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 <span class="hlt">order</span> 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>
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<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA029655','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA029655"><span><span class="hlt">Approximations</span> of Two-Attribute Utility Functions</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>1976-09-01</p>
<p>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 <span class="hlt">approximations</span> of u... study will elicit additional interest in the topic. 2. REMARKS ON <span class="hlt">APPROXIMATION</span> THEORY This section outlines a few basic ideas of <span class="hlt">approximation</span> theory</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19850022413','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19850022413"><span>On the convergence of difference <span class="hlt">approximations</span> to scalar conservation laws</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Osher, S.; Tadmor, E.</p>
<p>1985-01-01</p>
<p>A unified treatment of explicit in time, two level, second <span class="hlt">order</span> resolution, total variation diminishing, <span class="hlt">approximations</span> to scalar conservation laws are presented. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced and results in terms of the latter are obtained. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first <span class="hlt">order</span> accurate in general. Convergence for total variation diminishing-second <span class="hlt">order</span> resolution schemes <span class="hlt">approximating</span> convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19850021314','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19850021314"><span>Intermediate boundary conditions for LOD, ADI and <span class="hlt">approximate</span> factorization methods</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Leveque, R. J.</p>
<p>1985-01-01</p>
<p>A general approach to determining the correct intermediate boundary conditions for dimensional splitting methods is presented. The intermediate solution U is viewed as a second <span class="hlt">order</span> accurate <span class="hlt">approximation</span> to a modified equation. Deriving the modified equation and using the relationship between this equation and the original equation allows us to determine the correct boundary conditions for U*. This technique is illustrated by applying it to locally one dimensional (LOD) and alternating direction implicit (ADI) methods for the heat equation in two and three space dimensions. The <span class="hlt">approximate</span> factorization method is considered in slightly more generality.</p>
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</ol>
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<ol class="result-class" start="201">
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19830025319','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19830025319"><span>Spline <span class="hlt">approximations</span> for nonlinear hereditary control systems</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Daniel, P. L.</p>
<p>1982-01-01</p>
<p>A sline-based <span class="hlt">approximation</span> scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The <span class="hlt">approximating</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=boltzmann&pg=2&id=EJ1017061','ERIC'); return false;" href="https://eric.ed.gov/?q=boltzmann&pg=2&id=EJ1017061"><span>Quirks of Stirling's <span class="hlt">Approximation</span></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>Macrae, Roderick M.; Allgeier, Benjamin M.</p>
<p>2013-01-01</p>
<p>Stirling's <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> leads to incorrect conclusions. In this article, the problem is first illustrated using a familiar "toy…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=calculus+AND+1&pg=2&id=EJ994590','ERIC'); return false;" href="https://eric.ed.gov/?q=calculus+AND+1&pg=2&id=EJ994590"><span>Computing Functions by <span class="hlt">Approximating</span> the Input</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>Goldberg, Mayer</p>
<p>2012-01-01</p>
<p>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 <span class="hlt">approximate</span>. In this work, we take the opposite approach: we show how to compute the values of some transcendental functions by <span class="hlt">approximating</span> the input to these functions, and obtaining exact answers for their…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JAGeo..12..139E','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JAGeo..12..139E"><span>Spline <span class="hlt">approximation</span>, Part 1: Basic methodology</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Ezhov, Nikolaj; Neitzel, Frank; Petrovic, Svetozar</p>
<p>2018-04-01</p>
<p>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 <span class="hlt">approximation</span> with a continuous mathematical function is required. In this paper the <span class="hlt">approximation</span> of 2D curves by means of splines is treated. Splines offer quite flexible and elegant solutions for interpolation or <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> and all of them have certain advantages and disadvantages depending on the <span class="hlt">approximation</span> problem. In a series of three articles spline <span class="hlt">approximation</span> is presented from a geodetic point of view. In this paper (Part 1) the basic methodology of spline <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> approaches as well as the utilization of splines for deformation detection will be investigated on numerical examples in Part 3.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19800016563','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19800016563"><span>Inversion and <span class="hlt">approximation</span> of Laplace transforms</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Lear, W. M.</p>
<p>1980-01-01</p>
<p>A method of inverting Laplace transforms by using a set of orthonormal functions is reported. As a byproduct of the inversion, <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> process is simple enough to be put on a programmable hand calculator.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=Base&pg=3&id=EJ1158997','ERIC'); return false;" href="https://eric.ed.gov/?q=Base&pg=3&id=EJ1158997"><span><span class="hlt">Approximations</span> of e and ?: An Exploration</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>Brown, Philip R.</p>
<p>2017-01-01</p>
<p>Fractional <span class="hlt">approximations</span> 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 <span class="hlt">approximations</span> is suggested for students and teachers as an entry point into mathematics research.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/1083968','DOE-PATENT-XML'); return false;" href="https://www.osti.gov/servlets/purl/1083968"><span><span class="hlt">Approximate</span> error conjugation gradient minimization methods</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/doepatents">DOEpatents</a></p>
<p>Kallman, Jeffrey S</p>
<p>2013-05-21</p>
<p>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 <span class="hlt">approximate</span> error using the subset of rays, and calculating a minimum in a conjugate gradient direction based on the <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> error using the subset of rays, and logic for calculating a minimum in a conjugate gradient direction based on the <span class="hlt">approximate</span> error. In other embodiments, computer program products, methods, and systems are described capable of using <span class="hlt">approximate</span> error in constrained conjugate gradient minimization problems.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19870011345','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19870011345"><span>Piecewise linear <span class="hlt">approximation</span> for hereditary control problems</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Propst, Georg</p>
<p>1987-01-01</p>
<p>Finite dimensional <span class="hlt">approximations</span> are presented for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The <span class="hlt">approximation</span> 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 <span class="hlt">approximate</span> 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 <span class="hlt">approximations</span> 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 <span class="hlt">approximations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JMP....59e1502G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JMP....59e1502G"><span><span class="hlt">Approximate</span> isotropic cloak for the Maxwell equations</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Ghosh, Tuhin; Tarikere, Ashwin</p>
<p>2018-05-01</p>
<p>We construct a regular isotropic <span class="hlt">approximate</span> 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 <span class="hlt">approximations</span> 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 <span class="hlt">approximations</span> to these regularized cloaks using homogenization techniques so that one obtains cloaking of arbitrary accuracy with regular and isotropic parameters.</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-<span class="hlt">order</span> 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/1434885-approximate-computing-techniques-iterative-graph-algorithms','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/1434885-approximate-computing-techniques-iterative-graph-algorithms"><span><span class="hlt">Approximate</span> Computing Techniques for Iterative Graph Algorithms</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Panyala, Ajay R.; Subasi, Omer; Halappanavar, Mahantesh</p>
<p></p>
<p><span class="hlt">Approximate</span> computing enables processing of large-scale graphs by trading off quality for performance. <span class="hlt">Approximate</span> 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 <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> 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</p>
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<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>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1286958-survey-techniques-approximate-computing','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1286958-survey-techniques-approximate-computing"><span>A Survey of Techniques for <span class="hlt">Approximate</span> Computing</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p>
<p>Mittal, Sparsh</p>
<p>2016-03-18</p>
<p><span class="hlt">Approximate</span> computing trades off computation quality with the effort expended and as rising performance demands confront with plateauing resource budgets, <span class="hlt">approximate</span> computing has become, not merely attractive, but even imperative. Here, we present a survey of techniques for <span class="hlt">approximate</span> computing (AC). We discuss strategies for finding <span class="hlt">approximable</span> 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</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19760011449','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19760011449"><span><span class="hlt">Approximation</span> concepts for efficient structural synthesis</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Schmit, L. A., Jr.; Miura, H.</p>
<p>1976-01-01</p>
<p>It is shown that efficient structural synthesis capabilities can be created by using <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> concepts are described along with the technical foundations of the ACCESS 1 computer program, which implements several <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19790023272','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19790023272"><span>Linear <span class="hlt">Approximation</span> SAR Azimuth Processing Study</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Lindquist, R. B.; Masnaghetti, R. K.; Belland, E.; Hance, H. V.; Weis, W. G.</p>
<p>1979-01-01</p>
<p>A segmented linear <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> SAR azimuth processor meeting requirements similar to those of the SEASAT-A SAR was developed. The study resulted in a design with <span class="hlt">approximately</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19880037825&hterms=levels+law&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DA%2Blevels%2Blaw','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19880037825&hterms=levels+law&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DA%2Blevels%2Blaw"><span>On the convergence of difference <span class="hlt">approximations</span> to scalar conservation laws</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Osher, Stanley; Tadmor, Eitan</p>
<p>1988-01-01</p>
<p>A unified treatment is given for time-explicit, two-level, second-<span class="hlt">order</span>-resolution (SOR), total-variation-diminishing (TVD) <span class="hlt">approximations</span> to scalar conservation laws. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced to obtain results in terms of the latter. The existence of a cell entropy inequality is discussed, and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first-<span class="hlt">order</span> accurate in general. Convergence for TVD-SOR schemes <span class="hlt">approximating</span> convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012JChPh.137l5102O','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012JChPh.137l5102O"><span>Counting statistics for genetic switches based on effective interaction <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>Ohkubo, Jun</p>
<p>2012-09-01</p>
<p>Applicability of counting statistics for a system with an infinite number of states is investigated. The counting statistics has been studied a lot for a system with a finite number of states. While it is possible to use the scheme in <span class="hlt">order</span> to count specific transitions in a system with an infinite number of states in principle, we have non-closed equations in general. A simple genetic switch can be described by a master equation with an infinite number of states, and we use the counting statistics in <span class="hlt">order</span> to count the number of transitions from inactive to active states in the gene. To avoid having the non-closed equations, an effective interaction <span class="hlt">approximation</span> is employed. As a result, it is shown that the switching problem can be treated as a simple two-state model <span class="hlt">approximately</span>, which immediately indicates that the switching obeys non-Poisson statistics.</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 <span class="hlt">order</span> and exactly at fifth <span class="hlt">order</span> 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('https://ntrs.nasa.gov/search.jsp?R=19720031591&hterms=hierarchy&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D40%26Ntt%3Dhierarchy','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19720031591&hterms=hierarchy&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D40%26Ntt%3Dhierarchy"><span>The closure <span class="hlt">approximation</span> in the hierarchy equations.</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Adomian, G.</p>
<p>1971-01-01</p>
<p>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 <span class="hlt">approximation</span> in widely used hierarchy methods and the ?self-consistent field' <span class="hlt">approximation</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA089726','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA089726"><span>A Discrete <span class="hlt">Approximation</span> Framework for Hereditary Systems.</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>1980-05-01</p>
<p>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 <span class="hlt">approximation</span> schemes for the solutions to the homogeneous AEE which in turn...rich classes of these schemes . In addition, two particular families of <span class="hlt">approximation</span> schemes included in the general framework are developed and</p>
</li>
</ol>
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<ol class="result-class" start="221">
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018IJMPB..3250008G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018IJMPB..3250008G"><span><span class="hlt">Approximate</span> analytic expression for the Skyrmions crystal</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Grandi, Nicolás; Sturla, Mauricio</p>
<p>2018-01-01</p>
<p>We find <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JPhCS1003a2054A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JPhCS1003a2054A"><span>On direct theorems for best polynomial <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>Auad, A. A.; AbdulJabbar, R. S.</p>
<p>2018-05-01</p>
<p>This paper is to obtain similarity for the best <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA215217','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA215217"><span>Rational <span class="hlt">Approximations</span> with Hankel-Norm Criterion</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>1980-01-01</p>
<p>REPORT TYPE ANDu DATES COVERED It) L. TITLE AND SLWUIlL Fi901 ia FUNDING NUMOIRS, RATIONAL <span class="hlt">APPROXIMATIONS</span> 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 <span class="hlt">APPROXIMATIONS</span> WITH HANKEL-NORM CRITERION* Y. Genin* Philips Research Lab. 2, avenue van</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhRvD..95h3526W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhRvD..95h3526W"><span>Double power series method for <span class="hlt">approximating</span> cosmological perturbations</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Wren, Andrew J.; Malik, Karim A.</p>
<p>2017-04-01</p>
<p>We introduce a double power series method for finding <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> oscillating solutions in linear <span class="hlt">order</span> 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 <span class="hlt">approximations</span> for the metric potential, radiation and matter perturbations. Comparison with numerical solutions of the perturbation equations shows that our <span class="hlt">approximations</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22472091-loop-approximation-nucleon-qcd-sum-rules','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22472091-loop-approximation-nucleon-qcd-sum-rules"><span>On the loop <span class="hlt">approximation</span> in nucleon QCD sum rules</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Drukarev, E. G., E-mail: drukarev@thd.pnpi.spb.ru; Ryskin, M. G.; Sadovnikova, V. A.</p>
<p></p>
<p>There was a general belief that the nucleon QCD sum rules which include only the quark loops and thus contain only the condensates of dimension d = 3 and d = 4 have only a trivial solution. We demonstrate that there is also a nontrivial solution. We show that it can be treated as the lowest <span class="hlt">order</span> <span class="hlt">approximation</span> to the solution which includes the higher terms of the Operator Product Expansion. Inclusion of the radiative corrections improves the convergence of the series.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20170010676','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20170010676"><span>Temporal Planning for Compilation of Quantum <span class="hlt">Approximate</span> Optimization Algorithm Circuits</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Venturelli, Davide; Do, Minh Binh; Rieffel, Eleanor Gilbert; Frank, Jeremy David</p>
<p>2017-01-01</p>
<p>We investigate the application of temporal planners to the problem of compiling quantum circuits to newly emerging quantum hardware. While our approach is general, we focus our initial experiments on Quantum <span class="hlt">Approximate</span> Optimization Algorithm (QAOA) circuits that have few <span class="hlt">ordering</span> constraints and allow highly parallel plans. We report on experiments using several temporal planners to compile circuits of various sizes to a realistic hardware. This early empirical evaluation suggests that temporal planning is a viable approach to quantum circuit compilation.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JQSRT.179..132G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JQSRT.179..132G"><span>Comments on localized and integral localized <span class="hlt">approximations</span> in spherical coordinates</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Gouesbet, Gérard; Lock, James A.</p>
<p>2016-08-01</p>
<p>Localized <span class="hlt">approximation</span> procedures are efficient ways to evaluate beam shape coefficients of laser beams, and are particularly useful when other methods are ineffective or inefficient. Comments on these procedures are, however, required in <span class="hlt">order</span> to help researchers make correct decisions concerning their use. This paper has the flavor of a short review and takes the opportunity to attract the attention of the readers to a required refinement of terminology.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2003JCoAM.161..469F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003JCoAM.161..469F"><span>Numerical scheme <span class="hlt">approximating</span> solution and parameters in a beam equation</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Ferdinand, Robert R.</p>
<p>2003-12-01</p>
<p>We present a mathematical model which describes vibration in a metallic beam about its equilibrium position. This model takes the form of a nonlinear second-<span class="hlt">order</span> (in time) and fourth-<span class="hlt">order</span> (in space) partial differential equation with boundary and initial conditions. A finite-element Galerkin <span class="hlt">approximation</span> scheme is used to estimate model solution. Infinite-dimensional model parameters are then estimated numerically using an inverse method procedure which involves the minimization of a least-squares cost functional. Numerical results are presented and future work to be done is discussed.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19850002436','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19850002436"><span>Legendre-Tau <span class="hlt">approximation</span> for functional differential equations. Part 3: Eigenvalue <span class="hlt">approximations</span> and uniform stability</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Ito, K.</p>
<p>1984-01-01</p>
<p>The stability and convergence properties of the Legendre-tau <span class="hlt">approximation</span> for hereditary differential systems are analyzed. A charactristic equation is derived for the eigenvalues of the resulting <span class="hlt">approximate</span> system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under <span class="hlt">approximation</span>. It is the key to obtaining the convergence of <span class="hlt">approximate</span> solutions of the algebraic Riccati equation in trace norm.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://rosap.ntl.bts.gov/view/dot/9918','DOTNTL'); return false;" href="https://rosap.ntl.bts.gov/view/dot/9918"><span>Derivation of a first <span class="hlt">order</span> <span class="hlt">approximation</span> of particulate matter from aircraft</span></a></p>
<p><a target="_blank" href="http://ntlsearch.bts.gov/tris/index.do">DOT National Transportation Integrated Search</a></p>
<p></p>
<p>2003-06-22</p>
<p>The mass of particulate matter (PM) emitted from aircraft must be predicted for major : actions at airports to comply with current federal regulations. However, this PM mass in : the jet exhaust has not been effectively quantified to permit accurate ...</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3511751','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3511751"><span>Averaging principle for second-<span class="hlt">order</span> <span class="hlt">approximation</span> of heterogeneous models with homogeneous models</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Fibich, Gadi; Gavious, Arieh; Solan, Eilon</p>
<p>2012-01-01</p>
<p>Typically, models with a heterogeneous property are considerably harder to analyze than the corresponding homogeneous models, in which the heterogeneous property is replaced by its average value. In this study we show that any outcome of a heterogeneous model that satisfies the two properties of differentiability and symmetry is O(ɛ2) equivalent to the outcome of the corresponding homogeneous model, where ɛ is the level of heterogeneity. We then use this averaging principle to obtain new results in queuing theory, game theory (auctions), and social networks (marketing). PMID:23150569</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23150569','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23150569"><span>Averaging principle for second-<span class="hlt">order</span> <span class="hlt">approximation</span> of heterogeneous models with homogeneous models.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Fibich, Gadi; Gavious, Arieh; Solan, Eilon</p>
<p>2012-11-27</p>
<p>Typically, models with a heterogeneous property are considerably harder to analyze than the corresponding homogeneous models, in which the heterogeneous property is replaced by its average value. In this study we show that any outcome of a heterogeneous model that satisfies the two properties of differentiability and symmetry is O(ε(2)) equivalent to the outcome of the corresponding homogeneous model, where ε is the level of heterogeneity. We then use this averaging principle to obtain new results in queuing theory, game theory (auctions), and social networks (marketing).</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhRvB..95s5158B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhRvB..95s5158B"><span>Convergence behavior of the random phase <span class="hlt">approximation</span> renormalized correlation energy</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Bates, Jefferson E.; Sensenig, Jonathon; Ruzsinszky, Adrienn</p>
<p>2017-05-01</p>
<p>Based on the random phase <span class="hlt">approximation</span> (RPA), RPA renormalization [J. E. Bates and F. Furche, J. Chem. Phys. 139, 171103 (2013), 10.1063/1.4827254] is a robust many-body perturbation theory that works for molecules and materials because it does not diverge as the Kohn-Sham gap approaches zero. Additionally, RPA renormalization enables the simultaneous calculation of RPA and beyond-RPA correlation energies since the total correlation energy is the sum of a series of independent contributions. The first-<span class="hlt">order</span> <span class="hlt">approximation</span> (RPAr1) yields the dominant beyond-RPA contribution to the correlation energy for a given exchange-correlation kernel, but systematically underestimates the total beyond-RPA correction. For both the homogeneous electron gas model and real systems, we demonstrate numerically that RPA renormalization beyond first <span class="hlt">order</span> converges monotonically to the infinite-<span class="hlt">order</span> beyond-RPA correlation energy for several model exchange-correlation kernels and that the rate of convergence is principally determined by the choice of the kernel and spin polarization of the ground state. The monotonic convergence is rationalized from an analysis of the RPA renormalized correlation energy corrections, assuming the exchange-correlation kernel and response functions satisfy some reasonable conditions. For spin-unpolarized atoms, molecules, and bulk solids, we find that RPA renormalization is typically converged to 1 meV error or less by fourth <span class="hlt">order</span> regardless of the band gap or dimensionality. Most spin-polarized systems converge at a slightly slower rate, with errors on the <span class="hlt">order</span> of 10 meV at fourth <span class="hlt">order</span> and typically requiring up to sixth <span class="hlt">order</span> to reach 1 meV error or less. Slowest to converge, however, open-shell atoms present the most challenging case and require many higher <span class="hlt">orders</span> to converge.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/22181475','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/22181475"><span>Construction and accuracy of partial differential equation <span class="hlt">approximations</span> to the chemical master equation.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Grima, Ramon</p>
<p>2011-11-01</p>
<p>The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-<span class="hlt">order</span> partial differential equation <span class="hlt">approximations</span> to the master equation. We here consider two different types of higher-<span class="hlt">order</span> partial differential <span class="hlt">approximations</span>, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation <span class="hlt">approximation</span> of <span class="hlt">order</span> Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to <span class="hlt">order</span> Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to <span class="hlt">order</span> Ω(-(2Q-5)/2), and of skewness accurate to <span class="hlt">order</span> Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation <span class="hlt">approximation</span> from the system-size expansion of <span class="hlt">approximate</span> <span class="hlt">order</span> 2Q. Hence, we conclude that partial differential <span class="hlt">approximations</span> based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than <span class="hlt">approximations</span> of the same <span class="hlt">order</span> derived from the system-size expansion.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018Metro..55..147P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018Metro..55..147P"><span><span class="hlt">Approximate</span> Bayesian evaluations of measurement uncertainty</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Possolo, Antonio; Bodnar, Olha</p>
<p>2018-04-01</p>
<p>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 <span class="hlt">approximate</span> evaluation of the associated standard uncertainty. This contribution presents <span class="hlt">approximate</span>, 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 <span class="hlt">approximate</span> Bayesian estimates and uncertainty evaluations that we present have a long history and illustrious pedigree, and provide sufficiently accurate <span class="hlt">approximations</span> 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 <span class="hlt">approximations</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19860048947&hterms=Gravitational+motion+system&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DGravitational%2Bmotion%2Bsystem','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19860048947&hterms=Gravitational+motion+system&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DGravitational%2Bmotion%2Bsystem"><span><span class="hlt">Approximation</span> methods in gravitational-radiation theory</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Will, C. M.</p>
<p>1986-01-01</p>
<p>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, <span class="hlt">approximation</span> methods play a crucial role. Recent developments are summarized in two areas in which <span class="hlt">approximations</span> 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 <span class="hlt">approximation</span> gives accurate estimates of the complex frequencies of the modes.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014JSMTE..02..009F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014JSMTE..02..009F"><span>Minimal entropy <span class="hlt">approximation</span> for cellular automata</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Fukś, Henryk</p>
<p>2014-02-01</p>
<p>We present a method for the construction of <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> 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 <span class="hlt">approximation</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/15732387','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/15732387"><span>Smooth function <span class="hlt">approximation</span> using neural networks.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Ferrari, Silvia; Stengel, Robert F</p>
<p>2005-01-01</p>
<p>An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented. In this paper, the approach is implemented for the <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> properties can be investigated via linear algebra. Four algorithms are developed to achieve exact or <span class="hlt">approximate</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2011EPJC...71.1831B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2011EPJC...71.1831B"><span>On Born <span class="hlt">approximation</span> in black hole scattering</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Batic, D.; Kelkar, N. G.; Nowakowski, M.</p>
<p>2011-12-01</p>
<p>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 <span class="hlt">approximation</span>. 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> is not very useful for the scattering of massless scalar, electromagnetic as well as gravitational waves from black holes.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19970041606','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19970041606"><span>Exponential <span class="hlt">Approximations</span> Using Fourier Series Partial Sums</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Banerjee, Nana S.; Geer, James F.</p>
<p>1997-01-01</p>
<p>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 <span class="hlt">approximate</span> 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 <span class="hlt">approximated</span> to within O(N(sup -M-2), and the associated jump of the k(sup th) derivative of f is <span class="hlt">approximated</span> 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 <span class="hlt">approximations</span> to f. Each of these new <span class="hlt">approximations</span> 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 <span class="hlt">approximations</span>, 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.</p>
</li>
</ol>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3844179','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3844179"><span>An Expansion Formula with Higher-<span class="hlt">Order</span> Derivatives for Fractional Operators of Variable <span class="hlt">Order</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>Almeida, Ricardo</p>
<p>2013-01-01</p>
<p>We obtain <span class="hlt">approximation</span> formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional <span class="hlt">order</span>. The <span class="hlt">approximations</span> involve integer-<span class="hlt">order</span> derivatives only. An estimation for the error is given. The efficiency of the <span class="hlt">approximation</span> method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type. PMID:24319382</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19790054041&hterms=Balancing+equations&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3DBalancing%2Bequations','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19790054041&hterms=Balancing+equations&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3DBalancing%2Bequations"><span><span class="hlt">Approximate</span> convective heating equations for hypersonic flows</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Zoby, E. V.; Moss, J. N.; Sutton, K.</p>
<p>1979-01-01</p>
<p>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 <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> heating methods are in good agreement with existing experimental results as well as boundary-layer and viscous-shock-layer solutions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018LMaPh.108..185B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018LMaPh.108..185B"><span>On the dipole <span class="hlt">approximation</span> with error estimates</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Boßmann, Lea; Grummt, Robert; Kolb, Martin</p>
<p>2018-01-01</p>
<p>The dipole <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=20020052603&hterms=congruence&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dcongruence','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=20020052603&hterms=congruence&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dcongruence"><span>Congruence <span class="hlt">Approximations</span> for Entrophy Endowed Hyperbolic Systems</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Barth, Timothy J.; Saini, Subhash (Technical Monitor)</p>
<p>1998-01-01</p>
<p>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 <span class="hlt">approximation</span>. 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 <span class="hlt">approximation</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/957262-solving-infeasible-trust-region-problem-using-approximations','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/957262-solving-infeasible-trust-region-problem-using-approximations"><span>Solving the infeasible trust-region problem using <span class="hlt">approximations</span>.</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Renaud, John E.; Perez, Victor M.; Eldred, Michael Scott</p>
<p>2004-07-01</p>
<p>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 <span class="hlt">approximations</span> for both cases is an alternative investigated by many researchers including the authors. When <span class="hlt">approximations</span> 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 <span class="hlt">approximate</span> 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 <span class="hlt">order</span> 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 <span class="hlt">approximate</span> optimization framework using homotopy ideas to relax the constraints. This algorithm called interior-point trust-region sequential <span class="hlt">approximate</span> optimization (IPTRSAO) presents some similarities to the two normal-tangential steps algorithms. In this paper, a description of the similarities is presented</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.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4652602','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4652602"><span>Fostering Formal Commutativity Knowledge with <span class="hlt">Approximate</span> Arithmetic</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Hansen, Sonja Maria; Haider, Hilde; Eichler, Alexandra; Godau, Claudia; Frensch, Peter A.; Gaschler, Robert</p>
<p>2015-01-01</p>
<p>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 <span class="hlt">approximate</span> 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. <span class="hlt">Approximate</span> 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span>. 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</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018InvPr..34b5005A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018InvPr..34b5005A"><span>Variational Gaussian <span class="hlt">approximation</span> for Poisson data</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Arridge, Simon R.; Ito, Kazufumi; Jin, Bangti; Zhang, Chen</p>
<p>2018-02-01</p>
