decay estimates for the wave equation in two dimensions abstract we establish strichartz estimates both reversed and some direct ones pointwise decay estimates and weighted decay estimates for the linear wave equation in dimension two with an almost scaling critical potential in the case when there is no resonance or eigenvalue at the edge of the spectrum we also prove some simple applications

the lp boundedness of wave operators for schrodinger operators with threshold singularities abstract let h v be a schrodinger operator on l 2 r n with real valued potential v for n 4 and let h 0 if v has sufficiently rapid decay the wave operators defined as w s lim t e i t h e i t h 0 are known to be bounded on l p r n for all 1 p if zero is not an eigenvalue and on 1 p n 2 if zero is an eigenvalue we show that these wave operators are also bounded on l 1 r n by direct examination of the integral kernel of the leading term furthermore if r n v x x d x 0 for all eigenfunctions then the wave operators are l p bounded for 1 p n if in addition r n x v x x d x 0 then the wave operators are bounded for 1 p

lp lp estimates for the schrodinger equation on the line and inverse scattering for the nonlinear schrodinger equation with a potential abstract in this paper i prove a l p l p estimate for the solutions to the one dimensional schrodinger equation with a potential in l 1 where in the generic case 3 2 and in the exceptional case i e when there is a half bound state of zero energy 5 2 i use this estimate to construct the scattering operator for the nonlinear schrodinger equation with a potential i prove moreover that the low energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity

dispersive estimates for schrodinger operators in dimension two we prove l1 2 l 2 for the two dimensional schrodinger operator v with the decay rate t 1 we assume that zero energy is neither an eigenvalue nor a resonance this condition is formulated as in the recent paper by jensen and nenciu on threshold expansions for the two dimensional resolvent

second order design sensitivity analysis of mechanical system dynamics dependence of dynamic response of nonlinear mechanical systems on design variables is analysed an adjoint variable method is used to derive first and second order derivatives of measures of dynamic response with respect to design variables a computational algorithm is presented for numerical calculation of first and second design derivatives a simple oscillator example is solved analytically and by the adjoint variable method with identical results a burst fire automatic weapon mechanism with linear and nonlinear damping is treated numerically it is shown that quadratic appriximations of dynamic response using results of second order design sensitivity analysis can be substantially better than conventional linear approximations

a weighted dispersive estimate for schrodinger operators in dimension two let h v where v is a real valued potential on mathbb r 2 satisfying v x lesssim langle x rangle 3 we prove that if zero is a regular point of the spectrum of h v then w 1 e ith p ac f l infty mathbb r 2 lesssim frac 1 t log 2 t w f l 1 mathbb r 2 t geq 2 with w x log 2 x 2 this decay rate was obtained by murata in the setting of weighted l 2 spaces with polynomially growing weights

l p boundedness of wave operators for two dimensional schrodinger operators let be the two dimensional schrodinger operator with the real valued potential v which satisfies the decay condition at infinity for we show that the wave operators are bounded in for any 1 p under the condition that h has no zero bound states or zero resonance extending the corresponding results for higher dimensions as w intertwine h 0 and the absolutely continuous part h p ac of h f h p ac w f h 0 w for any borel function f on 1 this reduces the various l p mapping properties of f h p ac to those of f h 0 the convolution operator by the fourier transform of the function f 2

multi granularity distance metric learning via neighborhood granule margin maximization learning a distance metric from training samples is often a crucial step in machine learning and pattern recognition locality compactness and consistency are considered as the key principles in distance metric learning however the existing metric learning methods just consider one or two of them in this paper we develop a multi granularity distance learning technique first a new index neighborhood granule margin which simultaneously considers locality compactness and consistency of neighborhood is introduced to evaluate a distance metric by maximizing neighborhood granule margin we formulate the distance metric learning problem as a sample pair classification problem which can be solved by standard support vector machine solvers then a set of distance metrics are learned in different granular spaces the weights of the granular spaces are learned through optimizing the margin distribution finally the decisions from different granular spaces are combined with weighted voting experiments on uci datasets gender classification and object categorization tasks show that the proposed method is superior to the state of the art distance metric learning algorithms

adjoint sensitivity analysis for differential algebraic equations the adjoint dae system and its numerical solution an adjoint sensitivity method is presented for parameter dependent differential algebraic equation systems daes the adjoint system is derived along with conditions for its consistent initialization for daes of index up to two hessenberg for stable linear daes stability of the adjoint system for semi explicit daes or of an augmented adjoint system for fully implicit daes is shown in addition it is shown for these systems that numerical stability is maintained for the adjoint system or for the augmented adjoint system

