The growth theorem of convex mappings on the unit ball in

Let be an arbitrary norm on . Let be a normalized biholomorphic convex mapping on the unit ball in with respect to the norm . We will give an upper bound of the growth of .

     

A counterexample to the Fredholm alternative for the -Laplacian

The following nonhomogeneous Dirichlet boundary value problem for the one-dimensional -Laplacian with is considered:

where with small enough. Solvability properties of Problem (*) with respect to the spectral parameter are investigated. We focus our attention on some fundamental differences between the cases and . For we give a counterexample to the classical Fredholm alternative (which is valid for the linear case ).

     

The distribution of solutions of the congruence

For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

     

Common fixed points of commuting holomorphic maps in the unit ball of

Let be the unit ball of (). We prove that if are holomorphic self-maps of such that , then and have a common fixed point (possibly at the boundary, in the sense of -limits). Furthermore, if and have no fixed points in , then they have the same Wolff point, unless the restrictions of and to the one-dimensional complex affine subset of determined by the Wolff points of and are commuting hyperbolic automorphisms of that subset.

     

The connected stable rank of the purely infinite simple -algebras

Suppose that is a unital purely infinite simple -algebra. If the class [1] of the unit 1 in has torsion, then ; if [1] is torsion-free in , then . If is a non-unital purely infinite simple -algebra, then .

     

Nonexpansive, -continuous antirepresentations have common fixed points

Let be a closed convex subset of a Banach (dual Banach) space . By we denote an antirepresentation of a semitopological semigroup as nonexpansive mappings on . Suppose that the mapping is jointly continuous when has the weak (weak*) topology and the Banach space of bounded right uniformly continuous functions on has a right invariant mean. If is weakly compact (for some the set is weakly* compact) and norm separable, then has a common fixed point in .

     

On the existence of maximal Cohen-Macaulay modules over root extensions

Let be an unramified regular local ring having mixed characteristic and the integral closure of in a th root extension of its quotient field. We show that admits a finite, birational module such that . In other words, admits a maximal Cohen-Macaulay module.

     

On zeros of characters of finite groups

We present several results connecting the number of conjugacy classes of a finite group on which an irreducible character vanishes, and the size of some centralizer of an element. For example, we show that if is a finite group such that , then has an element , such that , where is the maximal number of zeros in a row of the character table of . Dual results connecting the number of irreducible characters which are zero on a fixed conjugacy class, and the degree of some irreducible character, are included too. For example, the dual of the above result is the following: Let be a finite group such that ; then has an irreducible character such that , where is the maximal number of zeros in a column of the character table of .

     

The Chevalley-Warning theorem and a combinatorial question on finite groups

Recently, W. D. Gao (1996) proved the following theorem: For a cyclic group of prime order, and any element in it, and an arbitrary sequence of elements from , the number of ways of writing as a sum of exactly of the 's is or modulo according as is zero or not. The dual purpose of this note is (i) to give an entirely different type of proof of this theorem; and (ii) to solve a conjecture of J. E. Olson (1976) by answering an analogous question affirmatively for solvable groups.

     

A factorization theorem for the derivative of a function in

We show that a function is the derivative of a function in the Hardy space of the unit disk for if and only if where and . Here, can be chosen to be non-vanishing, , and . As an application, we characterize positive measures on the unit disk such that the operator is bounded from the tent space to , where .

     

Remark about heat diffusion on periodic spaces

Let be a complete Riemannian manifold with a free cocompact -action. Let be the heat kernel on . We compute the asymptotics of in the limit in which and . We show that in this limit, the heat diffusion is governed by an effective Euclidean metric on coming from the Hodge inner product on .

     

Fixed point iteration for pseudocontractive maps

Let be a compact convex subset of a real Hilbert space, ; a continuous pseudocontractive map. Let and be real sequences in [0,1] satisfying appropriate conditions. For arbitrary define the sequence iteratively by where are arbitrary sequences in . Then, converges strongly to a fixed point of . A related result deals with the convergence of to a fixed point of when is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.

     

-identities on associative algebras

Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

     

Associative rings satisfying the Engel condition

Let be a commutative ring, and let be an associative -algebra generated by elements . We show that if satisfies the Engel condition of degree , then is upper Lie nilpotent of class bounded by a function that depends only on and . We deduce that the Engel condition in an arbitrary associative ring is inherited by its group of units, and implies a semigroup identity.

     

Harmonic maps with noncontact boundary values

Every rank one symmetric space , of noncompact type, admits a compactification by attaching a sphere at infinity. If does not have constant sectional curvature, then admits a natural contact structure. This paper presents a number of harmonic maps , from to , which extend continuously to , and have noncontact boundary values. If the boundary values are assumed continuously differentiable, then the contact structure must be preserved.

     

New facts

We use ``iterated square sequences' to show that there is an -definable partition such that if is an inner model not containing :

(a)

For some is stationary.

(b)

For each there is a generic extension of in which does not exist and is non-stationary.

This result is then applied to show that if is an inner model without , then some sentence not true in can be forced over .

     

Some corollaries of Frobenius' normal -complement theorem

For a prime divisor of the order of a finite group , we present the set of -subgroups generating . In particular, we present the set of primary subgroups of generating the last member of the lower central series of . The proof is based on the Frobenius Normal -Complement Theorem and basic properties of minimal nonnilpotent groups. Let be a group and a group-theoretic property inherited by subgroups and epimorphic images such that all minimal non--subgroups (-subgroups) of are not nilpotent. Then (see the lemma), if is generated by all -subgroups of it follows that is a -group.

     

The structure of some virtually free pro- groups

We prove two conjectures on pro- groups made by Herfort, Ribes and Zalesskii. The first says that a finitely generated pro- group which has an open free pro- subgroup of index is a free pro- product , where the are free pro- of finite rank and the are cyclic of order . The second says that if is a free pro- group of finite rank and is a finite -group of automorphisms of , then is a free factor of . The proofs use cohomology, and in particular a ``Brown theorem' for profinite groups.

     

A description of Hilbert -modules in which all closed submodules are orthogonally closed

Let , be -algebras and a full Hilbert --bimodule such that every closed right submodule is orthogonally closed, i.e., . Then there are families of Hilbert spaces , such that and are isomorphic to -direct sums , resp. , and is isomorphic to the outer direct sum .