Extinction times for a birth–death process with weak competition

We consider a birth–death process with birth rates iλ and death rates iμ+i(i?1)θ, where i is the current state of the process. A positive competition rate θ is assumed to be small. In the supercritical case where λ > μ, this process can be viewed as a demographic model for a population with high carrying capacity around (λ?μ)/θ. The article reports in a self-contained manner on the asymptotic properties of the time to extinction for this logistic branching process as θ → 0. All three reproduction regimes λ > μ, λ < μ, and λ = μ are studied.     

A polynomial–exponential equation related to the Ramanujan–Nagell equation

For any given odd prime p and a fixed positive integer D prime to p, we study the equation \(x^2+D^m=p^n\) in positive integers x, m and n. We use a classical work of Dem’janenko in 1965 on a certain quadratic Diophantine equation together with some results concerning the existence of primitive divisors of Lucas sequences to examine our equation when D is a product of \(p-1\) and a square.     

A new approach to the Assad–Kirk fixed point theorem

In the present paper, considering the Wardowski’s technique, we give a new approach to the Assad–Kirk fixed point theorem on metrically convex metric spaces.We also provide a nontrivial example showing that our result is a proper extension of the Assad–Kirk fixed point theorem.     

Polynomial birth–death processes and the second conjecture of Valent

The conjecture of Valent about the type of Jacobi matrices with polynomially growing weights is proved.     

A Geometric approximation to the euler equations: the Vlasov–Monge–Ampère system

This paper studies the Vlasov–Monge–Ampère system (VMA), a fully non-linear version of the Vlasov–Poisson system (VP) where the (real) Monge–Ampère equation det ²/x_i x_j = substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak solutions and local existence of smooth solutions are obtained. Links between the VMA system, the VP system and the Euler equations are established through rigorous asymptotic analysis.     

A bivariate non-homogeneous birth and death model for predator–prey interactions

We propose a bivariate non-homogeneous birth and death process as a model for predator–prey interactions. Its expectation is periodic, as it is a solution to the classical Lotka–Volterra system. Moreover, the mean age at extinction, as defined in Kendall (1948), is infinite.     

Compressible Navier–Stokes approximation to the Boltzmann equation

Even though the system of the compressible Navier–Stokes equations is not a limiting system of the Boltzmann equation when the Knudsen number tends to zero, it is the second order approximation by applying the Chapman–Enskog expansion. The purpose of this paper is to justify this approximation rigorously in mathematics. That is, if the difference between the initial data for the compressible Navier–Stokes equations and the Boltzmann equation is of the second order of the Knudsen number, so is the difference between two solutions for all time. The analysis is based on a refined energy method for a fluid-type system using the techniques for the system of viscous conservation laws.     

A Modica–Mortola approximation for the Steiner Problem

In this note we present a way to approximate the Steiner Problem by a family of elliptic energies of Modica–Mortola type, with an additional term relying on a weighted geodesic distance which takes care of the connectedness constraint.     

A cubic defining algebra for the Links–Gould polynomial

We define a finite-dimensional cubic quotient of the group algebra of the braid group, endowed with a (essentially unique) Markov trace which affords the Links–Gould invariant of knots and links. We investigate several of its properties, and state several conjectures about its structure.     

Irrotational approximation to the one-dimensional bipolar Euler–Poisson system

Under the assumptions that initial data have sufficiently small total variation and that the initial data are supersonic (or are subsonic respectively), we prove that in any bounded domain the L¹$L^1$ norm of the difference between the local solutions of the one-dimensional bipolar Euler–Poisson system and the potential flow system of the one-dimensional bipolar Euler–Poisson system with the same initial data can be bounded by the cube of the total variation of the initial data.     

A Signed Generalization of the Bernoulli–Laplace Diffusion Model

We bound the rate of convergence to stationarity for a signed generalization of the Bernoulli–Laplace diffusion model; this signed generalization is a Markov chain on the homogeneous space ( ₂S _n )/(S _r ×S _n–r ). Specifically, for r not too far from n/2, we determine that, to first order in n, n log n steps are both necessary and sufficient for total variation distance to become small. Moreover, for r not too far from n/2, we show that our signed generalization also exhibits the cutoff phenomenon.     

On the cycle maximum of birth–death processes and networks of queues

This paper considers the cycle maximum in birth–death processes as a stepping stone to characterisation of the asymptotic behaviour of the maximum number of customers in single queues and open Kelly–Whittle networks of queues. For positive recurrent birth–death processes we show that the sequence of sample maxima is stochastically compact. For transient birth–death processes we show that the sequence of sample maxima conditioned on the maximum being finite is stochastically compact.We show that the Markov chain recording the total number of customers in a Kelly–Whittle network is a birth–death process with birth and death rates determined by the normalising constants in a suitably defined sequence of closed networks. Explicit or asymptotic expressions for these normalising constants allow asymptotic evaluation of the birth and death rates, which, in turn, allows characterisation of the cycle maximum in a single busy cycle, and convergence of the sequence of sample maxima for Kelly–Whittle networks of queues.     

A polynomial time interior–point path–following algorithm for LCP based on Chen–Harker–Kanzow smoothing techniques

$$Osqrt{n}L$$ ); otherwise, the complexity bound is O(nL). The relations between our search direction and the one used in the standard interior-point algorithm are also discussed. Received September 9, 1996 / Revised version received December 22, 1997? Published online March 16, 1999     

A Charlier–Parseval approach to poisson approximation and its applications

We propose a new approach to Poisson approximation. The basic idea is very simple and based on properties of the Charlier polynomials and the Parseval identity. Such an approach quickly leads to new effective bounds for several Poisson approximation problems. We also give a selected survey on diverse Poisson approximation results.     

An algebraic approach to finding the Fermat–Torricelli point

Using a calculus and an algebraic approach, the Cartesian coordinates of the Fermat–Torricelli point are deduced for triangles with no internal angle greater than 120°. Although in theory, the deduction of these coordinates could be made ‘by hand’, in practice it is very laborious to obtain them without the aid of mathematical computer software, but with human guidance, since there are mathematical artifices not yet incorporated into the software. It is also shown that these coordinates can be conveniently expressed in terms of the side lengths and the area of the triangle. These coordinates are contrasted with the coordinates of a similar point: one whose sum of the squares of the distances to the vertices of an arbitrary triangle is a minimum.     

A particle approximation of the solution of the Kushner–Stratonovitch equation

We construct a sequence of branching particle systems α _n convergent in measure to the solution of the Kushner–Stratonovitch equation. The algorithm based on this result can be used to solve numerically the filtering problem. We prove that the rate of convergence of the algorithm is of order n ^?. This paper is the third in a sequence, and represents the most efficient algorithm we have identified so far. Received: 4 February 1997 / Revised version: 26 October 1998