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

extension of noncommutative soliton hierarchies a linear system which generates a moyal deformed two dimensional soliton equation as an integrability condition can be extended to a three dimensional linear system treating the deformation parameter as an additional coordinate the supplementary integrability conditions result in a first order differential equation with respect to the deformation parameter the flow of which commutes with the flow of the deformed soliton equation in this way a deformed soliton hierarchy can be extended to a bigger hierarchy by including the corresponding deformation equations we prove the extended hierarchy properties for the deformed akns hierarchy and specialize to the cases of deformed nls kdv and mkdv hierarchies corresponding results are also obtained for the deformed kp hierarchy a deformation equation determines a kind of seiberg witten map from classical solutions to solutions of the respective noncommutative deformed equation

linearizable mappings and the low growth criterion we examine a family of discrete second order systems which are integrable through reduction to a linear system these systems were previously identified using the singularity confinement criterion here we analyse them using the more stringent criterion of nonexponential growth of the degrees of the iterates we show that the linearizable mappings are characterized by a very special degree growth the ones linearizable by reduction to projective systems exhibit zero growth i e they behave like linear systems while the remaining ones derivatives of riccati and gambier mapping lead to linear growth this feature may well serve as a detector of integrability through linearization

alternate discrete painleve equations from limits of q pvi and d pv abstract we study the special limits of discrete painleve equations belonging to the q p vi and d p v families when the independent variable goes to 1 or 0 respectively we obtain discrete systems which are shown to be either contiguities of solutions of continuous painleve equations usually of p vi but also of the other painleve equations except p i which has no parameters and so no possibility of contiguity or linearisable mappings

negativity compensation in the nonnegative inverse eigenvalue problem abstract if a set of complex numbers can be partitioned as 1 s in such a way that each i is realized as the spectrum of a nonnegative matrix say a i then is trivially realized as the spectrum of the nonnegative matrix a a i in linear algebra appl 369 2003 169 it was shown that in some cases a real set can be realized even if some of the i are not realizable themselves here we systematize and extend these results in particular allowing the sets to be complex the leading idea is that one can associate to any nonrealizable set a certain negativity n and to any realizable set a certain positivity m then under appropriate conditions if m n we can conclude that is the spectrum of a nonnegative matrix additionally we prove a complex generalization of suleimanova s theorem

integrable systems and reductions of the self dual yang mills equations many integrable equations are known to be reductions of the self dual yang mills equations this article discusses some of the well known reductions including the standard soliton equations the classical painleve equations and integrable generalizations of the darboux halphen system and chazy equations the chazy equation first derived in 1909 is shown to correspond to the equations studied independently by ramanujan in 1916

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

conserved quantities and symmetries of kp hierarchy conserved quantities and symmetries of the kp equation from the point of view of the sato theory that provides a unifying approach to soliton equations is studied conserved quantities are derived from the generalized lax equations some reductions of the kp hierarchy such as kdv boussinesq a coupled kdv and sawada kotera equation are also considered by expansion of the squared eigenfunctions of the lax equations in terms of the function symmetries of the kp equations are obtained the relationship of this procedure to the two dimensional recursion operator newly found by fokas and santini is discussed

explorations of the extended nckp hierarchy a recently obtained extension xnckp of the moyal deformed kp hierarchy nckp hierarchy by a set of evolution equations in the moyal deformation parameters is further explored formulae are derived to compute these equations efficiently reductions of the xnckp hierarchy are treated in particular to the extended nckdv and ncboussinesq hierarchies furthermore a good part of the sato formalism for the kp hierarchy is carried over to the generalized framework in particular the well known bilinear identity theorem for the kp hierarchy expressed in terms of the formal baker akhiezer function extends to the xnckp hierarchy moreover it is demonstrated that n soliton solutions of the nckp equation are also solutions of the first few deformation equations this is shown to be related to the existence of certain families of algebraic identities

