# Marek Jerzy SzopaUniversity of Economics in Katowice · Department of Operations Research

Marek Jerzy Szopa

prof. dr hab.

## About

81

Publications

8,333

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

445

Citations

Introduction

I am interested in developing classical games: matching markets, mechanism design, fair share, and quantum extensions of games with special concern to their applications in decision making theory. Also in quantum communication, cryptography and applications of entanglement.

Additional affiliations

Education

October 1981 - June 1986

October 1977 - June 1982

## Publications

Publications (81)

The Prisoner's Dilemma [PD] is the best known example of a two-person simultaneous game for which the Nash equilibrium is far from Pareto optimal result. In this paper we define a quantum PD, for which players' strategies are defined as rotations of the SU(2) group, parameterized by three angles. Quantum strategies are correlated through the mechan...

Jazz band is a 3 player superadditive game in characteristic function form. Three players have to divide the payoff they can get, while being in a grand coalition, provided their individual and duo coalitions payoffs are known. Assumptions of individual and collective rationality lead to the notion of the core of the game. We discuss offers that ca...

Decision making by the two negotiating parties is simulated by a prisoner’s dilemma game. The game is formulated in a quantum manner, where players strategies are unitary transformations of qubits built over the basis of opposite decision options. Quantum strategies are correlated through the mechanism of quantum entanglement and the result of the...

Nash equilibria and correlated equilibria of classical and quantum games are investigated in the context of their Pareto efficiency. The examples of the prisoner's dilemma, battle of the sexes and the game of chicken are studied. Correlated equilibria usually improve Nash equilibria of games but require a trusted correlation device susceptible to m...

The principle of minimum Fisher information states that in the set of acceptable probability distributions characterizing the given system, it is best done by the one that minimizes the corresponding Fisher information. This principle can be applied to transaction processes, the dynamics of which can be interpreted as the market tendency to minimiz...

The principle of minimum Fisher information states that in the set of acceptable probability distributions characterizing the given system, it is best done by the one that minimizes the corresponding Fisher information. This principle can be applied to transaction processes, the dynamics of which can be interpreted as the market tendency to minimiz...

The aim of the paper is to study the problem of absentminded driver in the quantum domain. In the classical case, it is a well-known example of a decision problem with imperfect recall that exhibits lack of equivalence between mixed and behavioral strategies. The optimal payoff outcome is significantly lower than the maximum pay-off appearing in th...

The aim of the paper is to investigate Nash equilibria and correlated equilibria of classical and quantum games in the context of their Pareto optimality. We study four games: the prisoner's dilemma, battle of the sexes and two versions of the game of chicken. The correlated equilibria usually improve Nash equilibria of games but require a trusted...

Celem pracy jest poszukiwanie równowag Nasha, które byłyby bliższe wynikom optymalnym w sensie Pareto, niż odpowiednie równowagi gier klasycznych. Zbadane zostały trzy gry: dylemat więźnia, walka płci i dwie wersje gry w cykora. Wiadomo, że dla każdej z tych gier istnieją tzw. równowagi skorelowane (Aumann, 1973), które poprawiają rezultaty wspomni...

Why Technology Enhanced Learning requires teachers’ and trainers’ involvement
Action speaks louder than words
How to proceed to realize a quantum leap in Technology Enhanced Learning

W pracy przedstawiono teorię stabilnego dopasowania algorytmu odroczonej akceptacji (AOA) oraz algorytmy TTC i TTCC wraz z ich zastosowaniami do np.: kojarzenia uczelni i studentów, domów i właścicieli czy dawców i biorców nerek do przeszczepu. Dzięki tym algorytmom można projektować tzw. rynki kojarzenia, dla których optymalna alokacja dóbr jest m...

The E × e Jahn-Teller Hamiltonian in the Bargmann-Fock representation gives rise to a system of two coupled first-order differential equations in the complex field, which may be rewritten in the Birkhoff standard form. General leapfrog recurrence relations are derived, from which the quantized solutions of these equations can be obtained. The resul...

