Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model

In 1980 Jimbo and Miwa evaluated the diagonal two-point correlation function of the square lattice Ising model as a $\tau$-function of the sixth Painlev\'e system by constructing an associated isomonodromic system within their theory of holonomic quantum fields. More recently an alternative isomonodromy theory was constructed based on bi-orthogonal polynomials on the unit circle with regular semi-classical weights, for which the diagonal Ising correlations arise as the leading coefficient of the polynomials specialised appropriately. Here we demonstrate that the next-to-diagonal correlations of the anisotropic Ising model are evaluated as one of the elements of this isomonodromic system or essentially as the Cauchy-Hilbert transform of one of the bi-orthogonal polynomials.Comment: 11 pages, 1 figur

Random Matrix Theory and the Sixth Painlev\'e Equation

A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realised by matrices with Gaussian entries leads to statistical quantities related to the eigenspectrum, such as the distribution of the largest eigenvalue, which can be expressed as multidimensional integrals or equivalently as determinants. These distributions are well known to be $\tau$-functions for Painlev\'e systems, allowing for the former to be characterised as the solution of certain nonlinear equations. We consider the random matrix ensembles for which the nonlinear equation is the $\sigma$ form of \PVI. Known results are reviewed, as is their implication by way of series expansions for the distributions. New results are given for the boundary conditions in the neighbourhood of the fixed singularities at $t=0,1,\infty$ of $\sigma$\PVI displayed by a generalisation of the generating function for the distributions. The structure of these expansions is related to Jimbo's general expansions for the $\tau$-function of $\sigma$\PVI in the neighbourhood of its fixed singularities, and this theory is itself put in its context of the linear isomonodromy problem relating to \PVI.Comment: Dedicated to the centenary of the publication of the Painlev\'e VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 190

{\bf τ\tau-Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}

It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact $\tau$-functions for certain Painlev\'e systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or derivable from the existing literature, are likewise $\tau$-functions for certain Painlev\'e systems. In the case of symplectic matrix ensembles all exact evaluations, either known or derivable from the existing literature, are identified as the mean of two $\tau$-functions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two $\tau$-functions gives the gap probability in the corresponding unitary symmetry case, while one of those $\tau$-functions is the gap probability in the corresponding orthogonal symmetry case.Comment: AMS-Late

Boundary conditions associated with the Painlev\'e III' and V evaluations of some random matrix averages

In a previous work a random matrix average for the Laguerre unitary ensemble, generalising the generating function for the probability that an interval $(0,s)$ at the hard edge contains $k$ eigenvalues, was evaluated in terms of a Painlev\'e V transcendent in $\sigma$-form. However the boundary conditions for the corresponding differential equation were not specified for the full parameter space. Here this task is accomplished in general, and the obtained functional form is compared against the most general small $s$ behaviour of the Painlev\'e V equation in $\sigma$-form known from the work of Jimbo. An analogous study is carried out for the the hard edge scaling limit of the random matrix average, which we have previously evaluated in terms of a Painlev\'e \IIId transcendent in $\sigma$-form. An application of the latter result is given to the rapid evaluation of a Hankel determinant appearing in a recent work of Conrey, Rubinstein and Snaith relating to the derivative of the Riemann zeta function

A Robust Measure of Tidal Circularization in Coeval Binary Populations: The solar-type spectroscopic Binary Population in The Open Cluster M35

We present a new homogeneous sample of 32 spectroscopic binary orbits in the young (~ 150 Myr) main-sequence open cluster M35. The distribution of orbital eccentricity vs. orbital period (e-log(P)) displays a distinct transition from eccentric to circular orbits at an orbital period of ~ 10 days. The transition is due to tidal circularization of the closest binaries. The population of binary orbits in M35 provide a significantly improved constraint on the rate of tidal circularization at an age of 150 Myr. We propose a new and more robust diagnostic of the degree of tidal circularization in a binary population based on a functional fit to the e-log(P) distribution. We call this new measure the tidal circularization period. The tidal circularization period of a binary population represents the orbital period at which a binary orbit with the most frequent initial orbital eccentricity circularizes (defined as e = 0.01) at the age of the population. We determine the tidal circularizationperiod for M35 as well as for 7 additional binary populations spanning ages from the pre main-sequence (~ 3 Myr) to late main-sequence (~ 10 Gyr), and use Monte Carlo error analysis to determine the uncertainties on the derived circularization periods. We conclude that current theories of tidal circularization cannot account for the distribution of tidal circularization periods with population age.Comment: 37 pages, 9 figures, to be published in The Astrophysical Journal, February 200

