Showing papers in "Computers & Mathematics With Applications in 1997"

Finite Fields: Encyclopedia of Mathematics and Its Applications.

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A predator-prey system with stage-structure for predator

Abstract: This paper studies the asymptotoc behavior of a predator-prey model with stage structure. It is found that an orbitally asymptotically stable periodic orbit exists in that model. When time delay due to gestation of predator and time delay from crowding effect of prey are incorporated, we establish the condition for the permanence of populations and sufficient conditions under which positive equilibrium of the model is globally stable.

An inequality of Ostrowski-Grüss' type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules

Abstract: In this paper we derive a new inequality of Ostrowski-Gruss' type and apply it to the estimation of error bounds for some special means and for some numerical quadrature rules.

Natural discretizations for the divergence, gradient, and curl on logically rectangular grids☆

Abstract: This is the first in series of papers creating a discrete analog of vector analysis on logically rectangular, nonorthogonal, nonsmooth grids . We introduce notations for 2-D logically rectangular grids, describe both cell-valued and nodal discretizations for scalar functions, and construct the natural discretizations of vector fields, using the vector components normal and tangential to the cell boundaries. We then define natural discrete analogs of the divergence, gradient, and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates, such as length of elements of coordinate lines, areas of elements of coordinate surfaces, and elementary volumes. We introduce the discrete volume integral of scalar functions, the discrete surface integral, and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects. These theorems include the following: the discrete analog of relationship div A → = 0 if and only if A → = curl B → ; curl A → = 0 if and only if A → = grad ϕ ; if A → = grad ϕ , then the line integral does not depend on path; and if the line integral of a vector function is equal to zero for any closed path, then this vector is the gradient of a scalar function. Last, we define the discrete operators DIV, GRAD, and CURL in terms of primitive differencing operators (based on forward and backward differences) and primitive metric operators (related to multiplications of discrete functions by length of edges, areas of surfaces, and volumes of 3-D cells). These formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints.

The pickup and delivery problem: Faces and branch-and-cut algorithm

Abstract: This paper formulates the pickup and delivery problem, also known as the dial-a-ride problem, as an integer program. Its polyhedral structure is explored and four classes of valid inequalities developed. The results of a branch-and-cut algorithm based on these constraints are presented.

Dependent-Chance Programming: A Class of Stochastic Optimization

Abstract: This paper provides a theoretical framework of dependent-chance programming, as well as dependent-chance multiobjective programming and dependent-chance goal programming which are new types of stochastic optimization A stochastic simulation based genetic algorithm is also designed for solving dependent-chance programming models

Extended fuzzy linear models and least squares estimates

Abstract: Least squares regression of the fuzzy linear model is extended to overcome and interpret the occurrence of negative spreads [1]. The idea is to introduce best fit difference models which exploit both the Hukuhara difference and the L2-metric distance. The fuzzy models use LR-fuzzy numbers. Fitted models are compared by using the coefficient of determination, in a similar way to its use in classical statistical least squares fitting. The non-LR-fuzzy case is also considered.

Iterative techniques for time dependent Stokes problems

Abstract: In this paper, we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent Stokes equations using a mixed finite element spatial discretization. At each time step, a two by two block system corresponding to a perturbed Stokes problem must be solved. Although there are a number of techniques for iteratively solving this type of block system, to be effective, they require a good preconditioner for the resulting pressure operator (Schur complement). In contrast to the time independent Stokes equations where the pressure operator is well conditioned, the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced (and/or the Reynolds number is increased). In this paper, we shall describe and analyze preconditioners for the resulting pressure systems. These preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and Reynolds number parameter k .

Measuring 2-D shape compactness using the contact perimeter

Abstract: A new perimeter for shapes composed of cells is defined. This perimeter is called the contact perimeter, which corresponds to the sum of the boundaries of neighboring cells of the shape. Also, a relation between the perimeter of the shape and the contact perimeter is presented. The contact perimeter corresponds to the measure of compactness proposed here called discrete compactness. In this case, the term compactness does not refer to point-set topology, but is related to intrinsic properties of objects.

