Advanced Mathematical Methods for Scientists and Engineers
About: This article is published in Nuclear Science and Engineering.The article was published on 1980-12-01. It has received 1242 citations till now.
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TL;DR: In this article, the authors present convergence properties of Multipole Expansion of Intermolecular Interaction Operator (MEI) and van der Waals constants (VWC).
Abstract: 1. First-Order (Heitler-London) Exchange 2. Exchange-Induction Interaction 3. Exchange-Dispersion Interaction D. Convergence Properties of Symmetry-Adapted Theories IV. Multipole Expansion of Interaction Energy A. General Asymptotic Expansion of Interaction Energy B. Multipole Expansion of Intermolecular Interaction Operator C. van der Waals Constants D. Convergence Properties of Multipole Expansion of Interaction Energy E. Angular Dependence of Interaction Energy F. Computations of van der Waals Constants Ill. Exchange Effects
2,298 citations
University of South Carolina1, Los Alamos National Laboratory2, Moscow State University3, Delhi Technological University4, University of Paris5, University of California, Davis6, Indian Institute of Technology (BHU) Varanasi7, University of Moratuwa8, University of Illinois at Urbana–Champaign9, California Polytechnic State University10, Sandia National Laboratories11, Max Planck Society12, Indian Institute of Technology Kharagpur13, French Institute for Research in Computer Science and Automation14, University of New Mexico15, Charles University in Prague16, Birla Institute of Technology and Science17, Indian Institute of Technology Bombay18, University of West Bohemia19
TL;DR: The architecture of SymPy is presented, a description of its features, and a discussion of select domain specific submodules are discussed, to become the standard symbolic library for the scientific Python ecosystem.
Abstract: SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.
1,126 citations
Cites methods from "Advanced Mathematical Methods for S..."
...To evaluate slowly converging limits and 473 infinite series, mpmath automatically tries Richardson extrapolation and the Shanks 474 transformation (Euler-Maclaurin summation can also be used) [3]....
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TL;DR: An alternative derivation of passive imaging of the ballistic wave that is not based on normal modes is presented, showing that the global requirement of the equipartitioning of normal modes can be relaxed to the local requirement that the scattered waves propagate on average isotropically near the receivers.
Abstract: The Green's function of waves that propagate between two receivers can be found by cross-correlating multiply scattered waves recorded at these receivers. This technique obviates the need for a source at one of these locations, and is therefore called "passive imaging." This principle has been explained by assuming that the normal modes of the system are uncorrelated and that all carry the same amount of energy (equipartitioning). Here I present an alternative derivation of passive imaging of the ballistic wave that is not based on normal modes. The derivation is valid for scalar waves in three dimensions, and for elastic surface waves. Passive imaging of the ballistic wave is based on the destructive interference of waves radiated from scatterers away from the receiver line, and the constructive interference of waves radiated from secondary sources near the receiver line. The derivation presented here shows that the global requirement of the equipartitioning of normal modes can be relaxed to the local requirement that the scattered waves propagate on average isotropically near the receivers.
1,089 citations
TL;DR: It is shown how a quantitative microscopic theory for directional ordering in a flock can be derived directly from field data, and the minimally structured (maximum entropy) model is constructed consistent with experimental correlations in large flocks of starlings.
Abstract: Flocking is a typical example of emergent collective behavior, where interactions between individuals produce collective patterns on the large scale. Here we show how a quantitative microscopic theory for directional ordering in a flock can be derived directly from field data. We construct the minimally structured (maximum entropy) model consistent with experimental correlations in large flocks of starlings. The maximum entropy model shows that local, pairwise interactions between birds are sufficient to correctly predict the propagation of order throughout entire flocks of starlings, with no free parameters. We also find that the number of interacting neighbors is independent of flock density, confirming that interactions are ruled by topological rather than metric distance. Finally, by comparing flocks of different sizes, the model correctly accounts for the observed scale invariance of long-range correlations among the fluctuations in flight direction.
647 citations
Cites background or methods from "Advanced Mathematical Methods for S..."
...[19] CM Bender & SA Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw–Hill, New York, 1978)....
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...Our constrained optimization problem can be solved using the method of the Lagrange multipliers [19]: we introduce a generalized entropy function,...
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06 Jul 2001
TL;DR: A quantum analog of the symmetric random walk, which the authors call the Hadamard walk, is analyzed, which has position that is nearly uniformly distributed in the range after steps, in sharp contrast to the classical random walk.
Abstract: We define and analyze quantum computational variants of random walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the Hadamard walk. Several striking differences between the quantum and classical cases are observed. For example, when unrestricted in either direction, the Hadamard walk has position that is nearly uniformly distributed in the range [-t/\sqrt 2, t/\sqrt 2] after t steps, which is in sharp contrast to the classical random walk, which has distance O(\sqrt t) from the origin with high probability. With an absorbing boundary immediately to the left of the starting position, the probability that the walk exits to the left is 2/&pgr, and with an additional absorbing boundary at location n, the probability that the walk exits to the left actually increases, approaching 1/\sqrt 2 in the limit. In the classical case both values are 1.