<p>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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span>, 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 <span class="hlt">approximation</span>. 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 <span class="hlt">approximation</span> and the algorithms.</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/2060/19850012840','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19850012840"><span>Alternative <span class="hlt">approximation</span> concepts for space frame synthesis</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Lust, R. V.; Schmit, L. A.</p>
<p>1985-01-01</p>
<p>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 <span class="hlt">approximation</span> 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 <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> 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 <span class="hlt">approximation</span> concepts options, the results could be compared with previously published work.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD0628888','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD0628888"><span>RATIONAL <span class="hlt">APPROXIMATIONS</span> TO GENERALIZED HYPERGEOMETRIC FUNCTIONS.</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p></p>
<p>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 <span class="hlt">approximations</span> which converge to the</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1992JChPh..96.6654Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1992JChPh..96.6654Z"><span>An <span class="hlt">approximate</span> classical unimolecular reaction rate theory</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Zhao, Meishan; Rice, Stuart A.</p>
<p>1992-05-01</p>
<p>We describe a classical theory of unimolecular reaction rate which is derived from the analysis of Davis and Gray by use of simplifying <span class="hlt">approximations</span>. These <span class="hlt">approximations</span> 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 <span class="hlt">approximation</span> is similar to but extends and improves the <span class="hlt">approximations</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19900019726','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19900019726"><span><span class="hlt">Approximation</span> algorithms for planning and control</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Boddy, Mark; Dean, Thomas</p>
<p>1989-01-01</p>
<p>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 <span class="hlt">approximate</span> those that would result from having unlimited time for computation, where the degree of the <span class="hlt">approximation</span> depends on the amount of time it actually has. There exist <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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.</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('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-<span class="hlt">order</span> <span class="hlt">approximation</span> significantly underestimates actual APD values. PMID:18080006</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016PhRvB..94g5158D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016PhRvB..94g5158D"><span>Revised Thomas-Fermi <span class="hlt">approximation</span> for singular potentials</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Dufty, James W.; Trickey, S. B.</p>
<p>2016-08-01</p>
<p><span class="hlt">Approximations</span> 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 <span class="hlt">approximation</span> 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. <span class="hlt">Approximate</span> expressions for the regularized potential are given to leading <span class="hlt">order</span> in a nonlocality parameter, and the limiting behavior at high and low temperatures is described. The noninteracting part of the free energy in this <span class="hlt">approximation</span> 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.</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-<span class="hlt">order</span> 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 <span class="hlt">order</span> 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/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/scitech">SciTech Connect</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 <span class="hlt">order</span> 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>
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<ol class="result-class" start="261">
<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/scitech">SciTech Connect</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 <span class="hlt">order</span> 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/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 <span class="hlt">order</span> 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('http://adsabs.harvard.edu/abs/2018JPhCS1015c2181P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JPhCS1015c2181P"><span>Investigation of <span class="hlt">approximate</span> models of experimental temperature characteristics of machines</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Parfenov, I. V.; Polyakov, A. N.</p>
<p>2018-05-01</p>
<p>This work is devoted to the investigation of various approaches to the <span class="hlt">approximation</span> of experimental data and the creation of simulation mathematical models of thermal processes in machines with the aim of finding ways to reduce the time of their field tests and reducing the temperature error of the treatments. The main methods of research which the authors used in this work are: the full-scale thermal testing of machines; realization of various approaches at <span class="hlt">approximation</span> of experimental temperature characteristics of machine tools by polynomial models; analysis and evaluation of modelling results (model quality) of the temperature characteristics of machines and their derivatives up to the third <span class="hlt">order</span> in time. As a result of the performed researches, rational methods, type, parameters and complexity of simulation mathematical models of thermal processes in machine tools are proposed.</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/scitech">SciTech Connect</a></p>
<p>Korsch, H.J.; Schinke, R.</p>
<p>1980-08-01</p>
<p>The infinite-<span class="hlt">order</span>-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('http://adsabs.harvard.edu/abs/2003APS..DFD.KG004H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003APS..DFD.KG004H"><span>A Nonlinear Interactions <span class="hlt">Approximation</span> Model for Large-Eddy Simulation</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Haliloglu, Mehmet U.; Akhavan, Rayhaneh</p>
<p>2003-11-01</p>
<p>A new approach to LES modelling is proposed based on direct <span class="hlt">approximation</span> of the nonlinear terms \\overlineu_iuj in the filtered Navier-Stokes equations, instead of the subgrid-scale stress, τ_ij. The proposed model, which we call the Nonlinear Interactions <span class="hlt">Approximation</span> (NIA) model, uses graded filters and deconvolution to parameterize the local interactions across the LES cutoff, and a Smagorinsky eddy viscosity term to parameterize the distant interactions. A dynamic procedure is used to determine the unknown eddy viscosity coefficient, rendering the model free of adjustable parameters. The proposed NIA model has been applied to LES of turbulent channel flows at Re_τ ≈ 210 and Re_τ ≈ 570. The results show good agreement with DNS not only for the mean and resolved second-<span class="hlt">order</span> turbulence statistics but also for the full (resolved plus subgrid) Reynolds stress and turbulence intensities.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JSMTE..11.3401T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JSMTE..11.3401T"><span>Typical performance of <span class="hlt">approximation</span> algorithms for NP-hard problems</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Takabe, Satoshi; Hukushima, Koji</p>
<p>2016-11-01</p>
<p>Typical performance of <span class="hlt">approximation</span> algorithms is studied for randomized minimum vertex cover problems. A wide class of random graph ensembles characterized by an arbitrary degree distribution is discussed with the presentation of a theoretical framework. Herein, three <span class="hlt">approximation</span> algorithms are examined: linear-programming relaxation, loopy-belief propagation, and the leaf-removal algorithm. The former two algorithms are analyzed using a statistical-mechanical technique, whereas the average-case analysis of the last one is conducted using the generating function method. These algorithms have a threshold in the typical performance with increasing average degree of the random graph, below which they find true optimal solutions with high probability. Our study reveals that there exist only three cases, determined by the <span class="hlt">order</span> of the typical performance thresholds. In addition, we provide some conditions for classification of the graph ensembles and demonstrate explicitly some examples for the difference in thresholds.</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/scitech">SciTech Connect</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 <span class="hlt">order</span> 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>
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<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/2008PhRvL.101n0503O','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008PhRvL.101n0503O"><span><span class="hlt">Approximate</span> Locality for Quantum Systems on Graphs</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Osborne, Tobias J.</p>
<p>2008-10-01</p>
<p>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 <span class="hlt">approximate</span> 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 <span class="hlt">approximately</span> local continuous-time quantum walk.</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>
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<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://hdl.handle.net/2060/20110023853','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20110023853"><span>Capacitor-Chain Successive-<span class="hlt">Approximation</span> ADC</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Cunningham, Thomas</p>
<p>2003-01-01</p>
<p>A proposed successive-<span class="hlt">approximation</span> analog-to-digital converter (ADC) would contain a capacitively terminated chain of identical capacitor cells. Like a conventional successive-<span class="hlt">approximation</span> 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, <span class="hlt">approximately</span> 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 <span class="hlt">approximately</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA551784','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA551784"><span>Optimal Budget Allocation for Sample Average <span class="hlt">Approximation</span></span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>2011-06-01</p>
<p>an optimization algorithm applied to the sample average problem. We examine the convergence rate of the estimator as the computing budget tends to...regime for the optimization algorithm . 1 Introduction Sample average <span class="hlt">approximation</span> (SAA) is a frequently used approach to solving stochastic programs...appealing due to its simplicity and the fact that a large number of standard optimization algorithms are often available to optimize the resulting sample</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015PhDT.......103N','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015PhDT.......103N"><span><span class="hlt">Approximation</span> methods in relativistic eigenvalue perturbation theory</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Noble, Jonathan Howard</p>
<p></p>
<p>In this dissertation, three questions, concerning <span class="hlt">approximation</span> methods for the eigenvalues of quantum mechanical systems, are investigated: (i) What is a pseudo--Hermitian Hamiltonian, and how can its eigenvalues be <span class="hlt">approximated</span> 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 <span class="hlt">approximated</span> 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 <span class="hlt">approximated</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19930007473','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19930007473"><span><span class="hlt">Approximation</span> methods for stochastic petri nets</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Jungnitz, Hauke Joerg</p>
<p>1992-01-01</p>
<p>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, <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> converges quickly and shows reasonable accuracy in most cases. The convergence of Marie's is slower, but the accuracy is generally better. Delay</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA049995','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA049995"><span><span class="hlt">Approximate</span> Solutions for Certain Optimal Stopping Problems</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>1978-01-05</p>
<p>one-armed bandit problem) has arisen in a number of statistical applications (Chernoff and Ray (1965);, Chernoff (±9&]), Mallik (1971)): Let X(t... Mallik (1971) and Chernoff (1972). These previous <span class="hlt">approximations</span> were determined without the benefit of the "correction for continuity" given in (5.1...Vol. 1, 3rd edition, John Wiley and Sons, Inc., New York» 7. Mallik , A.K» (1971), "Sequential estimation of the common mean of two normal</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/17126801','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/17126801"><span><span class="hlt">Approximation</span> for limit cycles and their isochrons.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Demongeot, Jacques; Françoise, Jean-Pierre</p>
<p>2006-12-01</p>
<p>Local analysis of trajectories of dynamical systems near an attractive periodic orbit displays the notion of asymptotic phase and isochrons. These notions are quite useful in applications to biosciences. In this note, we give an expression for the first <span class="hlt">approximation</span> of equations of isochrons in the setting of perturbations of polynomial Hamiltonian systems. This method can be generalized to perturbations of systems that have a polynomial integral factor (like the Lotka-Volterra equation).</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA225292','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA225292"><span><span class="hlt">Approximations</span> of Thermoelastic and Viscoelastic Control Systems</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>1990-06-01</p>
<p>parabolic partial differential equations. The development of computational algorithms for designing controllers for such systems is an Immenselv complex...hereditary differential system on Rr , then <span class="hlt">approximate</span> the "’historv" or -’memory- term (i.e.. the integral term in i.S)). In this paper we will use a... variation introduced by Fabiano and Ito ([FI]) of the averaging scheme considered by Banks and Burns ([BB]) for the second stage. The idea of the "’AVE</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24206291','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24206291"><span>Block correlated second <span class="hlt">order</span> perturbation theory with a generalized valence bond reference function.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Xu, Enhua; Li, Shuhua</p>
<p>2013-11-07</p>
<p>The block correlated second-<span class="hlt">order</span> perturbation theory with a generalized valence bond (GVB) reference (GVB-BCPT2) is proposed. In this approach, each geminal in the GVB reference is considered as a "multi-orbital" block (a subset of spin orbitals), and each occupied or virtual spin orbital is also taken as a single block. The <span class="hlt">zeroth-order</span> Hamiltonian is set to be the summation of the individual Hamiltonians of all blocks (with explicit two-electron operators within each geminal) so that the GVB reference function and all excited configuration functions are its eigenfunctions. The GVB-BCPT2 energy can be directly obtained without iteration, just like the second <span class="hlt">order</span> Mo̸ller-Plesset perturbation method (MP2), both of which are size consistent. We have applied this GVB-BCPT2 method to investigate the equilibrium distances and spectroscopic constants of 7 diatomic molecules, conformational energy differences of 8 small molecules, and bond-breaking potential energy profiles in 3 systems. GVB-BCPT2 is demonstrated to have noticeably better performance than MP2 for systems with significant multi-reference character, and provide reasonably accurate results for some systems with large active spaces, which are beyond the capability of all CASSCF-based methods.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29671043','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29671043"><span>Local <span class="hlt">approximation</span> of a metapopulation's equilibrium.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Barbour, A D; McVinish, R; Pollett, P K</p>
<p>2018-04-18</p>
<p>We consider the <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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.</p>
</li>
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<ol class="result-class" start="281">
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/9783204','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/9783204"><span>Compression of strings with <span class="hlt">approximate</span> repeats.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Allison, L; Edgoose, T; Dix, T I</p>
<p>1998-01-01</p>
<p>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 <span class="hlt">approximate</span> 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 <span class="hlt">approximation</span> algorithm is also given. The model is further extended to include <span class="hlt">approximate</span> reverse complementary repeats when analyzing DNA strings. Tests include the recovery of parameter estimates from known sources and applications to real DNA strings.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2009LNCS.5687...98E','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2009LNCS.5687...98E"><span>New Hardness Results for Diophantine <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>Eisenbrand, Friedrich; Rothvoß, Thomas</p>
<p></p>
<p>We revisit simultaneous Diophantine <span class="hlt">approximation</span>, 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 <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> within a factor 2 n unless {textrm{P}} = NP.</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('http://adsabs.harvard.edu/abs/2016JHyd..535..340A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JHyd..535..340A"><span>Green-Ampt <span class="hlt">approximations</span>: A comprehensive analysis</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Ali, Shakir; Islam, Adlul; Mishra, P. K.; Sikka, Alok K.</p>
<p>2016-04-01</p>
<p>Green-Ampt (GA) model and its modifications are widely used for simulating infiltration process. Several explicit <span class="hlt">approximate</span> solutions to the implicit GA model have been developed with varying degree of accuracy. In this study, performance of nine explicit <span class="hlt">approximations</span> 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 <span class="hlt">approximate</span> GA models for solving variety of hydrological problems.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017APS..MARC25005B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017APS..MARC25005B"><span>A Gaussian <span class="hlt">Approximation</span> Potential for Silicon</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Bernstein, Noam; Bartók, Albert; Kermode, James; Csányi, Gábor</p>
<p></p>
<p>We present an interatomic potential for silicon using the Gaussian <span class="hlt">Approximation</span> Potential (GAP) approach, which uses the Gaussian process regression method to <span class="hlt">approximate</span> the reference potential energy surface as a sum of atomic energies. Each atomic energy is <span class="hlt">approximated</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5331172','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5331172"><span>Ranking Support Vector Machine with Kernel <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>Dou, Yong</p>
<p>2017-01-01</p>
<p>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 <span class="hlt">approximation</span> to avoid computing the kernel matrix. We explore two types of kernel <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span>. 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</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28293256','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28293256"><span>Ranking Support Vector Machine with Kernel <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, Kai; Li, Rongchun; Dou, Yong; Liang, Zhengfa; Lv, Qi</p>
<p>2017-01-01</p>
<p>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 <span class="hlt">approximation</span> to avoid computing the kernel matrix. We explore two types of kernel <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span>. 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016IAUS..308...97N','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016IAUS..308...97N"><span>An Origami <span class="hlt">Approximation</span> to the Cosmic Web</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Neyrinck, Mark C.</p>
<p>2016-10-01</p>
<p>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 <span class="hlt">approximation</span>, 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 <span class="hlt">approximation</span> 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-<span class="hlt">approximation</span> models giving velocity fields for the first time.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26587963','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26587963"><span><span class="hlt">Approximate</span> number and <span class="hlt">approximate</span> time discrimination each correlate with school math abilities in young children.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Odic, Darko; Lisboa, Juan Valle; Eisinger, Robert; Olivera, Magdalena Gonzalez; Maiche, Alejandro; Halberda, Justin</p>
<p>2016-01-01</p>
<p>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 <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> number discrimination task, an <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> number precision and extend those findings by showing that math abilities also correlate with <span class="hlt">approximate</span> time precision. But, contrary to <span class="hlt">approximate</span> number and time sharing common representations, we find that each of these dimensions uniquely correlates with formal math: <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> number and <span class="hlt">approximate</span> time and further clarify the relationship between quantity representations and mathematics. Copyright </p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/19151097','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/19151097"><span>Biochemical simulations: stochastic, <span class="hlt">approximate</span> stochastic and hybrid approaches.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Pahle, Jürgen</p>
<p>2009-01-01</p>
<p>Computer simulations have become an invaluable tool to study the sometimes counterintuitive temporal dynamics of (bio-)chemical systems. In particular, stochastic simulation methods have attracted increasing interest recently. In contrast to the well-known deterministic approach based on ordinary differential equations, they can capture effects that occur due to the underlying discreteness of the systems and random fluctuations in molecular numbers. Numerous stochastic, <span class="hlt">approximate</span> stochastic and hybrid simulation methods have been proposed in the literature. In this article, they are systematically reviewed in <span class="hlt">order</span> to guide the researcher and help her find the appropriate method for a specific problem.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2638628','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2638628"><span>Biochemical simulations: stochastic, <span class="hlt">approximate</span> stochastic and hybrid approaches</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p></p>
<p>2009-01-01</p>
<p>Computer simulations have become an invaluable tool to study the sometimes counterintuitive temporal dynamics of (bio-)chemical systems. In particular, stochastic simulation methods have attracted increasing interest recently. In contrast to the well-known deterministic approach based on ordinary differential equations, they can capture effects that occur due to the underlying discreteness of the systems and random fluctuations in molecular numbers. Numerous stochastic, <span class="hlt">approximate</span> stochastic and hybrid simulation methods have been proposed in the literature. In this article, they are systematically reviewed in <span class="hlt">order</span> to guide the researcher and help her find the appropriate method for a specific problem. PMID:19151097</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017ZaMP...68...82C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017ZaMP...68...82C"><span>Asymptotic <span class="hlt">approximations</span> for pure bending of thin cylindrical shells</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Coman, Ciprian D.</p>
<p>2017-08-01</p>
<p>A simplified partial wrinkling scenario for in-plane bending of thin cylindrical shells is explored by using several asymptotic strategies. The eighth-<span class="hlt">order</span> boundary eigenvalue problem investigated here originates in the Donnel-Mushtari-Vlasov shallow shell theory coupled with a linear membrane pre-bifurcation state. It is shown that the corresponding neutral stability curve is amenable to a detailed asymptotic analysis based on the method of multiple scales. This is further complemented by an alternative WKB <span class="hlt">approximation</span> that provides comparable information with significantly less effort.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012APS..MARB54007S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012APS..MARB54007S"><span>Asymptotically inspired moment-closure <span class="hlt">approximation</span> for adaptive networks</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Shkarayev, Maxim; Shaw, Leah</p>
<p>2012-02-01</p>
<p>Adaptive social networks, in which nodes and network structure co-evolve, are often described using a mean-field system of equations for the density of node and link types. These equations constitute an open system due to dependence on higher <span class="hlt">order</span> topological structures. We propose a moment-closure <span class="hlt">approximation</span> based on the analytical description of the system in an asymptotic regime. We apply the proposed approach to two examples of adaptive networks: recruitment to a cause model and epidemic spread model. We show a good agreement between the improved mean-field prediction and simulations of the full network system.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2013APS..MARZ29005S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2013APS..MARZ29005S"><span>Asymptotically inspired moment-closure <span class="hlt">approximation</span> for adaptive networks</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Shkarayev, Maxim</p>
<p>2013-03-01</p>
<p>Dynamics of adaptive social networks, in which nodes and network structure co-evolve, are often described using a mean-field system of equations for the density of node and link types. These equations constitute an open system due to dependence on higher <span class="hlt">order</span> topological structures. We propose a systematic approach to moment closure <span class="hlt">approximation</span> based on the analytical description of the system in an asymptotic regime. We apply the proposed approach to two examples of adaptive networks: recruitment to a cause model and adaptive epidemic model. We show a good agreement between the mean-field prediction and simulations of the full network system.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016IAUS..308...69H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016IAUS..308...69H"><span>The Zeldovich & Adhesion <span class="hlt">approximations</span> and applications to the local universe</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Hidding, Johan; van de Weygaert, Rien; Shandarin, Sergei</p>
<p>2016-10-01</p>
<p>The Zeldovich <span class="hlt">approximation</span> (ZA) predicts the formation of a web of singularities. While these singularities may only exist in the most formal interpretation of the ZA, they provide a powerful tool for the analysis of initial conditions. We present a novel method to find the skeleton of the resulting cosmic web based on singularities in the primordial deformation tensor and its higher <span class="hlt">order</span> derivatives. We show that the A 3 lines predict the formation of filaments in a two-dimensional model. We continue with applications of the adhesion model to visualise structures in the local (z < 0.03) universe.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19830044831&hterms=averaged+lagrangian&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Daveraged%2Blagrangian','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19830044831&hterms=averaged+lagrangian&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Daveraged%2Blagrangian"><span>Microscopic Lagrangian description of warm plasmas. IV - Macroscopic <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>Kim, H.; Crawford, F. W.</p>
<p>1983-01-01</p>
<p>The averaged-Lagrangian method is applied to linear wave propagation and nonlinear three-wave interaction in a warm magnetoplasma, in the macroscopic <span class="hlt">approximation</span>. The microscopic Lagrangian treated by Kim and Crawford (1977) and by Galloway and Crawford (1977) is first expanded to third <span class="hlt">order</span> in perturbation. Velocity integration is then carried out, before applying Hamilton's principle to obtain a general description of wave propagation and coupling. The results are specialized to the case of interaction between two electron plasma waves and an Alfven wave. The method is shown to be more powerful than the alternative possibility of working from the beginning with a macroscopic Lagrangian density.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015JPhB...48d5503A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015JPhB...48d5503A"><span><span class="hlt">Approximate</span> analytical solutions of a pair of coupled anharmonic oscillators</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Alam, Nasir; Mandal, Swapan; Öhberg, Patrik</p>
<p>2015-02-01</p>
<p>The Hamiltonian and the corresponding equations of motion involving the field operators of two quartic anharmonic oscillators indirectly coupled via a linear oscillator are constructed. The <span class="hlt">approximate</span> analytical solutions of the coupled differential equations involving the non-commuting field operators are solved up to the second <span class="hlt">order</span> in the anharmonic coupling. In the absence of nonlinearity these solutions are used to calculate the second <span class="hlt">order</span> variances and hence the squeezing in pure and in mixed modes. The higher <span class="hlt">order</span> quadrature squeezing and the amplitude squared squeezing of various field modes are also investigated where the squeezing in pure and in mixed modes are found to be suppressed. Moreover, the absence of a nonlinearity prohibits the higher <span class="hlt">order</span> quadrature and higher <span class="hlt">ordered</span> amplitude squeezing of the input coherent states. It is established that the mere coupling of two oscillators through a third one is unable to produce any squeezing effects of input coherent light, but the presence of a nonlinear interaction may provide squeezed states and other nonclassical phenomena.</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>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22375874-strong-washout-approximation-resonant-leptogenesis','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22375874-strong-washout-approximation-resonant-leptogenesis"><span>Strong washout <span class="hlt">approximation</span> to resonant leptogenesis</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Garbrecht, Björn; Gautier, Florian; Klaric, Juraj, E-mail: garbrecht@tum.de, E-mail: florian.gautier@tum.de, E-mail: juraj.klaric@tum.de</p>
<p></p>
<p>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 <span class="hlt">approximated</span> 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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</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>
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<ol class="result-class" start="301">
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA194685','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA194685"><span><span class="hlt">Approximations</span> and Implementations of Nonlinear Filtering Schemes.</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>1988-02-01</p>
<p>17) 0 0 3) P(fn) - (pf)n 4) Pf v0 - (Po <-> dp - (p0 dm is invariant under f (i.e. for all measurable A: (f’l(A)) - p(A) Remark: The Perron - Frobenius ...invariant density of the map f is then nothing else than the fixed point of the Perron - Frobenius operator. The following theorem by Lasota and Yorke [8...transition matrix R is defined. With this construct, the Perron - Frobenius operator is effectively 39 A A . w7 <span class="hlt">approximated</span> (exact for Markov Maps)by</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhRvE..96d2139K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhRvE..96d2139K"><span>Stochastic maps, continuous <span class="hlt">approximation</span>, and stable distribution</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Kessler, David A.; Burov, Stanislav</p>
<p>2017-10-01</p>
<p>A continuous <span class="hlt">approximation</span> framework for general nonlinear stochastic as well as deterministic discrete maps is developed. For the stochastic map with uncorelated Gaussian noise, by successively applying the Itô lemma, we obtain a Langevin type of equation. Specifically, we show how nonlinear maps give rise to a Langevin description that involves multiplicative noise. The multiplicative nature of the noise induces an additional effective force, not present in the absence of noise. We further exploit the continuum description and provide an explicit formula for the stable distribution of the stochastic map and conditions for its existence. Our results are in good agreement with numerical simulations of several maps.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JMP....58h3506Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JMP....58h3506Z"><span>Localization and stationary phase <span class="hlt">approximation</span> on supermanifolds</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Zakharevich, Valentin</p>
<p>2017-08-01</p>
<p>Given an odd vector field Q on a supermanifold M and a Q-invariant density μ on M, under certain compactness conditions on Q, the value of the integral ∫Mμ is determined by the value of μ on any neighborhood of the vanishing locus N of Q. We present a formula for the integral in the case where N is a subsupermanifold which is appropriately non-degenerate with respect to Q. In the process, we discuss the linear algebra necessary to express our result in a coordinate independent way. We also extend the stationary phase <span class="hlt">approximation</span> and the Morse-Bott lemma to supermanifolds.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JMP....59a2107A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JMP....59a2107A"><span>Born-Oppenheimer <span class="hlt">approximation</span> for a singular system</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Akbas, Haci; Turgut, O. Teoman</p>
<p>2018-01-01</p>
<p>We discuss a simple singular system in one dimension, two heavy particles interacting with a light particle via an attractive contact interaction and not interacting among themselves. It is natural to apply the Born-Oppenheimer <span class="hlt">approximation</span> to this problem. We present a detailed discussion of this approach; the advantage of this simple model is that one can estimate the error terms self-consistently. Moreover, a Fock space approach to this problem is presented where an expansion can be proposed to get higher <span class="hlt">order</span> corrections. A slight modification of the same problem in which the light particle is relativistic is discussed in a later section by neglecting pair creation processes. Here, the second quantized description is more challenging, but with some care, one can recover the first <span class="hlt">order</span> expression exactly.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018EPJP..133...37A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018EPJP..133...37A"><span>A new numerical <span class="hlt">approximation</span> of the fractal ordinary differential equation</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Atangana, Abdon; Jain, Sonal</p>
<p>2018-02-01</p>
<p>The concept of fractal medium is present in several real-world problems, for instance, in the geological formation that constitutes the well-known subsurface water called aquifers. However, attention has not been quite devoted to modeling for instance, the flow of a fluid within these media. We deem it important to remind the reader that the concept of fractal derivative is not to represent the fractal sharps but to describe the movement of the fluid within these media. Since this class of ordinary differential equations is highly complex to solve analytically, we present a novel numerical scheme that allows to solve fractal ordinary differential equations. Error analysis of the method is also presented. Application of the method and numerical <span class="hlt">approximation</span> are presented for fractal <span class="hlt">order</span> differential equation. The stability and the convergence of the numerical schemes are investigated in detail. Also some exact solutions of fractal <span class="hlt">order</span> differential equations are presented and finally some numerical simulations are presented.</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/scitech">SciTech Connect</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 <span class="hlt">order</span> 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('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 <span class="hlt">orders</span> 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('http://adsabs.harvard.edu/abs/2013CoPhC.184.1045J','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2013CoPhC.184.1045J"><span>ASP: Automated symbolic computation of <span class="hlt">approximate</span> symmetries of differential equations</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Jefferson, G. F.; Carminati, J.</p>
<p>2013-03-01</p>
