on the shape of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of v i v i v d i v d i coincide we denote this common dimension by i in this paper we prove that i 0 d i for 0 i d it is already known that 0 1 if k is algebraically closed

a nonstandard numerical scheme of predictor corrector type for epidemic models in this paper we construct and develop a competitive nonstandard finite difference numerical scheme of predictor corrector type for the classical sir epidemic model this proposed scheme is designed with the aim of obtaining dynamical consistency between the discrete solution and the solution of the continuous model the nonstandard finite difference scheme with conservation law nsfdcl developed here satisfies some important properties associated with the considered sir epidemic model such as positivity boundedness monotonicity stability and conservation of frequency of the oscillations numerical comparisons between the nsfdcl numerical scheme developed here and runge kutta type schemes show its effectiveness

how many clusters are best an experiment this paper reports the results of a monte carlo study on the relative effectiveness of two internal indices in estimating the true number of clusters in multivariate data the two indices are the davies and bouldin 1 index and a new modification of the hubert statistic 2 data in d dimensions are clustered to create sequences of partitions estimates are based on plots of the indices as functions of the number of recovered clusters neither index uses a priori information the effects of sample size dimensionality cluster spread number of true clusters and sampling window are examined clustered data are generated to assure a given number of distinct clusters the degree of clustering in the data is verified by a separate monte carlo study based on the jaccard and corrected rand indices that exhibits the importance of correcting external indices for chance the modified hubert index proposed here for the first time is shown to perform better than the davies bouldin index under all experimental conditions recovery of the true number of clusters gets better as the number of true clusters decreases and as the number of dimensions increases no effect occurs due to sampling window the complete link clustering method and a square error clustering method recognize the true number of clusters consistently better than the single link method this study demonstrates the difficulty inherent in estimating the number of clusters

the conservative characteristic fd methods for atmospheric aerosol transport problems in the paper we develop the new conservative characteristic finite difference methods c cfd for the atmospheric aerosol transport problems we propose the time second order and spatial high order conservative characteristic finite difference methods for the aerosol vertical advection diffusion process and the two dimensional conservative characteristic finite difference methods for aerosol horizontal transport process in the second order splitting algorithm based on the characteristic form of advection diffusion equations tracking back along the characteristic curve we treat the integrals over the tracking cells at the previous time level by the conservative interpolations and propose to treat the diffusion terms by the average along the characteristics where the high order discrete fluxes are obtained by approximating the cumulative mass function and are continuous at the tracking points the important feature is that the proposed c cfd schemes preserve mass and have second order accuracy in time and high order accuracy in space numerical tests are taken to show the accuracy in time and space and mass conservation of our c cfd schemes compared with the standard cfd method a real case of air quality modelling during the 2008 beijing olympics and a severe haze in north china are further simulated and analyzed by using our c cfd algorithm simulated results are in good agreement with observations the developed c cfd algorithm can be used for efficiently solving large scale atmospheric aerosol transport problems

a fully adaptive reaction diffusion integration scheme with applications to systems biology adaptive integration schemes for ode systems typically function by adjusting the time step size so as to keep the truncation error below some desired value for adaptive integration of pde systems involving coupled kinetic reaction and diffusion operations truncation error arises not only from the individual propagators but also from their method of coupling a common second order accurate method for coupling operators is strang s method of operator splitting we derive an expression for the truncation error resulting from strang splitting reaction and diffusion operators for an arbitrary number of spatial dimensions and demonstrate its use in adaptive time step algorithms in addition we present explanations of the second order implicit reaction and diffusion operators and their individual error calculations used in our implementation of the scheme finally using example simulations we discuss the use of this calculation for problems in systems biology

a classification of sharp tridiagonal pairs let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide the pair a a is called sharp whenever dimv0 1 it is known that if f is algebraically closed then a a is sharp in this paper we classify up to isomorphism the sharp tridiagonal pairs as a corollary we classify up to isomorphism the tridiagonal pairs over an algebraically closed field we obtain these classifications by proving the conjecture

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

existence and boundedness of solutions for a singular phase field system abstract this paper is devoted to the mathematical analysis of a thermo mechanical model describing phase transitions in terms of the entropy and order structure balance law we consider a macroscopic description of the phenomenon and make a presentation of the model then the initial and boundary value problem is addressed for the related pde system which contains some nonlinear and singular terms with respect to the temperature variable existence of the solution is shown along with the boundedness of both phase variable and absolute temperature finally uniqueness is proved in the framework of a source term depending lipschitz continuously on

