Joanna Albin,
Properties of shell models of Coulombic systems

Assemblies of Coulombically interacting particles conned by highly symmetrical external potentials are encountered in diverse systems of importance to theoretical and experimental physics and chemistry. First part of the presentation will deal with to assemblies of equicharged particles subject to spherically symmetric power-law conning potentials. Accurate rigorous upper bounds to energies of these systems, which are amenable to detailed mathematical analysis, will be shown to comprise terms with smooth, oscillatory, and uctuating dependences on N. Asymptotic equivalence of the shell-model and local-density (LDA) descriptions of Coulombic systems conned by radially symmetric potentials in two and three dimensions will be subject of the second part of presentation.

Zbigniew Bartoszewski,
Numerical solution of threshold problems for delay differential equations

The talk will be devoted to numerical solution of threshold problems for delay-differential equations of the form

(1)  

, where: , the delay function is unknown and has to be determined from the threshold condition

(2)  

with given threshold and

is a given operator (which usually is an integral operator). Such problems appear while modeling various problems in epidemics and population dynamics.

To solve this problem numerically an embedded pair of continuous Runge Kutta method of order and discrete Runge-Kutta method of order is employed. The latter one is used for the estimation of local discretization errors. For solving the threshold condition the bisection method is used.

Numerical examples will be presented which illustrate the effectiveness of the proposed method.

Łukasz Bratek,
Generalized Hamiltonian dynamics for geometric models of paticles with spin.

A relativistic spinning particle may be constructed on the clasical level as a system consisting of a world-line and associated with it collection of four-vectors introduced to describe the particle’s internal structure. Such systems can be neatly described in the framework of Dirac’s Generalized Hamiltonian Dynamics. Of particular significance are models for which the Casimir invariants of the Poincare group are independently fixed parameters. In this context, the concept of relativistic spherical top of Hanson and Regge can be used to provide an interesting example.

Dariusz Chruściński,
Non-Markovian evolution of quantum open systems

Quantum evolution is described in terms of the family of dynamical maps. We provide basic introduction to the subject. Moreover we analyze a notion of Markovianity based on the concept of divisibility. The general non-Markovian evolution is analyzed in terms of time-local generators and non-local memory kernels. Our discussion is illustrated by simple examples of qubit dynamics.

Marek Czachor:
The Dirac delta vanishing at zero and the problem of field quantization

Tomasz Derda,
Statistics of Avalanches and Critical Loads in Fiber Bundle Model of Nanopillar Arrays

The phenomena of fracture and failure of materials are common but usually undesirable. Under growing load materials undergo a damage process that leads to a failure if load increase is not terminated as, e.g., in the case of axially loaded nanopillar arrays, studied here. Nanopillar arrays are encountered in numerous areas of nanotechnology and reveal the potential applicability as components for the fabrication of electro-mechanical sense devices. Thus it is important to study the failure progress in such systems and we conduct our study in the framework of transfer load models. For this purpose a set of pillars is subjected to an external load which is increased quasi-statically. When the load carried by a pillar exceeds its strength-threshold the pillar crashes and its load is redistributed among intact pillars according to a given load transfer rule. Under increasing external load destruction of the system takes the form of avalanches of damaged pillars. When the load exceeds a certain critical value the avalanche becomes self-sustained until the system is completely damaged. We examine the distributions of such catastrophic avalanches and critical loads inducing these avalanches.

Adam Doliwa, Discrete KP equation with self-consistent sources

We show that the discrete KadomtsevPetviashvili (KP) equation with sources obtained recently by the source generalization method can be incorporated into the squared eigenfunctions symmetry extension procedure. Moreover, using the known correspondence between Darboux-type transformations and additional independent variables, we demonstrate that the equation with sources can be derived from Hirota’s discrete KP equations but in a space of bigger dimension. In this way we uncover the origin of the source terms as coming from multidimensional consistency of the Hirota system itself.

