On the dependence of the reflection operator on boundary conditions for biharmonic functions

The biharmonic equation arises in areas of continuum mechanics including linear elasticity theory and the Stokes flows, as well as in a radar imaging problem. We discuss the reflection formulas for the biharmonic functions u(x,y)∈R² subject to different boundary conditions on a real-analytic curve in the plane. The obtained formulas, generalizing the celebrated Schwarz symmetry principle for harmonic functions, have different structures. In particular, in the special case of the boundary, Γ₀:={y=0}, reflections are point-to-point when the given on Γ₀ conditions are u=n_∂u=0, u=Δu=0 or n_∂u=n_∂Δu=0, and point to a continuous set when u=n_∂Δu=0 or n_∂u=Δu=0 on Γ₀.     

Three hierarchies of simple games parameterized by “resource” parameters

This paper contributes to the program of numerical characterization and classification of simple games outlined in the classic monograph of von Neumann and Morgenstern. We suggest three possible ways to classify simple games beyond the classes of weighted and roughly weighted games. To this end we introduce three hierarchies of games and prove some relationships between their classes. We prove that our hierarchies are true (i.e., infinite) hierarchies. In particular, they are strict in the sense that more of the key “resource” (which may, for example, be the size or structure of the “tie-breaking” region where the weights of the different coalitions are considered so close that we are allowed to specify either winningness or nonwinningness of the coalition) yields the flexibility to capture strictly more games.     

Optimal control of vibrationally excited locomotion systems

Optimal controls are constructed for two types of mobile systems propelling themselves due to relative oscillatory motions of their parts. The system of the first type is modelled by a rigid body (main body) to which two links are attached by revolute joints. All three bodies interact with the environment with the forces depending on the velocity of motion of these bodies relative to the environment. The system is controlled by high-frequency periodic angular oscillations of the links relative to the main body. The system of the other type consists of two bodies, one of which (the main body) interacts with the environment and with the other body (internal body), which interacts with the main body but does not interact with the environment. The system is controlled by periodic oscillations of the internal body relative to the main body. For both systems, the motions with the main body moving along a horizontal straight line are considered. Optimal control laws that maximize the average velocity of the main body are found.     

Subset currents on free groups

We introduce and study the space ${{\mathcal{S}{\rm Curr} (F_N)}}$ of subset currents on the free group F _N , and, more generally, on a word-hyperbolic group. A subset current on F _N is a positive F _N -invariant locally finite Borel measure on the space ${{\mathfrak{C}_N}}$ of all closed subsets of ?F _N consisting of at least two points. The well-studied space Curr(F _N ) of geodesics currents–positive F _N -invariant locally finite Borel measures defined on pairs of different boundary points–is contained in the space of subset currents as a closed ${{\mathbb{R}}}$ -linear Out(F _N )-invariant subspace. Much of the theory of Curr(F _N ) naturally extends to the ${{\mathcal{S}\;{\rm Curr} (F_N)}}$ context, but new dynamical, geometric and algebraic features also arise there. While geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in F _N . If a free basis A is fixed in F _N , subset currents may be viewed as F _N -invariant measures on a “branching” analog of the geodesic flow space for F _N , whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of F _N with respect to A. Similarly to the case of geodesics currents, there is a continuous Out(F _N )-invariant “co-volume form” between the Outer space cv _N and the space ${{\mathcal{S}\;{\rm Curr} (F_N)}}$ of subset currents. Given a tree ${{T \in {\rm cv}_N}}$ and the “counting current” ${{\eta_H \in \mathcal{S}\;{\rm Curr} (F_N)}}$ corresponding to a finitely generated nontrivial subgroup H ≤  F _N , the value ${{\langle T, \eta_H \rangle}}$ of this intersection form turns out to be equal to the co-volume of H, that is the volume of the metric graph T _H /H, where ${{T_H \subseteq T}}$ is the unique minimal H-invariant subtree of T. However, unlike in the case of geodesic currents, the co-volume form ${{{\rm cv}_N \times \mathcal{S}\;{\rm Curr}(F_N)\; \to [0,\infty)}}$ does not extend to a continuous map ${{\overline{{\rm cv}}_N \times \mathcal{S}\; {\rm Curr} (F_N) \to [0,\infty)}}$ .     

The Destruction of Carbon Tetrachloride Dissolved in Water in a Dielectric Barrier Discharge in Oxygen

The kinetics of decomposition of tetrachloromethane (TCM) in its aqueous solutions and the kinetics of decomposition products formation was investigated under the action of DBD at atmospheric pressure in oxygen in a falling-flow reactor. The range of initial concentrations of TCM was 25–325 μmol/l, the discharge power—2–11 W and O₂ flow rates—1–3 cm³/s. It is shown that the kinetics of the TCM decomposition can be described by the equation of pseudo-first kinetic order. The rate constant depended weakly on the discharge parameters and was?~?5 s^?1. The energy efficiency of the decomposition, depending on the parameters, was 0.1–1.3 molecules per 100 eV. When the residence time of the solution with the discharge zone is more than 1 s, it is possible to achieve almost 100% degree of TCM decomposition. It is shown that the main products of the TCM decomposition in the liquid phase are aldehydes and Cl^? ions, and in the gas phase—the molecules CO and CO₂. The results for energy efficiency are compared with the results obtained in other AOP’s processes (Fenton process, photocatalytic process, the radiation process by the action of high-energy electron flux). It is shown that the action of the DBD is more effective than the action of the above processes.

     

Characterization of polylactic acid green composites and its biodegradation in a bacterial environment

The growing interest in the preservation of our environment is pushing for solutions to develop less impacting materials. Thus, the development of biocomposites and is recyclable and compostable end-of-life resources seem an interesting alternative. In this study, the characterization of Polylactic acid (PLA) reinforced with treated and untreated Olive husk flour (OHF) were investigated. More then, their biodegradation with a Bacillus sp. has been evaluated. The main results show that the bacteria degraded both the PLA and the composite. This degradation was confirmed by the release of reducing sugars as well as increasing weight loss of PLA matrix and composites.     

Yang–Mills instantons and dyons on group manifolds

We consider Euclidean SU(N)$\mathrm{SU}(N)$ Yang–Mills theory on the space G×R$G\times \mathbb{R}$, where G is a compact semisimple Lie group, and introduce first-order BPS-type equations which imply the full Yang–Mills equations. For gauge fields invariant under the adjoint G  -action these BPS equations reduce to first-order matrix equations, to which we give instanton solutions. In the case of G=SU(2)≅S³$G=\mathrm{SU}(2)\cong S^3$, our matrix equations are recast as Nahm equations, and a further algebraic reduction to the Toda chain equations is presented and solved for the SU(3) example. Finally, we change the metric on G×R$G\times \mathbb{R}$ to Minkowski and construct finite-energy dyon-type Yang–Mills solutions. The special case of G=SU(2)×SU(2)$G=\mathrm{SU}(2)\times \mathrm{SU}(2)$ may be used in heterotic flux compactifications.     

Effects of external global harmonic influence on chimera states

Nonlinear Dynamics - The effects of the global harmonic force on an ensemble of nonlocally coupled chaotic Rössler oscillators is studied. The autonomous ensemble without external force...