adjoint sensitivity analysis of regional air quality models the task of providing an optimal analysis of the state of the atmosphere requires the development of efficient computational tools that facilitate an efficient integration of observational data into models in a variational approach the data assimilation problem is posed as a minimization problem which requires the sensitivity derivatives of a cost functional with respect to problem parameters the direct decoupled method has been extensively applied for sensitivity studies of air pollution adjoint sensitivity is a complementary approach which efficiently calculates the derivatives of a functional with respect to a large number of parameters in this paper we discuss the mathematical foundations of the adjoint sensitivity method applied to air pollution models and present a complete set of computational tools for performing three dimensional adjoint sensitivity studies numerical examples show that three dimensional adjoint sensitivity analysis provides information on influence areas which cannot be obtained solely by an inverse analysis of the meteorological fields several illustrative data assimilation results in a twin experiments framework as well as the assimilation of a real data set are also presented

a numerical study of singularly perturbed generalized burgers huxley equation using three step taylor galerkin method in the present work a numerical study has been carried out for the singularly perturbed generalized burgers huxley equation using a three step taylor galerkin finite element method a burgers huxley equation represents the traveling wave phenomena in singular perturbed problems a very small positive parameter e called the singular perturbation parameter is multiplied with the highest order derivative term as this parameter tends towards zero the problem exhibits boundary layers the traditional methods fail to capture the boundary layers when e becomes very small in this paper a three step taylor galerkin finite element method is used to capture the boundary layers the method is third order accurate and has inbuilt upwinding stability analysis has been carried out and the numerical results show that the method is efficient in capturing the boundary layers

a modified bhattacharya exponential method to approximate positive and bounded solutions of the burgers fisher equation in this work we investigate numerically the classical burgers fisher equation using a modified bhattacharya method the partial differential equation under investigation possesses nonnegative and bounded solutions and under some suitable parameter conditions these solutions are traveling waves it is well known that the use of the bhattacharya approach leads to the design of numerical techniques that are sensitive to zero solutions however in this manuscript we provide a correction of that technique in order to approximate solutions of the burgers fisher equation that are bounded in 0 1 the proposed methodology is explicit and we establish thoroughly the capability of the technique to preserve the non negativity the boundedness and the monotonicity of the numerical approximations as well as the constant solutions of the continuous model the new class of methods introduced in this work considers the presence of a free parameter and we show that this family tends to an explicit and standard discretization of the burgers fisher equation when the free parameter tends to infinity some simulations illustrate the main features of the method

nonparametric maximum likelihood estimation of features in spatial point processes using voronoi tessellation abstract this article addresses the problem of estimating the support domain of a bounded point process in presence of background noise this situation occurs for example in the detection of a minefield from aerial observations a maximum likelihood estimator for a mixture of uniform point processes is derived using a natural partition of the space defined by the data themselves the voronoi tessellation the methodology is tested on simulations and compared to a model based clustering technique

a positive numerical scheme for a mixed type partial differential equation model for fungal growth we describe a numerical scheme for the solution of a mixed type pde system which arises in the modelling of fungal growth given the application conservation of mass and preservation of positivity are of paramount importance the scheme employs a method of lines approach in which the system is split into hyperbolic and parabolic parts positivity and conservation of mass are ensured by the use of generalised flux functions and in particular flux limiters the spatial discretisation results in stiff and non stiff components which are solved using implicit and explicit methods respectively properties of the scheme are investigated via comparison with experimental data and performance is compared with another method of solution

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

tridiagonal pairs and the q tetrahedron algebra abstract the q tetrahedron algebra q was recently introduced and has been studied in connection with tridiagonal pairs in this paper we further develop this connection let k denote an algebraically closed field and let q denote a nonzero scalar in k that is not a root of unity let v denote a vector space over k with finite positive dimension and let a a denote a tridiagonal pair on v let i i 0 d resp i i 0 d denote a standard ordering of the eigenvalues of a resp a ito and terwilliger have shown that when i q 2 i d and i q d 2 i 0 i d there exists an irreducible q module structure on v such that the q generators x 01 x 23 act as a a respectively in this paper we examine the case in which there exists a nonzero scalar c in k such that i q 2 i d and i q 2 i d cq d 2 i for 0 i d in this case we associate to a a a polynomial p in one variable and prove the following theorem as our main result theorem the following are equivalent 1 i there exists a q module structure on v such that x 01 acts as a and x 30 cx 23 acts as a where x 01 x 30 x 23 are standard generators for q 2 ii p q 2 d 2 q q 1 2 0 suppose i and ii hold then the q module structure on v is unique and irreducible

codeword distribution for frequency sensitive competitive learning with one dimensional input data we study the codeword distribution for a conscience type competitive learning algorithm frequency sensitive competitive learning fscl using one dimensional input data we prove that the asymptotic codeword density in the limit of large number of codewords is given by a power law of the form q x c spl middot p x sup spl alpha where p x is the input data density and spl alpha depends on the algorithm and the form of the distortion measure to be minimized we further show that the algorithm can be adjusted to minimize any l sub p distortion measure with p ranging in 0 2

finite difference schemes for nonlinear wave equation that inherit energy conservation property abstract we propose two general finite difference schemes that inherit energy conservation property from nonlinear wave equations such as the nonlinear klein gordon equation nlkge one of proposed schemes is implicit and another is explicit many studies exist on fdss that inherit energy conservation property from nlkge and we can derive all of their schemes from the proposed general schemes in this paper the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly because of this the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions we make new fdss for fermi pasta ulam equation string vibration equation shimoji kawai equation ske and ebihara equation and verify numerically the inheritance of the energy conservation property for nlkge and ske

gaussian mixture density modeling decomposition and applications we present a new approach to the modeling and decomposition of gaussian mixtures by using robust statistical methods the mixture distribution is viewed as a contaminated gaussian density using this model and the model fitting mf estimator we propose a recursive algorithm called the gaussian mixture density decomposition gmdd algorithm for successively identifying each gaussian component in the mixture the proposed decomposition scheme has advantages that are desirable but lacking in most existing techniques in the gmdd algorithm the number of components does not need to be specified a priori the proportion of noisy data in the mixture can be large the parameter estimation of each component is virtually initial independent and the variability in the shape and size of the component densities in the mixture is taken into account gaussian mixture density modeling and decomposition has been widely applied in a variety of disciplines that require signal or waveform characterization for classification and recognition we apply the proposed gmdd algorithm to the identification and extraction of clusters and the estimation of unknown probability densities probability density estimation by identifying a decomposition using the gmdd algorithm that is a superposition of normal distributions is successfully applied to automated cell classification computer experiments using both real data and simulated data demonstrate the validity and power of the gmdd algorithm for various models and different noise assumptions

a bounded linear integrator for some diffusive nonlinear time dependent partial differential equations we propose a numerical method to approximate the solutions of generalized forms of two bi dimensional models of mathematical physics namely the burgers fisher and the burgers huxley equations in one dimensional form the literature in the area gives account of the existence of analytical solutions for both models in the form of traveling wave fronts bounded within an interval i of the real numbers motivated by this fact we propose a finite difference methodology that guarantees that under certain analytical conditions on the model and computer parameters estimates within i will evolve discretely into new estimates which are likewise bounded within i additionally we establish the preservation in the discrete domain of the skew symmetry of the solutions of the models under study our computational implementation of the method confirms numerically that the properties of positivity and boundedness are preserved under the analytical constraints derived theoretically our simulations show a good agreement between the analytical solutions derived in the present work and the corresponding numerical approximations

a highly robust estimator through partially likelihood function modeling and its application in computer vision the authors present a highly robust estimator known as the model fitting mf estimator for general regression they explain that high robustness becomes possible through partially but completely modeling the unknown log likelihood function the partial modeling takes place by taking the bayesian statistical decision rule and a number of important heuristics into consideration while maximizing the log likelihood function applications include the automatic selection of multiple thresholds single rigid motion estimation or multiple rigid motion segmentation and estimation from two perspective views it is believed that the proposed mf estimator will aid in solving many robust estimation problems that demand an estimator that is either highly robust or capable of handling contaminated gaussian mixture models

robust clustering methods a unified view clustering methods need to be robust if they are to be useful in practice in this paper we analyze several popular robust clustering methods and show that they have much in common we also establish a connection between fuzzy set theory and robust statistics and point out the similarities between robust clustering methods and statistical methods such as the weighted least squares technique the m estimator the minimum volume ellipsoid algorithm cooperative robust estimation minimization of probability of randomness and the epsilon contamination model by gleaning the common principles upon which the methods proposed in the literature are based we arrive at a unified view of robust clustering methods we define several general concepts that are useful in robust clustering state the robust clustering problem in terms of the defined concepts and propose generic algorithms and guidelines for clustering noisy data we also discuss why the generalized hough transform is a suboptimal solution to the robust clustering problem

solitary wave solutions of the generalised burgers huxley equation exact solitary wave solutions of the generalised burgers huxley equation delta u delta t alpha udelta delta u delta chi delta 2u delta chi 2 beta u 1 udelta udelta gamma are obtained by using the relevant nonlinear transformations the results obtained are the generalisation of former work the method in this paper can also be applied to the burgers fisher equation

the structure of a tridiagonal pair let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of k linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide in this paper we show that the following i iv hold provided that k is algebraically closed i each of v0 v0 vd vd has dimension 1 ii there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for all u v v iii there exists a unique anti automorphism of end v that fixes each of a a iv the pair a a is determined up to isomorphism by the data i i 0d i i 0d i i 0d where i resp i is the eigenvalue of a resp a on vi resp vi and i i 0d is the split sequence of a a corresponding to i i 0d and i i 0d

persistence of travelling wave solutions of a fourth order diffusion system in this paper the extended burgers huxley equation with the fourth order derivative is considered first the convergence to the uniform steady state is proved which means the solution of the equation with positive initial data will remain positive for time t sufficiently large then the persistence of the travelling wave solution for the extended equation on the unbounded domain is investigated we have proved that this solution will persist under small perturbation of the equation

traveling wave solutions for nonlinear equations using symbolic computation abstract in this paper a direct and unified algorithm for constructing multiple travelling wave solutions of nonlinear partial differential equations pdes is presented and implemented in a computer algebraic system the key idea of this method is to take full advantage of a riccati equation involving a parameter and use its solutions to replace the tanh function in the tanh method it is quite interesting that the sign of the parameter can be used to exactly judge the number and types of such travelling wave solutions in this way we can successfully recover the previously known solitary wave solutions that had been found by the tanh method and other more sophisticated methods more importantly for some equations with no extra effect we also find other new and more general solutions at the same time by introducing appropriate transformations our method is further extended to the nonlinear pdes whose balancing numbers may be any nonzero real numbers the efficiency of the method can be demonstrated on a large variety of nonlinear pdes such as those considered in this paper burgers huxley equation coupled korteweg de vries equation caudrey dodd gibbon kawada equation active dissipative dispersive media equation generalized fisher equation and nonlinear heat conduction equation

a skew symmetry preserving computational technique for obtaining the positive and the bounded solutions of a time delayed advection diffusion reaction equation abstract in this work we consider a one dimensional time delayed advective version of the well known fisher kolmogorov petrovsky piscounov equation from population dynamics which extends several models from mathematical physics including the classical wave equation the nonlinear klein gordon equation a fitzhugh nagumo equation from electrodynamics and the burgers huxley equation and the newell whitehead segel equation from fluid mechanics we propose a skew symmetry preserving finite difference scheme for approximating the solutions of the model under investigation and establish conditions on the model coefficients and the numerical parameters under which the method provides positive or bounded approximations for initial data which are likewise positive or bounded respectively the derivation of the conditions under which the positivity and the boundedness of the approximations is guaranteed is based on the properties of the inverses of m matrices in fact the conditions obtained here assure that the iterative method is described in vector form through the multiplication by a matrix of this type we provide simulations in order to show that the technique is indeed conditionally positivity preserving and boundedness preserving

a non standard numerical scheme for a generalized gause type predator prey model abstract a non standard finite difference scheme is constructed to simulate a predator prey model of gause type with a functional response using fixed point analysis it is shown that the scheme preserves the physical properties of the model and gives results that are qualitatively equivalent to the real dynamics of the model it is also shown that the scheme undergoes a supercritical hopf bifurcation for a specific value of the bifurcation parameter k0 this leads to the existence of a stable limit cycle created by the scheme when the bifurcation parameter passes through k0 as predicted by the continuous model the scheme is used to simulate the model with the functional responses of holling types ii and iii the simulation results generated by the non standard finite difference scheme are compared with those obtained from the standard methods such as forward euler and runge kutta methods these comparisons show that the standard methods give erroneous results that disagree with the theoretical predictions of the model however it is proved that the proposed non standard finite difference scheme is consistent with the asymptotic dynamics of the model numerical simulations are presented to support these facts

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

tridiagonal pairs of shape 1 2 1 abstract let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfies the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and that for 0 i d the dimensions of v i v d i v i v d i coincide we denote this common value by i the sequence i i 0 d is called the shape of the pair in this paper we assume the shape is 1 2 1 and obtain the following results we describe six bases for v one diagonalizes a another diagonalizes a and the other four underlie the split decompositions for a a we give the action of a and a on each basis for each ordered pair of bases among the six we give the transition matrix at the end we classify the tridiagonal pairs of shape 1 2 1 in terms of a sequence of scalars called the parameter array

an optimal graph theoretic approach to data clustering theory and its application to image segmentation a novel graph theoretic approach for data clustering is presented and its application to the image segmentation problem is demonstrated the data to be clustered are represented by an undirected adjacency graph g with arc capacities assigned to reflect the similarity between the linked vertices clustering is achieved by removing arcs of g to form mutually exclusive subgraphs such that the largest inter subgraph maximum flow is minimized for graphs of moderate size approximately 2000 vertices the optimal solution is obtained through partitioning a flow and cut equivalent tree of g which can be efficiently constructed using the gomory hu algorithm 1961 however for larger graphs this approach is impractical new theorems for subgraph condensation are derived and are then used to develop a fast algorithm which hierarchically constructs and partitions a partially equivalent tree of much reduced size this algorithm results in an optimal solution equivalent to that obtained by partitioning the complete equivalent tree and is able to handle very large graphs with several hundred thousand vertices the new clustering algorithm is applied to the image segmentation problem the segmentation is achieved by effectively searching for closed contours of edge elements equivalent to minimum cuts in g which consist mostly of strong edges while rejecting contours containing isolated strong edges this method is able to accurately locate region boundaries and at the same time guarantees the formation of closed edge contours

