a relativistic hypergeometric function we survey our work on a function generalizing 2f1 this function is a joint eigenfunction of four askey wilson type hyperbolic difference operators reducing to the askey wilson polynomials for certain discrete values of the variables it is defined by a contour integral generalizing the barnes representation of 2f1 it has various symmetries including a hidden d4 symmetry in the parameters by means of the associated hilbert space transform the difference operators can be promoted to self adjoint operators provided the parameters vary over a certain polytope in the parameter space for a dense subset of parameter shifts give rise to an explicit evaluation in terms of rational functions of exponentials hyperbolic functions and plane waves

ratio coordinates for higher teichmuller spaces we define new coordinates for fock goncharov s higher teichmuller spaces for a surface with holes which are the moduli spaces of representations of the fundamental group into a reductive lie group g some additional data on the boundary leads to two closely related moduli spaces the mathscr x space and the mathscr a space forming a cluster ensemble fock and goncharov gave nice descriptions of the coordinates of these spaces in the cases of g mathrm pgl m and g mathrm sl m together with poisson structures we consider new coordinates for higher teichmuller spaces given as ratios of the coordinates of the mathscr a space for g mathrm sl m which are generalizations of kashaev s ratio coordinates in the case m 2 using kashaev s quantization for m 2 we suggest a quantization of the system of these new ratio coordinates which may lead to a new family of projective representations of mapping class groups these ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle and the key point of the quantization is to guarantee certain consistency under a change of such choices we prove this consistency for m 3 and for completeness we also give a full proof of the presentation of kashaev s groupoid of decorated ideal triangulations

relativistic lame functions the special case g 2 we study a class of eigenfunctions of an analytic difference operator generalizing the special lam operator paying particular attention to quantum mechanical aspects we show that in a suitable scaling limit the pertinent eigenfunctions lead to the eigenfunctions of the operator in a finite volume we establish various orthogonality and non orthogonality results by direct calculations generalize the one gap picture associated with the above lam operator and obtain duality properties for the hyperbolic trigonometric and rational specializations

invariance principle for the random conductance model with unbounded conductances we study a continuous time random walk x in an environment of i i d random conductances e 1 we obtain heat kernel bounds and prove a quenched invariance principle for x this holds even when e e

elliptic hypergeometry of supersymmetric dualities ii orthogonal groups knots and vortices we consider seiberg electric magnetic dualities for 4d mathcal n 1 sym theories with so n gauge group for all such known theories we construct superconformal indices scis in terms of elliptic hypergeometric integrals equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals in particular we describe a number of new elliptic beta integrals associated with the s confining theories with the spinor matter fields reductions of some dualities from sp 2n to so 2n or so 2n 1 gauge groups are described interrelation of scis and the witten anomaly is briefly discussed possible applications of the elliptic hypergeometric integrals to a two parameter deformation of 2d conformal field theory and related matrix models are indicated connections of the reduced scis with the state integrals of knot theory generalized agt duality for 3 3 d theories and a 2d vortex partition function are described

universal constructions in teichmuller theory we study a new model tess of a universal teichmuller space which is defined to be the collection tess of all ideal tesselations of the poincare disk together with a distinguished oriented edge modulo the natural action of the modbius group mob applying arguments from the decorated teichmuller theory of punctured surfaces which was developed elsewhere we find that tess is an infinite dimensional frehet manifold with canonical global coordinates and there are a universal sympletic form on tess which generalizes the classical weil petersson kahler two forms and a group g which generalizes the classical mapping class groups acting on tess and preserving in fact tess is isomorphic to the right coset space of mob in the group homeo of all orientation preserving homeomorphisms of the circle and one sees easily that tess naturally generalizes the other well known models of a universal teichmuller space namely bers universal teichmuller space and the right coset space of mob in the subgroup of c smooth elements of homeo which is of interest in physics the latter model supports the kirillov kostant two form and we show that the universal symplectic form pulls back to this two form implicit in our discussion so far is the fact that homeo is isomorphic to tess and one is led to ask for the characterization of various families of homeomorphisms in terms of the corresponding tesselation reporting on joint work with dennis sullivan we partially solve one such problem here using the ahlfors bers theory furthermore as in the classical decorated teichmuller theory there is an associated universal decorated teichmuller space tess which is actually of central concern in this work tess is a frechet manifold on which g acts and there is a g invariant two form just above for tess there is moreover a g invariant cell decomposition of tess which generalizes the cell decompositions of the classical decorated teichmullerspaces thus truly all of the structure of the decorated teichmuller spaces of punctured surfaces which is known from previous work arises as the restriction of a corresponding universal structure on tess

