It is proven that if k ≥ 2 is an integer and d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then d = 4k or d = 64k 5−48k 3+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k − 1, k + 1, c, d} are regular.

Let be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in with two independent approximation functions; that is if a certain sum converges then the set of all points (x,y) on the curve which satisfy simultaneously the inequalities ||qx|| < ψ1(q) and ||qy|| < ψ2(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for multiplicative approximation ||qx|| ||qy|| < ψ(q) we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.

Let H be the n-dimensional hyperbolic space of constant sectional curvature –1 and let G be the identity component of the isometry group of H. We find all the G-invariant pseudo-Riemannian metrics on the space of oriented geodesics of H (modulo orientation preserving reparametrizations). We characterize the null, time- and space-like curves, providing a relationship between the geometries of and H. Moreover, we show that is Kähler and find an orthogonal almost complex structure on .

In this paper, the authors consider the behavior on BMO() and Campanato spaces for the higher-dimensional Marcinkiewicz integral operator which is defined by where Ω is homogeneous of degree zero, has mean value zero and is integrable on the unit sphere. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO() or to a certain Campanato space, then [μΩ(f)]2 is either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness is also obtained. The corresponding Lusin area integral is also considered.

Second order nonlinear delay differential equations with positive delays are considered, and sufficient conditions are given that guarantee the existence of positive increasing solutions on the half-line with first order derivatives tending to zero at infinity. The approach is elementary and is essentially based on an old idea which appeared in the author's paper Arch. Math. (Basel)36 (1981), 168–178. The application of the result obtained to second order Emden-Fowler type differential equations with constant delays and, especially, to second order linear differential equations with constant delays, is also presented. Moreover, some (general or specific) examples demonstrating the applicability of the main result are given.

In this paper we deal with some boundary value problems related with diffusion processes in the presence of lower and upper solutions. Singularities as well as non local boundary conditions are allowed. We also prove the existence of extremal solutions and the uniqueness of solution for a particular case.

We prove that if G is a locally finite group admitting an automorphism φ of order four such that C G (φ) is Chernikov, then G has a soluble subgroup of finite index.

We extend the well-known Paley and Paley-Kahane-Khintchine inequalities on lacunary series to the unit polydisk of . Then we apply them to obtain sharp estimates for the mean growth in weighted spaces h(p, α), h(p, log(α)) of Hardy–Bloch type, consisting of functions n-harmonic in the polydisk. These spaces are closely related to the Bloch and mixed norm spaces and naturally arise as images under some fractional operators.

For a fixed integer e and prime p we construct the p-adic order bounded group valuations for a given abelian group G. These valuations give Hopf orders inside the group ring KG where K is an extension of with ramification index e. The orders are given explicitly when G is a p-group of order p or p 2. An example is given when G is not abelian.

The concept of a metaplectic form was introduced about 40 years ago by T. Kubota. He showed how Jacobi-Legendre symbols of arbitrary order give rise to characters of arithmetic groups. Metaplectic forms are the automorphic forms with these characters. Kubota also showed how higher analogues of the classical theta functions could be constructed using Selberg's theory of Eisenstein series. Unfortunately many aspects of these generalized theta series are still unknown, for example, their Fourier coefficients. The analogues in the case of function fields over finite fields can in principle be calculated explicitly and this was done first by J. Hoffstein in the case of a rational function field. Here we shall return to his calculations and clarify a number of aspects of them, some of which are important for recent developments.

In this paper we solve partially an open problem raised by B. Ricceri (Bull. London Math. Soc.33 (2001), 331–340). Infinitely many solutions for a Neumann problem are obtained through a direct variational approach where the nonlinearity has an oscillatory behaviour at infinity.

Certain generalized Samelson products of SO(3) are calculated and applications to the homotopy of gauge groups are given.

There is a commutative algebra of differential-difference operators, acting on polynomials on , associated with the reflection group B 2. This paper presents an integral transform which intertwines this algebra, allowing one free parameter, with the algebra of partial derivatives. The method of proof depends on properties of a certain class of balanced terminating hypergeometric series of 4 F 3-type. These properties are in the form of recurrence and contiguity relations and are proved herein.

In this paper, we define multi-normed spaces, and investigate some properties of multi-bounded mappings on multi-normed spaces. Moreover, we prove a generalized Hyers–Ulam–Rassias stability theorem associated to the Cauchy additive equation for mappings from linear spaces into multi-normed spaces.