a hybrid method for hydrodynamic kinetic flow part ii coupling of hydrodynamic and kinetic models in this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic kinetic problems in which the knudsen number may span from equilibrium to highly rarefied regimes our approach is characterized by using the full boltzmann equation for the kinetic regime the compressible euler equations for equilibrium with a buffer zone in which the bgk es equation is used to represent the transition between fully kinetic to equilibrium flows in this fashion the boltzmann solver is used only when the collision integral is non stiff and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities thus in principle the same mesh size and time steps can be used in the whole computation moreover the time step is limited only by convective terms since the boltzmann solver is applied only in wholly kinetic regimes we use the reduced noise dsmc scheme we have proposed in part i of the present work this ensures a smooth exchange of information across the different domains with a natural way to construct interface numerical fluxes several tests comparing our hybrid scheme with full boltzmann dsmc computations show the good agreement between the two solutions on a wide range of knudsen numbers

fast full search equivalent encoding algorithms for image compression using vector quantization three fast search routines to be used in the encoding phase of vector quantization vq image compression systems are presented these routines which are based on geometric considerations provide the same results as an exhaustive or full search examples show that the proposed algorithms need only 3 20 of the number of mathematical operations required by a full search and fewer than 50 of the operations required by recently proposed alternatives

a comparison of several vector quantization codebook generation approaches a review and a performance comparison of several often used vector quantization vq codebook generation algorithms are presented the codebook generation algorithms discussed include the linde buzo gray lbg binary splitting algorithm the pairwise nearest neighbor algorithm the simulated annealing algorithm and the fuzzy c means clustering analysis algorithm a new directed search binary splitting method which reduces the complexity of the lbg algorithm is presented also a new initial codebook selection method which can obtain a good initial codebook is presented by using this initial codebook selection algorithm the overall lbg codebook generation time can be reduced by a factor of 1 5 2

non autonomous he non heiles systems abstract scaling similarity solutions of three integrable pdes namely the sawada kotera fifth order kdv and kaup kupershmidt equations are considered it is shown that the resulting odes may be written as non autonomous hamiltonian equations which are time dependent generalizations of the well known integrable henon heiles systems the time dependent hamiltonians are given by logarithmic derivatives of the tau functions inherited from the original pdes the odes for the similarity solutions also have inherited backlund transformations which may be used to generate sequences of rational solutions as well as other special solutions related to the first painleve transcendent

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

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

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

numerical comparison between the boltzmann and es bgk models for rarefied gases abstract rarefied gas flows obey the boltzmann equation but numerical simulations of this equation are not always possible so that simpler models have been introduced the es bgk equation is one of these models it gives the correct transport coefficients for the navier stokes approximation so that boltzmann or es bgk simulations are expected to give the same results for dense gases but in the case of a rarefied flow complete numerical comparisons are needed in this paper we present numerical comparisons between the two models in transitional regimes where the es bgk model is expected to be useful for reentry flows around a compression ramp and a plate we also emphasize that the es bgk model gives flow predictions closer to the boltzmann result than the simpler bgk model

universal characters integrable chains and the painleve equations abstract the universal character is a generalization of the schur polynomial attached to a pair of partitions see adv math 74 1989 57 we prove that the universal character solves the darboux chain the n periodic closing of the chain is equivalent to the painleve equation of type a n 1 1 consequently we obtain an expression of rational solutions of the painleve equations in terms of the universal characters

casorati determinant solutions for the discrete painleve ii equation we present a class of solutions to the discrete painleve ii equation for particular values of its parameters it is shown that these solutions can be expressed in terms of casorati determinants whose entries are discrete airy functions the analogy between the tau function for the discrete pii and that of the discrete toda molecule equation is pointed out

a new asymptotic preserving scheme based on micro macro formulation for linear kinetic equations in the diffusion limit we propose a new numerical scheme for linear transport equations it is based on a decomposition of the distribution function into equilibrium and nonequilibrium parts we also use a projection technique that allows us to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic density and a kinetic equation for the nonequilibrium part by using a suitable time semi implicit discretization our scheme is able to accurately approximate the solution in both kinetic and diffusion regimes it is asymptotic preserving in the following sense when the mean free path of the particles is small our scheme is asymptotically equivalent to a standard numerical scheme for the limit diffusion model a uniform stability property is proved for the simple telegraph model various boundary conditions are studied our method is validated in one dimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes

dsrf a flexible trajectory descriptor for articulated human action recognition abstract in this paper we propose a novel skeletal representation that models the human body as the articulated interconnections of multiple rigid bodies an articulated human action can be viewed as the combination of multiple rigid body motion trajectories the dual square root function dsrf descriptor is firstly introduced by calculating the gradient based invariants for representing 6 d rigid body motion trajectories which can offer substantial advantages over raw data to effectively incorporate the dsrf descriptors in the skeletal representation a skeleton is decomposed into five parts and the most informative part method is raised for offering compact representation besides two rigid body configurations are investigated for representing the movement of each part in the recognition stage we first follow the simple template matching strategy with the nearest neighbor classifier a robust distance measure between two skeletal representations is designed in addition the dsrf based skeletal representation can be encoded as a sparse histogram by the bag of words approach then a support vector machine classifier with chi square kernel is trained for multiclass recognition tasks experimental results on three benchmark datasets demonstrate that our proposed approach outperforms existing skeleton representations in terms of recognition accuracy

