arXiv subjects
Publications and source records attributed to Tomasz Stachowiak.
The binary EML operator yields all (transcendental) elementary functions by recursive application, or a binary tree. The structure of the operator itself carries two distinct ingredients: that of an abelian group, and of functional inverse, which reveal a constructive path to many distinct functional families.
We apply the political stress index as introduced by Goldstone (1991) and implemented by Turchin (2013), to the case study of Poland. The approach quantifies political and social unrest as a single quantity based on a multitude of economic and demographic variables. The present-day data allow us to directly apply index without the need of simulating the elite component, as was done previously. Neither model version shows appreciable unrest levels for the present, while the simulated model applied to partial historical data yields the index in remarkable agreement with the fall of communism in Poland. We next analyze the model's sensitive dependence on its parameters (the hallmark of chaos), which limits its utility and application to other countries. The original equations cannot, by construction, describe the elite fraction for longer time-periods; and we propose a modification to remedy this problem. The model still holds some predictive power, but we argue that some components should be reinterpreted if one wants to keep its dynamical equations.
An application of (iterated) Bauer-Muir acceleration can give an Apéry-like continued fraction for $π$ with irrational coefficients, and much faster convergence. It can be considered a generalized continued fraction with the same matrix representation as the standard case, but the dimension increased due to irrationality. The construction is also given for $\ln(2)$ and $\sqrt[3]{2}$.
The analog of the Kepler system defined on the Heisenberg group introduced by Montgomery and Shanbrom in [Fields Inst. Commun., Vol. 73, Springer, New York, 2015, 319-342, arXiv:1212.2713] is integrable on the zero level of the Hamiltonian. We show that in all other cases the system is not Liouville integrable due to the lack of additional meromorphic first integrals. We prove that the analog of the two-body problem on the Heisenberg group is not integrable in the Liouville sense.
We explore the spectral problem of the two photon Rabi model from the point of view of complex differential equations in the Bargmann representation. The wave-functions are automatically entire but to ensure finite norm one has to effectively construct asymptotic expansions. This is achieved by means of the Mellin transformation and convergent factorial series, which allow direct computation of the spectral determinant. By further analysing the differential equation satisfied by the Mellin transform, we obtain a new form of the spectral conditions -- in terms of holonomy matrices and contour integrals.
An infinite family of exact solutions of the two-photon Rabi model was found by investigating the differential algebraic properties of the Hamiltonian. This family corresponds to energy level crossings not covered by the Juddian class, which is given by elemetary functions. In contrast, the new states are expressible in terms of parabolic cylinder or Bessel functions. We discuss three approaches for discovering this hidden structure: factorization of differential equations, Kimura transformation, and a doubly-infinite, transcendental basis of the Bargmann space.
The autonomous Duffing oscillator, and its van der Pol modification, are known to admit time-dependent first integrals for specific values of parameters. This corresponds to the existence of Darboux polynomials, and in fact more can be shown: that there exist Liouvillian first integrals which do not depend on time. They can be expressed in terms of the Gauss and Kummer hypergeometric functions, and are neither analytic, algebraic nor meromorphic. A criterion for this to happen in a general dynamical system is formulated as well.
We investigate the dynamical complexity of Cournot oligopoly dynamics of three firms by using the qualitative methods of dynamical systems to study the phase structure of this model. The phase space is organized with one-dimensional and two-dimensional invariant submanifolds (for the monopoly and duopoly) and unique stable node (global attractor) in the positive quadrant of the phase space (Cournot equilibrium). We also study the integrability of the system. We demonstrate the effectiveness of the method of the Darboux polynomials in searching for first integrals of the oligopoly. The general method as well as examples of adopting this method are presented. We study Darboux non-integrability of the oligopoly for linear demand functions and find first integrals of this system for special classes of the system, in particular, rational integrals can be found for a quite general set of model parameters. We show how first integral can be useful in lowering the dimension of the system using the example of $n$ almost identical firms. This first integral also gives information about the structure of the phase space and the behaviour of trajectories in the neighbourhood of a Nash equilibrium
I propose an alternative $f(R)$ theory of gravity constructed by applying the function $f$ directly to the Ricci tensor instead of the Ricci scalar. The main goal of this study is to derive the resulting modified Einstein equations for the metric case with Levi-Civita connection, as well as for the general nonmetric connection with torsion. The modification is then applied to the Robertson-Walker metric so that the cosmological evolution corresponding to the standard model can be studied. An appealing feature is that even in the vacuum case, scenarios without initial singularity and exponential expansion can be recovered. Finally, formulae for possible observational tests are given.