Joanna Gonera,
-conformal Newton-Hooke symmetry and superintegrability of Pais-Uhlenbeck model

The maximal (Noetherian) symmetry of free motion is described by the Schroedinger group which is the rst member of the whole family of so-called -conformal Galilean groups, l being integer or half-integer. For half-integer they are maximal symmetry groups of free motion described by the equation of th order. The symmetry group of harmonic oscillator is the Newton-Hook group. We show that the higher-dimensional extensions of the latter, the -conformal Newton-Hooke groups, describe the higher-order generalizations of harmonic oscillator – the Pais-Uhlenbeck oscillators provided the relevant frequencies obey denite relation. We explain that this restriction results from the structure of nite-dimensional representations of . The resulting oscillator is superintegrable. We nd the integrals of motion and explain the relationship between l-conformal NH symmetry and superintegrability.

Leszek Hadasz,
Super-Liouville – Double Liouville correspondence

W wystąpieniu zaprezentuję – motywowany relacją AGT – związek między iloczynem tensorowym supersymetrycznej teorii pola Liouville’a i urojonego, swobodnego fermionu oraz rzutowanego iloczynu tensorowego rzeczywistej i „urojonej” teorii pola Liouville’a. Przedstawię dowód ich równoważności w sektorze Neveu-Schwarza.

Gunnar Hornig,
Topological measures of complexity in divergence-free fields I,II

Many vector fields in physics are divergence-free, the most prominent examples being magnetic fields and vorticity fields. These fields are often embedded in a fluid or plasma where they undergo constant topological deformations due to the flow of these fluids. They also often exhibit complex, knotted or braided, structures. The existence of these complex structures is important for the equilibria these fields can attain and hence one would like to be able to quantify this complexity. These two lectures will show how knot theory, differential geometry and index theory can be used to describe different aspects of the complexity of these fields. We will give examples to show that these measures are relevant for the dynamics and discuss open questions.
Lecture 1 will give an introduction into the dynamics of magnetic fields in plasmas and vorticity fields in hydrodynamics. We will discuss the importance of having stable, quantitative measures to describe the complexity of these fields and show that magnetic helicity is a prime example of such a measure. We will then explain how the concept of helicity can be generalised to higher link invariants using Massey products, and discuss the problems and open questions related to this.
Lecture 2 will look at divergence-free fields from the dynamical systems point of view. We define Poincaré mappings, discuss the topological entropy of these mappings and use index theorems to extract certain information about the complexity. The applicability of the approach is demonstrated with applications taken from the relaxation of braided magnetic fields in plasmas.

Jacek Jezierski,
Conformal Yano-Killing tensors in General Relativity

We show some general properties of the conformal Yano-Killing tensor on a Riemannian manifold. Several differential and algebraic equations are derived. The examples of CYK tensors in Minkowski, Kerr, Taub-NUT, de Sitter and anti-de Sitter spacetimes are discussed. Conformal rescaling of conformal Yano-Killing tensors together with the pullback of these objects to a submanifold is used to construct all solutions of a CYK equation in anti-de Sitter and de Sitter spacetimes. The relation between spin-2 fields and solutions to the Maxwell equations is used in the construction of a new conserved quantity which is quadratic in terms of the Weyl tensor. The obtained formulae are similar to the functionals obtained from the Bel-Robinson tensor.

Jerzy Kijowski, Geometric structure of the arrival time operator (Struktura geometryczna operatora czasu przybycia w mechanice kwantowej)

Uniqueness of the construction of the quantum arrival time operator, which I have proposed 40 years ago, is discussed. For this purpose geometric aspects of quantum mechanics is analyzed. In particular, the „quantization through Lie derivative” procedure is defined.

Władysław Adam Majewski,
On applications of Orlicz spaces to Statistical Physics.