elliptic hypergeometric series on root systems abstract we derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems a n c n and d n in the special cases of classical and q series our approach leads to new elementary proofs of the corresponding identities

relativistic lame functions revisited a basis for the eigenfunctions of the relativistic elliptic two particle calogero moser hamiltonian is known for a dense set in the natural parameter space we study the question whether an interpolating basis exists employing an asymptotic power series ansatz for the hyperbolic specialization we determine all of the coefficients explicitly which gives rise to formal interpolating eigenfunctions for the elliptic case we also need a power series ansatz for the eigenvalues we obtain the first few coefficients explicitly thus obtaining evidence for the existence of interpolating formal eigenfunctions and eigenvalues

zygmund vector fields hilbert transform and fourier coefficients in shear coordinates we parametrize the space cal z of zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the farey tesselation then we express the hilbert transform and the fourier coefficients of the zygmund vector fields in terms of the above parametrization by infinitesimal shear functions finally we compute the weil petersson metric on the teichm uller space of a punctured surface in terms of shears

the decorated teichmuller space of punctured surfaces a principal 5 bundle over the usual teichmuller space of ans times punctured surface is introduced the bundle is mapping class group equivariant and admits an invariant foliation several coordinatizations of the total space of the bundle are developed there is furthermore a natural cell decomposition of the bundle finally we compute the coordinate action of the mapping class group on the total space the total space is found to have a rich equivariant geometric structure we sketch some connections with arithmetic groups diophantine approximations and certain problems in plane euclidean geometry furthermore these investigations lead to an explicit scheme of integration over the moduli spaces and to the construction of a universal teichmuller space which we hope will provide a formalism for understanding some connections between the teichmuller theory the kp hierarchy and the virasoro algebra these latter applications are pursued elsewhere

properties of generalized univariate hypergeometric functions based on spiridonov s analysis of elliptic generalizations of the gauss hypergeometric function we develop a common framework for 7 parameter families of generalized elliptic hyperbolic and trigonometric univariate hypergeometric functions in each case we derive the symmetries of the generalized hypergeometric function under the weyl group of type e 7 elliptic hyperbolic and of type e 6 trigonometric using the appropriate versions of the nassrallah rahman beta integral and we derive contiguous relations using fundamental addition formulas for theta and sine functions the top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with ruijsenaars relativistic hypergeometric function and the askey wilson function respectively we show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions we also describe an intimate connection between the hyperbolic and trigonometric theory which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions

on the symplectic geometry of deformations of a hyperbolic surface let r be a riemann surface in this manuscript we consider a geometry on the moduli space x r for r which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of r classically the space of flat metrics for a torus is the locally symmetric space 0 2 sl 2 r sl 2 z we shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative euler characteristic the teichmiiller space t r a covering of the moduli space is a complex kihler manifold a kuhler metric for t r defined in terms of the petersson product for automorphic forms was introduced by weil 1 the weil petersson metric is invariant under the covering transformations and so projects to the moduli space x r the metric provides a link between the function theory of r and the geometry of x r in the fenchel nielsen manuscript 8 a deformation based on an amalgamation construction for fuchsian groups is introduced the deformation is defined geometrically by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position the hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface choose a free homotopy class a on the surface r then for each marked surface r realize a by the closed geodesic ar the fenchel nielsen deformations for the athen define a 1 parameter group of diffeomorphisms of t r whose infinitesimal generator by definition is the fenchel nielsen vector field ta in 21 the fenchel nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t was begun the fenchel nielsen vector fields were found to be related to the geodesic length functions 1 introduced by fricke klein to provide coordinates for t r

unitary representations of uq 2 ℝ the modular double and the multiparticle q deformed toda chain the paper deals with the analytic theory of the quantum q deformed toda chains the technique used combines the methods of representation theory and the quantum inverse scattering method the key phenomenon which is under scrutiny is the role of the modular duality concept first discovered by l faddeev in the representation theory of noncompact semisimple quantum groups explicit formulae for the whittaker vectors are presented in terms of the double sine functions and the wave functions of the n particle q deformed open toda chain are given as a multiple integral of the mellin barnes type for the periodic chain the two dual baxter equations are derived

