relativistic chern simons higgs vortex equations an existence theorem is established for the solutions to the non abelian relativistic chern simons higgs vortex equations over a doubly periodic domain when the gauge group g assumes the most general and important prototype form g su n mathematics subject classification 2010 35c08 35j50 35q70 81e13

the relativistic non abelian chern simons equations we study ther xr system of nonlinear elliptic equations delta u a lambda sum limits b 1 r k ab e u b lambda sum limits b 1 r sum limits c 1 r e u b k ab e u c k bc 4 pi sum limits j 1 n a delta p aj a 1 2 r x r 2 where 0 is a constant parameter k kab is the cartan matrix of a semi simple lie algebra and p is the dirac measure concentrated atp r 2 this system of equations arises in the relativistic non abelian chern simons theory and may be viewed as a nonintegrable deformation of the integrable toda system we establish the existence of a class of solutions known as topological multivortices the crucial step in our method is the use of the decomposition theorem of cholesky for positive definite matrices so that a variational principle can be formulated

generalized drinfeld sokolov reductions and kdv type hierarchies generalized drinfeld sokolov ds hierarchies are constructed through local reductions of hamiltonian flows generated by monodromy invariants on the dual of a loop algebra following earlier work of de groot et al reductions based upon graded regular elements of arbitrary heisenberg subalgebras are considered we show that in the case of the nontwisted loop algebra l gl n graded regular elements exist only in those heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn pr or to equal numbers plus onen pr 1 we prove that the reduction belonging to the grade 1 regular elements in the casen pr yields thep p matrix version of the gelfand dickeyr kdv hierarchy generalizing the scalar casep 1 considered by ds the methods of ds are utilized throughout the analysis but formulating the reduction entirely within the hamiltonian framework provided by the classical r matrix approach leads to some simplifications even forp 1

on the τ functions of the degasperis procesi equation the degasperis procesi dp equation is investigated from the point of view of determinant pfaffian identities the reciprocal link between the dp equation and the pseudo 3 reduction of the c two dimensional toda system is used to construct the n soliton solution of the dp equation the n soliton solution of the dp equation is presented in the form of pfaffian through a hodograph reciprocal transformation the bilinear equations the identities between determinants and pfaffians and the functions of the dp equation are obtained from the pseudo 3 reduction of the c two dimensional toda system

the prolongation structures of quasi polynomial flows we look closely at the process of finding a wahlquist estabrook prolongation structure for a given system of nonlinear evolution equation s there are two main steps in this calculation the first to reduce the problem to the investigation of a finitely generated free lie algebra with constraints the second to find a finite dimensional linear representation of these generators we discuss some of the difficulties that arise in this calculation for quasi polynomial flows defined later we give an algorithm for the first step we do not totally solve the problems of the second step but do give an algebraic framework and a number of techniques that are quite generally applicable we illustrate these methods with many examples several of which are new

the unified approach to integrable relativistic equations soliton solutions over nonvanishing backgrounds ii in part i of this work the n soliton solution has been constructed for the generic system associated with the sl 2 c case of the scheme for the unified description of integrable relativistic massive fields here solutions are isolated for reductions of this system including the conventional complex sine gordon equation the massive thirring model and another complexification of the sine gordon equation defined by the lagrangian l 2 1 2 2 j 2 2 2 1 2 j i the latter model is shown to exhibit decays and fusion of subluminal solitons the reduction to the conventional complex sine gordon appears to be even more interesting as it cannot be defined by simply restricting the linear problem to some real form of sl 2 c algebra and the relevant involution turns out to be quite nontrivial when the background is flat this involution degenerates and so the n kink solution for the reduction cannot be extracted from the generic n kink solution directly this difficulty is

vertex operator representation of some quantum tori lie algebras we are defining the trigonometric lie subalgebras in bar x infty bar a infty bar b infty bar c infty bar d infty which are the natural generalization of the well known sin lie algebra the embedding formulas into bar x infty are introduced these algebras can be considered as some lie algebras of quantum tori an irreducible representation ofa b series of trigonometric lie algebras is constructed special cases of the trigonometric lie factor algebras which can be considered as a quantum preserving lie algebra structure deformation of the kac moody algebras are considered