W rozdziale omówiono negocjacje jako proces podejmowania decyzji, ze zwróceniem szczególnej uwagi na dwa ich aspekty: twardy (formalny) oraz miękki (behawioralny). Przedyskutowano wzajemne przenikanie się obu aspektów oraz ich niezbędność do właściwego prowadzenia negocjacji. Przedstawiono podział na negocjacje dystrybutywne i integracyjne oraz ade...

W publikacji przedstawiono zagadnienia analizy i wspomagania podejmowania decyzji negocjacyjnych. Autorzy zwracają uwagę, że do pomyślnego prowadzenia negocjacji nie wystarczą tylko tzw. umiejętności miękkie. Równie ważne, a w niektórych negocjacjach nawet ważniejsze, są twarde umiejętności negocjacyjne, obejmujące analizę problemu pod kątem jego u...

Persistent currents in a narrow mesoscopic ring with several distortions threaded by the magnetic flux of the Aharonow – Bohm type are investigated. The ring distortions are modeled by an appropriate potential term and the transfer matrix method is used to find the energy spectrum. It is shown that in the ring with N distortions, under some geometr...

WYGRAC W NEGOCJACJACH, czyli sztuka ustępowania Rozegraj trudne negocjacje jak mistrz szachowy partię życia – obserwuj drugą stronę, przewiduj kolejne ruchy, bądd przygotowany do odpowiedzi i wygrywaj.

Istotą negocjacji jest przewidywanie tego, co może zrobić druga strona i dopasowanie do tego naszych propozycji, ofert, zachowań. Zasadniczą rolę odgrywa więc w nich myślenie strategiczne - jego typowym przykładem jest "strategia szachowa", czyli sztuka zarządznia ustępstwami.

The Prisoner's Dilemma [PD] is the best known example of a two-person, simultaneous game, for which the Nash equilibrium is far from Pareto-optimal solutions. In this paper we define a quantum PD, for which player’s strategies are defined as rotations of the SU(2) group, parameterized by three angles. Quantum strategies are correlated through the m...

Istotą myślenia strategicznego jest przewidywanie tego, co może zrobić druga strona, oraz optymalne dopasowanie do tego swoich zachowań i propozycji. Takie myślenie, obok kwestii psychologicznych, odgrywa zasadniczą rolę w negocjacjach. Świetnym przykładem myślenia strategicznego w negocjacjach jest "strategia szachowa" czyli sztuka przygotowania s...

The ability to control the quantum state of a single electrons in a quantum ring made of a semiconductor is at the heart of recent developments towards a scalable quantum computer. A peculiar dispersion relation of quantum rings allows to steer the ground state properties by the magnetic flux and offers spin and orbital degrees of freedom for quant...

Persistent currents in distorted narrow mesoscopic rings threaded by the magnetic flux of the Aharonow Bohm type are investigated. It is shown that the ring distortions can be modelled by an appropriate potential term. The cases with a single and multiple distortions are considered. The single distortion opens a gap in the electron energy spectrum...

Persistent currents in distorted narrow mesoscopic rings threaded by the magnetic flux of the Aharonow Bohm type are investigated. It is shown that the ring distortions can be modelled by an appropriate potential term. The cases with a single and multiple distortions are considered. The single distortion opens a gap in the electron energy spectrum...

The possibility of making a flux qubit on nonsuperconducting mesoscopic ballistic quasi 1D ring is discussed. We showed that such ring can be effectively reduced to a two-state system with two external control parameters. The two states carry opposite persistent currents and are coupled by tunneling which leads to a quantum superposition of states....

Na przełomie XX i XXI wieku odkryto nowe odmiany alotropowych węgla: fulereny i nanorurki. Artykuł opisuje historię ich odkrycia i szczegóły geometryczne ich unikalnej budowy. Drobne różnice w ułożeniu i defekty heksagonelnej sieci węglowej skutkują wielkim bogactwem typów i rodzajów nanorurek i fulerenów. Omawiane są również możliwe zastosowania t...

