Explicit Free Parameterization of the Modified Tetrahedron Equation

The Modified Tetrahedron Equation (MTE) with affine Weyl quantum variables at N-th root of unity is solved by a rational mapping operator which is obtained from the solution of a linear problem. We show that the solutions can be parameterized in terms of eight free parameters and sixteen discrete phase choices, thus providing a broad starting point for the construction of 3-dimensional integrable lattice models. The Fermat curve points parameterizing the representation of the mapping operator in terms of cyclic functions are expressed in terms of the independent parameters. An explicit formula for the density factor of the MTE is derived. For the example N=2 we write the MTE in full detail. We also discuss a solution of the MTE in terms of bosonic continuum functions.Comment: 28 pages, 3 figure

Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations

In this paper we consider three-dimensional quantum q-oscillator field theory without spectral parameters. We construct an essentially big set of eigenstates of evolution with unity eigenvalue of discrete time evolution operator. All these eigenstates belong to a subspace of total Hilbert space where an action of evolution operator can be identified with quantized discrete BKP equations (synonym Miwa equations). The key ingredients of our construction are specific eigenstates of a single three-dimensional R-matrix. These eigenstates are boundary states for hidden three-dimensional structures of U_q(B_n^1) and U_q(D_n^1)$.Comment: 13 page

Quantum 2+1 evolution model

A quantum evolution model in 2+1 discrete space - time, connected with 3D fundamental map R, is investigated. Map R is derived as a map providing a zero curvature of a two dimensional lattice system called "the current system". In a special case of the local Weyl algebra for dynamical variables the map appears to be canonical one and it corresponds to known operator-valued R-matrix. The current system is a kind of the linear problem for 2+1 evolution model. A generating function for the integrals of motion for the evolution is derived with a help of the current system. The subject of the paper is rather new, and so the perspectives of further investigations are widely discussed.Comment: LaTeX, 37page

Topological insulators and geometry of vector bundles

For a long time, band theory of solids has focused on the energy spectrum, or Hamiltonian eigenvalues. Recently, it was realized that the collection of eigenvectors also contains important physical information. The local geometry of eigenspaces determines the electric polarization, while their global twisting gives rise to the metallic surface states in topological insulators. These phenomena are central topics of the present notes. The shape of eigenspaces is also responsible for many intriguing physical analogies, which have their roots in the theory of vector bundles. We give an informal introduction to the geometry and topology of vector bundles and describe various physical models from this mathematical perspective.Comment: v2: revised and extended version, submission to SciPos

On invariance of specific mass increment in the case of non-equilibrium growth

It is the first time invariance of specific mass increments of crystalline structures that co-exist in the case of non-equilibrium growth is grounded using the maximum entropy production principle. Based on the hypothesis of the existence of a universal growth equation, with the use of dimensional analysis, an explicit form of the dependence of specific mass increment on time is proposed. Applicability of the obtained results for describing growth in animate nature is discussed.Comment: 5 page

The method of restoring the musculoskeletal system after physical activity with relaxation

The paper discusses the theme of the restoration of the musculoskeletal system in athletes, engaged in physical activity with relaxation. The relevance of the chosen topic in that people after exercise pay little time for recovery and relaxation method few people know, this method is a good helper in the recovery of the bodyВ работе рассмотрена тема восстановление опорно-двигательного аппарата у спортсменов, занимающихся физическими нагрузками при помощи релаксации. Актуальность выбранной темы в том, что люди после физических нагрузок мало уделяют времени восстановлению, а метод релаксации мало кому знаком, именно этот метод хороший помощник при восстановление организм

Evolutionary Synthesis of Dynamical Object Emulator Based on RBF Neural Network

The combination of Genetic Algorithms (GAs) and Artificial Neural Networks (ANNs) has already resulted in researchers advancing in quite a few real world applications but it is in control that this alliance yields much appreciable benefit. The paper reports a Radial Basis Function (RBF) network training technique which joins together global strategy of GAs and a local adjusting procedure typical for RBF networks. While activation function window centres and widths are processed via a "slow" numeric GA, output-layer neurone synaptic weights are defined by a "fast" analytical method. The technique allows to minimize not only the network hidden-layer size but also the pattern set required for training the adequate dynamical object neuroemulator