Enriched Reedy categories

We define the notion of an enriched Reedy category and show that if is a -Reedy category for some symmetric monoidal model category and is a -model category, the category of -functors and -natural transformations from to is again a model category.

     

Counting cusps of subgroups of

Let be a number field with real places and complex places, and let be the ring of integers of . The quotient has cusps, where is the class number of . We show that under the assumption of the generalized Riemann hypothesis that if is not or an imaginary quadratic field and if , then has infinitely many maximal subgroups with cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

     

Representation of measures with polynomial denseness in , , and its application to determinate moment problems

It has been proved that algebraic polynomials are dense in the space , , iff the measure is representable as with a finite non-negative Borel measure and an upper semi-continuous function such that is a dense subset of the space    as equipped with the seminorm . The similar representation ( ) with the same and ( , and is also a dense

subset of ) corresponds to all those measures (supported by ) that are uniquely determined by their moments on ( ). The proof is based on de Branges' theorem (1959) on weighted polynomial approximation. A more general question on polynomial denseness in a separable Fréchet space in the sense of Banach has also been examined.

     

On the triple jump of the set of atoms of a Boolean algebra

We prove the following result concerning the degree spectrum of the atom relation on a computable Boolean algebra. Let be a computable Boolean algebra with infinitely many atoms and be the Turing degree of the atom relation of . If is a c.e. degree such that , then there is a computable copy of where the atom relation has degree . In particular, for every c.e. degree , any computable Boolean algebra with infinitely many atoms has a computable copy where the atom relation has degree .

     

Dynamics of the function and the Green-Tao theorem on arithmetic progressions in the primes

Let be the set of all positive integers , where are primes and possibly two, but not all three of them are equal. For any , define a function by where is the largest prime factor of . It is clear that if , then . For any , define , for . An element is semi-periodic if there exists a nonnegative integer and a positive integer such that . We use ind to denote the least such nonnegative integer . Wushi Goldring [Dynamics of the function and primes, J. Number Theory 119(2006), 86-98] proved that any element is semi-periodic. He showed that there exists such that , ind, and conjectured that ind can be arbitrarily large.

In this paper, it is proved that for any we have ind , and the Green-Tao Theorem on arithmetic progressions in the primes is employed to confirm Goldring's above conjecture.

     

Spectra of operators with Bishop's property

Let be a Banach space and let be the class that consists of all operators such that for every , the range of has a finite-codimension when it is closed. For an integer , we define the class as an extension of . We then study spectral properties of such operators, and we extend some known results of multi-cyclic operators with .

     

Nonabelian theta functions of positive genus

Let be a smooth projective irreducible curve over of genus and let be a set of distinct points on . We fix a nonnegative integer and denote by the moduli space of parabolic semistable vector bundles of rank on with trivial determinant and fixed parabolic structure of type at , where each weight is in . On there is a canonical line bundle , whose global sections are called generalized parabolic -theta functions of order . In this paper we prove the existence of such nonzero nonabelian theta functions, thus establishing a part of higher genus generalizations of the celebrated saturation conjectures.

     

Self-commutators of automorphic composition operators on the Dirichlet space

Let be a conformal automorphism on the unit disk and be the composition operator on the Dirichlet space induced by . In this article we completely determine the point spectrum, spectrum, essential spectrum and essential norm of the operators and self-commutators of , which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.

     

The depth of an ideal with a given Hilbert function

Let denote the polynomial ring in variables over a field with each . Let be a homogeneous ideal of with and the Hilbert function of the quotient algebra . Given a numerical function satisfying for some homogeneous ideal of , we write for the set of those integers such that there exists a homogeneous ideal of with and with . It will be proved that one has either for some or .

     

A rigidity theorem for holomorphic generators on the Hilbert ball

We present a rigidity property of holomorphic generators on the open unit ball of a Hilbert space . Namely, if is the generator of a one-parameter continuous semigroup on such that for some boundary point , the admissible limit - , then vanishes identically on .

     

On diophantine approximation along algebraic curves

Let be a quadratic form such that the associated algebraic curve contains a rational point. Here we show that there exists a domain such that for almost all , there exists an infinite sequence of nonzero integer triples satisfying the following two properties: (i) For each , is an excellent rational approximation to , in the sense that

and (ii) is a rational point on the curve . In addition, we give explicit values of for which both (i) and (ii) hold, and produce a similar result for a certain class of cubic curves.

     

The effective Chebotarev density theorem and modular forms modulo

Suppose that (resp. ) is a modular form of integral (resp. half-integral) weight with coefficients in the ring of integers of a number field . For any ideal , we bound the first prime for which (resp. ) is zero ( ). Applications include the solution to a question of Ono (2001) concerning partitions.

     

Stable algebras of entire functions

Suppose that and belong to the algebra generated by the rational functions and an entire function of finite order on and that has algebraic polar variety. We show that either or , where is a polynomial and are rational functions. In the latter case, belongs to the algebra generated by the rational functions, and .

The stability property is related to the problem of algebraic dependence of entire functions over the ring of polynomials. The case of algebraic dependence over of two entire or meromorphic functions on is completely resolved in this paper.

     

Rademacher multiplicator spaces equal to

Let be a rearrangement invariant function space on [0,1]. We consider the Rademacher multiplicator space of measurable functions such that for every a.e. converging series , where are the Rademacher functions. We characterize the situation when . We also discuss the behaviour of partial sums and tails of Rademacher series in function spaces.

     

Numerical peak points and numerical Šilov boundary for holomorphic functions

In this paper, we characterize the numerical and numerical strong-peak points for when is the complex space or . We also prove that for all is the numerical Šilov boundary for

     

Radial limits of inner functions and Bloch spaces

Let be an inner function in the unit ball , . Assume that

where and is the radial derivative. Then, for all , the set has a non-zero real Hausdorff -content, and it has a non-zero complex Hausdorff -content.

     

On the residual finiteness and other properties of (relative) one-relator groups

A relative one-relator presentation has the form where is a set, is a group, and is a word on . We show that if the word on obtained from by deleting all the terms from has what we call the unique max-min property, then the group defined by is residually finite if and only if is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the relative asphericity of presentations of the form (Theorem 6).

     

Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if and are operator algebras, then any bounded epimorphism of onto is completely bounded provided that contains a norming -subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a -algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a -algebra is similar to a -representation precisely when the image operator algebra -norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras of -diagonals () satisfying extends uniquely to a -isomorphism of the -algebras generated by and ; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.

     

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.