Showing papers in "Siam Journal on Applied Mathematics in 1981"
TL;DR: In Part I, a generalization of the Fuzzy c-Means and FuzzY c-Lines algorithms are developed and shown to be special cases of a more general class of fuzzy algorithms, the fuzzy scatter matrices.
Abstract: In Part I, a generalization of the Fuzzy c-Means (or Fuzzy ISODATA) clustering algorithms is developed. Necessary conditions for minimization of a generalized total weighted squared orthogonal error objective function lead to a Picard iteration scheme which generates simultaneously (i) c fuzzy clusters in the data; (ii) a set of c prototypical straight lines in feature space which best fit the data in a well-defined sense; (iii) a set of c prototpyical centers of mass (on the c lines) which characterize the “core” of each linear fuzzy cluster. Theoretical optimization is achieved using principal components of generalized within cluster fuzzy scatter matrices. A convergence theorem for each algorithm in the infinite family is given. The algorithms are exemplified by five numerical examples using both real and artificial data sets having essentially “linear” substructure. In Part II, the Fuzzy c-Means and Fuzzy c-Lines algorithms are shown to be special cases of a more general class of fuzzy algorithms, the...
337 citations
TL;DR: This work presents a complete analysis of the response to periodic input of an integrate-and-fire model, which is a simplified version of the Hodgkin–Huxley theory for space clamped nerves.
Abstract: Nerve membranes exhibit curious responses to alternating current stimulation, among which are phase locking, as well as responses without apparent periodic pattern. We investigate these phenomena by presenting a complete analysis of the response to periodic input of an integrate-and-fire model, which is a simplified version of the Hodgkin–Huxley theory for space clamped nerves.
287 citations
TL;DR: This work generalizes the straight line prototype of a cluster developed in Part I to any r-dimensional linear variety of $R^s ,( 0\leqq r < s)$ and considers a distance functional which utilizes convex combinations of the distance functionals developed here and in Part II.
Abstract: In Part I [SIAM J. Appl. Math., 40 (1981), pp. 339–357], Fuzzy c-Lines was introduced as an algorithm for detection and characterization of linearly clustered data. In Part II, we address two extensions of the theory in Part I. Specifically, we will first generalize the straight line prototype of a cluster developed in Part I to any r-dimensional linear variety of $R^s ,( 0\leqq r < s)$; secondly, we will consider a distance functional which utilizes convex combinations of the distance functionals developed here and in Part I. All of the notation and symbols used here are unchanged from Part I.
196 citations
TL;DR: In this article, the problem of identifying a spatially varying diffusion coefficient using an observed solution to the forward problem is treated as a first order hyperbolic equation in the unknown coefficient.
Abstract: We consider the problem of identifying a spatially varying diffusion coefficient using an observed solution to the forward problem. Under appropriate conditions, this problem can be treated as a first order hyperbolic equation in the unknown coefficient. We develop some continuous dependence results for this problem, and also propose a particularly advantageous set of test conditions for observing the forward solution.
179 citations
TL;DR: In this paper, a constant-parameter epidemic model for a closed population is analyzed to determine whether time delays can give rise to periodic oscillations, and Hopf bifurcation techniques are used to show the existence of locally asymptotically stable periodic solutions for certain parameter values.
Abstract: Constant-parameter epidemic models for a closed population (described by integral and integrodifferential equations) are analyzed to determine whether time delays can give rise to periodic oscillations. Delays can destabilize the steady state in one finite delay case and in one infinite delay case considered. Hopf bifurcation techniques are used to show the existence of locally asymptotically stable periodic solutions for certain parameter values. Inclusion of time delays in another related model does not alter the local asymptotic stability of the endemic steady state.
168 citations
TL;DR: A qualitative model for studying shock-wave chemistry interactions in combustion theory is introduced in this article, which bears the analogous relationship to reacting gas flow as Burgers' equation does to ordinary compressible fluid flow.
Abstract: A qualitative model for studying shock-wave chemistry interactions in combustion theory is introduced. The model which we study bears the analogous relationship to reacting gas flow as Burgers’ equation does to ordinary compressible fluid flow. When the corresponding physical assumptions of the Chapman–Jouget and von Neumann–Zeldovich–Doring theories are introduced in this model, explicit and completely analogous phenomena occur. Without any approximations, combustion profiles with finite reaction rate and finite diffusion are examined in detail. In the context of this model, the validity of the approximate theories mentioned above depends on the relative size of two critical parameters—the energy liberated by chemical reaction and a parameter which measures the ratio of the width of the shock layer to the reaction zone.
