The univalent functions in the diagonalBesov space A_{p}, where 1<p<\infty ,are characterized in terms of the distance from the boundary of a point in theimage domain. Here A_{2} is the Dirichlet space. A consequenceis that there exist functions in A_{p},\ p>2, for which thearea of the complement of the image of the unit disc iszero.

1991 Mathematics Subject Classification 30C99, 46E35.

Let X be acomplex infinite dimensional Banach space. We use σ_a(T) andσ_{ea}(T), respectively, to denote the approximate point spectrum and theessential approximate point spectrum of a bounded operator T onX. Also, \pi _a(T) denotes the set <$>{\rm{iso}σ_a(T)\backslash σ_{ea}(T)}<$>. An operator T onX obeys the a-Browder's theorem provided that<$>σ_{ea}(T) =σ_a(T\,)\backslash π_a(T)<$>. We investigateconnections between the Browder's theorems, the spectral mapping theorem and spectralcontinuity.

In this paper, weshow new results on slant submanifolds of an almost contact metric manifold. Westudy and characterize slant submanifolds of K-contact andSasakian manifolds. We also study the special class of three-dimensional slantsubmanifolds. We give several examples of slantsubmanifolds.

1991 Mathematics Subject Classification 53C15, 53C40.

For a C1function f:ℝ^n →ℝ\;(n \ge 2), we consider the least numberk of distinct critical points that f must possess whenrestricted to the sphere S=\{x\in ℝ^n: \Vert x\Vert =1\}. Clearly k\ge 2 (for f attains its absolute minimum and maximum onS), and a result of Lusternik and Schnirelmann establishes thatk=n if f is even. Here we prove that k=n if,for a given orthonormal system (e_i), \max\limits_{S \capV_i}\,f<\min\limits_{S \cap V_i^\bot}\,f, for all i=1, …n-1,where V_i is the subspace spanned by e_1, …, e_i andV_i^\bot its orthogonal complement. It is shown that this criterion is satisfied bysuitably restricted perturbations of quadratic forms having n distincteigenvalues.

In its originalformulation Lang's theorem referred to a semilinear map on an n-dimensional vectorspace over the algebraic closure of GF(p): it fixes the vectors of a copy ofV(n, p^h). In other words, every semilinear map defined over a finite field isequivalent by change of coordinates to a map induced by a field automorphism. We provide an elementaryproof of the theorem independent of the theory of algebraic groups and, as a by-product of ourinvestigation, obtain a convenient normal form for semilinear maps. We apply our theorem to classicalgroups and to projective geometry. In the latter application we uncover three simple yet surprisingresults.

Inthis paper, we prove that if M^2 is a complete maximalspacelike surface of an anti-de Sitter space {\bfH}^{4}_{2}(c) with constant scalar curvature, then S=0,S={-10c\over 11}, S={-4c\over 3} or S=-2c,where S is the squared norm of the second fundamental form ofM^{2}. Also

(1) S=0 if and only if M^2 is the totallygeodesic surface {\bf H}^2(c);

(2) S={-4c\over3} if and only if M^2 is the hyperbolic Veronesesurface;

(3) S=-2c if and only if M^2 is the hyperboliccylinder of the totally geodesic

surface {\bfH}^{3}_{1}(c) of {\bfH}^{4}_{2}(c).

1991 Mathematics Subject Classifaction 53C40, 53C42.

We provethat if n \geq 2 then A_nd cannot be a maximal left ideal of then-th Weyl algebra A_n over ℂ for almostevery operator d of degree 2 of A_n.

An identity property defined for a pair of 2-complexes (Y,X) first arose in 1993 within a strategy for constructing a counterexample of infinite type to Whitehead's Asphericity Conjecture. In this note we make use of the theory of pictures to characterize a more general right N-identity property, where N < \pi _1Y. We also define combinatorial asphericity (CA) for the pair (Y,X) and determine a test for (CA) in the case that Y is obtained from X by the addition of a single 2-cell. This test can be used to determine an explicit generating set for \pi _2Y.