Showing papers in "Siam Journal on Applied Mathematics in 1975"

Nonlinear Aspects of Competition Between Three Species

Abstract: It is shown that for three competitors, the classic Gause–Lotka–Volterra equations possess a special class of periodic limit cycle solutions, and a general class of solutions in which the system exhibits nonperiodic population oscillations of bounded amplitude but ever increasing cycle time. Biologically, the result is interesting as a caricature of the complexities that nonlinearities can introduce even into the simplest equations of population biology ; mathematically, the model illustrates some novel tactical tricks and dynamical peculiarities for 3-dimensional nonlinear systems.

Minimal Mutation Trees of Sequences

Abstract: Given a finite tree, some of whose vertices are identified with given finite sequences, we show how to construct sequences for all the remaining vertices simultaneously, so as to minimize the total edge-length of the tree. Edge-length is calculated by a metric whose biological significance is the mutational distance between two sequences.

Social Choice Scoring Functions

Abstract: Let a committee of voters be considering a finite set $A = \{ {a_1 ,a_2 , \cdots ,a_m } \}$ of alternatives for election. Each voter is assumed to rank the alternatives according to his preferences in a strict linear order. A social choice function is a rule which, to every finite committee of voters with specified preference orders, assigns a nonempty subset of A, interpreted as the set of “winners”. A social choice function is consistent if, whenever two disjoint committees meeting separately choose the same winner(s), then the committees meeting jointly choose precisely these winner(s). The function is symmetric if it does not depend on the names of the various voters and the various alternatives. It is shown that every symmetric, consistent social choice function is obtained (except for ties) in the following way: there is a sequence $s_1 ,s_2 , \cdots $, $s_m $ of m real numbers such that if every voter gives score $s_i $ to his ith most preferred alternative, then the alternative with highest score ...

On sequences of pairs of dependent random variables

Abstract: The generalized random variables $( {x,y} )$ have a given joint distribution. Pairs $( {x_i ,y_i } )$ are drawn independently. The observer of $( {x_1 , \cdots ,x_n } )$ and the observer of $( {y_1 , \cdots ,y_n } )$ each make a binary decision, of entropy bounded away from zero, with probability of disagreement $\varepsilon _n $. It is shown that $\varepsilon _n $ can be made to approach zero as $n \to \infty $ if and only if the maximum correlation of x and y is unity. Under a compactness condition, satisfied in particular when x and/or y takes only finitely many values, this occurs if and only if the joint distribution decomposes, that is when $\varepsilon _1 $ can be made to vanish by nontrivial decisions, as had been conjectured.Results are also obtained for nonidentically distributed pairs, for randomized decisions, for multivalued decisions and for decisions based on the infinite sequences.The question arose in the transmission of data from two dependent sources to two receivers. The results of Gac...

Wave Breakdown and Turbulence

Abstract: Some new ideas on the dynamics of shear flow turbulence are presented. Central to these is the phenomenon of wave breakdown. This is defined as the onset of a violent small-scale secondary instability developing on a large-scale primary disturbance of wave-like traveling type. It is suggested that breakdown together with the ensuing violent mixing process (a turbulent “burst”) constitutes the dominant nonlinear mechanism for the fluctuating velocity field in a turbulent boundary layer. A burst regeneration mechanism is proposed whereby one breakdown can excite large velocity defects in the shear flow which then may trigger a new breakdown, thus leading to self-maintenance of the turbulence.

Shorted operators. ii

Abstract: For a positive operator A acting on a Hilbert space, the shorted operator $\mathcal{L}( A )$ is defined to be the supremum of all positive operators which are smaller than A and which have range lying in a fixed subspace S. This maximization problem arises naturally in electrical network theory. In this paper we prove that the shorted operator exists, and develop various properties, including a relation to parallel addition [Anderson and Duffin, J. Math. Anal. Appl., 11 (1969), pp. 576–594]. The basic properties of the shorted operator were developed for finite-dimensional spaces by Anderson [this Journal, 20 (1971), pp. 520–525] ; some of these theorems remain true in all Hilbert spaces, but the proofs are different.

Coloring a Family of Circular Arcs

Abstract: This paper presents a collection of results about coloring a family of circular arcs. We prove that the strong perfect graph conjecture is valid for circular-arc graphs. We give some upper bounds on the number of colors needed to color various families of arcs. Finally, we convert the problem of determining whether a family of arcs can be q-colored into a multicommodity flow problem.

