parameter uniform convergence of a numerical method for a coupled system of singularly perturbed semilinear reaction diffusion equations with boundary and interior layers abstract in this paper we consider a coupled system of m 2 singularly perturbed semilinear reaction diffusion equations with a discontinuous source term having a discontinuity at a point in the interior of the domain the diffusion term of each equation is multiplied by small singular perturbation parameter but these parameters are assumed to be different in magnitude a numerical method is constructed on a variant of shishkin mesh the approximations generated by this method are shown to be almost second order uniformly convergent with respect to all perturbation parameters numerical results are in support of the theoretical results

parameter uniformly convergent exponential spline difference scheme for singularly perturbed semilinear reaction diffusion problems abstract we consider a dirichlet boundary value problem for singularly perturbed semilinear reaction diffusion equation the problem is discretized using an exponential spline difference scheme derived on the basis of splines in tension on piecewise uniform shishkin type mesh the convergence analysis is given and the method is shown to be almost second order accurate in the discrete maximum norm uniformly in the perturbation parameter e numerical experiments are conducted to demonstrate the theoretical results

solving singularly perturbed boundary value problems by spline in tension the difference scheme via spline in tension for the problem y p x y f x p x s 0 y 0 0 y 1 1 is derived the error of the form o h min h is obtained when p x p const the corresponding spline in tension achieves the second order of the global uniform convergence

parameter uniform hybrid difference scheme for solutions and derivatives in singularly perturbed initial value problems a second order initial value problem with a small parameter multiplying the first and second order derivatives is considered the precise knowledge about the behavior of the exact solution is analyzed based on the behavior of the exact solution a hybrid finite difference scheme on a shishkin mesh is proposed which is a combination of the second order difference scheme on the fine mesh and the modified midpoint upwind scheme on the coarse mesh by applying the truncation error estimate techniques and a difference analogue of gronwalls inequality we prove that the scheme is almost second order convergent for numerical solutions and scaled numerical derivatives numerical experiments support these theoretical results and indicate that the estimates are sharp

on a semilinear elliptic neumann problem with asymmetric nonlinearities abstract we consider the neumann problem 1 1 below we extend therange of applicability of the sharp nonresonance condition derived in go om so as in particular allow asymmetric nonlinearities 1 introduction this paper is mainly concerned with the neumann problem i au g u h x inq 1 1 du du 0 on oil here a is the laplacian operator q is a bounded domain of rn n 1 with c1 1 boundary and unit exterior normal v g is a continuous function on r and h is taken in l f2 or in lp q in a previous work go om we showed that if g s 2g s 1 2 lim sup oo s s oo s or s oo then a necessary and sufficient condition for 1 1 to be solvable for any h l f2 is that the range of g be all of k here kx 0 k2 denote the distinct eigenvalues of a on q with neumann boundary condition andg s jo g t dt the surjectivity of g thus provides a nonresonance condi tion at the eigenvalue xx 0 we refer to go om for historical background

monotone methods for higher order partial difference equations abstract a nonlinear difference equation of the order n m in two independent variables nx myu x y u together with appropriate initial and periodic boundary conditions is considered several comparison results for these problems are established these results are then used to provide sufficient conditions for the convergence of monotone iterative methods numerical illustrations dwelling upon the importance of our results are also included

local superlinearity and sublinearity for the p laplacian we study the existence nonexistence and multiplicity of positive solutions for a family of problems updelta p u f lambda x u u in w 0 1 p upomega where is a bounded domain in mathbb r n n p and 0 is a parameter the family we consider includes the well known nonlinearities of ambrosetti brezis cerami type in a more general form namely lambda a x u q b x u r where 0 leqslant q p 1 r leqslant p 1 here the coefficient a x is assumed to be nonnegative but b x is allowed to change sign even in the critical case preliminary results of independent interest include the extension to the p laplacian context of the brezis nirenberg result on local minimization in w 0 1 p text and c 0 1 a c 1 alpha estimate for equations of the form updelta p u h x u with h of critical growth a strong comparison result for the p laplacian and a variational approach to the method of upper lower solutions for the p laplacian