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

an extended noncommutative kp hierarchy introducing squared eigenfunctions in the moyal deformed lax equations generates an extended noncommutative kp nckp hierarchy the compatibility equations between the nckp flows and the extended nckp flows and the compatibility equations among the extended nckp flows themselves are constructed the n reductions and the k constraints are analyzed and thus the nckp hierarchy with self consistent sources the constrained nckp hierarchy and the constrained nckp hierarchy with self consistent sources are presented various illustrative examples are worked out together with their lax representations involving pseudodifferential lax operators

lax pair and darboux transformation of a noncommutative u n principal chiral model we present a noncommutative generalization of the lax formalism of the u n principal chiral model in terms of a one parameter family of flat connections the lax formalism is further used to derive a set of parametric noncommutative backlund transformations and an infinite set of conserved quantities from the lax pair we derive a noncommutative version of the darboux transformation of the model

non local continuity equations and binary darboux transformation of noncommutative anti self dual yang mills equations we present an infinite number of non local continuity equations of noncommutative anti self dual yang mills nc a sdym equations using the induction method of brezin et al 1979 phys lett b 82 442 and relate it to the lax pair and the parametric backlund transformation of the system from the lax pair we derive a binary darboux transformation to generate solutions of the nc a sdym equations

strongly asymmetric discrete painleve equations the additive case we examine a class of discrete painleve equations which present a strong asymmetry these equations can be written as a system of two equations the right hand sides of which do not have the same functional form we limit here our investigation to two canonical families of the quispel roberts thompson qrt classification both of which lead to difference equations several new integrable discrete systems are identified

on the structure of 2 1 dimensional commutative and noncommutative integrable equations we develop the symbolic representation method to derive the hierarchies of 2 1 dimensional integrable equations from the scalar lax operators and to study their properties globally the method applies to both commutative and noncommutative cases in the sense that the dependent variable takes its values in c or a noncommutative associative algebra we prove that these hierarchies are indeed quasi local in the commutative case as conjectured by mikhailov and yamilov j phys a 31 6707 1998 we propose a ring extension in the noncommutative case based on the symbolic representation as examples we give noncommutative versions of kadomtsev petviashvili kp modified kadomtsev petviashvili mkp and boussinesq equations

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

distribution of the first particle in discrete orthogonal polynomial ensembles we show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure if the difference or q log derivative of the weight function is rational in a number of classical special cases the recurrence procedure is equivalent to the difference and q painleve equations of 10 17 our approach is based on the formalism of discrete integrable operators and discrete riemann hilbert problems developed in 3 4

special solutions of discrete painleve equations through direct linearisation abstract we study the special solutions of discrete painleve equations which can be expressed through the discrete equivalent of quadratures the discrete painleve equations examined are the ones which appear in the classification based on affine weyl groups and which can be written as a system of two first order mappings the solutions obtained are the discrete equivalents of solutions which have been shown to exist for the continuous painleve equations

the nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial ebl digraphs the nonnegative inverse eigenvalue problem niep is given a family of complex numbers 1 n find necessary and sufficient conditions for the existence of a nonnegative matrix a of order n with spectrum loewy and london r loewy d london a note on the inverse eigenvalue problems for nonnegative matrices linear and multilinear algebra 6 1978 83 90 resolved it for n 3 and for n 4 when the spectrum is real in our way of handling the niep we focus our attention on the coefficients of the characteristic polynomial of a thus the niep that we consider is given k1 k2 kn real numbers find necessary and sufficient conditions for the existence of a nonnegative matrix a of order n with characteristic polynomial xn k1xn 1 k2xn 2 kn the coefficients of the characteristic polynomial are closely related to the cyclic structure of the weighted digraph with adjacency matrix a we introduce a special type of digraph structure that we shall call ebl in which this relation is specially simple we give some results that show the interest of ebl structures we completely solve the niep from the coefficients of the characteristic polynomial for n 4 we also solve a special case of the niep for n 2p 1 with k1 kp 1 0 and p 2

