Bruhat Order in the Full Symmetric sln\mathfrak{sl}_n Toda Lattice on Partial Flag Space

In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described the asymptotic behaviour of trajectories of the full symmetric $\mathfrak{sl}_n$ Toda lattice in the case of distinct eigenvalues of the Lax matrix. It turned out that it is completely determined by the Bruhat order on the permutation group. In the present paper we extend this result to the case when some eigenvalues of the Lax matrix coincide. In that case the trajectories are described in terms of the projection to a partial flag space where the induced dynamical system verifies the same properties as before: we show that when $t\to\pm\infty$ the trajectories of the induced dynamical system converge to a finite set of points in the partial flag space indexed by the Schubert cells so that any two points of this set are connected by a trajectory if and only if the corresponding cells are adjacent. This relation can be explained in terms of the Bruhat order on multiset permutations

Elliptic Schlesinger system and Painlev{\'e} VI

We construct an elliptic generalization of the Schlesinger system (ESS) with positions of marked points on an elliptic curve and its modular parameter as independent variables (the parameters in the moduli space of the complex structure). ESS is a non-autonomous Hamiltonian system with pair-wise commuting Hamiltonians. The system is bihamiltonian with respect to the linear and the quadratic Poisson brackets. The latter are the multi-color generalization of the Sklyanin-Feigin-Odeskii classical algebras. We give the Lax form of the ESS. The Lax matrix defines a connection of a flat bundle of degree one over the elliptic curve with first order poles at the marked points. The ESS is the monodromy independence condition on the complex structure for the linear systems related to the flat bundle. The case of four points for a special initial data is reduced to the Painlev{\'e} VI equation in the form of the Zhukovsky-Volterra gyrostat, proposed in our previous paper.Comment: 16 pages; Dedicated to the centenary of the publication of the Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 190

ДИНАМИКА СНЕГОЗАПАСОВ НА РАВНИННОЙ ТЕРРИТОРИИ РОССИИ В ЛЕСУ И В ПОЛЕ ПРИ КЛИМАТИЧЕСКИХ ИЗМЕНЕНИЯХ

In 1966-2010, snow surveys were simultaneously performed in forests and fields on the Russian plain territory. This made it possible to analyze characteristics of snow storages on fields and in forests as well as a dynamics of them under the present-day climate changes. Data of 81 weather stations located on the territory were used. According to data of these stations for the period 2001–2010 we obtained the following estimates for the maximal snow storage values, on average: for 20 stations located on the European territory of Russia to the north of 60° N –167 mmin forests and 162 on fields; for 44 stations to the south of 60° N – 118 and116 mm, respectively; for 10 stations in the south of West Siberia – 125 and107 mm; and for 7 stations in the East Siberia – 64 and70 mm. As one can see the last region is characterized by the opposite relation between forest and field conditions. Comparison of these values with similar data for the period 1966-2000 demonstrated that maximal snow storages decreased in forests by 7% but in fields they increased by 2%. The ratio of the maximum snow storage in the forest to their value in the field (i.e. a coefficient of snow reserve) for the periods 1981–1990, 1991–2000, and 2001–2010 are 1.15; 1.11 and 1.03, respectively. One of the reasons for the equalization of snow storage in forest and field may be changes of intensity and duration of snowstorms. In the calendar winters of 2001-2010, the average number of observations at weather stations with wind speeds over 10 m/s decreased relative to 1966-2010: in the European part of Russia by factor of 8.9 times, and in Western and Eastern Siberia – by 2.0 and 1.9 times, respectively. In the European part ofRussia, the number of observation periods when wind speed from 6 to 10 m/s was observed decreased by 1.9 times.Проведено сравнение снегозапасов в лесу и в поле за десятилетия с 1981 по 2010  г. Для метеостанций с наибольшей изменчивостью коэффициента снегозапасов на Европейской части России, в Западной и Восточной Сибири установлены изменения по десятилетиям числа наблюдений на метеостанциях со скоростями ветра в диапазоне 6–10 м/с и более

Fuchs versus Painlev\'e

We briefly recall the Fuchs-Painlev\'e elliptic representation of Painlev\'e VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the complete elliptic integral of the first and second kind, $K$ and $E$, is a straight consequence of the fact that the differential operators corresponding to the entries of Toeplitz-like determinants, are equivalent to the second order operator $L_E$ which has $E$ as solution (or, for off-diagonal correlations to the direct sum of $L_E$ and $d/dt$). We show that this can be generalized, mutatis mutandis, to the anisotropic Ising model. The singled-out second order linear differential operator $L_E$ being replaced by an isomonodromic system of two third-order linear partial differential operators associated with $\Pi_1$, the Jacobi's form of the complete elliptic integral of the third kind (or equivalently two second order linear partial differential operators associated with Appell functions, where one of these operators can be seen as a deformation of $L_E$). We finally explore the generalizations, to the anisotropic Ising models, of the links we made, in two previous papers, between Painlev\'e non-linear ODE's, Fuchsian linear ODE's and elliptic curves. In particular the elliptic representation of Painlev\'e VI has to be generalized to an ``Appellian'' representation of Garnier systems.Comment: Dedicated to the : Special issue on Symmetries and Integrability of Difference Equations, SIDE VII meeting held in Melbourne during July 200

Introduction to Integral Discriminants

The simplest partition function, associated with homogeneous symmetric forms S of degree r in n variables, is integral discriminant J_{n|r}(S) = \int e^{-S(x_1 ... x_n)} dx_1 ... dx_n. Actually, S-dependence remains the same if e^{-S} in the integrand is substituted by arbitrary function f(S), i.e. integral discriminant is a characteristic of the form S itself, and not of the averaging procedure. The aim of the present paper is to calculate J_{n|r} in a number of non-Gaussian cases. Using Ward identities -- linear differential equations, satisfied by integral discriminants -- we calculate J_{2|3}, J_{2|4}, J_{2|5} and J_{3|3}. In all these examples, integral discriminant appears to be a generalized hypergeometric function. It depends on several SL(n) invariants of S, with essential singularities controlled by the ordinary algebraic discriminant of S.Comment: 36 pages, 19 figure

The RNA acetyltransferase driven by ATP hydrolysis synthesizes N4-acetylcytidine of tRNA anticodon

The wobble base of Escherichia coli elongator tRNAMet is modified to N4-acetylcytidine (ac4C), which is thought to ensure the precise recognition of the AUG codon by preventing misreading of near-cognate AUA codon. By employing genome-wide screen of uncharacterized genes in Escherichia coli (‘ribonucleome analysis'), we found the ypfI gene, which we named tmcA (tRNAMet cytidine acetyltransferase), to be responsible for ac4C formation. TmcA is an enzyme that contains a Walker-type ATPase domain in its N-terminal region and an N-acetyltransferase domain in its C-terminal region. Recombinant TmcA specifically acetylated the wobble base of E. coli elongator tRNAMet by utilizing acetyl-coenzyme A (CoA) and ATP (or GTP). ATP/GTP hydrolysis by TmcA is stimulated in the presence of acetyl-CoA and tRNAMet. A mutation study revealed that E. coli TmcA strictly discriminates elongator tRNAMet from the structurally similar tRNAIle by mainly recognizing the C27–G43 pair in the anticodon stem. Our findings reveal an elaborate mechanism embedded in tRNAMet and tRNAIle for the accurate decoding of AUA/AUG codons on the basis of the recognition of wobble bases by the respective RNA-modifying enzymes