discrete painleve equations coalescences limits and degeneracies abstract starting from the standard form of the five discrete painleve equations we show how one can obtain through appropriate limits a host of new equations which are also the discrete analogues of the continuous painleve equations a particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n dependence and introduces more new discrete painleve equations

special functions arising from discrete painleve equations a survey this article is a survey on recent studies on special solutions of the discrete painleve equations especially on hypergeometric solutions of the q painleve equations the main part of this survey is based on the joint work k kajiwara t masuda m noumi y ohta y yamada hypergeometric solutions to the q painleve equations imrn 2004 47 2004 2497 2521 k kajiwara t masuda m noumi y ohta y yamada construction of hypergeometric solutions to the q painleve equations imrn 2005 24 2005 1439 1463 with kajiwara masuda ohta and yamada after recalling some basic facts concerning painleve equations for comparison we give an overview of the present status of studies on difference discrete painleve equations as a source of special functions

pade approximants and noise rational functions abstract we continue a study of pade approximants pa for a series perturbed by random noise this time we consider more general rational functions we begin with the simple case of a sum of two geometric series and then show how these considerations can be extended to a general rational function we do not study the most general case but rather concentrate on demonstrating how our results for geometric series extend to new situations encountered when a general rational function is considered we show that froissart doublets are a universal feature and we construct an analog of the froissart polynomial introduced in the earlier paper

asymptotic zero distribution of orthogonal polynomials in sinusoidal frequency estimation it is shown that the zeros of polynomials orthogonal with respect to certain positive measures on the unit circle have an asymptotic distribution which is uniform on a circle of radius less than or equal to one these results explain some phenomena observed when various linear prediction methods are used to estimate sinusoidal frequencies they also describe the asymptotic zero distribution of the prediction error filter polynomials for a class of time series including autoregressive moving average arma models

surfaces immersed in lie algebras obtained from the sigma models we study some geometrical aspects of two dimensional orientable surfaces arising from the study of sigma models to this aim we employ an identification of with the lie algebra by means of which we construct a generalized weierstrass formula for immersion of such surfaces the structural elements of the surface like its moving frame the gauss weingarten and the gauss codazzi ricci equations are expressed in terms of the solution of the model defining it further the first and second fundamental forms the gaussian curvature the mean curvature vector the willmore functional and the topological charge of surfaces are expressed in terms of this solution we present detailed implementation of these results for surfaces immersed in and lie algebras

the weierstrass enneper system for constant mean curvature surfaces and the completely integrable sigma model the integrability of a system which describes constant mean curvature surfaces by means of the adapted weierstrass enneper inducing formula is studied this is carried out by using a specific transformation which reduces the initial system to the completely integrable two dimensional euclidean nonlinear sigma model through the use of the apparatus of differential forms and cartan theory of systems in involution it is demonstrated that the general analytic solutions of both systems possess the same degree of freedom furthermore a new linear spectral problem equivalent to the initial weierstrass enneper system is derived via the method of differential constraints a new procedure for constructing solutions to this system is proposed and illustrated by several elementary examples including a multi soliton solution

multiplicative equations related to the affine weyl group e8 we derive integrable equations starting from autonomous mappings with a general form inspired by the multiplicative systems associated with the affine weyl group e8 1 five such systems are obtained three of which turn out to be linearisable and the remaining two are integrable in terms of elliptic functions in the case of the linearisable mappings we derive non autonomous forms which contain a free function of the independent variable and we present the linearisation in each case the two remaining systems are deautonomised to new discrete painleve equations we show that these equations are in fact special forms of much richer systems associated with the affine weyl groups e7 1 and e8 1 respectively

degeneration through coalescence of the q painleve vi equation starting from the q discrete form of the painleve vi equation we obtain its degenerate forms by applying the procedure of coalescence of singularities the whole cascade of degenerate forms is thus obtained leading to new forms for the discrete painleve iv and v equations the lax pairs of these discrete painleve equations are explicitly constructed thus confirming their integrability

ultradiscrete painleve ii equation and a special function solution ultradiscretization with parity variables which keeps the information of original variables sign is applied to the q painleve ii equation of type a6 q pii a special solution of the resulting ultradiscrete system which corresponds to the special function solution of q pii is constructed ultradiscrete analogues of the q airy equation and its special solutions are also discussed in the process