The standard tight-binding dispersion relation for graphene and carbon nanotubes has an electron-hole symmetry. This symmetry has not been observed experimentally until recently. We discuss here the effect of the overlap between π-orbitals at neighbouring sites, belonging to different sublattices, on the dispersion relation. The e-h asymmetry incre...

In this paper we consider a mesoscopic 1D, ballistic, metallic ring with a potential barrier. We show that the coherent coupling between two distinct quantum states with different winding numbers can lead to a formation of a qubit. We discuss the possible realizations of such a ring, the adjustment of a potential barrier parameters and the possible...

The magnetic field parallel to the symmetry axis of a system manifests itself mostly via the Aharonov-Bohm effect which causes the so-called persistent currents to flow around the tube. A direct consequence of these currents is the appearance of an orbital magnetic moment in the nanotube. The magnitude of this moment depends strongly on the size of...

The unusual band structure of carbon nanotubes (CNs) results in their remarkable magnetic properties. The application of magnetic field parallel to the tube axis can change the conducting properties of the CN from metallic to semiconducting and vice versa. Apart from that B induces (via the Bohm-Aharonov effect) orbital magnetic moments $\mu_{orb}$...

We address the objective of the generation of finite magnetic flux out of unbiased thermal current fluctuations in a collection of identical mesoscopic cylinders which are coupled via mutual inductances. The influence of thermal Nyquist fluctuations are described in terms of a set of Langevin equations or a corresponding Fokker–Planck equation, res...

Persistent currents in mesoscopic rings or cylinders are very sensitive to non-classical properties of the driving magnetic flux. In the paper we investigate the influence of the quantum and classical noise on the properties of currents. We also study the current characteristics in the presence of maximally entangled states (Bell states) of the non...

The carbon tori and nanotubes, either pure or hole-doped, are examined. In the two achiral (zigzag and armchair) systems the momentum lines are parallel to the edges of the Brillouin zone or the γ = –γ (γ ≈ 3.03 eV is the overlap integral for graphene) contour. The amplitude and shape of the persistent currents in these cases is predicted by an ana...

This issue contains the Proceedings of the 28th International Conference of Theoretical Physics, ICTP2004 — Electron Correlations in Nano- and Macrosystems, which was held in Ustron, Poland, from 2-7 September 2004. ICTP2004 followed the series of conferences organized by the Institute of Physics of the University of Silesia in Katowice, devoted bi...

This issue contains the Proceedings of the 28th International Conference of Theoretical Physics, ICTP2004 – Electron Correlations in Nano- and Macrosystems, which was held in Ustroń, Poland, from 2–7 September 2004. ICTP2004 followed the series of conferences organized by the Institute of Physics of the University of Silesia in Katowice, devoted bi...

Collective phenomena due to persistent currents in carbon multiwall nanotubes are studied. The formula for persistent currents minimising free energy and conditions for the stability of persistent currents in multiwall nanotubes in magnetic field are derived. Numerical calculations performed show the possibility of obtaining spontaneous currents in...

Persistent currents in multiwalled carbon nanotubes (MWNT's), driven by the magnetic field parallel to the tube axis are studied. The geometrical structure and possibility of the existence of MWNT's with shells in various chiral configurations are explored. The currents are calculated considering a possible Fermi energy shift by hole doping. The in...

Persistent currents in mesoscopic rings and cylinders threaded by a magnetostatic flux and also by monochromatic nonclassical electromagnetic fields are considered. The results depend on the quantum state of the nonclassical electromagnetic fields. It is shown that quantum and thermal noise in the field reduces the current and can change its charac...

A semi-phenomenological model is proposed to study dynamics and stedy states of magnetic fluxes and currents in mesoscopic rings and cylinders at non-zero temperature. The model is based on a Langevin equation for flux subject to zero-mean thermal equilibrium Nyquist noise. Quenched randomness, which mimics disorder, is included via the fluctuating...

Persistent currents in mesoscopic nonsuperconducting rings and cylinders
are a manifestation of quantum coherence. In this paper the possibility
of self-sustaining persistent currents in thick mesoscopic cylinders is
disscussed. The long-range magnetostatic (current-current) interactions
are taken into account by the method of self-consistent field...