<p>A recent paper (Pakdemirli et al. (2004) [12]) compared three methods of determining <span class="hlt">approximate</span> symmetries of differential equations. Two of these methods are well known and involve either a perturbation of the classical Lie symmetry generator of the differential system (Baikov, Gazizov and Ibragimov (1988) [7], Ibragimov (1996) [6]) or a perturbation of the dependent variable/s and subsequent determination of the classical Lie point symmetries of the resulting coupled system (Fushchych and Shtelen (1989) [11]), both up to a specified <span class="hlt">order</span> in the perturbation parameter. The third method, proposed by Pakdemirli, Yürüsoy and Dolapçi (2004) [12], simplifies the calculations required by Fushchych and Shtelen's method through the assignment of arbitrary functions to the non-linear components prior to computing symmetries. All three methods have been implemented in the new MAPLE package ASP (Automated Symmetry Package) which is an add-on to the MAPLE symmetry package DESOLVII (Vu, Jefferson and Carminati (2012) [25]). To our knowledge, this is the first computer package to automate all three methods of determining <span class="hlt">approximate</span> symmetries for differential systems. Extensions to the theory have also been suggested for the third method and which generalise the first method to systems of differential equations. Finally, a number of <span class="hlt">approximate</span> symmetries and corresponding solutions are compared with results in the literature.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19870046420&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=19870046420&hterms=rational+better&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3Drational%2Bbetter"><span>Nonlinear programming extensions to rational function <span class="hlt">approximations</span> of unsteady aerodynamics</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Tiffany, Sherwood H.; Adams, William M., Jr.</p>
<p>1987-01-01</p>
<p>This paper deals with <span class="hlt">approximating</span> unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft. Two methods of formulating these <span class="hlt">approximations</span> are extended to include both the same flexibility in constraining them and the same methodology in optimizing nonlinear parameters as another currently used 'extended least-squares' method. Optimal selection of 'nonlinear' parameters is made in each of the three methods by use of the same nonlinear (nongradient) optimizer. The objective of the nonlinear optimization is to obtain rational <span class="hlt">approximations</span> to the unsteady aerodynamics whose state-space realization is of lower <span class="hlt">order</span> than that required when no optimization of the nonlinear terms is performed. The free 'linear' parameters are determined using least-squares matrix techniques on a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from the different approaches are described, and results are presented which show comparative evaluations from application of each of the extended methods to a numerical example. The results obtained for the example problem show a significant (up to 63 percent) reduction in the number of differential equations used to represent the unsteady aerodynamic forces in linear time-invariant equations of motion as compared to a conventional method in which nonlinear terms are not optimized.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20020038859','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20020038859"><span>Variable <span class="hlt">Order</span> and Distributed <span class="hlt">Order</span> Fractional Operators</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Lorenzo, Carl F.; Hartley, Tom T.</p>
<p>2002-01-01</p>
<p>Many physical processes appear to exhibit fractional <span class="hlt">order</span> behavior that may vary with time or space. The continuum of <span class="hlt">order</span> in the fractional calculus allows the <span class="hlt">order</span> of the fractional operator to be considered as a variable. This paper develops the concept of variable and distributed <span class="hlt">order</span> fractional operators. Definitions based on the Riemann-Liouville definitions are introduced and behavior of the operators is studied. Several time domain definitions that assign different arguments to the <span class="hlt">order</span> q in the Riemann-Liouville definition are introduced. For each of these definitions various characteristics are determined. These include: time invariance of the operator, operator initialization, physical realization, linearity, operational transforms. and memory characteristics of the defining kernels. A measure (m2) for memory retentiveness of the <span class="hlt">order</span> history is introduced. A generalized linear argument for the <span class="hlt">order</span> q allows the concept of "tailored" variable <span class="hlt">order</span> fractional operators whose a, memory may be chosen for a particular application. Memory retentiveness (m2) and <span class="hlt">order</span> dynamic behavior are investigated and applications are shown. The concept of distributed <span class="hlt">order</span> operators where the <span class="hlt">order</span> of the time based operator depends on an additional independent (spatial) variable is also forwarded. Several definitions and their Laplace transforms are developed, analysis methods with these operators are demonstrated, and examples shown. Finally operators of multivariable and distributed <span class="hlt">order</span> are defined in their various applications are outlined.</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('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('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-<span class="hlt">order</span> <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://adsabs.harvard.edu/abs/2008JMAA..345..845C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008JMAA..345..845C"><span><span class="hlt">Approximate</span> controllability of a system of parabolic equations with delay</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Carrasco, Alexander; Leiva, Hugo</p>
<p>2008-09-01</p>
<p>In this paper we give necessary and sufficient conditions for the <span class="hlt">approximate</span> controllability of the following system of parabolic equations with delay: where [Omega] is a bounded domain in , D is an n×n nondiagonal matrix whose eigenvalues are semi-simple with nonnegative real part, the control and B[set membership, variant]L(U,Z) with , . The standard notation zt(x) defines a function from [-[tau],0] to (with x fixed) by zt(x)(s)=z(t+s,x), -[tau][less-than-or-equals, slant]s[less-than-or-equals, slant]0. Here [tau][greater-or-equal, slanted]0 is the maximum delay, which is supposed to be finite. We assume that the operator is linear and bounded, and [phi]0[set membership, variant]Z, [phi][set membership, variant]L2([-[tau],0];Z). To this end: First, we reformulate this system into a standard first-<span class="hlt">order</span> delay equation. Secondly, the semigroup associated with the first-<span class="hlt">order</span> delay equation on an appropriate product space is expressed as a series of strongly continuous semigroups and orthogonal projections related with the eigenvalues of the Laplacian operator (); this representation allows us to reduce the controllability of this partial differential equation with delay to a family of ordinary delay equations. Finally, we use the well-known result on the rank condition for the <span class="hlt">approximate</span> controllability of delay system to derive our main result.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19750022560','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19750022560"><span>Spline function <span class="hlt">approximation</span> techniques for image geometric distortion representation. [for registration of multitemporal remote sensor imagery</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Anuta, P. E.</p>
<p>1975-01-01</p>
<p>Least squares <span class="hlt">approximation</span> techniques were developed for use in computer aided correction of spatial image distortions for registration of multitemporal remote sensor imagery. Polynomials were first used to define image distortion over the entire two dimensional image space. Spline functions were then investigated to determine if the combination of lower <span class="hlt">order</span> polynomials could <span class="hlt">approximate</span> a higher <span class="hlt">order</span> distortion with less computational difficulty. Algorithms for generating <span class="hlt">approximating</span> functions were developed and applied to the description of image distortion in aircraft multispectral scanner imagery. Other applications of the techniques were suggested for earth resources data processing areas other than geometric distortion representation.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2005AGUSMGP22A..06G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2005AGUSMGP22A..06G"><span>New Tests of the Fixed Hotspot <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>Gordon, R. G.; Andrews, D. L.; Horner-Johnson, B. C.; Kumar, R. R.</p>
<p>2005-05-01</p>
<p>We present new methods for estimating uncertainties in plate reconstructions relative to the hotspots and new tests of the fixed hotspot <span class="hlt">approximation</span>. We find no significant motion between Pacific hotspots, on the one hand, and Indo-Atlantic hotspots, on the other, for the past ~ 50 Myr, but large and significant apparent motion before 50 Ma. Whether this motion is truly due to motion between hotspots or alternatively due to flaws in the global plate motion circuit can be tested with paleomagnetic data. These tests give results consistent with the fixed hotspot <span class="hlt">approximation</span> and indicate significant misfits when a relative plate motion circuit through Antarctica is employed for times before 50 Ma. If all of the misfit to the global plate motion circuit is due to motion between East and West Antarctica, then that motion is 800 ± 500 km near the Ross Sea Embayment and progressively less along the Trans-Antarctic Mountains toward the Weddell Sea. Further paleomagnetic tests of the fixed hotspot <span class="hlt">approximation</span> can be made. Cenozoic and Cretaceous paleomagnetic data from the Pacific plate, along with reconstructions of the Pacific plate relative to the hotspots, can be used to estimate an apparent polar wander (APW) path of Pacific hotspots. An APW path of Indo-Atlantic hotspots can be similarly estimated (e.g. Besse & Courtillot 2002). If both paths diverge in similar ways from the north pole of the hotspot reference frame, it would indicate that the hotspots have moved in unison relative to the spin axis, which may be attributed to true polar wander. If the two paths diverge from one another, motion between Pacific hotspots and Indo-Atlantic hotspots would be indicated. The general agreement of the two paths shows that the former is more important than the latter. The data require little or no motion between groups of hotspots, but up to ~10 mm/yr of motion is allowed within uncertainties. The results disagree, in particular, with the recent extreme interpretation of</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015PhRvE..92b3012F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015PhRvE..92b3012F"><span>Projection-free <span class="hlt">approximate</span> balanced truncation of large unstable systems</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Flinois, Thibault L. B.; Morgans, Aimee S.; Schmid, Peter J.</p>
<p>2015-08-01</p>
<p>In this article, we show that the projection-free, snapshot-based, balanced truncation method can be applied directly to unstable systems. We prove that even for unstable systems, the unmodified balanced proper orthogonal decomposition algorithm theoretically yields a converged transformation that balances the Gramians (including the unstable subspace). We then apply the method to a spatially developing unstable system and show that it results in reduced-<span class="hlt">order</span> models of similar quality to the ones obtained with existing methods. Due to the unbounded growth of unstable modes, a practical restriction on the final impulse response simulation time appears, which can be adjusted depending on the desired <span class="hlt">order</span> of the reduced-<span class="hlt">order</span> model. Recommendations are given to further reduce the cost of the method if the system is large and to improve the performance of the method if it does not yield acceptable results in its unmodified form. Finally, the method is applied to the linearized flow around a cylinder at Re = 100 to show that it actually is able to accurately reproduce impulse responses for more realistic unstable large-scale systems in practice. The well-established <span class="hlt">approximate</span> balanced truncation numerical framework therefore can be safely applied to unstable systems without any modifications. Additionally, balanced reduced-<span class="hlt">order</span> models can readily be obtained even for large systems, where the computational cost of existing methods is prohibitive.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28287440','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28287440"><span><span class="hlt">Approximate</span> Sensory Data Collection: A Survey.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Cheng, Siyao; Cai, Zhipeng; Li, Jianzhong</p>
<p>2017-03-10</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 approximatedatacollectionalgorithms. Weclassifythemintothreecategories: themodel-basedones, 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/21067438-bloch-approximation-periodically-perforated-media','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/21067438-bloch-approximation-periodically-perforated-media"><span>The Bloch <span class="hlt">Approximation</span> in Periodically Perforated Media</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Conca, C.; Gomez, D., E-mail: gomezdel@unican.es; Lobo, M.</p>
<p>2005-06-15</p>
<p>We consider a periodically heterogeneous and perforated medium filling an open domain {omega} of R{sup N}. Assuming that the size of the periodicity of the structure and of the holes is O({epsilon}),we study the asymptotic behavior, as {epsilon} {sup {yields}} 0, of the solution of an elliptic boundary value problem with strongly oscillating coefficients posed in {omega}{sup {epsilon}}({omega}{sup {epsilon}} being {omega} minus the holes) with a Neumann condition on the boundary of the holes. We use Bloch wave decomposition to introduce an <span class="hlt">approximation</span> of the solution in the energy norm which can be computed from the homogenized solution and themore » first Bloch eigenfunction. We first consider the case where {omega}is R{sup N} and then localize the problem for abounded domain {omega}, considering a homogeneous Dirichlet condition on the boundary of {omega}.« less</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017AIPC.1905c0017I','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017AIPC.1905c0017I"><span><span class="hlt">Approximate</span> analytical modeling of leptospirosis infection</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Ismail, Nur Atikah; Azmi, Amirah; Yusof, Fauzi Mohamed; Ismail, Ahmad Izani</p>
<p>2017-11-01</p>
<p>Leptospirosis is an infectious disease carried by rodents which can cause death in humans. The disease spreads directly through contact with feces, urine or through bites of infected rodents and indirectly via water contaminated with urine and droppings from them. Significant increase in the number of leptospirosis cases in Malaysia caused by the recent severe floods were recorded during heavy rainfall season. Therefore, to understand the dynamics of leptospirosis infection, a mathematical model based on fractional differential equations have been developed and analyzed. In this paper an <span class="hlt">approximate</span> analytical method, the multi-step Laplace Adomian decomposition method, has been used to conduct numerical simulations so as to gain insight on the spread of leptospirosis infection.</p>
</li>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3257063','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3257063"><span>Is probabilistic bias analysis <span class="hlt">approximately</span> Bayesian?</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>MacLehose, Richard F.; Gustafson, Paul</p>
<p>2011-01-01</p>
<p>Case-control studies are particularly susceptible to differential exposure misclassification when exposure status is determined following incident case status. Probabilistic bias analysis methods have been developed as ways to adjust standard effect estimates based on the sensitivity and specificity of exposure misclassification. The iterative sampling method advocated in probabilistic bias analysis bears a distinct resemblance to a Bayesian adjustment; however, it is not identical. Furthermore, without a formal theoretical framework (Bayesian or frequentist), the results of a probabilistic bias analysis remain somewhat difficult to interpret. We describe, both theoretically and empirically, the extent to which probabilistic bias analysis can be viewed as <span class="hlt">approximately</span> Bayesian. While the differences between probabilistic bias analysis and Bayesian approaches to misclassification can be substantial, these situations often involve unrealistic prior specifications and are relatively easy to detect. Outside of these special cases, probabilistic bias analysis and Bayesian approaches to exposure misclassification in case-control studies appear to perform equally well. PMID:22157311</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016APS..APR.L1045A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016APS..APR.L1045A"><span>Comparing numerical and analytic <span class="hlt">approximate</span> gravitational waveforms</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Afshari, Nousha; Lovelace, Geoffrey; SXS Collaboration</p>
<p>2016-03-01</p>
<p>A direct observation of gravitational waves will test Einstein's theory of general relativity under the most extreme conditions. The Laser Interferometer Gravitational-Wave Observatory, or LIGO, began searching for gravitational waves in September 2015 with three times the sensitivity of initial LIGO. To help Advanced LIGO detect as many gravitational waves as possible, a major research effort is underway to accurately predict the expected waves. In this poster, I will explore how the gravitational waveform produced by a long binary-black-hole inspiral, merger, and ringdown is affected by how fast the larger black hole spins. In particular, I will present results from simulations of merging black holes, completed using the Spectral Einstein Code (black-holes.org/SpEC.html), including some new, long simulations designed to mimic black hole-neutron star mergers. I will present comparisons of the numerical waveforms with analytic <span class="hlt">approximations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JMP....58j2104F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JMP....58j2104F"><span>Some remarks concerning the centrifugal term <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>Ferreira, F. J. S.; Bezerra, V. B.</p>
<p>2017-10-01</p>
<p>We generalize the Pekeris <span class="hlt">approximation</span> for the centrifugal term potential, l/(l +1 ) r2 , and use this to obtain the solutions of the radial Schrödinger equation for the arbitrary angular quantum number, l, of the Hulthén potential. We also obtain the expressions for the bound state energies corresponding to this potential and calculate their values for different states and compare with other results presented in the literature. We also consider some models of physical potentials, namely, the Eckart potential, the Poschl-Teller potentials, the Rosen-Morse potential, the Woods-Saxon potential, and the Manning-Rosen potential. Thus, following straightforward the example corresponding to the Hulthén potential, we show what the radial solutions and the energy spectra for these potentials are.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/5287255-approximate-flavor-symmetries-lepton-sector','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/5287255-approximate-flavor-symmetries-lepton-sector"><span><span class="hlt">Approximate</span> flavor symmetries in the lepton sector</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Rasin, A.; Silva, J.P.</p>
<p>1994-01-01</p>
<p><span class="hlt">Approximate</span> flavor symmetries in the quark sector have been used as a handle on physics beyond the standard model. Because of the great interest in neutrino masses and mixings and the wealth of existing and proposed neutrino experiments it is important to extend this analysis to the leptonic sector. We show that in the seesaw mechanism the neutrino masses and mixing angles do not depend on the details of the right-handed neutrino flavor symmetry breaking, and are related by a simple formula. We propose several [ital Ansa]$[ital uml]---[ital tze] which relate different flavor symmetry-breaking parameters and find that the MSWmore » solution to the solar neutrino problem is always easily fit. Further, the [nu][sub [mu]-][nu][sub [tau</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19890009103&hterms=Agreement&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DAgreement','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19890009103&hterms=Agreement&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DAgreement"><span>Accelerated convergence for synchronous <span class="hlt">approximate</span> agreement</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Kearns, J. P.; Park, S. K.; Sjogren, J. A.</p>
<p>1988-01-01</p>
<p>The protocol for synchronous <span class="hlt">approximate</span> agreement presented by Dolev et. al. exhibits the undesirable property that a faulty processor, by the dissemination of a value arbitrarily far removed from the values held by good processors, may delay the termination of the protocol by an arbitrary amount of time. Such behavior is clearly undesirable in a fault tolerant dynamic system subject to hard real-time constraints. A mechanism is presented by which editing data suspected of being from Byzantine-failed processors can lead to quicker, predictable, convergence to an agreement value. Under specific assumptions about the nature of values transmitted by failed processors relative to those transmitted by good processors, a Monte Carlo simulation is presented whose qualitative results illustrate the trade-off between accelerated convergence and the accuracy of the value agreed upon.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014JPO....44.2433B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014JPO....44.2433B"><span><span class="hlt">Approximate</span> Stokes Drift Profiles in Deep Water</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Breivik, Øyvind; Janssen, Peter A. E. M.; Bidlot, Jean-Raymond</p>
<p>2014-09-01</p>
<p>A deep-water <span class="hlt">approximation</span> to the Stokes drift velocity profile is explored as an alternative to the monochromatic profile. The alternative profile investigated relies on the same two quantities required for the monochromatic profile, viz the Stokes transport and the surface Stokes drift velocity. Comparisons with parametric spectra and profiles under wave spectra from the ERA-Interim reanalysis and buoy observations reveal much better agreement than the monochromatic profile even for complex sea states. That the profile gives a closer match and a more correct shear has implications for ocean circulation models since the Coriolis-Stokes force depends on the magnitude and direction of the Stokes drift profile and Langmuir turbulence parameterizations depend sensitively on the shear of the profile. The alternative profile comes at no added numerical cost compared to the monochromatic profile.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JEI....27a3011L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JEI....27a3011L"><span>Local facet <span class="hlt">approximation</span> for image stitching</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Li, Jing; Lai, Shiming; Liu, Yu; Wang, Zhengming; Zhang, Maojun</p>
<p>2018-01-01</p>
<p>Image stitching aims at eliminating multiview parallax and generating a seamless panorama given a set of input images. This paper proposes a local adaptive stitching method, which could achieve both accurate and robust image alignments across the whole panorama. A transformation estimation model is introduced by <span class="hlt">approximating</span> the scene as a combination of neighboring facets. Then, the local adaptive stitching field is constructed using a series of linear systems of the facet parameters, which enables the parallax handling in three-dimensional space. We also provide a concise but effective global projectivity preserving technique that smoothly varies the transformations from local adaptive to global planar. The proposed model is capable of stitching both normal images and fisheye images. The efficiency of our method is quantitatively demonstrated in the comparative experiments on several challenging cases.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20150000116','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20150000116"><span>Exact and <span class="hlt">Approximate</span> Probabilistic Symbolic Execution</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Luckow, Kasper; Pasareanu, Corina S.; Dwyer, Matthew B.; Filieri, Antonio; Visser, Willem</p>
<p>2014-01-01</p>
<p>Probabilistic software analysis seeks to quantify the likelihood of reaching a target event under uncertain environments. Recent approaches compute probabilities of execution paths using symbolic execution, but do not support nondeterminism. Nondeterminism arises naturally when no suitable probabilistic model can capture a program behavior, e.g., for multithreading or distributed systems. In this work, we propose a technique, based on symbolic execution, to synthesize schedulers that resolve nondeterminism to maximize the probability of reaching a target event. To scale to large systems, we also introduce <span class="hlt">approximate</span> algorithms to search for good schedulers, speeding up established random sampling and reinforcement learning results through the quantification of path probabilities based on symbolic execution. We implemented the techniques in Symbolic PathFinder and evaluated them on nondeterministic Java programs. We show that our algorithms significantly improve upon a state-of- the-art statistical model checking algorithm, originally developed for Markov Decision Processes.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19960022271','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19960022271"><span>Spline <span class="hlt">Approximation</span> of Thin Shell Dynamics</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>delRosario, R. C. H.; Smith, R. C.</p>
<p>1996-01-01</p>
<p>A spline-based method for <span class="hlt">approximating</span> thin shell dynamics is presented here. While the method is developed in the context of the Donnell-Mushtari thin shell equations, it can be easily extended to the Byrne-Flugge-Lur'ye equations or other models for shells of revolution as warranted by applications. The primary requirements for the method include accuracy, flexibility and efficiency in smart material applications. To accomplish this, the method was designed to be flexible with regard to boundary conditions, material nonhomogeneities due to sensors and actuators, and inputs from smart material actuators such as piezoceramic patches. The accuracy of the method was also of primary concern, both to guarantee full resolution of structural dynamics and to facilitate the development of PDE-based controllers which ultimately require real-time implementation. Several numerical examples provide initial evidence demonstrating the efficacy of the method.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19910023149','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19910023149"><span>An <span class="hlt">approximate</span> Riemann solver for hypervelocity flows</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Jacobs, Peter A.</p>
<p>1991-01-01</p>
<p>We describe an <span class="hlt">approximate</span> Riemann solver for the computation of hypervelocity flows in which there are strong shocks and viscous interactions. The scheme has three stages, the first of which computes the intermediate states assuming isentropic waves. A second stage, based on the strong shock relations, may then be invoked if the pressure jump across either wave is large. The third stage interpolates the interface state from the two initial states and the intermediate states. The solver is used as part of a finite-volume code and is demonstrated on two test cases. The first is a high Mach number flow over a sphere while the second is a flow over a slender cone with an adiabatic boundary layer. In both cases the solver performs well.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19900042199&hterms=1060&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3D%2526%25231060','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19900042199&hterms=1060&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3D%2526%25231060"><span>Improved <span class="hlt">approximations</span> for control augmented structural synthesis</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Thomas, H. L.; Schmit, L. A.</p>
<p>1990-01-01</p>
<p>A methodology for control-augmented structural synthesis is presented for structure-control systems which can be modeled as an assemblage of beam, truss, and nonstructural mass elements augmented by a noncollocated direct output feedback control system. Truss areas, beam cross sectional dimensions, nonstructural masses and rotary inertias, and controller position and velocity gains are treated simultaneously as design variables. The structural mass and a control-system performance index can be minimized simultaneously, with design constraints placed on static stresses and displacements, dynamic harmonic displacements and forces, structural frequencies, and closed-loop eigenvalues and damping ratios. Intermediate design-variable and response-quantity concepts are used to generate new <span class="hlt">approximations</span> for displacements and actuator forces under harmonic dynamic loads and for system complex eigenvalues. This improves the overall efficiency of the procedure by reducing the number of complete analyses required for convergence. Numerical results which illustrate the effectiveness of the method are given.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19720053619&hterms=habibi&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dhabibi','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19720053619&hterms=habibi&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dhabibi"><span>Estimation of correlation functions by stochastic <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>Habibi, A.; Wintz, P. A.</p>
<p>1972-01-01</p>
<p>Consideration of the autocorrelation function of a zero-mean stationary random process. The techniques are applicable to processes with nonzero mean provided the mean is estimated first and subtracted. Two recursive techniques are proposed, both of which are based on the method of stochastic <span class="hlt">approximation</span> and assume a functional form for the correlation function that depends on a number of parameters that are recursively estimated from successive records. One technique uses a standard point estimator of the correlation function to provide estimates of the parameters that minimize the mean-square error between the point estimates and the parametric function. The other technique provides estimates of the parameters that maximize a likelihood function relating the parameters of the function to the random process. Examples are presented.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19790012619','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19790012619"><span><span class="hlt">Approximate</span> maximum likelihood decoding of block codes</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Greenberger, H. J.</p>
<p>1979-01-01</p>
<p><span class="hlt">Approximate</span> maximum likelihood decoding algorithms, based upon selecting a small set of candidate code words with the aid of the estimated probability of error of each received symbol, can give performance close to optimum with a reasonable amount of computation. By combining the best features of various algorithms and taking care to perform each step as efficiently as possible, a decoding scheme was developed which can decode codes which have better performance than those presently in use and yet not require an unreasonable amount of computation. The discussion of the details and tradeoffs of presently known efficient optimum and near optimum decoding algorithms leads, naturally, to the one which embodies the best features of all of them.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2000PhyA..280..566B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2000PhyA..280..566B"><span>Markovian <span class="hlt">approximation</span> in foreign exchange markets</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Baviera, Roberto; Vergni, Davide; Vulpiani, Angelo</p>
<p>2000-06-01</p>
<p>In this paper, using the exit-time statistic, we study the structure of the price variations for the high-frequency data set of the bid-ask Deutschemark/US dollar exchange rate quotes registered by the inter-bank Reuters network over the period October 1, 1992 to September 30, 1993. Having rejected random-walk models for the returns, we propose a Markovian model which reproduce the available information of the financial series. Besides the usual correlation analysis we have verified the validity of this model by means of other tools all inspired by information theory. These techniques are not only severe tests of the <span class="hlt">approximation</span> but also evidence of some aspects of the data series which have a clear financial relevance.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2009IEITI..92..149H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2009IEITI..92..149H"><span><span class="hlt">Approximation</span> Preserving Reductions among Item Pricing Problems</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Hamane, Ryoso; Itoh, Toshiya; Tomita, Kouhei</p>
<p></p>
<p>When a store sells items to customers, the store wishes to determine the prices of the items to maximize its profit. Intuitively, if the store sells the items with low (resp. high) prices, the customers buy more (resp. less) items, which provides less profit to the store. So it would be hard for the store to decide the prices of items. Assume that the store has a set V of n items and there is a set E of m customers who wish to buy those items, and also assume that each item i ∈ V has the production cost di and each customer ej ∈ E has the valuation vj on the bundle ej ⊆ V of items. When the store sells an item i ∈ V at the price ri, the profit for the item i is pi = ri - di. The goal of the store is to decide the price of each item to maximize its total profit. We refer to this maximization problem as the item pricing problem. In most of the previous works, the item pricing problem was considered under the assumption that pi ≥ 0 for each i ∈ V, however, Balcan, et al. [In Proc. of WINE, LNCS 4858, 2007] introduced the notion of “loss-leader, ” and showed that the seller can get more total profit in the case that pi < 0 is allowed than in the case that pi < 0 is not allowed. In this paper, we derive <span class="hlt">approximation</span> preserving reductions among several item pricing problems and show that all of them have algorithms with good <span class="hlt">approximation</span> ratio.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006JPRS...60..239L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006JPRS...60..239L"><span><span class="hlt">Approximate</span> direct georeferencing in national coordinates</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Legat, Klaus</p>
<p></p>
<p>Direct georeferencing has gained an increasing importance in photogrammetry and remote sensing. Thereby, the parameters of exterior orientation (EO) of an image sensor are determined by GPS/INS, yielding results in a global geocentric reference frame. Photogrammetric products like digital terrain models or orthoimages, however, are often required in national geodetic datums and mapped by national map projections, i.e., in "national coordinates". As the fundamental mathematics of photogrammetry is based on Cartesian coordinates, the scene restitution is often performed in a Cartesian frame located at some central position of the image block. The subsequent transformation to national coordinates is a standard problem in geodesy and can be done in a rigorous manner-at least if the formulas of the map projection are rigorous. Drawbacks of this procedure include practical deficiencies related to the photogrammetric processing as well as the computational cost of transforming the whole scene. To avoid these problems, the paper pursues an alternative processing strategy where the EO parameters are transformed prior to the restitution. If only this transition was done, however, the scene would be systematically distorted. The reason is that the national coordinates are not Cartesian due to the earth curvature and the unavoidable length distortion of map projections. To settle these distortions, several corrections need to be applied. These are treated in detail for both passive and active imaging. Since all these corrections are <span class="hlt">approximations</span> only, the resulting technique is termed "<span class="hlt">approximate</span> direct georeferencing". Still, the residual distortions are usually very low as is demonstrated by simulations, rendering the technique an attractive approach to direct georeferencing.</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 <span class="hlt">order</span> 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 <span class="hlt">order</span> numerator and second <span class="hlt">order</span> denominator polynomials.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19950023722&hterms=time-dependent&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DTitle%26N%3D0%26No%3D80%26Ntt%3Dtime-dependent','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19950023722&hterms=time-dependent&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DTitle%26N%3D0%26No%3D80%26Ntt%3Dtime-dependent"><span>Second-<span class="hlt">order</span> numerical solution of time-dependent, first-<span class="hlt">order</span> hyperbolic equations</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Shah, Patricia L.; Hardin, Jay</p>
<p>1995-01-01</p>
<p>A finite difference scheme is developed to find an <span class="hlt">approximate</span> solution of two similar hyperbolic equations, namely a first-<span class="hlt">order</span> plane wave and spherical wave problem. Finite difference <span class="hlt">approximations</span> are made for both the space and time derivatives. The result is a conditionally stable equation yielding an exact solution when the Courant number is set to one.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2011ThCFD..25..357V','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2011ThCFD..25..357V"><span>Stratified wakes, the high Froude number <span class="hlt">approximation</span>, and potential flow</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Vasholz, David P.</p>
