span class="hlt">APPROXIMATING</span> SYMMETRIC POSITIVE SEMIDEFINITE TENSORS OF EVEN ORDER*</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>BARMPOUTIS, ANGELOS; JEFFREY, HO; VEMURI, BABA C.</p> <p>2012-01-01</p> <p>Tensors of various orders can be used for modeling physical quantities such as strain and diffusion as well as curvature and other quantities of geometric origin. Depending on the physical properties of the modeled quantity, the estimated tensors are often required to satisfy the positivity constraint, which can be satisfied only with tensors of even order. Although the space P02m of 2mth-order symmetric positive semi-definite tensors is known to be a convex cone, enforcing positivity constraint directly on P02m is usually not straightforward computationally because there is no known analytic description of P02m for m > 1. In this paper, we propose a novel approach for enforcing the positivity constraint on even-order tensors by <span class="hlt">approximating</span> the cone P02m for the cases 0 < m < 3, and presenting an explicit characterization of the <span class="hlt">approximation</span> Σ2m ⊂ Ω2m for m ≥ 1, using the subset Ω2m⊂P02m of semi-definite tensors that can be written as a sum of squares of tensors of order m. Furthermore, we show that this <span class="hlt">approximation</span> leads to a non-negative linear least-squares (NNLS) optimization problem with the complexity that equals the number of generators in Σ2m. Finally, we experimentally validate the proposed approach and we present an application for computing 2mth-order diffusion tensors from Diffusion Weighted Magnetic Resonance Images. PMID:23285313</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=visual+AND+acuity+AND+old+AND+adult&id=EJ1008108','ERIC'); return false;" href="https://eric.ed.gov/?q=visual+AND+acuity+AND+old+AND+adult&id=EJ1008108"><span>Developmental Change in the Acuity of <span class="hlt">Approximate</span> Number and Area Representations</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Odic, Darko; Libertus, Melissa E.; Feigenson, Lisa; Halberda, Justin</p> <p>2013-01-01</p> <p>From very early in life, humans can <span class="hlt">approximate</span> the number and surface area of objects in a scene. The ability to discriminate between 2 <span class="hlt">approximate</span> quantities, whether number or area, critically depends on the ratio between the quantities, with the most difficult ratio that a participant can reliably discriminate known as the Weber fraction.…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016OPhy...14...70C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016OPhy...14...70C"><span>Quasi-planar elemental clusters in pair interactions <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Chkhartishvili, Levan</p> <p>2016-01-01</p> <p>The pair-interactions <span class="hlt">approximation</span>, when applied to describe elemental clusters, only takes into account bonding between neighboring atoms. According to this approach, isomers of wrapped forms of 2D clusters - nanotubular and fullerene-like structures - and truly 3D clusters, are generally expected to be more stable than their quasi-planar counterparts. This is because quasi-planar clusters contain more peripheral atoms with dangling bonds and, correspondingly, fewer atoms with saturated bonds. However, the differences in coordination numbers between central and peripheral atoms lead to the polarization of bonds. The related corrections to the molar binding energy can make small, quasi-planar clusters more stable than their 2D wrapped allotropes and 3D isomers. The present work provides a general theoretical frame for studying the relative stability of small elemental clusters within the pair interactions <span class="hlt">approximation</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2860321','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2860321"><span>Speech Enhancement, Gain, and Noise Spectrum Adaptation Using <span class="hlt">Approximate</span> Bayesian Estimation</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Hao, Jiucang; Attias, Hagai; Nagarajan, Srikantan; Lee, Te-Won; Sejnowski, Terrence J.</p> <p>2010-01-01</p> <p>This paper presents a new <span class="hlt">approximate</span> Bayesian estimator for enhancing a noisy speech signal. The speech model is assumed to be a Gaussian mixture model (GMM) in the log-spectral domain. This is in contrast to most current models in frequency domain. Exact signal estimation is a computationally intractable problem. We derive three <span class="hlt">approximations</span> to enhance the efficiency of signal estimation. The Gaussian <span class="hlt">approximation</span> transforms the log-spectral domain GMM into the frequency domain using minimal Kullback–Leiber (KL)-divergency criterion. The frequency domain Laplace method computes the maximum a posteriori (MAP) estimator for the spectral amplitude. Correspondingly, the log-spectral domain Laplace method computes the MAP estimator for the log-spectral amplitude. Further, the gain and noise spectrum adaptation are implemented using the expectation–maximization (EM) algorithm within the GMM under Gaussian <span class="hlt">approximation</span>. The proposed algorithms are evaluated by applying them to enhance the speeches corrupted by the speech-shaped noise (SSN). The experimental results demonstrate that the proposed algorithms offer improved signal-to-noise ratio, lower word recognition error rate, and less spectral distortion. PMID:20428253</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24288668','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24288668"><span>Selecting summary statistics in <span class="hlt">approximate</span> Bayesian computation for calibrating stochastic models.