monodromies at infinity of confluent a hypergeometric functions abstract we study the monodromies at infinity of confluent a hypergeometric functions introduced by adolphson 2 in particular we extend the result of 38 for non confluent a hypergeometric functions to the confluent case the integral representation by rapid decay homology cycles proved in 9 will play a central role in the proof

critical behaviour of a fluid in a random shear flow renormalization group analysis of a simplified model critical behaviour of a fluid binary mixture or liquid crystal subjected to strongly anisotropic turbulent mixing is studied by means of the field theoretic renormalization group as a simplified model relaxational stochastic dynamics of a non conserved scalar order parameter coupled to a random velocity field with prescribed statistics is considered the velocity is taken gaussian white in time with a correlation function of the form t t k d where k is the component of the wave vector perpendicular to the distinguished direction direction of the flow the d dimensional generalization of the ensemble introduced by avellaneda and majda 1990 commun math phys 131 381 within the context of passive scalar advection it is shown that depending on the relation between the exponent and the space dimensionality d the system exhibits various types of large scale self similar behaviour associated with different infrared attractive fixed points of the renormalization group equations in addition to well known asymptotic regimes model a of equilibrium critical dynamics and a passively advected scalar with no self interaction the existence of a new non equilibrium and strongly anisotropic type of critical behaviour universality class is established and the corresponding critical dimensions are calculated to the second order of the double expansion in and e 4 d two loop approximation the most realistic values of the model parameters for example d 3 and the kolmogorov exponent 4 3 belong to this class the scaling behaviour appears anisotropic in the sense that the critical dimensions related to the directions parallel and perpendicular to the flow are essentially different the results are in qualitative agreement with the results obtained in experiments and simulations of fluid systems subjected to various kinds of regular and chaotic anisotropic flows

a mirror symmetric solution to the quantum toda lattice we give a representation theoretic proof of a conjecture from rietsch adv math 217 2401 2442 2008 providing integral formulas for solutions to the quantum toda lattice in general type this result generalizes work of givental for sl n b in a uniform way to arbitrary type and can be interpreted as a kind of mirror theorem for the full flag variety g b we also prove the existence of a totally positive and totally negative critical point of the superpotential in every mirror fiber

multivariate hypergeometric functions as τ functions of toda lattice and kadomtsev petviashvili equation we present the q deformed multivariate hypergeometric functions related to schur polynomials as tau functions of the kp and of the two dimensional toda lattice hierarchies the variables of the hypergeometric functions are the higher times of those hierarchies the discrete toda lattice variable shifts parameters of hypergeometric functions the role of additional symmetries in generating hypergeometric tau functions is explained

periods for irregular singular connections on surfaces given an integrable connection on a smooth quasi projective algebraic surface u over a subfield k of the complex numbers we define rapid decay homology groups with respect to the associated analytic connection which pair with the algebraic de rham cohomology in terms of period integrals these homology groups generalize the analogous groups in the same situation over curves defined by bloch and esnault in dimension two however new features appear in this context which we explain in detail assuming a conjecture of sabbah on the formal classification of meromorphic connections on surfaces known to be true if the rank is lower than or equal to 5 we prove perfectness of the period pairing in dimension two

1d action and partition function for the 2d ising model with a boundary magnetic field in this paper we present an alternative method to that developed by mccoy and wu to obtain some exact results for the 2d ising model with a general boundary magnetic field and for a finite size system this method is a generalization of ideas from plechko presented for the 2d ising model in zero field based on the representation of the ising model using a grassmann algebra a gaussian 1d action is obtained for a general configuration of the boundary magnetic field when the magnetic field is homogeneous we check that our results are in agreement with mccoy and wu s previous work this 1d action is used to compute in an efficient way the free energy in the special case of inhomogeneous boundary magnetic field this method is useful to obtain new exact results for interesting boundary problems such as wetting transitions

