the discrete zak transform application to time frequency analysis and synthesis of nonstationary signals algorithms to compute coefficients of the finite double sum expansion of time varying nonstationary signals and to synthesize them from a finite number of expansion coefficients are presented the algorithms are based on the computation of the discrete zak transform dzt fast algorithms to compute dzt are presented modifications to the algorithms which increase robustness are given

the undersampled discrete gabor transform conventional studies on discrete gabor transforms have generally been confined to the cases of critical sampling and oversampling in which the gabor families span the whole signal space in this paper we investigate undersampled discrete gabor transforms for an undersampled gabor triple g a b i e a spl middot b n we show that the associated generalized dual gabor window gdgw function is the same as the one associated with the oversampled g n b n a except for the constant factor ab n computations of undersampled gabor transforms are made possible by applying the methods algorithms developed in oversampled settings the undersampled gdgw is determined then we are able to obtain the best approximation of a signal x by linear combinations of vectors in the gabor family

discrete gabor structures and optimal representations the idea of gabor s 1946 signal expansion is to represent a signal in terms of a discrete set of time shifted and frequency modulated signals that are localized in the time frequency or phase space we present detailed descriptions of the block and banded structures for the gabor matrices based on the explicit descriptions of the sparsity of such matrices we can establish the sparse form of the gabor matrix and obtain the dual gabor atom mother wavelet the inverse of the gabor frame operator and carry out the discrete finite gabor transform in a very efficient way some explicit sufficient and also necessary conditions are derived for a gabor atom g to generate a gabor frame with respect to a tf lattice a b

the traveling salesman problem an overview of exact and approximate algorithms in this paper some of the main known algorithms for the traveling salesman problem are surveyed the paper is organized as follows 1 definition 2 applications 3 complexity analysis 4 exact algorithms 5 heuristic algorithms 6 conclusion

complete gabor transformation for signal representation properties of the gabor transformation used for image representation are discussed the properties can be expressed in matrix notation and the complete gabor coefficients can be found by multiplying the inverse of the gabor 1946 matrix and the signal vector the gabor matrix can be decomposed into the product of a sparse constant complex matrix and another sparse matrix that depends only on the window function a fast algorithm is suggested to compute the inverse of the window function matrix enabling discrete signals to be transformed into generalized nonorthogonal gabor representations efficiently a comparison is made between this method and the analytical method the relation between the window function matrix and the biorthogonal functions is demonstrated a numerical computation method for the biorthogonal functions is proposed

an efficient transformation of the generalized traveling salesman problem into the traveling salesman problem on digraphs abstract the generalized traveling salesman problem gtsp is a generalization of the traveling salesman problem tsp one of the outstanding intractable combinatorial optimization problems in the gtsp problem we are given n cities that are grouped into mutually disjoint districts clusters and nonnegative distances between the cities in different districts a traveling salesman has to find the shortest tour that visits exactly one city in each district in this paper we study the asymmetric version of this problem and describe a transformation by which an instance of the gtsp can be transformed into an instance of the asymmetric tsp with 2 n cities this compares favorably with the transformation proposed by lien and ma in 5 in which the resulting tsp has more than 3 n vertices we show that any optimal solution of the tsp instance corresponds to a unique optimal solution of the gtsp instance of no greater length thus the described transformation enables to solve the gtsp by applying some of the numerous tsp s heuristics or optimal approaches

a random key genetic algorithm for the generalized traveling salesman problem the generalized traveling salesman problem is a variation of the well known traveling salesman problem in which the set of nodes is divided into clusters the objective is to find a minimum cost tour passing through one node from each cluster we present an effective heuristic for this problem the method combines a genetic algorithm ga with a local tour improvement heuristic solutions are encoded using random keys which circumvent the feasibility problems encountered when using traditional ga encodings on a set of 41 standard test problems with symmetric distances and up to 442 nodes the heuristic found solutions that were optimal in most cases and were within 1 of optimality in all but the largest problems with computation times generally within 10 seconds the heuristic is competitive with other heuristics published to date in both solution quality and computation time

