Entropy Inequalities for Stable Densities and Strengthened Central Limit Theorems

NASA Astrophysics Data System (ADS)

Toscani, Giuseppe

2016-10-01

We consider the central limit theorem for stable laws in the case of the standardized sum of independent and identically distributed random variables with regular probability density function. By showing decay of different entropy functionals along the sequence we prove convergence with explicit rate in various norms to a Lévy centered density of parameter λ >1 . This introduces a new information-theoretic approach to the central limit theorem for stable laws, in which the main argument is shown to be the relative fractional Fisher information, recently introduced in Toscani (Ricerche Mat 65(1):71-91, 2016). In particular, it is proven that, with respect to the relative fractional Fisher information, the Lévy density satisfies an analogous of the logarithmic Sobolev inequality, which allows to pass from the monotonicity and decay to zero of the relative fractional Fisher information in the standardized sum to the decay to zero in relative entropy with an explicit decay rate.

Central Limit Theorems for the Shrinking Target Problem

NASA Astrophysics Data System (ADS)

Haydn, Nicolai; Nicol, Matthew; Vaienti, Sandro; Zhang, Licheng

2013-12-01

Suppose B i := B( p, r i ) are nested balls of radius r i about a point p in a dynamical system ( T, X, μ). The question of whether T i x∈ B i infinitely often (i.o.) for μ a.e. x is often called the shrinking target problem. In many dynamical settings it has been shown that if diverges then there is a quantitative rate of entry and for μ a.e. x∈ X. This is a self-norming type of strong law of large numbers. We establish self-norming central limit theorems (CLT) of the form (in distribution) for a variety of hyperbolic and non-uniformly hyperbolic dynamical systems, the normalization constants are . Dynamical systems to which our results apply include smooth expanding maps of the interval, Rychlik type maps, Gibbs-Markov maps, rational maps and, in higher dimensions, piecewise expanding maps. For such central limit theorems the main difficulty is to prove that the non-stationary variance has a limit in probability.

Central limit theorems under special relativity

PubMed Central

McKeague, Ian W.

2015-01-01

Several relativistic extensions of the Maxwell–Boltzmann distribution have been proposed, but they do not explain observed lognormal tail-behavior in the flux distribution of various astrophysical sources. Motivated by this question, extensions of classical central limit theorems are developed under the conditions of special relativity. The results are related to CLTs on locally compact Lie groups developed by Wehn, Stroock and Varadhan, but in this special case the asymptotic distribution has an explicit form that is readily seen to exhibit lognormal tail behavior. PMID:25798020

Central limit theorems under special relativity.

PubMed

McKeague, Ian W

2015-04-01

Several relativistic extensions of the Maxwell-Boltzmann distribution have been proposed, but they do not explain observed lognormal tail-behavior in the flux distribution of various astrophysical sources. Motivated by this question, extensions of classical central limit theorems are developed under the conditions of special relativity. The results are related to CLTs on locally compact Lie groups developed by Wehn, Stroock and Varadhan, but in this special case the asymptotic distribution has an explicit form that is readily seen to exhibit lognormal tail behavior.

The Central Limit Theorem for Supercritical Oriented Percolation in Two Dimensions

NASA Astrophysics Data System (ADS)

Tzioufas, Achillefs

2018-04-01

We consider the cardinality of supercritical oriented bond percolation in two dimensions. We show that, whenever the the origin is conditioned to percolate, the process appropriately normalized converges asymptotically in distribution to the standard normal law. This resolves a longstanding open problem pointed out to in several instances in the literature. The result applies also to the continuous-time analog of the process, viz. the basic one-dimensional contact process. We also derive general random-indices central limit theorems for associated random variables as byproducts of our proof.

The Central Limit Theorem for Supercritical Oriented Percolation in Two Dimensions

NASA Astrophysics Data System (ADS)

Tzioufas, Achillefs

2018-06-01

We consider the cardinality of supercritical oriented bond percolation in two dimensions. We show that, whenever the the origin is conditioned to percolate, the process appropriately normalized converges asymptotically in distribution to the standard normal law. This resolves a longstanding open problem pointed out to in several instances in the literature. The result applies also to the continuous-time analog of the process, viz. the basic one-dimensional contact process. We also derive general random-indices central limit theorems for associated random variables as byproducts of our proof.

Randomized central limit theorems: A unified theory.

PubMed

Eliazar, Iddo; Klafter, Joseph

2010-08-01

The central limit theorems (CLTs) characterize the macroscopic statistical behavior of large ensembles of independent and identically distributed random variables. The CLTs assert that the universal probability laws governing ensembles' aggregate statistics are either Gaussian or Lévy, and that the universal probability laws governing ensembles' extreme statistics are Fréchet, Weibull, or Gumbel. The scaling schemes underlying the CLTs are deterministic-scaling all ensemble components by a common deterministic scale. However, there are "random environment" settings in which the underlying scaling schemes are stochastic-scaling the ensemble components by different random scales. Examples of such settings include Holtsmark's law for gravitational fields and the Stretched Exponential law for relaxation times. In this paper we establish a unified theory of randomized central limit theorems (RCLTs)-in which the deterministic CLT scaling schemes are replaced with stochastic scaling schemes-and present "randomized counterparts" to the classic CLTs. The RCLT scaling schemes are shown to be governed by Poisson processes with power-law statistics, and the RCLTs are shown to universally yield the Lévy, Fréchet, and Weibull probability laws.

Randomized central limit theorems: A unified theory

NASA Astrophysics Data System (ADS)

Eliazar, Iddo; Klafter, Joseph

2010-08-01

The central limit theorems (CLTs) characterize the macroscopic statistical behavior of large ensembles of independent and identically distributed random variables. The CLTs assert that the universal probability laws governing ensembles’ aggregate statistics are either Gaussian or Lévy, and that the universal probability laws governing ensembles’ extreme statistics are Fréchet, Weibull, or Gumbel. The scaling schemes underlying the CLTs are deterministic—scaling all ensemble components by a common deterministic scale. However, there are “random environment” settings in which the underlying scaling schemes are stochastic—scaling the ensemble components by different random scales. Examples of such settings include Holtsmark’s law for gravitational fields and the Stretched Exponential law for relaxation times. In this paper we establish a unified theory of randomized central limit theorems (RCLTs)—in which the deterministic CLT scaling schemes are replaced with stochastic scaling schemes—and present “randomized counterparts” to the classic CLTs. The RCLT scaling schemes are shown to be governed by Poisson processes with power-law statistics, and the RCLTs are shown to universally yield the Lévy, Fréchet, and Weibull probability laws.

Laws of Large Numbers and Langevin Approximations for Stochastic Neural Field Equations

PubMed Central

2013-01-01

In this study, we consider limit theorems for microscopic stochastic models of neural fields. We show that the Wilson–Cowan equation can be obtained as the limit in uniform convergence on compacts in probability for a sequence of microscopic models when the number of neuron populations distributed in space and the number of neurons per population tend to infinity. This result also allows to obtain limits for qualitatively different stochastic convergence concepts, e.g., convergence in the mean. Further, we present a central limit theorem for the martingale part of the microscopic models which, suitably re-scaled, converges to a centred Gaussian process with independent increments. These two results provide the basis for presenting the neural field Langevin equation, a stochastic differential equation taking values in a Hilbert space, which is the infinite-dimensional analogue of the chemical Langevin equation in the present setting. On a technical level, we apply recently developed law of large numbers and central limit theorems for piecewise deterministic processes taking values in Hilbert spaces to a master equation formulation of stochastic neuronal network models. These theorems are valid for processes taking values in Hilbert spaces, and by this are able to incorporate spatial structures of the underlying model. Mathematics Subject Classification (2000): 60F05, 60J25, 60J75, 92C20. PMID:23343328

Central Limit Theorem: New SOCR Applet and Demonstration Activity

PubMed Central

Dinov, Ivo D.; Christou, Nicolas; Sanchez, Juana

2011-01-01

Modern approaches for information technology based blended education utilize a variety of novel instructional, computational and network resources. Such attempts employ technology to deliver integrated, dynamically linked, interactive content and multifaceted learning environments, which may facilitate student comprehension and information retention. In this manuscript, we describe one such innovative effort of using technological tools for improving student motivation and learning of the theory, practice and usability of the Central Limit Theorem (CLT) in probability and statistics courses. Our approach is based on harnessing the computational libraries developed by the Statistics Online Computational Resource (SOCR) to design a new interactive Java applet and a corresponding demonstration activity that illustrate the meaning and the power of the CLT. The CLT applet and activity have clear common goals; to provide graphical representation of the CLT, to improve student intuition, and to empirically validate and establish the limits of the CLT. The SOCR CLT activity consists of four experiments that demonstrate the assumptions, meaning and implications of the CLT and ties these to specific hands-on simulations. We include a number of examples illustrating the theory and applications of the CLT. Both the SOCR CLT applet and activity are freely available online to the community to test, validate and extend (Applet: http://www.socr.ucla.edu/htmls/SOCR_Experiments.html and Activity: http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_GeneralCentralLimitTheorem). PMID:21833159

Central Limit Theorem: New SOCR Applet and Demonstration Activity.

PubMed

Dinov, Ivo D; Christou, Nicolas; Sanchez, Juana

2008-07-01

Modern approaches for information technology based blended education utilize a variety of novel instructional, computational and network resources. Such attempts employ technology to deliver integrated, dynamically linked, interactive content and multifaceted learning environments, which may facilitate student comprehension and information retention. In this manuscript, we describe one such innovative effort of using technological tools for improving student motivation and learning of the theory, practice and usability of the Central Limit Theorem (CLT) in probability and statistics courses. Our approach is based on harnessing the computational libraries developed by the Statistics Online Computational Resource (SOCR) to design a new interactive Java applet and a corresponding demonstration activity that illustrate the meaning and the power of the CLT. The CLT applet and activity have clear common goals; to provide graphical representation of the CLT, to improve student intuition, and to empirically validate and establish the limits of the CLT. The SOCR CLT activity consists of four experiments that demonstrate the assumptions, meaning and implications of the CLT and ties these to specific hands-on simulations. We include a number of examples illustrating the theory and applications of the CLT. Both the SOCR CLT applet and activity are freely available online to the community to test, validate and extend (Applet: http://www.socr.ucla.edu/htmls/SOCR_Experiments.html and Activity: http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_GeneralCentralLimitTheorem).

Sanov and central limit theorems for output statistics of quantum Markov chains

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

Horssen, Merlijn van, E-mail: merlijn.vanhorssen@nottingham.ac.uk; Guţă, Mădălin, E-mail: madalin.guta@nottingham.ac.uk

2015-02-15

In this paper, we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov’s theorem for the multi-site empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this, we obtain a central limit theorem for the empirical measure. Suchmore » higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction, we give an example of a finite system whose level-1 (empirical mean) rate function is independent of a model parameter while the level-2 (empirical measure) rate is not.« less

STABILITY OF GAS CLOUDS IN GALACTIC NUCLEI: AN EXTENDED VIRIAL THEOREM

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

Chen, Xian; Cuadra, Jorge; Amaro-Seoane, Pau, E-mail: xchen@astro.puc.cl, E-mail: jcuadra@astro.puc.cl, E-mail: Pau.Amaro-Seoane@aei.mpg.de

2016-03-10

Cold gas entering the central 1–10{sup 2} pc of a galaxy fragments and condenses into clouds. The stability of the clouds determines whether they will be turned into stars or can be delivered to the central supermassive black hole (SMBH) to turn on an active galactic nucleus (AGN). The conventional criteria to assess the stability of these clouds, such as the Jeans criterion and Roche (or tidal) limit, are insufficient here, because they assume the dominance of self-gravity in binding a cloud, and neglect external agents, such as pressure and tidal forces, which are common in galactic nuclei. We formulatemore » a new scheme for judging this stability. We first revisit the conventional Virial theorem, taking into account an external pressure, to identify the correct range of masses that lead to stable clouds. We then extend the theorem to further include an external tidal field, which is equally crucial for the stability in the region of our interest—in dense star clusters, around SMBHs. We apply our extended Virial theorem to find new solutions to controversial problems, namely, the stability of the gas clumps in AGN tori, the circum-nuclear disk in the Galactic Center, and the central molecular zone of the Milky Way. The masses we derive for these structures are orders of magnitude smaller than the commonly used Virial masses (equivalent to the Jeans mass). Moreover, we prove that these clumps are stable, contrary to what one would naively deduce from the Roche (tidal) limit.« less

A fermionic de Finetti theorem

NASA Astrophysics Data System (ADS)

Krumnow, Christian; Zimborás, Zoltán; Eisert, Jens

2017-12-01

Quantum versions of de Finetti's theorem are powerful tools, yielding conceptually important insights into the security of key distribution protocols or tomography schemes and allowing one to bound the error made by mean-field approaches. Such theorems link the symmetry of a quantum state under the exchange of subsystems to negligible quantum correlations and are well understood and established in the context of distinguishable particles. In this work, we derive a de Finetti theorem for finite sized Majorana fermionic systems. It is shown, much reflecting the spirit of other quantum de Finetti theorems, that a state which is invariant under certain permutations of modes loses most of its anti-symmetric character and is locally well described by a mode separable state. We discuss the structure of the resulting mode separable states and establish in specific instances a quantitative link to the quality of the Hartree-Fock approximation of quantum systems. We hint at a link to generalized Pauli principles for one-body reduced density operators. Finally, building upon the obtained de Finetti theorem, we generalize and extend the applicability of Hudson's fermionic central limit theorem.

Likelihood-based confidence intervals for estimating floods with given return periods

NASA Astrophysics Data System (ADS)

Martins, Eduardo Sávio P. R.; Clarke, Robin T.

1993-06-01

This paper discusses aspects of the calculation of likelihood-based confidence intervals for T-year floods, with particular reference to (1) the two-parameter gamma distribution; (2) the Gumbel distribution; (3) the two-parameter log-normal distribution, and other distributions related to the normal by Box-Cox transformations. Calculation of the confidence limits is straightforward using the Nelder-Mead algorithm with a constraint incorporated, although care is necessary to ensure convergence either of the Nelder-Mead algorithm, or of the Newton-Raphson calculation of maximum-likelihood estimates. Methods are illustrated using records from 18 gauging stations in the basin of the River Itajai-Acu, State of Santa Catarina, southern Brazil. A small and restricted simulation compared likelihood-based confidence limits with those given by use of the central limit theorem; for the same confidence probability, the confidence limits of the simulation were wider than those of the central limit theorem, which failed more frequently to contain the true quantile being estimated. The paper discusses possible applications of likelihood-based confidence intervals in other areas of hydrological analysis.

Steady states of OQBM: Central Limit Theorem, Gaussian and non-Gaussian behavior

NASA Astrophysics Data System (ADS)

Petruccione, Francesco; Sinayskiy, Ilya

Open Quantum Brownian Motion (OQBM) describes a Brownian particle with an additional internal quantum degree of freedom. Originally, it was introduced as a scaling limit of Open Quantum Walks (OQWs). Recently, it was noted, that for the model of free OQBM with a two-level system as an internal degree of freedom and decoherent coupling to a dissipative environment, one could use weak external driving of the internal degree of freedom to manipulate the steady-state position of the walker. This observation establishes a useful connection between controllable parameters of the OQBM, e.g. driving strengths and magnitude of detuning, and its steady state properties. Although OQWs satisfy a central limit theorem (CLT), it is known, that OQBM, in general, does not. The aim of this work is to derive steady states for some particular OQBMs and observe possible transitions from Gaussian to non-Gaussian behavior depending on the choice of quantum coin and as a function of diffusion coefficient and dissipation strength.

Discrepancy-based error estimates for Quasi-Monte Carlo III. Error distributions and central limits

NASA Astrophysics Data System (ADS)

Hoogland, Jiri; Kleiss, Ronald

1997-04-01

In Quasi-Monte Carlo integration, the integration error is believed to be generally smaller than in classical Monte Carlo with the same number of integration points. Using an appropriate definition of an ensemble of quasi-random point sets, we derive various results on the probability distribution of the integration error, which can be compared to the standard Central Limit Theorem for normal stochastic sampling. In many cases, a Gaussian error distribution is obtained.

Nonrelativistic limits of colored gravity in three dimensions

NASA Astrophysics Data System (ADS)

Joung, Euihun; Li, Wenliang

2018-05-01

The three-dimensional nonrelativistic isometry algebras, namely Galilei and Newton-Hooke algebras, are known to admit double central extensions, which allows for nondegenerate bilinear forms hence for action principles through Chern-Simons formulation. In three-dimensional colored gravity, the same central extension helps the theory evade the multigraviton no-go theorems by enlarging the color-decorated isometry algebra. We investigate the nonrelativistic limits of three-dimensional colored gravity in terms of generalized İnönü-Wigner contractions.

Power-law exponent of the Bouchaud-Mézard model on regular random networks

NASA Astrophysics Data System (ADS)

Ichinomiya, Takashi

2013-07-01

We study the Bouchaud-Mézard model on a regular random network. By assuming adiabaticity and independency, and utilizing the generalized central limit theorem and the Tauberian theorem, we derive an equation that determines the exponent of the probability distribution function of the wealth as x→∞. The analysis shows that the exponent can be smaller than 2, while a mean-field analysis always gives the exponent as being larger than 2. The results of our analysis are shown to be in good agreement with those of the numerical simulations.

Central Limit Theorem: New SOCR Applet and Demonstration Activity

ERIC Educational Resources Information Center

Dinov, Ivo D.; Christou, Nicholas; Sanchez, Juana

2008-01-01

Modern approaches for information technology based blended education utilize a variety of novel instructional, computational and network resources. Such attempts employ technology to deliver integrated, dynamically linked, interactive content and multi-faceted learning environments, which may facilitate student comprehension and information…

Counting Penguins.

ERIC Educational Resources Information Center

Perry, Mike; Kader, Gary

1998-01-01

Presents an activity on the simplification of penguin counting by employing the basic ideas and principles of sampling to teach students to understand and recognize its role in statistical claims. Emphasizes estimation, data analysis and interpretation, and central limit theorem. Includes a list of items for classroom discussion. (ASK)

Eigenvector method for umbrella sampling enables error analysis

PubMed Central

Thiede, Erik H.; Van Koten, Brian; Weare, Jonathan; Dinner, Aaron R.

2016-01-01

Umbrella sampling efficiently yields equilibrium averages that depend on exploring rare states of a model by biasing simulations to windows of coordinate values and then combining the resulting data with physical weighting. Here, we introduce a mathematical framework that casts the step of combining the data as an eigenproblem. The advantage to this approach is that it facilitates error analysis. We discuss how the error scales with the number of windows. Then, we derive a central limit theorem for averages that are obtained from umbrella sampling. The central limit theorem suggests an estimator of the error contributions from individual windows, and we develop a simple and computationally inexpensive procedure for implementing it. We demonstrate this estimator for simulations of the alanine dipeptide and show that it emphasizes low free energy pathways between stable states in comparison to existing approaches for assessing error contributions. Our work suggests the possibility of using the estimator and, more generally, the eigenvector method for umbrella sampling to guide adaptation of the simulation parameters to accelerate convergence. PMID:27586912

A simple low-computation-intensity model for approximating the distribution function of a sum of non-identical lognormals for financial applications

NASA Astrophysics Data System (ADS)

Messica, A.

2016-10-01

The probability distribution function of a weighted sum of non-identical lognormal random variables is required in various fields of science and engineering and specifically in finance for portfolio management as well as exotic options valuation. Unfortunately, it has no known closed form and therefore has to be approximated. Most of the approximations presented to date are complex as well as complicated for implementation. This paper presents a simple, and easy to implement, approximation method via modified moments matching and a polynomial asymptotic series expansion correction for a central limit theorem of a finite sum. The method results in an intuitively-appealing and computation-efficient approximation for a finite sum of lognormals of at least ten summands and naturally improves as the number of summands increases. The accuracy of the method is tested against the results of Monte Carlo simulationsand also compared against the standard central limit theorem andthe commonly practiced Markowitz' portfolio equations.

Fish: A New Computer Program for Friendly Introductory Statistics Help

ERIC Educational Resources Information Center

Brooks, Gordon P.; Raffle, Holly

2005-01-01

All introductory statistics students must master certain basic descriptive statistics, including means, standard deviations and correlations. Students must also gain insight into such complex concepts as the central limit theorem and standard error. This article introduces and describes the Friendly Introductory Statistics Help (FISH) computer…

STUDIES IN RESEARCH METHODOLOGY. IV. A SAMPLING STUDY OF THE CENTRAL LIMIT THEOREM AND THE ROBUSTNESS OF ONE-SAMPLE PARAMETRIC TESTS,

DTIC Science & Technology

iconoclastic . Even at N=1024 these departures were quite appreciable at the testing tails, being greatest for chi-square and least for Z, and becoming worse in all cases at increasingly extreme tail areas. (Author)

A Unifying Probability Example.

ERIC Educational Resources Information Center

Maruszewski, Richard F., Jr.

2002-01-01

Presents an example from probability and statistics that ties together several topics including the mean and variance of a discrete random variable, the binomial distribution and its particular mean and variance, the sum of independent random variables, the mean and variance of the sum, and the central limit theorem. Uses Excel to illustrate these…

The Importance of Introductory Statistics Students Understanding Appropriate Sampling Techniques

ERIC Educational Resources Information Center

Menil, Violeta C.

2005-01-01

In this paper the author discusses the meaning of sampling, the reasons for sampling, the Central Limit Theorem, and the different techniques of sampling. Practical and relevant examples are given to make the appropriate sampling techniques understandable to students of Introductory Statistics courses. With a thorough knowledge of sampling…

Sky Radiance Distributions for Thermal Imaging Backgrounds.

DTIC Science & Technology

1987-12-01

background noise limited system. In infrared devices we have a spectral discrimination which is due to the spectral response of the detector /filter...cannot apply the central limit theorem [Ref.]- because the detector can capture only a few shots of the cloud form and the characteristics of the...objects most infrared systems can be used as detectors or target designators. Since infrared systems are passive the advantages of such systems are enormous

Does the central limit theorem always apply to phase noise? Some implications for radar problems

NASA Astrophysics Data System (ADS)

Gray, John E.; Addison, Stephen R.

2017-05-01

The phase noise problem or Rayleigh problem occurs in all aspects of radar. It is an effect that a radar engineer or physicist always has to take into account as part of a design or in attempt to characterize the physics of a problem such as reverberation. Normally, the mathematical difficulties of phase noise characterization are avoided by assuming the phase noise probability distribution function (PDF) is uniformly distributed, and the Central Limit Theorem (CLT) is invoked to argue that the superposition of relatively few random components obey the CLT and hence the superposition can be treated as a normal distribution. By formalizing the characterization of phase noise (see Gray and Alouani) for an individual random variable, the summation of identically distributed random variables is the product of multiple characteristic functions (CF). The product of the CFs for phase noise has a CF that can be analyzed to understand the limitations CLT when applied to phase noise. We mirror Kolmogorov's original proof as discussed in Papoulis to show the CLT can break down for receivers that gather limited amounts of data as well as the circumstances under which it can fail for certain phase noise distributions. We then discuss the consequences of this for matched filter design as well the implications for some physics problems.

Zero-Bounded Limits as a Special Case of the Squeeze Theorem for Evaluating Single-Variable and Multivariable Limits

ERIC Educational Resources Information Center

Gkioulekas, Eleftherios

2013-01-01

Many limits, typically taught as examples of applying the "squeeze" theorem, can be evaluated more easily using the proposed zero-bounded limit theorem. The theorem applies to functions defined as a product of a factor going to zero and a factor that remains bounded in some neighborhood of the limit. This technique is immensely useful…

Bootstrapping Cox’s Regression Model.

DTIC Science & Technology

1985-11-01

crucial points a multivariate martingale central limit theorem. Involved in this is a p x p covariance matrix Z with elements T j2= f {2(s8 ) - s(l)( s ,8o...1980). The statistical analaysis of failure time data. Wiley, New York. Meyer, P.-A. (1971). Square integrable martingales, a survey. Lecture Notes

How Sample Size Affects a Sampling Distribution

ERIC Educational Resources Information Center

Mulekar, Madhuri S.; Siegel, Murray H.

2009-01-01

If students are to understand inferential statistics successfully, they must have a profound understanding of the nature of the sampling distribution. Specifically, they must comprehend the determination of the expected value and standard error of a sampling distribution as well as the meaning of the central limit theorem. Many students in a high…

Planetary Accretion, Oxygen Isotopes and the Central Limit Theorem

NASA Technical Reports Server (NTRS)

Nuth, Joseph A., III; Hill, Hugh G. M.; Vondrak, Richard R. (Technical Monitor)

2001-01-01

The accumulation of presolar dust into increasingly larger aggregates (CAIs and Chondrules, Asteroids, Planets) should result in a very drastic reduction in the numerical spread in oxygen isotopic composition between bodies of similar size, in accord with the Central Limit Theorem. Observed variations in oxygen isotopic composition are many orders of magnitude larger than would be predicted by a simple, random accumulation model that begins in a well-mixed nebula - no matter which size-scale objects are used as the beginning or end points of the calculation. This discrepancy implies either that some as yet unspecified process acted on the solids in the Solar Nebula to increase the spread in oxygen isotopic composition during each and every stage of accumulation or that the nebula was heterogeneous and maintained this heterogeneity throughout most of nebular history. Large-scale nebular heterogeneity would have significant consequences for many areas of cosmochemistry, including the application of some well-known isotopic systems to the dating of nebular events or the prediction of bulk compositions of planetary bodies on the basis of a uniform cosmic abundance.

Bertrand's theorem and virial theorem in fractional classical mechanics

NASA Astrophysics Data System (ADS)

Yu, Rui-Yan; Wang, Towe

2017-09-01

Fractional classical mechanics is the classical counterpart of fractional quantum mechanics. The central force problem in this theory is investigated. Bertrand's theorem is generalized, and virial theorem is revisited, both in three spatial dimensions. In order to produce stable, closed, non-circular orbits, the inverse-square law and the Hooke's law should be modified in fractional classical mechanics.

Bayesian Probability Theory

NASA Astrophysics Data System (ADS)

von der Linden, Wolfgang; Dose, Volker; von Toussaint, Udo

2014-06-01

Preface; Part I. Introduction: 1. The meaning of probability; 2. Basic definitions; 3. Bayesian inference; 4. Combinatrics; 5. Random walks; 6. Limit theorems; 7. Continuous distributions; 8. The central limit theorem; 9. Poisson processes and waiting times; Part II. Assigning Probabilities: 10. Transformation invariance; 11. Maximum entropy; 12. Qualified maximum entropy; 13. Global smoothness; Part III. Parameter Estimation: 14. Bayesian parameter estimation; 15. Frequentist parameter estimation; 16. The Cramer-Rao inequality; Part IV. Testing Hypotheses: 17. The Bayesian way; 18. The frequentist way; 19. Sampling distributions; 20. Bayesian vs frequentist hypothesis tests; Part V. Real World Applications: 21. Regression; 22. Inconsistent data; 23. Unrecognized signal contributions; 24. Change point problems; 25. Function estimation; 26. Integral equations; 27. Model selection; 28. Bayesian experimental design; Part VI. Probabilistic Numerical Techniques: 29. Numerical integration; 30. Monte Carlo methods; 31. Nested sampling; Appendixes; References; Index.

Statistical Inference and Simulation with StatKey

ERIC Educational Resources Information Center

Quinn, Anne

2016-01-01

While looking for an inexpensive technology package to help students in statistics classes, the author found StatKey, a free Web-based app. Not only is StatKey useful for students' year-end projects, but it is also valuable for helping students learn fundamental content such as the central limit theorem. Using StatKey, students can engage in…

Pedagogical Simulation of Sampling Distributions and the Central Limit Theorem

ERIC Educational Resources Information Center

Hagtvedt, Reidar; Jones, Gregory Todd; Jones, Kari

2007-01-01

Students often find the fact that a sample statistic is a random variable very hard to grasp. Even more mysterious is why a sample mean should become ever more Normal as the sample size increases. This simulation tool is meant to illustrate the process, thereby giving students some intuitive grasp of the relationship between a parent population…

The Sampling Distribution and the Central Limit Theorem: What They Are and Why They're Important.

ERIC Educational Resources Information Center

Kennedy, Charlotte A.

The use of and emphasis on statistical significance testing has pervaded educational and behavioral research for many decades in spite of criticism by prominent researchers in this field. Much of the controversy is caused by lack of understanding or misinterpretations. This paper reviews criticisms of statistical significance testing and discusses…

Averaging in SU(2) open quantum random walk

NASA Astrophysics Data System (ADS)

Clement, Ampadu

2014-03-01

We study the average position and the symmetry of the distribution in the SU(2) open quantum random walk (OQRW). We show that the average position in the central limit theorem (CLT) is non-uniform compared with the average position in the non-CLT. The symmetry of distribution is shown to be even in the CLT.

Possible Potentials Responsible for Stable Circular Relativistic Orbits

ERIC Educational Resources Information Center

Kumar, Prashant; Bhattacharya, Kaushik

2011-01-01

Bertrand's theorem in classical mechanics of the central force fields attracts us because of its predictive power. It categorically proves that there can only be two types of forces which can produce stable, circular orbits. In this paper an attempt has been made to generalize Bertrand's theorem to the central force problem of relativistic…

Universal Hitting Time Statistics for Integrable Flows

NASA Astrophysics Data System (ADS)

Dettmann, Carl P.; Marklof, Jens; Strömbergsson, Andreas

2017-02-01

The perceived randomness in the time evolution of "chaotic" dynamical systems can be characterized by universal probabilistic limit laws, which do not depend on the fine features of the individual system. One important example is the Poisson law for the times at which a particle with random initial data hits a small set. This was proved in various settings for dynamical systems with strong mixing properties. The key result of the present study is that, despite the absence of mixing, the hitting times of integrable flows also satisfy universal limit laws which are, however, not Poisson. We describe the limit distributions for "generic" integrable flows and a natural class of target sets, and illustrate our findings with two examples: the dynamics in central force fields and ellipse billiards. The convergence of the hitting time process follows from a new equidistribution theorem in the space of lattices, which is of independent interest. Its proof exploits Ratner's measure classification theorem for unipotent flows, and extends earlier work of Elkies and McMullen.

Limits of predictions in thermodynamic systems: a review

NASA Astrophysics Data System (ADS)

Marsland, Robert, III; England, Jeremy

2018-01-01

The past twenty years have seen a resurgence of interest in nonequilibrium thermodynamics, thanks to advances in the theory of stochastic processes and in their thermodynamic interpretation. Fluctuation theorems provide fundamental constraints on the dynamics of systems arbitrarily far from thermal equilibrium. Thermodynamic uncertainty relations bound the dissipative cost of precision in a wide variety of processes. Concepts of excess work and excess heat provide the basis for a complete thermodynamics of nonequilibrium steady states, including generalized Clausius relations and thermodynamic potentials. But these general results carry their own limitations: fluctuation theorems involve exponential averages that can depend sensitively on unobservably rare trajectories; steady-state thermodynamics makes use of a dual dynamics that lacks any direct physical interpretation. This review aims to present these central results of contemporary nonequilibrium thermodynamics in such a way that the power of each claim for making physical predictions can be clearly assessed, using examples from current topics in soft matter and biophysics.

Two Universality Properties Associated with the Monkey Model of Zipf's Law

NASA Astrophysics Data System (ADS)

Perline, Richard; Perline, Ron

2016-03-01

The distribution of word probabilities in the monkey model of Zipf's law is associated with two universality properties: (1) the power law exponent converges strongly to $-1$ as the alphabet size increases and the letter probabilities are specified as the spacings from a random division of the unit interval for any distribution with a bounded density function on $[0,1]$; and (2), on a logarithmic scale the version of the model with a finite word length cutoff and unequal letter probabilities is approximately normally distributed in the part of the distribution away from the tails. The first property is proved using a remarkably general limit theorem for the logarithm of sample spacings from Shao and Hahn, and the second property follows from Anscombe's central limit theorem for a random number of i.i.d. random variables. The finite word length model leads to a hybrid Zipf-lognormal mixture distribution closely related to work in other areas.

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

Wollaber, Allan Benton

This is a powerpoint presentation which serves as lecture material for the Parallel Computing summer school. It goes over the fundamentals of the Monte Carlo calculation method. The material is presented according to the following outline: Introduction (background, a simple example: estimating π), Why does this even work? (The Law of Large Numbers, The Central Limit Theorem), How to sample (inverse transform sampling, rejection), and An example from particle transport.

Rocks: A Concrete Activity That Introduces Normal Distribution, Sampling Error, Central Limit Theorem and True Score Theory

ERIC Educational Resources Information Center

Van Duzer, Eric

2011-01-01

This report introduces a short, hands-on activity that addresses a key challenge in teaching quantitative methods to students who lack confidence or experience with statistical analysis. Used near the beginning of the course, this activity helps students develop an intuitive insight regarding a number of abstract concepts which are key to…

Distortions in Distributions of Impact Estimates in Multi-Site Trials: The Central Limit Theorem Is Not Your Friend

ERIC Educational Resources Information Center

May, Henry

2014-01-01

Interest in variation in program impacts--How big is it? What might explain it?--has inspired recent work on the analysis of data from multi-site experiments. One critical aspect of this problem involves the use of random or fixed effect estimates to visualize the distribution of impact estimates across a sample of sites. Unfortunately, unless the…

Central limit theorem for recurrent random walks on a strip with bounded potential

NASA Astrophysics Data System (ADS)

Dolgopyat, D.; Goldsheid, I.

2018-07-01

We prove that the recurrent random walk (RW) in random environment (RE) on a strip in bounded potential satisfies the central limit theorem (CLT). The key ingredients of the proof are the analysis of the invariant measure equation and construction of a linearly growing martingale for walks in bounded potential. Our main result implies a complete classification of recurrent i.i.d. RWRE on the strip. Namely the walk either exhibits the Sinai behaviour in the sense that converges, as , to a (random) limit (the Sinai law) or, it satisfies the CLT. Another application of our main result is the CLT for the quasiperiodic environments with Diophantine frequencies in the recurrent case. We complement this result by proving that in the transient case the CLT holds for all uniquely ergodic environments. We also investigate the algebraic structure of the environments satisfying the CLT. In particular, we show that there exists a collection of proper algebraic subvarieties in the space of transition probabilities, such that: • If RE is stationary and ergodic and the transition probabilities are con-centrated on one of subvarieties from our collection then the CLT holds. • If the environment is i.i.d then the above condition is also necessary forthe CLT. All these results are valid for one-dimensional RWRE with bounded jumps as a particular case of the strip model.

Different approach to the modeling of nonfree particle diffusion

NASA Astrophysics Data System (ADS)

Buhl, Niels

2018-03-01

A new approach to the modeling of nonfree particle diffusion is presented. The approach uses a general setup based on geometric graphs (networks of curves), which means that particle diffusion in anything from arrays of barriers and pore networks to general geometric domains can be considered and that the (free random walk) central limit theorem can be generalized to cover also the nonfree case. The latter gives rise to a continuum-limit description of the diffusive motion where the effect of partially absorbing barriers is accounted for in a natural and non-Markovian way that, in contrast to the traditional approach, quantifies the absorptivity of a barrier in terms of a dimensionless parameter in the range 0 to 1. The generalized theorem gives two general analytic expressions for the continuum-limit propagator: an infinite sum of Gaussians and an infinite sum of plane waves. These expressions entail the known method-of-images and Laplace eigenfunction expansions as special cases and show how the presence of partially absorbing barriers can lead to phenomena such as line splitting and band gap formation in the plane wave wave-number spectrum.

Continuous-time quantum walks on star graphs

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

Salimi, S.

2009-06-15

In this paper, we investigate continuous-time quantum walk on star graphs. It is shown that quantum central limit theorem for a continuous-time quantum walk on star graphs for N-fold star power graph, which are invariant under the quantum component of adjacency matrix, converges to continuous-time quantum walk on K{sub 2} graphs (complete graph with two vertices) and the probability of observing walk tends to the uniform distribution.

Non-parametric methods for cost-effectiveness analysis: the central limit theorem and the bootstrap compared.

PubMed

Nixon, Richard M; Wonderling, David; Grieve, Richard D

2010-03-01

Cost-effectiveness analyses (CEA) alongside randomised controlled trials commonly estimate incremental net benefits (INB), with 95% confidence intervals, and compute cost-effectiveness acceptability curves and confidence ellipses. Two alternative non-parametric methods for estimating INB are to apply the central limit theorem (CLT) or to use the non-parametric bootstrap method, although it is unclear which method is preferable. This paper describes the statistical rationale underlying each of these methods and illustrates their application with a trial-based CEA. It compares the sampling uncertainty from using either technique in a Monte Carlo simulation. The experiments are repeated varying the sample size and the skewness of costs in the population. The results showed that, even when data were highly skewed, both methods accurately estimated the true standard errors (SEs) when sample sizes were moderate to large (n>50), and also gave good estimates for small data sets with low skewness. However, when sample sizes were relatively small and the data highly skewed, using the CLT rather than the bootstrap led to slightly more accurate SEs. We conclude that while in general using either method is appropriate, the CLT is easier to implement, and provides SEs that are at least as accurate as the bootstrap. (c) 2009 John Wiley & Sons, Ltd.

Renyi entropy measures of heart rate Gaussianity.

PubMed

Lake, Douglas E

2006-01-01

Sample entropy and approximate entropy are measures that have been successfully utilized to study the deterministic dynamics of heart rate (HR). A complementary stochastic point of view and a heuristic argument using the Central Limit Theorem suggests that the Gaussianity of HR is a complementary measure of the physiological complexity of the underlying signal transduction processes. Renyi entropy (or q-entropy) is a widely used measure of Gaussianity in many applications. Particularly important members of this family are differential (or Shannon) entropy (q = 1) and quadratic entropy (q = 2). We introduce the concepts of differential and conditional Renyi entropy rate and, in conjunction with Burg's theorem, develop a measure of the Gaussianity of a linear random process. Robust algorithms for estimating these quantities are presented along with estimates of their standard errors.

Super Generalized Central Limit Theorem —Limit Distributions for Sums of Non-identical Random Variables with Power Laws—

NASA Astrophysics Data System (ADS)

Shintani, Masaru; Umeno, Ken

2018-04-01

The power law is present ubiquitously in nature and in our societies. Therefore, it is important to investigate the characteristics of power laws in the current era of big data. In this paper we prove that the superposition of non-identical stochastic processes with power laws converges in density to a unique stable distribution. This property can be used to explain the universality of stable laws that the sums of the logarithmic returns of non-identical stock price fluctuations follow stable distributions.

Trivial constraints on orbital-free kinetic energy density functionals

NASA Astrophysics Data System (ADS)

Luo, Kai; Trickey, S. B.

2018-03-01

Approximate kinetic energy density functionals (KEDFs) are central to orbital-free density functional theory. Limitations on the spatial derivative dependencies of KEDFs have been claimed from differential virial theorems. We identify a central defect in the argument: the relationships are not true for an arbitrary density but hold only for the minimizing density and corresponding chemical potential. Contrary to the claims therefore, the relationships are not constraints and provide no independent information about the spatial derivative dependencies of approximate KEDFs. A simple argument also shows that validity for arbitrary v-representable densities is not restored by appeal to the density-potential bijection.

Proof of factorization using background field method of QCD

NASA Astrophysics Data System (ADS)

Nayak, Gouranga C.

2010-02-01

Factorization theorem plays the central role at high energy colliders to study standard model and beyond standard model physics. The proof of factorization theorem is given by Collins, Soper and Sterman to all orders in perturbation theory by using diagrammatic approach. One might wonder if one can obtain the proof of factorization theorem through symmetry considerations at the lagrangian level. In this paper we provide such a proof.

Proof of factorization using background field method of QCD

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

Nayak, Gouranga C.

Factorization theorem plays the central role at high energy colliders to study standard model and beyond standard model physics. The proof of factorization theorem is given by Collins, Soper and Sterman to all orders in perturbation theory by using diagrammatic approach. One might wonder if one can obtain the proof of factorization theorem through symmetry considerations at the lagrangian level. In this paper we provide such a proof.

WASP (Write a Scientific Paper) using Excel - 8: t-Tests.

PubMed

Grech, Victor

2018-06-01

t-Testing is a common component of inferential statistics when comparing two means. This paper explains the central limit theorem and the concept of the null hypothesis as well as types of errors. On the practical side, this paper outlines how different t-tests may be performed in Microsoft Excel, for different purposes, both statically as well as dynamically, with Excel's functions. Copyright © 2018 Elsevier B.V. All rights reserved.

Statistical Inferences from the Topology of Complex Networks

DTIC Science & Technology

2016-10-04

stable, does not lose any information, has continuous and discrete versions, and obeys a strong law of large numbers and a central limit theorem. The...paper (with J.A. Scott) “Categorification of persistent homology” [7] in the journal Discrete and Computational Geome- try and the paper “Metrics for...Generalized Persistence Modules” (with J.A. Scott and V. de Silva) in the journal Foundations of Computational Math - ematics [5]. These papers develop

Inverse Problems and Imaging (Pitman Research Notes in Mathematics Series Number 245)

DTIC Science & Technology

1991-01-01

Multiparamcter spectral theory in Hilbert space functional differential cquations B D Sleeman F Kappel and W Schappacher 24 Mathematical modelling...techniques 49 Sequence spaces R Aris W 11 Ruckle 25 Singular points of smooth mappings 50 Recent contributions to nonlinear C G Gibson partial...of convergence in the central limit T Husain theorem 86 Hamilton-Jacobi equations in Hilbert spaces Peter Hall V Barbu and G Da Prato 63 Solution of

Probabilistic Modeling and Simulation of Metal Fatigue Life Prediction

DTIC Science & Technology

2002-09-01

distribution demonstrate the central limit theorem? Obviously not! This is much the same as materials testing. If only NBA basketball stars are...60 near the exit of a NBA locker room. There would obviously be some pseudo-normal distribution with a very small standard deviation. The mean...completed, the investigators must understand how the midgets and the NBA stars will affect the total solution. D. IT IS MUCH SIMPLER TO MODEL THE

Chemical Equilibrium and Polynomial Equations: Beware of Roots.

ERIC Educational Resources Information Center

Smith, William R.; Missen, Ronald W.

1989-01-01

Describes two easily applied mathematical theorems, Budan's rule and Rolle's theorem, that in addition to Descartes's rule of signs and intermediate-value theorem, are useful in chemical equilibrium. Provides examples that illustrate the use of all four theorems. Discusses limitations of the polynomial equation representation of chemical…

The Cr dependence problem of eigenvalues of the Laplace operator on domains in the plane

NASA Astrophysics Data System (ADS)

Haddad, Julian; Montenegro, Marcos

2018-03-01

The Cr dependence problem of multiple Dirichlet eigenvalues on domains is discussed for elliptic operators by regarding C r + 1-smooth one-parameter families of C1 perturbations of domains in Rn. As applications of our main theorem (Theorem 1), we provide a fairly complete description for all eigenvalues of the Laplace operator on disks and squares in R2 and also for its second eigenvalue on balls in Rn for any n ≥ 3. The central tool used in our proof is a degenerate implicit function theorem on Banach spaces (Theorem 2) of independent interest.

About normal distribution on SO(3) group in texture analysis

NASA Astrophysics Data System (ADS)

Savyolova, T. I.; Filatov, S. V.

2017-12-01

This article studies and compares different normal distributions (NDs) on SO(3) group, which are used in texture analysis. Those NDs are: Fisher normal distribution (FND), Bunge normal distribution (BND), central normal distribution (CND) and wrapped normal distribution (WND). All of the previously mentioned NDs are central functions on SO(3) group. CND is a subcase for normal CLT-motivated distributions on SO(3) (CLT here is Parthasarathy’s central limit theorem). WND is motivated by CLT in R 3 and mapped to SO(3) group. A Monte Carlo method for modeling normally distributed values was studied for both CND and WND. All of the NDs mentioned above are used for modeling different components of crystallites orientation distribution function in texture analysis.

[Longevity, disease, and duration of disability].

PubMed

Matsushita, S

1996-12-01

Disability and the resulting lowered quality of life are serious issues accompanying increased longevity. Active life expectancy #(8) can be to used to distinguish the number of years without disability from the number with disability; increases were found in both in longevity #(9, 19). With the same rate of age-related new disability in the cohorts between 1970 and 1990, the total disability increased three fold #(11). In elderly patients I showed that 1) the duration of disability of those at a specific age at death (predeath) #(1) increased with age, and it decreased in those who remained without disability, 2) the cumulative number of days of disability for patients who died at a specific age (a convolution function of predeath and mortality) #(2), approached a normal distribution, which is consistent with the central limit theorem, 3) competing risk with chronic disease in a patient greatly affects the incidence and duration of disability, 4) using the central limit theorem we can predict that preventing dementia will retard premature rectangularization of the disability-free survival curve, and will thus reduce the total disability, 5) disability is an example of how variation and selection of chronic diseases (disease Darwinism) can alter population structure. Insights into the evolution of senescence #(14-21), pleiotropy, and slower rates of molecular evolution in the core than at the border #(26, 27), reveal that the central nervous system is relatively robust and conservative for pleiotropy and may senesce relatively slowly, which support a new way of thinking #(3, 4) about old age. To minimize disability, public knowledge and education about an ideal lifestyle and the evolution of senescence is essential.

Slow Lévy flights

NASA Astrophysics Data System (ADS)

Boyer, Denis; Pineda, Inti

2016-02-01

Among Markovian processes, the hallmark of Lévy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that Lévy laws, as well as Gaussian distributions, can also be the limit distributions of processes with long-range memory that exhibit very slow diffusion, logarithmic in time. These processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the central limit theorem is modified in this context, keeping the usual distinction between analytic and nonanalytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.

From the Law of Large Numbers to Large Deviation Theory in Statistical Physics: An Introduction

NASA Astrophysics Data System (ADS)

Cecconi, Fabio; Cencini, Massimo; Puglisi, Andrea; Vergni, Davide; Vulpiani, Angelo

This contribution aims at introducing the topics of this book. We start with a brief historical excursion on the developments from the law of large numbers to the central limit theorem and large deviations theory. The same topics are then presented using the language of probability theory. Finally, some applications of large deviations theory in physics are briefly discussed through examples taken from statistical mechanics, dynamical and disordered systems.

Using Agent-Based Distillations to Explore Logistics Support to Urban, Humanitarian Assistance/Disaster Relief Operations

DTIC Science & Technology

2003-09-01

environments is warranted. The author’s initial concept was to set up the same scenario in three different PA agent-based programs, MANA, PYTHAGORAS ...of the consolidation as the result for that particular set of runs. This technique also allowed us to invoke the Central Limit Theorem . D...capabilities in the SOCRATES modeling environment. We encourage MANA and PYTHAGORAS to add this functionality to their products as well. We

On Nonlinear Functionals of Random Spherical Eigenfunctions

NASA Astrophysics Data System (ADS)

Marinucci, Domenico; Wigman, Igor

2014-05-01

We prove central limit theorems and Stein-like bounds for the asymptotic behaviour of nonlinear functionals of spherical Gaussian eigenfunctions. Our investigation combines asymptotic analysis of higher order moments for Legendre polynomials and, in addition, recent results on Malliavin calculus and total variation bounds for Gaussian subordinated fields. We discuss applications to geometric functionals like the defect and invariant statistics, e.g., polyspectra of isotropic spherical random fields. Both of these have relevance for applications, especially in an astrophysical environment.

Measurement of Hubble constant: non-Gaussian errors in HST Key Project data

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

Singh, Meghendra; Gupta, Shashikant; Pandey, Ashwini

2016-08-01

Assuming the Central Limit Theorem, experimental uncertainties in any data set are expected to follow the Gaussian distribution with zero mean. We propose an elegant method based on Kolmogorov-Smirnov statistic to test the above; and apply it on the measurement of Hubble constant which determines the expansion rate of the Universe. The measurements were made using Hubble Space Telescope. Our analysis shows that the uncertainties in the above measurement are non-Gaussian.

Asymptotic Linear Spectral Statistics for Spiked Hermitian Random Matrices

NASA Astrophysics Data System (ADS)

Passemier, Damien; McKay, Matthew R.; Chen, Yang

2015-07-01

Using the Coulomb Fluid method, this paper derives central limit theorems (CLTs) for linear spectral statistics of three "spiked" Hermitian random matrix ensembles. These include Johnstone's spiked model (i.e., central Wishart with spiked correlation), non-central Wishart with rank-one non-centrality, and a related class of non-central matrices. For a generic linear statistic, we derive simple and explicit CLT expressions as the matrix dimensions grow large. For all three ensembles under consideration, we find that the primary effect of the spike is to introduce an correction term to the asymptotic mean of the linear spectral statistic, which we characterize with simple formulas. The utility of our proposed framework is demonstrated through application to three different linear statistics problems: the classical likelihood ratio test for a population covariance, the capacity analysis of multi-antenna wireless communication systems with a line-of-sight transmission path, and a classical multiple sample significance testing problem.

Consciousness, crosstalk, and the mereological fallacy: An evolutionary perspective

NASA Astrophysics Data System (ADS)

Wallace, Rodrick

2012-12-01

The cross-sectional decontextualization afflicting contemporary neuroscience - attributing to ‘the brain’ what is the province of the whole organism - is mirrored by an evolutionary decontextualization exceptionalizing consciousness. The living state is characterized by cognitive processes at all scales and levels of organization. Many can be associated with dual information sources that ‘speak’ a ‘language’ of behavior-in-context. Shifting global broadcasts analogous to consciousness, albeit far slower - wound healing, tumor control, immune function, gene expression, etc. - have emerged through repeated evolutionary exaptation of the crosstalk and noise inherent to all information transmission. These recruit ‘unconscious’ cognitive modules into tunable arrays as needed to meet threats and opportunities across multiple frames of reference. The development is straightforward, based on the powerful necessary conditions imposed by the asymptotic limit theorems of communication theory, in the same sense that the Central Limit Theorem constrains sums of stochastic variates. Recognition of information as a form of free energy instantiated by physical processes that consume free energy permits analogs to phase transition and nonequilibrium thermodynamic arguments, leading to ‘dynamic regression models’ useful for data analysis.