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Burr, Tom; Skurikhin, Alexei</p> <p>2013-01-01</p> <p><span class="hlt">Approximate</span> Bayesian computation (ABC) is an approach for using measurement data to calibrate stochastic computer models, which are common in biology applications. ABC is becoming the "go-to" option when the data and/or parameter dimension is large because it relies on user-chosen summary statistics rather than the full data and is therefore computationally feasible. One technical challenge with ABC is that the quality of the <span class="hlt">approximation</span> to the posterior distribution of model parameters depends on the user-chosen summary statistics. In this paper, the user requirement to choose effective summary statistics in order to accurately estimate the posterior distribution of model parameters is investigated and illustrated by example, using a model and corresponding real data of mitochondrial DNA population dynamics. We show that for some choices of summary statistics, the posterior distribution of model parameters is closely <span class="hlt">approximated</span> and for other choices of summary statistics, the posterior distribution is not closely <span class="hlt">approximated</span>. A strategy to choose effective summary statistics is suggested in cases where the stochastic computer model can be run at many trial parameter settings, as in the example.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1993JIMSS...4...98B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1993JIMSS...4...98B"><span><span class="hlt">Approximation</span> methods for control of structural acoustics models with piezoceramic actuators</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Banks, H. T.; Fang, W.; Silcox, R. J.; Smith, R. C.</p> <p>1993-01-01</p> <p>The active control of acoustic pressure in a 2-D cavity with a flexible boundary (a beam) is considered. Specifically, this control is implemented via piezoceramic patches on the beam which produces pure bending moments. The incorporation of the feedback control in this manner leads to a system with an unbounded input term. <span class="hlt">Approximation</span> methods in this manner leads to a system with an unbounded input term. <span class="hlt">Approximation</span> methods in this manner leads to a system with an unbounded input team. <span class="hlt">Approximation</span> methods in the context of linear quadratic regulator (LQR) state space control formulation are discussed and numerical results demonstrating the effectiveness of this approach in computing feedback controls for noise reduction are presented.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li class="active"><span>24</span></li> <li><a href="#" onclick='return showDiv("page_25");'>25</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_24 --> <div id="page_25" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li class="active"><span>25</span></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="481"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006PhDT........54N','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006PhDT........54N"><span>Binary collision <span class="hlt">approximations</span> for the memory function for density fluctuations in equilibrium atomic liquids</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Noah, Joyce E.</p> <p></p> <p>Time correlation functions of density fluctuations of liquids at equilibrium can be used to relate the microscopic dynamics of a liquid to its macroscopic transport properties. Time correlation functions are especially useful since they can be generated in a variety of ways, from scattering experiments to computer simulation to analytic theory. The kinetic theory of fluctuations in equilibrium liquids is an analytic theory for calculating correlation functions using memory functions. In this work, we use a diagrammatic formulation of the kinetic theory to develop a series of binary collision <span class="hlt">approximations</span> for the collisional part of the memory function. We define binary collisions as collisions between two distinct density fluctuations whose identities are fixed during the duration of a collsion. R <span class="hlt">approximations</span> are for the short time part of the memory function, and build upon the work of Ranganathan and Andersen. These <span class="hlt">approximations</span> have purely repulsive interactions between the fluctuations. The