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

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

Cold air outbreaks and their signature in the ozonometric data at the mountain station near Kislovodsk, Russia

International audienceTwo cases of cold air outbreak in November 2001 are analyzed on the basis of ozone concentration measurements and weather data at the mountain station (43°44' N, 42°43 E, 2070 m a.s.l.) near Kislovodsk, North Caucasus, Russia. Two cold fronts, with fast clearing up in the rear of the cloud zone, passed the station in the morning of 12 and 16 November. In both cases, the ozone concentration drops down to 7?8 ppb within the frontal cloud zones, under 100% relative humidity, RH, and then sharply increases to 43?45 ppb in dry (RH about 50%), cold air during several hours. After a warm front passage (12 November) or RH growth in the cold post-frontal airmass (16 November), the ozone concentration decreases again to its average values near 30 ppb. Neither diurnal cycle nor photochemical generation could produce in the ozonometric data this specific "drop-and-peak'' signature which is considered to be caused by cold outbreak with post-frontal flow of "dry air stream'' type. Synoptic situation and tropopause topography are analyzed using the objective analysis data and METEOSAT images in the water vapor radiation band. It is concluded that the ozone growth in the rear of the cold fronts is associated with sinking of substratospheric or stratospherically influenced air from the areas of tropopause folding. Motion of cold surges with their frontal zones, jet streams, and tropopause folds is analyzed, including development of wave disturbances at the streamers and formation of substratospheric air "tongues'' underflowing the high tropopause and representing "aged'' intrusions

Uncertainties in Arctic sea ice thickness and volume: new estimates and implications for trends

Sea ice volume has decreased in the last decades, evoked by changes in sea ice area and thickness. Estimates of sea ice area and thickness rely on a number of geophysical parameters which introduce large uncertainties. To quantify these uncertainties we use freeboard retrievals from ICESat and investigate different assumptions about snow depth, sea ice density and area. We find that uncertainties in ice area are of minor importance for the estimates of sea ice volume during the cold season in the Arctic basin. The choice of mean ice density used when converting sea ice freeboard into thickness mainly influences the resulting mean sea ice thickness, while snow depth on top of the ice is the main driver for the year-to-year variability, particularly in late winter. The absolute uncertainty in the mean sea ice thickness is 0.28 m in February/March and 0.21 m in October/November. The uncertainty in snow depth contributes up to 70% of the total uncertainty and the ice density 30–35%, with higher values in October/November. We find large uncertainties in the total sea ice volume and trend. The mean total sea ice volume is 10 120 ± 1280 km³ in October/November and 13 250 ± 1860 km³ in February/March for the time period 2005–2007. Based on these uncertainties we obtain trends in sea ice volume of −1450 ± 530 km³ a^−1 in October/November and −880 ± 260 km³ a^−1 in February/March over the ICESat period (2003–2008). Our results indicate that, taking into account the uncertainties, the decline in sea ice volume in the Arctic between the ICESat (2003–2008) and CryoSat-2 (2010–2012) periods may have been less dramatic than reported in previous studies. However, more work and validation is required to quantify these changes and analyse possible unresolved biases in the freeboard retrievals

Supernova Ia: a Converging Delayed Detonation Wave

A model of a carbon-oxygen (C--O) presupernova core with an initial mass 1.33 M_\odot, an initial carbon mass fraction 0.27, and with an average mass growth-rate 5 x 10^{-7} M_\odot/yr due to accretion in a binary system was evolved from initial central density 10^9 g/cm^3, and temperature 2.05 x 10^8 K through convective core formation and its subsequent expansion to the carbon runaway at the center. The only thermonuclear reaction contained in the equations of evolution and runaway was the carbon burning reaction 12C + 12C with an energy release corresponding to the full transition of carbon and oxygen (with the same rate as carbon) into 56Ni. As a parameter we take \alpha_c - a ratio of a mixing length to the size of the convective zone. In spite of the crude assumptions, we obtained a pattern of the runaway acceptable for the supernova theory with the strong dependence of its duration on \alpha_c. In the variants with large enough values of \alpha_c=4.0 x 10^{-3} and 3.0 x 10^{-3} the fuel combustion occurred from the very beginning as a prompt detonation. In the range of 2.0 x 10^{-3} >= \alpha_c >= 3.0 x 10^{-4} the burning started as a deflagration with excitation of stellar pulsations with growing amplitude. Eventually, the detonation set in, which was activated near the surface layers of the presupernova (with m about 1.33 M_\odot) and penetrated into the star down to the deflagration front. Excitation of model pulsations and formation of a detonation front are described in detail for the variant with \alpha_c=1.0 x 10^{-3}.Comment: 13 pages, 11 figures, to appear in Astronomy Letter

The molecular aspects of personalized anticancer treatment

Only 25% of cancer patients, on average, benefit from therapy. Even in the cases of complete clinical response the tumor progression is an event of high level expectation. The main reasons for tumor progression are: intratumor heterogeneity resulted from clonal evolution, drug resistance, and tumor-promoting microenvironment. The reprogramming of microenvironmental stromal-inflammatory components is expected to allow tumor phenotype reversion. So, to find the new effective markers of tumor progression, drug response and targets for therapy, it could be promising to take into account the tumor-microenvironment heterogeneity and tumor clonal evolution