discrete gabor transform a feasible algorithm for implementing the gabor expansion the coefficients of which are computed by the discrete gabor transform dgt is presented for a given synthesis window and sampling pattern computing the auxiliary biorthogonal function of the dgt is nothing more than solving a linear system the dgt presented applies for both finite as well as infinite sequences by exploiting the nonuniqueness of the auxiliary biorthogonal function at oversampling an orthogonal like dgt is obtained as the discrete fourier transform dft is a discrete realization of the continuous time fourier transform similarly the dgt introduced provides a feasible vehicle to implement the useful gabor expansion

oversampling in the gabor scheme a method for calculating the coefficients of the gabor expansion in the case of oversampling is presented the method is based on the concept of frames and utilizes the zak transform as such the zak transform highlights the meaning and importance of frames in the context of oversampling and other aspects of signal representation by means of nonorthogonal bases

computationally attractive real gabor transforms we present a gabor transform for real discrete signals and present a computationally attractive method for computing the transform for the critically sampled case we derive a biorthogonal function which is very localized in the time domain therefore truncation of this biorthogonal function allows us to compute approximate expansion coefficients with significantly reduced computational requirements further truncation does not degrade the numerical stability of the transform we present a tight upper bound on the reconstruction error incurred due to use of a truncated biorthogonal function and summarize computational savings for example the expense of transforming a length 2048 signal using length 16 blocks is reduced by a factor of 26 over similar fft based methods with at most 0 04 squared error in the reconstruction

ricci curvature bounds and einstein metrics on compact manifolds where f mo m1 is a homeomorphism and dil f is the dilatation of f given by dil f supx x2 dist f x f x2 dist x1 x2 if mo and m1 are not homeomorphic define dl mo mi oo gromov 20 proves the remarkable result that the space of compact riemannian manifolds f a 3 d of sectional curvature iki 3 0 and diameter dm v and diameter dm c iki dm vm1 in particular gromov s compactness theorem may be strengthened to the statement that f a v d is c1 l compact in the lipschitz topology in this paper we study the question of lipschitz convergence of compact riemannian manifolds with bounds imposed on the ricci curvature ric in

the generalized gabor transform the generalized gabor transform for image representation is discussed for a given function f t t spl isin r the generalized gabor transform finds a set of coefficients a sub mr such that f t spl sigma sub m spl infin sup spl infin spl sigma sub r spl infin sup spl infin spl alpha sub mr g t mt exp i2 spl pi rt t the original gabor transform proposed by d gabor 1946 is the special case of t t the computation of the generalized gabor transform with biorthogonal functions is discussed the optimal biorthogonal functions are discussed a relation between a window function and its optimal biorthogonal function is presented based on the zak 1967 transform when t t is rational the finite discrete generalized gabor transform is also derived methods of computation for the biorthogonal function are discussed the relation between a window function and its optimal biorthogonal function derived for the continuous variable generalized gabor transform can be extended to the finite discrete case efficient algorithms for the optimal biorthogonal function and generalized gabor transform for the finite discrete case are proposed

optimal biorthogonal analysis window function for discrete gabor transform like the dft spl minus a discrete version of the fourier transform spl minus the recently developed discrete gabor transform dgt provides a feasible vehicle to implement the very useful gabor expansion in general the choice of the biorthogonal window function spl gamma i is not unique the authors discuss the solution of spl gamma i that is optimally close to an arbitrary desired function d i when the synthesis window h i and sampling pattern are given for d i h i the resulting spl gamma sub opt i directly leads to so called orthogonal like dgt combining the dgt they believe that the result presented in this paper is rather significant for digital signal processing

