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

Dressing Symmetries of Holomorphic BF Theories

We consider holomorphic BF theories, their solutions and symmetries. The equivalence of Cech and Dolbeault descriptions of holomorphic bundles is used to develop a method for calculating hidden (nonlocal) symmetries of holomorphic BF theories. A special cohomological symmetry group and its action on the solution space are described.Comment: 14 pages, LaTeX2

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

Hidden Symmetries of the Open N=2 String

It is known for ten years that self-dual Yang-Mills theory is the effective field theory of the open N=2 string in 2+2 dimensional spacetime. We uncover an infinite set of abelian rigid string symmetries, corresponding to the symmetries and integrable hierarchy of the self-dual Yang-Mills equations. The twistor description of the latter naturally connects with the BRST approach to string quantization, providing an interpretation of the picture phenomenon in terms of the moduli space of string backgrounds.Comment: 24 pages, no figures; v2: typos correcte

Particle size distribution analysis of pine sawdust: comparison of traditional oscillating screen method and photo-optical analysis

ArticleParticle size and particle size distribution (PSD) are crucial parameters which affect properties of particulate and agglomerated materials, and have an impact on a quality and utilization of a final product. The aim of this paper was to determine PSD as well as to assess dimensional features of pine sawdust fractions via mechanical sieve analysis and photo-optical analysis. The first one is a traditional and standard method taking into account only one parameter of particle shape and the second one is a modern method based on a digital image processing that considers also irregular shapes of biomass particles. Pine sawdust was grinded into three fractions: 4, 8 and 12 mm and analysed using two mentioned methods. A horizontal vibrating sieve shaker comprising 11 sieves and a bottom pan was used, and the obtained data of retained particles on each sieve were evaluated. For comparison, a computerized photo-optical particle analyser was applied with max Feret’s diameter as a measurement algorithm for a particle length, and PSD was analyzed by grouping the particles according to their distinct lengths adjusted to the sieves’ sizes used in the screening method. Moreover, additional results in dimensions and parameters of PSD were obtained and evaluated through the photo-optical method. Pine sawdust particles can be described as non-uniform, mainly prolonged, finer particles dominated in all fraction samples. The study showed differences in the results, inaccuracy and other drawbacks of the conventional sieving method such as clogging and falling-through phenomena as well as the limitations of the machine vision. Strong sides of both methods were discussed, too. Overall, the results contributed to a better knowledge of the material properties and different methods of PSD analysis