restoring discrete painleve equations from an e8 1 associated one we present a systematic method for the construction of discrete painleve equations the method dubbed restoration allows one to obtain all discrete painleve equations that share a common autonomous limit up to homographic transformations starting from any one of those limits as the restoration process crucially depends on the classification of canonical forms for the mappings in the quispel roberts thompson qrt family it can in principle only be applied to mappings that belong to that family however as we show in this paper it is still possible to obtain the results of the restoration even when the initial mapping is not of the qrt type at least for the system at hand but we believe our approach to be of much wider applicability for the equations derived in this paper we also show how starting from a form where the independent variable advances one step at a time one can obtain versions corresponding to multistep evolutions we present a systematic method for the construction of discrete painleve equations the method dubbed restoration allows one to obtain all discrete painleve equations that share a common autonomous limit up to homographic transformations starting from any one of those limits as the restoration process crucially depends on the classification of canonical forms for the mappings in the quispel roberts thompson qrt family it can in principle only be applied to mappings that belong to that family however as we show in this paper it is still possible to obtain the results of the restoration even when the initial mapping is not of the qrt type at least for the system at hand but we believe our approach to be of much wider applicability for the equations derived in this paper we also show how starting from a form where the independent variable advances one step at a time one can obtain versions corresponding to multistep evolutions

noncommutative burgers equation we present a noncommutative version of the burgers equation which possesses the lax representation and discuss the integrability in detail we find a noncommutative version of the cole hopf transformation and succeed in the linearization of it the linearized equation is the noncommutative diffusion equation and exactly solved we also discuss the properties of some exact solutions the result shows that the noncommutative burgers equation is completely integrable even though it contains infinite number of time derivatives furthermore we derive the noncommutative burgers equation from the noncommutative anti self dual yang mills equation by reduction which is an evidence for the noncommutative ward conjecture finally we present a noncommutative version of the burgers hierarchy by both the lax pair generating technique and sato s approach

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

about the self dual chern simons system and toda field theories on the noncommutative plane the relation of the noncommutative self dual chern simons ncsdcs system to the noncommutative generalizations of toda and of affine toda field theories is investigated more deeply this paper continues the programme initiated in cabrera carnero i 2005 j high energy phys jhep10 2005 071 where it was presented how it is possible to define toda field theories through second order differential equation systems starting from the ncsdcs system here we show that using the connection of the ncsdcs to the noncommutative chiral model exact solutions of the toda field theories can also be constructed by means of the noncommutative extension of the uniton method proposed in lee k m 2004 j high energy phys jhep08 2004 054 by lee particularly some specific solutions of the nc liouville model are explicitly constructed

from continuous painleve iv to the asymmetric discrete painleve i we examine the discrete equation that is obtained from the schlesinger transformations of the continuous painleve iv equation the use of the schlesinger transformations naturally establishes the lax pair of the discrete equation which is an asymmetric form of d p and has as continuous limit we analyse the transformations of the discrete equation and show that they lead to a self dual description where the discrete equation and the schlesinger s play the same role finally using the relation between and asymmetric d we examine in detail the special solutions of both equations

fiber dependent deautonomization of integrable 2d mappings and discrete painleve equations it is well known that two dimensional mappings preserving a rational elliptic fibration like the quispel roberts thompson mappings can be deautonomized to discrete painleve equations however the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated in this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber starting from a single autonomous mapping but varying the type of a chosen fiber we obtain different types of discrete painleve equations using this deautonomization procedure we also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the picard group and some additional geometric data this technique allows us to obtain factorized expressions of discrete painleve equations including the elliptic case further by imposing certain restrictions on such non autonomous mappings we obtain new and simple elliptic difference painleve equations including examples whose symmetry groups do not appear explicitly in sakai s classification

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

existence and construction of nonnegative matrices with complex spectrum abstract the following inverse spectrum problem for nonnegative matrices is considered given a set of complex numbers 1 2 n find necessary and sufficient conditions for the existence of an n n nonnegative matrix a with spectrum our work is motivated by a relevant theoretical result of guo wuwen linear algebra appl 266 1997 261 theorem 2 1 there exists a real parameter 0 max 2 j n j such that the problem has a solution if and only if 1 0 in particular we discuss how to compute 0 and the solution matrix a for certain class of matrices a sufficient condition for the problem to have a solution is also derived

noncommutative instantons revisited we find a new gauge in which u 1 noncommutative instantons are explicitly non singular on noncommutative r 4 we also present a pedagogical introduction into noncommutative gauge theories

