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 .

     

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 .

     

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.

     

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 .

     

Normalizers of the congruence subgroups of the Hecke group

Let cos and let be the Hecke group associated to . In this article, we show that for a prime ideal in , the congruence subgroups of are self-normalized in .

     

Ample divisors on the blow up of at points

Fix integers with and ; if assume . Let be general points of the complex projective space and let be the blow up of at with exceptional divisors , . Set . Here we prove that the divisor is ample if and only if , i.e. if and only 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 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 .

     

Positive differentials, theta functions and Hardy kernels

Let be a planar regular region whose Schottky double has genus and set . For fixed we determine the range of the function where is the Riemann theta function on . Also we introduce two weighted Hardy spaces to study the problem when the matrix is positive definite. The proof relies on new theta identities using Fay's trisecants formula.

     

Catenarity in module-finite algebras

The main theorem says that any module-finite (but not necessarily commutative) algebra over a commutative Noetherian universally catenary ring is catenary. Hence the ring is catenary if is Cohen-Macaulay. When is local and is a Cohen-Macaulay -module, we have that is a catenary ring, for any , and the equality holds true for any pair of prime ideals in and for any saturated chain of prime ideals between and .

     

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 ).

     

Quadratic base change for as a theta correspondence I: Occurrence

The local theta correspondence is considered for reductive dual pairs where is a -adic field of characteristic zero and is the orthogonal group attached to a quaternary quadratic form with coefficients in and of Witt rank one over . It is shown that certain representations of occur in the correspondence.

     

On the product of two generalized derivations

Two elements and in a ring determine a generalized derivation on by setting for any in . We characterize when the product is a generalized derivation in the cases when the ring is the algebra of all bounded operators on a Banach space , and when is a -algebra . We use these characterizations to compute the commutant of the range of .

     

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 .

     

Extremal points of a functional on the set of convex functions

We investigate the extremal points of a functional , for a convex or concave function . The admissible functions are convex themselves and satisfy a condition . We show that the extremal points are exactly and if these functions are convex and coincide on the boundary . No explicit regularity condition is imposed on , , or . Subsequently we discuss a number of extensions, such as the case when or are non-convex or do not coincide on the boundary, when the function also depends on , etc.

     

Finite time blow-up for the inhomogeneous equation

We consider the inhomogeneous equation

where , and , and give criteria on , and which determine whether for all and all the solution blows up in finite time or whether for and sufficiently small, the solution exists for all time.

     

Projections from

Let and be two infinite dimensional real Banach spaces. The following question is classical and long-standing. Are the following properties equivalent?

a) There exists a projection from the space of continuous linear operators onto the space of compact linear operators.

b) .

The answer is positive in certain cases, in particular if or has an unconditional basis. It seems that there are few results in the direction of a general solution. For example, suppose that and are reflexive and that or has the approximation property. Then, if , there is no projection of norm 1, from onto . In this paper, one obtains, in particular, the following result:

Theorem. Let be a real Banach space which is reflexive (resp. with a separable dual), of infinite dimension, and such that has the approximation property. Let be a real scalar with . Then can be equivalently renormed such that, for any projection from onto , one has . One gives also various results with two spaces and .

     

On a theorem of Scott and Swarup

Let be an exact sequence of hyperbolic groups induced by an automorphism of the free group . Let be a finitely generated distorted subgroup of . Then there exist and a free factor of such that the conjugacy class of is preserved by and contains a finite index subgroup of a conjugate of . This is an analog of a theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.

     

Behaviour of holomorphic automorphisms on equicontinuous subsets of the space

Consider a compact Hausdorff topological space , a -triple and , the -triple of all continuous -valued functions with the pointwise operations and the norm of the supremum. Let be the group of all holomorphic automorphisms of the unit ball of that map every equicontinuous subset lying strictly inside into another such a set. The real Banach-Lie group and its Lie algebra are investigated. The identity connected component of is identified when has the strong Banach-Stone property. This extends to the infinite dimensional setting a well known result concerning the case .

     

On the irrationality of a certain series

We prove that if is an integer greater than one, and are any positive rationals such that for all integers , then

is irrational and is not a Liouville number.