mock tridiagonal systems we introduce the notion of a mock tridiagonal system this is a generalization of a tridiagonal system in which the irreducibility assumption is replaced by a certain nonvanishing condition we show how mock tridiagonal systems can be used to construct tridiagonal systems that meet certain specifications this paper is part of our ongoing project to classify the tridiagonal systems up to isomorphism

transition maps between the 24 bases for a leonard pair abstract let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below 1 i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal 2 ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v in an earlier paper we described 24 special bases for v one feature of these bases is that with respect to each of them the matrices that represent a and a are i diagonal and irreducible tridiagonal or ii irreducible tridiagonal and diagonal or iii lower bidiagonal and upper bidiagonal or iv upper bidiagonal and lower bidiagonal for each ordered pair of bases among the 24 there exists a unique linear transformation from v to v that sends the first basis to the second basis we call this the transition map in this paper we find each transition map explicitly as a polynomial in a a

travelling wave solutions for the generalized burgers huxley equation abstract in this paper travelling wave solutions for the generalized burgers huxley equation are studied by using the first integral method which is based on the ring theory of commutative algebra we obtain a class of travelling solitary wave solutions for the generalized burgers huxley equation a minor error in the literature is clarified

two stage clustering via neural networks this paper presents a two stage approach that is effective for performing fast clustering first a competitive neural network cnn that can harmonize mean squared error and information entropy criteria is employed to exploit the substructure in the input data by identifying the local density centers a gravitation neural network gnn then takes the locations of these centers as initial weight vectors and undergoes an unsupervised update process to group the centers into clusters each node called gravi node in the gnn is associated with a finite attraction radius and would be attracted to a nearby centroid simultaneously during the update process creating the gravitation like behavior without incurring complicated computations this update process iterates until convergence and the converged centroid corresponds to a cluster compared to other clustering methods the proposed clustering scheme is free of initialization problem and does not need to pre specify the number of clusters the two stage approach is computationally efficient and has great flexibility in implementation a fully parallel hardware implementation is very possible

a petrov galerkin formulation for advection reaction diffusion problems abstract in this work we present a new method called su c pg to solve advection reaction diffusion scalar equations by the finite element method fem the supg for streamline upwind petrov galerkin method is currently one of the most popular methods for advection diffusion problems due to its inherent consistency and efficiency in avoiding the spurious oscillations obtained from the plain galerkin method when there are discontinuities in the solution following this ideas tezduyar and park treated the more general advection reaction diffusion problem and they developed a stabilizing term for advection reaction problems without significant diffusive boundary layers in this work an supg extension for all situations is performed covering the whole plane represented by the peclet number and the dimensionless reaction number the scheme is based on the extension of the super convergence feature through the inclusion of an additional perturbation function and a corresponding proportionality constant both proportionality constants that one corresponding to the standard perturbation function from supg and the new one introduced here are selected in order to verify the super convergence feature i e exact nodal values are obtained for a restricted class of problems uniform mesh no source term constant physical properties it is also shown that the su c pg scheme verifies the discrete maximum principle dmp that guarantees uniform convergence of the finite element solution moreover it is shown that super convergence is closely related to the dmp motivating the interest in developing numerical schemes that extend the super convergence feature to a broader class of problems

discrete maximum principle and adequate discretizations of linear parabolic problems in this paper we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with dirichlet type boundary condition first we formulate the qualitative properties for the differential equations and shed light on their relations then we show how the well known discretization schemes can be written in the form of a one step iterative process we give necessary and sufficient conditions of the main qualitative properties of these iterations we apply the results to the finite difference and galerkin finite element solutions of linear parabolic problems in our main result we show that the nonnegativity preservation property is equivalent to the maximum minimum principle and they imply the maximum norm contractivity in one two and three dimensions we list sufficient a priori conditions that ensure the required qualitative properties finally we demonstrate the above results on numerical examples

tridiagonal pairs and the μ conjecture let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide we say the pair a a is sharp whenever dimv0 1 it is known that if f is algebraically closed then a a is sharp a conjectured classification of the sharp tridiagonal pairs was recently introduced by t ito and the second author we present a result which supports the conjecture given scalars i i 0d i i 0d in f that satisfy the known constraints on the eigenvalues of a tridiagonal pair we define an f algebra t by generators and relations we consider the f algebra e0 te0 for a certain idempotent e0 t let f x1 xd denote the polynomial algebra over f involving d mutually commuting indeterminates we display a surjective f algebra homomorphism f x1 xd e0 te0 we conjecture that is an isomorphism we show that this conjecture implies the classification conjecture and that the conjecture holds for d 5

comparison of some finite element methods for solving the diffusion convection reaction equation abstract in this paper we describe several finite element methods for solving the diffusion convection reaction equation none of them is new although the presentation is non standard in an effort to emphasize the similarities and differences between them in particular it is shown that the classical supg method is very similar to an explicit version of the characteristic galerkin method whereas the taylor galerkin method has a stabilization effect similar to a sub grid scale model which is in turn related to the introduction of bubble functions

nearest neighbor clutter removal for estimating features in spatial point processes abstract we consider the problem of detecting features in spatial point processes in the presence of substantial clutter one example is the detection of minefields using reconnaissance aircraft images that identify many objects that are not mines our solution uses kth nearest neighbor distances of points in the process to classify them as clutter or otherwise the observed kth nearest neighbor distances are modeled as a mixture distribution the parameters of which are estimated by a simple em algorithm this method allows for detection of generally shaped features that need not be path connected in the minefield example this method yields high detection and low false positive rates another application to outlining seismic faults is considered with some success the method works well in high dimensions the method can also be used to produce very high breakdown point robust estimators of a covariance matrix

the numerical solution of a generalized burgers huxley equation through a conditionally bounded and symmetry preserving method in this article we propose a non standard finite difference scheme to approximate the solutions of a generalized burgers huxley equation from fluid dynamics our numerical method preserves the skew symmetry of the partial differential equation under study and under some analytical constraints of the model constants and the computational parameters involved it is capable of preserving the boundedness and the positivity of the solutions in the linear regime the scheme is consistent to first order in time due partially to the inclusion of a tuning parameter in the approximation of a temporal derivative and to second order in space we compare the results of our computational technique against the exact solutions of some particular initial boundary value problems our simulations indicate that the method presented in this work approximates well the theoretical solutions and moreover that the method preserves the boundedness of solutions within the analytical constraints derived here in the problem of approximating solitary wave solutions of the model under consideration we present numerical evidence on the existence of an optimum value of the tuning parameter of our technique for which a minimum relative error is achieved finally we linearly perturb a steady state solution of the partial differential equation under investigation and show that our simulations still converge to the same constant solution establishing thus robustness of our method in this sense

analysis of operator splitting for advection diffusion reaction problems from air pollution modelling abstract operator or time splitting is often used in the numerical solution of initial boundary value problems for differential equations it is for example standard practice in computational air pollution modelling where we encounter systems of three dimensional time dependent partial differential equations of the advection diffusion reaction type for such systems little attention has been devoted to the analysis of splitting and to the question why splitting can work so well from the theoretical point of view the success of splitting is primarily determined by the splitting error this paper presents an analysis of operator splitting aimed at providing insight into the splitting error using the lie operator formalism a general expression is derived for a three term strang splitting in the pure initial value case for a class of advection diffusion reaction problems the splitting error is analyzed in greater detail a special case is discussed in which the splitting error can be reduced also some attention is paid to the use of operator splitting in initial boundary value problems

robust clustering with applications in computer vision a clustering algorithm based on the minimum volume ellipsoid mve robust estimator is proposed the mve estimator identifies the least volume region containing h percent of the data points the clustering algorithm iteratively partitions the space into clusters without prior information about their number at each iteration the mve estimator is applied several times with values of h decreasing from 0 5 a cluster is hypothesized for each ellipsoid the shapes of these clusters are compared with shapes corresponding to a known unimodal distribution by the kolmogorov smirnov test the best fitting cluster is then removed from the space and a new iteration starts constrained random sampling keeps the computation low the clustering algorithm was successfully applied to several computer vision problems formulated in the feature space paradigm multithresholding of gray level images analysis of the hough space and range image segmentation

a convergent and dynamically consistent finite difference method to approximate the positive and bounded solutions of the classical burgers fisher equation in this work we investigate both analytically and numerically a spatially two dimensional advection diffusion reaction equation that generalizes the burgers and the fisher s equations the partial differential equation of interest is a nonlinear model for which the existence and the uniqueness of positive and bounded solutions are analytically established here at the same time we propose an exact finite difference discretization of the burgers fisher model of interest and show that as the continuous counterpart the method proposed is capable of preserving the positivity and the boundedness of the numerical approximations as well as the temporal and spatial monotonicity of the discrete initial boundary conditions it is shown that the method is convergent with first order in time and second order in space we provide some simulations that illustrate the fact that the proposed technique preserves the positivity the boundedness and the monotonicity