quantum line operators from lax pairs motivated by the realization of the yang baxter equation of 2d integrable models in the 4d gauge theory of costello witten yamazaki cwy we study the embedding of integrable 2d toda field models inside this construction this is done by using the lax formulation of 2d integrable systems and by thinking of the standard lax pair l in terms of components of cwy gauge connection propagating along particular directions in the gauge bundle we also use the results of the cwy theory to build quantum line operators for 2d toda models and compute the one loop contribution of two intersecting lines exchanging one gluon other features such as local symmetries and comments on extension of our method to other 2d integrable models are also discussed we also comment on some basic points that still need a refinement before discussing about a fully consistent embedding of lax pairs into cwy theory

on simple polytopes letp be a simpled polytope ind dimensional euclidean space mathbb e d and let p be the subalgebra of the polytope algebra generated by the classes of summands ofp it is shown that the dimensions of the weight spaces r p of p are theh numbers ofp which describe the dehn sommerville equations between the numbers of faces ofp and reflect the duality between r p and d r p moreover p admits a lefschetz decomposition under multiplication by the element of 1 p corresponding top itself which yields a proof of the necessity of mcmullen s conditions in theg theorem on thef vectors of simple polytopes the lefschetz decomposition is closely connected with the new hodge riemann minkowski quadratic inequalities between mixed volumes which generalize minkowski s second inequality also proved are analogous generalizations of the aleksandrov fenchel inequalities a striking feature is that these are obtained without using brunn minkowski theory indeed the brunn minkowski theorem without characterization of the cases of equality can be deduced from them the connexion found between p and the face ring of the dual simplicial polytopep enables this ring to be looked at in two ways and a conjectured formulation of theg theorem in terms of a gale diagram ofp is also established

classical and quantum integrable systems with boundaries we study two dimensional classically integrable field theory with independent boundary conditions at each end and obtain three possible generating functions for integrals of motion when this model is an ultralocal one classically integrable boundary conditions can be found when solving boundary k equations in the quantum case we also find that the unitarity condition of the quantum r matrix is sufficient to construct commutative quantities with boundaries and its reflection equations are obtained

simplicial moves on balanced complexes abstract we introduce a notion of cross flips local moves that transform a balanced i e properly d 1 colored triangulation of a combinatorial d manifold into another balanced triangulation these moves form a natural analog of bistellar flips also known as pachner moves specifically we establish the following theorem any two balanced triangulations of a closed combinatorial d manifold can be connected by a sequence of cross flips along the way we prove that for every m d 2 and any closed combinatorial d manifold m two m colored triangulations of m can be connected by a sequence of bistellar flips that preserve the vertex colorings

the generalised zakharov shabat system and the gauge group action the generalized zakharov shabat systems with complex valued non regular cartan elements and the systems studied by caudrey beals and coifman cbc systems and their gauge equivalent are studied this study includes the properties of fundamental analytical solutions for the gauge equivalent to cbc systems and the minimal set of scattering data the description of the class of nonlinear evolutionary equations solvable by the inverse scattering method and the recursion operator related to such systems the hierarchies of hamiltonian structures the results are illustrated in the example of the multi component nonlinear schrodinger equations and the corresponding gauge equivalent multi component heisenberg ferromagnetic type models related to so 5 c algebra

a note on affine toda couplings the author proves the clebsch gordan property for the couplings of an untwisted affine toda field theory utilizing dorey s 1991 conjecture

two dimensional generalized toda lattice the zero curvature representation is obtained for the two dimensional generalized toda lattices connected with semisimple lie algebras the reduction group and conservation laws are found and the mass spectrum is calculated

discretization of hyperbolic type darboux integrable equations preserving integrability a method of integrable discretization of the liouville type nonlinear partial differential equations based on integrals is suggested new examples of the discrete liouville type models are presented

more non abelian loop toda solitons we find new solutions including soliton like ones for a special case of non abelian loop toda equations associated with complex general linear groups we use the method of rational dressing based on an appropriate block matrix representation suggested by the gradation under consideration we present solutions in the form of a direct matrix generalization of hirota s soliton solution already well known in the case of abelian loop toda systems