Melt block copolymerization of ε-caprolactone and L-lactide

AB block copolymers of ε-caprolactone and (L)-lactide could be prepared by ring-opening polymerization in the melt at 110°C using stannous octoate as a catalyst and ethanol as an initiator provided ε-caprolactone was polymerized first. Ethanol initiated the polymerization of ε-caprolactone producing a polymer with ε-caprolactone derived hydroxyl end groups which after addition of L-lactide in the second step of the polymerization initiated the ring-opening copolymerization of L-lactide. The number-average molecular weights of the poly(ε-caprolactone) blocks varied from 1.5 to 5.2 × 103, while those of the poly(L-lactide) blocks ranged from 17.4 to 49.7 × 103. The polydispersities of the block copolymers varied from 1.16 to 1.27. The number-average molecular weights of the polymers were controlled by the monomer/hydroxyl group ratio, and were independent on the monomer/stannous octoate ratio within the range of experimental conditions studied. When L-lactide was polymerized first, followed by copolymerization of ε-caprolactone, random copolymers were obtained. The formation of random copolymers was attributed to the occurrence of transesterification reactions. These side reactions were caused by the ε-caprolactone derived hydroxyl end groups generated during the copolymerization of ε-caprolactone with pre-polymers of L-lactide. The polymerization proceeds through an ester alcoholysis reaction mechanism, in which the stannous octoate activated ester groups of the monomers react with hydroxyl groups

Fine Selmer Groups and Isogeny Invariance

We investigate fine Selmer groups for elliptic curves and for Galois representations over a number field. More specifically, we discuss Conjecture A, which states that the fine Selmer group of an elliptic curve over the cyclotomic extension is a finitely generated $\mathbb{Z}_p$-module. The relationship between this conjecture and Iwasawa's classical $\mu=0$ conjecture is clarified. We also present some partial results towards the question whether Conjecture A is invariant under isogenies.Comment: 20 page

Tidal dissipation in rotating giant planets

[Abridged] Tides may play an important role in determining the observed distributions of mass, orbital period, and eccentricity of the extrasolar planets. In addition, tidal interactions between giant planets in the solar system and their moons are thought to be responsible for the orbital migration of the satellites, leading to their capture into resonant configurations. We treat the underlying fluid dynamical problem with the aim of determining the efficiency of tidal dissipation in gaseous giant planets. In cases of interest, the tidal forcing frequencies are comparable to the spin frequency of the planet but small compared to its dynamical frequency. We therefore study the linearized response of a slowly and possibly differentially rotating planet to low-frequency tidal forcing. Convective regions of the planet support inertial waves, while any radiative regions support generalized Hough waves. We present illustrative numerical calculations of the tidal dissipation rate and argue that inertial waves provide a natural avenue for efficient tidal dissipation in most cases of interest. The resulting value of Q depends in a highly erratic way on the forcing frequency, but we provide evidence that the relevant frequency-averaged dissipation rate may be asymptotically independent of the viscosity in the limit of small Ekman number. In short-period extrasolar planets, if the stellar irradiation of the planet leads to the formation of a radiative outer layer that supports generalized Hough modes, the tidal dissipation rate can be enhanced through the excitation and damping of these waves. These dissipative mechanisms offer a promising explanation of the historical evolution and current state of the Galilean satellites as well as the observed circularization of the orbits of short-period extrasolar planets.Comment: 74 pages, 12 figures, submitted to The Astrophysical Journa

Антибіотикопрофілактика в хірургії

Наук. кер.: М.Г. КононенкоГнійно-запальні післяопераційні ускладнення за останні десятиріччя набувають все більшої актуальності. Це вже стає проблемою. Такі ускладнення необхідно попереджувати. Для забезпечення тканин операційного поля антибіотиком у ефективній (бактерицидній) концентрації на весь період хірургічного втручання проводять антибіотикопрофілактику (АБП). Вона є складовою частиною комплексної профілактики гнійно-запальних ускладнень. При цитуванні документа, використовуйте посилання http://essuir.sumdu.edu.ua/handle/123456789/2734