asymptotically optimal quantizers for detection of i i d the asymptotic probability of error for quantization in maximum likelihood tests is analyzed the authors assume quantizers with large numbers of levels generated from a companding function a theorem that relates the companding function to the asymptotic probability of error is proved the companding function is then optimized

harmonics in gaussian multiplicative and additive noise cramer rao bounds the concern here is retrieval of single and multiple tone harmonics observed in white gaussian multiplicative and additive noise computable cramer rao bound crb expressions are derived on the frequency and phase estimates as well as on the sample mean or variance of the multiplicative noise processes the zero and nonzero mean multiplicative noise cases are addressed separately and are shown to yield distinct crbs on the frequency and phase estimates tight lower and upper bounds on the crbs themselves are developed which relative to the crbs are intuitively more appealing and easier to implement well established formulas on the achievable accuracy for estimates of constant amplitude harmonics turn put to be special cases of our results numerical studies support our claims

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

a weak formulation of roe s approximate riemann solver abstract this paper presents a weak formulation of roe s approximate riemann solver based on a notion of weak solution to nonlinear hyperbolic systems in nonconservative form this weak formulation is applied to the euler equations for real gases and the homogeneous equilibrium two phase flow model numerical results for a two phase flow heat addition problem are given

a godunov type method for the shallow water equations with discontinuous topography in the resonant regime abstract we investigate the riemann problem for the shallow water equations with variable and possibly discontinuous topography and provide a complete description of the properties of its solutions existence uniqueness in the non resonant regime multiple solutions in the resonant regime this analysis leads us to a numerical algorithm that provides one with a riemann solver next we introduce a godunov type scheme based on this riemann solver which is well balanced and of quasi conservative form finally we present numerical experiments which demonstrate the convergence of the proposed scheme even in the resonance regime except in the limiting situation when riemann data precisely belong to the resonance hypersurface

on multicommutators for simple algebraic groups abstract there are several examples of groups for which any pair of commutators can be written such that both of them have a common entry and one can look for a similar property for n tuples of commutators here we answer for simple algebraic groups over any field the weaker question under which condition the set of n tuples of commutators with one common entry is zariski dense in the set of all n tuples of commutators surprisingly there is a uniform bound on n in terms of the so called coxeter number of g to answer the question positively an analogue result is proved for lie algebras of simple and simply connected algebraic groups

a well balanced approach for flows over mobile bed with high sediment transport in this paper we deal with the numerical computation of one dimensional unsteady free surface flows over mobile bed we focus on flows characterized by high concentration of sediments and strong interaction between flow and bottom dynamics as in hyper concentrated and debris flows these features are fully considered in the adopted system of equations challenging in its numerical approximation is the preservation of the coupling and the treatment of a non conservative flux in the momentum equation in order to devise a new godunov type approach we analyzed in detail the riemann problem associated with the mobile bed phenomena and the peculiar features of its wave relations the scheme we developed is based on two supports well balanced treatment of the variable updating at the new time level and flux evaluation by three wave approximations of the intercell riemann problem that without any split embody the effect of the non conservative term the properties of the new numerical strategy named awb are assessed by comparison with exact solutions of riemann problems built by handling an inverse technique finally awb has been applied to cases of practical interest where wave interaction and friction effects makes the flow more complex the obtained results point out that the new method is able to predict faithfully the overall behaviour of the solution and of any type of waves the use of awb in this one dimensional frame is therefore fostered in representing rapid transients in river torrent flows with movable bed

on the conjectures of j thompson and o ore if g is a finite simple group of lie type over a field containing more than 8 elements for twisted groups xn ql we require q 8 except for 2b2 q2 2g2 q2 and 2 f4 q2 where we assume q2 8 then g is the square of some conjugacy class and consequently every element in g is a commutator

