Associated primes of local cohomology modules

Let be an ideal of a commutative Noetherian ring and a finitely generated -module. Let be a natural integer. It is shown that there is a finite subset of , such that is contained in union with the union of the sets , where and . As an immediate consequence, we deduce that the first non- -cofinite local cohomology module of with respect to has only finitely many associated prime ideals.

     

-additive families and the invariance of Borel classes

We prove that any -additive family of sets in an absolutely Souslin metric space has a -discrete refinement provided every partial selector set for is -discrete. As a corollary we obtain that every mapping of a metric space onto an absolutely Souslin metric space, which maps -sets to -sets and has complete fibers, admits a section of the first class. The invariance of Borel and Souslin sets under mappings with complete fibers, which preserves -sets, is shown as an application of the previous result.

     

Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space

Let be a real Hilbert space. Let , be bounded monotone mappings with , where and are closed convex subsets of satisfying certain conditions. Suppose the equation has a solution in . Then explicit iterative methods are constructed that converge strongly to such a solution. No invertibility assumption is imposed on , and the operators and need not be defined on compact subsets of .

     

Bordism groups of special generic mappings

The Pontrjagin-Thom construction expresses a relation between the oriented bordism groups of framed immersions , and the stable homotopy groups of spheres. We apply the Pontrjagin-Thom construction to the oriented bordism groups of mappings n$">, with mildest singularities. Recently, O. Saeki showed that for , the group is isomorphic to the group of smooth structures on the sphere of dimension . Generalizing, we prove that is isomorphic to the -th stable homotopy group , , where is the group of oriented auto-diffeomorphisms of the sphere and is the group of rotations of .

     

-function of an operator: A white noise approach

Let be the canonical framework of white noise analysis over the Gel'fand triple and be the space of continuous linear operators from to . Let be a self-adjoint operator in with spectral representation . In this paper, it is proved that under appropriate conditions upon , there exists a unique linear mapping such that for each . The mapping is then naturally used to define as , where is the Dirac -function. Finally, properties of the mapping are investigated and several results are obtained.

     

There exist no gaps between Gevrey differentiable and nowhere Gevrey differentiable

We verify that there exist no gaps between Gevrey differentiable and nowhere Gevrey differentiable in the sense that for given 1$">, there is a nowhere Gevrey differentiable function on of order that is Gevrey differentiable of order for any s$">, which also gives a strong example that is Gevrey differentiable but nowhere analytic.

     

The Fefferman-Stein type inequality for the Kakeya maximal operator in Wolff's range

Let , , be the Kakeya (Nikodým) maximal operator defined as the supremum of averages over tubes of eccentricity . The (so-called) Fefferman-Stein type inequality:

is shown in the range , where and are some constants depending only on and the dimension and is a weight. The result is a sharp bound up to -factors.

     

New properties of Mrowka's space

We extend the technique of Mrowka to show that his space has the property that dim while ind , assuming his extra set-theoretic hypothesis. We also show that is compact, so assuming the extra axiom, there is an compact metric space with no compact completion.

     

Stable rank of corner rings

B. Blackadar recently proved that any full corner in a unital C*-algebra has K-theoretic stable rank greater than or equal to the stable rank of . (Here is a projection in , and fullness means that .) This result is extended to arbitrary (unital) rings in the present paper: If is a full idempotent in , then . The proofs rely partly on algebraic analogs of Blackadar's methods and partly on a new technique for reducing problems of higher stable rank to a concept of stable rank one for skew (rectangular) corners . The main result yields estimates relating stable ranks of Morita equivalent rings. In particular, if where is a finitely generated projective generator, and can be generated by elements, then .

     

Examples concerning heredity problems of WCG Banach spaces

We present two examples of WCG spaces that are not hereditarily WCG. The first is a space with an unconditional basis, and the second is a space such that is WCG and does not contain . The non-WCG subspace of has the additional property that is not WCG and is reflexive.

     

The content of a Gaussian polynomial is invertible

Let be an integral domain and let be a nonzero polynomial in . The content of is the ideal generated by the coefficients of . The polynomial is called Gaussian if for all . It is well known that if is an invertible ideal, then is Gaussian. In this note we prove the converse.

     

Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings

Let be a real Banach space with a uniformly Gâteaux differentiable norm possessing uniform normal structure, be a nonempty closed convex and bounded subset of , be an asymptotically nonexpansive mapping with sequence . Let be fixed, be such that , , and . Define the sequence iteratively by , n= 0, 1, 2, ..._. $"> It is proved that, for each integer , there is a unique such that If, in addition, and , then converges strongly to a fixed point of .

     

A spectral characterization of the -torus by the first stability eigenvalue

Let be a compact hypersurface with constant mean curvature immersed into the unit Euclidean sphere . In this paper we derive a sharp upper bound for the first eigenvalue of the stability operator of in terms of the mean curvature and the length of the total umbilicity tensor of the hypersurface. Moreover, we prove that this bound is achieved only for the so-called -tori in , with . This extends to the case of constant mean curvature hypersurfaces previous results given by Wu (1993) and Perdomo (2002) for minimal hypersurfaces.

     

Self-adjointness of the perturbed wave operator on

We give classes of unbounded real-valued for which is self-adjoint on , , where is the wave operator defined on .

     

Multiplicatively spectrum-preserving maps of function algebras

Let be a compact Hausdorff space and a function algebra. Assume that is the maximal ideal space of . Denoting by the spectrum of an , which in this case coincides with the range of , a result of Molnár is generalized by our Main Theorem: If is a surjective map with the property for every pair of functions , then there exists a homeomorphism such that

for every and every with .

     

Algebraic functions with even monodromy

Let be a compact Riemann surface of genus and be an integer. We show that admits meromorphic functions with monodromy group equal to the alternating group

     

Finite -arc transitive Cayley graphs and flag-transitive projective planes

In this paper, a characterisation is given of finite -arc transitive Cayley graphs with . In particular, it is shown that, for any given integer with and , there exists a finite set (maybe empty) of -transitive Cayley graphs with such that all -transitive Cayley graphs of valency are their normal covers. This indicates that -arc transitive Cayley graphs with are very rare. However, it is proved that there exist 4-arc transitive Cayley graphs for each admissible valency (a prime power plus one). It is then shown that the existence of a flag-transitive non-Desarguesian projective plane is equivalent to the existence of a very special arc transitive normal Cayley graph of a dihedral group.

     

Transference for amenable actions

Suppose acts amenably on a measure space with quasi-invariant -finite measure . Let be an isometric representation of on and a finite Radon measure on . We show that the operator has -operator norm not exceeding the -operator norm of the convolution operator defined by . We shall also prove an analogous result for the maximal function associated to a countable family of Radon measures .

     

A character of the gradient estimate for diffusion semigroups

Let be the semigroup of the diffusion process generated by on . It is proved that there exists and an -valued function such that holds for all 0$"> and all if and only if satisfies the formula for all

     

Local dual and poly-scale refinability

For a compactly supported function , let , be the space of all finite linear combinations of . In this paper, we consider the explicit construction of local duals of and the poly-scale refinability of functions in when is an -refinable function. We show that for any -refinable function , there exists a local dual of in for some , and that any function in is poly-scale refinable.