Simulated annealing simulated

Abstract: The performance of simulated annealing methods for finding a global minimum point of a function is studied.

Hyper-Hamilton Laceable and Caterpillar-Spannable Product Graphs

Abstract: We define a Hamilton laceable graph G to be hyper-Hamilton laceable (hyper HL) if either 1. (a) G is equitable, and if v is any node of G , then G − v is HL, or 2. (b) G is nearly equitable, and if v is any node in its larger color set, then G - v is HL. In particular, the hypercube Q n is hyper HL. A graph G is caterpillar-spannable (CS) if it has a spanning tree which is a caterpillar. We present several theorems concerning products of CS graphs. It is shown that the product of two CS graphs such that at least one of them has maximum degree 3 is CS.

Generalized tanh method with symbolic computation and generalized shallow water wave equation

Abstract: We report that symbolic computation with the generalized tanh method leads to new soliton-like solutions for a (2+1)-dimensional generalization of the shallow water wave equations.

Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval

Abstract: Polynomial interpolation between large numbers of arbitrary nodes does notoriously not, in general, yield useful approximations of continuous functions Following [1], we suggest to determine for each set of n + 1 nodes another denominator of degree n , and then to interpolate every continuous function by a rational function with that same denominator, so that the resulting interpolation process remains a linear projection The optimal denominator is chosen so as to minimize the Lebesgue constant for the given nodes It has to be computed numerically For that purpose, the barycentric representation of rational interpolants, which displays the linearity of the interpolation and reduces the determination of the denominator to that of the barycentric weights, is used The optimal weights can then be computed by solving an optimization problem with simple bounds which could not be solved accurately by the first author in [1] We show here how to do so, and we present numerical results

Estimations of domains with cycles

Abstract: A method for estimating domains with limit cycles and finding surfaces with the traces of all cycles is proposed. Corresponding estimations of domains with cycles for Lorenz and Rossler systems are indicated.

Characterization and generation of α-dense curves

Abstract: If a curve ranges through a cube H = Π n i =1 [ a i , b i ] so that, for every point x ∈ H , there is at least a point in the curve at a distance not greater than a positive number α, the curve is called α-dense in H . It is very important in the construction of such curves to find algorithms of approximation for global optimization and, in fact, there are several methods to do it, e.g., [1–3]. In this paper, we give a characterization of such curves in terms of γ-stochastically independent functions, as well as a constructive method to generate them by means of only one function ϕ called γ-uniformly distributed. Finally, in the above class, i.e., in the set of the γ-uniformly distributed functions, we will choose one of them, will construct the corresponding curve, and will compute its theoretic calculation time.

VLSI architecture of a cellular automata machine

Abstract: In the past, Cellular Automata based models and machines [1] have been proposed for simulation of physical systems without any analytical insight into the behaviour of the underlying simulation machine. This paper makes a significant departure from this traditional approach. An elegant mathematical model using simple matrix algebra is reported in this paper for characterizing the behaviour of two-dimensional nearest neighbourhood linear cellular automata with null and periodic boundary conditions. Based on this mathematical model, a VLSI architecture of a Cellular Automata Machine (CAM) has been proposed. Interesting applications of CAM in the fields of image analysis and fractal image generation are also reported.

Inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices

Abstract: Matrices represented as a sum of diagonal and semiseparable ones are considered here. These matrices belong to the class of structured matrices which arises in numerous applications. Fast O ( N ) algorithms for their inversion were developed before under additional restrictions which were a source of instability. Our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms. In this paper, the case of semiseparable matrices of order one is considered.