635 citations
Cites background or methods from "Advanced Mathematical Methods for S..."
...We use extensively the Method of Stationary Phase [4] to extract the asymptotic properties of the resulting wave function....
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...thank Mike Saks for discussions on the asymptotic behavior of integrals and for directing us to [4]....
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...Using the Method of Stationary Phase [4, 5], it is possible to derive the asymptotic form of the amplitudes from their integral representation, and hence also the form of the probability distribution P (n; t)....
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...There is a well-developed theory of the asymptotic expansion of integrals that allows us to determine the behavior of the wave function in the limit [4, 5]....
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References
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TL;DR: In this paper, a review of the Kuramoto model of coupled phase oscillators is presented, with a rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years.
Abstract: Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included.
2,864 citations
TL;DR: In this article, the authors present convergence properties of Multipole Expansion of Intermolecular Interaction Operator (MEI) and van der Waals constants (VWC).
Abstract: 1. First-Order (Heitler-London) Exchange 2. Exchange-Induction Interaction 3. Exchange-Dispersion Interaction D. Convergence Properties of Symmetry-Adapted Theories IV. Multipole Expansion of Interaction Energy A. General Asymptotic Expansion of Interaction Energy B. Multipole Expansion of Intermolecular Interaction Operator C. van der Waals Constants D. Convergence Properties of Multipole Expansion of Interaction Energy E. Angular Dependence of Interaction Energy F. Computations of van der Waals Constants Ill. Exchange Effects
2,298 citations
University of South Carolina1, Los Alamos National Laboratory2, Moscow State University3, Delhi Technological University4, University of Paris5, University of California, Davis6, Indian Institute of Technology (BHU) Varanasi7, University of Moratuwa8, University of Illinois at Urbana–Champaign9, California Polytechnic State University10, Sandia National Laboratories11, Max Planck Society12, Indian Institute of Technology Kharagpur13, French Institute for Research in Computer Science and Automation14, University of New Mexico15, Charles University in Prague16, Birla Institute of Technology and Science17, Indian Institute of Technology Bombay18, University of West Bohemia19
TL;DR: The architecture of SymPy is presented, a description of its features, and a discussion of select domain specific submodules are discussed, to become the standard symbolic library for the scientific Python ecosystem.
Abstract: SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.
1,126 citations
TL;DR: An alternative derivation of passive imaging of the ballistic wave that is not based on normal modes is presented, showing that the global requirement of the equipartitioning of normal modes can be relaxed to the local requirement that the scattered waves propagate on average isotropically near the receivers.
Abstract: The Green's function of waves that propagate between two receivers can be found by cross-correlating multiply scattered waves recorded at these receivers. This technique obviates the need for a source at one of these locations, and is therefore called "passive imaging." This principle has been explained by assuming that the normal modes of the system are uncorrelated and that all carry the same amount of energy (equipartitioning). Here I present an alternative derivation of passive imaging of the ballistic wave that is not based on normal modes. The derivation is valid for scalar waves in three dimensions, and for elastic surface waves. Passive imaging of the ballistic wave is based on the destructive interference of waves radiated from scatterers away from the receiver line, and the constructive interference of waves radiated from secondary sources near the receiver line. The derivation presented here shows that the global requirement of the equipartitioning of normal modes can be relaxed to the local requirement that the scattered waves propagate on average isotropically near the receivers.
1,089 citations
TL;DR: The response of a model microelectrochemical system to a time-dependent applied voltage is analyzed, including electrochemistry, colloidal science, and microfluidics, including surface conduction, multicomponent electrolytes, and Faradaic processes.
Abstract: The response of a model microelectrochemical system to a time-dependent applied voltage is analyzed. The article begins with a fresh historical review including electrochemistry, colloidal science, and microfluidics. The model problem consists of a symmetric binary electrolyte between parallel-plate blocking electrodes, which suddenly apply a voltage. Compact Stern layers on the electrodes are also taken into account. The Nernst-Planck-Poisson equations are first linearized and solved by Laplace transforms for small voltages, and numerical solutions are obtained for large voltages. The "weakly nonlinear" limit of thin double layers is then analyzed by matched asymptotic expansions in the small parameter epsilon= lambdaD/L, where lambdaD is the screening length and L the electrode separation. At leading order, the system initially behaves like an RC circuit with a response time of lambdaDL/D (not lambdaD2/D), where D is the ionic diffusivity, but nonlinearity violates this common picture and introduces multiple time scales. The charging process slows down, and neutral-salt adsorption by the diffuse part of the double layer couples to bulk diffusion at the time scale, L2/D. In the "strongly nonlinear" regime (controlled by a dimensionless parameter resembling the Dukhin number), this effect produces bulk concentration gradients, and, at very large voltages, transient space charge. The article concludes with an overview of more general situations involving surface conduction, multicomponent electrolytes, and Faradaic processes.
938 citations