remarks on endpoint strichartz estimates for schrodinger equations with the critical inverse square potential abstract the purpose of this paper is to study the validity of global in time strichartz estimates for the schrodinger equation on r n n 3 with the negative inverse square potential x 2 in the critical case n 2 2 4 it turns out that the situation is different from the subcritical case n 2 2 4 in which the full range of strichartz estimates is known to be hold more precisely splitting the solution into the radial and non radial parts we show that i the radial part satisfies a weak type endpoint estimate which can be regarded as an extension to higher dimensions of the endpoint strichartz estimate with radial data for the two dimensional free schrodinger equation ii other endpoint estimates in lorentz spaces for the radial part fail in general iii the non radial part satisfies the full range of strichartz estimates

sensitivity theory for nonlinear systems ii extensions to additional classes of responses this work extends a recent functional analytic formulation of sensitivity theory to include treatment of additional types of responses there are physical systems where a critical point of a function that depends on the system s state vector and parameters defines the location in phase space where the response functional is evaluated the g teaux differentials giving the sensitivities of both the functional and the critical point to changes in the system s parameters are obtained by alternative formalisms the foward sensitivity formalism is the simpler and more general but may be prohibitively expensive for problems with large data bases the adjoint sensitivity formalism although less generally applicable and requiring several adjoint calculations is likely to be the only practical approach sensitivity theory is also extended to include treatment of general operators acting on the system s state vector and parameters as response in this case the forward sensitivity formalism is the same as for functional responses but the adjoint sensitivity formalism is considerably different the adjoint sensitivity formalism requires expanding the indirect effect term an element of a hilbert space in terms of elements of an orthonormal basis since as many calculations of adjoint functions are required as there are nonzero terms in this expansion careful consideration of truncating the expansion is needed to assess the advantages of the adjoint sensitivity formalism over the forward sensitivity formalism

time decay estimates for the wave equation with potential in dimension two abstract we study the wave equation with potential u t t u v u 0 in two spatial dimensions with v a real valued decaying potential with h v we study a variety of mapping estimates of the solution operators cos t h and sin t h h under the assumption that zero is a regular point of the spectrum of h we prove a dispersive estimate with a time decay rate of t 1 2 a polynomially weighted dispersive estimate which attains a faster decay rate of t 1 log t 2 for t 2 finally we prove dispersive estimates if zero is not a regular point of the spectrum of h

on the fourth order schrodinger equation in four dimensions dispersive estimates and zero energy resonances abstract we study the fourth order schrodinger operator h 2 v for a decaying potential v in four dimensions in particular we show that the t 1 decay rate holds in the l 1 l setting if zero energy is regular furthermore if the threshold energies are regular then a faster decay rate of t 1 log t 2 is attained for large t at the cost of logarithmic spatial weights zero is not regular for the free equation hence the free evolution does not satisfy this bound due to the presence of a resonance at the zero energy we provide a full classification of the different types of zero energy resonances and study the effect of each type on the time decay in the dispersive bounds

output bounds for reduced order approximations of elliptic partial differential equations abstract we present an a posteriori finite element procedure that provides inexpensive rigorous accurate and constant free lower and upper bounds for the error in the outputs engineering quantities of interest predicted by lagrangian reduced order approximations to coercive elliptic partial differential equations the bound calculation requires i the reduced order approximation for the primal and dual field variables ii a lower bound for the minimum eigenvalue of the symmetric part of the operator and iii the solution of purely local symmetric neumann subproblems defined on small decoupled nodal overlapping patches there are two critical components to the neumann subproblems a partition of unity attenuated local residual which eliminates hybrid fluxes from both the construction and analysis of the resulting estimators while simultaneously preserving the global bound property and a l 2 regularization term which provides stability despite the absence of local nodal equilibrium of the reduced order primal and dual solutions the estimator bounding property and optimal convergence rate as the reduced order basis is enriched are proven and corroborating numerical results are presented for two examples a heat conduction fin symmetric problem and a conjugate advection diffusion multi material heat transfer non symmetric problem

asymptotic expansions in time for solutions of schrodinger type equations let ia i p d v be a dissipative operator in l2 rn where p d is an elliptic differential operator of order m with real constant coefficients and v is a compact operator from the weighted sobolev space hm s rn to h0 p s rn s r for some m 1 let r z be the resolvent of a then an asymptotic expansion of r z as z approaches a critical value of the polynomial p is given the coefficient operators in the expansion are computed explicitly by using the resolvent expansion and the results of m murata 9 an asymptotic expansion of e ita as t is given