second type of <span class="hlt">approximation</span>, RA <span class="hlt">approximations</span>, is for the longer time part of the memory function, where the density fluctuations now interact via repulsive and attractive forces. Although RA <span class="hlt">approximations</span> are a natural extension of R <span class="hlt">approximations</span>, they permit two density fluctuations to become trapped in the wells of the interaction potential, leading to long-lived oscillatory behavior, which is unphysical. Therefore we consider S <span class="hlt">approximations</span> which describe binary particles which experience the random effect of the surroundings while interacting via repulsive or repulsive and attractive interactions. For each of these <span class="hlt">approximations</span> for the memory function we numerically solve the kinetic equation to generate correlation functions. These results are compared to molecular dynamics results for the correlation functions. Comparing the successes and failures of the different <span class="hlt">approximations</span>, we conclude that R <span class="hlt">approximations</span> give more accurate intermediate and</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/21038975','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/21038975"><span>Rational <span class="hlt">approximations</span> to rational models: alternative algorithms for category learning.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Sanborn, Adam N; Griffiths, Thomas L; Navarro, Daniel J</p> <p>2010-10-01</p> <p>Rational models of cognition typically consider the abstract computational problems posed by the environment, assuming that people are capable of optimally solving those problems. This differs from more traditional formal models of cognition, which focus on the psychological processes responsible for behavior. A basic challenge for rational models is thus explaining how optimal solutions can be <span class="hlt">approximated</span> by psychological processes. We outline a general strategy for answering this question, namely to explore the psychological plausibility of <span class="hlt">approximation</span> algorithms developed in computer science and statistics. In particular, we argue that Monte Carlo methods provide a source of rational process models that connect optimal solutions to psychological processes. We support this argument through a detailed example, applying this approach to Anderson's (1990, 1991) rational model of categorization (RMC), which involves a particularly challenging computational problem. Drawing on a connection between the RMC and ideas from nonparametric Bayesian statistics, we propose 2 alternative algorithms for <span class="hlt">approximate</span> inference in this model. The algorithms we consider include Gibbs sampling, a procedure appropriate when all stimuli are presented simultaneously, and particle filters, which sequentially <span class="hlt">approximate</span> the posterior distribution with a small number of samples that are updated as new data become available. Applying these algorithms to several existing datasets shows that a particle filter with a single particle provides a good description of human inferences.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2004OptCo.238..351G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2004OptCo.238..351G"><span>General properties and analytical <span class="hlt">approximations</span> of photorefractive solitons</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Geisler, A.; Homann, F.; Schmidt, H.-J.</p> <p>2004-08-01</p> <p>We investigate general properties of spatial 1-dimensional bright photorefractive solitons and discuss various analytical <span class="hlt">approximations</span> for the soliton profile and the half width, both depending on an intensity parameter r. The case of dark solitons is also briefly addressed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19760027049&hterms=Nonlinear+equations&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D90%26Ntt%3DNonlinear%2Bequations','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19760027049&hterms=Nonlinear+equations&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D90%26Ntt%3DNonlinear%2Bequations"><span>An <span class="hlt">approximation</span> technique for predicting the transient response of a second order nonlinear equation</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Laurenson, R. M.; Baumgarten, J. R.</p> <p>1975-01-01</p> <p>An <span class="hlt">approximation</span> technique has been developed for determining the transient response of a nonlinear dynamic system. The nonlinearities in the system which has been considered appear in the system's dissipation function. This function was expressed as a second order polynomial in the system's velocity. The developed <span class="hlt">approximation</span> is an extension of the classic Kryloff-Bogoliuboff technique. Two examples of the developed <span class="hlt">approximation</span> are presented for comparative purposes with other <span class="hlt">approximation</span> methods.