In this paper we present a formula for the determinant of a matrixvalued elliptic differential operator of even order on a line segment [0, T] with boundary conditions.

For quantum integrable models with elliptic R-matrix, we construct the Baxter Q-operator in infinite-dimensional representations of the algebra of observables.

We prove that the single layer potential operator of planar linear elastostatics is elliptic in [H(Γ)], if the domain Ω is scaled appropriately.

In this article, we will study polynomial growth solutions to a uniformly elliptic operator of non-divergence form defined on R n .

Solution methods for nonlinear boundary value problems form one of the most important topics in applied mathematics and, similarly to linear equations, preconditioned iterative methods are the most efficient tools to solve such problems. For linear equations, the theory of equivalent operators in Hilbert space has proved an efficient organized framework for the study of preconditioners [6, 9], ...

The localization of vibrations is a widely observed, but little understood physical phenomenon. Roughly speaking, the effect of localization is a confinement of some eigenfunctions of an elliptic operator to a small portion of the original domain in the presence of irregularities of the boundary or of the coefficients of the underlying operator. Until recently, there have been essentially no ma...

The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and is closely related to analysis o...

1. The SpinC-Dirac operator and the SpinC-quantization In this section we reformulate the “SpinC-quantization commutes with cobordism” principal in a more analytic language. In Subsection 1.1, we briefly recall the notion of SpinC-Dirac operator. We refer the reader to [2, 6] for details. We also express the SpinC-quantization of an orbifold M = X/G in terms of the index of the lift of the Spin...

There are known methods of approximating the solution of parabolic 2 nd order systems by solving stochastic diierential equations instead. The main idea is based on the fact that stochastic diierential equation deenes a diiusion process, generated by an elliptic diierential operator on R d. We propose a diierence scheme for the ellip-tic operator, which possesses the structure of Markov (jump) ...