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.

     

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 .

     

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 .

     

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 .

     

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.

     

-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 .

     

Abstract blowing down

Assume that is a surface over an algebraically closed field . Let be obtained from by blowing up a smooth point and let be the exceptional curve. Let be the category of coherent sheaves on . In this note we show how to recover from , if we know the object .

     

Iwasawa invariants and class numbers of quadratic fields for the prime

Let be a square-free integer with and . Put and . For the cyclotomic -extension of , we denote by the -th layer of over . We prove that the -Sylow subgroup of the ideal class group of is trivial for all integers if and only if the class number of is not divisible by the prime . This enables us to show that there exist infinitely many real quadratic fields in which splits and whose Iwasawa -invariant vanishes.

     

Annihilating a subspace of with the sign of a continuous function

Let be a -finite, nonatomic, Baire measure space. Let be a finite dimensional subspace of . There is a bounded, continuous function, , defined on , such that

(1) for all , and (2) almost everywhere.

     

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 .

     

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 .

     

is a CS-algebra

In this paper, it is shown that the group algebra is right CS if and only if . Moreover, when , then is also CS as a module over its center.

     

Conjugate -connections and holonomy groups

In this article we show that when the structure group of the reducible principal bundle is and is an -subbundle of , the rank of the holonomy group of a connection which is gauge equivalent to its conjugate connection is less than or equal to , and use the estimate to show that for all odd prime , if the holonomy group of the irreducible connection as above is simple and is not isomorphic to , , or , then it is isomorphic to .

     

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 .

     

The Furuta inequality in Banach -algebras

Let be real numbers with and Furuta (1987) proved that if bounded linear operators on a Hilbert space satisfy , then . This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach -algebra with continuous involution.

     

type free resolutions of monomial ideals

Let be a monomial ideal of . Bayer-Peeva-Sturmfels studied a subcomplex of the Taylor resolution, defined by a simplicial complex . They proved that if is generic (i.e., no variable appears with the same non-zero exponent in two distinct monomials which are minimal generators), then is the minimal free resolution of , where is the Scarf complex of . In this paper, we prove the following: for a generic (in the above sense) monomial ideal and each integer , there is an embedded prime of . Thus a generic monomial ideal with no embedded primes is Cohen-Macaulay (in this case, is shellable). We also study a non-generic monomial ideal whose minimal free resolution is for some . In particular, we prove that if all associated primes of have the same height, then is Cohen-Macaulay and is pure and strongly connected.

     

Collapsibility of and some related CW complexes

Let denote the order complex of the partition lattice. The natural -action on the set induces an -action on . We show that the regular CW complex is collapsible. Even more, we show that is collapsible, where is a suitable type selection of the partition lattice. This allows us to generalize and reprove in a conceptual way several previous results regarding the multiplicity of the trivial character in the -representation on .

     

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 .

     

Invertible completions of upper triangular operator matrices

In this note we prove that if

is a upper triangular operator matrix acting on the Banach space , then is invertible for some if and only if and satisfy the following conditions:

(i)

is left invertible;

(ii)

is right invertible;

(iii)

.

Furthermore we show that , where is the union of certain of the holes in which happen to be subsets of .

     

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.