Automorphic-differential identities and actions of pointed coalgebras on rings

In this paper, we prove the following two results which generalize the theorem concerning automorphic-differential endomorphisms asserted by J. Bergen. Let be a ring, its left Martindale quotient ring and a right ideal of having no nonzero left annihilator. (1) Let be a pointed coalgebra which measures such that the group-like elements of act as automorphisms of . If is prime and for , then . Furthermore, if the action of extends to and if such that , then . (2) Let be an endomorphism of given as a sum of composition maps of left multiplications, right multiplications, automorphisms and skew-derivations. If is semiprime and , then .

     

On the Deleted Product Criterion For Embeddability in

For a space let . Let act on and on by exchanging factors and antipodes respectively. We present a new short proof of the following theorem by Weber: For an -polyhedron and , if there exists an equivariant map , then is embeddable in . We also prove this theorem for a peanian continuum and . We prove that the theorem is not true for the 3-adic solenoid and .

     

A decreasing operator function associated with the Furuta inequality

Let with and let and . As a generalization of a result due to Furuta, it is shown that the operator function

is decreasing for and if . Moreover, if and , then is decreasing for and . The latter result is an extension of an earlier result of Furuta.

     

Norms of embeddings of logarithmic Bessel potential spaces

Let be a subset of with finite volume, let and let be a Young function with for large . We show that the norm on the Orlicz space is equivalent to

We also obtain estimates of the norms of the embeddings of certain logarithmic Bessel potential spaces in which are sharp in their dependences on provided that is large enough.

     

-convexity of Orlicz-Bochner spaces

A characterization of -convexity of arbitrary Banach space is given. Moreover, it is proved that the Orlicz-Bochner function space is P-convex if and only if both spaces and are -convex. In particular, the Lebesgue-Bochner space with is -convex iff is -convex.

     

Certain invariant subspace structure of

In this note, we study certain structure of an invariant subspace of . Considering the largest -invariant (resp. -invariant) subspace in the wandering subspace of with respect to the shift operator , we give an alternative characterization of Beurling-type invariant subspaces. Furthermore, we consider a certain class of invariant subspaces.

     

Vector bundles on log terminal varieties

Let be an -dimensional variety and an ample vector bundle on of rank . We give a complete classification of pairs , with log terminal and such that is not ample. The results we obtain were conjectured by Fujita, and recently by Zhang.

     

Optimal estimation of shell thickness in Cutland's construction of Wiener measure

In Cutland's construction of Wiener measure, he used the product of Gaussian measures on , where is an infinite integer. It is mentioned by Cutland and Ng that for the product measure ,

where and with any positive infinite number. We prove here that may be replaced by with any positive infinite number. This is the optimal estimation for the shell thickness. It is also proved that . And for the *Lebesgue measure , is finite and not infinitesimal iff with finite, while for the *Lebesgue area of the sphere , should be .

     

First order differential operators in white noise analysis

Let be the space of test white noise functionals. We first introduce a family of products on including Wiener and Wick products, and then show that with each product , we can associate a first order differential operator, called a first order -differential operator. We next show that a first order -differential operator is indeed a continuous derivation under the product . We finally characterize by means of rotation-invariance and continuous derivation under the product . Here and are the Gross Laplacian and the number operator on , respectively.

     

Conjugate Hardy's inequalities with decreasing weights

We prove that for a decreasing weight on , the conjugate Hardy transform is bounded on () if and only if it is bounded on the cone of all decreasing functions of . This property does not depend on .

     

Some extremal problems in

Fix a positive integer and . We provide expressions for the weighted distance

where is normalized Lebesgue measure on the unit circle, is a nonnegative integrable function, and ranges over the trigonometric polynomials with frequencies in

or

These distances are related to other extremal problems, and are shown to be positive if and only if is integrable. In some cases they are expressed in terms of the series coefficients of the outer functions associated with .

     

Remarks on the results by Koskela concerning the radial uniqueness for Sobolev functions

In this note we aim to complete the results by Koskela concerning the radial uniqueness for Sobolev functions.

Let be a positive nonincreasing function on the interval , and let denote the unit ball of . Consider a -precise function on such that

where . We give conditions on which assure that whenever has vanishing fine boundary limits on a set of positive -capacity.

We are also concerned with the sharpness.

     

Ideal and subalgebra coefficients

For an ideal or -subalgebra of , consider subfields , where is generated - as ideal or -subalgebra - by polynomials in . It is a standard result for ideals that there is a smallest such . We give an algorithm to find it. We also prove that there is a smallest such for -subalgebras. The ideal results use reduced Gröbner bases. For the subalgebra results we develop and then use subduced SAGBI (bases), the analog to reduced Gröbner bases.

     

Some results on finite Drinfeld modules

Let be a global function field, a degree one prime divisor of and let be the Dedekind domain of functions in regular outside . Let be the Hilbert class field of , the integral closure of in . Let be a rank one normalized Drinfeld -module and let be a prime ideal in . We explicitly determine the finite -module structure of . In particular, if , is an odd prime number and is the Carlitz -module, then the finite -module is always cyclic.

     

Remarks on the non-Cohen-Macaulay locus of Noetherian schemes

In this paper we give a notion of polynomial type of a Noetherian scheme and define the function by for all Then we show that if admits a dualizing complex and is equidimensional, is (lower) semicontinuous; moreover, in that case, the non-Cohen-Macaulay locus nCM is not Cohen-Macaulay} is biequidimensional iff is constant on nCM

     

A weak-type inequality of subharmonic functions

We prove the weak-type inequality , , between a non-negative subharmonic function and an -valued smooth function , defined on an open set containing the closure of a bounded domain in a Euclidean space , satisfying , and , where is a constant. Here is the harmonic measure on with respect to 0. This inequality extends Burkholder's inequality in which and , a Euclidean space.

     

Integration on a convex polytope

We present an exact formula for integrating a (positively) homogeneous function on a convex polytope . We show that it suffices to integrate the function on the -dimensional faces of , thus reducing the computational burden. Further properties are derived when has continuous higher order derivatives. This result can be used to integrate a continuous function after approximation via a polynomial.

     

On the Mergelyan approximation property on pseudoconvex domains in

Let be a smoothly bounded pseudoconvex domain of finite type in . We prove the Mergelyan approximation property in various topologies on when the estimates for -equation are known in the corresponding topologies.

     

Eigenvalues of the form valued Laplacian for Riemannian submersions

Let be a Riemannian submersion of closed manifolds. Let be an eigen -form of the Laplacian on with eigenvalue which pulls back to an eigen -form of the Laplacian on with eigenvalue . We are interested in when the eigenvalue can change. We show that , so the eigenvalue can only increase; and we give some examples where , so the eigenvalue changes. If the horizontal distribution is integrable and if is simply connected, then , so the eigenvalue does not change.