Handbook Of Mathematical Functions
01 Jan 2016-
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Citations
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University of Manchester1, KEK2, CERN3, Complutense University of Madrid4, SLAC National Accelerator Laboratory5, Toyama College6, Lebedev Physical Institute7, Fermilab8, University of Paris-Sud9, Lawrence Livermore National Laboratory10, National Research Nuclear University MEPhI11, Queen's University Belfast12, Korea Institute of Science and Technology Information13, Istituto Nazionale di Fisica Nucleare14, Northeastern University15, University of Seville16, National University of Cordoba17, Saint Joseph University18, Joint Institute for Nuclear Research19, University of Wollongong20, Illawarra Health & Medical Research Institute21, Hampton University22, TRIUMF23, ETH Zurich24, Centre national de la recherche scientifique25, University of Bordeaux26, University of Helsinki27, National Technical University of Athens28, Johns Hopkins University School of Medicine29, University of Notre Dame30, Ashikaga Institute of Technology31, Kobe University32, Intelligence and National Security Alliance33, University of Trieste34, University of Warwick35, University of Belgrade36, Instituto Superior Técnico37, European Space Agency38, Varian Medical Systems39, George Washington University40, Ritsumeikan University41, Ton Duc Thang University42, Université Paris-Saclay43, Idaho State University44, Naruto University of Education45
01 Nov 2016-Nuclear Instruments & Methods in Physics Research Section A-accelerators Spectrometers Detectors and Associated Equipment
TL;DR: Geant4 as discussed by the authors is a software toolkit for the simulation of the passage of particles through matter, which is used by a large number of experiments and projects in a variety of application domains, including high energy physics, astrophysics and space science, medical physics and radiation protection.
Abstract: Geant4 is a software toolkit for the simulation of the passage of particles through matter. It is used by a large number of experiments and projects in a variety of application domains, including high energy physics, astrophysics and space science, medical physics and radiation protection. Over the past several years, major changes have been made to the toolkit in order to accommodate the needs of these user communities, and to efficiently exploit the growth of computing power made available by advances in technology. The adaptation of Geant4 to multithreading, advances in physics, detector modeling and visualization, extensions to the toolkit, including biasing and reverse Monte Carlo, and tools for physics and release validation are discussed here.
2,260 citations
TL;DR: It is argued that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation.
Abstract: The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a d-sparse Hamiltonian H[over ^] for time-interval t with error e is O[td∥H[over ^]∥_{max}+log(1/e)/loglog(1/e)], which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (i) transducing eigenvalues of H[over ^] into a single ancilla qubit, (ii) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii) projecting this ancilla with near unity success probability.
543 citations
Cites methods from "Handbook Of Mathematical Functions"
...1 |k|! ∣∣ τ 2 ∣∣|k| [30] decays rapidly with k, good approxima-...
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...which is provided by the Jacobi-Anger expansion [30]...
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TL;DR: An accurate, non-binning MI estimator for the case of one discrete data set and one continuous data set is presented, which applies when measuring the relationship between base sequence and gene expression level, or the effect of a cancer drug on patient survival time.
Abstract: Mutual information (MI) is a powerful method for detecting relationships between data sets. There are accurate methods for estimating MI that avoid problems with “binning” when both data sets are discrete or when both data sets are continuous. We present an accurate, non-binning MI estimator for the case of one discrete data set and one continuous data set. This case applies when measuring, for example, the relationship between base sequence and gene expression level, or the effect of a cancer drug on patient survival time. We also show how our method can be adapted to calculate the Jensen–Shannon divergence of two or more data sets.
511 citations
TL;DR: A quadratic approximation without a linear term, precision-effect estimate with standard error (PEESE), is shown to have the smallest bias and mean squared error in most cases and to outperform conventional meta-analysis estimators, often by a great deal.
