Maximal functions with polynomial densities in lacunary directions

Given a real polynomial in one variable such that , we consider the maximal operator in ,

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We prove that is bounded on for 1$"> with bounds that only depend on the degree of .

     

Higher Weierstrass points on

We study the arithmetic properties of higher Weierstrass points on modular curves for primes . In particular, for , we obtain a relationship between the reductions modulo of the collection of -Weierstrass points on and the supersingular locus in characteristic .

     

On the inversion of the convolution and Laplace transform

We present a new inversion formula for the classical, finite, and asymptotic Laplace transform of continuous or generalized functions . The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of evaluated on a Müntz set of real numbers. The inversion sequence converges in the strongest possible sense. The limit is uniform if is continuous, it is in if , and converges in an appropriate norm or Fréchet topology for generalized functions . As a corollary we obtain a new constructive inversion procedure for the convolution transform ; i.e., for given and we construct a sequence of continuous functions such that .

     

An extension theorem for separately holomorphic functions with pluripolar singularities

Let be a pseudoconvex domain and let be a locally pluriregular set, . Put

Let be an open neighborhood of and let be a relatively closed subset of . For let be the set of all for which the fiber is not pluripolar. Assume that are pluripolar. Put

Then there exists a relatively closed pluripolar subset of the ``envelope of holomorphy' of such that:

,

for every function separately holomorphic on there exists exactly one function holomorphic on with on , and

is singular with respect to the family of all functions .

     

A black-box group algorithm for recognizing finite symmetric and alternating groups, I

We present a Las Vegas algorithm which, for a given black-box group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and permutation groups.

In this paper, we handle the case when the degree of the standard permutation representation is part of the input. In a sequel, we shall treat the case when the value of is not known in advance.

As an important ingredient in the theoretical basis for the algorithm, we prove the following result about the orders of elements of : the conditional probability that a random element is an -cycle, given that , is at least .

     

Limits of interpolatory processes

Given distinct real numbers and a positive approximation of the identity , which converges weakly to the Dirac delta measure as goes to zero, we investigate the polynomials which solve the interpolation problem

with prescribed data . More specifically, we are interested in the behavior of when the data is of the form for some prescribed function . One of our results asserts that if is sufficiently nice and has sufficiently well-behaved moments, then converges to a limit which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is , where is an arbitrary finite subset of the integer lattice , as their degree goes to infinity.

     

Hyperbolic mean growth of bounded holomorphic functions in the ball

We consider the hyperbolic Hardy class , . It consists of holomorphic in the unit complex ball for which and

where denotes the hyperbolic distance of the unit disc. The hyperbolic version of the Littlewood-Paley type -function and the area function are defined in terms of the invariant gradient of , and membership of is expressed by the property of the functions. As an application, we can characterize the boundedness and the compactness of the composition operator , defined by , from the Bloch space into the Hardy space .

     

Analytic models for commuting operator tuples on bounded symmetric domains

For a domain in and a Hilbert space of analytic functions on which satisfies certain conditions, we characterize the commuting -tuples of operators on a separable Hilbert space  such that is unitarily equivalent to the restriction of to an invariant subspace, where is the operator -tuple on the Hilbert space tensor product  . For the unit disc and the Hardy space , this reduces to a well-known theorem of Sz.-Nagy and Foias; for a reproducing kernel Hilbert space on such that the reciprocal of its reproducing kernel is a polynomial in and  , this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spaces for which ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation) on a Cartan domain corresponding to the parameter in the continuous Wallach set, and reproducing kernel Hilbert spaces for which is a rational function. Further, we treat also the more general problem when the operator is replaced by ,  being a certain generalization of a unitary operator tuple. For the case of the spaces on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on , which seems to be of an independent interest.

     

On the Clifford algebra of a binary form

The Clifford algebra of a binary form of degree is the -algebra , where is the ideal generated by . has a natural homomorphic image that is a rank Azumaya algebra over its center. We prove that the center is isomorphic to the coordinate ring of the complement of an explicit -divisor in , where is the curve and is the genus of .

     

On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds

The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold where is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and a discrete subgroup of acting on by left translations. For this purpose, we shall first show that if is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric then the restriction of to the center of is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group . If is one such group, we prove that no timelike geodesic in can be translated by an element of By the way, we rediscover that the Heisenberg Lie group admits a flat left-invariant Lorentz metric if and only if

     

Poset block equivalence of integral matrices

Given square matrices and with a poset-indexed block structure (for which an block is zero unless ), when are there invertible matrices and with this required-zero-block structure such that ? We give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain . As one application, when is a field we classify such matrices up to similarity by matrices respecting the block structure. We also give complete invariants for equivalence under the additional requirement that the diagonal blocks of and have determinant . The invariants involve an associated diagram (the ``-web') of -module homomorphisms. The study is motivated by applications to symbolic dynamics and -algebras.