Using integer linear programming for instruction scheduling and register allocation in multi-issue processors

Abstract: Instruction scheduling and register allocation are two very important optimizations in modern compilers for advanced processors. These two optimizations must be performed simultaneously in order to maximize the instruction-level parallelism and to fully utilize the registers [1]. In this paper, we solve register allocation and instruction scheduling simultaneously using integer linear programming (ILP). We have successfully worked out the ILP formulations for the problem with and without register spilling. Two kinds of optimizations are considered: 1. (1) fix the number of free registers and then solve the minimum number of cycles to execute the instructions, or 2. (2) fix the maximum execution cycles for the instructions and solve the minimum number of registers needed. Besides being theoretically interesting, our solution serves as a reference point for other heuristic solutions. The formulations are also applicable to high-level synthesis of ASICs and designs for embedded processors. In these application domains, the code quality is more important than the compilation time.

Quasiattractors and Homoclinic Tangencies

Abstract: Recent results describing nontrivial dynamical phenomena in systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so-called quasiattractors of these systems may exhibit rather nontrivial features which are in a sharp distinction, with that one could expect in analogy with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors. Keywords-Attractor, Homoclinic bifurcations, Dynamical models.

Feature saliency measures

Abstract: This paper presents a survey of feature saliency measures used in artificial neural networks. Saliency measures can be used for assessing a feature's relative importance. In this paper, we contrast two basic philosophies for measuring feature saliency or importance within a feed-forward neural network. One philosophy is to evaluate each feature with respect to relative changes in either the neural network's output or the neural network's probability of error. We refer to this as a derivative-based philosophy of feature saliency. Using the derivative-based philosophy, we propose a new and more efficient probability of error measure. A second philosophy is to measure the relative size of the weight vector emanating from each feature. We refer to this as a weight-based philosophy of feature saliency. We derive several unifying relationships which exist within the derivative-based feature saliency measures, as well as between the derivative and the weight-based feature saliency measures. We also report experimental results for an target recognition problem using a number of derivative-based and weight-based saliency measures.

Shape Preserving Approximations by Polynomials and Splines

Abstract: A review of shape preserving approximation methods and algorithms for approxi- mating univariate functions or discrete data is given. The notion of 'shape' refers to the geometrical behavior of a function's or approximant's graph, and usually includes positivity, monotonicity, and/or convexity. But, in the recent literature, the broader concept of shape also includes symmetry, gen- eralized convexity, unimodality, Lipschitz property, possessing peaks or discontinuities, etc. Special stress is put on shape preserving interpolation methods by polynomials and splines. Of course, this text has no pretensions to be complete. geywords--Shape preserving approximation, Approximation of univariate functions, Approxima- tion of discrete data, Shape preserving interpolation, Polynomial, Spline, Positivity, Monotonicity, Convexity, Generalized convexity, Unimodality, Lipschitz property.

Discrete-time Geo1, Geo2/G/1 retrial queueing systems with two types of calls

Abstract: We consider a discrete-time Geo1, Geo2/G/1 retrial queue with two types of calls. When arriving calls are blocked due to the server being busy, Type I calls are queued in the priority queue with infinite capacity whereas, Type II calls enter the retrial group in order to try service again after a random amount of time. We find the joint generating function of the number of calls in the priority queue and the number of calls in the retrial group in a closed form. It is shown that our results are consistent with those already known for special cases.

Analytic solutions of an iterative functional differential equation

Abstract: This paper is concerned with an iterative functional differential equation x ′( z ) = x ( m ) ( z ), where x ( m ) ( z ) = x ( x (… x ( z ))) is the m th iterate of the function x ( z ). By constructing a convergent power series solution y ( z ) of a companion equation of the form αy ′( αz ) = y ′( z ) y ( α m z ), analytic solutions of the form y ( αy −1 ( z )) for the original differential equation are obtained.

An Erdős-Ko-Rado Theorem For Signed Sets

Abstract: A signed r-set on [n] = {1,…,n} is a pair (A, f), where A ⊂ [n] is an r-set and f is a function from A to {−1, 1}. A family A of signed r-sets is intersecting if for any (A, f), (B, g) ∈ A there exists x ∈ A ∩ B such that f(x) = g(x). In this note, we prove that if A is an intersecting family of signed r-sets on [n], then | A | ≤ 2r−1 (r−1n−1). We also present an application of this result to a diameter problem in the grid.

