Searcharxiv⌕ Search

arXiv subjects

Jan L. Cieśliński

Publications and source records attributed to Jan L. Cieśliński.

At least 19 recordsLinked to original sources
arXiv2025

An explicit integrator uniform in the true anomaly and exactly preserving all integrals of motion in the three-dimensional Kepler problem

We develop a numerical scheme for the Kepler problem that preserves exactly all first integrals: angular momentum, total energy, and the Laplace-Runge-Lenz vector. This property ensures that orbital trajectories retain their precise shape and orientation over long times, avoiding the spurious precession typical of many standard methods. The scheme uses an adaptive time step derived from a constant angular increment. Analytical considerations and numerical experiments demonstrate that the algorithm combines high accuracy with long-term stability.

math.NA↗
arXiv2023

On geometric interpretation of Euler's substitutions

We consider a classial case of irrational integrals containing a square root of a quadratic polynomial. It is well known that they can be expressed in terms of elementary functions by one of three Euler's substitutions. It is less known that the Euler substittutions have a beautiful geometric interpretation. In the framework of this interpretation one can see that the number 3 is not the most suitable. We show that it is natural to introduce the fourth Euler substitution. By the way, it is not clear who was the first to attribute these three substitutions to Euler. In his original treatise Leonhard Euler uses two substitutions which are sufficient to cover all cases.

math.HO↗
arXiv2020

On the famous unpublished preprint "Methods of integration which preserve the contact transformation property of the Hamilton equations" by René De Vogelaere

The well known 1956 unpublished report by De Vogelaere, motivated by the physics of particle accelerators, is the first demonstration of the existence of numerical integrators (now called symplectic integrators) that preserve a basic property of Hamiltonian systems. The purpose of this short note is to introduce René De Vogelaere, list his pioneering accomplishments, and relate them to other early work on symplectic integrators. The preprint itself, typset in LaTeX, is attached as an appendix.

math.NA↗
arXiv2019

Determination of topological properties of thin samples by the van der Pauw method

We solve the problem of determining basic topological properties of flat samples by performing measurements on their outer edge. The global maximum of four probe resistances shows a characteristic behaviour, which is dependent on the genus (i.e., the number of holes) of the domain. An extension of the van der Pauw method on domains having zero, one, or two holes is presented and discussed. A possibility of measuring topological properties of condensed matter is demonstrated. Experimental results for triply connected domains are presented and explained by continuous symmetry breaking caused by the presence of two holes. The results are consistent with the topological theorem of Hurwitz on the number of automorphisms of Riemann surfaces.

physics.class-ph↗
arXiv2018

Improving the accuracy of the fast inverse square root algorithm

We present improved algorithms for fast calculation of the inverse square root for single-precision floating-point numbers. The algorithms are much more accurate than the famous fast inverse square root algorithm and have the same or similar computational cost. The main idea of our work consists in modifying the Newton-Raphson method and demanding that the maximal error is as small as possible. Such modification is possible when the distribution of Newton-Raphson corrections is not symmetric (e.g., if they are non-positive functions).

math.NA↗
arXiv2016

Group interpretation of the spectral parameter. The case of isothermic surfaces

It is well known that in some cases the spectral parameter has a group interpretation. We discuss in detail the case of Gauss-Codazzi equations for isothermic surfaces immersed in $E^3$. The algebra of Lie point symmetries is 4-dimensional and all these symmetries are also symmetries of the Gaus-Weingarten equations (which can be considered as so(3)-valued non-parametric linear problem). In order to obtain a non-removable spectral parameter one has to consider so(4,1)-valued linear problem which has a 3-dimensional algebra of Lie point symmetries. The missing symmetry introduces a non-removable parameter. In the second part of the paper we extend these results on the case of isothermic immersions in arbitrary multidimensional Euclidean spaces. In order to simplify calculations the problem was formulated in terms of a Clifford algebra.

nlin.SI↗
arXiv2015

On geometry of the scator space

We consider the scator space - a hypercomplex, non-distributive hyperbolic algebra introduced by Fernández-Guasti and Zaldívar. We discuss isometries of the scator space and find consequent method for treating them algebraically, along with scators themselves. It occurs that introduction of zero divisors cannot be avoided while dealing with these isometries. The scator algebra may be endowed with a nice physical interpretation, although it suffers from lack of some physically demanded important features. Despite that, there arises some open questions, e.g., whether hypothetical tachyons can be considered as usual particles possessing time-like trajectories.

math-ph↗
arXiv2015

Determination of the Riemann modulus and sheet resistivity by a six-point generalization of the van der Pauw method