periodic solutions of a second order ordinary differential equation a necessary and sufficient condition for nonresonance the nonlinearity g in 1 1 is a continuous function from r to r and the forcing term h is taken in l o 27r nonresonance means that 1 1 admits at least one solution x for any given h integrating eq 1 1 over a period one immediately sees that a necessary condition for nonresonance is that the function g be unbounded from above and from below on r it will appear later cf 1 6 that this unboundedness can be looked at as a condition relating the behaviour at infinity of the nonlinearity g with respect to the first eigenvalue 1 0 of the associated linear problem xf ax in 0 27r x 0 x 27r x 0 x1 271 1 2

an almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction diffusion problem this paper is concerned with a high order convergent discretization for the semilinear reaction diffusion problem varepsilon 2 u b x u 0 for x in 0 1 subject to u 0 u 1 0 where varepsilon in 0 1 we assume that b u x u b 0 2 0 on 0 1 times bbb r 1 which guarantees uniqueness of a solution to the problem asymptotic properties of this solution are discussed we consider a polynomial based three point difference scheme on a simple piecewise equidistant mesh of shishkin type existence and local uniqueness of a solution to the scheme are analysed we prove that the scheme is almost fourth order accurate in the discrete maximum norm uniformly in the perturbation parameter varepsilon we present numerical results in support of this result

a spline method for second order singularly perturbed boundary value problems we consider a difference scheme based on cubic spline in compression for second order singularly perturbed boundary value problem of the form ey p x y q x y r x y a 0 y b 1 the method is shown to have second and fourth order convergence depending on the choice of parameters 1 and 2 involved in the method the method is tested on an example and the results found to be in agreement with the theory

stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations we prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation dot y t a t y t b t y t tau f t by a periodic boundary value approach this equation arises e g in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations we obtain convergence results for the standard collocation algorithm and for two variants in particular estimates of the difference between the collocation solution and the true solution are derived for the standard collocation scheme the convergence results are unconditional that is they do not require mesh ratio restrictions numerical results that support the theoretical findings are also given

positivity and lower and upper solutions for fourth order boundary value problems abstract this paper is devoted to the study the boundary value problem u 4 t f t u t for all t i 0 1 u 0 u 1 u 0 u 1 0 we prove the existence of at least one two or three solutions in the presence of a pair of not necessarily ordered lower and upper solutions the proof follows from maximum principles related to the operator u 4 m u and amann s three solutions theorem

existence and multiplicity of positive solutions for classes of singular elliptic pdes we consider the boundary value problem u g u u in u 0on where rn is a bounded domain is a nonnegative function in l such that 0 on some subset of of positive measure and g 0 r is continuous we establish the existence of three positive solutions when g 0 0 positone the graph of s 1g s is roughly s shaped and 0 we also prove that there exists at least one positive solution when g 0 0 and 0 1 we employ the method of sub super solutions to prove our results

locally linear discriminant analysis for multimodally distributed classes for face recognition with a single model image we present a novel method of nonlinear discriminant analysis involving a set of locally linear transformations called locally linear discriminant analysis llda the underlying idea is that global nonlinear data structures are locally linear and local structures can be linearly aligned input vectors are projected into each local feature space by linear transformations found to yield locally linearly transformed classes that maximize the between class covariance while minimizing the within class covariance in face recognition linear discriminant analysis lida has been widely adopted owing to its efficiency but it does not capture nonlinear manifolds of faces which exhibit pose variations conventional nonlinear classification methods based on kernels such as generalized discriminant analysis gda and support vector machine svm have been developed to overcome the shortcomings of the linear method but they have the drawback of high computational cost of classification and overfitting our method is for multiclass nonlinear discrimination and it is computationally highly efficient as compared to gda the method does not suffer from overfitting by virtue of the linear base structure of the solution a novel gradient based learning algorithm is proposed for finding the optimal set of local linear bases the optimization does not exhibit a local maxima problem the transformation functions facilitate robust face recognition in a low dimensional subspace under pose variations using a single model image the classification results are given for both synthetic and real face data

graded mesh difference schemes for singularly perturbed two point boundary value problems etude de l approximation numerique des operateurs differentiels sans points de retournement par des schemas aux differences finies compacts analyse de la stabilite

nonresonance to the right of the first eigenvalue for the one dimensional p laplacian a quasilinear two point boundary value problem was considered this problem is given by the equation