Ivan Matychyn,
Control of systems with fractional dynamics under uncertainty I,II

There exist multiple definitions of fractional differentiation operators each having its advantages and shortcomings. The classical Riemann–Liouville fractional derivative has singularity at the origin. Therefore, the trajectories of corresponding differential system start at the infinity, which seems not always justified from the physical point of view. That is why Caputo regularized derivatives appeared, in which this defect was eliminated. This means that the trajectories of corresponding systems do not arrive at the infinity at any finite moment of time. However, both the Riemann–Liouville and the Caputo fractional operators do not possess neither semigroup nor commutative properties, which are inherent to the derivatives of integer order. A workaround for this problem is provided by the Miller–Ross sequential derivatives, which, in particular, make it feasible to lower the order of a system of differential equations by increasing their number. Hilfer fractional derivatives provide a framework that encompasses both the Riemann–Liouville and Caputo derivatives as particular cases. Here we also introduce sequential derivatives of special form. Their relation to the Riemann–Liouville and Caputo fractional derivatives and to each other is established.
Initial value problems (IVPs) for systems of fractional differential equations (FDEs) involving Riemann–Liouville, Caputo, Miller–Ross, and Hilfer derivatives are treated here on the basis of the Laplace transform method. Analytical solutions to these IVPs are obtained in terms of the Mittag-Leffler generalized matrix functions.
Further we examine problems of control of dynamic systems governed by FDEs from a differential game-theoretic perspective and consider the uncertainty as caused by opponent player’s control. In so doing, we apply the method of resolving functions sometimes called the method of the inverse Minkowski functionals. This method employs theorems on L  B-measurability and resulting superpositional measurability of certain set-valued mappings and their selections to substantiate a measurable choice of the control function.
The theoretical results are illustrated by an example of a system governed by Bagley–Torvik equation, which describes motion of a rigid plate immersed in a Newtonian fluid.

Łukasz Płociniczak,
Nonlinear anomalous diffusion: model and approximate solutions

Diffusion is one of the most common and important natural phenomena. It is a “random” spatial dispersion of some particles in its medium. The remarkable fact about the diffusion is its universality – almost on every scale some kind of diffusion is observed. We see it in fragrant particles dispersion, bacteria movement, contaminant transport and mergers of galaxies at cosmological scales. The diffusion equation is a classical description of these phenomena. Albeit of a spectacular precision and accuracy of the classical diffusion equation, a number of researches have lately reported about some discrepancies between theory and experiment. Especially this anomalous diffusion has been observed in porous media like those in building materials. In that case a self-similar profile has been observed only for a different space-time scaling. To model this feature a number of methods have been proposed, notably by the use of fractional derivative. We have obtained a simple, analytical approximation to the equation of nonlinear fractional diffusion. The high accuracy of the formula has been verified numerically. The method that we used is based on a series approximation of the Erdelyi-Kober fractional operator. After series truncation an ordinary differential equation can be obtained an solved. Our results are almost identical to those obtained numerically by other Authors, but our analytical form is an additional advantage that is very valuable in experimental work.

Jacek Pozorski,
Modelling the effects of subfilter flow scales on dispersed particles in large-eddy simulations of turbulent flows

Turbulent two-phase flows with solid particles are of industrial importance in power, chemical, and process engineering. At the same time, the modelling of such flows remains an open issue. Recently, the Large-Eddy Simulation (LES) approach has become an alternative to Reynolds-based approaches, at least for geometrically-simpler cases. It is based on a spatial filtering (smoothing) of the governing flow equations. There has been a long discussion about the impact of subfilter, or subgrid, flow scales (SGS) on the dynamics of the dispersed phase and there is an agreement now that these effects are important for smaller-inertia particles, including preferential concentration statistics, turbulent kinetic energy, collision rates or deposition velocity in wall-bounded flows.

In the proposed contribution, the state-of-the art in the modelling of the SGS particle dispersion coupled to LES of the carrier phase will be recalled. Then, proposals and results to date concerning the models based on the stochastic diffusion processes will be presented. At last, so-called structural models will be addressed.