Showing papers in "Siam Journal on Applied Mathematics in 1971"

On the Nucleolus of a Characteristic Function Game

Abstract: Introduction. The nucleolus is a solution concept for games in characteristic function form. It was introduced by D. Schmeidler in [1]. This paper is concerned with some of the properties of the nucleolus. Some of the results are new (Theorems 2, 4, 5) whereas others are known (Theorems l and 3), but the proofs given here are, in a sense, simpler than those in [1]. A characteristic function game is a pair (N, v) consisting of a set N = {1, 2, , n} of n players, and a characteristic function v, which maps each subset S of N, called a coalition, to a number v(S). In addition, it is assumed that v(N) _ 0, and that v(S) = 0 for all one-person coalitions, as well as for the empty coalition. A payoff vector is an n-tuple x = (x1, , xn) such that xi > 0 for all i = 1, , n, and x(N) = v(N), where for each coalition S, x(S) denotes Zi,eS Xi. X is the set of all payoff vectors. Of course, X depends on the particular game v. For any x in X, let q(x) be a vector in E2n, the components of which are the numbers v(S) x(S), arranged in order of decreasing magnitude, where S runs over all the coalitions of N. We shall say that x is at least as acceptable as y (with respect to v), and write x > y, if q(x) is not greater than q(y), in the lexicographical order on E2n. If q(x) is smaller than q(y), we shall say that x is more acceptable than y and write x >y. The nucleolus of the game is the set of those points in X which are "most acceptable," that is, {x E X :x > y for all y E X}. In [1], Schmeidler proved the basic result that every game possesses a nonempty nucleolus. He further showed that the nucleolus actually consists of just one point, and that-viewed as a point function of the game-it is continuous. Our aim is to give new proofs of the uniqueness and of the continuity of the nucleolus. To this end, we will have to introduce a few definitions and to prove an auxiliary theorem (Theorem 2), which may have some interest in its own right. DEFINITION. Let bo, b1, ... , bp be a sequence of sets whose elements are coalitions of N. This sequence is a coalition array whenever: (i) every coalition of N is contained in exactly one of the sets b1, bp, (ii) bo contains only one-element coalitions. For every game v and payoff vector x, let b1(x, v) be the set of those S c N for which max {v(S) x(S): S c N} is attained. Similarly, b2(x, v) is the set of those S c N where max {v(S) x(S) :S ? b1(x, v)} is attained, and so on. Finally, let bo(x) = {{i} : xi = 0}. It is obvious that bo(x), b1(x, v), . . , bp(x, v) is a coalition array. We shall say that it is the array that belongs to (v, x). DEFINITION 2. A coalition array bo, , bp has property I if for all k= 1,2,...,pandanyyinE n, (1) y(S) _ 0forall Sebo, (2) y(S) ? O for all Se b1 U U bk, and (3) y(N) = 0

Optimal Computer Search Trees and Variable-Length Alphabetical Codes

Abstract: An algorithm is given for constructing an alphabetic binary tree of minimum weighted path length (for short, an optimal alphabetic tree). The algorithm needs $4n^2 + 2n$ operations and $4n$ storage locations, where n is the number of terminal nodes in the tree. A given binary tree corresponds to a computer search procedure, where the given files or letters (represented by terminal nodes) are partitioned into two parts successively until a particular file or letter is finally identified. If the files or letters are listed alphabetically, such as a dictionary, then the binary tree must have, from left to right, the terminal nodes consecutively. Since different letters have different frequencies (weights) of occurring, an alphabetic tree of minimum weighted path length corresponds to a computer search tree with minimum-mean search time. A binary tree also represents a (variable-length) binary code. In an alphabetic binary code, the numerical binary order of the code words corresponds to the alphabetical orde...

A Generalized Negative Binomial Distribution

Abstract: A generalized negative binomial (GNB) distribution with an additional parameter $\beta $ has been obtained by using Lagrange’s expansion. The parameter is such that both mean and variance tend to increase or decrease with an increase or decrease in its value but the variance increases or decreases faster than the mean. For $\beta = \frac{1} {2}$, the mean and variance are approximately equal and so the GNB distribution resembles the Poisson distribution. When $\beta = 0$ or 1, the GNB distribution reduces to the binomial or negative binomial distribution respectively. It has been shown that the generalized negative binomial distribution converges to a Poisson-type distribution in which the variance may be more or less than the mean, depending upon the value of a parameter. Expected frequencies have been calculated for a number of examples to show that the distribution provides a very satisfactory fit in different practical situations. Its convolution property together with other properties are quite inter...