We consider the system of two material points that interact by elastic forces according to Hooke's law and their motion is restricted to certain curves lying on the plane. The nonintegrability of this system and idea of the proof are communicated. Moreover, the analysis of global dynamics by means of Poincaré cross sections is given and local analysis in the neighborhood of an equilibrium is performed by applying the Birkhoff normal form. Conditions of linear stability are determined and some particular periodic solutions are identified.
We consider two special types of double pendula, with the motion of masses restricted to various surfaces. In order to get quick insight into the dynamics of the considered systems the Poincaré cross sections as well as bifurcation diagrams have been used. The numerical computations show that both models are chaotic which suggest that they are not integrable. We give an analytic proof of this fact checking the properties of the differential Galois group of the system's variational equations along a particular non-equilibrium solution.
We investigate a generalisation of the Rabi system in the Bargmann-Fock representation. In this representation the eigenproblem of the considered quantum model is described by a system of two linear differential equations with one independent variable. The system has only one irregular singular point at infinity. We show how the quantisation of the model is related to asymptotic behaviour of solutions in a vicinity of this point. The explicit formulae for the spectrum and eigenfunctions of the model follow from an analysis of the Stokes phenomenon. An interpretation of the obtained results in terms of differential Galois group of the system is also given.
An implicit formula for the spectrum of the two-photon Rabi model was presented by Travenec in Phys. Rev. A 85, 043805 (2012) in analogy to the method of Braak [Phys. Rev. Lett. 107, 100401 (2011)]. The spectrum is given by common zeros of four functions $G_c(z, E)$ taken with a fixed value of $z\in\mathbb{C}$. We point out that all the $G_c(z,E)$ functions defined by Travenec are identically zero, thus the spectrum is not in fact given by their roots.
We formulate a universal method for solving an arbitrary quantum system which, in the Bargmann representation, is described by a system of linear equations with one independent variable, such as one- and multi-photon Rabi models, or $N$ level systems interacting with a single mode of the electromagnetic field and their various generalizations. We explain three types of conditions that determine the spectrum and show their usage for two deformations of the Rabi model. We prove that the spectra of both models are just zeros of transcendental functions, which in one case are given explicitly in terms of confluent Heun functions.
It is shown that in the Rabi model, for an integer value of the spectral parameter $x$, in addition to the finite number of the classical Judd states there exist infinitely many possible eigenstates. These eigenstates exist if the parameters of the problem are zeros of a certain transcendental function; in other words, there are infinitely many possible choices of parameters for which integer $x$ belongs to the spectrum. Morover, it is shown that the classical Judd eigenstates appear as degenerate cases of the confluent Heun function.
The accuracy of the numerical findings of Lukes-Gerakopoulos PRD 86, 044013 (2012), regarding the existence of additional first integrals in the Zipoy-Voorhees space-time, was recently questioned by Maciejewski et al. PRD 88, 064003 (2013). In this comment, it is shown that the discrepancy between the results of Lukes-Gerakopoulos (2012) and Maciejewski et al. (2013) is not due to issues related to numerical accuracy, as claimed in Maciejewski et al (2013), but due to different choice of coordinates used in Maciejewski et al.(2013).
We show that the geodesic motion in the Zipoy-Voorhees space-time is not Liouville integrable, in that there does not exist an additional first integral meromorphic in the phase space variables.
The one dimensional Dirac equation with a rational potential is reducible to an ordinary differential equation with a Riccati-like coefficient. Its integrability can be studied with the help of differential Galois theory, although the results have to be stated with recursive relations, because in general the equation is of Heun type. The inverse problem of finding integrable rational potentials based on the properties of the singular points is also presented; in particular, a general class of integrable potentials leading to the Whittaker equation is found.