Six point generalization of the van der Pauw method is presented. The method is applicable for two dimensional homogeneous systems with an isolated hole. A single measurement performed on the contacts located arbitrarily on the sample edge allows to determine the specific resistivity and a dimensionless parameter related to the hole, known as the Riemann modulus. The parameter is invariant under conformal mappings of the sample shape. The hole can be regarded as a high resistivity defect. Therefore the method can be applied for experimental determination of the sample inhomogeneity.

physics.ins-det↗
arXiv2015

Determination of the Riemann modulus and sheet resistance of a sample with a hole by the van der Pauw method

The van der Pauw method for two-dimensional samples of arbitrary shape with an isolated hole is considered. Correlations between extreme values of the resistances allow one to determine the specific resistivity of the sample and the dimensionless parameter related to the geometry of the isolated hole, known as the Riemann modulus. The parameter is invariant under conformal mappings. Experimental verification of the method is presented.

physics.class-ph↗
arXiv2014

The sine-Gordon equation on time scales

We formulate and discuss integrable analogue of the sine-Gordon equation on arbitrary time scales. This unification contains the sine-Gordon equation, discrete sine-Gordon equation and the Hirota equation (doubly discrete sine-Gordon equation) as special cases. We present the Lax pair, check compatibility conditions and construct the Darboux-Bäcklund transformation. Finally, we obtain a soliton solution on arbitrary time scale. The solution is expressed by the so called Cayley exponential function.

nlin.SI↗
arXiv2014

Discrete approximations on functional classes for the integrable nonlinear Schrödinger dynamical system: a symplectic finite-dimensional reduction approach

We investigate discretizations of the integrable discrete nonlinear Schrödinger dynamical system and related symplectic structures. We develop an effective scheme of invariant reducing the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter. We construct a finite set of recurrent algebraic regular relations allowing to generate solutions of the discrete nonlinear Schrödinger dynamical system and we discuss the related functional spaces of solutions. Finally, we discuss the Fourier transform approach to studying the solution set of the discrete nonlinear Schrödinger dynamical system and its functional-analytical aspects.

nlin.SI↗
arXiv2013

Improving the accuracy of the AVF method

The Average Vector Field (AVF) method is a B-series scheme of the second order. As a discrete gradient method it preserves exactly the energy integral for any canonical Hamiltonian system. We present and discuss two locally exact and energy-preserving modifications of the AVF method: AVF-LEX (of the third order) and AVF-SLEX (of the fourth order). Applications to spherically symmetric potentials are given, including a compact explicit expression for the AVF scheme for the Coulomb-Kepler problem.

math.NA↗
arXiv2013

Locally exact modifications of discrete gradient schemes

Locally exact integrators preserve linearization of the original system at every point. We construct energy-preserving locally exact discrete gradient schemes for arbitrary multidimensional canonical Hamiltonian systems by modifying classical discrete gradient schemes. Modifications of this kind are found for any discrete gradient.

physics.comp-ph↗
arXiv2013

Fast exact digital differential analyzer for circle generation

In the first part of the paper we present a short review of applications of digital differential analyzers (DDA) to generation of circles showing that they can be treated as one-step numerical schemes. In the second part we present and discuss a novel fast algorithm based on a two-step numerical scheme (explicit midpoint rule). Although our algorithm is as cheap as the simplest one-step DDA algoritm (and can be represented in terms of shifts and additions), it generates circles with maximal accuracy, i.e., it is exact up to round-off errors.

cs.GR↗
arXiv2013

Locally exact modifications of numerical schemes

We present a new class of exponential integrators for ordinary differential equations: locally exact modifications of known numerical schemes. Local exactness means that they preserve the linearization of the original system at every point. In particular, locally exact integrators preserve all fixed points and are A-stable. We apply this approach to popular schemes including Euler schemes, implicit midpoint rule and trapezoidal rule. We found locally exact modifications of discrete gradient schemes (for symmetric discrete gradients and coordinate increment discrete gradients) preserving their main geometric property: exact conservation of the energy integral (for arbitrary multidimensional Hamiltonian systems in canonical coordinates). Numerical experiments for a 2-dimensional anharmonic oscillator show that locally exact schemes have very good accuracy in the neighbourhood of stable equilibrium, much higher than suggested by the order of new schemes (locally exact modification sometimes increases the order but in many cases leaves it unchanged).

math.NA↗
arXiv2013

Van der Pauw method on a sample with an isolated hole

Explicit results of the van der Pauw method for a sample containing an isolated hole are presented together with experimental confirmation. Results of measurements and numerical analysis strongly suggest that four probe resistivities obey inequality similar in the form to the famous van der Pauw equation. The inequality seems to be valid for any sample with an isolated hole and contacts located on the same edge, however rigorous proof is not given. The inequality can be used for experimental detection of the sample quality.

cond-mat.mes-hall↗