sensitivity theory for nonlinear systems i nonlinear functional analysis approach concepts of nonlinear functional analysis are employed to investigate the mathematical foundations underlying sensitivity theory this makes it possible not only to ascertain the limitations inherent in existing analytical approaches to sensitivity analysis but also to rigorously formulate a considerably more general sensitivity theory for physical problems characterized by systems of nonlinear equations and by nonlinear functionals as responses two alternative formalisms labeled the forward sensitivity formalism and the adjoint sensitivity formalism are developed in order to evaluate the sensitivity of the response to variations in the system parameters the forward sensitivity formalism is formulated in normed linear spaces and the existence of the g teaux differentials of the operators appearing in the problem is shown to be both necessary and sufficient for its validity this formalism is conceptually straightforward and can be advantageously used to assess the effects of relatively few parameter alterations on many responses on the other hand for problems involving many parameter alterations or a large data base and comparatively few functional type responses the alternative adjoint sensitivity formalism is computationally more economical however it is shown that this formalism can be developed only under conditions that are more restrictive than those underlying the validity of the forward sensitivity formalism in particular the requirement that operators acting on the state vector and on the system parameters must admit densely defined g teaux derivatives is shown to be of fundamental importance for the validity of this formalism the present analysis significantly extends the scope of sensitivity theory and provides a basis for still further generalizations

additive logistic regression a statistical view of boosting boosting is one of the most important recent developments in classification methodology boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced for many classification algorithms this simple strategy results in dramatic improvements in performance we show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles namely additive modeling and maximum likelihood for the two class problem boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum bernoulli likelihood as a criterion we develop more direct approximations and show that they exhibit nearly identical results to boosting direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most situations and far superior in some we suggest a minor modification to boosting that can reduce computation often by factors of 10 to 50 finally we apply these insights to produce an alternative formulation of boosting decision trees this approach based on best first truncated tree induction often leads to better performance and can provide interpretable descriptions of the aggregate decision rule it is also much faster computationally making it more suitable to large scale data mining applications

dispersive estimates for higher dimensional schrodinger operators with threshold eigenvalues i the odd dimensional case we investigate l1 rn l rn dispersive estimates for the schrodinger operator h v when there is an eigenvalue at zero energy and n 5 is odd in particular we show that if there is an eigenvalue at zero energy then there is a time dependent rank one operator ft satisfying ft l1 l t 2 n2 for t 1 such that eithpac ft l1 l t 1 n2 for t 1 with stronger decay conditions on the potential it is possible to generate an operator valued expansion for the evolution taking the form eithpac h t 2 n2a 2 t 1 n2a 1 t n2a0 with a 2 and a 1 finite rank operators mapping l1 rn to l rn while a0 maps weighted l1 spaces to weighted l spaces the leading order terms a 2 and a 1 vanish when certain orthogonality conditions between the potential v and the zero energy eigenfunctions are satisfied we show that under the same orthogonality conditions the remaining t n2a0 term also exists as a map from l1 rn to l rn hence eithpac h satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero

dispersive estimates for schrodinger equations with threshold resonance and eigenvalue let h v x be a three dimensional schrodinger operator we study the time decay in lp spaces of scattering solutions e ithpcu where pc is the orthogonal projection onto the continuous spectral subspace of l2 r3 for h under suitable decay assumptions on v x it is shown that they satisfy the so called lp lq estimates e ithpcu p 4 t 3 1 2 1 p u q for all 1 q 2 p with 1 p 1 q 1 if h has no threshold resonance and eigenvalue and for all 3 2 q 2 p 3 if otherwise

on the lp boundedness of wave operators for two dimensional schrodinger operators with threshold obstructions abstract let h v be a schrodinger operator on l 2 r 2 with real valued potential v and let h 0 if v has sufficient pointwise decay the wave operators w s lim t e i t h e i t h 0 are known to be bounded on l p r 2 for all 1 p if zero is not an eigenvalue or resonance we show that if there is an s wave resonance or an eigenvalue only at zero then the wave operators are bounded on l p r 2 for 1 p this result stands in contrast to results in higher dimensions where the presence of zero energy obstructions is known to shrink the range of valid exponents p

software and algorithms for sensitivity analysis of large scale differential algebraic systems sensitivity analysis for dae systems is important in many engineering and scientific applications the information contained in the sensitivity trajectories is useful for parameter estimation optimization model reduction and experimental design in this paper we present algorithms and software for sensitivity analysis of large scale dae systems of index up to two the new software provides for consistent initialization of the solutions and the sensitivities interfaces seamlessly with automatic differentiation for the accurate evaluation of the sensitivity equations and is capable via mpi of exploiting the natural parallelism of sensitivity analysis as well as providing an efficient solution in sequential computations