optimality of klt for high rate transform coding of gaussian vector scale mixtures application to reconstruction estimation and classification the karhunen loeacuteve transform klt is known to be optimal for high rate transform coding of gaussian vectors for both fixed rate and variable rate encoding the klt is also known to be suboptimal for some non gaussian models this paper proves high rate optimality of the klt for variable rate encoding of a broad class of non gaussian vectors gaussian vector scale mixtures gvsm which extend the gaussian scale mixture gsm model of natural signals a key concavity property of the scalar gsm same as the scalar gvsm is derived to complete the proof optimality holds under a broad class of quadratic criteria which include mean squared error mse as well as generalized f divergence loss in estimation and binary classification systems finally the theory is illustrated using two applications signal estimation in multiplicative noise and joint optimization of classification reconstruction systems

commutators in classical lie algebras abstract if the field k has enough elements or if its characteristic is suitably restricted then for a given element a of a perfect classical lie algebra l it is possible to compute in polynomial time x y l such that a x y

exact solution of the riemann problem for the shallow water equations with discontinuous bottom geometry in this paper we present the exact solution of the riemann problem for the non linear shallow water equations with a step like bottom the solution has been obtained by solving an enlarged system that includes an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step the resulting solution is unique and satisfies the principle of dissipation of energy across the shock wave we provide examples of possible wave patterns numerical solution of a first order dissipative scheme as well as an implementation of our riemann solver in the second order upwind method are compared with the proposed exact riemann problem solution a practical implementation of the proposed exact riemann solver in the framework of a second order upwind tvd method is also illustrated

word maps word maps with constants and representation varieties of one relator groups abstract we consider word maps and word maps with constants on a simple algebraic group g we present results on the images of such maps in particular we prove a theorem on the dominance of general word maps with constants which can be viewed as an analogue of a well known theorem of borel on the dominance of genuine word maps besides we establish a relationship between the existence of unipotents in the image of the map induced by w f m and the structure of the representation variety r w g of the group w f m w

a lagrangian formulation of zador s entropy constrained quantization theorem zador s 1963 1966 classic result for the asymptotic high rate behavior of entropy constrained vector quantization is recast in a lagrangian form which better matches the lloyd algorithm used to optimize such quantizers the equivalence of the two formulations is shown and the result is proved for source distributions that are absolutely continuous with respect to the lebesgue measure which satisfy an entropy condition thereby generalizing the conditions stated by zador under which the result holds

the riemann problem for the one dimensional free surface shallow water equations with a bed step theoretical analysis and numerical simulations in this paper the solution of the riemann problem for the one dimensional free surface shallow water equations over a bed step is analyzed both from a theoretical and a numerical point of view particular attention has been paid to the wave that is generated at the location of the bed discontinuity starting from the classical shallow water equations considering the bed level as an additional variable and adding to the system an equation imposing its time invariance we show that this wave is a contact wave across which one of the riemann invariants namely the energy is not constant this is due to the fact that the relevant problem is nonconservative we demonstrate that in this type of system riemann invariants do not generally hold in contact waves furthermore we show that in this case the equations that link the flow variables across the contact wave are the generalized rankine hugoniot relations and we obtain these for the specific problem from the numerical point of view we present an accurate and efficient solver for the step riemann problem to be used in a finite volume godunov type framework through a two step predictor corrector procedure the solver is able to provide solutions with any desired accuracy the predictor step uses a well balanced generalized roe solver while the corrector step solves the exact nonlinear system of equations that consitutes the problem by means of an iterative procedure that starts from the predictor solution in order to show the effectiveness and the accuracy of the proposed approach we consider several step riemann problems and compare the exact solutions with the numerical results obtained by using a standard roe approach far from the step and the novel two step algorithm for the fluxes over the step achieving good results

automorphic orbits in free groups abstract let fn be the free group of a finite rank n we study orbits orb u where u is an element of the group fn under the action of an automorphism if an orbit like that is finite we determine precisely what its cardinality can be if u runs through the whole group fn and runs through the whole group aut fn another problem that we address here is related to whitehead s algorithm that determines whether or not a given element of a free group of finite rank is an automorphic image of another given element it is known that the first part of this algorithm reducing a given free word to a free word of minimum possible length by elementary whitehead automorphisms is fast of quadratic time with respect to the length of the word on the other hand the second part of the algorithm applied to two words of the same minimum length was always considered very slow we give here an improved algorithm for the second part and we believe this algorithm always terminates in polynomial time with respect to the length of the words we prove that this is indeed the case if the free group has rank 2

sums of orbits of algebraic groups i abstract let g be a simple algebraic group which is defined and split over a field k and let l be a corresponding lie algebra further let r be the corresponding root system we prove here that if char k 0 or a very good prime for r and k r then there exists an orbit o l l k with respect to the adjoint action of g g k such that l o o this is an analogue of the corresponding result for chevalley groups the thompson problem see e w ellers n gordeev on the conjecture of j thompson and o ore trans amer math soc 350 1998 3657 3671 we also prove that if char k 2 for r b r c r f 4 and char k 3 for r g 2 then the zariski closure of a sum of any r r b r r 3 d r e r f 4 r 1 r a r b 3 g 2 2r r c r g orbits of elements of l z l coincides with l for the group g sl 2 k where k is an algebraically closed field of characteristic zero we list all cases of rational k g modules v which have an orbit o v such that o o v

combining image compression and classification using vector quantization we describe a method of combining classification and compression into a single vector quantizer by incorporating a bayes risk term into the distortion measure used in the quantizer design algorithm once trained the quantizer can operate to minimize the bayes risk weighted distortion measure if there is a model providing the required posterior probabilities or it can operate in a suboptimal fashion by minimizing the squared error only comparisons are made with other vector quantizer based classifiers including the independent design of quantization and minimum bayes risk classification and kohonen s lvq a variety of examples demonstrate that the proposed method can provide classification ability close to or superior to learning vq while simultaneously providing superior compression performance

information theoretic bounds on target recognition performance based on degraded image data this paper derives bounds on the performance of statistical object recognition systems wherein an image of a target is observed by a remote sensor detection and recognition problems are modeled as composite hypothesis testing problems involving nuisance parameters we develop information theoretic performance bounds on target recognition based on statistical models for sensors and data and examine conditions under which these bounds are tight in particular we examine the validity of asymptotic approximations to probability of error in such imaging problems problems involving gaussian poisson and multiplicative noise and random pixel deletions are considered as well as least favorable gaussian clutter a sixth application involving compressed sensor image data is considered in some detail this study provides a systematic and computationally attractive framework for analytically characterizing target recognition performance under complicated non gaussian models and optimizing system parameters

the riemann problem for the shallow water equations with discontinuous topography the wet dry case abstract in this paper we consider riemann problems for the shallow water equations with discontinuous topography whose initial conditions correspond to a wet dry front at time t 0 there is vacuum on the right or on the left of the step besides the theoretical interest of this analysis the results may be useful to design numerical methods and or to produce reference solutions to compare different schemes we show that depending on the state at the wet side 0 1 or 2 self similar solutions can be constructed by composing simple waves in problems with 0 solutions the step acts as an obstacle for the fluid and physically meaningful solutions can be constructed by interpreting the problem as a partial riemann problems for the homogeneous shallow water system some numerical results are shown where different numerical methods are compared in particular it is shown that in the non uniqueness cases the numerical solutions can converge to one or to the other solution what is the reason that explains the huge differences observed when different numerical methods are applied to the shallow water system with abrupt changes in the bottom moreover problems with zero solutions will be reinterpreted as partial riemann problems for the homogeneous system what will allow us to build a physically solution when one side of the step is wet and the other one is dry we will specify the regions where we can find zero one or two solutions giving the form of the solution when it is possible and giving an alternative when it is not possible

self adjusting grid methods for one dimensional hyperbolic conservation laws it is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws the grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals the resulting algorithm is a simple extension of many currently popular godunov type methods computer codes using one of these methods can be easily modified to add the moving mesh as an option numerical examples are given that illustrate the improved accuracy of godunov s and roe s methods on a self adjusting mesh

multiresolution graph fourier transform for compression of piecewise smooth images piecewise smooth pws images e g depth maps or animation images contain unique signal characteristics such as sharp object boundaries and slowly varying interior surfaces leveraging on recent advances in graph signal processing in this paper we propose to compress the pws images using suitable graph fourier transforms gfts to minimize the total signal representation cost of each pixel block considering both the sparsity of the signal s transform coefficients and the compactness of transform description unlike fixed transforms such as the discrete cosine transform we can adapt gft to a particular class of pixel blocks in particular we select one among a defined search space of gfts to minimize total representation cost via our proposed algorithms leveraging on graph optimization techniques such as spectral clustering and minimum graph cuts furthermore for practical implementation of gft we introduce two techniques to reduce computation complexity first at the encoder we low pass filter and downsample a high resolution hr pixel block to obtain a low resolution lr one so that a lr gft can be employed at the decoder upsampling and interpolation are performed adaptively along hr boundaries coded using arithmetic edge coding so that sharp object boundaries can be well preserved second instead of computing gft from a graph in real time via eigen decomposition the most popular lr gfts are pre computed and stored in a table for lookup during encoding and decoding using depth maps and computer graphics images as examples of the pws images experimental results show that our proposed multiresolution gft scheme outperforms h 264 intra by 6 8 db on average in peak signal to noise ratio at the same bit rate

information theoretic analysis of interscale and intrascale dependencies between image wavelet coefficients this paper presents an information theoretic analysis of statistical dependencies between image wavelet coefficients the dependencies are measured using mutual information which has a fundamental relationship to data compression estimation and classification performance mutual information is computed analytically for several statistical image models and depends strongly on the choice of wavelet filters in the absence of an explicit statistical model a method is studied for reliably estimating mutual information from image data the validity of the model based and data driven approaches is assessed on representative real world photographic images our results are consistent with empirical observations that coding schemes exploiting inter and intrascale dependencies alone perform very well whereas taking both into account does not significantly improve coding performance a similar observation applies to other image processing applications

products of cyclic similarity classes in the groups gln f abstract let g gln f be the group of all n n invertible matrices over a field f where f 4 and n 3 let j1 j2 be two prescribed cyclic elements of g in jordan canonical form and assume that all the eigenvalues of j1 or j2 lie in f it is shown that any nonscalar matrix in g can be written as a product of two matrices having the prescribed jordan canonical forms j1 j2 subject only to the obvious 0 determinant condition as a corollary we show that if f 4 the group psln f has a conjugacy class c such that c2 psln f another corollary states that every nonscalar matrix in g can be written as a product of two cyclic unipotent matrices

finite subgroups of algebraic groups generalizing a classical theorem of jordan to arbitrary characteristic we prove that every finite subgroup of gln over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of lie type in characteristic p a commutative group of order prime to p and a p group while this statement can be deduced from the classification of finite simple groups our proof is self contained and uses methods only from algebraic geometry and the theory of linear algebraic groups we believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics

reliability of first order numerical schemes for solving shallow water system over abrupt topography we compare some first order well balanced numerical schemes for shallow water system with special interest in applications where there are abrupt variations of the topography we show that the space step required to obtain a prescribed error depends on the method moreover the solutions given by the numerical scheme can be significantly different if not enough space resolution is used we shall pay special attention to the well known hydrostatic reconstruction technique where it is shown that the effect of large bottom discontinuities might be missed and a modification is proposed to avoid this problem

arbitrarily tight upper and lower bounds on the bayesian probability of error this paper presents new upper and lower bounds on the minimum probability of error of bayesian decision systems for the two class problem these bounds can be made arbitrarily close to the exact minimum probability of error making them tighter than any previously known bounds

fine quantization in signal detection and estimation the performance lost to data quantization in signal detection and estimation procedures is considered the performance is measured by f divergences which are useful indices of discrimination information between statistical hypotheses some properties of the f divergences are briefly discussed and a general result for the loss in divergence due to uniform data quantization is obtained several applications of this result in specific problems of signal detection and estimation being developed and some numerical results that illustrate the asymptotic behavior of the divergence in these applications are given the divergence lost to finite nonuniform quantization through the use of a companding model is considered and the asymptotically optimum nonuniform quantizer within this class is derived some interesting problems that remain open are discussed

