Showing papers in "Siam Journal on Applied Mathematics in 1977"
TL;DR: The problem of determining the minimum length of an optimum rectilinear Steiner tree for a set A of points in the plane is shown to be NP-complete and the emphasis of the literature on heuristics and special case algorithms is well justified.
Abstract: An optimum rectilinear Steiner tree for a set A of points in the plane is a tree which interconnects A using horizontal and vertical lines of shortest possible total length. Such trees correspond to single net wiring patterns on printed backplanes which minimize total wire length. We show that the problem of determining this minimum length, given A, is NP-complete. Thus the problem of finding optimum rectilinear Steiner trees is probably computationally hopeless, and the emphasis of the literature for this problem on heuristics and special case algorithms is well justified. A number of intermediary lemmas concerning the NP-completeness of certain graph-theoretic problems are proved and may be of independent interest.
1,151 citations
TL;DR: It is shown that the problem of computing Steiner minimal trees for general planar point sets is inherently at least as difficult as any of the $NP$-complete problems.
Abstract: It is shown that the problem of computing Steiner minimal trees for general planar point sets is inherently at least as difficult as any of the $NP$-complete problems (a well known class of computa...
553 citations
TL;DR: In this article, a Condorcet social choice function elects the candidate that beats every other candidate under simple majority when such a candidate exists, and several extensions of the simple majority principle have been proposed.
Abstract: A Condorcet social choice function elects the candidate that beats every other candidate under simple majority when such a candidate exists. Various extensions of Condorcet’s simple majority principle that deal with situations that have no simple majority winner have been proposed.Nine Condorcet social choice functions are analyzed and compared on the basis of how well they satisfy a number of conditions for social choice functions. The conditions include several generalizations of Condorcet’s Principle. Remarks on the relative merits of the nine basic functions are included.
549 citations
TL;DR: The paper studies a mathematical model, based on Michaelis-Menten kinetics, for one substrate and n competing species, and answers the basic question as to which species survive and which do not, and determines the limiting behaviors.
Abstract: The continuous culture of micro-organisms using the chemostat is an important research technique in microbiology and population biology. It offers advantages in the form of economical production of micro-organisms for the industrial microbiologist and is a laboratory idealization of nature for population studies. The paper studies a mathematical model, based on Michaelis-Menten kinetics, for one substrate and n competing species. Given the parameters of the system, we answer the basic question as to which species survive and which do not, and determine the limiting behaviors. The primary conclusion is that the species will survive whose Michaelis-Menten constant is smallest in comparison with its intrinsic rate of natural increase.
466 citations
TL;DR: The cumulative effect on dynamical systems, of even very small random perturbations, may be considerable after sufficiently long times as mentioned in this paper, even if the corresponding deterministic system h...
Abstract: The cumulative effect on dynamical systems, of even very small random perturbations, may be considerable after sufficiently long times. For example, even if the corresponding deterministic system h...
257 citations
TL;DR: In this paper, the general system of differential equations describing predator-prey dynamics is modified by the assumption that the coefficients are periodic functions of time and it is shown that this system has a periodic solution (in place of an equilibrium) provided the long term time average, of the predator's net, unihibited death rate is in a suitable range.
Abstract: The general system of differential equations describing predator-prey dynamics is modified by the assumption that the coefficients are periodic functions of time. By use of standard techniques of bifurcation theory, as well as a recent global result of Rabinowitz, it is shown that this system has a periodic solution (in place of an equilibrium) provided the long term time average, of the predator’s net, unihibited death rate is in a suitable range. The bifurcation is from the periodic solution of the time-dependent logistic equation for the prey (which results in the absence of any predator). Numerical results which clearly show this bifurcation phenomenon are briefly discussed.
233 citations
TL;DR: In this paper, the problem of determining the optimal ordering policies under stochastic demand is examined when two supply options, air and surface, are available, with different costs and different delivery times.
Abstract: The problem of determining the optimal ordering policies under stochastic demand is examined when two supply options, air and surface, are available, with different costs and different delivery times. Assuming mild conditions on the holding-penalty cost functions, linear ordering costs and backlogging, some sufficient conditions are obtained to show when it is optimal to order nothing by air and others to show when it is optimal to order nothing by surface. Explicit formulae are derived for the optimal orders in the case when air delivery time is $\kappa $ periods and surface delivery time is $\kappa + 1$ periods, respectively, under more general conditions than before.
227 citations
TL;DR: A decomposition procedure for evaluating the reliability of a network whose elements fail independently with known probabilities is presented and it is shown that computing network reliability is NP-hard, so that efficiency in special cases is all that can be achieved.
