Elliptic

This is the first book on elliptic quantum groups, i.e., quantum groups associated to elliptic solutions of the Yang-Baxter equation. Based on research by the author and his collaborators, the book presents a comprehensive survey on the subject including a brief history of formulations and applications, a detailed formulation of the elliptic quantum group in the Drinfeld realization, explicit construction of both finite and infinite-dimensional representations, and a construction of the vertex operators as intertwining operators of these representations.

This comprehensive yet concise book deals with nonlocal elliptic differential operators, whose coefficients involve shifts generated by diffeomorophisms of the manifold on which the operators are defined. The main goal of the study is to relate analytical invariants (in particular, the index) of such elliptic operators to topological invariants of the manifold itself. This problem can be solved by modern methods of noncommutative geometry.

After two decades of research and development, elliptic curve cryptography now has widespread exposure and acceptance. Industry, banking, and government standards are in place to facilitate extensive deployment of this efficient public-key mechanism.

The aim of our book is the investigation of the behavior of strong and weak solutions to the regular oblique derivative problems for second order elliptic equations, linear and quasi-linear, in the neighborhood of the boundary singularities. The main goal is to establish the precise exponent of the solution decrease rate and under the best possible conditions. The question on the behavior of solutions of elliptic boundary value problems near boundary singularities is of great importance for its many applications, e.g., in hydrodynamics, aerodynamics, fracture mechanics, in the geodesy etc. Only few works are devoted to the regular oblique derivative problems for second order elliptic equations in non-smooth domains. All results are given with complete proofs. The monograph will be of interest to graduate students and specialists in elliptic boundary value problems and their applications.

This handbook is inspired by occasional questions from my stu­ dents and coworkers as to how they can obtain easily the best network functions from which they can complete their filter design projects to satisfy certain criteria. They don't need any help to design the filter. They need only the network function. It appears that this crucial step can be a bottleneck to designers. This handbook is meant to supply the information for those who need a quick answer to a simple question of this kind. There are three most useful basic standard low-pass magnitude characteristics used in filter design. These are the Butterworth, the Chebyshev, and the elliptic characteristics. The Butterworth charac­ teristic is maximally flat at the origin. The Chebyshev characteristic gives equal-ripple variation in the pass band. The elliptic character­ istic gives equal-ripple variation in both the pass band and the stop band. The Butterworth and the Chebyshev characteristics are fairly easy to use, and formulas for their parameters are widely available and fairly easy to apply. The theory and derivation of formulas for the elliptic characteristic, however, are much more difficult to handle and understand. This is chiefly because their original development made use of the Jacobian elliptic functions, which are not familiar to most electrical engineers. Although there are several other methods of developing this characteristic, such as the potential analogy, the Chebyshev rational functions, and numerical techniques, most filter designers are as unfamiliar with these methods as they are with the elliptic functions.

This monograph explores applications of Carleman estimates in the study of stabilization and controllability properties of partial differential equations, including the stabilization property of the damped wave equation and the null-controllability of the heat equation. All analysis is performed in the case of open sets in the Euclidean space; a second volume will extend this treatment to Riemannian manifolds.

This monograph explores applications of Carleman estimates in the study of stabilization and controllability properties of partial differential equations, including quantified unique continuation, logarithmic stabilization of the wave equation, and null-controllability of the heat equation. Where the first volume derived these estimates in regular open sets in Euclidean space and Dirichlet boundary conditions, here they are extended to Riemannian manifolds and more general boundary conditions.

The study of qualitative aspects of PDE's has always attracted much attention from the early beginnings. More recently, once basic issues about PDE's, such as existence, uniqueness and stability of solutions, have been understood quite well, research on topological and/or geometric properties of their solutions has become more intense.