an affine weyl group approach to the eight parameter discrete painleve equation we present a geometrical construction of the eight parameter discrete painleve equations our starting point is the e 1 8 affine weyl group we assume that the multi dimensional function lives on the vertices of the weight lattice of this group we derive the bilinear equations related to the discrete painleve equation in the form of nonautonomous hirota miwa equations and the elementary miura transformations the compatibility condition of the various miura s that can be written leads to three types of equations difference multiplicative q and another type where the parameters and the independent variable enter through the arguments of elliptic functions we write explicitly the discrete equations in the first two cases and produce their degeneration through coalescence of parameters

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

constant mean curvature surfaces via an integrable dynamical system it is shown that the equation which describes constant mean curvature surfaces via the generalized weierstrass enneper induction has hamiltonian form its simplest finite dimensional reduction is the integrable hamiltonian system with two degrees of freedom this finite dimensional system admits action and classes of equivalence of its trajectories are in one to one correspondence with different helicoidal constant mean curvature surfaces thus the interpretation of well known delaunay and do carmo dajczer surfaces via an integrable finite dimensional hamiltonian system is established

a systematic approach to reductions of type q abs equations we present a class of reductions of mobius type for the lattice equations known as q1 q2 and q3 from the abs list the deautonomized form of one particular reduction of q3 is shown to exist on the surface which belongs to the multiplicative type of rational surfaces in sakai s classification of painleve systems using the growth of degrees of iterates all other mappings that result from the class of reductions considered here are shown to be linearizable any possible linearizations are calculated explicitly by constructing a birational transformation defined by invariant curves in the blown up space of initial values for each reduction

time reversal symmetry and random polynomials we analyse the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed deterministic phase the density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients this means that contrary to earlier belief eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation moreover a class of time reverse invariant quantum systems is shown for which spectra display fluctuations characteristic of orthogonal ensemble while eigenvectors confer to predictions of unitary ensemble

elliptic discrete painlev equations we present explicit forms of discrete painleve equations in which the independent variable and the parameter enter through the arguments of elliptic functions these equations have eight degrees of freedom and the geometry of their transformations is described by the affine weyl group e 1 8

the complex zeros of random polynomials mark kac gave an explicit formula for the expectation of the number vn a of zeros of a random polynomial n i pn z e tj j o in any measurablc subset q of the reals here in i are independent standard normal random variables in fact for each n 1 he obtained an explicit intensity function gn for which e vn l jgn x dx here we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset q of the complex plane c namely we show that e vn ki j hn x y dxdy j gn x dx where hn is an explicit intensity function we also study the asymptotics of hn showing that for large n its mass lies close to and is uniformly distributed around the unit circle

10e9 solution to the elliptic painleve equation a function formalism for sakai s elliptic painleve equation is presented this establishes the equivalence between the two formulations by sakai and by ohta ramani grammaticos we also give a simple geometric description of the elliptic painleve equation as a non autonomous deformation of the addition formula on elliptic curves by using these formulations we construct a particular solution of the elliptic painleve equation expressed in terms of the elliptic hypergeometric function 10e9

constant curvature surfaces of the supersymmetric ℂpn 1 sigma model constant curvature surfaces are constructed from the finite action solutions of the supersymmetric pn 1 sigma model it is shown that there is a unique holomorphic solution which leads to constant curvature surfaces the generalized veronese curve we give a general criterion to construct non holomorphic solutions of the model we extend our analysis to general supersymmetric grassmannian models

szego polynomials associated with wiener levinson filters abstract szego polynomials are studied in connection with wiener levinson filters formed from discrete signals xn xn k n 1k 0 our main interest is in the frequency analysis problem of finding the unknown frequencies j when the signal is a trigonometric polynomial x n k j i i j e i j k associated with this signal is the sequence of monic szego polynomials n n z n 0 orthogonal on the unit circle with respect to a distribution function n explicit expressions for the weight function n and associated szego function dn z are given in terms of the z transform xn z of the signal xn several theorems are given to support the following conjecture which was suggested by numerical experiments as n and n increase the 2i 1 zeros of n n z of largest modulus approach the points ei j we conclude by showing that the reciprocal polynomials n n z z n n n 1 z are pade numerators for pade approximants of fixed denominator degree to a meromorphic function related to dn z