iterative algorithms of domain decomposition for the solution of a quasilinear elliptic problem this paper deals with iterative algorithms for domain decomposition applied to the solution of a quasilinear elliptic problem two iterative algorithms are examined the first one is the schwarz alternating procedure and the second algorithm is suitable for parallel computing convergence results are established in the two domain and multidomain decomposition cases some issues of parallel implementation of these algorithms are discussed

a robust layer resolving spline collocation method for a convection diffusion problem abstract we consider finite difference approximation of a singularly perturbed one dimensional convection diffusion two point boundary value problem the problem is numerically treated by a quadratic spline collocation method on a piecewise uniform slightly modified shishkin mesh the position of collocation points is chosen so that the obtained scheme satisfies the discrete minimum principle we prove pointwise convergence of order o n 2 ln 2 n inside the boundary layer and second order convergence elsewhere the uniform convergence of the approximate continual solution is also given further we approximate normalized flux and give estimates of the error at the mesh points and between them the numerical experiments presented in the paper confirm our theoretical results

solvability of three point boundary value problems for second order differential equations with deviating arguments the monotone iterative technique is used to boundary problems for second order ordinary differential equations with deviating arguments corresponding results are formulated when the problem has extremal solutions or weakly coupled extremal quasi solutions

lower and upper solutions for elliptic problems in nonsmooth domains abstract in this paper we prove some existence results of semilinear dirichlet problems in nonsmooth domains in presence of lower and upper solutions well ordered or not we first prove existence results in an abstract setting using degree theory we secondly apply them for domains with conical points

the three solutions theorem for p laplacian boundary value problems abstract the three solutions theorem is established for p laplacian boundary value problems in the presence of two pairs of lower and upper solutions which are well ordered by the relationship between the lower and upper solutions and degree theory in addition an application of the three solutions theorem is given

discrete galerkin method for fredholm integro differential equations with weakly singular kernels approximations to a solution and its derivatives of a boundary value problem of an nth order linear fredholm integro differential equation with weakly singular or other nonsmooth kernels are determined these approximations are piecewise polynomial functions on special graded grids for their finding a discrete galerkin method and an integral equation reformulation of the boundary value problem are used optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained to illustrate the theoretical results a collection of numerical results of a test problem is presented

solutions for an operator equation under the conditions of pairs of paralleled lower and upper solutions abstract in this paper under the condition of two pairs of strict lower and upper solutions and using the concept of e 1 b limit increasing operator some multiplicity results for an operator equation are obtained by the method of the fixed point index

monotonicity and discretization error estimates precise bounds of supremum norms are given for the inverse of monotone matrices this result sharpens an earlier result by varga matrix iterative analysis prentice hall englewood cliffs nj 1962 for h matrices that are monotone various algebraic approaches to prove monotonicity based on an extended version of the weak regular splitting theorem are presented and applied to some important examples of finite difference and finite element matrices this together with the above mentioned bounds of the norm of the inverse operator shows that the discretization error in supremum and least square norms for self adjoint problems can be bounded by a constant times the norm of the truncation error where the best constant is available

numerical solution of a singularly perturbed two point boundary value problem using equidistribution analysis of convergence abstract adaptive grid methods are becoming established as valuable computational techniques for the numerical solution of differential equations with near singular solutions adaptive methods are equally effective in approximating solutions of problems with boundary layers or interior layers see for example mulholland et al siam j sci comput 19 4 1998 1261 1289 much is now being done in developing error analyes for methods that are based on adaptivity in this paper we present a rigorous error analysis for the solution of a singularly perturbed two point boundary value problem on a grid that is constructed adaptively from a knowledge of the exact solution the discrete solutions are generated by an upwind finite difference scheme and the grid is formed by equidistributing a monitor function based on arc length an error analysis shows that the discrete solutions are uniformly convergent with respect to the perturbation parameter epsilon the epsilon uniform convergence is confirmed by numerical computations

linearized domain decomposition approaches for nonlinear boundary value problems abstract linearized schwarz domain decomposition approaches for nonlinear boundary value problems of the form u f x u u subject to dirichlet boundary conditions are proposed and analyzed in the presence of subsolutions and supersolutions we construct a globally convergent linear monotone iteration suitable for implementation in a distributed computing environment these iterations provide an alternative to the typical locally convergent approach of discretizing and solving the resulting non linear algebraic equations using a newton iteration the work also extends previous results obtained in the case where f has no dependence on the derivative of the solution the schwarz iteration is first proposed and studied in detail on two subdomains the result is then generalized to an arbitrary number of subdomains both alternating and parallel schwarz iterations are analyzed numerical results are provided to demonstrate the theory and the utility of the proposed iterations