<p>2011-12-01</p>
<p>Properties of a steady wake generated by a body moving uniformly at constant depth through a stratified fluid are studied as a function of two parameters inserted into the linearized equations of motion. The first parameter, μ, multiplies the along-track gradient term in the source equation. When formal solutions for an arbitrary buoyancy frequency profile are written as eigenfunction expansions, one finds that the limit μ → 0 corresponds to a high Froude number <span class="hlt">approximation</span> accompanied by a substantial reduction in the complexity of the calculation. For μ = 1, upstream effects are present and the eigenvalues correspond to critical speeds above which transverse waves disappear for any given mode. For sufficiently high modes, the high Froude number <span class="hlt">approximation</span> is valid. The second tracer multiplies the square of the buoyancy frequency term in the linearized conservation of mass equation and enables direct comparisons with the limit of potential flow. Detailed results are given for the simplest possible profile, in which the buoyancy frequency is independent of depth; emphasis is placed upon quantities that can, in principle, be experimentally measured in a laboratory experiment. The vertical displacement field is written in terms of a stratified wake form factor {{H}} , which is the sum of a wavelike contribution that is non-zero downstream and an evanescent contribution that appears symmetrically upstream and downstream. First- and second-<span class="hlt">order</span> cross-track moments of {{H}} are analyzed. First-<span class="hlt">order</span> results predict enhanced upstream vertical displacements. Second-<span class="hlt">order</span> results expand upon previous predictions of wavelike resonances and also predict evanescent resonance effects.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29357078','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29357078"><span>Risk <span class="hlt">approximation</span> in decision making: <span class="hlt">approximative</span> numeric abilities predict advantageous decisions under objective risk.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Mueller, Silke M; Schiebener, Johannes; Delazer, Margarete; Brand, Matthias</p>
<p>2018-01-22</p>
<p>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 <span class="hlt">approximate</span> number processing skills. The current study investigates the effects of <span class="hlt">approximative</span> 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. <span class="hlt">Approximative</span> 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 <span class="hlt">approximation</span> skills and <span class="hlt">approximate</span> number processing to be subject of future investigations on decision making under risk.</p>
</li>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23546248','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23546248"><span>Limitations of the paraxial Debye <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>Sheppard, Colin J R</p>
<p>2013-04-01</p>
<p>In the paraxial form of the Debye integral for focusing, higher <span class="hlt">order</span> defocus terms are ignored, which can result in errors in dealing with aberrations, even for low numerical aperture. These errors can be avoided by using a different integration variable. The aberrations of a glass slab, such as a coverslip, are expanded in terms of the new variable, and expressed in terms of Zernike polynomials to assist with aberration balancing. Tube length error is also discussed.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017AIPC.1895k0003K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017AIPC.1895k0003K"><span>Numerical investigation of sixth <span class="hlt">order</span> Boussinesq equation</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Kolkovska, N.; Vucheva, V.</p>
<p>2017-10-01</p>
<p>We propose a family of conservative finite difference schemes for the Boussinesq equation with sixth <span class="hlt">order</span> dispersion terms. The schemes are of second <span class="hlt">order</span> of <span class="hlt">approximation</span>. The method is conditionally stable with a mild restriction τ = O(h) on the step sizes. Numerical tests are performed for quadratic and cubic nonlinearities. The numerical experiments show second <span class="hlt">order</span> of convergence of the discrete solution to the exact one.</p>
</li>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008NuPhB.789..525B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008NuPhB.789..525B"><span>Analytical <span class="hlt">approximation</span> schemes for solving exact renormalization group equations in the local 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>Bervillier, C.; Boisseau, B.; Giacomini, H.</p>
<p>2008-02-01</p>
<p>The relation between the Wilson-Polchinski and the Litim optimized ERGEs in the local potential <span class="hlt">approximation</span> is studied with high accuracy using two different analytical approaches based on a field expansion: a recently proposed genuine analytical <span class="hlt">approximation</span> scheme to two-point boundary value problems of ordinary differential equations, and a new one based on <span class="hlt">approximating</span> 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).</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/826901','SCIGOV-STC'); return false;" href="https://www.osti.gov/servlets/purl/826901"><span>Consistent Yokoya-Chen <span class="hlt">Approximation</span> to Beamstrahlung(LCC-0010)</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Peskin, M</p>
<p>2004-04-22</p>
<p>I reconsider the Yokoya-Chen <span class="hlt">approximate</span> evolution equation for beamstrahlung and modify it slightly to generate simple, consistent analytical <span class="hlt">approximations</span> for the electron and photon energy spectra. I compare these <span class="hlt">approximations</span> to previous ones, and to simulation data.I reconsider the Yokoya-Chen <span class="hlt">approximate</span> evolution equation for beamstrahlung and modify it slightly to generate simple, consistent analytical <span class="hlt">approximations</span> for the electron and photon energy spectra. I compare these <span class="hlt">approximations</span> to previous ones, and to simulation data.</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/1235211','SCIGOV-STC'); return false;" href="https://www.osti.gov/servlets/purl/1235211"><span><span class="hlt">Approximate</span> Model for Turbulent Stagnation Point Flow.</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Dechant, Lawrence</p>
<p>2016-01-01</p>
<p>Here we derive an <span class="hlt">approximate</span> turbulent self-similar model for a class of favorable pressure gradient wedge-like flows, focusing on the stagnation point limit. While the self-similar model provides a useful gross flow field estimate this approach must be combined with a near wall model is to determine skin friction and by Reynolds analogy the heat transfer coefficient. The combined approach is developed in detail for the stagnation point flow problem where turbulent skin friction and Nusselt number results are obtained. Comparison to the classical Van Driest (1958) result suggests overall reasonable agreement. Though the model is only valid near themore » stagnation region of cylinders and spheres it nonetheless provides a reasonable model for overall cylinder and sphere heat transfer. The enhancement effect of free stream turbulence upon the laminar flow is used to derive a similar expression which is valid for turbulent flow. Examination of free stream enhanced laminar flow suggests that the rather than enhancement of a laminar flow behavior free stream disturbance results in early transition to turbulent stagnation point behavior. Excellent agreement is shown between enhanced laminar flow and turbulent flow behavior for high levels, e.g. 5% of free stream turbulence. Finally the blunt body turbulent stagnation results are shown to provide realistic heat transfer results for turbulent jet impingement problems.« less</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/10104879-simple-approximate-model-parachute-inflation','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/10104879-simple-approximate-model-parachute-inflation"><span>A simple, <span class="hlt">approximate</span> model of parachute inflation</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Macha, J.M.</p>
<p>1992-11-01</p>
<p>A simple, <span class="hlt">approximate</span> model of parachute inflation is described. The model is based on the traditional, practical treatment of the fluid resistance of rigid bodies in nonsteady flow, with appropriate extensions to accommodate the change in canopy inflated shape. Correlations for the steady drag and steady radial force as functions of the inflated radius are required as input to the dynamic model. In a novel approach, the radial force is expressed in terms of easily obtainable drag and reefing fine tension measurements. A series of wind tunnel experiments provides the needed correlations. Coefficients associated with the added mass of fluidmore » are evaluated by calibrating the model against an extensive and reliable set of flight data. A parameter is introduced which appears to universally govern the strong dependence of the axial added mass coefficient on motion history. Through comparisons with flight data, the model is shown to realistically predict inflation forces for ribbon and ringslot canopies over a wide range of sizes and deployment conditions.« less</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/7068077-simple-approximate-model-parachute-inflation','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/7068077-simple-approximate-model-parachute-inflation"><span>A simple, <span class="hlt">approximate</span> model of parachute inflation</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Macha, J.M.</p>
<p>1992-01-01</p>
<p>A simple, <span class="hlt">approximate</span> model of parachute inflation is described. The model is based on the traditional, practical treatment of the fluid resistance of rigid bodies in nonsteady flow, with appropriate extensions to accommodate the change in canopy inflated shape. Correlations for the steady drag and steady radial force as functions of the inflated radius are required as input to the dynamic model. In a novel approach, the radial force is expressed in terms of easily obtainable drag and reefing fine tension measurements. A series of wind tunnel experiments provides the needed correlations. Coefficients associated with the added mass of fluidmore » are evaluated by calibrating the model against an extensive and reliable set of flight data. A parameter is introduced which appears to universally govern the strong dependence of the axial added mass coefficient on motion history. Through comparisons with flight data, the model is shown to realistically predict inflation forces for ribbon and ringslot canopies over a wide range of sizes and deployment conditions.« less</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/9541868','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/9541868"><span><span class="hlt">Approximation</span> algorithms for a genetic diagnostics problem.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Kosaraju, S R; Schäffer, A A; Biesecker, L G</p>
<p>1998-01-01</p>
<p>We define and study a combinatorial problem called WEIGHTED DIAGNOSTIC COVER (WDC) that models the use of a laboratory technique called genotyping in the diagnosis of an important class of chromosomal aberrations. An optimal solution to WDC would enable us to define a genetic assay that maximizes the diagnostic power for a specified cost of laboratory work. We develop <span class="hlt">approximation</span> algorithms for WDC by making use of the well-known problem SET COVER for which the greedy heuristic has been extensively studied. We prove worst-case performance bounds on the greedy heuristic for WDC and for another heuristic we call directional greedy. We implemented both heuristics. We also implemented a local search heuristic that takes the solutions obtained by greedy and dir-greedy and applies swaps until they are locally optimal. We report their performance on a real data set that is representative of the options that a clinical geneticist faces for the real diagnostic problem. Many open problems related to WDC remain, both of theoretical interest and practical importance.</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1420280-planetary-ices-linear-mixing-approximation','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1420280-planetary-ices-linear-mixing-approximation"><span>Planetary Ices and the Linear Mixing <span class="hlt">Approximation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p>
<p>Bethkenhagen, M.; Meyer, Edmund Richard; Hamel, S.; ...</p>
<p>2017-10-10</p>
<p>Here, the validity of the widely used linear mixing <span class="hlt">approximation</span> (LMA) for the equations of state (EOSs) of planetary ices is investigated at pressure–temperature conditions typical for the interiors of Uranus and Neptune. The basis of this study is ab initio data ranging up to 1000 GPa and 20,000 K, calculated via density functional theory molecular dynamics simulations. In particular, we determine a new EOS for methane and EOS data for the 1:1 binary mixtures of methane, ammonia, and water, as well as their 2:1:4 ternary mixture. Additionally, the self-diffusion coefficients in the ternary mixture are calculated along three different Uranus interior profiles and compared to the values of the pure compounds. We find that deviations of the LMA from the results of the real mixture are generally small; for the thermal EOSs they amount to 4% or less. The diffusion coefficients in the mixture agree with those of the pure compounds within 20% or better. Finally, a new adiabatic model of Uranus with an inner layer of almost pure ices is developed. The model is consistent with the gravity field data and results in a rather cold interior (more » $${T}_{\\mathrm{core}}\\sim 4000$$ K).« less</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/1376569','SCIGOV-STC'); return false;" href="https://www.osti.gov/servlets/purl/1376569"><span>Stopping power beyond the adiabatic <span class="hlt">approximation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Caro, M.; Correa, A. A.; Artacho, E.</p>
<p>2017-06-01</p>
<p>Energetic ions traveling in solids deposit energy in a variety of ways, being nuclear and electronic stopping the two avenues in which dissipation is usually treated. This separation between electrons and ions relies on the adiabatic <span class="hlt">approximation</span> in which ions interact via forces derived from the instantaneous electronic ground state. In a more detailed view, in which non-adiabatic effects are explicitly considered, electronic excitations alter the atomic bonding, which translates into changes in the interatomic forces. In this work, we use time dependent density functional theory and forces derived from the equations of Ehrenfest dynamics that depend instantaneously on themore » time-dependent electronic density. With them we analyze how the inter-ionic forces are affected by electronic excitations in a model of a Ni projectile interacting with a Ni target, a metallic system with strong electronic stopping and shallow core level states. We find that the electronic excitations induce substantial modifications to the inter-ionic forces, which translate into nuclear stopping power well above the adiabatic prediction. Particularly, we observe that most of the alteration of the adiabatic potential in early times comes from the ionization of the core levels of the target ions, not readily screened by the valence electrons.« less</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27532639','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27532639"><span>When Density Functional <span class="hlt">Approximations</span> Meet Iron Oxides.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Meng, Yu; Liu, Xing-Wu; Huo, Chun-Fang; Guo, Wen-Ping; Cao, Dong-Bo; Peng, Qing; Dearden, Albert; Gonze, Xavier; Yang, Yong; Wang, Jianguo; Jiao, Haijun; Li, Yongwang; Wen, Xiao-Dong</p>
<p>2016-10-11</p>
<p>Three density functional <span class="hlt">approximations</span> (DFAs), PBE, PBE+U, and Heyd-Scuseria-Ernzerhof screened hybrid functional (HSE), were employed to investigate the geometric, electronic, magnetic, and thermodynamic properties of four iron oxides, namely, α-FeOOH, α-Fe 2 O 3 , Fe 3 O 4 , and FeO. Comparing our calculated results with available experimental data, we found that HSE (a = 0.15) (containing 15% "screened" Hartree-Fock exchange) can provide reliable values of lattice constants, Fe magnetic moments, band gaps, and formation energies of all four iron oxides, while standard HSE (a = 0.25) seriously overestimates the band gaps and formation energies. For PBE+U, a suitable U value can give quite good results for the electronic properties of each iron oxide, but it is challenging to accurately get other properties of the four iron oxides using the same U value. Subsequently, we calculated the Gibbs free energies of transformation reactions among iron oxides using the HSE (a = 0.15) functional and plotted the equilibrium phase diagrams of the iron oxide system under various conditions, which provide reliable theoretical insight into the phase transformations of iron oxides.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20100036788','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20100036788"><span>Configuring Airspace Sectors with <span class="hlt">Approximate</span> Dynamic Programming</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Bloem, Michael; Gupta, Pramod</p>
<p>2010-01-01</p>
<p>In response to changing traffic and staffing conditions, supervisors dynamically configure airspace sectors by assigning them to control positions. A finite horizon airspace sector configuration problem models this supervisor decision. The problem is to select an airspace configuration at each time step while considering a workload cost, a reconfiguration cost, and a constraint on the number of control positions at each time step. Three algorithms for this problem are proposed and evaluated: a myopic heuristic, an exact dynamic programming algorithm, and a rollouts <span class="hlt">approximate</span> dynamic programming algorithm. On problem instances from current operations with only dozens of possible configurations, an exact dynamic programming solution gives the optimal cost value. The rollouts algorithm achieves costs within 2% of optimal for these instances, on average. For larger problem instances that are representative of future operations and have thousands of possible configurations, excessive computation time prohibits the use of exact dynamic programming. On such problem instances, the rollouts algorithm reduces the cost achieved by the heuristic by more than 15% on average with an acceptable computation time.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20110016648','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20110016648"><span>Improved Discrete <span class="hlt">Approximation</span> of Laplacian of Gaussian</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Shuler, Robert L., Jr.</p>
<p>2004-01-01</p>
<p>An improved method of computing a discrete <span class="hlt">approximation</span> of the Laplacian of a Gaussian convolution of an image has been devised. The primary advantage of the method is that without substantially degrading the accuracy of the end result, it reduces the amount of information that must be processed and thus reduces the amount of circuitry needed to perform the Laplacian-of- Gaussian (LOG) operation. Some background information is necessary to place the method in context. The method is intended for application to the LOG part of a process of real-time digital filtering of digitized video data that represent brightnesses in pixels in a square array. The particular filtering process of interest is one that converts pixel brightnesses to binary form, thereby reducing the amount of information that must be performed in subsequent correlation processing (e.g., correlations between images in a stereoscopic pair for determining distances or correlations between successive frames of the same image for detecting motions). The Laplacian is often included in the filtering process because it emphasizes edges and textures, while the Gaussian is often included because it smooths out noise that might not be consistent between left and right images or between successive frames of the same image.</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/6637686-approximate-stoichiometry-rich-hydrocarbon-mixtures','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/6637686-approximate-stoichiometry-rich-hydrocarbon-mixtures"><span><span class="hlt">Approximate</span> stoichiometry for rich hydrocarbon mixtures</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Beans, E.W.</p>
<p>1993-03-01</p>
<p>The stoichiometry of lean mixtures can readily and accurately be determined from the assumption that all the carbon oxidizes to carbon dioxide and all the hydrogen oxidizes to water. This assumption is valid up to an equivalence ratio ([sigma]) of 0.8 and can be used with little error up to [sigma] = 1. The composition of the products of a hydrocarbon burnt in air under the foregoing assumption can be obtained from simple carbon, hydrogen, oxygen and nitrogen balances. Given the composition, one can determine the energy released and/or the adiabatic flame temperature. For rich mixtures, the foregoing assumption, ofmore » course, is not valid. Hence, there is no easy way to determine the stoichiometry of the products of a rich mixture. The objective of this note is to present an equation' which will allow one to readily determine the composition of the products of rich hydrocarbon mixtures. The equation is based on equilibrium composition calculations and some assumptions regarding the characteristics of hydrocarbons. The equation gives <span class="hlt">approximate</span> results. However, the results are sufficiently accurate for many situations. If more accuracy is wanted, one should use an equilibrium combustion program like the one by Gordon and McBride.« less</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014SPIE.9093E..0BM','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014SPIE.9093E..0BM"><span>Born <span class="hlt">approximation</span>, multiple scattering, and butterfly algorithm</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Martinez, Alex; Qiao, Zhijun</p>
<p>2014-06-01</p>
<p>Many imaging algorithms have been designed assuming the absence of multiple scattering. In the 2013 SPIE proceeding, we discussed an algorithm for removing high <span class="hlt">order</span> scattering components from collected data. In this paper, our goal is to continue this work. First, we survey the current state of multiple scattering in SAR. Then, we revise our method and test it. Given an estimate of our target reflectivity, we compute the multi scattering effects in our target region for various frequencies. Furthermore, we propagate this energy through free space towards our antenna, and remove it from the collected data.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950012948','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950012948"><span>Comparison of dynamical <span class="hlt">approximation</span> schemes for non-linear gravitational clustering</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Melott, Adrian L.</p>
<p>1994-01-01</p>
<p>We have recently conducted a controlled comparison of a number of <span class="hlt">approximations</span> for gravitational clustering against the same n-body simulations. These include ordinary linear perturbation theory (Eulerian), the adhesion <span class="hlt">approximation</span>, the frozen-flow <span class="hlt">approximation</span>, the Zel'dovich <span class="hlt">approximation</span> (describable as first-<span class="hlt">order</span> Lagrangian perturbation theory), and its second-<span class="hlt">order</span> generalization. In the last two cases we also created new versions of <span class="hlt">approximation</span> by truncation, i.e., smoothing the initial conditions by various smoothing window shapes and varying their sizes. The primary tool for comparing simulations to <span class="hlt">approximation</span> schemes was crosscorrelation of the evolved mass density fields, testing the extent to which mass was moved to the right place. The Zel'dovich <span class="hlt">approximation</span>, 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-<span class="hlt">order</span> generalization worked slightly better. All other schemes, including those proposed as generalizations of the Zel'dovich <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span>, depending upon the purpose. The theoretical implication is that pancaking is implicit in all cosmological gravitational clustering, at least from Gaussian initial conditions, even</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27679480','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27679480"><span>Finding <span class="hlt">approximate</span> gene clusters with Gecko 3.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Winter, Sascha; Jahn, Katharina; Wehner, Stefanie; Kuchenbecker, Leon; Marz, Manja; Stoye, Jens; Böcker, Sebastian</p>
<p>2016-11-16</p>
<p>Gene-<span class="hlt">order</span>-based comparison of multiple genomes provides signals for functional analysis of genes and the evolutionary process of genome organization. Gene clusters are regions of co-localized genes on genomes of different species. The rapid increase in sequenced genomes necessitates bioinformatics tools for finding gene clusters in hundreds of genomes. Existing tools are often restricted to few (in many cases, only two) genomes, and often make restrictive assumptions such as short perfect conservation, conserved gene <span class="hlt">order</span> or monophyletic gene clusters. We present Gecko 3, an open-source software for finding gene clusters in hundreds of bacterial genomes, that comes with an easy-to-use graphical user interface. The underlying gene cluster model is intuitive, can cope with low degrees of conservation as well as misannotations and is complemented by a sound statistical evaluation. To evaluate the biological benefit of Gecko 3 and to exemplify our method, we search for gene clusters in a dataset of 678 bacterial genomes using Synechocystis sp. PCC 6803 as a reference. We confirm detected gene clusters reviewing the literature and comparing them to a database of operons; we detect two novel clusters, which were confirmed by publicly available experimental RNA-Seq data. The computational analysis is carried out on a laptop computer in <40 min. © The Author(s) 2016. Published by Oxford University Press on behalf of Nucleic Acids Research.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19770007977','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19770007977"><span>Test particle propagation in magnetostatic turbulence. 2: The local <span class="hlt">approximation</span> method</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Klimas, A. J.; Sandri, G.; Scudder, J. D.; Howell, D. R.</p>
<p>1976-01-01</p>
<p>An <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> method. Simplification of the integrodifferential equation is achieved through <span class="hlt">approximation</span> to the Laplace transform of its kernel. The <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> method. The n'th level of <span class="hlt">approximation</span> is constructed formally, and the first five levels of <span class="hlt">approximation</span> are calculated explicitly. It is shown that each level of <span class="hlt">approximation</span> is governed by an inhomogeneous partial differential equation in time with time independent operator coefficients. The <span class="hlt">order</span> in time of these partial differential equations is found to increase as n does. At n = 0 the most local first <span class="hlt">order</span> partial differential equation which governs the Markovian limit is regained.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016Galax...4....4Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016Galax...4....4Z"><span>Padé <span class="hlt">Approximant</span> and Minimax Rational <span class="hlt">Approximation</span> in Standard Cosmology</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Zaninetti, Lorenzo</p>
<p>2016-02-01</p>
<p>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 <span class="hlt">approximant</span>. A comparison with a known analytical solution shows that the Pad\\'e <span class="hlt">approximant</span> 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximant</span>. A comparison of $\\Lambda$CDM with other cosmologies is done adopting a statistical point of view.</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://pubs.er.usgs.gov/publication/70020693','USGSPUBS'); return false;" href="https://pubs.er.usgs.gov/publication/70020693"><span>A comparison of zero-<span class="hlt">order</span>, first-<span class="hlt">order</span>, and monod biotransformation models</span></a></p>
<p><a target="_blank" href="http://pubs.er.usgs.gov/pubs/index.jsp?view=adv">USGS Publications Warehouse</a></p>
<p>Bekins, B.A.; Warren, E.; Godsy, E.M.</p>
<p>1998-01-01</p>
<p>Under some conditions, a first-<span class="hlt">order</span> kinetic model is a poor representation of biodegradation in contaminated aquifers. Although it is well known that the assumption of first-<span class="hlt">order</span> kinetics is valid only when substrate concentration, S, is much less than the half-saturation constant, K(s), this assumption is often made without verification of this condition. We present a formal error analysis showing that the relative error in the first-<span class="hlt">order</span> <span class="hlt">approximation</span> is S/K(S) and in the zero-<span class="hlt">order</span> <span class="hlt">approximation</span> the error is K(s)/S. We then examine the problems that arise when the first-<span class="hlt">order</span> <span class="hlt">approximation</span> is used outside the range for which it is valid. A series of numerical simulations comparing results of first- and zero-<span class="hlt">order</span> rate <span class="hlt">approximations</span> to Monod kinetics for a real data set illustrates that if concentrations observed in the field are higher than K(s), it may better to model degradation using a zero-<span class="hlt">order</span> rate expression. Compared with Monod kinetics, extrapolation of a first-<span class="hlt">order</span> rate to lower concentrations under-predicts the biotransformation potential, while extrapolation to higher concentrations may grossly over-predict the transformation rate. A summary of solubilities and Monod parameters for aerobic benzene, toluene, and xylene (BTX) degradation shows that the a priori assumption of first-<span class="hlt">order</span> degradation kinetics at sites contaminated with these compounds is not valid. In particular, out of six published values of KS for toluene, only one is greater than 2 mg/L, indicating that when toluene is present in concentrations greater than about a part per million, the assumption of first-<span class="hlt">order</span> kinetics may be invalid. Finally, we apply an existing analytical solution for steady-state one-dimensional advective transport with Monod degradation kinetics to a field data set.A formal error analysis is presented showing that the relative error in the first-<span class="hlt">order</span> <span class="hlt">approximation</span> is S/KS and in the zero-<span class="hlt">order</span> <span class="hlt">approximation</span> the error is KS/S where S is the substrate</p>
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<ol class="result-class" start="361">
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28793122','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28793122"><span>Scenario <span class="hlt">approximation</span> in a phenomenological study in Mexico: experience report.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Guerrero-Castañeda, Raúl Fernando; Menezes, Tânia Maria de Oliva; Vargas, Ma Guadalupe Ojeda</p>
<p>2017-01-01</p>
<p>To report our experience using scenario <span class="hlt">approximation</span> in a phenomenological study of nursing in Mexico. Experience report on scenario <span class="hlt">approximation</span> to coexist with elderly in <span class="hlt">order</span> to select the participants of a phenomenological study. During a four-month period in 2016, visits were carried out two groups of elderly individuals where several activities were carried out. Coexistence with the elderly throughout accompaniment in the groups' activities together with joint dialogue allowed selection of those who corresponded to the characteristics of the study objective. Scenario <span class="hlt">approximation</span> is necessary in phenomenological studies, not only for creating empathy among the participants but also for the researchers to immerse themselves in the phenomenon under study, as shown by the first approaches of the researcher. Relatar la experiencia del acercamiento al escenario de un estudio fenomenológico en enfermería en México. Relato de experiencia sobre el acercamiento al escenario de estudio para convivir con adultos mayores con la finalidad de seleccionar a los participantes de un estudio fenomenológico. Se llevaron a cabo visitas durante el año 2016, en un periodo de cuatro meses a dos grupos de adultos mayores en donde se realizaron diversas actividades. La convivencia con los adultos mayores a través del acompañamiento en las actividades que realizaban en los grupos y el diálogo conjunto permitió seleccionar a aquellos que respondían a las características del objeto de estudio. Es necesaria la aproximación al escenario de estudios fenomenológicos, no sólo con la finalidad de ganar empatía de los participantes sino para sumergirse en el fenómeno de estudio, mismo que se va mostrando desde los primeros acercamientos del investigador.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1987SPIE..754..117B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1987SPIE..754..117B"><span>Differential Geometry Applied To Least-Square Error Surface <span class="hlt">Approximations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Bolle, Ruud M.; Sabbah, Daniel</p>
<p>1987-08-01</p>
<p>This paper focuses on extraction of the parameters of individual surfaces from noisy depth maps. The basis for this are least-square error polynomial <span class="hlt">approximations</span> to the range data and the curvature properties that can be computed from these <span class="hlt">approximations</span>. The curvature properties are derived using the invariants of the Weingarten Map evaluated at the origin of local coordinate systems centered at the range points. The Weingarten Map is a well-known concept in differential geometry; a brief treatment of the differential geometry pertinent to surface curvature is given. We use the curvature properties for extracting certain surface parameters from the curvature properties of the <span class="hlt">approximations</span>. Then we show that curvature properties alone are not enough to obtain all the parameters of the surfaces; higher <span class="hlt">order</span> properties (information about change of curvature) are needed to obtain full parametric descriptions. This surface parameter estimation problem arises in the design of a vision system to recognize 3D objects whose surfaces are composed of planar patches and patches of quadrics of revolution. (Quadrics of revolution are quadrics that are surfaces of revolution.) A significant portion of man-made objects can be modeled using these surfaces. The actual process of recognition and parameter extraction is framed as a set of stacked parameter space transforms. The transforms are "stacked" in the sense that any one transform computes only a partial geometric description that forms the input to the next transform. Those who are interested in the organization and control of the recognition and parameter recognition process are referred to [Sabbah86], this paper briefly touches upon the organization, but concentrates mainly on geometrical aspects of the parameter extraction.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012IJSyS..43.1642L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012IJSyS..43.1642L"><span>New realisation of Preisach model using adaptive polynomial <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>Liu, Van-Tsai; Lin, Chun-Liang; Wing, Home-Young</p>
<p>2012-09-01</p>
<p>Modelling system with hysteresis has received considerable attention recently due to the increasing accurate requirement in engineering applications. The classical Preisach model (CPM) is the most popular model to demonstrate hysteresis which can be represented by infinite but countable first-<span class="hlt">order</span> reversal curves (FORCs). The usage of look-up tables is one way to approach the CPM in actual practice. The data in those tables correspond with the samples of a finite number of FORCs. This approach, however, faces two major problems: firstly, it requires a large amount of memory space to obtain an accurate prediction of hysteresis; secondly, it is difficult to derive efficient ways to modify the data table to reflect the timing effect of elements with hysteresis. To overcome, this article proposes the idea of using a set of polynomials to emulate the CPM instead of table look-up. The polynomial <span class="hlt">approximation</span> requires less memory space for data storage. Furthermore, the polynomial coefficients can be obtained accurately by using the least-square <span class="hlt">approximation</span> or adaptive identification algorithm, such as the possibility of accurate tracking of hysteresis model parameters.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/1337238-efficient-approximation-algorithms-weighted-matching','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/1337238-efficient-approximation-algorithms-weighted-matching"><span>Efficient <span class="hlt">Approximation</span> Algorithms for Weighted $b$-Matching</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Khan, Arif; Pothen, Alex; Mostofa Ali Patwary, Md.</p>
<p>2016-01-01</p>
