the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

betti numbers of holomorphic symplectic quotients via arithmetic fourier transform a fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field this technique in turn gives information on betti numbers of holomorphic symplectic quotients as a consequence simple unified proofs are obtained for formulas of poincare polynomials of toric hyperkahler varieties recovering results of bielawski dancer and hausel sturmfels poincare polynomials of hilbert schemes of points and twisted atiyah drinfeld hitchin manin adhm spaces of instantons on 2 recovering results of nakajima yoshioka and poincare polynomials of all nakajima quiver varieties as an application a proof of a conjecture of kac on the number of absolutely indecomposable representations of a quiver is announced

counting representations of quivers over finite fields abstract by counting the numbers of isomorphism classes of representations indecomposable or absolutely indecomposable of quivers over finite fields with fixed dimension vectors we obtain a multi variable formal identity if the quiver has no edge loops this identity turns out to be a q analogue of the kac denominator identity modulus a conjecture of kac

a computational criterion for the kac conjecture abstract we give a criterion for the kac conjecture asserting that the free term of the polynomial counting the absolutely indecomposable representations of a quiver over a finite field of given dimension coincides with the corresponding root multiplicity of the associated kac moody algebra our criterion suits very well for computer tests

symmetry solutions for reaction diffusion equations with spatially dependent diffusivity nonclassical and classical symmetry techniques are employed to analyse a reaction diffusion equation with a cubic source term here the diffusivity diffusion term is assumed to be an arbitrary function of the spatial variable classification using lie point and nonclassical symmetries is performed it turns out that the diffusivity needs to be given as a quadratic function of the spatial variable for the given governing equation to admit nonclassical symmetries both nonclassical and classical symmetries are used to construct some group invariant exact solutions the results are applied to models arising in population dynamics

arithmetic harmonic analysis on character and quiver varieties ii we study connections between the topology of generic character varieties of fundamental groups of punctured riemann surfaces macdonald polynomials quiver representations hilbert schemes on c x x c x modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups c 2012 elsevier inc all rights reserved

iterations of the non classical symmetries method and conditional lie backlund symmetries iterations of the non classical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the original equation invariant solutions of these heir equations supply new solutions of the original equation we show that particular cases of such invariant solutions correspond to zhdanov s conditional lie backlund symmetries

motivic donaldson thomas invariants of the conifold and the refined topological vertex abstract we compute the motivic donaldson thomas theory of the resolved conifold in all chambers of the space of stability conditions of the corresponding quiver the answer is a product formula whose terms depend on the position of the stability vector generalizing known results for the corresponding numerical invariants our formulae imply in particular a motivic form of the dt pt correspondence for the resolved conifold the answer for the motivic pt series is in full agreement with the prediction of the refined topological vertex formalism

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

on the integrability of homogeneous scalar evolution equations abstract we determine the existence of infinitely many symmetries for equations of the formut uk f u uk 1 when they are homogeneous with respect to the scalinguk k with 0 algorithms are given to determine whether a system has a symmetry also independent oftandx if it has one generalized symmetry we prove it has infinitely many and these can be found using recursion operators or master symmetries the method of proof uses the symbolic method and results from diophantine approximation theory we list the 10 integrable hierarchies the methods can in principle be applied to the 0 case as we illustrate for one example with 0 which seems to be new in principle they can also be used for systems of evolution equations but so far this has only been demonstrated for one class of examples

exact polynomial solutions to some nonlinear diffusion equations abstract a class of non group invariant exact solutions to a number of nonlinear diffusion equations is constructed expressed in the appropriate variables these solutions are polynomials in the spatial variable the method is mainly applicable to equations which may be written in an appropriate quadratically nonlinear form a number of equivalence transformations between various nonlinear diffusion equations are also derived

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

group properties of generalized quasi linear wave equations abstract in this paper complete group classification of a class of 1 1 dimensional generalized quasi linear wave equations is performed by using the lie ovsiannikov method additional equivalent transformation and furcate split method lie reductions of some truly variable coefficient wave equations which are singled out from the classification results are investigated some classes of exact solutions of these variable coefficient wave equations are constructed by means of both the reductions and the additional equivalent transformations the nonclassical symmetries to the generalized quasi linear wave equation are also studied this enabled to obtain some exact solutions of the wave equations which are invariant under certain conditional symmetries

