Linear difference equations with transition points

Two linearly independent asymptotic solutions are constructed for the second-order linear difference equation

where and have power series expansions of the form

with . Our results hold uniformly for in an infinite interval containing the transition point given by . As an illustration, we present an asymptotic expansion for the monic polynomials which are orthogonal with respect to the modified Jacobi weight , , where , -1$"> and is real analytic and strictly positive on .

     

Maps into complex space

If the dimension of is denoted by or , then a generic map satisfies , while in certain cases there is no map that satisfies .

     

Homology manifold bordism

The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact homology manifolds of dimension is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.

First, we establish homology manifold transversality for submanifolds of dimension : if is a map from an -dimensional homology manifold to a space , and is a subspace with a topological -block bundle neighborhood, and , then is homology manifold -cobordant to a map which is transverse to , with an -dimensional homology submanifold.

Second, we obtain a codimension splitting obstruction in the Wall -group for a simple homotopy equivalence from an -dimensional homology manifold to an -dimensional Poincaré space with a codimension Poincaré subspace with a topological normal bundle, such that if (and for only if) splits at up to homology manifold -cobordism.

Third, we obtain the multiplicative structure of the homology manifold bordism groups .

     

Generalizations of Gonçalves' inequality

Let be a polynomial with complex coefficients and roots , ..., , let denote its norm over the unit circle, and let denote Mahler's measure of . Gonçalves' inequality asserts that

We prove that

for , where is an explicit constant, and that

for . We also establish additional lower bounds on the norms of a polynomial in terms of its coefficients.

     

A perturbed elementary operator and range-kernel orthogonality

Let denote the algebra of operators on a Hilbert . If and are commuting normal operators, and and are commuting quasi-nilpotents such that , then define and by , , and . It is proved that and , where is some scalar and is the quasi-nilpotent part of the operator .

     

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.

     

On the normal bundle of submanifolds of

Let be a submanifold of dimension of the complex projective space . We prove results of the following type.i) If is irregular and , then the normal bundle is indecomposable. ii) If is irregular, and , then is not the direct sum of two vector bundles of rank . iii) If , and is decomposable, then the natural restriction map is an isomorphism (and, in particular, if is embedded Segre in , then is indecomposable). iv) Let and , and assume that is a direct sum of line bundles; if assume furthermore that is simply connected and is not divisible in . Then is a complete intersection. These results follow from Theorem 2.1 below together with Le Potier's vanishing theorem. The last statement also uses a criterion of Faltings for complete intersection. In the case when this fact was proved by M. Schneider in 1990 in a completely different way.

     

On the minimal free resolution of general forms

Let and let be the ideal of generically chosen forms of degrees . We give the precise graded Betti numbers of in the following cases:

• ;
• and is even;
• , is odd and ;
• is even and all generators have the same degree, , which is even;
• is even and ;
• is odd, is even, and .

We give very good bounds on the graded Betti numbers in many other cases. We also extend a result of M. Boij by giving the graded Betti numbers for a generic compressed Gorenstein algebra (i.e., one for which the Hilbert function is maximal, given and the socle degree) when is even and the socle degree is large. A recurring theme is to examine when and why the minimal free resolution may be forced to have redundant summands. We conjecture that if the forms all have the same degree, then there are no redundant summands, and we present some evidence for this conjecture.

     

A defect relation for non-Archimedean analytic curves in arbitrary projective varieties

If is a non-Archimedean analytic curve in a projective variety embedded in and if are hypersurfaces of in general position with then we prove the defect relation:

     

Traces and Sobolev extension domains

Assume that is a bounded domain and its boundary is -regular, . We show that if there exists a bounded trace operator , and , and -Hölder continuous functions are dense in , , then the domain is a -extension domain.

     

Two kinds of strong pseudoprimes up to

Let be an odd composite integer. Write with odd. If either mod or mod for some , then we say that is a strong pseudoprime to base , or spsp() for short. Define to be the smallest strong pseudoprime to all the first prime bases. If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the are known for . Conjectured values of were given by us in our previous papers (Math. Comp. 72 (2003), 2085-2097; 74 (2005), 1009-1024).