dromion solutions of noncommutative davey stewartson equations we consider a noncommutative version of the davey stewartson equations and derive two families of quasideterminant solution via darboux and binary darboux transformations these solutions can be verified by direct substitution we then calculate the dromion solutions of the equations and obtain computer plots in a noncommutative setting

with a cole hopf transformation to solutions of the noncommutative kp hierarchy in terms of wronski matrices in the case of the kp hierarchy where the dependent variable takes values in an arbitrary associative algebra it is known that there are solutions which can be expressed in terms of quasideterminants of a wronski matrix which solves the linear heat hierarchy we obtain these solutions without the help of quasideterminants in a simple way via solutions of matrix kp hierarchies over and by use of a cole hopf transformation for this class of exact solutions we work out a correspondence with weakly nonassociative algebras

rough clustering using generalized fuzzy clustering algorithm in this paper we present a rough k means clustering algorithm based on minimizing the dissimilarity which is defined in terms of the squared euclidean distances between data points and their closest cluster centers this approach is referred to as generalized rough fuzzy k means grfkm algorithm the proposed method solves the divergence problem of available approaches where the cluster centers may not be converged to their final positions and reduces the number of user defined parameters the presented method is shown to be converged experimentally compared to available rough k means clustering algorithms the proposed method provides less computing time unlike available approaches the convergence of the proposed method is independent of the used threshold value moreover it yields better clustering results than rfkm for the handwritten digits data set landsat satellite data set and synthetic data set in terms of validity indices compared to mrkm and rfkm grfkm can reduce the value of xie beni index using the handwritten digits data set where a lower xie beni index value implies the better clustering quality the proposed method can be applied to handle real life situations needing reasoning with uncertainty

generating function associated with the rational solutions of the painleve ii equation we consider the hankel determinant representation for the rational solutions of the painleve ii equation we give an explicit formula for the generating function of the entries in terms of logarithmic derivative of the airy function

constructing solutions to the ultradiscrete painleve equations we investigate the nature of particular solutions to the ultradiscrete painleve equations we start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions for the ultradiscrete painleve equations ii and iii we present special solutions reminiscent of the casorati determinant ones which exist in the continuous and discrete cases finally we analyse the discrete painleve equation i and show how it contains both the continuous and the ultradiscrete ones as particular limits

microscopically implicit macroscopically explicit schemes for the bgk equation in this work a new class of numerical methods for the bgk model of kinetic equations is introduced the schemes proposed are implicit with respect to the distribution function while the macroscopic moments are evolved explicitly in this fashion the stability condition on the time step coincides with a macroscopic cfl evaluated using estimated values for the macroscopic velocity and sound speed thus the stability restriction does not depend on the relaxation time and it does not depend on the microscopic velocity of energetic particles either with the technique proposed here the updating of the distribution function requires the solution of a linear system of equations even though the bgk model is highly non linear thus the proposed schemes are particularly effective for high or moderate mach numbers where the macroscopic cfl condition is comparable to accuracy requirements we show results for schemes of order 1 and 2 and the generalization to higher order is sketched

noncommutative toda chains hankel quasideterminants and the painleve ii equation we construct solutions of an infinite toda system and an analog of the painleve ii equation over noncommutative differential division rings in terms of quasideterminants of hankel matrices

application of the τ function theory of painleve equations to random matrices piv pii and the gue tracy and widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite gue in terms of a piv and pii transcendent respectively we generalise these results to the evaluation of where for and otherwise and the average is with respect to the joint eigenvalue distribution of the gue as well as to the evaluation of of particular interest are and f n 2 and their scaled limits which give the distribution of the largest eigenvalue and the density respectively our results are obtained by applying the okamoto function theory of piv and pii for which we give a self contained presentation based on the recent work of noumi and yamada we point out that the same approach can be used to study the quantities and f n a for the other classical matrix ensembles

noncommutative painleve equations and systems of calogero type all painleve equations can be written as a time dependent hamiltonian system and as such they admit a natural generalization to the case of several particles with an interaction of calogero type rational trigonometric or elliptic recently these systems of interacting particles have been proved to be relevant in the study of beta models an almost two decade old open question by takasaki asks whether these multi particle systems can be understood as isomonodromic equations thus extending the painleve correspondence in this paper we answer in the affirmative by displaying explicitly suitable isomonodromic lax pair formulations as an application of the isomonodromic representation we provide a construction based on discrete schlesinger transforms to produce solutions for these systems for special values of the coupling constants starting from uncoupled ones the method is illustrated for the case of the second painleve equation

a fast vq codebook generation algorithm using codeword displacement in this paper we present a fast codebook generation algorithm called cgaucd codebook generation algorithm using codeword displacement by making use of the codeword displacement between successive partition processes by implementing a fast search algorithm named mfaupi modified fast algorithm using projection and inequality for vq encoding in the partition step of cgaucd the codebook generation time can be further reduced significantly using mfaupi the computing time of cgaucd can be reduced by a factor of 4 7 7 6 compared to generalized lloyd algorithm gla our proposed method can reduce the codebook generation time by a factor of 35 9 121 2 compared to the best codebook generation algorithm to our knowledge our approach can further reduce the corresponding computing time by 26 0 32 8 it is noted that our proposed algorithm can generate the same codebook as that produced by the gla the superiority of our method is more remarkable when a larger codebook is generated