Computing Equilibria of N-Person Games

Abstract: The algorithm of Lemke and Howson for finding an equilibrium of a 2-person game is extended to provide a constructive procedure for finding an equilibrium of an N-person game by finding in succession an equilibrium for each of certain related k-person games, $1\leqq k\leqq N$.

A necessary and sufficient qualification for constrained optimization

Abstract: A weak qualification is given which insures that a broad class of constrained optimization problems satisfies the analogue of the Kuhn–Tucker conditions at optimality. The qualification is shown to be necessary and sufficient for these conditions to be valid for any objective function which is differentiable at the optimum.

Multiple stable solutions of nonlinear boundary value problems arising in chemical reactor theory

Abstract: This paper is concerned with the nonlinear boundary value problem (1) $\beta u''-u'+f(u)=0$, (2) $u'(0)-au(0)=0,u'(1)=0$, where $f(u)=b(c-u)\exp(-k/(1+u))$ and $\beta,a,b,c,k$ are constants. First a formal singular perturbation procedure is applied to reveal the possibility of multiple solutions of (1) and (2). Then an iteration procedure is introduced which yields sequences converging to the maximal solution from above and the minimal solution from below. A criterion for a unique solution of (1), (2) is given. It is mentioned that for certain values of the parameters multiple solutions have been found numerically. Finally, the stability of solutions of (1), (2) is discussed for certain values of the parameters. A solution $u(x)$ of (1), (2) is said to be stable if the first eigenvalue $\sigma$ of the variational equations $(1)' \beta v''-v'+[\sigma\beta+f'(u)]v=0$ and $(2)' v'(0)-av(0)=0, v'(1)=0$, is positive.

Wave Propagation in a One-Dimensional Random Medium

Abstract: Wave propagation in a slab of random medium is considered. The index of refraction is assumed to fluctuate randomly about a mean value, the fluctuations being small. Using a recent result of Hashminskii we give a description of the statistical characteristics of the reflection and transmission coefficients.

Stochastic Differential Equations with Applications to Random Harmonic Oscillators and Wave Propagation in Random Media

Abstract: The two-time method is used to obtain an expansion, valid for $\varepsilon $ small and t large, of the vector solution $u( {t,\varepsilon } )$ of an abstract ordinary differential equation involving $\varepsilon $. The same method is used to get expansions of functions of u. The results are shown to apply to the solutions of stochastic equations. They are used to find the first two moments and the transition probability of the displacement of a harmonic oscillator with spring constant a random function of t. The result contains the condition for mean square stability due to Stratonovich. The results are also applied to one-dimensional wave propagation through a layer with refractive index a random function of position. They are used to find the mean square amplitude reflection and transmission coefficients, which are just the mean power reflection and transmission coefficients. A graph of the mean square transmission coefficient as a function of layer thickness is presented. The results are also compared ...

On a Generalization of the Lemke–Howson Algorithm to Noncooperative N-Person Games

Abstract: In [1] it has been shown that the existence of equilibrium points in a bimatrix game can be proved without a fixed-point theorem. If the game is nondegenerated, the number of equilibrium points is odd and all equilibrium points are obtained by a computational procedure in finitely many steps.The purpose of this note is to show that nondegeneracy can be defined for general N-person games and that for such games also there exist an odd number of equilibrium points. The algorithm developed in [1] can be extended to nondegenerated games even for more than 2 players.

An Application of the Correspondence Principle of Linear Viscoelasticity Theory

Abstract: A numerical solution procedure is given for plane boundary value problems of linear viscoelasticity theory, based on Kelvin’s point force solution to the field equations of elasticity theory. Conditions for applicability of the correspondence principle are assumed; otherwise rather general material properties, domain shapes, and boundary conditions are considered. Systems of singular integral equations are formed and solved numerically in the Laplace transform space. Transform inversion is considered separately. Data for a familiar problem of an elastically reinforced viscoelastic cylinder are given for illustration. The procedure may be extended to include anisotropic viscoelastic behavior; also it is not inherently limited to plane problems.

Minimum Value for c in the Sobolev Inequality $\| {\phi ^3 } \|\leqq c\| {\nabla \phi } \|^3 $

Abstract: It is shown that the minimum value for c is $4/3 \sqrt 3 \pi ^2 \cong .0780$ for $\phi $ of function class $C^0 $ piecewise $C^2 $ in real Euclidean 3-space.