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/22121507','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/22121507"><span>Configurational entropy: an improvement of the quasiharmonic <span class="hlt">approximation</span> using configurational temperature.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Nguyen, Phuong H; Derreumaux, Philippe</p> <p>2012-01-14</p> <p>One challenge in computational biophysics and biology is to develop methodologies able to estimate accurately the configurational entropy of macromolecules. Among many methods, the quasiharmonic <span class="hlt">approximation</span> (QH) is most widely used as it is simple in both theory and implementation. However, it has been shown that this method becomes inaccurate by overestimating entropy for systems with rugged free energy landscapes. Here, we propose a simple method to improve the QH <span class="hlt">approximation</span>, i.e., to reduce QH entropy. We <span class="hlt">approximate</span> the potential energy landscape of the system by an effective harmonic potential, and request that this potential must produce exactly the configurational temperature of the system. Due to this constraint, the force constants associated with the effective harmonic potential are increased, or equivalently, entropy of motion governed by this effective harmonic potential is reduced. We also introduce the effective configurational temperature concept which can be used as an indicator to check the anharmonicity of the free energy landscape. To validate the new method we compare it with the recently developed expansion <span class="hlt">approximate</span> method by calculating entropy of one simple model system and two peptides with 3 and 16 amino acids either in gas phase or in explicit solvent. We show that the new method appears to be a good choice in practice as it is a compromise between accuracy and computational speed. A modification of the expansion <span class="hlt">approximate</span> method is also introduced and advantages are discussed in some detail.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JPhA...50i3001S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JPhA...50i3001S"><span><span class="hlt">Approximation</span> and inference methods for stochastic biochemical kinetics—a tutorial review</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Schnoerr, David; Sanguinetti, Guido; Grima, Ramon</p> <p>2017-03-01</p> <p>Stochastic fluctuations of molecule numbers are ubiquitous in biological systems. Important examples include gene expression and enzymatic processes in living cells. Such systems are typically modelled as chemical reaction networks whose dynamics are governed by the chemical master equation. Despite its simple structure, no analytic solutions to the chemical master equation are known for most systems. Moreover, stochastic simulations are computationally expensive, making systematic analysis and statistical inference a challenging task. Consequently, significant effort has been spent in recent decades on the development of efficient <span class="hlt">approximation</span> and inference methods. This article gives an introduction to basic modelling concepts as well as an overview of state of the art methods. First, we motivate and introduce deterministic and stochastic methods for modelling chemical networks, and give an overview of simulation and exact solution methods. Next, we discuss several <span class="hlt">approximation</span> methods, including the chemical Langevin equation, the system size expansion, moment closure <span class="hlt">approximations</span>, time-scale separation <span class="hlt">approximations</span> and hybrid methods. We discuss their various properties and review recent advances and remaining challenges for these methods. We present a comparison of several of these methods by means of a numerical case study and highlight some of their respective advantages and disadvantages. Finally, we discuss the problem of inference from experimental data in the Bayesian framework and review recent methods developed the literature. In summary, this review gives a self-contained introduction to modelling, <span class="hlt">approximations</span> and inference methods for stochastic chemical kinetics.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19930007574','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19930007574"><span>The role of under-determined <span class="hlt">approximations</span> in engineering and science application</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Carpenter, William C.