an asymptotic preserving automatic domain decomposition method for the vlasov poisson bgk system with applications to plasmas in this work we present an efficient strategy to deal with plasma physics simulations in which localized departures from thermodynamical equilibrium are present the method is based on the introduction of intermediate regions which allows smooth transitions between kinetic and fluid zones in this paper we extend domain decomposition techniques obtained through dynamic coupling and buffer zones to the study of plasmas and moreover we combine them with asymptotic preserving and asymptotically accurate strategies for the time integration we use a hybrid scheme in which both kinetic and fluid descriptions are considered and coupled together while the kinetic model is solved by asymptotic preserving and accurate methods in order to guarantee high efficiency and accuracy in all regimes the numerical scheme is validated and its performances are analyzed by numerical simulations

a semi discrete kadomtsev petviashvili equation and its coupled integrable system we establish connections between two cascades of integrable systems generated from the continuum limits of the hirota miwa equation and its remarkable nonlinear counterpart under the miwa transformation respectively among these equations we are mainly concerned with the semi discrete bilinear kadomtsev petviashvili kp equation which is seldomly studied in literature we present both of its casorati and grammian determinant solutions through the pfaffianization procedure proposed by hirota and ohta we are able to derive the coupled integrable system for the semi discrete kp equation

efficient asymptotic preserving ap schemes for some multiscale kinetic equations many kinetic models of the boltzmann equation have a diffusive scaling that leads to the navier stokes type parabolic equations as the small scaling parameter approaches zero in practical applications it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter from the rarefied kinetic regimes to the hydrodynamic diffusive regimes an earlier approach in s jin l pareschi and g toscani siam j numer anal 35 1998 pp 2405 2439 reformulates such systems into the common hyperbolic relaxation system by jin and xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time splitting method to solve the relaxation system here we observe that the combination of the two time split steps may yield hyperbolic parabolic systems that are more advantageous in both stability and efficiency for numerical computations we show that such an approach yields a class of asymptotic preserving ap schemes which are suitable for the computation of multiscale kinetic problems we use the goldstein taylor and carleman models to illustrate this approach

on a direct approach to quasideterminant solutions of a noncommutative kp equation a noncommutative version of the modified kp equation and a family of its solutions expressed as quasideterminants are discussed the origin of these solutions is explained by means of darboux transformations and the solutions are verified directly we also verify directly an explicit connection between quasideterminant solutions of the noncommutative mkp equation and the noncommutative kp equation arising from the miura transformation

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

high order asymptotic preserving nodal discontinuous galerkin imex schemes for the bgk equation in this paper we develop high order asymptotic preserving ap schemes for the bgk equation in a hyperbolic scaling which leads to the macroscopic models such as the euler and compressible navier stokes equations in the asymptotic limit our approaches are based on the so called micro macro formulation of the kinetic equation which involves a natural decomposition of the problem to the equilibrium and the non equilibrium parts the proposed methods are formulated for the bgk equation with constant or spatially variant knudsen number the new ingredients for the proposed methods to achieve high order accuracy are the following we introduce discontinuous galerkin dg discretization of arbitrary order of accuracy with nodal lagrangian basis functions in space we employ a high order globally stiffly accurate implicit explicit imex runge kutta rk scheme as time discretization two versions of the schemes are proposed scheme i is a direct formulation based on the micro macro decomposition of the bgk equation while scheme ii motivated by the asymptotic analysis for the continuous problem utilizes certain properties of the projection operator compared with scheme i scheme ii not only has better computational efficiency the computational cost is reduced by half roughly but also allows the establishment of a formal asymptotic analysis specifically it is demonstrated that when 0 e 1 scheme ii up to o e 2 becomes a local dg discretization with an explicit rk method for the macroscopic compressible navier stokes equations a method in a similar spirit to the ones in bassi and rebay 1997 3 cockburn and shu 1998 16 numerical results are presented for a wide range of knudsen number to illustrate the effectiveness and high order accuracy of the methods

quantization based clustering algorithm in this paper a quantization based clustering algorithm qbca is proposed to cluster a large number of data points efficiently unlike previous clustering algorithms qbca places more emphasis on the computation time of the algorithm specifically qbca first assigns the data points to a set of histogram bins by a quantization function then it determines the initial centers of the clusters according to this point distribution finally qbca performs clustering at the histogram bin level rather than the data point level we also propose two approaches to improve the performance of qbca further i a shrinking process is performed on the histogram bins to reduce the number of distance computations and ii a hierarchical structure is constructed to perform efficient indexing on the histogram bins finally we analyze the performance of qbca theoretically and experimentally and show that the approach 1 can be easily implemented 2 identifies the clusters effectively and 3 outperforms most of the current state of the art clustering approaches in terms of efficiency

coalescence cascades and special function solutions for the continuous and discrete painleve equations we present a systematization of the special function type solutions of continuous and discrete painleve equations our method is to start from and construct its special solutions from the solutions of the hypergeometric equation and by coalescence obtain the lower painleve equations together with their special solutions in the discrete case we study the symmetric one component three point mapping forms of discrete painleve equations starting from d