a systematic method for constructing discrete painleve equations in the degeneration cascade of the e8 group we present a systematic and quite elementary method for constructing discrete painleve equations in the degeneration cascade for e8 1 starting from the invariant for the autonomous limit of the e8 1 equation one wishes to study the method relies on choosing simple homographies that will cast this invariant into certain judiciously chosen canonical forms these new invariants lead to mappings the deautonomisations of which allow us to build up the entire degeneration cascade of the original mapping we explain the method on three examples two symmetric mappings and an asymmetric one and we discuss the link between our results and the known geometric structure of these mappings

gravitational instantons with faster than quadratic curvature decay iii this is our third paper in a series on the gravitational instantons in this paper we classify alg and alh gravitational instantons in alg case we extend tian yau hein s construction slightly and show that it is the only alg gravitational instanton in alh case we prove a torelli type theorem

on the lattice geometry and birational group of the six point multi ratio equation the inherent self consistency properties of the six point multi ratio equation allow it to be considered on a domain associated with a t shaped coxeter dynkin diagram this extends the kadomtsev pe

on a novel representation of discrete painleve equations we introduce a new representation for the discrete painleve equations it is based on the trihomographic form previously used in conjunction with discrete painleve equations associated with the affine weyl group e8 1 we show here that all discrete painleve equations do possess a trihomographic form the latter is often more convenient than the usual representation in particular when it comes to the application of the singularity confinement integrability criterion we show how trihomographic forms can be combined in order to construct generic discrete painleve equations or limits thereof a trihomographic form for the generic elliptic painleve equation is also derived

distribution of roots of random polynomials we consider polynomials of high degree with random coefficients which appear in the context of quantum chaotic dynamics and investigate various conditions under which their roots tend to concentrate near the unit circle in the complex plane correlation functions of roots are computed analytically we also investigate a certain class of random polynomials whose roots cover in a uniform way the riemann sphere special emphasis is devoted to the influence of symmmetries

elliptic painleve equations from next nearest neighbor translations on the e_8 1 lattice the well known elliptic discrete painleve equation of sakai is constructed by a standard translation on the lattice given by nearest neighbor vectors in this paper we give a new elliptic discrete painleve equation obtained by translations along next nearest neighbor vectors this equation is a generic 8 parameter version of a 2 parameter elliptic difference equation found by reduction from adler s partial difference equation the so called q4 equation we also provide a projective reduction of the well known equation of sakai

a geometrical description of the discrete painleve vi and v equations we present a geometrical approach for the discrete painleve equations based on weyl groups the method relies on the bilinear formalism and assumes that the multidimensional function lives on the weight lattice of the appropriate affine weyl group the equations for the function a system of nonautonomous hirota miwa equations govern the evolution along the independent variable and the parameters of the equation the latter evolution induced by the schlesinger transformations in the present paper we analyse the case of the e 1 7 group using the geometrical description we derive the nonlinear discrete equations we find that in the case of the e 1 7 group these are the asymmetric q pvi and d pv that were recently proposed

singularity confinement and full deautonomisation a discrete integrability criterion abstract we present a new approach to singularity confinement which makes it an efficient and reliable discrete integrability detector our method is based on the full deautonomisation procedure which consists in analysing non autonomous extensions of a given discrete system obtained by adding terms that are initially absent but whose presence does not alter the singularity pattern a justification for this approach is given through an algebro geometric analysis we also introduce the notions of early and late confinement while the former is a confinement that may exist already for the autonomous system the latter corresponds to a singularity pattern longer than that of the autonomous case late confinement will be shown to play an important role in the singularity analysis of systems with non trivial gauge freedom for which the existence of an undetected gauge in conjunction with a sketchy analysis might lead to erroneous conclusions as to their integrability an algebro geometric analysis of the role of late confinement in this context is also offered this novel type of singularity confinement analysis will be shown to allow for the exact calculation of the algebraic entropy of a given mapping

tropical geometric interpretation of ultradiscrete singularity confinement using the interpretation of the ultradiscretization procedure as a non archimedean valuation we use results of tropical geometry to show how roots and poles manifest themselves in piece wise linear systems as points of non differentiability this will allow us to demonstrate a correspondence between singularity confinement for discrete integrable systems and ultradiscrete singularity confinement for ultradiscrete integrable systems