Fluctuation theorem: A critical review

NASA Astrophysics Data System (ADS)

Malek Mansour, M.; Baras, F.

2017-10-01

Fluctuation theorem for entropy production is revisited in the framework of stochastic processes. The applicability of the fluctuation theorem to physico-chemical systems and the resulting stochastic thermodynamics were analyzed. Some unexpected limitations are highlighted in the context of jump Markov processes. We have shown that these limitations handicap the ability of the resulting stochastic thermodynamics to correctly describe the state of non-equilibrium systems in terms of the thermodynamic properties of individual processes therein. Finally, we considered the case of diffusion processes and proved that the fluctuation theorem for entropy production becomes irrelevant at the stationary state in the case of one variable systems.

Finite-size analysis of continuous-variable measurement-device-independent quantum key distribution

NASA Astrophysics Data System (ADS)

Zhang, Xueying; Zhang, Yichen; Zhao, Yijia; Wang, Xiangyu; Yu, Song; Guo, Hong

2017-10-01

We study the impact of the finite-size effect on the continuous-variable measurement-device-independent quantum key distribution (CV-MDI QKD) protocol, mainly considering the finite-size effect on the parameter estimation procedure. The central-limit theorem and maximum likelihood estimation theorem are used to estimate the parameters. We also analyze the relationship between the number of exchanged signals and the optimal modulation variance in the protocol. It is proved that when Charlie's position is close to Bob, the CV-MDI QKD protocol has the farthest transmission distance in the finite-size scenario. Finally, we discuss the impact of finite-size effects related to the practical detection in the CV-MDI QKD protocol. The overall results indicate that the finite-size effect has a great influence on the secret-key rate of the CV-MDI QKD protocol and should not be ignored.

A Functional Central Limit Theorem for the Becker-Döring Model

NASA Astrophysics Data System (ADS)

Sun, Wen

2018-04-01

We investigate the fluctuations of the stochastic Becker-Döring model of polymerization when the initial size of the system converges to infinity. A functional central limit problem is proved for the vector of the number of polymers of a given size. It is shown that the stochastic process associated to fluctuations is converging to the strong solution of an infinite dimensional stochastic differential equation (SDE) in a Hilbert space. We also prove that, at equilibrium, the solution of this SDE is a Gaussian process. The proofs are based on a specific representation of the evolution equations, the introduction of a convenient Hilbert space and several technical estimates to control the fluctuations, especially of the first coordinate which interacts with all components of the infinite dimensional vector representing the state of the process.

Some functional limit theorems for compound Cox processes

NASA Astrophysics Data System (ADS)

Korolev, Victor Yu.; Chertok, A. V.; Korchagin, A. Yu.; Kossova, E. V.; Zeifman, Alexander I.

2016-06-01

An improved version of the functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes (compound Cox processes) to Lévy processes in the Skorokhod space under more realistic moment conditions. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Lévy processes with variance-mean mixed normal distributions, in particular, to stable Lévy processes.

Some functional limit theorems for compound Cox processes

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

Korolev, Victor Yu.; Institute of Informatics Problems FRC CSC RAS; Chertok, A. V.

2016-06-08

An improved version of the functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes (compound Cox processes) to Lévy processes in the Skorokhod space under more realistic moment conditions. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Lévy processes with variance-mean mixed normal distributions, in particular, to stable Lévy processes.

A short walk in quantum probability

NASA Astrophysics Data System (ADS)

Hudson, Robin

2018-04-01

This is a personal survey of aspects of quantum probability related to the Heisenberg commutation relation for canonical pairs. Using the failure, in general, of non-negativity of the Wigner distribution for canonical pairs to motivate a more satisfactory quantum notion of joint distribution, we visit a central limit theorem for such pairs and a resulting family of quantum planar Brownian motions which deform the classical planar Brownian motion, together with a corresponding family of quantum stochastic areas. This article is part of the themed issue `Hilbert's sixth problem'.

Statistical Mechanics and Applications in Condensed Matter

NASA Astrophysics Data System (ADS)

Di Castro, Carlo; Raimondi, Roberto

2015-08-01

Preface; 1. Thermodynamics: a brief overview; 2. Kinetics; 3. From Boltzmann to Gibbs; 4. More ensembles; 5. The thermodynamic limit and its thermodynamic stability; 6. Density matrix and quantum statistical mechanics; 7. The quantum gases; 8. Mean-field theories and critical phenomena; 9. Second quantization and Hartree-Fock approximation; 10. Linear response and fluctuation-dissipation theorem in quantum systems: equilibrium and small deviations; 11. Brownian motion and transport in disordered systems; 12. Fermi liquids; 13. The Landau theory of the second order phase transitions; 14. The Landau-Wilson model for critical phenomena; 15. Superfluidity and superconductivity; 16. The scaling theory; 17. The renormalization group approach; 18. Thermal Green functions; 19. The microscopic foundations of Fermi liquids; 20. The Luttinger liquid; 21. Quantum interference effects in disordered electron systems; Appendix A. The central limit theorem; Appendix B. Some useful properties of the Euler Gamma function; Appendix C. Proof of the second theorem of Yang and Lee; Appendix D. The most probable distribution for the quantum gases; Appendix E. Fermi-Dirac and Bose-Einstein integrals; Appendix F. The Fermi gas in a uniform magnetic field: Landau diamagnetism; Appendix G. Ising and gas-lattice models; Appendix H. Sum over discrete Matsubara frequencies; Appendix I. Hydrodynamics of the two-fluid model of superfluidity; Appendix J. The Cooper problem in the theory of superconductivity; Appendix K. Superconductive fluctuations phenomena; Appendix L. Diagrammatic aspects of the exact solution of the Tomonaga Luttinger model; Appendix M. Details on the theory of the disordered Fermi liquid; References; Author index; Index.

The Variation Theorem Applied to H-2+: A Simple Quantum Chemistry Computer Project

ERIC Educational Resources Information Center

Robiette, Alan G.

1975-01-01

Describes a student project which requires limited knowledge of Fortran and only minimal computing resources. The results illustrate such important principles of quantum mechanics as the variation theorem and the virial theorem. Presents sample calculations and the subprogram for energy calculations. (GS)

Generalized virial theorem and pressure relation for a strongly correlated Fermi gas

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

Tan, Shina

2008-12-15

For a two-component Fermi gas in the unitarity limit (i.e., with infinite scattering length), there is a well-known virial theorem, first shown by J.E. Thomas et al. A few people rederived this result, and extended it to few-body systems, but their results are all restricted to the unitarity limit. Here I show that there is a generalized virial theorem for FINITE scattering lengths. I also generalize an exact result concerning the pressure to the case of imbalanced populations.

Orbit-averaged quantities, the classical Hellmann-Feynman theorem, and the magnetic flux enclosed by gyro-motion

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

Perkins, R. J., E-mail: rperkins@pppl.gov; Bellan, P. M.

Action integrals are often used to average a system over fast oscillations and obtain reduced dynamics. It is not surprising, then, that action integrals play a central role in the Hellmann-Feynman theorem of classical mechanics, which furnishes the values of certain quantities averaged over one period of rapid oscillation. This paper revisits the classical Hellmann-Feynman theorem, rederiving it in connection to an analogous theorem involving the time-averaged evolution of canonical coordinates. We then apply a modified version of the Hellmann-Feynman theorem to obtain a new result: the magnetic flux enclosed by one period of gyro-motion of a charged particle inmore » a non-uniform magnetic field. These results further demonstrate the utility of the action integral in regards to obtaining orbit-averaged quantities and the usefulness of this formalism in characterizing charged particle motion.« less

The Implicit Function Theorem and Non-Existence of Limit of Functions of Several Variables

ERIC Educational Resources Information Center

dos Santos, A. L. C.; da Silva, P. N.

2008-01-01

We use the Implicit Function Theorem to establish a result of non-existence of limit to a certain class of functions of several variables. We consider functions given by quotients such that both the numerator and denominator functions are null at the limit point. We show that the non-existence of the limit of such function is related with the…

A short walk in quantum probability.

PubMed

Hudson, Robin

2018-04-28

This is a personal survey of aspects of quantum probability related to the Heisenberg commutation relation for canonical pairs. Using the failure, in general, of non-negativity of the Wigner distribution for canonical pairs to motivate a more satisfactory quantum notion of joint distribution, we visit a central limit theorem for such pairs and a resulting family of quantum planar Brownian motions which deform the classical planar Brownian motion, together with a corresponding family of quantum stochastic areas.This article is part of the themed issue 'Hilbert's sixth problem'. © 2018 The Author(s).

ACIRF User’s Guide for the General Model (Version 3.5)

DTIC Science & Technology

1992-06-01

61 3c Example ACIRF formatted output for the frozen-in model (summary of measured realization statistics for antenr.. 2...must be delta correlated in angle, delay, and Doppler frequency: < z(KL,O)o) *(Kijj",o) = S(K±,T, O)D) 5(KL-K’) 8(T-r’) 8(0D-Oab) .( 61 ) The first-order... 61 , and the central limit theorem could be invoked to argue that h(p,r,t) and hA(p,rt) are zero- mean, normally-distributed complex quantities. Indeed

Simple proof of the concavity of the entropy power with respect to Gaussian noise

NASA Technical Reports Server (NTRS)

Dembo, Amir

1989-01-01

A very simple proof of M. H. Costa's result that the entropy power of Xt = X + N (O, tI) is concave in t, is derived as an immediate consequence of an inequality concerning Fisher information. This relationship between Fisher information and entropy is found to be useful for proving the central limit theorem. Thus, one who seeks new entropy inequalities should try first to find new inequalities about Fisher information, or at least to exploit the existing ones in new ways.

Large deviation theory for the kinetics and energetics of turnover of enzyme catalysis in a chemiostatic flow.

PubMed

Das, Biswajit; Gangopadhyay, Gautam

2018-05-07

In the framework of large deviation theory, we have characterized nonequilibrium turnover statistics of enzyme catalysis in a chemiostatic flow with externally controllable parameters, like substrate injection rate and mechanical force. In the kinetics of the process, we have shown the fluctuation theorems in terms of the symmetry of the scaled cumulant generating function (SCGF) in the transient and steady state regime and a similar symmetry rule is reflected in a large deviation rate function (LDRF) as a property of the dissipation rate through boundaries. Large deviation theory also gives the thermodynamic force of a nonequilibrium steady state, as is usually recorded experimentally by a single molecule technique, which plays a key role responsible for the dynamical symmetry of the SCGF and LDRF. Using some special properties of the Legendre transformation, here, we have provided a relation between the fluctuations of fluxes and dissipation rates, and among them, the fluctuation of the turnover rate is routinely estimated but the fluctuation in the dissipation rate is yet to be characterized for small systems. Such an enzymatic reaction flow system can be a very good testing ground to systematically understand the rare events from the large deviation theory which is beyond fluctuation theorem and central limit theorem.

Large deviation theory for the kinetics and energetics of turnover of enzyme catalysis in a chemiostatic flow

NASA Astrophysics Data System (ADS)

Das, Biswajit; Gangopadhyay, Gautam

2018-05-01

In the framework of large deviation theory, we have characterized nonequilibrium turnover statistics of enzyme catalysis in a chemiostatic flow with externally controllable parameters, like substrate injection rate and mechanical force. In the kinetics of the process, we have shown the fluctuation theorems in terms of the symmetry of the scaled cumulant generating function (SCGF) in the transient and steady state regime and a similar symmetry rule is reflected in a large deviation rate function (LDRF) as a property of the dissipation rate through boundaries. Large deviation theory also gives the thermodynamic force of a nonequilibrium steady state, as is usually recorded experimentally by a single molecule technique, which plays a key role responsible for the dynamical symmetry of the SCGF and LDRF. Using some special properties of the Legendre transformation, here, we have provided a relation between the fluctuations of fluxes and dissipation rates, and among them, the fluctuation of the turnover rate is routinely estimated but the fluctuation in the dissipation rate is yet to be characterized for small systems. Such an enzymatic reaction flow system can be a very good testing ground to systematically understand the rare events from the large deviation theory which is beyond fluctuation theorem and central limit theorem.

Generalized energy measurements and modified transient quantum fluctuation theorems

NASA Astrophysics Data System (ADS)

Watanabe, Gentaro; Venkatesh, B. Prasanna; Talkner, Peter

2014-05-01

Determining the work which is supplied to a system by an external agent provides a crucial step in any experimental realization of transient fluctuation relations. This, however, poses a problem for quantum systems, where the standard procedure requires the projective measurement of energy at the beginning and the end of the protocol. Unfortunately, projective measurements, which are preferable from the point of view of theory, seem to be difficult to implement experimentally. We demonstrate that, when using a particular type of generalized energy measurements, the resulting work statistics is simply related to that of projective measurements. This relation between the two work statistics entails the existence of modified transient fluctuation relations. The modifications are exclusively determined by the errors incurred in the generalized energy measurements. They are universal in the sense that they do not depend on the force protocol. Particularly simple expressions for the modified Crooks relation and Jarzynski equality are found for Gaussian energy measurements. These can be obtained by a sequence of sufficiently many generalized measurements which need not be Gaussian. In accordance with the central limit theorem, this leads to an effective error reduction in the individual measurements and even yields a projective measurement in the limit of infinite repetitions.

Deterministic Approach to the Kinetic Theory of Gases

NASA Astrophysics Data System (ADS)

Beck, József

2010-02-01

In the so-called Bernoulli model of the kinetic theory of gases, where (1) the particles are dimensionless points, (2) they are contained in a cube container, (3) no attractive or exterior forces are acting on them, (4) there is no collision between the particles, (5) the collision against the walls of the container are according to the law of elastic reflection, we deduce from Newtonian mechanics two local probabilistic laws: a Poisson limit law and a central limit theorem. We also prove some global law of large numbers, justifying that "density" and "pressure" are constant. Finally, as a byproduct of our research, we prove the surprising super-uniformity of the typical billiard path in a square.

Generalised Central Limit Theorems for Growth Rate Distribution of Complex Systems

NASA Astrophysics Data System (ADS)

Takayasu, Misako; Watanabe, Hayafumi; Takayasu, Hideki

2014-04-01

We introduce a solvable model of randomly growing systems consisting of many independent subunits. Scaling relations and growth rate distributions in the limit of infinite subunits are analysed theoretically. Various types of scaling properties and distributions reported for growth rates of complex systems in a variety of fields can be derived from this basic physical model. Statistical data of growth rates for about 1 million business firms are analysed as a real-world example of randomly growing systems. Not only are the scaling relations consistent with the theoretical solution, but the entire functional form of the growth rate distribution is fitted with a theoretical distribution that has a power-law tail.

Exact infinite-time statistics of the Loschmidt echo for a quantum quench.

PubMed

Campos Venuti, Lorenzo; Jacobson, N Tobias; Santra, Siddhartha; Zanardi, Paolo

2011-07-01

The equilibration dynamics of a closed quantum system is encoded in the long-time distribution function of generic observables. In this Letter we consider the Loschmidt echo generalized to finite temperature, and show that we can obtain an exact expression for its long-time distribution for a closed system described by a quantum XY chain following a sudden quench. In the thermodynamic limit the logarithm of the Loschmidt echo becomes normally distributed, whereas for small quenches in the opposite, quasicritical regime, the distribution function acquires a universal double-peaked form indicating poor equilibration. These findings, obtained by a central limit theorem-type result, extend to completely general models in the small-quench regime.

Confidence intervals for the population mean tailored to small sample sizes, with applications to survey sampling.

PubMed

Rosenblum, Michael A; Laan, Mark J van der

2009-01-07

The validity of standard confidence intervals constructed in survey sampling is based on the central limit theorem. For small sample sizes, the central limit theorem may give a poor approximation, resulting in confidence intervals that are misleading. We discuss this issue and propose methods for constructing confidence intervals for the population mean tailored to small sample sizes. We present a simple approach for constructing confidence intervals for the population mean based on tail bounds for the sample mean that are correct for all sample sizes. Bernstein's inequality provides one such tail bound. The resulting confidence intervals have guaranteed coverage probability under much weaker assumptions than are required for standard methods. A drawback of this approach, as we show, is that these confidence intervals are often quite wide. In response to this, we present a method for constructing much narrower confidence intervals, which are better suited for practical applications, and that are still more robust than confidence intervals based on standard methods, when dealing with small sample sizes. We show how to extend our approaches to much more general estimation problems than estimating the sample mean. We describe how these methods can be used to obtain more reliable confidence intervals in survey sampling. As a concrete example, we construct confidence intervals using our methods for the number of violent deaths between March 2003 and July 2006 in Iraq, based on data from the study "Mortality after the 2003 invasion of Iraq: A cross sectional cluster sample survey," by Burnham et al. (2006).

Logical errors on proving theorem

NASA Astrophysics Data System (ADS)

Sari, C. K.; Waluyo, M.; Ainur, C. M.; Darmaningsih, E. N.

2018-01-01

In tertiary level, students of mathematics education department attend some abstract courses, such as Introduction to Real Analysis which needs an ability to prove mathematical statements almost all the time. In fact, many students have not mastered this ability appropriately. In their Introduction to Real Analysis tests, even though they completed their proof of theorems, they achieved an unsatisfactory score. They thought that they succeeded, but their proof was not valid. In this study, a qualitative research was conducted to describe logical errors that students made in proving the theorem of cluster point. The theorem was given to 54 students. Misconceptions on understanding the definitions seem to occur within cluster point, limit of function, and limit of sequences. The habit of using routine symbol might cause these misconceptions. Suggestions to deal with this condition are described as well.

A Diffusion Approximation Based on Renewal Processes with Applications to Strongly Biased Run-Tumble Motion.

PubMed

Thygesen, Uffe Høgsbro

2016-03-01

We consider organisms which use a renewal strategy such as run-tumble when moving in space, for example to perform chemotaxis in chemical gradients. We derive a diffusion approximation for the motion, applying a central limit theorem due to Anscombe for renewal-reward processes; this theorem has not previously been applied in this context. Our results extend previous work, which has established the mean drift but not the diffusivity. For a classical model of tumble rates applied to chemotaxis, we find that the resulting chemotactic drift saturates to the swimming velocity of the organism when the chemical gradients grow increasingly steep. The dispersal becomes anisotropic in steep gradients, with larger dispersal across the gradient than along the gradient. In contrast to one-dimensional settings, strong bias increases dispersal. We next include Brownian rotation in the model and find that, in limit of high chemotactic sensitivity, the chemotactic drift is 64% of the swimming velocity, independent of the magnitude of the Brownian rotation. We finally derive characteristic timescales of the motion that can be used to assess whether the diffusion limit is justified in a given situation. The proposed technique for obtaining diffusion approximations is conceptually and computationally simple, and applicable also when statistics of the motion is obtained empirically or through Monte Carlo simulation of the motion.

A uniform Tauberian theorem in dynamic games

NASA Astrophysics Data System (ADS)

Khlopin, D. V.

2018-01-01

Antagonistic dynamic games including games represented in normal form are considered. The asymptotic behaviour of value in these games is investigated as the game horizon tends to infinity (Cesàro mean) and as the discounting parameter tends to zero (Abel mean). The corresponding Abelian-Tauberian theorem is established: it is demonstrated that in both families the game value uniformly converges to the same limit, provided that at least one of the limits exists. Analogues of one-sided Tauberian theorems are obtained. An example shows that the requirements are essential even for control problems. Bibliography: 31 titles.

On the symmetry foundation of double soft theorems

NASA Astrophysics Data System (ADS)

Li, Zhi-Zhong; Lin, Hung-Hwa; Zhang, Shun-Qing

2017-12-01

Double-soft theorems, like its single-soft counterparts, arises from the underlying symmetry principles that constrain the interactions of massless particles. While single soft theorems can be derived in a non-perturbative fashion by employing current algebras, recent attempts of extending such an approach to known double soft theorems has been met with difficulties. In this work, we have traced the difficulty to two inequivalent expansion schemes, depending on whether the soft limit is taken asymmetrically or symmetrically, which we denote as type A and B respectively. The soft-behaviour for type A scheme can simply be derived from single soft theorems, and are thus non-perturbatively protected. For type B, the information of the four-point vertex is required to determine the corresponding soft theorems, and thus are in general not protected. This argument can be readily extended to general multi-soft theorems. We also ask whether unitarity can be emergent from locality together with the two kinds of soft theorems, which has not been fully investigated before.

The HVT technique and the 'uncertainty' relation for central potentials

NASA Astrophysics Data System (ADS)

Grypeos, M. E.; Koutroulos, C. G.; Oyewumi, K. J.; Petridou, Th

2004-08-01

The quantum mechanical hypervirial theorems (HVT) technique is used to treat the so-called 'uncertainty' relation for quite a general class of central potential wells, including the (reduced) Poeschl-Teller and the Gaussian one. It is shown that this technique is quite suitable in deriving an approximate analytic expression in the form of a truncated power series expansion for the dimensionless product Pnl equiv langr2rangnllangp2rangnl/planck2, for every (deeply) bound state of a particle moving non-relativistically in the well, provided that a (dimensionless) parameter s is sufficiently small. Attention is also paid to a number of cases, among the limited existing ones, in which exact analytic or semi-analytic expressions for Pnl can be derived. Finally, numerical results are given and discussed.

Wealth distribution on complex networks

NASA Astrophysics Data System (ADS)

Ichinomiya, Takashi

2012-12-01

We study the wealth distribution of the Bouchaud-Mézard model on complex networks. It is known from numerical simulations that this distribution depends on the topology of the network; however, no one has succeeded in explaining it. Using “adiabatic” and “independent” assumptions along with the central-limit theorem, we derive equations that determine the probability distribution function. The results are compared to those of simulations for various networks. We find good agreement between our theory and the simulations, except for the case of Watts-Strogatz networks with a low rewiring rate due to the breakdown of independent assumption.

On spurious detection of linear response and misuse of the fluctuation-dissipation theorem in finite time series

NASA Astrophysics Data System (ADS)

Gottwald, Georg A.; Wormell, J. P.; Wouters, Jeroen

2016-09-01

Using a sensitive statistical test we determine whether or not one can detect the breakdown of linear response given observations of deterministic dynamical systems. A goodness-of-fit statistics is developed for a linear statistical model of the observations, based on results for central limit theorems for deterministic dynamical systems, and used to detect linear response breakdown. We apply the method to discrete maps which do not obey linear response and show that the successful detection of breakdown depends on the length of the time series, the magnitude of the perturbation and on the choice of the observable. We find that in order to reliably reject the assumption of linear response for typical observables sufficiently large data sets are needed. Even for simple systems such as the logistic map, one needs of the order of 106 observations to reliably detect the breakdown with a confidence level of 95 %; if less observations are available one may be falsely led to conclude that linear response theory is valid. The amount of data required is larger the smaller the applied perturbation. For judiciously chosen observables the necessary amount of data can be drastically reduced, but requires detailed a priori knowledge about the invariant measure which is typically not available for complex dynamical systems. Furthermore we explore the use of the fluctuation-dissipation theorem (FDT) in cases with limited data length or coarse-graining of observations. The FDT, if applied naively to a system without linear response, is shown to be very sensitive to the details of the sampling method, resulting in erroneous predictions of the response.

Critical Behavior of the Annealed Ising Model on Random Regular Graphs

NASA Astrophysics Data System (ADS)

Can, Van Hao

2017-11-01

In Giardinà et al. (ALEA Lat Am J Probab Math Stat 13(1):121-161, 2016), the authors have defined an annealed Ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2-regular graphs. Then in Can (Annealed limit theorems for the Ising model on random regular graphs, arXiv:1701.08639, 2017), we generalized their results to the class of all random regular graphs. In this paper, we study the critical behavior of this model. In particular, we determine the critical exponents and prove a non standard limit theorem stating that the magnetization scaled by n^{3/4} converges to a specific random variable, with n the number of vertices of random regular graphs.

Analytical study of bound states in graphene nanoribbons and carbon nanotubes: The variable phase method and the relativistic Levinson theorem

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

Miserev, D. S., E-mail: d.miserev@student.unsw.edu.au, E-mail: erazorheader@gmail.com

2016-06-15

The problem of localized states in 1D systems with a relativistic spectrum, namely, graphene stripes and carbon nanotubes, is studied analytically. The bound state as a superposition of two chiral states is completely described by their relative phase, which is the foundation of the variable phase method (VPM) developed herein. Based on our VPM, we formulate and prove the relativistic Levinson theorem. The problem of bound states can be reduced to the analysis of closed trajectories of some vector field. Remarkably, the Levinson theorem appears as the Poincaré index theorem for these closed trajectories. The VPM equation is also reducedmore » to the nonrelativistic and semiclassical limits. The limit of a small momentum p{sub y} of transverse quantization is applicable to an arbitrary integrable potential. In this case, a single confined mode is predicted.« less

Limit theorems for Lévy walks in d dimensions: rare and bulk fluctuations

NASA Astrophysics Data System (ADS)

Fouxon, Itzhak; Denisov, Sergey; Zaburdaev, Vasily; Barkai, Eli

2017-04-01

We consider super-diffusive Lévy walks in d≥slant 2 dimensions when the duration of a single step, i.e. a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d-dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of the central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have a universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called ‘infinite density’, describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of the PDF of velocity in one step. A typical realization of the walk consists of anomalous diffusive motion (described by anisotropic d-dimensional Lévy distribution) interspersed with long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distributions the d-dimensional process can be reduced to a one-dimensional Lévy walk. We briefly discuss the consequences of non-universality for the d  >  1 dimensional fractional diffusion equation, in particular the non-uniqueness of the fractional Laplacian.

Heuristic analogy in Ars Conjectandi: From Archimedes' De Circuli Dimensione to Bernoulli's theorem.

PubMed

Campos, Daniel G

2018-02-01

This article investigates the way in which Jacob Bernoulli proved the main mathematical theorem that undergirds his art of conjecturing-the theorem that founded, historically, the field of mathematical probability. It aims to contribute a perspective into the question of problem-solving methods in mathematics while also contributing to the comprehension of the historical development of mathematical probability. It argues that Bernoulli proved his theorem by a process of mathematical experimentation in which the central heuristic strategy was analogy. In this context, the analogy functioned as an experimental hypothesis. The article expounds, first, Bernoulli's reasoning for proving his theorem, describing it as a process of experimentation in which hypothesis-making is crucial. Next, it investigates the analogy between his reasoning and Archimedes' approximation of the value of π, by clarifying both Archimedes' own experimental approach to the said approximation and its heuristic influence on Bernoulli's problem-solving strategy. The discussion includes some general considerations about analogy as a heuristic technique to make experimental hypotheses in mathematics. Copyright © 2018 Elsevier Ltd. All rights reserved.

Restoring the consistency with the contact density theorem of a classical density functional theory of ions at a planar electrical double layer.

PubMed

Gillespie, Dirk

2014-11-01

Classical density functional theory (DFT) of fluids is a fast and efficient theory to compute the structure of the electrical double layer in the primitive model of ions where ions are modeled as charged, hard spheres in a background dielectric. While the hard-core repulsive component of this ion-ion interaction can be accurately computed using well-established DFTs, the electrostatic component is less accurate. Moreover, many electrostatic functionals fail to satisfy a basic theorem, the contact density theorem, that relates the bulk pressure, surface charge, and ion densities at their distances of closest approach for ions in equilibrium at a smooth, hard, planar wall. One popular electrostatic functional that fails to satisfy the contact density theorem is a perturbation approach developed by Kierlik and Rosinberg [Phys. Rev. A 44, 5025 (1991)PLRAAN1050-294710.1103/PhysRevA.44.5025] and Rosenfeld [J. Chem. Phys. 98, 8126 (1993)JCPSA60021-960610.1063/1.464569], where the full free-energy functional is Taylor-expanded around a bulk (homogeneous) reference fluid. Here, it is shown that this functional fails to satisfy the contact density theorem because it also fails to satisfy the known low-density limit. When the functional is corrected to satisfy this limit, a corrected bulk pressure is derived and it is shown that with this pressure both the contact density theorem and the Gibbs adsorption theorem are satisfied.

Closer look at time averages of the logistic map at the edge of chaos

NASA Astrophysics Data System (ADS)

Tirnakli, Ugur; Tsallis, Constantino; Beck, Christian

2009-05-01

The probability distribution of sums of iterates of the logistic map at the edge of chaos has been recently shown [U. Tirnakli , Phys. Rev. E 75, 040106(R) (2007)] to be numerically consistent with a q -Gaussian, the distribution which—under appropriate constraints—maximizes the nonadditive entropy Sq , which is the basis of nonextensive statistical mechanics. This analysis was based on a study of the tails of the distribution. We now check the entire distribution, in particular, its central part. This is important in view of a recent q generalization of the central limit theorem, which states that for certain classes of strongly correlated random variables the rescaled sum approaches a q -Gaussian limit distribution. We numerically investigate for the logistic map with a parameter in a small vicinity of the critical point under which conditions there is convergence to a q -Gaussian both in the central region and in the tail region and find a scaling law involving the Feigenbaum constant δ . Our results are consistent with a large number of already available analytical and numerical evidences that the edge of chaos is well described in terms of the entropy Sq and its associated concepts.

Volumes of critical bubbles from the nucleation theorem

NASA Astrophysics Data System (ADS)

Wilemski, Gerald

2006-09-01

A corollary of the nucleation theorem due to Kashchiev [Nucleation: Basic Theory with Applications (Butterworth-Heinemann, Oxford, 2000)] allows the volume V* of a critical bubble to be determined from nucleation rate measurements. The original derivation was limited to one-component, ideal gas bubbles with a vapor density much smaller than that of the ambient liquid. Here, an exact result is found for multicomponent, nonideal gas bubbles. Provided a weak density inequality holds, this result reduces to Kashchiev's simple form which thus has a much broader range of applicability than originally expected. Limited applications to droplets are also mentioned, and the utility of the pT,x form of the nucleation theorem as a sum rule is noted.

Trust Method for Multi-Agent Consensus

DTIC Science & Technology

2012-03-22

irreducible10 and by Lemma 1, is also stochastic. And according to the Perron - Frobenius theorem, the fact that has an eigenvalue of 1 with a positive...if the limit lim→∞ exists. According to the Perron - Frobenius theorem, this limit exists for primitive matrices and according to Lemma 2, ...and > 0. Here, is known as the Perron matrix of graph with parameter . If we substitute the normalized Laplacian for in

Paretian Poisson Processes

NASA Astrophysics Data System (ADS)

Eliazar, Iddo; Klafter, Joseph

2008-05-01

Many random populations can be modeled as a countable set of points scattered randomly on the positive half-line. The points may represent magnitudes of earthquakes and tornados, masses of stars, market values of public companies, etc. In this article we explore a specific class of random such populations we coin ` Paretian Poisson processes'. This class is elemental in statistical physics—connecting together, in a deep and fundamental way, diverse issues including: the Poisson distribution of the Law of Small Numbers; Paretian tail statistics; the Fréchet distribution of Extreme Value Theory; the one-sided Lévy distribution of the Central Limit Theorem; scale-invariance, renormalization and fractality; resilience to random perturbations.

A Perron-Frobenius Type of Theorem for Quantum Operations

NASA Astrophysics Data System (ADS)

Lagro, Matthew; Yang, Wei-Shih; Xiong, Sheng

2017-10-01

We define a special class of quantum operations we call Markovian and show that it has the same spectral properties as a corresponding Markov chain. We then consider a convex combination of a quantum operation and a Markovian quantum operation and show that under a norm condition its spectrum has the same properties as in the conclusion of the Perron-Frobenius theorem if its Markovian part does. Moreover, under a compatibility condition of the two operations, we show that its limiting distribution is the same as the corresponding Markov chain. We apply our general results to partially decoherent quantum random walks with decoherence strength 0 ≤ p ≤ 1. We obtain a quantum ergodic theorem for partially decoherent processes. We show that for 0 < p ≤ 1, the limiting distribution of a partially decoherent quantum random walk is the same as the limiting distribution for the classical random walk.

Spheres, charges, instantons, and bootstrap: A five-dimensional odyssey

NASA Astrophysics Data System (ADS)

Chang, Chi-Ming; Fluder, Martin; Lin, Ying-Hsuan; Wang, Yifan

2018-03-01

We combine supersymmetric localization and the conformal bootstrap to study five-dimensional superconformal field theories. To begin, we classify the admissible counter-terms and derive a general relation between the five-sphere partition function and the conformal and flavor central charges. Along the way, we discover a new superconformal anomaly in five dimensions. We then propose a precise triple factorization formula for the five-sphere partition function, that incorporates instantons and is consistent with flavor symmetry enhancement. We numerically evaluate the central charges for the rank-one Seiberg and Morrison-Seiberg theories, and find strong evidence for their saturation of bootstrap bounds, thereby determining the spectra of long multiplets in these theories. Lastly, our results provide new evidence for the F-theorem and possibly a C-theorem in five-dimensional superconformal theories.

Quantum spaces, central extensions of Lie groups and related quantum field theories

NASA Astrophysics Data System (ADS)

Poulain, Timothé; Wallet, Jean-Christophe

2018-02-01

Quantum spaces with su(2) noncommutativity can be modelled by using a family of SO(3)-equivariant differential *-representations. The quantization maps are determined from the combination of the Wigner theorem for SU(2) with the polar decomposition of the quantized plane waves. A tracial star-product, equivalent to the Kontsevich product for the Poisson manifold dual to su(2) is obtained from a subfamily of differential *-representations. Noncommutative (scalar) field theories free from UV/IR mixing and whose commutative limit coincides with the usual ϕ 4 theory on ℛ3 are presented. A generalization of the construction to semi-simple possibly non simply connected Lie groups based on their central extensions by suitable abelian Lie groups is discussed. Based on a talk presented by Poulain T at the XXVth International Conference on Integrable Systems and Quantum symmetries (ISQS-25), Prague, June 6-10 2017.

Brownian motion properties of optoelectronic random bit generators based on laser chaos.

PubMed

Li, Pu; Yi, Xiaogang; Liu, Xianglian; Wang, Yuncai; Wang, Yongge

2016-07-11

The nondeterministic property of the optoelectronic random bit generator (RBG) based on laser chaos are experimentally analyzed from two aspects of the central limit theorem and law of iterated logarithm. The random bits are extracted from an optical feedback chaotic laser diode using a multi-bit extraction technique in the electrical domain. Our experimental results demonstrate that the generated random bits have no statistical distance from the Brownian motion, besides that they can pass the state-of-the-art industry-benchmark statistical test suite (NIST SP800-22). All of them give a mathematically provable evidence that the ultrafast random bit generator based on laser chaos can be used as a nondeterministic random bit source.

Lognormal-like statistics of a stochastic squeeze process

NASA Astrophysics Data System (ADS)

Shapira, Dekel; Cohen, Doron

2017-10-01

We analyze the full statistics of a stochastic squeeze process. The model's two parameters are the bare stretching rate w and the angular diffusion coefficient D . We carry out an exact analysis to determine the drift and the diffusion coefficient of log(r ) , where r is the radial coordinate. The results go beyond the heuristic lognormal description that is implied by the central limit theorem. Contrary to the common "quantum Zeno" approximation, the radial diffusion is not simply Dr=(1 /8 ) w2/D but has a nonmonotonic dependence on w /D . Furthermore, the calculation of the radial moments is dominated by the far non-Gaussian tails of the log(r ) distribution.

Random Numbers and Monte Carlo Methods

NASA Astrophysics Data System (ADS)

Scherer, Philipp O. J.

Many-body problems often involve the calculation of integrals of very high dimension which cannot be treated by standard methods. For the calculation of thermodynamic averages Monte Carlo methods are very useful which sample the integration volume at randomly chosen points. After summarizing some basic statistics, we discuss algorithms for the generation of pseudo-random numbers with given probability distribution which are essential for all Monte Carlo methods. We show how the efficiency of Monte Carlo integration can be improved by sampling preferentially the important configurations. Finally the famous Metropolis algorithm is applied to classical many-particle systems. Computer experiments visualize the central limit theorem and apply the Metropolis method to the traveling salesman problem.

A Hybrid Common Fixed Point Theorem under Certain Recent Properties

PubMed Central

Imdad, Mohammad

2014-01-01

We prove a common fixed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings via common limit range property. Our result improves some results from the existing literature, especially the ones contained in Sintunavarat and Kumam (2009). Some illustrative and interesting examples to highlight the realized improvements are also furnished. PMID:24592191

The Star-Forming Main Sequence as a Natural Consequence of the Central Limit Theorem

NASA Astrophysics Data System (ADS)

Kelson, Daniel David

2015-08-01

Star-formation rates (SFR) of disk galaxies correlate with stellar mass, with a small dispersion in SSFR at fixed mass, sigma~0.3 dex. With such scatter this star-formation main sequence (SFMS) has been interpreted as deterministic and fundamental. Here I demonstrate that such a correlation arises naturally from the central limit theorem. The derivation begins by approximating in situ stellar mass growth as a stochastic process, much like a random walk, where the expectation of SFR at any time is equal to the SFR at the previous time. The SFRs of real galaxies, however, do not experience wholly random stochastic changes over time, but change in a highly correlated fashion due to the long reach of gravity and the correlation of structure in the universe. We therefore generalize the results for star-formation as a stochastic process that has random correlations over random and potentially infinite timescales. For unbiased samples of (disk) galaxies we derive expectation values for SSFR and its scatter, such that =2/T, and Sig[SFR/M]=. Note that this relative scatter is independent of mass and time. This derived correlation between SFR and stellar mass, and its evolution, matches published data to z=10 with sufficient accuracy to constrain cosmological parameters from the data. This statistical approach to the diversity of star-formation histories reproduces several important observables, including: the scatter in SSFR at fixed mass; the forms of SFHs of nearby dwarf galaxies and the Milky Way. At least one additional process beyond a single one responsible for in situ stellar mass growth will be required to match the evolution of the stellar mass function, and we discuss ways to generalize the framework. The implied dispersion in SFHs, and the SFMS's insensitivity to timescales of stochasticity, thus substantially limits the ability to connect massive galaxies to their progenitors over long cosmic baselines. Such analytical work shows promise for statisically modeling distributions of galaxies over cosmic time, in a manner particularly indpendent of the thorny uncertainties in sub-grid astrophysics of modern cosmological simulations.

New dimensions for wound strings: The modular transformation of geometry to topology

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

McGreevy, John; Silverstein, Eva; Starr, David

2007-02-15

We show, using a theorem of Milnor and Margulis, that string theory on compact negatively curved spaces grows new effective dimensions as the space shrinks, generalizing and contextualizing the results in E. Silverstein, Phys. Rev. D 73, 086004 (2006).. Milnor's theorem relates negative sectional curvature on a compact Riemannian manifold to exponential growth of its fundamental group, which translates in string theory to a higher effective central charge arising from winding strings. This exponential density of winding modes is related by modular invariance to the infrared small perturbation spectrum. Using self-consistent approximations valid at large radius, we analyze this correspondencemore » explicitly in a broad set of time-dependent solutions, finding precise agreement between the effective central charge and the corresponding infrared small perturbation spectrum. This indicates a basic relation between geometry, topology, and dimensionality in string theory.« less

Two proposed convergence criteria for Monte Carlo solutions

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

Forster, R.A.; Pederson, S.P.; Booth, T.E.

1992-01-01

The central limit theorem (CLT) can be applied to a Monte Carlo solution if two requirements are satisfied: (1) The random variable has a finite mean and a finite variance; and (2) the number N of independent observations grows large. When these two conditions are satisfied, a confidence interval (CI) based on the normal distribution with a specified coverage probability can be formed. The first requirement is generally satisfied by the knowledge of the Monte Carlo tally being used. The Monte Carlo practitioner has a limited number of marginal methods to assess the fulfillment of the second requirement, such asmore » statistical error reduction proportional to 1/[radical]N with error magnitude guidelines. Two proposed methods are discussed in this paper to assist in deciding if N is large enough: estimating the relative variance of the variance (VOV) and examining the empirical history score probability density function (pdf).« less

The limit distribution in the q-CLT for q\\,\\geqslant \\,1 is unique and can not have a compact support

NASA Astrophysics Data System (ADS)

Umarov, Sabir; Tsallis, Constantino

2016-10-01

In a paper by Umarov et al (2008 Milan J. Math. 76 307-28), a generalization of the Fourier transform, called the q-Fourier transform, was introduced and applied for the proof of a q-generalized central limit theorem (q-CLT). Subsequently, Hilhorst illustrated (2009 Braz. J. Phys. 39 371-9 2010 J. Stat. Mech. P10023) that the q-Fourier transform for q\\gt 1, is not invertible in the space of density functions. Indeed, using an invariance principle, he constructed a family of densities with the same q-Fourier transform and noted that ‘as a consequence, the q-CLT falls short of achieving its stated goal’. The distributions constructed there have compact support. We prove now that the limit distribution in the q-CLT is unique and can not have a compact support. This result excludes all the possible counterexamples which can be constructed using the invariance principle and fills the gap mentioned by Hilhorst.

The Baetylus Theorem-the central disconnect driving consumer behavior and investment returns in Wearable Technologies.

PubMed

Levine, James A

2016-08-01

The Wearable Technology market may increase fivefold by the end of the decade. There is almost no academic investigation as to what drives the investment hypothesis in wearable technologies. This paper seeks to examine this issue from an evidence-based perspective. There is a fundamental disconnect in how consumers view wearable sensors and how companies market them; this is called The Baetylus Theorem where people believe (falsely) that by buying a wearable sensor they will receive health benefit; data suggest that this is not the case. This idea is grounded social constructs, psychological theories and marketing approaches. A marketing proposal that fails to recognize The Baetylus Theorem and how it can be integrated into a business offering has not optimized its competitive advantage. More importantly, consumers should not falsely believe that purchasing a wearable technology, improves health.

Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics

NASA Astrophysics Data System (ADS)

Simon, R.; Mukunda, N.; Chaturvedi, S.; Srinivasan, V.

2008-11-01

In quantum theory, symmetry has to be defined necessarily in terms of the family of unit rays, the state space. The theorem of Wigner asserts that a symmetry so defined at the level of rays can always be lifted into a linear unitary or an antilinear antiunitary operator acting on the underlying Hilbert space. We present two proofs of this theorem which are both elementary and economical. Central to our proofs is the recognition that a given Wigner symmetry can, by post-multiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis often focuses on the behaviour of certain two-dimensional subspaces of the Hilbert space under the action of a given Wigner symmetry, but the relevance of this behaviour to the larger picture of the whole Hilbert space is made transparent at every stage.

Dynamic inversion enables external magnets to concentrate ferromagnetic rods to a central target.

PubMed

Nacev, A; Weinberg, I N; Stepanov, P Y; Kupfer, S; Mair, L O; Urdaneta, M G; Shimoji, M; Fricke, S T; Shapiro, B

2015-01-14

The ability to use magnets external to the body to focus therapy to deep tissue targets has remained an elusive goal in magnetic drug targeting. Researchers have hitherto been able to manipulate magnetic nanotherapeutics in vivo with nearby magnets but have remained unable to focus these therapies to targets deep within the body using magnets external to the body. One of the factors that has made focusing of therapy to central targets between magnets challenging is Samuel Earnshaw's theorem as applied to Maxwell's equations. These mathematical formulations imply that external static magnets cannot create a stable potential energy well between them. We posited that fast magnetic pulses could act on ferromagnetic rods before they could realign with the magnetic field. Mathematically, this is equivalent to reversing the sign of the potential energy term in Earnshaw's theorem, thus enabling a quasi-static stable trap between magnets. With in vitro experiments, we demonstrated that quick, shaped magnetic pulses can be successfully used to create inward pointing magnetic forces that, on average, enable external magnets to concentrate ferromagnetic rods to a central location.

Quantization of Chirikov Map and Quantum KAM Theorem.

NASA Astrophysics Data System (ADS)

Shi, Kang-Jie

KAM theorem is one of the most important theorems in classical nonlinear dynamics and chaos. To extend KAM theorem to the regime of quantum mechanics, we first study the quantum Chirikov map, whose classical counterpart provides a good example of KAM theorem. Under resonance condition 2pihbar = 1/N, we obtain the eigenstates of the evolution operator of this system. We find that the wave functions in the coherent state representation (CSR) are very similar to the classical trajectories. In particular, some of these wave functions have wall-like structure at the locations of classical KAM curves. We also find that a local average is necessary for a Wigner function to approach its classical limit in the phase space. We then study the general problem theoretically. Under similar conditions for establishing the classical KAM theorem, we obtain a quantum extension of KAM theorem. By constructing successive unitary transformations, we can greatly reduce the perturbation part of a near-integrable Hamiltonian system in a region associated with a Diophantine number {rm W}_{o}. This reduction is restricted only by the magnitude of hbar.. We can summarize our results as follows: In the CSR of a nearly integrable quantum system, associated with a Diophantine number {rm W}_ {o}, there is a band near the corresponding KAM torus of the classical limit of the system. In this band, a Gaussian wave packet moves quasi-periodically (and remain close to the KAM torus) for a long time, with possible diffusion in both the size and the shape of its wave packet. The upper bound of the tunnelling rate out of this band for the wave packet can be made much smaller than any given power of hbar, if the original perturbation is sufficiently small (but independent of hbar). When hbarto 0, we reproduce the classical KAM theorem. For most near-integrable systems the eigenstate wave function in the above band can either have a wall -like structure or have a vanishing amplitude. These conclusions agree with the numerical results of the quantum Chirikov map.

Gaussification and entanglement distillation of continuous-variable systems: a unifying picture.

PubMed

Campbell, Earl T; Eisert, Jens

2012-01-13

Distillation of entanglement using only Gaussian operations is an important primitive in quantum communication, quantum repeater architectures, and distributed quantum computing. Existing distillation protocols for continuous degrees of freedom are only known to converge to a Gaussian state when measurements yield precisely the vacuum outcome. In sharp contrast, non-Gaussian states can be deterministically converted into Gaussian states while preserving their second moments, albeit by usually reducing their degree of entanglement. In this work-based on a novel instance of a noncommutative central limit theorem-we introduce a picture general enough to encompass the known protocols leading to Gaussian states, and new classes of protocols including multipartite distillation. This gives the experimental option of balancing the merits of success probability against entanglement produced.

Enhancement effects in polarimetric radar returns: Phase difference statistics

NASA Technical Reports Server (NTRS)

Lang, R. H.; Khadr, N.

1993-01-01

The probability density functions (pdfs) of the co- and cross-polarized phase differences are derived for backscatter from vegetation using the coherent and incoherent scattering theories. Unlike previous derivations, no assumptions or observations other than the applicability of the Central Limit Theorem (CLT), the low fractional volume of the medium, the reciprocity of the scatterers, and the azimuthal symmetry of the scatterer's orientation statistics are employed. Everything else follows logically via the mathematics. The difference between the coherent theory and the incoherent theory is referred to as the backscatter enhancement effect. The influence of this enhancement effect on the phase difference pdfs is examined and found to be important under combined conditions of scatterer anisotropy and appropriate reflection coefficient values.

Adiabatic Theorem for Quantum Spin Systems

NASA Astrophysics Data System (ADS)

Bachmann, S.; De Roeck, W.; Fraas, M.

2017-08-01

The first proof of the quantum adiabatic theorem was given as early as 1928. Today, this theorem is increasingly applied in a many-body context, e.g., in quantum annealing and in studies of topological properties of matter. In this setup, the rate of variation ɛ of local terms is indeed small compared to the gap, but the rate of variation of the total, extensive Hamiltonian, is not. Therefore, applications to many-body systems are not covered by the proofs and arguments in the literature. In this Letter, we prove a version of the adiabatic theorem for gapped ground states of interacting quantum spin systems, under assumptions that remain valid in the thermodynamic limit. As an application, we give a mathematical proof of Kubo's linear response formula for a broad class of gapped interacting systems. We predict that the density of nonadiabatic excitations is exponentially small in the driving rate and the scaling of the exponent depends on the dimension.

Computability, Gödel's incompleteness theorem, and an inherent limit on the predictability of evolution

PubMed Central

Day, Troy

2012-01-01

The process of evolutionary diversification unfolds in a vast genotypic space of potential outcomes. During the past century, there have been remarkable advances in the development of theory for this diversification, and the theory's success rests, in part, on the scope of its applicability. A great deal of this theory focuses on a relatively small subset of the space of potential genotypes, chosen largely based on historical or contemporary patterns, and then predicts the evolutionary dynamics within this pre-defined set. To what extent can such an approach be pushed to a broader perspective that accounts for the potential open-endedness of evolutionary diversification? There have been a number of significant theoretical developments along these lines but the question of how far such theory can be pushed has not been addressed. Here a theorem is proven demonstrating that, because of the digital nature of inheritance, there are inherent limits on the kinds of questions that can be answered using such an approach. In particular, even in extremely simple evolutionary systems, a complete theory accounting for the potential open-endedness of evolution is unattainable unless evolution is progressive. The theorem is closely related to Gödel's incompleteness theorem, and to the halting problem from computability theory. PMID:21849390

Mixing rates and limit theorems for random intermittent maps

NASA Astrophysics Data System (ADS)

Bahsoun, Wael; Bose, Christopher

2016-04-01

We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps {{T}α} using the full parameter range 0<α <∞ , in general. We derive a number of results around a common theme that illustrates in detail how the constituent map that is fastest mixing (i.e. smallest α) combined with details of the randomizing process, determines the asymptotic properties of the random transformation. Our key result (theorem 1.1) establishes sharp estimates on the position of return time intervals for the quenched dynamics. The main applications of this estimate are to limit laws (in particular, CLT and stable laws, depending on the parameters chosen in the range 0<α <1 ) for the associated skew product; these are detailed in theorem 3.2. Since our estimates in theorem 1.1 also hold for 1≤slant α <∞ we study a second class of random transformations derived from piecewise affine Gaspard-Wang maps, prove existence of an infinite (σ-finite) invariant measure and study the corresponding correlation asymptotics. To the best of our knowledge, this latter kind of result is completely new in the setting of random transformations.