Abstract: We present a decomposition procedure for evaluating the reliability of a network whose elements fail independently with known probabilities. A subnetwork is “solved” by computing the probability for classes of failure patterns within which the network performance is constant. We then iterate, combining solved subnetworks until the entire network has been evaluated. The running time for each iteration depends superexponentially on the subnetwork boundary size, but for treelike or “thin” networks total time grows only linearly with total network size. Details of our decomposition scheme are presented for a wide variety of measures of performance. For measures where computing the distribution of performance is too slow we present a quicker method which finds the average performance. Finally, we show that computing network reliability is NP-hard, so that efficiency in special cases is all that can be achieved.
188 citations
Journal Article•
144 citations
TL;DR: In this paper, sufficient conditions are established for the convergence of the biorthogonal series solving edge problems which arise in elasticity and in Stokes flow in cavities, which are problems previously considered to be intractable to analysis.
Abstract: Sufficient conditions are established for the convergence of the biorthogonal series solving edge problems which arise in elasticity and in Stokes flow in cavities. These conditions and those given in Part I, (D. D. Joseph, The convergence of biorthogonal series for biharmonic and Stokes flow edge problems, SIAM, J. Appl. Math., 33(1977), pp. 337–347) include all those which are likely to arise in applications. Examples of conditional convergence of the series to step functions and to ramp functions are presented. Problems previously considered to be intractable to analysis are solved by analysis.
140 citations
TL;DR: In this article, the bounds for the index of block triangular matrices are given in terms of the indices of the diagonal blocks, and an explicit formula for the Drazin inverse of M is given which is a function of $A,B,C,A^D $ and $C^D$ only.
Abstract: Bounds for the index of block triangular matrices are given in terms of the indices of the diagonal blocks. For a matrix of the form \[ M = \left[ {\begin{array}{*{20}c} A &\vline & B \\ \hline 0 &\vline & C \\ \end{array} } \right] \] where A and C are square, an explicit formula for the Drazin inverse, $M^D $, of M is given which is a function of $A,B,C,A^D $ and $C^D $ only.
TL;DR: In this article, a class of appropriately fast algorithms for reconstructing the internal structure of the cross section from such data is rigorously derived starting from the classical Radon inversion formula for divergent beams is given as a singlular integral.
Abstract: An arrangement of a single x-ray source and a circular array (centered at the source) of detectors allows rapid collection of a number of x-ray projections of a cross section of an object A class of appropriately fast algorithms for reconstructing the internal structure of the cross section from such data is rigorously derived starting from the classical Radon inversion formula A new Radon inversion formula for divergent beams is given as a singlular integral It is shown that this singular integral can be regularized a number of ways, each one of which leads to an efficient numerical evaluation The structure, and hence the speed, of the resulting algorithms is similar to the so-called “convolution reconstruction techniques” which have been previously derived for data slowly collected along parallel rays and have been found both fast and accurate It is demonstrated that, provided the same number of x-ray photons is used, the method of this paper gives as accurate reconstructions for diverging x-rays a
TL;DR: In this paper, the authors consider the inverse problem of determining small variations in propagation speed from observations of signals which pass through the medium of interest and are then observed remotely and show that the variation satisfies an integral equation of integral transform type with an atypical kernel.
Abstract: We consider the inverse problem of determining small variations in propagation speed from observations of signals which pass through the medium of interest and are then observed remotely. We show that the variation satisfies an integral equation of integral transform type with an atypical kernel. In a variety of examples for the scalar and vector wave equation (Maxwell’s equations) and the equations of linear elasticity, we solve this integral equation by elementary means.
TL;DR: Equations are formulated that link the probability of a bacterium suddenly changing its direction with temporal change in the number of attractant molecules that are bound to the bacteria’s receptor molecules.
Abstract: Equations are formulated that link the probability of a bacterium suddenly changing its direction with temporal change in the number of attractant molecules that are bound to the bacteria’s receptor molecules. “Memory” arising from the noninstantaneous nature of the attractant-receptor chemical reaction is taken into account in this one-dimensional model by considering separate populations of left- and right-moving bacteria. Previously formulated phenomenological equations for bacterial chemotaxis are recovered when a perturbation analysis is permitted by sufficiently shallow gradients. A detailed comparison is provided with certain relevant experiments.
TL;DR: In this paper, a new probability distribution based on the H-function of Fox is introduced, which is shown to be a generalization of most common nonnegative (Pr $\Pr ([X < 0] = 0)$ distributions.
Abstract: This paper introduces a new probability distribution based on the H-function of Fox. The distribution is shown to be a generalization of most common “nonnegative” (Pr $\Pr ([X < 0] = 0)$ distributions. Furthermore, it is proved that products, quotients and powers of H-function variates are H-function variates. Several examples are given.