a fast exact gla based on code vector activity detection this paper introduces a new method for reducing the number of distance calculations in the generalized lloyd algorithm gla which is a widely used method to construct a codebook in vector quantization reduced comparison search detects the activity of the code vectors and utilizes it on the classification of the training vectors for training vectors whose current code vector has not been modified we calculate distances only to the active code vectors a large proportion of the distance calculations can be omitted without sacrificing the optimality of the partition the new method is included in several fast gla variants reducing their running times over 50 on average

rational solutions of the discrete time toda lattice and the alternate discrete painleve ii equation the yablonskii vorob ev polynomials yn t which are defined by a second order bilinear differential difference equation provide rational solutions of the toda lattice they are also polynomial tau functions for the rational solutions of the second painleve equation pii here we define two variable polynomials yn t h on a lattice with spacing h by considering rational solutions of the discrete time toda lattice as introduced by suris these polynomials are shown to have many properties that are analogous to those of the yablonskii vorob ev polynomials to which they reduce when h 0 they also provide rational solutions for a particular discretization of pii namely the so called alternate discrete pii and this connection leads to an expression in terms of the umemura polynomials for the third painleve equation piii it is shown that the backlund transformation for the alternate discrete painleve equation is a symplectic map and the shift in time is also symplectic finally we present a lax pair for the alternate discrete pii which recovers jimbo and miwa s lax pair for pii in the continuum limit h 0

monodromy of the matrix schrodinger equations and darboux transformations a schrodinger operator with a matrix valued rational potential u z is said to have trivial monodromy if all the solutions of the corresponding schrodinger equations are single valued in the complex plane for any a local criterion of this property in terms of the laurent coefficients of the potential u near its singularities which are assumed to be regular is found it is proved that any such operator with a potential vanishing at infinity can be obtained by a matrix analogue of the darboux transformation from the schrodinger operator this generalizes the well known duistermaat grunbaum result to the matrix case and gives the explicit description of the schrodinger operators with trivial monodromy in this case

a class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources in this paper we propose a general time discrete framework to design asymptotic preserving schemes for initial value problem of the boltzmann kinetic and related equations numerically solving these equations are challenging due to the nonlinear stiff collision source terms induced by small mean free or relaxation time we propose to penalize the nonlinear collision term by a bgk type relaxation term which can be solved explicitly even if discretized implicitly in time moreover the bgk type relaxation operator helps to drive the density distribution toward the local maxwellian thus naturally imposes an asymptotic preserving scheme in the euler limit the scheme so designed does not need any nonlinear iterative solver or the use of wild sum it is uniformly stable in terms of the possibly small knudsen number and can capture the macroscopic fluid dynamic euler limit even if the small scale determined by the knudsen number is not numerically resolved it is also consistent to the compressible navier stokes equations if the viscosity and heat conductivity are numerically resolved the method is applicable to many other related problems such as hyperbolic systems with stiff relaxation and high order parabolic equations

a fast k means clustering algorithm using cluster center displacement in this paper we present a fast k means clustering algorithm fkmcucd using the displacements of cluster centers to reject unlikely candidates for a data point the computing time of our proposed algorithm increases linearly with the data dimension d whereas the computational complexity of major available kd tree based algorithms increases exponentially with the value of d theoretical analysis shows that our method can reduce the computational complexity of full search by a factor of sf and sf is independent of vector dimension the experimental results show that compared to full search our proposed method can reduce computational complexity by a factor of 1 37 4 39 using the data set from six real images compared with the filtering algorithm which is among the available best algorithms of k means clustering our algorithm can effectively reduce the computing time it is noted that our proposed algorithm can generate the same clusters as that produced by hard k means clustering the superiority of our method is more remarkable when a larger data set with higher dimension is used

image restoration of compressed image using classified vector quantization abstract to reduce communication bandwidth or storage space image compression is needed however the subjective quality of compressed images may be unacceptable and the improvement of quality for compressed images may be desirable this paper extends and modifies classified vector quantization cvq to improve the quality of compressed images the process consists of two phases the encoding phase and the decoding phase the encoding procedure needs a codebook for the encoder which transforms a compressed image to a set of codeword indices the decoding phase also requires a different codebook for the decoder which enhances a compressed image from a set of codeword indices using cvq to improve a compressed image s quality is different from the existing algorithm which cannot reconstruct the high frequency components for compressed images the experimental results show that the image quality is improved dramatically for images in the training set the improvement of psnr is about 3 db for images which are outside the training set the improvement of psnr is about 0 57 db which is comparable to the existing method

riemann equations in bidifferential calculus we consider equations that formally resemble a matrix riemann or hopf equation in the framework of bidifferential calculus with different choices of a first order bidifferential calculus we obtain a variety of equations including a semi discrete and a fully discrete version of the matrix riemann equation a corresponding universal solution generating method then either yields a continuous or discrete cole hopf transformation or leaves us with the problem of solving riemann equations hence an application of the hodograph method if the bidifferential calculus extends to second order solutions of a system of riemann equations are also solutions of an equation that arises on the universal level of bidifferential calculus as an integrability condition depending on the choice of bidifferential calculus the latter can represent a number of prominent integrable equations like self dual yang mills as well as matrix versions of the two dimensional toda lattice hirota s bilinear difference equation 2 1 dimensional nonlinear schrodinger nls kadomtsev petviashvili kp equation and davey stewartson equations for all of them a recent non isospectral binary darboux transformation result in bidifferential calculus applies which can be specialized to generate solutions of the associated riemann equations for the latter we clarify the relation between these specialized binary darboux transformations and the aforementioned solution generating method from arbitrary size matrix versions of the riemann equations associated with an integrable equation possessing a bidifferential calculus formulation multi soliton type solutions of the latter can be generated this includes breaking multi soliton type solutions of the self dual yang mills and the 2 1 dimensional nls equation which are parametrized by solutions of riemann equations

a generalization of determinant formulae for the solutions of painleve ii and xxxiv equations a generalization of determinant formulae for the classical solutions of painleve xxxiv and painleve ii equations are constructed using the technique of darboux transformation and hirota s bilinear formalism it is shown that the solutions admit determinant formulae even for the transcendental case

a hybrid method for hydrodynamic kinetic flow part i a particle grid method for reducing stochastic noise in kinetic regimes in this work we present a hybrid particle grid monte carlo method for the boltzmann equation which is characterized by a significant reduction of the stochastic noise in the kinetic regime the hybrid method is based on a first order splitting in time to separate the transport from the relaxation step the transport step is solved by a deterministic scheme while a hybrid dsmc based method is used to solve the collision step such a hybrid scheme is based on splitting the solution in a collisional and a non collisional part at the beginning of the collision step and the dsmc method is used to solve the relaxation step for the collisional part of the solution only this is accomplished by sampling only the fraction of particles candidate for collisions from the collisional part of the solution performing collisions as in a standard dsmc method and then projecting the particles back onto a velocity grid to compute a piecewise constant reconstruction for the collisional part of the solution the latter is added to a piecewise constant reconstruction of the non collisional part of the solution which in fact remains unchanged during the relaxation step numerical results show that the stochastic noise is significantly reduced at large knudsen numbers with respect to the standard dsmc method indeed in this algorithm the particle scheme is applied only on the collisional part of the solution so only this fraction of the solution is affected by stochastic fluctuations but since the collisional part of the solution reduces as the knudsen number increases stochastic noise reduces as well at large knudsen numbers

universal characters and an extension of the kp hierarchy the universal character is a polynomial attached to a pair of partitions and is a generalization of the schur polynomial in this paper we define vertex operators which play roles of raising operators for the universal character by means of the vertex operators we obtain a series of non linear partial differential equations of infinite order called the uc hierarchy we regard it as an extension of the kp hierarchy we investigate also solutions of the uc hierarchy the totality of the space of solutions forms a direct product of two infinite dimensional grassmann manifolds and its infinitesimal transformations are described in terms of the lie algebra

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

a consistent bgk type model for gas mixtures we introduce a relaxation collision operator for a mixture of gases which satisfies several fundamental properties different bgk type collision operators for gas mixtures have been introduced earlier but none of them could satisfy all the basic physical properties positivity correct exchange coefficients entropy inequality indifferentiability principle we show that all those properties are verified for our model and we derive its navier stokes limit by a chapman enskog expansion

multivariate orthogonal polynomials and integrable systems multivariate orthogonal polynomials in d real dimensions are considered from the perspective of the cholesky factorization of a moment matrix the approach allows for the construction of corresponding multivariate orthogonal polynomials associated second kind functions jacobi type matrices and associated three term relations and also christoffel darboux formulae the multivariate orthogonal polynomials their second kind functions and the corresponding christoffel darboux kernels are shown to be quasi determinants as well as schur complements of bordered truncations of the moment matrix quasi tau functions are introduced it is proven that the second kind functions are multivariate cauchy transforms of the multivariate orthogonal polynomials discrete and continuous deformations of the measure lead to toda type integrable hierarchy being the corresponding flows described through lax and zakharov shabat equations bilinear equations are found varying size matrix nonlinear partial difference and differential equations of the 2d toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials the discrete flows which are shown to be connected with a gauss borel factorization of the jacobi type matrices and its quasi determinants lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi tau matrices which generalize to the multidimensional realm those that relate the baker and adjoint baker functions to ratios of miwa shifted tau functions in the 1d scenario in this context the multivariate extension of the elementary darboux transformation is given in terms of quasi determinants of matrices built up by the evaluation at a poised set of nodes lying in an appropriate hyperplane in r d of the multivariate orthogonal polynomials the multivariate christoffel formula for the iteration of m elementary darboux transformations is given as a quasi determinant it is shown using congruences in the space of semi infinite matrices that the discrete and continuous flows are intimately connected and determine nonlinear partial difference differential equations that involve only one site in the integrable lattice behaving as a kadomstev petviashvili type system finally a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given in particular it is shown that the toda times that preserve the invariance condition lay in a secant variety of the veronese variety of the fixed point set of the linear isometry

non commutative q painleve vi equation by applying suitable centrality conditions to non commutative non isospectral lattice modified gel fand dikii type systems we obtain the corresponding non autonomous equations then we derive the non commutative q discrete painleve vi equation with full range of parameters as the 2 2 similarity reduction of the non commutative non isospectral and non autonomous lattice modified korteweg de vries equation we also comment on the fact that in making the analogous reduction starting from the schwarzian korteweg de vries equation no such non isospectral generalization is needed

