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Piotr Stachura

Publications and source records attributed to Piotr Stachura.

14 recordsLinked to original sources
arXiv2025

On regularizations of the integral representation of Dirac delta function -- elementary approach

The article presents, in an elementary way, but with mathematical precision and without harm to the intuition, the path from the integral representation to the Dirac delta, starting with Schwartz's functional approach. Next, the considered representation is presented in a more intuitive sequential approach, formulated by Mikusiński and Sikorski. Finally, we present the third regularization that can be related to Sato's approach to distributions.

math.HO↗
arXiv2024

A quantum space of Euclidean lines

This article presents a differential groupoid with ``coaction'' of the groupoid underlying the Quantum Euclidean Group (i.e. its $C^*$-algebra is the $C^*$-algebra of this quantum group). The dual of the Lie algebroid is a Poisson manifold that can be identified with the space of oriented lines in Euclidean space equipped with a Poisson action of the Poisson-Lie Euclidean group.

math.QA↗
arXiv2018

The $κ$-Poincaré Group on a $C^*$-level

The $C^*$-algebraic $κ$-Poincaré Group is constructed. The construction uses groupoid algebras of differential groupoids associated to Lie group decomposition. It turns out the underlying $C^*$-algebra is the same as for "$κ$-Euclidean Group" but a comultiplication is twisted by some unitary multiplier. Generators and commutation relations among them are presented.

math.OA↗
arXiv2017

Sensitivity of Love and quasi-Rayleigh waves to model parameters

We examine the sensitivity of the Love and the quasi-Rayleigh waves to model parameters. Both waves are guided waves that propagate in the same model of an elastic layer above an elastic halfspace. We study their dispersion curves without any simplifying assumptions, beyond the standard approach of elasticity theory in isotropic media. We examine the sensitivity of both waves to elasticity parameters, frequency and layer thickness, for varying frequency and different modes. In the case of Love waves, we derive and plot the absolute value of a dimensionless sensitivity coefficient in terms of partial derivatives, and perform an analysis to find the optimum frequency for determining the layer thickness. For a coherency of the background information, we briefly review the Love-wave dispersion relation and provide details of the less common derivation of the quasi-Rayleigh relation in an appendix. We compare that derivation to past results in the literature, finding certain discrepancies among them.

physics.geo-ph↗
arXiv2016

Bogolyubov inequality for the ground state and its application to interacting rotor systems

We have formulated and proved the Bogolyubov inequality for operators at zero temperature. So far this inequality has been known for matrices, and we were able to extend it to certain class of operators. We have also applied this inequality to the system of interacting rotors. We have shown that if: {\em i)} the dimension of the lattice is 1 or 2, {\em ii)} the interaction decreases sufficiently fast with a distance, and {\em iii)} there is an energy gap over the ground state, then the spontaneous magnetization in the ground state is zero, i.e. there is no LRO in the system. We present also heuristic arguments (of perturbation-theoretic nature) suggesting that one- and two-dimensional system of interacting rotors has the energy gap independent of the system size if the interaction is sufficiently small.

cond-mat.stat-mech↗
arXiv2016

Forward problem for Love and quasi-Rayleigh waves: Exact dispersion relations and their sensitivities

We examine two types of guided waves: the Love and the quasi-Rayleigh waves. Both waves propagate in the same model of an elastic isotropic layer above an elastic isotropic halfspace. From their dispersion relations, we calculate their speeds as functions of the elasticity parameters, mass densities, frequency and layer thickness. We examine the sensitivity of these relations to the model and wave properties.

physics.geo-ph↗
arXiv2015

Operator reflection positivity inequalities and their applications to interacting quantum rotors

In the Reflection Positivity theory and its application to statistical mechanical systems, certain matrix inequalities play a central role. The Dyson-Lieb-Simon and Kennedy-Lieb-Shastry-Schupp inequalities constitute prominent examples. In this paper we extend the KLS-S inequality to the case where matrices are replaced by certain operators. As an application, we prove the occurrence of the long range order in the ground state of two-dimensional quantum rotors.

math-ph↗
arXiv2013

On the quantum 'ax+b'group

The more detailed description of the quantum 'ax+b' group of Baaj and Skandalis is presented. In particular we give generators and present formulae for action of the comultiplication on them; it is also shown that this group is a quantization of a Poisson-Lie structure on a classical 'ax+b' group.

math.QA↗
arXiv2005

Towards a topological (dual of) quantum $κ$-Poincaré group

We argue that the $κ$-deformation is related to a factorization of a Lie group, therefore {\em an approproate version of $κ$-Poincaré does exist on the $C^*$-algebraic level}. The explict form of this factorization is computed that leads to an ``action'' of the Lorentz group (with space reflections) considered in Doubly Special Relativity theory. The orbit structure is found and ``the momentum manifold'' is extended in a way that removes singularities of the ``action'' and results in a true action. Some global properties of this manifold are investigated

hep-th↗
arXiv2002

From Double Lie Groups to Quantum Groups

It is shown that there is a $C^*$-algebraic quantum group related to any double Lie group. An algebra underlying this quantum group is an algebra of a differential groupoid naturally associated with a double Lie group

math.QA↗
arXiv1999

Differential groupoids and $C^*$-algebras

The construction of a C*-algebra of a differential groupoid is presented. It is shown that it defines a covariant functor from the category of differential groupoids in a sense of S. Zakrzewski to the category of C*-algebras.

math.QA↗