167 citations
TL;DR: In this paper, an ordinary differential equation model of two species competing for a single essential nutrient in a periodic environment is studied, and the existence of the two species is demonstrated when the amplitude of the forcing is not too small nor too large.
Abstract: An ordinary differential equation model of two species competing for a single essential nutrient in a periodic environment is studied. Coexistence of the two species is demonstrated when the amplitude of the forcing is not too small nor too large. The results explain some of the results obtained from numerical simulation of chemostat experiments [9].
156 citations
TL;DR: Stochastic integrals of two types are introduced and studied for pth order processes and in particular for symmetric stable processes, where the “covariation” plays a role analogous to the covariance.
Abstract: This work extends to processes with finite moments of order $p,1 < p < 2$ and to symmetric $\alpha $-stable processes, $1 < \alpha < 2$, some of the basic linear theory known for processes with finite second moments $(p = 2)$ and for Gaussian processes $(\alpha = 2)$. Here the “covariation” plays a role analogous to the covariance. Specifically, stochastic integrals of two types are introduced and studied for pth order processes and in particular for symmetric stable processes. Regression estimates and linear estimates on certain symmetric stable processes are evaluated, including regression and linear filtering of signal in noise. Also, for certain symmetric stable inputs, the identification of a linear system from the input covariation and the input–output cross covariation is considered, and the way the distribution of the output depends on the linear system is studied.
115 citations
TL;DR: A new formulation is proposed for the problem of parameter estimation of dynamic systems with both process and measurement noise that gives estimates that are maximum likelihood asymptotically in time.
Abstract: A new formulation is proposed for the problem of parameter estimation of dynamic systems with both process and measurement noise. The formulation gives estimates that are maximum likelihood asymptotically in time. The means used to overcome the difficulties encountered by previous formulations are discussed. It is then shown how the proposed formulation can be efficiently implemented in a computer program. A computer program using the proposed formulation is available in a form suitable for routine application. Examples with simulated and real data are given to illustrate that the program works well.
99 citations
TL;DR: In this article, the authors studied the exploitative competition of two microorganisms for two complementary nutrients in the continuous culture, and the predicted biological conditions which should give rise to each of the possible competitive outcomes are presented in detail.
Abstract: This paper concerns the exploitative competition of two microorganisms for two complementary nutrients in the continuous culture. Consumption of the limiting resources follows the Holling Type II functional response or, equivalently, Michaelis–Menten kinetics, generalized to the two-resource situation. The predicted biological conditions which should give rise to each of the possible competitive outcomes are presented in detail and analyzed globally. A major conclusion is that each of the four outcomes of classical Lotka–Volterra two-species competition theory has multiple mechanistic origins in terms of consumer-resource interactions. It is also shown that all four classical outcomes, including the case in which winning depends on the initial abundances of the competitors, can arise for this purely exploitative competition. Moreover, the outcomes of this exploitative competition can be predicted, in advance of actual competition, from measurements made on each species grown by itself on the resources.
92 citations
TL;DR: In the continuous-time selection model for a single locus with several alleles, a stable polymorphism persists under mutation and global stability can be shown for two and three alleles.
Abstract: In the continuous-time selection model for a single locus with several alleles, a stable polymorphism persists under mutation. Global stability can be shown for two and three alleles.For the discrete-time case there follows existence and uniqueness of a polymorphism.
TL;DR: In this article, the diffusion process Z is studied in the context of heavy traffic theory for K-station networks of queues, with attention restricted to the case $k = 2$ for simplicity.
Abstract: Let $Z = \{ Z(t),t > 0\} $ be a reflected Brownian motion on the K-dimensional nonnegative orthant, with the direction of reflection constant over each boundary surface. Such processes arise in heavy traffic theory for K-station networks of queues. This paper continues our study of the diffusion process Z,with attention restricted to the case $k = 2$ for simplicity. (Most of the results extend directly to higher dimensions; our notation and style of argument are designed to suggest appropriate generalizations for arbitrary K wherever possible.) The backward equation for the transition density function (with boundary and initial conditions), the corresponding forward equation and the equation for the steady-state distribution are all derived informally. Also presented are various calculations relating to steady-state distributions, including a moment formula and the derivation of a condition (involving the drifts and directions of reflection), that we conjecture to be necessary and sufficient for existence...
TL;DR: In this paper, a system of reaction-diffusion equations describing the propagation of combustion waves along a thermally insulated cylindrical sample of solid fuel is considered, and uniform propagation of a plane combustion wave is subjected to a linear stability analysis.