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

on the stability of residual free bubbles for convection diffusion problemsand their approximation by a two level finite element method we consider the galerkin finite element method for partial diffferential equations in two dimensions where the finite dimensional space used consists of piecewise isoparametric polynomials enriched with bubble functions writing l for the differential operator we show that for elliptic convection diffusion problems the component of the bubble enrichment that stabilizes the method is essentially equivalent to a petrov galerkin method with an l spline exponentially fitted trial space and piecewise polynomial test space the remaining component of the bubble influences the accuracy of the method a stability inequality recently obtained by brezzi franca and russo for a limiting case of bubbles applied to convection diffusion problems is shown to be slightly weaker than the standard stability inequality that is obtained for the sdfem supg method thereby demonstrating that the bubble approach is in general slightly less stable than the streamline diffusion method when the trial functions are piecewise linear we show that residual free bubbles are as stable as sdfem supg and we extend this stability inequality to include positive mesh peclet numbers in the convection dominated regime approximate computations of the residual free bubbles are performed using a two level finite element method

a positive and bounded finite element approximation of the generalized burgers huxley equation abstract we present a finite element scheme capable of preserving the nonnegative and bounded solutions of the generalized burgers huxley equation proofs of existence and uniqueness of a solution to the continuous problem together with some results concerning the boundedness and the nonnegativity of the solution are given under appropriate conditions on the mesh and the initial and boundary data boundedness and nonnegativity of the finite element approximation are established an a priori error estimate for the approximation is also derived numerical experiments are presented which support the derived theoretical results

tridiagonal pairs of q racah type abstract let f denote an algebraically closed field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that a w w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d for 0 i d let i resp i denote the eigenvalue of a resp a associated with v i resp v i the pair a a is said to have q racah type whenever i a b q 2 i d c q d 2 i and i a b q 2 i d c q d 2 i for 0 i d where q a b c a b c are scalars in f with q b c b c nonzero and q 2 1 1 this type is the most general one we classify up to isomorphism the tridiagonal pairs over f that have q racah type our proof involves the representation theory of the quantum affine algebra u q sl 2

an energy based computational method in the analysis of the transmission of energy in a chain of coupled oscillators in this paper we study the phenomenon of nonlinear supratransmission in a semi infinite discrete chain of coupled oscillators described by modified sine gordon equations with constant external and internal damping and subject to harmonic external driving at the end we develop a consistent and conditionally stable finite difference scheme in order to analyze the effect of damping in the amount of energy injected in the chain of oscillators numerical bifurcation analyses to determine the dependence of the amplitude at which supratransmission first occurs with respect to the frequency of the driving oscillator are carried out in order to show the consequences of damping on harmonic phonon quenching and the delay of appearance of critical amplitude

discretizations of nonlinear differential equations using explicit nonstandard methods abstract in a recent paper chen and solis investigated the appearance of spurious solutions when first order odes are discretized using runge kutta schemes they concluded that the reliability of the numerical solutions to a particular ode could be verified only by constructing several discrete models and comparing their numerical results with the known properties of the exact solutions we demonstrate that by using nonstandard schemes all the difficulties found by chen and solis can be eliminated and that qualitatively correct numerical solutions are obtained for all values of the step size we illustrate these issues by applying nonstandard finite difference techniques to the logistic sine cubic and monod equations

neural gas network for vector quantization and its application to time series prediction a neural network algorithm based on a soft max adaptation rule is presented this algorithm exhibits good performance in reaching the optimum minimization of a cost function for vector quantization data compression the soft max rule employed is an extension of the standard k means clustering procedure and takes into account a neighborhood ranking of the reference weight vectors it is shown that the dynamics of the reference weight vectors during the input driven adaptation procedure are determined by the gradient of an energy function whose shape can be modulated through a neighborhood determining parameter and resemble the dynamics of brownian particles moving in a potential determined by the data point density the network is used to represent the attractor of the mackey glass equation and to predict the mackey glass time series with additional local linear mappings for generating output values the results obtained for the time series prediction compare favorably with the results achieved by backpropagation and radial basis function networks

preserving nonnegativity of an affine finite element approximation for a convection diffusion reaction problem an affine finite element scheme approximation of a time dependent linear convection diffusion reaction problem in 2d and 3d is presented for these equations which do not satisfy an underlying maximum principle sufficient conditions are given in terms of the coefficient functions the computational grid and the discretization parameters to ensure that the nonnegativity property of the true solution is also satisfied by its approximation numerical examples are given which confirm the necessity and sufficiency of the discretization conditions to guarantee the nonnegativity of the approximation we present a monotone fem scheme for a convection diffusion reaction problem in 2 3d the considered equation does not possess an underlying maximum principle sufficient conditions are given to ensure the nonnegativity of the approximations numerical examples confirm the necessity and sufficiency of the conditions

