The language of Einstein spoken by optical instruments

Einstein had to learn the mathematics of Lorentz transformations in order to complete his covariant formulation of Maxwell's equations. The mathematics of Lorentz transformations, called the Lorentz group, continues playing its important role in optical sciences. It is the basic mathematical language for coherent and squeezed states. It is noted that the six-parameter Lorentz group can be represented by two-by-two matrices. Since the beam transfer matrices in ray optics is largely based on two-by-two matrices or $ABCD$ matrices, the Lorentz group is bound to be the basic language for ray optics, including polarization optics, interferometers, lens optics, multilayer optics, and the Poincar\'e sphere. Because the group of Lorentz transformations and ray optics are based on the same two-by-two matrix formalism, ray optics can perform mathematical operations which correspond to transformations in special relativity. It is shown, in particular, that one-lens optics provides a mathematical basis for unifying the internal space-time symmetries of massive and massless particles in the Lorentz-covariant world.Comment: LaTex 8 pages, presented at the 10th International Conference on Quantum Optics (Minsk, Belarus, May-June 2004), to be published in the proceeding

Non-Smooth Spatio-Temporal Coordinates in Nonlinear Dynamics

This paper presents an overview of physical ideas and mathematical methods for implementing non-smooth and discontinuous substitutions in dynamical systems. General purpose of such substitutions is to bring the differential equations of motion to the form, which is convenient for further use of analytical and numerical methods of analyses. Three different types of nonsmooth transformations are discussed as follows: positional coordinate transformation, state variables transformation, and temporal transformations. Illustrating examples are provided.Comment: 15 figure

Lie Groupoids in Classical Field Theory I: Noether's Theorem

In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are combined with space-time transformations. Here, we outline the basis of the program and, as a first step, show how to (re)formulate Noether's theorem about the connection between symmetries and conservation laws in this approach.Comment: 38 pages, new final section adde

Householder transformations and optimal linear combinations

Several theorems related to the Householder transformation and separability criteria are proven. Orthogonal transformations, topology, divergence, mathematical matrices, and group theory are discussed

Type-Directed Program Transformations for the Working Functional Programmer

We present preliminary research on Deuce+, a set of tools integrating plain text editing with structural manipulation that brings the power of expressive and extensible type-directed program transformations to everyday, working programmers without a background in computer science or mathematical theory. Deuce+ comprises three components: (i) a novel set of type-directed program transformations, (ii) support for syntax constraints for specifying "code style sheets" as a means of flexibly ensuring the consistency of both the concrete and abstract syntax of the output of program transformations, and (iii) a domain-specific language for specifying program transformations that can operate at a high level on the abstract (and/or concrete) syntax tree of a program and interface with syntax constraints to expose end-user options and alleviate tedious and potentially mutually inconsistent style choices. Currently, Deuce+ is in the design phase of development, and discovering the right usability choices for the system is of the highest priority

Product Integrals and Wilson loops

Using product integrals we review the unambiguous mathematical representation of Wilson line and Wilson loop operators, including their behavior under gauge transformations and the non-abelian Stokes theorem. Interesting consistency conditions among Wilson lines are also presented.Comment: 3 pages LaTeX. Write-up of talk at DPF2000, Columbus, OH, August 11, 200