gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

random matrix theory and the sixth painleve 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 realized 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 functions for painleve systems allowing for the former to be characterized as the solution of certain nonlinear equations we consider the random matrix ensembles for which the nonlinear equation is the 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 of pvi displayed by a generalization of the generating function for the distributions the structure of these expansions is related to jimbo s general expansions for the function of 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

canonical structure and symmetries of the schlesinger equations the schlesinger equations s n m describe monodromy preserving deformations of order m fuchsian systems with n 1 poles they can be considered as a family of commuting time dependent hamiltonian systems on the direct product of n copies of m m matrix algebras equipped with the standard linear poisson bracket in this paper we present a new canonical hamiltonian formulation of the general schlesinger equations s n m for all n m and we compute the action of the symmetries of the schlesinger equations in these coordinates

discrete velocity and lattice boltzmann models for binary mixtures of nonideal fluids in this paper a discrete velocity model and a lattice boltzmann model are proposed for binary mixtures of nonideal fluids based on the enskog theory the velocity space of the enskog equation for each component is first discretized by applying a gaussian quadrature resulting in a discrete velocity model that can be solved by suitable numerical schemes a lattice boltzmann model is then derived from the discrete velocity model with a slightly modified equilibrium the hydrodynamics of each model are also derived through the chapmann enskog procedure

green function painleve vi equation and eisenstein series of weight one we study the problem how many singular points of a solution t to the painleve vi equation with parameter 8 1 8 1 8 3 8 might have in c 0 1 here t0 c 0 1 is called a singular point of t if t0 0 1 t0 based on hitchin s formula we explore the connection of this problem with green function and the eisenstein series of weight one among other things we prove i there are only three solutions which have no singular points in c 0 1 ii for a special type of solutions called real solutions here any branch of a solution has at most two singular points in particular at most one pole in c 0 1 iii any riccati solution has singular points in c 0 1 iv for each n 5 and n 6 6 we calculate the number of the real j values of zeros of the eisenstein series en 1 k1 k2 of weight one where k1 k2 runs over 0 n 1 2 with gcd k1 k2 n 1 the geometry of the critical points of the green function on a flat torus e as varies in the modulim1 plays a fundamental role in our analysis of the painleve iv equation in particular the conjectures raised in 22 on the shape of the domain 5 m1 which consists of tori whose green function has extra pair of critical points are completely solved here

picard and chazy solutions to the painleve vi equation we study the solutions of a particular family of painleve vi equations with parameters beta gamma 0 delta frac 1 2 and 2 alpha 2 mu 1 2 for 2 mu in mathbb z we show that in the case of half integer mu all solutions can be written in terms of known functions and they are of two types a two parameter family of solutions found by picard and a new one parameter family of classical solutions which we call chazy solutions we give explicit formulae for them and completely determine their asymptotic behaviour near the singular points 0 1 infty and their nonlinear monodromy we study the structure of analytic continuation of the solutions to the pvi mu equation for any mu such that 2 mu in mathbb z as an application we classify all the algebraic solutions for mu half integer we show that they are in one to one correspondence with regular polygons or star polygons in the plane for mu integer we show that all algebraic solutions belong to a one parameter family of rational solutions

from klein to painleve via fourier laplace and jimbo we describe a method for constructing explicit algebraic solutions to the sixth painleve equation generalising that of dubrovin and mazzocco there are basically two steps first we explain how to construct finite braid group orbits of triples of elements of rm sl 2 rm c out of triples of generators of three dimensional complex reflection groups this involves the fourier laplace transform for certain irregular connections then we adapt a result of jimbo to produce the painleve vi solutions in particular this solves a riemann hilbert problem explicitly each step is illustrated using the complex reflection group associated to klein s simple group of order 168 this leads to a new algebraic solution with seven branches we also prove that unlike the algebraic solutions of dubrovin and mazzocco and hitchin this solution is not equivalent to any solution coming from a finite subgroup of rm sl 2 rm c the results of this paper also yield a simple proof of a recent theorem of inaba iwasaki and saito on the action of okamoto s affine rm d 4 symmetry group as well as the correct connection formulae for generic painleve vi equations

quantum painleve calogero correspondence for painleve vi this paper is a continuation of our previous paper where the painleve calogero correspondence has been extended to auxiliary linear problems associated with painleve equations we have proved for the first five equations from the painleve list that one of the linear problems can be recast in the form of the non stationary schrodinger equation whose hamiltonian is a natural quantization of the classical calogero like hamiltonian for the corresponding painleve equation in the present paper we establish the quantum painleve calogero correspondence for the most general case the painleve vi equation we also show how the desired special gauge and the needed choice of variables can be derived starting from the corresponding schlesinger system with rational spectral parameter

finite branch solutions to painleve vi around a fixed singular point every finite branch local solution to the sixth painleve equation around a fixed singular point is an algebraic branch solution in particular a global solution is an algebraic solution if and only if it is finitely many valued globally the proof of this result relies on algebraic geometry of painleve vi riemann hilbert correspondence geometry and dynamics on cubic surfaces resolutions of kleinian singularities and power geometry of algebraic differential equations in the course of the proof we are also able to classify all finite branch solutions up to backlund transformations

