renormalization group analysis of differential equations subject to slowly modulated perturbations abstract the application of renormalization group rg theory to the asymptotic analysis of differential equations is considered it is found that there is a class of small structural perturbations whose effects cannot be systematically treated using the gell mann low rg approach applied in this context guided by a reinterpretation of the calculational procedure employed in this approach whose motivation is provided by the unnecessarily complicated nature of its standard interpretation we formulate a generalized perturbative analysis and an rg approach which naturally and systematically treat equations subject to perturbations of this class this formulation of rg theory is demonstrated with a number of examples for which the gell mann low formulation fails to provide a systematic theoretical framework for one representative example it is found that the wilson rg formulation also fails the implications of this failure are discussed

travelling waves of auto catalytic chemical reaction of general order an elliptic approach abstract in this paper we study the existence and non existence of travelling wave to parabolic system of the form a t a x x a f b b t d b x x a f b with f a degenerate nonlinearity in the context of an auto catalytic chemical reaction a is the density of a chemical species called reactant a b that of another chemical species b called auto catalyst and d d b d a 0 is the ratio of diffusion coefficients d b of b and d a of a respectively such a system also arises from isothermal combustion the nonlinearity is called degenerate since f 0 f 0 0 one case of interest in this article is the propagating wave fronts in an isothermal auto catalytic chemical reaction of order n a n b n 1 b with 1 n 2 and d 1 due to different molecular weights and or sizes of a and b the resulting nonlinearity is f b b n explicit bounds v and v that depend on d are derived such that there is a unique travelling wave of every speed v v and there does not exist any travelling wave of speed v v new to the literature it is shown that v v d when d 1 furthermore when d 1 it is shown rigorously that there exists a v min such that there is a travelling wave of speed v if and only if v v min estimates on v min improve significantly that of early works another case in which two different orders of isothermal auto catalytic chemical reactions are involved is also studied with interesting new results proved

stability of moving fronts in the ginzburg landau equation we use renormalization group ideas to study stability of moving fronts in the ginzburg landau equation in one spatial dimension in particular we prove stability of the real fronts under complex perturbations this extends the results of aronson and weinberger to situations where the maximum principle is inapplicable and constitutes a step in proving the general marginal stability hypothesis for the ginzburg landau equation

a method for training finite mixture models under a fuzzy clustering principle in this paper we establish a novel regard towards fuzzy clustering showing it provides a sound framework for fitting finite mixture models we propose a novel fuzzy clustering type methodology for finite mixture model fitting effected by utilizing a regularized form of the fuzzy c means fcm algorithm and introducing a proper dissimilarity functional for the algorithm with respect to the probabilistic properties of the model being treated we apply the proposed methodology in a number of popular finite mixture models and the corresponding expressions of the fuzzy model fitting algorithm are derived we examine the efficacy of our novel approach in both clustering and classification applications of benchmark data sets and we demonstrate the advantages of the proposed approach over maximum likelihood

a convergence proof for bird s direct simulation monte carlo method for the boltzmann equation bird s direct simulation monte carlo method for the boltzmann equation is considered the limit as the number of particles tends to infinity of the random empirical measures associated with the bird algorithm is shown to be a deterministic measure valued function satisfying an equation close in a certain sense to the boltzmann equation a markov jump process is introduced which is related to bird s collision simulation procedure via a random time transformation convergence is established for the markov process and the random time transformation these results together with some general properties concerning the convergence of random measures make it possible to characterize the limiting behavior of the bird algorithm

