An algebra of meromorphic corner symbols

Operators on manifolds with corners that have base configurations with geometric singularities can be analysed in the frame of a conormal symbolic structure which is in spirit similar to the one for conical singularities of Kondrat'ev's work. Solvability of elliptic equations and asymptotics of solutions are determined by meromorphic conormal symbols. We study the case when the base has edge singularities which is a natural assumption in a number of applications. There are new phenomena, caused by a specific kind of higher degeneracy of the underlying symbols. We introduce an algebra of meromorphic edge operators that depend on complex parameters and investigate meromorphic inverses in the parameter-dependent elliptic case. Among the examples are resolvents of elliptic differential operators on manifolds with edges.     

Minimal Bar Tableaux

Motivated by recent results of Stanley, we generalize the rank of a partition λ to the rank of a shifted partition S(λ). We show that the number of bars required in a minimal bar tableau of S(λ) is max(o, e + (ℓ(λ) mod 2)), where o and e are the number of odd and even rows of λ. As a consequence we show that the irreducible projective characters of S_n vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur’s Q_λ symmetric functions in terms of the power sum symmetric functions. Received November 20, 2003     

Exact controllability of infinite dimensional systems persists under small perturbations

This paper studies the concept of controllability for infinite-dimensional linear control systems in Banach spaces. First, we prove that the set of admissible control operators for the semigroup generator is unchanged if we perturb the generator by the Desch–Schappacher perturbations. Second we show that exact controllability is not changed by such perturbations.     

Bounded Positive Solutions of Schrödinger Equations in Two-Dimensional Exterior Domains

We prove under quite general assumptions the existence of a bounded positive solution to the semilinear Schrödinger equation in a two-dimensional exterior domain.     

Nonlinear Free Boundary Problems with Singular Source Terms

We prove the existence of solutions to nonlinear free boundary problem with singularities at given points.     

On the Long-Time Behavior of the Quantum Fokker-Planck Equation

We analyze the long-time behavior of transport equations for a class of dissipative quantum systems with Fokker-planck type diffusion operator, subject to confining potentials of harmonic oscillator type. We establish the existence and uniqueness of a non-equilibrium steady state for the corresponding dynamics. Further, using a (classical) convex Sobolev inequality, we prove an optimal exponential rate of decay towards this state and additionally give precise dispersion estimates in those cases, where no stationary state exists.     

Construction of the fundamental solution of disturbed parabolic equation

In present paper the parabolic equation solution is built. The construction is reduced to iterative procedure. And convergence of the latter is proven.     

Drift transforms and Green function estimates for discontinuous processes

In this paper, we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed process are comparable to those of the original process. As an application of the general results, the drift transform of symmetric stable processes is studied in detail. In particular, we show that the relativistic α-stable process in a bounded C^1,1-smooth open set D can be obtained from symmetric α-stable process in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable.     

Crossover from high-temperature vortex-liquid to low-temperature vortex-glass behavior in single phase HgBa2Ca2Cu3O8+δ superconductor

The ρ-T curves in our single phase HgBa₂Ca₂Cu₃O_8+δ superconductor were measured as a function of temperature and magnetic field, ρ=ρ₀exp(−U_eff/κ_BT). It can be transformed to another form d(lnρ)/d(1/T)=−U_eff+TdU_eff/dT, then this becomes a plot of the activation energy U_eff as a function of temperature. Our data plotted in these ways show a clear crossover from high-temperature two-dimensional vortex-liquid to a critical region associated with the low-temperature three-dimensional vortex-glass phase transition. The critical exponents v(z−1)=3.9±1.9 in this system are little different with previous measurements in BSCCO and YBCO systems.