A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation

A bijective map $r: X^2 \longrightarrow X^2$, where $X = \{x_1, ..., x_n \}$ is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter equation} (YBE) if the braid relation $r_{12}r_{23}r_{12} = r_{23}r_{12}r_{23}$ holds in $X^3.$ A non-degenerate involutive solution $(X,r)$ satisfying $r(xx)=xx$, for all $x \in X$, is called \emph{square-free solution}. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions $(X,r)$ and the associated Yang-Baxter algebraic structures -- the semigroup $S(X,r)$, the group $G(X,r)$ and the $k$- algebra $A(k, X,r)$ over a field $k$, generated by $X$ and with quadratic defining relations naturally arising and uniquely determined by $r$. We study the properties of the associated Yang-Baxter structures and prove a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, a semigroup of I type, and a semigroup of skew-polynomial type, are equivalent. This implies that the Yang-Baxter algebra $A(k, X,r)$ is Poincar\'{e}-Birkhoff-Witt type algebra, with respect to some appropriate ordering of $X$. We conjecture that every square-free solution of YBE is retractable, in the sense of Etingof-Schedler.Comment: 34 page

ENERGY TRANSFER EVALUATION IN THE “STRIKER-BALL” SYSTEM

Classical Newton’s model has been used to analyze the energy transfer mechanisms in the “striker-ball” system during impact. The velocity restitution coefficient and velocity transfer index were determined from the laboratory experiments and in natural sports playing conditions (volleyball, football, tennis). It was found that a living system controls the energy transfer in the impact phase depending on the motion task of the stroke

Innovations and ISO standards management in the organizations

The turbulent environment in which organizations must function nowadays necessitates the continuous search for opportunities to innovate. Innovations require the availability of various resources, including knowledge, information, financial and organizational conditions, in order to be implemented. Building an innovation management system is of leading importance for achieving better results, increasing its advantages and strengthening the organization's place among its competitors on the market. The development, implementation and maintenance of such a system have the role of a framework for systematic innovation management practices. An important point is the quality level of the management system, which is why the International Organization for Standardization (ISO) is developing a special series of ISO standards based on CEN/TS 16555. The purpose of this article is to characterize the features of the ISO 56000 series of standards for innovation management in organizations and to bring their benefits into a strategic aspect

Time dependent correlations in marine stratocumulus cloud base height records

The scaling ranges of time correlations in the cloud base height records of marine boundary layer stratocumulus are studied applying the Detrended Fluctuation Analysis statistical method. We have found that time dependent variations in the evolution of the $\alpha$ exponent reflect the diurnal dynamics of cloud base height fluctuations in the marine boundary layer. In general, a more stable structure of the boundary layer corresponds to a lower value of the $\alpha$ - indicator, i.e. larger anti-persistence, thus a set of fluctuations tending to induce a greater stability of the stratocumulus. In contrast, during periods of higher instability in the marine boundary, less anti-persistent (more persistent like) behavior of the system drags it out of equilibrium, corresponding to larger $\alpha$ values. From an analysis of the frequency spectrum, the stratocumulus base height evolution is found to be a non-stationary process with stationary increments. The occurrence of these statistics in cloud base height fluctuations suggests the usefulness of similar studies for the radiation transfer dynamics modeling.Comment: 12 pages, 6 figures; to appear in Int. J. Mod. Phys. C, Vol. 13, No. 2 (2002