Abstract: We consider a system of reaction-diffusion equations describing the propagation of combustion waves along a thermally insulated cylindrical sample of solid fuel. Uniform propagation of a plane combustion wave is subjected to a linear stability analysis. It is shown that, if the activation energy is sufficiently high and the diameter of the sample sufficiently large, then the experimentally observed spinning propagation of combustion waves appears as a Hopf-type bifurcation of the solution corresponding to a plane wave.The possibility of similar phenomena in gas combustion is discussed.
TL;DR: In this paper, the authors consider the motion of a Brownian particle in an infinite potential field, and the rate of approach to equilibrium is determined by the second eigenvalue of the stationary Fokker-Planck operator.
Abstract: We consider the motion of a Brownian particle in an infinite potential field. The rate of approach to equilibrium is determined by the second eigenvalue of the stationary Fokker–Planck operator. The inverse of this eigenvalue is the expected time for the particle to overcome the potential barriers on its way to the deepest potential well. The height of the largest potential barrier is termed the activation energy, and the eigenvalue is computed asymptotically for large activation energies. Applications to the calculation of chemical reaction rates and ionic conductance in crystals are given.
TL;DR: In this paper, the authors apply the techniques of canonical transforms to equations of the type (A(t)P2 +B(t){IPQ + UPI+ C(t),02 +D(t,Q +E(t)/P +F(t))I)f(q, t) = -iat,(q, t), where 0 and P are the quantum position and momentum operators.
Abstract: We apply the techniques of canonical transforms to equations of the type (A(t)P2 +B(t){IPQ + UPI+ C(t)02 +D(t)Q +E(t)P +F(t)I)f(q, t) = -iat,(q, t), where 0 and P are the quantum position and momentum operators. The time-dependent parameters of the W A SL (2, R) evolution operator are found through linear differential equations. In terms of these we give explicitly the Green's function, all separating coordinates and similarity solutions of the equation. We analyze the behavior of Gaussian and coherent-state initial conditions in closed form and present a new interpretation of all the Lewis-Riesenfeld constants of motion. 1. Introduction. There has been sustained interest in the description of quantum systems with time-dependent Hamiltonians. These systems have been used to model, for example, the motion of charged particles in time-dependent electromagnetic fields and coherent states in lasers. (See the list of references given in (3) and (11).) Gunther (5), (6) and Leach (9)-(14) have used time-dependent canonical trans- formations to reduce some of the above problems to time-independent ones, mainly for classical mechanics. They have been able to extend their methods to quantum systems for the cases when the canonical transformation is linear and real. In quantum mechanics, one has to be aware (9), (10) that not all Hamiltonians can be mapped meaningfully into each other, not even all quadratic ones: there exist distinct orbits in the vector space of the latter under the action of real linear canonical transformations. These orbits are characterized by (among other things) the spectrum of the operators in each equivalence class. Here, we take up their suggestion that the techniques of canonical transforms which we developed in (19), (20), (22), (24), (25) can be used to extend and simplify the analysis of differential equations of the type (l.l1a) H(t)o(q, t) =-iato(q, t), q, t (- ,
TL;DR: In this paper, a branch of periodic solutions which exhibit the alternatives of the global Hopf bifurcation theorem is calculated for two general systems of differential equations, i.e., systems of the Hopf type.
Abstract: Branches of periodic solutions which exhibit the alternatives of the global Hopf bifurcation theorem are calculated for two general systems of differential equations. In the first, a branch of solu...
TL;DR: In this paper, the Frank-Kamenetskii parameter was used to show that the temperature perturbation becomes unbounded as $t \to t_B \leqq \infty $ for values of the Frank Kamanetski parameter greater than the critical value.
Abstract: The equations describing the induction period process for a super-critical, high-activation energy thermal explosion in a bounded domain are studied. A formal proof, based on comparison techniques, is used to show that the temperature perturbation becomes unbounded as $t \to t_B \leqq \infty $ for values of the Frank-Kamenetski parameter $\delta $ greater than the critical value. Upper and lower bound estimates for $t_B $ are found by using specified comparison equations. A comparison of these bounds with values of $t_B $ obtained from numerical solutions of the basic equations shows that the estimates provide an excellent prediction of the escape time when $\delta $ is greater than 2–3 times the critical value. For smaller values of $\delta $, where heat loss is more significant, a better comparison equation is required.
TL;DR: The direct scattering problem of Zakharov and Shabat as discussed by the authors for the nonlinear Schrodinger equation is solved for the initial condition $u = {\text{sech}} x$ and the asymptotic solution is a bound state of N centered solitons (all moving at the same speed).