monotone method for second order periodic boundary value problems in recent years there has been an extensive study of the existence of periodic solutions l s 11 14 151 in 8 111 the existence of solutions of first and second order pbvp period boundary value problems has been obtained by a novel approach of combining the classical method of lower and upper solutions and the method of alternative problems lyapunovschmidt method which provide conditions that are easily verifiable and which covers previous known results of other authors as a constructive method for obtaining extremal solutions of initial and boundary value problems the monotone iterative procedure has been employed by several researchers 5 7 11 13 151 the objective of this paper is to employ this useful technique for second order pbvp to obtain the minimal and maximal solutions as limits of monotone iterates our method can be used to study semilinear parabolic initial boundary value problems and other problems at resonance

piecewise polynomial collocation for fredholm integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of initial or boundary value problems of linear fredholm integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of convergence of numerical solutions for all values of the nonuniformity parameter of the underlying grid

a constructive monotone iterative method for second order bvp in the presence of lower and upper solutions this paper concerns the monotone approximations of solutions of boundary value problems such as u f t u u 0 u 0 u 1 0 we consider linear iterative scheme in case f is lipschitz in u and satisfies a one sided lipschitz condition in u the initial approximations are lower and upper solutions which can be ordered one way a email protected we also consider the periodic and the dirichlet problems

convergence analysis of a finite difference scheme for a riemann liouville fractional derivative two point boundary value problem on an adaptive grid abstract a two point boundary value problem whose highest order term is a riemann liouville fractional derivative of order 2 with 0 1 is considered such problem is reformulated as a volterra integral equation of the second kind an integral discrete scheme is developed for this volterra integral equation on an adaptive grid that is constructed adaptively from a knowledge of the exact solution it is shown from a rigorous priori error analysis that the discrete solutions are uniformly convergent with respect to the parameter besides in order to establish the parameter of the volterra integral equation we construct a nonlinear optimization problem which is solved by the nelder mead simplex method numerical results are given to demonstrate the performance of presented method

parameter uniform numerical method for global solution and global normalized flux of singularly perturbed boundary value problems using grid equidistribution in this paper we present the analysis of an upwind scheme for obtaining the global solution and the normalized flux for a convection diffusion two point boundary value problem the solution of the upwind scheme is obtained on a suitable nonuniform mesh which is formed by equidistributing the arc length monitor function it is shown that the discrete solution obtained by the upwind scheme and the global solution obtained via interpolation converges uniformly with respect to the perturbation parameter in addition we prove the uniform first order convergence of the weighted derivative of the numerical solution on this nonuniform mesh and the uniform convergence of the global normalized flux on the whole domain numerical results are presented that demonstrate the sharpness of our results

monotone method for the neumann problem with lower and upper solutions in the reverse order this work is devoted to the study of the existence and approximation of solutions of the problem u f t u u u a u b 0 in the presence of lower and upper solutions in the reverse order to this end we prove anti maximum principles for the nonlinear operator u 2k u email protected

on linear monotone iteration and schwarz methods for nonlinear elliptic pdes the schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains the solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain in this paper proofs of convergence of some schwarz alternating methods for nonlinear elliptic problems which are known to have solutions by the monotone method also known as the method of subsolutions and supersolutions are given in particular an additive schwarz method for scalar as well some coupled nonlinear pdes are shown to converge to some solution on finitely many subdomains even when multiple solutions are possible in the coupled system case each subdomain pde is linear decoupled and can be solved concurrently with other subdomain pdes these results are applicable to several models in population biology

a numerical approach to nonlinear two point boundary value problems for odes in this paper we propose a numerical approach to solve some problems connected with the implementation of the newton type methods for the resolution of the nonlinear system of equations related to the discretization of a nonlinear two point bvps for odes with mixed linear boundary conditions by using the finite difference method

global convergence method for singularly perturbed boundary value problems we give uniformly convergent splines difference scheme for singularly perturbed boundary value problems eu p x u q x u f x u a 0 u b 1 1 by using splines fitted with delta sequence due to the very stiff nature of the problem under consideration we prove the o min h2 e2 order of uniform convergence with respect to small parameter e at nodes on uniform mesh and o min h e order of uniform global convergence with respect to the approximate solution given by s x i 1n s i x h xi x where h is the heaviside function which is the approximation for the closed form of the exact solution

