Egorov’s theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associ...

We study Fredholmness of nonlocal boundary value problems for fourth order elliptic differential-operator equations in UMD Banach spaces. The main condition is given in terms of R-boundedness of some families of bounded operators generated by inverse of the characteristic operator pencil of the equation. Then we prove an isomorphism of the problem on the semi-axis, for some special boundary con...

Maxwell theory can be studied in a gauge which is invariant under conformal rescalings of the metric, and first proposed by Eastwood and Singer. This paper studies the corresponding quantization in flat Euclidean 4-space. The resulting ghost operator is a fourth-order elliptic operator, while the operator P on perturbations Aμ of the potential is a sixth-order elliptic operator. The operator P ...

In the paper, we present a family of multivariate compactly supported scaling functions, which we call as elliptic scaling functions. The elliptic scaling functions are the convolution of elliptic splines, which correspond to homogeneous elliptic differential operators, with distributions. The elliptic scaling functions satisfy refinement relations with real isotropic dilation matrices. The ell...

We define a theta operator between spaces of Siegel modular forms (mod p), generalizing the theta operator for (elliptic) modular forms.

A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations Abstract This paper presents constructive a posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations (PDEs) on a bounded domain. This type of estimates plays an important role in the numerical verification of the...

1. Review of Differential Geometry 2 2. Definition of an Elliptic Operator 5 3. Properties of Elliptic Operators 7 4. Example of an Elliptic Operator 9 5. Example: The Euler Characteristic 12 6. Example: The Signature Invariant 14 7. A Theorem of Atiyah, Frank and Mayer 18 8. Clifford Algebras 20 9. A Diversion: Constructing Vector Fields on Spheres using Clifford Algebras 23 10. Topological In...

We develop the general a priori error analysis of residual-free bubble nite element approximations to non-self-adjoint elliptic problems of the form ("A+C)u = f subject to homogeneous Dirichlet boundary condition, where A is a symmetric second-order elliptic operator, C is a skew-symmetric rst-order diierential operator, and " is a positive parameter. Optimal-order error bounds are derived in v...

We give the infinite-dimensional representation for the elliptic K-operator satisfying the boundary Yang-Baxter equation. By restricting the functional space to finite-dimensional space, we construct the elliptic K-matrix associated to Belavin's completely Z-symmetric R-matrix.