Approximation from locally finite-dimensional shift-invariant spaces

After exploring some topological properties of locally finite-dimensional shift-invariant subspaces of , we show that if provides approximation order , then it provides the corresponding simultaneous approximation order. In the case is generated by a compactly supported function in , it is proved that provides approximation order in the -norm with if and only if the generator is a derivative of a compactly supported function that satisfies the Strang-Fix conditions.

     

Asymptotic behaviour of certain sets of prime ideals

Let be ideals of the commutative ring , let be a Noetherian -module and let be a submodule of ; also let be an Artinian -module and let be a submodule of . It is shown that, whenever is a sequence of -tuples of non-negative integers which is non-decreasing in the sense that for all and all , then Ass is independent of for all large , and also Att is independent of for all large . These results are proved without any regularity conditions on the ideals , and so (a special case of) the first answers in the affirmative a question raised by S. McAdam.

     

On contravariant finiteness of subcategories of modules of projective dimension

Let be an artin algebra. This paper presents a sufficient condition for the subcategory of to be contravariantly finite in , where is the subcategory of consisting of --modules of projective dimension less than or equal to . As an application of this condition it is shown that is contravariantly finite in for each when is stably equivalent to a hereditary algebra.

     

Minimal prime ideals in enveloping algebras of Lie superalgebras

Let be a finite dimensional Lie superalgebra over a field of characteristic zero. Let be the enveloping algebra of . We show that when , then is not semiprime, but it has a unique minimal prime ideal; it follows then that when is classically simple, has a unique minimal prime ideal. We further show that when is a finite dimensional nilpotent Lie superalgebra, then has a unique minimal prime ideal.

     

Generalized rational identities of subnormal subgroups of skew fields

Let be a skew field with infinite center such that , and let be a non-central subnormal subgroup of the multiplicative group of . Then there are no non-trivial generalized rational identities of . This generalizes a theorem proved by Makar-Limanov.

     

-Fréchet-Urysohn property of Franklin compact spaces

Franklin compact spaces defined by maximal almost disjoint families of subsets of are considered from the view of its -sequentiality and -Fréchet-Urysohn-property for ultrafilters . Our principal results are the following: CH implies that for every -point there are a Franklin compact -Fréchet-Urysohn space and a Franklin compact space which is not -Fréchet-Urysohn; and, assuming CH, for every Franklin compact space there is a -point such that it is not -Fréchet-Urysohn. Some new problems are raised.

     

Representations at fixed points of smooth actions of compact connected Lie groups

Let be a compact connected Lie group acting smoothly on a connected closed manifold with nonempty fixed point set . In this paper, we study the relation between the cohomology of or and the equivalent representations of at fixed points.

     

Bounds for the operator norms of some Nörlund matrices

Suppose is a non-increasing sequence of non-negative numbers with , , , and is the lower triangular matrix defined by , , and , . We show that the operator norm of as a linear operator on is no greater than , for ; this generalizes, yet again, Hardy's inequality for sequences, and simplifies and improves, in this special case, more generally applicable results of D. Borwein, Cass, and Kratz. When the tend to a positive limit, the operator norm of on is exactly . We also give some cases when the operator norm of on is less than .

     

On the prime model property

Assume is superstable, is a formula over , is countable and is countable and . We investigate models in assuming has the prime model property. We prove some corollaries on the number of models in . We show an example of an -stable and with having exactly 3 models.

     

Stable splittings of

Using the Snaith-Mitchell-Priddy splittings of and , we can give new stable splittings of and
respectively.

     

Almost disjoint permutation groups

A permutation group on a set of (infinite) cardinality is almost disjoint if no element of except the identity has fixed points, i.e., if is an almost disjoint family of subsets of . We show how almost disjoint permutation groups can be constructed from almost disjoint families of sets.

     

Discrete valuation overrings of Noetherian domains

We show that, given a chain of prime ideals in a Noetherian domain , there exist a finitely generated overring of and a saturated chain of primes in contracting term by term to the given chain. We further show that there is a discrete rank valuation overring of whose primes contract to those of the given chain.

     

Rational nodal curves with no smooth Weierstrass points

Let denote the rational curve with nodes obtained from the Riemann sphere by identifying 0 with and with for , where is a primitive th root of unity. We show that if is even, then has no smooth Weierstrass points, while if is odd, then has smooth Weierstrass points.

     

A reciprocity law for certain Frobenius extensions

Let be a finite Galois extension of algebraic number fields with Galois group . Assume that is a Frobenius group and is a Frobenius complement of . Let be the maximal normal nilpotent subgroup of . If is nilpotent, then every Artin L-function attached to an irreducible representation of arises from an automorphic representation over , i.e., the Langlands' reciprocity conjecture is true for such Galois extensions.

     

The range of a ring homomorphism from a commutative -algebra

We prove that if a commutative semi-simple Banach algebra is the range of a ring homomorphism from a commutative -algebra, then is -equivalent, i.e. there are a commutative -algebra and a bicontinuous algebra isomorphism between and . In particular, it is shown that the group algebras , and the disc algebra are not ring homomorphic images of -algebras.

     

Stability of semigroups commuting with a compact operator

It is proved that if are bounded -semigroups on Banach spaces and , resp., and , are bounded operators with dense ranges such that intertwines with and commutes with , then is strongly stable provided ---the generator of ---does not have eigenvalue on . An analogous result holds for power-bounded operators.

     

On the perturbation theory of -accretive operators in Banach spaces

Let be a real Banach space. Let be -accretive with compact. Let be such that is condensing for some Let and assume that there exists a bounded open set and such that is bounded and

for all Then A basic homotopy result of the degree theory for with condensing and possibly unbounded, is used to improve and/or extend recent results by Hirano and Kalinde.

     

Differences of vector-valued functions on topological groups

Let be a locally compact group equipped with right Haar measure. The right differences of functions on are defined by for . Let and suppose for some and all . We prove that is a right uniformly continuous function of . If is abelian and the Beurling spectrum does not contain the unit of the dual group , then we show . These results have analogues for functions , where is a separable or reflexive Banach space. Finally, we apply our methods to vector-valued right uniformly continuous differences and to absolutely continuous elements of left Banach -modules.

     

Critical points of real entire functions and a conjecture of Pólya

Let be a nonconstant real entire function of genus and assume that all the zeros of are distributed in some infinite strip , . It is shown that (1) if has nonreal zeros in the region , and has nonreal zeros in the same region, and if the points and are located outside the Jensen disks of , then has exactly critical zeros in the closed interval , (2) if is at most of order , , and minimal type, then for each positive constant there is a positive integer such that for all has only real zeros in the region , and (3) if is of order less than , then has just as many critical points as couples of nonreal zeros.

     

Invariant subspaces of the harmonic Dirichlet space with large co-dimension

In this paper, we comment on the complexity of the invariant subspaces (under the bilateral Dirichlet shift ) of the harmonic Dirichlet space . Using the sampling theory of Seip and some work on invariant subspaces of Bergman spaces, we will give examples of invariant subspaces with , . We will also generalize this to the Dirichlet classes , , as well as the Besov classes , , .