Methods Of Mathematical Physics Research Articles

Program of "30 Years of Mathematical Methods in High Energy Physics" in honor of Professor Tohru Eguchi's 60th birthday March 17-19, 2008 RIMS, Kyoto, Japan

Program of "30 Years of Mathematical Methods in High Energy Physics" in honor of Professor Tohru Eguchi's 60th birthday March 17-19, 2008 RIMS, Kyoto, Japan

30 Years of Mathematical Methods in High Energy Physics

30 Years of Mathematical Methods in High Energy Physics

Numerical modeling of the radiation by a submerged fluid-filled cylindrical shell: Observation of the S0, A0, and A waves

We consider a submerged fluid-filled cylindrical shell subjected to an external acoustic pulse and analyze the structure of the field radiated by the shell into the fluids, both external and internal. We first propose a computationally efficient semi-analytical model of the interaction based on the Reissner-Mindlin shell theory combining some of the classical methods of mathematical physics with the finite-difference methodology, and then use the model to simulate the interaction. We demonstrate that the model accurately reproduces the wave structure of the radiated fields seen in the experiments for submerged evacuated shells, namely, both the symmetric Lamb waves S0 and the pseudo-Rayleigh waves A0. It is further observed that the internal and external wave patterns associated with the A0 waves exhibit the same alternation of the equiphase lines as the one seen in the experiments for a plate loaded by the fluid on both sides, a result that seems to be particularly relevant in the context of very limited number of experimental images of the radiated field for shells loaded by fluid from both inside and outside. Not less interestingly, we also demonstrate that the Scholte-Stoneley, or A, wave is also reproduced by the model, and we offer some insights into the non-observability of this wave for certain types of cylindrical shells reported in earlier experimental studies.

Methods of Mathematical Physics, vol. 1 (575 pp, April 1989) Methods of Mathematical Physics, vol. 2, Differential Equations (852 pages, April 1989), by Richard Courant and David Hilbert − WILEY-VCH Book review

Methods of Mathematical Physics, vol. 1 (575 pp, April 1989) Methods of Mathematical Physics, vol. 2, Differential Equations (852 pages, April 1989), by Richard Courant and David Hilbert − WILEY-VCH Book review

Economic Phenomena from the Viewpoint of the Mechanics of Materials

The article describes the increasing use of physical and mathematical models to describe and analyze economic processes based upon the analogies found between physical and economic processes during the formation of the Lausanne and Cambridge Schools of Economics. In brief, one of the necessary outcomes of the application of physical-mathematical methods in economics has been the gradual creation of conditions in basic and applied economic research during the course of the twentieth century, which at the turn of the millennium gave rise to econophysics. Contributions of the Czech School of Economics are documented, particularly the analogy between the length of fatigue cracks in metals exposed to periodic loads over time and the fall of market prices for commodities over time.

Editors’ preface for the topical issue “Finite dimensional integrable systems, dynamics, and Lie theoretic methods in Geometry and Mathematical Physics”

The articles collected in this issue are related to two workshops that took place in the summer of 2011. The workshop Cartan Connections, Geometry of Homogeneous Spaces, and Dynamics was organized by Andreas Cap (University of Vienna), Charles Frances (Universite Paris-Sud), and Karin Melnick (University of Maryland) from July 10–23, 2011, at the research platform International Erwin Schrodinger Institute for Mathematical Physics (ESI) of the University of Vienna. The conference FDIS2011 – Finite Dimensional Integrable Systems in Geometry and Physics, was organized by Vladimir Matveev, Christian Richter, and Konrad Schobel (all University of Jena) in Jena from July 26–29, 2011.

Web Search Queries Can Predict Stock Market Volumes

We live in a computerized and networked society where many of our actions leave a digital trace and affect other people’s actions. This has lead to the emergence of a new data-driven research field: mathematical methods of computer science, statistical physics and sociometry provide insights on a wide range of disciplines ranging from social science to human mobility. A recent important discovery is that search engine traffic (i.e., the number of requests submitted by users to search engines on the www) can be used to track and, in some cases, to anticipate the dynamics of social phenomena. Successful examples include unemployment levels, car and home sales, and epidemics spreading. Few recent works applied this approach to stock prices and market sentiment. However, it remains unclear if trends in financial markets can be anticipated by the collective wisdom of on-line users on the web. Here we show that daily trading volumes of stocks traded in NASDAQ-100 are correlated with daily volumes of queries related to the same stocks. In particular, query volumes anticipate in many cases peaks of trading by one day or more. Our analysis is carried out on a unique dataset of queries, submitted to an important web search engine, which enable us to investigate also the user behavior. We show that the query volume dynamics emerges from the collective but seemingly uncoordinated activity of many users. These findings contribute to the debate on the identification of early warnings of financial systemic risk, based on the activity of users of the www.

MASA : a library for verification using manufactured and analytical solutions

In this paper we introduce the Manufactured Analytical Solution Abstraction ( MASA ) library for applying the method of manufactured solutions to the verification of software used for solving a large class of problems stemming from numerical methods in mathematical physics including nonlinear equations, systems of algebraic equations, and ordinary and partial differential equations. We discuss the process of scientific software verification, manufactured solution generation using symbolic manipulation with computer algebra systems such as Maple™ or SymPy, and automatic differentiation for forcing function evaluation. We discuss a hierarchic methodology that can be used to alleviate the combinatorial complexity in generating symbolic manufactured solutions for systems of equations based on complex physics. Finally, we detail the essential features and examples of the Application Programming Interface behind MASA , an open source library designed to act as a central repository for manufactured and analytical solutions over a diverse range of problems.

The Symplectic Method for the Bending of Symmetrical Laminated Composite Plates with Clamped Boundary

The symplectic method is applied to study analytically the stress distributions of composite laminated plates in this paper applies. The dual equations were presented by variation principle and introducing separation of variables used. Then in the symplectic space which consists of the original variables and their dual variables, the problem can be solved via effective mathematical physics methods such as the method of separation of variables and Eigen function vector expansion. The equation and Eigen-vector are deduced. The results of cross-ply laminated graphite–epoxy composite plate with clamped boundary are shown, which is compared with established results. The parameters’ influences on mechanical property are also discussed.

Recent Advances in Analytical Methods in Mathematical Physics

1 Division of Mechanics, Faculty of Civil Engineering, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey 2 Department of Plasma Theory, Institute for Nuclear Research, National Academy of Sciences of Ukraine, Kyiv 03680, Ukraine 3 Department of Mathematics and Statistics, Dalhousie University, Halifax NS, Canada B3H 4R2 4 Research Laboratory for Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 5 Department of Mechanical Engineering, Faculty of Engineering, National University of Singapore, Singapore 117576 6 MITRE Corporation, 3101 Virginia Ave, Silver Spring, MD 20910, USA 7 Department of Electrical and Electronics Engineering, Faculty of Engineering, Trakya University, 22030 Edirne, Turkey

Symplectic Solutions in Singularity Problems of Anisotropic Beams

Based on the two-dimensional elasticity,the symplectic method is applied to study analytically the stress distributions of anisotropic beam.Using variation principle and introducing separation of variables, dual equations were presented.Then in the symplectic space which consists of the original variables and their dual variables,the problem can be solved via effective mathematical physics methods such as the method of sepatation of variables and eigenfunction vector expansion.So the original problems come down to solve the eigensolutions of zero eigenvalue and non-zeroes eigenvalue that describe the exponentially decaying localized solutions usually ignored by Saint-Venant's principle. Completed numerical examples are newly given to compare with established results.

