Showing papers in "Siam Journal on Applied Mathematics in 1969"

The Nucleolus of a Characteristic Function Game

Abstract: : In RM 23, a proof was given that the nucleolus is continuous as a function of the characteristic function. This proof is not correct; the author, at least, does not know how to complete it. In the paper a correct proof for this fact is given. The proof is based on an alternative definition of the nucleolus, which is of some interest in its own right. (Author)

Conditions for Positive and Nonnegative Definiteness in Terms of Pseudoinverses

Abstract: Matrix pseudoinverses producing necessary and sufficient conditions for positive and nonnegative definiteness

Regularity Propeties of Flows Through Porous Media

Abstract: where u is essentially the density of the gas and A is the Laplace operator. Note that, apart from constants, grad um 1 is the velocity vector and u grad um1 is the flux vector. Thus, in particular, um1 is essentially the pressure which, by Darcy's law, is also the velocity potential. For m > 1, (1) is a nonlinear equation which is parabolic for u > 0, but which degenerates when u = 0. The most striking manifestation of the degeneracy of this equation is the finite speed of propagation of disturbances. Thus, if at some instant of time a solution u of (1) has compact support, then it will continue to have compact support for all later times. In general, the transition from a region where u > 0 to one where u = 0 is not smooth and it is therefore necessary to interpret the term "solution of (1)" in some generalized sense. Our object in this paper is to study the regularity properties of a class of generalized solutions of the Cauchy problem for (1). We show that, with respect to the space variables, the velocity potential is Lipschitz continuous, the flux is continuous, and the density is Holder continuous.

Approximate Solutions of Systems of Singular Integral Equations

Abstract: Using the properties of the related orthogonal polynomials, approximate solutions of systems of simultaneous singular integral equations are obtained, in which the essential features of the singularity of the unknown functions are preserved In the system of integral equations of the first kind, the fundamental solution is the weight function of the Chebyshev polynomials of first or second kind In the system of singular integral equations of the second kind with constant coefficients, the elements of the fundamental matrix are the weights of Jacobi polynomials A direct method is introduced to obtain the fundamental matrix of the system The approximate solution is then expressed as the fundamental function, representing the singular behavior of the unknown functions, multiplied by a series of proper orthogonal polynomials with unknown coefficients The techniques of deriving the system of algebraic equations to determine these coefficients are described In order to have an idea about the effectiveness

Optimal Sequencing of Two Equivalent Processors

Abstract: This paper presents an efficient algorithm for a class of sequencing problems in which n tasks with an arbitrary precedence relation have to be processed by two processors of equal ability, and each task requires one unit of time.

On Minimizing the Number of Multiplications Necessary for Matrix Multiplication

Abstract: This paper develops an algorithm to multiply a px2 matrix by a 2xn matrix in $\lceil (3pn+max(n,p))/2 \rceil$ multiplications for matrix multiplication without commutativity The algorithm minimizes the number of multiplications for matrix multiplication without commutativity for the special cases p=1 or 2, n=1,2, $\cdots$ and p = 3, n = 3 It is shown that with commutativity fewer multiplications are required

On Best Linear Estimation and General Gauss-Markov Theorem in Linear Models with Arbitrary Nonnegative Covariance Structure

Abstract: Given the general linear model $y = X\beta + e$ having the covariance matrix $\sigma ^2 V$ of the errors, with $\sigma ^2 > 0$, V known and nonnegative (possibly singular), we specify the complete nonempty class $\mathcal{V}$ of conditional inverses of V such that, for any estimable parametric function $\lambda '\beta $ and any $V^ * $ in $\mathcal{V}$, a best linear unbiased estimator of $\lambda '\beta $ is given by $\lambda '\hat \beta $, where $\hat \beta $ is any solution to the general normal equations $X'V^ * X\beta = X'V^ * y$. Properties of the solutions $\hat \beta $ are presented. It is further verified that if y is distributed as a multivariate normal variable then $\lambda '\hat \beta $ is the maximum likelihood estimator of $\lambda '\beta $. A procedure for testing hypotheses, using solutions to the general normal equations, is also presented.