On Minimizing Nonseparable Functions Defined on the Integers with an Inventory Application

Abstract: The general form of the problem considered is: \[ {\text{Find}}\,{\text{the}}\,{\text{infimum}}\,{\text{of}}\,f:X \to R, \] where X is a discrete rectangle, that is to say, \[ X = \{ {x:x \in I^n ,a_i \leqq x_i \leqq b_i ,i = 1, \cdots ,n} \}, \]I is the set of integers $\{ { \cdots , - 2, - 1,0, + 1, \cdots } \}$, $a_i $, $b_i $ are integers or infinite, and R is the real line.A condition called discrete convexity is developed for functions f which is a sufficient condition for a local optimum of f to be a global optimum of f. A general solution strategy is given, and computational results are presented which show the possibility of solving this problem in cases where n is in the hundreds.

Necessary and Sufficient Conditions for Almost Sure Sample Stability of Linear Ito Equations

Abstract: This paper is devoted to the study of the stability properties of two simple but nontrivial linear Ito stochastic differential equations. By applying a recent result due to Khasminskii, what appear to be the first explicit and exact regions of stability for linear stochastic differential equations are obtained.

On Diffusion in a Fractured Medium

Abstract: A “narrow fracture” approximation is applied to the diffusion equation on an inhomogeneous region in order to convert the parabolic interface problem into a boundary value problem with tangential derivatives on part of the boundary. This boundary value problem is shown to have a unique weak solution. Finally, a particular steady state model is solved, essentially in closed form, with a “Galerkin by lines” technique.

The Post-Buckling Problem for Thin Elastic Shells

Abstract: An asymptotic integration technique is developed to describe the post-buckling behavior of thin elastic shells. The introduction of slow space and time scales directly into the shell differential equations permits a modeling of dynamic effects and a means of accounting for finite boundaries. In most cases, quadratic nonlinear interactions dominate the dynamics and lead to hexagonal shaped patterns in the buckled shell. The form of the initial imperfection is crucial in determining the critical buckling load, as, for a given plan form, there are two separated branches in the load-displacement curve. The critical points on each branch occur at different values of the load parameter. Finally, a minimal principle is derived which exhibits the consistency of the present approach with the variational procedures of Koiter and others.

Computing Kakutani Fixed Points

Abstract: Let C be a compact convex subset of $R^m $, let $C^ * $ be the set of compact convex subsets of C, and let $f:C \to C^ * $ be a closed (i.e., upper semicontinuous) point-to-set map. An algorithm is specified which generates a sequence of points in C such that every cluster point x is a fixed point of f (i.e., $x \in f( x ) $).

Passage Through Resonance for a One-Dimensional Oscillator with Slowly Varying Frequency

Abstract: This paper concerns a model problem illustrating the techniques needed to handle passage through resonance for oscillatory systems with slowly varying frequencies. The model consists of a linear oscillator with a slowly varying frequency which at some time coincides with the constant frequency of the forcing function. It is shown that the solution can be constructed by matching the two asymptotic expansions which one obtains for oscillations near and away from resonance. As each of these asymptotic expansions depends simultaneously on two time scales, this example combines use of the two principal techniques of singular perturbations. The results show that the amplitude increases, then decreases as the natural frequency passes through the resonant value. Extension of the techniques to systems of differential equations is also indicated.

On Vector Spaces Associated with a Graph

Abstract: For a given graph G, let $V({\bf Q})$, $V({\bf B})$, and $V(G)$ be the vector spaces associated with the sets of cutsets, circuits, and subgraphs of G, respectively. It is shown that $V({\bf Q})$ and $V({\bf B})$ are orthogonal complements of $V(G)$ if and only if G has an even number of forests. Also, it is shown that the intersection of $V({\bf Q})$ and $V({\bf B})$ is the null space of the matrix whose rows correspond to the bases of $V({\bf Q})$ and $V({\bf B})$.

Asymptotic Behavior of Solutions of the Porous Media Equation

Abstract: A degenerate quasi-linear parabolic equation which arises in the study of fluid flow through porous media is investigated. It is shown that the solutions of a class of mixed initial boundary value problems in the domain $x > 0$, $t > 0$ converge upon a suitably chosen similarity solution as $t \to \infty $. The rate of convergence is discussed.

Optimal Search Using Uninterrupted Contact Investigation

Abstract: This paper finds the optimal search plan (in the sense of minimizing mean time to find the target) for a class of searches for a stationary target. In this class the target must be contacted by one sensor and identified by another. Complicating the search is the possibility of false targets which must be identified to be distinguished from the target. Search plans are described by a search effort density function and a policy for investigating contacts. The existence of false targets is determined by a known density function for the mean number of false targets in a region.Under the condition that contact investigation, once begun, must not be interrupted until the contact is identified, it is shown that the optimal plan, in a specified class, is to allocate search effort according to a Neyman–Pearson type of allocation and to investigate contacts immediately. It is also shown that if at any time an optimal search is stopped and it is decided to replan the search, the optimal plan is to continue the origi...