a positive operator approach to the neumann problem for a second order ordinary differential equation abstract in this paper the solvability of the second order neumann problem u t cu t f t u t 0 qquad u 0 a u r b is studied we suppose that there is a lower solution and an upper solution in the reversed order and we obtain optimal conditions infto assure the existence of a solution lying between and

finite element boundary value integration of wheeler feynman electrodynamics the electromagnetic two body problem is solved as a boundary value problem associated to an action functional we show that the functional is frechet differentiable and that its conditions for criticality are the mixed type neutral differential delay equations with state dependent delay of wheeler feynman electrodynamics we construct a finite element method that finds c 1 smooth solutions when suitable past and future positions of the particles are given as boundary data the numerical trajectories satisfy a variational problem defined in a finite dimensional hermite functional space of c 1 piecewise polynomials the numerical variational problem is solved using a combination of newton s method intercalated with boundary adjustments to ensure that the velocity of the solution is continuous with the boundary data we recover the known circular orbits and compute several other novel trajectories of the wheeler feynman electrodynamics we also discuss the local convexity of the functional close to the new found trajectories and the possibility of solutions with less regularity

a moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction diffusion problems with arbitrary small diffusion terms abstract in this paper a system of time dependent boundary layer originated reaction dominated problems with diffusion parameters of different magnitudes is considered for numerical analysis the presence of these parameters lead to the boundary layer phenomena here an optimal order uniformly accurate boundary layer adaptive method moving mesh method is proposed this method is able to capture the layer phenomena without using a priori information of the solution the problem is discretized by a modified implicit euler scheme in time direction for the present system adaptive mesh generation is required in space due to the singularly perturbed nature of the problem for this purpose a positive error monitor function is used whose equidistribution will move the mesh points towards the boundary layers parameter uniform error estimates are derived to show that the convergence rate is optimal with respect to the problem discretization numerical experiments strongly verify the theoretical findings and confirm the efficiency and accuracy of the proposed method

higher order global solution and normalized flux for singularly perturbed reaction diffusion problems in this paper a computational technique is proposed for obtaining a higher order global solution and global normalized flux of singularly perturbed reaction diffusion two point boundary value problems the hoc higher order compact finite difference scheme developed in gracia et al 2001 4 and which is constructed on an appropriate piecewise uniform shishkin mesh has been considered to find an almost fourth order convergent solution at mesh points using these values piecewise cubic interpolants based approximations for solution and normalized flux in whole domain have been defined it has been shown that the global solution and the global normalized flux are also uniformly convergent moreover for the global solution the order of uniform convergence in the whole domain is optimal i e it is the same as this one obtained at mesh points whereas for the global normalized flux the uniform convergence is almost third order except at midpoints of the mesh where it is also almost fourth order theoretical error bounds have been provided along with some numerical examples which corroborate the efficiency of the proposed technique to find good approximations to the global solution and the global normalized flux

unstable periodic solutions of a parabolic problem in the presence of non well ordered lower and upper solutions abstract we prove the solvability of the dirichlet periodic problem for a semilinear parabolic equation assuming the existence of a lower solution and an upper solution which do not satisfy the ordering condition our results yield the existence of unstable solutions localized by means of and requiring weak regularity of the coefficients of the equation

minimization of eigenvalues for a quasilinear elliptic neumann problem with indefinite weight we investigate the minimization of the positive principal eigenvalue of the problem pu m u p 2u in u 0 on over a class of sign changing weights m with m 0 it is proved that minimizers exist and satisfy a bang bang type property in dimension one we obtain a complete description of the minimizers this problem is motivated by applications from population dynamics

domain decomposition approaches for mesh generation via the equidistribution principle moving mesh methods based on the equidistribution principle are powerful techniques for the space time adaptive solution of evolution problems solving the resulting coupled system of equations namely the original pde and the mesh pde however is challenging in parallel we propose in this paper several schwarz domain decomposition algorithms for this task we then study in detail the convergence properties of these algorithms applied to the nonlinear mesh pde in one spatial dimension we prove convergence for classical transmission conditions and optimal and optimized variants for the generation of steady equidistributing grids a classical parallel schwarz algorithm is presented and analyzed for the generation of time dependent moving equidistributing grids we conclude our study with numerical experiments

infinite max margin factor analysis via data augmentation this paper addresses the bayesian estimation of the discriminative probabilistic latent models especially the mixture models we develop the max margin factor analysis mmfa model which utilizes the latent variable support vector machine lvsvm as the classification criterion in the latent space to learn a discriminative subspace with max margin constraint furthermore to deal with multimodally distributed data we further extend mmfa to infinite gaussian mixture model and develop the infinite max margin factor analysis immfa model via the consideration of dirichlet process mixtures dpm it jointly learns clustering max margin classifiers and the discriminative latent space in a united framework to improve the prediction performance moreover both of mmfa and immfa are natural to handle outlier rejection problem since the observations are described by a single or a mixture of gaussian distributions additionally thanks to the conjugate property the parameters in the two models can be inferred efficiently via the simple gibbs sampler finally we implement our models on synthesized and real world data including multimodally distributed datasets and measured radar echo data to validate the classification and rejection performance of the proposed models jointly learning fa and svm mmfa is proposed to get a discriminative subspace clustering the dataset in the subspace by dpm mmfa is extended to immfa thanks to the jointly learning framework they gain good prediction performance having the data description ability the proposed models can reject outlier samples in bayesian framework parameters can be inferred efficiently by the gibbs sampler

replacing the finite difference methods for nonlinear two point boundary value problems by successive application of the linear shooting method abstract this paper studies numerical solution of nonlinear two point boundary value problems for second order ordinary differential equations first it establishes a connection between the finite difference method and the quasi linearization method we prove that using finite differences to discretize the sequence of linear differential equations arising from quasi linearization newton method on operator level leads to the usual iteration formula of the newton finite difference method from the provided derivation it can easily be inferred that such a relation holds also for the picard and the constant slope methods based on this result we propose a way of replacing the newton picard and constant slope finite difference methods by respective successive application of the linear shooting method this approach has a number of advantages it removes the necessity of solving systems of algebraic equations hence working with matrices altogether compared to the usual finite difference method with general solver it reduces the number of computational operations from o n 3 where n is the number of mesh points to only o n

face recognition using laplacianfaces we propose an appearance based face recognition method called the laplacianface approach by using locality preserving projections lpp the face images are mapped into a face subspace for analysis different from principal component analysis pca and linear discriminant analysis lda which effectively see only the euclidean structure of face space lpp finds an embedding that preserves local information and obtains a face subspace that best detects the essential face manifold structure the laplacianfaces are the optimal linear approximations to the eigenfunctions of the laplace beltrami operator on the face manifold in this way the unwanted variations resulting from changes in lighting facial expression and pose may be eliminated or reduced theoretical analysis shows that pca lda and lpp can be obtained from different graph models we compare the proposed laplacianface approach with eigenface and fisherface methods on three different face data sets experimental results suggest that the proposed laplacianface approach provides a better representation and achieves lower error rates in face recognition

existence and localization of solution for second order elliptic bvp in presence of lower and upper solutions without any order abstract we prove an existence and localization result for a solution of a second order elliptic equations with dirichlet and regular oblique derivative boundary conditions the main result is obtained assuming the existence of a lower and an upper solution without particular order and a positively homogeneous asymptotic behaviour of the right hand member of the equation

monotone methods for periodic solutions of second order scalar functional differential equations an iterative method is presented which starting from a lower or from an upper periodic solution provides a monotone sequence converging to a periodic solution of 1 with some restrictions on the growth off the method extends to functional differential equations of type 1 two numerical examples with an a posteriori error analysis are given

face recognition from a single training image under arbitrary unknown lighting using spherical harmonics in this paper we propose two novel methods for face recognition under arbitrary unknown lighting by using spherical harmonics illumination representation which require only one training image per subject and no 3d shape information our methods are based on the result which demonstrated that the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace we provide two methods to estimate the spherical harmonic basis images spanning this space from just one image our first method builds the statistical model based on a collection of 2d basis images we demonstrate that by using the learned statistics we can estimate the spherical harmonic basis images from just one image taken under arbitrary illumination conditions if there is no pose variation compared to the first method the second method builds the statistical models directly in 3d spaces by combining the spherical harmonic illumination representation and a 3d morphable model of human faces to recover basis images from images across both poses and illuminations after estimating the basis images we use the same recognition scheme for both methods we recognize the face for which there exists a weighted combination of basis images that is the closest to the test face image we provide a series of experiments that achieve high recognition rates under a wide range of illumination conditions including multiple sources of illumination our methods achieve comparable levels of accuracy with methods that have much more onerous training data requirements comparison of the two methods is also provided

