We present several new inequalities for Euler's beta function, B(x, y). One of our results states that the beta function can be approximated on (0, 1] × (0, 1] by rational functions as follows, with the best possible constants α = 1 and β = ⅔π2 − 4 = 2.579 73 ….

We derive Povzner–Wienholtz-type self-adjointness results for m × m matrix-valued Sturm–Liouville operators in L2((a, b);R dx)m, m ∈ N, for (a, b) a half-line or R.

This paper is devoted to the proof of the well posedness of a class of ordinary differential equations (ODEs). The vector field depends on the solution to a scalar conservation law. Forward uniqueness of Filippov solutions is obtained, as well as their Hölder continuous dependence on the initial data of the ODE. Furthermore, we prove the continuous dependence in C0 of the solution to the ODE from the initial data of the conservation law in L1.

This problem is motivated by a model of traffic flow.

This paper studies the overall evolution of fronts propagating with a normal velocity that depends on position, υn = f(x), where f is rapidly oscillating and periodic. A level-set formulation is used to rewrite this problem as the periodic homogenization of a Hamilton–Jacobi equation. The paper presents a series of variational characterization (formulae) of the effective Hamiltonian or effective normal velocity. It also examines the situation when f changes sign.

Uniform asymptotic expansions are obtained for the associated Legendre functions and , and the Ferrers functions and , as the order μ → ∞. The approximations are uniformly valid for 0 ≤ ν + ½ ≤ μ(1 − δ), where δ ∈ (0, 1) is fixed, x ∈ (−1, 1) in the real-variable case and Re z ≥ 0 in the complex-variable case. Explicit error bounds are available for all approximations. In the complex-variable case, expansions are obtained by an application of two existing general asymptotic theories to the associated Legendre differential equation: the first case (in which ν is fixed) applies to regions containing an isolated simple pole; and the second case (in which 0 ≤ ν + ½ ≤ μ(1 − δ)) applies to regions containing a coalescing turning point and double pole. In both cases, the expansions involve modified Bessel functions. In the real-variable case (in which 0 ≤ ν + ½ ≤ μ(1 − δ)), asymptotic expansions of Liouville–Green type are obtained, which involve elementary functions.

There is a large catalogue of decompositions of conditioned superprocesses in terms of an ‘immortal backbone’ or ‘skeleton’ along the branches of which mass is constantly immigrated. We add to this with a study of the (infinite-variance) (1 + β)-superprocess, conditioned on survival until some fixed time T. As one would expect, we see a Poisson number of immortal trees (conditioned on there being at least one), along which mass (conditioned to die before time T) is immigrated. However, here we see a new source of immigration. Not only is mass immigrated along the branches of the immortal trees, but also there is an extra burst of immigration whenever the immortal tree branches. Moreover, the rate of immigration along the branches is no longer deterministic. In the limit as T → ∞, the immortal trees degenerate to the Evans immortal particle and the immigration (of unconditioned mass) along the particle is dictated by a stable subordinator.

A sequence of integers S is called Glasner if, given any ε > 0 and any infinite subset A of T = R/Z, and given y in T, we can find an integer n ∈ S such that there is an element of {nx : x ∈ A} whose distance to y is not greater than ε. In this paper we show that if a sequence of integers is uniformly distributed in the Bohr compactification of the integers, then it is also Glasner. The theorem is proved in a quantitative form.

In this paper, we prove that the dimension of the space of bounded energy-finite solutions for the Schrödinger operator is invariant under rough isometries between complete Riemannian manifolds satisfying the local volume condition, the local Poincaré inequality and the local Sobolev inequality. We also prove that the dimension of the space of bounded harmonic functions with finite Dirichlet integral is invariant under rough isometries between complete Riemannian manifolds satisfying the same local conditions. These results generalize those of Kanai, Grigor'yan, the second author, and Li and Tam.

We consider random perturbations of two-dimensional Navier–Stokes equations. Under some natural conditions on random forces, we study asymptotic properties of solutions and stationary measures.

The paper deals with linear pencils N − λP of ordinary differential operators on a finite interval with λ-dependent boundary conditions. Three different problems of this form arising in elasticity and hydrodynamics are considered. So-called linearization pairs (W, T) are constructed for the problems in question. More precisely, functional spaces W densely embedded in L2 and linear operators T acting in W are constructed such that the eigenvalues and the eigen- and associated functions of T coincide with those of the original problems. The spectral properties of the linearized operators T are studied. In particular, it is proved that the eigen- and associated functions of all linearizations (and hence of the corresponding original problems) form Riesz bases in the spaces W and in other spaces which are obtained by interpolation between D(T) and W.

Ratio-dependent predator–prey models are favoured by many animal ecologists recently as they better describe predator–prey interactions where predation involves a searching process. When densities of prey and predator are spatially homogeneous, the so-called Michaelis–Menten ratio-dependent predator–prey system, which is an ordinary differential system, has been studied by many authors. The present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous Neumann boundary condition. Its main purpose is to study qualitative properties of solutions to this reaction-diffusion (partial differential) system. In particular, we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary-differential-equation dynamics, non-constant positive steady states exist for the partial-differential-equation model. This demonstrates that stationary patterns arise as a result of diffusion.

We study a connection between the L2 average decay of the Fourier transform of functions with respect to a given measure and the Hausdorff behaviour of that measure.

This paper deals with scalar delay differential equations with dominant delayed terms. Sufficient conditions are obtained for uniform stability, uniformly asymptotic stability and globally asymptotic stability of the equations. The criteria extend and improve some existing ones. The main results are applied to two physiological models. Some counterexamples are also given to show the invalidity of some existing results.

An initial–boundary-value problem for the nonlinear equations of real compressible viscous heat-conducting flow with general large initial data is investigated. The main point is to study the real flow for which the pressure and internal energy have nonlinear dependence on temperature, unlike the linear dependence for ideal flow, and the viscosity coefficients and heat conductivity are also functions of density and/or temperature. The shear viscosity is also presented. The existence, uniqueness and regularity of global solutions are established with large initial data in H1. It is shown that there is no shock wave, vacuum, mass concentration, or heat concentration (hot spots) developed in a finite time, although the solutions have large oscillations.