conservation laws and symmetries of generalized sine gordon equations we study some systems of non linear pde s eqs 1 1 below which can be regarded either as generalizations of the sine gordon equation or as two dimensional versions of the toda lattice equations we show that these systems have an infinite number of non trivial conservation laws and an infinite number of symmetries the second result is deduced from the first by a variant of the hamiltonian formalism for evolution equations we also consider some specializations of the systems

kac moody groups and integrability of soliton equations a new approach to integrability of affine toda field theories and closely related to them kdv hierarchies is proposed the flows of a hierarchy are explicity identified with infinitesimal action of the principal abelian subalgebra of the corresponding affine kac moody algebra on a homogeneous space

random evolution processes with feedback a general random evolution markov process is constructed which switches back and forth at random among a given collection of markov processes modes of evolution defined on a common evolution state space and indexed by an evolution rule space feedback is incorporated by allowing the path of the evolution component to influence the changes in evolution rule the semigroup of the random evolution process is studied and is used to compare the process with the operator random evolutions of griego and hersh using deterministic modes of evolution we generalize the markov processes constructed by erickson and by heath we also study new random evolution processes constructed from brownian motions and from regular step processes

comments on defects in the ar toda field theories a simple basic argument is given based solely on energy momentum considerations to recover conditions under which ar affine or conformal toda field theories can support defects of integrable type associated triangle relations are solved to provide expressions for transmission matrices that generalize previously known examples calculated for the sine gordon model and the a2 affine toda model

the generalized additional symmetries of the two toda lattice hierarchy the generalized additional symmetries of the two toda lattice hierarchy are investigated in this paper the algebraic structure of this generalized additional symmetry is showed as w w and the actions of the generalized additional symmetries on function are also discussed by restricting the two toda lattice hierarchy to the semi infinite case

local conservation laws for the two dimensional periodic su n 1 toda lattices an infinite number of local conservation laws are derived for the two dimensional periodic su n 1 toda lattice equations following the well known method for the sine gordon equation

hidden quantum group symmetry and integrable perturbations of conformal field theories the hidden quantum group symmetry in the quantum sine gordon model is found this symmetry provides the possibility to restrict the operator algebra of the model to subalgebras it is shown that these subalgebras are massive deformations of minimal conformal field theories

complex network view of evolving manifolds we study complex networks formed by triangulations and higher dimensional simplicial complexes representing closed evolving manifolds in particular for triangulations the set of possible transformations of these networks is restricted by the condition that at each step all the faces must be triangles stochastic application of these operations leads to random networks with different architectures we perform extensive numerical simulations and explore the geometries of growing and equilibrium complex networks generated by these transformations and their local structural properties this characterization includes the hausdorff and spectral dimensions of the resulting networks their degree distributions and various structural correlations our results reveal a rich zoo of architectures and geometries of these networks some of which appear to be small worlds while others are finite dimensional with hausdorff dimension equal or higher than the original dimensionality of their simplices the range of spectral dimensions of the evolving triangulations turns out to be from about 1 4 to infinity our models include simplicial complexes representing manifolds with evolving topologies for example an h holed torus with a progressively growing number of holes this evolving graph demonstrates features of a small world network and has a particularly heavy tailed degree distribution

on the squared eigenfunction symmetry of the toda lattice hierarchy the squared eigenfunction symmetry for the toda lattice hierarchy is explicitly constructed in the form of the kronecker product of the vector eigenfunction and the vector adjoint eigenfunction which can be viewed as the generating function for the additional symmetries when the eigenfunction and the adjoint eigenfunction are the wave function and the adjoint wave function respectively then after the fay like identities and some important relations about the wave functions are investigated the action of the squared eigenfunction related to the additional symmetry on the tau function is derived which is equivalent to the adler shiota van moerbeke formulas

non crystallographic reduction of generalized calogero moser models we apply a recently introduced reduction procedure based on the embedding of non crystallographic coxeter groups into crystallographic ones to calogero moser systems for rational potentials the familiar generalized calogero hamiltonian is recovered for the hamiltonians of trigonometric hyperbolic and elliptic types we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio we explicitly show for the simplest of these non crystallographic models how the corresponding classical equations of motion can be derived from a lie algebraic lax pair based on the larger crystallographic coxeter group

