Von Neumann Theory Research Articles

Laminar, cellular, transverse, and multiheaded pulsating detonations in condensed phase energetic materials from molecular dynamics simulations.

The development of condensed-phase detonation instabilities is simulated using moving window molecular dynamics and a generic AB model of a high explosive. It is found that an initially planar detonation front with one-dimensional flow can become unstable through development of transverse perturbations resulting in highly inhomogeneous and complex two- and three-dimensional distributions of pressure and other variables within the detonation front. Chemical reactions are initiated in localized transverse shock fronts and Mach stems with a pressure and temperature higher than those predicted by classic Zel'dovich, von Neumann, and Doering detonation theory. The two-dimensional cellular and transverse and three-dimensional pulsating detonation structures are found by varying the physico-chemical properties of AB energetic material, sample geometry, and boundary conditions. The different regimes of condensed-phase detonation that can develop from instabilities within a planar detonation front exhibit structures, although at a much smaller scale, that are similar to those observed in gases and diluted liquids.

Self-adjoint extensions for discrete linear Hamiltonian systems

This paper is concerned with self-adjoint extensions for a class of discrete linear Hamiltonian systems. By applying the generalized von Neumann theory and the GKN theory for Hermitian subspaces, a complete characterization of all the self-adjoint subspace extensions for the systems are obtained in terms of boundary conditions via linear independent square summable solutions. As a consequence, characterizations of all the self-adjoint subspace extensions are given in the limit point and limit circle cases. In addition, some sufficient conditions for the corresponding minimal subspace to be an operator or a densely defined operator are given, and consequently, characterizations of all the self-adjoint operator extensions for the system are obtained. In particular, even-order formally self-adjoint vector difference equations are discussed as a special case.

William W. Cooper: Innovator, Fighter, and Scholar

W illiam Wager Cooper was born on July 23, 1914 in Birmingham, Alabama, where his father was a bookkeeper and later a distributor for Anheuser-Busch. Bill’s family moved to Chicago when he was three years old. His father owned a chain of gasoline stations, but lost them during the Great Depression. With his father in ill health and no family revenue, Bill left high school after his sophomore year to work at whatever jobs he could find. These included spotting pins in bowling alleys, caddying at golf courses, and even professional boxing, where his record was 58 wins, 3 losses, and 2 draws. While hitchhiking to the golf course one day, Bill met Eric L. Kohler, a principal with Arthur Andersen & Co., who became his life-long mentor. Kohler loaned him money to enroll as a non-degree student at The University of Chicago. Bill liked the academic atmosphere and took the college entrance examinations, intending to become a physical chemist. During his studies, Kohler asked him to review the mathematics used in a patent infringement suit. After Cooper found errors in the mathematics used by the plaintiff’s engineers, Andersen hired him part time during the school year and full time in the summer. This awakened Cooper’s interest in accounting, and he switched his major to economics. While at Chicago, Bill became good friends with a political science student, Herbert A. Simon. Cooper graduated Phi Beta Kappa with his A.B. in 1938.

Schrödinger and Dirac operators with the Aharonov–Bohm and magnetic-solenoid fields

We construct all self-adjoint Schrödinger and Dirac operators (Hamiltonians) with both the pure Aharonov–Bohm (AB) field and the so-called magnetic-solenoid field (a collinear superposition of the AB field and a constant magnetic field). We perform a spectral analysis for these operators, which includes finding spectra and spectral decompositions, or inversion formulae. In constructing the Hamiltonians and performing their spectral analysis, we follow, respectively, the von Neumann theory of self-adjoint extensions of symmetric operators and the Krein method of guiding functionals.

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Prospect Theory, Indifference Curves, and Hedging Risks

The prospect theory is one of the most popular decision-making theories. It is based on S-shaped utility functions, unlike the von Neumann and Morgenstern (NM) theory, which is based on concave utility functions. The S-shaped functions bring challenges, and extensions and generalizations of the NM theory into the prospect theory are not always possible. For example, in the prospect theory, the monotonicity of indifference curves depends on the underlying mean, unlike in the NM theory. Risk-hedging decisions also become more complex within the prospect theory. In this paper, we discuss these topics and establish general results concerning certain covariances from which we can in turn infer properties of indifference curves and hedging decisions within the prospect theory.

Electron capture and scaling anomaly in polar molecules

We present a new analysis of the electron capture mechanism in polar molecules, based on von Neumann's theory of self-adjoint extensions. Our analysis suggests that it is theoretically possible for polar molecules to form bound states with electrons, even with dipole moments smaller than the critical value D 0 = 1.63 × 10 −18 esu cm . We argue that the quantum mechanical scaling anomaly is responsible for the formation of these bound states.