two variations on a3 a1 a1 1 type discrete painlev equations by considering the normalizers of reflection subgroups of types a 1 1 and a 1 3 in w d5 1 two subgroups w a3 a1 1 w a1 1 and w a1 a1 1 w a3 1 can be constructed from a a3a1a1 1 ty

riccati solutions of discrete painleve equations with weyl group symmetry of type e8 1 we present special solutions of the discrete painleve equations associated with a0 1 a0 1 and a0 1 surfaces these solutions can be expressed by solutions of linear difference equations here the a0 1 surface discrete painleve equation is the most generic difference equation as all discrete painleve equations can be obtained by its degeneration limit these special solutions exist when the parameters of the discrete painleve equation satisfy a particular constraint we consider that these special functions belong to the hypergeometric family although they seem to go beyond the known discrete and q discrete hypergeometric functions we also discuss the degeneration scheme of these solutions

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

distribution of roots of random real generalized polynomials the average density of zeros for monic generalized polynomials p n z phi z sum nolimits k 1 n c k f k z with real holomorphic f k and real gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative we obtain compact expressions for both the regular component generated by the complex roots and the singular one real roots of the average density of roots the density of the regular component goes to zero in the vicinity of the real axis like lmz we present the low and high disorder asymptotic behaviors then we particularize to the large n limit of the average density of complex roots of monic algebraic polynomials of the form p n z z n sum nolimits k 1 n c k z n k with real independent identically distributed gaussian coefficients having zero mean and dispersion delta 1 sqrt n lambda the average density tends to a simple universal function of 2nlog z and in the domain coth 2 n sin arg z where nearly all the roots are located for largen

discretising the painleve equations a la hirota mickens we present a systematic method for discretising the painleve equations inspired by the method of hirota while extending it and by that of mickens by specifying it to the case at hand we derive various discrete analogues of painleve i and ii we obtain forms that have been previously derived as well as new ones in particular equations with a geometry described by the affine weyl group e8 1 as a by product we obtain also linearisable equations some of which are new

a classification of two dimensional integrable mappings and rational elliptic surfaces we classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces while the qrt mappings can be restricted on each fiber there exist several classes of integrable mappings which exchange fibers we also show an equivalent condition when a generalized halphen surface becomes a halphen surface of index m

singularity analysis for difference painleve equations associated with the affine weyl group e 8 we propose a new expression for the generic additive discrete painleve equation associated with the affine weyl group this expression makes the singularity analysis of the mapping perfectly straightforward and allows the implementation of the singularity confinement criterion in an extremely simple way using this new approach we derive new classes of discrete painleve equations related to

hypergeometric solutions to the q painleve equation of type a 1 4 we consider the q painleve equation of type a 1 4 a version of q painleve v equation and construct a family of solutions expressible in terms of certain basic hypergeometric series we also present the determinant formula for the solutions

globally nilpotent differential operators and the square ising model we recall various multiple integrals with one parameter related to the isotropic square ising model and corresponding respectively to the n particle contributions of the magnetic susceptibility to the lattice form factors to the two point correlation functions and to their extensions the univariate analytic functions defined by these integrals are holonomic and even g functions they satisfy fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties we recall the explicit forms found in previous work of these fuchsian equations as well as their russian doll and direct sum structures these differential operators are selected fuchsian linear differential operators and their remarkable properties have a deep geometrical origin they are all globally nilpotent or sometimes even have zero p curvature we also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known focusing on the factorized parts of all these operators we find out that the global nilpotence of the factors resp p curvature nullity corresponds to a set of selected structures of algebraic geometry elliptic curves modular curves curves of genus five six and even a remarkable weight 1 modular form emerging in the three particle contribution 3 of the magnetic susceptibility of the square ising model noticeably this associated weight 1 modular form is also seen in the factors of the differential operator for another n fold integral of the ising class 3 h for the staircase polygons counting and in ap ry s study of 3 g functions naturally occur as solutions of globally nilpotent operators in the case where we do not have g functions but hamburger functions one irregular singularity at 0 or that correspond to the confluence of singularities in the scaling limit the p curvature is also found to verify new structures associated with simple deformations of the nilpotent property