coarse variables of autonomous ode systems and their evolution abstract given an autonomous system of ordinary differential equations ode we consider developing practical models for the deterministic slow coarse behavior of the ode system two types of coarse variables are considered the first type consists of running finite time averages of phase functions approaches to construct the coarse evolution equation for this type are discussed and implemented on a forced lorenz system and a singularly perturbed system whose fast flow does not necessarily converge to an equilibrium we explore two strategies in one we compute locally invariant manifolds of the fast dynamics parameterized by the slow variables in the other the choice of our coarse variables automatically guarantees them to be slow in a precise sense this allows their evolution to be phrased in terms of averaging utilizing limit measures probability distributions of the fast flow coarse evolution equations are constructed based on these approaches and tested against coarse response of the microscopic models the second type of coarse variables are defined as non trivial scalar state functions that are required by design to evolve autonomously to the extent possible with the goal of being candidate state functions for unambiguously initializable coarse dynamics the question motivates a mathematical restatement in terms of a first order pde a computational approximation is developed and tested on the lorenz system and the hald hamiltonian system

pattern propagation in nonlinear dissipative systems abstract we discuss the problem of pattern selection in situations where a stable nonuniform state of a nonlinear dissipative system propagates into an initially unstable homogeneous region our strategy is to consider this process as a generalization of front propagation in a nonlinear diffusion problem for which rigorous results are known and we point out that these known properties are consistent with a marginal stability hypothesis that has been suggested in the theory of dendritic crystal growth we then describe a more general interpretation of the marginal stability hypothesis and finally present numerical evidence for its validity from three different pattern forming models

methods of centre manifold and multiple scales in the theory of weakly nonlinear stability for fluid motions the equivalence between the method of centre manifold and the method of multiple scales was examined when they are applied to the weakly nonlinear stability analysis for fluid motions the relationship between the landau equations of the seventh order which were reduced from the navier stokes equations by applying these two methods is clarified how we should define the disturbance amplitude to obtain the complete equivalence between them is discussed

on the turing patterns in one dimensional gradient skew gradient dissipative systems in this article fundamental properties concerning the turing patterns are considered in one dimensional dissipative systems with gradient skew gradient structure introduced in m kuwamura and e yanagida phys d 175 2003 pp 185 195 it is a natural extension of free energy which covers reaction diffusion systems of activator inhibitor type the theory based on this concept provides a new perspective on a fundamental problem of what unique turing pattern is to be selected among many

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

a random discrete velocity model and approximation of the boltzmann equation an approximation procedure for the boltzmann equation based on random choices of collision pairs from a fixed velocity set and on discrete velocity models is designed in a suitable limit the procedure is shown to converge to the time discretized and spatially homogeneous boltzmann equation

l 1 stability of multi dimensional discrete velocity boltzmann equations we study the l 1 stability of multi dimensional discrete velocity boltzmann equations under suitable smallness assumption on initial data we show that bounded mild solutions are l 1 stable for a stability estimate we employ bony s multi dimensional analysis for total interactions over characteristic planes

renormalisation group theory for two problems in linear continuum mechanics abstract we consider the problems of 1 the stress field in an infinite wedge when a moment is applied at the tip and 2 the flow of an inviscid incompressible fluid past an infinite wedge both problems are linear but nevertheless exhibit anomalous dimensions for wedge angles larger than a critical value c in 1 it is the stress field and in 2 it is the velocity potential which can have anomalous power law behaviour in distance from the tip we use these problems as simple examples to illustrate how partial differential equations can be solved using the renormalisation group

invariant manifolds in mechanics ii zero dimensional elastic bodies with directors application de la methode des varietes invariantes a un cas concret etablissement a partir de l elasticite tridimensionnelle d une theorie des corps de dimension 0 avec trois directeurs

travelling kinks in schlogl s second model for non equilibrium phase transitions using the separation technique developed by osborne and stuart 1980 travelling wave solutions are written down for the nonlinear reaction diffusion equation of schlogl s second model assuming that the necessary conditions for the static phase coexistence are satisfied

speed selection for traveling wave solutions to the diffusion reaction equation with cubic reaction term and burgers nonlinear convection the problem of traveling wave tw speed selection for solutions to a generalized murray burgers kpp fisher parabolic equation with a strictly positive cubic reaction term is considered theoretically and the initial boundary value problem is numerically solved in order to support obtained analytical results depending on the magnitude of a parameter inherent in the reaction term i the term is either a concave function or a function with the inflection point and ii transition from pulled to pushed tw solution occurs due to interplay of two nonlinear terms the reaction term and the burgers convection term explicit pushed tw solutions are derived it is shown that physically observable tw solutions i e solutions obtained by solving the initial boundary value problem with a sufficiently steep initial condition can be determined by seeking the tw solution characterized by the maximum decay rate at its leading edge in the appendix the developed approach is applied to a non linear diffusion reaction equation that is widely used to model premixed turbulent combustion