</p> <p>1992-01-01</p> <p>There is currently a great deal of interest in using response surfaces in the optimization of aircraft performance. The objective function and/or constraint equations involved in these optimization problems may come from numerous disciplines such as structures, aerodynamics, environmental engineering, etc. In each of these disciplines, the mathematical complexity of the governing equations usually dictates that numerical results be obtained from large computer programs such as a finite element method program. Thus, when performing optimization studies, response surfaces are a convenient way of transferring information from the various disciplines to the optimization algorithm as opposed to bringing all the sundry computer programs together in a massive computer code. Response surfaces offer another advantage in the optimization of aircraft structures. A characteristic of these types of optimization problems is that evaluation of the objective function and response equations (referred to as a functional evaluation) can be very expensive in a computational sense. Because of the computational expense in obtaining functional evaluations, the present study was undertaken to investigate under-determinined <span class="hlt">approximations</span>. An under-determined <span class="hlt">approximation</span> is one in which there are fewer training pairs (pieces of information about a function) than there are undetermined parameters (coefficients or weights) associated with the <span class="hlt">approximation</span>. Both polynomial <span class="hlt">approximations</span> and neural net <span class="hlt">approximations</span> were examined. Three main example problems were investigated: (1) a function of one design variable was considered; (2) a function of two design variables was considered; and (3) a 35 bar truss with 4 design variables was considered.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.cs.cornell.edu/~yexiang/publications/Xiaojian-Wu-et-al-aaai-2018-final.pdf','USGSPUBS'); return false;" href="https://www.cs.cornell.edu/~yexiang/publications/Xiaojian-Wu-et-al-aaai-2018-final.pdf"><span>Efficiently <span class="hlt">approximating</span> the Pareto frontier: Hydropower dam placement in the Amazon basin</span></a></p> <p><a target="_blank" href="http://pubs.er.usgs.gov/pubs/index.jsp?view=adv">USGS Publications Warehouse</a></p> <p>Wu, Xiaojian; Gomes-Selman, Jonathan; Shi, Qinru; Xue, Yexiang; Garcia-Villacorta, Roosevelt; Anderson, Elizabeth; Sethi, Suresh; Steinschneider, Scott; Flecker, Alexander; Gomes, Carla P.</p> <p>2018-01-01</p> <p>Real–world problems are often not fully characterized by a single optimal solution, as they frequently involve multiple competing objectives; it is therefore important to identify the so-called Pareto frontier, which captures solution trade-offs. We propose a fully polynomial-time <span class="hlt">approximation</span> scheme based on Dynamic Programming (DP) for computing a polynomially succinct curve that <span class="hlt">approximates</span> the Pareto frontier to within an arbitrarily small > 0 on treestructured networks. Given a set of objectives, our <span class="hlt">approximation</span> scheme runs in time polynomial in the size of the instance and 1/. We also propose a Mixed Integer Programming (MIP) scheme to <span class="hlt">approximate</span> the Pareto frontier. The DP and MIP Pareto frontier approaches have complementary strengths and are surprisingly effective. We provide empirical results showing that our methods outperform other approaches in efficiency and accuracy. Our work is motivated by a problem in computational sustainability concerning the proliferation of hydropower dams throughout the Amazon basin. Our goal is to support decision-makers in evaluating impacted ecosystem services on the full scale of the Amazon basin. Our work is general and can be applied to <span class="hlt">approximate</span> the Pareto frontier of a variety of multiobjective problems on tree-structured networks.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19880020071','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19880020071"><span>Stability of semidiscrete <span class="hlt">approximations</span> for hyperbolic initial-boundary-value problems: Stationary modes</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Warming, Robert F.; Beam, Richard M.</p> <p>1988-01-01</p> <p>Spatially discrete difference <span class="hlt">approximations</span> for hyperbolic initial-boundary-value problems (IBVPs) require numerical boundary conditions in addition to the analytical boundary conditions specified for the differential equations. Improper treatment of a numerical boundary condition can cause instability of the discrete IBVP even though the <span class="hlt">approximation</span> is stable for the pure initial-value or Cauchy problem. In the discrete IBVP stability literature there exists a small class of discrete <span class="hlt">approximations</span> called borderline cases. For nondissipative <span class="hlt">approximations</span>, borderline cases are unstable according to the theory of the Gustafsson, Kreiss, and Sundstrom (GKS) but they may be Lax-Richtmyer stable or unstable in the L sub 2 norm on a finite domain. It is shown that borderline <span class="hlt">approximation</span> can be characterized by the presence of a stationary mode for the finite-domain problem. A stationary mode has the property that it does not decay with time and a nontrivial stationary mode leads to algebraic growth of the solution norm with mesh refinement. An analytical condition is given which makes it easy to detect a stationary mode; several examples of numerical boundary conditions are investigated corresponding to borderline cases.