lie superalgebraic approach to super toda lattice and generalized super kdv equations we propose a super lax type equation based on a certain class of lie superalgebra as a supersymmetric extension of generalized modified kdv hierarchy we are able to construct an infinite set of conservation laws and the consistent time evolution generators for generalized modified super kdv equations thefirst few of the conserved currents the modified super kdv equation and the super miura transformation are worked out explicitly in the case of twisted affine lie superalgebraosp 2 2 2

linearizing integral transform for the multicomponent lattice kp abstract we present a generalization of the direct linearization method for the kadomtsev petviashvili kp equation and its lattice analogues such as the two dimensional toda equation the generalization consists of a riemann hilbert type of integral transform which relates solutions of the associated spectral problem to any given solution of the spectral problem by dimensional reduction we obtain the integral transform for the korteweg de vries kdv equation and its lattice analogues the integral transform is well suited for the investigation of the finite matrix generalizations of the above mentioned equations such finite matrix generalizations may be of use in connection with partial difference equations for spin correlation functions in the two dimensional ising model

toda lattice mass transport in lagrangian mechanics and in a two dimensional system abstract this paper explores the connection between the hydrodynamic mass transport description and the thermodynamic description for a nonlinear range of the toda lattices particular attention is paid to the broken isotropy in the kdv and burgers equations the flow variable representation is established from the lagrangian mechanics for hydrodynamic mass transport based on the inverse scattering transform the gel fand levitan marchenko glm equation is formulated from the kdv equation expressed by the flow variable representation we found that a kernel of the glm equation is given by the concentration variable q x t a lagrangian is formulated for the kdv equation in state space q x t k x t next an extension of the flow variable representation is sought in a two dimensional system the lhs of the kadomtsev petviashvili kp equation takes the same form as in the second formalism of the kdv equation by setting up the flow variable representation of the kp equation the burgers equation in two dimensions is formulated these results contribute to an understanding of the broken isotropy for the nonlinear mass transport equations these results provide physical insight into various consequences of the generalized form of the kawasaki ohta equation from the viewpoint of mass transport

supersymmetric two dimensional toda lattice the two dimensional toda lattice connected with contragradient lie superalgebras is studied the systems of linear equations associated with the models for which the inverse scattering method is applicable are written down the reduction group is calculated

projectivities in simplicial complexes and colorings of simple polytopes for each strongly connected finite dimensional pure simplicial complex delta we construct a finite group pi delta the group of projectivities of delta which is a combinatorial but not a topological invariant of delta this group is studied for combinatorial manifolds and in particular for polytopal simplicial spheres the results are applied to a coloring problem for simplicial or dually simple polytopes which arises in the area of toric manifolds

toda lattice hierarchy and conservation laws time evolutions of the toda lattice hierarchies of ueno and takasaki are induced by hamiltonians which are conservation laws for the original one and two dimensional toda lattice obtained by olive and turok moreover these hamiltonians for two dimensional toda lattice hierarchy are also conserved quantities of the two component kp hierarchy in which that system is embedded the one dimensional toda lattice hierarchy is characterized by the bilinear relations and a new version of the one dimensional toda lattice hierarchy is constructed generalized toda lattice hierarchies associated to all affine lie algebras are presented

the determinant representation for quantum correlation functions of the sinh gordon model we consider the quantum sinh gordon model in this paper form factors in this model were calculated by mussardo and colleagues we sum up all contributions of form factors and obtain a closed expression for a correlation function this expression is a determinant of an integral operator similar determinant representations have been proven to be useful not only in the theory of correlation functions but also in matrix models

reductions of integrable equations on a iii type symmetric spaces we study a class of integrable nonlinear differential equations related to the a iii type symmetric spaces these spaces are realized as factor groups of the form su n s u n k u k we use the cartan involution corresponding to this symmetric space as an element of the reduction group and restrict generic lax operators to this symmetric space the symmetries of the lax operator are inherited by the fundamental analytic solutions and give a characterization of the corresponding riemann hilbert data

specializations of integrable systems and affine lie algebras we analyze a number of restrictions on the evolution systems associated with the zero curvature equations corresponding to the extended dynkin diagram a 1 n 1 the resulting specialized evolution systems contain exponential terms like a third order differential equation previously derived in ref 5