Physics and the Measurement of Continuous Variables

This paper addresses the doubts voiced by Wigner about the physical rele- vance of the concept of geometrical points by exploiting some facts known to all but honored by none: Almost all real numbers are transcendental; the explicit represen- tation of any one will require an infinite amount of physical resources. An instrument devised to measure a continuous real variable will need a continuum of internal states to achieve perfect resolution. Consequently, a laboratory instrument for measuring a continuous variable in a finite time can report only a finite number of values, each of which is constrained to be a rational number. It does not matter whether the variable is classical or quantum-mechanical. Now, in von Neumann's measurement theory (von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton Uni- versity Press, Princeton, 1955), an operator A with a continuous spectrum—which has no eigenvectors—cannot be measured, but it can be approximated by operators with discrete spectra which are measurable. The measurable approximant F( A) is not canonically determined; it has to be chosen by the experimentalist. It is argued that this operator can always be chosen in such a way that Sewell's results (Sewell in Rep. Math. Phys. 56: 271, 2005; Sewell, Lecture given at the J.T. Lewis Memorial Conference, Dublin, 2005) on the measurement of a hermitian operator on a finite- dimensional vector space (described in Sect. 3.2) constitute an adequate resolution of the measurement problem in this theory. From this follows our major conclusion, which is that the notion of a geometrical point is as meaningful in nonrelativistic quantum mechanics as it is in classical physics. It is necessary to be sensitive to the fact that there is a gap between theoretical and experimental physics, which reveals itself tellingly as an error inherent in the measurement of a continuous variable.

Meso comes to markets: Comment on ‘Markets come to bits’

Mirowski's research programme of markomata formally sets out to model ‘the diversity of market forms by making use of the mathematical theory of automata, and then to taxonomize and organize these entities by means of a theory of evolution’ (Mirowski, this issue). This suggests a new sort of economics in which it is markets that evolve not people, systems that evolve not agents, prosthesis that evolve not preferences, and rules that evolve not information. Economies evolve; people do not. This proposition is both obvious and true, making it an excellent foundation for an evolutionary social science. So, if we are to take Mirowski's program seriously as a general framework for economic analysis, which we think we certainly should, then it is important to recognize that although Mirowski's unified rubric of markomata elides agency and is instead based upon John von Neumann's theory of automata as a model for the study of social institutions, the separate lines of research into markets as computational entities are all still very much but a microeconomics. Markomata is a theory of the core microeconomic building blocks of an economic system as well as a theory of their evolutionary principles (i.e. a microeconomics) but it is not yet a theory of the whole economy. Yet, it is our view that Mirowski's markomata program can indeed be the basis for such a general theory when analytically expounded in terms of the new evolutionary economic framework of micro meso macro (Dopfer et al., 2004 and Dopfer, 2005) in which market automata (Mirowski and Somefun, 1998 and Potts, 2001) are abstracted as a special, central instance of generic meso rules (Dopfer and Potts, 2004). Which is to suggest that the Mirowski framework of markomata can be yet further developed

The von Neumann way to treat systems of mixed dimensionality

We show how the von Neumann theory of self-adjoint extensions can be used to investigatequantum systems the configuration space of which can be decomposed into parts of different dimensionalities. The method can be applied in many situations; we illustrate it on examples including point contact spectroscopy, nanotube systems, microwave resonators, or spin conductance oscillations.

The basic George B. Dantzig

The late George B. Dantzig , widely known as the father of linear programming, was a major influence in mathematics, operations research, and economics. As Professor Emeritus at Stanford University, he continued his decades of research on linear programming and related subjects. Dantzig was awarded eight honorary doctorates, the National Medal of Science, and the John von Neumann Theory Prize from the Institute for Operations Research and the Management Sciences.The 24 chapters of this volume highlight the amazing breadth and enduring influence of Dantzig's research. Short, non-technical summaries at the opening of each major section introduce a specific research area and discuss the current significance of Dantzig's work in that field. Among the topics covered are mathematical statistics, the Simplex Method of linear programming, economic modeling, network optimization, and nonlinear programming. The book also includes a complete bibliography of Dantzig's writings.

Dirac equation in magnetic-solenoid field

We consider the Dirac equation in the magnetic-solenoid field (the field of a solenoid and a collinear uniform magnetic field). For the case of Aharonov-Bohm solenoid, we construct self-adjoint extensions of the Dirac Hamiltonian using von Neumann's theory of deficiency indices. We find self-adjoint extensions of the Dirac Hamiltonian in both above dimensions and boundary conditions at the AB solenoid. Besides, for the first time, solutions of the Dirac equation in the magnetic-solenoid field with a finite radius solenoid were found. We study the structure of these solutions and their dependence on the behavior of the magnetic field inside the solenoid. Then we exploit the latter solutions to specify boundary conditions for the magnetic-solenoid field with Aharonov-Bohm solenoid.