k local hyperplane distance nearest neighbor classifier oriented local discriminant analysis k local hyperplane distance nearest neighbor hknn classifier is an improved k nearest neighbor knn algorithm that has been successfully applied to pattern classification this paper embeds the decision rule of hknn classifier into the discriminant analysis model to develop a new feature extractor the obtained feature extractor is called k local hyperplane distance nearest neighbor classifier oriented local discriminant analysis holda in which a regularization item is imposed on the original hknn algorithm to obtain a more reliable distance metric based on this distance metric the homo class and hetero class local scatters are characterized in holda by maximizing the ratio of the hetero class local scatter to the homo class local scatter we obtain a subspace which is suitable for feature extraction and classification in general this paper provides a framework for building a feature extractor from the decision rule of a classifier by this means the feature extractor and classifier can be seamlessly integrated experimental results on four databases demonstrate that the integrated pattern recognition system is effective

a remark on lp boundedness of wave operators for two dimensional schrodinger operators let h v be a two dimensional schrodinger operator with a real potential v x satisfying the decay condition 6 let h0 we show that the wave operators are bounded in lp r2 under the condition that h has no zero resonances or bound states in this paper the condition imposed in a previous paper k yajima commun math phys 208 125 152 1999 is removed

dispersive estimates for schrodinger operators in dimensions one and three we consider l 1 l estimates for the time evolution of hamiltonians h v in dimensions d 1 and d 3 with bound t 2 we require decay of the potentials but no regularity in d 1 the decay assumption is 1 x v x dx whereas in d 3 it is v x c 1 x 3

efficient computation of sensitivities for ordinary differential equation boundary value problems for models described by ordinary differential equation boundary value problems ode bvps we derive adjoint equations for sensitivity analysis giving explicit forms for the boundary conditions of the adjoint boundary value problem the solutions of the adjoint equations are used to efficiently compute gradients of both integral form and pointwise constraints existence and stability results are given for the adjoint system and its numerical solution the use of the method is demonstrated for a simple example where it is seen that the method is particularly advantageous for problems with more than a few parameters

solving ill conditioned and singular linear systems a tutorial on regularization it is shown that the basic regularization procedures for finding meaningful approximate solutions of ill conditioned or singular linear systems can be phrased and analyzed in terms of classical linear algebra that can be taught in any numerical analysis course apart from rewriting many known results in a more elementary form we also derive a new two parameter family of merit functions for the determination of the regularization parameter the traditional merit functions from generalized cross validation gcv and generalized maximum likelihood gml are recovered as special cases

time decay for solutions of schrodinger equations with rough and time dependent potentials in this paper we establish dispersive estimates for solutions to the linear schrodinger equation in three dimensions frac 1 i partial t psi triangle psi v psi 0 qquad psi s f 0 1 where v t x is a time dependent potential that satisfies the conditions sup t v t cdot l frac 3 2 mathbb r 3 sup x in mathbb r 3 int mathbb r 3 int infty infty frac v hat tau x x y d tau dy c 0 here c0 is some small constant and v hat tau x denotes the fourier transform with respect to the first variable we show that under these conditions 0 1 admits solutions l t l2x 3 l2t l6x 3 for any f l2 3 satisfying the dispersive inequality psi t infty le c t s frac32 f 1 text for all times t s 0 2 for the case of time independent potentials v x 0 2 remains true if int mathbb r 6 frac v x v y x y 2 dxdy 4 pi 2 text and v mathcal k sup x in mathbb r 3 int mathbb r 3 frac v y x y dy 4 pi we also establish the dispersive estimate with an e loss for large energies provided v mathcal k v 2 infty

dispersive estimates for the three dimensional schrodinger equation with rough potentials the three dimensional schrodinger propogator e ith h v is a bounded map from l 1 to l with norm controlled by t 3 2 provided the potential satisfies two conditions an integrability condition limiting the singularities and decay of v and a zero energy spectral condition on h this is shown by expressing the spectral measure of h in terms of its resolvents and proving a family of l p mapping estimates for the resolvents previous results in this direction had required v to satisfy explicit pointwise bounds

twin mahalanobis distance based support vector machines for pattern recognition twin support vector machines tsvms achieve fast training speed and good performance for data classification however tsvms do not take full advantage of the statistical information in data such as the covariance of each class of data this paper proposes a new twin mahalanobis distance based support vector machine tmsvm classifier in which two mahalanobis distance based kernels are constructed according to the covariance matrices of two classes of data for optimizing the nonparallel hyperplanes tmsvms have a special case of tsvms when the covariance matrices in a reproducing kernel hilbert space are degenerated to the identity ones tmsvms are suitable for many real problems especially for the case that the covariance matrices of two classes of data are obviously different the experimental results on several artificial and benchmark datasets indicate that tmsvms not only possess fast learning speed but also obtain better generalization than tsvms and other methods