Resolution of the EPR Paradox for Fermion Spin Correlations

NASA Astrophysics Data System (ADS)

Close, Robert

2011-10-01

The EPR paradox addresses the question of whether a physical system can have a definite state independent of its measurement. Bell's Theorem places limits on correlations between local measurements of particles whose properties are established prior to measurement. Experimental violation of Bell's theorem has been regarded as evidence against the existence of a definite state prior to measurement. We model fermions as having a spatial distribution of spin values, so that a Stern-Gerlach device samples the spin distribution differently at different orientations. The computed correlations agree with quantum mechanical predictions and experimental observations. Bell's Theorem is not applicable because for any sampling of angles, different points on the sphere have different density of states.

Surpassing the no-cloning limit with a heralded hybrid linear amplifier for coherent states

PubMed Central

Haw, Jing Yan; Zhao, Jie; Dias, Josephine; Assad, Syed M.; Bradshaw, Mark; Blandino, Rémi; Symul, Thomas; Ralph, Timothy C.; Lam, Ping Koy

2016-01-01

The no-cloning theorem states that an unknown quantum state cannot be cloned exactly and deterministically due to the linearity of quantum mechanics. Associated with this theorem is the quantitative no-cloning limit that sets an upper bound to the quality of the generated clones. However, this limit can be circumvented by abandoning determinism and using probabilistic methods. Here, we report an experimental demonstration of probabilistic cloning of arbitrary coherent states that clearly surpasses the no-cloning limit. Our scheme is based on a hybrid linear amplifier that combines an ideal deterministic linear amplifier with a heralded measurement-based noiseless amplifier. We demonstrate the production of up to five clones with the fidelity of each clone clearly exceeding the corresponding no-cloning limit. Moreover, since successful cloning events are heralded, our scheme has the potential to be adopted in quantum repeater, teleportation and computing applications. PMID:27782135

Distributions of underdense meteor trail amplitudes and its application to meteor scatter communication system design

NASA Astrophysics Data System (ADS)

Weitzen, J. A.; Bourque, S.; Ostergaard, J. C.; Bench, P. M.; Baily, A. D.

1991-04-01

Analysis of data from recent experiments leads to the observation that distributions of underdense meteor trail peak signal amplitudes differ from classic predictions. In this paper the distribution of trail amplitudes in decibels relative 1 W (dBw) is considered, and it is shown that Lindberg's theorem can be used to apply central limit arguments to this problem. It is illustrated that a Gaussian model for the distribution of the logarithm of the peak received signal level of underdense trails provides a better fit to data than classic approaches. Distributions of underdense meteor trail amplitudes at five frequencies are compared to a Gaussian distribution and the classic model. Implications of the Gaussian assumption on the design of communication systems are discussed.

Compounding approach for univariate time series with nonstationary variances

NASA Astrophysics Data System (ADS)

Schäfer, Rudi; Barkhofen, Sonja; Guhr, Thomas; Stöckmann, Hans-Jürgen; Kuhl, Ulrich

2015-12-01

A defining feature of nonstationary systems is the time dependence of their statistical parameters. Measured time series may exhibit Gaussian statistics on short time horizons, due to the central limit theorem. The sample statistics for long time horizons, however, averages over the time-dependent variances. To model the long-term statistical behavior, we compound the local distribution with the distribution of its parameters. Here, we consider two concrete, but diverse, examples of such nonstationary systems: the turbulent air flow of a fan and a time series of foreign exchange rates. Our main focus is to empirically determine the appropriate parameter distribution for the compounding approach. To this end, we extract the relevant time scales by decomposing the time signals into windows and determine the distribution function of the thus obtained local variances.

Large-deviation probabilities for correlated Gaussian processes and intermittent dynamical systems

NASA Astrophysics Data System (ADS)

Massah, Mozhdeh; Nicol, Matthew; Kantz, Holger

2018-05-01

In its classical version, the theory of large deviations makes quantitative statements about the probability of outliers when estimating time averages, if time series data are identically independently distributed. We study large-deviation probabilities (LDPs) for time averages in short- and long-range correlated Gaussian processes and show that long-range correlations lead to subexponential decay of LDPs. A particular deterministic intermittent map can, depending on a control parameter, also generate long-range correlated time series. We illustrate numerically, in agreement with the mathematical literature, that this type of intermittency leads to a power law decay of LDPs. The power law decay holds irrespective of whether the correlation time is finite or infinite, and hence irrespective of whether the central limit theorem applies or not.

Stochastic arbitrage return and its implication for option pricing

NASA Astrophysics Data System (ADS)

Fedotov, Sergei; Panayides, Stephanos

2005-01-01

The purpose of this work is to explore the role that random arbitrage opportunities play in pricing financial derivatives. We use a non-equilibrium model to set up a stochastic portfolio, and for the random arbitrage return, we choose a stationary ergodic random process rapidly varying in time. We exploit the fact that option price and random arbitrage returns change on different time scales which allows us to develop an asymptotic pricing theory involving the central limit theorem for random processes. We restrict ourselves to finding pricing bands for options rather than exact prices. The resulting pricing bands are shown to be independent of the detailed statistical characteristics of the arbitrage return. We find that the volatility “smile” can also be explained in terms of random arbitrage opportunities.

A CLT on the SNR of Diagonally Loaded MVDR Filters

NASA Astrophysics Data System (ADS)

Rubio, Francisco; Mestre, Xavier; Hachem, Walid

2012-08-01

This paper studies the fluctuations of the signal-to-noise ratio (SNR) of minimum variance distorsionless response (MVDR) filters implementing diagonal loading in the estimation of the covariance matrix. Previous results in the signal processing literature are generalized and extended by considering both spatially as well as temporarily correlated samples. Specifically, a central limit theorem (CLT) is established for the fluctuations of the SNR of the diagonally loaded MVDR filter, under both supervised and unsupervised training settings in adaptive filtering applications. Our second-order analysis is based on the Nash-Poincar\\'e inequality and the integration by parts formula for Gaussian functionals, as well as classical tools from statistical asymptotic theory. Numerical evaluations validating the accuracy of the CLT confirm the asymptotic Gaussianity of the fluctuations of the SNR of the MVDR filter.

Compounding approach for univariate time series with nonstationary variances.

PubMed

Schäfer, Rudi; Barkhofen, Sonja; Guhr, Thomas; Stöckmann, Hans-Jürgen; Kuhl, Ulrich

2015-12-01

A defining feature of nonstationary systems is the time dependence of their statistical parameters. Measured time series may exhibit Gaussian statistics on short time horizons, due to the central limit theorem. The sample statistics for long time horizons, however, averages over the time-dependent variances. To model the long-term statistical behavior, we compound the local distribution with the distribution of its parameters. Here, we consider two concrete, but diverse, examples of such nonstationary systems: the turbulent air flow of a fan and a time series of foreign exchange rates. Our main focus is to empirically determine the appropriate parameter distribution for the compounding approach. To this end, we extract the relevant time scales by decomposing the time signals into windows and determine the distribution function of the thus obtained local variances.

Thermodynamic phase transitions for Pomeau-Manneville maps

NASA Astrophysics Data System (ADS)

Venegeroles, Roberto

2012-08-01

We study phase transitions in the thermodynamic description of Pomeau-Manneville intermittent maps from the point of view of infinite ergodic theory, which deals with diverging measure dynamical systems. For such systems, we use a distributional limit theorem to provide both a powerful tool for calculating thermodynamic potentials as also an understanding of the dynamic characteristics at each instability phase. In particular, topological pressure and Rényi entropy are calculated exactly for such systems. Finally, we show the connection of the distributional limit theorem with non-Gaussian fluctuations of the algorithmic complexity proposed by Gaspard and Wang [Proc. Natl. Acad. Sci. USA10.1073/pnas.85.13.4591 85, 4591 (1988)].

From the necessary to the possible: the genesis of the spin-statistics theorem

NASA Astrophysics Data System (ADS)

Blum, Alexander

2014-12-01

The spin-statistics theorem, which relates the intrinsic angular momentum of a single particle to the type of quantum statistics obeyed by a system of many such particles, is one of the central theorems in quantum field theory and the physics of elementary particles. It was first formulated in 1939/40 by Wolfgang Pauli and his assistant Markus Fierz. This paper discusses the developments that led up to this first formulation, starting from early attempts in the late 1920s to explain why charged matter particles obey Fermi-Dirac statistics, while photons obey Bose-Einstein statistics. It is demonstrated how several important developments paved the way from such general philosophical musings to a general (and provable) theorem, most notably the use of quantum field theory, the discovery of new elementary particles, and the generalization of the notion of spin. It is also discussed how the attempts to prove a spin-statistics connection were driven by Pauli from formal to more physical arguments, culminating in Pauli's 1940 proof. This proof was a major success for the beleaguered theory of quantum field theory and the methods Pauli employed proved essential for the renaissance of quantum field theory and the development of renormalization techniques in the late 1940s.

Matching factorization theorems with an inverse-error weighting

NASA Astrophysics Data System (ADS)

Echevarria, Miguel G.; Kasemets, Tomas; Lansberg, Jean-Philippe; Pisano, Cristian; Signori, Andrea

2018-06-01

We propose a new fast method to match factorization theorems applicable in different kinematical regions, such as the transverse-momentum-dependent and the collinear factorization theorems in Quantum Chromodynamics. At variance with well-known approaches relying on their simple addition and subsequent subtraction of double-counted contributions, ours simply builds on their weighting using the theory uncertainties deduced from the factorization theorems themselves. This allows us to estimate the unknown complete matched cross section from an inverse-error-weighted average. The method is simple and provides an evaluation of the theoretical uncertainty of the matched cross section associated with the uncertainties from the power corrections to the factorization theorems (additional uncertainties, such as the nonperturbative ones, should be added for a proper comparison with experimental data). Its usage is illustrated with several basic examples, such as Z boson, W boson, H0 boson and Drell-Yan lepton-pair production in hadronic collisions, and compared to the state-of-the-art Collins-Soper-Sterman subtraction scheme. It is also not limited to the transverse-momentum spectrum, and can straightforwardly be extended to match any (un)polarized cross section differential in other variables, including multi-differential measurements.

Matching factorization theorems with an inverse-error weighting

DOE PAGES

Echevarria, Miguel G.; Kasemets, Tomas; Lansberg, Jean-Philippe; ...

2018-04-03

We propose a new fast method to match factorization theorems applicable in different kinematical regions, such as the transverse-momentum-dependent and the collinear factorization theorems in Quantum Chromodynamics. At variance with well-known approaches relying on their simple addition and subsequent subtraction of double-counted contributions, ours simply builds on their weighting using the theory uncertainties deduced from the factorization theorems themselves. This allows us to estimate the unknown complete matched cross section from an inverse-error-weighted average. The method is simple and provides an evaluation of the theoretical uncertainty of the matched cross section associated with the uncertainties from the power corrections tomore » the factorization theorems (additional uncertainties, such as the nonperturbative ones, should be added for a proper comparison with experimental data). Its usage is illustrated with several basic examples, such as Z boson, W boson, H 0 boson and Drell–Yan lepton-pair production in hadronic collisions, and compared to the state-of-the-art Collins–Soper–Sterman subtraction scheme. In conclusion, it is also not limited to the transverse-momentum spectrum, and can straightforwardly be extended to match any (un)polarized cross section differential in other variables, including multi-differential measurements.« less

Matching factorization theorems with an inverse-error weighting

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

Echevarria, Miguel G.; Kasemets, Tomas; Lansberg, Jean-Philippe

We propose a new fast method to match factorization theorems applicable in different kinematical regions, such as the transverse-momentum-dependent and the collinear factorization theorems in Quantum Chromodynamics. At variance with well-known approaches relying on their simple addition and subsequent subtraction of double-counted contributions, ours simply builds on their weighting using the theory uncertainties deduced from the factorization theorems themselves. This allows us to estimate the unknown complete matched cross section from an inverse-error-weighted average. The method is simple and provides an evaluation of the theoretical uncertainty of the matched cross section associated with the uncertainties from the power corrections tomore » the factorization theorems (additional uncertainties, such as the nonperturbative ones, should be added for a proper comparison with experimental data). Its usage is illustrated with several basic examples, such as Z boson, W boson, H 0 boson and Drell–Yan lepton-pair production in hadronic collisions, and compared to the state-of-the-art Collins–Soper–Sterman subtraction scheme. In conclusion, it is also not limited to the transverse-momentum spectrum, and can straightforwardly be extended to match any (un)polarized cross section differential in other variables, including multi-differential measurements.« less

Cosmological singularity theorems and splitting theorems for N-Bakry-Émery spacetimes

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

Woolgar, Eric, E-mail: ewoolgar@ualberta.ca; Wylie, William, E-mail: wwylie@syr.edu

We study Lorentzian manifolds with a weight function such that the N-Bakry-Émery tensor is bounded below. Such spacetimes arise in the physics of scalar-tensor gravitation theories, including Brans-Dicke theory, theories with Kaluza-Klein dimensional reduction, and low-energy approximations to string theory. In the “pure Bakry-Émery” N = ∞ case with f uniformly bounded above and initial data suitably bounded, cosmological-type singularity theorems are known, as are splitting theorems which determine the geometry of timelike geodesically complete spacetimes for which the bound on the initial data is borderline violated. We extend these results in a number of ways. We are able tomore » extend the singularity theorems to finite N-values N ∈ (n, ∞) and N ∈ (−∞, 1]. In the N ∈ (n, ∞) case, no bound on f is required, while for N ∈ (−∞, 1] and N = ∞, we are able to replace the boundedness of f by a weaker condition on the integral of f along future-inextendible timelike geodesics. The splitting theorems extend similarly, but when N = 1, the splitting is only that of a warped product for all cases considered. A similar limited loss of rigidity has been observed in a prior work on the N-Bakry-Émery curvature in Riemannian signature when N = 1 and appears to be a general feature.« less

Generalized Fourier slice theorem for cone-beam image reconstruction.

PubMed

Zhao, Shuang-Ren; Jiang, Dazong; Yang, Kevin; Yang, Kang

2015-01-01

The cone-beam reconstruction theory has been proposed by Kirillov in 1961, Tuy in 1983, Feldkamp in 1984, Smith in 1985, Pierre Grangeat in 1990. The Fourier slice theorem is proposed by Bracewell 1956, which leads to the Fourier image reconstruction method for parallel-beam geometry. The Fourier slice theorem is extended to fan-beam geometry by Zhao in 1993 and 1995. By combining the above mentioned cone-beam image reconstruction theory and the above mentioned Fourier slice theory of fan-beam geometry, the Fourier slice theorem in cone-beam geometry is proposed by Zhao 1995 in short conference publication. This article offers the details of the derivation and implementation of this Fourier slice theorem for cone-beam geometry. Especially the problem of the reconstruction from Fourier domain has been overcome, which is that the value of in the origin of Fourier space is 0/0. The 0/0 type of limit is proper handled. As examples, the implementation results for the single circle and two perpendicular circle source orbits are shown. In the cone-beam reconstruction if a interpolation process is considered, the number of the calculations for the generalized Fourier slice theorem algorithm is O(N^4), which is close to the filtered back-projection method, here N is the image size of 1-dimension. However the interpolation process can be avoid, in that case the number of the calculations is O(N5).

Cosmological singularity theorems and splitting theorems for N-Bakry-Émery spacetimes

NASA Astrophysics Data System (ADS)

Woolgar, Eric; Wylie, William

2016-02-01

We study Lorentzian manifolds with a weight function such that the N-Bakry-Émery tensor is bounded below. Such spacetimes arise in the physics of scalar-tensor gravitation theories, including Brans-Dicke theory, theories with Kaluza-Klein dimensional reduction, and low-energy approximations to string theory. In the "pure Bakry-Émery" N = ∞ case with f uniformly bounded above and initial data suitably bounded, cosmological-type singularity theorems are known, as are splitting theorems which determine the geometry of timelike geodesically complete spacetimes for which the bound on the initial data is borderline violated. We extend these results in a number of ways. We are able to extend the singularity theorems to finite N-values N ∈ (n, ∞) and N ∈ (-∞, 1]. In the N ∈ (n, ∞) case, no bound on f is required, while for N ∈ (-∞, 1] and N = ∞, we are able to replace the boundedness of f by a weaker condition on the integral of f along future-inextendible timelike geodesics. The splitting theorems extend similarly, but when N = 1, the splitting is only that of a warped product for all cases considered. A similar limited loss of rigidity has been observed in a prior work on the N-Bakry-Émery curvature in Riemannian signature when N = 1 and appears to be a general feature.

Glueball spectrum and hadronic processes in low-energy QCD

NASA Astrophysics Data System (ADS)

Frasca, Marco

2010-10-01

Low-energy limit of quantum chromodynamics (QCD) is obtained using a mapping theorem recently proved. This theorem states that, classically, solutions of a massless quartic scalar field theory are approximate solutions of Yang-Mills equations in the limit of the gauge coupling going to infinity. Low-energy QCD is described by a Yukawa theory further reducible to a Nambu-Jona-Lasinio model. At the leading order one can compute glue-quark interactions and one is able to calculate the properties of the σ and η-η mesons. Finally, it is seen that all the physics of strong interactions, both in the infrared and ultraviolet limit, is described by a single constant Λ arising in the ultraviolet by dimensional transmutation and in the infrared as an integration constant.

Closed-form solutions and scaling laws for Kerr frequency combs

PubMed Central

Renninger, William H.; Rakich, Peter T.

2016-01-01

A single closed-form analytical solution of the driven nonlinear Schrödinger equation is developed, reproducing a large class of the behaviors in Kerr-comb systems, including bright-solitons, dark-solitons, and a large class of periodic wavetrains. From this analytical framework, a Kerr-comb area theorem and a pump-detuning relation are developed, providing new insights into soliton- and wavetrain-based combs along with concrete design guidelines for both. This new area theorem reveals significant deviation from the conventional soliton area theorem, which is crucial to understanding cavity solitons in certain limits. Moreover, these closed-form solutions represent the first step towards an analytical framework for wavetrain formation, and reveal new parameter regimes for enhanced Kerr-comb performance. PMID:27108810

Constraints on the symmetry noninheriting scalar black hole hair

NASA Astrophysics Data System (ADS)

Smolić, Ivica

2017-01-01

Any recipe to grow black hole hair has to circumvent no-hair theorems by violating some of their assumptions. Recently discovered hairy black hole solutions exist due to the fact that their scalar fields don't inherit the symmetries of the spacetime metric. We present here a general analysis of the constraints which limit the possible forms of such a hair, for both the real and the complex scalar fields. These results can be taken as a novel piece of the black hole uniqueness theorems or simply as a symmetry noninheriting Ansätze guide. In addition, we introduce new classification of the gravitational field equations which might prove useful for various generalizations of the theorems about spacetimes with symmetries.

Photoelectric effect from observer's mathematics point of view

NASA Astrophysics Data System (ADS)

Khots, Boris; Khots, Dmitriy

2014-12-01

When we consider and analyze physical events with the purpose of creating corresponding models we often assume that the mathematical apparatus used in modeling is infallible. In particular, this relates to the use of infinity in various aspects and the use of Newton's definition of a limit in analysis. We believe that is where the main problem lies in contemporary study of nature. This work considers Physical aspects in a setting of arithmetic, algebra, geometry, analysis, topology provided by Observer's Mathematics (see www.mathrelativity.com). Certain results and communications pertaining to solution of these problems are provided. In particular, we prove the following Theorems, which give Observer's Mathematics point of view on Einstein photoelectric effect theory and Lamb-Scully and Hanbury-Brown-Twiss experiments: Theorem 1. There are some values of light intensity where anticorrelation parameter A ∈ [0,1). Theorem 2. There are some values of light intensity where anticorrelation parameter A = 1. Theorem 3. There are some values of light intensity where anticorrelation parameter A > 1.

Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem

NASA Astrophysics Data System (ADS)

Li, Lei; Liu, Jian-Guo; Lu, Jianfeng

2017-10-01

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.

Generalized fluctuation-dissipation theorem as a test of the Markovianity of a system

NASA Astrophysics Data System (ADS)

Willareth, Lucian; Sokolov, Igor M.; Roichman, Yael; Lindner, Benjamin

2017-04-01

We study how well a generalized fluctuation-dissipation theorem (GFDT) is suited to test whether a stochastic system is not Markovian. To this end, we simulate a stochastic non-equilibrium model of the mechanosensory hair bundle from the inner ear organ and analyze its spontaneous activity and response to external stimulation. We demonstrate that this two-dimensional Markovian system indeed obeys the GFDT, as long as i) the averaging ensemble is sufficiently large and ii) finite-size effects in estimating the conjugated variable and its susceptibility can be neglected. Furthermore, we test the GFDT also by looking only at a one-dimensional projection of the system, the experimentally accessible position variable. This reduced system is certainly non-Markovian and the GFDT is somewhat violated but not as drastically as for the equilibrium fluctuation-dissipation theorem. We explore suitable measures to quantify the violation of the theorem and demonstrate that for a set of limited experimental data it might be difficult to decide whether the system is Markovian or not.

Darboux theorems and Wronskian formulas for integrable systems I. Constrained KP flows

NASA Astrophysics Data System (ADS)

Oevel, W.

1993-05-01

Generalizations of the classical Darboux theorem are established for pseudo-differential scattering operators of the form L = limit∑i=0N u i∂ i + limitΣi=1m Φ i∂ -1limitΨi†i. Iteration of the Darboux transformations leads to a gauge transformed operator with coefficients given by Wronskian formulas involving a set of eigenfunctions of L. Nonlinear integrable partial differential equations are associated with the scattering operator L which arise as a symmetry reduction of the multicomponent KP hierarchy. With a suitable linear time evolution for the eigenfunctions the Darboux transformation is used to obtain solutions of the integrable equations in terms of Wronskian determinants.

Revisiting the analogue of the Jebsen-Birkhoff theorem in Brans-Dicke gravity

NASA Astrophysics Data System (ADS)

Faraoni, Valerio; Hammad, Fayçal; Cardini, Adriana M.; Gobeil, Thomas

2018-04-01

We report the explicit form of the general static, spherically symmetric, and asymptotically flat solution of vacuum Brans-Dicke gravity in the Jordan frame, assuming that the Brans-Dicke scalar field has no singularities or zeros (except possibly for a central singularity). This general solution is conformal to the Fisher-Wyman geometry of Einstein theory and its nature depends on a scalar charge parameter. Apart from the Schwarzschild black hole, only wormhole throats and central naked singularities are possible.

Infinite densities for Lévy walks.

PubMed

Rebenshtok, A; Denisov, S; Hänggi, P; Barkai, E

2014-12-01

Motion of particles in many systems exhibits a mixture between periods of random diffusive-like events and ballistic-like motion. In many cases, such systems exhibit strong anomalous diffusion, where low-order moments 〈|x(t)|(q)〉 with q below a critical value q(c) exhibit diffusive scaling while for q>q(c) a ballistic scaling emerges. The mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion, e.g., the known diffusion equations and central limit theorems fail to describe both aspects. In this paper we resolve this problem by resorting to the concept of infinite density. Using the widely applicable Lévy walk model, we find a general expression for the corresponding non-normalized density which is fully determined by the particles velocity distribution, the anomalous diffusion exponent α, and the diffusion coefficient K(α). We explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data.

A generalization of Bertrand's theorem to surfaces of revolution

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

Zagryadskii, Oleg A; Kudryavtseva, Elena A; Fedoseev, Denis A

We prove a generalization of Bertrand's theorem to the case of abstract surfaces of revolution that have no 'equators'. We prove a criterion for exactly two central potentials to exist on this type of surface (up to an additive and a multiplicative constant) for which all bounded orbits are closed and there is a bounded nonsingular noncircular orbit. We prove a criterion for the existence of exactly one such potential. We study the geometry and classification of the corresponding surfaces with the aforementioned pair of potentials (gravitational and oscillatory) or unique potential (oscillatory). We show that potentials of the requiredmore » form do not exist on surfaces that do not belong to any of the classes described. Bibliography: 33 titles.« less

Bell's "Theorem": loopholes vs. conceptual flaws

NASA Astrophysics Data System (ADS)

Kracklauer, A. F.

2017-12-01

An historical overview and detailed explication of a critical analysis of what has become known as Bell's Theorem to the effect that, it should be impossible to extend Quantum Theory with the addition of local, real variables so as to obtain a version free of the ambiguous and preternatural features of the currently accepted interpretations is presented. The central point on which this critical analysis, due originally to Edwin Jaynes, is that Bell incorrectly applied probabilistic formulas involving conditional probabilities. In addition, mathematical technicalities that have complicated the understanding of the logical or mathematical setting in which current theory and experimentation are embedded, are discussed. Finally, some historical speculations on the sociological environment, in particular misleading aspects, in which recent generations of physicists lived and worked are mentioned.

An Estimation of the Logarithmic Timescale in Ergodic Dynamics

NASA Astrophysics Data System (ADS)

Gomez, Ignacio S.

An estimation of the logarithmic timescale in quantum systems having an ergodic dynamics in the semiclassical limit, is presented. The estimation is based on an extension of the Krieger’s finite generator theorem for discretized σ-algebras and using the time rescaling property of the Kolmogorov-Sinai entropy. The results are in agreement with those obtained in the literature but with a simpler mathematics and within the context of the ergodic theory. Moreover, some consequences of the Poincaré’s recurrence theorem are also explored.

On Leighton's comparison theorem

NASA Astrophysics Data System (ADS)

Ghatasheh, Ahmed; Weikard, Rudi

2017-06-01

We give a simple proof of a fairly flexible comparison theorem for equations of the type -(p (u‧ + su)) ‧ + rp (u‧ + su) + qu = 0 on a finite interval where 1 / p, r, s, and q are real and integrable. Flexibility is provided by two functions which may be chosen freely (within limits) according to the situation at hand. We illustrate this by presenting some examples and special cases which include Schrödinger equations with distributional potentials as well as Jacobi difference equations.

The critical domain size of stochastic population models.

PubMed

Reimer, Jody R; Bonsall, Michael B; Maini, Philip K

2017-02-01

Identifying the critical domain size necessary for a population to persist is an important question in ecology. Both demographic and environmental stochasticity impact a population's ability to persist. Here we explore ways of including this variability. We study populations with distinct dispersal and sedentary stages, which have traditionally been modelled using a deterministic integrodifference equation (IDE) framework. Individual-based models (IBMs) are the most intuitive stochastic analogues to IDEs but yield few analytic insights. We explore two alternate approaches; one is a scaling up to the population level using the Central Limit Theorem, and the other a variation on both Galton-Watson branching processes and branching processes in random environments. These branching process models closely approximate the IBM and yield insight into the factors determining the critical domain size for a given population subject to stochasticity.

Non-Gaussian Nature of Fracture and the Survival of Fat-Tail Exponents

NASA Astrophysics Data System (ADS)

Tallakstad, Ken Tore; Toussaint, Renaud; Santucci, Stephane; Måløy, Knut Jørgen

2013-04-01

We study the fluctuations of the global velocity Vl(t), computed at various length scales l, during the intermittent mode-I propagation of a crack front. The statistics converge to a non-Gaussian distribution, with an asymmetric shape and a fat tail. This breakdown of the central limit theorem (CLT) is due to the diverging variance of the underlying local crack front velocity distribution, displaying a power law tail. Indeed, by the application of a generalized CLT, the full shape of our experimental velocity distribution at large scale is shown to follow the stable Levy distribution, which preserves the power law tail exponent under upscaling. This study aims to demonstrate in general for crackling noise systems how one can infer the complete scale dependence of the activity—and extreme event distributions—by measuring only at a global scale.

A geometric theory for Lévy distributions

NASA Astrophysics Data System (ADS)

Eliazar, Iddo

2014-08-01

Lévy distributions are of prime importance in the physical sciences, and their universal emergence is commonly explained by the Generalized Central Limit Theorem (CLT). However, the Generalized CLT is a geometry-less probabilistic result, whereas physical processes usually take place in an embedding space whose spatial geometry is often of substantial significance. In this paper we introduce a model of random effects in random environments which, on the one hand, retains the underlying probabilistic structure of the Generalized CLT and, on the other hand, adds a general and versatile underlying geometric structure. Based on this model we obtain geometry-based counterparts of the Generalized CLT, thus establishing a geometric theory for Lévy distributions. The theory explains the universal emergence of Lévy distributions in physical settings which are well beyond the realm of the Generalized CLT.

On the generation of log-Lévy distributions and extreme randomness

NASA Astrophysics Data System (ADS)

Eliazar, Iddo; Klafter, Joseph

2011-10-01

The log-normal distribution is prevalent across the sciences, as it emerges from the combination of multiplicative processes and the central limit theorem (CLT). The CLT, beyond yielding the normal distribution, also yields the class of Lévy distributions. The log-Lévy distributions are the Lévy counterparts of the log-normal distribution, they appear in the context of ultraslow diffusion processes, and they are categorized by Mandelbrot as belonging to the class of extreme randomness. In this paper, we present a natural stochastic growth model from which both the log-normal distribution and the log-Lévy distributions emerge universally—the former in the case of deterministic underlying setting, and the latter in the case of stochastic underlying setting. In particular, we establish a stochastic growth model which universally generates Mandelbrot’s extreme randomness.

A geometric theory for Lévy distributions

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

Eliazar, Iddo, E-mail: eliazar@post.tau.ac.il

2014-08-15

Lévy distributions are of prime importance in the physical sciences, and their universal emergence is commonly explained by the Generalized Central Limit Theorem (CLT). However, the Generalized CLT is a geometry-less probabilistic result, whereas physical processes usually take place in an embedding space whose spatial geometry is often of substantial significance. In this paper we introduce a model of random effects in random environments which, on the one hand, retains the underlying probabilistic structure of the Generalized CLT and, on the other hand, adds a general and versatile underlying geometric structure. Based on this model we obtain geometry-based counterparts ofmore » the Generalized CLT, thus establishing a geometric theory for Lévy distributions. The theory explains the universal emergence of Lévy distributions in physical settings which are well beyond the realm of the Generalized CLT.« less

Exponential Thurston maps and limits of quadratic differentials

NASA Astrophysics Data System (ADS)

Hubbard, John; Schleicher, Dierk; Shishikura, Mitsuhiro

2009-01-01

We give a topological characterization of postsingularly finite topological exponential maps, i.e., universal covers g\\colon{C}to{C}setminus\\{0\\} such that 0 has a finite orbit. Such a map either is Thurston equivalent to a unique holomorphic exponential map λ e^z or it has a topological obstruction called a degenerate Levy cycle. This is the first analog of Thurston's topological characterization theorem of rational maps, as published by Douady and Hubbard, for the case of infinite degree. One main tool is a theorem about the distribution of mass of an integrable quadratic differential with a given number of poles, providing an almost compact space of models for the entire mass of quadratic differentials. This theorem is given for arbitrary Riemann surfaces of finite type in a uniform way.

Quantum interval-valued probability: Contextuality and the Born rule

NASA Astrophysics Data System (ADS)

Tai, Yu-Tsung; Hanson, Andrew J.; Ortiz, Gerardo; Sabry, Amr

2018-05-01

We present a mathematical framework based on quantum interval-valued probability measures to study the effect of experimental imperfections and finite precision measurements on defining aspects of quantum mechanics such as contextuality and the Born rule. While foundational results such as the Kochen-Specker and Gleason theorems are valid in the context of infinite precision, they fail to hold in general in a world with limited resources. Here we employ an interval-valued framework to establish bounds on the validity of those theorems in realistic experimental environments. In this way, not only can we quantify the idea of finite-precision measurement within our theory, but we can also suggest a possible resolution of the Meyer-Mermin debate on the impact of finite-precision measurement on the Kochen-Specker theorem.

Scaling Limits and Generic Bounds for Exploration Processes

NASA Astrophysics Data System (ADS)

Bermolen, Paola; Jonckheere, Matthieu; Sanders, Jaron

2017-12-01

We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become blocked. Given an initial number of vertices N growing to infinity, we study statistical properties of the proportion of explored (active or blocked) nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the jamming constant, through a diffusion approximation for the exploration process which can be described as a unidimensional process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e., random sequential adsorption. As opposed to exploration processes on homogeneous random graphs, these do not allow for such a dimensional reduction. Instead we derive a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and we obtain generic bounds for the fluid limit and jamming constant: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, using coupling techinques, we give trajectorial interpretations of the generic bounds.

Bringing Definitions into High Definition

ERIC Educational Resources Information Center

Mason, John

2010-01-01

Why do definitions play such a central role in mathematics? It may seem obvious that precision about the terms one uses is necessary in order to use those terms reasonably (while reasoning). Definitions are chosen so as to be definite about the terms one uses, but also to make both the statement of, and the reasoning to justify, theorems as…

Stock network stability in times of crisis

NASA Astrophysics Data System (ADS)

Heiberger, Raphael H.

2014-01-01

Despite many efforts crises on financial markets are in large part still scientific black-boxes. In this paper, we use a winner-take-all approach to construct a longitudinal network of S&P 500 companies and their correlations between 2000 and 2012. A comparison to complex ecosystems is drawn, especially whether the May-Wigner theorem can describe real-world economic phenomena. The results confirm the utility of the May-Wigner theorem as a stability indicator for the US stock market, since its development matches with the two major crises of this period, the dot-com bubble and, particularly, the financial crisis. In those times of financial turmoil, the stock network changes its composition, but unlike ecological systems it tightens and the disassortative structure of prosperous markets transforms into a more centralized topology.

Thermal transport in the Falicov-Kimball model

NASA Astrophysics Data System (ADS)

Freericks, J. K.; Zlatić, V.

2001-12-01

We prove the Jonson-Mahan theorem for the thermopower of the Falicov-Kimball model by solving explicitly for correlation functions in the large dimensional limit. We prove a similar result for the thermal conductivity. We separate the results for thermal transport into the pieces of the heat current that arise from the kinetic energy and those that arise from the potential energy. Our method of proof is specific to the Falicov-Kimball model, but illustrates the near cancellations between the kinetic- and potential-energy pieces of the heat current implied by the Jonson-Mahan theorem.

Algorithms for the Equilibration of Matrices and Their Application to Limited-Memory Quasi-Newton Methods

DTIC Science & Technology

2010-05-01

irreducible, by the Perron - Frobenius theorem (see, for example, Theorem 8.4.4 in [28]), the eigenvalue 1 is simple. Next, the rank-one matrix Q has the...We refer to (2.1) as the scaling equation. Although algorithms must use A, existence and unique- ness theory need consider only the nonnegative matrix...B. If p = 1 and A is nonnegative , then A = B. We reserve the term binormalization for the case p = 2. We say A is scalable if there exists x > 0

Generalized parametric down conversion, many particle interferometry, and Bell's theorem

NASA Technical Reports Server (NTRS)

Choi, Hyung Sup

1992-01-01

A new field of multi-particle interferometry is introduced using a nonlinear optical spontaneous parametric down conversion (SPDC) of a photon into more than two photons. The study of SPDC using a realistic Hamiltonian in a multi-mode shows that at least a low conversion rate limit is possible. The down converted field exhibits many stronger nonclassical phenomena than the usual two photon parametric down conversion. Application of the multi-particle interferometry to a recently proposed many particle Bell's theorem on the Einstein-Podolsky-Rosen problem is given.

Weak interaction probes of light nuclei

NASA Astrophysics Data System (ADS)

Towner, I. S.

1986-03-01

Experimental evidence for pion enhancement in axial charge transitions as predicted by softpion theorems is reviewed. Corrections from non-soft-pion terms seem to be limited. For transitions involving the space part of the axial-vector current, soft-pion theorems are powerless. Meson-exchange currents then involve a complicated interplay among competing process. Explicit calculations in the hard-pion model for closed-shell-plus (or minus)-one nuclei, A=15 and A= =17, are in reasonable agreement with experiment. Quenching in the off-diagonal spin-flip matrix element is larger than in the diagonal matrix element.

A model of high-rate indentation of a cylindrical striking pin into a deformable body

NASA Astrophysics Data System (ADS)

Zalazinskaya, E. A.; Zalazinsky, A. G.

2017-12-01

Mathematical modeling of an impact and high-rate indentation to a significant depth of a flat-faced hard cylindrical striking pin into a massive deformable target body is carried out. With the application of the kinematic extreme theorem of the plasticity theory and the kinetic energy variation theorem, the phase trajectories of the striking pin are calculated, the initial velocity of the striking pin in the body, the limit values of the inlet duct length, and the depth of striking pin penetration into the target are determined.

Using Bayes' theorem for free energy calculations

NASA Astrophysics Data System (ADS)

Rogers, David M.

Statistical mechanics is fundamentally based on calculating the probabilities of molecular-scale events. Although Bayes' theorem has generally been recognized as providing key guiding principals for setup and analysis of statistical experiments [83], classical frequentist models still predominate in the world of computational experimentation. As a starting point for widespread application of Bayesian methods in statistical mechanics, we investigate the central quantity of free energies from this perspective. This dissertation thus reviews the basics of Bayes' view of probability theory, and the maximum entropy formulation of statistical mechanics before providing examples of its application to several advanced research areas. We first apply Bayes' theorem to a multinomial counting problem in order to determine inner shell and hard sphere solvation free energy components of Quasi-Chemical Theory [140]. We proceed to consider the general problem of free energy calculations from samples of interaction energy distributions. From there, we turn to spline-based estimation of the potential of mean force [142], and empirical modeling of observed dynamics using integrator matching. The results of this research are expected to advance the state of the art in coarse-graining methods, as they allow a systematic connection from high-resolution (atomic) to low-resolution (coarse) structure and dynamics. In total, our work on these problems constitutes a critical starting point for further application of Bayes' theorem in all areas of statistical mechanics. It is hoped that the understanding so gained will allow for improvements in comparisons between theory and experiment.

A novel model for the chaotic dynamics of superdiffusion

NASA Astrophysics Data System (ADS)

Cushman, J. H.; Park, M.; O'Malley, D.

2009-04-01

Previously we've shown that by modeling the convective velocity in a turbulent flow field as Brownian, one obtains Richardson super diffusion where the expected distance between pairs of particles scales with time cubed. By proving generalized central limit type theorems it's possible to show that modeling the velocity or the acceleration as α-stable Levy gives rise to more general scaling laws that can easily explain other super diffusive regimes. The problem with this latter approach is that the mean square displacement of a particle is infinite. Here we provide an alternate approach that gives a power law mean square displacement of any desired order. We do so by constructing compressed and stretched extensions to Brownian motion. The finite size Lyapunov exponent, the underlying stochastic differential equation and its corresponding Fokker-Planck equations are derived. The fractal dimension of these processes turns out to be the same as that of classical Brownian motion.

Dependence of exponents on text length versus finite-size scaling for word-frequency distributions

NASA Astrophysics Data System (ADS)

Corral, Álvaro; Font-Clos, Francesc

2017-08-01

Some authors have recently argued that a finite-size scaling law for the text-length dependence of word-frequency distributions cannot be conceptually valid. Here we give solid quantitative evidence for the validity of this scaling law, using both careful statistical tests and analytical arguments based on the generalized central-limit theorem applied to the moments of the distribution (and obtaining a novel derivation of Heaps' law as a by-product). We also find that the picture of word-frequency distributions with power-law exponents that decrease with text length [X. Yan and P. Minnhagen, Physica A 444, 828 (2016), 10.1016/j.physa.2015.10.082] does not stand with rigorous statistical analysis. Instead, we show that the distributions are perfectly described by power-law tails with stable exponents, whose values are close to 2, in agreement with the classical Zipf's law. Some misconceptions about scaling are also clarified.

Fourier Spot Volatility Estimator: Asymptotic Normality and Efficiency with Liquid and Illiquid High-Frequency Data

PubMed Central

2015-01-01

The recent availability of high frequency data has permitted more efficient ways of computing volatility. However, estimation of volatility from asset price observations is challenging because observed high frequency data are generally affected by noise-microstructure effects. We address this issue by using the Fourier estimator of instantaneous volatility introduced in Malliavin and Mancino 2002. We prove a central limit theorem for this estimator with optimal rate and asymptotic variance. An extensive simulation study shows the accuracy of the spot volatility estimates obtained using the Fourier estimator and its robustness even in the presence of different microstructure noise specifications. An empirical analysis on high frequency data (U.S. S&P500 and FIB 30 indices) illustrates how the Fourier spot volatility estimates can be successfully used to study intraday variations of volatility and to predict intraday Value at Risk. PMID:26421617

Goldstone mode and pair-breaking excitations in atomic Fermi superfluids

NASA Astrophysics Data System (ADS)

Hoinka, Sascha; Dyke, Paul; Lingham, Marcus G.; Kinnunen, Jami J.; Bruun, Georg M.; Vale, Chris J.

2017-10-01

Spontaneous symmetry breaking is a central paradigm of elementary particle physics, magnetism, superfluidity and superconductivity. According to Goldstone's theorem, phase transitions that break continuous symmetries lead to the existence of gapless excitations in the long-wavelength limit. These Goldstone modes can become the dominant low-energy excitation, showing that symmetry breaking has a profound impact on the physical properties of matter. Here, we present a comprehensive study of the elementary excitations in a homogeneous strongly interacting Fermi gas through the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) of molecules using two-photon Bragg spectroscopy. The spectra exhibit a discrete Goldstone mode, associated with the broken-symmetry superfluid phase, as well as pair-breaking single-particle excitations. Our techniques yield a direct determination of the superfluid pairing gap and speed of sound in close agreement with strong-coupling theories.

Non-linear programming in shakedown analysis with plasticity and friction

NASA Astrophysics Data System (ADS)

Spagnoli, A.; Terzano, M.; Barber, J. R.; Klarbring, A.

2017-07-01

Complete frictional contacts, when subjected to cyclic loading, may sometimes develop a favourable situation where slip ceases after a few cycles, an occurrence commonly known as frictional shakedown. Its resemblance to shakedown in plasticity has prompted scholars to apply direct methods, derived from the classical theorems of limit analysis, in order to assess a safe limit to the external loads applied on the system. In circumstances where zones of plastic deformation develop in the material (e.g., because of the large stress concentrations near the sharp edges of a complete contact), it is reasonable to expect an effect of mutual interaction of frictional slip and plastic strains on the load limit below which the global behaviour is non dissipative, i.e., both slip and plastic strains go to zero after some dissipative load cycles. In this paper, shakedown of general two-dimensional discrete systems, involving both friction and plasticity, is discussed and the shakedown limit load is calculated using a non-linear programming algorithm based on the static theorem of limit analysis. An illustrative example related to an elastic-plastic solid containing a frictional crack is provided.

Does Conceptual Understanding of Limit Partially Lead Students to Misconceptions?

NASA Astrophysics Data System (ADS)

Mulyono, B.; Hapizah

2017-09-01

This article talks about the result of preliminary research of my dissertation, which will investigate student’s retention of conceptual understanding. In my preliminary research, I surveyed 73 students of mathematics education program by giving some questions to test their retention of conceptual understanding of limits. Based on the results of analyzing of students’ answers I conclude that most of the students have problems with their retention of conceptual understanding and they also have misconception of limits. The first misconception I identified is that students always used the substitution method to determine a limit of a function at a point, but they did not check whether the function is continue or not at the point. It means that they only use the substitution theorem partially, because they do not consider that the substitution theorem \\mathop{{lim}}\\limits\\text{x\\to \\text{c}}f(x)=f(c) works only if f(x) is defined at χ = c. The other misconception identified is that some students always think there must be available of variables χ in a function to determine the limit of the function. I conjecture that conceptual understanding of limit partially leads students to misconceptions.

Large fluctuations of the macroscopic current in diffusive systems: a numerical test of the additivity principle.

PubMed

Hurtado, Pablo I; Garrido, Pedro L

2010-04-01

Most systems, when pushed out of equilibrium, respond by building up currents of locally conserved observables. Understanding how microscopic dynamics determines the averages and fluctuations of these currents is one of the main open problems in nonequilibrium statistical physics. The additivity principle is a theoretical proposal that allows to compute the current distribution in many one-dimensional nonequilibrium systems. Using simulations, we validate this conjecture in a simple and general model of energy transport, both in the presence of a temperature gradient and in canonical equilibrium. In particular, we show that the current distribution displays a Gaussian regime for small current fluctuations, as prescribed by the central limit theorem, and non-Gaussian (exponential) tails for large current deviations, obeying in all cases the Gallavotti-Cohen fluctuation theorem. In order to facilitate a given current fluctuation, the system adopts a well-defined temperature profile different from that of the steady state and in accordance with the additivity hypothesis predictions. System statistics during a large current fluctuation is independent of the sign of the current, which implies that the optimal profile (as well as higher-order profiles and spatial correlations) are invariant upon current inversion. We also demonstrate that finite-time joint fluctuations of the current and the profile are well described by the additivity functional. These results suggest the additivity hypothesis as a general and powerful tool to compute current distributions in many nonequilibrium systems.

A full-angle Monte-Carlo scattering technique including cumulative and single-event Rutherford scattering in plasmas

NASA Astrophysics Data System (ADS)

Higginson, Drew P.

2017-11-01

We describe and justify a full-angle scattering (FAS) method to faithfully reproduce the accumulated differential angular Rutherford scattering probability distribution function (pdf) of particles in a plasma. The FAS method splits the scattering events into two regions. At small angles it is described by cumulative scattering events resulting, via the central limit theorem, in a Gaussian-like pdf; at larger angles it is described by single-event scatters and retains a pdf that follows the form of the Rutherford differential cross-section. The FAS method is verified using discrete Monte-Carlo scattering simulations run at small timesteps to include each individual scattering event. We identify the FAS regime of interest as where the ratio of temporal/spatial scale-of-interest to slowing-down time/length is from 10-3 to 0.3-0.7; the upper limit corresponds to Coulomb logarithm of 20-2, respectively. Two test problems, high-velocity interpenetrating plasma flows and keV-temperature ion equilibration, are used to highlight systems where including FAS is important to capture relevant physics.

Modular design and implementation of field-programmable-gate-array-based Gaussian noise generator

NASA Astrophysics Data System (ADS)

Li, Yuan-Ping; Lee, Ta-Sung; Hwang, Jeng-Kuang

2016-05-01

The modular design of a Gaussian noise generator (GNG) based on field-programmable gate array (FPGA) technology was studied. A new range reduction architecture was included in a series of elementary function evaluation modules and was integrated into the GNG system. The approximation and quantisation errors for the square root module with a first polynomial approximation were high; therefore, we used the central limit theorem (CLT) to improve the noise quality. This resulted in an output rate of one sample per clock cycle. We subsequently applied Newton's method for the square root module, thus eliminating the need for the use of the CLT because applying the CLT resulted in an output rate of two samples per clock cycle (>200 million samples per second). Two statistical tests confirmed that our GNG is of high quality. Furthermore, the range reduction, which is used to solve a limited interval of the function approximation algorithms of the System Generator platform using Xilinx FPGAs, appeared to have a higher numerical accuracy, was operated at >350 MHz, and can be suitably applied for any function evaluation.

Generalized quantum no-go theorems of pure states

NASA Astrophysics Data System (ADS)

Li, Hui-Ran; Luo, Ming-Xing; Lai, Hong

2018-07-01

Various results of the no-cloning theorem, no-deleting theorem and no-superposing theorem in quantum mechanics have been proved using the superposition principle and the linearity of quantum operations. In this paper, we investigate general transformations forbidden by quantum mechanics in order to unify these theorems. First, we prove that any useful information cannot be created from an unknown pure state which is randomly chosen from a Hilbert space according to the Harr measure. And then, we propose a unified no-go theorem based on a generalized no-superposing result. The new theorem includes the no-cloning theorem, no-anticloning theorem, no-partial-erasure theorem, no-splitting theorem, no-superposing theorem or no-encoding theorem as a special case. Moreover, it implies various new results. Third, we extend the new theorem into another form that includes the no-deleting theorem as a special case.

Semi-classical analysis and pseudo-spectra

NASA Astrophysics Data System (ADS)

Davies, E. B.

We prove an approximate spectral theorem for non-self-adjoint operators and investigate its applications to second-order differential operators in the semi-classical limit. This leads to the construction of a twisted FBI transform. We also investigate the connections between pseudo-spectra and boundary conditions in the semi-classical limit.

Pythagoras Triples Explained via Central Squares

ERIC Educational Resources Information Center

Gomes, Luis Teia

2015-01-01

Very much like today, the Old Babylonians (20th to 16th centuries BC) had the need to understand and use what is now called the Pythagoras' theorem x[superscript 2] + y[superscript 2] = z[superscript 2]. They applied it in very practical problems such as to determine how the height of a cane leaning against a wall changes with its inclination. In…

Four Theorems on the Psychometric Function

PubMed Central

May, Keith A.; Solomon, Joshua A.