period determinant of an irregular connection over an elliptic curve abstract in this article we calculate the period of an irregular singular connection d d y on the legendre curve u y 2 x x 1 x we calculate its de rham cohomology and the cycles in the homology of the dual connection and describe the period matrix terasoma s work is introduced to approximate the direct image connection by a sequence of regular connections as an intermediate step where u a 1 spec k y x y y finally we will make the comparison of the period obtained by this approximation and that over u

fermionic construction of tau functions and random processes abstract tau functions expressed as fermionic expectation values e date m jimbo m kashiwara t miwa transformation groups for soliton equations in m jimbo t miwa eds nonlinear integrable systems classical theory and quantum theory world scientific 1983 pp 39 120 are shown to provide a natural and straightforward description of a number of random processes and statistical models involving hard core configurations of identical particles on the integer lattice like a discrete version simple exclusion processes asep nonintersecting random walkers lattice coulomb gas models and others as well as providing a powerful tool for combinatorial calculations involving paths between pairs of partitions we study the decay of the initial step function within the discrete asep d asep model as an example for constant hopping rates we obtain vershik kerov type of asymptotic configuration of particles

a riemann roch theorem for flat bundles with values in the algebraic chern simons theory our purpose in this paper is to continue the algebraic study of complex local systems on complex algebraic varieties we prove a riemann roch theorem for these objects using algebraic chern simons characteristic classes a complex local system e on a smooth projective complex variety x gives rise to a locally free analytic sheaf ean e c oan x which using gaga admits a canonical algebraic structure e the tautological analytic connection on e co x induces an integrable algebraic connection e e x combining gaga with the poincare lemma we see that the analytic cohomology of the local system can be identified with the hypercohomology of the algebraic de rham complex

integrals over grassmannians and random permutations testing the independence of two gaussian populations involves the distribution of the sample canonical correlation coefficients given that the actual correlation is zero the laplace transform as a function of x of this distribution is not only an integral over the grassmannian gr p f n of p dimensional planes in real complex or quaternion n space f n but is also related to a generalized hypergeometric function such integrals are solutions of painleve like equations in the complex case they are solutions to genuine painleve equations these integrals over gr p c n have remarkable expansions in x related to random words of length l formed with an alphabet of p letters 1 p the coefficients of these expansions are given by the probability that a word i contains a subsequence of letters p p 1 1 in that order and ii that the maximal length of the disjoint union of p 1 increasing subsequences of the word is less than or equal tok where k refers to the power of x note that if each letter appears in the word then the maximal length of the disjoint union of p increasing subsequences of the word is automatically l and is thus trivial c 2003 elsevier inc all rights reserved

confluent a hypergeometric functions and rapid decay homology cycles we study confluent a hypergeometric functions introduced by adolphson in particular we give their integral representations by using rapid decay homology cycles of hien the method of toric compactifications introduced in our previous works will be used to prove our main theorem moreover we apply it to obtain a formula for the asymptotic expansions at infinity of confluent a hypergeometric functions

periods of hodge structures and special values of the gamma function at the end of the 1970s gross and deligne conjectured that periods of geometric hodge structures with multiplication by an abelian number field are products of values of the gamma function at rational arguments with exponents determined by the hodge decomposition we prove an alternating variant of this conjecture for smooth projective varieties acted upon by an automorphism of finite order thus improving previous results of maillot and rossler the proof relies on a product formula for periods of regular singular connections due to saito and terasoma

periods for flat algebraic connections in previous work hien math ann 337 631 669 2007 we established a duality between the algebraic de rham cohomology of a flat algebraic connection on a smooth quasi projective surface over the complex numbers and the rapid decay homology of the dual connection relying on a conjecture by c sabbah in the present article we generalize this result to arbitrary dimensions unconditionally using recent results of t mochizuki

