Preface

<正>These two special issues of Acta Mathematica Scientia are dedicated to ProfessorJames Glimm on the occasion of his seventy-fifth birthday.     

樊■——为数学而生,数学乃是他的生命

本文概括与综述樊■先生在拓扑群、非线性与凸分析、不动点理论、算子理论、线性代数与矩阵论、数学规划、逼近论等分支领域的重大学术成就与奠基性贡献,展现樊■在数学上那炉火纯青、出神入化的境界与独特的风格:条件自然,论断凝练,证明优美,应用广泛.     

Contributions of Lo Yang to Value Distribution Theory

Professor Lo Yang is a world famous mathematician of our country. He made a lot of outstanding achievements in the value distribution theory of function theory, which are highly rated and widely quoted by domestic and foreign scholars. He also did a lot of work to develop Chinese mathematics. It can be said that Professor Yang is one of the mathematicians who made main influences on the mathematical development in modern China. This paper briefly introduces Professor Yang’s life, mainly discusses his academic achievement and influence, and briefly describes his contributions to the Chinese mathematics community.     

James Mills Peirce and the cult of quaternions

Quaternions furnish an example of the rise and fall in popularity of a mathematical theory. Without attempting to trace its mathematical and sociological causes, this note highlights some moments in the history, of this phenomenon, as well as some of the scientists involved, especially the neglected mathematician, James Mills Peirce.     

A proportional view: The mathematics of James Glenie (1750–1817)

The mathematical work of James Glenie (1750–1817) was published at irregular intervals during a turbulent life. His ideas, mostly deriving from his time as an Assistant in Mathematics at St Andrews University in Scotland, were developed intermittently over a period of thirty-seven years. His mathematical achievements, underestimated by previous historians, were deeply rooted in Euclidean geometry and his own generalized theory of proportion. Among them are many new geometrical constructions and proofs, a novel demonstration of the binomial theorem, and an alternative approach to the differential calculus.     

对冯·诺意曼“量子概率论”的两点质疑

本文对于公认的量子力学数学基础—冯·诺意曼概率论提出了两点质疑。从物理上讲,它将导致“粒子的势能以一定的概率超过总能量”这种违背能量守恒原理的结论。从数学上讲,它在某些方面还不如玻恩的概率解释(即常被物理学家称为“经典概率论”的数学意义的概率论)。因此,量子力学的数学基础似乎需要进一步发展和完善。     

Laplace transform of generalized functions of slow growth

We present certain special sections of the theory of functions of several variables and of the theory of generalized functions, which are the basic mathematical tools for certain directions in modern quantum field theory. In passing we study the properties of certain special algebras of analytic functions. A considerable part of the results in the article are due to the author.     

Statistical Algebraic Approach to Quantum Mechanics

A scheme for constructing quantum mechanics not based on the Hilbert space and linear operators as primary elements of the theory is proposed. A particular variant of the algebraic approach is discussed. The elements of a noncommutative algebra (i.e., the observables) and the nonlinear functionals on this algebra (i.e., the physical states) serve as the primary components of the theory. The functionals are associated with the results of a single measurement. The ensembles of physical states are suggested for the role of quantum states in the standard quantum mechanics. It is shown that the mathematical formalism of the standard quantum mechanics can be fully recovered within this scheme.     

Z 分次表示理论

箭图Hecke 代数的Z 分次表示理论是“代数群、量子群及Hecke 代数” 领域中当前最活跃的研究方向之一. 箭图Hecke 代数及其分圆版本产生于对量子群及其可积最高权表示的范畴化的研究, 它们与数学及数学物理的许多不同分支如Lie 代数、量子群、Kazhdan-Lusztig 理论、代数几何(箭图簇,反常层)、扭结理论、拓扑量子场论(TQFT) 等都有着紧密的联系与相互作用. 本文详细介绍了该方向的最新进展、前沿以及研究前景.     

A Survey of the Matrix Exponential Formulae with Some Applications

The matrix exponential formulae are very important tools in diverse fields of mathematical physics, in particular, they are also useful in studying quantum information theory. In this paper, we survey results related to matrix exponential thoroughly. The purpose of the article is pedagogical.     

Book Reviews

Abstract

The Mathematical and Computational Sciences in Emerging Manufacturing Technologies and Management Practices by Avner Friedman, James Glimm, and John Lavery. Philadelphia: Society of Industrial and Applied Mathematics, 1992, 86 pp., free (ISBN 0-89871-307-2). Reviewed by Michael M. Meyer

The Physics of Chance by Charles Ruhla. Oxford, U.K.: Oxford University Press, 1992, 222 pp., $55 (hardcover), $27.95 (paperback) (ISBN 0-19-853960-6). Translated by G. Barton. Reviewed by Michael M. Meyer

The Future of the Survey of Income and Program Participation edited by Constance F. Citro and Graham Kalton. Washington, DC: National Academy Press, 1993, 284 pp. Reviewed by Michael M. Meyer     

Locality problem in quantum theory

We discuss the locality problem in relativistic and nonrelativistic quantum theory. We show that there exists a formulation of quantum theory that, on one hand, preserves the mathematical apparatus of the standard quantum mechanics and, on the other hand, ensures the satisfaction of the locality condition for each individual event including the measurement procedure. As an example, we consider the scattering from two slits. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 327–343, May, 2008.     

Global existence of shock front solution to axially symmetric piston problem in compressible flow

In this paper we study the global existence of BV solution to two-dimensional piston problem in fluid dynamics. Different from previous results on related problems we remove the restriction on the strength of the leading shock and require the velocity of the piston is rather fast or the density is quite small instead. The main tool in our proof is Glimm Scheme with some improvement. To define the Glimm functional we derive more precise estimates for the interaction of elementary waves, particularly in the region near the leading shock.     