2013-01-01

In a 2-alternative forced-choice (2AFC) discrimination task, observers choose which of two stimuli has the higher value. The psychometric function for this task gives the probability of a correct response for a given stimulus difference, . This paper proves four theorems about the psychometric function. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient general expression for the psychometric function. Discrimination data are often fitted with a Weibull function. Theorem 2 proves that the Weibull “slope” parameter, , can be approximated by , where is the of the Weibull function that fits best to the cumulative noise distribution, and depends on the transducer. We derive general expressions for and , from which we derive expressions for specific cases. One case that follows naturally from our general analysis is Pelli's finding that, when , . We also consider two limiting cases. Theorem 3 proves that, as sensitivity improves, 2AFC performance will usually approach that for a linear transducer, whatever the actual transducer; we show that this does not apply at signal levels where the transducer gradient is zero, which explains why it does not apply to contrast detection. Theorem 4 proves that, when the exponent of a power-function transducer approaches zero, 2AFC performance approaches that of a logarithmic transducer. We show that the power-function exponents of 0.4–0.5 fitted to suprathreshold contrast discrimination data are close enough to zero for the fitted psychometric function to be practically indistinguishable from that of a log transducer. Finally, Weibull reflects the shape of the noise distribution, and we used our results to assess the recent claim that internal noise has higher kurtosis than a Gaussian. Our analysis of for contrast discrimination suggests that, if internal noise is stimulus-independent, it has lower kurtosis than a Gaussian. PMID:24124456

Statistical characterization of speckle noise in coherent imaging systems

NASA Astrophysics Data System (ADS)

Yaroslavsky, Leonid; Shefler, A.

2003-05-01

Speckle noise imposes fundamental limitation on image quality in coherent radiation based imaging and optical metrology systems. Speckle noise phenomena are associated with properties of objects to diffusely scatter irradiation and with the fact that in recording the wave field, a number of signal distortions inevitably occur due to technical limitations inherent to hologram sensors. The statistical theory of speckle noise was developed with regard to only limited resolving power of coherent imaging devices. It is valid only asymptotically as much as the central limit theorem of the probability theory can be applied. In applications this assumption is not always applicable. Moreover, in treating speckle noise problem one should also consider other sources of the hologram deterioration. In the paper, statistical properties of speckle due to the limitation of hologram size, dynamic range and hologram signal quantization are studied by Monte-Carlo simulation for holograms recorded in near and far diffraction zones. The simulation experiments have shown that, for limited resolving power of the imaging system, widely accepted opinion that speckle contrast is equal to one holds only for rather severe level of the hologram size limitation. For moderate limitations, speckle contrast changes gradually from zero for no limitation to one for limitation to less than about 20% of hologram size. The results obtained for the limitation of the hologram sensor"s dynamic range and hologram signal quantization reveal that speckle noise due to these hologram signal distortions is not multiplicative and is directly associated with the severity of the limitation and quantization. On the base of the simulation results, analytical models are suggested.

Quantum calculus of classical vortex images, integrable models and quantum states

NASA Astrophysics Data System (ADS)

Pashaev, Oktay K.

2016-10-01

From two circle theorem described in terms of q-periodic functions, in the limit q→1 we have derived the strip theorem and the stream function for N vortex problem. For regular N-vortex polygon we find compact expression for the velocity of uniform rotation and show that it represents a nonlinear oscillator. We describe q-dispersive extensions of the linear and nonlinear Schrodinger equations, as well as the q-semiclassical expansions in terms of Bernoulli and Euler polynomials. Different kind of q-analytic functions are introduced, including the pq-analytic and the golden analytic functions.

Determination of the priority indexes for the oil refinery wastewater treatment process

NASA Astrophysics Data System (ADS)

Chesnokova, M. G.; Myshlyavtsev, A. V.; Kriga, A. S.; Shaporenko, A. P.; Markelov, V. V.

2017-08-01

The wastewater biological treatment intensity and effectiveness are influenced by many factors: temperature, pH, presence and concentration of toxic substances, the biomass concentration et al. Regulation of them allows controlling the biological treatment process. Using the Bayesian theorem the link between changes was determined and the wastewater indexes normative limits exceeding influence for activated sludge characteristics alteration probability was evaluated. The estimation of total, or aposterioric, priority index presence probability, which characterizes the wastewater treatment level, is an important way to use the Bayesian theorem in activated sludge swelling prediction at the oil refinery biological treatment unit.

A Program Certification Assistant Based on Fully Automated Theorem Provers

NASA Technical Reports Server (NTRS)

Denney, Ewen; Fischer, Bernd

2005-01-01

We describe a certification assistant to support formal safety proofs for programs. It is based on a graphical user interface that hides the low-level details of first-order automated theorem provers while supporting limited interactivity: it allows users to customize and control the proof process on a high level, manages the auxiliary artifacts produced during this process, and provides traceability between the proof obligations and the relevant parts of the program. The certification assistant is part of a larger program synthesis system and is intended to support the deployment of automatically generated code in safety-critical applications.

Soft theorems for shift-symmetric cosmologies

NASA Astrophysics Data System (ADS)

Finelli, Bernardo; Goon, Garrett; Pajer, Enrico; Santoni, Luca

2018-03-01

We derive soft theorems for single-clock cosmologies that enjoy a shift symmetry. These so-called consistency conditions arise from a combination of a large diffeomorphism and the internal shift symmetry and fix the squeezed limit of all correlators with a soft scalar mode. As an application, we show that our results reproduce the squeezed bispectrum for ultra-slow-roll inflation, a particular shift-symmetric, nonattractor model which is known to violate Maldacena's consistency relation. Similar results have been previously obtained by Mooij and Palma using background-wave methods. Our results shed new light on the infrared structure of single-clock cosmological spacetimes.

Inferring energy dissipation from violation of the fluctuation-dissipation theorem

NASA Astrophysics Data System (ADS)

Wang, Shou-Wen

2018-05-01

The Harada-Sasa equality elegantly connects the energy dissipation rate of a moving object with its measurable violation of the Fluctuation-Dissipation Theorem (FDT). Although proven for Langevin processes, its validity remains unclear for discrete Markov systems whose forward and backward transition rates respond asymmetrically to external perturbation. A typical example is a motor protein called kinesin. Here we show generally that the FDT violation persists surprisingly in the high-frequency limit due to the asymmetry, resulting in a divergent FDT violation integral and thus a complete breakdown of the Harada-Sasa equality. A renormalized FDT violation integral still well predicts the dissipation rate when each discrete transition produces a small entropy in the environment. Our study also suggests a way to infer this perturbation asymmetry based on the measurable high-frequency-limit FDT violation.

A Perron-Frobenius type of theorem for quantum operations

NASA Astrophysics Data System (ADS)

Lagro, Matthew

Quantum random walks are a generalization of classical Markovian random walks to a quantum mechanical or quantum computing setting. Quantum walks have promising applications but are complicated by quantum decoherence. We prove that the long-time limiting behavior of the class of quantum operations which are the convex combination of norm one operators is governed by the eigenvectors with norm one eigenvalues which are shared by the operators. This class includes all operations formed by a coherent operation with positive probability of orthogonal measurement at each step. We also prove that any operation that has range contained in a low enough dimension subspace of the space of density operators has limiting behavior isomorphic to an associated Markov chain. A particular class of such operations are coherent operations followed by an orthogonal measurement. Applications of the convergence theorems to quantum walks are given.

Math majors' visual proofs in a dynamic environment: the case of limit of a function and the ɛ-δ approach

NASA Astrophysics Data System (ADS)

Caglayan, Günhan

2015-08-01

Despite few limitations, GeoGebra as a dynamic geometry software stood as a powerful instrument in helping university math majors understand, explore, and gain experiences in visualizing the limits of functions and the ɛ - δ formalism. During the process of visualizing a theorem, the order mattered in the sequence of constituents. Students made use of such rich constituents as finger-hand gestures and cursor gestures in an attempt to keep a record of visual demonstration in progress, while being aware of the interrelationships among these constituents and the transformational aspect of the visually proving process. Covariational reasoning along with interval mapping structures proved to be the key constituents in the visualizing and sense-making of a limit theorem using the delta-epsilon formalism. Pedagogical approaches and teaching strategies based on experimental mathematics - mindtool - consituential visual proofs trio would permit students to study, construct, and meaningfully connect the new knowledge to the previously mastered concepts and skills in a manner that would make sense for them.

Fan beam image reconstruction with generalized Fourier slice theorem.

PubMed

Zhao, Shuangren; Yang, Kang; Yang, Kevin

2014-01-01

For parallel beam geometry the Fourier reconstruction works via the Fourier slice theorem (or central slice theorem, projection slice theorem). For fan beam situation, Fourier slice can be extended to a generalized Fourier slice theorem (GFST) for fan-beam image reconstruction. We have briefly introduced this method in a conference. This paper reintroduces the GFST method for fan beam geometry in details. The GFST method can be described as following: the Fourier plane is filled by adding up the contributions from all fanbeam projections individually; thereby the values in the Fourier plane are directly calculated for Cartesian coordinates such avoiding the interpolation from polar to Cartesian coordinates in the Fourier domain; inverse fast Fourier transform is applied to the image in Fourier plane and leads to a reconstructed image in spacial domain. The reconstructed image is compared between the result of the GFST method and the result from the filtered backprojection (FBP) method. The major differences of the GFST and the FBP methods are: (1) The interpolation process are at different data sets. The interpolation of the GFST method is at projection data. The interpolation of the FBP method is at filtered projection data. (2) The filtering process are done in different places. The filtering process of the GFST is at Fourier domain. The filtering process of the FBP method is the ramp filter which is done at projections. The resolution of ramp filter is variable with different location but the filter in the Fourier domain lead to resolution invariable with location. One advantage of the GFST method over the FBP method is in short scan situation, an exact solution can be obtained with the GFST method, but it can not be obtained with the FBP method. The calculation of both the GFST and the FBP methods are at O(N^3), where N is the number of pixel in one dimension.

Are reconstruction filters necessary?

NASA Astrophysics Data System (ADS)

Holst, Gerald C.

2006-05-01

Shannon's sampling theorem (also called the Shannon-Whittaker-Kotel'nikov theorem) was developed for the digitization and reconstruction of sinusoids. Strict adherence is required when frequency preservation is important. Three conditions must be met to satisfy the sampling theorem: (1) The signal must be band-limited, (2) the digitizer must sample the signal at an adequate rate, and (3) a low-pass reconstruction filter must be present. In an imaging system, the signal is band-limited by the optics. For most imaging systems, the signal is not adequately sampled resulting in aliasing. While the aliasing seems excessive mathematically, it does not significantly affect the perceived image. The human visual system detects intensity differences, spatial differences (shapes), and color differences. The eye is less sensitive to frequency effects and therefore sampling artifacts have become quite acceptable. Indeed, we love our television even though it is significantly undersampled. The reconstruction filter, although absolutely essential, is rarely discussed. It converts digital data (which we cannot see) into a viewable analog signal. There are several reconstruction filters: electronic low-pass filters, the display media (monitor, laser printer), and your eye. These are often used in combination to create a perceived continuous image. Each filter modifies the MTF in a unique manner. Therefore image quality and system performance depends upon the reconstruction filter(s) used. The selection depends upon the application.

The Lyapunov-Krasovskii theorem and a sufficient criterion for local stability of isochronal synchronization in networks of delay-coupled oscillators

NASA Astrophysics Data System (ADS)

Grzybowski, J. M. V.; Macau, E. E. N.; Yoneyama, T.

2017-05-01

This paper presents a self-contained framework for the stability assessment of isochronal synchronization in networks of chaotic and limit-cycle oscillators. The results were based on the Lyapunov-Krasovskii theorem and they establish a sufficient condition for local synchronization stability of as a function of the system and network parameters. With this in mind, a network of mutually delay-coupled oscillators subject to direct self-coupling is considered and then the resulting error equations are block-diagonalized for the purpose of studying their stability. These error equations are evaluated by means of analytical stability results derived from the Lyapunov-Krasovskii theorem. The proposed approach is shown to be a feasible option for the investigation of local stability of isochronal synchronization for a variety of oscillators coupled through linear functions of the state variables under a given undirected graph structure. This ultimately permits the systematic identification of stability regions within the high-dimensionality of the network parameter space. Examples of applications of the results to a number of networks of delay-coupled chaotic and limit-cycle oscillators are provided, such as Lorenz, Rössler, Cubic Chua's circuit, Van der Pol oscillator and the Hindmarsh-Rose neuron.

Math Majors' Visual Proofs in a Dynamic Environment: The Case of Limit of a Function and the ?-d Approach

ERIC Educational Resources Information Center

Caglayan, Günhan

2015-01-01

Despite few limitations, GeoGebra as a dynamic geometry software stood as a powerful instrument in helping university math majors understand, explore, and gain experiences in visualizing the limits of functions and the ?-d formalism. During the process of visualizing a theorem, the order mattered in the sequence of constituents. Students made use…

A Limit Theorem on the Cores of Large Standard Exchange Economies

PubMed Central

Brown, Donald J.; Robinson, Abraham

1972-01-01

This note introduces a new mathematical tool, nonstandard analysis, for the analysis of an important class of problems in mathematical economics—the relation between bargaining and the competitive price system. PMID:16591988

Chiral Luttinger liquids and a generalized Luttinger's theorem in fractional quantum Hall edges via finite-entanglement scaling

NASA Astrophysics Data System (ADS)

Varjas, Daniel; Zaletel, Michael; Moore, Joel

2014-03-01

We use bosonic field theories and the infinite system density matrix renormalization group (iDMRG) method to study infinite strips of fractional quantum Hall (FQH) states starting from microscopic Hamiltonians. Finite-entanglement scaling allows us to accurately measure chiral central charge, edge mode exponents and momenta without finite-size errors. We analyze states in the first and second level of the standard hierarchy and compare our results to predictions of the chiral Luttinger liquid (χLL) theory. The results confirm the universality of scaling exponents in chiral edges and demonstrate that renormalization is subject to universal relations in the non-chiral case. We prove a generalized Luttinger's theorem involving all singularities in the momentum-resolved density, which naturally arises when mapping Landau levels on a cylinder to a fermion chain and deepens our understanding of non-Fermi liquids in 1D.

Chiral Luttinger liquids and a generalized Luttinger theorem in fractional quantum Hall edges via finite-entanglement scaling

NASA Astrophysics Data System (ADS)

Varjas, Dániel; Zaletel, Michael P.; Moore, Joel E.

2013-10-01

We use bosonic field theories and the infinite system density matrix renormalization group method to study infinite strips of fractional quantum Hall states starting from microscopic Hamiltonians. Finite-entanglement scaling allows us to accurately measure chiral central charge, edge-mode exponents, and momenta without finite-size errors. We analyze states in the first and second levels of the standard hierarchy and compare our results to predictions of the chiral Luttinger liquid theory. The results confirm the universality of scaling exponents in chiral edges and demonstrate that renormalization is subject to universal relations in the nonchiral case. We prove a generalized Luttinger theorem involving all singularities in the momentum-resolved density, which naturally arises when mapping Landau levels on a cylinder to a fermion chain and deepens our understanding of non-Fermi liquids in one dimension.

The Baetylus Theorem—the central disconnect driving consumer behavior and investment returns in Wearable Technologies

PubMed Central

Levine, James A.

2016-01-01

The Wearable Technology market may increase fivefold by the end of the decade. There is almost no academic investigation as to what drives the investment hypothesis in wearable technologies. This paper seeks to examine this issue from an evidence-based perspective. There is a fundamental disconnect in how consumers view wearable sensors and how companies market them; this is called The Baetylus Theorem where people believe (falsely) that by buying a wearable sensor they will receive health benefit; data suggest that this is not the case. This idea is grounded social constructs, psychological theories and marketing approaches. A marketing proposal that fails to recognize The Baetylus Theorem and how it can be integrated into a business offering has not optimized its competitive advantage. More importantly, consumers should not falsely believe that purchasing a wearable technology, improves health. PMID:27617162

A No-Go Theorem for the Continuum Limit of a Periodic Quantum Spin Chain

NASA Astrophysics Data System (ADS)

Jones, Vaughan F. R.

2018-01-01

We show that the Hilbert space formed from a block spin renormalization construction of a cyclic quantum spin chain (based on the Temperley-Lieb algebra) does not support a chiral conformal field theory whose Hamiltonian generates translation on the circle as a continuous limit of the rotations on the lattice.

Analytic boosted boson discrimination

DOE PAGES

Larkoski, Andrew J.; Moult, Ian; Neill, Duff

2016-05-20

Observables which discriminate boosted topologies from massive QCD jets are of great importance for the success of the jet substructure program at the Large Hadron Collider. Such observables, while both widely and successfully used, have been studied almost exclusively with Monte Carlo simulations. In this paper we present the first all-orders factorization theorem for a two-prong discriminant based on a jet shape variable, D 2, valid for both signal and background jets. Our factorization theorem simultaneously describes the production of both collinear and soft subjets, and we introduce a novel zero-bin procedure to correctly describe the transition region between thesemore » limits. By proving an all orders factorization theorem, we enable a systematically improvable description, and allow for precision comparisons between data, Monte Carlo, and first principles QCD calculations for jet substructure observables. Using our factorization theorem, we present numerical results for the discrimination of a boosted Z boson from massive QCD background jets. We compare our results with Monte Carlo predictions which allows for a detailed understanding of the extent to which these generators accurately describe the formation of two-prong QCD jets, and informs their usage in substructure analyses. In conclusion, our calculation also provides considerable insight into the discrimination power and calculability of jet substructure observables in general.« less

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

Larkoski, Andrew J.; Moult, Ian; Neill, Duff

Observables which discriminate boosted topologies from massive QCD jets are of great importance for the success of the jet substructure program at the Large Hadron Collider. Such observables, while both widely and successfully used, have been studied almost exclusively with Monte Carlo simulations. In this paper we present the first all-orders factorization theorem for a two-prong discriminant based on a jet shape variable, D 2, valid for both signal and background jets. Our factorization theorem simultaneously describes the production of both collinear and soft subjets, and we introduce a novel zero-bin procedure to correctly describe the transition region between thesemore » limits. By proving an all orders factorization theorem, we enable a systematically improvable description, and allow for precision comparisons between data, Monte Carlo, and first principles QCD calculations for jet substructure observables. Using our factorization theorem, we present numerical results for the discrimination of a boosted Z boson from massive QCD background jets. We compare our results with Monte Carlo predictions which allows for a detailed understanding of the extent to which these generators accurately describe the formation of two-prong QCD jets, and informs their usage in substructure analyses. In conclusion, our calculation also provides considerable insight into the discrimination power and calculability of jet substructure observables in general.« less

Noether symmetries and the Swinging Atwood Machine

NASA Astrophysics Data System (ADS)

Moreira, I. C.; Almeida, M. A.

1991-07-01

In this work we apply the Noether theorem with generalised symmetries for discussing the integrability of the Swinging Atwood Machine (SAM) model. We analyse also the limitations of this procedure and compare it with the Yoshida method.

An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations

NASA Astrophysics Data System (ADS)

Drivas, Theodore D.; Eyink, Gregory L.

2017-12-01

We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier-Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have non-negative anomalous entropy production and kinetic energy dissipation, with both vanishing when solutions are above the critical degree of Besov regularity. Stationary, planar shocks in Euclidean space with an ideal-gas equation of state provide simple examples that satisfy the conditions of our theorems and which demonstrate sharpness of our L 3-based conditions. These conditions involve space-time Besov regularity, but we show that they are satisfied by Euler solutions that possess similar space regularity uniformly in time.

The mechanical problems on additive manufacturing of viscoelastic solids with integral conditions on a surface increasing in the growth process

NASA Astrophysics Data System (ADS)

Parshin, D. A.; Manzhirov, A. V.

2018-04-01

Quasistatic mechanical problems on additive manufacturing aging viscoelastic solids are investigated. The processes of piecewise-continuous accretion of such solids are considered. The consideration is carried out in the framework of linear mechanics of growing solids. A theorem about commutativity of the integration over an arbitrary surface increasing in the solid growing process and the time-derived integral operator of viscoelasticity with a limit depending on the solid point is proved. This theorem provides an efficient way to construct on the basis of Saint-Venant principle solutions of nonclassical boundary-value problems for describing the mechanical behaviour of additively formed solids with integral satisfaction of boundary conditions on the surfaces expanding due to the additional material influx to the formed solid. The constructed solutions will retrace the evolution of the stress-strain state of the solids under consideration during and after the processes of their additive formation. An example of applying the proved theorem is given.

Understanding band gaps of solids in generalized Kohn-Sham theory.

PubMed

Perdew, John P; Yang, Weitao; Burke, Kieron; Yang, Zenghui; Gross, Eberhard K U; Scheffler, Matthias; Scuseria, Gustavo E; Henderson, Thomas M; Zhang, Igor Ying; Ruzsinszky, Adrienn; Peng, Haowei; Sun, Jianwei; Trushin, Egor; Görling, Andreas

2017-03-14

The fundamental energy gap of a periodic solid distinguishes insulators from metals and characterizes low-energy single-electron excitations. However, the gap in the band structure of the exact multiplicative Kohn-Sham (KS) potential substantially underestimates the fundamental gap, a major limitation of KS density-functional theory. Here, we give a simple proof of a theorem: In generalized KS theory (GKS), the band gap of an extended system equals the fundamental gap for the approximate functional if the GKS potential operator is continuous and the density change is delocalized when an electron or hole is added. Our theorem explains how GKS band gaps from metageneralized gradient approximations (meta-GGAs) and hybrid functionals can be more realistic than those from GGAs or even from the exact KS potential. The theorem also follows from earlier work. The band edges in the GKS one-electron spectrum are also related to measurable energies. A linear chain of hydrogen molecules, solid aluminum arsenide, and solid argon provide numerical illustrations.

Heat fluctuations of Brownian oscillators in nonstationary processes: Fluctuation theorem and condensation transition

NASA Astrophysics Data System (ADS)

Crisanti, A.; Sarracino, A.; Zannetti, M.

2017-05-01

We study analytically the probability distribution of the heat released by an ensemble of harmonic oscillators to the thermal bath, in the nonequilibrium relaxation process following a temperature quench. We focus on the asymmetry properties of the heat distribution in the nonstationary dynamics, in order to study the forms taken by the fluctuation theorem as the number of degrees of freedom is varied. After analyzing in great detail the cases of one and two oscillators, we consider the limit of a large number of oscillators, where the behavior of fluctuations is enriched by a condensation transition with a nontrivial phase diagram, characterized by reentrant behavior. Numerical simulations confirm our analytical findings. We also discuss and highlight how concepts borrowed from the study of fluctuations in equilibrium under symmetry-breaking conditions [Gaspard, J. Stat. Mech. (2012) P08021, 10.1088/1742-5468/2012/08/P08021] turn out to be quite useful in understanding the deviations from the standard fluctuation theorem.

Entanglement conservation, ER=EPR, and a new classical area theorem for wormholes

DOE PAGES

Remmen, Grant N.; Bao, Ning; Pollack, Jason

2016-07-11

We consider the question of entanglement conservation in the context of the ER=EPR correspondence equating quantum entanglement with wormholes. In quantum mechanics, the entanglement between a system and its complement is conserved under unitary operations that act independently on each; ER=EPR suggests that an analogous statement should hold for wormholes. We accordingly prove a new area theorem in general relativity: for a collection of dynamical wormholes and black holes in a spacetime satisfying the null curvature condition, the maximin area for a subset of the horizons (giving the largest area attained by the minimal cross section of the multi-wormhole throatmore » separating the subset from its complement) is invariant under classical time evolution along the outermost apparent horizons. The evolution can be completely general, including horizon mergers and the addition of classical matter satisfying the null energy condition. In conclusion, this theorem is the gravitational dual of entanglement conservation and thus constitutes an explicit characterization of the ER=EPR duality in the classical limit.« less

Understanding band gaps of solids in generalized Kohn–Sham theory

PubMed Central

Perdew, John P.; Yang, Weitao; Burke, Kieron; Yang, Zenghui; Gross, Eberhard K. U.; Scheffler, Matthias; Scuseria, Gustavo E.; Henderson, Thomas M.; Zhang, Igor Ying; Ruzsinszky, Adrienn; Peng, Haowei; Sun, Jianwei; Trushin, Egor; Görling, Andreas

2017-01-01

The fundamental energy gap of a periodic solid distinguishes insulators from metals and characterizes low-energy single-electron excitations. However, the gap in the band structure of the exact multiplicative Kohn–Sham (KS) potential substantially underestimates the fundamental gap, a major limitation of KS density-functional theory. Here, we give a simple proof of a theorem: In generalized KS theory (GKS), the band gap of an extended system equals the fundamental gap for the approximate functional if the GKS potential operator is continuous and the density change is delocalized when an electron or hole is added. Our theorem explains how GKS band gaps from metageneralized gradient approximations (meta-GGAs) and hybrid functionals can be more realistic than those from GGAs or even from the exact KS potential. The theorem also follows from earlier work. The band edges in the GKS one-electron spectrum are also related to measurable energies. A linear chain of hydrogen molecules, solid aluminum arsenide, and solid argon provide numerical illustrations. PMID:28265085

Entanglement conservation, ER=EPR, and a new classical area theorem for wormholes

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

Remmen, Grant N.; Bao, Ning; Pollack, Jason

We consider the question of entanglement conservation in the context of the ER=EPR correspondence equating quantum entanglement with wormholes. In quantum mechanics, the entanglement between a system and its complement is conserved under unitary operations that act independently on each; ER=EPR suggests that an analogous statement should hold for wormholes. We accordingly prove a new area theorem in general relativity: for a collection of dynamical wormholes and black holes in a spacetime satisfying the null curvature condition, the maximin area for a subset of the horizons (giving the largest area attained by the minimal cross section of the multi-wormhole throatmore » separating the subset from its complement) is invariant under classical time evolution along the outermost apparent horizons. The evolution can be completely general, including horizon mergers and the addition of classical matter satisfying the null energy condition. In conclusion, this theorem is the gravitational dual of entanglement conservation and thus constitutes an explicit characterization of the ER=EPR duality in the classical limit.« less

A full-angle Monte-Carlo scattering technique including cumulative and single-event Rutherford scattering in plasmas [Theory of cumulative large-angle collisions in plasmas

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

Higginson, Drew P.

Here, we describe and justify a full-angle scattering (FAS) method to faithfully reproduce the accumulated differential angular Rutherford scattering probability distribution function (pdf) of particles in a plasma. The FAS method splits the scattering events into two regions. At small angles it is described by cumulative scattering events resulting, via the central limit theorem, in a Gaussian-like pdf; at larger angles it is described by single-event scatters and retains a pdf that follows the form of the Rutherford differential cross-section. The FAS method is verified using discrete Monte-Carlo scattering simulations run at small timesteps to include each individual scattering event.more » We identify the FAS regime of interest as where the ratio of temporal/spatial scale-of-interest to slowing-down time/length is from 10 -3 to 0.3–0.7; the upper limit corresponds to Coulomb logarithm of 20–2, respectively. Two test problems, high-velocity interpenetrating plasma flows and keV-temperature ion equilibration, are used to highlight systems where including FAS is important to capture relevant physics.« less

Statistical genetics and evolution of quantitative traits

NASA Astrophysics Data System (ADS)

Neher, Richard A.; Shraiman, Boris I.

2011-10-01

The distribution and heritability of many traits depends on numerous loci in the genome. In general, the astronomical number of possible genotypes makes the system with large numbers of loci difficult to describe. Multilocus evolution, however, greatly simplifies in the limit of weak selection and frequent recombination. In this limit, populations rapidly reach quasilinkage equilibrium (QLE) in which the dynamics of the full genotype distribution, including correlations between alleles at different loci, can be parametrized by the allele frequencies. This review provides a simplified exposition of the concept and mathematics of QLE which is central to the statistical description of genotypes in sexual populations. Key results of quantitative genetics such as the generalized Fisher’s “fundamental theorem,” along with Wright’s adaptive landscape, are shown to emerge within QLE from the dynamics of the genotype distribution. This is followed by a discussion under what circumstances QLE is applicable, and what the breakdown of QLE implies for the population structure and the dynamics of selection. Understanding the fundamental aspects of multilocus evolution obtained through simplified models may be helpful in providing conceptual and computational tools to address the challenges arising in the studies of complex quantitative phenotypes of practical interest.

Predicting the performance of linear optical detectors in free space laser communication links

NASA Astrophysics Data System (ADS)

Farrell, Thomas C.

2018-05-01

While the fundamental performance limit for optical communications is set by the quantum nature of light, in practical systems background light, dark current, and thermal noise of the electronics also degrade performance. In this paper, we derive a set of equations predicting the performance of PIN diodes and linear mode avalanche photo diodes (APDs) in the presence of such noise sources. Electrons generated by signal, background, and dark current shot noise are well modeled in PIN diodes as Poissonian statistical processes. In APDs, on the other hand, the amplifying effects of the device result in statistics that are distinctly non-Poissonian. Thermal noise is well modeled as Gaussian. In this paper, we appeal to the central limit theorem and treat both the variability of the signal and the sum of noise sources as Gaussian. Comparison against Monte-Carlo simulation of PIN diode performance (where we do model shot noise with draws from a Poissonian distribution) validates the legitimacy of this approximation. On-off keying, M-ary pulse position, and binary differential phase shift keying modulation are modeled. We conclude with examples showing how the equations may be used in a link budget to estimate the performance of optical links using linear receivers.

A full-angle Monte-Carlo scattering technique including cumulative and single-event Rutherford scattering in plasmas [Theory of cumulative large-angle collisions in plasmas

DOE PAGES

Higginson, Drew P.

2017-08-12

Here, we describe and justify a full-angle scattering (FAS) method to faithfully reproduce the accumulated differential angular Rutherford scattering probability distribution function (pdf) of particles in a plasma. The FAS method splits the scattering events into two regions. At small angles it is described by cumulative scattering events resulting, via the central limit theorem, in a Gaussian-like pdf; at larger angles it is described by single-event scatters and retains a pdf that follows the form of the Rutherford differential cross-section. The FAS method is verified using discrete Monte-Carlo scattering simulations run at small timesteps to include each individual scattering event.more » We identify the FAS regime of interest as where the ratio of temporal/spatial scale-of-interest to slowing-down time/length is from 10 -3 to 0.3–0.7; the upper limit corresponds to Coulomb logarithm of 20–2, respectively. Two test problems, high-velocity interpenetrating plasma flows and keV-temperature ion equilibration, are used to highlight systems where including FAS is important to capture relevant physics.« less

TARGETED SEQUENTIAL DESIGN FOR TARGETED LEARNING INFERENCE OF THE OPTIMAL TREATMENT RULE AND ITS MEAN REWARD.

PubMed

Chambaz, Antoine; Zheng, Wenjing; van der Laan, Mark J

2017-01-01

This article studies the targeted sequential inference of an optimal treatment rule (TR) and its mean reward in the non-exceptional case, i.e. , assuming that there is no stratum of the baseline covariates where treatment is neither beneficial nor harmful, and under a companion margin assumption. Our pivotal estimator, whose definition hinges on the targeted minimum loss estimation (TMLE) principle, actually infers the mean reward under the current estimate of the optimal TR. This data-adaptive statistical parameter is worthy of interest on its own. Our main result is a central limit theorem which enables the construction of confidence intervals on both mean rewards under the current estimate of the optimal TR and under the optimal TR itself. The asymptotic variance of the estimator takes the form of the variance of an efficient influence curve at a limiting distribution, allowing to discuss the efficiency of inference. As a by product, we also derive confidence intervals on two cumulated pseudo-regrets, a key notion in the study of bandits problems. A simulation study illustrates the procedure. One of the corner-stones of the theoretical study is a new maximal inequality for martingales with respect to the uniform entropy integral.

Quality correction factors of composite IMRT beam deliveries: theoretical considerations.

PubMed

Bouchard, Hugo

2012-11-01

In the scope of intensity modulated radiation therapy (IMRT) dosimetry using ionization chambers, quality correction factors of plan-class-specific reference (PCSR) fields are theoretically investigated. The symmetry of the problem is studied to provide recommendable criteria for composite beam deliveries where correction factors are minimal and also to establish a theoretical limit for PCSR delivery k(Q) factors. The concept of virtual symmetric collapsed (VSC) beam, being associated to a given modulated composite delivery, is defined in the scope of this investigation. Under symmetrical measurement conditions, any composite delivery has the property of having a k(Q) factor identical to its associated VSC beam. Using this concept of VSC, a fundamental property of IMRT k(Q) factors is demonstrated in the form of a theorem. The sensitivity to the conditions required by the theorem is thoroughly examined. The theorem states that if a composite modulated beam delivery produces a uniform dose distribution in a volume V(cyl) which is symmetric with the cylindrical delivery and all beams fulfills two conditions in V(cyl): (1) the dose modulation function is unchanged along the beam axis, and (2) the dose gradient in the beam direction is constant for a given lateral position; then its associated VSC beam produces no lateral dose gradient in V(cyl), no matter what beam modulation or gantry angles are being used. The examination of the conditions required by the theorem lead to the following results. The effect of the depth-dose gradient not being perfectly constant with depth on the VSC beam lateral dose gradient is found negligible. The effect of the dose modulation function being degraded with depth on the VSC beam lateral dose gradient is found to be only related to scatter and beam hardening, as the theorem holds also for diverging beams. The use of the symmetry of the problem in the present paper leads to a valuable theorem showing that k(Q) factors of composite IMRT beam deliveries are close to unity under specific conditions. The theoretical limit k(Q(pcsr),Q(msr) ) (f(pcsr),f(msr) )=1 is determined based on the property of PCSR deliveries to provide a uniform dose in the target volume. The present approach explains recent experimental observations and proposes ideal conditions for IMRT reference dosimetry. The result of this study could potentially serve as a theoretical basis for reference dosimetry of composite IMRT beam deliveries or for routine IMRT quality assurance.

Four theorems on the psychometric function.

PubMed

May, Keith A; Solomon, Joshua A

2013-01-01

In a 2-alternative forced-choice (2AFC) discrimination task, observers choose which of two stimuli has the higher value. The psychometric function for this task gives the probability of a correct response for a given stimulus difference, Δx. This paper proves four theorems about the psychometric function. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient general expression for the psychometric function. Discrimination data are often fitted with a Weibull function. Theorem 2 proves that the Weibull "slope" parameter, β, can be approximated by β(Noise) x β(Transducer), where β(Noise) is the β of the Weibull function that fits best to the cumulative noise distribution, and β(Transducer) depends on the transducer. We derive general expressions for β(Noise) and β(Transducer), from which we derive expressions for specific cases. One case that follows naturally from our general analysis is Pelli's finding that, when d' ∝ (Δx)(b), β ≈ β(Noise) x b. We also consider two limiting cases. Theorem 3 proves that, as sensitivity improves, 2AFC performance will usually approach that for a linear transducer, whatever the actual transducer; we show that this does not apply at signal levels where the transducer gradient is zero, which explains why it does not apply to contrast detection. Theorem 4 proves that, when the exponent of a power-function transducer approaches zero, 2AFC performance approaches that of a logarithmic transducer. We show that the power-function exponents of 0.4-0.5 fitted to suprathreshold contrast discrimination data are close enough to zero for the fitted psychometric function to be practically indistinguishable from that of a log transducer. Finally, Weibull β reflects the shape of the noise distribution, and we used our results to assess the recent claim that internal noise has higher kurtosis than a Gaussian. Our analysis of β for contrast discrimination suggests that, if internal noise is stimulus-independent, it has lower kurtosis than a Gaussian.

The Great Emch Closure Theorem and a combinatorial proof of Poncelet's Theorem

NASA Astrophysics Data System (ADS)

Avksentyev, E. A.

2015-11-01

The relations between the classical closure theorems (Poncelet's, Steiner's, Emch's, and the zigzag theorems) and some of their generalizations are discussed. It is known that Emch's Theorem is the most general of these, while the others follow as special cases. A generalization of Emch's Theorem to pencils of circles is proved, which (by analogy with the Great Poncelet Theorem) can be called the Great Emch Theorem. It is shown that the Great Emch and Great Poncelet Theorems are equivalent and can be derived one from the other using elementary geometry, and also that both hold in the Lobachevsky plane as well. A new closure theorem is also obtained, in which the construction of closure is slightly more involved: closure occurs on a variable circle which is tangent to a fixed pair of circles. In conclusion, a combinatorial proof of Poncelet's Theorem is given, which deduces the closure principle for an arbitrary number of steps from the principle for three steps using combinatorics and number theory. Bibliography: 20 titles.

Feynman amplitudes and limits of heights

NASA Astrophysics Data System (ADS)

Amini, O.; Bloch, S. J.; Burgos Gil, J. I.; Fresán, J.

2016-10-01

We investigate from a mathematical perspective how Feynman amplitudes appear in the low-energy limit of string amplitudes. In this paper, we prove the convergence of the integrands. We derive this from results describing the asymptotic behaviour of the height pairing between degree-zero divisors, as a family of curves degenerates. These are obtained by means of the nilpotent orbit theorem in Hodge theory.

Central Configurations of the Curved N-Body Problem

NASA Astrophysics Data System (ADS)

Diacu, Florin; Stoica, Cristina; Zhu, Shuqiang

2018-06-01

We consider the N-body problem of celestial mechanics in spaces of nonzero constant curvature. Using the concept of effective potential, we define the moment of inertia for systems moving on spheres and hyperbolic spheres and show that we can recover the classical definition in the Euclidean case. After proving some criteria for the existence of relative equilibria, we find a natural way to define the concept of central configuration in curved spaces using the moment of inertia and show that our definition is formally similar to the one that governs the classical problem. We prove that, for any given point masses on spheres and hyperbolic spheres, central configurations always exist. We end with results concerning the number of central configurations that lie on the same geodesic, thus extending the celebrated theorem of Moulton to hyperbolic spheres and pointing out that it has no straightforward generalization to spheres, where the count gets complicated even for two bodies.

A new test of multivariate nonlinear causality

PubMed Central

Bai, Zhidong; Jiang, Dandan; Lv, Zhihui; Wong, Wing-Keung; Zheng, Shurong

2018-01-01

The multivariate nonlinear Granger causality developed by Bai et al. (2010) (Mathematics and Computers in simulation. 2010; 81: 5-17) plays an important role in detecting the dynamic interrelationships between two groups of variables. Following the idea of Hiemstra-Jones (HJ) test proposed by Hiemstra and Jones (1994) (Journal of Finance. 1994; 49(5): 1639-1664), they attempt to establish a central limit theorem (CLT) of their test statistic by applying the asymptotical property of multivariate U-statistic. However, Bai et al. (2016) (2016; arXiv: 1701.03992) revisit the HJ test and find that the test statistic given by HJ is NOT a function of U-statistics which implies that the CLT neither proposed by Hiemstra and Jones (1994) nor the one extended by Bai et al. (2010) is valid for statistical inference. In this paper, we re-estimate the probabilities and reestablish the CLT of the new test statistic. Numerical simulation shows that our new estimates are consistent and our new test performs decent size and power. PMID:29304085

A new test of multivariate nonlinear causality.

PubMed

Bai, Zhidong; Hui, Yongchang; Jiang, Dandan; Lv, Zhihui; Wong, Wing-Keung; Zheng, Shurong

2018-01-01

The multivariate nonlinear Granger causality developed by Bai et al. (2010) (Mathematics and Computers in simulation. 2010; 81: 5-17) plays an important role in detecting the dynamic interrelationships between two groups of variables. Following the idea of Hiemstra-Jones (HJ) test proposed by Hiemstra and Jones (1994) (Journal of Finance. 1994; 49(5): 1639-1664), they attempt to establish a central limit theorem (CLT) of their test statistic by applying the asymptotical property of multivariate U-statistic. However, Bai et al. (2016) (2016; arXiv: 1701.03992) revisit the HJ test and find that the test statistic given by HJ is NOT a function of U-statistics which implies that the CLT neither proposed by Hiemstra and Jones (1994) nor the one extended by Bai et al. (2010) is valid for statistical inference. In this paper, we re-estimate the probabilities and reestablish the CLT of the new test statistic. Numerical simulation shows that our new estimates are consistent and our new test performs decent size and power.

Determining the speed of sound in the air by sound wave interference

NASA Astrophysics Data System (ADS)

Silva, Abel A.

2017-07-01

Mechanical waves propagate through material media. Sound is an example of a mechanical wave. In fluids like air, sound waves propagate through successive longitudinal perturbations of compression and decompression. Audible sound frequencies for human ears range from 20 to 20 000 Hz. In this study, the speed of sound v in the air is determined using the identification of maxima of interference from two synchronous waves at frequency f. The values of v were correct to 0 °C. The experimental average value of {\\bar{ν }}\\exp =336 +/- 4 {{m}} {{{s}}}-1 was found. It is 1.5% larger than the reference value. The standard deviation of 4 m s-1 (1.2% of {\\bar{ν }}\\exp ) is an improved value by the use of the concept of the central limit theorem. The proposed procedure to determine the speed of sound in the air aims to be an academic activity for physics classes of scientific and technological courses in college.

Log-normal distribution from a process that is not multiplicative but is additive.

PubMed

Mouri, Hideaki

2013-10-01

The central limit theorem ensures that a sum of random variables tends to a Gaussian distribution as their total number tends to infinity. However, for a class of positive random variables, we find that the sum tends faster to a log-normal distribution. Although the sum tends eventually to a Gaussian distribution, the distribution of the sum is always close to a log-normal distribution rather than to any Gaussian distribution if the summands are numerous enough. This is in contrast to the current consensus that any log-normal distribution is due to a product of random variables, i.e., a multiplicative process, or equivalently to nonlinearity of the system. In fact, the log-normal distribution is also observable for a sum, i.e., an additive process that is typical of linear systems. We show conditions for such a sum, an analytical example, and an application to random scalar fields such as those of turbulence.

Computing and analyzing the sensitivity of MLP due to the errors of the i.i.d. inputs and weights based on CLT.

PubMed

Yang, Sheng-Sung; Ho, Chia-Lu; Siu, Sammy

2010-12-01

In this paper, we propose an algorithm based on the central limit theorem to compute the sensitivity of the multilayer perceptron (MLP) due to the errors of the inputs and weights. For simplicity and practicality, all inputs and weights studied here are independently identically distributed (i.i.d.). The theoretical results derived from the proposed algorithm show that the sensitivity of the MLP is affected by the number of layers and the number of neurons adopted in each layer. To prove the reliability of the proposed algorithm, some experimental results of the sensitivity are also presented, and they match the theoretical ones. The good agreement between the theoretical results and the experimental results verifies the reliability and feasibility of the proposed algorithm. Furthermore, the proposed algorithm can also be applied to compute precisely the sensitivity of the MLP with any available activation functions and any types of i.i.d. inputs and weights.

The Existence of Periodic Orbits and Invariant Tori for Some 3-Dimensional Quadratic Systems

PubMed Central

Jiang, Yanan; Han, Maoan; Xiao, Dongmei

2014-01-01

We use the normal form theory, averaging method, and integral manifold theorem to study the existence of limit cycles in Lotka-Volterra systems and the existence of invariant tori in quadratic systems in ℝ3. PMID:24982980

Unified quantum no-go theorems and transforming of quantum pure states in a restricted set

NASA Astrophysics Data System (ADS)

Luo, Ming-Xing; Li, Hui-Ran; Lai, Hong; Wang, Xiaojun

2017-12-01

The linear superposition principle in quantum mechanics is essential for several no-go theorems such as the no-cloning theorem, the no-deleting theorem and the no-superposing theorem. In this paper, we investigate general quantum transformations forbidden or permitted by the superposition principle for various goals. First, we prove a no-encoding theorem that forbids linearly superposing of an unknown pure state and a fixed pure state in Hilbert space of a finite dimension. The new theorem is further extended for multiple copies of an unknown state as input states. These generalized results of the no-encoding theorem include the no-cloning theorem, the no-deleting theorem and the no-superposing theorem as special cases. Second, we provide a unified scheme for presenting perfect and imperfect quantum tasks (cloning and deleting) in a one-shot manner. This scheme may lead to fruitful results that are completely characterized with the linear independence of the representative vectors of input pure states. The upper bounds of the efficiency are also proved. Third, we generalize a recent superposing scheme of unknown states with a fixed overlap into new schemes when multiple copies of an unknown state are as input states.

The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a Nikishin system

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

Rakhmanov, E A; Suetin, S P

2013-09-30

The distribution of the zeros of the Hermite-Padé polynomials of the first kind for a pair of functions with an arbitrary even number of common branch points lying on the real axis is investigated under the assumption that this pair of functions forms a generalized complex Nikishin system. It is proved (Theorem 1) that the zeros have a limiting distribution, which coincides with the equilibrium measure of a certain compact set having the S-property in a harmonic external field. The existence problem for S-compact sets is solved in Theorem 2. The main idea of the proof of Theorem 1 consists in replacing a vector equilibrium problem in potentialmore » theory by a scalar problem with an external field and then using the general Gonchar-Rakhmanov method, which was worked out in the solution of the '1/9'-conjecture. The relation of the result obtained here to some results and conjectures due to Nuttall is discussed. Bibliography: 51 titles.« less

Analogues of Chernoff's theorem and the Lie-Trotter theorem

NASA Astrophysics Data System (ADS)

Neklyudov, Alexander Yu

2009-10-01

This paper is concerned with the abstract Cauchy problem \\dot x=\\mathrm{A}x, x(0)=x_0\\in\\mathscr{D}(\\mathrm{A}), where \\mathrm{A} is a densely defined linear operator on a Banach space \\mathbf X. It is proved that a solution x(\\,\\cdot\\,) of this problem can be represented as the weak limit \\lim_{n\\to\\infty}\\{\\mathrm F(t/n)^nx_0\\}, where the function \\mathrm F\\colon \\lbrack 0,\\infty)\\mapsto\\mathscr L(\\mathrm X) satisfies the equality \\mathrm F'(0)y=\\mathrm{A}y, y\\in\\mathscr{D}(\\mathrm{A}), for a natural class of operators. As distinct from Chernoff's theorem, the existence of a global solution to the Cauchy problem is not assumed. Based on this result, necessary and sufficient conditions are found for the linear operator \\mathrm{C} to be closable and for its closure to be the generator of a C_0-semigroup. Also, we obtain new criteria for the sum of two generators of C_0-semigroups to be the generator of a C_0-semigroup and for the Lie-Trotter formula to hold. Bibliography: 13 titles.

Prediction of HR/BP response to the spontaneous breathing trial by fluctuation dissipation theory

NASA Astrophysics Data System (ADS)

Chen, Man

2014-03-01

We applied the non-equilibrium fluctuation dissipation theorem to predict how critically-ill patients respond to treatment, based on both heart rate data and blood pressure data collected by standard hospital monitoring devices. The non-equilibrium fluctuation dissipation theorem relates the response of a system to a perturbation to the fluctuations in the stationary state of the system. It is shown that the response of patients to a standard procedure performed on patients, the spontaneous breathing trial (SBT), can be predicted by the non-equilibrium fluctuation dissipation approach. We classify patients into different groups according to the patients' characteristics. For each patient group, we extend the fluctuation dissipation theorem to predict interactions between blood pressure and beat-to-beat dynamics of heart rate in response to a perturbation (SBT), We also extract the form of the perturbation function directly from the physiological data, which may help to reduce the prediction error. We note this method is not limited to the analysis of the heart rate dynamics, but also can be applied to analyze the response of other physiological signals to other clinical interventions.

Implications of the Corotation Theorem on the MRI in Axial Symmetry

NASA Astrophysics Data System (ADS)

Montani, G.; Cianfrani, F.; Pugliese, D.

2016-08-01

We analyze the linear stability of an axially symmetric ideal plasma disk, embedded in a magnetic field and endowed with a differential rotation. This study is performed by adopting the magnetic flux function as the fundamental dynamical variable, in order to outline the role played by the corotation theorem on the linear mode structure. Using some specific assumptions (e.g., plasma incompressibility and propagation of the perturbations along the background magnetic field), we select the Alfvénic nature of the magnetorotational instability, and, in the geometric optics limit, we determine the dispersion relation describing the linear spectrum. We show how the implementation of the corotation theorem (valid for the background configuration) on the linear dynamics produces the cancellation of the vertical derivative of the disk angular velocity (we check such a feature also in the standard vector formalism to facilitate comparison with previous literature, in both the axisymmetric and three-dimensional cases). As a result, we clarify that the unstable modes have, for a stratified disk, the same morphology, proper of a thin-disk profile, and the z-dependence has a simple parametric role.

Cook-Levin Theorem Algorithmic-Reducibility/Completeness = Wilson Renormalization-(Semi)-Group Fixed-Points; ``Noise''-Induced Phase-Transitions (NITs) to Accelerate Algorithmics (``NIT-Picking'') REPLACING CRUTCHES!!!: Models: Turing-machine, finite-state-models, finite-automata

NASA Astrophysics Data System (ADS)

Young, Frederic; Siegel, Edward

Cook-Levin theorem theorem algorithmic computational-complexity(C-C) algorithmic-equivalence reducibility/completeness equivalence to renormalization-(semi)-group phase-transitions critical-phenomena statistical-physics universality-classes fixed-points, is exploited via Siegel FUZZYICS =CATEGORYICS = ANALOGYICS =PRAGMATYICS/CATEGORY-SEMANTICS ONTOLOGY COGNITION ANALYTICS-Aristotle ``square-of-opposition'' tabular list-format truth-table matrix analytics predicts and implements ''noise''-induced phase-transitions (NITs) to accelerate versus to decelerate Harel [Algorithmics (1987)]-Sipser[Intro.Thy. Computation(`97)] algorithmic C-C: ''NIT-picking''(!!!), to optimize optimization-problems optimally(OOPO). Versus iso-''noise'' power-spectrum quantitative-only amplitude/magnitude-only variation stochastic-resonance, ''NIT-picking'' is ''noise'' power-spectrum QUALitative-type variation via quantitative critical-exponents variation. Computer-''science''/SEANCE algorithmic C-C models: Turing-machine, finite-state-models, finite-automata,..., discrete-maths graph-theory equivalence to physics Feynman-diagrams are identified as early-days once-workable valid but limiting IMPEDING CRUTCHES(!!!), ONLY IMPEDE latter-days new-insights!!!

Discontinuous Galerkin Methods for NonLinear Differential Systems

NASA Technical Reports Server (NTRS)

Barth, Timothy; Mansour, Nagi (Technical Monitor)

2001-01-01

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

Sufficient conditions for uniqueness of the weak value

NASA Astrophysics Data System (ADS)

Dressel, J.; Jordan, A. N.