harnack inequalities on weighted graphs and some applications to the random conductance model we establish elliptic and parabolic harnack inequalities on graphs with unbounded weights as an application we prove a local limit theorem for a continuous time random walk x in an environment of ergodic random conductances taking values in 0 infty satisfying some moment conditions

a generalized hypergeometric function satisfying four analytic difference equations of askey wilson type the hypergeometric function 2 f 1 can be written in terms of a contour integral involving gamma functions we generalize this barnes representation by using a certain generalized gamma function as a building block in this way we obtain a new 2 f 1 generalization with various symmetry features we determine the analyticity properties of the r function in all of its eight arguments and show that it is a joint eigenfunction of four distinct askey wilson type difference operators two acting on v and two on the askey wilson polynomials can be obtained by a suitable discretization of v or

random walks on supercritical percolation clusters we obtain gaussian upper and lower bounds on the transition density qt x y of the continuous time simple random walk on a supercritical percolation cluster c1 in the euclidean lattice the bounds analogous to aronsen s bounds for uniformly elliptic divergence form diusions hold with constants ci depending only on p the percolation probability and d the irregular nature of the medium means that the bound for qt x only holds for t sx where the constant sx depends on the percolation congura tion

some mathematical constants the theory of multiple gamma functions was studied in about 1900 and has been revived in the study of determinants of laplacians in the middle or so of 1980s there are some classes of mathematical constants involved naturally in the gamma and multiple gamma functions here we introduce those known mathematical constants associated with the gamma and multiple gamma functions and will show how they are involved if possible

invariance principle for the random conductance model we study a continuous time random walk x in an environment of i i d random conductances mu e in 0 infty in mathbb z d we assume that mathbb p mu e 0 p c so that the bonds with strictly positive conductances percolate but make no other assumptions on the law of the e we prove a quenched invariance principle for x and obtain green s functions bounds and an elliptic harnack inequality

the lie algebra of homeomorphisms of the circle we define and study an infinite dimensional lie algebrahomeo which is shown to be naturally associated to thetopologicallie grouphomeo of all orientation preserving homeomorphisms of the circle roughly we rely on the universal decorated teichmuller theory developed before as motivation to provide frechet coordinates on the homogeneous space given byhomeo modulo the group of real fractional linear transformations whose corresponding vector fields on the circle we then extend by the usual lie algebrasl2of real traceless two by two matrices in order to definehomeo surprisingly homeo turns out to be equal to the algebra of all vector fields on the circle which are piecewisesl2 in the obvious sense it is evidently important to consider the relationship between our new frechet coordinates and the usual trigonometric functions on the circle and we undertake here both natural infinitesimal calculations we finally apply some further previous work in order to give sufficient conditions on the fourier coefficients of a certain class of homeomorphisms of the circle which arises naturally in topology and number theory

surface order large deviations for high density percolation we derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a bernoulli percolation process restricted to a big finite box with sufficiently large parameter we also establish a useful version of the isoperimetric inequality which is the main tool of our proofs

sensitivity analysis of flexible multibody systems using composite materials components in this paper a general formulation for the computation of the first order analytical sensitivities based on the direct method using automatic differentiation of flexible multibody systems is presented the direct method for sensitivity calculation is obtained by differentiating the equations that define the response of the flexible multibody systems of composite materials with respect to the design variables which are the ply orientations of the laminated in order to appraise the benefits of the approach suggested and to highlight the risks of the procedure the analytical sensitivities are compared with the numerical results obtained by using the finite difference method for the beam composite material elements the section properties and their sensitivities are found using an asymptotic procedure that involves a two dimensional 2 d finite element analysis of their cross section the equations of the sensitivities are obtained by automatic differentiation and integrated in time simultaneously with the equations of motion of the multibody systems the equations of motion and the sensitivities of the flexible multibody system are solved and the accelerations velocities and the sensitivities of accelerations and velocities are integrated in time using a multi step multi order integration algorithm through the application of the methodology to two simple flexible multibody systems the difficulties and benefits of the procedure with respect to finite difference approaches or to the direct implementation of the analytic sensitivities are discussed copyright 2008 john wiley sons ltd