discrete painleve equations from singularity patterns the asymmetric trihomographic case we derive the discrete painleve equations associated with the affine weyl group e8 1 that can be represented by an in the quispel roberts thompson sense asymmetric trihomographic system the method used in this paper is based on singularity confinement we start by obtaining all possible singularity patterns for a general asymmetric trihomographic system and discard those patterns that cannot lead to confined singularities working with the remaining ones we implement the confinement conditions and derive the corresponding discrete painleve equations which involve two variables by eliminating either of these variables we obtain a symmetric equation examining all these equations of a single variable we find that they coincide exactly with those derived in previous works of ours thereby establishing the completeness of our results

cp n 1 harmonic maps and the weierstrass problem a weierstrass type system of equations corresponding to the cpn 1 harmonic maps is presented the system constitutes a further generalization of our previous construction j math phys 44 328 2003 it consists of four first order equations for three complex functions which are shown to be equivalent to the cpn 1 harmonic maps when the harmonic maps are holomorphic or antiholomorphic one of the functions vanishes and the system reduces to the previously given generalization of the weierstrass problem we also discuss a possible interpretation of our results and show that in our new case the induced metric is proportional to the total energy density of the map and not only to its holomorphic part as was the case in the previous generalizations

general solutions of the supersymmetric ℂp2 sigma model and its generalisation to ℂpn 1 a new approach for the construction of finite action solutions of the supersymmetric pn 1 sigma model is presented we show that this approach produces more non holomorphic solutions than those obtained in previous approaches we study the p2 model in detail and present its solutions in an explicit form we also show how to generalise this construction to n 3

discrete dynamical systems associated with the configuration space of 8 points in ℙ3 ℂ a 3 dimensional analogue of sakai s theory concerning the relation between rational surfaces and discrete painleve equations is studied for a family of rational varieties obtained by blow ups at 8 points in general position in 3 we define its symmetry group using the inner product that is associated with the intersection numbers and show that the group is isomorphic to the weyl group of type e7 1 by parametrizing the configuration space by means of elliptic curves the action of the weyl group and the dynamical system associated with a translation are explicitly described as a result it is found that the action of the weyl group on 3 preserves a one parameter family of quadratic surfaces and that it can therefore be reduced to the action on 1 1

on roots of random polynomials we study the distribution of the complex roots of random polynomials of degree n with i i d coefficients using techniques related to rice s treatment of the real roots question we derive under appropriate moment and regularity conditions an exact formula for the average density of this distribution which yields appropriate limit average densities further using a different technique we prove limit distribution results for coefficients in the domain of attraction of the stable law

quantum discrete dubrovin equations the discrete equations of motion for the quantum mappings of kdv type are given in terms of the sklyanin variables which are also known as quantum separated variables both temporal discrete time evolutions and spatial along the lattice at a constant time level evolutions are considered in the classical limit the temporal equations reduce to the classical discrete dubrovin equations as given in a previous publication nijhoff f w 2000 chaos solitons fractals 11 19 28 the reconstruction of the original dynamical variables in terms of the sklyanin variables is also achieved

real roots of random polynomials universality close to accumulation points we identify the scaling region of a width o n 1 in the vicinity of the accumulation points t 1 of the real roots of a random kac like polynomial of large degree n we argue that the density of the real roots in this region tends to a universal form shared by all polynomials with independent identically distributed coefficients ci as long as the second moment e c2i is finite in particular we reveal a gradual in contrast to the previously reported abrupt and quite nontrivial suppression of the number of real roots for coefficients with a nonzero mean value n e ci scaled as n n 1 2

on analytic descriptions of two dimensional surfaces associated with the bb c p n 1 sigma model we study analytic descriptions of conformal immersions of the riemann sphere s2 into the sigma model in particular an explicit expression for two dimensional 2d surfaces obtained from the generalized weierstrass formula is given it is also demonstrated that these surfaces coincide with those obtained from the sym tafel formula these two approaches correspond to parametrizations of one and the same surface in

miura transformations for discrete painleve equations coming from the affine e8 weyl group we derive integrable equations starting from autonomous mappings with a general form inspired by the additive systems associated to the affine weyl group e8 1 by deautonomisation we obtain two hitherto unknown systems one of which turns out to be a linearisable one and we show that both these systems arise from the deautonomisation of a non qrt quispel roberts thompson mapping in order to unambiguously prove the integrability of these nonautonomous systems we introduce a series of miura transformations which allows us to prove that one of these systems is indeed a discrete painleve equation related to the affine weyl group e7 1 and to cast it in canonical form a similar sequence of miura transformations allows us to effectively linearise the second system we obtain an interesting off shoot of our calculations is that the series of miura transformations when applied at the autonomous limit allows one to transform a non qrt invariant into a qrt one

studies on the painlev equations iii second and fourth painlev equations p ii andp iv etude des transformations birationnelles canoniques du systeme de painleve associe a chacune des equations

elliptic and multiplicative discrete painleve equations from the affine weyl group e8 pursuing our study of discrete painlev equations associated with the affine weyl group we present our results on multiplicative and elliptic equations our method is based on the trihomographic representation of discrete painlev equations which was shown to be equally rich as the more customary representations the relation of equations to ones associated with affine weyl groups situated in the degeneration cascade of is also presented since it constitutes a precious aid for the derivation of the systems

solution of the equivalence problem for the third painleve equation we find necessary conditions for a second order ordinary differential equation to be equivalent to the painleve iii equation under a general point transformation their sufficiency is established by reduction to known results for the equations of the form y f x y we consider separately the generic case and the case of reducibility to an autonomous equation the results are illustrated by the primary resonance equation

new airy type solutions of the ultradiscrete painleve ii equation with parity variables the q difference painleve ii equation admits special solutions written in terms of determinants whose entries are the general solution of the q airy equation an ultradiscrete limit of the special solutions is studied using the procedure of ultradiscretization with parity variables then we obtain new airy type solutions of the ultradiscrete painleve ii equation with parity variables and the solutions have a richer structure than the known solutions

an algebraic geometric approach to integrable maps of the plane we show that the dynamics of a birational map on an elliptic curve over a field is typically conjugate to addition by a point under the associated group law when the field is taken to be the function field of rational complex functions of one variable this amounts to an algebraic geometric version of the arnold liouville integrability theorem for planar integrable maps by products of this approach are that birational maps preserving foliations are necessarily the composition of two involutions and that relationships between birational maps preserving the same foliation can be described in terms of the respective points they add on the corresponding weierstrass curves when the result is applied to finite fields it helps explain some universal features of the periodic orbit distribution function for the reductions of integrable maps

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

discrete painleve equations associated with the affine weyl group e8 we examine a family of discrete systems which define analogues of the painleve equations and which are related to the affine weyl group a domain hardly explored to date using the method of deautonomization we derive new integrable mappings the coefficients of which involve periodic functions with periods 14 24 and 30 our method is based on a novel representation of these discrete painleve equations dubbed trihomographic which we show to be generic

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

discrete painleve equations with maximal periodicities we present the discrete painleve equations the coefficients of which have the maximal periodicity allowed by the geometric description in terms of affine weyl groups for each equation of the degeneration pattern obtained by sakai and which starts at the group we give the corresponding discrete painleve equation some of these systems were already known but here we proceed in a systematic way filling in all the gaps in particular we derive the equations with maximal periodicity associated to the affine weyl group which is the richest system in the sakai classification and which has yet to be studied in detail

on the limits of discrete painleve equations associated with the affine weyl group e8 we study the discrete painleve equations that can be obtained as limits from the equations associated with the affine weyl group e8 1 we obtain equations associated with the groups e7 1 and e6 1 as well as linearisable systems in the e7 1 and e6 1 cases we obtain several new discrete painleve equations along with equations which can be related to the ones already known the same is true for linearisable systems in the case of new linearisable mappings we present their explicit linearisation

affine weyl groups discrete dynamical systems and painleve equations a new class of representations of affine weyl groups on rational functions are constructed in order to formulate discrete dynamical systems associated with affine root systems as an application some examples of difference and differential systems of painleve type are discussed

how to find the conserved quantities of nonlinear discrete equations a method of finding conserved quantities of nonlinear ordinary difference equations is briefly discussed the method is based on the hypothesis that the conserved quantities are expressed by a ratio of two polynomials of dependent variables we obtain the conserved quantities of the discrete equation of motion of an anharmonic oscillator and a nonlinear difference equation of third order