<p>We describe a half-<span class="hlt">approximation</span> algorithm, b-Suitor, for computing a b-Matching of maximum weight in a graph with weights on the edges. b-Matching is a generalization of the well-known Matching problem in graphs, where the objective is to choose a subset of M edges in the graph such that at most a specified number b(v) of edges in M are incident on each vertex v. Subject to this restriction we maximize the sum of the weights of the edges in M. We prove that the b-Suitor algorithm computes the same b-Matching as the one obtained by the greedy algorithm for themore » problem. We implement the algorithm on serial and shared-memory parallel processors, and compare its performance against a collection of <span class="hlt">approximation</span> algorithms that have been proposed for the Matching problem. Our results show that the b-Suitor algorithm outperforms the Greedy and Locally Dominant edge algorithms by one to two <span class="hlt">orders</span> of magnitude on a serial processor. The b-Suitor algorithm has a high degree of concurrency, and it scales well up to 240 threads on a shared memory multiprocessor. The b-Suitor algorithm outperforms the Locally Dominant edge algorithm by a factor of fourteen on 16 cores of an Intel Xeon multiprocessor.« less</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19960008825','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19960008825"><span>On High-<span class="hlt">Order</span> Radiation Boundary Conditions</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>1995-01-01</p>
<p>In this paper we develop the theory of high-<span class="hlt">order</span> radiation boundary conditions for wave propagation problems. In particular, we study the convergence of sequences of time-local <span class="hlt">approximate</span> conditions to the exact boundary condition, and subsequently estimate the error in the solutions obtained using these <span class="hlt">approximations</span>. We show that for finite times the Pade <span class="hlt">approximants</span> proposed by Engquist and Majda lead to exponential convergence if the solution is smooth, but that good long-time error estimates cannot hold for spatially local conditions. Applications in fluid dynamics are also discussed.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018NIMPB.418...87S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018NIMPB.418...87S"><span>SEE rate estimation based on diffusion <span class="hlt">approximation</span> of charge collection</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Sogoyan, Armen V.; Chumakov, Alexander I.; Smolin, Anatoly A.</p>
<p>2018-03-01</p>
<p>The integral rectangular parallelepiped (IRPP) method remains the main approach to single event rate (SER) prediction for aerospace systems, despite the growing number of issues impairing method's validity when applied to scaled technology nodes. One of such issues is uncertainty in parameters extraction in the IRPP method, which can lead to a spread of several <span class="hlt">orders</span> of magnitude in the subsequently calculated SER. The paper presents an alternative approach to SER estimation based on diffusion <span class="hlt">approximation</span> of the charge collection by an IC element and geometrical interpretation of SEE cross-section. In contrast to the IRPP method, the proposed model includes only two parameters which are uniquely determined from the experimental data for normal incidence irradiation at an ion accelerator. This approach eliminates the necessity of arbitrary decisions during parameter extraction and, thus, greatly simplifies calculation procedure and increases the robustness of the forecast.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27118889','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27118889"><span>Universal recovery map for <span class="hlt">approximate</span> Markov chains.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Sutter, David; Fawzi, Omar; Renner, Renato</p>
<p>2016-02-01</p>
<p>A central question in quantum information theory is to determine how well lost information can be reconstructed. Crucially, the corresponding recovery operation should perform well without knowing the information to be reconstructed. In this work, we show that the quantum conditional mutual information measures the performance of such recovery operations. More precisely, we prove that the conditional mutual information I ( A : C | B ) of a tripartite quantum state ρ ABC can be bounded from below by its distance to the closest recovered state [Formula: see text], where the C -part is reconstructed from the B -part only and the recovery map [Formula: see text] merely depends on ρ BC . One particular application of this result implies the equivalence between two different approaches to define topological <span class="hlt">order</span> in quantum systems.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/2058450','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/2058450"><span>[Bladder endometriosis. Diagnostic and therapeutic <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>Monllor Gisbert, J; Merino Hernaez, C; Olivier Gómez, C; Carballido Rodríguez, J</p>
<p>1991-01-01</p>
<p>Endometriosis is defined by the presence of functionally active endometria in ectopic position; so when its clinical behaviour adopts the characteristics of tumoration, endometrioma is accepted as an alternative name. Location of this pathology in the urinary apparatus is uncommon in terms of incidence, since the bladder is a selected site. This condition implies a high morbidity rate because it requires a high level of suspicion in <span class="hlt">order</span> to be diagnosed. It is acknowledged that no image study (CAT, Echography, MNR, etc.) is pathognomonic for endometriosis and it is necessary to perform endoscopy and biopsy to achieve a correct diagnosis. The paper includes the strategy to follow, both with regard to diagnosis and the current therapeutic approaches.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4841654','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4841654"><span>Universal recovery map for <span class="hlt">approximate</span> Markov chains</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Sutter, David; Fawzi, Omar; Renner, Renato</p>
<p>2016-01-01</p>
<p>A central question in quantum information theory is to determine how well lost information can be reconstructed. Crucially, the corresponding recovery operation should perform well without knowing the information to be reconstructed. In this work, we show that the quantum conditional mutual information measures the performance of such recovery operations. More precisely, we prove that the conditional mutual information I(A:C|B) of a tripartite quantum state ρABC can be bounded from below by its distance to the closest recovered state RB→BC(ρAB), where the C-part is reconstructed from the B-part only and the recovery map RB→BC merely depends on ρBC. One particular application of this result implies the equivalence between two different approaches to define topological <span class="hlt">order</span> in quantum systems. PMID:27118889</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-<span class="hlt">order</span> 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-<span class="hlt">order</span> 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('http://adsabs.harvard.edu/abs/2018PhRvD..97a6016B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018PhRvD..97a6016B"><span>Born-Oppenheimer <span class="hlt">approximation</span> in an effective field theory language</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Brambilla, Nora; Krein, Gastão; Tarrús Castellà, Jaume; Vairo, Antonio</p>
<p>2018-01-01</p>
<p>The Born-Oppenheimer <span class="hlt">approximation</span> is the standard tool for the study of molecular systems. It is founded on the observation that the energy scale of the electron dynamics in a molecule is larger than that of the nuclei. A very similar physical picture can be used to describe QCD states containing heavy quarks as well as light-quarks or gluonic excitations. In this work, we derive the Born-Oppenheimer <span class="hlt">approximation</span> for QED molecular systems in an effective field theory framework by sequentially integrating out degrees of freedom living at energies above the typical energy scale where the dynamics of the heavy degrees of freedom occurs. In particular, we compute the matching coefficients of the effective field theory for the case of the H2+ diatomic molecule that are relevant to compute its spectrum up to O (m α5). Ultrasoft photon loops contribute at this <span class="hlt">order</span>, being ultimately responsible for the molecular Lamb shift. In the effective field theory the scaling of all the operators is homogeneous, which facilitates the determination of all the relevant contributions, an observation that may become useful for high-precision calculations. Using the above case as a guidance, we construct under some conditions an effective field theory for QCD states formed by a color-octet heavy quark-antiquark pair bound with a color-octet light-quark pair or excited gluonic state, highlighting the similarities and differences between the QED and QCD systems. Assuming that the multipole expansion is applicable, we construct the heavy-quark potential up to next-to-leading <span class="hlt">order</span> in the multipole expansion in terms of nonperturbative matching coefficients to be obtained from lattice QCD.</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 <span class="hlt">orders</span> 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://ntrs.nasa.gov/search.jsp?R=19930043559&hterms=clustering&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3Dclustering','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19930043559&hterms=clustering&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3Dclustering"><span>Testing <span class="hlt">approximations</span> for non-linear gravitational clustering</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Coles, Peter; Melott, Adrian L.; Shandarin, Sergei F.</p>
<p>1993-01-01</p>
<p>The accuracy of various analytic <span class="hlt">approximations</span> for following the evolution of cosmological density fluctuations into the nonlinear regime is investigated. The Zel'dovich <span class="hlt">approximation</span> is found to be consistently the best <span class="hlt">approximation</span> scheme. It is extremely accurate for power spectra characterized by n = -1 or less; when the <span class="hlt">approximation</span> is 'enhanced' by truncating highly nonlinear Fourier modes the <span class="hlt">approximation</span> is excellent even for n = +1. The performance of linear theory is less spectrum-dependent, but this <span class="hlt">approximation</span> is less accurate than the Zel'dovich one for all cases because of the failure to treat dynamics. The lognormal <span class="hlt">approximation</span> generally provides a very poor fit to the spatial pattern.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19890028795&hterms=discrete+mathematics&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Ddiscrete%2Bmathematics','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19890028795&hterms=discrete+mathematics&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Ddiscrete%2Bmathematics"><span>Singular perturbation and time scale approaches in discrete control systems</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Naidu, D. S.; Price, D. B.</p>
<p>1988-01-01</p>
<p>After considering a singularly perturbed discrete control system, a singular perturbation approach is used to obtain outer and correction subsystems. A time scale approach is then applied via block diagonalization transformations to decouple the system into slow and fast subsystems. To a <span class="hlt">zeroth-order</span> <span class="hlt">approximation</span>, the singular perturbation and time-scale approaches are found to yield equivalent results.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19940032991','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19940032991"><span><span class="hlt">Approximate</span> analysis for repeated eigenvalue problems with applications to controls-structure integrated design</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Kenny, Sean P.; Hou, Gene J. W.</p>
<p>1994-01-01</p>
<p>A method for eigenvalue and eigenvector <span class="hlt">approximate</span> analysis for the case of repeated eigenvalues with distinct first derivatives is presented. The <span class="hlt">approximate</span> analysis method developed involves a reparameterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-<span class="hlt">order</span> <span class="hlt">approximations</span> to changes in the eigenvalues and the eigenvectors associated with the repeated eigenvalue problem. This work also presents a numerical technique that facilitates the definition of an eigenvector derivative for the case of repeated eigenvalues with repeated eigenvalue derivatives (of all <span class="hlt">orders</span>). Examples are given which demonstrate the application of such equations for sensitivity and <span class="hlt">approximate</span> analysis. Emphasis is placed on the application of sensitivity analysis to large-scale structural and controls-structures optimization problems.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22676078-testing-approximate-predictions-displacements-cosmological-dark-matter-halos','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22676078-testing-approximate-predictions-displacements-cosmological-dark-matter-halos"><span>Testing <span class="hlt">approximate</span> predictions of displacements of cosmological dark matter halos</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Munari, Emiliano; Monaco, Pierluigi; Borgani, Stefano</p>
<p></p>
<p>We present a test to quantify how well some <span class="hlt">approximate</span> methods, designed to reproduce the mildly non-linear evolution of perturbations, are able to reproduce the clustering of DM halos once the grouping of particles into halos is defined and kept fixed. The following methods have been considered: Lagrangian Perturbation Theory (LPT) up to third <span class="hlt">order</span>, Truncated LPT, Augmented LPT, MUSCLE and COLA. The test runs as follows: halos are defined by applying a friends-of-friends (FoF) halo finder to the output of an N-body simulation. The <span class="hlt">approximate</span> methods are then applied to the same initial conditions of the simulation, producing formore » all particles displacements from their starting position and velocities. The position and velocity of each halo are computed by averaging over the particles that belong to that halo, according to the FoF halo finder. This procedure allows us to perform a well-posed test of how clustering of the matter density and halo density fields are recovered, without asking to the <span class="hlt">approximate</span> method an accurate reconstruction of halos. We have considered the results at z =0,0.5,1, and we have analysed power spectrum in real and redshift space, object-by-object difference in position and velocity, density Probability Distribution Function (PDF) and its moments, phase difference of Fourier modes. We find that higher LPT <span class="hlt">orders</span> are generally able to better reproduce the clustering of halos, while little or no improvement is found for the matter density field when going to 2LPT and 3LPT. Augmentation provides some improvement when coupled with 2LPT, while its effect is limited when coupled with 3LPT. Little improvement is brought by MUSCLE with respect to Augmentation. The more expensive particle-mesh code COLA outperforms all LPT methods, and this is true even for mesh sizes as large as the inter-particle distance. This test sets an upper limit on the ability of these methods to reproduce the clustering of halos, for the cases when these</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JCAP...07..050M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JCAP...07..050M"><span>Testing <span class="hlt">approximate</span> predictions of displacements of cosmological dark matter halos</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Munari, Emiliano; Monaco, Pierluigi; Koda, Jun; Kitaura, Francisco-Shu; Sefusatti, Emiliano; Borgani, Stefano</p>
<p>2017-07-01</p>
<p>We present a test to quantify how well some <span class="hlt">approximate</span> methods, designed to reproduce the mildly non-linear evolution of perturbations, are able to reproduce the clustering of DM halos once the grouping of particles into halos is defined and kept fixed. The following methods have been considered: Lagrangian Perturbation Theory (LPT) up to third <span class="hlt">order</span>, Truncated LPT, Augmented LPT, MUSCLE and COLA. The test runs as follows: halos are defined by applying a friends-of-friends (FoF) halo finder to the output of an N-body simulation. The <span class="hlt">approximate</span> methods are then applied to the same initial conditions of the simulation, producing for all particles displacements from their starting position and velocities. The position and velocity of each halo are computed by averaging over the particles that belong to that halo, according to the FoF halo finder. This procedure allows us to perform a well-posed test of how clustering of the matter density and halo density fields are recovered, without asking to the <span class="hlt">approximate</span> method an accurate reconstruction of halos. We have considered the results at z=0,0.5,1, and we have analysed power spectrum in real and redshift space, object-by-object difference in position and velocity, density Probability Distribution Function (PDF) and its moments, phase difference of Fourier modes. We find that higher LPT <span class="hlt">orders</span> are generally able to better reproduce the clustering of halos, while little or no improvement is found for the matter density field when going to 2LPT and 3LPT. Augmentation provides some improvement when coupled with 2LPT, while its effect is limited when coupled with 3LPT. Little improvement is brought by MUSCLE with respect to Augmentation. The more expensive particle-mesh code COLA outperforms all LPT methods, and this is true even for mesh sizes as large as the inter-particle distance. This test sets an upper limit on the ability of these methods to reproduce the clustering of halos, for the cases when these objects are</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/scitech">SciTech Connect</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-<span class="hlt">order</span> 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('https://www.osti.gov/biblio/839007-corrections-one-photon-approximation-gt-transition','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/839007-corrections-one-photon-approximation-gt-transition"><span>Corrections to the one-photon <span class="hlt">approximation</span> in the 0+-->2+ transition of 12C</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Paul Gueye; Madeleine Bernheim; J. F. Danel</p>
<p>2001-04-18</p>
<p>Contribution of higher <span class="hlt">order</span> effects to the one-photon exchange <span class="hlt">approximation</span> were studied in the first excited state of 12C by comparing inclusive inelastic scattering cross sections of electrons and positrons obtained at the Saclay Linear Accelerator. The data were compared to a distorted wave Born <span class="hlt">approximation</span> (DWBA)calculation. The results indicate an effect less than 2% within 2sigma, compatible with what was observed in recent elastic scattering measurements.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/20856423','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/20856423"><span><span class="hlt">Approximate</span> effective nonlinear coefficient of second-harmonic generation in KTiOPO(4).</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Asaumi, K</p>
<p>1993-10-20</p>
<p>A simplified <span class="hlt">approximate</span> expression for the effective nonlinear coefficient of type-II second-harmonicgeneration in KTiOPO(4) was obtained by observing that the difference between the refractive indices n(x) and n(y) is 1 <span class="hlt">order</span> of magnitude smaller than the difference between n(z) and n(y) (or n(x)). The agreement of this <span class="hlt">approximate</span> equation with the true definition is good, with a maximum discrepancy of 4%.</p>
</li>
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<ol class="result-class" start="381">
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19890014937','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19890014937"><span>Aeroservoelastic modeling and applications using minimum-state <span class="hlt">approximations</span> of the unsteady aerodynamics</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Tiffany, Sherwood H.; Karpel, Mordechay</p>
<p>1989-01-01</p>
<p>Various control analysis, design, and simulation techniques for aeroelastic applications require the equations of motion to be cast in a linear time-invariant state-space form. Unsteady aerodynamics forces have to be <span class="hlt">approximated</span> as rational functions of the Laplace variable in <span class="hlt">order</span> to put them in this framework. For the minimum-state method, the number of denominator roots in the rational <span class="hlt">approximation</span>. Results are shown of applying various <span class="hlt">approximation</span> enhancements (including optimization, frequency dependent weighting of the tabular data, and constraint selection) with the minimum-state formulation to the active flexible wing wind-tunnel model. The results demonstrate that good models can be developed which have an <span class="hlt">order</span> of magnitude fewer augmenting aerodynamic equations more than traditional approaches. This reduction facilitates the design of lower <span class="hlt">order</span> control systems, analysis of control system performance, and near real-time simulation of aeroservoelastic phenomena.</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-<span class="hlt">order</span> 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-<span class="hlt">order</span> 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-<span class="hlt">order</span> Taylor expansion in a natural way, and it is of significant faster convergence rate.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19900013242','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19900013242"><span>Differential equation based method for accurate <span class="hlt">approximations</span> in optimization</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Pritchard, Jocelyn I.; Adelman, Howard M.</p>
<p>1990-01-01</p>
<p>A method to efficiently and accurately <span class="hlt">approximate</span> 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 <span class="hlt">approximations</span>, hence, the method is denoted the Differential Equation Based (DEB) method. <span class="hlt">Approximations</span> were developed for vibration frequencies, mode shapes and static displacements. The DEB <span class="hlt">approximation</span> method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series <span class="hlt">approximations</span> 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 <span class="hlt">approximation</span>. The method is applicable to simultaneous perturbation of several design variables. Also, the <span class="hlt">approximations</span> may be used to calculate other system response quantities. For example, the <span class="hlt">approximations</span> for displacements are used to <span class="hlt">approximate</span> bending stresses.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19950044221&hterms=kolb&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Dkolb','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19950044221&hterms=kolb&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Dkolb"><span>First-<span class="hlt">order</span> inflation</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Kolb, Edward W.</p>
<p>1991-01-01</p>
<p>In the original proposal, inflation occurred in the process of a strongly first-<span class="hlt">order</span> phase transition. This model was soon demonstrated to be fatally flawed. Subsequent models for inflation involved phase transitions that were second-<span class="hlt">order</span>, or perhaps weakly first-<span class="hlt">order</span>; some even involved no phase transition at all. Recently the possibility of inflation during a strongly first-<span class="hlt">order</span> phase transition has been revived. In this talk I will discuss some models for first-<span class="hlt">order</span> inflation, and emphasize unique signatures that result if inflation is realized in a first-<span class="hlt">order</span> transition. Before discussing first-<span class="hlt">order</span> inflation, I will briefly review some of the history of inflation to demonstrate how first-<span class="hlt">order</span> inflation differs from other models.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4330682','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4330682"><span>The measurement of psychological literacy: a first <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>Roberts, Lynne D.; Heritage, Brody; Gasson, Natalie</p>
<p>2015-01-01</p>
<p>Psychological literacy, the ability to apply psychological knowledge to personal, family, occupational, community and societal challenges, is promoted as the primary outcome of an undergraduate education in psychology. As the concept of psychological literacy becomes increasingly adopted as the core business of undergraduate psychology training courses world-wide, there is urgent need for the construct to be accurately measured so that student and institutional level progress can be assessed and monitored. Key to the measurement of psychological literacy is determining the underlying factor-structure of psychological literacy. In this paper we provide a first <span class="hlt">approximation</span> of the measurement of psychological literacy by identifying and evaluating self-report measures for psychological literacy. Multi-item and single-item self-report measures of each of the proposed nine dimensions of psychological literacy were completed by two samples (N = 218 and N = 381) of undergraduate psychology students at an Australian university. Single and multi-item measures of each dimension were weakly to moderately correlated. Exploratory and confirmatory factor analyses of multi-item measures indicated a higher <span class="hlt">order</span> three factor solution best represented the construct of psychological literacy. The three factors were reflective processes, generic graduate attributes, and psychology as a helping profession. For the measurement of psychological literacy to progress there is a need to further develop self-report measures and to identify/develop and evaluate objective measures of psychological literacy. Further <span class="hlt">approximations</span> of the measurement of psychological literacy remain an imperative, given the construct's ties to measuring institutional efficacy in teaching psychology to an undergraduate audience. PMID:25741300</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016GeoJI.204...59V','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016GeoJI.204...59V"><span>The impact of <span class="hlt">approximations</span> and arbitrary choices on geophysical images</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Valentine, Andrew P.; Trampert, Jeannot</p>
<p>2016-01-01</p>
<p>Whenever a geophysical image is to be constructed, a variety of choices must be made. Some, such as those governing data selection and processing, or model parametrization, are somewhat arbitrary: there may be little reason to prefer one choice over another. Others, such as defining the theoretical framework within which the data are to be explained, may be more straightforward: typically, an `exact' theory exists, but various <span class="hlt">approximations</span> may need to be adopted in <span class="hlt">order</span> to make the imaging problem computationally tractable. Differences between any two images of the same system can be explained in terms of differences between these choices. Understanding the impact of each particular decision is essential if images are to be interpreted properly-but little progress has been made towards a quantitative treatment of this effect. In this paper, we consider a general linearized inverse problem, applicable to a wide range of imaging situations. We write down an expression for the difference between two images produced using similar inversion strategies, but where different choices have been made. This provides a framework within which inversion algorithms may be analysed, and allows us to consider how image effects may arise. In this paper, we take a general view, and do not specialize our discussion to any specific imaging problem or setup (beyond the restrictions implied by the use of linearized inversion techniques). In particular, we look at the concept of `hybrid inversion', in which highly accurate synthetic data (typically the result of an expensive numerical simulation) is combined with an inverse operator constructed based on theoretical <span class="hlt">approximations</span>. It is generally supposed that this offers the benefits of using the more complete theory, without the full computational costs. We argue that the inverse operator is as important as the forward calculation in determining the accuracy of results. We illustrate this using a simple example, based on imaging the</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20150003441','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20150003441"><span>Atmospheric Turbulence Modeling for Aerospace Vehicles: Fractional <span class="hlt">Order</span> Fit</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Kopasakis, George (Inventor)</p>
<p>2015-01-01</p>
<p>An improved model for simulating atmospheric disturbances is disclosed. A scale Kolmogorov spectral may be scaled to convert the Kolmogorov spectral into a finite energy von Karman spectral and a fractional <span class="hlt">order</span> pole-zero transfer function (TF) may be derived from the von Karman spectral. Fractional <span class="hlt">order</span> atmospheric turbulence may be <span class="hlt">approximated</span> with an integer <span class="hlt">order</span> pole-zero TF fit, and the <span class="hlt">approximation</span> may be stored in memory.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2001PhRvD..63i6003M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2001PhRvD..63i6003M"><span>Resumming the large-N <span class="hlt">approximation</span> for time evolving 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>Mihaila, Bogdan; Dawson, John F.; Cooper, Fred</p>
<p>2001-05-01</p>
<p>In this paper we discuss two methods of resumming the leading and next to leading <span class="hlt">order</span> 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 <span class="hlt">approximations</span> 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 <span class="hlt">approximations</span> 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span>, which we call the dynamic Debye screening <span class="hlt">approximation</span>, makes the further <span class="hlt">approximation</span> of replacing the exact x2 propagator by its value at leading <span class="hlt">order</span> in the 1/N expansion. These two <span class="hlt">approximations</span> are compared with exact numerical simulations for the quantum roll problem. The bare vertex <span class="hlt">approximation</span> captures the physics at large and modest N better than the dynamic Debye screening <span class="hlt">approximation</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19840042981&hterms=Nonlinear+equations&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D60%26Ntt%3DNonlinear%2Bequations','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19840042981&hterms=Nonlinear+equations&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D60%26Ntt%3DNonlinear%2Bequations"><span>Difference equation state <span class="hlt">approximations</span> for nonlinear hereditary control problems</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Rosen, I. G.</p>
<p>1984-01-01</p>
<p>Discrete <span class="hlt">approximation</span> schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve <span class="hlt">approximation</span> by a sequence of optimal control problems in which the original infinite dimensional state equation has been <span class="hlt">approximated</span> by a finite dimensional discrete difference equation. Convergence of the state <span class="hlt">approximations</span> is argued using linear semigroup theory and is then used to demonstrate that solutions to the <span class="hlt">approximating</span> optimal control problems in some sense <span class="hlt">approximate</span> solutions to the original control problem. Two schemes, one based upon piecewise constant <span class="hlt">approximation</span>, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available <span class="hlt">approximation</span> methods for the solution of hereditary control problems. Previously announced in STAR as N83-33589</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19830025318','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19830025318"><span>Difference equation state <span class="hlt">approximations</span> for nonlinear hereditary control problems</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Rosen, I. G.</p>
<p>1982-01-01</p>
<p>Discrete <span class="hlt">approximation</span> schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve <span class="hlt">approximation</span> by a sequence of optimal control problems in which the original infinite dimensional state equation has been <span class="hlt">approximated</span> by a finite dimensional discrete difference equation. Convergence of the state <span class="hlt">approximations</span> is argued using linear semigroup theory and is then used to demonstrate that solutions to the <span class="hlt">approximating</span> optimal control problems in some sense <span class="hlt">approximate</span> solutions to the original control problem. Two schemes, one based upon piecewise constant <span class="hlt">approximation</span>, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available <span class="hlt">approximation</span> methods for the solution of hereditary control problems.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19740011605','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19740011605"><span>Periodicity of high-<span class="hlt">order</span> neural functions</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Kellaway, P.; Borda, R. P.; Frost, J. D.; Carrie, J. R. G.; Coats, A. C.</p>
<p>1973-01-01</p>
<p>The results of recent studies on higher <span class="hlt">order</span>, integrative processes in the central nervous system are reported. Attempts were made to determine whether these processes exhibit any ongoing rhythmicity which might manifest itself in alterations of attention and alertness. Experiments were also designed to determine if a periodicity <span class="hlt">approximating</span> that of the REM could be detected in various parameters of brain electrical activity.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JPhCS.901a2173G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JPhCS.901a2173G"><span>Born <span class="hlt">approximation</span> in linear-time invariant system</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Gumjudpai, Burin</p>
<p>2017-09-01</p>
<p>An alternative way of finding the LTI’s solution with the Born <span class="hlt">approximation</span>, is investigated. We use Born <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> are explored. Transforming the LTI system into Helmholtz equation, <span class="hlt">approximated</span> general solution can be found for any given forms of force with its initial value.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018MNRAS.475..244A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018MNRAS.475..244A"><span>Second-<span class="hlt">order</span> singular pertubative theory for gravitational lenses</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Alard, C.</p>
<p>2018-03-01</p>
<p>The extension of the singular perturbative approach to the second <span class="hlt">order</span> is presented in this paper. The general expansion to the second <span class="hlt">order</span> is derived. The second-<span class="hlt">order</span> expansion is considered as a small correction to the first-<span class="hlt">order</span> expansion. Using this approach, it is demonstrated that in practice the second-<span class="hlt">order</span> expansion is reducible to a first <span class="hlt">order</span> expansion via a re-definition of the first-<span class="hlt">order</span> pertubative fields. Even if in usual applications the second-<span class="hlt">order</span> correction is small the reducibility of the second-<span class="hlt">order</span> expansion to the first-<span class="hlt">order</span> expansion indicates a potential degeneracy issue. In general, this degeneracy is hard to break. A useful and simple second-<span class="hlt">order</span> <span class="hlt">approximation</span> is the thin source <span class="hlt">approximation</span>, which offers a direct estimation of the correction. The practical application of the corrections derived in this paper is illustrated by using an elliptical NFW lens model. The second-<span class="hlt">order</span> pertubative expansion provides a noticeable improvement, even for the simplest case of thin source <span class="hlt">approximation</span>. To conclude, it is clear that for accurate modelization of gravitational lenses using the perturbative method the second-<span class="hlt">order</span> perturbative expansion should be considered. In particular, an evaluation of the degeneracy due to the second-<span class="hlt">order</span> term should be performed, for which the thin source <span class="hlt">approximation</span> is particularly useful.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19900014691','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19900014691"><span>Explicitly solvable complex Chebyshev <span class="hlt">approximation</span> problems related to sine polynomials</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Freund, Roland</p>
<p>1989-01-01</p>
<p>Explicitly solvable real Chebyshev <span class="hlt">approximation</span> problems on the unit interval are typically characterized by simple error curves. A similar principle is presented for complex <span class="hlt">approximation</span> problems with error curves induced by sine polynomials. As an application, some new explicit formulae for complex best <span class="hlt">approximations</span> are derived.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=estimator&pg=6&id=EJ1109033','ERIC'); return false;" href="https://eric.ed.gov/?q=estimator&pg=6&id=EJ1109033"><span>Meta-Regression <span class="hlt">Approximations</span> to Reduce Publication Selection Bias</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>Stanley, T. D.; Doucouliagos, Hristos</p>
<p>2014-01-01</p>
<p>Publication selection bias is a serious challenge to the integrity of all empirical sciences. We derive meta-regression <span class="hlt">approximations</span> to reduce this bias. Our approach employs Taylor polynomial <span class="hlt">approximations</span> to the conditional mean of a truncated distribution. A quadratic <span class="hlt">approximation</span> without a linear term, precision-effect estimate with…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19810006883','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19810006883"><span>ACCESS 3. <span class="hlt">Approximation</span> concepts code for efficient structural synthesis: User's guide</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Fleury, C.; Schmit, L. A., Jr.</p>
<p>1980-01-01</p>