hyperbolic beta integrals abstract hyperbolic beta integrals are analogues of euler s beta integral in which the role of euler s gamma function is taken over by ruijsenaars hyperbolic gamma function they may be viewed as q q bibasic analogues of the beta integral in which the two bases q and q are interrelated by modular inversion and they entail q analogues of the beta integral for q 1 the integrals under consideration are the hyperbolic analogues of the ramanujan integral the askey wilson integral and the nassrallah rahman integral we show that the hyperbolic nassrallah rahman integral is a formal limit case of spiridonov s elliptic nassrallah rahman integral

quenched invariance principle for simple random walk on percolation clusters we consider the simple random walk on the unique infinite cluster of super critical bond percolation in d with d 2 we prove that for almost every percolation configuration the path distribution of the walk converges weakly to that of non degenerate isotropic brownian motion our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square integrable martingale the size of the deformation expressed by the so called corrector is estimated by means of ergodicity arguments

direct method to construct integrals for nth order autonomous ordinary difference equations a direct method to construct integrals for an nth order autonomous ordinary difference equation oe wnnf wn wnn1 is presented as an illustration we first consider third order autonomous oe wn3

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

on hilbert fourier and wavelet transforms several new families of wavelets which are derived from earlier work by the author on teichmuller theory are introduced and studied these wavelets are associated with the modular group rather than the affine group of the real line as for classical wavelets in further contrast to the usual setting there are explicit and essentially algebraic formulas relating wavelet and fourier expansions furthermore rather than relying on equally spaced multiscale sampling methods the wavelet expansions introduced here rely instead on an explicit nonuniformly spaced multiscale sampling method that depends upon the farey sequence of classical number theory these ingredients combine to give novel methods for calculating wavelet fourier and hilbert transforms with potential applications to myriad problems in electrical engineering by employing a new characteristic length associated to 2 2 matrices finiteness and convergence properties of the wavelet expansions are derived for functions with more than 5 2 derivatives in l2 one of our wavelet families forms a frame and the hilbert transform is calculable in hyperbolic coordinates 2002 wiley periodicals inc

krull s theory for the double gamma function abstract the barnes g function g x 1 2 satisfies the functional equation log g x 1 log g x log x we complement w krull s work in bemerkungen zur differenzengleichung g x 1 g x x math nachrichten 1 1948 365 376 with additional results that yield a different characterization of the function g new expansions and sharp bounds for g on x 0 in terms of gamma and digamma functions a new expansion for the gamma function and summation formulae with polygamma functions

the dilogarithmic central extension of the ptolemy thompson group via the kashaev quantization abstract quantization of universal teichmuller space provides projective representations of the ptolemy thompson group which is isomorphic to the thompson group t this yields certain central extensions of t by z called dilogarithmic central extensions we compute a presentation of the dilogarithmic central extension t kash of t resulting from the kashaev quantization and show that it corresponds to 6 times the euler class in h 2 t z meanwhile the braided ptolemy thompson groups t t of funar kapoudjian are extensions of t by the infinite braid group b and by abelianizing the kernel b one constructs central extensions t ab t ab of t by z which are of topological nature we show t kash t ab our result is analogous to that of funar and sergiescu who computed a presentation of another dilogarithmic central extension t cf of t resulting from the chekhov fock goncharov quantization and thus showed that it corresponds to 12 times the euler class and that t cf t ab in addition we suggest a natural relationship between the two quantizations in the level of projective representations

gaussian bounds and parabolic harnack inequality on locally irregular graphs a well known theorem of delmotte is that gaussian bounds parabolic harnack inequality and the combination of volume doubling and poincare inequality are equivalent for graphs in this paper we consider graphs for which these conditions hold but only for sufficiently large balls and prove a similar equivalence

