Operator Covariant Transform and Local Principle

We describe connections between the localization technique introduced by I. B. Simonenko and operator covariant transform produced by nilpotent Lie groups

Calculus of operators: covariant transform and relative convolutions

The paper outlines a covariant theory of operators related to groups and homogeneous spaces. A methodical use of groups and their representations allows to obtain results of algebraic and analytical nature. The consideration is systematically illustrated by a representative collection of examples

p-mechanics as a physical theory: an introduction

This paper provides an introduction to p-mechanics, which is a consistent physical theory suitable for a simultaneous description of classical and quantum mechanics. p-mechanics naturally provides a common ground for several different approaches to quantization (geometric, Weyl, coherent states, Berezin, deformation, Moyal, etc) and has a potential for expansions into field and string theories. The backbone of p-mechanics is solely the representation theory of the Heisenberg group

Paley-Wiener Theorem for Nilpotent Lie Groups

A Paley-Wiener-type theorem is proved for connected and simply connected Lie groups

Poincaré extension of Möbius transformations

Given sphere preserving (Möbius) transformations in n-dimensional Euclidean space one can use the Poincaré extension to obtain sphere preserving transformations in a half-space of n+1 dimensions. The Poincaré extension is usually provided either by an explicit formula or by some geometric construction. We investigate its algebraic background and describe all available options. The solution is given in terms of one-parameter subgroups of Möbius transformations acting on triples of quadratic forms. To focus on the concepts, this paper deals with the Möbius transformations of the real line only

Quantum and Classical Brackets

We describe an p-mechanical (see funct-an/9405002 and quant-ph/9610016) brackets which generate quantum (commutator) and classic (Poisson) brackets in corresponding representations of the Heisenberg group. We do not use any kind of semiclassic approximation or limiting procedures for h->0. Harmonic oscillator considered within the approach

Computation and Dynamics: Classical and Quantum

We discuss classical and quantum computations in terms of corresponding Hamiltonian dynamics. This allows us to introduce quantum computations which involve parallel processing of both: the data and programme instructions. Using mixed quantum‐classical dynamics we look for a full cost of computations on quantum computers with classical terminals

An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library

We propose to consider ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. “to be orthogonal”, “to be tangent”, etc.), as new objects in an extended Möbius–Lie geometry. It was recently demonstrated in several related papers, that such ensembles of cycles naturally parameterize many other conformally-invariant families of objects, e.g. loxodromes or continued fractions. The paper describes a method, which reduces a collection of conformally invariant geometric relations to a system of linear equations, which may be accompanied by one fixed quadratic relation. To show its usefulness, the method is implemented as a C++ library. It operates with numeric and symbolic data of cycles in spaces of arbitrary dimensionality and metrics with any signatures. Numeric calculations can be done in exact or approximate arithmetic. In the two- and three-dimensional cases illustrations and animations can be produced. An interactive Python wrapper of the library is provided as well

Asymptotic of 'rigid-body' motions for nonlinear dynamics: physical insight and methodologies

The purpose of the present work is to show that an adequate basis for understanding the essentially nonlinear phenomena must also be essentially nonlinear however still simple enough to play the role of a basis. It is shown that such types of 'elementary' nonlinear models can be revealed by tracking the hidden links between analytical tools of analyses and subgroups of the rigid-body motions or, in other terms, rigid Euclidean transformation. While the subgroup of rotations is linked with linear and weakly nonlinear vibrations, the translations with reflections can be viewed as a geometrical core of the strongly nonlinear dynamics associated with the so-called vibro-impact behaviors. It is shown that the corresponding analytical approach develops through non-smooth temporal substitutions generated by the impact models.Comment: Presented at 12th DSTA Conference, December 2-5, 2013 {\L}\'od\'z, Polan

Monogenic Calculus as an Intertwining Operator

We revise a monogenic calculus for several non-commuting operators, which is defined through group representations. Instead of an algebraic homomorphism we use group covariance. The related notion of joint spectrum and spectral mapping theorem are discussed. The construction is illustrated by a simple example of calculus and joint spectrum of two non-commuting selfadjoint (n\times n) matrices