A Conversation with Eugenio Regazzini

Eugenio Regazzini was born on August 12, 1946 in Cremona (Italy), and took his degree in 1969 at the University “L. Bocconi” of Milano. He has held positions at the universities of Torino, Bologna and Milano, and at the University “L. Bocconi” as assistant professor and lecturer from 1974 to 1980, and then professor since 1980. He is currently professor in probability and mathematical statistics at the University of Pavia. In the periods 1989–2001 and 2006–2009 he was head of the Institute for Applications of Mathematics and Computer Science of the Italian National Research Council (C.N.R.) in Milano and head of the Department of Mathematics at the University of Pavia, respectively. For twelve years between 1989 and 2006, he served as a member of the Scientific Board of the Italian Mathematical Union (U.M.I.). In 2007, he was elected Fellow of the IMS and, in 2001, Fellow of the “Istituto Lombardo—Accademia di Scienze e Lettere.” His research activity in probability and statistics has covered a wide spectrum of topics, including finitely additive probabilities, foundations of the Bayesian paradigm, exchangeability and partial exchangeability, distribution of functionals of random probability measures, stochastic integration, history of probability and statistics. Overall, he has been one of the most authoritative developers of de Finetti’s legacy. In the last five years, he has extended his scientific interests to probabilistic methods in mathematical physics; in particular, he has studied the asymptotic behavior of the solutions of equations, which are of interest for the kinetic theory of gases. The present interview was taken in occasion of his 65th birthday.

Symplectic Solutions in Singularity Problems of Anisotropic Beams

Based on the two-dimensional elasticity,the symplectic method is applied to study analytically the stress distributions of anisotropic beam.Using variation principle and introducing separation of variables, dual equations were presented.Then in the symplectic space which consists of the original variables and their dual variables,the problem can be solved via effective mathematical physics methods such as the method of sepatation of variables and eigenfunction vector expansion.So the original problems come down to solve the eigensolutions of zero eigenvalue and non-zeroes eigenvalue that describe the exponentially decaying localized solutions usually ignored by Saint-Venant's principle. Completed numerical examples are newly given to compare with established results.

The Bending of Symmetrical Laminated Composite Plate Based on Symplectic Method

This paper applies a symplectic method to study analytically the stress distributions of Composite laminated plates. Using variation principle and introducing separation of variables, dual equations were presented. Then in the symplectic space which consists of the original variables and their dual variables,the problem can be solved via effective mathematical physics methods such as the method of sepatation of variables and eigenfunction vector expansion. The transcendental equation and eigen-vector are deduced. The results of cross-ply and angle-ply laminated graphite–epoxy composite plate are shown, which is compared with established results. The parameters’ influences on mechanical property are also discussed.

Asymptotic solutions for Mathieu instability under random parametric excitation and nonlinear damping

A theoretical analysis is presented of the response of a lightly and nonlinearly damped mass–spring system in which the spring constant contains a small randomly fluctuating component. Damping is represented by a combination of linear and nonlinear power-law damping. System response to some initial disturbance at time zero is described by a sinusoidal wave whose amplitude and phase vary slowly and randomly with time. Leading order formulations for the equations of amplitude and phase are obtained through the application of methods of stochastic averaging of Stratonovich. The equations of amplitude and phase are given in two versions: Fokker–Planck equations for transient probability and Langevin equations for response in the time-domain. Solutions in closed-form of these equations are derived by methods of mathematical and theoretical physics involving higher transcendental functions. They are used to study the behavior of system response for ever increasing time applying asymptotic methods of analysis such as the method of steepest descent or saddle-point method. It is found that system behavior depends on the power density of the parametric excitation at twice the natural frequency and on the magnitude and form of the damping. Depending on these parameters different types of system behavior are found to be possible: response which decays exponentially to zero, response which leads to a stationary state of random behavior, and response which can either grow unboundedly or which approaches zero in a finite time.