     

Resolutions of ideals of quasiuniform fat point subschemes of

The notion of a quasiuniform fat point subscheme is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal defining are put forward. In a large range of cases, it is shown that the Hilbert function conjecture implies the resolution conjecture. In addition, the main result gives the first determination of the resolution of the th symbolic power of an ideal defining general points of when both and are large (in particular, for infinitely many for each of infinitely many , and for infinitely many for every 2$">). Resolutions in other cases, such as ``fat points with tails', are also given. Except where an explicit exception is made, all results hold for an arbitrary algebraically closed field . As an incidental result, a bound for the regularity of is given which is often a significant improvement on previously known bounds.

     

A classification and examples of rank one chain domains

A chain order of a skew field is a subring of so that implies Such a ring has rank one if , the Jacobson radical of is its only nonzero completely prime ideal. We show that a rank one chain order of is either invariant, in which case corresponds to a real-valued valuation of or is nearly simple, in which case and are the only ideals of or is exceptional in which case contains a prime ideal that is not completely prime. We use the group of divisorial of with the subgroup of principal to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index of in Using the covering group of and the result that the group ring is embeddable into a skew field for a skew field, examples of rank one chain orders are constructed for each possible exceptional case.

     

The periodic Euler-Bernoulli equation

We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem

where the functions and are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by and . Here we develop a theory analogous to the theory of the Hill operator .

We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or -spectrum.

Our new analysis begins with a detailed study of the zeros of the function , for any given ``quasimomentum' , where is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to is , where is the discriminant and the period of ). We show that the multiplicity of any zero of can be one or two and (for some ) if and only if is also a zero of another entire function , independent of . Furthermore, we show that has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each -gap. If is a double zero of , it may happen that there is only one Floquet solution with quasimomentum ; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree.

Next we show that if is an open -gap of the pseudospectrum (i.e., ), then the Floquet matrix has a specific Jordan anomaly at and .

We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by the eigenvalues of this multipoint problem and show that is also characterized as the set of values of for which there is a proper Floquet solution such that .

We also show (Theorem 7) that each gap of the -spectrum contains exactly one and each -gap of the pseudospectrum contains exactly two 's, counting multiplicities. Here when we say ``gap' or ``-gap' we also include the endpoints (so that when two consecutive bands or -bands touch, the in-between collapsed gap, or -gap, is a point). We believe that can be used to formulate the associated inverse spectral problem.

As an application of Theorem 7, we show that if is a collapsed (``closed') -gap, then the Floquet matrix is diagonalizable.

Some of the above results were conjectured in our previous works. However, our conjecture that if all the -gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.

     

A new variational characterization of -dimensional space forms

A Riemannian manifold is associated with a Schouten -tensor which is a naturally defined Codazzi tensor in case is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional defined on , where is the space of smooth Riemannian metrics on a compact smooth manifold and is the elementary symmetric functions of the eigenvalues of with respect to . We prove that if and a conformally flat metric is a critical point of with , then must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of with characterized the three-dimensional space forms.

     

The space for nondoubling measures in terms of a grand maximal operator

Let be a Radon measure on , which may be nondoubling. The only condition that must satisfy is the size condition , for some fixed . Recently, some spaces of type and were introduced by the author. These new spaces have properties similar to those of the classical spaces and defined for doubling measures, and they have proved to be useful for studying the boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space in terms of a maximal operator is given. It is shown that belongs to if and only if , and , as in the usual doubling situation.

     

Stratified transversality by isotopy

For an abstract stratified set or a -regular stratification, hence for any -, - or -regular stratification, we prove that after stratified isotopy of , a stratified subspace of , or a stratified map , can be made transverse to a fixed stratified map .

     

The free entropy dimension of hyperfinite von Neumann algebras

Suppose is a hyperfinite von Neumann algebra with a normal, tracial state and is a set of selfadjoint generators for . We calculate , the modified free entropy dimension of . Moreover, we show that depends only on and . Consequently, is independent of the choice of generators for . In the course of the argument we show that if is a set of selfadjoint generators for a von Neumann algebra with a normal, tracial state and has finite-dimensional approximants, then for any hyperfinite von Neumann subalgebra of Combined with a result by Voiculescu, this implies that if has a regular diffuse hyperfinite von Neumann subalgebra, then .