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

invariance principle for the random conductance model in a degenerate ergodic environment we study a continuous time random walk x on zd in an environment of random conductances taking values in 0 we assume that the law of the conductances is ergodic with respect to space shifts we prove a quenched invariance principle for x under some moment conditions of the environment the key result on the sublinearity of the corrector is obtained by moser s iteration scheme

new elliptic solutions of the yang baxter equation we consider finite dimensional reductions of an integral operator with the elliptic hypergeometric kernel describing the most general known solution of the yang baxter equation with a rank 1 symmetry algebra the reduced r operators reproduce at their bottom the standard baxter s r matrix for the 8 vertex model and sklyanin s l operator the general formula has a remarkably compact form and yields new elliptic solutions of the yang baxter equation based on the finite dimensional representations of the elliptic modular double the same result is also derived using the fusion formalism

disorder entropy and harmonic functions we study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on z d we prove that the vector space of harmonic functions growing at most linearly is d 1 dimensional almost surely in particular there are no non constant sublinear harmonic functions thus implying the uniqueness of the corrector the main ingredient of the proof is a quantitative annealed version of the kaimanovich vershik entropy argument this also provides bounds on the derivative of the heat kernel simplifying and generalizing existing results the argument applies to many different environments even reversibility is not necessary we also mention several open problems and conjectures on the behavior of harmonic functions on stationary random graphs

new solutions of the star triangle relation with discrete and continuous spin variables a new solution to the star triangle relation is given for an ising type model of interacting spins containing integer and real valued components boltzmann weights of the model are given in terms of the lens elliptic gamma function and are based on yamazaki s recently obtained solution of the star star relation the star triangle relation given here implies seiberg duality for the 4 d index of the su 2 quiver gauge theory and the corresponding two component spin case of the star star relation of yamazaki a proof of the star triangle relation is given resulting in a new elliptic hypergeometric summation integration identity the star triangle relation in this paper contains the master solution of bazhanov and sergeev as a special case two other limiting cases are considered one of which gives a new star triangle relation in terms of ratios of infinite q products while the other case gives a new way of deriving a star triangle relation that was previously obtained by the author

boundary quantum knizhnik zamolodchikov equations and bethe vectors solutions to boundary quantum knizhnik zamolodchikov equations are constructed as bilateral sums involving off shell bethe vectors in case the reflection matrix is diagonal and only the 2 dimensional representation of mathcal u q widehat mathfrak sl 2 is involved we also consider their rational and classical degenerations

inversions of integral operators and elliptic beta integrals on root systems we prove a novel type of inversion formula for elliptic hypergeometric integrals associated to a pair of root systems using the a c inversion formula to invert one of the known c type elliptic beta integrals we obtain a new elliptic beta integral for the root system of type a validity of this integral is established by a different method as well

dynamic analysis of spatial flexible multibody systems using joint co ordinates this paper presents a systematic method for deriving the minimum number of equations of motion for spatial flexible multibody systems relative kinematics are developed using relative joint co ordinates and relative joint velocities to formulate a minimum number of equations of motion the present method takes advantage of the simplicity of the absolute co ordinate formulation and computational efficiency of the relative joint co ordinate formulation the system equations of motion are first formulated in terms of a coupled set of absolute reference co ordinates and elastic modal co ordinates these equations are transformed to the joint co ordinate space by use of a velocity transformation matrix including the elastic modal velocities a computer algorithm is proposed and extended to closed loop mechanisms one example of a flexible vehicle is presented and the results are discussed to illustrate the computational efficiency of the method

inverse z transform by mobius inversion and the error bounds of aliasing in sampling a general algorithm based on two special mobius inversion formulae is developed to compute the inverse z transform this approach to fourier analysis uses what is called the arithmetic fourier transform aft with the new aft algorithm one can compute the inverse z transform of an infinite two sided sequence it is compared with the conventional dft approach both methods have aliasing errors due to sampling the error bounds of the aliasing effects in the dft and the new proposed method are established and compared in general the aft algorithm is not so vulnerable to the aliasing errors in the high frequency components as the dft approach

on chemical distances and shape theorems in percolation models with long range correlations in this paper we provide general conditions on a one parameter family of random infinite subsets of z d to contain a unique infinite connected component for which the chemical distances are comparable to the euclidean distance in addition we show that these conditions also imply a shape theorem for the corresponding infinite connected component by verifying these conditions for specific models we obtain novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the gaussian free field as a byproduct we obtain alternative proofs to the corresponding results for random interlacements in the work of cern and popov on the internal distance in the interlacement set electron j probab 17 29 1 25 2012 and while our main interest is in percolation models with long range correlations we also recover results in the spirit of the work of antal and pisztora on the chemical distance for supercritical bernoulli percolation ann probab 24 2 1036 1048 1996 for bernoulli percolation finally as a corollary we derive new results about the chemical diameter of the largest connected component in the complement of the trace of the random walk on the torus