The Kepler map in the three-body problem

The Kepler map was derived by Petrosky (1986) and Chirikov and Vecheslavov (1986) as a tool for description of the long-term chaotic orbital behaviour of the comets in nearly parabolic motion. It is a two-dimensional area-preserving map, describing the motion of a comet in terms of energy and time. Its second equation is based on Kepler’s third law, hence the title of the map. Since 1980s the Kepler map has become paradigmatic in a number of applications in celestial mechanics and atomic physics. It represents an important kind of general separatrix maps. Petrosky and Broucke (1988) used refined methods of mathematical physics to derive analytical expressions for its single parameter. These methods became available only in the second half of the 20th century, and it may seem that the map is inherently a very modern mathematical tool. With the help of the Jacobi integral I show that the Kepler map, including analytical formulae for its parameter, can be derived by quite elementary methods. The prehistory and applications of the Kepler map are considered and discussed.

Bernhard Riemann’s legacy of 1859

The German version of Riemann’s Collected Works is confined to a single volume of 690 pages. Even so, this volume has had an abiding and profound impact on modern mathematics and physics, as we shall see. In fifteen years of activity, from 1851, when he gained his doctorate at the University of Göttingen, to his death in 1866, two months short of his fortieth birthday, Riemann contributed to almost all areas of mathematics. He perceived mathematics from the analytic point of view and used analysis to illuminate subjects as diverse as number theory and geometry. Although regarded principally as a mathematician Riemann had an abiding interest in physics and researched significantly in the methods of mathematical physics, particularly in the area of partial differential equations.

The mechanism of intermolecular energy transmission and perception of ultraweak actions by chemical and biological systems

A novel concept of intermolecular energy transfer and reception of the ultralow action in living systems is proposed. The concept is based on the methods of nonlinear mathematical physics used in description of energy movement along molecular chains and on quantum mechanical ideas concerning signal formation in anisotropic media. A concept of a molecular cell as an indivisible structural unit and a constituent of a biological (chemical) system has been put forward and substantiated, which manifests collective features of the unity of molecules, physical fields, and energetically strained bound water media in processes of energy transfer and reception. Both intermolecular energy transfer and amplification of the ultralow action has been shown to be the components of a unified energy process in a living system, and the physical basis of both processes is the unity of molecules and water-field media in a molecular cell.

On the role of mathematics in biology

It has been widely acknowledged that a key ingredient in the flourishing of Physics in the XX Century was the use of sophisticated mathematical techniques. In consequence, the term “Mathematical Physics” has enjoyed an unrivalled prestige for most of that period, and monographs like Courant and Hilbert s “Methods of Mathematical Physics” have been acclaimed as milestones by mathematicians and physicists alike.

Philipp Frank, Richard von Mises, and the Frank-Mises

The theoretical physicist Philipp Frank (1884–1966) and the applied mathematician Richard von Mises (1883–1953) both received their university education in Vienna shortly after 1900 and became friends at the latest during the Great War.They were attached to the Vienna Circle of Logical Positivists and wrote an influential two-part work on the differential and integral equations of mechanics and physics, the Frank-Mises, of 1925 and 1927, with its second edition following in 1930 and 1935.This work originated in the lectures that the mathematician Bernhard Riemann (1826–1866) delivered on partial differential equations and their applications to physical questions at the University of Gottingen between 1854 and 1862, which were edited and published posthumously in1869 by the physicist Karl Hattendorff (1834–1882).The immediate precursor of the Frank-Mises, however, was the extensive revision of Hattendorff’s edition of Riemann’s lectures that the mathematician Heinrich Weber (1842–1913) published in two volumes, the Riemann-Weber, of 1900 and 1901, with its second edition following in 1910 and 1912. I trace this historical lineage, explore the nature and contents of the Frank-Mises, and discuss its complementary relationship to the first volume of the text that the mathematicians Richard Courant (1888–1972) and David Hilbert (1862–1943) published on the methods of mathematical physics in 1924, the Courant-Hilbert,which, when it and its second volume of 1937 were translated into English and extensively revised in 1953 and 1961, eclipsed the classic Frank-Mises.