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

on noncommutative merons and instantons the yang mills ym and self dual yang mills sdym equations on the noncommutative euclidean four dimensional space are considered we introduce an ansatz for a gauge potential reducing the noncommutative sdym equations to a difference form of the nahm equations by constructing solutions to the difference nahm equations we obtain solutions of the noncommutative sdym equations they are noncommutative generalizations of the known solutions to the sdym equations such as the minkowski solution the one instanton solution and others using the noncommutative deformation of the corrigan fairlie t hooft wilzek ansatz we reduce the noncommutative ym equations to equations on a scalar field which have meron solutions in the commutative limit and show that they have no such solutions in the noncommutative case to overcome this difficulty another ansatz reducing the noncommutative ym equations to a system of difference equations on matrix valued functions is used for self dual configurations this system is reduced to the difference nahm equations

the nonnegative inverse eigenvalue problem abstract let 1 n be a list of complex numbers the nonnegative inverse eigenvalue problem niep is the problem of determining necessary and sufficient conditions in order that be the spectrum of an entrywise nonnegative n n matrix our purpose is to give an overview of the niep its history modern results and subproblems with particular emphasis on symmetric realizing matrices a substantial bibliography is included

a new statement about the theorem determining the region of eigenvalues of stochastic matrices abstract let mn denote the set of points in the complex plane that are eigenvalues of n dimensional stochastic matrices the set mn is completely determined by the karpelevich theorem the statement of which however is lengthy and intricate the paper shortens the presentation of the theorem

continual lie algebras and noncommutative counterparts of exactly solvable models noncommutative counterparts of exactly solvable models are introduced on the basis of a generalization of saveliev vershik continual lie algebras examples of noncommutative liouville and sin h gordon equations are given the simplest soliton solution to the noncommutative sine gordon equation is found

on a transcendental equation related to painleve iii and its discrete forms we examine the canonical forms of painlev equations and argue that the equation for piii in which one parameter is taken to be equal to zero should be considered as a canonical form different from the standard piii our argument is based on the fact that the value of this parameter cannot be modified through auto b cklund transformations we investigate the possible discrete forms of this equation and produce two of them one is of a difference type where the independent variable enters linearly while the second one is of q type where the independent variable enters in a multiplicative way the properties of these discrete equations are also studied

an algebraic scheme associated with the noncommutative kp hierarchy and some of its extensions a well known ansatz trace method for soliton solutions turns the equations of the non commutative kp hierarchy and those of certain extensions into families of algebraic sum identities we develop an algebraic formalism in particular involving a mixable shuffle product to explore their structure more precisely we show that the equations of the non commutative kp hierarchy and its extension xnckp in the case of a moyal deformed product as derived in previous work correspond to identities in this algebra furthermore the moyal product is replaced by a more general associative product this leads to a new even more general extension of the non commutative kp hierarchy relations with rota baxter algebras are established

conserved quantities in the noncommutative principal chiral model with wess zumino term we construct a noncommutative extension of the u n principal chiral model with wess zumino term and obtain an infinite set of local and non local conserved quantities for the model using the iterative procedure of brezin et al 1979 phys lett b 82 442 we also present the equivalent description as a lax formalism of the model we expand the fields perturbatively and derive zeroth and first order equations of motion zero curvature condition iteration method lax formalism local and non local conserved quantities

from continuous to discrete painleve equations abstract discrete analogues of the painleve equations have recently been derived using either a method based on orthogonal polynomials or the singularity confinement method here we analyze additional methods for deriving integrable non autonomous discrete equations these methods are based on the discrete akns approach on the use of backlund and schlesinger transforms of the continuous painleve equations and on discrete analogues of miura transformations

