Showing papers in "Studies in Applied Mathematics in 1977"
TL;DR: In this article, the Hartree-Fock theory of a plasma was used to prove existence and uniqueness of a minimization of the functional function of an electron trapped in its own hole.
Abstract: The equation dealt with in this paper is in three dimensions. It comes from minimizing the functional which, in turn, comes from an approximation to the Hartree-Fock theory of a plasma. It describes an electron trapped in its own hole. The interesting mathematical aspect of the problem is that & is not convex, and usual methods to show existence and uniqueness of the minimum do not apply. By using symmetric decreasing rearrangement inequalities we are able to prove existence and uniqueness (modulo translations) of a minimizing Φ. To prove uniqueness a strict form of the inequality, which we believe is new, is employed.
1,039 citations
280 citations
TL;DR: In this paper, a method for isolating certain nonlinear partial difference equations which are solvable by the method of inverse scattering is discussed, characterized by the dispersion relation (amplification factor) of the associated linearized equation.
Abstract: A systematic method for isolating certain nonlinear partial difference equations which are solvable by the method of inverse scattering is discussed. The equations are characterized by the dispersion relation (amplification factor) of the associated linearized equation. Corresponding to every neutrally stable linear scheme is a nonlinear difference equation which converges to an exactly solvable nonlinear evolution equation in the continuum limit. Explicit solition solutions are given.
247 citations
TL;DR: The long-time asymptotic solution of the Korteweg-deVries equation, corresponding to initial data which decay rapidly as |x|∞ and produce no solitons, is found to be considerably more complicated than previously reported.
Abstract: The long-time asymptotic solution of the Korteweg-deVries equation, corresponding to initial data which decay rapidly as |x|∞ and produce no solitons, is found to be considerably more complicated than previously reported. In general, the asymptotic solution consists of exponential decay, similarity, rapid oscillations and a “collisionless shock” layer. The wave amplitude in this layer decays as [(lnt)/t]2/3. Only for very special initial conditions is the shock layer absent from the solution.
218 citations
TL;DR: In this paper, an inviscid, incompressible, stably stratified fluid occupies a horizontal channel, along which an internal gravity-wave packet is propagating and the wave induced mean motions are calculated, and the equations describing the evolution of the wave amplitude derived.
Abstract: An inviscid, incompressible, stably stratified fluid occupies a horizontal channel, along which an internal gravity-wave packet is propagating. The wave induced mean motions are calculated, and the equations describing the evolution of the wave amplitude derived. When the group velocity of the wave packet coincides with a long-wave speed there is a resonance, and the equations describing this resonance are derived.
181 citations
139 citations
TL;DR: In this article, reaction-diffusion equations which, locally in space and time, are close to plane wave trains are investigated and the structure of these layers, which are mathematically analogous to gas-dynamic shocks, is investigated on the basis of the full equations.
Abstract: Solutions of reaction-diffusion equations which, locally in space and time, are close to plane wave trains are investigated Apart from certain exceptional (and localized) regions, such solutions are approximately described by solutions of a Hamilton-Jacobi equation obtained from the properties of ideal plane waves This provides a fairly simple description of many features of the initial value problem, but implies that transition layers in which the approximate description is invalid will generally arise The structure of these layers, which are mathematically analogous to gas-dynamic shocks, is investigated on the basis of the full equations
113 citations
TL;DR: In this paper, a van der Pol relaxation oscillator that is subjected to external sinusoidal forcing can exhibit stable and unstable periodic and almost periodic responses, and it even happens that two stable subharmonics having different periods may coexist.
Abstract: : A van der Pol relaxation oscillator that is subjected to external sinusoidal forcing can exhibit stable and unstable periodic and almost periodic responses. For some forcing amplitudes it even happens that two stable subharmonics having different periods may coexist. The stable responses of such forced oscillators are investigated. By numerically computing the rotation number of stable oscillations for various values of the forcing amplitude and oscillator tuning, descriptions are obtained of regions of phase locking, successive bifurcation of stable subharmonic and almost periodic oscillations, and overlap regions where two distinct stable oscillations can coexist.
101 citations
TL;DR: In this paper, it was shown that if an integer linear program in n variables has more than 2n linear inequality constraints, then either some of the constraints are unnecessary or there is at least one feasible integer point.
Abstract: This note shows that if an integer linear program in n variables has more than 2n linear inequality constraints, then either some of the constraints are unnecessary or there is at least one feasible integer point.
90 citations
80 citations
TL;DR: In this article, the authors explain combinatorially the occurrence of certain classical sequences of orthogonal polynomials as sequences of rook polynomial, and give some new examples related to general stairstep boards.
Abstract: We explain combinatorially the occurrence of certain classical sequences of orthogonal polynomials as sequences of rook polynomials, and we give some new examples related to general stairstep boards.