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

how many zeros of a random polynomial are real we provide an elementary geometric derivation of the kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients we show that the expected number of real zeros is simply the length of the moment curve 1 t tn projected onto the surface of the unit sphere divided by the probability density of the real zeros is proportional to how fast this curve is traced out we then relax kac s assumptions by considering a variety of random sums series and distributions and we also illustrate such ideas as integral geometry and the fubini study metric

2 parameter tau τ function for the first painleve equation topological recursion and direct monodromy problem via exact wkb analysis we show that a 2 parameter family of tau functions for the first painleve equation can be constructed by the discrete fourier transform of the topological recursion partition function for a family of elliptic curves we also perform an exact wkb theoretic computation of the stokes multipliers of associated isomonodromy system assuming certain conjectures

from trihomographic to elliptic painleve equations we use the trihomographic representation of discrete painleve equations in order to derive a new factorised form for equations associated with the affine weyl group introducing an ancillary dependent variable we are able to obtain factorised forms for the general additive multiplicative and elliptic discrete related painleve equations the advantage of these factorised forms is that they make the singularity analysis of the straightforward in par with the situation for equations associated with lower affine weyl groups

on the autonomous limit of discrete painleve equations abstract we consider the autonomous limit of the known families of discrete painleve equations we show how these autonomous systems can be written as special cases of the qrt mapping for the latter we present in detail its integration in both the symmetric and asymmetric cases

canonical surfaces associated with projectors in grassmannian sigma models we discuss the construction of higher dimensional surfaces based on the harmonic maps of s2 into cpn 1 and other grassmannians we show that there are two ways of implementing this procedure both based on the use of the relevant projectors we study various properties of such projectors and show that the gaussian curvature of these surfaces in general is not constant we look in detail at the surfaces corresponding to the veronese sequence of such maps and show that for all of them this curvature is constant but its value depends on which mapping is used in the construction of the surface

the gambier mapping revisited abstract we examine critically the gambier equation and show that it is the generic linearisable equation containing as reductions all the second order equations which are integrable through linearisation we then introduce the general discrete form of this equation the gambier mapping and present conditions for its integrability finally we obtain the reductions of the gambier mapping identify their integrable forms and compute their continuous limits

bilinear equations and q discrete painleve equations satisfied by variables and coefficients in cluster algebras we construct cluster algebras the variables and coefficients of which satisfy the discrete mkdv equation the discrete toda equation and other integrable bilinear equations several of which lead to q discrete painleve equations these cluster algebras are obtained from quivers with an infinite number of vertices or with the mutation period property we will also show that a suitable transformation of quivers corresponds to a reduction of the difference equation

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

special function solutions of the discrete painleve equations in this paper we present a review of the special solutions of discrete difference and q discrete painleve equations in terms of discrete special functions these solutions exist whenever the parameters of the painleve equation satisfy a particular constraint the discrete special functions belong to the hypergeometric family although in some cases they seem to go beyond the known discrete and q discrete hypergeometric functions the equations studied in this paper are chosen on the basis of our recent classification of discrete painleve equation with the help of affine weyl groups

conformally parametrized surfaces associated with protect bb c p n 1 sigma models two dimensional parametrized surfaces immersed in the su n algebra are investigated the focus is on surfaces parametrized by solutions of the equations for the sigma model the lie point symmetries of the model are computed for arbitrary n the weierstrass formula for immersion is determined and an explicit formula for a moving frame on a surface is constructed this allows us to determine the structural equations and geometrical properties of surfaces in the fundamental forms gaussian and mean curvatures willmore functional and topological charge of surfaces are given explicitly in terms of any holomorphic solution of the model the approach is illustrated through several examples including surfaces immersed in low dimensional su n algebras

explicit formulas for the distribution of complex zeros of a family of random sums abstract the present paper provides an explicit formula for the average intensity of the distribution of complex zeros of a family of random sums of the form s n z j 0 n j f j z where z is the complex variable x i y j j i j and j j 0 n and j j 0 n are sequences of standard normal independent random variables and f j j 0 n is a sequence of given analytic functions that are real valued on the real number line in addition the numerical computations of the intensity functions and the empirical distributions for the special cases of random weyl polynomials random taylor polynomials and random truncated fourier cosine series are included as examples

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions