We consider a nonlinear hemivariational inequality with the p-Laplacian at resonance. Using an extension of the nonsmooth mountain pass theorem of Chang, which makes use of the Cerami compactness condition, we prove the existence of a nontrivial solution. Our existence results here extends a recent theorem on resonant hemivariational inequalities, by the authors in 1999.

We derive an explicit formula for the number of elements in the symplectic similitude group GSp(2n, q) with given trace and determinant.

An analogue of cyclotomic number fields for function fields over the finite field q, was investigated by L. Carlitz in 1935 and has been studied recently by D. Hayes, M. Rosen, S. Galovich and others. For each nonzero polynomial M in q [T], we denote by k (ΛM) the cyclotomic function field associated with M, where k = q(T). Replacing T by 1/T in k and considering the cyclotomic function field Fv that corresponds to (1/T)v+1 gets us an extension of k, denoted by Lv, which is the fixed field of Fv modulo . We define a (v, n, M)-extension to be the composite N = knk (Λm) Lv where kn is the constant field of degree n over k. In this paper we give analytic class number formulas for (v, n, M)-extensions when M has a nonzero constant term.

We prove a Marcinkiewicz-type multiplier theorem on the Heisenberg group: for 1 < p < ∞, we establish the boundedness on Lp (ℍn) of spectral multipliers m (ℒ1,…,ℒn, iT) of the n partial sub-Laplacians ℒ1,…,ℒn and iT, where m satisfies an (n + l)-fold Marcinkiewicz-type condition. We also establish regularity and cancellation conditions which the convolution kernels of these Marcinkiewicz multipliers m (ℒ1,…,ℒn,iT) satisfy.

In response to a question posed by P. Erdös, B. H. Neumann showed that in a group with every subset of pairwise noncommuting elements finite there is a bound on the size of these sets. Recently, H. E. Bell, A. A. Klein and the first author showed that a similar result holds for rings. However in the case of semigroups, finiteness of subsets of pairwise noncommuting elements does not assure the existence of a bound for their size. The largest class of semigroups in which we found Neumann's result valid are cancellative semigroups.

Permutation polynomials of the form xτf (x3) over a finite field give rise to group permutation polynomials. We give a group theoretic criterion and some other criteria in terms of symmetric functions and power functions.

We give a sufficient condition for normal structure more general than the well known ɛ0(X) < 1. Moreover we obtain sufficient conditions for the fixed point property for some B-convex Banach spaces.

A fundamental result in the theory of black holes due to Hawking asserts that the event horizon of a black hole in the stationary space-time is a 2-sphere topologically. In this article we prove the Riemannian analogue of Hawking's result. In other words, we prove that each bolt of a 4-dimensional complete noncompact Einstein manifold of zero scalar curvature admitting a semifree isometric circle action is a 2-sphere topologically. We also study the structure of the orbit space of an Einstein manifold admitting a free isometric circle action.

We characterise restricted weak upper semicontinuity of the subdifferential of convex functions in terms of the Fenchel biconjugate mapping.

In this paper we prove that a completely regular pseudocompact space with a quasi-regular-Gδ-diagonal is metrisable.

In this paper finite versions of Hardy's discrete, Hardy's integral and the Levin–Cochran–Lee inequalities will be considered and some new generalisations of these inequalities will be derived. Moreover, it will be shown that the constant factors involved in the right-hand sides of the integral results obtained are the best possible.

Let A be a finite Abelian group written additively. For two positive integers k, l with k ≠ l, we say that a subset S ⊂ A is of type (k, l) or is a (k, l) -set if the equation x1 + x2 + … + xk − xk+1−… − xk+1 = 0 has no solution in the set S. In this paper we determine the largest possible cardinality of a (k, l)-set of the cyclic group ℤP where p is an odd prime. We also determine the number of (k, l)-sets of ℤp which are in arithmetic progression and have maximum cardinality.

Suppose that A is a closed linear operator in a Fréchet space X. We show that there always is a maximal subspace Z containing all x ∈ X for which the abstract Cauchy problem has a mild solution, which is a Fréchet space for a stronger topology. The space Z is isomorphic to a quotient of a Fréchet space F, and the part Az of A in Z is similar to the quotient of a closed linear operator B on F for which the abstract Cauchy problem is well-posed. If mild solutions of the Cauchy problem for A in X are unique it is not necessary to pass to a quotient, and we reobtain a result due to R. deLaubenfels.

Moreover, we obtain a continuous selection operator for mild solutions of the inhomogeneous equation.

Assume that a parallelotope P is inscribed in a three-dimensional convex body C A conjecture says that , where Wi is the ratio of the width of C to the width of P for the direction perpendicular to the i-th pair of parallel facets of P. We prove three weaker inequalities. One of them is , where a3 denotes the related axial diameter of C.

The dimension of an ordered set is invariant with respect to any subdivision of its completion. This may be applied to support the conjecture (still open) that the problem to determine the (order) dimension of an N-free ordered set is NP-complete.

Large deviation theorems are proved for non-degenerate U-statistical sums of degree m with kernel h (x1, …, xm) = x1 … xm under the Cramér condition and under the Linnik condition. The method of proof uses truncation and the contraction technique.

It is proved that the monoid RN of all partial recursive functions of one variable is finitely generated, and that RN × RNis a cyclic (left and right) RN-act (under the natural diagonal actions s (a, b) = (sa, sb), (a, b) s = (as, bs)). We also construct a finitely presented monoid S such that S × S is a cyclic left and right S-act, and study further interesting properties of diagonal acts and their relationship with power monoids.