Higher Level Affine Crystals and Young Walls

Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals for quantum affine algebras of type , , , , , and . The irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls. The notion of slices and splitting of blocks plays a crucial role in the construction of crystals.Presented by Peter Littelman.     

Regularity in Capacity and the Dirichlet Laplacian

Given an open set in , we prove that every function in is zero everywhere on the boundary if and only if is regular in capacity. If in addition is bounded, then it is regular in capacity if and only if the mapping from into is injective, where denotes the Perron solution of the Dirichlet problem. Let be the set of all open subsets of which are regular in capacity. Then one can define metrics and on only involving the resolvent of the Dirichlet Laplacian. Convergence in those metrics will be defined to be the local/global uniform convergence of the resolvent of the Dirichlet Laplacian applied to the constant function . We prove that the spaces and are complete and contain the set of all open sets which are regular in the sense of Wiener (or Dirichlet regular) as a closed subset.     

Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds

Let be a one-parameter family of positive integral operators on a locally compact space . For a possibly non-uniform partition of define a finite measure on the path space by using a) for the transition between any two consecutive partition times of distance and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary normalize the result to get a probability measure. We prove a version of Chernoff's theorem of semigroup theory and tightness results which yield convergence in law of such measures as the partition gets finer. In particular let be a closed smooth submanifold of a manifold . We prove convergence of Brownian motion on , conditioned to visit at all partition times, to a process on whose law has a density with respect to Brownian motion on which contains scalar, mean and sectional curvatures terms. Various approximation schemes for Brownian motion on are also given.      

Existence and Stability Results for Renormalized Solutions to Noncoercive Nonlinear Elliptic Equations with Measure Data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is

Uniform Estimates of the Fundamental Solution for a Family of Hypoelliptic Operators

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: , in where is the Laplace operator, , and the limit operator is hypoelliptic. It is well known that admits a fundamental solution . Here we establish some a priori estimates uniform in of it, using a modification of the lifting technique of Rothschild and Stein. As a consequence we deduce some a priori estimates uniform in , for solutions of the approximated equation . These estimates can be used in particular while studying regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.     

Quasi-regular Dirichlet Forms and L^p-resolvents on Measurable Spaces

We prove that for any semi-Dirichlet form on a measurable Lusin space E there exists a Lusin topology with the given -algebra as the Borel -algebra so that becomes quasi-regular. However one has to enlarge E by a zero set. More generally a corresponding result for arbitrary -resolvents is proven.     

Intrinsic Ultracontractivity for Schrödinger Operators with Mixed Boundary Conditions

Let be a domain in , . Let be a divergence form uniformly elliptic operator with Dirichlet boundary conditions on and Neumann boundary conditions on , where is a closed subset of . We prove intrinsic ultracontractivity for the semigroup associated to the Schrödinger operator , where is a potential in the Kato class, provided that is locally Lipschitz and is given by the boundary of either a Hölder domain of order or a uniformly Hölder domain of order , . Our results extend to the mixed boundary case the results of Bañuelos, Bass and Burdzy, Bass and Hsu, and Davies and Simon.     

Two-parameter p, q-variation Paths and Integrations of Local Times

In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter -variation path integrals. Our condition of locally bounded -variation is more natural and easy to verify than those of Young. This result can be easily generalized to multi-parameter case. The second result is to define the integral of local time pathwise and then give generalized It’s formula when is only of bounded -variation in . In the case that is of locally bounded variation in , the integral is the Lebesgue–Stieltjes integral and was used by Elworthy, Truman and Zhao. When is of only locally -variation, where , , and , the integral is a two-parameter Young integral of -variation rather than a Lebesgue–Stieltjes integral. In the special case that is independent of , we give a new condition for Meyer's formula and is defined pathwise as a Young integral. For this we prove the local time is of -variation in for each , for each almost surely (-variation in the sense of Lyons and Young, i.e. ).     

The Zaremba Problem with Singular Interfaces as a Corner Boundary Value Problem

We study mixed boundary value problems for an elliptic operator on a manifold with boundary , i.e., in on , where is subdivided into subsets with an interface and boundary conditions on that are Shapiro–Lopatinskij elliptic up to from the respective sides. We assume that is a manifold with conical singularity . As an example we consider the Zaremba problem, where is the Laplacian and Dirichlet, Neumann conditions. The problem is treated as a corner boundary value problem near which is the new point and the main difficulty in this paper. Outside the problem belongs to the edge calculus as is shown in Bull. Sci. Math. (to appear).With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.     

