Kernels of Hankel operators and hyponormality of Toeplitz operators

We give a formula for and describe when . We explore the hyponormality of Toeplitz operators whose symbols are of circulant type and some more general types. In addition, we discuss formulas for and estimates of the rank of the self-commutator of a hyponormal Toeplitz operator. Received September 17, 1999 / Revised May 25, 2000 / Published online December 8, 2000     

Quasilinearity below the 1st eigenvalue

This paper establishes the existence of two nontrivial weak solutions for a quasilinear Dirichlet problem below the first eigenvalue via the mountain pass theorem.

     

Trees, parking functions, syzygies, and deformations of monomial ideals

For a graph , we construct two algebras whose dimensions are both equal to the number of spanning trees of . One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to -parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.

     

Local solvability of PDE with constant coefficients

The fundamental result of the paper consists in the following assertion: Let D be an open set in Rⁿ,f £ L₂(D), and let be a distribution in Rⁿ with a compact support, whose Fourier transform satisfies the condition

On the Bergman Theory for Solenoidal and Irrotational Vector Fields. II. Conformal Covariance and Invariance of the Main Objects

This is a continuation of our work (González-Cervantes et al. in On the Bergman theory for solenoidal and irrotational vector fields. I. General theory. Operator theory: advances and applications. Birkhauser, accepted) where for solenoidal and irrotational vector fields theory as well as for the Moisil–Théodoresco quaternionic analysis we introduced the notions of the Bergman space and the Bergman reproducing kernel and studied their main properties. In particular, we described the behavior of the Bergman theory for a given domain whenever the domain is transformed by a conformal map. The formulas obtained hint that the corresponding objects (spaces, operators, etc.) can be characterized as conformally covariant or invariant, and in the present paper we construct a series of categories and functors which allow us to give such characterizations in precise terms.     

Automatic structures,rational growth,and geometrically finite hyperbolic groups

Summary We show that the set of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic groupG is dense in the product of the sets over all maximal parabolic subgroupsP. The set of equivalence classes of biautomatic structures onG is isomorphic to the product of the sets over the cusps (conjugacy classes of maximal parabolic subgroups) ofG. Each maximal parabolicP is a virtually abelian group, so and were computed in [NS1].We show that any geometrically finite hyperbolic group has a generating set for which the full language of geodesics forG is regular. Moreover, the growth function ofG with respect to this generating set is rational. We also determine which automatic structures on such a group are equivalent to geodesic ones. Not all are, though all biautomatic structures are.Oblatum 14-VI-1993 & 4-I-1994Both authors acknowledge support from the NSF for this research.     

First search for the flavor changing neutral current decay D0-->gammagamma

Using 13.8 fb(-1) of data collected at or just below the Upsilon(4S) with the CLEO detector, we report the result of a search for the flavor changing neutral current process D0-->gammagamma. We observe no significant signal for this decay mode and determine 90% confidence level upper limits on the branching fractions B(D0-->gammagamma)/B(D0-->pi(0)pi(0))<0.033 and B(D0-->gammagamma)<2.9 x 10(-5).     

Hydrodynamics of resonance oscillations of columns of inelastic particles

We study oscillations of a one-dimensional (1D) column of N slightly inelastic particles, produced by a piston vibrating at one end of a closed tube. It is found that for large enough vibrational amplitudes of the piston, the column oscillates periodically with the period equal to the vibrational period. The oscillation patterns are governed by the shock waves propagating across the column. The averaged kinetic energy per particle is shown to be proportional to the square of the vibrational frequency, omega. This energy also strongly depends on the vibrational amplitude. The maximal value of this kinetic energy achievable by these external vibrations is found to be of order omega(2)L(2), where L is the total volume (length) of the tube free of particles. The above results on the column resonance oscillations are also predicted by a 3D hydrodynamic model of an inelastic granular gas.