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

uniformly stable numerical schemes for the boltzmann equation preserving the compressible navier stokes asymptotics in this paper we develop a numerical method to solve boltzmann like equations of kinetic theory which is able to capture the compressible navier stokes dynamics at small knudsen numbers our approach is based on the micro macro decomposition technique which applies to general collision operators this decomposition is performed in all the phase space and leads to an equivalent formulation of the boltzmann or bgk equation that couples a kinetic equation with macroscopic ones this new formulation is then discretized with a semi implicit time scheme combined with a staggered grid space discretization finally several numerical tests are presented in order to illustrate the efficiency of our approach incidentally we also introduce in this paper a modification of a standard splitting method that allows to preserve the compressible navier stokes asymptotics in the case of the simplified bgk model up to our knowledge this property is not known for general collision operators

rarefied flow computations using nonlinear model boltzmann equations high resolution finite difference schemes for solving the nonlinear model boltzmann equations are presented for the computations of rarefied gas flows the discrete ordinate method is first applied to remove the velocity space dependency of the distribution function which renders the model boltzmann equation in phase space to a set of hyperbolic conservation laws with source terms in physical space then a high order essentially nonoscillatory method due to harten et al j comput phys 71 231 1987 is adapted and extended to solve them explicit methods using operator splitting and implicit methods using the lower upper factorization are described to treat multidimensional problems the methods are tested for both steady and unsteady rarefied gas flows to illustrate its potential use the computed results using model boltzmann equations are found to compare well both with those using the direct simulation monte carlo results in the transitional regime flows and those with the continuum navier stokes calculations in near continuum regime flows

a unified gas kinetic scheme for continuum and rarefied flows iv fluid dynamic equations are valid in their respective modeling scales such as the particle mean free path scale of the boltzmann equation and the hydrodynamic scale of the navier stokes ns equations with a variation of the modeling scales theoretically there should have a continuous spectrum of fluid dynamic equations even though the boltzmann equation is claimed to be valid in all scales many boltzmann solvers including direct simulation monte carlo method require the cell resolution to the order of particle mean free path scale therefore they are still single scale methods in order to study multiscale flow evolution efficiently the dynamics in the computational fluid has to be changed with the scales a direct modeling of flow physics with a changeable scale may become an appropriate approach the unified gas kinetic scheme ugks is a direct modeling method in the mesh size scale and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path the cell size of ugks is not limited by the particle mean free path with the variation of the ratio between the numerical cell size and local particle mean free path the ugks recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale the previous ugks is mostly constructed from the evolution solution of kinetic model equations even though the ugks is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time it still has space to develop more accurate flow solver in the region where the time step is comparable with the local particle mean free time in such a scale there is dynamic difference from the full boltzmann collision term and the model equations this work is about the further development of the ugks with the implementation of the full boltzmann collision term in the region where it is needed the central ingredient of the ugks is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled the newly developed ugks has the asymptotic preserving ap property of recovering the ns solutions in the continuum flow regime and the full boltzmann solution in the rarefied regime in the mostly unexplored transition regime the ugks itself provides a valuable tool for the non equilibrium flow study the mathematical properties of the scheme such as stability accuracy and the asymptotic preserving will be analyzed in this paper as well

numerical solution of the boltzmann equation i spectrally accurate approximation of the collision operator in this paper we show that the use of spectral galerkin methods for the approximation of the boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost we prove that the spectral algorithm preserves the total mass and approximates with infinite order accuracy momentum and energy consistency of the method is also proved and a stability result for a smoothed positive scheme is given we demonstrate that the fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method

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

matrix toda and volterra lattices abstract we consider matrix toda and volterra lattice equations and their relation with matrix biorthogonal polynomials from that relation we give a method for constructing a new solution of these systems from another given one an illustrative example is presented

casorati determinant solutions for the discrete painleve iii equation the discrete painleve iii equation is investigated based on the bilinear formalism it is shown that it admits the solutions expressed by the casorati determinant whose entries are given by the discrete bessel functions moreover based on the observation that these discrete bessel functions are transformed to the q bessel functions by a simple variable transformation we present a q difference analog of the painleve iii equation

darboux transformation and exact multisolitons of cpn nonlinear sigma model abstract exact multisolitons of c p n sigma model are obtained by using darboux transformation in terms of quasideterminant darboux matrix the multisoliton solutions are shown to be expressed in terms of quasideterminants of gelfand and retakh 1991 10 the method of quasideterminant darboux matrix has been shown to be related with the dressing method that has already been used to compute iteratively exact multisolitons of the c p n sigma model by means of nonlinear superposition principle of backlund transformations explicit expressions of one two and three soliton solutions of c p 1 model have been computed by using properties of quasideterminants

implicit gas kinetic unified algorithm based on multi block docking grid for multi body reentry flows covering all flow regimes based on the previous researches of the gas kinetic unified algorithm gkua for flows from highly rarefied free molecule transition to continuum a new implicit scheme of cell centered finite volume method is presented for directly solving the unified boltzmann model equation covering various flow regimes in view of the difficulty in generating the single block grid system with high quality for complex irregular bodies a multi block docking grid generation method is designed on the basis of data transmission between blocks and the data structure is constructed for processing arbitrary connection relations between blocks with high efficiency and reliability as a result the gas kinetic unified algorithm with the implicit scheme and multi block docking grid has been firstly established and used to solve the reentry flow problems around the multi bodies covering all flow regimes with the whole range of knudsen numbers from 10 to 3 7e6 the implicit and explicit schemes are applied to computing and analyzing the supersonic flows in near continuum and continuum regimes around a circular cylinder with careful comparison each other it is shown that the present algorithm and modelling possess much higher computational efficiency and faster converging properties the flow problems including two and three side by side cylinders are simulated from highly rarefied to near continuum flow regimes and the present computed results are found in good agreement with the related dsmc simulation and theoretical analysis solutions which verify the good accuracy and reliability of the present method it is observed that the spacing of the multi body is smaller the cylindrical throat obstruction is greater with the flow field of single body asymmetrical more obviously and the normal force coefficient bigger while in the near continuum transitional flow regime of near space flying surroundings the spacing of the multi body increases to six times of the diameter of the single body the interference effects of the multi bodies tend to be negligible the computing practice has confirmed that it is feasible for the present method to compute the aerodynamics and reveal flow mechanism around complex multi body vehicles covering all flow regimes from the gas kinetic point of view of solving the unified boltzmann model velocity distribution function equation

global existence to the bgk model of boltzmann equation abstract we present an existence and a stability proof for solutions to the bgk model of boltzmann equation f v x f f m f t 0 x r n v r n m f p 2 t n 2 exp v u 2 2t p pu p u 2 pt 1 x r n 1 v v 2 f t x v dv it relies on the strong compactness of u t and on a new estimate on the third moment of v dv we also prove the entropy relation for 1

backlund transformations and the atiyah ward ansatz for non commutative anti self dual yang mills equations we present backlund transformations for the non commutative anti self dual yang mills equation where the gauge group is g gl 2 and use it to generate a series of exact solutions from a simple seed solution the solutions generated by this approach are represented in terms of quasi determinants we also explain the origins of all the ingredients of the backlund transformations within the framework of non commutative twistor theory in particular we show that the generated solutions belong to a non commutative version of the atiyah ward ansatz

numerical passage from kinetic to fluid equations a numerical way to pass from the bhatnagar gross krook model of the boltzmann equation to compressible euler equations is presented in order to do so a stable discretization of the kinetic equati

speeding up fuzzy c means using a hierarchical data organisation to control the precision of membership calculation we examine the run time behaviour of conventional fuzzy c means implementations investigating into fcm termination conditions and membership update equations we derive an approximative fcm that yields the same results as a conventional implementation within a given precision we incorporate additional information about the data set by reorganizing the set as a tree our modification leads to an fcm algorithm with a significantly different run time behaviour the gain of using the modified implementation increases with an increasing number of data objects and especially an increasing number of clusters but is also sensitive to the chosen fuzzifier

determinant structure of the rational solutions for the painleve iv equation rational solutions for the painleve iv equation are investigated by hirota bilinear formalism it is shown that the solutions in one hierarchy are expressed by 3 reduced schur functions and those in another two hierarchies by casorati determinants of the hermite polynomials or by a special case of the schur polynomials

functional representations of integrable hierarchies we consider a general framework for integrable hierarchies in lax form and derive certain universal equations from which functional representations of particular hierarchies such as kp discrete kp mkp akns i e formulations in terms of functional equations are systematically and quite easily obtained the formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative typically matrix algebra the obtained functional representations can be understood as noncommutative analogues of fay identities for the kp hierarchy

fredholm determinants and pole free solutions to the noncommutative painleve ii equation we extend the formalism of integrable operators a la its izergin korepin slavnov to matrix valued convolution operators on a semi infinite interval and to matrix integral operators with a kernel of the form frac e 1 t lambda e 2 mu lambda mu thus proving that their resolvent operators can be expressed in terms of solutions of some specific riemann hilbert problems we also describe some applications mainly to a noncommutative version of painleve ii recently introduced by retakh and rubtsov and a related noncommutative equation of painleve type we construct a particular family of solutions of the noncommutative painleve ii that are pole free for real values of the variables and hence analogous to the hastings mcleod solution of commutative painleve ii such a solution plays the same role as its commutative counterpart relative to the tracy widom theorem but for the computation of the fredholm determinant of a matrix version of the airy kernel

determinant structure of the rational solutions for the painleve ii equation two types of determinant representations of the rational solutions for the painleve ii equation are discussed by using the bilinear formalism one of them is a representation by the devisme polynomials and another one is hankel determinant representation they are derived from the determinant solutions of the kp hierarchy and toda lattice respectively

a smooth transition model between kinetic and hydrodynamic equations this paper presents a model which provides a smooth transition between a kinetic and a hydrodynamic domain the idea is to use a buffer zone in which both hydrodynamics and kinetic equations will be solved the solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations we use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone thus no boundary condition is needed at the transition point consequently this model avoids the delicate issue of finding the interface condition in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations a simple kinetic scheme is developed to discretize our model and numerical examples are used to validate the method