<p>A user's guide is presented for ACCESS-3, a research oriented program which combines dual methods and a collection of <span class="hlt">approximation</span> concepts to achieve excellent efficiency in structural synthesis. The finite element method is used for structural analysis and dual algorithms of mathematical programming are applied in the design optimization procedure. This program retains all of the ACCESS-2 capabilities and the data preparation formats are fully compatible. Four distinct optimizer options were added: interior point penalty function method (NEWSUMT); second <span class="hlt">order</span> primal projection method (PRIMAL2); second <span class="hlt">order</span> Newton-type dual method (DUAL2); and first <span class="hlt">order</span> gradient projection-type dual method (DUAL1). A pure discrete and mixed continuous-discrete design variable capability, and zero <span class="hlt">order</span> <span class="hlt">approximation</span> of the stress constraints are also included.</p>
</li>
<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 <span class="hlt">order</span> 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 <span class="hlt">order</span> 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://adsabs.harvard.edu/abs/2014PhDT........34R','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014PhDT........34R"><span>Dynamic Analyses of Result Quality in Energy-Aware <span class="hlt">Approximate</span> Programs</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>RIngenburg, Michael F.</p>
<p></p>
<p>Energy efficiency is a key concern in the design of modern computer systems. One promising approach to energy-efficient computation, <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> computations. We propose three styles of tools: prototyping tools that allow developers to experiment with <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> constructs to the language, an <span class="hlt">approximation</span> 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 <span class="hlt">approximate</span> operations. The second tracks the number of <span class="hlt">approximate</span> 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 <span class="hlt">approximation</span>. Online monitors can be used to perform real time adjustments to energy usage in <span class="hlt">order</span> 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 <span class="hlt">approximation</span> strategies and identify new strategies, our online tools succeed in providing new insights into the effects of <span class="hlt">approximation</span> on output quality, and our monitors succeed in</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19980232922','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19980232922"><span><span class="hlt">Approximate</span> Analytical Solutions for Hypersonic Flow Over Slender Power Law Bodies</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Mirels, Harold</p>
<p>1959-01-01</p>
<p><span class="hlt">Approximate</span> analytical solutions are presented for two-dimensional and axisymmetric hypersonic flow over slender power law bodies. Both zero <span class="hlt">order</span> (M approaches infinity) and first <span class="hlt">order</span> (small but nonvanishing values of 1/(M(Delta)(sup 2) solutions are presented, where M is free-stream Mach number and Delta is a characteristic slope. These solutions are compared with exact numerical integration of the equations of motion and appear to be accurate particularly when the shock is relatively close to the body.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2000NuAlg..23..263A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2000NuAlg..23..263A"><span>A result about scale transformation families in <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>Apprato, Dominique; Gout, Christian</p>
<p>2000-06-01</p>
<p>Scale transformations are common in <span class="hlt">approximation</span>. In surface <span class="hlt">approximation</span> from rapidly varying data, one wants to suppress, or at least dampen the oscillations of the <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> method which allows some control over the behavior of the <span class="hlt">approximation</span>. More precisely, we study the convergence of an <span class="hlt">approximation</span> from a data set of , while using scale transformations on the values before and after classical <span class="hlt">approximation</span>. In addition, the construction of scale transformations is also given. The algorithm is presented with some numerical examples.</p>
</li>
</ol>
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<ol class="result-class" start="401">
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/21867342','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/21867342"><span><span class="hlt">Approximation</span> for discrete Fourier transform and application in study of three-dimensional interacting electron gas.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Yan, Xin-Zhong</p>
<p>2011-07-01</p>
<p>The discrete Fourier transform is <span class="hlt">approximated</span> by summing over part of the terms with corresponding weights. The <span class="hlt">approximation</span> reduces significantly the requirement for computer memory storage and enhances the numerical computation efficiency with several <span class="hlt">orders</span> without losing accuracy. As an example, we apply the algorithm to study the three-dimensional interacting electron gas under the renormalized-ring-diagram <span class="hlt">approximation</span> where the Green's function needs to be self-consistently solved. We present the results for the chemical potential, compressibility, free energy, entropy, and specific heat of the system. The ground-state energy obtained by the present calculation is compared with the existing results of Monte Carlo simulation and random-phase <span class="hlt">approximation</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017imap.book.....V','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017imap.book.....V"><span>Introduction to Methods of <span class="hlt">Approximation</span> in Physics and Astronomy</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>van Putten, Maurice H. P. M.</p>
<p>2017-04-01</p>
<p> secular behavior. For instance, secular evolution of orbital parameters may derive from averaging over essentially periodic behavior on relatively short, orbital periods. When the original number of degrees of freedom is large, averaging over dynamical time scales may lead to a formulation in terms of a system in <span class="hlt">approximately</span> thermodynamic equilibrium subject to evolution on a secular time scale by a regular or singular perturbation. In modern astrophysics and cosmology, gravitation is being probed across an increasingly broad range of scales and more accurately so than ever before. These observations probe weak gravitational interactions below what is encountered in our solar system by many <span class="hlt">orders</span> of magnitude. These observations hereby probe (curved) spacetime at low energy scales that may reveal novel properties hitherto unanticipated in the classical vacuum of Newtonian mechanics and Minkowski spacetime. Dark energy and dark matter encountered on the scales of galaxies and beyond, therefore, may be, in part, revealing our ignorance of the vacuum at the lowest energy scales encountered in cosmology. In this context, our application of Newtonian mechanics to globular clusters, galaxies and cosmology is an <span class="hlt">approximation</span> assuming a classical vacuum, ignoring the potential for hidden low energy scales emerging on cosmological scales. Given our ignorance of the latter, this poses a challenge in the potential for unknown systematic deviations. If of quantum mechanical origin, such deviations are often referred to as anomalies. While they are small in traditional, macroscopic Newtonian experiments in the laboratory, they same is not a given in the limit of arbitrarily weak gravitational interactions. We hope this selection of introductory material is useful and kindles the reader's interest to become a creative member of modern astrophysics and cosmology.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018PhRvB..97l5428P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018PhRvB..97l5428P"><span>Surface waves on multilayer hyperbolic metamaterials: Operator approach to effective medium <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>Popov, Vladislav; Lavrinenko, Andrei V.; Novitsky, Andrey</p>
<p>2018-03-01</p>
<p>In this paper, we elaborate on the operator effective medium <span class="hlt">approximation</span> developed recently in Popov et al. [Phys. Rev. B 94, 085428 (2016), 10.1103/PhysRevB.94.085428] to get insight into the surface polariton excitation at the interface of a multilayer hyperbolic metamaterial (HMM). In particular, we find that HMMs with bilayer unit cells support the TE- and TM-polarized surface waves beyond the Maxwell Garnett <span class="hlt">approximation</span> due to the spatial dispersion interpreted as effective magnetoelectric coupling. The latter is also responsible for the dependence of surface wave propagation on the <span class="hlt">order</span> of layers in the unit cell. Elimination of the magnetoelectric coupling in three-layer unit cells complying with inversion symmetry restores the qualitative regularity of the Maxwell Garnett <span class="hlt">approximation</span>, as well as strongly suppresses the influence of the <span class="hlt">order</span> of layers in the unit cell.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4134886','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4134886"><span>Roughness in Lattice <span class="hlt">Ordered</span> Effect Algebras</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Xin, Xiao Long; Hua, Xiu Juan; Zhu, Xi</p>
<p>2014-01-01</p>
<p>Many authors have studied roughness on various algebraic systems. In this paper, we consider a lattice <span class="hlt">ordered</span> effect algebra and discuss its roughness in this context. Moreover, we introduce the notions of the interior and the closure of a subset and give some of their properties in effect algebras. Finally, we use a Riesz ideal induced congruence and define a function e(a, b) in a lattice <span class="hlt">ordered</span> effect algebra E and build a relationship between it and congruence classes. Then we study some properties about <span class="hlt">approximation</span> of lattice <span class="hlt">ordered</span> effect algebras. PMID:25170523</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24320847','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24320847"><span><span class="hlt">Approximate</span>, computationally efficient online learning in Bayesian spiking neurons.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Kuhlmann, Levin; Hauser-Raspe, Michael; Manton, Jonathan H; Grayden, David B; Tapson, Jonathan; van Schaik, André</p>
<p>2014-03-01</p>
<p>Bayesian spiking neurons (BSNs) provide a probabilistic interpretation of how neurons perform inference and learning. Online learning in BSNs typically involves parameter estimation based on maximum-likelihood expectation-maximization (ML-EM) which is computationally slow and limits the potential of studying networks of BSNs. An online learning algorithm, fast learning (FL), is presented that is more computationally efficient than the benchmark ML-EM for a fixed number of time steps as the number of inputs to a BSN increases (e.g., 16.5 times faster run times for 20 inputs). Although ML-EM appears to converge 2.0 to 3.6 times faster than FL, the computational cost of ML-EM means that ML-EM takes longer to simulate to convergence than FL. FL also provides reasonable convergence performance that is robust to initialization of parameter estimates that are far from the true parameter values. However, parameter estimation depends on the range of true parameter values. Nevertheless, for a physiologically meaningful range of parameter values, FL gives very good average estimation accuracy, despite its <span class="hlt">approximate</span> nature. The FL algorithm therefore provides an efficient tool, complementary to ML-EM, for exploring BSN networks in more detail in <span class="hlt">order</span> to better understand their biological relevance. Moreover, the simplicity of the FL algorithm means it can be easily implemented in neuromorphic VLSI such that one can take advantage of the energy-efficient spike coding of BSNs.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015JPS...278..274U','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015JPS...278..274U"><span>Battery state-of-charge estimation using <span class="hlt">approximate</span> least squares</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Unterrieder, C.; Zhang, C.; Lunglmayr, M.; Priewasser, R.; Marsili, S.; Huemer, M.</p>
<p>2015-03-01</p>
<p>In recent years, much effort has been spent to extend the runtime of battery-powered electronic applications. In <span class="hlt">order</span> to improve the utilization of the available cell capacity, high precision estimation approaches for battery-specific parameters are needed. In this work, an <span class="hlt">approximate</span> least squares estimation scheme is proposed for the estimation of the battery state-of-charge (SoC). The SoC is determined based on the prediction of the battery's electromotive force. The proposed approach allows for an improved re-initialization of the Coulomb counting (CC) based SoC estimation method. Experimental results for an implementation of the estimation scheme on a fuel gauge system on chip are illustrated. Implementation details and design guidelines are presented. The performance of the presented concept is evaluated for realistic operating conditions (temperature effects, aging, standby current, etc.). For the considered test case of a GSM/UMTS load current pattern of a mobile phone, the proposed method is able to re-initialize the CC-method with a high accuracy, while state-of-the-art methods fail to perform a re-initialization.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017APS..DFDL31003D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017APS..DFDL31003D"><span>Autonomic Closure for Turbulent Flows Using <span class="hlt">Approximate</span> Bayesian Computation</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Doronina, Olga; Christopher, Jason; Hamlington, Peter; Dahm, Werner</p>
<p>2017-11-01</p>
<p>Autonomic closure is a new technique for achieving fully adaptive and physically accurate closure of coarse-grained turbulent flow governing equations, such as those solved in large eddy simulations (LES). Although autonomic closure has been shown in recent a priori tests to more accurately represent unclosed terms than do dynamic versions of traditional LES models, the computational cost of the approach makes it challenging to implement for simulations of practical turbulent flows at realistically high Reynolds numbers. The optimization step used in the approach introduces large matrices that must be inverted and is highly memory intensive. In <span class="hlt">order</span> to reduce memory requirements, here we propose to use <span class="hlt">approximate</span> Bayesian computation (ABC) in place of the optimization step, thereby yielding a computationally-efficient implementation of autonomic closure that trades memory-intensive for processor-intensive computations. The latter challenge can be overcome as co-processors such as general purpose graphical processing units become increasingly available on current generation petascale and exascale supercomputers. In this work, we outline the formulation of ABC-enabled autonomic closure and present initial results demonstrating the accuracy and computational cost of the approach.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2001MNRAS.325.1463F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2001MNRAS.325.1463F"><span>Axisymmetric modes of rotating relativistic stars in the Cowling <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>Font, José A.; Dimmelmeier, Harald; Gupta, Anshu; Stergioulas, Nikolaos</p>
<p>2001-08-01</p>
<p>Axisymmetric pulsations of rotating neutron stars can be excited in several scenarios, such as core collapse, crust- and core-quakes or binary mergers, and could become detectable in either gravitational waves or high-energy radiation. Here, we present a comprehensive study of all low-<span class="hlt">order</span> axisymmetric modes of uniformly and rapidly rotating relativistic stars. Initial stationary configurations are appropriately perturbed and are numerically evolved using an axisymmetric, non-linear relativistic hydrodynamics code, assuming time-independence of the gravitational field (Cowling <span class="hlt">approximation</span>). The simulations are performed using a high-resolution shock-capturing finite-difference scheme accurate enough to maintain the initial rotation law for a large number of rotational periods, even for stars at the mass-shedding limit. Through Fourier transforms of the time evolution of selected fluid variables, we compute the frequencies of quasi-radial and non-radial modes with spherical harmonic indices l=0, 1, 2 and 3, for a sequence of rotating stars from the non-rotating limit to the mass-shedding limit. The frequencies of the axisymmetric modes are affected significantly by rotation only when the rotation rate exceeds about 50 per cent of the maximum allowed. As expected, at large rotation rates, apparent mode crossings between different modes appear. In addition to the above modes, several axisymmetric inertial modes are also excited in our numerical evolutions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017APS..DPPG11031S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017APS..DPPG11031S"><span>Convergence of the Ponderomotive Guiding Center <span class="hlt">approximation</span> in the LWFA</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Silva, Thales; Vieira, Jorge; Helm, Anton; Fonseca, Ricardo; Silva, Luis</p>
<p>2017-10-01</p>
<p>Plasma accelerators arose as potential candidates for future accelerator technology in the last few decades because of its predicted compactness and low cost. One of the proposed designs for plasma accelerators is based on Laser Wakefield Acceleration (LWFA). However, simulations performed for such systems have to solve the laser wavelength which is <span class="hlt">orders</span> of magnitude lower than the plasma wavelength. In this context, the Ponderomotive Guiding Center (PGC) algorithm for particle-in-cell (PIC) simulations is a potent tool. The laser is <span class="hlt">approximated</span> by its envelope which leads to a speed-up of around 100 times because the laser wavelength is not solved. The plasma response is well understood, and comparison with the full PIC code show an excellent agreement. However, for LWFA, the convergence of the self-injected beam parameters, such as energy and charge, was not studied before and has vital importance for the use of the algorithm in predicting the beam parameters. Our goal is to do a thorough investigation of the stability and convergence of the algorithm in situations of experimental relevance for LWFA. To this end, we perform simulations using the PGC algorithm implemented in the PIC code OSIRIS. To verify the PGC predictions, we compare the results with full PIC simulations. This project has received funding from the European Union's Horizon 2020 research and innovation programme under Grant agreement No 653782.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/17568146','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/17568146"><span><span class="hlt">Approximate</span> labeling via graph cuts based on linear programming.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Komodakis, Nikos; Tziritas, Georgios</p>
<p>2007-08-01</p>
<p>A new framework is presented for both understanding and developing graph-cut-based combinatorial algorithms suitable for the <span class="hlt">approximate</span> optimization of a very wide class of Markov Random Fields (MRFs) that are frequently encountered in computer vision. The proposed framework utilizes tools from the duality theory of linear programming in <span class="hlt">order</span> to provide an alternative and more general view of state-of-the-art techniques like the \\alpha-expansion algorithm, which is included merely as a special case. Moreover, contrary to \\alpha-expansion, the derived algorithms generate solutions with guaranteed optimality properties for a much wider class of problems, for example, even for MRFs with nonmetric potentials. In addition, they are capable of providing per-instance suboptimality bounds in all occasions, including discrete MRFs with an arbitrary potential function. These bounds prove to be very tight in practice (that is, very close to 1), which means that the resulting solutions are almost optimal. Our algorithms' effectiveness is demonstrated by presenting experimental results on a variety of low-level vision tasks, such as stereo matching, image restoration, image completion, and optical flow estimation, as well as on synthetic problems.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016ChPhB..25l4307D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016ChPhB..25l4307D"><span>Higher-<span class="hlt">order</span> harmonics of general limited diffraction Bessel beams</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Ding, De-Sheng; Huang, Jin-Huang</p>
<p>2016-12-01</p>
<p>In this paper, we extensively study the higher-<span class="hlt">order</span> harmonic generation of the general limited diffraction m-th-<span class="hlt">order</span> Bessel beam. The analysis is based on successive <span class="hlt">approximations</span> of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation. Asymptotic expansions are presented for higher-<span class="hlt">order</span> harmonic Bessel beams in near and far fields. The validity of asymptotic <span class="hlt">approximation</span> is also analyzed. The higher-<span class="hlt">order</span> harmonic of the Bessel beam with the lowest zero-<span class="hlt">order</span> is taken as a special example. Project supported by the National Natural Science Foundation of China (Grant Nos. 11074038 and 11374051).</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=What+AND+soil+AND+conservation&pg=3&id=ED260969','ERIC'); return false;" href="https://eric.ed.gov/?q=What+AND+soil+AND+conservation&pg=3&id=ED260969"><span>Bioregions and World <span class="hlt">Order</span>.</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>Breakthrough, 1985</p>
<p>1985-01-01</p>
<p>What bioregions can do to contribute to world <span class="hlt">order</span> and security is discussed in this newsletter. A bioregion is defined as an identifiable geographical area of interacting life-systems that is relatively self-sustaining in the ever-renewing processes of nature. Articles included are: "Bioregionalism and World <span class="hlt">Order</span>" (Gerald Mische);…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29173456','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29173456"><span>Recognition of computerized facial <span class="hlt">approximations</span> by familiar assessors.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Richard, Adam H; Monson, Keith L</p>
<p>2017-11-01</p>
<p>Studies testing the effectiveness of facial <span class="hlt">approximations</span> typically involve groups of participants who are unfamiliar with the <span class="hlt">approximated</span> individual(s). This limitation requires the use of photograph arrays including a picture of the subject for comparison to the facial <span class="hlt">approximation</span>. While this practice is often necessary due to the difficulty in obtaining a group of assessors who are familiar with the <span class="hlt">approximated</span> subject, it may not accurately simulate the thought process of the target audience (friends and family members) in comparing a mental image of the <span class="hlt">approximated</span> subject to the facial <span class="hlt">approximation</span>. As part of a larger process to evaluate the effectiveness and best implementation of the ReFace facial <span class="hlt">approximation</span> software program, the rare opportunity arose to conduct a recognition study using assessors who were personally acquainted with the subjects of the <span class="hlt">approximations</span>. ReFace facial <span class="hlt">approximations</span> were generated based on preexisting medical scans, and co-workers of the scan donors were tested on whether they could accurately pick out the <span class="hlt">approximation</span> of their colleague from arrays of facial <span class="hlt">approximations</span>. 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 <span class="hlt">approximated</span> subjects. Out of 220 recognition tests only 10.5% resulted in the assessor selecting the correct <span class="hlt">approximation</span> (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 <span class="hlt">approximations</span> the assessors felt resembled the target individual, the overall sensitivity for ReFace <span class="hlt">approximations</span> 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 <span class="hlt">approximated</span> subjects. Some possible explanations for this disparity in</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4142747','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4142747"><span><span class="hlt">Approximation</span> Set of the Interval Set in Pawlak's Space</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Wang, Jin; Wang, Guoyin</p>
<p>2014-01-01</p>
<p>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 <span class="hlt">approximations</span> (i.e., upper <span class="hlt">approximation</span> set R¯(Z) and lower <span class="hlt">approximation</span> set R_(Z)) are presented, respectively. The disadvantages of using upper-<span class="hlt">approximation</span> set R¯(Z) or lower-<span class="hlt">approximation</span> set R_(Z) as <span class="hlt">approximation</span> sets of the uncertain set (uncertain concept) Z are analyzed, and a new method for looking for a better <span class="hlt">approximation</span> set of the interval set Z is proposed. The conclusion that the <span class="hlt">approximation</span> set R 0.5(Z) is an optimal <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950017128','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950017128"><span>Fourth <span class="hlt">order</span> difference methods for hyperbolic IBVP's</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Gustafsson, Bertil; Olsson, Pelle</p>
<p>1994-01-01</p>
<p>Fourth <span class="hlt">order</span> difference <span class="hlt">approximations</span> of initial-boundary value problems for hyperbolic partial differential equations are considered. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics, the second one for modeling shocks and rarefaction waves. The time discretization is done with a third <span class="hlt">order</span> Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second <span class="hlt">order</span> viscosity. In case of the non-linear Burger's equation we use a flux splitting technique that results in an energy estimate for certain different <span class="hlt">approximations</span>, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth <span class="hlt">order</span> methods with a standard second <span class="hlt">order</span> one and with a third <span class="hlt">order</span> TVD-method. The results show that the fourth <span class="hlt">order</span> methods are the only ones that give good results for all the considered test problems.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2011AGUFM.A33B0195M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2011AGUFM.A33B0195M"><span><span class="hlt">Approximating</span> Reflectance and Transmittance of Vegetation Using Multiple Spectral Invariants</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Mottus, M.</p>
<p>2011-12-01</p>
<p>Canopy spectral invariants, eigenvalues of the radiative transfer equation and photon recollision probability are some of the new theoretical tools that have been applied in remote sensing of vegetation and atmosphere. The theoretical approach based on spectral invariants, informally also referred to as the p-theory, owns its attractivity to several factors. Firstly, it provides a rapid and physically-based way of describing canopy scattering. Secondly, the p-theory aims at parameterizing canopy structure in reflectance models using a simple and intuitive concept which can be applied at various structural levels, from shoot to tree crown. The theory has already been applied at scales from the molecular level to forest stands. The most important shortcoming of the p-theory lies in its inability to predict the directionality of scattering. The theory is currently based on only one physical parameter, the photon recollision probability p. It is evident that one parameter cannot contain enough information to reasonably predict the observed complex reflectance patterns produced by natural vegetation canopies. Without estimating scattering directionality, however, the theory cannot be compared with even the most simple (and well-tested) two-stream vegetation reflectance models. In this study, we evaluate the possibility to use additional parameters to fit the measured reflectance and transmittance of a vegetation stand. As a first step, the parameters are applied to separate canopy scattering into reflectance and transmittance. New parameters are introduced following the general approach of eigenvector expansion. Thus, the new parameters are coined higher-<span class="hlt">order</span> spectral invariants. Calculation of higher-<span class="hlt">order</span> invariants is based on separating first-<span class="hlt">order</span> scattering from total scattering. Thus, the method explicitly accounts for different view geometries with different fractions of visible sunlit canopy (e.g., hot-spot). It additionally allows to produce different</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/21913750','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/21913750"><span>Second-<span class="hlt">order</span> perturbation theory with a density matrix renormalization group self-consistent field reference function: theory and application to the study of chromium dimer.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Kurashige, Yuki; Yanai, Takeshi</p>
<p>2011-09-07</p>
<p>We present a second-<span class="hlt">order</span> perturbation theory based on a density matrix renormalization group self-consistent field (DMRG-SCF) reference function. The method reproduces the solution of the complete active space with second-<span class="hlt">order</span> perturbation theory (CASPT2) when the DMRG reference function is represented by a sufficiently large number of renormalized many-body basis, thereby being named DMRG-CASPT2 method. The DMRG-SCF is able to describe non-dynamical correlation with large active space that is insurmountable to the conventional CASSCF method, while the second-<span class="hlt">order</span> perturbation theory provides an efficient description of dynamical correlation effects. The capability of our implementation is demonstrated for an application to the potential energy curve of the chromium dimer, which is one of the most demanding multireference systems that require best electronic structure treatment for non-dynamical and dynamical correlation as well as large basis sets. The DMRG-CASPT2/cc-pwCV5Z calculations were performed with a large (3d double-shell) active space consisting of 28 orbitals. Our approach using large-size DMRG reference addressed the problems of why the dissociation energy is largely overestimated by CASPT2 with the small active space consisting of 12 orbitals (3d4s), and also is oversensitive to the choice of the <span class="hlt">zeroth-order</span> Hamiltonian. © 2011 American Institute of Physics</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/16268689','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/16268689"><span>On the origins of <span class="hlt">approximations</span> for stochastic chemical kinetics.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Haseltine, Eric L; Rawlings, James B</p>
<p>2005-10-22</p>
<p>This paper considers the derivation of <span class="hlt">approximations</span> 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 <span class="hlt">approximate</span>, partitioned master equations, which are Markovian in nature, from the original master equation. Under different conditions dictated by relaxation time arguments, such <span class="hlt">approximations</span> give rise to both the equilibrium and hybrid (deterministic or Langevin equations coupled with discrete stochastic simulation) <span class="hlt">approximations</span> 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 <span class="hlt">approximate</span> master equations. Two simple examples illustrate situations in which these two <span class="hlt">approximate</span> methods are applicable and demonstrate the two methods' efficiencies.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018ResPh...8.1234M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018ResPh...8.1234M"><span>Precise analytic <span class="hlt">approximations</span> for the Bessel function J1 (x)</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Maass, Fernando; Martin, Pablo</p>
<p>2018-03-01</p>
<p>Precise and straightforward analytic <span class="hlt">approximations</span> for the Bessel function J1 (x) have been found. Power series and asymptotic expansions have been used to determine the parameters of the <span class="hlt">approximation</span>, 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 <span class="hlt">Approximation</span> technique have been performed. Two procedures have been used to determine the parameters of the <span class="hlt">approximations</span>. The maximum absolute errors are in both cases smaller than 0.01. The zeros of the <span class="hlt">approximation</span> are also very precise with less than 0.04 per cent for the first one. A second <span class="hlt">approximation</span> has been also determined using two more parameters, and in this way the accuracy has been increased to less than 0.001.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD1023882','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD1023882"><span>Convergence Rates of Finite Difference Stochastic <span class="hlt">Approximation</span> Algorithms</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>2016-06-01</p>
<p>dfferences as gradient <span class="hlt">approximations</span>. 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 <span class="hlt">approximations</span>. 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 <span class="hlt">approximations</span>. It</p>
</li>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.gpo.gov/fdsys/pkg/FR-2010-09-28/pdf/2010-24285.pdf','FEDREG'); return false;" href="https://www.gpo.gov/fdsys/pkg/FR-2010-09-28/pdf/2010-24285.pdf"><span>75 FR 59743 - Public Land <span class="hlt">Order</span> No. 7750; Partial Revocation of Secretarial <span class="hlt">Order</span> dated January 20, 1910; Oregon</span></a></p>
<p><a target="_blank" href="http://www.gpo.gov/fdsys/browse/collection.action?collectionCode=FR">Federal Register 2010, 2011, 2012, 2013, 2014</a></p>
<p></p>
<p>2010-09-28</p>
<p>... partially revokes a Secretarial <span class="hlt">Order</span> dated January 20, 1910, insofar as it affects <span class="hlt">approximately</span> 9,001.84... is no longer needed for the purpose for which it was withdrawn. DATES: Effective Date: September 28..., it is <span class="hlt">ordered</span> as follows: The Secretarial <span class="hlt">Order</span> dated January 20, 1910, which withdrew land on behalf...</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=flight&pg=6&id=EJ726148','ERIC'); return false;" href="https://eric.ed.gov/?q=flight&pg=6&id=EJ726148"><span>Court <span class="hlt">Ordered</span> Desegregation</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>Reber, Sarah J.</p>
<p>2005-01-01</p>
<p>The effect of the court <span class="hlt">ordered</span> desegregation plans, on trends in segregation and white flight, are estimated. The effect of availability of school districts and other factors on the white flight across districts is also mentioned.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.epa.gov/laws-regulations/laws-and-executive-orders','PESTICIDES'); return false;" href="https://www.epa.gov/laws-regulations/laws-and-executive-orders"><span>Laws and Executive <span class="hlt">Orders</span></span></a></p>
<p><a target="_blank" href="http://www.epa.gov/pesticides/search.htm">EPA Pesticide Factsheets</a></p>
<p></p>
<p></p>
<p>Lists and links to descriptions of the major laws and executive <span class="hlt">orders</span> that EPA administers and/or that guide EPA rulemakings, including the Clean Air Act, the Toxic Substance Control Act, CERCLA or Superfund, and the Clean Water Act.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=computers+AND+optical&pg=4&id=EJ245729','ERIC'); return false;" href="https://eric.ed.gov/?q=computers+AND+optical&pg=4&id=EJ245729"><span>Automated <span class="hlt">Ordering</span> System.</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>Jones, Richard M.</p>
<p>1981-01-01</p>
<p>A computer program that utilizes an optical scanning machine is used for <span class="hlt">ordering</span> supplies in a Louisiana school system. The program provides savings in time and labor, more accurate data, and easy-to-use reports. (Author/MLF)</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/15065716','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/15065716"><span>Geometrical-optics <span class="hlt">approximation</span> of forward scattering by coated particles.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Xu, Feng; Cai, Xiaoshu; Ren, Kuanfang</p>
<p>2004-03-20</p>
<p>By means of geometrical optics we present an <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> precision. This method proves effective for transparent and tiny absorbent particles with size parameters larger than 75 but fails to give good <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> is effective only for forward small angles, typically less than 10 degrees or so.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/476036-better-approximation-guarantees-job-shop-scheduling','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/476036-better-approximation-guarantees-job-shop-scheduling"><span>Better <span class="hlt">approximation</span> guarantees for job-shop scheduling</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Goldberg, L.A.; Paterson, M.; Srinivasan, A.</p>
<p>1997-06-01</p>
<p>Job-shop scheduling is a classical NP-hard problem. Shmoys, Stein & Wein presented the first polynomial-time <span class="hlt">approximation</span> algorithm for this problem that has a good (polylogarithmic) <span class="hlt">approximation</span> guarantee. We improve the <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> guarantees for some NP-hard special cases.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19860059247&hterms=Legendre+polynomials&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3DLegendre%2Bpolynomials','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19860059247&hterms=Legendre+polynomials&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3DLegendre%2Bpolynomials"><span>Legendre-tau <span class="hlt">approximations</span> for functional differential equations</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>1986-01-01</p>
<p>The numerical <span class="hlt">approximation</span> 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 <span class="hlt">approximate</span> 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 <span class="hlt">approximations</span>. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline <span class="hlt">approximation</span> is made.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19870002588','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19870002588"><span>Sensitivity analysis and <span class="hlt">approximation</span> methods for general eigenvalue problems</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Murthy, D. V.; Haftka, R. T.</p>
<p>1986-01-01</p>
<p>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 <span class="hlt">approximate</span> reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and <span class="hlt">approximate</span> 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 <span class="hlt">approximation</span> methods for eigenvalues are proposed and evaluated. Linear and quadratic <span class="hlt">approximations</span> are based directly on the Taylor series. Several <span class="hlt">approximation</span> methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. <span class="hlt">Approximation</span> methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the <span class="hlt">approximations</span> are given. General recommendations are made for the selection of appropriate <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> is sought.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19830021837','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19830021837"><span>Legendre-Tau <span class="hlt">approximations</span> for functional differential equations</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>1983-01-01</p>
<p>The numerical <span class="hlt">approximation</span> 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 <span class="hlt">approximate</span> 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 <span class="hlt">approximations</span>. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline <span class="hlt">approximations</span> is made.</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('https://ntrs.nasa.gov/search.jsp?R=19900042212&hterms=equations+differential&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D90%26Ntt%3Dequations%2Bdifferential','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19900042212&hterms=equations+differential&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D90%26Ntt%3Dequations%2Bdifferential"><span>Differential equation based method for accurate <span class="hlt">approximations</span> in optimization</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Pritchard, Jocelyn I.; Adelman, Howard M.</p>
<p>1990-01-01</p>
<p>This paper describes a method to efficiently and accurately <span class="hlt">approximate</span> 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 <span class="hlt">approximations</span>, hence, the method is denoted the Differential Equation Based (DEB) method. <span class="hlt">Approximations</span> were developed for vibration frequencies, mode shapes and static displacements. The DEB <span class="hlt">approximation</span> method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series <span class="hlt">approximations</span> 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 <span class="hlt">approximation</span>. The method is applicable to simultaneous perturbation of several design variables. Also, the <span class="hlt">approximations</span> may be used to calculate other system response quantities. For example, the <span class="hlt">approximations</span> for displacement are used to <span class="hlt">approximate</span> bending stresses.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018PhRvD..97j5002B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018PhRvD..97j5002B"><span>Thin-wall <span class="hlt">approximation</span> in vacuum decay: A lemma</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Brown, Adam R.</p>
<p>2018-05-01</p>
<p>The "thin-wall <span class="hlt">approximation</span>" gives a simple estimate of the decay rate of an unstable quantum field. Unfortunately, the <span class="hlt">approximation</span> is uncontrolled. In this paper I show that there are actually two different thin-wall <span class="hlt">approximations</span> 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 <span class="hlt">approximations</span> converge. In the presence of gravity, a generalization of this lemma provides a simple sufficient condition for nonperturbative vacuum instability.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/6174646-convergence-strong-potential-born-approximation-sub-less-than-sub-greater-than','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/6174646-convergence-strong-potential-born-approximation-sub-less-than-sub-greater-than"><span>Convergence of the strong-potential-Born <span class="hlt">approximation</span> in Z/sub less-than//Z/sub greater-than/</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>McGuire, J.H.; Sil, N.C.</p>
<p>1986-01-01</p>
<p>Convergence of the strong-potential Born (SPB) <span class="hlt">approximation</span> as a function of the charges of the projectile and target is studied numerically. Time-reversal invariance (or detailed balance) is satisfied at sufficiently high velocities even when the charges are asymmetric. This demonstarates that the SPB <span class="hlt">approximation</span> converges to the correct result even when the charge of the ''weak'' potential, which is kept to first <span class="hlt">order</span>, is larger than the charge of the ''strong'' potential, which is retained to all <span class="hlt">orders</span>. Consequently, the SPB <span class="hlt">approximation</span> is valid for systems of arbitrary charge symmetry (including symmetric systems) at sufficiently high velocities.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19930018571','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19930018571"><span>Embedding impedance <span class="hlt">approximations</span> in the analysis of SIS mixers</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Kerr, A. R.; Pan, S.-K.; Withington, S.</p>
<p>1992-01-01</p>
<p>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 <span class="hlt">approximation</span> to Tucker's quantum mixer theory. This paper examines the adequacy of three <span class="hlt">approximations</span> to Tucker's theory: (1) the usual three-frequency <span class="hlt">approximation</span> which assumes a sinusoidal LO voltage at the junction, and a short-circuit at all frequencies above the upper sideband; (2) a five-frequency <span class="hlt">approximation</span> which allows two LO voltage harmonics and five small-signal sidebands; and (3) a quasi five-frequency <span class="hlt">approximation</span> 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 <span class="hlt">approximations</span> 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 <span class="hlt">approximations</span> 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 <span class="hlt">approximation</span> is a considerable improvement over the three-frequency <span class="hlt">approximation</span>, and should be suitable for much design work. For the realistic SIS mixers considered here, the five-frequency <span class="hlt">approximation</span> gives results very close to those of the eight-harmonic solution. Use of these <span class="hlt">approximations</span>, 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29197243','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29197243"><span>Automated facial recognition of manually generated clay facial <span class="hlt">approximations</span>: Potential application in unidentified persons data repositories.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Parks, Connie L; Monson, Keith L</p>
<p>2018-01-01</p>
<p>This research examined how accurately 2D images (i.e., photographs) of 3D clay facial <span class="hlt">approximations</span> were matched to corresponding photographs of the <span class="hlt">approximated</span> individuals using an objective automated facial recognition system. Irrespective of search filter (i.e., blind, sex, or ancestry) or rank class (R 1 , R 10 , R 25 , and R 50 ) employed, few operationally informative results were observed. In only a single instance of 48 potential match opportunities was a clay <span class="hlt">approximation</span> matched to a corresponding life photograph within the top 50 images (R 50 ) of a candidate list, even with relatively small gallery sizes created from the application of search filters (e.g., sex or ancestry search restrictions). Increasing the candidate lists to include the top 100 images (R 100 ) resulted in only two additional instances of correct match. Although other untested variables (e.g., <span class="hlt">approximation</span> method, 2D photographic process, and practitioner skill level) may have impacted the observed results, this study suggests that 2D images of manually generated clay <span class="hlt">approximations</span> are not readily matched to life photos by automated facial recognition systems. Further investigation is necessary in <span class="hlt">order</span> to identify the underlying cause(s), if any, of the poor recognition results observed in this study (e.g., potential inferior facial feature detection and extraction). Additional inquiry exploring prospective remedial measures (e.g., stronger feature differentiation) is also warranted, particularly given the prominent use of clay <span class="hlt">approximations</span> in unidentified persons casework. Copyright © 2017. Published by Elsevier B.V.</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 <span class="hlt">orders</span> 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://adsabs.harvard.edu/abs/1999NIMPB.153....1S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1999NIMPB.153....1S"><span>A unitary convolution <span class="hlt">approximation</span> for the impact-parameter dependent electronic energy loss</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Schiwietz, G.; Grande, P. L.</p>
<p>1999-06-01</p>
<p>In this work, we propose a simple method to calculate the impact-parameter dependence of the electronic energy loss of bare ions for all impact parameters. This perturbative convolution <span class="hlt">approximation</span> (PCA) is based on first-<span class="hlt">order</span> perturbation theory, and thus, it is only valid for fast particles with low projectile charges. Using Bloch's stopping-power result and a simple scaling, we get rid of the restriction to low charge states and derive the unitary convolution <span class="hlt">approximation</span> (UCA). Results of the UCA are then compared with full quantum-mechanical coupled-channel calculations for the impact-parameter dependent electronic energy loss.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27006884','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27006884"><span>Laplace transform homotopy perturbation method for the <span class="hlt">approximation</span> of variational problems.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Filobello-Nino, U; Vazquez-Leal, H; Rashidi, M M; Sedighi, H M; Perez-Sesma, A; Sandoval-Hernandez, M; Sarmiento-Reyes, A; Contreras-Hernandez, A D; Pereyra-Diaz, D; Hoyos-Reyes, C; Jimenez-Fernandez, V M; Huerta-Chua, J; Castro-Gonzalez, F; Laguna-Camacho, J R</p>
<p>2016-01-01</p>
<p>This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in <span class="hlt">order</span> to find analytical <span class="hlt">approximate</span> solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the <span class="hlt">approximate</span> solutions, belongs to the interval [0.001918936920, 0.06334882582], which confirms the accuracy of the proposed methods, taking into account the complexity and difficulty of variational problems.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19880014221','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19880014221"><span>Complex space monofilar <span class="hlt">approximation</span> of diffraction currents on a conducting half plane</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Lindell, I. V.</p>
<p>1987-01-01</p>
<p>Simple <span class="hlt">approximation</span> of diffraction surface currents on a conducting half plane, due to an incoming plane wave, is obtained with a line current (monofile) in complex space. When compared to an <span class="hlt">approximating</span> current at the edge, the diffraction pattern is seen to improve by an <span class="hlt">order</span> of magnitude for a minimal increase of computation effort. Thus, the inconvient Fresnel integral functions can be avoided for quick calculations of diffracted fields and the accuracy is good in other directions than along the half plane. The method can be applied to general problems involving planar metal edges.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017CPL...690...20K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017CPL...690...20K"><span>Restoring the Pauli principle in the random phase <span class="hlt">approximation</span> ground state</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Kosov, D. S.</p>
<p>2017-12-01</p>
<p>Random phase <span class="hlt">approximation</span> ground state contains electronic configurations where two (and more) identical electrons can occupy the same molecular spin-orbital violating the Pauli exclusion principle. This overcounting of electronic configurations happens due to quasiboson <span class="hlt">approximation</span> in the treatment of electron-hole pair operators. We describe the method to restore the Pauli principle in the RPA wavefunction. The proposed theory is illustrated by the calculations of molecular dipole moments and electronic kinetic energies. The Hartree-Fock based RPA, which is corrected for the Pauli principle, gives the results of comparable accuracy with Møller-Plesset second <span class="hlt">order</span> perturbation theory and coupled-cluster singles and doubles method.</p>
</li>
</ol>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24479023','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24479023"><span>Birth <span class="hlt">order</span> and psychopathology.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Risal, Ajay; Tharoor, Hema</p>
<p>2012-07-01</p>
<p>Ordinal position the child holds within the sibling ranking of a family is related to intellectual functioning, personality, behavior, and development of psychopathology. To study the association between birth <span class="hlt">order</span> and development of psychopathology in patients attending psychiatry services in a teaching hospital. Hospital-based cross-sectional study. Retrospective file review of three groups of patients was carried out. Patient-related variables like age of onset, birth <span class="hlt">order</span>, family type, and family history of mental illness were compared with psychiatry diagnosis (ICD-10) generated. SPSS 13; descriptive statistics and one-way analysis of variance (ANOVA) were used. Mean age of onset of mental illness among the adult general psychiatry patients (group I, n = 527) was found to be 33.01 ± 15.073, while it was 11.68 ± 4.764 among the child cases (group II, n = 47) and 26.74 ± 7.529 among substance abuse cases (group III, n = 110). Among group I patients, commonest diagnosis was depression followed by anxiety and somatoform disorders irrespective of birth <span class="hlt">order</span>. Dissociative disorders were most prevalent in the first born child (36.7%) among group II patients. Among group III patients, alcohol dependence was maximum diagnosis in all birth <span class="hlt">orders</span>. Depression and alcohol dependence was the commonest diagnosis in adult group irrespective of birth <span class="hlt">order</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27242648','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27242648"><span>Arguments from Developmental <span class="hlt">Order</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Stöckle-Schobel, Richard</p>
<p>2016-01-01</p>
<p>In this article, I investigate a special type of argument regarding the role of development in theorizing about psychological processes and cognitive capacities. Among the issues that developmental psychologists study, discovering the ontogenetic trajectory of mechanisms or capacities underpinning our cognitive functions ranks highly. The <span class="hlt">order</span> in which functions are developed or capacities are acquired is a matter of debate between competing psychological theories, and also philosophical conceptions of the mind - getting the role and the significance of the different steps in this <span class="hlt">order</span> right could be seen as an important virtue of such theories. Thus, a special kind of strategy in arguments between competing philosophical or psychological theories is using developmental <span class="hlt">order</span> in arguing for or against a given psychological claim. In this article, I will introduce an analysis of arguments from developmental <span class="hlt">order</span>, which come in two general types: arguments emphasizing the importance of the early cognitive processes and arguments emphasizing the late cognitive processes. I will discuss their role in one of the central tools for evaluating scientific theories, namely in making inferences to the best explanation. I will argue that appeal to developmental <span class="hlt">order</span> is, by itself, an insufficient criterion for theory choice and has to be part of an argument based on other core explanatory or empirical virtues. I will end by proposing a more concerted study of philosophical issues concerning (cognitive) development, and I will present some topics that also pertain to a full-fledged 'philosophy of development.'</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/501518','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/501518"><span>The birth <span class="hlt">order</span> puzzle.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Zajonc, R B; Markus, H; Markus, G B</p>
<p>1979-08-01</p>
<p>Studies relating intellectual performance to birth <span class="hlt">order</span> report conflicting results, some finding intellectual scores to increase, others to decrease with birth <span class="hlt">order</span>. In contrast, the relationship between intellectual performance and family size is stable and consistently replicable. Why do these two highly related variables generate such divergent results? This birth <span class="hlt">order</span> puzzle is resolved by means of the confluence model that quantifies the influences upon intellectual growth arising within the family context. At the time of a new birth, two opposing influences act upon intellectual growth of the elder sibling: (a) his or her intellectual environment is "diluted" and (b) he or she loses the "last-born's handicap" and begins serving as an intellectual resource to the younger sibling. Since these opposite effects are not equal in magnitude, the differences in intellectual performance among birth ranks are shown to be age dependent. While elder children may surpass their younger siblings in intellectual performance at some ages, they may be overtaken by them at others. Thus when age is taken into consideration, the birth <span class="hlt">order</span> literature loses its chaotic character and an <span class="hlt">orderly</span> pattern of results emerges.</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22572343-samba-sparse-approximation-moment-based-arbitrary-polynomial-chaos','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22572343-samba-sparse-approximation-moment-based-arbitrary-polynomial-chaos"><span>SAMBA: Sparse <span class="hlt">Approximation</span> of Moment-Based Arbitrary Polynomial Chaos</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Ahlfeld, R., E-mail: r.ahlfeld14@imperial.ac.uk; Belkouchi, B.; Montomoli, F.</p>
<p>2016-09-01</p>
<p>A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrixmore » is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse <span class="hlt">Approximation</span> of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher <span class="hlt">order</span> convergence and accuracy for a set of nonlinear test functions with 2, 5 and</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3663730','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3663730"><span>Cluster and propensity based <span class="hlt">approximation</span> of a network</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p></p>
<p>2013-01-01</p>
<p>Background The models in this article generalize current models for both correlation networks and multigraph networks. Correlation networks are widely applied in genomics research. In contrast to general networks, it is straightforward to test the statistical significance of an edge in a correlation network. It is also easy to decompose the underlying correlation matrix and generate informative network statistics such as the module eigenvector. However, correlation networks only capture the connections between numeric variables. An open question is whether one can find suitable decompositions of the similarity measures employed in constructing general networks. Multigraph networks are attractive because they support likelihood based inference. Unfortunately, it is unclear how to adjust current statistical methods to detect the clusters inherent in many data sets. Results Here we present an intuitive and parsimonious parametrization of a general similarity measure such as a network adjacency matrix. The cluster and propensity based <span class="hlt">approximation</span> (CPBA) of a network not only generalizes correlation network methods but also multigraph methods. In particular, it gives rise to a novel and more realistic multigraph model that accounts for clustering and provides likelihood based tests for assessing the significance of an edge after controlling for clustering. We present a novel Majorization-Minimization (MM) algorithm for estimating the parameters of the CPBA. To illustrate the practical utility of the CPBA of a network, we apply it to gene expression data and to a bi-partite network model for diseases and disease genes from the Online Mendelian Inheritance in Man (OMIM). Conclusions The CPBA of a network is theoretically appealing since a) it generalizes correlation and multigraph network methods, b) it improves likelihood based significance tests for edge counts, c) it directly models higher-<span class="hlt">order</span> relationships between clusters, and d) it suggests novel clustering</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016CMaPh.346..397B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016CMaPh.346..397B"><span>Local Random Quantum Circuits are <span class="hlt">Approximate</span> Polynomial-Designs</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.; Horodecki, Michał</p>
<p>2016-09-01</p>
<p>We prove that local random quantum circuits acting on n qubits composed of O( t 10 n 2) many nearest neighbor two-qubit gates form an <span class="hlt">approximate</span> unitary t-design. Previously it was unknown whether random quantum circuits were a t-design for any t > 3. The proof is based on an interplay of techniques from quantum many-body theory, representation theory, and the theory of Markov chains. In particular we employ a result of Nachtergaele for lower bounding the spectral gap of frustration-free quantum local Hamiltonians; a quasi-orthogonality property of permutation matrices; a result of Oliveira which extends to the unitary group the path-coupling method for bounding the mixing time of random walks; and a result of Bourgain and Gamburd showing that dense subgroups of the special unitary group, composed of elements with algebraic entries, are ∞-copy tensor-product expanders. We also consider pseudo-randomness properties of local random quantum circuits of small depth and prove that circuits of depth O( t 10 n) constitute a quantum t-copy tensor-product expander. The proof also rests on techniques from quantum many-body theory, in particular on the detectability lemma of Aharonov, Arad, Landau, and Vazirani. We give applications of the results to cryptography, equilibration of closed quantum dynamics, and the generation of topological <span class="hlt">order</span>. In particular we show the following pseudo-randomness property of generic quantum circuits: Almost every circuit U of size O( n k ) on n qubits cannot be distinguished from a Haar uniform unitary by circuits of size O( n ( k-9)/11) that are given oracle access to U.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012SPIE.8499E..1IS','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012SPIE.8499E..1IS"><span>OCR enhancement through neighbor embedding and fast <span class="hlt">approximate</span> nearest neighbors</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Smith, D. C.</p>
<p>2012-10-01</p>
<p>Generic optical character recognition (OCR) engines often perform very poorly in transcribing scanned low resolution (LR) text documents. To improve OCR performance, we apply the Neighbor Embedding (NE) single-image super-resolution (SISR) technique to LR scanned text documents to obtain high resolution (HR) versions, which we subsequently process with OCR. For comparison, we repeat this procedure using bicubic interpolation (BI). We demonstrate that mean-square errors (MSE) in NE HR estimates do not increase substantially when NE is trained in one Latin font style and tested in another, provided both styles belong to the same font category (serif or sans serif). This is very important in practice, since for each font size, the number of training sets required for each category may be reduced from dozens to just one. We also incorporate randomized k-d trees into our NE implementation to perform <span class="hlt">approximate</span> nearest neighbor search, and obtain a 1000x speed up of our original NE implementation, with negligible MSE degradation. This acceleration also made it practical to combine all of our size-specific NE Latin models into a single Universal Latin Model (ULM). The ULM eliminates the need to determine the unknown font category and size of an input LR text document and match it to an appropriate model, a very challenging task, since the dpi (pixels per inch) of the input LR image is generally unknown. Our experiments show that OCR character error rates (CER) were over 90% when we applied the Tesseract OCR engine to LR text documents (scanned at 75 dpi and 100 dpi) in the 6-10 pt range. By contrast, using k-d trees and the ULM, CER after NE preprocessing averaged less than 7% at 3x (100 dpi LR scanning) and 4x (75 dpi LR scanning) magnification, over an <span class="hlt">order</span> of magnitude improvement. Moreover, CER after NE preprocessing was more that 6 times lower on average than after BI preprocessing.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015PhRvE..92f2806S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015PhRvE..92f2806S"><span><span class="hlt">Approximating</span> frustration scores in complex networks via perturbed Laplacian spectra</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Savol, Andrej J.; Chennubhotla, Chakra S.</p>
<p>2015-12-01</p>
<p>Systems of many interacting components, as found in physics, biology, infrastructure, and the social sciences, are often modeled by simple networks of nodes and edges. The real-world systems frequently confront outside intervention or internal damage whose impact must be predicted or minimized, and such perturbations are then mimicked in the models by altering nodes or edges. This leads to the broad issue of how to best quantify changes in a model network after some type of perturbation. In the case of node removal there are many centrality metrics which associate a scalar quantity with the removed node, but it can be difficult to associate the quantities with some intuitive aspect of physical behavior in the network. This presents a serious hurdle to the application of network theory: real-world utility networks are rarely altered according to theoretic principles unless the kinetic impact on the network's users are fully appreciated beforehand. In pursuit of a kinetically interpretable centrality score, we discuss the f-score, or frustration score. Each f-score quantifies whether a selected node accelerates or inhibits global mean first passage times to a second, independently selected target node. We show that this is a natural way of revealing the dynamical importance of a node in some networks. After discussing merits of the f-score metric, we combine spectral and Laplacian matrix theory in <span class="hlt">order</span> to quickly <span class="hlt">approximate</span> the exact f-score values, which can otherwise be expensive to compute. Following tests on both synthetic and real medium-sized networks, we report f-score runtime improvements over exact brute force approaches in the range of 0 to 400 % with low error (<3 % ).</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JCoPh.320....1A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JCoPh.320....1A"><span>SAMBA: Sparse <span class="hlt">Approximation</span> of Moment-Based Arbitrary Polynomial Chaos</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Ahlfeld, R.; Belkouchi, B.; Montomoli, F.</p>
<p>2016-09-01</p>
<p>A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse <span class="hlt">Approximation</span> of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher <span class="hlt">order</span> convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10</p>
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<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>
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<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=precalculus&id=EJ1127893','ERIC'); return false;" href="https://eric.ed.gov/?q=precalculus&id=EJ1127893"><span><span class="hlt">Approximating</span> Exponential and Logarithmic Functions Using Polynomial Interpolation</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>Gordon, Sheldon P.; Yang, Yajun</p>
<p>2017-01-01</p>
<p>This article takes a closer look at the problem of <span class="hlt">approximating</span> the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial <span class="hlt">approximations</span> at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=calculus+AND+8&pg=7&id=EJ952945','ERIC'); return false;" href="https://eric.ed.gov/?q=calculus+AND+8&pg=7&id=EJ952945"><span>Finding the Best Quadratic <span class="hlt">Approximation</span> of a Function</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>Yang, Yajun; Gordon, Sheldon P.</p>
<p>2011-01-01</p>
<p>This article examines the question of finding the best quadratic function to <span class="hlt">approximate</span> 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 <span class="hlt">approximations</span> at various points in the interval under consideration, the other based on the fact…</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24807446','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24807446"><span>Function <span class="hlt">approximation</span> using combined unsupervised and supervised learning.</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>2014-03-01</p>
<p>Function <span class="hlt">approximation</span> is one of the core tasks that are solved using neural networks in the context of many engineering problems. However, good <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span>. We compare the <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> using the original high-dimensional data.</p>
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<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/scitech">SciTech Connect</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>
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<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=filters&pg=5&id=EJ950130','ERIC'); return false;" href="https://eric.ed.gov/?q=filters&pg=5&id=EJ950130"><span>Reply to Steele & Ferrer: Modeling Oscillation, <span class="hlt">Approximately</span> or Exactly?</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>Oud, Johan H. L.; Folmer, Henk</p>
<p>2011-01-01</p>
<p>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 <span class="hlt">approximate</span> estimation procedure applied. This procedure is the latent version of the local linear <span class="hlt">approximation</span> procedure…</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.gpo.gov/fdsys/pkg/CFR-2010-title36-vol2/pdf/CFR-2010-title36-vol2-sec254-11.pdf','CFR'); return false;" href="https://www.gpo.gov/fdsys/pkg/CFR-2010-title36-vol2/pdf/CFR-2010-title36-vol2-sec254-11.pdf"><span>36 CFR 254.11 - Exchanges at <span class="hlt">approximately</span> equal value.</span></a></p>
<p><a target="_blank" href="http://www.gpo.gov/fdsys/browse/collectionCfr.action?selectedYearFrom=2010&page.go=Go">Code of Federal Regulations, 2010 CFR</a></p>
<p></p>
<p>2010-07-01</p>
<p>... 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 <span class="hlt">approximately</span> equal value. (a) The authorized officer may exchange lands which are of <span class="hlt">approximately</span> equal value upon a determination that: (1...</p>
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<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>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2007JMAA..334..909Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2007JMAA..334..909Z"><span>Best uniform <span class="hlt">approximation</span> to a class of rational functions</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Zheng, Zhitong; Yong, Jun-Hai</p>
<p>2007-10-01</p>
<p>We explicitly determine the best uniform polynomial <span class="hlt">approximation</span> 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 <span class="hlt">approximating</span> 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 <span class="hlt">approximation</span> error by a trigonometric function. Moreover, we formulate an ansatz which offers a heuristic strategies to determine the best <span class="hlt">approximating</span> 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 <span class="hlt">approximation</span> to some more functions.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2003JCoAM.155..359B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003JCoAM.155..359B"><span>Laplace <span class="hlt">approximation</span> for Bessel functions of matrix argument</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Butler, Ronald W.; Wood, Andrew T. A.</p>
<p>2003-06-01</p>
<p>We derive Laplace <span class="hlt">approximations</span> 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 <span class="hlt">approximations</span>. On the theoretical side, it is shown that the Laplace <span class="hlt">approximations</span> to A[nu], B[nu] and [Psi] given here, together with the Laplace <span class="hlt">approximations</span> to the matrix argument functions 1F1 and 2F1 presented in Butler and Wood (Laplace <span class="hlt">approximations</span> to hyper-geometric functions with matrix argument, Ann. Statist. (2002)), satisfy all the important confluence relations and symmetry relations enjoyed by the original functions.</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1188218-properties-boltzmann-equation-classical-approximation','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1188218-properties-boltzmann-equation-classical-approximation"><span>Properties of the Boltzmann equation in the classical <span class="hlt">approximation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p>
<p>Epelbaum, Thomas; Gelis, François; Tanji, Naoto; ...</p>
<p>2014-12-30</p>
<p>We examine the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical <span class="hlt">approximation</span>. 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 <span class="hlt">approximations</span>. This cutoff dependence in the classical <span class="hlt">approximations</span> of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical <span class="hlt">approximation</span> 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-<span class="hlt">approximated</span> result for comparison.« less</p>
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<ol class="result-class" start="461">
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19910012800','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19910012800"><span>Recent advances in <span class="hlt">approximation</span> concepts for optimum structural design</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Barthelemy, Jean-Francois M.; Haftka, Raphael T.</p>
<p>1991-01-01</p>
<p>The basic <span class="hlt">approximation</span> 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 <span class="hlt">approximations</span>; it covers functions <span class="hlt">approximations</span> and problem <span class="hlt">approximations</span>. 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 <span class="hlt">approximations</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=ionic&pg=5&id=EJ826728','ERIC'); return false;" href="https://eric.ed.gov/?q=ionic&pg=5&id=EJ826728"><span>Concomitant <span class="hlt">Ordering</span> and Symmetry Lowering</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>Boo, William O. J.; Mattern, Daniell L.</p>
<p>2008-01-01</p>
<p>Examples of concomitant <span class="hlt">ordering</span> include magnetic <span class="hlt">ordering</span>, Jahn-Teller cooperative <span class="hlt">ordering</span>, electronic <span class="hlt">ordering</span>, ionic <span class="hlt">ordering</span>, and <span class="hlt">ordering</span> of partially-filled sites. Concomitant <span class="hlt">ordering</span> sets in when a crystal is cooled and always lowers the degree of symmetry of the crystal. Concomitant <span class="hlt">ordering</span> concepts can also be productively applied to…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2003CPL...371..433P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003CPL...371..433P"><span>Hierarchically <span class="hlt">ordered</span> carbon tubes</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Pan, Zheng-Wei; Zhu, Hao-Guo; Zhang, Zong-Tao; Im, Hee-Jung; Dai, Sheng; Beach, David B.; Lowndes, Douglas H.</p>
<p>2003-04-01</p>
<p>Micropatterns of hierarchically <span class="hlt">ordered</span> carbon tubes (i.e., <span class="hlt">ordered</span> carbon microtubes composed of aligned carbon nanotubes) were grown on a film-like iron/silica substrate consisting of ring-like catalyst patterns. The substrates were prepared by a combined technique, in which the sol-gel method was used to prepare catalyst film and transmission electron microscope grids were used as a shadow mask. In comparison with other techniques that involve sophisticated lithography, this approach represents a simple and low-cost way to the micropatterning of aligned carbon nanotubes.</p>
