Showing papers in "Siam Review in 1991"

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Abstract: 5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems

Abstract: An algorithm is described for solving large-scale instances of the Symmetric Traveling Salesman Problem (STSP) to optimality. The core of the algorithm is a “polyhedral” cutting-plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph. The cuts are generated by several identification procedures that have been described in a companion paper. Whenever the cutting-plane procedure does not terminate with an optimal solution the algorithm uses a tree-search strategy that, as opposed to branch-and-bound, keeps on producing cuts after branching. The algorithm has been implemented in FORTRAN. Two different linear programming (LP) packages have been used as the LP solver. The implementation of the algorithm and the interface with one of the LP solvers is described in sufficient detail to permit the replication of our experiments. Computational results are reported with up to 42 STSPs with sizes rangin...

The fractional Fourier transform and applications

Abstract: This paper describes the “fractional Fourier transform,” which admits computation by an algorithm that has complexity proportional to the fast Fourier transform algorithm. Whereas the discrete Fourier transform (DFT) is based on integral roots of unity $e^{{{ - 2\pi i} / n}} $, the fractional Fourier transform is based on fractional roots of unity $e^{ - 2\pi i\alpha } $ where $\alpha $ is arbitrary. The fractional Fourier transform and the corresponding fast algorithm are useful for such applications as computing DFTs of sequences with prime lengths, computing DFTs of sparse sequences, analyzing sequences with noninteger periodicities, performing high-resolution trigonometric interpolation, detecting lines in noisy images, and detecting signals with linearly drifting frequencies. In many cases, the resulting algorithms are faster by arbitrarily large factors than conventional techniques.

Nilpotent and high-order approximations of vector field systems

Abstract: This paper is concerned with the study of structurally stable properties associated with systems described by one, or several, real analytic vector fields via approximating systems which retain inf...

Parallel algorithms for sparse linear systems

Abstract: This paper surveys recent progress in the development of parallel algorithms for solving sparse linear systems on computer architectures having multiple processors. Attention is focused on direct methods for solving sparse symmetric positive definite systems, specifically by Cholesky factorization. Recent progress on parallel algorithms is surveyed for all phases of the solution process, including ordering, symbolic factorization, numeric factorization, and triangular solution.

Frequency content of randomly scattered signals

Abstract: The statistical properties of acoustic signals reflected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it. The asymptotic analy...

The role of the strengthened Cauchy-Buniakowskii-Schwarz inequality in multilevel methods

Abstract: The basic theory of the strengthened Cauchy–Buniakowskii–Schwarz inequality and its applications in multilevel methods for the solution of linear systems arising from finite element or finite difference discretisation of elliptic partial differential equations is surveyed. Proofs are given both in a finite element context, and in purely algebraic form.

Inertia-controlling methods for general quadratic programming

Abstract: Active-set quadratic programming (QP) methods use a working set to define the search direction and multiplier estimates. In the method proposed by Fletcher in 1971, and in several subsequent mathematically equivalent methods, the working set is chosen to control the inertia of the reduced Hessian, which is never permitted to have more than one nonpositive eigenvalue. (Such methods will be called inertia-controlling.) This paper presents an overview of a generic inertia-controlling QP method, including the equations satisfied by the search direction when the reduced Hessian is positive definite, singular and indefinite. Recurrence relations are derived that define the search direction and Lagrange multiplier vector through equations related to the Karush–Kuhn–Tucker system. Discussion is included of connections with inertia-controlling methods that maintain an explicit factorization of the reduced Hessian matrix.

Vorticity, turbulence, and acoustics in fluid flow

Abstract: This paper presents recent and ongoing research in mathematical fluid dynamics and emphasizes the interdisciplinary interaction of ideas from large-scale computation, asymptotic methods, and mathem...

A survey of consistency properties in cooperative game theory

Abstract: The main purpose of this survey paper is to review the axiomatic characterizations of the Shapley value, the prekernel, the prenucleolus, and the core by means of a consistency property in terms of the reduced games. Whenever possible, new results and new proofs are added.

Some recent advances in network flows

Abstract: The literature on network flow problems is extensive, and over the past 40 years researchers have made continuous improvements to algorithms for solving several classes of problems. However, the surge of activity concerning the algorithmic aspects of network flow problems over the past few years has been particularly striking. Several techniques have proven to be very successful in permitting researchers to make these recent contributions: (i) scaling of the problem data; (ii) improved analysis of algorithms, especially amortized worst-case performance and the use of potential functions; and (iii) enhanced data structures. This survey illustrates some of these techniques and their usefulness in developing faster network flow algorithms. The discussion focuses on the design of faster algorithms from the worst-case perspective, and is limited to the following fundamental problems: the shortest path problem, the maximum flow problem, and the minimum cost flow problem. Several representative algorithms from e...

A generalized birthday problem

Abstract: A generalized version of the birthday problem is as follows. Suppose each member of a population independently receives a number randomly selected from $\{ {1,2,3, \cdots ,x} \}$, and a random sample of size n is to be taken. If $0 < p < 1$, what is the smallest value of n so that the probability that at least two of the sample have the same number is at least p? Both empirical modeling and approximation techniques are used to determine n as a function of x when p is fixed. An error analysis of the approximation is presented.

Nonlinear and stochastic phenomena: the grand challenge for partial differential equations

Abstract: The major problems for partial differential equations are either nonlinear or stochastic or both. Considerable progress has occurred recently in these areas, much of which has resulted from a striking interplay among theory, computation, and applications.The nonlinear structures emphasized here are the nonlinear hyperbolic waves, which occur in the solutions of nonlinear hyperbolic conservation laws and associated dissipative equations. These equations are those most commonly used to model physical and chemical processes in continuum systems. In one space dimension these nonlinear waves form traveling waves, for which the methods of dynamical systems and bifurcation theory are very useful. Established points of view toward mathematical theories have been challenged by recent work (e.g., uniqueness of solutions, entropy conditions, and elliptic regions in hyperbolic equations). A type of geometry is defined in the state space of the conservation law by the nonlinear waves. This geometry is generically sing...

The Simplex and Projective Scaling Algorithms as Iteratively Reweighted Least Squares Methods

Abstract: The projective scaling algorithm in Karmarkar [Combinatorica, 4 (1984), pp. 373–395] and Dantzig’s simplex method (see [Linear Programming and Extensions, Princeton University Press, 1963]) are usually thought of as fundamentally different approaches to linear programming. When viewed in Dantzig’s column space geometry, however, both algorithms turn out to be iteratively reweighted least squares methods. The projective scaling algorithm (and the affine scaling method in Dikin [Soviet Math. Dokl., 8 (1967), pp. 674–675]) can then be derived as a natural generalization of the simplex method. This derivation shows how the dual variables arise in the interior methods. The insight is essentially geometric; suitable figures are provided.

Searching for new mathematics

Abstract: Nonmathematicians interested in keeping up with contemporary mathematical research face daunting obstacles. The high level of abstraction, unenlightening notation, and the format of journal papers all conspire to make any foray into the mathematical literature frustrating and unrewarding. Mathematicians can do more to make their research accessible to a wider audience.

Choosing an inner product that separates variables

Abstract: As has been often observed, a physically natural inner product will exist that orthogonally decouples the time and space eigenmodes under the assumptions most frequently encountered in linear distributed problems This tutorial practices this approach by working through several practical examples