A new class of integrals involving the confluent hypergeometric function 1F1(a;c;z) and the Riemann Ξ-function is considered. It generalizes a class containing some integrals of Ramanujan, Hardy and Ferrar and gives, as by-products, transformation formulae of the form F(z, α) = F(iz, β), where αβ = 1. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulae of Ferrar and Hardy. A one-variable generalization of a well-known identity of Ramanujan is also given. We conclude with a generalization of a conjecture due to Ramanujan, Hardy and Littlewood involving infinite series of the Möbius function.

In this paper we present a result of existence of infinitely many arbitrarily small positive solutions to the following Dirichlet problem involving the p-Laplacian, where Ω ∈ RN is a bounded open set with sufficiently smooth boundary ∂Ω, p > 1, λ > 0, and f: Ω × R → R is a Carathéodory function satisfying the following condition: there exists t̄ > 0 such that Precisely, our result ensures the existence of a sequence of a.e. positive weak solutions to the above problem, converging to zero in L∞(Ω).

This paper provides a topological and ergodic analysis of random linear Hamiltonian systems. We consider a class of Hamiltonian equations presenting absolutely continuous dynamics and prove the existence of the radial limits of the Weyl M-functions in the L1-topology. The proof is based on previous ergodic relations obtained for the Floquet coefficient. The second part of the paper is devoted to the qualitative description of disconjugate linear Hamiltonian equations. We show that the principal solutions at ±∞ define singular ergodic measures, and determine an invariant region in the Lagrange bundle which concentrates the essential dynamical information. We apply this theory to the study of the n-dimensional Schrödinger equation at the first point of the spectrum.

The method of artificial insemination in the mouse, which has been used successfully only once previously (Mark and Long, 1911), has made it possible to collect exact data on the duration of life and the fertilising capacity of spermatozoa in the female genital tract. Earlier results concerning the conditions under which spermatozoa from the male genital tract attain their maximum activity (Merton, II) and the exact knowledge of the time of parturition (Merton, I) were helpful in carrying out artificial insemination during the following œstrous period.

In this paper we consider mathematical models inspired by the mechanisms of biological evolution. We take populations which are subject to interaction and mutation. In the cases we consider, the interaction is through competition or through a prey-predator relationship. The models consider the specific characteristics as taking values in real intervals and the equations are of the integro—differential type. In the case of competition, thanks to the fact that some of the equations have solutions which are quite explicit, we succeed in proving the existence of attracting stationary solutions. In the case of prey and predator, using techniques of dynamical systems in infinite-dimensional spaces, we succeed in showing the existence of a global attractor, which in some instances reduces to a point. Our analysis takes into account having δ distributions, corresponding to all individuals having the same characteristics, as possible populations.

This paper is concerned with the existence of topological multivortex solutions in (2 + 1) self-dual gauge theories such as the classical abelian Higgs model or the Chern–Simons Higgs gauge theories. A general form of topological multivortex equations is presented with the inclusion of antivortices. Two kinds of solutions are considered; topological vortex solutions and topological vortex–antivortex solutions. We construct these solutions by super and subsolution methods, and derive the exponential decay of solutions at infinity and the quantized integral formula. As an application, we prove the existence of a topological multivortex solutions in a generalized Chern–Simons Higgs theory.

Limit-point and limit-circle criteria are given for the generalised Sturm-Liouville differential expression

where

(i) p, q, and w are real-valued on [a, b),

(ii) p−1, q, w are locally Lebesgue integrable on [a, b),

(iii) w > 0 almost everywhere on [a, b) and the principal coefficient p is allowed toassume both positive and negative values.

For regular symmetric ordinary differential expressions we show that (i) the minimal operator is bounded below and (ii) the Friedrichs extension is determined by Dirichlet boundary conditions. Both proofs are based on elementary inequalities.

The subject of this paper is steady motion of vortices.

1. Extended definition of “steady motion.” The motion of any system of solid or fluid or solid and fluid matter is said to be steady when its configuration remains equal and similar, and the velocities of homologous particles equal, however the configuration may move in space, and however distant individual material particles may at one time be from the points homologous to their positions at another time.

We study the existence and the asymptotic behaviour of global solutions of the semilinear parabolic equation u(0) = ϧwhere a, b ∈ℝ, q > 1, p > 1. Forq=2p/(p+1) and ½ 1(p-1)>1 (equivalently, q > (n + 2)/(n + 1)), we prove the existence of mild global solutions for small initial data with respect to some norm. Some of those solutions are asymptotically self-similar.

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

Abnormalities of the digits occur under a great variety of forms, and with such frequency that most medical men interested in the subject are more or less familiar with them.

In studying such cases one cannot fail to be struck with the fact that, although the tendency to abnormality may be inherited, the inheritance is not always “true,” i.e. the precise form is not accurately reproduced in the offspring. Especially, perhaps, is this the case where the abnormality consists in a deficiency of digits. One individual of a family may have four fingers, another three, and another two, or even one, and that one very imperfect. Such cases have often been recorded (see B.M.J., 1886, vol. ii. p. 976).

We study the problem of the rigorous derivation of one-dimensional models for a thin curved beam starting from three-dimensional nonlinear elasticity. We describe the limiting models obtained for different scalings of the energy. In particular, we prove that the limit functional corresponding to higher scalings coincides with the one derived by dimension reduction starting from linearized elasticity.

We study the periodic and Neumann boundary-value problems associated with the second-order nonlinear differential equation

where is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when

and λ > 0 is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.

Contracted inverse semigroup rings are studied subject to the restriction that the semilattices of the given inverse semigroups satisfy a certain finiteness condition, introduced in 1980 by Teply, Turman and Quesada. Results are obtained on semiprimitivity, primitivity, primeness, decomposition into a direct sum of ideals, and chain conditions on one-sided ideals.

We study weak solutions for a class of free-boundary problems which includes as a special case the classical problem of travelling gravity waves on water of finite depth. We show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions. We also prove that in very general situations the free boundary is necessarily the graph of a function.

When a sheet of paper is crumpled in the hands and then crushed flat against a desk-top, the pattern of creases so formed is governed by certain simple rules. These rules generalize to theorems on folding Riemannian manifolds isometrically into one another. The most interesting results apply to the case in which domain and codomain have the same dimension. The main technique of proof combines the notion of volume with Hopf's concept of the degree of a map.

This article is concerned with eigenvalue problems of the form Au = λTu in a Hilbert space H, where Ais a selfadjoint positive operator generated by a second-order Sturm-Liouville differential expression and T a selfadjoint indefinite multiplicative operator which is one-to-one. Emphasis is on the full-range and partial-range expansionproperties of the eigenfunctions.

In this paper we employ some new ideas and more delicate estimates, as well as some ideas from previous works, to establish the existence of universal attractors for a nonlinear one-dimensional heat-conductive viscous real gas in the bounded domain Ω = (0,1).

Quasi-differential expressions with matrix-valued coefficients, which generalize those of Shin and Zettl, are considered with regard to equivalence, adjoints and symmetry. The characterization results imply that in the scalar case the class of quasi-differential expressions considered here coincides with that of Shin and is equivalent to that of Zettl. Furthermore polynomials in quasi-differential expressions are defined as expressions of the same kind and shown to coincide with the usual ones. Finally it is indicated that the known general results for the deficiency indices carry over to quasi-differential expressions.