on razamat s a 2 and a 3 kernel identities in recent work on superconformal quantum field theories razamat arrived at elliptic kernel identities of a new and striking character they relate solely to the root systems a 2 and a 3 and have no coupling type parameters cite ra18 the pertinent 2 and 3 variable hamiltonians are analytic difference operators and the kernel functions are built from the elliptic gamma function razamat presented compelling evidence for the validity of these identities and checked them to a certain order in a power series expansion this paper is mainly concerned with analytical proofs of these identities more specifically we furnish a complete proof of the identities for the a 2 elliptic case and for the a 3 hyperbolic case and consider several specializations we also discuss the implications the kernel identities might have for a hilbert space scenario involving common eigenvectors of the hamiltonians and the integral operators associated with the kernel functions

on the growth of logarithmic differences difference quotients and logarithmic derivatives of meromorphic functions a crucial ingredient in the recent discovery by ablowitz halburd herbst and korhonen 2000 2007 that a connection exists between discrete painleve equations and finite order nevanlinna theory is an estimate of the integrated average of log f z 1 f z on z r we obtained essentially the same estimate in our previous paper 2008 independent of halburd et al 2006 we continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences difference quotients and logarithmic derivatives for finite order meromorphic functions in addition to the potential applications of our new estimates in integrable systems they are also of independent interest in particular our findings show that there are marked differences between the growth of meromorphic functions with nevanlinna order less than and greater than one we have established a difference analogue of the classical wiman valiron type estimates for meromorphic functions with order less than one which allow us to prove that all entire solutions of linear difference equations with polynomial coefficients of order less than one must have positive rational order of growth we have also established that any entire solution to a first order algebraic difference equation with polynomial coefficients must have a positive order of growth which is a difference analogue of a classical result of polya

q special functions with q 1 and their application to discrete integrable systems solutions for certain discrete integrable systems are constructed by using integral solutions for hypergeometric q difference systems with q 1 we apply a solution for lauricella s d type hypergeometric q difference system with q 1 to construct a solution for the discrete kp hierarchy furthermore an integral solution of a q difference analogue of bessel s equation is newly introduced and applied to construct a solution for a q difference analogue of the cylindrical toda equation with q 1

random walks on infinite percolation clusters in models with long range correlations for a general class of percolation models with long range correlations on zdzd d 2d 2 introduced in j math phys 55 2014 083307 we establish regularity conditions of barlow ann probab 32 2004 3024 3084 that mesoscopic subballs of all large enough balls in the unique infinite percolation cluster have regular volume growth and satisfy a weak poincare inequality as immediate corollaries we deduce quenched heat kernel bounds parabolic harnack inequality and finiteness of the dimension of harmonic functions with at most polynomial growth heat kernel bounds and the quenched invariance principle of probab theory related fields 166 2016 619 657 allow to extend various other known results about bernoulli percolation by mimicking their proofs for instance the local central limit theorem of electron j probab 14 209 1 27 or the result of ann probab 43 2015 2332 2373 that the dimension of at most linear harmonic functions on the infinite cluster is d 1d 1 in terms of specific models all these results are new for random interlacements at every level in any dimension d 3d 3 as well as for the vacant set of random interlacements ann of math 2 171 2010 2039 2087 comm pure appl math 62 2009 831 858 and the level sets of the gaussian free field comm math phys 320 2013 571 601 in the regime of the so called local uniqueness which is believed to coincide with the whole supercritical regime for these models

an addition type formula for the elliptic digamma function abstract eisenstein derived addition formulas for the weierstrass zeta function from the addition formula for the cotangent function and the fact that the weierstrass zeta function can be represented as an infinite sum of the cotangent functions in this paper we apply this idea of eisenstein s to the addition type formula for the double cotangent function established by the author we show that the elliptic digamma function defined by the logarithmic derivative of the elliptic gamma function satisfies an addition type formula this formula includes the addition formula for the weierstrass zeta function evaluation formulas for the double eisenstein series introduced by tsumura and the double shuffle relations for the double eisenstein series proved by gangl kaneko zagier

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models