Boundary-Layer Separation in Unsteady Flow

Abstract: Extension of the familiar concept of boundary-layer separation to flow along moving walls and unsteady flows is a subject that attracted some interest in the 1950’s and has been investigated further in the past few years The well-known criterion of vanishing wall-shear does not apply in such flows, and therefore the definition of the phenomenon becomes more difficult than in the simpler types of flow considered by Prandtl The practical importance of extending the concept is discussed and arguments in favor of its definition in terms of Goldstein’s singularity are reviewed The model proposed by the present authors in 1971 is described, together with available numerical and experimental evidence that supports it Numerical studies of steady and unsteady separating boundary-layer flows are reported, in which singularities of Goldstein’s type are detected and comparisons can be made between the position-vs-time curves of the singularity and of the point of vanishing wall-shear One of these studies involv

Algorithmic aspects of vertex elimination on directed graphs.

Abstract: We consider a graph-theoretic elimination process which is related to performing Gaussian elimination on sparse systems of linear eauations. We give efficient algorithms to: (1) calculate the fill-in produced by any elimination ordering; (2) find a perfect elimination ordering if one exists; and (3) find a minimal elimination ordering. We also show that problems (1) and (2) are at least as time-consuming as testing whether a directed graph is transitive, and that the problem of finding a minimum ordering is NP-complete.

The Effect of Strain Rate on Diffusion Flames

Abstract: Several steady state and time-dependent solutions to the compressible conservation laws describing direct one-step near-equilibrium irreversible exothermic burning of initially unmixed gaseous reactants, with Lewis-Semenov number unity, are presented. The quantitative investigation first establishes the Burke-Schumann thin-flame solution using the Shvab-Zeldovich formulation. Real flames do not have the indefinitely thin reaction zone associated with the Burke-Schumann solution. Singular perturbation analysis is used to provide a modification of the thin-flame solution which includes a more realistic reaction zone of small but finite thickness. The particular geometry emphasized is the un bounded counterflow such that there exists a spatially constant rate of strain along the flame. While the solutions for diffusion flames under a finite tangential strain rate may be of interest in and of themselves for laminar flow, the problems are motivated by the authors' belief that they are pertinent to the study of diffusion-flame burning in transitional and turbulent shear flows.

Finite Element Models for Ocean Circulation Problems

Abstract: In this paper a finite element model is presented for a limited region ocean circulation problem. Stability and aliasing error analysis are given as well as the conservation laws that are intrinsic to the finite element scheme.

Preference Structures. II: Distances Between Asymmetric Relations

Abstract: In this paper the results of Preference structures I are extended to not necessarily transitive relations. The extension of the city block metric of [3] leads to a discussion of the relation of majority rule to the median of a collection of relations. The extension of the Euclidean metric of [3] leads to a discussion of the mean of a collection of relations and methods for finding means. Finally, the extension of the Euclidean metric leads to a discussion of statistical inference about random variables whose ranges are preference relations.

Generators for Certain Alternating Groups with Applications to Cryptography

Abstract: A set of block ciphers is described which can readily be adapted for computer encipherment of data. The subgroup of permutations generated by any one of these ciphers is shown to be the alternating group, in all cases of interest, suggesting that such systems have a high level of security.

The Existence of Oscillatory Solutions in the Field–Noyes Model for the Belousov–Zhabotinskii Reaction

Abstract: The cerium ion catalyzed oxidation of malonic acid by bromate in a sulphuric acid solution is essentially the Belousov–Zhabotinskii reaction. Belousov [1] observed temporal oscillations in the concentrations of the cerium ions Ce(III) and Ce(IV) when the reagent was stirred. The quantitative model for the chemical mechanism suggested by Field and Noyes [3] is examined here and it is shown that the solutions of the model equations are oscillatory, of finite amplitude and must possess at least one periodic solution.

Two-Graphs, Switching Classes and Euler Graphs are Equal in Number

Abstract: Seidel has shown that the number $t_n $ of two-graphs on n nodes is equal to the number of switching classes of graphs on n nodes. Robinson, and independently Liskovec, have given an explicit formula for the number $e_n $ of Euler graphs on n nodes. It is shown here that $t_n = e_n $ for all n.

Disturbances in a Boundary Layer Introduced by a Low Intensity Array of Vortices

Abstract: To acquire insight into the role of free-stream turbulence on laminar-turbulent transition, the interaction between a boundary layer and an array of single-wavenumber vortices convected at the mean free-stream velocity is studied analytically and numerically. For small amplitudes, the effect of a spectrum can be obtained by superposition. The flow field is taken to be the sum of the steady laminar field (Blasius) plus a flow field ascribable to the effects of the vortex array. This latter flow field is further subdivided into the portion that exists in the absence of the plate (the vortex array itself) plus a flow field representing the alteration to that array due to the shearing mean flow and no-slip and impermeability conditions at the plate surface. This last portion of the flow field is described by a nonhomogeneous Orr–Sommerfeld equation with phase speed unity and real wavenumber. The forcing function depends on the mean flow and on the free-stream disturbance array. The problem is not an eigenvalu...