Abstract: The direct scattering problem of Zakharov and Shabat for the nonlinear Schrodinger equation $iu_t + u_{xx} + \kappa | u |^2 u = 0$ is solved for the initial condition $u = {\text{sech}} x$ The asymptotic solution $(t \uparrow \infty )$ is a “bound state” of N centered solitons (all moving at the same speed), where \[N\] is the integral part of $(\kappa /2)^{1/2} $
TL;DR: In this paper, a simple model based on the FitzHugh equations is developed to simulate the phenomenon of recurrent neural feedback, which occurs when a neuron excites a second neuron which in turn excites or inhibits the first neuron.
Abstract: A very simple model based on the FitzHugh equations is developed to simulate the phenomenon of recurrent neural feedback. This phenomenon, which is ubiquitous in the vertebrate nervous system, occurs when a neuron excites a second neuron which in turn excites or inhibits the first neuron. Since the excitation or inhibition occurs only after conduction and synaptic delays, the model involves a system of differential-difference equations. Conditions for the existence of a Hopf bifurcation are derived, and formulas for the stability of the bifurcation are given. Some numerical results for large amplitude solutions are presented. A discussion of the applicability of the model is given.
TL;DR: In this article, a nonlinear diffusion equation with a power law diffusion coefficient is studied, and the theory of homology is introduced and used to generate classes of equations, the solutions of which are related through Backlund transformations.
Abstract: A nonlinear diffusion equation with a power law diffusion coefficient is studied. Self-similar and partially invariant solutions are shown to be identical. The theory of homology is introduced and used to generate classes of equations, the solutions of which are related through Backlund transformations. An example of an application is given, which generalizes the solution of Storm (J. Appl. Phys., 22 (1951) pp. 940–951) on heat conduction in metals.
TL;DR: In this article, the authors employ a model for an edge-cooled flat flame burner to obtain expressions for the flame speed, flame temperature, standoff distance as well as the quenching distance for a plane flame front.
Abstract: We employ a model for an edge-cooled flat flame burner to obtain expressions for the flame speed, flame temperature, standoff distance as well as the quenching distance for a plane flame front.For a given standoff distance there is a low-temperature as well as a high-temperature solution. We show by a linear stability analysis of the plane front that the high-temperature solution is unstable when the Lewis number is sufficiently large and the inflow velocity sufficiently less than the adiabatic flame speed. We also show that this instability is the type that will lead to a bifurcating time-periodic solution describing a pulsating flame.
TL;DR: The concept of Liouville distribution of the first and second kind provides respective generalizations to the multivariate gamma distribution and the Dirichlet distribution as mentioned in this paper, and these new classes of multivariate distributions defined by functional forms can be used to generate some well-known statistical distributions.
Abstract: The concept of Liouville distribution of the first and second kind provides respective generalizations to the multivariate gamma distribution and the Dirichlet distribution. It is shown that these new classes of multivariate distributions defined by functional forms can be used to generate some well-known statistical distributions.
TL;DR: In this paper, the problem of identifying an unknown diffusion coefficient in a nonlinear diffusion equation from overspecified data measured on the boundary is considered and conditions ensuring compatibility of the data are derived.
Abstract: We consider the problem of identifying an unknown diffusion coefficient in a nonlinear diffusion equation from overspecified data measured on the boundary. Conditions ensuring compatibility of the data are derived and it is shown that the mapping which carries the diffusion coefficient into the data is isotonic. This result is then used to derive a uniqueness result for the inverse problem and for an associated identification problem. Some results of numerical experiments are presented to illustrate the various results.
TL;DR: In this paper, the authors describe the dynamics of transition from the extinguished to the ignited state as the reaction-rate parameter is slowly varied through the critical value, and the asymptotic analysis is based on the largeness of two parameters, one characterizing the activation energy and the other the slow passage.
Abstract: Reactive systems involving Arrhenius kinetics often exhibit multiple steady states A typical response is an S-curve, whose turnaround points correspond to ignition or extinction This paper describes the dynamics of transition from the extinguished to the ignited state as the reaction-rate parameter is slowly varied through the critical value Both lumped and spatially distributed models are studied The asymptotic analysis is based on the largeness of two parameters, one characterizing the activation energy and the other the slow passage
TL;DR: In this paper, the joint density for location of the object sought and unsuccessful search is derived for the case of constant diffusion and drift parameters and piecewise linear searching paths, and the interpretation of the terms in the approximate solution is discussed.