TL;DR: An asymptotic procedure for deriving equations governing the passage of a weakly coupled nonlinear system of oscillators is discussed in this article, which avoids an inner-outer-matching technique and is valid when the small coupling and detuning parameters are arbitrary Resonance is permitted to occur at one or several instances of time or to last for a finite length of time
Abstract: An asymptotic procedure for deriving equations governing the passage of a weakly coupled nonlinear system of oscillators is discussed The procedure avoids an inner-outer-matching technique and is valid when the small coupling and detuning parameters are arbitrary Resonance is permitted to occur at one or several instances of time or to last for a finite length of time Numerical results are discussed
TL;DR: In this article, the effects of transverse shear and normal stress deformation for circular cylindrical shells are supplemented by upper and lower bound determinations of influence coefficients, using minimum-potential and complementary energy principles in conjunction with asymptoticexpansion results.
Abstract: Known results on asymptotic two-dimensional equations for circular cylindrical shells, including the effects of transverse shear and normal stress deformation, are supplemented by upper- and lower-bound determinations of influence coefficients, using minimum-potential and complementary energy principles in conjunction with asymptotic-expansion results. The new bound analysis shows that the consequences of the asymptotic two-dimensional theory are in exact agreement, except for terms which are small of higher order, with the corresponding consequences of three-dimensional theory, for some classes of edge conditions. The analysis also shows the nature of the differences between results of two- and three-dimensional theory, as a function of geometrical and elastic parameters, where this difference is of importance because of the effect of a St. Venant boundary layer.
TL;DR: In this paper, it was shown that if n is odd, the number of crossings can be any integer up to n(n−4)/2+1, except n (n−3)/2−1.
Abstract: We address the following question: In drawing a cycle on n vertices (or a graph all of whose degrees are 2) in the plane with straight line arcs, how many crossings can there be? A complete answer is given; namely, if n is odd, the number of crossings can be anything up to n(n−3)/2 except n(n−3)/2−1. For n even, the number of crossings in one cycle can be any integer up to n(n−4)/2+1; general bivalent even graphs can achieve any integer up to n(2n−7)/4 as the number of crossings.
TL;DR: In this paper, it was shown that a result of Katona can be made into a three part Sperner theorem which is independent of the best previously known such theorem, in that neither hypothesis implies the other.
Abstract: We show that a result of Katona can be made into a three part Sperner theorem which is independent of the best previously known such theorem, in that neither hypothesis implies the other. These three part theorems are stated in terms of a three dimensional rectangular integer lattice L, and give sufficient conditions for F ⊆ L, containing no two points on a line, to be no larger in size than the set of points of middle rank in L. The theorems apply to the more general problem in which L is the product of three symmetric chain orders and F ⊆ L contains no two points equal in two components and ordered in the third.
TL;DR: The set of dual pairs of any norm v equivalent to a Hilbert norm is shown to be naturally homeomorphic to the sphere of the Hilbert space in this article, where the authors show that the arc-connectedness of the numerical range determined by v is the sum of two v-dual vectors.
Abstract: The set of dual pairs of any norm v equivalent to a Hilbert norm is shown to be naturally homeomorphic to the sphere of the Hilbert space. The proof begins with a known result showing the representability of every vector as a sum of two orthogonal vectors, one coming from a cone and the other from its dual (a generalization of representation by orthogonal subspaces). The key theorem, showing that every non-zero vector has a positive multiple which is the sum of two v-dual vectors, follows from this and in turn provides the required homeomorphism. One consequence of this topological equivalence is the arc-connectedness of the numerical range determined by v.
TL;DR: The notions of binary string and binary symmetric function are introduced, and basic results presented for binary addition and multiplication are presented.
Abstract: The notions of binary string and binary symmetric function are introduced, and basic results presented. Boolean algorithms are given for binary addition and multiplication. An analysis of the redundancies involved is straightforward. The examination of carry propagation which arises in the Boolean analysis of functions may lead to a new interpretation of the notion of computational complexity.
TL;DR: In this article, a resonance of a magnetic impurity with a quantum spin-chain is studied and a Fresnel diffraction pattern is found, which is similar to the one described in this paper.
Abstract: A resonance of a magnetic impurity with a quantum spin-chain is studied. A Fresnel diffraction pattern is found.
TL;DR: In this article, an axially uniform vortex is developed by visualizing a circular tube with uniform surface suction of fluid possessing circulation at infinity, which is then perturbed by a uniform flow past it.
Abstract: It is believed that the flow past a tornado causes the formation of smaller vortices which produce the “suction spots” observed along the path of destruction. Here we develop a greatly simplified mathematical model to investigate this phenomenon. An axially uniform vortex is developed by visualizing a circular tube with uniform surface suction of fluid possessing circulation at infinity. This vortex is then perturbed by a uniform flow past it. An inner asymptotic expansion of an E1/3 radial boundary layer is matched to an outer expansion to obtain a solution. The results show that a stagnation point developing into a secondary vortex is formed in a free shear layer at critical flow conditions. However, it is difficult to apply our results quantitatively because of the difficulty in comparing the axially uniform vortex with a real tornado vortex.