On zeros of Eisenstein series for genus zero Fuchsian groups

Let SL be a genus zero Fuchsian group of the first kind with as a cusp, and let be the holomorphic Eisenstein series of weight on that is nonvanishing at and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on and on a choice of a fundamental domain , we prove that all but possibly of the nontrivial zeros of lie on a certain subset of . Here is a constant that does not depend on the weight, is the upper half-plane, and is the canonical hauptmodul for

     

The -invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal is not principal.

     

planes in

We establish a homeomorphism between the moduli space of ordered -tuples of 2-dimensional linear subspaces (mod ) and the quotient by simultaneous conjugation of a certain open subset . For , this leads to an explicit computation of the moduli space of central 2-arrangements in mod and its subspace of those classes that contain a complex hyperplane arrangement.

     

On a conjecture about MRA Riesz wavelet bases

Let be a compactly supported refinable function in such that the shifts of are stable and for a -periodic trigonometric polynomial . A wavelet function can be derived from by . If is an orthogonal refinable function, then it is well known that generates an orthonormal wavelet basis in . Recently, it has been shown in the literature that if is a -spline or pseudo-spline refinable function, then always generates a Riesz wavelet basis in . It was an open problem whether can always generate a Riesz wavelet basis in for any compactly supported refinable function in with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function does not generate a Riesz wavelet basis in . Our proof is based on some necessary and sufficient conditions on the -periodic functions and in such that the wavelet function , defined by , generates a Riesz wavelet basis in .

     

Maximality theorems for Fréchet algebras

The algebra of unbounded holomorphic functions that is contained in the algebra is studied. For in but not in , we show that the algebra generated by and is dense in for all .

     

Necessary conditions for Schatten class localization operators

We study time-frequency localization operators of the form , where is the symbol of the operator and are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for , the Schatten class of order , is that belongs to the modulation space and the window functions to the modulation space . Here we prove a partial converse: if for every pair of window functions with a uniform norm estimate, then the corresponding symbol must belong to the modulation space . In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For and , we recapture earlier results, which were obtained by different methods.

     

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 the symplectic phase space of KdV

We prove that the Birkhoff map for KdV constructed on can be interpolated between and . In particular, the symplectic phase space can be described in terms of Birkhoff coordinates.

     

Arens-Michael enveloping algebras and analytic smash products

Let be a finite-dimensional complex Lie algebra, and let be its universal enveloping algebra. We prove that if , the Arens-Michael envelope of is stably flat over (i.e., if the canonical homomorphism is a localization in the sense of Taylor (1972), then is solvable. To this end, given a cocommutative Hopf algebra and an -module algebra , we explicitly describe the Arens-Michael envelope of the smash product as an ``analytic smash product' of their completions w.r.t. certain families of seminorms.

     

On homeomorphisms and quasi-isometries of the real line

We show that the group of piecewise-linear homeomorphisms of having bounded slopes surjects onto the group of all quasi-isometries of . We prove that the following groups can be imbedded in : the group of compactly supported piecewise-linear homeomorphisms of , the Richard Thompson group , and the free group of continuous rank.

     

Linear functionals on the Cuntz algebra

For a pure state on , which is an extension of a pure state on with the property that if is a corresponding representation, then , induces a unital shift of of the Powers index . We describe states on by using sequences of unit vectors in . We study the linear functionals on the Cuntz algebra whose restrictions are the product pure state on . We find conditions on the sequence of unit vectors for which the corresponding linear functionals on become states under these conditions.

     

Hilbert modules over a class of semicrossed products

Given the disk algebra and an automorphism , there is associated a non-self-adjoint operator algebra called the semicrossed product of with . Buske and Peters showed that there is a one-to-one correspondence between the contractive Hilbert modules over and pairs of contractions and on satisfying . In this paper, we show that the orthogonally projective and Shilov Hilbert modules over correspond to pairs of isometries on satisfying . The problem of commutant lifting for is left open, but some related results are presented.     

Conservativeness of diffusion processes with drift

We show the conservativeness of the Girsanov transformed diffusion process by drift with or 4d/(d+2)$">, or if is of the Hardy class with sufficiently small coefficient of energy . Here 0$"> is the lower bound of the symmetric measurable matrix-valued function appearing in the given Dirichlet form. In particular, our result improves the conservativeness of the transformed process by when .

     

A counterexample to the maximality of toric varieties

We present a counterexample to the conjecture of Bihan, Franz, McCrory, and van Hamel concerning the maximality of toric varieties. There exists a six dimensional projective toric variety with the sum of the Betti numbers of strictly less than the sum of the Betti numbers of .

     

Sobolev spaces and the Cayley transform

The generalised Cayley transform  from an Iwasawa -group into the corresponding real unit sphere  induces isomorphisms between suitable Sobolev spaces and . We study the differential of  , and we obtain a criterion for a function to be in  .

     

Whitney property in two dimensional Sobolev spaces

For , we prove that all the functions of satisfy the Whitney property; i.e., if is such that (in the sense of capacity) on a connected set , then is constant on .

     

Class numbers of quadratic fields and

Let be a square free integer. We shall show that there exist infinitely many positive fundamental discriminants 0$"> with a positive density such that the class numbers of quadratic fields and are both not divisible by 3.

     

Pointwise convergence of bounded cascade sequences

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For an initial function , a cascade sequence is constructed by the iteration where is defined by In this paper, under a condition that the sequence is bounded in , we prove that the following three statements are equivalent: (i) converges . (ii) For , there exist a positive constant and a constant such that (iii) For some converges in . An example is presented to illustrate our result.

     

Monoid of self-equivalences and free loop spaces

Let be a simply-connected closed oriented -dimensional manifold. We prove that for any field of coefficients there exists a natural homomorphism of commutative graded algebras where is the loop algebra defined by Chas and Sullivan. As usual denotes the monoid of self-equivalences homotopic to the identity, and the space of based loops. When is of characteristic zero, yields isomorphisms where denotes the Hodge decomposition on .

     

The yellow cake

In this paper we consider the following property:

For every function there are functions

(for ) such that

We show that, despite some expectation suggested by S. Shelah (1997), does not imply . Next, we introduce cardinal characteristics of the continuum responsible for the failure of .