propagating fronts and the center manifold theorem we prove the existence of propagating front solutions for the swift hohenberg equation sh using the center manifold theorem we reduce the problem to a two dimensional system of ordinary differential equations they describe stationary solutions and front solutions of the partial differential equation sh we identify the well known amplitude equation as the lowest order approximation to the equation of motion on the center manifold

renormalization group and the ginzburg landau equation we use renormalization group methods to prove detailed long time asymptotics for the solutions of the ginzburg landau equations with initial data approaching asx different spiraling stationary solutions a universal pattern is formed depending only on this asymptotics at spatial infinity

existence of traveling waves of invasion for ginzburg landau type problems in infinite cylinders we study a class of systems of reaction diffusion equations in infinite cylinders which arise within the context of ginzburg landau theories and describe the kinetics of phase transformation in second order or weakly first order phase transitions with non conserved order parameters we use a variational characterization to study the existence of a special class of traveling wave solutions which are characterized by a fast exponential decay in the direction of propagation our main result is a simple verifiable criterion for existence of these traveling waves under the very general assumptions of non linearities we also prove boundedness regularity and some other properties of the obtained solutions as well as several sufficient conditions for existence or non existence of such traveling waves and give rigorous upper and lower bounds for their speed in addition we prove that the speed of the obtained solutions gives a sharp upper bound for the propagation speed of a class of disturbances which are initially sufficiently localized we give a sample application of our results using a computer assisted approach

approximation of the boltzmann equation by discrete velocity models two convergence results related to the approximation of the boltzmann equation by discrete velocity models are presented first we construct a sequence of deterministic discrete velocity models and prove convergence as the number of discrete velocities tends to infinity of their solutions to the solution of a spatially homogeneous boltzmann equation second we introduce a sequence of markov jump processes interpreted as random discrete velocity models and prove convergence as the intensity of jumps tends to infinity of these processes to the solution of a deterministic discrete velocity model

existence and nonexistence of travelling waves of a reaction diffusion system abstract in this paper we study the existence and nonexistence of travelling waves of the reaction diffusion system a t a x x e a x a b k a b q b t d b x x d e b x a b k a b q where a b are concentration densities of the two chemical species d 0 is the ratio of diffusion coefficients of the two chemical species involved e is the strength of the electric field applied k 0 is a parameter and q 1 the system arises from the study of two dimensional iodate arsenous acid iaa reaction with a constant electric field applied iaa reaction is very much studied in experiments and demonstrates the interesting phenomenon of instability driven by buoyancy of travelling waves by using a novel approach explicit bounds c d e and c d e are derived such that there is a unique travelling wave of every speed c c d e and there does not exist any travelling wave of speed c c d e with q being fixed in addition c d e bears a close resemblance of the minimum travelling wave speed of the classical case of d 1 and e 0 when 0 e 1 and d 1

discrete velocity models of the boltzmann equation a survey on the mathematical aspects of the theory discrete velocity models of the boltzmann equation are of considerable conceptual interest in the kinetic theory of gases and at the same time a fascinating mathematical subject the last decade has seen a lot of work on this subject which is reviewed here we put an emphasis on mathematical aspects in particular the global existence question qualitative properties of solutions explicit solutions and the fluid dynamical limit the mechanical background is discussed more concisely in part i numerous contributions concerning other questions are outlined we try to give a complete reference list up to and including 1985