the uniqueness of the exact solution of the riemann problem for the shallow water equations with discontinuous bottom abstract the riemann problem for the shallow water equations with discontinuous topography is considered in a general case the exact solution of this problem is not unique which complicates the application of an exact riemann solver in numerical methods since it is not clear which solution should be chosen in the present work it is shown that involving an additional physical assumption makes it possible to prove the existence and uniqueness of the solution the assumption is that the discharge at the bottom discontinuity should continuously depend on the initial conditions the proven uniqueness opens up a possibility to use an exact riemann solver for a numerical solution of the shallow water equations with complex discontinuous topography

new error bounds and optimum quantization for multisensor distributed signal detection the binary signal detection problem is considered when a distributed system of sensors operates in a decentralized fashion local processing at each sensor is performed using chernoff s large deviation theorems the author considers as a criterion the rate of convergence of the error probability to zero it is shown that the optimum quantizer of blocks of data under the above criterion is the likelihood ratio quantizer a lower bound to the error probability is also developed the question of how many coarsely quantized sensors can replace the infinitely quantized one is also answered the main result given is the structure of the optimum quantizer consisting of the calculation of the likelihood ratio concatenated by a scalar quantizer

modelling the transition between fixed and mobile bed conditions in two phase free surface flows in a two phase free surface flow the transition from a mobile bed condition to a fixed bed one and vice versa occurs at a sharp interface across which the relevant system of partial differential equations changes abruptly this leads to the possibility of conceiving a new type of riemann problem rp which we have called composite riemann problem crp where not only the initial constant values of the variables but also the system of equations change from left to right of a discontinuity in this paper we present a strategy for solving a crp by reducing it to a standard rp of a single composite system of equations this can be obtained by combining the two original systems by means of a suitable weighting function namely the erodibility variable and the introduction of an appropriate differential equation for this quantity in this way the crp problem can be analyzed theoretically with standard methods and the features of the solutions can be clearly identified in particular a stationary contact wave is able to correctly describe the sharp transition between mobile and fixed bed conditions a finite volume scheme based on the multiple averages generalized roe approach rosatti and begnudelli 2013 22 was used to numerically solve the fixed mobile crp several test cases demonstrate the effectiveness exact well balanceness and high accuracy of the scheme when applied to problems that fall within the physical range of applicability of the relevant mathematical model

diameters of finite simple groups sharp bounds and applications let g be a finite simple group and let s be a normal subset of g we determine the diameter of the cayley graph r g s associated with g and s up to a multiplicative constant many applications follow for example we deduce that there is a constant c such that every element of g is a product of c involutions and we generalize this to elements of arbitrary order we also show that for any word w w xl xd there is a constant c c w such that for any simple group g on which w does not vanish every element of g is a product of c values of w from this we deduce that every verbal subgroup of a semisimple profinite group is closed other applications concern covering numbers expanders and random walks on finite simple groups

the waring problem for finite simple groups the classical waring problem deals with expressing every natural number as a sum of g k k th powers recently there has been considerable interest in similar questions for non abelian groups and simple groups in particular here the k th power word can be replaced by an arbitrary group word w 6 1 and the goal is to express group elements as short products of

criteria for equidistribution of solutions of word equations on sl 2 abstract we study equidistribution of solutions of word equations of the form w x y g in the family of finite groups sl 2 q we provide criteria for equidistribution in terms of the trace polynomial of w this allows us to get an explicit description of certain classes of words possessing the equidistribution property and show that this property is generic within these classes

the covering number of the grouppsln f abstract let g be the group psl n f where n 3 f is a field and f 4 assume further that if n 3 then f is either finite or algebraically closed given an integer k and a subset a g denote a k a 1 a 2 a k a 1 a 2 a k a denote by cn g the minimal value of k such that c k g for every nontrivial conjugacy class c of g it is shown that cn g n related results on factorizations of matrices in gl n f and sl n f are also discussed