numerical bifurcation analysis of delay differential equations numerical methods for the bifurcation analysis of delay differential equations ddes have only recently received much attention partially because the theory of ddes smoothness boundedness stability of solutions is more complicated and less established than the corresponding theory of ordinary differential equations as a consequence no established software packages exist at present for the bifurcation analysis of ddes we outline existing numerical methods for the computation and stability analysis of steady state solutions and periodic solutions of systems of ddes with several constant delays

a monotone method for constructing extremal solutions of second order scalar boundary value problems a monotone iterative method is given for the construction of extremal solutions to second order scalar boundary value problems starting from a pair of upper and lower solutions the fully nonlinear case is considered

a uniformly convergent finite difference scheme for a singularly perturbed semilinear equation boundary value problems for singularly perturbed semilinear elliptic equations are considered special piecewise uniform meshes are constructed which yield accurate numerical solutions irrespective of the value of the small parameter numerical methods composed of standard monotone finite difference operators and these piecewise uniform meshes are shown theoretically to be uniformly with respect to the singular perturbation parameter convergent numerical results are also presented which indicate that in practice the method is first order accurate

fredholm alternative for the p laplacian in higher dimensions abstract in this paper we characterize the set of all right hand sides h c formula for which the boundary value problem p u 1 u p 2 u h in u 0 on has at least one weak solution u w01 p here 1 0 is the first eigenvalue of the p laplacian in particular we prove that for h 1 0 this problem is solvable and the energy functional e h u 1 p u p 1 p u p hu is unbounded from below

the numerical solution of forward backward differential equations decomposition and related issues this paper focuses on the decomposition by numerical methods of solutions to mixed type functional differential equations mfdes into sums of forward solutions and backward solutions we consider equations of the form x t ax t bx t 1 cx t 1 and develop a numerical approach using a central difference approximation which leads to the desired decomposition and propagation of the solution we include illustrative examples to demonstrate the success of our method along with an indication of its current limitations

face authentication for multiple subjects using eigenflow abstract in this paper we present a novel scheme for face authentication to deal with variations such as facial expressions and registration errors with which traditional intensity based methods do not perform well we propose the eigenflow approach in this approach the optical flow and the optical flow residue between a test image and an image in the training set are first computed the optical flow is then fitted to a model that is pre trained by applying principal component analysis to optical flows resulting from facial expressions and registration errors for the subject the eigenflow residue optimally combined with the optical flow residue using linear discriminant analysis determines the authenticity of the test image an individual modeling method and a common modeling method are described we also present a method to optimally choose the threshold for each subject for a multiple subject authentication system experimental results show that the proposed scheme outperforms the traditional methods in the presence of facial expression variations and registration errors

a uniformly accurate spline collocation method for a normalized flux we are concerned with a two point boundary value problem for a semilinear singularly perturbed reaction diffusion equation with a singular perturbation parameter e our goal is to construct global e uniform approximations of the solution y x and the normalized flux p x e d dx y x using the collocation with the classical quadratic splines u x c1 i on a slightly modified piecewise uniform mesh of shishkin type the constructed approximate solution and normalized flux converge e uniformly with the rate o n 2 ln2 n and o n 1 ln n respectively on the shishkin type mesh and with o n 1 ln 2 n and o ln 3 n when the mesh has to be modified we present numerical experiments in support of these results

acquiring linear subspaces for face recognition under variable lighting previous work has demonstrated that the image variation of many objects human faces in particular under variable lighting can be effectively modeled by low dimensional linear spaces even when there are multiple light sources and shadowing basis images spanning this space are usually obtained in one of three ways a large set of images of the object under different lighting conditions is acquired and principal component analysis pca is used to estimate a subspace alternatively synthetic images are rendered from a 3d model perhaps reconstructed from images under point sources and again pca is used to estimate a subspace finally images rendered from a 3d model under diffuse lighting based on spherical harmonics are directly used as basis images in this paper we show how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low dimensional linear space and that this subspace is close to those acquired by the other methods more specifically there exist configurations of k point light source directions with k typically ranging from 5 to 9 such that by taking k images of an object under these single sources the resulting subspace is an effective representation for recognition under a wide range of lighting conditions since the subspace is generated directly from real images potentially complex and or brittle intermediate steps such as 3d reconstruction can be completely avoided nor is it necessary to acquire large numbers of training images or to physically construct complex diffuse harmonic light fields we validate the use of subspaces constructed in this fashion within the context of face recognition