2012-01-01

We review and clarify the sufficient conditions for uniquely defining the generalized weak value as the weak limit of a conditioned average using the contextual values formalism introduced in Dressel, Agarwal and Jordan (2010 Phys. Rev. Lett. 104 240401). We also respond to criticism of our work by Parrott (arXiv:1105.4188v1) concerning a proposed counter-example to the uniqueness of the definition of the generalized weak value. The counter-example does not satisfy our prescription in the case of an underspecified measurement context. We show that when the contextual values formalism is properly applied to this example, a natural interpretation of the measurement emerges and the unique definition in the weak limit holds. We also prove a theorem regarding the uniqueness of the definition under our sufficient conditions for the general case. Finally, a second proposed counter-example by Parrott (arXiv:1105.4188v6) is shown not to satisfy the sufficiency conditions for the provided theorem.

Einstein-Podolsky-Rosen paradox implies a minimum achievable temperature

NASA Astrophysics Data System (ADS)

Rogers, David M.

2017-01-01

This work examines the thermodynamic consequences of the repeated partial projection model for coupling a quantum system to an arbitrary series of environments under feedback control. This paper provides observational definitions of heat and work that can be realized in current laboratory setups. In contrast to other definitions, it uses only properties of the environment and the measurement outcomes, avoiding references to the "measurement" of the central system's state in any basis. These definitions are consistent with the usual laws of thermodynamics at all temperatures, while never requiring complete projective measurement of the entire system. It is shown that the back action of measurement must be counted as work rather than heat to satisfy the second law. Comparisons are made to quantum jump (unravelling) and transition-probability based definitions, many of which appear as particular limits of the present model. These limits show that our total entropy production is a lower bound on traditional definitions of heat that trace out the measurement device. Examining the master equation approximation to the process at finite measurement rates, we show that most interactions with the environment make the system unable to reach absolute zero. We give an explicit formula for the minimum temperature achievable in repeatedly measured quantum systems. The phenomenon of minimum temperature offers an explanation of recent experiments aimed at testing fluctuation theorems in the quantum realm and places a fundamental purity limit on quantum computers.

The Study of Spherical Cores with a Toroidal Magnetic Field Configuration

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

Gholipour, Mahmoud

Observational studies of the magnetic fields in molecular clouds have significantly improved the theoretical models developed for the structure and evolution of dense clouds and for the star formation process as well. The recent observational analyses on some cores indicate that there is a power-law relationship between magnetic field and density in the molecular clouds. In this study, we consider the stability of spherical cores with a toroidal magnetic field configuration in the molecular clouds. For this purpose, we model a spherical core that is in magnetostatic equilibrium. Herein, we propose an equation of density structure, which is a modifiedmore » form of the isothermal Lane–Emden equation in the presence of the toroidal magnetic field. The proposed equation describes the effect of the toroidal magnetic field on the cloud structure and the mass cloud. Furthermore, we found an upper limit for this configuration of magnetic field in the molecular clouds. Then, the virial theorem is used to consider the cloud evolution leading to an equation in order to obtain the lower limit of the field strength in the molecular cloud. However, the results show that the field strength of the toroidal configuration has an important effect on the cloud structure, whose upper limit is related to the central density and field gradient. The obtained results address some regions of clouds where the cloud decomposition or star formation can be seen.« less

A Decomposition Theorem for Finite Automata.

ERIC Educational Resources Information Center

Santa Coloma, Teresa L.; Tucci, Ralph P.

1990-01-01

Described is automata theory which is a branch of theoretical computer science. A decomposition theorem is presented that is easier than the Krohn-Rhodes theorem. Included are the definitions, the theorem, and a proof. (KR)

Quality correction factors of composite IMRT beam deliveries: Theoretical considerations

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

Bouchard, Hugo

2012-11-15

Purpose: In the scope of intensity modulated radiation therapy (IMRT) dosimetry using ionization chambers, quality correction factors of plan-class-specific reference (PCSR) fields are theoretically investigated. The symmetry of the problem is studied to provide recommendable criteria for composite beam deliveries where correction factors are minimal and also to establish a theoretical limit for PCSR delivery k{sub Q} factors. Methods: The concept of virtual symmetric collapsed (VSC) beam, being associated to a given modulated composite delivery, is defined in the scope of this investigation. Under symmetrical measurement conditions, any composite delivery has the property of having a k{sub Q} factor identicalmore » to its associated VSC beam. Using this concept of VSC, a fundamental property of IMRT k{sub Q} factors is demonstrated in the form of a theorem. The sensitivity to the conditions required by the theorem is thoroughly examined. Results: The theorem states that if a composite modulated beam delivery produces a uniform dose distribution in a volume V{sub cyl} which is symmetric with the cylindrical delivery and all beams fulfills two conditions in V{sub cyl}: (1) the dose modulation function is unchanged along the beam axis, and (2) the dose gradient in the beam direction is constant for a given lateral position; then its associated VSC beam produces no lateral dose gradient in V{sub cyl}, no matter what beam modulation or gantry angles are being used. The examination of the conditions required by the theorem lead to the following results. The effect of the depth-dose gradient not being perfectly constant with depth on the VSC beam lateral dose gradient is found negligible. The effect of the dose modulation function being degraded with depth on the VSC beam lateral dose gradient is found to be only related to scatter and beam hardening, as the theorem holds also for diverging beams. Conclusions: The use of the symmetry of the problem in the present paper leads to a valuable theorem showing that k{sub Q} factors of composite IMRT beam deliveries are close to unity under specific conditions. The theoretical limit k{sub Q{sub p{sub c{sub s{sub r,Q{sub m{sub s{sub r}{sup f{sub p}{sub c}{sub s}{sub r},f{sub m}{sub s}{sub r}}}}}}}}}=1 is determined based on the property of PCSR deliveries to provide a uniform dose in the target volume. The present approach explains recent experimental observations and proposes ideal conditions for IMRT reference dosimetry. The result of this study could potentially serve as a theoretical basis for reference dosimetry of composite IMRT beam deliveries or for routine IMRT quality assurance.« less

Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

NASA Technical Reports Server (NTRS)

Mostrel, M. M.

1988-01-01

New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

Ground-state energies of the nonlinear sigma model and the Heisenberg spin chains

NASA Technical Reports Server (NTRS)

Zhang, Shoucheng; Schulz, H. J.; Ziman, Timothy

1989-01-01

A theorem on the O(3) nonlinear sigma model with the topological theta term is proved, which states that the ground-state energy at theta = pi is always higher than the ground-state energy at theta = 0, for the same value of the coupling constant g. Provided that the nonlinear sigma model gives the correct description for the Heisenberg spin chains in the large-s limit, this theorem makes a definite prediction relating the ground-state energies of the half-integer and the integer spin chains. The ground-state energies obtained from the exact Bethe ansatz solution for the spin-1/2 chain and the numerical diagonalization on the spin-1, spin-3/2, and spin-2 chains support this prediction.

Bandwidth efficient coding: Theoretical limits and real achievements. Error control techniques for satellite and space communications

NASA Technical Reports Server (NTRS)

Costello, Daniel J., Jr.; Courturier, Servanne; Levy, Yannick; Mills, Diane G.; Perez, Lance C.; Wang, Fu-Quan

1993-01-01

In his seminal 1948 paper 'The Mathematical Theory of Communication,' Claude E. Shannon derived the 'channel coding theorem' which has an explicit upper bound, called the channel capacity, on the rate at which 'information' could be transmitted reliably on a given communication channel. Shannon's result was an existence theorem and did not give specific codes to achieve the bound. Some skeptics have claimed that the dramatic performance improvements predicted by Shannon are not achievable in practice. The advances made in the area of coded modulation in the past decade have made communications engineers optimistic about the possibility of achieving or at least coming close to channel capacity. Here we consider the possibility in the light of current research results.

Index theorem and universality properties of the low-lying eigenvalues of improved staggered quarks.

PubMed

Follana, E; Hart, A; Davies, C T H

2004-12-10

We study various improved staggered quark Dirac operators on quenched gluon backgrounds in lattice QCD generated using a Symanzik-improved gluon action. We find a clear separation of the spectrum into would-be zero modes and others. The number of would-be zero modes depends on the topological charge as expected from the index theorem, and their chirality expectation value is large ( approximately 0.7). The remaining modes have low chirality and show clear signs of clustering into quartets and approaching the random matrix theory predictions for all topological charge sectors. We conclude that improvement of the fermionic and gauge actions moves the staggered quarks closer to the continuum limit where they respond correctly to QCD topology.

Slowly changing potential problems in Quantum Mechanics: Adiabatic theorems, ergodic theorems, and scattering

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

Fishman, S., E-mail: fishman@physics.technion.ac.il; Soffer, A., E-mail: soffer@math.rutgers.edu

2016-07-15

We employ the recently developed multi-time scale averaging method to study the large time behavior of slowly changing (in time) Hamiltonians. We treat some known cases in a new way, such as the Zener problem, and we give another proof of the adiabatic theorem in the gapless case. We prove a new uniform ergodic theorem for slowly changing unitary operators. This theorem is then used to derive the adiabatic theorem, do the scattering theory for such Hamiltonians, and prove some classical propagation estimates and asymptotic completeness.

The Non-Signalling theorem in generalizations of Bell's theorem

NASA Astrophysics Data System (ADS)

Walleczek, J.; Grössing, G.

2014-04-01

Does "epistemic non-signalling" ensure the peaceful coexistence of special relativity and quantum nonlocality? The possibility of an affirmative answer is of great importance to deterministic approaches to quantum mechanics given recent developments towards generalizations of Bell's theorem. By generalizations of Bell's theorem we here mean efforts that seek to demonstrate the impossibility of any deterministic theories to obey the predictions of Bell's theorem, including not only local hidden-variables theories (LHVTs) but, critically, of nonlocal hidden-variables theories (NHVTs) also, such as de Broglie-Bohm theory. Naturally, in light of the well-established experimental findings from quantum physics, whether or not a deterministic approach to quantum mechanics, including an emergent quantum mechanics, is logically possible, depends on compatibility with the predictions of Bell's theorem. With respect to deterministic NHVTs, recent attempts to generalize Bell's theorem have claimed the impossibility of any such approaches to quantum mechanics. The present work offers arguments showing why such efforts towards generalization may fall short of their stated goal. In particular, we challenge the validity of the use of the non-signalling theorem as a conclusive argument in favor of the existence of free randomness, and therefore reject the use of the non-signalling theorem as an argument against the logical possibility of deterministic approaches. We here offer two distinct counter-arguments in support of the possibility of deterministic NHVTs: one argument exposes the circularity of the reasoning which is employed in recent claims, and a second argument is based on the inconclusive metaphysical status of the non-signalling theorem itself. We proceed by presenting an entirely informal treatment of key physical and metaphysical assumptions, and of their interrelationship, in attempts seeking to generalize Bell's theorem on the basis of an ontic, foundational interpretation of the non-signalling theorem. We here argue that the non-signalling theorem must instead be viewed as an epistemic, operational theorem i.e. one that refers exclusively to what epistemic agents can, or rather cannot, do. That is, we emphasize that the non-signalling theorem is a theorem about the operational inability of epistemic agents to signal information. In other words, as a proper principle, the non-signalling theorem may only be employed as an epistemic, phenomenological, or operational principle. Critically, our argument emphasizes that the non-signalling principle must not be used as an ontic principle about physical reality as such, i.e. as a theorem about the nature of physical reality independently of epistemic agents e.g. human observers. One major reason in favor of our conclusion is that any definition of signalling or of non-signalling invariably requires a reference to epistemic agents, and what these agents can actually measure and report. Otherwise, the non-signalling theorem would equal a general "no-influence" theorem. In conclusion, under the assumption that the non-signalling theorem is epistemic (i.e. "epistemic non-signalling"), the search for deterministic approaches to quantum mechanics, including NHVTs and an emergent quantum mechanics, continues to be a viable research program towards disclosing the foundations of physical reality at its smallest dimensions.

Consistency of the adiabatic theorem.

PubMed

Amin, M H S

2009-06-05

The adiabatic theorem provides the basis for the adiabatic model of quantum computation. Recently the conditions required for the adiabatic theorem to hold have become a subject of some controversy. Here we show that the reported violations of the adiabatic theorem all arise from resonant transitions between energy levels. In the absence of fast driven oscillations the traditional adiabatic theorem holds. Implications for adiabatic quantum computation are discussed.

Extended optical theorem in isotropic solids and its application to the elastic radiation force

NASA Astrophysics Data System (ADS)

Leão-Neto, J. P.; Lopes, J. H.; Silva, G. T.

2017-04-01

In this article, we derive the extended optical theorem for the elastic-wave scattering by a spherical inclusion (with and without absorption) in a solid matrix. This theorem expresses the extinction cross-section, i.e., the time-averaged power extracted from the incoming beam per its intensity, regarding the partial-wave expansion coefficients of the incident and scattered waves. We also establish the connection between the optical theorem and the elastic radiation force by a plane wave in a linear and isotropic solid. We obtain the absorption, scattering, and extinction efficiencies (the corresponding power per characteristic incident intensity per sphere cross-section area) for a plane wave and a spherically focused beam. We discuss to which extent the radiation force theory for plane waves can be used to the focused beam case. Considering an iron sphere embedded in an aluminum matrix, we numerically compute the scattering and elastic radiation force efficiencies. The radiation force on a stainless steel sphere embedded in a tissue-like medium (soft solid) is also computed. In this case, resonances are observed in the force as a function of the sphere size parameter (the wavenumber times the sphere radius). Remarkably, the relative difference between our findings and previous lossless liquid models is about 100% in the long-wavelength limit. Regarding some applications, the obtained results have a direct impact on ultrasound-based elastography techniques and ultrasonic nondestructive testing, as well as implantable devices activated by ultrasound.

Optimal no-go theorem on hidden-variable predictions of effect expectations

NASA Astrophysics Data System (ADS)

Blass, Andreas; Gurevich, Yuri

2018-03-01

No-go theorems prove that, under reasonable assumptions, classical hidden-variable theories cannot reproduce the predictions of quantum mechanics. Traditional no-go theorems proved that hidden-variable theories cannot predict correctly the values of observables. Recent expectation no-go theorems prove that hidden-variable theories cannot predict the expectations of observables. We prove the strongest expectation-focused no-go theorem to date. It is optimal in the sense that the natural weakenings of the assumptions and the natural strengthenings of the conclusion make the theorem fail. The literature on expectation no-go theorems strongly suggests that the expectation-focused approach is more general than the value-focused one. We establish that the expectation approach is not more general.

Using Pictures to Enhance Students' Understanding of Bayes' Theorem

ERIC Educational Resources Information Center

Trafimow, David

2011-01-01

Students often have difficulty understanding algebraic proofs of statistics theorems. However, it sometimes is possible to prove statistical theorems with pictures in which case students can gain understanding more easily. I provide examples for two versions of Bayes' theorem.

A hierarchical generalization of the acoustic reciprocity theorem involving higher-order derivatives and interaction quantities.

PubMed

Lin, Ju; Li, Jie; Li, Xiaolei; Wang, Ning

2016-10-01

An acoustic reciprocity theorem is generalized, for a smoothly varying perturbed medium, to a hierarchy of reciprocity theorems including higher-order derivatives of acoustic fields. The standard reciprocity theorem is the first member of the hierarchy. It is shown that the conservation of higher-order interaction quantities is related closely to higher-order derivative distributions of perturbed media. Then integral reciprocity theorems are obtained by applying Gauss's divergence theorem, which give explicit integral representations connecting higher-order interactions and higher-order derivative distributions of perturbed media. Some possible applications to an inverse problem are also discussed.

Bivariate tensor product [Formula: see text]-analogue of Kantorovich-type Bernstein-Stancu-Schurer operators.

PubMed

Cai, Qing-Bo; Xu, Xiao-Wei; Zhou, Guorong

2017-01-01

In this paper, we construct a bivariate tensor product generalization of Kantorovich-type Bernstein-Stancu-Schurer operators based on the concept of [Formula: see text]-integers. We obtain moments and central moments of these operators, give the rate of convergence by using the complete modulus of continuity for the bivariate case and estimate a convergence theorem for the Lipschitz continuous functions. We also give some graphs and numerical examples to illustrate the convergence properties of these operators to certain functions.

The purpose of adaptation

PubMed Central

2017-01-01

A central feature of Darwin's theory of natural selection is that it explains the purpose of biological adaptation. Here, I: emphasize the scientific importance of understanding what adaptations are for, in terms of facilitating the derivation of empirically testable predictions; discuss the population genetical basis for Darwin's theory of the purpose of adaptation, with reference to Fisher's ‘fundamental theorem of natural selection'; and show that a deeper understanding of the purpose of adaptation is achieved in the context of social evolution, with reference to inclusive fitness and superorganisms. PMID:28839927

The purpose of adaptation.

PubMed

Gardner, Andy

2017-10-06

A central feature of Darwin's theory of natural selection is that it explains the purpose of biological adaptation. Here, I: emphasize the scientific importance of understanding what adaptations are for, in terms of facilitating the derivation of empirically testable predictions; discuss the population genetical basis for Darwin's theory of the purpose of adaptation, with reference to Fisher's 'fundamental theorem of natural selection'; and show that a deeper understanding of the purpose of adaptation is achieved in the context of social evolution, with reference to inclusive fitness and superorganisms.

Sines and Cosines. Part 3 of 3

NASA Technical Reports Server (NTRS)

Apostol, Tom M. (Editor)

1994-01-01

In this 'Project Mathematics' series video, the addition formulas of sines and cosines are explained and their real life applications are demonstrated. Both film footage and computer animation is used. Several mathematical concepts are discussed and include: Ptolemy's theorem concerned with quadrilaterals; the difference between a central angle and an inscribed angle; sines and chord lengths; special angles; subtraction formulas; and a application to simple harmonic motion. A brief history of the city Alexandria, its mathematicians, and their contribution to the field of mathematics is shown.

Eigenvalue Problems.

DTIC Science & Technology

1987-06-01

Vibration of an Elastic Bar We are interested in studying the small, longitudinal vibra- tions of a longitudinally loaded, elastically supported, elastic...u 2 + + 2u O(( m,Q Uk .(J- MO In the study of eigenvalue problems, central use will be made of Rellich’s theorem (cf. Agmon [19651), which states...H , where a > 0. Sufficient conditions for (4.2) - (4.4) to hold were given in Section 3; cf. (3.15) -(3.17). For the study of (4.1) it is useful to

Similarity and Congruence.

ERIC Educational Resources Information Center

Herman, Daniel L.

This instructional unit is an introduction to the common properties of similarity and congruence. Manipulation of objects leads to a recognition of these properties. The ASA, SAS, and SSS theorems are not mentioned. Limited use is made in the application of the properties of size and shape preserved by similarity or congruence. A teacher's guide…

BAT - The Bayesian analysis toolkit

NASA Astrophysics Data System (ADS)

Caldwell, Allen; Kollár, Daniel; Kröninger, Kevin

2009-11-01

We describe the development of a new toolkit for data analysis. The analysis package is based on Bayes' Theorem, and is realized with the use of Markov Chain Monte Carlo. This gives access to the full posterior probability distribution. Parameter estimation, limit setting and uncertainty propagation are implemented in a straightforward manner.

Annealed Scaling for a Charged Polymer

NASA Astrophysics Data System (ADS)

Caravenna, F.; den Hollander, F.; Pétrélis, N.; Poisat, J.

2016-03-01

This paper studies an undirected polymer chain living on the one-dimensional integer lattice and carrying i.i.d. random charges. Each self-intersection of the polymer chain contributes to the interaction Hamiltonian an energy that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. The focus is on the annealed free energy per monomer in the limit as the length of the polymer chain tends to infinity. We derive a spectral representation for the free energy and use this to prove that there is a critical curve in the parameter plane of charge bias versus inverse temperature separating a ballistic phase from a subballistic phase. We show that the phase transition is first order. We prove large deviation principles for the laws of the empirical speed and the empirical charge, and derive a spectral representation for the associated rate functions. Interestingly, in both phases both rate functions exhibit flat pieces, which correspond to an inhomogeneous strategy for the polymer to realise a large deviation. The large deviation principles in turn lead to laws of large numbers and central limit theorems. We identify the scaling behaviour of the critical curve for small and for large charge bias. In addition, we identify the scaling behaviour of the free energy for small charge bias and small inverse temperature. Both are linked to an associated Sturm-Liouville eigenvalue problem. A key tool in our analysis is the Ray-Knight formula for the local times of the one-dimensional simple random walk. This formula is exploited to derive a closed form expression for the generating function of the annealed partition function, and for several related quantities. This expression in turn serves as the starting point for the derivation of the spectral representation for the free energy, and for the scaling theorems. What happens for the quenched free energy per monomer remains open. We state two modest results and raise a few questions.

Approaching Cauchy's Theorem

ERIC Educational Resources Information Center

Garcia, Stephan Ramon; Ross, William T.

2017-01-01

We hope to initiate a discussion about various methods for introducing Cauchy's Theorem. Although Cauchy's Theorem is the fundamental theorem upon which complex analysis is based, there is no "standard approach." The appropriate choice depends upon the prerequisites for the course and the level of rigor intended. Common methods include…

Early Vector Calculus: A Path through Multivariable Calculus

ERIC Educational Resources Information Center

Robertson, Robert L.

2013-01-01

The divergence theorem, Stokes' theorem, and Green's theorem appear near the end of calculus texts. These are important results, but many instructors struggle to reach them. We describe a pathway through a standard calculus text that allows instructors to emphasize these theorems. (Contains 2 figures.)

Pick's Theorem: What a Lemon!

ERIC Educational Resources Information Center

Russell, Alan R.

2004-01-01

Pick's theorem can be used in various ways just like a lemon. This theorem generally finds its way in the syllabus approximately at the middle school level and in fact at times students have even calculated the area of a state considering its outline with the help of the above theorem.

Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

NASA Astrophysics Data System (ADS)

Belinschi, Serban; Nowak, Maciej A.; Speicher, Roland; Tarnowski, Wojciech

2017-03-01

We extend the so-called ‘single ring theorem’ (Feinberg and Zee 1997 Nucl. Phys. B 504 579), also known as the Haagerup-Larsen theorem (Haagerup and Larsen 2000 J. Funct. Anal. 176 331). We do this by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix X, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of X. We show that this object allows the calculation of the conditional expectation of the squared eigenvalue condition number. We give examples and provide a cross-check of the analytic prediction by the large scale numerics.

Gauss Modular-Arithmetic Congruence = Signal X Noise PRODUCT: Clock-model Archimedes HYPERBOLICITY Centrality INEVITABILITY: Definition: Complexity= UTTER-SIMPLICITY: Natural-Philosophy UNITY SIMPLICITY Redux!!!

NASA Astrophysics Data System (ADS)

Kummer, E. E.; Siegel, Edward Carl-Ludwig

2011-03-01

Clock-model Archimedes [http://linkage.rockeller.edu/ wli/moved.8.04/ 1fnoise/ index. ru.html] HYPERBOLICITY inevitability throughout physics/pure-maths: Newton-law F=ma, Heisenberg and classical uncertainty-principle=Parseval/Plancherel-theorems causes FUZZYICS definition: (so miscalled) "complexity" = UTTER-SIMPLICITY!!! Watkins[www.secamlocal.ex.ac.uk/people/staff/mrwatkin/]-Hubbard[World According to Wavelets (96)-p.14!]-Franklin[1795]-Fourier[1795;1822]-Brillouin[1922] dual/inverse-space(k,w) analysis key to Fourier-unification in Archimedes hyperbolicity inevitability progress up Siegel cognition hierarchy-of-thinking (HoT): data-info.-know.-understand.-meaning-...-unity-simplicity = FUZZYICS!!! Frohlich-Mossbauer-Goldanskii-del Guidice [Nucl.Phys.B:251,375(85);275,185 (86)]-Young [arXiv-0705.4678y2, (5/31/07] theory of health/life=aqueous-electret/ ferroelectric protoplasm BEC = Archimedes-Siegel [Schrodinger Cent.Symp.(87); Symp.Fractals, MRS Fall Mtg.(89)-5-pprs] 1/w-"noise" Zipf-law power-spectrum hyperbolicity INEVITABILITY= Chi; Dirac delta-function limit w=0 concentration= BEC = Chi-Quong.

Some new results on the statistics of radio wave scintillation. I - Empirical evidence for Gaussian statistics

NASA Technical Reports Server (NTRS)

Rino, C. L.; Livingston, R. C.; Whitney, H. E.

1976-01-01

This paper presents an analysis of ionospheric scintillation data which shows that the underlying statistical structure of the signal can be accurately modeled by the additive complex Gaussian perturbation predicted by the Born approximation in conjunction with an application of the central limit theorem. By making use of this fact, it is possible to estimate the in-phase, phase quadrature, and cophased scattered power by curve fitting to measured intensity histograms. By using this procedure, it is found that typically more than 80% of the scattered power is in phase quadrature with the undeviated signal component. Thus, the signal is modeled by a Gaussian, but highly non-Rician process. From simultaneous UHF and VHF data, only a weak dependence of this statistical structure on changes in the Fresnel radius is deduced. The signal variance is found to have a nonquadratic wavelength dependence. It is hypothesized that this latter effect is a subtle manifestation of locally homogeneous irregularity structures, a mathematical model proposed by Kolmogorov (1941) in his early studies of incompressible fluid turbulence.

LD-SPatt: large deviations statistics for patterns on Markov chains.

PubMed

Nuel, G

2004-01-01

Statistics on Markov chains are widely used for the study of patterns in biological sequences. Statistics on these models can be done through several approaches. Central limit theorem (CLT) producing Gaussian approximations are one of the most popular ones. Unfortunately, in order to find a pattern of interest, these methods have to deal with tail distribution events where CLT is especially bad. In this paper, we propose a new approach based on the large deviations theory to assess pattern statistics. We first recall theoretical results for empiric mean (level 1) as well as empiric distribution (level 2) large deviations on Markov chains. Then, we present the applications of these results focusing on numerical issues. LD-SPatt is the name of GPL software implementing these algorithms. We compare this approach to several existing ones in terms of complexity and reliability and show that the large deviations are more reliable than the Gaussian approximations in absolute values as well as in terms of ranking and are at least as reliable as compound Poisson approximations. We then finally discuss some further possible improvements and applications of this new method.

Multiplicative processes in visual cognition

NASA Astrophysics Data System (ADS)

Credidio, H. F.; Teixeira, E. N.; Reis, S. D. S.; Moreira, A. A.; Andrade, J. S.

2014-03-01

The Central Limit Theorem (CLT) is certainly one of the most important results in the field of statistics. The simple fact that the addition of many random variables can generate the same probability curve, elucidated the underlying process for a broad spectrum of natural systems, ranging from the statistical distribution of human heights to the distribution of measurement errors, to mention a few. An extension of the CLT can be applied to multiplicative processes, where a given measure is the result of the product of many random variables. The statistical signature of these processes is rather ubiquitous, appearing in a diverse range of natural phenomena, including the distributions of incomes, body weights, rainfall, and fragment sizes in a rock crushing process. Here we corroborate results from previous studies which indicate the presence of multiplicative processes in a particular type of visual cognition task, namely, the visual search for hidden objects. Precisely, our results from eye-tracking experiments show that the distribution of fixation times during visual search obeys a log-normal pattern, while the fixational radii of gyration follow a power-law behavior.

The effect of signal variability on the histograms of anthropomorphic channel outputs: factors resulting in non-normally distributed data

NASA Astrophysics Data System (ADS)

Elshahaby, Fatma E. A.; Ghaly, Michael; Jha, Abhinav K.; Frey, Eric C.

2015-03-01

Model Observers are widely used in medical imaging for the optimization and evaluation of instrumentation, acquisition parameters and image reconstruction and processing methods. The channelized Hotelling observer (CHO) is a commonly used model observer in nuclear medicine and has seen increasing use in other modalities. An anthropmorphic CHO consists of a set of channels that model some aspects of the human visual system and the Hotelling Observer, which is the optimal linear discriminant. The optimality of the CHO is based on the assumption that the channel outputs for data with and without the signal present have a multivariate normal distribution with equal class covariance matrices. The channel outputs result from the dot product of channel templates with input images and are thus the sum of a large number of random variables. The central limit theorem is thus often used to justify the assumption that the channel outputs are normally distributed. In this work, we aim to examine this assumption for realistically simulated nuclear medicine images when various types of signal variability are present.

Generalized Optical Theorem Detection in Random and Complex Media

NASA Astrophysics Data System (ADS)

Tu, Jing

The problem of detecting changes of a medium or environment based on active, transmit-plus-receive wave sensor data is at the heart of many important applications including radar, surveillance, remote sensing, nondestructive testing, and cancer detection. This is a challenging problem because both the change or target and the surrounding background medium are in general unknown and can be quite complex. This Ph.D. dissertation presents a new wave physics-based approach for the detection of targets or changes in rather arbitrary backgrounds. The proposed methodology is rooted on a fundamental result of wave theory called the optical theorem, which gives real physical energy meaning to the statistics used for detection. This dissertation is composed of two main parts. The first part significantly expands the theory and understanding of the optical theorem for arbitrary probing fields and arbitrary media including nonreciprocal media, active media, as well as time-varying and nonlinear scatterers. The proposed formalism addresses both scalar and full vector electromagnetic fields. The second contribution of this dissertation is the application of the optical theorem to change detection with particular emphasis on random, complex, and active media, including single frequency probing fields and broadband probing fields. The first part of this work focuses on the generalization of the existing theoretical repertoire and interpretation of the scalar and electromagnetic optical theorem. Several fundamental generalizations of the optical theorem are developed. A new theory is developed for the optical theorem for scalar fields in nonhomogeneous media which can be bounded or unbounded. The bounded media context is essential for applications such as intrusion detection and surveillance in enclosed environments such as indoor facilities, caves, tunnels, as well as for nondestructive testing and communication systems based on wave-guiding structures. The developed scalar optical theorem theory applies to arbitrary lossless backgrounds and quite general probing fields including near fields which play a key role in super-resolution imaging. The derived formulation holds for arbitrary passive scatterers, which can be dissipative, as well as for the more general class of active scatterers which are composed of a (passive) scatterer component and an active, radiating (antenna) component. Furthermore, the generalization of the optical theorem to active scatterers is relevant to many applications such as surveillance of active targets including certain cloaks, invisible scatterers, and wireless communications. The latter developments have important military applications. The derived theoretical framework includes the familiar real power optical theorem describing power extinction due to both dissipation and scattering as well as a reactive optical theorem related to the reactive power changes. Meanwhile, the developed approach naturally leads to three optical theorem indicators or statistics, which can be used to detect changes or targets in unknown complex media. In addition, the optical theorem theory is generalized in the time domain so that it applies to arbitrary full vector fields, and arbitrary media including anisotropic media, nonreciprocal media, active media, as well as time-varying and nonlinear scatterers. The second component of this Ph.D. research program focuses on the application of the optical theorem to change detection. Three different forms of indicators or statistics are developed for change detection in unknown background media: a real power optical theorem detector, a reactive power optical theorem detector, and a total apparent power optical theorem detector. No prior knowledge is required of the background or the change or target. The performance of the three proposed optical theorem detectors is compared with the classical energy detector approach for change detection. The latter uses a mathematical or functional energy while the optical theorem detectors are based on real physical energy. For reference, the optical theorem detectors are also compared with the matched filter approach which (unlike the optical theorem detectors) assumes perfect target and medium information. The practical implementation of the optical theorem detectors is based for certain random and complex media on the exploitation of time reversal focusing ideas developed in the past 20 years in electromagnetics and acoustics. In the final part of the dissertation, we also discuss the implementation of the optical theorem sensors for one-dimensional propagation systems such as transmission lines. We also present a new generalized likelihood ratio test for detection that exploits a prior data constraint based on the optical theorem. Finally, we also address the practical implementation of the optical theorem sensors for optical imaging systems, by means of holography. The later is the first holographic implementation the optical theorem for arbitrary scenes and targets.

Experimental Test of the Differential Fluctuation Theorem and a Generalized Jarzynski Equality for Arbitrary Initial States

NASA Astrophysics Data System (ADS)

Hoang, Thai M.; Pan, Rui; Ahn, Jonghoon; Bang, Jaehoon; Quan, H. T.; Li, Tongcang

2018-02-01

Nonequilibrium processes of small systems such as molecular machines are ubiquitous in biology, chemistry, and physics but are often challenging to comprehend. In the past two decades, several exact thermodynamic relations of nonequilibrium processes, collectively known as fluctuation theorems, have been discovered and provided critical insights. These fluctuation theorems are generalizations of the second law and can be unified by a differential fluctuation theorem. Here we perform the first experimental test of the differential fluctuation theorem using an optically levitated nanosphere in both underdamped and overdamped regimes and in both spatial and velocity spaces. We also test several theorems that can be obtained from it directly, including a generalized Jarzynski equality that is valid for arbitrary initial states, and the Hummer-Szabo relation. Our study experimentally verifies these fundamental theorems and initiates the experimental study of stochastic energetics with the instantaneous velocity measurement.

Generalized virial theorem for massless electrons in graphene and other Dirac materials

NASA Astrophysics Data System (ADS)

Sokolik, A. A.; Zabolotskiy, A. D.; Lozovik, Yu. E.

2016-05-01

The virial theorem for a system of interacting electrons in a crystal, which is described within the framework of the tight-binding model, is derived. We show that, in the particular case of interacting massless electrons in graphene and other Dirac materials, the conventional virial theorem is violated. Starting from the tight-binding model, we derive the generalized virial theorem for Dirac electron systems, which contains an additional term associated with a momentum cutoff at the bottom of the energy band. Additionally, we derive the generalized virial theorem within the Dirac model using the minimization of the variational energy. The obtained theorem is illustrated by many-body calculations of the ground-state energy of an electron gas in graphene carried out in Hartree-Fock and self-consistent random-phase approximations. Experimental verification of the theorem in the case of graphene is discussed.

The geometric Mean Value Theorem

NASA Astrophysics Data System (ADS)

de Camargo, André Pierro

2018-05-01

In a previous article published in the American Mathematical Monthly, Tucker (Amer Math Monthly. 1997; 104(3): 231-240) made severe criticism on the Mean Value Theorem and, unfortunately, the majority of calculus textbooks also do not help to improve its reputation. The standard argument for proving it seems to be applying Rolle's theorem to a function like Although short and effective, such reasoning is not intuitive. Perhaps for this reason, Tucker classified the Mean Value Theorem as a technical existence theorem used to prove intuitively obvious statements. Moreover, he argued that there is nothing obvious about the Mean Value Theorem without the continuity of the derivative. Under so unfair discrimination, we felt the need to come to the defense of this beautiful theorem in order to clear up these misunderstandings.

A note on generalized Weyl's theorem

NASA Astrophysics Data System (ADS)

Zguitti, H.

2006-04-01

We prove that if either T or T* has the single-valued extension property, then the spectral mapping theorem holds for B-Weyl spectrum. If, moreover T is isoloid, and generalized Weyl's theorem holds for T, then generalized Weyl's theorem holds for f(T) for every . An application is given for algebraically paranormal operators.

On the addition theorem of spherical functions

NASA Astrophysics Data System (ADS)

Shkodrov, V. G.

The addition theorem of spherical functions is expressed in two reference systems, viz., an inertial system and a system rigidly fixed to a planet. A generalized addition theorem of spherical functions and a particular addition theorem for the rigidly fixed system are derived. The results are applied to the theory of a planetary potential.

The dynamics of a harvested predator-prey system with Holling type IV functional response.

PubMed

Liu, Xinxin; Huang, Qingdao

2018-05-31

The paper aims to investigate the dynamical behavior of a predator-prey system with Holling type IV functional response in which both the species are subject to capturing. We mainly consider how the harvesting affects equilibria, stability, limit cycles and bifurcations in this system. We adopt the method of qualitative and quantitative analysis, which is based on the dynamical theory, bifurcation theory and numerical simulation. The boundedness of solutions, the existence and stability of equilibrium points of the system are further studied. Based on the Sotomayor's theorem, the existence of transcritical bifurcation and saddle-node bifurcation are derived. We use the normal form theorem to analyze the Hopf bifurcation. Simulation results show that the first Lyapunov coefficient is negative and a stable limit cycle may bifurcate. Numerical simulations are performed to make analytical studies more complete. This work illustrates that using the harvesting effort as control parameter can change the behaviors of the system, which may be useful for the biological management. Copyright © 2018 Elsevier B.V. All rights reserved.

A rapid-pressure correlation representation consistent with the Taylor-Proudman theorem materially-frame-indifferent in the 2D limit

NASA Technical Reports Server (NTRS)

Ristorcelli, J. R.; Lumley, J. L.; Abid, R.

1994-01-01

A nonlinear representation for the rapid-pressure correlation appearing in the Reynolds stress equations, consistent with the Taylor-Proudman theorem, is presented. The representation insures that the modeled second-order equations are frame-invariant with respect to rotation when the flow is two-dimensional in planes perpendicular to the axis of rotation. The representation satisfies realizability in a new way: a special ansatz is used to obtain analytically, the values of coefficients valid away from the realizability limit: the model coefficients are functions of the state of the turbulence that are valid for all states of the mechanical turbulence attaining their constant limiting values only when the limit state is achieved. Utilization of all the mathematical constraints are not enough to specify all the coefficients in the model. The unspecified coefficients appear as free parameters which are used to insure that the representation is asymptotically consistent with the known equilibrium states of a homogeneous sheared turbulence. This is done by insuring that the modeled evolution equations have the same fixed points as those obtained from computer and laboratory experiments for the homogeneous shear. Results of computations of the homogeneous shear, with and without rotation, and with stabilizing and destabilizing curvature, are shown. Results are consistently better, in a wide class of flows which the model not been calibrated, than those obtained with other nonlinear models.

Aspects of Higher-Spin Conformal Field Theories and Their Renormalization Group Flows

NASA Astrophysics Data System (ADS)

Diab, Kenan S.

In this thesis, we study conformal field theories (CFTs) with higher-spin symmetry and the renormalization group flows of some models with interactions that weakly break the higher-spin symmetry. When the higher-spin symmetry is exact, we will present CFT analogues of two classic results in quantum field theory: the Coleman-Mandula theorem, which is the subject of chapter 2, and the Weinberg-Witten theorem, which is the subject of chapter 3. Schematically, our Coleman-Mandula analogue states that a CFT that contains a symmetric conserved current of spin s > 2 in any dimension d > 3 is effectively free, and our Weinberg-Witten analogue states that the presence of certain short, higher-spin, "sufficiently asymmetric" representations of the conformal group is either inconsistent with conformal symmetry or leads to free theories in d = 4 dimensions. In both chapters, the basic strategy is to solve certain Ward identities in convenient kinematical limits and thereby show that the number of solutions is very limited. In the latter chapter, Hofman-Maldacena bounds, which constrain one-point functions of the stress tensor in general states, play a key role. Then, in chapter 4, we will focus on the particular examples of the O(N) and Gross-Neveu model in continuous dimensions. Using diagrammatic techniques, we explicitly calculate how the coefficients of the two-point function of a U(1) current and the two-point function of the stress tensor (CJ and CT, respectively) are renormalized in the 1/N and epsilon expansions. From the higher-spin perspective, these models are interesting since they are related via the AdS/CFT correspondence to Vasiliev gravity. In addition to checking and extending a number of previously-known results about CT and CJ in these theories, we find that in certain dimensions, CJ and CT are not monotonic along the renormalization group flow. Although it was already known that certain supersymmetric models do not satisfy a "CJ"- or " CT"-theorem, this shows that such a theorem is unlikely to hold even under more restrictive assumptions.

Discovering the Theorem of Pythagoras

NASA Technical Reports Server (NTRS)

Lattanzio, Robert (Editor)

1988-01-01

In this 'Project Mathematics! series, sponsored by the California Institute of Technology, Pythagoraus' theorem a(exp 2) + b(exp 2) = c(exp 2) is discussed and the history behind this theorem is explained. hrough live film footage and computer animation, applications in real life are presented and the significance of and uses for this theorem are put into practice.

Guided Discovery of the Nine-Point Circle Theorem and Its Proof

ERIC Educational Resources Information Center

Buchbinder, Orly

2018-01-01

The nine-point circle theorem is one of the most beautiful and surprising theorems in Euclidean geometry. It establishes an existence of a circle passing through nine points, all of which are related to a single triangle. This paper describes a set of instructional activities that can help students discover the nine-point circle theorem through…

Aspects of AdS/CFT: Conformal Deformations and the Goldstone Equivalence Theorem

NASA Astrophysics Data System (ADS)

Cantrell, Sean Andrew

The AdS/CFT correspondence provides a map from the states of theories situated in AdSd+1 to those in dual conformal theories in a d-dimensional space. The correspondence can be used to establish certain universal properties of some theories in one space by examining the behave of general objects in the other. In this thesis, we develop various formal aspects of AdS/CFT. Conformal deformations manifest in the AdS/CFT correspondence as boundary conditions on the AdS field. Heretofore, double-trace deformations have been the primary focus in this context. To better understand multitrace deformations, we revisit the relationship between the generating AdS partition function for a free bulk theory and the boundary CFT partition function subject to arbitrary conformal deformations. The procedure leads us to a formalism that constructs bulk fields from boundary operators. We independently replicate the holographic RG flow narrative to go on to interpret the brane used to regulate the AdS theory as a renormalization scale. The scale-dependence of the dilatation spectrum of a boundary theory in the presence of general deformations can be thus understood on the AdS side using this formalism. The Goldstone equivalence theorem allows one to relate scattering amplitudes of massive gauge fields to those of scalar fields in the limit of large scattering energies. We generalize this theorem under the framework of the AdS/CFT correspondence. First, we obtain an expression of the equivalence theorem in terms of correlation functions of creation and annihilation operators by using an AdS wave function approach to the AdS/CFT dictionary. It is shown that the divergence of the non-conserved conformal current dual to the bulk gauge field is approximately primary when computing correlators for theories in which the masses of all the exchanged particles are sufficiently large. The results are then generalized to higher spin fields. We then go on to generalize the theorem using conformal blocks in two and four-dimensional CFTs. We show that when the scaling dimensions of the exchanged operators are large compared to both their spins and the dimension of the current, the conformal blocks satisfy an equivalence theorem.

Evaluation of air quality zone classification methods based on ambient air concentration exposure.

PubMed

Freeman, Brian; McBean, Ed; Gharabaghi, Bahram; Thé, Jesse

2017-05-01

Air quality zones are used by regulatory authorities to implement ambient air standards in order to protect human health. Air quality measurements at discrete air monitoring stations are critical tools to determine whether an air quality zone complies with local air quality standards or is noncompliant. This study presents a novel approach for evaluation of air quality zone classification methods by breaking the concentration distribution of a pollutant measured at an air monitoring station into compliance and exceedance probability density functions (PDFs) and then using Monte Carlo analysis with the Central Limit Theorem to estimate long-term exposure. The purpose of this paper is to compare the risk associated with selecting one ambient air classification approach over another by testing the possible exposure an individual living within a zone may face. The chronic daily intake (CDI) is utilized to compare different pollutant exposures over the classification duration of 3 years between two classification methods. Historical data collected from air monitoring stations in Kuwait are used to build representative models of 1-hr NO 2 and 8-hr O 3 within a zone that meets the compliance requirements of each method. The first method, the "3 Strike" method, is a conservative approach based on a winner-take-all approach common with most compliance classification methods, while the second, the 99% Rule method, allows for more robust analyses and incorporates long-term trends. A Monte Carlo analysis is used to model the CDI for each pollutant and each method with the zone at a single station and with multiple stations. The model assumes that the zone is already in compliance with air quality standards over the 3 years under the different classification methodologies. The model shows that while the CDI of the two methods differs by 2.7% over the exposure period for the single station case, the large number of samples taken over the duration period impacts the sensitivity of the statistical tests, causing the null hypothesis to fail. Local air quality managers can use either methodology to classify the compliance of an air zone, but must accept that the 99% Rule method may cause exposures that are statistically more significant than the 3 Strike method. A novel method using the Central Limit Theorem and Monte Carlo analysis is used to directly compare different air standard compliance classification methods by estimating the chronic daily intake of pollutants. This method allows air quality managers to rapidly see how individual classification methods may impact individual population groups, as well as to evaluate different pollutants based on dosage and exposure when complete health impacts are not known.

Classical noise, quantum noise and secure communication

NASA Astrophysics Data System (ADS)

Tannous, C.; Langlois, J.

2016-01-01

Secure communication based on message encryption might be performed by combining the message with controlled noise (called pseudo-noise) as performed in spread-spectrum communication used presently in Wi-Fi and smartphone telecommunication systems. Quantum communication based on entanglement is another route for securing communications as demonstrated by several important experiments described in this work. The central role played by the photon in unifying the description of classical and quantum noise as major ingredients of secure communication systems is highlighted and described on the basis of the classical and quantum fluctuation dissipation theorems.

Characterization of Generalized Young Measures Generated by Symmetric Gradients

NASA Astrophysics Data System (ADS)

De Philippis, Guido; Rindler, Filip

2017-06-01

This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer-Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The "local" proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti's rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences.

The Poincaré-Hopf Theorem for line fields revisited

NASA Astrophysics Data System (ADS)

Crowley, Diarmuid; Grant, Mark

2017-07-01

A Poincaré-Hopf Theorem for line fields with point singularities on orientable surfaces can be found in Hopf's 1956 Lecture Notes on Differential Geometry. In 1955 Markus presented such a theorem in all dimensions, but Markus' statement only holds in even dimensions 2 k ≥ 4. In 1984 Jänich presented a Poincaré-Hopf theorem for line fields with more complicated singularities and focussed on the complexities arising in the generalized setting. In this expository note we review the Poincaré-Hopf Theorem for line fields with point singularities, presenting a careful proof which is valid in all dimensions.

Common fixed point theorems for maps under a contractive condition of integral type

NASA Astrophysics Data System (ADS)

Djoudi, A.; Merghadi, F.

2008-05-01

Two common fixed point theorems for mapping of complete metric space under a general contractive inequality of integral type and satisfying minimal commutativity conditions are proved. These results extend and improve several previous results, particularly Theorem 4 of Rhoades [B.E. Rhoades, Two fixed point theorems for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 63 (2003) 4007-4013] and Theorem 4 of Sessa [S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math. (Beograd) (N.S.) 32 (46) (1982) 149-153].

Nonlinear system theory: another look at dependence.

PubMed

Wu, Wei Biao

2005-10-04

Based on the nonlinear system theory, we introduce previously undescribed dependence measures for stationary causal processes. Our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems. The proposed dependence measures provide a natural framework for a limit theory for stationary processes. In particular, under conditions with quite simple forms, we present limit theorems for partial sums, empirical processes, and kernel density estimates. The conditions are mild and easily verifiable because they are directly related to the data-generating mechanisms.

Thermal Noise Limit in Frequency Stabilization of Lasers with Rigid Cavities

NASA Technical Reports Server (NTRS)

Numata, Kenji; Kemery, Amy; Camp, Jordan

2005-01-01

We evaluated thermal noise (Brownian motion) in a rigid reference cavity Used for frequency stabilization of lasers, based on the mechanical loss of cavity materials and the numerical analysis of the mirror-spacer mechanics with the direct application of the fluctuation dissipation theorem. This noise sets a fundamental limit for the frequency stability achieved with a rigid frequency-reference cavity of order 1 Hz/rtHz at 10mHz at room temperature. This level coincides with the world-highest level stabilization results.

A Converse of the Mean Value Theorem Made Easy

ERIC Educational Resources Information Center

Mortici, Cristinel

2011-01-01

The aim of this article is to discuss some results about the converse mean value theorem stated by Tong and Braza [J. Tong and P. Braza, "A converse of the mean value theorem", Amer. Math. Monthly 104(10), (1997), pp. 939-942] and Almeida [R. Almeida, "An elementary proof of a converse mean-value theorem", Internat. J. Math. Ed. Sci. Tech. 39(8)…

Recurrence theorems: A unified account

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

Wallace, David, E-mail: david.wallace@balliol.ox.ac.uk

I discuss classical and quantum recurrence theorems in a unified manner, treating both as generalisations of the fact that a system with a finite state space only has so many places to go. Along the way, I prove versions of the recurrence theorem applicable to dynamics on linear and metric spaces and make some comments about applications of the classical recurrence theorem in the foundations of statistical mechanics.

A variational theorem for creep with applications to plates and columns

NASA Technical Reports Server (NTRS)

Sanders, J Lyell, Jr; Mccomb, Harvey G , Jr; Schlechte, Floyd R

1958-01-01

A variational theorem is presented for a body undergoing creep. Solutions to problems of the creep behavior of plates, columns, beams, and shells can be obtained by means of the direct methods of the calculus of variations in conjunction with the stated theorem. The application of the theorem is illustrated for plates and columns by the solution of two sample problems.