Coupling genetic algorithm with a grid search method to solve mixed integer nonlinear programming problems

Abstract: A new hybrid algorithm is being introduced for solving Mixed Integer Nonlinear Programming ( minlp ) problems which arise from study of many real-life engineering problems such as the minimum cost development of oil fields and the optimization of a multiproduct batch plant. This new algorithm employs both the Genetic Algorithm and a modified grid search method interfacing in such a way that the resulting hybrid algorithm is capable of solving many minlp problems efficiently and accurately. Testings indicate that this algorithm is efficient and robust even for some ill-conditioned problems with nonconvex constraints.

Numerical study of Lorenz's equation by the Adomian method

Abstract: In this paper, we use the decomposition method (of Adomian) for solving differential systems coming from physics (meteorology). We also give a comparison between the Runge-Kutta method and the decomposition technique. Furthermore, we reconfirm the famous “Butterfly effect.”

Simple Bifurcations Leading to Hyperbolic Attractors

Abstract: we prove the existence of a new stability boundary of periodic orbits in a high- dimensional case, thereby resolving the problem on a "blue sky catastrophe" in a general one- parameter family. We additionally establish the existence of a codimension-one boundary which separates the MorseSmale systems from systems with hyperbolic SmaleWilliams-like attractors. The route across this boundary is accomplished by the disappearance of a saddle-node periodic orbit. We also study the principal bifurcations of a torus breakdown which lead to Anosov attractors and to multidimensional solenoids.

The resolution of the Gibbs phenomenon for "spliced" functions in one and two dimensions

Abstract: In this paper we study approximation methods for analytic functions that have been “spliced” into nonintersecting subdomains. We assume that we are given the first 2N + 1 Fourier coefficients for the functions in each subdomain. The objective is to approximate the “spliced” function in each subdomain and then to “glue” the approximations together in order to recover the original function in the full domain. The Fourier partial sum approximation in each subdomain yields poor results, as the convergence is slow and spurious oscillations occur at the boundaries of each subdomain. Thus once we “glue” the subdomain approximations back together, the approximation for the function in the full domain will exhibit oscillations throughout the entire domain. Recently methods have been developed that successfully eliminate the Gibbs phenomenon for analytic but nonperiodic functions in one dimension. These methods are based on the knowledge of the first 2N + 1 Fourier coefficients and use either the Gegenbauer polynomials (Gottlieb et al.) or the Bernoulli polynomials (Abarbanel, Gottlieb, Cai et al., and Eckhoff). We propose a way to accurately reconstruct a “spliced” function in a full domain by extending the current methods to eliminate the Gibbs phenomenon in each nonintersecting subdomain and then “gluing” the approximations back together. We solve this problem in both one and two dimensions. In the one-dimensional case we provide two alternative options, the Bernoulli method and the Gegenbauer method, as well as a new hybrid method, the Gegenbauer-Bernoulli method. In the two-dimensional case we prove, for the very first time, exponential convergence of the Gegenbauer method, and then we apply it to solve the “spliced” function problem.

Block predictor-corrector schemes for the parallel solution of ODEs

Abstract: A fourth-order block method based on the composite Simpson rule is developed for the parallel solution of ordinary differential equations. Like the block scheme based on the composite Trapezoidal Rule, its principal error term is linear in the block size while the increased order and stability allow a modest increase in parallelism without further computational complexity. Numerical results confirm the enhanced properties of the higher-order method.

A heuristic approach to generating file spanning trees for reliability analysis of distributed computing systems

Abstract: The reliability of Distributed Computing Systems (DCS) in terms of Distributed Program Reliability (DPR) and Distributed System Reliability (DSR) has been studied intensively. Current reliability algorithms available for the analysis of DPR and DSR include MFST, FARE, FST, and FST-SPR. This paper presents a reliability algorithm, called HRFST, that eliminates the need to search a spanning tree during each subgraph generation. The HRFST algorithm reduces both the number of subgraphs (or trees) generated and the actual execution time required for analysis of DPR and DSR. Examination of several sample cases shows that the HRFST algorithm is more efficient than the FST-SPR algorithm.