General One-Phase Stefan Problems and Free Boundary Problems for the Heat Equation with Cauchy Data Prescribed on the Free Boundary

Abstract: where u(x, t) and the free boundary s(t) are to be determined. Here f, g, h, A, p, and a are the data of problem (1.1), with f, g, h, A defined for t > 0 and (p(x) defined for 0 ? x 0 and a > 0 and compatibility and regularity conditions described in Theorem 1 of ? 2. In Theorem 1 we prove existence and uniqueness of the solution of (1.1). If a is the supremum of those t for which (1.1) has a solution, then we prove, in Theorem 1, that either a = oo, or a is finite with s(a-) 0 and s(t) > 0 for t < a, or a is finite, s(t) does not tend to 0 as t T a, and lim inf s'(t) = lim inf ux(s(t), t) = oo as t T a. We describe the Stefan problem (1.1) as general because there are no sign restrictions on f, g, h and (p. We may interpret (1.1) as a problem of heat conduction with melting when

Generalized Weighted Mean Programming

Abstract: Two algorithms for nonlinear programming are given. The idea behind these algorithms is the arithmetic-geometric mean inequality. Two examples are shown together with the necessary proofs of convergence.

Transient Behavior of Unstable Nonlinear Systems with Applications to the Bénard and Taylor Problems

Abstract: A nonlinear initial value problem containing two parameters, $\varepsilon $ and $\lambda $, is considered for a function $u(t)$. For each t, $u(t)$ is a vector in a Hilbert space. When $\varepsilon = 0$ the problem has a steady state solution $u_0 $ for any value of $\lambda $. This solution is assumed to be linearly stable for $\lambda \lambda _c $ for some value $\lambda _c $ . This means that the linear problem for the derivative $u_\varepsilon (t)$ at $\varepsilon = 0$ has exponentially growing solutions for $\lambda > \lambda _c $ but not for $\lambda < \lambda _c $. The solution $u(t)$ is found for $\varepsilon $ small and $\lambda $ slightly larger than $\lambda _c $. It is found that u contains one unstable mode which initially increases exponentially but ultimately approaches a constant, while all other modes decay, up to terms of order $\varepsilon ^2 $. To this order $u(t)$ approaches a steady state solution $u(\infty )$ different from $u_0 $,...

On the Invulnerability of the Regular Complete k-Partite Graphs

Abstract: The size of the smallest edge cut-set (cohesion) and the size of the smallest node cut-set (connectivity) have been used by some authors as vulnerability measures of a graph. Generalizations of the cohesion and the connectivity are introduced herein as new vulnerability criteria. It is shown that the regular complete k-partite graph is optimal with respect to connectivity, cohesion and the generalized cohesion criteria. It is then shown that this graph also has the smallest total number of distinct minimum size edge cut-sets and the smallest total number of distinct minimum size node cut-sets.

Weighted Pseudoinverses with Singular Weights

Abstract: This paper develops a singular weighted pseudoinverse which generalizes the product generalized inverse of a matrix. A condition to insure uniqueness of this pseudoinverse and a computational formula are obtained.

Asymptotic Behavior of Stekloff Eigenvalues and Eigenfunctions

Abstract: We consider solutions $u_n (x,y)$ of Laplace’s equation which are regular in the interior of a smooth closed plane curve c, and the boundary conditions $\partial u_n /\partial v = \lambda _n gu_n $, where g is sufficiently smooth, positive, periodic and a prescribed function of arclength, and $u_n $, $\lambda _n $ are eigenfunctions and eigenvalues to be determined. For large $\lambda _n $ we show that $\lambda _n = O(n)$, n a large integer, and that $u_n $ is trigonometric, asymptotically. The method employed is the reduction of the problem to a boundary integral equation and the studying of that equation. The results are confirmed in a separable case which arises by a special choice of $g(s)$ and the curve c.

A Variation of the Topological Method of Ważewski

Abstract: A theorem of Wazewski type is proved for the existence of solutions of the vector system $y' = f(t,y)$ without assuming that solutions of initial value problems are unique. The theorem is then used to prove the existence of solutions of certain types of boundary value problems for the equation $x'' = h(t,x,x')$.