The main purpose of this paper is to give exact values of for ; to give a lower bound of : ; and to give reasons and numerical evidence of K2- and -spsp's to support the following conjecture: for any , where (resp. ) is the smallest K2- (resp. -) strong pseudoprime to all the first prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp's to the first 9 prime bases. The entire calculation took about 4000 hours on a PC Pentium IV/1.8GHz. (Recall that a K2-spsp is an spsp of the form: with primes and ; and that a -spsp is an spsp and a Carmichael number of the form: with each prime factor mod .)

     

Iterating the Cesàro operators

The discrete Cesàro operator associates to a given complex sequence the sequence , where . When is a convergent sequence we show that converges under the sup-norm if, and only if, . For its adjoint operator , we establish that converges for any .

The continuous Cesàro operator, , has two versions: the finite range case is defined for and the infinite range case for . In the first situation, when is continuous we prove that converges under the sup-norm to the constant function . In the second situation, when is a continuous function having a limit at infinity, we prove that converges under the sup-norm if, and only if, .

     

Ramsey families of subtrees of the dyadic tree

We show that for every rooted, finitely branching, pruned tree of height there exists a family which consists of order isomorphic to subtrees of the dyadic tree with the following properties: (i) the family is a subset of ; (ii) every perfect subtree of contains a member of ; (iii) if is an analytic subset of , then for every perfect subtree of there exists a perfect subtree of such that the set either is contained in or is disjoint from .

     

Radicals and Plotkin's problem concerning geometrically equivalent groups

If and are groups and is a normal subgroup of , then the -closure of in is the normal subgroup of . In particular, is the -radical of . Plotkin calls two groups and geometrically equivalent, written , if for any free group of finite rank and any normal subgroup of the -closure and the -closure of in are the same. Quasi-identities are formulas of the form for any words in a free group. Generally geometrically equivalent groups satisfy the same quasi-identities. Plotkin showed that nilpotent groups and satisfy the same quasi-identities if and only if and are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample.

     

Local dual spaces of Banach spaces of vector-valued functions

We show that is a local dual of , and is a local dual of , where is a Banach space. A local dual space of a Banach space is a subspace of so that we have a local representation of in satisfying the properties of the representation of in provided by the principle of local reflexivity.

     

Strong local homogeneity and coset spaces

We prove that for every homogeneous and strongly locally homogeneous separable metrizable space there is a metrizable compactification of such that, among other things, for all there is a homeomorphism such that . This implies that is a coset space of some separable metrizable topological group .

     

-hyponormal operators are subscalar

We prove that if are injective, then is subscalar if and only if is subscalar. As corollaries, it is shown that -hyponormal operators and log-hyponormal operators are subscalar; also w-hyponormal operators with Ker Kerand their generalized Aluthge transformations are subscalar.

     

A note on -estimates for stable integrals with drift

Let be of the form where is a symmetric stable process of index with . We obtain various -estimates for the process . In particular, for and any measurable, nonnegative function we derive the inequality

As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation for any initial value .

     

Gröbner bases of associative algebras and the Hochschild cohomology

We give an algorithmic way to construct a free bimodule resolution of an algebra admitting a Gröbner base. It enables us to compute the Hochschild (co)homology of the algebra. Let be a finitely generated algebra over a commutative ring with a (possibly infinite) Gröbner base on a free algebra , that is, is the quotient with the ideal of generated by . Given a Gröbner base for an -subbimodule of the free -bimodule generated by a set , we have a morphism of -bimodules from the free -bimodule generated by to sending the generator to the element . We construct a Gröbner base on for the -subbimodule Ker() of , and with this we have the free -bimodule generated by and an exact sequence . Applying this construction inductively to the -bimodule itself, we have a free -bimodule resolution of .