A global pinching theorem for surfaces with constant mean curvature in

Let be a compact immersed surface in the unit sphere with constant mean curvature . Denote by the linear map from into , , where is the linear map associated to the second fundamental form and is the identity map. Let denote the square of the length of . We prove that if , then is either totally umbilical or an -torus, where is a constant depending only on the mean curvature .

     

On the maximal inequalities for martingales involving two functions

Let and be nonnegative convex functions, and let and be the right continuous derivatives of and respectively. In this paper, we prove the equivalence of the following three conditions: (i) (ii) and (iii) s_0,$">where and are the Orlicz martingale spaces. As a corollary, we get a sufficient and necessary condition under which the extension of Doob's inequality holds. We also discuss the converse inequalities.

     

Low-dimensional unitary representations of

We characterize all simple unitarizable representations of the braid group on complex vector spaces of dimension . In particular, we prove that if and denote the two generating twists of , then a simple representation (for ) is unitarizable if and only if the eigenvalues of are distinct, satisfy and 0$"> for , where the are functions of the eigenvalues, explicitly described in this paper.

     

On the predictability of discrete dynamical systems

Let be a metric space. A function is said to be non-sensitive at a point if for every 0$"> there is a 0$"> such that for any choice of points , , , we have that for every . Let be the set of all homeomorphisms from onto endowed with the topology of uniform convergence. The main goal of the present paper is to prove that for certain spaces , ``most' functions in are non-sensitive at ``most' points of .

     

Radicals and Plotkin's problem concerning geometrically equivalent groups

If and are groups and is a normal subgroup of , then the -closure of in is the normal subgroup of . In particular, is the -radical of . Plotkin calls two groups and geometrically equivalent, written , if for any free group of finite rank and any normal subgroup of the -closure and the -closure of in are the same. Quasi-identities are formulas of the form for any words in a free group. Generally geometrically equivalent groups satisfy the same quasi-identities. Plotkin showed that nilpotent groups and satisfy the same quasi-identities if and only if and are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample.

     

An extremal problem for trigonometric polynomials

Let be a trigonometric polynomial of degree The problem of finding the largest value for in the inequality is studied. We find exactly provided is the conjugate of an even integer and For general we get an interval estimate for where the interval length tends to as tends to

     

On the volume and the Gauss map image of spacelike hypersurfaces in de Sitter space

In this paper we prove that a complete spacelike hypersurface in de Sitter space such that its image under the Gauss map is contained in a hyperbolic geodesic ball of radius is necessarily compact and its -dimensional volume satisfies , where denotes the volume of a unitary round -sphere. We also characterize the case where these inequalities become equalities. As an application of our result, we also conclude that Goddard's conjecture is true under the assumption that the hyperbolic image of the hypersurface is bounded.

     

Numerical bounds for critical exponents of crossing Brownian motion

We consider -dimensional crossing Brownian motion in a truncated Poissonian potential conditioned to reach a fixed hyperplane at distance from the starting point. The transverse fluctuation of the path is expected to be of order . We prove that for : . As a second critical exponent we introduce , which describes the fluctuations of naturally defined distance functions for crossing Brownian motion. The numerical bound we obtain is an improvement of Corollary 3.1 in Scaling identity for crossing Brownian motion in a Poissonian potential (Probab. Theory Related Fields 112 (1998), 299-319), resulting in if and if the killing rate is strictly positive (0$">).

     

A question of B. Plotkin about the semigroup of endomorphisms of a free group

Let be a free group of finite rank , let be the semigroup of endomorphisms of , and let be the group of automorphisms of .

Theorem. If is an automorphism of , then there is an such that for all .

     

On the absolute continuity of a class of invariant measures

Let be a compact connected subset of , let , be contractive self-conformal maps on a neighborhood of , and let be a family of positive continuous functions on . We consider the probability measure that satisfies the eigen-equation

for some 0$">. We prove that if the attractor is an -set and is absolutely continuous with respect to , the Hausdorff -dimensional measure restricted on the attractor , then is absolutely continuous with respect to (i.e., they are equivalent). A special case of the result was considered by Mauldin and Simon (1998). In another direction, we also consider the -property of the Radon-Nikodym derivative of and give a condition for which is unbounded.

     

Smooth diameter and eigenvalue rigidity in positive Ricci curvature

A recent injectivity radius estimate and previous sphere theorems yield the following smooth diameter sphere theorem for manifolds of positive Ricci curvature: For any given and there exists a positive constant 0$">such that any -dimensional complete Riemannian manifold with Ricci curvature , sectional curvature and diameter is Lipschitz close and diffeomorphic to the standard unit -sphere. A similar statement holds when the diameter is replaced by the first eigenvalue of the Laplacian.

     

Metacompact subspaces of products of ordinals

Let be a subspace of the product of finitely many ordinals. is countably metacompact, and is metacompact iff has no closed subset homeomorphic to a stationary subset of a regular uncountable cardinal. A theorem generalizing these two results is: is -metacompact iff has no closed subset homeomorphic to a -stationary set where .

     

On a semilinear Schrödinger equation with critical Sobolev exponent

We consider the semilinear Schrödinger equation , , where , are periodic in for , 0$">, is of subcritical growth and 0 is in a gap of the spectrum of . We show that under suitable hypotheses this equation has a solution . In particular, such a solution exists if and .

     

On the theorem of Hayman and Wu

We show that the Hayman-Wu constant Øis strictly smaller than Previously it has been shown that Ø A main tool in our proof is an analysis of the hyperbolic geodesic curvature of straight lines in simply connected domains.

     

On one problem of uniqueness of meromorphic functions concerning small functions

In this paper, we show that if two non-constant meromorphic functions and satisfy for , where are five distinct small functions with respect to and , and is a positive integer or with , then . As a special case this also answers the long-standing problem on uniqueness of meromorphic functions concerning small functions.

     

-Weyl's theorem for operator matrices

If is a upper triangular matrix on the Hilbert space , then -Weyl's theorem for and need not imply -Weyl's theorem for , even when . In this note we explore how -Weyl's theorem and -Browder's theorem survive for operator matrices on the Hilbert space.

     

A new statistic for the problem

A finite -trajectory is a sequence of positive integers such that if is odd, if is even, 1$"> if and . For such a sequence we define . We prove that if is odd and . Histograms suggest that may have an interesting limiting distribution.     

LCM-splitting sets in some ring extensions

Let be a saturated multiplicative set of an integral domain . Call an lcm splitting set if and are principal ideals for every and . We show that if is an -stable overring of (that is, if whenever and is principal, it follows that and if is an lcm splitting set of , then the saturation of in is an lcm splitting set in . Consequently, if is Noetherian and is a (nonzero) prime element, then is also a prime element of the integral closure of . Also, if is Noetherian, is generated by prime elements of and if the integral closure of is a UFD, then so is the integral closure of .

     

On a problem of J. P. Williams

Let be the algebra of all bounded operators on a Hilbert space . Let be a continuous function on the closed disk and let

for some 0,$"> for all and all with . Then is differentiable on . The paper shows that the function may have a discontinuous derivative.

     

The homogeneous spectrum of a graded commutative ring

Suppose is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a -graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring have Noetherian spectrum. If , we show that has Noetherian spectrum, while for each we establish existence of an example where the homogeneous spectrum of is not Noetherian.