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

the perron generalized eigenspace and the spectral cone of a cone preserving map a unified treatment is offered to reprove known results on the following four highlights of the combinatorial spectral theory of nonnegative matrices or to extend or partly extend the results to the setting of a linear map preserving a polyhedral proper or proper cone the preferred basis theorem equivalent conditions for equality of the graph theoretic level characteristic and the spectral height characteristic the majorization relation between the two characteristics and the relation between the combinatorial properties of a nonnegative matrix and the positivity of the individual entries in its principal components this is achieved by employing the new concept of spectral cone of a cone preserving map and combining the cone theoretic methods developed in our previous papers on the geometric spectral theory of cone preserving maps with the algebraic analytic method introduced by hartwig neumann and rose and further exploited by neumann and schneider for nonnegative matrices

geometric mapping properties of semipositive matrices abstract semipositive matrices map a positive vector to a positive vector and as such they are a very broad generalization of the irreducible nonnegative matrices nevertheless the ensuing geometric mapping properties of semipositive matrices result in several parallels to the theory of cone preserving and cone mapping matrices it is shown that for a semipositive matrix a there exist a proper polyhedral cone k 1 of nonnegative vectors and a polyhedral cone k 2 of nonnegative vectors such that a k 1 k 2 the set of all nonnegative vectors mapped by a to the nonnegative orthant is a proper polyhedral cone as a consequence a belongs to a proper polyhedral cone comprising semipositive matrices when the powers a k k 0 1 have a common semipositivity vector then a has a positive eigenvalue if a has a sole peripheral eigenvalue and the powers of a have a common semipositivity vector with a non vanishing term in the direction of the left eigenspace of then a leaves a proper cone invariant

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

reduction of the number of particles in the stochastic weighted particle method for the boltzmann equation different ideas for reducing the number of particles in the stochastic weighted particle method for the boltzmann equation are described and discussed the corresponding error bounds are obtained and numerical tests for the spatially homogeneous boltzmann equation presented it is shown that if an appropriate reduction procedure is used then any effect on the accuracy of the numerical scheme is negligible

chern simons solitons toda theories and the chiral model the two dimensional self dual chern simons equations are equivalent to the conditions for static zero energy solutions of the 2 1 dimensional gauged nonlinear schrodinger equation with chern simons matter gauge dynamics in this paper we classify all finite chargesu n solutions by first transforming the self dual chern simons equations into the two dimensional chiral model or harmonic map equations and then using the uhlenbeck wood classification of harmonic maps into the unitary groups this construction also leads to a new relationship between thesu n toda andsu n chiral model solutions

principal pivot transforms properties and applications the principal pivot transform ppt is a transformation of the matrix of a linear system tantamount to exchanging unknowns with the corresponding entries of the right hand side of the system the notion of the ppt is encountered in mathematical programming statistics and numerical analysis among other areas the purpose of this paper is to draw attention to the main properties and uses of ppts make some new observations and motivate further applications of ppts in matrix theory special consideration is given to ppts of matrices whose principal minors are positive 2000 elsevier science inc all rights reserved

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

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

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

linear equations over cones and collatz wielandt numbers let k be a proper cone in r n let a be an n x n real matrix that satisfies ak k let b be a given vector of k and let be a given positive real number the following two linear equations are considered in this paper i i n a x b x k and ii a l n x b x k we obtain several equivalent conditions for the solvability of the first equation for the second equation we give an equivalent condition for its solvability in case when b a and we also find a necessary condition when b a and also when b a sufficiently close to b a where b a denotes the local spectral radius of a at b with fixed we also consider the questions of when the set a l n k k equals 0 or k and what the face of k generated by the set is then we derive some new results about local spectral radii and collatz wielandt sets or numbers associated with a cone preserving map and extend a known characterization of m matrices among z matrices in terms of alternating sequences