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19840022749','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19840022749"><span>Legendre-tau <span class="hlt">approximation</span> for functional differential equations. Part 2: The linear quadratic optimal control problem</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Ito, K.; Teglas, R.</p> <p>1984-01-01</p> <p>The numerical scheme based on the Legendre-tau <span class="hlt">approximation</span> is proposed to <span class="hlt">approximate</span> the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good <span class="hlt">approximations</span> at low orders and provides an <span class="hlt">approximation</span> technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline <span class="hlt">approximations</span>) is made.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2002PhRvB..66u4206G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2002PhRvB..66u4206G"><span>Phonons in random alloys: The itinerant coherent-potential <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ghosh, Subhradip; Leath, P. L.; Cohen, Morrel H.</p> <p>2002-12-01</p> <p>We present the itinerant coherent-potential <span class="hlt">approximation</span> (ICPA), an analytic, translationally invariant, and tractable form of augmented-space-based multiple-scattering theory18 in a single-site <span class="hlt">approximation</span> for harmonic phonons in realistic random binary alloys with mass and force-constant disorder. We provide expressions for quantities needed for comparison with experimental structure factors such as partial and average spectral functions and derive the sum rules associated with them. Numerical results are presented for Ni55Pd45 and Ni50Pt50 alloys which serve as test cases, the former for weak force-constant disorder and the latter for strong. We present results on dispersion curves and disorder-induced widths. Direct comparisons with the single-site coherent potential <span class="hlt">approximation</span> (CPA) and experiment are made which provide insight into the physics of force-constant changes in random alloys. The CPA accounts well for the weak force-constant disorder case but fails for strong force-constant disorder where the ICPA succeeds.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23005202','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23005202"><span>Analytical <span class="hlt">approximations</span> for the collapse of an empty spherical bubble.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Obreschkow, D; Bruderer, M; Farhat, M</p> <p>2012-06-01</p> <p>The Rayleigh equation 3/2R+RR+pρ(-1)=0 with initial conditions R(0)=R(0), R(0)=0 models the collapse of an empty spherical bubble of radius R(T) in an ideal, infinite liquid with far-field pressure p and density ρ. The solution for r≡R/R(0) as a function of time t≡T/T(c), where R(T(c))≡0, is independent of R(0), p, and ρ. While no closed-form expression for r(t) is known, we find that r(0)(t)=(1-t(2))(2/5) <span class="hlt">approximates</span> r(t) with an error below 1%. A systematic development in orders of t(2) further yields the 0.001% <span class="hlt">approximation</span> r(*)(t)=r(0)(t)[1-a(1)Li(2.21)(t(2))], where a(1)≈-0.01832099 is a constant and Li is the polylogarithm. The usefulness of these <span class="hlt">approximations</span> is demonstrated by comparison to high-precision cavitation data obtained in microgravity.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/7766771','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/7766771"><span>An <span class="hlt">approximate</span> generalized linear model with random effects for informative missing data.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Follmann, D; Wu, M</p> <p>1995-03-01</p> <p>This paper develops a class of models to deal with missing data from longitudinal studies. We assume that separate models for the primary response and missingness (e.g., number of missed visits) are linked by a common random parameter. Such models have been developed in the econometrics (Heckman, 1979, Econometrica 47, 153-161) and biostatistics (Wu and Carroll, 1988, Biometrics 44, 175-188) literature for a Gaussian primary response. We allow the primary response, conditional on the random parameter, to follow a generalized linear model and <span class="hlt">approximate</span> the generalized linear model by conditioning on the data that describes missingness. The resultant <span class="hlt">approximation</span> is a mixed generalized linear model with possibly heterogeneous random effects. An example is given to illustrate the <span class="hlt">approximate</span> approach, and simulations are performed to critique the adequacy of the <span class="hlt">approximation</span> for repeated binary data.