Vol. 9, No. 4 (2020): Regular Issue

We consider the continuity equation $\partial_t \mu_t + \text{div}(\mathbf{b} \mu_t) = 0$, where $\{\mu_t\}_{t \in \mathbb R}$ is a measurable family of (possibily signed) Borel measures on $\mathbb R^d$ and $\mathbf{b} \colon \mathbb R \times \mathbb R^d \to \mathbb R^d$ is a bounded Borel vector field (and the equation is understood in the sense of distributions). We discuss some uniqueness and non-uniqueness results for this equation: in particular, we report on some counterexamples in which uniqueness of the flow of the vector field holds but one can construct non-trivial signed measure-valued solutions to the continuity equation with zero initial data. This is based on a joint work with N.A. Gusev [1].

We solve completely the problem of Liouvillian integrability for cubic Li\'{e}nard differential equations with quadratic damping. %Our results are applicable for a wide family of dynamical systems. Our main tool is the method of Puiseux series. We find necessary and sufficient conditions for equations under consideration to have Jacobi last multipliers of a special form. It turns out that some particular sub--families being Liouvillian non--integrable possess Jacobi last multipliers. The Jacobi last multipliers give rise to non--standard Lagrangians and it is an interesting property of these dynamical systems. In addition, we prove that cubic Li\'{e}nard differential equations with quadratic damping do not have algebraic limit cycles.

We give an analytic and topological description of the Uhlenbeck manifold, that is a manifold of triples (a symmetric operator, an eigenvector, an eigenvalue), for the finite-dimensional symmetric matrices and the family of stationary periodic Schrodinger operators. Then, we describe an uplifting of Lax vector fields to these manifolds.

We study $C^1$-smooth maps obtained by small perturbations of $C^1$-smooth skew products of maps of an interval with $\Omega$-stable quotients and present results on the coexistence of periods of periodic orbits for maps under consideration. In particular, $C^1$-smooth $\Omega$-stable maps of an interval do not contain maps of type $2^{\infty}$, i.e. maps that have the unbounded set of (the least) periods of periodic orbits $\tau$ for $\tau=\{2^i\}_{i\geq 0}$. We prove here that analogously to $C^1$-smooth skew products of maps of an interval with $\Omega$-stable quotients there exist the maps under consideration with $\tau=\{2^i\}_{i\geq 0}.$

We consider the space of finite divergence-free Borel vector measures on~$\mathbb R^d$, endowed with the total variation norm. For $d=2$ we present a characterization of the extreme points of the unit ball in this space. This allows one to decompose (for $d=2$) any finite divergence-free vector measure into measures induced by closed Lipschitz curves. The results are based on a joint work with P. Bonicatto.

The difference approximation of the one-dimensional Liouville equation for the sample distribution function density of the non-stationary time series estimated by the histogram is considered. The scheme with semi-group property conservation is constructed for evolution model of this sample distribution function density. We investigate the problem of appropriate Liouville equation construction for given initial and final distributions. We prove the necessary and sufficient condition of such a representation, which is a strong positivity of the initial density distribution in the inner class intervals. The determination of the corresponding Liouville velocity algorithm is constructed and its mechanical-statistical meaning is shown. The dynamical system, associated with this Liouville equation, is considered. We interpret the Liouville statistical velocity as a corresponding velocity of dynamical system, according to representation of statistical mechanics. We show, that this interpretation leads to monotonic discrete dynamical system with stationary point, corresponding to equality of initial and final distribution functions.

Let $X$ be a dendrite, $f:X\to X$ be a monotone map. In the article the relationship between a structure of a dendrite $X$ and dynamical properties of $f$ is studied. Namely the relation between a structure of $X$ and a structure of the sets of periodic points of $f$, non-wandering points of $f$, $\omega$-limit sets of trajectories is established. The structure of dendrites on which there exist monotone pointwise chain recurrent maps is characterized. The structure of dendrites on which there exist monotone maps with homoclinic points is described.

A modification of the Ekeland variational principle for functions unbounded from below is obtained. For a wide class of differentiable functions not necessarily bounded below, it is shown that there exists a minimizing sequence satisfying the first-order necessary conditions, up to any desired approximation.