a robust competitive clustering algorithm with applications in computer vision this paper addresses three major issues associated with conventional partitional clustering namely sensitivity to initialization difficulty in determining the number of clusters and sensitivity to noise and outliers the proposed robust competitive agglomeration rca algorithm starts with a large number of clusters to reduce the sensitivity to initialization and determines the actual number of clusters by a process of competitive agglomeration noise immunity is achieved by incorporating concepts from robust statistics into the algorithm rca assigns two different sets of weights for each data point the first set of constrained weights represents degrees of sharing and is used to create a competitive environment and to generate a fuzzy partition of the data set the second set corresponds to robust weights and is used to obtain robust estimates of the cluster prototypes by choosing an appropriate distance measure in the objective function rca can be used to find an unknown number of clusters of various shapes in noisy data sets as well as to fit an unknown number of parametric models simultaneously several examples such as clustering mixture decomposition line plane fitting segmentation of range images and estimation of motion parameters of multiple objects are shown

on a positivity preserving numerical model for a linearized hyperbolic fisher kolmogorov petrovski piscounov equation abstract in this work we introduce a nonstandard finite difference scheme to approximate positive solutions of a modified hyperbolic fisher kolmogorov petrovski piscounov equation appearing in the investigation of the dynamics of some populations the technique has a consistency of second order and it provides nonnegative approximations for nonnegative initial profiles the stability analysis of the method is carried out in detail and the scheme is validated against known analytical solutions for some initial boundary value problems

numerical treatment of the spherically symmetric solutions of a generalized fisher kolmogorov petrovsky piscounov equation in the present work the connection of the generalized fisher kpp equation to physical and biological fields is noted radially symmetric solutions to the generalized fisher kpp equation are considered and analytical results for the positivity and asymptotic stability of solutions to the corresponding time independent elliptic differential equation are quoted an energy analysis of the generalized theory is carried out with further physical applications in mind and a numerical method that consistently approximates the energy of the system and its rate of change is presented the method is thoroughly tested against analytical and numerical results on the classical fisher kpp equation the heaviside equation and the generalized fisher kpp equation with logistic nonlinearity and heaviside initial profile obtaining as a result that our method is highly stable and accurate even in the presence of discontinuities as an application we establish numerically that under the presence of suitable initial conditions there exists a threshold for the relaxation time with the property that solutions to the problems considered are nonnegative if and only if the relaxation time is below a critical value an analytical prediction is provided for the heaviside equation against which we verify the validity of our computational code and numerical approximations are provided for several generalized fisher kpp problems

high order multi implicit spectral deferred correction methods for problems of reactive flow models for reacting flow are typically based on advection diffusion reaction a d r partial differential equations many practical cases correspond to situations where the relevant time scales associated with each of the three sub processes can be widely different leading to disparate time step requirements for robust and accurate time integration in particular interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection the numerical strategy introduced in this paper is a general procedure to account for this time scale disparity the proposed methods are high order multi implicit generalizations of spectral deferred correction methods misdc methods constructed for the temporal integration of a d r equations spectral deferred correction methods compute a high order approximation to the solution of a differential equation by using a simple low order numerical method to solve a series of correction equations each of which increases the order of accuracy of the approximation the key feature of misdc methods is their flexibility in handling several sub processes implicitly but independently while avoiding the splitting errors present in traditional operator splitting methods and also allowing for different time steps for each process the stability accuracy and efficiency of misdc methods are first analyzed using a linear model problem and the results are compared to semi implicit spectral deferred correction methods furthermore numerical tests on simplified reacting flows demonstrate the expected convergence rates for misdc methods of orders three four and five the gain in efficiency by independently controlling the sub process time steps is illustrated for nonlinear problems where reaction and diffusion are much stiffer than advection although the paper focuses on this specific time scales ordering the generalization to any ordering combination is straightforward

conservative multi implicit spectral deferred correction methods for reacting gas dynamics in most models of reacting gas dynamics the characteristic time scales of chemical reactions are much shorter than the hydrodynamic and diffusive time scales rendering the reaction part of the model equations stiff moreover non linear forcings may introduce into the solutions sharp gradients or shocks the robust behavior and correct propagation of which require the use of specialized spatial discretization procedures this study presents high order conservative methods for the temporal integration of model equations of reacting flows by means of a method of lines discretization on the flux difference form of the equations these methods compute approximations to the cell averaged or finite volume solution the temporal discretization is based on a multi implicit generalization of spectral deferred correction methods the advection term is integrated explicitly and the diffusion and reaction terms are treated implicitly but independently with the splitting errors reduced via the spectral deferred correction procedure to reduce computational cost different time steps may be used to integrate processes with widely differing time scales numerical results show that the conservative nature of the methods allows a robust representation of discontinuities and sharp gradients the results also demonstrate the expected convergence rates for the methods of orders three four and five for smooth problems