A Determinant Maximization Problem Occurring in the Theory of Data Communication

Abstract: Let H be a Hermitian $m \times m$ matrix, with eigenvalues $\lambda _1 \geqq \lambda _2 \geqq \cdots \geqq \lambda _m \geqq 0$ of which $m^ + \geqq 1$ are positive. Then the maximum of $\det ( {I_n + GHG^ * } )$ over all complex $n \times m$ matrices G) satisfying trace $GG^ * \leqq 1$ is \[ r^{ - r} \left( {1 + \mathop \Sigma \limits_{i = 1}^r \lambda _i^{ - 1} } \right)^r \mathop \Pi \limits_{j = 1}^r \lambda _j , \] where r is the largest integer satisfying $r\leqq n,r\leqq m^ + $, and $r< \lambda _r ( {1 + \Sigma _{i = 1}^r \lambda _i^{ - 1} } )$ . The matrices G for which the maximum is attained are characterized. They have rank r. A special case of this problem arises in the optimization of certain data communication systems.

Computer Extension of Perturbation Series in Fluid Mechanics

Abstract: In a regular perturbation expansion, the routine labor of calculating higher approximations can often effectively be delegated to a computer. Then dozens or even hundreds of terms can typically be found. These may suffice to permit the structure of the solution to be analyzed, and the series then recast to improve its utility. Although special languages are being developed for algebraic and other nonnumerical manipulation, effective computations can be carried out using simply FORTRAN or the like. A number of recent applications in fluid mechanics are surveyed. Details are presented of two new computations inspired by work of Sydney Goldstein: impulsive heating of a flat plate in boundary-layer flow and the decay of an array of cubical vortices.

On the Highest and Other Solitary Waves

Abstract: For solitary waves an expansion parameter which differs from those previously employed permits calculation to extremely high order. The observed behavior of the coefficients entering the power series for the position or velocity field yields much information about the nature of singularities in the solution. The principal conclusions are, (i) the wave of maximum amplitude has a nondimensional amplitude $ \cong 3\sqrt {3}/2\pi $ and a nondimensional speed (Froude number) $ \cong (3\sqrt {3}/\pi)^{1/2} $. (ii) All previous theories employing an expansion parameter are incomplete. (iii) The various series relating Froude number to amplitude, recently advanced to the ninth order, are asymptotic.For undular bores direct numerical calculations show that (i) the relationship between relative elevation and relative velocity given by long wave theory is approached for the “ahead of” and “behind” an undular bore even when the bore is generated in ways which violate the conditions of the long wave theory, (ii) the a...

Duality for Stochastic Programming Interpreted as L. P. in $L_p $-Space

Abstract: The linearly constrained multistage stochastic programming problem is interpreted as a programming problem in $L_p $-space, linear if the stochastic problem is linear, and a duality theory is developed from the general results of Rockafellar [16] The duality is symmetric for linear problems, provided that the stochastic model is suitably generalized, and can be given an economic interpretation If a certain set $\mathcal{C}$, closely related to the epigraph of the perturbation function, is closed, then the stochastic programming problem attains its minimum, which equals the supremum of the dual problem The closedness of $\mathcal{C}$ follows from simple conditions on the technology matrix A for the problem

On the continuity of the optimal policy set for linear programs

Abstract: For a linear program it is shown that the optimal policy set behaves continuously if the constraint vector changes on the set for which the program has a solution. The result implies that there exists a continuous optimal policy function for which a construction is indicated.

Vortex wakes of bluff cylinders in shear flow

Abstract: A curved gauze was used to produce a shear flow in a wind tunnel, with a nearly linear variation of velocity across the working section. Experiments were made with cylinders of circular and other bluff section spanning the wind tunnel, with their axes normal to the vorticity vector. Measurements of the frequency of vortex shedding in the wake showed a number of “cells” across the span of the cylinder, each with a different frequency. The flows were strongly influenced by the boundaries at the ends of the cylinders, and with some forms of end plate, the positions of the cell boundaries fluctuated with time.

Approximation of Stochastic Processes by Gaussian Diffusions, and Applications to Wright-Fisher Genetic Models

Abstract: For each $N\geqq 1$, let $\{ {X_n^N ,n\geq 0} \}$ be a discrete-time stochastic process, and let $\Delta X_n^N = X_{n + 1}^N - X_n^N $. Suppose that $E( {\Delta X_n^N | {X_n^N } } ) = O( {\varepsilon ^N } )$ and $\operatorname{var} ( {\Delta X_n^N | {X_n^N } } ) = O( {\tau ^N } )$, where $\varepsilon ^N \to 0$ and ${\tau ^N / \varepsilon ^N \to 0}$ as $N \to \infty $. Conditions are given under which there are constants $\gamma _n^N $ such that $Z_n^N ( {X_n^N - \gamma _n^N } )( \varepsilon ^N / \tau ^N )^{1 / 2} $ can be approximated by a Gaussian diffusion when N is large. It is shown that these conditions are satisfied by the Wright–Fisher models for fluctuations in gene frequency under theinfluence of mutation, selection and random drift. For these models, N is the population size and the constants $\gamma _n^N $ are the gene frequencies specified by Haldane’s deterministic theory of evolution.