Global entropy solutions to a class of quasi-linear wave equations with large time-oscillating sources

This research explores the Cauchy problem for a class of quasi-linear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality.     

Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for flows with strong discontinuities

Typical nonlinear wave interaction problems involve strong waves moving through a background of weak disturbance. Previous existence theorems and error analysis for computations are usually restricted to more idealized situations such as small data or single equations. We consider here the problem of a single strong discontinuity interacting with a weak background for general hyperbolic systems of conservation laws. We obtain the stability, consistency theorems and upper bounds of the truncation errors for the Glimm scheme and for a front tracking method. The major error in the Glimm scheme is the error generated by the strong discontinuity. This error is reduced when a front tracking method is applied to follow the location of the strong discontinuity. This demonstrates an advantage of front tracking methods in one-space dimension.     

The possibility of reconciling quantum mechanics with classical probability theory

We describe a scheme for constructing quantum mechanics in which a quantum system is considered as a collection of open classical subsystems. This allows using the formal classical logic and classical probability theory in quantum mechanics. Our approach nevertheless allows completely reproducing the standard mathematical formalism of quantum mechanics and identifying its applicability limits. We especially attend to the quantum state reduction problem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 457–472, December, 2006.     

BOOK REVIEWS

Book reviewed in this article: America at Work Series: Machines for America; Power for America; Wings for America, by Marshall Dunn, Author of “Science and Modern Progress” and “Up to Civilization,” and Lloyd N. Morrisett, Professor of Education, University of California, Los Angeles. Factors Affecting Student Achievement and Change in a Physical Science Survey Course, by Waldo Lyle Brewer, Ph.D. The Practical Outline of Mechanical Trades for Home Study, edited by William L. Schaff, Assistant Professor of Education, Brooklyn College. Practical Physics, by Marsh W. White, Ph.D., Editor, Professor of Physics; Kenneth V. Manning, Ph.D., Assistant Supervisor of Physics Extension; Robert L. Weber, Ph.D., Assistant Professor of Physics; R. Orin Cornett, Ph.D., Lecturer in Electronics, Harvard University Heat and Thermodynamics, by Mark W. Zemansky, Ph.D. Mathematics for the Sheet Metal Worker, by Clayton E. Buell, B.S., M.Ed. Educators Guide to Free Films, Third Edition, Compiled and Edited by Mary Foley Horkheimer; and John W. Differ, M.A. Methods of Teaching in Town and Rural Schools, by E. L. Ritter and L. A. Shephers Radio Education Pioneering in the Mid-West, by Albert A. Reed, LL.D., Formerly Deputy State Superintendent of Public Instruction, Lincoln, Nebraska; Director-Emeritus, University Extension Division, University of Nebraska; Professor Emeritus, Secondary Education, University of Nebraska. Piloting and Maneuvering of Ships, by Lyman M. Kells, Ph.D., Associate Professor of Mathematics; Willis F. Kern, Associate Professor of Mathematics; and James R. Bland, Associate Professor of Mathematics Navigation, by Lyman M. Keils, Ph.D., Professor of Mathematics; Willis F. Kern, Associate Professor of Mathematics; and James R. Bland, Associate Professor of Mathematics     

A Quantum Dynamical Approach to Matrix Khrushchev's Formulas

Khrushchev's formula is the cornerstone of the so‐called Khrushchev theory, a body of results which has revolutionized the theory of orthogonal polynomials on the unit circle. This formula can be understood as a factorization of the Schur function for an orthogonal polynomial modification of a measure on the unit circle. No such formula is known in the case of matrix‐valued measures. This constitutes the main obstacle to generalize Khrushchev theory to the matrix‐valued setting, which we overcome in this paper. It was recently discovered that orthogonal polynomials on the unit circle and their matrix‐valued versions play a significant role in the study of quantum walks, the quantum mechanical analogue of random walks. In particular, Schur functions turn out to be the mathematical tool which best codify the return properties of a discrete time quantum system, a topic in which Khrushchev's formula has profound and surprising implications. We will show that this connection between Schur functions and quantum walks is behind a simple proof of Khrushchev's formula via “quantum” diagrammatic techniques for CMV matrices. This does not merely give a quantum meaning to a known mathematical result, since the diagrammatic proof also works for matrix‐valued measures. Actually, this path‐counting approach is so fruitful that it provides different matrix generalizations of Khrushchev's formula, some of them new even in the case of scalar measures. Furthermore, the path‐counting approach allows us to identify the properties of CMV matrices which are responsible for Khrushchev's formula. On the one hand, this helps to formalize and unify the diagrammatic proofs using simple operator theory tools. On the other hand, this is the origin of our main result which extends Khrushchev's formula beyond the CMV case, as a factorization rule for Schur functions related to general unitary operators.© 2016 Wiley Periodicals, Inc.     

Magnetic Bloch theory and noncommutative geometry

An interpretation of the magnetic Bloch theory in terms of noncommutative geometry is given. As an application we obtain a mathematical interpretation of the quantum Hall effect.     

Quantum Probability: New Perspectives for the Laws of Chance

The main philosophical successes of quantum probability is the discovery that all the so-called quantum paradoxes have the same conceptual root and that such root is of probabilistic nature. This discovery marks the birth of quantum probability not as a purely mathematical (noncommutative) generalization of a classical theory, but as a conceptual turning point on the laws of chance, made necessary by experimental results.