Global co-stationarity of the ground model from a new countable length sequence

Suppose are models of ZFC with the same ordinals, and that for all regular cardinals in , satisfies . If contains a sequence for some ordinal , then for all cardinals in with regular in and , is stationary in . That is, a new -sequence achieves global co-stationarity of the ground model.

     

Large groups and their periodic quotients

We first give a short group theoretic proof of the following result of Lackenby. If is a large group, is a finite index subgroup of admitting an epimorphism onto a non-cyclic free group, and are elements of , then the quotient of by the normal subgroup generated by is large for all but finitely many . In the second part of this note we use similar methods to show that for every infinite sequence of primes , there exists an infinite finitely generated periodic group with descending normal series , such that and is either trivial or abelian of exponent .

     

Approximation of holomorphic maps with a lower bound on the rank

Let be a closed polydisc or ball in , and let be a quasi-projective algebraic manifold which is Zariski locally equivalent to , or a complement of an algebraic subvariety of codimension in such a manifold. If is an integer satisfying , then every holomorphic map from a neighborhood of to with rank at every point of can be approximated uniformly on by entire maps with rank at every point of .

     

Julia sets converging to the unit disk

We consider the family of rational maps , where and is small. If is equal to 0, the limiting map is and the Julia set is the unit circle. We investigate the behavior of the Julia sets of when tends to 0, obtaining two very different cases depending on and . The first case occurs when ; here the Julia sets of converge as sets to the closed unit disk. In the second case, when one of or is larger than , there is always an annulus of some fixed size in the complement of the Julia set, no matter how small is.

     

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.

     

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.

     

Tauberian type theorem for operators with interpolation spectrum for Hölder classes

We consider an invertible operator on a Banach space whose spectrum is an interpolating set for Hölder classes. We show that if , , with and , then for all , assuming that satisfies suitable regularity conditions. When is a Hilbert space and (i.e. is a contraction), we show that under the same assumptions, is unitary and this is sharp.

     

Algebraic characterizations of measure algebras

We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean -algebra. For instance, a Boolean -algebra is a measure algebra if and only if is the union of a chain of sets such that for every ,

(i)

every antichain in has at most elements (for some integer ),

(ii)

if is a sequence with for each , then , and

(iii)

for every , if is a sequence with , then for eventually all , .

The chain is essentially unique.

     

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.

     

Detecting completeness from Ext-vanishing

Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let be a commutative noetherian ring and an ideal in the Jacobson radical of . Let be the -adic completion of . If is a finitely generated -module such that for all , then is -adically complete.

     

A lower bound for the equilateral number of normed spaces

We show that if the Banach-Mazur distance between an -dimensional normed space and is at most , then there exist equidistant points in . By a well-known result of Alon and Milman, this implies that an arbitrary -dimensional normed space admits at least equidistant points, where is an absolute constant. We also show that there exist equidistant points in spaces sufficiently close to , .

     

Uniqueness of structures for connective covers

We refine our earlier work on the existence and uniqueness of structures on -theoretic spectra to show that the connective versions of real and complex -theory as well as the connective Adams summand at each prime have unique structures as commutative -algebras. For the -completion we show that the McClure-Staffeldt model for is equivalent as an ring spectrum to the connective cover of the periodic Adams summand . We establish a Bousfield equivalence between the connective cover of the Lubin-Tate spectrum and .

     

Free resolutions of parameter ideals for some rings with finite local cohomology

Let be a -dimensional local ring, with maximal ideal , containing a field and let be a system of parameters for . If and the local cohomology module is finitely generated, then there exists an integer such that the modules have the same Betti numbers, for all .

     

Finite rank Toeplitz operators on the Bergman space

Given a complex Borel measure with compact support in the complex plane the sesquilinear form defined on analytic polynomials and by , determines an operator from the space of such polynomials to the space of linear functionals on . This operator is called the Toeplitz operator with symbol . We show that has finite rank if and only if is a finite linear combination of point masses. Application to Toeplitz operators on the Bergman space is immediate.

     

On the compactness of the product of Hankel operators on the sphere

Consider Hankel operators and on the unit sphere in . If , then a necessary condition for to be compact is . We show that when , this condition is no longer necessary for to be compact.

     

A new construction of the unstable manifold for the measure-preserving Hénon map

Let denote the measure-preserving Hénon map with the parameter . The map has a hyperbolic fixed point . The main result of this paper is that the unstable mainfold of is the iterated limit of a very simple set. Informally,

where is the line and denotes the unstable manifold of .

     

Solutions to arithmetic convolution equations

In the -algebra of arithmetic functions , endowed with the usual pointwise linear operations and the Dirichlet convolution, let denote the convolution power with factors . We investigate the solvability of polynomial equations of the form

with fixed coefficients . In some cases the solutions have specific properties and can be determined explicitly. We show that the property of the coefficients to belong to convergent Dirichlet series transfers to those solutions , whose values are simple zeros of the polynomial . We extend this to systems of convolution equations, which need not be of polynomial-type.

     

Codes over rings of size four, Hermitian lattices, and corresponding theta functions

Let be an imaginary quadratic field with ring of integers , where is a square free integer such that , and let is a linear code defined over . The level theta function of is defined on the lattice , where is the natural projection. In this paper, we prove that:

i) for any such that , and have the same coefficients up to ,

ii) for , determines the code uniquely,

iii) for , there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to .

     

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 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 .