The aim of this paper is to investigate the existence of the stationary states of current in the mesoscopic cylinder. The dynamics of the flux is governed by a stochastic differential equation. We discuss both the influence of equilibrium (thermal) and non-equilibrium noise sources.

A brief review of the remarkable properties of carbon nanotubes is presented. The manifestation of mesoscopic coherent transport are persistent currents flowing in carbon nanotubes in the presence of static magnetic field. Persistent currents as a function of hole doping are investigated. The possibility of flux expulsion and flux trapping in a set...

We study magnetic fluxes and currents in a set of mesoscopic rings which
form a cylinder. We investigate the noiseless system as well as the
influence of equilibrium and non-equilibrium fluctuations on the
properties of selfsustaining currents. Thermal equilibrium Nyquist noise
does not destroy selfsustaining currents up to temperatures of the same...

The idea of nondissipative, persistent currents in mesoscopic metallic or semiconducting rings and cyclinders appears counterintuitive, because it contradicts our experience with currents in macroscopic metals. On the other hand such orbital currents are well known properties of atoms. A comparative study of nondissipative ring currents in differen...

The generalization of geometric phase for the quantum systems described by quaternionic quantum mechanics is given. The geometry of the quantum cyclic evolution is studied and the quaternionic Berry phase is shown to be given by the holonomy of the suitably defined fiber bundle.

Three different topological structures built on the basis of a graphene lattice are investigated. Clusters of the (3,6) type homeomorphic with the sphere, toroidal, and cylindrical carbon nanotubes are shown to have the same dispersion relation, inherited from the planar graphene lattice. The persistent currents in axially symmetric structures, the...

A semiphenomenological model is proposed to study magnetic fluxes and currents in mesoscopic rings at nonzero temperature. The model is based on a Langevin equation for a flux subject to thermal equilibrium Nyquist noise. Quenched randomness, which mimics disorder, is included via the fluctuating parameter method. It is shown that self-sustaining a...

Examples of quasi-exactly solvable hamiltonians in the Bargmann—Fock representation are given. The existence of an invariant subspace is studied as a result of hidden symmetry of the spectral problem.

The role played by the magnetostatic interaction in mesoscopic multichannel systems is discussed. We show that the interaction of currents from different channels, when taken in the selfconsistent mean field approximation, leads to selfinductance terms in the Hamiltonian producing an internal magnetic flux. Such multichannel systems can exhibit spo...

Persistent currents driven by a static magnetic flux parallel to the carbon nanotube axis are investigated. Owing to the hexagonal symmetry of graphene the Fermi contour expected for a 2D-lattice reduces to two points. However the electron or hole doping shifts the Fermi energy upwards or downwards and as a result, the shape of the Fermi surface ch...

An infinite series of (3, 6) cages is defined by trivalent carbon polyhedra composed of hexagonal and four triangular rings. A zone-folding construction is applied to the graphene band structure to yield explicit expressions for the pi-molecular orbitals, energies, and symmetries of the cages that depend only on four indices m, n, p, and q. Leapfro...

An infinite series of 3,6 cages is defined by trivalent carbon polyhedra composed of hexagonal and four triangular rings. A zone-folding construction is applied to the graphene band structure to yield explicit expressions for the-molecular orbitals, energies, and symmetries of the cages that depend only on four indices m, n, p, and q. Leapfrog memb...

The construction of convex trivalent cages, whose polyhedra are homeomorphic with the sphere is discussed from the geometrical and topological points of view. The classification of such cages with respect to the number of l-gonal faces, where l ⋜ 6 is presented. The tetrahedrally symmetric (3,6) cages are studied in detail and the method of finding...

Mesoscopic metal rings can carry persistent currents driven by a
constant magnetic field. The geometrical structure of a toroidal carbon
nanotube can be characterized by four independent parameters. We derive
the formula for persistent currents driven by a constant Bohm--Aharonov
type of field perpendicular to the plane of the torus. The dependenci...

