Amenable actions and almost invariant sets

In this paper, we study the connections between properties of the action of a countable group on a countable set and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of on , where is a measure space. In particular, we show that the action of on is amenable iff the shift has almost invariant sets.

     

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 .

     

A landing theorem for periodic rays of exponential maps

For the family of exponential maps , we show the following analog of a theorem of Douady and Hubbard concerning polynomials. Suppose that is a periodic dynamic ray of an exponential map with nonescaping singular value. Then lands at a repelling or parabolic periodic point. We also show that there are periodic dynamic rays landing at all periodic points of such an exponential map, with the exception of at most one periodic orbit.

     

Large cardinals with few measures

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly many normal measures on the least measurable cardinal . This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of strong compactness or supercompactness measures on can be exactly if is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.

     

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 .

     

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).

     

Row and column finite matrices

Consider the ring of all column finite matrices over a ring . We prove that each such matrix is conjugate to a row and column finite matrix if and only if is right Noetherian and is countable. We then demonstrate that one can perform this conjugation on countably many matrices simultaneously. Some applications and limitations are given.

     

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.

     

On the distribution of Kloosterman sums

For a prime , we consider Kloosterman sums

over a finite field of elements. It is well known that due to results of Deligne, Katz and Sarnak, the distribution of the sums when runs through is in accordance with the Sato-Tate conjecture. Here we show that the same holds where runs through the sums for , for any two sufficiently large sets .

We also improve a recent bound on the nonlinearity of a Boolean function associated with the sequence of signs of Kloosterman sums.

     

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.

     

Differential structure of the Thompson components of selfadjoint operators

Different equivalence relations are defined in the set of selfadjoint operators of a Hilbert space in order to extend a very well known relation in the cone of positive operators. As in the positive case, for the equivalence class admits a differential structure, which is compatible with a complete metric defined on . This metric coincides with the Thompson metric when is positive.

     

A note on equilibrium points of Green's function

We answer a question raised by Ahmet Sebbar and Thérèse Falliero (2007) by showing that for every finitely connected planar domain there exists a compact subset , independent of , containing all critical points of Green's function of with pole at .

     

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.

     

Perturbation of Wigner matrices and a conjecture

Let be an arbitrary self-adjoint matrix and be an (random) Wigner matrix. We show that is positive definite in the average. This partially answers a long-standing conjecture. On the basis of asymptotic freeness our result implies that is positive definite whenever the noncommutative random variables and are in free relation, with semicircular.

     

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.

     

Strong compactness and a partition property

We show that if holds for every , then is strongly compact.

     

Sharp bounds for the valence of certain harmonic polynomials

In Khavinson and Swiatek (2002) it was proved that harmonic polynomials , where is a holomorphic polynomial of degree , have at most complex zeros. We show that this bound is sharp for all by proving a conjecture of Sarason and Crofoot about the existence of certain extremal polynomials . We also count the number of equivalence classes of these polynomials.

     

On Generic differential -extensions

Let be an algebraically closed field with trivial derivation and let denote the differential rational field , with , , , , differentially independent indeterminates over . We show that there is a Picard-Vessiot extension for a matrix equation , with differential Galois group , with the property that if is any differential field with field of constants , then there is a Picard-Vessiot extension with differential Galois group if and only if there are with well defined and the equation giving rise to the extension .

     

Directional derivative estimates for Berezin's operator calculus

Directional derivative estimates for Berezin symbols of bounded operators on Bergman spaces of arbitrary bounded domains in are obtained. These estimates also hold in the setting of the Segal-Bargmann space on . It is also shown that our estimates are sharp at every point of by exhibiting the optimizers explicitly.

     

On the plane section of an integral curve in positive characteristic

If is an integral curve and an algebraically closed field of characteristic 0, it is known that the points of the general plane section of are in uniform position. From this it follows easily that the general minimal curve containing is irreducible. If char, the points of may not be in uniform position. However, we prove that the general minimal curve containing is still irreducible.