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

remarks on the lower and upper solutions method for second and third order periodic boundary value problems we study the solvability and the approximation of the solutions by monotone iteration of the second and third order periodic boundary value problems bvps u cu t u u 0 u email protected u 0 u email protected and email protected au bu t u u 0 u email protected u 0 u email protected u 0 u email protected in the presence of lower and upper solutions which may not satisfy the usual ordering condition to this end some maximum and antimaximum principles are stated and proved for certain linear differential operators naturally associated with the previously stated problems

positive solutions of a diffusive predator prey model with modified leslie gower and holling type ii schemes abstract in this paper we investigate the existence multiplicity and stability of positive solutions to a prey predator model with modified leslie gower and holling type ii schemes p u u a 1 b u c 1 v u k 1 in v v a 2 c 2 v u k 2 in u 0 v 0 in u v 0 on where r n n 1 is a bounded domain with a smooth boundary the parameters a i b c i k i i 1 2 are positive numbers u and v are the respective populations of prey and predator here we say u v with u v 0 is a positive solution of problem p if u v is a solution of p and u v 0 in

uniform fourth order difference scheme for a singular perturbation problem the numerical solution of a nonlinear singularly perturbed two point boundary value problem is studied the developed method is based on hermitian approximation of the second derivative on special discretization mesh numerical examples which demonstrate the effectiveness of the method are presented

oscillation of delay parabolic differential equations with several coefficients some new sufficient conditions are established for the oscillation of delay parabolic differential equations of the form u x t t a t u i 1n pi x t u x t i j 1m qj x t u x t j x t t0 g where is a bounded domain in rn with a piecewise smooth boundary and is the laplacian in euclidean n space rn with three different boundary conditions our results improve the well known oscillation result of e when n m 1 an example is considered to illustrate our main results

an abstract framework in the numerical solution of boundary value problems for neutral functional differential equations we consider the numerical solution of boundary value problems for general neutral functional differential equations the problems are restated in an abstract form and then a general discretization of the abstract form is introduced and a convergence analysis of this discretization is developed

collocation methods for the computation of periodic solutions of delay differential equations in this paper we investigate collocation methods for the computation of periodic solutions of autonomous delay differential equations ddes periodic solutions are found by solving a periodic two point boundary value problem which is an infinite dimensional problem for ddes in contrast to the case of ordinary differential equations we investigate three collocation methods based on piecewise polynomials we discuss computational issues and show numerical orders of convergence using an extensive number of tests we compare our numerical results with known theoretical convergence results for initial value problems for ddes in particular we show how superconvergence at the mesh points can be lost or recovered depending on the dde model under consideration and on the choice of collocation discretization we end with a brief discussion of adaptive mesh selection

class dependent factor analysis and its application to face recognition we propose a class dependent factor analysis model cdfa which can be used in the general face recognition task under certain variations the model utilizes the class information in a supervised manner to define a separate manifold for each class inside each manifold a mixture of gaussians is designated to handle the variation the proposed model learns the system parameters in a probabilistic framework allowing a bayesian decision model a manifold embedding technique is incorporated to handle the nonlinearity introduced by the variation hence a novel connection between manifold learning and probabilistic generative models is proposed cdfa has better recognition accuracy and scalability over a classical factor analysis model experimental evaluations on the face recognition under changing illumination conditions and facial expressions indicate the ability of the proposed model to handle different types of variation the achieved recognition rates are comparable to the state of art results while it is also shown that the recognition rate does not decrease critically as the number of gallery identities increases

positive solutions of elliptic problems with locally oscillating nonlinearities abstract we prove the existence of two unbounded sequences of strictly positive solutions obtained respectively as minimum and saddle points of the associated action functional of the elliptic problem u f x u in u 0 on assuming that an oscillatory behaviour at of s 2 0 s f x d occurs locally in