Irreducible symmetry representations characterize topological invariants of point group molecules. In the present paper this result is extended to periodic structures in two and three dimensions, using plane and space symmetry groups. Applications include a symmetry analysis of the band structure for the graphite sheet and related nanotubes of the...

Different expressions for calculating the Berry phase of adiabatic processes are reviewed and their limitations are discussed. These expressions are then applied to the case of a circuit surrounding a triple degeneracy. It is shown that the most general formula for the Berry phase requires the use of the full SU(3) invariance group of a T state cou...

The Jahn - Teller Hamiltonian in its Bargmann - Fock representation is transformed by the Birkhoff method into a canonical form in which all regular singularities between zero and infinity have been removed. The resulting equation is of the Kummer type and identical to the previously obtained canonical form of the Rabi Hamiltonian (Szopa M and Ceul...

The Racah-Wigner approach is applied to demonstrate various group-theoretic labelling schemes for modes of oscillations and elasticity parameters of the cluster of nodes of a cube. It is shown that there exists a fundamental basis in the configuration space, closed under the action of the group Oh-the geometric symmetry group of the equilibrium. It...

Equidistributive eigenvectors of the adjacency matrix of a symmetric graph have equal weight on equivalent vertices. Real and complex coefficients are in general insufficient to achieve this property for all components of a degenerate manifold of an arbitrary graph. Quaternionic vector coefficients are shown to be necessary for equidistributivity o...

The persistent currents driven by the pure Aharonov-Bohm type magnetic field in mesoscopic normal metal or semiconducting cylinders are studied. A two-dimensional (2D) Fermi surfaces are characterized by four parameters. Several conditions for the coherence and enhancement of currents are discussed. These results are then generalized to a three-dim...

Physical properties of cages and clusters obey symmetry rules that are extensions of the celebrated Euler-Poincaré theorem on polyhedra. A connection is established between this result and a fundamental topological relationship in the theory of homology groups. The connection allows us to assign symmetry representations to physically relevant topol...

The Rabi Hamiltonian, describing the coupling of a two-level system to a single quantized boson mode, is studied in the Bargmann-Fock representation. The corresponding system of differential equations is transformed into a canonical form in which all regular singularities between zero and infinity have been removed. The canonical or Birkhoff-transf...

Persistent currents in mesoscopic cylinders made of a very clean metal and with nearly flat Fermi surface are studied. It is shown that the inclusion of the orbital magnetic interaction between electrons can lead to spontaneous currents (spontaneous fluxes) and to flux trapping if the number of interacting electrons is large enough. The free energy...

In a system of mesoscopic rings the influence of orbital magnetic interaction between the electrons is investigated. At a critical temperature Tc the system undergoes a phase transition into a current carrying state. Tc depends strongly on geometry of the system and/or its Fermi-surface, and on the quantum size gap at the Fermi level. Elastic scatt...

It is shown that a mesoscopic metallic system can exhibit a phase transition to a low temperature state with a spontaneous orbital current and with the corresponding spontaneous magnetic flux in zero external field, if it is sufficiently free of elastic defect scattering. The ground state of such a system is then a flux phase state, with a real mag...

It is shown that the ground state of a sufficiently pure mesoscopic metallic in ring is in general a flux phase state, i.e. a state with a spontaneous magnetic flux øSP generated by a spontaneous orbital current. The magnitude of spontaneous current depends strongly on the geometry of the system and of the Fermi surface.

It is shown that a sufficiently pure mesoscopic metallic ring will exhibit a magnetic phase transition to a low-temperature state with a persistent orbital current in zero external field (to a flux-phase state). The transition temperature and the spontaneous persistent current depend strongly on the geometry of the ring and of the Fermi surface.

Persistent currents in mesoscopic rings threaded by a magnetic field are studied. Due to the mesoscopic dimensions the system has a discrete energy spectrum and the level spacing at the Fermi surface is large compared with the thermal excitations at sufficiently low temperatures. The inclusion of the self-inductance of the ring gives a self-consist...