pushed global modes in weakly inhomogeneous subcritical flows abstract a new type of nonlinear global mode or fully nonlinear synchronized solution arising in the dynamics of open shear flows which behave as oscillators annu rev fluid mech 22 1990 473 and in optical parametric oscillators j opt soc am b 17 2000 997 is identified in the context of the real subcritical ginzburg landau equation with slowly varying coefficients the nonlinear global modes satisfy a boundary condition accounting for the inlet of the flow we show that the spatial structure of these new nonlinear global modes consists of a localized state limited by a pushed upstream front that withstand the mean advection and a fast return to zero downstream achieved either by a second stationary front facing backward or by a saddle node bifurcation driven by the non parallelism of the flow we derive scaling laws for the slope of the nonlinear global modes at the inlet and for the position of the maximum amplitude which are in agreement with similar scaling obtained in experiments with a shear layer in a hele shaw cell phys rev lett 82 1999 1442 extension to the complex ginzburg landau equation is discussed in a large region of parameters the nonlinear global modes have the same spatial structure as in the real case and oscillate at a global frequency selected at threshold by the pushed front in parameters region where these pushed global modes are not selected new states with a non periodic in time behavior are exhibited

a holistic finite difference approach models linear dynamics consistently i prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small using such tools of dynamical systems theory gives new assurances about the quality of finite difference models under nonlinear and other perturbations on grids with finite spacing for example the linear advection diffusion equation is found to be stably modelled for all advection speeds and all grid spacings the theorems establish an extremely good form for the artificial internal boundary conditions that need to be introduced to apply centre manifold theory when numerically solving nonlinear partial differential equations this approach can be used to systematically derive finite difference models which automatically have excellent characteristics their good performance for finite grid spacing implies that fewer grid points may be used and consequently there will be less difficulties with stiff rapidly decaying modes in continuum problems

initial conditions for models of dynamical systems abstract the long time behaviour of many dynamical systems may be effectively predicted by a low dimensional model that describes the evolution of a reduced set of variables we consider the question of how to equip such a low dimensional model with appropriate initial conditions so that it faithfully reproduces the long term behaviour of the orginal high dimensional dynamical system our method involves putting the dynamical system into normal form which not only generates the low dimensional model but also provides the correct initital conditions for the model we illustrate the method with several examples

instabilities and propagation properties in a fourth order reaction diffusion equation in this paper we investigate the instability and the propagation properties of a class of reaction diffusion equations of fourth order two examples are introduced the extended fisher kolmogorov equation efk and the swift hohenberg equation sh both have been studied before by related methods see for example peletier and rottschafer 2004 19 van saarloos 2003 24 but the analysis here will support the introduced linear mechanism in front selection these two equations support a patterned front solutions and the double eigenvalue mechanism is used to provide evidence for that and to determine a minimal front speed

galerkin petrov approach for the boltzmann equation abstract in this work we propose a new galerkin petrov method for the numerical solution of the classical spatially homogeneous boltzmann equation this method is based on an approximation of the distribution function by associated laguerre polynomials and spherical harmonics and test in a variational manner with globally defined three dimensional polynomials a numerical realisation of the algorithm is presented the algorithmic developments are illustrated with the help of several numerical tests

mixtures of probabilistic principal component analyzers principal component analysis pca is one of the most popular techniques for processing compressing and visualizing data although its effectiveness is limited by its global linearity while nonlinear variants of pca have been proposed an alternative paradigm is to capture data complexity by a combination of local linear pca projections however conventional pca does not correspond to a probability density and so there is no unique way to combine pca models therefore previous attempts to formulate mixture models for pca have been ad hoc to some extent in this article pca is formulated within a maximum likelihood framework based on a specific form of gaussian latent variable model this leads to a well defined mixture model for probabilistic principal component analyzers whose parameters can be determined using an expectationmaximization algorithm we discuss the advantages of this model in the context of clustering density modeling and local dimensionality reduction and we demonstrate its application to image compression and handwritten digit recognition

nonlinear functionals of multi d discrete velocity boltzmann equations in this paper we study nonlinear functionals measuring potential interactions and l1 distance between two mild solutions for the multi dimensional discrete velocity boltzmann equations when the initial data are a small perturbation of a vacuum we employ bony s dispersion estimates to show that these functionals satisfy lyapunov type estimates which are useful for the study of time asymptotics and l1 stability of mild solutions