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/1111451-approximate-fold-cross-validation-least-squares-svm-kernel-ridge-regression','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/1111451-approximate-fold-cross-validation-least-squares-svm-kernel-ridge-regression"><span><span class="hlt">Approximate</span> l-fold cross-validation with Least Squares SVM and Kernel Ridge Regression</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Edwards, Richard E; Zhang, Hao; Parker, Lynne Edwards</p> <p>2013-01-01</p> <p>Kernel methods have difficulties scaling to large modern data sets. The scalability issues are based on computational and memory requirements for working with a large matrix. These requirements have been addressed over the years by using low-rank kernel <span class="hlt">approximations</span> or by improving the solvers scalability. However, Least Squares Support VectorMachines (LS-SVM), a popular SVM variant, and Kernel Ridge Regression still have several scalability issues. In particular, the O(n^3) computational complexity for solving a single model, and the overall computational complexity associated with tuning hyperparameters are still major problems. We address these problems by introducing an O(n log n) <span class="hlt">approximate</span> l-foldmore » cross-validation method that uses a multi-level circulant matrix to <span class="hlt">approximate</span> the kernel. In addition, we prove our algorithm s computational complexity and present empirical runtimes on data sets with <span class="hlt">approximately</span> 1 million data points. We also validate our <span class="hlt">approximate</span> method s effectiveness at selecting hyperparameters on real world and standard benchmark data sets. Lastly, we provide experimental results on using a multi-level circulant kernel <span class="hlt">approximation</span> to solve LS-SVM problems with hyperparameters selected using our method.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19930092184','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19930092184"><span>Analytical Derivation and Experimental Evaluation of Short-Bearing <span class="hlt">Approximation</span> for Full Journal Bearing</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Dubois, George B; Ocvirk, Fred W</p> <p>1953-01-01</p> <p>An <span class="hlt">approximate</span> analytical solution including the effect of end leakage from the oil film of short plain bearings is presented because of the importance of endwise flow in sleeve bearings of the short lengths commonly used. The analytical <span class="hlt">approximation</span> is supported by experimental data, resulting in charts which facilitate analysis of short plain bearings. The analytical <span class="hlt">approximation</span> includes the endwise flow and that part of the circumferential flow which is related to surface velocity and film thickness but neglects the effect of film pressure on the circumferential flow. In practical use, this <span class="hlt">approximation</span> applies best to bearings having a length-diameter ratio up to 1, and the effects of elastic deflection, inlet oil pressure, and changes of clearance with temperature minimize the relative importance of the neglected term. The analytical <span class="hlt">approximation</span> was found to be an extension of a little-known pressure-distribution function originally proposed by Michell and Cardullo.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD1007427','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD1007427"><span><span class="hlt">Approximation</span> of Quantum Stochastic Differential Equations for Input-Output Model Reduction</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>2016-02-25</p> <p><span class="hlt">Approximation</span> of Quantum Stochastic Differential Equations for Input-Output Model Reduction We have completed a short program of theoretical research...on dimensional reduction and <span class="hlt">approximation</span> of models based on quantum stochastic differential equations. Our primary results lie in the area of...2211 quantum probability, quantum stochastic differential equations REPORT DOCUMENTATION PAGE 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 10. SPONSOR</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JChPh.145t4117M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JChPh.145t4117M"><span>The derivation and <span class="hlt">approximation</span> of coarse-grained dynamics from Langevin dynamics</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ma, Lina; Li, Xiantao; Liu, Chun</p> <p>2016-11-01</p> <p>We present a derivation of a coarse-grained description, in the form of a generalized Langevin equation, from the Langevin dynamics model that describes the dynamics of bio-molecules. The focus is placed on the form of the memory kernel function, the colored noise, and the second fluctuation-dissipation theorem that connects them. Also presented is a hierarchy of <span class="hlt">approximations</span> for the memory and random noise terms, using rational <span class="hlt">approximations</span> in the Laplace domain. These <span class="hlt">approximations</span> offer increasing accuracy. More importantly, they eliminate the need to evaluate the integral associated with the memory term at each time step. Direct sampling of the colored noise can also be avoided within this framework. Therefore, the numerical implementation of the generalized Langevin equation is much more efficient.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=20040174931&hterms=Free+energy&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3DFree%2Benergy','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=20040174931&hterms=Free+energy&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3DFree%2Benergy"><span><span class="hlt">Approximate</span> scaling properties of RNA free energy landscapes</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Baskaran, S.; Stadler, P. F.; Schuster, P.</p> <p>1996-01-01</p> <p>RNA free energy landscapes are analysed by means of "time-series" that are obtained from random walks restricted to excursion sets. The power spectra, the scaling of the jump size distribution, and the scaling of the curve length measured with different yard stick lengths are used to describe the structure of these "time series". Although they are stationary by construction, we find that their local behavior is consistent with both AR(1) and self-affine processes. Random walks confined to excursion sets (i.e., with the restriction that the fitness value exceeds a certain threshold at each step) exhibit essentially the same statistics as free random walks. We find that an AR(1) time series is in general <span class="hlt">approximately</span> self-affine on timescales up to <span class="hlt">approximately</span> the correlation length. We present an empirical relation between the correlation parameter rho of the AR(1) model and the exponents characterizing self-affinity.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1994ApJ...423...22C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1994ApJ...423...22C"><span>Applying the Zel'dovich <span class="hlt">approximation</span> to general relativity</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Croudace, K. M.; Parry, J.; Salopek, D. S.; Stewart, J. M.</p> <p>1994-03-01</p> <p>Starting from general relativity, we give a systematic derivation of the Zel'dovich <span class="hlt">approximation</span> describing the nonlinear evolution of collisionless dust. We begin by evolving dust along world lines, and we demonstrate that the Szekeres line element is an exact but apparently unstable solution of the evolution equations describing pancake collapse. Next, we solve the Einstein field equations by employing Hamilton-Jacobi techniques and a spatial gradient expansion. We give a prescription for evolving a primordial or 'seed' metric up to the formation of pancakes, and demonstrate its validity by rederiving the Szekeres solution <span class="hlt">approximately</span> at third order and exactly at fifth order in spatial gradients. Finally we show that the range of validity of the expansion can be improved quite significantly if one notes that the 3-metric must have nonnegative eigenvalues. With this improvement the exact Szekeres solution is obtained after only one iteration.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20120017356','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20120017356"><span>Spectrally-Invariant <span class="hlt">Approximation</span> Within Atmospheric Radiative Transfer</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Marshak, A.; Knyazikhin, Y.; Chiu, J. C.; Wiscombe, W. J.</p> <p>2011-01-01</p> <p>Certain algebraic combinations of single scattering albedo and solar radiation reflected from, or transmitted through, vegetation canopies do not vary with wavelength. These "spectrally invariant relationships" are the consequence of wavelength independence of the extinction coefficient and scattering phase function in vegetation. In general, this wavelength independence does not hold in the atmosphere, but in clouddominated atmospheres the total extinction and total scattering phase function vary only weakly with wavelength. This paper identifies the atmospheric conditions under which the spectrally invariant <span class="hlt">approximation</span> can accurately describe the extinction. and scattering properties of cloudy atmospheres. The validity of the assumptions and the accuracy of the <span class="hlt">approximation</span> are tested with ID radiative transfer calculations using publicly available radiative transfer models: Discrete Ordinate Radiative Transfer (DISORT) and Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART). It is shown for cloudy atmospheres with cloud optical depth above 3, and for spectral intervals that exclude strong water vapor absorption, that the spectrally invariant relationships found in vegetation canopy radiative transfer are valid to better than 5%. The physics behind this phenomenon, its mathematical basis, and possible applications to remote sensing and climate are discussed.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li class="active"><span>25</span></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_25 --> <div class="footer-extlink text-muted" style="margin-bottom:1rem; text-align:center;">Some links on this page may take you to non-federal websites. 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