A Note on Kronecker Matrix Products and Matrix Equation Systems

Abstract: where B' is the transpose of B. It has been shown that Definition 1 and Theorem 1 can fruitfully be applied to problems of matrix differentiation [2]. In this note it will be shown that they can be applied to a more general class of linear matrix equations, including linear matrix differential equations. Firstly, four standard properties of Kronecker products have to be related, all of which may be proved in an elementary fashion [1, p. 223 if.]. The matrices involved can have any appropriate orders. In Property 4 it is assumed that A and B are square of order m and s, respectively. (The same order assumption will be made in Theorems 2 and 3, which will be presented further on.) PROPERTY 1. (A B)(C D) =(AC) (BD). PROPERTY 2. (A 0 B)' = A' 0 B'. PROPERTY. 3. (A + B) 09 (C + D) = A C + A D + B (& C + B X D. PROPERTY 4. If A has characteristic roots oci, i = 1, -*, m, and if B has characteristic roots /IB, j = 1,... , s, then A 0 B has characteristic roots oci,Bj. Further, Is 0 A + B 0) Im has characteristic roots oci + f3B. For the treatment of differential equations we need the matrix exponential

Higher order averaging and related methods for perturbed periodic and quasi-periodic systems

Abstract: for 0 ? t 0. Following the basic work of Krylov and Bogoliubov [9] in 1934, the first systematic and comprehensive account of the method of averaging was given by Bogoliubov and Mitropolski [1] in 1958. In this account, the authors were primarily concerned with first and second order approximations. They laid the mathematical foundation for the method which established the validity of the first order approximation over a time interval of 0(1/E). Several authors have subsequently considered higher order averaging, e.g., Volosov [20]-[22], but have only explicitly treated first and second order averaging. In this way they establish an algorithm for the general Nth order method; but they do not establish it as an Nth order asymptotic method on 0 0. In 1964, Mitropolski [12] indicated how to establish Nth order averaging as an Nth order asymptotic method on 0 ? t 2 simply by conjecturing the general structure of the fundamental differential inequality [12, p. 339] from which the Nth order error estimate follows. More recently, Zabreiko and Ledovskaja [23] gave a set of equations defining the general Nth order method of averaging and a set of conditions which they claim are sufficient to establish Nth order averaging as an Nth order asymptotic method on 0 ? t < Llr as E -* 0. In particular, they state [23, p. 1455] that these conditions are satisfied if the right-hand side f(t, x, E) is an almost periodic function of t (and has sufficiently many derivatives in x and ? which are bounded). This statement is in general not true, the reason being that the almost periodicity off(t, x, e) in t does not, in general, imply the boundedness of the function p1(t, x) or of its first partial derivative with respect to x, where

Extensions of Lagrange Multipliers in Nonlinear Programming

Abstract: defined for x > 0, u > 0. A saddle point of (1.2) is defined to be a point (x*, u*), x* > Q, u* > 0 such that +(x, u*) < +(x*, u*) _ +(x*, u) for all x _ 0, u ? 0. One of the Kuhn-Tucker results is that a saddle point of(1.2) provides a solution to (1.1). Under cer-eain additional regularity assumptions the equivalence of the saddle point problem and the program (1.1) has been shown [1], [10]. In particular, if the feasible set in (1.1) has an interior point (a nonnegative x such that g(x) < b),3 and if the objective function is concave and the constraint functions are convex, then x* is a solution to (1.1) if and only if there is a nonnegative u* such that (x*, u*) is a saddle point of (1.2). This is the so-called Kuhn-Tucker equivalence theorem of nonlinear programming. In this paper Lagrange multipliers are generalized from the usual constants to (possibly nonlinear) multiplier functions. This leads to the statement of several general equivalence results under quite weak assumptions. From a somewhat different point of view, Everett [4] drops differentiability assumptions and discusses a procedure for solving nonlinear programs in terms of

The Streetwalker’s Dilemma: A Job Shop Model

Abstract: We consider the problem of maximizing the long-run average return in a single server traffic reward system in which the customer’s offer, a joint distribution of reward and of service time required to earn this reward, is independent of the renewal process which governs customer arrivals. After formulating the problem as a semi-Markov decision process, we characterize the form of an optimal policy. When the renewal process is Poisson, the characterization is easily stated : accept a customer if and only if the ratio of his expected reward to his expected service time is larger than g, the long-run average return. When the arrival process is Poisson, g is easily found. Next, batch arrivals are permitted, and further results are obtained.

Subdefinite Matrices and Quadratic Forms

Abstract: Recent results in quasiconvex and pseudoconvex programming [1], [3], [4] call the attention to these classes of functions. Little work has been done, however, in order to make easier the recognition of these types of functions. The definitions of quasiconvexity and pseudoconvexity do not offer a useful test from a practical point of view. In this paper one of the simplest problems of this kind is investigated, that of a quasiconvex, or pseudoconvex quadratic form. The connection of this problem with quadratic programming is obvious. The problem leads to a new class of real symmetric matrices, which we call subdefinite. After introducing the main definitions in the first section, we look for the characteristics of merely subdefinite (not semidefinite) matrices (? 2). The third section deals with the convexity properties of subdefinite quadratic forms. A few numerical examples are attached.

Hyperbolic Pairs in the Method of Conjugate Gradients

Abstract: (starting from an arbitrary x1 and r1 = b Ax1, Pt = r1) converges to the solution of(1) in n steps or less. Theoretically, the process (2) can be carried out in such a manner that only one multiplication of A times a vector is required at each step, although in practice two such multiplications are often employed. In many applications we are led to equations of the form (1) where A is nonsingular and symmetric but not positive definite. For example the problem of minimizing the quadratic form

Asymptotically Almost Periodic Solutions of Differential Equations

Abstract: In this paper we examine asymptotically almost periodic solutions of an almost autonomous differential equation. Solutions of this type have been examined previously for plane systems by Wong and Burton [8] and Utz and Waltman [6]. In both of the above papers solutions of the almost autonomous equation spiral to a periodic orbit of the limiting equations and are thus shown to be asymptotically almost periodic. This device is also exploited in the theorems proved here, although the solutions will not be required to approach periodic orbits of the limiting equation in such a uniform manner. Also, conditions will be given so that solutions satisfy a somewhat more standard type of asymptotic almost periodicity than the type discussed in [8] and [6].

Asymptotic Theory of Unsteady Three-Dimensional Waves in a Channel of Arbitrary Cross Section

Abstract: A mathematically tractable theory is developed for the study of unsteady three-dimensional gravitating waves in a channel of arbitrary cross section. Three-dimensional effects due to bottom topography on the lateral change of free surface elevation are fully accounted for, and a unified approach to the derivation of various expansions is achieved. This paper deals mainly with finite-amplitude waves in a homogeneous incompressible inviscid fluid. Three cases are studied, which correspond to different orders of magnitude of a parameter representing the product of the square of the axial length scale of the motion with the amplitude of the derivation of surface elevation from that of conventional shallow-water theory. The bottom topography has a pronounced effect on the surface elevation when that parameter is of the same order as the cube of the hydraulic radius. Asymptotic equations governing finite-amplitude waves are then derived, and a Neumann problem defined on a variable domain is posed for determinin...

Some Results on Matching in Bipartite Graphs

Abstract: Let A, , Am and B1, ... , B, be partitions of A and B respectively into -nonempty disjoint subsets. Let G' denote the quotient bipartite graph with sets of vertices A' = {Al,... , Am} and B' = {B1, * * , B4} and a set of edges E' defined such that {Ai,Bj} eE' if and only if {a,b} eE for some aAi, bBj. Of course, VA, and VB' denote the natural induced measures on A' and B', and D' is defined in the obvious way. It is important to note that if all vertices have weight 1 and with a e A we associate the subset D({a}) c B, then by the marriage theorem of P. Hall (cf. [3]) VB dominates VA if and only if there exists a system of distinct representatives (cf. [3]) for these subsets.2 If b(a) is the representative chosen from D({a}), then the mapping a -b(a) is a matching of A into B, i.e., a 1 1 mapping of A into B such that {a, b(a)} is an edge of G. It is this application of our results which motivated the present study (cf. Examples 1 and 2). Our main object in this note is to develop several theorems which will enable one to show that VB dominates VA by examining the structure of the (hopefully much simpler) quotient graph G'.