commuting flows and conservation laws for noncommutative lax hierarchies we discuss commuting flows and conservation laws for lax hierarchies on noncommutative spaces in the framework of the sato theory on commutative spaces the sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the lax hierarchies in terms of pseudodifferential operators noncommutative extension of the sato theory has been already studied by the author and toda and the existence of various noncommutative lax hierarchies are guaranteed in this paper we present conservation laws for the noncommutative lax hierarchies with both space space and space time noncommutativities and prove the existence of infinite number of conserved densities we also give the explicit representations of them in terms of lax operators our results include noncommutative versions of kp kdv boussinesq coupled kdv sawada kotera modified kdv equation and so on

bicomplexes and backlund transformations a bicomplex is a simple mathematical structure in particular associated with completely integrable models the conditions defining a bicomplex are a special form of a parameter dependent zero curvature condition we generalize the concept of a darboux matrix to bicomplexes and use it to derive backlund transformations for several models the method also works for moyal deformed equations with a corresponding deformed bicomplex

reduced systems of 2 2 pseudo euclidean noncommutative self dual yang mills theories self dual yang mills equations on noncommutative spaces associated with pseudo euclidean space of signature 2 2 are shown to be related via dimensional reductions to noncommutative formulations of toda equations of generalized nonlinear schrodinger ns equations of the super korteweg de vries super kdv as well as of the matrix kdv equations the noncommutative extensions of their linear systems and bicomplexes associated with conserved quantities are discussed as well a q plane version of the kdv equation with linear system is also shown

bicomplex formulation and moyal deformation of 2 1 dimensional fordy kulish systems using bicomplex formalism we construct generalizations of fordy kulish systems of matrix nonlinear schrodinger equations on two dimensional space time in two respects firstly we obtain corresponding equations in three space time dimensions secondly a moyal deformation is applied to the space time coordinates and the ordinary product of functions replaced by the moyal product in a suitable way both generalizations preserve the existence of an infinite set of conservation laws

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

the niep for four dimensional leslie and doubly stochastic matrices with zero trace from the coefficients of the characteristic polynomial abstract the constraints on the coefficients of the characteristic polynomials of four dimensional doubly stochastic matrices with zero trace and leslie stochastic matrices with zero trace are determined then on the basis of these constraints the regions of the complex plane consisting of those points which can serve as characteristic roots of four dimensional doubly stochastic matrices with zero trace and leslie stochastic matrices with zero trace are characterized

factorization methods for noncommutative kp and toda hierarchy we show that the solution space of the noncommutative kp hierarchy is the same as that of the commutative kp hierarchy owing to the birkhoff decomposition of groups over the noncommutative algebra the noncommutative toda hierarchy is introduced we derive the bilinear identities for the baker akhiezer functions and calculate the n soliton solutions of the noncommutative toda hierarchy

an inequality for the spectra of nonnegative matrices abstract let a be a n n entrywise nonnegative matrix and let s k trace a k k 1 2 3 it is shown that if n 1 then n 2 s 3 3 ns 1 s 2 2 s 1 3 n 2 n 1 ns 2 s 1 2 3 2 is nonnegative the result is used to show that if 1 2 2 4 n is the spectrum of a nonnegative matrix where 2 is nonreal and 1 max j j 1 n then 1 t 2 t 2 t 4 n need not be realizable for all t 0 even when re 2 0

a geometrical representation of the spectra of four dimensional nonnegative matrices abstract a geometrical representation of the set of four complex numbers which are the spectrum of 4 dimensional entrywise nonnegative real matrices is provided the characterization is based on the result for the nonnegative inverse eigenvalue problem niep from the coefficients of the characteristic polynomial given in 16

from the continuous pv to discrete painleve equations we study the discrete transformations that are associated with the auto backlund of the continuous pv equation we show that several two parameter discrete painleve equations can be obtained as contiguity relations of pv among them we find the asymmetric d pii equation which is a well known form of discrete piii the relation between the ternary pi previously obtained through the discrete dressing approach and pv is also established a new discrete painleve equation is also derived

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

an inverse eigenvalue problem for symmetric and normal matrices abstract some new sufficient conditions are found for n real numbers to be the spectrum of some n n symmetric nonnegative matrix and for n complex numbers to be the spectrum of some n n normal nonnegative matrix