Correcting Duporcq's theorem☆

PubMed Central

Nawratil, Georg

2014-01-01

In 1898, Ernest Duporcq stated a famous theorem about rigid-body motions with spherical trajectories, without giving a rigorous proof. Today, this theorem is again of interest, as it is strongly connected with the topic of self-motions of planar Stewart–Gough platforms. We discuss Duporcq's theorem from this point of view and demonstrate that it is not correct. Moreover, we also present a revised version of this theorem. PMID:25540467

Dynamic Investigation of Triangles Inscribed in a Circle, Which Tend to an Equilateral Triangle

ERIC Educational Resources Information Center

Stupel, Moshe; Oxman, Victor; Sigler, Avi

2017-01-01

We present a geometrical investigation of the process of creating an infinite sequence of triangles inscribed in a circle, whose areas, perimeters and lengths of radii of the inscribed circles tend to a limit in a monotonous manner. First, using geometrical software, we investigate four theorems that represent interesting geometrical properties,…

Bi-centenary of successes of Fourier theorem: its power and limitations in optical system designs

NASA Astrophysics Data System (ADS)

Roychoudhuri, Chandrasekhar

2007-09-01

We celebrate the two hundred years of successful use of the Fourier theorem in optics. However, there is a great enigma associated with the Fourier transform integral. It is one of the most pervasively productive and useful tool of physics and optics because its foundation is based on the superposition of harmonic functions and yet we have never declared it as a principle of physics for valid reasons. And, yet there are a good number of situations where we pretend it to be equivalent to the superposition principle of physics, creating epistemological problems of enormous magnitude. The purpose of the paper is to elucidate the problems while underscoring the successes and the elegance of the Fourier theorem, which are not explicitly discussed in the literature. We will make our point by taking six major engineering fields of optics and show in each case why it works and under what restricted conditions by bringing in the relevant physics principles. The fields are (i) optical signal processing, (ii) Fourier transform spectrometry, (iii) classical spectrometry of pulsed light, (iv) coherence theory, (v) laser mode locking and (vi) pulse broadening. We underscore that mathematical Fourier frequencies, not being physical frequencies, cannot generate real physical effects on our detectors. Appreciation of this fundamental issue will open up ways to be innovative in many new optical instrument designs. We underscore the importance of always validating our design platforms based on valid physics principles (actual processes undergoing in nature) captured by an appropriate hypothesis based on diverse observations. This paper is a comprehensive view of the power and limitations of Fourier Transform by summarizing a series of SPIE conference papers presented during 2003-2007.

Voronovskaja's theorem revisited

NASA Astrophysics Data System (ADS)

Tachev, Gancho T.

2008-07-01

We represent a new quantitative variant of Voronovskaja's theorem for Bernstein operator. This estimate improves the recent quantitative versions of Voronovskaja's theorem for certain Bernstein-type operators, obtained by H. Gonska, P. Pitul and I. Rasa in 2006.

Riemannian and Lorentzian flow-cut theorems

NASA Astrophysics Data System (ADS)

Headrick, Matthew; Hubeny, Veronika E.

2018-05-01

We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut (MFMC) theorem for boundary regions, applied recently to develop a ‘bit-thread’ interpretation of holographic entanglement entropies. We also prove various properties of the max flow and min cut, including respective nesting properties. In the Lorentzian setting, we prove the analogous MFMC theorem, which states that the volume of a maximal slice equals the flux of a minimal flow, where a flow is defined as a divergenceless timelike vector field with norm at least 1. This theorem includes as a special case a continuum version of Dilworth’s theorem from the theory of partially ordered sets. We include a brief review of the necessary tools from the theory of convex optimization, in particular Lagrangian duality and convex relaxation.

An Introduction to Kristof's Theorem for Solving Least-Square Optimization Problems Without Calculus.

PubMed

Waller, Niels

2018-01-01

Kristof's Theorem (Kristof, 1970 ) describes a matrix trace inequality that can be used to solve a wide-class of least-square optimization problems without calculus. Considering its generality, it is surprising that Kristof's Theorem is rarely used in statistics and psychometric applications. The underutilization of this method likely stems, in part, from the mathematical complexity of Kristof's ( 1964 , 1970 ) writings. In this article, I describe the underlying logic of Kristof's Theorem in simple terms by reviewing four key mathematical ideas that are used in the theorem's proof. I then show how Kristof's Theorem can be used to provide novel derivations to two cognate models from statistics and psychometrics. This tutorial includes a glossary of technical terms and an online supplement with R (R Core Team, 2017 ) code to perform the calculations described in the text.

Formal reasoning about systems biology using theorem proving

PubMed Central

Hasan, Osman; Siddique, Umair; Tahar, Sofiène

2017-01-01

System biology provides the basis to understand the behavioral properties of complex biological organisms at different levels of abstraction. Traditionally, analysing systems biology based models of various diseases have been carried out by paper-and-pencil based proofs and simulations. However, these methods cannot provide an accurate analysis, which is a serious drawback for the safety-critical domain of human medicine. In order to overcome these limitations, we propose a framework to formally analyze biological networks and pathways. In particular, we formalize the notion of reaction kinetics in higher-order logic and formally verify some of the commonly used reaction based models of biological networks using the HOL Light theorem prover. Furthermore, we have ported our earlier formalization of Zsyntax, i.e., a deductive language for reasoning about biological networks and pathways, from HOL4 to the HOL Light theorem prover to make it compatible with the above-mentioned formalization of reaction kinetics. To illustrate the usefulness of the proposed framework, we present the formal analysis of three case studies, i.e., the pathway leading to TP53 Phosphorylation, the pathway leading to the death of cancer stem cells and the tumor growth based on cancer stem cells, which is used for the prognosis and future drug designs to treat cancer patients. PMID:28671950

The scalar glueball operator, the a-theorem, and the onset of conformality

NASA Astrophysics Data System (ADS)

Nunes da Silva, T.; Pallante, E.; Robroek, L.

2018-03-01

We show that the anomalous dimension γG of the scalar glueball operator contains information on the mechanism that leads to the onset of conformality at the lower edge of the conformal window in a non-Abelian gauge theory. In particular, it distinguishes whether the merging of an UV and an IR fixed point - the simplest mechanism associated to a conformal phase transition and preconformal scaling - does or does not occur. At the same time, we shed light on new analogies between QCD and its supersymmetric version. In SQCD, we derive an exact relation between γG and the mass anomalous dimension γm, and we prove that the SQCD exact beta function is incompatible with merging as a consequence of the a-theorem; we also derive the general conditions that the latter imposes on the existence of fixed points, and prove the absence of an UV fixed point at nonzero coupling above the conformal window of SQCD. Perhaps not surprisingly, we then show that an exact relation between γG and γm, fully analogous to SQCD, holds for the massless Veneziano limit of large-N QCD. We argue, based on the latter relation, the a-theorem, perturbation theory and physical arguments, that the incompatibility with merging may extend to QCD.

Double soft graviton theorems and Bondi-Metzner-Sachs symmetries

NASA Astrophysics Data System (ADS)

Anupam, A. H.; Kundu, Arpan; Ray, Krishnendu

2018-05-01

It is now well understood that Ward identities associated with the (extended) BMS algebra are equivalent to single soft graviton theorems. In this work, we show that if we consider nested Ward identities constructed out of two BMS charges, a class of double soft factorization theorems can be recovered. By making connections with earlier works in the literature, we argue that at the subleading order, these double soft graviton theorems are the so-called consecutive double soft graviton theorems. We also show how these nested Ward identities can be understood as Ward identities associated with BMS symmetries in scattering states defined around (non-Fock) vacua parametrized by supertranslations or superrotations.

Strange kinetics of bulk-mediated diffusion on lipid bilayers

PubMed Central

Campagnola, Grace; Nepal, Kanti; Peersen, Olve B.

2016-01-01

Diffusion at solid-liquid interfaces is crucial in many technological and biophysical processes. Although its behavior seems deceivingly simple, recent studies showing passive superdiffusive transport suggest diffusion on surfaces may hide rich complexities. In particular, bulk-mediated diffusion occurs when molecules are transiently released from the surface to perform three-dimensional excursions into the liquid bulk. This phenomenon bears the dichotomy where a molecule always return to the surface but the mean jump length is infinite. Such behavior is associated with a breakdown of the central limit theorem and weak ergodicity breaking. Here, we use single-particle tracking to study the statistics of bulk-mediated diffusion on a supported lipid bilayer. We find that the time-averaged mean square displacement (MSD) of individual trajectories, the archetypal measure in diffusion processes, does not converge to the ensemble MSD but it remains a random variable, even in the long observation-time limit. The distribution of time averages is shown to agree with a Lévy flight model. Our results also unravel intriguing anomalies in the statistics of displacements. The time averaged MSD is shown to depend on experimental time and investigations of fractional moments show a scaling 〈|r(t)|q〉 ∼ tqv(q) with non-linear exponents, i.e. v(q) ≠ const. This type of behavior is termed strong anomalous diffusion and is rare among experimental observations. PMID:27095275

Two-component Jaffe models with a central black hole - I. The spherical case

NASA Astrophysics Data System (ADS)

Ciotti, Luca; Ziaee Lorzad, Azadeh

2018-02-01

Dynamical properties of spherically symmetric galaxy models where both the stellar and total mass density distributions are described by the Jaffe (1983) profile (with different scalelengths and masses) are presented. The orbital structure of the stellar component is described by Osipkov-Merritt anisotropy, and a black hole (BH) is added at the centre of the galaxy; the dark matter halo is isotropic. First, the conditions required to have a nowhere negative and monotonically decreasing dark matter halo density profile are derived. We then show that the phase-space distribution function can be recovered by using the Lambert-Euler W function, while in absence of the central BH only elementary functions appears in the integrand of the inversion formula. The minimum value of the anisotropy radius for consistency is derived in terms of the galaxy parameters. The Jeans equations for the stellar component are solved analytically, and the projected velocity dispersion at the centre and at large radii are also obtained analytically for generic values of the anisotropy radius. Finally, the relevant global quantities entering the Virial Theorem are computed analytically, and the fiducial anisotropy limit required to prevent the onset of Radial Orbit Instability is determined as a function of the galaxy parameters. The presented models, even though highly idealized, represent a substantial generalization of the models presented in Ciotti, and can be useful as starting point for more advanced modelling, the dynamics and the mass distribution of elliptical galaxies.

Visual Theorems.

ERIC Educational Resources Information Center

Davis, Philip J.

1993-01-01

Argues for a mathematics education that interprets the word "theorem" in a sense that is wide enough to include the visual aspects of mathematical intuition and reasoning. Defines the term "visual theorems" and illustrates the concept using the Marigold of Theodorus. (Author/MDH)

Note on the theorems of Bjerknes and Crocco

NASA Technical Reports Server (NTRS)

Theodorsen, Theodore

1946-01-01

The theorems of Bjerknes and Crocco are of great interest in the theory of flow around airfoils at Mach numbers near and above unity. A brief note shows how both theorems are developed by short vector transformations.

Analysis of non locality proofs in Quantum Mechanics

NASA Astrophysics Data System (ADS)

Nisticò, Giuseppe

2012-02-01

Two kinds of non-locality theorems in Quantum Mechanics are taken into account: the theorems based on the criterion of reality and the quite different theorem proposed by Stapp. In the present work the analyses of the theorem due to Greenberger, Horne, Shimony and Zeilinger, based on the criterion of reality, and of Stapp's argument are shown. The results of these analyses show that the alleged violations of locality cannot be considered definitive.

PYGMALION: A Creative Programming Environment

DTIC Science & Technology

1975-06-01

iiiiiimimmmimm wm^m^mmm’ wi-i ,»■»’■’.■- v* 26 Examples of Purely Iconic Reasoning 1-H Pythagoras ’ original proof of the Pythagorean Theorem ... Theorem Proving Machine񓟋. His program employed properties of the representation to guide the proof of theorems . His simple heruristic "Reject...one theorem the square of the hypotenuse. "Every proposition is presented as a self-contained fact relying on its own intrinsic evidence. Instead

A Maximal Element Theorem in FWC-Spaces and Its Applications

PubMed Central

Hu, Qingwen; Miao, Yulin

2014-01-01

A maximal element theorem is proved in finite weakly convex spaces (FWC-spaces, in short) which have no linear, convex, and topological structure. Using the maximal element theorem, we develop new existence theorems of solutions to variational relation problem, generalized equilibrium problem, equilibrium problem with lower and upper bounds, and minimax problem in FWC-spaces. The results represented in this paper unify and extend some known results in the literature. PMID:24782672

Generalized Bloch theorem and topological characterization

NASA Astrophysics Data System (ADS)

Dobardžić, E.; Dimitrijević, M.; Milovanović, M. V.

2015-03-01

The Bloch theorem enables reduction of the eigenvalue problem of the single-particle Hamiltonian that commutes with the translational group. Based on a group theory analysis we present a generalization of the Bloch theorem that incorporates all additional symmetries of a crystal. The generalized Bloch theorem constrains the form of the Hamiltonian which becomes manifestly invariant under additional symmetries. In the case of isotropic interactions the generalized Bloch theorem gives a unique Hamiltonian. This Hamiltonian coincides with the Hamiltonian in the periodic gauge. In the case of anisotropic interactions the generalized Bloch theorem allows a family of Hamiltonians. Due to the continuity argument we expect that even in this case the Hamiltonian in the periodic gauge defines observables, such as Berry curvature, in the inverse space. For both cases we present examples and demonstrate that the average of the Berry curvatures of all possible Hamiltonians in the Bloch gauge is the Berry curvature in the periodic gauge.

The probability density function (PDF) of Lagrangian Turbulence

NASA Astrophysics Data System (ADS)

Birnir, B.

2012-12-01

The statistical theory of Lagrangian turbulence is derived from the stochastic Navier-Stokes equation. Assuming that the noise in fully-developed turbulence is a generic noise determined by the general theorems in probability, the central limit theorem and the large deviation principle, we are able to formulate and solve the Kolmogorov-Hopf equation for the invariant measure of the stochastic Navier-Stokes equations. The intermittency corrections to the scaling exponents of the structure functions require a multiplicative (multipling the fluid velocity) noise in the stochastic Navier-Stokes equation. We let this multiplicative noise, in the equation, consists of a simple (Poisson) jump process and then show how the Feynmann-Kac formula produces the log-Poissonian processes, found by She and Leveque, Waymire and Dubrulle. These log-Poissonian processes give the intermittency corrections that agree with modern direct Navier-Stokes simulations (DNS) and experiments. The probability density function (PDF) plays a key role when direct Navier-Stokes simulations or experimental results are compared to theory. The statistical theory of turbulence is determined, including the scaling of the structure functions of turbulence, by the invariant measure of the Navier-Stokes equation and the PDFs for the various statistics (one-point, two-point, N-point) can be obtained by taking the trace of the corresponding invariant measures. Hopf derived in 1952 a functional equation for the characteristic function (Fourier transform) of the invariant measure. In distinction to the nonlinear Navier-Stokes equation, this is a linear functional differential equation. The PDFs obtained from the invariant measures for the velocity differences (two-point statistics) are shown to be the four parameter generalized hyperbolic distributions, found by Barndorff-Nilsen. These PDF have heavy tails and a convex peak at the origin. A suitable projection of the Kolmogorov-Hopf equations is the differential equation determining the generalized hyperbolic distributions. Then we compare these PDFs with DNS results and experimental data.

Super central configurations of the n-body problem

NASA Astrophysics Data System (ADS)

Xie, Zhifu

2010-04-01

In this paper, we consider the inverse problem of central configurations of the n-body problem. For a given q =(q1,q2,…,qn)ε(Rd)n, let S(q ) be the admissible set of masses by S(q )={m =(m1,…,mn)∣miεR+, q is a central configurationfor m}. For a given m εS(q), let Sm(q) be the permutational admissible set about m =(m1,m2,…,mn) by Sm(q)={m'∣m'εS(q), m'≠m and m' is apermutation of m}. Here, q is called a super central configuration if there exists m such that Sm(q) is nonempty. For any q in the planar four-body problem, q is not a super central configuration as an immediate consequence of a theorem proved by MacMillan and Bartky ["Permanent configurations in the problem of four bodies," Trans. Am. Math. Soc. 34, 838 (1932)]. The main discovery in this paper is the existence of super central configurations in the collinear three-body problem. We proved that for any q in the collinear three-body problem and any m εS(q), Sm(q) has at most one element and the detailed classification of Sm(q) is provided.

Revisiting Ramakrishnan's approach to relatively. [Velocity addition theorem uniqueness

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

Nandi, K.K.; Shankara, T.S.

The conditions under which the velocity addition theorem (VAT) is formulated by Ramakrishnan gave rise to doubts about the uniqueness of the theorem. These conditions are rediscussed with reference to their algebraic and experimental implications. 9 references.

General Theorems about Homogeneous Ellipsoidal Inclusions

ERIC Educational Resources Information Center

Korringa, J.; And Others

1978-01-01

Mathematical theorems about the properties of ellipsoids are developed. Included are Poisson's theorem concerning the magnetization of a homogeneous body of ellipsoidal shape, the polarization of a dielectric, the transport of heat or electricity through an ellipsoid, and other problems. (BB)

A no-hair theorem for black holes in f(R) gravity

NASA Astrophysics Data System (ADS)

Cañate, Pedro

2018-01-01

In this work we present a no-hair theorem which discards the existence of four-dimensional asymptotically flat, static and spherically symmetric or stationary axisymmetric, non-trivial black holes in the frame of f(R) gravity under metric formalism. Here we show that our no-hair theorem also can discard asymptotic de Sitter stationary and axisymmetric non-trivial black holes. The novelty is that this no-hair theorem is built without resorting to known mapping between f(R) gravity and scalar–tensor theory. Thus, an advantage will be that our no-hair theorem applies as well to metric f(R) models that cannot be mapped to scalar–tensor theory.

Generalized Browder's and Weyl's theorems for Banach space operators

NASA Astrophysics Data System (ADS)

Curto, Raúl E.; Han, Young Min

2007-12-01

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem. We also prove that the spectral mapping theorem holds for the Drazin spectrum and for analytic functions on an open neighborhood of [sigma](T). As applications, we show that if T is algebraically M-hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f(T), where f[set membership, variant]H((T)), the space of functions analytic on an open neighborhood of [sigma](T). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f(T), for each f[set membership, variant]H([sigma](T)).

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

Stein, Leo C.; Yagi, Kent; Yunes, Nicolás, E-mail: leostein@astro.cornell.edu

The gravitational field outside of astrophysical black holes is completely described by their mass and spin frequency, as expressed by the no-hair theorems. These theorems assume vacuum spacetimes, and thus they apply only to black holes and not to stars. Despite this, we analytically find that the gravitational potential of arbitrarily rapid, rigidly rotating stars can still be described completely by only their mass, spin angular momentum, and quadrupole moment. Although these results are obtained in the nonrelativistic limit (to leading order in a weak-field expansion of general relativity, GR), they are also consistent with fully relativistic numerical calculations ofmore » rotating neutron stars. This description of the gravitational potential outside the source in terms of just three quantities is approximately universal (independent of equation of state). Such universality may be used to break degeneracies in pulsar and future gravitational wave observations to extract more physics and test GR in the strong-field regime.« less

Symmetry for the duration of entropy-consuming intervals.

PubMed

García-García, Reinaldo; Domínguez, Daniel

2014-05-01

We introduce the violation fraction υ as the cumulative fraction of time that a mesoscopic system spends consuming entropy at a single trajectory in phase space. We show that the fluctuations of this quantity are described in terms of a symmetry relation reminiscent of fluctuation theorems, which involve a function Φ, which can be interpreted as an entropy associated with the fluctuations of the violation fraction. The function Φ, when evaluated for arbitrary stochastic realizations of the violation fraction, is odd upon the symmetry transformations that are relevant for the associated stochastic entropy production. This fact leads to a detailed fluctuation theorem for the probability density function of Φ. We study the steady-state limit of this symmetry in the paradigmatic case of a colloidal particle dragged by optical tweezers through an aqueous solution. Finally, we briefly discuss possible applications of our results for the estimation of free-energy differences from single-molecule experiments.

The derivative-free Fourier shell identity for photoacoustics.

PubMed

Baddour, Natalie

2016-01-01

In X-ray tomography, the Fourier slice theorem provides a relationship between the Fourier components of the object being imaged and the measured projection data. The Fourier slice theorem is the basis for X-ray Fourier-based tomographic inversion techniques. A similar relationship, referred to as the 'Fourier shell identity' has been previously derived for photoacoustic applications. However, this identity relates the pressure wavefield data function and its normal derivative measured on an arbitrary enclosing aperture to the three-dimensional Fourier transform of the enclosed object evaluated on a sphere. Since the normal derivative of pressure is not normally measured, the applicability of the formulation is limited in this form. In this paper, alternative derivations of the Fourier shell identity in 1D, 2D polar and 3D spherical polar coordinates are presented. The presented formulations do not require the normal derivative of pressure, thereby lending the formulas directly adaptable for Fourier based absorber reconstructions.

Brane surgery: energy conditions, traversable wormholes, and voids

NASA Astrophysics Data System (ADS)

Barceló1, C.; Visser, M.

2000-09-01

Branes are ubiquitous elements of any low-energy limit of string theory. We point out that negative tension branes violate all the standard energy conditions of the higher-dimensional spacetime they are embedded in; this opens the door to very peculiar solutions of the higher-dimensional Einstein equations. Building upon the (/3+1)-dimensional implementation of fundamental string theory, we illustrate the possibilities by considering a toy model consisting of a (/2+1)-dimensional brane propagating through our observable (/3+1)-dimensional universe. Developing a notion of ``brane surgery'', based on the Israel-Lanczos-Sen ``thin shell'' formalism of general relativity, we analyze the dynamics and find traversable wormholes, closed baby universes, voids (holes in the spacetime manifold), and an evasion (not a violation) of both the singularity theorems and the positive mass theorem. These features appear generic to any brane model that permits negative tension branes: This includes the Randall-Sundrum models and their variants.

Path integrals, supersymmetric quantum mechanics, and the Atiyah-Singer index theorem for twisted Dirac

NASA Astrophysics Data System (ADS)

Fine, Dana S.; Sawin, Stephen

2017-01-01

Feynman's time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ensure that these products converge. The limit defines a path integral which agrees pointwise with the heat kernel for a generalized Laplacian. The result is a rigorous construction of the propagator for supersymmetric quantum mechanics, with potential, as a path integral. Further, the class of Laplacians includes the square of the twisted Dirac operator, which corresponds to an extension of N = 1/2 supersymmetric quantum mechanics. General results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the Atiyah-Singer index theorem.

Reviews Book: The Babylonian Theorem Video Game: BrainBox360 (Physics Edition) Book: Teaching and Learning Science: Towards a Personalized Approach Book: Good Practice in Science Teaching: What Research Has to Say Equipment: PAPERSHOW Equipment: SEP Steady State Bottle Kit Equipment: Sciencescope Datalogging Balance Equipment: USB Robot Arm Equipment: Sciencescope Spectrophotometer Web Watch

NASA Astrophysics Data System (ADS)

2010-07-01

WE RECOMMEND Good Practice in Science Teaching: What Research Has to Say Book explores and summarizes the research Steady State Bottle Kit Another gem from SEP Sciencescope Datalogging Balance Balance suits everyday use Sciencescope Spectrophotometer Device displays clear spectrum WORTH A LOOK The Babylonian Theorem Text explains ancient Egyptian mathematics BrainBox360 (Physics Edition) Video game tests your knowledge Teaching and Learning Science: Towards a Personalized Approach Book reveals how useful physics teachers really are PAPERSHOW Gadget kit is useful but has limitations Robotic Arm Kit with USB PC Interface Robot arm teaches programming WEB WATCH Simple applets teach complex topics

Proposal for founding mistrustful quantum cryptography on coin tossing

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

Kent, Adrian; Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ,

2003-07-01

A significant branch of classical cryptography deals with the problems which arise when mistrustful parties need to generate, process, or exchange information. As Kilian showed a while ago, mistrustful classical cryptography can be founded on a single protocol, oblivious transfer, from which general secure multiparty computations can be built. The scope of mistrustful quantum cryptography is limited by no-go theorems, which rule out, inter alia, unconditionally secure quantum protocols for oblivious transfer or general secure two-party computations. These theorems apply even to protocols which take relativistic signaling constraints into account. The best that can be hoped for, in general, aremore » quantum protocols which are computationally secure against quantum attack. Here a method is described for building a classically certified bit commitment, and hence every other mistrustful cryptographic task, from a secure coin-tossing protocol. No security proof is attempted, but reasons are sketched why these protocols might resist quantum computational attack.« less

Research in Stochastic Processes.

DTIC Science & Technology

1983-10-01

increases. A more detailed investigation for the exceedances themselves (rather than Just the cluster centers) was undertaken, together with J. HUsler and...J. HUsler and M.R. Leadbetter, Compoung Poisson limit theorems for high level exceedances by stationary sequences, Center for Stochastic Processes...stability by a random linear operator. C.D. Hardin, General (asymmetric) stable variables and processes. T. Hsing, J. HUsler and M.R. Leadbetter, Compound

The dynamical behaviour of our planetary system. Proceedings. 4th Alexander von Humboldt Colloquium on Celestial Mechanics, Ramsau (Austria), 17 - 23 Mar 1996.

NASA Astrophysics Data System (ADS)

Dvorak, R.; Henrard, J.

1996-03-01

The following topics were dealt with: celestial mechanics, dynamical astronomy, planetary systems, resonance scattering, Hamiltonian mechanics non-integrability, irregular periodic orbits, escape, dynamical system mapping, fast Fourier method, precession-nutation, Nekhoroshev theorem, asteroid dynamics, the Trojan problem, planet-crossing orbits, Kirkwood gaps, future research, human comprehension limitations.

Formally Generating Adaptive Security Protocols

DTIC Science & Technology

2013-03-01

User Interfaces for Theorem Provers, 2012. [9] Xiaoming Liu, Christoph Kreitz, Robbert van Renesse, Jason J. Hickey, Mark Hayden, Ken- neth Birman, and...Constable, Mark Hayden, Jason Hickey, Christoph Kreitz, Robbert van Renesse, Ohad Rodeh, and Werner Vogels. The Horus and Ensemble projects: Accom...plishments and limitations. In DARPA Information Survivability Conference and Exposition (DISCEX 2000), pages 149–161, Hilton Head, SC, 2000. IEEE

Asymptotic Safety Guaranteed in Supersymmetry

NASA Astrophysics Data System (ADS)

Bond, Andrew D.; Litim, Daniel F.

2017-11-01

We explain how asymptotic safety arises in four-dimensional supersymmetric gauge theories. We provide asymptotically safe supersymmetric gauge theories together with their superconformal fixed points, R charges, phase diagrams, and UV-IR connecting trajectories. Strict perturbative control is achieved in a Veneziano limit. Consistency with unitarity and the a theorem is established. We find that supersymmetry enhances the predictivity of asymptotically safe theories.

Lanchester-Type Models of Warfare. Volume II

DTIC Science & Technology

1980-10-01

the so-called PERRON - FROBENIUS theorem50 for nonnegative matrices that one can guarantee that (without any further assumptions about A and B) there...always exists a vector of nonnegative values such that, for example, (7.18.6) holds. Before we state the PERRON - FROBENIUS theorem for nonnegative...a proof of this important theorem). THEOREM .5.-1.1 ( PERRON [121] and FROBENIUS [60]): Let C z 0 be an n x n matrix. Then, 1. C has a nonnegative real

A remark on the energy conditions for Hawking's area theorem

NASA Astrophysics Data System (ADS)

Lesourd, Martin

2018-06-01

Hawking's area theorem is a fundamental result in black hole theory that is universally associated with the null energy condition. That this condition can be weakened is illustrated by the formulation of a strengthened version of the theorem based on an energy condition that allows for violations of the null energy condition. With the semi-classical context in mind, some brief remarks pertaining to the suitability of the area theorem and its energy condition are made.

Gibbs-Curie-Wulff Theorem in Organic Materials: A Case Study on the Relationship between Surface Energy and Crystal Growth.

PubMed

Li, Rongjin; Zhang, Xiaotao; Dong, Huanli; Li, Qikai; Shuai, Zhigang; Hu, Wenping

2016-02-24

The equilibrium crystal shape and shape evolution of organic crystals are found to follow the Gibbs-Curie-Wulff theorem. Organic crystals are grown by the physical vapor transport technique and exhibit exactly the same shape as predicted by the Gibbs-Curie-Wulff theorem under optimal conditions. This accordance provides concrete proof for the theorem. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Quantum capacity of quantum black holes

NASA Astrophysics Data System (ADS)

Adami, Chris; Bradler, Kamil

2014-03-01

The fate of quantum entanglement interacting with a black hole has been an enduring mystery, not the least because standard curved space field theory does not address the interaction of black holes with matter. We discuss an effective Hamiltonian of matter interacting with a black hole that has a precise analogue in quantum optics and correctly reproduces both spontaneous and stimulated Hawking radiation with grey-body factors. We calculate the quantum capacity of this channel in the limit of perfect absorption, as well as in the limit of a perfectly reflecting black hole (a white hole). We find that the white hole is an optimal quantum cloner, and is isomorphic to the Unruh channel with positive quantum capacity. The complementary channel (across the horizon) is entanglement-breaking with zero capacity, avoiding a violation of the quantum no-cloning theorem. The black hole channel on the contrary has vanishing capacity, while its complement has positive capacity instead. Thus, quantum states can be reconstructed faithfully behind the black hole horizon, but not outside. This work sheds new light on black hole complementarity because it shows that black holes can both reflect and absorb quantum states without violating the no-cloning theorem, and makes quantum firewalls obsolete.

Control analysis for autonomously oscillating biochemical networks.

PubMed Central

Reijenga, Karin A; Westerhoff, Hans V; Kholodenko, Boris N; Snoep, Jacky L

2002-01-01

It has hitherto not been possible to analyze the control of oscillatory dynamic cellular processes in other than qualitative ways. The control coefficients, used in metabolic control analyses of steady states, cannot be applied directly to dynamic systems. We here illustrate a way out of this limitation that uses Fourier transforms to convert the time domain into the stationary frequency domain, and then analyses the control of limit cycle oscillations. In addition to the already known summation theorems for frequency and amplitude, we reveal summation theorems that apply to the control of average value, waveform, and phase differences of the oscillations. The approach is made fully operational in an analysis of yeast glycolytic oscillations. It follows an experimental approach, sampling from the model output and using discrete Fourier transforms of this data set. It quantifies the control of various aspects of the oscillations by the external glucose concentration and by various internal molecular processes. We show that the control of various oscillatory properties is distributed over the system enzymes in ways that differ among those properties. The models that are described in this paper can be accessed on http://jjj.biochem.sun.ac.za. PMID:11751299

A Note on a Sampling Theorem for Functions over GF(q)n Domain

NASA Astrophysics Data System (ADS)

Ukita, Yoshifumi; Saito, Tomohiko; Matsushima, Toshiyasu; Hirasawa, Shigeichi

In digital signal processing, the sampling theorem states that any real valued function ƒ can be reconstructed from a sequence of values of ƒ that are discretely sampled with a frequency at least twice as high as the maximum frequency of the spectrum of ƒ. This theorem can also be applied to functions over finite domain. Then, the range of frequencies of ƒ can be expressed in more detail by using a bounded set instead of the maximum frequency. A function whose range of frequencies is confined to a bounded set is referred to as bandlimited function. And a sampling theorem for bandlimited functions over Boolean domain has been obtained. Here, it is important to obtain a sampling theorem for bandlimited functions not only over Boolean domain (GF(q)n domain) but also over GF(q)n domain, where q is a prime power and GF(q) is Galois field of order q. For example, in experimental designs, although the model can be expressed as a linear combination of the Fourier basis functions and the levels of each factor can be represented by GF(q)n, the number of levels often take a value greater than two. However, the sampling theorem for bandlimited functions over GF(q)n domain has not been obtained. On the other hand, the sampling points are closely related to the codewords of a linear code. However, the relation between the parity check matrix of a linear code and any distinct error vectors has not been obtained, although it is necessary for understanding the meaning of the sampling theorem for bandlimited functions. In this paper, we generalize the sampling theorem for bandlimited functions over Boolean domain to a sampling theorem for bandlimited functions over GF(q)n domain. We also present a theorem for the relation between the parity check matrix of a linear code and any distinct error vectors. Lastly, we clarify the relation between the sampling theorem for functions over GF(q)n domain and linear codes.

High-Order Semi-Discrete Central-Upwind Schemes for Multi-Dimensional Hamilton-Jacobi Equations

NASA Technical Reports Server (NTRS)

Bryson, Steve; Levy, Doron; Biegel, Bryan (Technical Monitor)

2002-01-01

We present the first fifth order, semi-discrete central upwind method for approximating solutions of multi-dimensional Hamilton-Jacobi equations. Unlike most of the commonly used high order upwind schemes, our scheme is formulated as a Godunov-type scheme. The scheme is based on the fluxes of Kurganov-Tadmor and Kurganov-Tadmor-Petrova, and is derived for an arbitrary number of space dimensions. A theorem establishing the monotonicity of these fluxes is provided. The spacial discretization is based on a weighted essentially non-oscillatory reconstruction of the derivative. The accuracy and stability properties of our scheme are demonstrated in a variety of examples. A comparison between our method and other fifth-order schemes for Hamilton-Jacobi equations shows that our method exhibits smaller errors without any increase in the complexity of the computations.

GPS FOM Chimney Analysis using Generalized Extreme Value Distribution

NASA Technical Reports Server (NTRS)

Ott, Rick; Frisbee, Joe; Saha, Kanan

2004-01-01

Many a time an objective of a statistical analysis is to estimate a limit value like 3-sigma 95% confidence upper limit from a data sample. The generalized Extreme Value Distribution method can be profitably employed in many situations for such an estimate. . .. It is well known that according to the Central Limit theorem the mean value of a large data set is normally distributed irrespective of the distribution of the data from which the mean value is derived. In a somewhat similar fashion it is observed that many times the extreme value of a data set has a distribution that can be formulated with a Generalized Distribution. In space shuttle entry with 3-string GPS navigation the Figure Of Merit (FOM) value gives a measure of GPS navigated state accuracy. A GPS navigated state with FOM of 6 or higher is deemed unacceptable and is said to form a FOM 6 or higher chimney. A FOM chimney is a period of time during which the FOM value stays higher than 5. A longer period of FOM of value 6 or higher causes navigated state to accumulate more error for a lack of state update. For an acceptable landing it is imperative that the state error remains low and hence at low altitude during entry GPS data of FOM greater than 5 must not last more than 138 seconds. I To test the GPS performAnce many entry test cases were simulated at the Avionics Development Laboratory. Only high value FoM chimneys are consequential. The extreme value statistical technique is applied to analyze high value FOM chimneys. The Maximum likelihood method is used to determine parameters that characterize the GEV distribution, and then the limit value statistics are estimated.

The Holographic Electron Density Theorem, de-quantization, re-quantization, and nuclear charge space extrapolations of the Universal Molecule Model

NASA Astrophysics Data System (ADS)

Mezey, Paul G.

2017-11-01

Two strongly related theorems on non-degenerate ground state electron densities serve as the basis of "Molecular Informatics". The Hohenberg-Kohn theorem is a statement on global molecular information, ensuring that the complete electron density contains the complete molecular information. However, the Holographic Electron Density Theorem states more: the local information present in each and every positive volume density fragment is already complete: the information in the fragment is equivalent to the complete molecular information. In other words, the complete molecular information provided by the Hohenberg-Kohn Theorem is already provided, in full, by any positive volume, otherwise arbitrarily small electron density fragment. In this contribution some of the consequences of the Holographic Electron Density Theorem are discussed within the framework of the "Nuclear Charge Space" and the Universal Molecule Model. In the Nuclear Charge Space" the nuclear charges are regarded as continuous variables, and in the more general Universal Molecule Model some other quantized parameteres are also allowed to become "de-quantized and then re-quantized, leading to interrelations among real molecules through abstract molecules. Here the specific role of the Holographic Electron Density Theorem is discussed within the above context.

Generalized Dandelin’s Theorem

NASA Astrophysics Data System (ADS)

Kheyfets, A. L.

2017-11-01

The paper gives a geometric proof of the theorem which states that in case of the plane section of a second-order surface of rotation (quadrics of rotation, QR), such conics as an ellipse, a hyperbola or a parabola (types of conic sections) are formed. The theorem supplements the well-known Dandelin’s theorem which gives the geometric proof only for a circular cone and applies the proof to all QR, namely an ellipsoid, a hyperboloid, a paraboloid and a cylinder. That’s why the considered theorem is known as the generalized Dandelin’s theorem (GDT). The GDT proof is based on a relatively unknown generalized directrix definition (GDD) of conics. The work outlines the GDD proof for all types of conics as their necessary and sufficient condition. Based on the GDD, the author proves the GDT for all QR in case of a random position of the cutting plane. The graphical stereometric structures necessary for the proof are given. The implementation of the structures by 3d computer methods is considered. The article shows the examples of the builds made in the AutoCAD package. The theorem is intended for the training course of theoretical training of elite student groups of architectural and construction specialties.

The B-field soft theorem and its unification with the graviton and dilaton

NASA Astrophysics Data System (ADS)

Di Vecchia, Paolo; Marotta, Raffaele; Mojaza, Matin

2017-10-01

In theories of Einstein gravity coupled with a dilaton and a two-form, a soft theorem for the two-form, known as the Kalb-Ramond B-field, has so far been missing. In this work we fill the gap, and in turn formulate a unified soft theorem valid for gravitons, dilatons and B-fields in any tree-level scattering amplitude involving the three massless states. The new soft theorem is fixed by means of on-shell gauge invariance and enters at the subleading order of the graviton's soft theorem. In contrast to the subsubleading soft behavior of gravitons and dilatons, we show that the soft behavior of B-fields at this order cannot be fully fixed by gauge invariance. Nevertheless, we show that it is possible to establish a gauge invariant decomposition of the amplitudes to any order in the soft expansion. We check explicitly the new soft theorem in the bosonic string and in Type II superstring theories, and furthermore demonstrate that, at the next order in the soft expansion, totally gauge invariant terms appear in both string theories which cannot be factorized into a soft theorem.

Abel's theorem in the noncommutative case

NASA Astrophysics Data System (ADS)

Leitenberger, Frank

2004-03-01

We define noncommutative binary forms. Using the typical representation of Hermite we prove the fundamental theorem of algebra and we derive a noncommutative Cardano formula for cubic forms. We define quantized elliptic and hyperelliptic differentials of the first kind. Following Abel we prove Abel's theorem.

Impossible colorings and Bell's theorem

NASA Astrophysics Data System (ADS)

Aravind, P. K.

1999-11-01

An argument due to Zimba and Penrose is generalized to show how all known non-coloring proofs of the Bell-Kochen-Specker (BKS) theorem can be converted into inequality-free proofs of Bell's nonlocality theorem. A compilation of many such inequality-free proofs is given.

Understanding Rolle's Theorem

ERIC Educational Resources Information Center

Parameswaran, Revathy

2009-01-01

This paper reports on an experiment studying twelfth grade students' understanding of Rolle's Theorem. In particular, we study the influence of different concept images that students employ when solving reasoning tasks related to Rolle's Theorem. We argue that students' "container schema" and "motion schema" allow for rich…

An Application of the Perron-Frobenius Theorem to a Damage Model Problem.

DTIC Science & Technology

1985-04-01

RO-RI6I 20B AN APPLICATION OF THE PERRON - FROBENIUS THEOREM TO A ill I DAMAGOE MODEL PR BLEM.. (U) PITTSBURGH UNIV PA CENTER FOR I MULTIYARIATE...any copyright notation herein. * . .r * j * :h ~ ** . . .~. ~ % *~’ :. ~ ~ v 4 .% % %~ AN APPLICATION OF THE PERRON - FROBENIUS THEOREM TO A DAMAGE...University of Sheffield, U.K. S ~ Summry Using the Perron - Frobenius theorem, it is established that if’ (X,Y) is a random vector of non-negative

International Conference on Fixed Point Theory and Applications (Colloque International Theorie Du Point Fixe et Applications)

DTIC Science & Technology

1989-06-09

Theorem and the Perron - Frobenius Theorem in matrix theory. We use the Hahn-Banach theorem and do not use any fixed-point related concepts. 179 A...games defined b’, tions 87 Isac G. Fixed point theorems on convex cones , generalized pseudo-contractive mappings and the omplementarity problem 89...and (II), af(x) ° denotes the negative polar cone ot of(x). This condition are respectively called "inward" and "outward". Indeed, when X is convex

Numerical solution of linear and nonlinear Fredholm integral equations by using weighted mean-value theorem.

PubMed

Altürk, Ahmet

2016-01-01

Mean value theorems for both derivatives and integrals are very useful tools in mathematics. They can be used to obtain very important inequalities and to prove basic theorems of mathematical analysis. In this article, a semi-analytical method that is based on weighted mean-value theorem for obtaining solutions for a wide class of Fredholm integral equations of the second kind is introduced. Illustrative examples are provided to show the significant advantage of the proposed method over some existing techniques.

Markov Property of the Conformal Field Theory Vacuum and the a Theorem.

PubMed

Casini, Horacio; Testé, Eduardo; Torroba, Gonzalo

2017-06-30

We use strong subadditivity of entanglement entropy, Lorentz invariance, and the Markov property of the vacuum state of a conformal field theory to give new proof of the irreversibility of the renormalization group in d=4 space-time dimensions-the a theorem. This extends the proofs of the c and F theorems in dimensions d=2 and d=3 based on vacuum entanglement entropy, and gives a unified picture of all known irreversibility theorems in relativistic quantum field theory.

A Polarimetric Extension of the van Cittert-Zernike Theorem for Use with Microwave Interferometers

NASA Technical Reports Server (NTRS)

Piepmeier, J. R.; Simon, N. K.

2004-01-01

The van Cittert-Zernike theorem describes the Fourier-transform relationship between an extended source and its visibility function. Developments in classical optics texts use scalar field formulations for the theorem. Here, we develop a polarimetric extension to the van Cittert-Zernike theorem with applications to passive microwave Earth remote sensing. The development provides insight into the mechanics of two-dimensional interferometric imaging, particularly the effects of polarization basis differences between the scene and the observer.

Nonlocal Quantum Information Transfer Without Superluminal Signalling and Communication

NASA Astrophysics Data System (ADS)

Walleczek, Jan; Grössing, Gerhard

2016-09-01

It is a frequent assumption that—via superluminal information transfers—superluminal signals capable of enabling communication are necessarily exchanged in any quantum theory that posits hidden superluminal influences. However, does the presence of hidden superluminal influences automatically imply superluminal signalling and communication? The non-signalling theorem mediates the apparent conflict between quantum mechanics and the theory of special relativity. However, as a `no-go' theorem there exist two opposing interpretations of the non-signalling constraint: foundational and operational. Concerning Bell's theorem, we argue that Bell employed both interpretations, and that he finally adopted the operational position which is associated often with ontological quantum theory, e.g., de Broglie-Bohm theory. This position we refer to as "effective non-signalling". By contrast, associated with orthodox quantum mechanics is the foundational position referred to here as "axiomatic non-signalling". In search of a decisive communication-theoretic criterion for differentiating between "axiomatic" and "effective" non-signalling, we employ the operational framework offered by Shannon's mathematical theory of communication, whereby we distinguish between Shannon signals and non-Shannon signals. We find that an effective non-signalling theorem represents two sub-theorems: (1) Non-transfer-control (NTC) theorem, and (2) Non-signification-control (NSC) theorem. Employing NTC and NSC theorems, we report that effective, instead of axiomatic, non-signalling is entirely sufficient for prohibiting nonlocal communication. Effective non-signalling prevents the instantaneous, i.e., superluminal, transfer of message-encoded information through the controlled use—by a sender-receiver pair —of informationally-correlated detection events, e.g., in EPR-type experiments. An effective non-signalling theorem allows for nonlocal quantum information transfer yet—at the same time—effectively denies superluminal signalling and communication.

Nongeostrophic theory of zonally averaged circulation. I - Formulation

NASA Technical Reports Server (NTRS)

Tung, Ka Kit

1986-01-01

A nongeostrophic theory of zonally averaged circulation is formulated using the nonlinear primitive equations (mass conservation, thermodynamics, and zonal momentum) on a sphere. The relationship between the mean meridional circulation and diabatic heating rate is studied. Differences between results of nongeostropic theory and the geostrophic formulation concerning the role of eddy forcing of the diabatic circulation and the nonlinear nearly inviscid limit versus the geostrophic limit are discussed. Consideration is given to the Eliassen-Palm flux divergence, the Eliassen-Palm pseudodivergence, the nonacceleration theorem, and the nonlinear nongeostrophic Taylor relationship.

Nonlinear system theory: Another look at dependence

PubMed Central

Wu, Wei Biao

2005-01-01

Based on the nonlinear system theory, we introduce previously undescribed dependence measures for stationary causal processes. Our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems. The proposed dependence measures provide a natural framework for a limit theory for stationary processes. In particular, under conditions with quite simple forms, we present limit theorems for partial sums, empirical processes, and kernel density estimates. The conditions are mild and easily verifiable because they are directly related to the data-generating mechanisms. PMID:16179388

Non-Markovian State-Dependent Networks in Critical Loading

DTIC Science & Technology

2015-02-04

available for gener- alized Jackson networks; see Reiman [19]. Such limit theorems are useful to obtain approximations to various quantities of...2.1d))—so the limit process is an unconstrained diffusion; see Mandelbaum, Massey, and Reiman [13], Pang, Talreja, and Whitt[16], and references therein...standard critical loading condition that (λn − Rμn)/√n → λ2 − μ2 as n → ∞; cf. Reiman [19]. Lemma 2.1. Let condition (A0) hold and maxi∈IK supx∈IRK+(λ n i

Stationary moments, diffusion limits, and extinction times for logistic growth with random catastrophes.

PubMed

Schlomann, Brandon H

2018-06-06

A central problem in population ecology is understanding the consequences of stochastic fluctuations. Analytically tractable models with Gaussian driving noise have led to important, general insights, but they fail to capture rare, catastrophic events, which are increasingly observed at scales ranging from global fisheries to intestinal microbiota. Due to mathematical challenges, growth processes with random catastrophes are less well characterized and it remains unclear how their consequences differ from those of Gaussian processes. In the face of a changing climate and predicted increases in ecological catastrophes, as well as increased interest in harnessing microbes for therapeutics, these processes have never been more relevant. To better understand them, I revisit here a differential equation model of logistic growth coupled to density-independent catastrophes that arrive as a Poisson process, and derive new analytic results that reveal its statistical structure. First, I derive exact expressions for the model's stationary moments, revealing a single effective catastrophe parameter that largely controls low order statistics. Then, I use weak convergence theorems to construct its Gaussian analog in a limit of frequent, small catastrophes, keeping the stationary population mean constant for normalization. Numerically computing statistics along this limit shows how they transform as the dynamics shifts from catastrophes to diffusions, enabling quantitative comparisons. For example, the mean time to extinction increases monotonically by orders of magnitude, demonstrating significantly higher extinction risk under catastrophes than under diffusions. Together, these results provide insight into a wide range of stochastic dynamical systems important for ecology and conservation. Copyright © 2018 Elsevier Ltd. All rights reserved.

Analytical and numerical analysis of inverse optimization problems: conditions of uniqueness and computational methods

PubMed Central

Zatsiorsky, Vladimir M.

2011-01-01

One of the key problems of motor control is the redundancy problem, in particular how the central nervous system (CNS) chooses an action out of infinitely many possible. A promising way to address this question is to assume that the choice is made based on optimization of a certain cost function. A number of cost functions have been proposed in the literature to explain performance in different motor tasks: from force sharing in grasping to path planning in walking. However, the problem of uniqueness of the cost function(s) was not addressed until recently. In this article, we analyze two methods of finding additive cost functions in inverse optimization problems with linear constraints, so-called linear-additive inverse optimization problems. These methods are based on the Uniqueness Theorem for inverse optimization problems that we proved recently (Terekhov et al., J Math Biol 61(3):423–453, 2010). Using synthetic data, we show that both methods allow for determining the cost function. We analyze the influence of noise on the both methods. Finally, we show how a violation of the conditions of the Uniqueness Theorem may lead to incorrect solutions of the inverse optimization problem. PMID:21311907

Two dissimilar approaches to dynamical systems on hyper MV -algebras and their information entropy

NASA Astrophysics Data System (ADS)

Mehrpooya, Adel; Ebrahimi, Mohammad; Davvaz, Bijan

2017-09-01

Measuring the flow of information that is related to the evolution of a system which is modeled by applying a mathematical structure is of capital significance for science and usually for mathematics itself. Regarding this fact, a major issue in concern with hyperstructures is their dynamics and the complexity of the varied possible dynamics that exist over them. Notably, the dynamics and uncertainty of hyper MV -algebras which are hyperstructures and extensions of a central tool in infinite-valued Lukasiewicz propositional calculus that models many valued logics are of primary concern. Tackling this problem, in this paper we focus on the subject of dynamical systems on hyper MV -algebras and their entropy. In this respect, we adopt two varied approaches. One is the set-based approach in which hyper MV -algebra dynamical systems are developed by employing set functions and set partitions. By the other method that is based on points and point partitions, we establish the concept of hyper injective dynamical systems on hyper MV -algebras. Next, we study the notion of entropy for both kinds of systems. Furthermore, we consider essential ergodic characteristics of those systems and their entropy. In particular, we introduce the concept of isomorphic hyper injective and hyper MV -algebra dynamical systems, and we demonstrate that isomorphic systems have the same entropy. We present a couple of theorems in order to help calculate entropy. In particular, we prove a contemporary version of addition and Kolmogorov-Sinai Theorems. Furthermore, we provide a comparison between the indispensable properties of hyper injective and semi-independent dynamical systems. Specifically, we present and prove theorems that draw comparisons between the entropies of such systems. Lastly, we discuss some possible relationships between the theories of hyper MV -algebra and MV -algebra dynamical systems.

Migdal's theorem and electron-phonon vertex corrections in Dirac materials

NASA Astrophysics Data System (ADS)

Roy, Bitan; Sau, Jay D.; Das Sarma, S.

2014-04-01

Migdal's theorem plays a central role in the physics of electron-phonon interactions in metals and semiconductors, and has been extensively studied theoretically for parabolic band electronic systems in three-, two-, and one-dimensional systems over the last fifty years. In the current work, we theoretically study the relevance of Migdal's theorem in graphene and Weyl semimetals which are examples of 2D and 3D Dirac materials, respectively, with linear and chiral band dispersion. Our work also applies to 2D and 3D topological insulator systems. In Fermi liquids, the renormalization of the electron-phonon vertex scales as the ratio of sound (vs) to Fermi (vF) velocity, which is typically a small quantity. In two- and three-dimensional quasirelativistic systems, such as undoped graphene and Weyl semimetals, the one loop electron-phonon vertex renormalization, which also scales as η =vs/vF as η →0, is, however, enhanced by an ultraviolet logarithmic divergent correction, arising from the linear, chiral Dirac band dispersion. Such enhancement of the electron-phonon vertex can be significantly softened due to the logarithmic increment of the Fermi velocity, arising from the long range Coulomb interaction, and therefore, the electron-phonon vertex correction does not have a logarithmic divergence at low energy. Otherwise, the Coulomb interaction does not lead to any additional renormalization of the electron-phonon vertex. Therefore, electron-phonon vertex corrections in two- and three-dimensional Dirac fermionic systems scale as vs/vF0, where vF0 is the bare Fermi velocity, and small when vs≪vF0. These results, although explicitly derived for the intrinsic undoped systems, should hold even when the chemical potential is tuned away from the Dirac points.

32 bit digital optical computer - A hardware update

NASA Technical Reports Server (NTRS)

Guilfoyle, Peter S.; Carter, James A., III; Stone, Richard V.; Pape, Dennis R.

1990-01-01

Such state-of-the-art devices as multielement linear laser diode arrays, multichannel acoustooptic modulators, optical relays, and avalanche photodiode arrays, are presently applied to the implementation of a 32-bit supercomputer's general-purpose optical central processing architecture. Shannon's theorem, Morozov's control operator method (in conjunction with combinatorial arithmetic), and DeMorgan's law have been used to design an architecture whose 100 MHz clock renders it fully competitive with emerging planar-semiconductor technology. Attention is given to the architecture's multichannel Bragg cells, thermal design and RF crosstalk considerations, and the first and second anamorphic relay legs.

Adiabiatic invariants of the Kepler problem: an elementary treatment

NASA Astrophysics Data System (ADS)

Borghi, Riccardo

2013-09-01

An elementary introduction to the adiabatic invariants of the Kepler problem is proposed. Unlike the other didactical expositions already present in the literature, which are based on the Hamilton-Jacobi theory of mechanics, our derivation is suitable to be grasped even by first-year undergraduates. A central role in the present analysis is played by an elementary proof of the virial theorem for the Kepler problem which is based on the chain rule for derivatives. As a byproduct of our analysis, an interpretation of Keplerian orbit eccentricities in terms of the time average of the position vector direction is also provided.

On Euler's Theorem for Homogeneous Functions and Proofs Thereof.

ERIC Educational Resources Information Center

Tykodi, R. J.

1982-01-01

Euler's theorem for homogenous functions is useful when developing thermodynamic distinction between extensive and intensive variables of state and when deriving the Gibbs-Duhem relation. Discusses Euler's theorem and thermodynamic applications. Includes six-step instructional strategy for introducing the material to students. (Author/JN)

Ergodic theorem, ergodic theory, and statistical mechanics

PubMed Central

Moore, Calvin C.

2015-01-01

This perspective highlights the mean ergodic theorem established by John von Neumann and the pointwise ergodic theorem established by George Birkhoff, proofs of which were published nearly simultaneously in PNAS in 1931 and 1932. These theorems were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60-y-old fundamental problem of the subject—namely, the rationale for the hypothesis that time averages can be set equal to phase averages. The evolution of this problem is traced from the origins of statistical mechanics and Boltzman's ergodic hypothesis to the Ehrenfests' quasi-ergodic hypothesis, and then to the ergodic theorems. We discuss communications between von Neumann and Birkhoff in the Fall of 1931 leading up to the publication of these papers and related issues of priority. These ergodic theorems initiated a new field of mathematical-research called ergodic theory that has thrived ever since, and we discuss some of recent developments in ergodic theory that are relevant for statistical mechanics. PMID:25691697

From Einstein's theorem to Bell's theorem: a history of quantum non-locality

NASA Astrophysics Data System (ADS)

Wiseman, H. M.

2006-04-01

In this Einstein Year of Physics it seems appropriate to look at an important aspect of Einstein's work that is often down-played: his contribution to the debate on the interpretation of quantum mechanics. Contrary to physics ‘folklore’, Bohr had no defence against Einstein's 1935 attack (the EPR paper) on the claimed completeness of orthodox quantum mechanics. I suggest that Einstein's argument, as stated most clearly in 1946, could justly be called Einstein's reality locality completeness theorem, since it proves that one of these three must be false. Einstein's instinct was that completeness of orthodox quantum mechanics was the falsehood, but he failed in his quest to find a more complete theory that respected reality and locality. Einstein's theorem, and possibly Einstein's failure, inspired John Bell in 1964 to prove his reality locality theorem. This strengthened Einstein's theorem (but showed the futility of his quest) by demonstrating that either reality or locality is a falsehood. This revealed the full non-locality of the quantum world for the first time.

The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum

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

Alpay, Daniel, E-mail: dany@math.bgu.ac.il; Kimsey, David P., E-mail: dpkimsey@gmail.com; Colombo, Fabrizio, E-mail: fabrizio.colombo@polimi.it

In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of spectral analysis of quaternionic operators. The S-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators. The lack of a suitable notion ofmore » spectrum was a major obstruction to fully understand the spectral theorem for quaternionic normal operators. A prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.« less

Estimation of critical behavior from the density of states in classical statistical models

NASA Astrophysics Data System (ADS)

Malakis, A.; Peratzakis, A.; Fytas, N. G.

2004-12-01

We present a simple and efficient approximation scheme which greatly facilitates the extension of Wang-Landau sampling (or similar techniques) in large systems for the estimation of critical behavior. The method, presented in an algorithmic approach, is based on a very simple idea, familiar in statistical mechanics from the notion of thermodynamic equivalence of ensembles and the central limit theorem. It is illustrated that we can predict with high accuracy the critical part of the energy space and by using this restricted part we can extend our simulations to larger systems and improve the accuracy of critical parameters. It is proposed that the extensions of the finite-size critical part of the energy space, determining the specific heat, satisfy a scaling law involving the thermal critical exponent. The method is applied successfully for the estimation of the scaling behavior of specific heat of both square and simple cubic Ising lattices. The proposed scaling law is verified by estimating the thermal critical exponent from the finite-size behavior of the critical part of the energy space. The density of states of the zero-field Ising model on these lattices is obtained via a multirange Wang-Landau sampling.

The Kolmogorov-Obukhov Statistical Theory of Turbulence

NASA Astrophysics Data System (ADS)

Birnir, Björn

2013-08-01

In 1941 Kolmogorov and Obukhov postulated the existence of a statistical theory of turbulence, which allows the computation of statistical quantities that can be simulated and measured in a turbulent system. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier-Stokes equation. The additive noise in the stochastic Navier-Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov-Obukhov 1962 scaling hypothesis with the She-Leveque intermittency corrections. Then we compute the invariant measure of turbulence, writing the stochastic Navier-Stokes equation as an infinite-dimensional Ito process, and solving the linear Kolmogorov-Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.

Culture and the Trajectories of Developmental Pathology: Insights from Control and Information Theories.

PubMed

Wallace, Rodrick

2018-06-01

Cognition in living entities-and their social groupings or institutional artifacts-is necessarily as complicated as their embedding environments, which, for humans, includes a particularly rich cultural milieu. The asymptotic limit theorems of information and control theories permit construction of a new class of empirical 'regression-like' statistical models for cognitive developmental processes, their dynamics, and modes of dysfunction. Such models may, as have their simpler analogs, prove useful in the study and re-mediation of cognitive failure at and across the scales and levels of organization that constitute and drive the phenomena of life. These new models particularly focus on the roles of sociocultural environment and stress, in a large sense, as both trigger for the failure of the regulation of bio-cognition and as 'riverbanks' determining the channels of pathology, with implications across life-course developmental trajectories. We examine the effects of an embedding cultural milieu and its socioeconomic implementations using the 'lenses' of metabolic optimization, control system theory, and an extension of symmetry-breaking appropriate to information systems. A central implication is that most, if not all, human developmental disorders are fundamentally culture-bound syndromes. This has deep implications for both individual treatment and public health policy.

The Weierstrassian movement patterns of snails

PubMed Central

Santini, Giacomo; Chelazzi, Guido; Focardi, Stefano

2017-01-01

Weierstrassian Lévy walks are the archetypical form of random walk that do not satisfy the central limit theorem and are instead characterized by scale invariance. They were originally regarded as a mathematical abstraction but subsequent theoretical studies showed that they can, in principle, at least, be generated by chaos. Recently, Weierstrassian Lévy walks have been found to provide accurate representations of the movement patterns of mussels (Mytilus edulis) and mud snails (Hydrobia ulvae) recorded in the laboratory under controlled conditions. Here, we tested whether Weierstrassian Lévy walks and chaos are present under natural conditions in intertidal limpets Patella vulgata and P. rustica, and found that both characteristics are pervasive. We thereby show that Weierstrassian Lévy walks may be fundamental to how molluscs experience and interact with the world across a wide range of ecological contexts. We also show in an easily accessible way how chaos can produce a wide variety of Weierstrassian Lévy walk movement patterns. Our findings support the Lévy flight foraging hypothesis that posits that because Lévy walks can optimize search efficiencies, natural selection should have led to adaptations for Lévy walks. PMID:28680656

Random-fractal Ansatz for the configurations of two-dimensional critical systems

NASA Astrophysics Data System (ADS)

Lee, Ching Hua; Ozaki, Dai; Matsueda, Hiroaki

2016-12-01

Critical systems have always intrigued physicists and precipitated the development of new techniques. Recently, there has been renewed interest in the information contained in the configurations of classical critical systems, whose computation do not require full knowledge of the wave function. Inspired by holographic duality, we investigated the entanglement properties of the classical configurations (snapshots) of the Potts model by introducing an Ansatz ensemble of random fractal images. By virtue of the central limit theorem, our Ansatz accurately reproduces the entanglement spectra of actual Potts snapshots without any fine tuning of parameters or artificial restrictions on ensemble choice. It provides a microscopic interpretation of the results of previous studies, which established a relation between the scaling behavior of snapshot entropy and the critical exponent. More importantly, it elucidates the role of ensemble disorder in restoring conformal invariance, an aspect previously ignored. Away from criticality, the breakdown of scale invariance leads to a renormalization of the parameter Σ in the random fractal Ansatz, whose variation can be used as an alternative determination of the critical exponent. We conclude by providing a recipe for the explicit construction of fractal unit cells consistent with a given scaling exponent.

A Dynamic Intrusion Detection System Based on Multivariate Hotelling's T2 Statistics Approach for Network Environments

PubMed Central

Avalappampatty Sivasamy, Aneetha; Sundan, Bose

2015-01-01

The ever expanding communication requirements in today's world demand extensive and efficient network systems with equally efficient and reliable security features integrated for safe, confident, and secured communication and data transfer. Providing effective security protocols for any network environment, therefore, assumes paramount importance. Attempts are made continuously for designing more efficient and dynamic network intrusion detection models. In this work, an approach based on Hotelling's T2 method, a multivariate statistical analysis technique, has been employed for intrusion detection, especially in network environments. Components such as preprocessing, multivariate statistical analysis, and attack detection have been incorporated in developing the multivariate Hotelling's T2 statistical model and necessary profiles have been generated based on the T-square distance metrics. With a threshold range obtained using the central limit theorem, observed traffic profiles have been classified either as normal or attack types. Performance of the model, as evaluated through validation and testing using KDD Cup'99 dataset, has shown very high detection rates for all classes with low false alarm rates. Accuracy of the model presented in this work, in comparison with the existing models, has been found to be much better. PMID:26357668

A Dynamic Intrusion Detection System Based on Multivariate Hotelling's T2 Statistics Approach for Network Environments.

PubMed

Sivasamy, Aneetha Avalappampatty; Sundan, Bose

2015-01-01

The ever expanding communication requirements in today's world demand extensive and efficient network systems with equally efficient and reliable security features integrated for safe, confident, and secured communication and data transfer. Providing effective security protocols for any network environment, therefore, assumes paramount importance. Attempts are made continuously for designing more efficient and dynamic network intrusion detection models. In this work, an approach based on Hotelling's T(2) method, a multivariate statistical analysis technique, has been employed for intrusion detection, especially in network environments. Components such as preprocessing, multivariate statistical analysis, and attack detection have been incorporated in developing the multivariate Hotelling's T(2) statistical model and necessary profiles have been generated based on the T-square distance metrics. With a threshold range obtained using the central limit theorem, observed traffic profiles have been classified either as normal or attack types. Performance of the model, as evaluated through validation and testing using KDD Cup'99 dataset, has shown very high detection rates for all classes with low false alarm rates. Accuracy of the model presented in this work, in comparison with the existing models, has been found to be much better.

Elastic scattering of X-rays and gamma rays by 2S electrons in ions and neutral atoms

NASA Astrophysics Data System (ADS)

Costescu, A.; Spânulescu, S.; Stoica, C.

2012-08-01

The nonrelativistic limit of Rayleigh scattering amplitude on 2s electrons of neutral and partially ionized atoms is obtained by making use of the Green Function method. The result takes into consideration the retardation, relativistic kinematics and screening effects. The spurious singularities introduced by the retardation in a nonrelativistic approach are cancelled by the relativistic kinematics. For neutral and partially ionized atoms, a screening model is considered with an effective charge obtained by fitting the Hartree-Fock charge distribution with pure Coulombian wave functions corresponding to a central potential of a nucleus with Zeff as the atomic number. The total cross section of the photoeffect on the 2s electrons is also calculated from the imaginary part of the forward scattering amplitude by means of the optical theorem. The numerical results obtained are in a good agreement (10%) with the ones obtained by Kissell for the Rayleigh amplitude and by Scofield for the Photoeffect total cross section on the 2s electrons, for atoms with atomic number 18 ≤ Z ≤ 92 and photon energies ω≤αZm. (α=1/137,... is the fine structure constant, m is the electron mass).

Hydrostatic equilibrium of stars without electroneutrality constraint

NASA Astrophysics Data System (ADS)

Krivoruchenko, M. I.; Nadyozhin, D. K.; Yudin, A. V.

2018-04-01

The general solution of hydrostatic equilibrium equations for a two-component fluid of ions and electrons without a local electroneutrality constraint is found in the framework of Newtonian gravity theory. In agreement with the Poincaré theorem on analyticity and in the context of Dyson's argument, the general solution is demonstrated to possess a fixed (essential) singularity in the gravitational constant G at G =0 . The regular component of the general solution can be determined by perturbation theory in G starting from a locally neutral solution. The nonperturbative component obtained using the method of Wentzel, Kramers and Brillouin is exponentially small in the inner layers of the star and grows rapidly in the outward direction. Near the surface of the star, both components are comparable in magnitude, and their nonlinear interplay determines the properties of an electro- or ionosphere. The stellar charge varies within the limits of -0.1 to 150 C per solar mass. The properties of electro- and ionospheres are exponentially sensitive to variations of the fluid densities in the central regions of the star. The general solutions of two exactly solvable stellar models without a local electroneutrality constraint are also presented.

Bring the Pythagorean Theorem "Full Circle"

ERIC Educational Resources Information Center

Benson, Christine C.; Malm, Cheryl G.

2011-01-01

Middle school mathematics generally explores applications of the Pythagorean theorem and lays the foundation for working with linear equations. The Grade 8 Curriculum Focal Points recommend that students "apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and…

Using Discovery in the Calculus Class

ERIC Educational Resources Information Center

Shilgalis, Thomas W.

1975-01-01

This article shows how two discoverable theorems from elementary calculus can be presented to students in a manner that assists them in making the generalizations themselves. The theorems are the mean value theorems for derivatives and for integrals. A conjecture is suggested by pictures and then refined. (Author/KM)

Three Lectures on Theorem-proving and Program Verification

NASA Technical Reports Server (NTRS)

Moore, J. S.

1983-01-01

Topics concerning theorem proving and program verification are discussed with particlar emphasis on the Boyer/Moore theorem prover, and approaches to program verification such as the functional and interpreter methods and the inductive assertion approach. A history of the discipline and specific program examples are included.

On the far-field computation of acoustic radiation forces.

PubMed

Martin, P A

2017-10-01

It is known that the steady acoustic radiation force on a scatterer due to incident time-harmonic waves can be calculated by evaluating certain integrals of velocity potentials over a sphere surrounding the scatterer. The goal is to evaluate these integrals using far-field approximations and appropriate limits. Previous derivations are corrected, clarified, and generalized. Similar corrections are made to textbook derivations of optical theorems.

The importance of being equivalent: Newton's two models of one-body motion

NASA Astrophysics Data System (ADS)

Pourciau, Bruce

2004-05-01

As an undergraduate at Cambridge, Newton entered into his "Waste Book" an assumption that we have named the Equivalence Assumption (The Younger): "If a body move progressively in some crooked line [about a center of motion] ..., [then this] crooked line may bee conceived to consist of an infinite number of streight lines. Or else in any point of the croked line the motion may bee conceived to be on in the tangent". In this assumption, Newton somewhat imprecisely describes two mathematical models, a "polygonal limit model" and a "tangent deflected model", for "one-body motion", that is, for the motion of a "body in orbit about a fixed center", and then claims that these two models are equivalent. In the first part of this paper, we study the Principia to determine how the elder Newton would more carefully describe the polygonal limit and tangent deflected models. From these more careful descriptions, we then create Equivalence Assumption (The Elder), a precise interpretation of Equivalence Assumption (The Younger) as it might have been restated by Newton, after say 1687. We then review certain portions of the Waste Book and the Principia to make the case that, although Newton never restates nor even alludes to the Equivalence Assumption after his youthful Waste Book entry, still the polygonal limit and tangent deflected models, as well as an unspoken belief in their equivalence, infuse Newton's work on orbital motion. In particular, we show that the persuasiveness of the argument for the Area Property in Proposition 1 of the Principia depends crucially on the validity of Equivalence Assumption (The Elder). After this case is made, we present the mathematical analysis required to establish the validity of the Equivalence Assumption (The Elder). Finally, to illustrate the fundamental nature of the resulting theorem, the Equivalence Theorem as we call it, we present three significant applications: we use the Equivalence Theorem first to clarify and resolve questions related to Leibniz's "polygonal model" of one-body motion; then to repair Newton's argument for the Area Property in Proposition 1; and finally to clarify and resolve questions related to the transition from impulsive to continuous forces in "De motu" and the Principia.

Generalized chaos synchronization theorems for bidirectional differential equations and discrete systems with applications

NASA Astrophysics Data System (ADS)

Ji, Ye; Liu, Ting; Min, Lequan

2008-05-01

Two constructive generalized chaos synchronization (GCS) theorems for bidirectional differential equations and discrete systems are introduced. Using the two theorems, one can construct new chaos systems to make the system variables be in GCS. Five examples are presented to illustrate the effectiveness of the theoretical results.

The Law of Cosines for an "n"-Dimensional Simplex

ERIC Educational Resources Information Center

Ding, Yiren

2008-01-01

Using the divergence theorem technique of L. Eifler and N.H. Rhee, "The n-dimensional Pythagorean Theorem via the Divergence Theorem" (to appear: Amer. Math. Monthly), we extend the law of cosines for a triangle in a plane to an "n"-dimensional simplex in an "n"-dimensional space.

When 95% Accurate Isn't: Exploring Bayes's Theorem

ERIC Educational Resources Information Center

CadwalladerOlsker, Todd D.

2011-01-01

Bayes's theorem is notorious for being a difficult topic to learn and to teach. Problems involving Bayes's theorem (either implicitly or explicitly) generally involve calculations based on two or more given probabilities and their complements. Further, a correct solution depends on students' ability to interpret the problem correctly. Most people…

Computer Algebra Systems and Theorems on Real Roots of Polynomials

ERIC Educational Resources Information Center

Aidoo, Anthony Y.; Manthey, Joseph L.; Ward, Kim Y.

2010-01-01

A computer algebra system is used to derive a theorem on the existence of roots of a quadratic equation on any bounded real interval. This is extended to a cubic polynomial. We discuss how students could be led to derive and prove these theorems. (Contains 1 figure.)

Fluctuation theorem for Hamiltonian Systems: Le Chatelier's principle

NASA Astrophysics Data System (ADS)

Evans, Denis J.; Searles, Debra J.; Mittag, Emil

2001-05-01

For thermostated dissipative systems, the fluctuation theorem gives an analytical expression for the ratio of probabilities that the time-averaged entropy production in a finite system observed for a finite time takes on a specified value compared to the negative of that value. In the past, it has been generally thought that the presence of some thermostating mechanism was an essential component of any system that satisfies a fluctuation theorem. In the present paper, we point out that a fluctuation theorem can be derived for purely Hamiltonian systems, with or without applied dissipative fields.

Nambu-Goldstone theorem and spin-statistics theorem

NASA Astrophysics Data System (ADS)

Fujikawa, Kazuo

On December 19-21 in 2001, we organized a yearly workshop at Yukawa Institute for Theoretical Physics in Kyoto on the subject of "Fundamental Problems in Field Theory and their Implications". Prof. Yoichiro Nambu attended this workshop and explained a necessary modification of the Nambu-Goldstone theorem when applied to nonrelativistic systems. At the same workshop, I talked on a path integral formulation of the spin-statistics theorem. The present essay is on this memorable workshop, where I really enjoyed the discussions with Nambu, together with a short comment on the color freedom of quarks.

Counting Heron Triangles with Constraints

DTIC Science & Technology

2013-01-25

Heron triangle is an integer, then b is even, say b = 2b1. By Pythagoras ’ theorem , a4 = h2 +4b21, and since in a Heron triangle, the heights are always...our first result, which follows an idea of [10, Theorem 2.3]. Theorem 4. Let a, b be two fixed integers, and let ab be factored as in (1). Then H(a, b...which we derive the result. Theorem 4 immediately offers us an interesting observation regarding a special class of fixed sides (a, b). Corollary 5. If

On Pythagoras Theorem for Products of Spectral Triples

NASA Astrophysics Data System (ADS)

D'Andrea, Francesco; Martinetti, Pierre

2013-05-01

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non-pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and we provide non-unital counter-examples inspired by K-homology.

Which symmetry? Noether, Weyl, and conservation of electric charge

NASA Astrophysics Data System (ADS)

Brading, Katherine A.

In 1918, Emmy Noether published a (now famous) theorem establishing a general connection between continuous 'global' symmetries and conserved quantities. In fact, Noether's paper contains two theorems, and the second of these deals with 'local' symmetries; prima facie, this second theorem has nothing to do with conserved quantities. In the same year, Hermann Weyl independently made the first attempt to derive conservation of electric charge from a postulated gauge symmetry. In the light of Noether's work, it is puzzling that Weyl's argument uses local gauge symmetry. This paper explores the relationships between Weyl's work, Noether's two theorems, and the modern connection between gauge symmetry and conservation of electric charge. This includes showing that Weyl's connection is essentially an application of Noether's second theorem, with a novel twist.

Time Evolution of the Dynamical Variables of a Stochastic System.

ERIC Educational Resources Information Center

de la Pena, L.

1980-01-01

By using the method of moments, it is shown that several important and apparently unrelated theorems describing average properties of stochastic systems are in fact particular cases of a general law; this method is applied to generalize the virial theorem and the fluctuation-dissipation theorem to the time-dependent case. (Author/SK)

A Generalization of the Prime Number Theorem

ERIC Educational Resources Information Center

Bruckman, Paul S.

2008-01-01

In this article, the author begins with the prime number theorem (PNT), and then develops this into a more general theorem, of which many well-known number theoretic results are special cases, including PNT. He arrives at an asymptotic relation that allows the replacement of certain discrete sums involving primes into corresponding differentiable…

A Fascinating Application of Steiner's Theorem for Trapezium: Geometric Constructions Using Straightedge Alone

ERIC Educational Resources Information Center

Stupel, Moshe; Ben-Chaim, David

2013-01-01

Based on Steiner's fascinating theorem for trapezium, seven geometrical constructions using straight-edge alone are described. These constructions provide an excellent base for teaching theorems and the properties of geometrical shapes, as well as challenging thought and inspiring deeper insight into the world of geometry. In particular, this…

Leaning on Socrates to Derive the Pythagorean Theorem

ERIC Educational Resources Information Center

Percy, Andrew; Carr, Alistair

2010-01-01

The one theorem just about every student remembers from school is the theorem about the side lengths of a right angled triangle which Euclid attributed to Pythagoras when writing Proposition 47 of "The Elements". Usually first met in middle school, the student will be continually exposed throughout their mathematical education to the…

Unpacking Rouché's Theorem

ERIC Educational Resources Information Center

Howell, Russell W.; Schrohe, Elmar

2017-01-01

Rouché's Theorem is a standard topic in undergraduate complex analysis. It is usually covered near the end of the course with applications relating to pure mathematics only (e.g., using it to produce an alternate proof of the Fundamental Theorem of Algebra). The "winding number" provides a geometric interpretation relating to the…

Geometry of the Adiabatic Theorem

ERIC Educational Resources Information Center

Lobo, Augusto Cesar; Ribeiro, Rafael Antunes; Ribeiro, Clyffe de Assis; Dieguez, Pedro Ruas

2012-01-01

We present a simple and pedagogical derivation of the quantum adiabatic theorem for two-level systems (a single qubit) based on geometrical structures of quantum mechanics developed by Anandan and Aharonov, among others. We have chosen to use only the minimum geometric structure needed for the understanding of the adiabatic theorem for this case.…

The Classical Version of Stokes' Theorem Revisited

ERIC Educational Resources Information Center

Markvorsen, Steen

2008-01-01

Using only fairly simple and elementary considerations--essentially from first year undergraduate mathematics--we show how the classical Stokes' theorem for any given surface and vector field in R[superscript 3] follows from an application of Gauss' divergence theorem to a suitable modification of the vector field in a tubular shell around the…

The Parity Theorem Shuffle

ERIC Educational Resources Information Center

Smith, Michael D.

2016-01-01

The Parity Theorem states that any permutation can be written as a product of transpositions, but no permutation can be written as a product of both an even number and an odd number of transpositions. Most proofs of the Parity Theorem take several pages of mathematical formalism to complete. This article presents an alternative but equivalent…

Virtual continuity of measurable functions and its applications

NASA Astrophysics Data System (ADS)

Vershik, A. M.; Zatitskii, P. B.; Petrov, F. V.

2014-12-01

A classical theorem of Luzin states that a measurable function of one real variable is `almost' continuous. For measurable functions of several variables the analogous statement (continuity on a product of sets having almost full measure) does not hold in general. The search for a correct analogue of Luzin's theorem leads to a notion of virtually continuous functions of several variables. This apparently new notion implicitly appears in the statements of embedding theorems and trace theorems for Sobolev spaces. In fact it reveals the nature of such theorems as statements about virtual continuity. The authors' results imply that under the conditions of Sobolev theorems there is a well-defined integration of a function with respect to a wide class of singular measures, including measures concentrated on submanifolds. The notion of virtual continuity is also used for the classification of measurable functions of several variables and in some questions on dynamical systems, the theory of polymorphisms, and bistochastic measures. In this paper the necessary definitions and properties of admissible metrics are recalled, several definitions of virtual continuity are given, and some applications are discussed. Bibliography: 24 titles.

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

Koenig, Robert; Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125; Mitchison, Graeme

In its most basic form, the finite quantum de Finetti theorem states that the reduced k-partite density operator of an n-partite symmetric state can be approximated by a convex combination of k-fold product states. Variations of this result include Renner's 'exponential' approximation by 'almost-product' states, a theorem which deals with certain triples of representations of the unitary group, and the result of D'Cruz et al. [e-print quant-ph/0606139;Phys. Rev. Lett. 98, 160406 (2007)] for infinite-dimensional systems. We show how these theorems follow from a single, general de Finetti theorem for representations of symmetry groups, each instance corresponding to a particular choicemore » of symmetry group and representation of that group. This gives some insight into the nature of the set of approximating states and leads to some new results, including an exponential theorem for infinite-dimensional systems.« less

The Levy sections theorem revisited

NASA Astrophysics Data System (ADS)

Figueiredo, Annibal; Gleria, Iram; Matsushita, Raul; Da Silva, Sergio

2007-06-01

This paper revisits the Levy sections theorem. We extend the scope of the theorem to time series and apply it to historical daily returns of selected dollar exchange rates. The elevated kurtosis usually observed in such series is then explained by their volatility patterns. And the duration of exchange rate pegs explains the extra elevated kurtosis in the exchange rates of emerging markets. In the end, our extension of the theorem provides an approach that is simpler than the more common explicit modelling of fat tails and dependence. Our main purpose is to build up a technique based on the sections that allows one to artificially remove the fat tails and dependence present in a data set. By analysing data through the lenses of the Levy sections theorem one can find common patterns in otherwise very different data sets.

Tutorial on Fourier space coverage for scattering experiments, with application to SAR

NASA Astrophysics Data System (ADS)

Deming, Ross W.

2010-04-01

The Fourier Diffraction Theorem relates the data measured during electromagnetic, optical, or acoustic scattering experiments to the spatial Fourier transform of the object under test. The theorem is well-known, but since it is based on integral equations and complicated mathematical expansions, the typical derivation may be difficult for the non-specialist. In this paper, the theorem is derived and presented using simple geometry, plus undergraduatelevel physics and mathematics. For practitioners of synthetic aperture radar (SAR) imaging, the theorem is important to understand because it leads to a simple geometric and graphical understanding of image resolution and sampling requirements, and how they are affected by radar system parameters and experimental geometry. Also, the theorem can be used as a starting point for imaging algorithms and motion compensation methods. Several examples are given in this paper for realistic scenarios.

Cellular compartmentation follows rules: The Schnepf theorem, its consequences and exceptions: A biological membrane separates a plasmatic from a non-plasmatic phase.

PubMed

Moog, Daniel; Maier, Uwe G

2017-08-01

Is the spatial organization of membranes and compartments within cells subjected to any rules? Cellular compartmentation differs between prokaryotic and eukaryotic life, because it is present to a high degree only in eukaryotes. In 1964, Prof. Eberhard Schnepf formulated the compartmentation rule (Schnepf theorem), which posits that a biological membrane, the main physical structure responsible for cellular compartmentation, usually separates a plasmatic form a non-plasmatic phase. Here we review and re-investigate the Schnepf theorem by applying the theorem to different cellular structures, from bacterial cells to eukaryotes with their organelles and compartments. In conclusion, we can confirm the general correctness of the Schnepf theorem, noting explicit exceptions only in special cases such as endosymbiosis and parasitism. © 2017 WILEY Periodicals, Inc.

Estimating the boundaries of a limit cycle in a 2D dynamical system using renormalization group

NASA Astrophysics Data System (ADS)

Dutta, Ayan; Das, Debapriya; Banerjee, Dhruba; Bhattacharjee, Jayanta K.

2018-04-01

While the plausibility of formation of limit cycle has been a well studied topic in context of the Poincare-Bendixson theorem, studies on estimates in regard to the possible size and shape of the limit cycle seem to be scanty in the literature. In this paper we present a pedagogical study of some aspects of the size of this limit cycle using perturbative renormalization group by doing detailed and explicit calculations upto second order for the Selkov model for glycolytic oscillations. This famous model is well known to lead to a limit cycle for certain ranges of values of the parameters involved in the problem. Within the tenets of the approximations made, reasonable agreement with the numerical plots can be achieved.

Guided discovery of the nine-point circle theorem and its proof

NASA Astrophysics Data System (ADS)

Buchbinder, Orly

2018-01-01

The nine-point circle theorem is one of the most beautiful and surprising theorems in Euclidean geometry. It establishes an existence of a circle passing through nine points, all of which are related to a single triangle. This paper describes a set of instructional activities that can help students discover the nine-point circle theorem through investigation in a dynamic geometry environment, and consequently prove it using a method of guided discovery. The paper concludes with a variety of suggestions for the ways in which the whole set of activities can be implemented in geometry classrooms.

Kato type operators and Weyl's theorem

NASA Astrophysics Data System (ADS)

Duggal, B. P.; Djordjevic, S. V.; Kubrusly, Carlos

2005-09-01

A Banach space operator T satisfies Weyl's theorem if and only if T or T* has SVEP at all complex numbers [lambda] in the complement of the Weyl spectrum of T and T is Kato type at all [lambda] which are isolated eigenvalues of T of finite algebraic multiplicity. If T* (respectively, T) has SVEP and T is Kato type at all [lambda] which are isolated eigenvalues of T of finite algebraic multiplicity (respectively, T is Kato type at all [lambda][set membership, variant]iso[sigma](T)), then T satisfies a-Weyl's theorem (respectively, T* satisfies a-Weyl's theorem).

Cooperation Among Theorem Provers

NASA Technical Reports Server (NTRS)

Waldinger, Richard J.

1998-01-01

In many years of research, a number of powerful theorem-proving systems have arisen with differing capabilities and strengths. Resolution theorem provers (such as Kestrel's KITP or SRI's SNARK) deal with first-order logic with equality but not the principle of mathematical induction. The Boyer-Moore theorem prover excels at proof by induction but cannot deal with full first-order logic. Both are highly automated but cannot accept user guidance easily. The purpose of this project, and the companion project at Kestrel, has been to use the category-theoretic notion of logic morphism to combine systems with different logics and languages.

Nambu-Goldstone theorem and spin-statistics theorem

NASA Astrophysics Data System (ADS)

Fujikawa, Kazuo

2016-05-01

On December 19-21 in 2001, we organized a yearly workshop at Yukawa Institute for Theoretical Physics in Kyoto on the subject of “Fundamental Problems in Field Theory and their Implications”. Prof. Yoichiro Nambu attended this workshop and explained a necessary modification of the Nambu-Goldstone theorem when applied to non-relativistic systems. At the same workshop, I talked on a path integral formulation of the spin-statistics theorem. The present essay is on this memorable workshop, where I really enjoyed the discussions with Nambu, together with a short comment on the color freedom of quarks.

Solving a Class of Spatial Reasoning Problems: Minimal-Cost Path Planning in the Cartesian Plane.

DTIC Science & Technology

1987-06-01

as in Figure 72. By the Theorem of Pythagoras : Z1 <a z 2 < C Yl(bl+b 2)uI, the cost of going along (a,b,c) is greater that the...preceding lemmas to an indefinite number of boundary-crossing episodes is accomplished by the following theorems . Theorem 1 extends the result of Lemma 1... Theorem 1: Any two Snell’s-law paths within a K-explored wedge defined by Snell’s-law paths RL and R. do not intersect within the K-explored portion of

Virtual Volatility, an Elementary New Concept with Surprising Stock Market Consequences

NASA Astrophysics Data System (ADS)

Prange, Richard; Silva, A. Christian

2006-03-01

Textbook investors start by predicting the future price distribution, PDF, of a candidate stock (or portfolio) at horizon T, e.g. a year hence. A (log)normal PDF with center (=drift =expected return) μT and width (=volatility) σT is often assumed on Central Limit Theorem grounds, i.e. by a random walk of daily (log)price increments δs. The standard deviation, stdev, of historical (ex post) δs `s is usually a fair predictor of the coming year's (ex ante) stdev(δs) = σdaily, but the historical mean E(δs) at best roughly limits the true, to be predicted, drift by μtrueT˜ μhistT ± σhistT. Textbooks take a PDF with σ ˜ σdaily and μ as somehow known, as if accurate predictions of μ were possible. It is elementary and presumably new to argue that an average of PDF's over a range of μ values should be taken, e.g. an average over forecasts by different analysts. We estimate that this leads to a PDF with a `virtual' volatility σ ˜ 1.3σdaily. It is indeed clear that uncertainty in the value of the expected gain parameter increases the risk of investment in that security by most measures, e. g. Sharpe's ratio μT/σT will be 30% smaller because of this effect. It is significant and surprising that there are investments which benefit from this 30% virtual increase in the volatility

A stochastic HMM-based forecasting model for fuzzy time series.

PubMed

Li, Sheng-Tun; Cheng, Yi-Chung

2010-10-01

Recently, fuzzy time series have attracted more academic attention than traditional time series due to their capability of dealing with the uncertainty and vagueness inherent in the data collected. The formulation of fuzzy relations is one of the key issues affecting forecasting results. Most of the present works adopt IF-THEN rules for relationship representation, which leads to higher computational overhead and rule redundancy. Sullivan and Woodall proposed a Markov-based formulation and a forecasting model to reduce computational overhead; however, its applicability is limited to handling one-factor problems. In this paper, we propose a novel forecasting model based on the hidden Markov model by enhancing Sullivan and Woodall's work to allow handling of two-factor forecasting problems. Moreover, in order to make the nature of conjecture and randomness of forecasting more realistic, the Monte Carlo method is adopted to estimate the outcome. To test the effectiveness of the resulting stochastic model, we conduct two experiments and compare the results with those from other models. The first experiment consists of forecasting the daily average temperature and cloud density in Taipei, Taiwan, and the second experiment is based on the Taiwan Weighted Stock Index by forecasting the exchange rate of the New Taiwan dollar against the U.S. dollar. In addition to improving forecasting accuracy, the proposed model adheres to the central limit theorem, and thus, the result statistically approximates to the real mean of the target value being forecast.

Discovering Theorems in Abstract Algebra Using the Software "GAP"

ERIC Educational Resources Information Center

Blyth, Russell D.; Rainbolt, Julianne G.

2010-01-01

A traditional abstract algebra course typically consists of the professor stating and then proving a sequence of theorems. As an alternative to this classical structure, the students could be expected to discover some of the theorems even before they are motivated by classroom examples. This can be done by using a software system to explore a…

Bell's Theorem and Einstein's "Spooky Actions" from a Simple Thought Experiment

ERIC Educational Resources Information Center

Kuttner, Fred; Rosenblum, Bruce

2010-01-01

In 1964 John Bell proved a theorem allowing the experimental test of whether what Einstein derided as "spooky actions at a distance" actually exist. We will see that they "do". Bell's theorem can be displayed with a simple, nonmathematical thought experiment suitable for a physics course at "any" level. And a simple, semi-classical derivation of…

Unique Factorization and the Fundamental Theorem of Arithmetic

ERIC Educational Resources Information Center

Sprows, David

2017-01-01

The fundamental theorem of arithmetic is one of those topics in mathematics that somehow "falls through the cracks" in a student's education. When asked to state this theorem, those few students who are willing to give it a try (most have no idea of its content) will say something like "every natural number can be broken down into a…

Viète's Formula and an Error Bound without Taylor's Theorem

ERIC Educational Resources Information Center

Boucher, Chris

2018-01-01

This note presents a derivation of Viète's classic product approximation of pi that relies on only the Pythagorean Theorem. We also give a simple error bound for the approximation that, while not optimal, still reveals the exponential convergence of the approximation and whose derivation does not require Taylor's Theorem.

A Physical Proof of the Pythagorean Theorem

ERIC Educational Resources Information Center

Treeby, David

2017-01-01

What proof of the Pythagorean theorem might appeal to a physics teacher? A proof that involved the notion of mass would surely be of interest. While various proofs of the Pythagorean theorem employ the circumcenter and incenter of a right-angled triangle, we are not aware of any proof that uses the triangle's center of mass. This note details one…

Research in Stochastic Processes

DTIC Science & Technology

1988-10-10

To appear in Proceedings Volume, Oberwolfach Conf. on Extremal Value Theory, Ed. J. HUsler and R. Reiss, Springer. 4. M.R. Leadbetter. The exceedance...Hsing, J. Husler and M.R. Leadbetter, On the exceedance point process for a stationary sequence, Probability Theor. Rel. Fields, 20, 1988, 97-112 Z.J...Oberwotfach Conf. on Extreme Value Theory. J. Husler and R. Reiss. eds.. Springer. to appear V. Mandrekar, On a limit theorem and invariance

The physics of the earth's core: An introduction

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

Melchior, P.

1986-01-01

This book is a reference text providing information on physical topics of recent developments in internal geophysics. The text summarizes papers covering theoretical geophysics. Basic formulae, definitions and theorems are not explained in detail due to the limited space. The contents include applications to geodesy, geophysics, astronomy, astrophysics, geophysics and planetary physics. The formal contents include: The Earth's model; Thermodynamics; Hydrodynamics; Geomagnetism; Geophysical implications in the Earth's core.

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

Rizwan-uddin

Recently, various branches of engineering and science have seen a rapid increase in the number of dynamical analyses undertaken. This modern phenomenon often obscures the fact that such analyses were sometimes carried out even before the current trend began. Moreover, these earlier analyses, which even now seem very ingenuous, were carried out at a time when the available information about dynamical systems was not as well disseminated as it is today. One such analysis, carried out in the early 1960s, showed the existence of stable limit cycles in a simple model for space-independent xenon dynamics in nuclear reactors. The authors,more » apparently unaware of the now well-known bifurcation theorem by Hopf, could not numerically discover unstable limit cycles, though they did find regions in parameter space where the fixed points are stable for small perturbations but unstable for very large perturbations. The analysis was carried out both analytically and numerically. As a tribute to these early nonlinear dynamicists in the field of nuclear engineering, in this paper, the Hopf theorem and its conclusions are briefly described, and then the solution of the space-independent xenon oscillation problem is presented, which was obtained using the bifurcation analysis BIFDD code. These solutions are presented along with a discussion of the earlier results.« less

Deliquescence and efflorescence of small particles.

PubMed

McGraw, Robert; Lewis, Ernie R

2009-11-21

We examine size-dependent deliquescence/efflorescence phase transformation for particles down to several nanometers in size. Thermodynamic properties of inorganic salt particles, coated with aqueous solution layers of varying thickness and surrounded by vapor, are analyzed. A thin layer criterion (TLC) is introduced to define a limiting deliquescence relative humidity (RH(D)) for small particles. This requires: (1) equality of chemical potentials between salt in an undissolved core, and thin adsorbed solution layer, and (2) equality of chemical potentials between water in the thin layer and vapor phase. The usual bulk deliquescence conditions are recovered in the limit of large dry particle size. Nanosize particles are found to deliquesce at relative humidity just below the RH(D) on crossing a nucleation barrier, located at a critical solution layer thickness. This barrier vanishes precisely at the RH(D) defined by the TLC. Concepts and methods from nucleation theory including the kinetic potential, self-consistent nucleation theory, nucleation theorems, and the Gibbs dividing surface provide theoretical foundation and point to unifying features of small particle deliquescence/efflorescence processes. These include common thermodynamic area constructions, useful for interpretation of small particle water uptake measurements, and a common free-energy surface, with constant RH cross sections describing deliquescence and efflorescence related through the nucleation theorem.

Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon

NASA Astrophysics Data System (ADS)

Kay, Bernard S.; Radzikowski, Marek J.; Wald, Robert M.

1997-02-01

We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, (M,g_{ab}), with a compactly generated Cauchy horizon. These theorems demonstrate the breakdown of the theory at certain base points of the Cauchy horizon, which are defined as 'past terminal accumulation points' of the horizon generators. Thus, the theorems may be interpreted as giving support to Hawking's 'Chronology Protection Conjecture', according to which the laws of physics prevent one from manufacturing a 'time machine'. Specifically, we prove: Theorem 1. There is no extension to (M,g_{ab}) of the usual field algebra on the initial globally hyperbolic region which satisfies the condition of F-locality at any base point. In other words, any extension of the field algebra must, in any globally hyperbolic neighbourhood of any base point, differ from the algebra one would define on that neighbourhood according to the rules for globally hyperbolic spacetimes. Theorem 2. The two-point distribution for any Hadamard state defined on the initial globally hyperbolic region must (when extended to a distributional bisolution of the covariant Klein-Gordon equation on the full spacetime) be singular at every base point x in the sense that the difference between this two point distribution and a local Hadamard distribution cannot be given by a bounded function in any neighbourhood (in M 2 M) of (x,x). In consequence of Theorem 2, quantities such as the renormalized expectation value of J2 or of the stress-energy tensor are necessarily ill-defined or singular at any base point. The proof of these theorems relies on the 'Propagation of Singularities' theorems of Duistermaat and Hörmander.

Enter the reverend: introduction to and application of Bayes' theorem in clinical ophthalmology.

PubMed

Thomas, Ravi; Mengersen, Kerrie; Parikh, Rajul S; Walland, Mark J; Muliyil, Jayprakash

2011-12-01

Ophthalmic practice utilizes numerous diagnostic tests, some of which are used to screen for disease. Interpretation of test results and many clinical management issues are actually problems in inverse probability that can be solved using Bayes' theorem. Use two-by-two tables to understand Bayes' theorem and apply it to clinical examples. Specific examples of the utility of Bayes' theorem in diagnosis and management. Two-by-two tables are used to introduce concepts and understand the theorem. The application in interpretation of diagnostic tests is explained. Clinical examples demonstrate its potential use in making management decisions. Positive predictive value and conditional probability. The theorem demonstrates the futility of testing when prior probability of disease is low. Application to untreated ocular hypertension demonstrates that the estimate of glaucomatous optic neuropathy is similar to that obtained from the Ocular Hypertension Treatment Study. Similar calculations are used to predict the risk of acute angle closure in a primary angle closure suspect, the risk of pupillary block in a diabetic undergoing cataract surgery, and the probability that an observed decrease in intraocular pressure is due to the medication that has been started. The examples demonstrate how data required for management can at times be easily obtained from available information. Knowledge of Bayes' theorem helps in interpreting test results and supports the clinical teaching that testing for conditions with a low prevalence has a poor predictive value. In some clinical situations Bayes' theorem can be used to calculate vital data required for patient management. © 2011 The Authors. Clinical and Experimental Ophthalmology © 2011 Royal Australian and New Zealand College of Ophthalmologists.

Thermal Noise Limit in Frequency Stabilization of Lasers with Rigid Cavities

NASA Technical Reports Server (NTRS)

Numata, Kenji; Kemery, Amy; Camp, Jordan

2004-01-01

We evaluated thermal noise (Brownian motion) in a rigid reference cavity used for frequency stabilization of lasers, based on the mechanical loss of cavity materials and the numerical analysis of the mirror-spacer mechanics with t.he direct application of the fluctuation dissipation theorem. This noise sets a fundamental limit for the frequency stability achieved with a rigid frequency- reference cavity of order 1 Hz/square root Hz(0.01 Hz/square root Hz) at 10 mHz (100 Hz) at room temperature. This level coincides with the world-highest level stabilization results.

On the adiabatic limit of Hadamard states

NASA Astrophysics Data System (ADS)

Drago, Nicolò; Gérard, Christian

2017-08-01

We consider the adiabatic limit of Hadamard states for free quantum Klein-Gordon fields, when the background metric and the field mass are slowly varied from their initial to final values. If the Klein-Gordon field stays massive, we prove that the adiabatic limit of the initial vacuum state is the (final) vacuum state, by extending to the symplectic framework the adiabatic theorem of Avron-Seiler-Yaffe. In cases when only the field mass is varied, using an abstract version of the mode decomposition method we can also consider the case when the initial or final mass vanishes, and the initial state is either a thermal state or a more general Hadamard state.

Communication. Kinetics of scavenging of small, nucleating clusters. First nucleation theorem and sum rules

DOE PAGES

Malila, Jussi; McGraw, Robert; Laaksonen, Ari; ...

2015-01-07

Despite recent advances in monitoring nucleation from a vapor at close-to-molecular resolution, the identity of the critical cluster, forming the bottleneck for the nucleation process, remains elusive. During past twenty years, the first nucleation theorem has been often used to extract the size of the critical cluster from nucleation rate measurements. However, derivations of the first nucleation theorem invoke certain questionable assumptions that may fail, e.g., in the case of atmospheric new particle formation, including absence of subcritical cluster losses and heterogeneous nucleation on pre-existing nanoparticles. Here we extend the kinetic derivation of the first nucleation theorem to give amore » general framework to include such processes, yielding sum rules connecting the size dependent particle formation and loss rates to the corresponding loss-free nucleation rate and the apparent critical size from a naïve application of the first nucleation theorem that neglects them.« less

A new blackhole theorem and its applications to cosmology and astrophysics

NASA Astrophysics Data System (ADS)

Wang, Shouhong; Ma, Tian

2015-04-01

We shall present a blackhole theorem and a theorem on the structure of our Universe, proved in a recently published paper, based on 1) the Einstein general theory of relativity, and 2) the cosmological principle that the universe is homogeneous and isotropic. These two theorems are rigorously proved using astrophysical dynamical models coupling fluid dynamics and general relativity based on a symmetry-breaking principle. With the new blackhole theorem, we further demonstrate that both supernovae explosion and AGN jets, as well as many astronomical phenomena including e.g. the recent reported are due to combined relativistic, magnetic and thermal effects. The radial temperature gradient causes vertical Benard type convection cells, and the relativistic viscous force (via electromagnetic, the weak and the strong interactions) gives rise to a huge explosive radial force near the Schwarzschild radius, leading e.g. to supernovae explosion and AGN jets.

Atiyah-Patodi-Singer index theorem for domain-wall fermion Dirac operator

NASA Astrophysics Data System (ADS)

Fukaya, Hidenori; Onogi, Tetsuya; Yamaguchi, Satoshi

2018-03-01

Recently, the Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. Although it is widely applied to physics, the mathematical set-up in the original APS index theorem is too abstract and general (allowing non-trivial metric and so on) and also the connection between the APS boundary condition and the physical boundary condition on the surface of topological material is unclear. For this reason, in contrast to the Atiyah-Singer index theorem, derivation of the APS index theorem in physics language is still missing. In this talk, we attempt to reformulate the APS index in a "physicist-friendly" way, similar to the Fujikawa method on closed manifolds, for our familiar domain-wall fermion Dirac operator in a flat Euclidean space. We find that the APS index is naturally embedded in the determinant of domain-wall fermions, representing the so-called anomaly descent equations.

The detailed balance principle and the reciprocity theorem between photocarrier collection and dark carrier distribution in solar cells

NASA Astrophysics Data System (ADS)

Rau, Uwe; Brendel, Rolf

1998-12-01

It is shown that a recently described general relationship between the local collection efficiency of solar cells and the dark carrier concentration (reciprocity theorem) directly follows from the principle of detailed balance. We derive the relationship for situations where transport of charge carriers occurs between discrete states as well as for the situation where electronic transport is described in terms of continuous functions. Combining both situations allows to extend the range of applicability of the reciprocity theorem to all types of solar cells, including, e.g., metal-insulator-semiconductor-type, electrochemical solar cells, as well as the inclusion of the impurity photovoltaic effect. We generalize the theorem further to situations where the occupation probability of electronic states is governed by Fermi-Dirac statistics instead of Boltzmann statistics as underlying preceding work. In such a situation the reciprocity theorem is restricted to small departures from equilibrium.

Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state.

PubMed

Gieseler, Jan; Quidant, Romain; Dellago, Christoph; Novotny, Lukas

2014-05-01

Fluctuation theorems are a generalization of thermodynamics on small scales and provide the tools to characterize the fluctuations of thermodynamic quantities in non-equilibrium nanoscale systems. They are particularly important for understanding irreversibility and the second law in fundamental chemical and biological processes that are actively driven, thus operating far from thermal equilibrium. Here, we apply the framework of fluctuation theorems to investigate the important case of a system relaxing from a non-equilibrium state towards equilibrium. Using a vacuum-trapped nanoparticle, we demonstrate experimentally the validity of a fluctuation theorem for the relative entropy change occurring during relaxation from a non-equilibrium steady state. The platform established here allows non-equilibrium fluctuation theorems to be studied experimentally for arbitrary steady states and can be extended to investigate quantum fluctuation theorems as well as systems that do not obey detailed balance.

Exploiting structure: Introduction and motivation

NASA Technical Reports Server (NTRS)

Xu, Zhong Ling

1994-01-01

This annual report summarizes the research activities that were performed from 26 Jun. 1993 to 28 Feb. 1994. We continued to investigate the Robust Stability of Systems where transfer functions or characteristic polynomials are affine multilinear functions of parameters. An approach that differs from 'Stability by Linear Process' and that reduces the computational burden of checking the robust stability of the system with multilinear uncertainty was found for low order, 2-order, and 3-order cases. We proved a crucial theorem, the so-called Face Theorem. Previously, we have proven Kharitonov's Vertex Theorem and the Edge Theorem by Bartlett. The detail of this proof is contained in the Appendix. This Theorem provides a tool to describe the boundary of the image of the affine multilinear function. For SPR design, we have developed some new results. The third objective for this period is to design a controller for IHM by the H-infinity optimization technique. The details are presented in the Appendix.

An Integrated Environment for Efficient Formal Design and Verification

NASA Technical Reports Server (NTRS)

1998-01-01

The general goal of this project was to improve the practicality of formal methods by combining techniques from model checking and theorem proving. At the time the project was proposed, the model checking and theorem proving communities were applying different tools to similar problems, but there was not much cross-fertilization. This project involved a group from SRI that had substantial experience in the development and application of theorem-proving technology, and a group at Stanford that specialized in model checking techniques. Now, over five years after the proposal was submitted, there are many research groups working on combining theorem-proving and model checking techniques, and much more communication between the model checking and theorem proving research communities. This project contributed significantly to this research trend. The research work under this project covered a variety of topics: new theory and algorithms; prototype tools; verification methodology; and applications to problems in particular domains.

Communication, Correlation and Complementarity

NASA Astrophysics Data System (ADS)

Schumacher, Benjamin Wade

1990-01-01

In quantum communication, a sender prepares a quantum system in a state corresponding to his message and conveys it to a receiver, who performs a measurement on it. The receiver acquires information about the message based on the outcome of his measurement. Since the state of a single quantum system is not always completely determinable from measurement, quantum mechanics limits the information capacity of such channels. According to a theorem of Kholevo, the amount of information conveyed by the channel can be no greater than the entropy of the ensemble of possible physical signals. The connection between information and entropy allows general theorems to be proved regarding the energy requirements of communication. For example, it can be shown that one particular quantum coding scheme, called thermal coding, uses energy with maximum efficiency. A close analogy between communication and quantum correlation can be made using Everett's notion of relative states. Kholevo's theorem can be used to prove that the mutual information of a pair of observables on different systems is bounded by the entropy of the state of each system. This confirms and extends an old conjecture of Everett. The complementarity of quantum observables can be described by information-theoretic uncertainty relations, several of which have been previously derived. These relations imply limits on the degree to which different messages can be coded in complementary observables of a single channel. Complementarity also restricts the amount of information that can be recovered from a given channel using a given decoding observable. Information inequalities can be derived which are analogous to the well-known Bell inequalities for correlated quantum systems. These inequalities are satisfied for local hidden variable theories but are violated by quantum systems, even where the correlation is weak. These information inequalities are metric inequalities for an "information distance", and their structure can be made exactly analogous to that of the familiar covariance Bell inequalities by introducing a "covariance distance". Similar inequalities derived for successive measurements on a single system are also violated in quantum mechanics.

Cosmic equilibration: A holographic no-hair theorem from the generalized second law

NASA Astrophysics Data System (ADS)

Carroll, Sean M.; Chatwin-Davies, Aidan

2018-02-01

In a wide class of cosmological models, a positive cosmological constant drives cosmological evolution toward an asymptotically de Sitter phase. Here we connect this behavior to the increase of entropy over time, based on the idea that de Sitter spacetime is a maximum-entropy state. We prove a cosmic no-hair theorem for Robertson-Walker and Bianchi I spacetimes that admit a Q-screen ("quantum" holographic screen) with certain entropic properties: If generalized entropy, in the sense of the cosmological version of the generalized second law conjectured by Bousso and Engelhardt, increases up to a finite maximum value along the screen, then the spacetime is asymptotically de Sitter in the future. Moreover, the limiting value of generalized entropy coincides with the de Sitter horizon entropy. We do not use the Einstein field equations in our proof, nor do we assume the existence of a positive cosmological constant. As such, asymptotic relaxation to a de Sitter phase can, in a precise sense, be thought of as cosmological equilibration.

Black-hole solutions with scalar hair in Einstein-scalar-Gauss-Bonnet theories

NASA Astrophysics Data System (ADS)

Antoniou, G.; Bakopoulos, A.; Kanti, P.

2018-04-01

In the context of the Einstein-scalar-Gauss-Bonnet theory, with a general coupling function between the scalar field and the quadratic Gauss-Bonnet term, we investigate the existence of regular black-hole solutions with scalar hair. Based on a previous theoretical analysis, which studied the evasion of the old and novel no-hair theorems, we consider a variety of forms for the coupling function (exponential, even and odd polynomial, inverse polynomial, and logarithmic) that, in conjunction with the profile of the scalar field, satisfy a basic constraint. Our numerical analysis then always leads to families of regular, asymptotically flat black-hole solutions with nontrivial scalar hair. The solution for the scalar field and the profile of the corresponding energy-momentum tensor, depending on the value of the coupling constant, may exhibit a nonmonotonic behavior, an unusual feature that highlights the limitations of the existing no-hair theorems. We also determine and study in detail the scalar charge, horizon area, and entropy of our solutions.

Time Scale for Adiabaticity Breakdown in Driven Many-Body Systems and Orthogonality Catastrophe

NASA Astrophysics Data System (ADS)

Lychkovskiy, Oleg; Gamayun, Oleksandr; Cheianov, Vadim

2017-11-01

The adiabatic theorem is a fundamental result in quantum mechanics, which states that a system can be kept arbitrarily close to the instantaneous ground state of its Hamiltonian if the latter varies in time slowly enough. The theorem has an impressive record of applications ranging from foundations of quantum field theory to computational molecular dynamics. In light of this success it is remarkable that a practicable quantitative understanding of what "slowly enough" means is limited to a modest set of systems mostly having a small Hilbert space. Here we show how this gap can be bridged for a broad natural class of physical systems, namely, many-body systems where a small move in the parameter space induces an orthogonality catastrophe. In this class, the conditions for adiabaticity are derived from the scaling properties of the parameter-dependent ground state without a reference to the excitation spectrum. This finding constitutes a major simplification of a complex problem, which otherwise requires solving nonautonomous time evolution in a large Hilbert space.

On the homotopy equivalence of simple AI-algebras

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

Aristov, O Yu

1999-02-28

Let A and B be simple unital AI-algebras (an AI-algebra is an inductive limit of C*-algebras of the form BigOplus{sub i}{sup k}C([0,1],M{sub N{sub i}}). It is proved that two arbitrary unital homomorphisms from A into B such that the corresponding maps K{sub 0}A{yields}K{sub 0}B coincide are homotopic. Necessary and sufficient conditions on the Elliott invariant for A and B to be homotopy equivalent are indicated. Moreover, two algebras in the above class having the same K-theory but not homotopy equivalent are constructed. A theorem on the homotopy of approximately unitarily equivalent homomorphisms between AI-algebras is used in the proof, whichmore » is deduced in its turn from a generalization to the case of AI-algebras of a theorem of Manuilov stating that a unitary matrix almost commuting with a self-adjoint matrix h can be joined to 1 by a continuous path consisting of unitary matrices almost commuting with h.« less

Sampling theory for asynoptic satellite observations. I Space-time spectra, resolution, and aliasing. II - Fast Fourier synoptic mapping

NASA Technical Reports Server (NTRS)

Salby, M. L.

1982-01-01

An evaluation of the information content of asynoptic data taken in the form of nadir sonde and limb scan observations is presented, and a one-to-one correspondence is established between the alias-free data and twice-daily synoptic maps. Attention is given to space and time limitations of sampling and the orbital geometry is discussed. The sampling pattern is demonstrated to determine unique space-time spectra at all wavenumbers and frequencies. Spectral resolution and aliasing are explored, while restrictions on sampling and information content are defined. It is noted that irregular sampling at high latitudes produces spurious contamination effects. An Asynoptic Sampling Theorem is thereby formulated, as is a Synoptic Retrieval Theorem, in the second part of the article. In the latter, a procedure is developed for retrieving the unique correspondence between the asymptotic data and the synoptic maps. Applications examples are provided using data from the Nimbus-6 satellite.

A mathematical theorem as the basis for the second law: Thomson's formulation applied to equilibrium

NASA Astrophysics Data System (ADS)

Allahverdyan, A. E.; Nieuwenhuizen, Th. M.

2002-03-01

There are several formulations of the second law, and they may, in principle, have different domains of validity. Here a simple mathematical theorem is proven which serves as the most general basis for the second law, namely the Thomson formulation (“cyclic changes cost energy”), applied to equilibrium. This formulation of the second law is a property akin to particle conservation (normalization of the wave function). It has been strictly proven for a canonical ensemble, and made plausible for a micro-canonical ensemble. As the derivation does not assume time-inversion invariance, it is applicable to situations where persistent currents occur. This clear-cut derivation allows to revive the “no perpetuum mobile in equilibrium” formulation of the second law and to criticize some assumptions which are widespread in literature. The result puts recent results devoted to foundations and limitations of the second law in proper perspective, and structurizes this relatively new field of research.

Why are para-hydrogen clusters superfluid? A quantum theorem of corresponding states study.

PubMed

Sevryuk, Mikhail B; Toennies, J Peter; Ceperley, David M

2010-08-14

The quantum theorem of corresponding states is applied to N=13 and N=26 cold quantum fluid clusters to establish where para-hydrogen clusters lie in relation to more and less quantum delocalized systems. Path integral Monte Carlo calculations of the energies, densities, radial and pair distributions, and superfluid fractions are reported at T=0.5 K for a Lennard-Jones (LJ) (12,6) potential using six different de Boer parameters including the accepted value for hydrogen. The results indicate that the hydrogen clusters are on the borderline to being a nonsuperfluid solid but that the molecules are sufficiently delocalized to be superfluid. A general phase diagram for the total and kinetic energies of LJ (12,6) clusters encompassing all sizes from N=2 to N=infinity and for the entire range of de Boer parameters is presented. Finally the limiting de Boer parameters for quantum delocalization induced unbinding ("quantum unbinding") are estimated and the new results are found to agree with previous calculations for the bulk and smaller clusters.

Systematic Approaches to Experimentation: The Case of Pick's Theorem

ERIC Educational Resources Information Center

Papadopoulos, Ioannis; Iatridou, Maria

2010-01-01

In this paper two 10th graders having an accumulated experience on problem-solving ancillary to the concept of area confronted the task to find Pick's formula for a lattice polygon's area. The formula was omitted from the theorem in order for the students to read the theorem as a problem to be solved. Their working is examined and emphasis is…

Topology and the Lay of the Land: A Mathematician on the Topographer's Turf.

ERIC Educational Resources Information Center

Shubin, Mikhail

1992-01-01

Presents a proof of Euler's Theorem on polyhedra by relating the theorem to the field of modern topology, specifically to the topology of relief maps. An analogous theorem involving the features of mountain summits, basins, and passes on a terrain is proved and related to the faces, vertices, and edges on a convex polyhedron. (MDH)

Weak Compactness and Control Measures in the Space of Unbounded Measures

PubMed Central

Brooks, James K.; Dinculeanu, Nicolae

1972-01-01

We present a synthesis theorem for a family of locally equivalent measures defined on a ring of sets. This theorem is then used to exhibit a control measure for weakly compact sets of unbounded measures. In addition, the existence of a local control measure for locally strongly bounded vector measures is proved by means of the synthesis theorem. PMID:16591980

A Layer Framework to Investigate Student Understanding and Application of the Existence and Uniqueness Theorems of Differential Equations

ERIC Educational Resources Information Center

Raychaudhuri, D.

2007-01-01

The focus of this paper is on student interpretation and usage of the existence and uniqueness theorems for first-order ordinary differential equations. The inherent structure of the theorems is made explicit by the introduction of a framework of layers concepts-conditions-connectives-conclusions, and we discuss the manners in which students'…

Erratum: Correction to: Information Transmission and Criticality in the Contact Process

NASA Astrophysics Data System (ADS)

Cassandro, M.; Galves, A.; Löcherbach, E.

2018-01-01

The original publication of the article unfortunately contained a mistake in the first sentence of Theorem 1 and in the second part of the proof of Theorem 1. The corrected statement of Theorem as well as the corrected proof are given below. The full text of the corrected version is available at http://arxiv.org/abs/1705.11150.

Optical theorem for acoustic non-diffracting beams and application to radiation force and torque

PubMed Central

Zhang, Likun; Marston, Philip L.

2013-01-01

Acoustical and optical non-diffracting beams are potentially useful for manipulating particles and larger objects. An extended optical theorem for a non-diffracting beam was given recently in the context of acoustics. The theorem relates the extinction by an object to the scattering at the forward direction of the beam’s plane wave components. Here we use this theorem to examine the extinction cross section of a sphere centered on the axis of the beam, with a non-diffracting Bessel beam as an example. The results are applied to recover the axial radiation force and torque on the sphere by the Bessel beam. PMID:24049681

Republication of: A theorem on Petrov types

NASA Astrophysics Data System (ADS)

Goldberg, J. N.; Sachs, R. K.

2009-02-01

This is a republication of the paper “A Theorem on Petrov Types” by Goldberg and Sachs, Acta Phys. Pol. 22 (supplement), 13 (1962), in which they proved the Goldberg-Sachs theorem. The article has been selected for publication in the Golden Oldies series of General Relativity and Gravitation. Typographical errors of the original publication were corrected by the editor. The paper is accompanied by a Golden Oldie Editorial containing an editorial note written by Andrzej Krasiński and Maciej Przanowski and Goldberg’s brief autobiography. The editorial note explains some difficult parts of the proof of the theorem and discusses the influence of results of the paper on later research.

A general Kastler-Kalau-Walze type theorem for manifolds with boundary

NASA Astrophysics Data System (ADS)

Wang, Jian; Wang, Yong

2016-11-01

In this paper, we establish a general Kastler-Kalau-Walze type theorem for any dimensional manifolds with boundary which generalizes the results in [Y. Wang, Lower-dimensional volumes and Kastler-Kalau-Walze type theorem for manifolds with boundary, Commun. Theor. Phys. 54 (2010) 38-42]. This solves a problem of the referee of [J. Wang and Y. Wang, A Kastler-Kalau-Walze type theorem for five-dimensional manifolds with boundary, Int. J. Geom. Meth. Mod. Phys. 12(5) (2015), Article ID: 1550064, 34 pp.], which is a general expression of the lower dimensional volumes in terms of the geometric data on the manifold.

Electrostatic Hellmann-Feynman theorem applied to long-range interatomic forces - The hydrogen molecule.

NASA Technical Reports Server (NTRS)

Steiner, E.

1973-01-01

The use of the electrostatic Hellmann-Feynman theorem for the calculation of the leading term in the 1/R expansion of the force of interaction between two well-separated hydrogen atoms is discussed. Previous work has suggested that whereas this term is determined wholly by the first-order wavefunction when calculated by perturbation theory, the use of the Hellmann-Feynman theorem apparently requires the wavefunction through second order. It is shown how the two results may be reconciled and that the Hellmann-Feynman theorem may be reformulated in such a way that only the first-order wavefunction is required.

A Benes-like theorem for the shuffle-exchange graph

NASA Technical Reports Server (NTRS)

Schwabe, Eric J.

1992-01-01

One of the first theorems on permutation routing, proved by V. E. Beness (1965), shows that given a set of source-destination pairs in an N-node butterfly network with at most a constant number of sources or destinations in each column of the butterfly, there exists a set of paths of lengths O(log N) connecting each pair such that the total congestion is constant. An analogous theorem yielding constant-congestion paths for off-line routing in the shuffle-exchange graph is proved here. The necklaces of the shuffle-exchange graph play the same structural role as the columns of the butterfly in Beness' theorem.

Tree-manipulating systems and Church-Rosser theorems.

NASA Technical Reports Server (NTRS)

Rosen, B. K.

1973-01-01

Study of a broad class of tree-manipulating systems called subtree replacement systems. The use of this framework is illustrated by general theorems analogous to the Church-Rosser theorem and by applications of these theorems. Sufficient conditions are derived for the Church-Rosser property, and their applications to recursive definitions, the lambda calculus, and parallel programming are discussed. McCarthy's (1963) recursive calculus is extended by allowing a choice between call-by-value and call-by-name. It is shown that recursively defined functions are single-valued despite the nondeterminism of the evaluation algorithm. It is also shown that these functions solve their defining equations in a 'canonical' manner.

Quantum voting and violation of Arrow's impossibility theorem

NASA Astrophysics Data System (ADS)

Bao, Ning; Yunger Halpern, Nicole

2017-06-01

We propose a quantum voting system in the spirit of quantum games such as the quantum prisoner's dilemma. Our scheme enables a constitution to violate a quantum analog of Arrow's impossibility theorem. Arrow's theorem is a claim proved deductively in economics: Every (classical) constitution endowed with three innocuous-seeming properties is a dictatorship. We construct quantum analogs of constitutions, of the properties, and of Arrow's theorem. A quantum version of majority rule, we show, violates this quantum Arrow conjecture. Our voting system allows for tactical-voting strategies reliant on entanglement, interference, and superpositions. This contribution to quantum game theory helps elucidate how quantum phenomena can be harnessed for strategic advantage.

Common fixed points in best approximation for Banach operator pairs with Ciric type I-contractions

NASA Astrophysics Data System (ADS)

Hussain, N.

2008-02-01

The common fixed point theorems, similar to those of Ciric [Lj.B. Ciric, On a common fixed point theorem of a Gregus type, Publ. Inst. Math. (Beograd) (N.S.) 49 (1991) 174-178; Lj.B. Ciric, On Diviccaro, Fisher and Sessa open questions, Arch. Math. (Brno) 29 (1993) 145-152; Lj.B. Ciric, On a generalization of Gregus fixed point theorem, Czechoslovak Math. J. 50 (2000) 449-458], Fisher and Sessa [B. Fisher, S. Sessa, On a fixed point theorem of Gregus, Internat. J. Math. Math. Sci. 9 (1986) 23-28], Jungck [G. Jungck, On a fixed point theorem of Fisher and Sessa, Internat. J. Math. Math. Sci. 13 (1990) 497-500] and Mukherjee and Verma [R.N. Mukherjee, V. Verma, A note on fixed point theorem of Gregus, Math. Japon. 33 (1988) 745-749], are proved for a Banach operator pair. As applications, common fixed point and approximation results for Banach operator pair satisfying Ciric type contractive conditions are obtained without the assumption of linearity or affinity of either T or I. Our results unify and generalize various known results to a more general class of noncommuting mappings.

Practical Aspects of Stabilized FEM Discretizations of Nonlinear Conservation Law Systems with Convex Extension

NASA Technical Reports Server (NTRS)

Barth, Timothy; Saini, Subhash (Technical Monitor)

1999-01-01

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the Galerkin least-squares (GLS) and the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the POE system. Central to the development of the simplified GLS and DG methods is the Degenerative Scaling Theorem which characterizes right symmetrizes of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobean matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler, Navier-Stokes, and magnetohydrodynamic (MHD) equations. Linear and nonlinear energy stability is proven for the simplified GLS and DG methods. Spatial convergence properties of the simplified GLS and DO methods are numerical evaluated via the computation of Ringleb flow on a sequence of successively refined triangulations. Finally, we consider a posteriori error estimates for the GLS and DG demoralization assuming error functionals related to the integrated lift and drag of a body. Sample calculations in 20 are shown to validate the theory and implementation.

Optimal Repairman Allocation Models

DTIC Science & Technology

1976-03-01

state X under policy ir. Then lim {k1’ lC0 (^)I) e.(X,k) - 0 k*0 *’-’ (3.1.1) Proof; The result is proven by induction on |CQ(X...following theorem. Theorem 3.1 D. Under the conditions of theorem 3.1 A, define g1[ 1) (X) - g^U), then lim k- lC0 W l-mle (XHkl00^ Ig*11 (X

Individual and Collective Analyses of the Genesis of Student Reasoning Regarding the Invertible Matrix Theorem in Linear Algebra

ERIC Educational Resources Information Center

Wawro, Megan Jean

2011-01-01

In this study, I considered the development of mathematical meaning related to the Invertible Matrix Theorem (IMT) for both a classroom community and an individual student over time. In this particular linear algebra course, the IMT was a core theorem in that it connected many concepts fundamental to linear algebra through the notion of…

A Converse of Fermat's Little Theorem

ERIC Educational Resources Information Center

Bruckman, P. S.

2007-01-01

As the name of the paper implies, a converse of Fermat's Little Theorem (FLT) is stated and proved. FLT states the following: if p is any prime, and x any integer, then x[superscript p] [equivalent to] x (mod p). There is already a well-known converse of FLT, known as Lehmer's Theorem, which is as follows: if x is an integer coprime with m, such…

Bayes' Theorem: An Old Tool Applicable to Today's Classroom Measurement Needs. ERIC/AE Digest.

ERIC Educational Resources Information Center

Rudner, Lawrence M.

This digest introduces ways of responding to the call for criterion-referenced information using Bayes' Theorem, a method that was coupled with criterion-referenced testing in the early 1970s (see R. Hambleton and M. Novick, 1973). To illustrate Bayes' Theorem, an example is given in which the goal is to classify an examinee as being a master or…

CONTRIBUTIONS TO RATIONAL APPROXIMATION,

DTIC Science & Technology

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

Conditioned Limit Theorems for Some Null Recurrent Markov Processes

DTIC Science & Technology

1976-08-01

Chapter 1 INTRODUCTION 1.1 Summary of Results Let (Vk, k ! 0) be a discrete time Markov process with state space EC(- , ) and let S be...explain our results in some detail. 2 We begin by stating our three basic assumptions: (1) vk s k 2 0 Is a Markov process with state space E C(-o,%); (Ii... 12 n 3. CONDITIONING ON T (, > n.................................1.9 3.1 Preliminary Results

Generalization of the Bogoliubov-Zubarev Theorem for Dynamic Pressure to the Case of Compressibility

NASA Astrophysics Data System (ADS)

Rudoi, Yu. G.

2018-01-01

We present the motivation, formulation, and modified proof of the Bogoliubov-Zubarev theorem connecting the pressure of a dynamical object with its energy within the framework of a classical description and obtain a generalization of this theorem to the case of dynamical compressibility. In both cases, we introduce the volume of the object into consideration using a singular addition to the Hamiltonian function of the physical object, which allows using the concept of the Bogoliubov quasiaverage explicitly already on a dynamical level of description. We also discuss the relation to the same result known as the Hellmann-Feynman theorem in the framework of the quantum description of a physical object.

Some constructions of biharmonic maps and Chen’s conjecture on biharmonic hypersurfaces

NASA Astrophysics Data System (ADS)

Ou, Ye-Lin

2012-04-01

We give several construction methods and use them to produce many examples of proper biharmonic maps including biharmonic tori of any dimension in Euclidean spheres (Theorem 2.2, Corollaries 2.3, 2.4 and 2.6), biharmonic maps between spheres (Theorem 2.9) and into spheres (Theorem 2.10) via orthogonal multiplications and eigenmaps. We also study biharmonic graphs of maps, derive the equation for a function whose graph is a biharmonic hypersurface in a Euclidean space, and give an equivalent formulation of Chen's conjecture on biharmonic hypersurfaces by using the biharmonic graph equation (Theorem 4.1) which paves a way for the analytic study of the conjecture.

Reciprocity relations in aerodynamics

NASA Technical Reports Server (NTRS)

Heaslet, Max A; Spreiter, John R

1953-01-01

Reverse flow theorems in aerodynamics are shown to be based on the same general concepts involved in many reciprocity theorems in the physical sciences. Reciprocal theorems for both steady and unsteady motion are found as a logical consequence of this approach. No restrictions on wing plan form or flight Mach number are made beyond those required in linearized compressible-flow analysis. A number of examples are listed, including general integral theorems for lifting, rolling, and pitching wings and for wings in nonuniform downwash fields. Correspondence is also established between the buildup of circulation with time of a wing starting impulsively from rest and the buildup of lift of the same wing moving in the reverse direction into a sharp-edged gust.

Fluctuation theorem for channel-facilitated membrane transport of interacting and noninteracting solutes.

PubMed

Berezhkovskii, Alexander M; Bezrukov, Sergey M

2008-05-15

In this paper, we discuss the fluctuation theorem for channel-facilitated transport of solutes through a membrane separating two reservoirs. The transport is characterized by the probability, P(n)(t), that n solute particles have been transported from one reservoir to the other in time t. The fluctuation theorem establishes a relation between P(n)(t) and P-(n)(t): The ratio P(n)(t)/P-(n)(t) is independent of time and equal to exp(nbetaA), where betaA is the affinity measured in the thermal energy units. We show that the same fluctuation theorem is true for both single- and multichannel transport of noninteracting particles and particles which strongly repel each other.

One-range addition theorems for derivatives of Slater-type orbitals.

PubMed

Guseinov, Israfil

2004-06-01

Using addition theorems for STOs introduced by the author with the help of complete orthonormal sets of psi(alpha)-ETOs (Guseinov II (2003) J Mol Model 9:190-194), where alpha=1, 0, -1, -2, ..., a large number of one-range addition theorems for first and second derivatives of STOs are established. These addition theorems are especially useful for computation of multicenter-multielectron integrals over STOs that arise in the Hartree-Fock-Roothaan approximation and also in the Hylleraas function method, which play a significant role for the study of electronic structure and electron-nuclei interaction properties of atoms, molecules, and solids. The relationships obtained are valid for arbitrary quantum numbers, screening constants and location of STOs.

Out-of-time-order fluctuation-dissipation theorem

NASA Astrophysics Data System (ADS)

Tsuji, Naoto; Shitara, Tomohiro; Ueda, Masahito

2018-01-01

We prove a generalized fluctuation-dissipation theorem for a certain class of out-of-time-ordered correlators (OTOCs) with a modified statistical average, which we call bipartite OTOCs, for general quantum systems in thermal equilibrium. The difference between the bipartite and physical OTOCs defined by the usual statistical average is quantified by a measure of quantum fluctuations known as the Wigner-Yanase skew information. Within this difference, the theorem describes a universal relation between chaotic behavior in quantum systems and a nonlinear-response function that involves a time-reversed process. We show that the theorem can be generalized to higher-order n -partite OTOCs as well as in the form of generalized covariance.

Some theorems and properties of multi-dimensional fractional Laplace transforms

NASA Astrophysics Data System (ADS)

Ahmood, Wasan Ajeel; Kiliçman, Adem

2016-06-01

The aim of this work is to study theorems and properties for the one-dimensional fractional Laplace transform, generalize some properties for the one-dimensional fractional Lapalce transform to be valid for the multi-dimensional fractional Lapalce transform and is to give the definition of the multi-dimensional fractional Lapalce transform. This study includes: dedicate the one-dimensional fractional Laplace transform for functions of only one independent variable with some of important theorems and properties and develop of some properties for the one-dimensional fractional Laplace transform to multi-dimensional fractional Laplace transform. Also, we obtain a fractional Laplace inversion theorem after a short survey on fractional analysis based on the modified Riemann-Liouville derivative.

A coupled mode formulation by reciprocity and a variational principle

NASA Technical Reports Server (NTRS)

Chuang, Shun-Lien

1987-01-01

A coupled mode formulation for parallel dielectric waveguides is presented via two methods: a reciprocity theorem and a variational principle. In the first method, a generalized reciprocity relation for two sets of field solutions satisfying Maxwell's equations and the boundary conditions in two different media, respectively, is derived. Based on the generalized reciprocity theorem, the coupled mode equations can then be formulated. The second method using a variational principle is also presented for a general waveguide system which can be lossy. The results of the variational principle can also be shown to be identical to those from the reciprocity theorem. The exact relations governing the 'conventional' and the new coupling coefficients are derived. It is shown analytically that the present formulation satisfies the reciprocity theorem and power conservation exactly, while the conventional theory violates the power conservation and reciprocity theorem by as much as 55 percent and the Hardy-Streifer (1985, 1986) theory by 0.033 percent, for example.

Does the Coase theorem hold in real markets? An application to the negotiations between waterworks and farmers in Denmark.

PubMed

Abildtrup, Jens; Jensen, Frank; Dubgaard, Alex

2012-01-01

The Coase theorem depends on a number of assumptions, among others, perfect information about each other's payoff function, maximising behaviour and zero transaction costs. An important question is whether the Coase theorem holds for real market transactions when these assumptions are violated. This is the question examined in this paper. We consider the results of Danish waterworks' attempts to establish voluntary cultivation agreements with Danish farmers. A survey of these negotiations shows that the Coase theorem is not robust in the presence of imperfect information, non-maximising behaviour and transaction costs. Thus, negotiations between Danish waterworks and farmers may not be a suitable mechanism to achieve efficiency in the protection of groundwater quality due to violations of the assumptions of the Coase theorem. The use of standard schemes or government intervention (e.g. expropriation) may, under some conditions, be a more effective and cost efficient approach for the protection of vulnerable groundwater resources in Denmark. Copyright © 2011 Elsevier Ltd. All rights reserved.

A Formally-Verified Decision Procedure for Univariate Polynomial Computation Based on Sturm's Theorem

NASA Technical Reports Server (NTRS)

Narkawicz, Anthony J.; Munoz, Cesar A.

2014-01-01

Sturm's Theorem is a well-known result in real algebraic geometry that provides a function that computes the number of roots of a univariate polynomial in a semiopen interval. This paper presents a formalization of this theorem in the PVS theorem prover, as well as a decision procedure that checks whether a polynomial is always positive, nonnegative, nonzero, negative, or nonpositive on any input interval. The soundness and completeness of the decision procedure is proven in PVS. The procedure and its correctness properties enable the implementation of a PVS strategy for automatically proving existential and universal univariate polynomial inequalities. Since the decision procedure is formally verified in PVS, the soundness of the strategy depends solely on the internal logic of PVS rather than on an external oracle. The procedure itself uses a combination of Sturm's Theorem, an interval bisection procedure, and the fact that a polynomial with exactly one root in a bounded interval is always nonnegative on that interval if and only if it is nonnegative at both endpoints.

Quantum Mechanics, Can It Be Consistent with Locality?

NASA Astrophysics Data System (ADS)

Nisticò, Giuseppe; Sestito, Angela

2011-07-01

We single out an alternative, strict interpretation of the Einstein-Podolsky-Rosen criterion of reality, and identify the implied extensions of quantum correlations. Then we prove that the theorem of Bell, and the non-locality theorems without inequalities, fail if the new extensions are adopted. Therefore, these theorems can be interpreted as arguments against the wide interpretation of the criterion of reality rather than as a violation of locality.

Specification Improvement Through Analysis of Proof Structure (SITAPS): High Assurance Software Development

DTIC Science & Technology

2016-02-01

proof in mathematics. For example, consider the proof of the Pythagorean Theorem illustrated at: http://www.cut-the-knot.org/ pythagoras / where 112...methods and tools have made significant progress in their ability to model software designs and prove correctness theorems about the systems modeled...assumption criticality” or “ theorem root set size” SITAPS detects potentially brittle verification cases. SITAPS provides tools and techniques that

Delaunay Refinement Mesh Generation

DTIC Science & Technology

1997-05-18

edge is locally Delaunay; thus, by Lemma 3, every edge is Delaunay. Theorem 5 Let V be a set of three or more vertices in the plane that are not all...this document. Delaunay triangulations are valuable in part because they have the following optimality properties. Theorem 6 Among all triangulations of...no locally Delaunay edges. By Theorem 5, a triangulation with no locally Delaunay edges is the Delaunay triangulation. The property of max-min

Development of a Dependency Theory Toolbox for Database Design.

DTIC Science & Technology

1987-12-01

published algorithms and theorems , and hand simulating these algorithms can be a tedious and error prone chore. Additionally, since the process of...to design and study relational databases exists in the form of published algorithms and theorems . However, hand simulating these algorithms can be a...published algorithms and theorems . Hand simulating these algorithms can be a tedious and error prone chore. Therefore, a toolbox of algorithms and

Field Computation and Nonpropositional Knowledge.

DTIC Science & Technology

1987-09-01

field computer It is based on xeneralization of Taylor’s theorem to continuous dimensional vector spaces. 20. DISTRIBUTION/AVAILABILITY OF ABSTRACT 21...generalization of Taylor’s theorem to continuous dimensional vector -5paces A number of field computations are illustrated, including several Lransforma...paradigm. The "old" Al has been quite successful in performing a number of difficult tasks, such as theorem prov- ing, chess playing, medical diagnosis and

Ignoring the Innocent: Non-combatants in Urban Operations and in Military Models and Simulations

DTIC Science & Technology

2006-01-01

such a model yields is a sufficiency theorem , a single run does not provide any information on the robustness of such theorems . That is, given that...often formally resolvable via inspection, simple differentiation, the implicit function theorem , comparative statistics, and so on. The only way to... Pythagoras , and Bactowars. For each, Grieger discusses model parameters, data collection, terrain, and other features. Grieger also discusses

Mean energy of some interacting bosonic systems derived by virtue of the generalized Hellmann-Feynman theorem

NASA Astrophysics Data System (ADS)

Fan, Hong-yi; Xu, Xue-xiang

2009-06-01

By virtue of the generalized Hellmann-Feynman theorem [H. Y. Fan and B. Z. Chen, Phys. Lett. A 203, 95 (1995)], we derive the mean energy of some interacting bosonic systems for some Hamiltonian models without proceeding with diagonalizing the Hamiltonians. Our work extends the field of applications of the Hellmann-Feynman theorem and may enrich the theory of quantum statistics.

Reduction theorems for optimal unambiguous state discrimination of density matrices

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

Raynal, Philippe; Luetkenhaus, Norbert; Enk, Steven J. van

2003-08-01

We present reduction theorems for the problem of optimal unambiguous state discrimination of two general density matrices. We show that this problem can be reduced to that of two density matrices that have the same rank n and are described in a Hilbert space of dimensions 2n. We also show how to use the reduction theorems to discriminate unambiguously between N mixed states (N{>=}2)

Formalization of the Integral Calculus in the PVS Theorem Prover

NASA Technical Reports Server (NTRS)

Butler, Ricky W.

2004-01-01

The PVS Theorem prover is a widely used formal verification tool used for the analysis of safety-critical systems. The PVS prover, though fully equipped to support deduction in a very general logic framework, namely higher-order logic, it must nevertheless, be augmented with the definitions and associated theorems for every branch of mathematics and Computer Science that is used in a verification. This is a formidable task, ultimately requiring the contributions of researchers and developers all over the world. This paper reports on the formalization of the integral calculus in the PVS theorem prover. All of the basic definitions and theorems covered in a first course on integral calculus have been completed.The theory and proofs were based on Rosenlicht's classic text on real analysis and follow the traditional epsilon-delta method. The goal of this work was to provide a practical set of PVS theories that could be used for verification of hybrid systems that arise in air traffic management systems and other aerospace applications. All of the basic linearity, integrability, boundedness, and continuity properties of the integral calculus were proved. The work culminated in the proof of the Fundamental Theorem Of Calculus. There is a brief discussion about why mechanically checked proofs are so much longer than standard mathematics textbook proofs.

The Knaster-Kuratowski-Mazurkiewicz theorem and abstract convexities

NASA Astrophysics Data System (ADS)

Cain, George L., Jr.; González, Luis

2008-02-01

The Knaster-Kuratowski-Mazurkiewicz covering theorem (KKM), is the basic ingredient in the proofs of many so-called "intersection" theorems and related fixed point theorems (including the famous Brouwer fixed point theorem). The KKM theorem was extended from Rn to Hausdorff linear spaces by Ky Fan. There has subsequently been a plethora of attempts at extending the KKM type results to arbitrary topological spaces. Virtually all these involve the introduction of some sort of abstract convexity structure for a topological space, among others we could mention H-spaces and G-spaces. We have introduced a new abstract convexity structure that generalizes the concept of a metric space with a convex structure, introduced by E. Michael in [E. Michael, Convex structures and continuous selections, Canad. J. MathE 11 (1959) 556-575] and called a topological space endowed with this structure an M-space. In an article by Shie Park and Hoonjoo Kim [S. Park, H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996) 173-187], the concepts of G-spaces and metric spaces with Michael's convex structure, were mentioned together but no kind of relationship was shown. In this article, we prove that G-spaces and M-spaces are close related. We also introduce here the concept of an L-space, which is inspired in the MC-spaces of J.V. Llinares [J.V. Llinares, Unified treatment of the problem of existence of maximal elements in binary relations: A characterization, J. Math. Econom. 29 (1998) 285-302], and establish relationships between the convexities of these spaces with the spaces previously mentioned.

A generalized measurement equation and van Cittert-Zernike theorem for wide-field radio astronomical interferometry

NASA Astrophysics Data System (ADS)

Carozzi, T. D.; Woan, G.

2009-05-01

We derive a generalized van Cittert-Zernike (vC-Z) theorem for radio astronomy that is valid for partially polarized sources over an arbitrarily wide field of view (FoV). The classical vC-Z theorem is the theoretical foundation of radio astronomical interferometry, and its application is the basis of interferometric imaging. Existing generalized vC-Z theorems in radio astronomy assume, however, either paraxiality (narrow FoV) or scalar (unpolarized) sources. Our theorem uses neither of these assumptions, which are seldom fulfiled in practice in radio astronomy, and treats the full electromagnetic field. To handle wide, partially polarized fields, we extend the two-dimensional (2D) electric field (Jones vector) formalism of the standard `Measurement Equation' (ME) of radio astronomical interferometry to the full three-dimensional (3D) formalism developed in optical coherence theory. The resulting vC-Z theorem enables full-sky imaging in a single telescope pointing, and imaging based not only on standard dual-polarized interferometers (that measure 2D electric fields) but also electric tripoles and electromagnetic vector-sensor interferometers. We show that the standard 2D ME is easily obtained from our formalism in the case of dual-polarized antenna element interferometers. We also exploit an extended 2D ME to determine that dual-polarized interferometers can have polarimetric aberrations at the edges of a wide FoV. Our vC-Z theorem is particularly relevant to proposed, and recently developed, wide FoV interferometers such as Low Frequency Array (LOFAR) and Square Kilometer Array (SKA), for which direction-dependent effects will be important.

Free time minimizers for the three-body problem

NASA Astrophysics Data System (ADS)

Moeckel, Richard; Montgomery, Richard; Sánchez Morgado, Héctor

2018-03-01

Free time minimizers of the action (called "semi-static" solutions by Mañe in International congress on dynamical systems in Montevideo (a tribute to Ricardo Mañé), vol 362, pp 120-131, 1996) play a central role in the theory of weak KAM solutions to the Hamilton-Jacobi equation (Fathi in Weak KAM Theorem in Lagrangian Dynamics Preliminary Version Number 10, 2017). We prove that any solution to Newton's three-body problem which is asymptotic to Lagrange's parabolic homothetic solution is eventually a free time minimizer. Conversely, we prove that every free time minimizer tends to Lagrange's solution, provided the mass ratios lie in a certain large open set of mass ratios. We were inspired by the work of Da Luz and Maderna (Math Proc Camb Philos Soc 156:209-227, 1980) which showed that every free time minimizer for the N-body problem is parabolic and therefore must be asymptotic to the set of central configurations. We exclude being asymptotic to Euler's central configurations by a second variation argument. Central configurations correspond to rest points for the McGehee blown-up dynamics. The large open set of mass ratios are those for which the linearized dynamics at each Euler rest point has a complex eigenvalue.

High-Dimensional Multivariate Repeated Measures Analysis with Unequal Covariance Matrices.

PubMed

Harrar, Solomon W; Kong, Xiaoli

2015-03-01

In this paper, test statistics for repeated measures design are introduced when the dimension is large. By large dimension is meant the number of repeated measures and the total sample size grow together but either one could be larger than the other. Asymptotic distribution of the statistics are derived for the equal as well as unequal covariance cases in the balanced as well as unbalanced cases. The asymptotic framework considered requires proportional growth of the sample sizes and the dimension of the repeated measures in the unequal covariance case. In the equal covariance case, one can grow at much faster rate than the other. The derivations of the asymptotic distributions mimic that of Central Limit Theorem with some important peculiarities addressed with sufficient rigor. Consistent and unbiased estimators of the asymptotic variances, which make efficient use of all the observations, are also derived. Simulation study provides favorable evidence for the accuracy of the asymptotic approximation under the null hypothesis. Power simulations have shown that the new methods have comparable power with a popular method known to work well in low-dimensional situation but the new methods have shown enormous advantage when the dimension is large. Data from Electroencephalograph (EEG) experiment is analyzed to illustrate the application of the results.

High-Dimensional Multivariate Repeated Measures Analysis with Unequal Covariance Matrices

PubMed Central

Harrar, Solomon W.; Kong, Xiaoli

2015-01-01

In this paper, test statistics for repeated measures design are introduced when the dimension is large. By large dimension is meant the number of repeated measures and the total sample size grow together but either one could be larger than the other. Asymptotic distribution of the statistics are derived for the equal as well as unequal covariance cases in the balanced as well as unbalanced cases. The asymptotic framework considered requires proportional growth of the sample sizes and the dimension of the repeated measures in the unequal covariance case. In the equal covariance case, one can grow at much faster rate than the other. The derivations of the asymptotic distributions mimic that of Central Limit Theorem with some important peculiarities addressed with sufficient rigor. Consistent and unbiased estimators of the asymptotic variances, which make efficient use of all the observations, are also derived. Simulation study provides favorable evidence for the accuracy of the asymptotic approximation under the null hypothesis. Power simulations have shown that the new methods have comparable power with a popular method known to work well in low-dimensional situation but the new methods have shown enormous advantage when the dimension is large. Data from Electroencephalograph (EEG) experiment is analyzed to illustrate the application of the results. PMID:26778861

Identifiability of tree-child phylogenetic networks under a probabilistic recombination-mutation model of evolution.

PubMed

Francis, Andrew; Moulton, Vincent

2018-06-07

Phylogenetic networks are an extension of phylogenetic trees which are used to represent evolutionary histories in which reticulation events (such as recombination and hybridization) have occurred. A central question for such networks is that of identifiability, which essentially asks under what circumstances can we reliably identify the phylogenetic network that gave rise to the observed data? Recently, identifiability results have appeared for networks relative to a model of sequence evolution that generalizes the standard Markov models used for phylogenetic trees. However, these results are quite limited in terms of the complexity of the networks that are considered. In this paper, by introducing an alternative probabilistic model for evolution along a network that is based on some ground-breaking work by Thatte for pedigrees, we are able to obtain an identifiability result for a much larger class of phylogenetic networks (essentially the class of so-called tree-child networks). To prove our main theorem, we derive some new results for identifying tree-child networks combinatorially, and then adapt some techniques developed by Thatte for pedigrees to show that our combinatorial results imply identifiability in the probabilistic setting. We hope that the introduction of our new model for networks could lead to new approaches to reliably construct phylogenetic networks. Copyright © 2018 Elsevier Ltd. All rights reserved.

A mathematical model for evolution and SETI.

PubMed

Maccone, Claudio

2011-12-01

Darwinian evolution theory may be regarded as a part of SETI theory in that the factor f(l) in the Drake equation represents the fraction of planets suitable for life on which life actually arose. In this paper we firstly provide a statistical generalization of the Drake equation where the factor f(l) is shown to follow the lognormal probability distribution. This lognormal distribution is a consequence of the Central Limit Theorem (CLT) of Statistics, stating that the product of a number of independent random variables whose probability densities are unknown and independent of each other approached the lognormal distribution when the number of factors increased to infinity. In addition we show that the exponential growth of the number of species typical of Darwinian Evolution may be regarded as the geometric locus of the peaks of a one-parameter family of lognormal distributions (b-lognormals) constrained between the time axis and the exponential growth curve. Finally, since each b-lognormal distribution in the family may in turn be regarded as the product of a large number (actually "an infinity") of independent lognormal probability distributions, the mathematical way is paved to further cast Darwinian Evolution into a mathematical theory in agreement with both its typical exponential growth in the number of living species and the Statistical Drake Equation.

Modelling Evolution and SETI Mathematically

NASA Astrophysics Data System (ADS)

Maccone, Claudio

2012-05-01

Darwinian evolution theory may be regarded as a part of SETI theory in that the factor fl in the Drake equation represents the fraction of planets suitable for life on which life actually arose. In this paper we firstly provide a statistical generalization of the Drake equation where the factor fl is shown to follow the lognormal probability distribution. This lognormal distribution is a consequence of the Central Limit Theorem (CLT) of Statistics, stating that the product of a number of independent random variables whose probability densities are unknown and independent of each other approached the lognormal distribution when the number of factor increased to infinity. In addition we show that the exponential growth of the number of species typical of Darwinian Evolution may be regarded as the geometric locus of the peaks of a one-parameter family of lognormal distributions constrained between the time axis and the exponential growth curve. Finally, since each lognormal distribution in the family may in turn be regarded as the product of a large number (actually "an infinity") of independent lognormal probability distributions, the mathematical way is paved to further cast Darwinian Evolution into a mathematical theory in agreement with both its typical exponential growth in the number of living species and the Statistical Drake Equation.

A Mathematical Model for Evolution and SETI

NASA Astrophysics Data System (ADS)

Maccone, Claudio

2011-12-01

Darwinian evolution theory may be regarded as a part of SETI theory in that the factor fl in the Drake equation represents the fraction of planets suitable for life on which life actually arose. In this paper we firstly provide a statistical generalization of the Drake equation where the factor fl is shown to follow the lognormal probability distribution. This lognormal distribution is a consequence of the Central Limit Theorem (CLT) of Statistics, stating that the product of a number of independent random variables whose probability densities are unknown and independent of each other approached the lognormal distribution when the number of factors increased to infinity. In addition we show that the exponential growth of the number of species typical of Darwinian Evolution may be regarded as the geometric locus of the peaks of a one-parameter family of lognormal distributions (b-lognormals) constrained between the time axis and the exponential growth curve. Finally, since each b-lognormal distribution in the family may in turn be regarded as the product of a large number (actually "an infinity") of independent lognormal probability distributions, the mathematical way is paved to further cast Darwinian Evolution into a mathematical theory in agreement with both its typical exponential growth in the number of living species and the Statistical Drake Equation.

Continuous Time Finite State Mean Field Games

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

Gomes, Diogo A., E-mail: dgomes@math.ist.utl.pt; Mohr, Joana, E-mail: joana.mohr@ufrgs.br; Souza, Rafael Rigao, E-mail: rafars@mat.ufrgs.br

In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study an N+1-player problem, which the mean field model attempts to approximate. Our main result is the convergence as N{yields}{infinity} of the mean field model and an estimatemore » of the rate of convergence. We end the paper with some further examples for potential mean field games.« less

Determination of the Thermal Noise Limit of Graphene Biotransistors.

PubMed

Crosser, Michael S; Brown, Morgan A; McEuen, Paul L; Minot, Ethan D

2015-08-12

To determine the thermal noise limit of graphene biotransistors, we have measured the complex impedance between the basal plane of single-layer graphene and an aqueous electrolyte. The impedance is dominated by an imaginary component but has a finite real component. Invoking the fluctuation-dissipation theorem, we determine the power spectral density of thermally driven voltage fluctuations at the graphene/electrolyte interface. The fluctuations have 1/f(p) dependence, with p = 0.75-0.85, and the magnitude of fluctuations scales inversely with area. Our results explain noise spectra previously measured in liquid-gated suspended graphene devices and provide realistic targets for future device performance.

Infinite Set of Soft Theorems in Gauge-Gravity Theories as Ward-Takahashi Identities

NASA Astrophysics Data System (ADS)

Hamada, Yuta; Shiu, Gary

2018-05-01

We show that the soft photon, gluon, and graviton theorems can be understood as the Ward-Takahashi identities of large gauge transformation, i.e., diffeomorphism that does not fall off at spatial infinity. We found infinitely many new identities which constrain the higher order soft behavior of the gauge bosons and gravitons in scattering amplitudes of gauge and gravity theories. Diagrammatic representations of these soft theorems are presented.