In this paper we prove the existence theorem for the implicit Darboux problem on the quarterplane x ≥ 0, y ≥ 0. Moreover, we study the topological structure of the solution set of this problem.

Let M be a complex manifold and L → M be a positive holomorphic line bundle over M equipped with a Hermitian metric h of class C2. If D ⊂⊂ M is a pseudoconvex domain which is relatively compact in M then there exists an integer r0 such that for every r ≥ r0 and for every connected holomorphic covering π: the covering is holomorphically convex with respect to holomorphic sections of .

In 1990 Allen Bell presented a sufficient condition for the primeness of the universal enveloping algebra of a Lie superalgebra. Let Q be a nonsingular bilinear form on a finite-dimensional vector space over a field of characteristic zero. In this paper we show that Bell's criterion applies to the Hamiltonian Cartan type superalgebras determined by Q, and hence that their enveloping algebras are semiprimitive.

We obtain some results about a conjecture of Hadamard's quotient of rational series in the case of several variables.

The aim of this note is to show how the trace formula of Arthur-Selberg can be used to derive index theorems for noncompact arithmetic manifolds. Of special interest is the question, under which circumstances there is an index formula without error term, that is, of the same shape as in the compact case. We shall thus present evidence for the hypothesis that the error term for the Euler operator vanishes in the case that the rational rank is smaller than the real rank.

Let T: B → B be a bounded linear operator on the complex Banach space B and let f(z) be analytic on a domain D containing the spectrum Sp(T) of T. Then f(T) is defined by

where C is a contour surrounding SP(T) and contained in D.

In this note we give a subdifferential mean value inequality for every continuous Gâteaux subdiferentiable function f in a Banach space which only requires a bound for one but not necessarily all of the subgradients of f at every point of its domain. We also give a subdifferential approximate Rolle's theorem satating that if a subdifferentiable function oscilllates between −ɛ and ɛ on the boundary of the unit ball then there exists a subgradient of the function at an interior point of the ball which has norm less than or equal to 2ɛ.

In this note known factorisation results for the Dickson polynomials of the second kind, fk(X, a), are used to obtain simple restrictions on those k for which fk(X, a) is a permutation polynomial over Fq.

An algorithmic approach to degree reduction of Bézier curves is presented. The algorithm is based on the matrix representations of the degree elevation and degree reduction processes. The control points of the approximation are obtained by the generalised least squares method. The computations are carried out by minimising the L2 and discrete l2 distance between the two curves.

For a local field of characteristic 0, the functional equations of zeta distributions of prehomogeneous vector spaces have been obtained by M. Sato, Shintani, Igusa, F. Sato and Gyoja. In this paper, we shall consider the case of local fields of characteristic p > 0.

Recently, Blažić, Bokan and Rakić, obtained some classes of 4-dimensional Osserman pseudo-Riemannian manifolds. One of these is the class of rank 2 locally symmetric space endowed with an integrable para-quaternionic structure. In this paper we give an explicit construction of an example of a space of that kind.