Equivalent Realizations of Linear Systems

Abstract: In this paper, we continue an investigation [7] of the case in which the initial system specification is a set of linear first order differential equations as might be derived from a physical description of the system. Since the impulse response matrix is not easily computable from this type of representation, our aim is to develop a theory of equivalence solely in terms of known coefficient matrices. Not surprisingly, it is not possible to include all linear time-variable systems in this theory. However, it is shown that a broad class of systems (including time-invariant and analytic systems) can be delineated which do admit a complete equivalence characterization of this type. In ? 2 some basic system concepts and definitions are briefly reviewed, and in ? 3 the class of "constant rank" system representations is introduced. It is shown that members of this class have a canonical structure whose components are also of constant rank and that the dimensions of these components can be computed explicitly from the given system coefficient matrices and a finite number of their derivatives. In ? 4 a complete equivalence characterization is developed for constant rank systems. Based on this characterization it is shown in ? 5 that a minimal zero-state equivalent of a specified constant rank system representation can be constructed without solving differential equations. The type of realization obtained by this procedure is shown in ? 5 and ? 6 to possess several properties which make it most natural for system analysis or synthesis. It is time-invariant (periodic) when the impulse response matrix is time-invariant (periodic) and, under appropriate conditions, has bounded coefficients and is uniformly asymptotically stable [8] when the impulse response matrix is bounded-input bounded-output stable [8].

Generalized Szász Operators for the Approximation in the Complex Domain

Abstract: converge (as u -oo) to f (x) at each point t = x > 0, where f (t) is continuous. J. J. Gergen, F. G. Dressel and W. H. Purcell [2] proved that for a certain class of analytic functions f (z) the operators P(u, f ) approximate these functions. E. W. Cheney and A. Sharma [1] showed that the operators P(n, f ), n = 1, 2, , are variation-diminishing in the sense of I. J. Schoenberg [8]. They also proved that if f is convex, then P(n, f) is decreasing in n, unless f is linear (in which case P(n,f) = P(n + 1,f) for all n). Recently A. Jakimovski and D. Leviatan [4] considered the following generalization of P(u, f ): Let g(z) Z= o aj? be an analytic function in the disk lzl 1, and suppose g(1) = 0. Define the Appell polynomials Pk(X) pk(X, g), k > 0, by g(u)eux Pk(X)Uk. k = O

Stochastic Programs with Recourse II: On the Continuity of the Objective

Abstract: : In an earlier paper, 'Stochastic Programs with Recourse,' the authors introduced a general class of stochastic (linear) programs and showed, among other things, that the objective of any such program is convex when considered as a function of the first-stage decision variables. In this paper it is shown that the objective is also lower semi-continuous. In the process of proving this result, a lemma of general interest in the theory of convex functions is established.

Some Theorems on Sorting

Abstract: It is a “well-known fact” that a lower bound for the average number of comparisons required to sort a table of N items is $\log _2 N!$, where the average is taken over all possible permutations of the table. In this paper a somewhat better lower bound is obtained, which in a way provides considerable insight into the theoretical limitations on methods of sorting by comparison.

A Connection Between Elementary Functions and Higher Transcendental Functions

Abstract: If $\,f$ is holomorphic on a domain D in the complex plane, an analogous function F of several complex variables is constructed by taking a weighted average of $\,f$ over the convex hull of $\{ z_1 ,z_2 , \cdots ,z_k \}$. Although F is defined at first only if the convex hull is contained in D, it is shown later that F can be continued analytically along any rectifiable arc in $D^k $, provided that singular points with $z_i = z_j $ (for some distinct i, j) are excluded if D is multiply connected. Taylor and Laurent series for f have single-series analogues for F, and the analogue of Cauchy’s integral formula is a representation of F by an integral around a contour in D encircling $z_1 ,z_2 , \cdots ,z_k $. The hypergeometric function ${}_2 F_1 (a,b;c;x)$ is an average of $z^{ - a } $ over the line segment joining $1-x$ and 1, the confluent hypergeometric function ${}_1 F_1 (b;c;x)$ is an average of $e^z $ over the line segment joining x and 0, and elliptic integrals are averages of a half-odd-integral pow...

The Design of Combinatorial Information Retrieval Systems for Files with Multiple-Valued Attributes

Abstract: The recent advent of large scale, high-speed computers has produced an “information revolution.” One of the consequences of this has been the need for the development of filing systems which are capable of handling large volumes of data and permitting efficient information retrieval. This research is concerned with a file structure in which parameters like number of attributes, number of levels of each attribute, and types of queries are known and fixed. For this type of situation, first a review is given for a number of different types of filing schemes which have been recently discussed in the literature. Then attention is turned to a general model and filing systems based on certain types of combinatorial configurations. The construction of one type of configuration is achieved through the development of suitable methods of extending some of the properties of certain small orthogonal arrays to larger schemes. This type of construction may be combined with certain other schemes to yield multistage filin...