Abstract: Publication selection bias is a serious challenge to the integrity of all empirical sciences We derive meta-regression approximations to reduce this bias Our approach employs Taylor polynomial approximations to the conditional mean of a truncated distribution A quadratic approximation without a linear term, precision-effect estimate with standard error (PEESE), is shown to have the smallest bias and mean squared error in most cases and to outperform conventional meta-analysis estimators, often by a great deal Monte Carlo simulations also demonstrate how a new hybrid estimator that conditionally combines PEESE and the Egger regression intercept can provide a practical solution to publication selection bias PEESE is easily expanded to accommodate systematic heterogeneity along with complex and differential publication selection bias that is related to moderator variables By providing an intuitive reason for these approximations, we can also explain why the Egger regression works so well and when it does not These meta-regression methods are applied to several policy-relevant areas of research including antidepressant effectiveness, the value of a statistical life, the minimum wage, and nicotine replacement therapy
486 citations
Cites background from "Handbook Of Mathematical Functions"
...Some of these approximations are in fact power series (Abramowitz and Stegun, 1964)....
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TL;DR: The aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.
Abstract: Given the square matrices $A, B, D, E$ and the matrix $C$ of conforming dimensions, we consider the linear matrix equation $A{\mathbf X} E+D{\mathbf X} B = C$ in the unknown matrix ${\mathbf X}$. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.
451 citations
Cites background from "Handbook Of Mathematical Functions"
...Here K and K ′ are the complete elliptic integrals of the first kind(4) [1]....
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...where dn is a Jacobian elliptic function, and K is the complete elliptic integral of the first kind, of modulus m [1]....
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References
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TL;DR: A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool as discussed by the authors, and it can be found in many libraries.
Abstract: A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool.
4,083 citations
TL;DR: In this article, Euler's Integral: A Historical Profile of the Gamma Function is presented. But the Gamma function is not considered in this paper, since it is not an integral function.
Abstract: (1959). Leonhard Euler's Integral: A Historical Profile of the Gamma Function. The American Mathematical Monthly: Vol. 66, No. 10, pp. 849-869.
191 citations
01 Apr 1967-Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics
TL;DR: In this paper, the authors describe a lion which is a ppli cable when s impl e rec urrence proce dures ca nnol be used becau se uf in sla bilil Y.
Abstract: A ne w al~orilhm is ~ive n fur cumpulin ~ Ihe soluliun of a ny secu nd-orde r lin ea r diffe re nce e qua lion which is a ppli cable when s impl e rec urrence proce dures ca nnol be used becau se uf in sla bilil Y. Co mpare d wilh Ih e we ll -knuwn lV1ill e r a l ~o rilhm Ih e ne w m elhod has Ih e advanla~es of (i) a Ul umal ic ally deLermining the currec t number of rec urrence steps. (ii ) app lying to illllOlllo~en eou s differe nce eqlla ~ li uns, (iii ) e nab lin g mure powerful e rrur a na lyses lu be co ns lru c led. The me lhud is illuSlra led by num e rica l comp ulalion s, ineludin~ e rror ana lyses. uf A n~e rWe be r . S iruve, a nd Besse l fun c li uns, and Ih e s u luli un of a differe nli a l e qualiun in C he bys he v se ri es.
124 citations
"Handbook Of Mathematical Functions" refers background in this paper
...In a paper published in 1967 [5], Olver provided the first (and only) stable algorithm for computing all types of solutions of a difference equation with three different kinds of behavior: strongly growing, strongly decaying, and showing moderate growth or decay....
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01 Oct 1967-Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics
TL;DR: The problem of the solution of a given se t of lin ear equations Ax = b on a high speed digital computer has been studied intensively, and there are a large number of method s, more or less sati sfactory, for carrying out such a solution as discussed by the authors.
Abstract: The problem of the solution of a given se t of lin ear equations Ax = b on a high-speed digital computer has been studied intensively, and there are a large number of method s, more or less sati sfactory , for carrying out such a solution. Nevertheless occasions arise when existing methods are inadequate, either because the solutions are required exactly, or because the coefficie nt matrix A is \"ill-conditioned.\" A notorious exam ple of the latter is furnished by the Hilbert matrices A = H\" given by
51 citations
"Handbook Of Mathematical Functions" refers methods in this paper
...This technique was first described in a paper by Newman in 1967 [7]; it was employed with great success in computing and checking the tables in Chapter 24 of the Handbook....
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