gersten s conjecture for thek theory with torsion coefficients of a discrete valuation ring between quillen k groups induced by the obvious inclusion functors are all zero in 6 quillen proved this conjecture if r is essentially of finite type over a field in the case of discrete valuation rings this conjecture reduces to conjecture let r be a discrete valuation ring with residue field k and field of fractions f then for all q z 0 the transfer map which is defined since every k vector space is of finite projective dimension as an r module k k k r is zero equivalently the map r ec f induced by the ring homomorphism r p f is injective a third equivalent formulation is that the boundary map

gauss manin determinants for rank 1 irregular connections on curves let f u to rm spec k be a smooth open curve over a field k supset k where k is an algebraically closed field of characteristic 0 let nabla l to l otimes omega 1 u k be a possibly irregular absolutely integrable connection on a line bundle l a formula is given for the determinant of de rham cohomology with its gaus manin connection big det rf l otimes omega 1 u k det nabla gm big the formula is expressed as a norm from the curve of a cocycle with values in a complex defining algebraic differential characters 7 and this cocycle is shown to exist for connections of arbitrary rank

generalized hirota bilinear identity and integrable q difference and lattice hierarchies abstract the hirota bilinear identity for the cauchy baker akhieser cba kernel is introduced as a basic tool to construct integrable hierarchies containing lattice and q difference times a determinant formula for the action of meromorphic function on the cba kernel is derived this formula gives the opportunity to construct generic solutions for integrable lattice equations

effects of mixing and stirring on the critical behaviour stochastic dynamics of a nonconserved scalar order parameter near its critical point subject to random stirring and mixing is studied using the field theoretic renormalization group the stirring and mixing are modelled by a random external gaussian noise with the correlation function t t k4 d y and the divergence free due to incompressibility velocity field governed by the stochastic navier stokes equation with a random gaussian force with the correlation function depending on the relations between the exponents y and y and the space dimensionality d the model reveals several types of scaling regimes some of them are well known model a of equilibrium critical dynamics and linear passive scalar field advected by a random turbulent flow but there are three new non equilibrium regimes universality classes associated with new nontrivial fixed points of the renormalization group equations the corresponding critical dimensions are calculated in the two loop approximation second order of the triple expansion in y y and e 4 d

symmetries of the hypergeometric function phantom _ _ 1 in this paper we show that the generalized hypergeometric function m f m 1 has a one parameter group of local symmetries which is a conjugation of a flow of a rational calogero mozer system we use the symmetry to construct fermionic fields on a complex torus which have linear algebraic properties similar to those of the local solutions of the generalized hypergeometric equation the fields admit a nontrivial action of the quaternions based on the above symmetry we use the similarity between the linear algebraic structures to introduce the quaternionic action on the direct sum of the space of solutions of the generalized hypergeometric equation and its dual as a side product we construct a good basis for the monodromy operators of the generalized hypergeometric equation inspired by the study of multiple flag varieties with finitely many orbits of the diagonal action of the general linear group by magyar weyman and zelevinsky as an example of computational effectiveness of the basis we give a proof of the existence of the monodromy invariant hermitian form on the space of solutions of the generalized hypergeometric equation in the case of real local exponents different from the proofs of beukers and heckman and of haraoka as another side product we prove an elliptic generalization of cauchy identity

toda hierarchies and their applications the 2d toda hierarchy occupies a central position in the family of integrable hierarchies of the toda type the 1d toda hierarchy and the ablowitz ladik aka relativistic toda hierarchy can be derived from the 2d toda hierarchy as reductions these integrable hierarchies have been applied to various problems of mathematics and mathematical physics since 1990s a recent example is a series of studies on models of statistical mechanics called the melting crystal model this research has revealed that the aforementioned two reductions of the 2d toda hierarchy underlie two different melting crystal models technical clues are a fermionic realization of the quantum torus algebra special algebraic relations therein called shift symmetries and a matrix factorization problem the two melting crystal models thus exhibit remarkable similarity with the hermitian and unitary matrix models for which the two reductions of the 2d toda hierarchy play the role of fundamental integrable structures