Abstract: After a brief discussion of the operational origin of search problems, the mathematical problem is formulated. The mathematical quantity of interest is the joint density for location of the object sought and unsuccessful search. When the object moves according to a diffusion process, this joint density satisfies a parabolic equation. After the introduction of scaled variables, the search equation can be approximately solved by the “ray method”. The interpretation of the terms in the approximate solution is discussed. The case of constant diffusion and drift parameters and piecewise linear searching paths arises often in operational situations. This case is considered in detail.
TL;DR: In this article, the waiting time density of a partially correlated generalization of $M/M/1 $ is shown to have the hyperexponential distribution, which can be constructed in many ways, including a modified Bessel function of order zero.
Abstract: Aligning service mechanism and demand is achieved essentially in two ways: either service and/or arrival parameters are managed to vary with system state, or consecutive inter-arrival intervals and service times are not assumed to be independent. The former is by now well studied. In the latter, a bivariate distribution with negative exponential marginals, which can be constructed in many ways, constitutes a first attempt. With a particular construction involving a modified Bessel function of order zero, the waiting time density (as well as its stationary counterpart) of such a partially correlated generalization of $M/ M/1 $ is shown to have the hyperexponential distribution.
TL;DR: In this article, the problem of determining the surface impedance of an obstacle from measurements of the far field pattern of a scattered time harmonic acoustic wave is considered, and a constructive method for determining the impedance is given through the use of the Backus-Gilbert method for solving improperly posed moment problems.
Abstract: We consider the problem of determining the surface impedance of an obstacle from measurements of the far field pattern of a scattered time harmonic acoustic wave. It is shown that this problem can be stabilized by assuming that the impedance is Holder continuous with an a priori bound given on the impedance and its Holder coefficient. A constructive method for determining the impedance is given through the use of the Backus–Gilbert method for solving improperly posed moment problems.
TL;DR: In this article, the authors describe the extremely rapid transient phase of a thermal explosion in a vessel with constant wall temperature, where a nearly inert, cool, reactant-rich conduction-controlled zone and a highly localized veritable fireball controlled by chemical reaction rates are separated by a distinct zone of time-invariant spatial structure.
Abstract: The extremely rapid transient phase of a thermal explosion in a vessel with constant wall temperature is described. The solution is developed in terms of a nearly-inert, cool, reactant-rich conduction-controlled zone and a highly localized veritable fireball controlled by chemical reaction rates, which are separated by a distinct zone of time-invariant spatial structure. In the first region the temperature is only slightly different from the initial value. In contrast, the variation in temperature in the fireball is large, ranging almost to the adiabatic explosion value. The rise in temperature is accompanied by significant fuel consumption. When the fireball formation is nearly complete, the fuel available therein is vanishingly small. The theoretical formulation provides a specific time scale for the rapid transient process, which is extremely short in comparison to that in the induction period which precedes the explosive event. The fireball size, found to depend on the material properties of the combu...
TL;DR: In this article, a linear stability analysis is carried out for the cellular instability of a nonadiabatic downward-propagating premixed flame, and it is shown that if the molecular weight of the deficient reactant is sufficiently small, then an increase in heat loss may lead to destabilization of an adiabatically stable flame (i.e., a flame which is stable in the absence of heat losses).
Abstract: A linear stability analysis is carried out for the cellular instability of a nonadiabatic downward-propagating premixed flame. It is shown that if the molecular weight of the deficient reactant is sufficiently small, then an increase in heat loss may lead to destabilization of an adiabatically stable flame (i.e., a flame which is stable in the absence of heat losses).
TL;DR: In this article, the Laplace transform of the first passage time density satisfies an ordinary differential equation in a finite interval with two point boundary conditions, and a formal asymptotic solution to this problem was constructed when the singular boundary point is absorbing and the regular boundary point was either absorbing or reflecting.
Abstract: The first passage problem is studied for a singular diffusion process arising in population biology with the deterministic part having a stable equilibrium point and small diffusion. The Laplace transform of the first passage time density satisfies an ordinary differential equation in a finite interval with two point boundary conditions. A formal asymptotic solution to this problem is constructed when the singular boundary point is absorbing and the regular boundary point is either absorbing or reflecting. The solution depends upon the initial value z of the process. For z near the equilibrium point the density is approximated by the Ornstein–Uhlenbeck equation, for z near the singular boundary point it is approximated by the diffusion limit of the linear birth-death process, and elsewhere it is approximated by the WKB solution. These different solutions are then connected together. Using these results, the process is shown to exit with probability one in the limit of small diffusion. The mean and varianc...