the application of centre manifolds to amplitude expansions i ordinary differential equations abstract the theory of centre manifolds for a system of ordinary differential equations is summarized and its relationship to amplitude expansions based upon multiple time scales is discussed emphasis is placed upon the practical computational aspects of applying centre manifold theory to near critical problems and in particular to the computation of the centre manifold the calculations are illustrated by a detailed analysis of two problems in fluid mechanics

invariant manifolds in mechanics i the general construction of coarse theories from fine theories la theorie dite fine fournit potentiellement des informations tres detaillees sur le comportement du materiau et generalement plus d informations que necessaire tandis que la theorie dite grossiere donne des informations brutes plus accessibles qui peuvent etre suffisantes dans des circonstances donnees presentation d une methode permettant de deduire les theories grossieres des theories fines

renormalization group derivation of phase equations abstract phase equations describing the evolution of large scale modulation of spatially periodic patterns in two dimensional systems are derived by employing the renormalization group method a general formula for phase diffusion coefficients is given under certain conditions

from a gaussian mixture model to nonadditive fuzzy systems this work explores how a kind of probabilistic system namely the gaussian mixture model gmm can be translated to an additive fuzzy system we will prove the mathematical equivalence between the conditional mean of a gmm and the defuzzified output of a generalized fuzzy model gfm the relationship between a gmm and a gfm and the conditions for gmm to gfm translation will be made explicit in the form of theorems the work will then extend to special cases of the gfm specifically the mamdani larsen and takagi sugeno fuzzy models the possibility of reverse translation that is from a gfm to a gmm will also be discussed finally we will consider the generality of a gmm specifically how it can approximate other distribution functions

travelling waves in radially symmetric parabolic systems on montre que toutes les solutions ondes planes de propagation ou stationnaire de certains systemes paraboliques a symetrie radiale peuvent etre trouvees en resolvant une equation differentielle scalaire associee

speed of wave front solutions to hyperbolic reaction diffusion equations the asymptotic speed problem of front solutions to hyperbolic reaction diffusion hrd equations is studied in detail we perform linear and variational analyses to obtain bounds for the speed in contrast to what has been done in previous work here we derive upper bounds in addition to lower ones in such a way that we can obtain improved bounds for some functions it is possible to determine the speed without any uncertainty this is also achieved for some systems of hrd i e time delayed lotka volterra equations that take into account the interaction among different species an analytical analysis is performed for several systems of biological interest and we find good agreement with the results of numerical simulations as well as with available observations for a system discussed recently s1063 651x 99 06211 x

the application of centre manifolds to amplitude expansions ii infinite dimensional problems abstract the theory of centre manifolds for infinite dimensional systems is described with emphasis on the practical computational aspects of applying the theory to near critical problems and in particular to computation of the centre manifold the calculations are illustrated by detailed analyses of two specific problems

fuzzy semi supervised co clustering for text documents in this paper we propose a new heuristic semi supervised fuzzy co clustering algorithm ss hfcr for categorization of large web documents in this approach the clustering process is carried out by incorporating some prior knowledge in the form of pair wise constraints provided by users into the fuzzy co clustering framework each constraint specifies whether a pair of documents must or cannot be clustered together moreover we formulate the competitive agglomeration cost function which is also able to make use of prior knowledge in the clustering process the experimental studies on a number of large benchmark datasets demonstrate the strength and potentials of ss hfcr in terms of accuracy stability and efficiency compared with some of the recent popular semi supervised clustering approaches

selection stability and renormalization abstract we illustrate how to extend the concept of structural stability through applying it to the front propagation speed selection problem this consideration leads us to a renormalization group study of the problem the study illustrates two very general conclusions 1 singular perturbations in applied mathematics are best understood as renormalized perturbation methods and 2 amplitude equations are renormalization group equations

fuzzy gaussian mixture models in this paper in order to improve both the performance and the efficiency of the conventional gaussian mixture models gmms generalized gmms are firstly introduced by integrating the conventional gmms and the active curve axis gmms for fitting non linear datasets and then two types of fuzzy gaussian mixture models fgmms with a faster convergence process are proposed based on the generalized gmms inspired from the mechanism of fuzzy c means fcms which introduces the degree of fuzziness on the dissimilarity function based on distances one is named as probability based fgmms defining the dissimilarity as the multiplicative inverse of probability density function and the other is distance based fgmms which define the dissimilarity function focusing the degree of fuzziness only on the distances between points and component centres different from fcms both of the proposed dissimilarity functions are based on the exponential function of the distance the fgmms are compared with the conventional gmms and the generalized gmms in terms of the fitting degree and convergence speed the experimental results show that the proposed fgmms not only possess the non linearity to fit datasets with curve manifolds but also have a much faster convergence process saving more than half computational cost than gmms

equations de type de boltzmann spatialement homogenes we study the construction of a nonhomogeneous n valued markov process whose laws at fixed times evolve according to a boltzmann type equation spatially homogeneous we then consider the problem of the asymptotic propagation of chaos for a system of a large number of interacting particles

a possibilistic clustering approach toward generative mixture models generative mixture models mms provide one of the most popular methodologies for unsupervised data clustering mms are formulated on the basis of the assumption that each observation derives from belongs to a single cluster however in many applications data may intuitively belong to multiple classes thus rendering the single cluster assignment assumptions of mms irrelevant furthermore even in applications where a single cluster data assignment is required the induced multinomial allocation of the modeled data points to the clusters derived by a mm imposing the constraint that the membership probabilities of a data point across clusters sum to one makes mms very vulnerable to the presence of outliers in the clustered data sets and renders them ineffective in discriminating between cases of equal evidence or ignorance to resolve these issues in this paper we introduce a possibilistic formulation of mms possibilistic clustering is a methodology that yields possibilistic data partitions with the obtained membership values being interpreted as degrees of possibility compatibilities of the data points with respect to the various clusters we provide an efficient maximum likelihood fitting algorithm for the proposed model and we conduct an objective evaluation of its efficacy using benchmark data

wild s solution of the nonlinear boltzmann equation the relaxation to equilibrium of a spatially uniform pseudo maxwellian gas is considered a modified wild expansion is defined for solving the nonlinear boltzmann equation the positivity and asymptotic conditions as well as the conservation rules are maintained at each truncation order some particular examples are evaluated the comparison with exact solutions illustrates the very fast convergence of this method

invariant painleve analysis and coherent structures of two families of reaction diffusion equations exact closed form coherent structures pulses fronts domain walls having the form of complicated traveling waves are constructed for two families of reaction diffusion equations by the use of invariant painleve analysis these analytical solutions which are derived directly from the underlying pde s are investigated in the light of restrictions imposed by the ode that any traveling wave reduction of the corresponding pde must satisfy in particular it is shown that the coherent structures a asymptotically satisfy the ode governing traveling wave reductions and b are accessible to the pde from compact support initial conditions the solutions are compared with each other and with previously known solutions of the equations

regularized linear fuzzy clustering and probabilistic pca mixture models fuzzy c means fcm type fuzzy clustering approaches are closely related to gaussian mixture models gmms and em like algorithms have been used in fcm clustering with regularized objective functions especially fcm with regularization by kullback leibler information klfcm is a fuzzy counterpart of gmms in this paper we propose to apply probabilistic principal component analysis pca mixture models to linear clustering following a discussion on the relationship between local pca and linear fuzzy clustering although the proposed method is a kind of the constrained model of klfcm the algorithm includes the fuzzy c varieties fcv algorithm as a special case and the algorithm can be regarded as a modified fcv algorithm with regularization by k l information numerical experiments demonstrate that the proposed clustering algorithm is more flexible than the maximum likelihood approaches and is useful for capturing local substructures properly