Self-adjoint extensions of the Dirac Hamiltonian in the magnetic-solenoid field and related exact solutions

We study solutions of Dirac equation in the field of Aharonov-Bohm solenoid and a collinear uniform magnetic field. On this base we construct self-adjoint extensions of the Dirac Hamiltonian using von Neumann's theory of deficiency indices. We reduce $(3+1)$-dimensional problem to $(2+1)$-dimensional one by a proper choice of spin operator. Then we study the problem doing a finite radius regularization of the solenoid field. We exploit solutions of the latter problem to specify boundary conditions in the singular case.

Machine Dreams: Economics Becomes a Cyborg Science. By Philip Mirowski. New York: Cambridge University Press, 2002. Pp. xiv, 655. $35.00, paper.

It is impossible to summarize and critique this book adequately in a short review. Its author is a leading historian of economic thought, skilled in many disciplines, imaginative and penetrating in analysis, and possessed of knowledge, brilliance, and pixie-like cynicism. His More Heat than Light (New York: Cambridge University Press, 1989) traced the origins of twentieth-century economics to a mimicking of nineteenth-century physics and mathematics in order to achieve the status of “science.” This book traces the social construction of high theory in economics during World War II and the Cold War to the influence of John von Neumann's general theory of automata and the military funding of basic economic science. Like his earlier book, this is history of economics on a massively ambitious scale—a theory of the entire postwar period.

Efficient Portfolios, Sparse Matrices, and Entities: A Retrospective

In 1989 I was pleased and honored to be awarded the ORSA/TIMS (now INFORMS) John von Neumann Theory Prize for my work in portfolio theory, sparse matrices, and SIMSCRIPT. The following is a retrospective on my work in these fields.

Comparison between Weak Mach Reflection and von Neumann Reflection.

The purpose of the present paper is to compare the characteristics of the weak Mach reflection (MR) and the so-called von Neumann reflection (vNR). A series of experiments was performed to investigate both reflections. Attention is focused on the flow properties around the triple point. By measuring the location of the triple point and the angle between the incident and reflected shocks, we estimated the approximate flow properties around the triple point. The results show that, while the triple point moves along an almost straight trajectory through a wedge tip, the angles of incidence and reflection vary as the incident shock propagates. As a result, the well-known self-similarity law does not hold for both reflections. For the von Neumann reflection, the present experimental data vary almost along the trivial solution curve of the von Neumann's three-shock theory, and an asymptotic state seems to exist on this curve. In contrast, for weak Mach reflections, these data do not lie on the trivial solution curve but reveal the so-called von Neumann paradox.

Quantum histories and measurements

We define a quantum measurement as destruction of coherence between certain components of the Schrödinger wavefunction Ψ( x, t), related to the particle's possible pasts. The approach generalises von Neumann theory to finite time and continuous measurements. Examples include the particle's mean position and continuous monitoring of the particle's momentum.

Kirchhoff's Rule for Quantum Wires. II: The Inverse Problem with Possible Applications to Quantum Computers

In this article we continue our investigations of one particle quantum scattering theory for Schrödinger operators on a set of connected (idealized one-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local Kirchhoff law). In “Kirchhoff's rule for quantum wires” [J. Phys. A: Math. Gen. 32, 595–630 (1999)] we provided an explicit algebraic expression for the resulting (on-shell) S-matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S-matrix at a fixed, prescribed energy. We show that any unitary n × n matrix may be realized as the S-matrix at a given energy by choosing appropriate (unique) boundary conditions. This might possibly be used for the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a given time associated to a quantum mechanical Hamiltonian. We also suggest an approach by which the S-matrix in our context may be obtained from “scattering experiments”, another aspect of the inverse problem. Finally we extend our previous discussion, how our approach is related to von Neumann's theory of selfadjoint extensions.

A parametric study of Mach reflection in steady flows

The flow fields associated with Mach reflection wave configurations in steady flows are analysed, and an analytical model for predicting the wave configurations is proposed. It is found that provided the flow field is free of far-field downstream influences, the Mach stem heights are solely determined by the set-up geometry for given incoming-flow Mach numbers. It is shown that the point at which the Mach stem height equals zero is exactly at the von Neumann transition. For some parameters, the flow becomes choked before the Mach stem height approaches zero. It is suggested that the existence of a Mach reflection not only depends on the strength and the orientation of the incident shock wave, as prevails in von Neumann's three-shock theory, but also on the set-up geometry to which the Mach reflection wave configuration is attached. The parameter domain, beyond which the flow gets choked and hence a Mach reflection cannot be established, is calculated. Predictions based on the present model are found to agree well both with experimental and numerical results.