Excess functions for cooperative games without sidepayments.

Abstract: A family of excess functions is defined for cooperative n-person nonsidepayment games. It is shown that by using these excess functions the $\varepsilon $-core, kernel, and nucleolus of a nonsidepayment game can be defined in a way that preserves a significant portion of the structure that these concepts exhibit in the sidepayment case. Some excess functions are extensions of the sidepayment games’ “excesses.” It is also shown that the set of nucleoli that exist for some excess functions is precisely the set of reasonable (in a core sense) outcomes of a game.

Modern Developments in Transonic Flow

Abstract: A survey is given of transonic small disturbance theory. Basic equations, shock relations, similarity laves, lift and drag integrals are derived., The airfoil boundary value problem is formulated. Finite difference methods and computational algorithms are described. Results are compared with other calculation methods and experiments.

Some Abstract Pivot Algorithms

Abstract: Several problems in the theory of combinatorial geometries (or matroids) are solved by means of algorithms which involve the notion of “abstract pivots” The main example is the Edmonds–Fulkerson partition theorem, which is applied to prove a number of generalized exchange properties for bases

Solutions of the navier-stokes equation at large reynolds number*

Abstract: The problem of two-dimensional incompressible laminar flow past a bluff body at large Reynolds number $( R )$ is discussed. The governing equations are the Navier–Stokes equations. For $R = \infty $, the Euler equations are obtained. A solution for R large should be obtained by a perturbation of an Euler solution. However, for given boundary conditions, the Euler solution is not unique. The solution to be perturbed is the relevant Euler solution, namely the one which is the Euler limit of the Navier–Stokes solution with the same boundary conditions. For certain semi-infinite or streamlined bodies, the relevant Euler solution represents potential flow. For flow inside a closed domain a theorem of Prandtl states the relevant Euler solution has constant vorticity in each vortex. In many cases it can be determined by simultaneously considering the boundary layer equations. For flow past a bluff body, the relevant Euler solution is not known, although the free streamline flow for which the free streamline deta...

Uniform Ray Theory of Surface, Internal and Acoustic Wave Propagation in a Rotating Ocean or Atmosphere

Abstract: First a short wavelength asymptotic expansion, employing rays, is presented for waves propagating in a rotating compressible fluid layer of nonuniform depth. This theory applies to surface, internal and acoustic waves in an ocean or atmosphere. It yields infinite amplitudes at caustics and shorelines. Therefore two different asymptotic expansions, uniform in a region containing a caustic and a shoreline respectively, are constructed. They yield the correct finite amplitude at the caustic and at the shoreline. In addition, expansions uniform in a region containing a caustic and a shoreline, or two or more caustics and shorelines, are constructed. Both general time-dependent waves and time-harmonic waves are considered. Linear theory is employed.

Interaction Between a Shock Wave and a Turbulent Boundary Layer in Transonic Flow

Abstract: Interaction between a shock wave and an unseparated turbulent boundary layer is considered. The method of matched asymptotic expansions is used, with solutions valid in the double limit as Reynolds number tends to infinity and Mach number tends to unity. The shock is weak enough that interaction effects can be considered as perturbations to the undisturbed flow; the case considered is that where the sonic line is near the outer edge of the boundary layer. The interaction region consists of four layers; the external flow region, and three boundary layer regions consisting of the two usual boundary layer regions (velocity defect and wall layers) and an intermediate Reynolds stress sublayer. It is shown that, with order estimates for Reynolds stress perturbations based on Ribner’s [17] calculations, the induced wall pressure distribution can be calculated using only the two outer regions, independent of a specific closure condition, and that this solution is in fact a turbulent free interaction solution. A d...

Ray-Optical Analysis of Reflection in an Open-Ended Parallel-Plane Waveguide. I: TM Case

Abstract: The reflection problem for a TM mode traveling toward the open end of a semi-infinite parallel-plane waveguide, is solved by ray methods. Unlike a previous solution due to Yee, Felsen and Kelley, the present ray-optical solution is a rigorous asymptotic result, i.e., it is identical with the asymptotic expansion of the exact solution when the width of the waveguide is large compared to the wavelength. Numerical results for the modal reflection coefficients are presented and are compared with calculations based on the exact solution. It is found that the agreement between ray-optical and exact values is excellent and even better than in the approach of Yee et al., especially in the vicinity of cutoff frequencies of higher order modes.