Equivalence of $n$-point Gauss–Chebyshev rule and $4n$-point midpoint rule in computing the period of a Lotka–Volterra system

Two integrals (3.6), (4.7) for the period of a periodic solution of the Lotka–Volterra system are presented in terms of two inverse functions of restricted on , , respectively. In computing this period numerically, the integral (3.6), which possesses a weak singularity of the square root type at each endpoint of the integration, is an excellent example of using the Gauss–Chebyshev integration rule of the first kind; while the integral (4.7), which is an integral of a smooth periodic function over its period , is an excellent example of using the midpoint rule, but not the trapezoidal rule, suggested by Waldvogel [39, 40], due to a removable singularity of the integrand at , , , , and , respectively. This paper shows, in computing the period of a periodic solution of the Lotka–Volterra system, the -point Gauss–Chebyshev integration rule of the first kind applied to the integral (3.6) becomes the -point midpoint rule to the integral (4.7). Dedicated to R. Bruce Kellogg on the occasion of his 75th birthday.     

Subordination of Regular Kernels and Abelian Theorems for Quotients

We prove general boundary limit theorems of abelian type for quotients of functions defined in the half space . The functions considered are defined as convolutions of a kernel with Borel measures defined on the boundary . Our theorems are of the form

Coherence for Product Monoids and their Actions

Let and be two monoids (algebras) in a monoidal category . Further let be a distributive law in the sense of [J. Beck, Lect. Notes Math., 80:119–140, 1969]; naturally yields a monoid . Consider a word in the symbols , , and . The first coherence theorem proved in this paper asserts that all morphisms coincide in , provided they arise as composites of morphisms which are -products of ’s ‘canonical’ structure morphisms, and of , , , , , , , and . Assume now that an object is endowed with both an -object structure , and an -object structure . Further assume that these two structures are compatible, in the sense that they naturally yield an -object . Let be a word in , , , and , which contains a single instance of , in the rightmost position. The second coherence theorem states that all morphisms coincide in , provided they arise as composites of morphisms which are -products of ’s ‘canonical’ structure morphisms, and of , , , , , , , , , and .     

Martingales on Frame Bundles

Let M be a smooth manifold endowed with a symmetric connection . There are two important ways of lift the connection of M to the frame bundle BM, the canonical lift and the horizontal lift . The aim of this work is determine the -martingales and the -martingales on BM. Our results allow to establish new characterizations of harmonic maps from Riemannian manifolds to frame bundles. The research of P. Catuogno is supported in part by FAPESP 01/13158-4 and S. Stelmastchuk is fully supported by FAPESP 02/12154-8.     

Branching Rules, Kostka–Foulkes Polynomials and q-multiplicities in Tensor Product for the Root Systems B n , C n and D n

The Kostka–Foulkes polynomials related to a root system can be defined as alternating sums running over the Weyl group associated to . By restricting these sums over the elements of the symmetric group when is of type or , we obtain again a class of Kostka–Foulkes polynomials. When is of type or there exists a duality between these polynomials and some natural -multiplicities and in tensor products [11]. In this paper we first establish identities for the which implies in particular that they can be decomposed as sums of Kostka–Foulkes polynomials with nonnegative integer coefficients. Moreover these coefficients are branching coefficients This allows us to clarify the connection between the -multiplicities and the polynomials defined by Shimozono and Zabrocki. Finally we show that and coincide up to a power of with the one dimension sum introduced by Hatayama and co-workers when all the parts of are equal to , which partially proves some conjectures of Lecouvey and Shimozono and Zabrocki.Presented by P. Littelmann.     

The Pluri-Fine Topology is Locally Connected

We prove that the pluri-fine topology on any open set in is locally connected. This answers a question by Fuglede.     

Hilbert Space Representations of Cross Product Algebras II

In this paper, we study and classify Hilbert space representations of cross product -algebras of the quantized enveloping algebra with the coordinate algebras of the quantum motion group and of the complex plane, and of the quantized enveloping algebra with the coordinate algebras of the quantum group and of the quantum disc. Invariant positive functionals and the corresponding Heisenberg representations are explicitly described.Presented by S.L. Woronowicz.     

Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

Generic Extensions of Prinjective Modules

Let be an algebraically closed field and let be a finite partially ordered set of finite prinjective type. We study generic extensions of prinjective modules over the incidence -algebra of . We prove that there exist generic extensions of prinjective -modules and describe properties of the monoid of generic extensions.Partially supported by Polish KBN Grant 5 P0 3A 015 21.     

On Non-M-Cosingular Completely ⊕-Supplemented Modules

In this paper, it is shown that any non--cosingular -supplemented module is if and only if has the summand intersection property. Let be any module such that has a coclosure in . Then we prove that is (completely) -supplemented if and only if for some submodule of such that and both are (completely) -supplemented.     

Symmetric Stable Processes on Unbounded Domains

Let be a symmetric -stable process in , , . We give necessary and sufficient condition under which the expectation of a very general function of the exit time from horns is finite. These domains include the symmetric domains given by increasing functions studied earlier by various authors. Our methods differ from those in earlier papers in that we obtain our results from estimates on the transition densities instead of harmonic measure. Some of this estimates are of independent interest.Supported in part by NSF grant #9700585-DMS and RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP.