AbstractWe shall study continuous-time Markov chains on the nonnegative integers which are both irreducible and transient, and which exhibit discernible stationarity before drift to infinity “sets in”. We will show how this ‘quasi’ stationary behaviour can be modelled using a limiting conditional distribution: specifically, the limiting state probabilities conditional on not having left 0 for the last time. By way of adualchain, obtained by killing the original process on last exit from 0, we invoke the theory of quasistationarity forabsorbingMarkov chains. We prove that the conditioned state probabilities of the original chain are equal to the state probabilities of its dual conditioned on non-absorption, thus allowing to establish the simultaneous existence and then equivalence, of their limiting conditional distributions. Although a limiting conditional distribution for the dual chain is always quasistationary distribution in the usual sense, a similar statement is not possible for the original chain.
In this article we consider modified search directions in the endgame of interior point methods for linear programming. In this stage, the normal equations determining the search directions become ill-conditioned. The modified search directions are computed by solving perturbed systems in which the systems may be solved efficiently by the preconditioned conjugate gradient solver. A variation of Cholesky factorization is presented for computing a better preconditioner when the normal equations are ill-conditioned. These ideas have been implemented successfully and the numerical results show that the algorithms enhance the performance of the preconditioned conjugate gradients-based interior point methods.
The Shannon system is generalized and the expansion of a function in the generalized Shannon system is considered. No study of a wavelet expansion exists without the assumption of 'fast' decay of wavelets. The wavelet psi which is associated with the generalized Shannon system has a 'slow' decay. The expansion of a function in the system is shown to converge at a point which satisfies the Lipschitz condition of order alpha> 0. On the other hand, there is a continuous function whose wavelet expansion in the generalized Shannon system diverges. An observation of Gibbs' phenomenon is also given.
This paper examines the predictions of shallow water theory for steady and unsteady withdrawal flows through an extended sink from fluid of finite depth. Two-dimensional plane flows and three-dimensional axi-symmetric flow through a circular drain are examined. Shallow water theory indicates the presence of limiting configurations, where the surface of the fluid collapses directly into the sink. In addition, this theory suggests that some previously computed steady solutions may be unstable.
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
AbstractSimple chemical reactions can be described by the Michaelis-Menten response curve relating the velocity V of the reaction and the concentration [S] of the substrate S. To handle more complicated reactions without introducing general polynomial response curves, the rate constants can be considered to be scale dependent. This leads to a new response curve with characteristic sigmoidal shape. But not all sigmoidal curves can be accurately fit with three parameters. In order to get an accurate fit, the lower part of the ∫ shaped curve cannot be too shallow and the upper part can't be too steep. This paper determines an exact mathematical expression for the steepness and shallowness allowed.
Abstract Let Δ denote a triangulation of a planar polygon Ω. For any positive integer 0 ≤ r < k, let denote the vector space of functions in Cr whose restrictions to each triangle of Δ are polynomials of total degree at most k. Such spaces, called bivariate spline spaces, have many applications in surface fitting, scattered data interpolation, function approximation and numerical solutions of partial differential equations. An important problem is to give the function expression. In this paper, we prove that, if (Δ, Ω) is type-X, then any bivariate spline function in can be expressed by a series of univariate polynomials and a special bivariate finite element function in satisfying a so-called integral conformality condition system. We also give a direct sum decomposition of the space . In addition, the dimension of for a kind of triangulation has been determined.
Abstract A three stage procedure for the analysis and least-cost design of looped water distribution networks is considered in this paper. The first stage detects spanning trees and identifies the true global optimum for the system. The second stage determines hydraulically feasible pipe flows for the network by the numerical solution of a set of non-linear simultaneous equations and shows that these solutions are contained within closed convex polygonal regions in the solution space bounded by singularities resulting from zero flows in individual pipes. Ideal pipe diameters, consistent with the pipe flows and the constant velocity constraint adopted to prevent the system degenerating into a branched network, are selected and costed. It is found that the most favourable optimum is in the vicinity of a vertex in the solution space corresponding to the minimum spanning tree. In the third stage, commercial pipes are specified and the design finalised. Upper bound formulae for the number of spanning trees and hydraulically feasible solutions in a network have also been proposed. The treatment of large networks by a heuristic procedure is described which is shown to result in significant economies compared with designs obtained by non-linear programming.
AbstractAn inequality involving the logarithmic mean is established. Specifically, we show thatwhere . Then several generalizations are given.
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
Consider the forced differential equation with variable delayx'(t) + b(t)x(t - tau(t)) = f(t), t greater than or equal to 0,wheref is an element of C([0, infinity)) and b, tau is an element of C([0, infinity), [0, infinity)).We establish a sufficient condition for every solution to tend to zero. We also obtain a sharper condition for every solution to tend to zero when integral(tau-tau(t))(t) b(s)ds is asymptotically constant.
AbstractA local existence and uniqueness result is proved for the three-dimensional Euler-Poisson system without a pressure term which arises in plasma physics.
By applying Laplace transform theory to solve first-order homogeneous differential-difference equations it is conjectured that a resulting infinite sum of a series may be expressed in closed form. The technique used in obtaining a series in closed form is then applied to other examples in teletraffic theory and renewal processes.
Exact solutions are developed for instantaneous point sources subject to nonlinear diffusion and loss or gain proportional to nth power of concentration, With n > 1. The solutions for the loss give, at large times, power-law decrease to zero of slug central concentration and logarithmic increase of slug semi-width. Those for gain give concentration decreasing initially, going through a minimum, and then increasing, with blow-up to infinite concentration in finite time. Slug semi-width increases with time to la finite maximum in finite time at a blow-up. Taken in conjunction with previous studies,:these new results provide an overall schema for instantaneous nonlinear diffusion point sources with nonlinear loss or gain for the total range n greater than or equal to 0. Six distinct regimes of behaviour of slug semi-width and concentration are identified, depending on the range of n, 0 less than or equal to n < 1, n = 1, or n > 1. Three of them are for loss, and three for gain. The classical Barenblatt-Pattle nonlinear instantaneous point-source solutions with material concentration occupy a central place in the total schema.
A general framework is developed for constructing higher order spectral refinement schemes for a simple eigenvalue. Well-known techniques for ordinary spectral refinement are carried over to higher order spectral refinement yielding faster rate's of convergence. Numerical examples are given by considering an integral operator.
Recently, several papers [2–4, 6] have been published concerning a pursuit problem which was apparently first posed explicitly by Leonardo da Vinci and which may have been present in earlier thinking about kinematics and geometry. Falconry appears to go back, in Europe, to the days of Pliny, Aristotle and Martial, and, in Asia, to 2000 BC [5].
RAFIKUL ALAM, REKHA P. KULKARNI and BALMOHAN V. LIMAYE: Accelerated spectral refinement. Part I: Simple eigenvalue 487 J. E. ALDRIDGE: See G. E. PRINCE I. ALI and S. KALLA: A generalized Hankel transform and its use for solving certain partial differential equations 105 MALCOLM ANDERSON: Near-field expansion of the metric due to a cosmic string 180 MALCOLM R. ANDERSON: See TZE-CHUEN TOH K. BALACHANDRAN: See R. SUBRAMANIAM J. C. BARTON and C. J. ELIEZER: On pursuit curves 358 CHARLES BU: Forced cubic Schrodinger equation with Robin boundary data: continuous dependency result 301 G. B. BYRNES: See G. E. PRINCE REYNALDO CASTILLO: See WAI KIN CHAN P. CERONE and A. SOFO: Summing series arising from integro-differential-difference equations 473 WAI KIN CHAN, REYNALDO CASTILLO and KING FAI LAI: Foliations in supergravity 161 E. W. M. CHOW and A. W.-C. LUN: Apparent horizons in vacuum Robinson-Trautman spacetimes 217 PAULINE COOLEN-SCHRIJNER, ANDREW HART and PHIL POLLETT: Quasistationarity of continuous-time Markov chains with positive drift 423 N. ELEZOVIC, M. MATIC, C. E. M. PEARCE and J. PECARIC: On two lemmas of Brown and Shepp having application to sum sets and fractals, III 329 C. J. ELIEZER: See J. C. BARTON J. F. Q. FERNANDES and A. W.-C. LUN: Integrability conditions for the Bianchi identities as transformations in Schwarzschild space-time 260 L. K. FORBES: See A. J. KOERBER HANG GAO and XUNJING LI: Necessary conditions for optimal control of elliptic systems 542 S. E. GODFREY: See G. E. PRINCE J. R. GRAEF and C. QIAN: Global attractivity in differential equations with variable delays 568 ANDREW HART: See PAULINE COOLEN-SCHRIJNER JACK HEIDEL and JOHN MALONEY: An analysis of a fractal Michaelis-Menten curve 410 JACK HEIDEL and JOHN MALONEY: When can sigmoidal data be fit to a Hill curve? 83 JACK HEIDEL: See JOHN MALONEY L. D. HIRD, P. F. SIEW and S. WANG: A numerical solution to the flow between eccentric rotating cylinders with a slotted sleeve 129 S. KALLA: See I. ALI HONG OH KIM and JONG HA PARK: The generalized Shannon system in wavelet space 386 JURGEN KLENK: Existence of stationary vacuum solutions of Einstein's equations in an exterior domain 231
The averaging of Hamilton-Jacobi equation with fast variables in viscosity solution sense in infinite dimensions is studied. It is proved that viscosity solution of the original equation converges to viscosity solution of the averaged equation and this result is applied to study the limit problem of the value function for optimal control problem with fast variables.
Abstract We improve some results of [17], which relate to key tools given in [7] for establishing canonical inequalities used in the analysis of sum sets and fractals.
This paper studies a system proposed by K. Gopalsamy and P. X. Weng to model a population growth with feedback control and time delays. Sufficient conditions are established under which the positive equilibrium of the system is globally attracting. The conjecture proposed by Gopalsamy and Weng is here confirmed and improved.