on the invariant faces associated with a cone preserving map for an n n nonnegative matrix p an isomorphism is obtained between the lattice of initial subsets of 1 n for p and the lattice of p invariant faces of the nonnegative orthant r motivated by this isomorphism we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone preserving map on a polyhedral cone formulated in terms of its invariant faces in particular we obtain the following extension of the famous rothblum index theorem for a nonnegative matrix if a leaves invariant a polyhedral cone k then for each distinguished eigenvalue ofa fork there is a chain of m distinct a invariant join irreducible faces of k each containing in its relative interior a generalized eigenvector of a corresponding to referred to as semi distinguished a invariant faces associated with where m is the maximal order of distinguished generalized eigenvectors of a corresponding to but there is no such chain with more than m members we introduce the important new concepts of semi distinguished a invariant faces and of spectral pairs of faces associated with a cone preserving map and obtain several properties of a cone preserving map that mostly involve these two concepts when the underlying cone is polyhedral perfect or strictly convex and or smooth or is the cone of all real polynomials of degree not exceeding n that are nonnegative on a closed interval plentiful illustrative examples are provided some open problems are posed at the end 1991 mathematics subject classification primary 15a48 secondary 47b65 47a25 46b42

invariant ellipsoidal cones abstract conditions on the spectrum of a matrix a which are equivalent to the existence of a proper convex cone k such that a is k nonnegative k positive k irreducible or k strongly nonnegative are known we study the effect of adding the requirement that k be ellipsoidal it is shown that the conditions are unchanged except in the k nonnegative case

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 the collatz wielandt sets associated with a cone preserving map abstract let k be a closed pointed full cone in r n in their treatment of perron frobenius theory for a linear map a preserving k by wielandt s approach barker and schneider introduced four sets namely 1 and 1 the collatz wielandt sets associated with a we determine the greatest lower bound and the least upper bound of these sets some known results for a nonnegative matrix relating its spectral radius to its upper and lower collatz wielandt number are extended to the setting of a cone preserving map applications of our results to the nonnegativity of solutions of linear inequalities associated with a nonnegative matrix are also considered

coupling of the boltzmann and euler equations with automatic domain decomposition a criterion for automatic domain decomposition has been introduced for the coupling of the boltzmann and euler equations this criterion is obtained by using the grad s 13 moments expansion method the boltzmann and euler equations have been solved by the particle method this criterion has been further tested for the spatial homogeneous relaxation case and the coupling of the boltzmann and euler equations in two dimensional physical space some numerical results obtained from the particle code of the full boltzmann equation are compared with those of coupling of the boltzmann and euler equations

perron frobenius theory for finite dimensional spaces with a hyperbolic cone abstract if x is a real n dimensional space provided with a subnorm then the inequality x defines a so called hyperbolic cone ine r x in this case the perron frobenius theory admits some special features a relevant characterization of nonnegative operators in a matrix form is given first auxiliary information from spectral theory and from the geometry of subnormed spaces is collected as preparation

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

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

on matrices leaving invariant a nontrivial convex set abstract a real square matrix a leaves a nontrivial convex set invariant if there exists a convex set c which is not a linear subspace such that a c c it is shown that this is equivalent to the statement that a has an eigenvalue with 0 or 1

two dimensional exactly and completely integrable dynamical systems monopoles instantons dual models relativistic strings lund regge model generalized toda lattice etc an investigation of two dimensional exactly and completely integrable dynamical systems associated with the local part of an arbitrary lie algebra mathfrak g whose grading is consistent with an arbitrary integral embedding of 3d subalgebra in mathfrak g has been carried out we have constructed in an explicit form the corresponding systems of nonlinear partial differential equations of the second order and obtained their general solutions in the sense of a goursat problem a method for the construction of a wide class of infinite dimensional lie algebras of finite growth has been proposed

numerical study of a stochastic weighted particle method for a model kinetic equation a stochastic weighted particle method is applied to a model nonlinear kinetic equation a detailed study of various numerical approximations is presented the main effect achieved by the new method is an artificial increase of the relative number of simulation particles with prescribed velocities

geometric conditions for the existence of positive eigenvalues of matrices finite dimensional theorems of perron frobenius type are proved for a e e nn and a nonnegative integer k we let w t a be the cone generated by ai ak l in en we show that a satisfies the perron schaefer condition if and only if the closure wk a of w t a is a pointed cone this theorem is closely related to several known results if k po a the index of the eigenvalue 0 in spec a we prove that a has a positive eigenvalue if and only if w t a is a pointed nonzero cone or equivalently w t a is not a real subspace of en our proofs are elementary and based on a method of birkhoff s we discuss the relation of this method to pringsheim s theorem