We discuss complex geometric properties of the universal Teichm\"uller space $\mathcal T$. It is a complex Banach manifold which name is motivated by the fact that all classical Teichm\"uller spaces $T(G)$, associated with compact Riemann surfaces, are contained in $\mathcal T$ as complex subvarieties. Another important subset of $\mathcal T$ is the space $\mathcal S$ of orientation-preserving diffeomorphisms of $S^1$ considered modulo M\"obius transforms. It is a K\"ahler Frechet manifold. Our interest in $\mathcal T$ was initially motivated by its relation to string theory which we have studied earlier in a series of papers.

Both quantum information features and irreversible quantum evolution of the models arising in physical systems in one-particle approximation are discussed. It is shown that the calculation of the reduced density matrix and entanglement analysis are considerably simplified in this case. The irreversible quantum evolution described by Gorini--Kossakowski--Sudarshan--Lindblad equations in the one-particle approximation could be defined by a solution of a Shroedinger equation with a dissipative generator. It simplifies the solution of the initial equation on the one side and gives a physical interpretation of such a Shroedinger equation with non-Hermitian Hamiltonian on the other side.

The paper is devoted to non-autonomous dynamics, which is generated by positive self-adjoint operator $A$ and a family of non-negative self-adjoint operators $\{B(t)\}_{t\geq 0}$ defined in a separable Hilbert space. It is shown that solution operator $\{U(t,s)\}_{0 \leq s \leq t}$ of the evolution equation can be approximated in the operator norm topology by a product formula that involves $A$ and $B(t)$. We also established the rate of convergence of the product formula to the solution operator. These results are proved using the evolution semigroup approach to non-autonomous dynamics.

The paper presents some weak compactness criterion for a subset~$M$ of the set $\mathfrak{RM}_b(T,\mathcal{G})$ of all positive bounded Radon measures on a Hausdorff topological space $(T,\mathcal{G})$ similar to the Prokhorov criterion for a complete separable metric space. Since for a general topological space the classical space $C_b(T,\mathcal{G})$ of all bounded continuous functions on~$T$ can be trivial and so does not separate points and closed sets, we consider instead of $C_b(T,\mathcal{G})$-weak compactness $S(T,\mathcal{G})$-weak compactness with respect to the new uniformly closed linear space $S(T,\mathcal{G})$ of all (symmetrizable) metasemicontinuous functions.

It has been recently proven that, under ad hoc conditions, the Cipher Block Chaining (CBC) mode of operation can behave chaotically according to the mathematical definition of Devaney, on the infinite discrete product set of finite memory coupled with (finite) media of unbounded size. This occurs when the chosen block cipher function satisfies some properties related to a well defined associated graph. Rudimentary examples taken from so-called transposition cipher methods have formerly been proposed as illustrative examples. In this paper, the same canvas will be followed by regarding the conditions under which the CBC mode behaves chaotically. But the encryption function is now the Rivest Cipher 5 (RC5) one, a very famous symmetric key block cipher algorithm. Therefore, our goal is to prove the chaotic behavior of RC5-CBC encryption algorithm according to the reputed Devaney's definition. Then, this unpredictability was checked in hardware through such sensitivity tests which allowed us to validate that RC5-CBC exhibits a high degree of randomness, key and plain sensitivity, in addition to the so-called avalanche effect.

In current study, vibration and stability analysis comparison of two nanostructures i.e. nanoshell (NS) and piezoelectric nanoshell (PENS) subjected to electrostatic excitation and Visco-Pasternak medium is investigated using the Gurtin--Murdoch surface/interface (S/I) theory. For this analysis, Hamilton's principles, the assumed mode method combined with Lagrange--Euler's and also Complex averaging method combined with Arc-length continuation are used. It can be seen that by changing the surface/interface densities and as a result, increasing or decreasing the system stiffness, the natural frequency can be less or greater than the state without taking into account the S/I effects. In both nanostructures (NS and PENS), considering the surface/interface effects increase the nonlinear behaviour compared with without S/I effects.

We consider a non-stationary mixed variational inequality problem involving an integrable mapping and a convex function, where only approximation sequences are known instead of exact values of the cost mapping and function, and feasible set. We apply a descent method and partial penalization to prove the convergence is attained without concordance of penalty, accuracy, and approximation parameters under coercivity type conditions.