a stable cox ingersoll ross model with restart we consider a stable cox ingersoll ross model in a domain d 0 which is killed at the boundary and which while alive jumps instantaneously at an exponentially distributed random time with intensity 0 to a new point according to a distribution from this new point it repeats the above behavior independently of what has transpired previously we give the infinitesimal generator of this new process and prove that the corresponding semigroup is compact and thus the spectrum of the generator consists exclusively of eigenvalues we further show that the principal eigenvalue gives the exponential rate of decay of not exiting the domain by time t finally we study some properties of the principal eigenvalue

on the liouville integrability of the periodic kostant toda flow on matrix loops of level k in this work we consider the periodic kostant toda flow on matrix loops in mathfrak s mathfrak l n mathbb c of level k which correspond to periodic infinite band matrices with period n with lower bandwidth equal to k and fixed upper bandwidth equal to 1 with 1 s on the first superdiagonal we show that the coadjoint orbits through the submanifold of such matrix loops can be identified with those of a finite dimensional lie group which appears in the form of a semi direct product we then characterize the generic coadjoint orbits and obtain an explicit global cross section for such orbits we also establish the liouville integrability of the periodic kostant toda flow on such orbits via the construction of action angle variables

form factors of boundary fields for the a2 affine toda field theory in this paper we carry out the boundary form factor program for the a2 affine toda field theory at the self dual point the latter is an integrable model consisting of a pair of particles which are conjugated to each other that is and possessing two bound states resulting from the scattering processes 1 1 2 and 2 2 1 we obtain solutions up to four particle form factors for two families of fields which can be identified with spinless and spin 1 fields of the bulk theory previously known as well as new bulk form factor solutions are obtained as a particular limit of ours minimal solutions of the boundary form factor equations for all an affine toda field theories are given which will serve as the starting point for a generalization of our results to higher rank algebras

nonlinear schrodinger equations and simple lie algebras we associate a system of integrable generalised nonlinear schrodinger nls equations with each hermitian symmetric space these nls equations are considered as reductions of more general systems this time associated with a reductive homogeneous space the nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space the hamiltonian structure is investigated using r matrix techniques and shown to be canonical for all these equations throughout the reduction procedure this hamiltonian structure does not degenerate each of the above systems of equations is gauge equivalent to a generalised ferromagnet reductions of the latter are discussed in terms of the corresponding nls type equations

harmonic functions and the dirichlet problem for revival markov processes harmonic functions and the dirichlet problem on an open set g are defined for a pieced out or revival markov process constructed from a contin uous base process a one to one correspondence is obtained between the bounded harmonic functions of the revival process and those of the base process the harmonic functions of the revival process are shown to be continuous on g under certain conditions and to coincide with the solutions of o f 0 on g where is the characteristic operator these results are uses a branching process to solve a type of nonlinear dirichlet problem in this paper harmonic functions and the dirichlet problem are studied for a general revival process with a continuous base process a bounded nearly borel function f is defined to be harmonic on an open set g if it satisfies the usual mean value property on open subsets of g but since the revival process is discontinuous it is necessary that f be defined on the entire state space not just on g and of course the classical theorems see 2 do not apply the main result of this paper is theorem 4 6 which gives a one to one correspondence between the harmonic functions of the base process and the harmonic functions of the revival process since the base process is continuous this correspondence is then used to obtain analogues of the classical theorems for the revival process in particular it is shown that under appropriate conditions the harmonic functions are continuous on g and coincide with the zeros of the characteristic operator on g and the dirichlet problem with a given exterior function is solved the results are applied to random evolution processes and branching processes the random evolution processes considered are more general than those used by previous authors and allow feedback and an uncountable number of evolution modes moreover the main theorem yields additional insights in the random evolution setting it gives a correspondence between solutions of the coupled system of equations and solutions of the associated uncoupled system in the branching process setting of nagasawa 11 it gives a correspondence between solutions of the nonlinear equation and solutions of the associated linear equation

prolongation structures and lie algebra real forms a technique useful as an aid for finding finite dimensional representations of a wahlquist estabrook prolongation algebra is introduced the technique is illustrated with an examination of equations of the form 2za x1 x2 fa zb

bethe ansatz equations for the classical a 1 _ n affine toda field theories we establish a correspondence between classical affine toda field theories and an bethe ansatz systems we show that the connection coefficients relating specific solutions of the associated classical linear problem satisfy functional relations of the type that appear in the context of the massive quantum integrable model