darboux transformation of the generalized coupled dispersionless integrable system the darboux transformation on matrix solutions to the generalized coupled dispersionless integrable system based on a non abelian lie group is studied and the solutions are shown to be expressed in terms of quasideterminants as an explicit example the darboux transformation on scalar solutions to the system based on the lie group su 2 is discussed in detail and the solutions are shown to be expressed as ratios of determinants

quasideterminant solutions of a non abelian hirota miwa equation a non abelian version of the hirota miwa equation is considered in an earlier paper of nimmo 2006 j phys a math gen 39 5053 65 it was shown how solutions expressed as quasideterminants could be constructed for this system by means of darboux transformations in this paper we discuss these solutions from a different perspective and show that the solutions are quasi plucker coordinates and that the non abelian hirota miwa equation may be written as a quasi plucker relation the special case of the matrix hirota miwa equation is also considered using a more traditional bilinear approach and the techniques are compared

on a non abelian hirota miwa equation a generalization of the hirota miwa equation to an abstract non abelian associative algebra is considered this system is integrable in the sense that it arises as compatibility condition for a linear system and has solutions constructed by means of the application of an arbitrary number of darboux transformations these solutions are in general expressed in terms of quasideterminants

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

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

special polynomials associated with rational solutions of the fifth painleve equation in this paper special polynomials associated with rational and algebraic solutions of the fifth painleve equation pv are studied these special polynomials defined by second order bilinear differential difference equations which are equivalent to toda equations the structure of the zeroes of these special polynomials which involve a parameter is investigated and it is shown that these have an intriguing symmetric and regular structure for large negative values of the parameter the zeroes have an approximate triangular structure as the parameter increases the zeroes coalesce for certain values and eventually for large positive values of the parameter the zeroes also have an approximate triangular structure though with the orientation reversed in fact the interaction of the zeroes is solitonic in nature since the same pattern reappears with its orientation reversed

the third painleve equation and associated special polynomials in this paper we are concerned with rational solutions algebraic solutions and associated special polynomials with these solutions for the third painleve equation piii these rational and algebraic solutions of piii are expressible in terms of special polynomials defined by second order bilinear differential difference equations which are equivalent to toda equations the structure of the roots of these special polynomials is studied and it is shown that these have an intriguing highly symmetric and regular structure using the hamiltonian theory for piii it is shown that these special polynomials satisfy pure difference equations fourth order bilinear differential equations as well as differential difference equations further representations of the associated rational solutions in the form of determinants through schur functions are given

a hybrid kinetic fluid model for solving the gas dynamics boltzmann bgk equation our purpose is to derive a hybrid model for particle systems which combines a kinetic description of the fast particles with a fluid description of the thermal ones in the present work fast particles will be described through a collisional kinetic equation of boltzmann bgk type while thermal particles will be modeled by means of a system of euler type equations then we construct a numerical scheme for this model this scheme satisfies exact conservation properties we validate the approach by presenting various numerical tests

discrete velocity models and numerical schemes for the boltzmann bgk equation in plane and axisymmetric geometries we present new numerical models for computing transitional or rarefied gas flows as described by the boltzmann bgk and bgk es equations we first propose a new discrete velocity model based on the entropy minimization principle this model satisfies the conservation laws and the entropy dissipation moreover the problem of conservation and entropy for axisymmetric flows is investigated we find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator then we propose some models that satisfy these constraints owing to these properties we obtain numerical schemes that are economic in terms of discretization and robust in particular we develop a linearized implicit scheme for computing stationary solutions of the discrete velocity bgk and bgk es models this scheme is the basis of a code which can compute high altitude hypersonic flows in 2d plane and axisymmetric geometries our results are analyzed and compared to other methods

a hierarchical uniformly high order dg imex scheme for the 1d bgk equation abstract a class of high order nodal discontinuous galerkin implicit explicit dg imex schemes with asymptotic preserving ap property has been developed for the one dimensional 1d bgk equation in xiong et al 2015 40 based on a micro macro reformulation the schemes are globally stiffly accurate and asymptotically consistent and as the knudsen number becomes small or goes to zero they recover first the compressible navier stokes cns and then the euler limit motivated by the recent work of filbet and rey 2015 27 and the references therein in this paper we propose a hierarchical high order ap method namely kinetic cns and euler solvers are automatically applied in regions where their corresponding models are appropriate the numerical solvers for different regimes are coupled naturally by interface conditions to the best of our knowledge the resulting scheme is the very first hierarchical one being proposed in the literature that enjoys ap property as well as uniform high order accuracy numerical experiments demonstrate the efficiency and effectiveness of the proposed approach as time evolves three different regimes are dynamically identified and naturally coupled leading to significant cpu time savings more than 80 for some of our test problems

bilinear discrete painleve ii and its particular solutions by analogy to the continuous painleve ii equation we present particular solutions of the discrete painleve ii d pii equation these solutions are of a rational and special function airy type our analysis is based on the bilinear formalism that allows us to obtain the tau function for d pii two different forms of bilinear d pii are obtained and we show that they can be related by a simple gauge transformation