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

on linear systems and τ functions associated with lame s equation and painleve s equation vi abstract painleve s transcendental differential equation p vi may be expressed as the consistency condition for a pair of linear differential equations with 2 2 matrix coefficients with rational entries by a construction due to tracy and widom this linear system is associated with certain kernels which give trace class operators on hilbert space this paper expresses such operators in terms of hankel operators of linear systems which are realised in terms of the laurent coefficients of the solutions of the differential equations let p t l 2 0 l 2 t be the orthogonal projection then the fredholm determinant t det i p t defines the function which is here expressed in terms of the solution of a matrix gelfand levitan equation for suitable values of the parameters solutions of the hypergeometric equation give a linear system with similar properties for meromorphic transfer functions that have poles on an arithmetic progression the corresponding hankel operator has a simple form with respect to an exponential basis in l 2 0 so det i p t can be expressed as a series of finite determinants this applies to elliptic functions of the second kind such as satisfy lame s equation with l 1

holonomy of the ising model form factors we study the ising model two point diagonal correlation function c n n by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of wu mccoy tracy and barouch we extend this expansion weighting by powers of a variable the j particle contributions f j n n in the form factor expansion the corresponding extension of the two point diagonal correlation function c n n is shown for arbitrary to be a solution of the sigma form of the painleve vi equation introduced by jimbo and miwa in their isomonodromic approach to the ising model fuchsian linear differential equations for the form factors f j n n are obtained for j 9 and shown to have both a russian doll nesting and a decomposition of the corresponding linear differential operators as a direct sum of operators equivalent to symmetric powers of the second order linear differential operator associated with the elliptic integral e from this we show that each f j n n is unexpectedly simple being expressed polynomially in terms of the elliptic integrals e and k in contrast we exhibit some mathematical objects built from these form factors f j n n which break the direct sum of symmetric powers decomposition with its associated polynomial expressions first we show that the scaling limit of these differential operators and form factors breaks the direct sum structure but not the russian doll structure secondly we show that the previous extension of two point diagonal correlation functions c n n is for singled out values cos m n m n integers also solutions of fuchsian linear differential equations these solutions of painleve vi are not polynomial in e and k but are actually algebraic functions being associated with modular curves

real root property of the spectral polynomial of the treibich verdier potential and related problems abstract we study the spectral polynomial of the treibich verdier potential such spectral polynomial which is a generalization of the classical lame polynomial plays fundamental roles in both the finite gap theory and the ode theory of heun s equation in this paper we prove that all the roots of such spectral polynomial are real and distinct under some assumptions the proof uses the classical concept of sturm sequence and isomonodromic theories we also prove an analogous result for a polynomial associated with a generalized lame equation where we apply a new approach based on the viewpoint of the monodromy data

fuchs versus painleve we briefly recall the fuchs painleve elliptic representation of painleve 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 kinds 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 is replaced by an isomonodromic system of two third order linear partial differential operators associated with 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 painleve nonlinear odes fuchsian linear odes and elliptic curves in particular the elliptic representation of painleve vi has to be generalized to an appellian representation of gamier systems

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

the logarithmic asymptotics of the sixth painleve equation we compute the monodromy group associated with the solutions of the sixth painleve equation with a logarithmic asymptotic behavior at a critical point and we characterize the asymptotic behavior in terms of the monodromy itself

stokes matrices and monodromy of the quantum cohomology of projective spaces in this paper we compute stokes matrices and monodromy of the quantum cohomology of projective spaces this problem can be formulated in a classical framework as the problem of computation of stokes matrices and monodromy of differential equations with regular and irregular singularities we prove that the stokes matrix of the quantum cohomology coincides with the gram matrix in the theory of derived categories of coherent sheaves we also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups

a family of solutions of a higher order pvi equation near a regular singularity restriction of the n dimensional garnier system to a complex line yields a system of second order nonlinear differential equations which may be regarded as a higher order version of the sixth painleve equation near a regular singularity of the system we present a 2n parameter family of solutions expanded into convergent series these solutions are constructed by iteration and their convergence is proved by using a kind of majorant series for simplicity we describe the proof in the case n 2

effects of temporal stimulus properties on the perception of across frequency asynchrony the role of temporal stimulus parameters in the perception of across frequency synchrony and asynchrony was investigated using pairs of 500 ms tones consisting of a 250 hz tone and a tone with a higher frequency of 1 2 4 or 6 khz subjective judgments suggested veridical perception of across frequency synchrony but with greater sensitivity to changes in asynchrony for pairs in which the lower frequency tone was leading than for pairs in which it was lagging consistent with the subjective judgments thresholds for the detection of asynchrony measured in a three alternative forced choice task were lower when the signal interval contained a pair with the low frequency tone leading than a pair with a high frequency tone leading a similar asymmetry was observed for asynchrony discrimination when the standard asynchrony was relatively small 20 ms but not for larger standard asynchronies independent manipulation of onset and offset ramp durations indicated a dominant role of onsets in the perception of across frequency asynchrony a physiologically inspired model involving broadly tuned monaural coincidence detectors that receive inputs from frequency selective onset detectors was able to accurately reproduce the asymmetric distributions of synchrony judgments the model provides testable predictions for future physiological investigations of responses to broadband stimuli with across frequency delays