TL;DR: This work discusses mechanical theories associated with the fluid mosaic model of the cell membrane and its applications in medicine and materials science.
Abstract: We discuss mechanical theories associated with the fluid mosaic model of the cell membrane [1].
TL;DR: In this paper, the conditions under which a nonlinear functional admits a Volterra-like integral representation, the class of systems that admit a Wiener representation and the meaning of such a representation were derived.
Abstract: Volterra and Wiener series provide a general representation for a wide class of nonlinear systems. In this paper we derive rigorous results concerning (a) the conditions under which a nonlinear functional admits a Volterra-like integral representation, (b) the class of systems that admit a Wiener representation and the meaning of such a representation, (c) some sufficient conditions providing a connection between the Volterra-like and the Wiener representations, (d) the mathematical validity of the method of Lee and Schetzen for identifying a nonlinear system.
TL;DR: In this paper, an asymptotic theory is presented to analyze perturbations of bifurcations of the solutions of nonlinear problems, which may result from imperfections, impurities, or other inhomogeneities in the corresponding physical problem.
Abstract: An asymptotic theory is presented to analyze perturbations of bifurcations of the solutions of nonlinear problems. The perturbations may result from imperfections, impurities, or other inhomogeneities in the corresponding physical problem. It is shown that for a wide class of problems the perturbations are singular. The method of matched asymptotic expansions is used to obtain asymptotic expansions of the solutions. Global representations of the solutions of the perturbed problem are obtained when the bifurcation solutions are known globally. This procedure also gives a quantitative method for analyzing singularities of nonlinear mappings and their unfoldings. Applications are given to a simple elasticity problem, and to nonlinear boundary value problems.
TL;DR: In this article, a system of differential equations is proposed as a model of dispersion between two populations in habitats separated by a barrier, and the effect of barrier strength on the limiting populations is analyzed.
Abstract: A system of differential equations is proposed as a model of dispersion between two populations in habitats separated by a barrier. Both strong and weak barriers are considered, and the effect of barrier strength on the limiting populations is analyzed. The case of a predator feeding indiscriminately on these populations is also considered and the effect of barrier strength on the limiting population analyzed.
TL;DR: In this article, necessary and sufficient conditions for the existence of a matrix with prescribed singular values and prescribed elements in its main diagonal are given, and several related results are also given.
Abstract: Necessary and sufficient conditions are given for the existence of a matrix with prescribed singular values and prescribed elements in its main diagonal. Numerous related results are also given.
TL;DR: In this article, the stability properties of uniform steady solutions to systems of reaction-diffusion equations subject to Neumann boundary data are investigated for three types of diffusion coefficients: equal positive diffusion coefficient, unequal positive coefficient, and unequal nonnegative coefficient.
Abstract: This paper primarily treats the stability properties of uniform steady solutions to systems of reaction-diffusion equations subject to Neumann boundary data. The importance of such stability questions for physical problems has been made by Turing, Othmer and Scriven, and others. The three cases of equal positive diffusion coefficients, unequal positive diffusion coefficients and unequal nonnegative diffusion coefficients are investigated. The possible destabilizing effects of unequal diffusion coefficients are clearly indicated. In the case of equal diffusion coefficients we estimate the extent of asymptotic stability. We also briefly treat the case when Neumann data is replaced by Dirichlet data. This latter case has been studied more extensively in the literature. 1. Introduction. This paper is primarily concerned with the stability proper- ties of uniform steady state solutions to a weakly coupled system of reaction diffusion equations. These equations have been very important in the modeling and study of population growth and nuclear and chemical reactions. A seminal paper in this field is A. M. Turing (11) who suggested that a system of reaction diffusion equations could describe, in some sense, morphogenesis. Later Othmer and Scriven (9) formally used linearization techniques to study the stability of constant steady states occurring in the system of reaction diffusion equations. Theorems 1 and 2 of this paper can be considered as justification of the mathematical procedures in (9). We are concerned with the system of reaction diffusion equations
TL;DR: This work first elucidate the stochastic structures of random nerve nets and derives macroscopic state equations which apply to a wide range of ensembles of random nets which are shown to hold in a weak sense.
Abstract: The brain is a large-scale system composed of an enormous number of neurons. In order to understand its functioning, we need to know the macroscopic behavior of a nerve net as a whole. Statistical neurodynamics treats an ensemble of nets of randomly connected neurons and derives macroscopic equations from the microscopic state transition laws of the nets. There arises, however, a theoretical difficulty in deriving the macroscopic state equations, because of possible correlations among the microscopic states. The situation is similar to that encountered in deriving the Boltzmann equation in statistical mechanics of gases.We first elucidate the stochastic structures of random nerve nets. We then derive macroscopic state equations which apply to a wide range of ensembles of random nets. These equations are shown to hold in a weak sense: we prove that the probability that these equations are valid within an arbitrarily small error and for an arbitrarily long time converges to 1 as the number n of the componen...