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

symmetries and heir equations for the laminar boundary layer model iterations of the nonclassical symmetries method applied to both the prandtl equation and prandtl system give rise to new nonlinear equations which inherit the lie symmetry algebra of the original prandtl equation system invariant solutions of these heir equations supply new solutions of the original equation system

aspects of the painleve property for partial differential equations abstract aspects of the painleve property are reviewed special attention is paid to the connection with hirota s method the formulation of a painleve type property suitable for differential difference equations and certain results concerning the self dual yang mills equations

kac s conjecture from nakajima quiver varieties we prove a generating function formula for the betti numbers of nakajima quiver varieties we prove that it is a q deformation of the weyl kac character formula in particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a certain weight in the corresponding kac moody algebra which was conjectured by kac in 1982

nonclassical symmetries of evolutionary partial differential equations and compatibility the determining equations for the nonclassical reductions of a general nth order evolutionary partial differential equations is considered it is shown that requiring compatibility with a first order quasilinear partial differential equation the determining equations are obtained burgers equation and the kdv equation and generalizations serve as examples illustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries

iterating the nonclassical symmetries method we show that iterations of the nonclassical symmetries method for any evolution equation give raise to new nonlinear equations which inherit the symmetry algebra of the given equation invariant solutions of these heir equations supply new solutions of the original equation the same method generates heir equations for hyperbolic and elliptic equations some examples are given

dilations and commutant lifting for jointly isometric operators a geometric approach abstract a tuple of commuting contractions t t 1 t 2 t n is called a joint isometry if t j t j i we give a geometric proof that joint isometries have a regular unitary dilation and that its commutant lifts we also show that t is subnormal and that its minimal normal extension is also jointly isometric

on the existence of nonclassical symmetries of partial differential equations relations between determining equations for nonclassical symmetries of bluman and cole and integrability conditions of differential equations supplemented by invariance conditions are studied it is shown that every system of equations with p independent variables admits p dimensional groups of such symmetries nonclassical symmetries of the eikonal equation in a curved space are investigated they are related to shear free congruences of null geodesics

nonclassical symmetry reductions for the kadomtsev petviashvili equation abstract in this paper we discuss dimensional reductions of the physically and mathematically significant kadomtsev petviashvili equation in particular we exhibit some new reductions of the kadomtsev petviashvili equation which are obtained by a generalization of the direct method for determining similarity reductions recently developed by clarkson and kruskal j math phys 30 1989 2201 new solutions of the kadomtsev petviashvili equation are obtained in terms of weierstrass elliptic functions solutions of the lame equations and the first second and fourth painleve transcendents a group theoretical explanation of the new solutions is provided in terms of conditional symmetries leaving only specific subsets of solutions of the kadomtsev petviashvili equation invariant

on the noncommutative donaldson thomas invariants arising from brane tilings abstract given a brane tiling that is a bipartite graph on a torus we can associate with it a quiver potential and a quiver potential algebra under certain consistency conditions on a brane tiling we prove a formula for the donaldson thomas type invariants of the moduli space of framed cyclic modules over the corresponding quiver potential algebra we relate this formula with the counting of perfect matchings of the periodic plane tiling corresponding to the brane tiling we prove that the same consistency conditions imply that the quiver potential algebra is a 3 calabi yau algebra we also formulate a rationality conjecture for the generating functions of the donaldson thomas type invariants

nonclassical symmetries as special solutions of heir equations in phys d 78 1994 124 we have found that iterations of the nonclassical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the given equation in the present paper we show that special solutions of the right order heir equation correspond to classical and nonclassical symmetries of the original equations an infinite number of nonlinear equations which possess nonclassical symmetries are derived

on the number of absolutely indecomposable representations of a quiver limburgs univ ctr dept wni b 3590 diepenbeek belgium sevenhant b limburgs univ ctr dept wni univ campus bldg d b 3590 diepenbeek belgium

nonclassical symmetries and backlund transformations abstract in this paper we show how to obtain well known backlund transformations of several equations by applying the nonclassical symmetries method in particular we derive the backlund transformation between the modified korteweg de vries and the korteweg de vries equations the backlund transformation between the calogero and the third order linear equations the auto backlund transformations of the potential korteweg de vries the potential modified korteweg de vries equation and the burgers equations as a consequence the n solitons solution of the korteweg de vries equation can be obtained by the nonclassical symmetries method

absolutely indecomposable representations and kac moody lie algebras a conjecture of kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated kac moody lie algebra in this paper we prove these conjectures for indivisible dimension vectors