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<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 <span class="hlt">order</span> 1 over epsilon squared time scale.</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25462030','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25462030"><span><span class="hlt">Approximate</span> number word knowledge before the cardinal principle.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Gunderson, Elizabeth A; Spaepen, Elizabet; Levine, Susan C</p>
<p>2015-02-01</p>
<p><span class="hlt">Approximate</span> number word knowledge-understanding the relation between the count words and the <span class="hlt">approximate</span> magnitudes of sets-is a critical piece of knowledge that predicts later math achievement. However, researchers disagree about when children first show evidence of <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> 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 <span class="hlt">approximate</span> number word knowledge on the verbal production task even when only examining set sizes above 4. Across studies, children's age predicted <span class="hlt">approximate</span> 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 <span class="hlt">approximation</span> ability. Thus, children can develop <span class="hlt">approximate</span> knowledge of number words up to 10 before learning the cardinal principle. Furthermore, <span class="hlt">approximate</span> 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.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=automation+AND+large+AND+data&pg=4&id=ED077516','ERIC'); return false;" href="https://eric.ed.gov/?q=automation+AND+large+AND+data&pg=4&id=ED077516"><span><span class="hlt">Order</span> Division Automated System.</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>Kniemeyer, Justin M.; And Others</p>
<p></p>
<p>This publication was prepared by the <span class="hlt">Order</span> Division Automation Project staff to fulfill the Library of Congress' requirement to document all automation efforts. The report was originally intended for internal use only and not for distribution outside the Library. It is now felt that the library community at-large may have an interest in the…</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=psychology+AND+birth+AND+order&pg=5&id=EJ219982','ERIC'); return false;" href="https://eric.ed.gov/?q=psychology+AND+birth+AND+order&pg=5&id=EJ219982"><span>The Birth <span class="hlt">Order</span> Puzzle.</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>Zajonc, R. B.; And Others</p>
<p>1979-01-01</p>
<p>Discusses the controversy of the relationship between birth <span class="hlt">order</span> and intellectual performance through a detailed evaluation of the confluence model which assumes that the rate of intellectual growth is a function of the intellectual environment within the family and associated with the special circumstances of last children. (CM)</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=global+AND+Governance&pg=4&id=EJ774301','ERIC'); return false;" href="https://eric.ed.gov/?q=global+AND+Governance&pg=4&id=EJ774301"><span>Education and World <span class="hlt">Order</span></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>Jones, Phillip W.</p>
<p>2007-01-01</p>
<p>The impact on educational analysis of mainstream international relations (IR) theories is yet to realize its full potential. The problem of education in relation to the construction of world <span class="hlt">order</span> is considered in relation to core developments in IR theory since the Second World War. In particular, the global architecture of education is seen as a…</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=proactive+AND+interference&pg=7&id=EJ396036','ERIC'); return false;" href="https://eric.ed.gov/?q=proactive+AND+interference&pg=7&id=EJ396036"><span>Memory for Serial <span class="hlt">Order</span>.</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>Lewandowsky, Stephan; Murdock, Bennet B., Jr.</p>
<p>1989-01-01</p>
<p>An extension to Murdock's Theory of Distributed Associative Memory, based on associative chaining between items, is presented. The extended theory is applied to several serial <span class="hlt">order</span> phenomena, including serial list learning, delayed recall effects, partial report effects, and buildup and release from proactive interference. (TJH)</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19730004865','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19730004865"><span><span class="hlt">Order</span>, topology and preference</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Sertel, M. R.</p>
<p>1971-01-01</p>
<p>Some standard <span class="hlt">order</span>-related and topological notions, facts, and methods are brought to bear on central topics in the theory of preference and the theory of optimization. Consequences of connectivity are considered, especially from the viewpoint of normally preordered spaces. Examples are given showing how the theory of preference, or utility theory, can be applied to social analysis.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=rodgers&pg=4&id=EJ628843','ERIC'); return false;" href="https://eric.ed.gov/?q=rodgers&pg=4&id=EJ628843"><span>Birth <span class="hlt">Order</span> Debate Resolved?</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>Zajonc, R. B.</p>
<p>2001-01-01</p>
<p>Critiques Rodgers et al.'s June 2000 research on the relation between birth <span class="hlt">order</span> and intelligence, which suggests that it is a methodological illusion. Explains how the intellectual environment and the teaching function (whereby older children tutor younger ones) contribute to the growth of intellectual maturity, the first negatively and the…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=deforestation&pg=6&id=ED219316','ERIC'); return false;" href="https://eric.ed.gov/?q=deforestation&pg=6&id=ED219316"><span>Land and World <span class="hlt">Order</span>.</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>Mische, Patricia, Ed.; And Others</p>
<p>1982-01-01</p>
<p>The papers in this publication discuss the land and how what happens to the land affects us. The publication is one in a series of monographs that examine the linkages between local and global concerns and explore alternative world futures. Examples of topics discussed in the papers follow. The paper "Land and World <span class="hlt">Order</span>" examines…</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22415855-similarity-transformed-perturbation-theory-top-truncated-local-coupled-cluster-solutions-theory-applications-intermolecular-interactions','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22415855-similarity-transformed-perturbation-theory-top-truncated-local-coupled-cluster-solutions-theory-applications-intermolecular-interactions"><span>Similarity-transformed perturbation theory on top of truncated local coupled cluster solutions: Theory and applications to intermolecular interactions</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Azar, Richard Julian, E-mail: julianazar2323@berkeley.edu; Head-Gordon, Martin, E-mail: mhg@cchem.berkeley.edu</p>
<p>2015-05-28</p>
<p>Your correspondents develop and apply fully nonorthogonal, local-reference perturbation theories describing non-covalent interactions. Our formulations are based on a Löwdin partitioning of the similarity-transformed Hamiltonian into a <span class="hlt">zeroth-order</span> intramonomer piece (taking local CCSD solutions as its <span class="hlt">zeroth-order</span> eigenfunction) plus a first-<span class="hlt">order</span> piece coupling the fragments. If considerations are limited to a single molecule, the proposed intermolecular similarity-transformed perturbation theory represents a frozen-orbital variant of the “(2)”-type theories shown to be competitive with CCSD(T) and of similar cost if all terms are retained. Different restrictions on the <span class="hlt">zeroth</span>- and first-<span class="hlt">order</span> amplitudes are explored in the context of large-computation tractability and elucidationmore » of non-local effects in the space of singles and doubles. To accurately <span class="hlt">approximate</span> CCSD intermolecular interaction energies, a quadratically growing number of variables must be included at <span class="hlt">zeroth-order</span>.« less</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950012943','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950012943"><span>A test of the adhesion <span class="hlt">approximation</span> for gravitational clustering</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Melott, Adrian L.; Shandarin, Sergei; Weinberg, David H.</p>
<p>1993-01-01</p>
<p>We quantitatively compare a particle implementation of the adhesion <span class="hlt">approximation</span> to fully non-linear, numerical 'N-body' simulations. Our primary tool, cross-correlation of N-body simulations with the adhesion <span class="hlt">approximation</span>, indicates good agreement, better than that found by the same test performed with the Zel-dovich <span class="hlt">approximation</span> (hereafter ZA). However, the cross-correlation is not as good as that of the truncated Zel-dovich <span class="hlt">approximation</span> (TZA), obtained by applying the Zel'dovich <span class="hlt">approximation</span> after smoothing the initial density field with a Gaussian filter. We confirm that the adhesion <span class="hlt">approximation</span> produces an excessively filamentary distribution. Relative to the N-body results, we also find that: (a) the power spectrum obtained from the adhesion <span class="hlt">approximation</span> 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.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19950048281&hterms=weinberg&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAuthor-Name%26N%3D0%26No%3D70%26Ntt%3Dweinberg','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19950048281&hterms=weinberg&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAuthor-Name%26N%3D0%26No%3D70%26Ntt%3Dweinberg"><span>A test of the adhesion <span class="hlt">approximation</span> for gravitational clustering</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Melott, Adrian L.; Shandarin, Sergei F.; Weinberg, David H.</p>
<p>1994-01-01</p>
<p>We quantitatively compare a particle implementation of the adhesion <span class="hlt">approximation</span> to fully nonlinear, numerical 'N-body' simulations. Our primary tool, cross-correlation of N-body simulations with the adhesion <span class="hlt">approximation</span>, indicates good agreement, better than that found by the same test performed with the Zel'dovich <span class="hlt">approximation</span> (hereafter ZA). However, the cross-correlation is not as good as that of the truncated Zel'dovich <span class="hlt">approximation</span> (TZA), obtained by applying the Zel'dovich <span class="hlt">approximation</span> after smoothing the initial density field with a Gaussian filter. We confirm that the adhesion <span class="hlt">approximation</span> produces an excessively filamentary distribution. Relative to the N-body results, we also find that: (a) the power spectrum obtained from the adhesion <span class="hlt">approximation</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19910021552','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19910021552"><span>The convergence rate of <span class="hlt">approximate</span> solutions for nonlinear scalar conservation laws</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Nessyahu, Haim; Tadmor, Eitan</p>
<p>1991-01-01</p>
<p>The convergence rate is discussed of <span class="hlt">approximate</span> solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the <span class="hlt">approximate</span> solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable <span class="hlt">approximate</span> solutions, that their Lip'convergence rate to the entropy solution is of the same <span class="hlt">order</span> as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018CG....113...23B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018CG....113...23B"><span>Valid <span class="hlt">approximation</span> of spatially distributed grain size distributions - A priori information encoded to a feedforward network</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Berthold, T.; Milbradt, P.; Berkhahn, V.</p>
<p>2018-04-01</p>
<p>This paper presents a model for the <span class="hlt">approximation</span> of multiple, spatially distributed grain size distributions based on a feedforward neural network. Since a classical feedforward network does not guarantee to produce valid cumulative distribution functions, a priori information is incor porated into the model by applying weight and architecture constraints. The model is derived in two steps. First, a model is presented that is able to produce a valid distribution function for a single sediment sample. Although initially developed for sediment samples, the model is not limited in its application; it can also be used to <span class="hlt">approximate</span> any other multimodal continuous distribution function. In the second part, the network is extended in <span class="hlt">order</span> to capture the spatial variation of the sediment samples that have been obtained from 48 locations in the investigation area. Results show that the model provides an adequate <span class="hlt">approximation</span> of grain size distributions, satisfying the requirements of a cumulative distribution function.</p>
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<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 <span class="hlt">order</span>, 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>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19840040799&hterms=Butterfly&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3DButterfly','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19840040799&hterms=Butterfly&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3DButterfly"><span>The uniform asymptotic swallowtail <span class="hlt">approximation</span> - Practical methods for oscillating integrals with four coalescing saddle points</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Connor, J. N. L.; Curtis, P. R.; Farrelly, D.</p>
<p>1984-01-01</p>
<p>Methods that can be used in the numerical implementation of the uniform swallowtail <span class="hlt">approximation</span> are described. An explicit expression for that <span class="hlt">approximation</span> is presented to the lowest <span class="hlt">order</span>, showing that there are three problems which must be overcome in practice before the <span class="hlt">approximation</span> can be applied to any given problem. It is shown that a recently developed quadrature method can be used for the accurate numerical evaluation of the swallowtail canonical integral and its partial derivatives. Isometric plots of these are presented to illustrate some of their properties. The problem of obtaining the arguments of the swallowtail integral from an analytical function of its argument is considered, describing two methods of solving this problem. The asymptotic evaluation of the butterfly canonical integral is addressed.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/7047033-boson-expansions-based-random-phase-approximation-representation','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/7047033-boson-expansions-based-random-phase-approximation-representation"><span>Boson expansions based on the random phase <span class="hlt">approximation</span> representation</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Pedrocchi, V.G.; Tamura, T.</p>
<p>1984-04-01</p>
<p>A new boson expansion theory based on the random phase <span class="hlt">approximation</span> is presented. The boson expansions are derived here directly in the random phase <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span> 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.</p>
</li>
</ol>
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<ol class="result-class" start="481">
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27802627','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27802627"><span>Minimax rational <span class="hlt">approximation</span> of the Fermi-Dirac distribution.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Moussa, Jonathan E</p>
<p>2016-10-28</p>
<p>Accurate rational <span class="hlt">approximations</span> of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known <span class="hlt">approximations</span> 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 <span class="hlt">approximation</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018MNRAS.476.2040D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018MNRAS.476.2040D"><span>Polynomial <span class="hlt">approximation</span> of the Lense-Thirring rigid precession frequency</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>De Falco, Vittorio; Motta, Sara</p>
<p>2018-05-01</p>
<p>We propose a polynomial <span class="hlt">approximation</span> of the global Lense-Thirring rigid precession frequency to study low-frequency quasi-periodic oscillations around spinning black holes. This high-performing <span class="hlt">approximation</span> 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 <span class="hlt">approximation</span>, showing that the computational times are reduced by a factor of ≈70 in the range of minutes.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1335130-minimax-rational-approximation-fermi-dirac-distribution','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1335130-minimax-rational-approximation-fermi-dirac-distribution"><span>Minimax rational <span class="hlt">approximation</span> of the Fermi-Dirac distribution</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p>
<p>Moussa, Jonathan E.</p>
<p>2016-10-27</p>
<p>Accurate rational <span class="hlt">approximations</span> of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known <span class="hlt">approximations</span> 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 <span class="hlt">approximation</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JChPh.145p4108M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JChPh.145p4108M"><span>Minimax rational <span class="hlt">approximation</span> of the Fermi-Dirac distribution</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Moussa, Jonathan E.</p>
<p>2016-10-01</p>
<p>Accurate rational <span class="hlt">approximations</span> of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known <span class="hlt">approximations</span> 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 <span class="hlt">approximation</span> 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.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017BrJPh..47...72T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017BrJPh..47...72T"><span>Reduced <span class="hlt">Order</span> Podolsky Model</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Thibes, Ronaldo</p>
<p>2017-02-01</p>
<p>We perform the canonical and path integral quantizations of a lower-<span class="hlt">order</span> derivatives model describing Podolsky's generalized electrodynamics. The physical content of the model shows an auxiliary massive vector field coupled to the usual electromagnetic field. The equivalence with Podolsky's original model is studied at classical and quantum levels. Concerning the dynamical time evolution, we obtain a theory with two first-class and two second-class constraints in phase space. We calculate explicitly the corresponding Dirac brackets involving both vector fields. We use the Senjanovic procedure to implement the second-class constraints and the Batalin-Fradkin-Vilkovisky path integral quantization scheme to deal with the symmetries generated by the first-class constraints. The physical interpretation of the results turns out to be simpler due to the reduced derivatives <span class="hlt">order</span> permeating the equations of motion, Dirac brackets and effective action.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhDT........77D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhDT........77D"><span>Essays on variational <span class="hlt">approximation</span> techniques for stochastic optimization problems</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Deride Silva, Julio A.</p>
<p></p>
<p>This dissertation presents five essays on <span class="hlt">approximation</span> and modeling techniques, based on variational analysis, applied to stochastic optimization problems. It is divided into two parts, where the first is devoted to equilibrium problems and maxinf optimization, and the second corresponds to two essays in statistics and uncertainty modeling. Stochastic optimization lies at the core of this research as we were interested in relevant equilibrium applications that contain an uncertain component, and the design of a solution strategy. In addition, every stochastic optimization problem relies heavily on the underlying probability distribution that models the uncertainty. We studied these distributions, in particular, their design process and theoretical properties such as their convergence. Finally, the last aspect of stochastic optimization that we covered is the scenario creation problem, in which we described a procedure based on a probabilistic model to create scenarios for the applied problem of power estimation of renewable energies. In the first part, Equilibrium problems and maxinf optimization, we considered three Walrasian equilibrium problems: from economics, we studied a stochastic general equilibrium problem in a pure exchange economy, described in Chapter 3, and a stochastic general equilibrium with financial contracts, in Chapter 4; finally from engineering, we studied an infrastructure planning problem in Chapter 5. We stated these problems as belonging to the maxinf optimization class and, in each instance, we provided an <span class="hlt">approximation</span> scheme based on the notion of lopsided convergence and non-concave duality. This strategy is the foundation of the augmented Walrasian algorithm, whose convergence is guaranteed by lopsided convergence, that was implemented computationally, obtaining numerical results for relevant examples. The second part, Essays about statistics and uncertainty modeling, contains two essays covering a convergence problem for a sequence</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5646186','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5646186"><span>The checkpoint <span class="hlt">ordering</span> problem</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Hungerländer, P.</p>
<p>2017-01-01</p>
<p>Abstract We suggest a new variant of a row layout problem: Find an <span class="hlt">ordering</span> of n departments with given lengths such that the total weighted sum of their distances to a given checkpoint is minimized. The Checkpoint <span class="hlt">Ordering</span> Problem (COP) is both of theoretical and practical interest. It has several applications and is conceptually related to some well-studied combinatorial optimization problems, namely the Single-Row Facility Layout Problem, the Linear <span class="hlt">Ordering</span> Problem and a variant of parallel machine scheduling. In this paper we study the complexity of the (COP) and its special cases. The general version of the (COP) with an arbitrary but fixed number of checkpoints is NP-hard in the weak sense. We propose both a dynamic programming algorithm and an integer linear programming approach for the (COP) . Our computational experiments indicate that the (COP) is hard to solve in practice. While the run time of the dynamic programming algorithm strongly depends on the length of the departments, the integer linear programming approach is able to solve instances with up to 25 departments to optimality. PMID:29170574</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4976340','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4976340"><span>Competing <span class="hlt">Orders</span> and Anomalies</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Moon, Eun-Gook</p>
<p>2016-01-01</p>
<p>A conservation law is one of the most fundamental properties in nature, but a certain class of conservation “laws” could be spoiled by intrinsic quantum mechanical effects, so-called quantum anomalies. Profound properties of the anomalies have deepened our understanding in quantum many body systems. Here, we investigate quantum anomaly effects in quantum phase transitions between competing <span class="hlt">orders</span> and striking consequences of their presence. We explicitly calculate topological nature of anomalies of non-linear sigma models (NLSMs) with the Wess-Zumino-Witten (WZW) terms. The non-perturbative nature is directly related with the ’t Hooft anomaly matching condition: anomalies are conserved in renormalization group flow. By applying the matching condition, we show massless excitations are enforced by the anomalies in a whole phase diagram in sharp contrast to the case of the Landau-Ginzburg-Wilson theory which only has massive excitations in symmetric phases. Furthermore, we find non-perturbative criteria to characterize quantum phase transitions between competing <span class="hlt">orders</span>. For example, in 4D, we show the two competing <span class="hlt">order</span> parameter theories, CP(1) and the NLSM with WZW, describe different universality class. Physical realizations and experimental implication of the anomalies are also discussed. PMID:27499184</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28368840','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28368840"><span>Minimal-<span class="hlt">Approximation</span>-Based Distributed Consensus Tracking of a Class of Uncertain Nonlinear Multiagent Systems With Unknown Control Directions.</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>2017-03-28</p>
<p>A minimal-<span class="hlt">approximation</span>-based distributed adaptive consensus tracking approach is presented for strict-feedback multiagent systems with unknown heterogeneous nonlinearities and control directions under a directed network. Existing <span class="hlt">approximation</span>-based consensus results for uncertain nonlinear multiagent systems in lower-triangular form have used multiple function <span class="hlt">approximators</span> in each local controller to <span class="hlt">approximate</span> unmatched nonlinearities of each follower. Thus, as the follower's <span class="hlt">order</span> increases, the number of the <span class="hlt">approximators</span> used in its local controller increases. However, the proposed approach employs only one function <span class="hlt">approximator</span> to construct the local controller of each follower regardless of the <span class="hlt">order</span> of the follower. The recursive design methodology using a new error transformation is derived for the proposed minimal-<span class="hlt">approximation</span>-based design. Furthermore, a bounding lemma on parameters of Nussbaum functions is presented to handle the unknown control direction problem in the minimal-<span class="hlt">approximation</span>-based distributed consensus tracking framework and the stability of the overall closed-loop system is rigorously analyzed in the Lyapunov sense.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016OcMod.107...21Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016OcMod.107...21Z"><span>A diffusion <span class="hlt">approximation</span> for ocean wave scatterings by randomly distributed ice floes</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Zhao, Xin; Shen, Hayley</p>
<p>2016-11-01</p>
<p>This study presents a continuum approach using a diffusion <span class="hlt">approximation</span> method to solve the scattering of ocean waves by randomly distributed ice floes. In <span class="hlt">order</span> to model both strong and weak scattering, the proposed method decomposes the wave action density function into two parts: the transmitted part and the scattered part. For a given wave direction, the transmitted part of the wave action density is defined as the part of wave action density in the same direction before the scattering; and the scattered part is a first <span class="hlt">order</span> Fourier series <span class="hlt">approximation</span> for the directional spreading caused by scattering. An additional <span class="hlt">approximation</span> is also adopted for simplification, in which the net directional redistribution of wave action by a single scatterer is assumed to be the reflected wave action of a normally incident wave into a semi-infinite ice cover. Other required input includes the mean shear modulus, diameter and thickness of ice floes, and the ice concentration. The directional spreading of wave energy from the diffusion <span class="hlt">approximation</span> is found to be in reasonable agreement with the previous solution using the Boltzmann equation. The diffusion model provides an alternative method to implement wave scattering into an operational wave model.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26093569','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26093569"><span>A consistent hierarchy of generalized kinetic equation <span class="hlt">approximations</span> to the master equation applied to surface catalysis.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Herschlag, Gregory J; Mitran, Sorin; Lin, Guang</p>
<p>2015-06-21</p>
<p>We develop a hierarchy of <span class="hlt">approximations</span> to the master equation for systems that exhibit translational invariance and finite-range spatial correlation. Each <span class="hlt">approximation</span> within the hierarchy is a set of ordinary differential equations that considers spatial correlations of varying lattice distance; the assumption is that the full system will have finite spatial correlations and thus the behavior of the models within the hierarchy will approach that of the full system. We provide evidence of this convergence in the context of one- and two-dimensional numerical examples. Lower levels within the hierarchy that consider shorter spatial correlations are shown to be up to three <span class="hlt">orders</span> of magnitude faster than traditional kinetic Monte Carlo methods (KMC) for one-dimensional systems, while predicting similar system dynamics and steady states as KMC methods. We then test the hierarchy on a two-dimensional model for the oxidation of CO on RuO2(110), showing that low-<span class="hlt">order</span> truncations of the hierarchy efficiently capture the essential system dynamics. By considering sequences of models in the hierarchy that account for longer spatial correlations, successive model predictions may be used to establish empirical <span class="hlt">approximation</span> of error estimates. The hierarchy may be thought of as a class of generalized phenomenological kinetic models since each element of the hierarchy <span class="hlt">approximates</span> the master equation and the lowest level in the hierarchy is identical to a simple existing phenomenological kinetic models.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/18516140','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/18516140"><span>Scattering from very rough layers under the geometric optics <span class="hlt">approximation</span>: further investigation.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Pinel, Nicolas; Bourlier, Christophe</p>
<p>2008-06-01</p>
<p>Scattering from very rough homogeneous layers is studied in the high-frequency limit (under the geometric optics <span class="hlt">approximation</span>) by taking the shadowing effect into account. To do so, the iterated Kirchhoff <span class="hlt">approximation</span>, recently developed by Pinel et al. [Waves Random Complex Media17, 283 (2007)] and reduced to the geometric optics <span class="hlt">approximation</span>, is used and investigated in more detail. The contributions from the higher <span class="hlt">orders</span> of scattering inside the rough layer are calculated under the iterated Kirchhoff <span class="hlt">approximation</span>. The method can be applied to rough layers of either very rough or perfectly flat lower interfaces, separating either lossless or lossy media. The results are compared with the PILE (propagation-inside-layer expansion) method, recently developed by Déchamps et al. [J. Opt. Soc. Am. A23, 359 (2006)], and accelerated by the forward-backward method with spectral acceleration. They highlight that there is very good agreement between the developed method and the reference numerical method for all scattering <span class="hlt">orders</span> and that the method can be applied to root-mean-square (RMS) heights at least down to 0.25lambda.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhRvD..95l3503P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhRvD..95l3503P"><span>Validity of the Born <span class="hlt">approximation</span> for beyond Gaussian weak lensing observables</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Petri, Andrea; Haiman, Zoltán; May, Morgan</p>
<p>2017-06-01</p>
<p>Accurate forward modeling of weak lensing (WL) observables from cosmological parameters is necessary for upcoming galaxy surveys. Because WL probes structures in the nonlinear regime, analytical forward modeling is very challenging, if not impossible. Numerical simulations of WL features rely on ray tracing through the outputs of N -body simulations, which requires knowledge of the gravitational potential and accurate solvers for light ray trajectories. A less accurate procedure, based on the Born <span class="hlt">approximation</span>, only requires knowledge of the density field, and can be implemented more efficiently and at a lower computational cost. In this work, we use simulations to show that deviations of the Born-<span class="hlt">approximated</span> convergence power spectrum, skewness and kurtosis from their fully ray-traced counterparts are consistent with the smallest nontrivial O (Φ3) post-Born corrections (so-called geodesic and lens-lens terms). Our results imply a cancellation among the larger O (Φ4) (and higher <span class="hlt">order</span>) terms, consistent with previous analytic work. We also find that cosmological parameter bias induced by the Born-<span class="hlt">approximated</span> power spectrum is negligible even for a LSST-like survey, once galaxy shape noise is considered. When considering higher <span class="hlt">order</span> statistics such as the κ skewness and kurtosis, however, we find significant bias of up to 2.5 σ . Using the LensTools software suite, we show that the Born <span class="hlt">approximation</span> saves a factor of 4 in computing time with respect to the full ray tracing in reconstructing the convergence.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=20040030512&hterms=Kohn&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3DKohn','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=20040030512&hterms=Kohn&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3DKohn"><span><span class="hlt">Approximating</span> the Helium Wavefunction in Positronium-Helium Scattering</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>DiRienzi, Joseph; Drachman, Richard J.</p>
<p>2003-01-01</p>
<p>In the Kohn variational treatment of the positronium- hydrogen scattering problem the scattering wave function is <span class="hlt">approximated</span> 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 <span class="hlt">approximation</span> to d e t e e the zero-energy scattering length for the Ps-He system, using a suite of <span class="hlt">approximate</span> target functions. The results we obtain will be compared with each other and with corresponding values found by other <span class="hlt">approximation</span> techniques.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.loc.gov/pictures/collection/hh/item/ca2206.photos.182659p/','SCIGOV-HHH'); return false;" href="https://www.loc.gov/pictures/collection/hh/item/ca2206.photos.182659p/"><span>6. NORTH SIDE, FROM <span class="hlt">APPROXIMATELY</span> 25 FEET SOUTHEAST OF SOUTHWEST ...</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>6. NORTH SIDE, FROM <span class="hlt">APPROXIMATELY</span> 25 FEET SOUTHEAST OF SOUTHWEST CORNER OF BUILDING 320, LOOKING SOUTH. - Oakland Naval Supply Center, Administration Building-Dental Annex-Dispensary, Between E & F Streets, East of Third Street, Oakland, Alameda County, CA</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5837704','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=5837704"><span>Fundamentals and Recent Developments in <span class="hlt">Approximate</span> Bayesian Computation</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Lintusaari, Jarno; Gutmann, Michael U.; Dutta, Ritabrata; Kaski, Samuel; Corander, Jukka</p>
<p>2017-01-01</p>
<p>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 <span class="hlt">approximate</span> quantitative answers are obtainable. <span class="hlt">Approximate</span> Bayesian computation (ABC) refers to a family of algorithms for <span class="hlt">approximate</span> 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; <span class="hlt">approximate</span> Bayesian computation; Bayesian inference; likelihood-free inference; phylogenetics; simulator-based models; stochastic simulation models; tree-based models.] PMID:28175922</p>
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<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>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.loc.gov/pictures/collection/hh/item/ca1808.photos.180885p/','SCIGOV-HHH'); return false;" href="https://www.loc.gov/pictures/collection/hh/item/ca1808.photos.180885p/"><span>15. Looking north from east bank of ditch, <span class="hlt">approximately</span> halfway ...</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>15. Looking north from east bank of ditch, <span class="hlt">approximately</span> halfway between cement pipe to north and burned irrigation pump station to south - Natomas Ditch System, Blue Ravine Segment, Juncture of Blue Ravine & Green Valley Roads, Folsom, Sacramento County, CA</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.loc.gov/pictures/collection/hh/item/co0042.photos.021480p/','SCIGOV-HHH'); return false;" href="https://www.loc.gov/pictures/collection/hh/item/co0042.photos.021480p/"><span>1. Rockwork <span class="hlt">approximately</span> 6 of a mile upstream from Keystone ...</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>1. Rockwork <span class="hlt">approximately</span> 6 of a mile upstream from Keystone Bridge. View looking south from a distance of 50 feet. - Denver & Rio Grande Rockwork, East of South Platte, Waterton, Jefferson County, CO</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016PhRvB..94p5149T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016PhRvB..94p5149T"><span><span class="hlt">Approximations</span> to the exact exchange potential: KLI versus semilocal</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Tran, Fabien; Blaha, Peter; Betzinger, Markus; Blügel, Stefan</p>
<p>2016-10-01</p>
<p>In the search for an accurate and computationally efficient <span class="hlt">approximation</span> 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) <span class="hlt">approximation</span>, which is a beyond-semilocal <span class="hlt">approximation</span>. It is shown that overall the KLI- and BJ-based potentials are the most reliable <span class="hlt">approximations</span> to the exact exchange potential, however, sizable differences, especially for the antiferromagnetic transition-metal oxides, can be obtained.</p>
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