TL;DR: In this article, a class of neural models is introduced in which the topology of the neural network has been generated by a controlled probability model, and it is shown that the resulting linear operator has a spectral measure that converges in probability to a universal one when the size of the net tends to infinity.
Abstract: A class of neural models is introduced in which the topology of the neural network has been generated by a controlled probability model. It is shown that the resulting linear operator has a spectral measure that converges in probability to a universal one when the size of the net tends to infinity: a law of large numbers for the spectra of such operators. The analytical treatment is accompanied by omputational experiments.
TL;DR: Canonical representation of a singular pencil given in Gantmacher [1] and a theorem of Kucera [3] for a special case are used to solve the matrix equation $AXB + CXD = E.
Abstract: Canonical representation of a singular pencil given in Gantmacher [1] and a theorem of Kucera [3] for a special case are used to solve the matrix equation $AXB + CXD = E$.
TL;DR: In this paper, it was shown that there exists a linear ordering of the points in Cartesian products of nonnegative integers or integers such that the first j points in this ordering is a configuration that minimizes the number of boundary points (points not in the set that have Euclidean distance one from the set) among sets of size j.
Abstract: It is shown that there exists a linear ordering of the points in $I_ + ^n $ or $I^n $ (Cartesian products of nonnegative integers or integers) such that the first j points in this ordering is a configuration that minimizes the number of boundary points (points not in the set that have Euclidean distance one from the set) among sets of size j. The relation of the $I_ + ^n $ result to Macaulay’s theorem is deduced also.
TL;DR: In this article, it was shown that the lattice of subspaces of finite-dimensional vector spaces over a finite field is a symmetric chain order, and that the same conclusion holds if the LYM property assumption is replaced by the condition that every element of rank k is ordered with the same number of elements of ranks $k - 1$ and $k + 1$.
Abstract: It is shown that a partial order satisfying the “LYM property”, symmetry of Whitney numbers under the inversion $j \to n - j$, and unimodality of Whitney numbers can be partitioned into symmetric chains. It follows that the same conclusion follows if the LYM property assumption is replaced by the condition that every element of rank k is ordered with the same number of elements of ranks $k - 1$ and $k + 1$. It further follows that the lattice of subspaces of finite-dimensional vector space over a finite field is a symmetric chain order.
TL;DR: In this article, the system of partial differential equations which describe the predator-prey interaction, where diffusion and spatial dependence are taken into account, is considered and conditions under which solutions are uniformly bounded, and all solutions asymptotically enter a fixed region in phase space.
Abstract: We consider the system of partial differential equations which describe the predator-prey interaction, where diffusion and spatial dependence are taken into account. The equations reduce to the Kolmogorov equations, when spatial dependence is neglected. We find conditions under which solutions are uniformly bounded, and we show that all solutions asymptotically enter a fixed region in phase space. We also consider the question of stability of those constant states which correspond to the extinction of one or both species.
TL;DR: In this article, necessary and sufficient conditions for an efficient solution to be properly efficient are developed, which are useful both in verifying that certain efficient solutions are properly efficient and in identifying efficient solutions that are not proper.
Abstract: Vector maximization problems arise when more than one objective function is to be maximized over a given feasibility region. While the concept of efficiency has played a useful role in the analysis of such problems, a slightly more restricted concept of efficiency, that of proper efficiency, has been proposed in order to eliminate efficient solutions of a certain anomalous type. In this paper, necessary and sufficient conditions for an efficient solution to be properly efficient are developed. These conditions relate the proper efficiency of a given solution to the stability of certain single-objective maximization problems. The conditions are useful both in verifying that certain efficient solutions are properly efficient and in identifying efficient solutions that are not proper. An immediate corollary of the theory is that all efficient solutions in linear vector maximization problems are properly efficient. Examples are given to illustrate our results.
TL;DR: In this article, a maximum principle for the nonlinear differential equation of torsional creep was derived and used to compute isoperimetric bounds for the maximum stress and the stiffness.
Abstract: In this paper we derive a maximum principle for the nonlinear differential equation of torsional creep and use this principle to compute isoperimetric bounds for the maximum stress and the torsional stiffness.
TL;DR: The result unifies the theory which to date has been developed independently for the two problems and gives rise to transformation rules of algorithms and theorems of one problem to ones of the other problem.
Abstract: This paper establishes a basic relationship between the max flow problem in networks with positive gains and the min-cost flow problem in pure networks. The result unifies the theory which to date has been developed independently for the two problems and gives rise to transformation rules of algorithms and theorems of one problem to ones of the other problem.