The first occurrence for the irreducible modules of general linear groups in the polynomial algebra

Let be a prime number and let be the group of all invertible matrices over the prime field . It is known that every irreducible -module can occur as a submodule of , the polynomial algebra with variables over . Given an irreducible -module , the purpose of this paper is to find out the first value of the degree of which occurs as a submodule of , the subset of consisting of homogeneous polynomials of degree . This generalizes Schwartz-Tri's result to the case of any prime .

     

A note on -bases of rings

Let be a tower of rings of characteristic . Suppose that is a finitely presented -module. We give necessary and sufficient conditions for the existence of -bases of over . Next, let be a polynomial ring where is a perfect field of characteristic , and let be a regular noetherian subring of containing such that . Suppose that is a free -module. Then, applying the above result to a tower of rings, we shall show that a polynomial of minimal degree in is a -basis of over .

     

Lomonosov's theorem cannot be extended to chains of four operators

We show that the celebrated Lomonosov theorem cannot be improved by increasing the number of commuting operators. Specifically, we prove that if is the operator without a non-trivial closed invariant subspace constructed by C. J. Read, then there are three operators , and (non-multiples of the identity) such that commutes with , commutes with , commutes with , and is compact. It is also shown that the commutant of contains only series of .

     

Every -regular

We prove the following: Theorem A. If is a -regular ultrafilter, then either

(a)

is -regular, or

(b)

the cofinality of the linear order is , and is -regular for all .

Corollary B. Suppose that is singular, and either is regular, or . Then every -regular ultrafilter is -regular.

We also discuss some consequences and variations.

     

Asymptotic behaviour of Castelnuovo-Mumford regularity

Let be a polynomial ring over a field. For a graded -module generated in degree at most , the Castelnuovo-Mumford regularity of each of (i) its symmetric power, (ii) its torsion-free symmetric power and (iii) the integral closure of its torsion-free symmetric power is bounded above by a linear function in with leading coefficient at most . For a graded ideal of , the regularity of is given by a linear function of for all sufficiently large . The leading coefficient of this function is identified.

     

A note on restricted weak-type estimates for Bochner-Riesz operators with negative index in

It is shown that the Bochner-Riesz operator on of negative order is of restricted weak type in the critical points and , where , for in the two-dimensional case and , for if .

     

Bloch radius, normal families and quasiregular mappings

Bloch's Theorem is extended to -quasiregular maps , where is the standard -dimensional sphere. An example shows that Bloch's constant actually depends on for .

     

Universal -lattices of minimal rank

Let be the minimal rank of -universal -lattices, by which we mean positive definite -lattices which represent all positive -lattices of rank . It is a well known fact that for . In this paper, we determine and find all -universal lattices of rank for .

     

On pointed Hopf algebras of dimension

In this note we describe nonsemisimple Hopf algebras of dimension with coradical isomorphic to , abelian of order , over an algebraically closed field of characteristic zero. If is cyclic or , then we also determine the number of isomorphism classes of such Hopf algebras.

     

On Tate-Shafarevich groups of abelian varieties

Let be a finite Galois extension of number fields with Galois group , let be an abelian variety defined over , and let and denote, respectively, the Tate-Shafarevich groups of over and of over . Assuming that these groups are finite, we derive, under certain restrictions on and , a formula for the order of the subgroup of of -invariant elements. As a corollary, we obtain a simple formula relating the orders of , and when is a quadratic extension and is the twist of by the non-trivial character of .

     

-sequences

A sequence of positive integers is called a -sequence if every integer has at most representations with all in and . A -sequence is also called a -sequence or Sidon sequence. The main result is the following

Theorem. Let be a -sequence and for an integer . Then there is a -sequence of size , where .

Corollary. Let . The interval then contains a -sequence of size , when .

     

Spectral types of skewed Bernoulli shift

For the transformation for , it is proved that a real-valued function of modulus is not a multiplicative coboundary if the discontinuities of are -adic points and . It is also proved that the weakly mixing skew product transformations arising from Bernoulli shifts have Lebesgue spectrum.

     

A characterization of total reflection orders

Let be a Coxeter system with set of reflections . It is known that if is a total reflection order for , then, for each , and its complement are stable under conjugation by . Moreover the upper and lower -conjugates of are still total reflection orders. For any total order on , say that is stable if is stable under conjugation by for each . We prove that if and all orders obtained from by successive lower or upper -conjugations are stable, then is a total reflection order.

     

Solving the -Laplacian on manifolds

We prove in this paper that the equation on a -hyperbolic manifold has a solution with -integrable gradient for any bounded measurable function with compact support.

     

A bound on the reduction number of a primary ideal

Let be a local ring of positive dimension and let be an -primary ideal. We denote the reduction number of by , which is the smallest integer such that for some reduction of In this paper we give an upper bound on in terms of numerical invariants which are related with the Hilbert coefficients of when is Cohen-Macaulay. If , it is known that where denotes the multiplicity of If in Corollary 1.5 we prove where is the first Hilbert coefficient of From this bound several results follow. Theorem 1.3 gives an upper bound on in a more general setting.

     

Yang index of the deleted product

For any a -dimensional polyhedron is constructed such that the Yang index of its deleted product equals . This answers a question of Izydorek and Jaworowski (1995). For any a -dimensional closed manifold with involution is constructed such that , but can be mapped into a -dimensional polyhedron without antipodal coincidence.

     

An uncertainty principle for convolution operators on discrete groups

Consider a discrete group and a bounded self-adjoint convolution operator on ; let be the spectrum of . The spectral theorem gives a unitary isomorphism between and a direct sum , where , and is a regular Borel measure supported on . Through this isomorphism corresponds to multiplication by the identity function on each summand. We prove that a nonzero function and its transform cannot be simultaneously concentrated on sets , such that and the cardinality of are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.

     

The spectral properties of certain linear operators and their extensions

Let be a Hilbert space with inner-product , and let be a bounded positive operator on which determines an inner-product, . Denote by the completion of with respect to the norm . In this paper, operators having certain relationships with are studied. In particular, if where , then has an extension , and and have essentially the same spectral and Fredholm properties.

     

Reflection and uniqueness theorems for harmonic functions

Suppose that is harmonic on an open half-ball in such that the origin 0 is the centre of the flat part of the boundary . If has non-negative lower limit at each point of and tends to 0 sufficiently rapidly on the normal to at 0, then has a harmonic continuation by reflection across . Under somewhat stronger hypotheses, the conclusion is that . These results strengthen recent theorems of Baouendi and Rothschild. While the flat boundary set can be replaced by a spherical surface, it cannot in general be replaced by a smooth -dimensional manifold.

     

On semisimple Hopf algebras of dimension

We consider the problem of the classification of semisimple Hopf algebras of dimension where are two prime numbers. First we prove that the order of the group of grouplike elements of is not , and that if it is , then . We use it to prove that if and its dual Hopf algebra are of Frobenius type, then is either a group algebra or a dual of a group algebra. Finally, we give a complete classification in dimension , and a partial classification in dimensions and .