On structure groups of set-theoretic solutions to the Yang-Baxter equation

This paper explores the structure groups $G_{(X,r)}$ of finite non-degenerate set-theoretic solutions $(X,r)$ to the Yang-Baxter equation. Namely, we construct a finite quotient $\overline{G}_{(X,r)}$ of $G_{(X,r)}$, generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if $X$ injects into $G_{(X,r)}$, then it also injects into $\overline{G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of $G_{(X,r)}$. We show that multipermutation solutions are the only involutive solutions with diffuse structure group; that only free abelian structure groups are biorderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: biorderable, left-orderable, abelian, free abelian, torsion free.Comment: 32 pages. Final version. Accepted for publication in Proc. Edinburgh Math. So

Cohomology and extensions of braces

Braces and linear cycle sets are algebraic structures playing a major role in the classification of involutive set-theoretic solutions to the Yang-Baxter equation. This paper introduces two versions of their (co)homology theories. These theories mix the Harrison (co)homology for the abelian group structure and the (co)homology theory for general cycle sets, developed earlier by the authors. Different classes of brace extensions are completely classified in terms of second cohomology groups.Comment: 16 pages. Final version. Accepted for publication in Pacific Journal of Mathematic

Interference Commensurate Oscillations in Q1D Conductors

We suggest an analytical theory to describe angular magnetic oscillations recently discovered in quasi-one-dimensional conductor (TMTSF)2PF6 [see Phys. Rev. B, 57, 7423 (1998)] and define the positions of the oscillation minima. The origin of these oscillations is related to interference effects resulting from an interplay of quasi-periodic and periodic ("commensurate") electron trajectories in an inclined magnetic field. We reproduce via calculations existing experimental data and predict some novel effects.Comment: 10 pages, 2 figure

Evolution of unstable system.

Scenario of appearance and development of instability in problem of a flow around a solid sphere at rest is discussed. The scenario was created by solutions to the multimoment hydrodynamics equations, which were applied to investigate the unstable phenomena. These solutions allow interpreting Stokes flow, periodic pulsations of the recirculating zone in the wake behind the sphere, the phenomenon of vortex shedding observed experimentally. In accordance with the scenario, system loses its stability when entropy outflow through surface confining the system cannot be compensated by entropy produced within the system. The system does not find a new stable position after losing its stability, that is, the system remains further unstable. As Reynolds number grows, one unstable flow regime is replaced by another. The replacement is governed tendency of the system to discover fastest path to depart from the state of statistical equilibrium. This striving, however, does not lead the system to disintegration. Periodically, reverse solutions to the multimoment hydrodynamics equations change the nature of evolution and guide the unstable system in a highly unlikely direction. In case of unstable system, unlikely path meets the direction of approaching the state of statistical equilibrium. Such behavior of the system contradicts the scenario created by solutions to the classic hydrodynamics equations. Unstable solutions to the classic hydrodynamics equations are not fairly prolonged along time to interpret experiment. Stable solutions satisfactorily reproduce all observed stable medium states. As Reynolds number grows one stable solution is replaced by another. They are, however, incapable of reproducing any of unstable regimes recorded experimentally. In particular, stable solutions to the classic hydrodynamics equations cannot put anything in correspondence to any of observed vortex shedding modes. In accordance with our interpretation, the reason for this is the classic hydrodynamics equations themselves

Remaining life realization of large-sized multicomponent products

З досліджуваного класу відновлюваних великогабаритних складених виробів, таких як зубчасті колеса, вальцювальні валки й універсальні шпинделі виділена друга група деталей - вальцювальні валки. З урахуванням режимів їхньої експлуатації розроблена методика із визначення залишкового ресурсу й міцності цих виробів після відпрацьовування ними ресурсу по робочій поверхні. Впровадження комплексу заходів щодо відновлення вальцювальних валків дозволяє істотно продовжити «життєвий» цикл їхньої експлуатації й розширити використання розроблених технічних рішень на інші вироби розглянутого класу. Реалізація технологічних процесів повторного використання деталей дозволяє забезпечити значний економічний ефект.From the investigated class of large-sized composite products such as gears, mill rolls and universal spindles, the mill rolls were separated out in the second group of parts. In compliance with their operating modes the procedure has been developed for determination of remaining life time and strength of these products after working out of their operating surface life cycle. Implementation of measures for mill rolls reconditioning allows to increase significantly their operating life cycle and to spread the application of elaborated technical solutions for the other parts of the class under consideration. Realization of technological procedures as for parts reuse allows providing significant economic benefit

A reliable method for the stability analysis of structures

A space truss is in an unstable configuration if it can displace incrementally without an incremental change in its loading and its supports. The load path which follows after an unstable configuration can be unique (snap-through), or there can be several possible load paths (bifurcation). This paper presents a method to detect nearly unstable configurations of a truss, a method to determine the loading, displacements and reactions of the unstable configuration and a method to traverse the load path which follows after the unstable configuration. The detection of structural configurations with singular tangent stiffness matrix is essential because they can be unstable. The secondary paths, especially in unstable buckling, can play the most important role in the loss of stability and collapse of the structure. A new method for reliable detection and accurate computation of singular points on load paths is presented and applied to a space truss subjected to symmetric and asymmetric snow loads.Keywords: space structure, stability, snap-through, bifurcation, nonlinear behaviou

Reflection equation as a tool for studying solutions to the Yang-Baxter equation

Given a right-non-degenerate set-theoretic solution $(X,r)$ to the Yang-Baxter equation, we construct a whole family of YBE solutions $r^{(k)}$ on $X$ indexed by its reflections $k$ (i.e., solutions to the reflection equation for $r$). This family includes the original solution and the classical derived solution. All these solutions induce isomorphic actions of the braid group/monoid on $X^n$. The structure monoids of $r$ and $r^{(k)}$ are related by an explicit bijective $1$-cocycle-like map. We thus turn reflections into a tool for studying YBE solutions, rather than a side object of study. In a different direction, we study the reflection equation for non-degenerate involutive YBE solutions, show it to be equivalent to (any of the) three simpler relations, and deduce from the latter systematic ways of constructing new reflections.Comment: 18 pages, 12 figures. Final versio