pfaffian systems of a hypergeometric equations i bases of twisted cohomology groups abstract we consider bases of pfaffian systems for a hypergeometric systems these are given by grobner deformations they also provide bases for twisted cohomology groups for a hypergeometric system associated with a class of order polytopes these bases have a combinatorial description the size of the bases associated with a subclass of the order polytopes has a growth rate of polynomial order

toda lattice hierarchy and generalized string equations string equations of thepth generalized kontsevich model and the compactifiedc 1 string theory are re examined in the language of the toda lattice hierarchy as opposed to a hypothesis postulated in the literature the generalized kontsevich model atp 1 does not coincide with thec 1 string theory at selfdual radius a broader family of solutions of the toda lattice hierarchy including these models is constructed and shown to satisfy generalized string equations the status of a variety ofc 1 string models is discussed in this new framework

ohta jasnow kawasaki approximation for nonconserved coarsening under shear we analytically study coarsening dynamics in a system with nonconserved scalar order parameter when a uniform time independent shear flow is present we use an anisotropic version of the ohta jasnow kawasaki approximation to calculate the growth exponents in two and three dimensions for d 3 the exponents we find are the same as expected on the basis of simple scaling arguments that is 3 2 in the flow direction and 1 2 in all the other directions while for d 2 we find an unusual behavior in that the domains experience an unlimited narrowing for very large times and a nontrivial dynamical scaling appears in addition we consider the case where an oscillatory shear is applied to a two dimensional system finding in this case a standard t 1 2 growth modulated by periodic oscillations we support our two dimensional results by means of numerical simulations and we propose to test our predictions by experiments on twisted nematic liquid crystals

e 1 degeneration of the irregular hodge filtration for a regular function f on a smooth complex quasi projective variety j d yu introduced in 35 a filtration the irregular hodge filtration on the de rham complex with twisted differential dc df extending a definition of deligne in the case of curves in this article we show the degeneration ate1 of the spectral sequence attached to the irregular hodge filtration by using the method of 26 we also make explicit the relation with a complex intro duced by m kontsevich and give details on his proof of the corresponding e1 degeneration by reduction to characteristicp when the pole divisor of the function is reduced with normal crossings in appendix e m saito gives a different proof of thee1 degeneration

structure des connexions meromorphes formelles de plusieurs variables et semi continuite de l irregularite we prove malgrange s conjecture on the absence of confluence phenomena for integrable meromorphic connections more precisely if y x is a complex analytic fibration by smooth curves z a hypersurface of y finite over x and an integrable meromorphic connection on y with poles along z then the function which attaches to x x the sum of the irregularities of the fiber x at the points of z x is lower semicontinuous

mirror symmetry for p 2 and tropical geometry abstract this paper explores the relationship between mirror symmetry for p 2 at the level of big quantum cohomology and tropical geometry the mirror of p 2 is typically taken to be c 2 w where w is a landau ginzburg potential of the form x y 1 x y the complex moduli space of the mirror is the universal unfolding of w and oscillatory integrals produce a frobenius manifold structure on this universal unfolding we show that w can be deformed by counting maslov index two tropical disks and the natural parameters appearing in this deformation are then the flat coordinates on the moduli space furthermore the oscillatory integrals are shown to read off directly tropical curve counts from the potential thus we show in fact that mirror symmetry for p 2 is equivalent in a strong sense to tropical curve counting formulas including tropical formulas for gravitational descendent invariants

quantum algebras and q special functions related to coherent states maps of the disc the quantum algebras generated by the coherent states maps of the disc are investigated it is shown that the analytic realization of these algebras leads to a generalized analysis which includes standard analysis as well as q analysis the applications of the analysis to star product quantizations and q special functions theory are given among others the meromorphic continuation of the generalized basic hypergeometric series is found and a reproducing measure is constructed when the series is treated as a reproducing kernel

character sums associated to finite coxeter groups we prove a character sum identity for coxeter arrangements which is a finite field analogue of macdonald s conjecture proved by opdam