a lagrangian based approach for the asymmetric generalized traveling salesman problem this paper presents an optimal approach for the asymmetric generalized traveling salesman problem gtsp the gtsp is defined on a directed graph in which the nodes are grouped into m predefined mutually exclusive and exhaustive sets with the arc set containing no intraset arcs the problem is to find a minimum cost m arc directed cycle which includes exactly one node from each set our approach employs a lagrangian relaxation to compute a lower bound on the total cost of an optimal solution the lower bound and a heuristically determined upper bound are used to identify and remove arcs and nodes which are guaranteed not to be in an optimal solution finally we use an efficient branch and bound procedure which exploits the multiple choice structure of the node sets we present computational results for the optimal approach tested on a series of randomly generated problems the results show success on a range of problems with up to 104 nodes

smoothing riemannian metrics with bounded ricci curvatures in dimension four abstract we obtain a local smoothing result for riemannian manifolds with bounded ricci curvatures in dimension four more precisely given a riemannian metric with bounded ricci curvature and small l 2 norm of curvature on a metric ball we can find a smooth metric with bounded curvature which is c 1 close to the original metric on a smaller ball but still of definite size

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

real analyticity of solutions of hamilton s equation il existe des constantes positives cn et sn dependant seulement de la dimension n de la variete m telles que pour une metrique initiale g 0 avec r g 0 1 l equation de hamilton a une solution jusqu a t sn et l estimation suivante est vraie pour la metrique solution en t o t sn k 0 t k k 1 12 k r 2 c n

transformation of the generalized traveling salesman problem into the standard traveling salesman problem abstract in the generalized traveling salesman problem gtsp we are given a set of cities that are grouped into possibly intersecting clusters the objective is to find a closed path of minimum cost that visits at least one city in each cluster given an instance g of the gtsp we first transform g into another instance g of the gtsp in which all the clusters are nonintersecting and then transform g into an instance g of the standard traveling salesman problem tsp we show that any feasible solution of the tsp instance g can be transformed into a feasible solution of the gtsp instance g of no greater cost and that any optimal solution of the tsp instance g can be transformed into an optimal solution of the gtsp instance g

the generalized gabor scheme of image representation in biological and machine vision a scheme suitable for visual information representation in a combined frequency position space is investigated through image decomposition into a finite set of two dimensional gabor elementary functions gef the scheme is generalized to account for the position dependent gabor sampling rate oversampling logarithmic frequency scaling and phase quantization characteristic of the visual system comparison of reconstructed signal highlights the advantages of the generalized gabor scheme in coding typical bandlimited images it is shown that there exists a tradeoff between the number of frequency components used per position and the number of such clusters sampling rate utilized along the spatial coordinate

generalized network design problems abstract network design problems ndp consist of identifying an optimal subgraph of a graph subject to side constraints in generalized ndp the vertex set is partitioned into clusters and the feasibility conditions are expressed in terms of the clusters several applications of generalized ndp arise in the fields of telecommunications transportation and biology the aim of this review article is to formally define generalized ndp to study their properties and to provide some applications

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

deforming the metric on complete riemannian manifolds soit m g ij x une variete de riemann a n dimensions complete non compacte de tenseur de complexe riemannien r ijkl satisfaisant r ijkl 2 k 0 sur m ou 0 0 dependant seulement de n et de k 0 telle que l equation d evolution g ij x t t 2r ij x t sur m g ij x o g ij x x m ait une solution lisse g ij x t 0 pour un temps court 0 t t u k 0 et satisfait l estimation suivante pour un entier m 0 il existe des constantes c m 0 dependant seulement de n m et k 0 telles que sup x m v m r ijkl x t 2 c m t m 0 t t u k 0

nilpotent structures and invariant metrics on collapsed manifolds let mn be a complete riemannian manifold of bounded curvature say iki 0 we put mn w n c u fn c where 1n e consists of those points at which the injectivity radius of the exponential map is e the complementary set n c is called the e collapsed part of mn if x e 4 n r e then the metric ball bx r is quasi isometric with small distortion to the flat ball b0 r in the euclidean space rn after slightly