on engel words on simple algebraic groups abstract let g be a simple algebraic group defined over an algebraically closed field k and let g g k we consider here the problem when an element g g can be presented in the form g e n g 1 g 2 g 1 g 1 g 1 n times g 2 for some g 1 g 2 g this can always be done if g is a semisimple or a unipotent element also it is known that this holds for g sl 2 k here we prove that g e n g 1 g 2 for every n if g is a group of type b 2 or g 2 and we prove g e 2 g 1 g 2 g 1 g 1 g 2 if g pgl 3 k we also show g e 2 g 1 g 2 g 1 g 1 g 2 if g is a regular element of g and g is a group of rank 3 for any simple group g we give a criterion for some general regular elements of g to be presented in the form g 1 g 1 g 2

higher secant varieties of the minimal adjoint orbit the adjoint group of a simple complex lie algebra g has a unique minimal orbit in the projective space pg whose pre image in g we denote by c we explicitly describe for every classical g and every natural number k the zariski closure k over bar c over bar of the union kc of all spaces spanned by k points on c the image of k over bar c over bar in pg is usually called the k 1 st secant variety of pc these higher secant varieties are known and easily determined for g sl n or g sp 2n for completeness we give short proofs of these results our main contribution is therefore the explicit description of k over bar c over bar for g o n where the embedding of pc into po n is isomorphic to the phicker embedding of the grassmannian of isotropic lines in pn 1 into p n 1 n 2 1 we show that the first and the second secant variety are then characterised by certain conditions on the eigenvalues of matrices in o n while the third and higher secant varieties coincide with those of the grassmannian of all projective lines finally unlike for g sl n or sp 2n the sets kc are not all closed in o n and we present a partial result on the nilpotent orbits contained in them c 2004 elsevier inc all rights reserved

bayes risk weighted vector quantization with posterior estimation for image compression and classification classification and compression play important roles in communicating digital information their combination is useful in many applications including the detection of abnormalities in compressed medical images in view of the similarities of compression and low level classification it is not surprising that there are many similar methods for their design because some of these methods are useful for designing vector quantizers it seems natural that vector quantization vq is explored for the combined goal we investigate several vq based algorithms that seek to minimize both the distortion of compressed images and errors in classifying their pixel blocks these algorithms are investigated with both full search and tree structured codes we emphasize a nonparametric technique that minimizes both error measures simultaneously by incorporating a bayes risk component into the distortion measure used for the design and encoding we introduce a tree structured posterior estimator to produce the class posterior probabilities required for the bayes risk computation in this design for two different image sources we demonstrate that this system provides superior classification while maintaining compression close or superior to that of several other vq based designs including kohonen s 1992 learning vector quantizer and a sequential quantizer classifier design

a closure independent generalized roe solver for free surface two phase flows over mobile bed several different natural phenomena can be studied in the framework of free surface two phase flows over mobile bed mathematically they can be described by the same set of highly nonlinear hyperbolic nonconservative pdes but they differ in the possible algebraic closure relations these affect significantly the relevant eigenvalues and consequently all finite volume numerical methods based on upwind godunov type fluxes in this work the generalized roe solver introduced in 29 for the case of a specific closure is reformulated in a complete closure independent way this gives the solver a quite general applicability to the class of problems previously mentioned moreover the new method maintains all the desirable features shown by the original one full two dimensionality and exact well balanceness this result is made possible thanks to the development of a novel multiple averages mas approach that allows a straightforward determination of the matrices required by the solver several tests show the capabilities of the proposed numerical strategy

high resolution source coding for non difference distortion measures multidimensional companding entropy coded vector quantization is studied using high resolution multidimensional companding over a class of nondifference distortion measures for distortion measures which are locally quadratic a rigorous derivation of the asymptotic distortion and entropy coded rate of multidimensional companders is given along with conditions for the optimal choice of the compressor function this optimum compressor when it exists depends on the distortion measure but not on the source distribution the rate distortion performance of the companding scheme is studied using an asymptotic expression for the rate distortion function which parallels the shannon lower bound for difference distortion measures it is proved that the high resolution performance of the scheme is arbitrarily close to the rate distortion limit for large quantizer dimensions if the compressor function and the lattice quantizer used in the companding scheme are optimal extending an analogous statement for entropy coded lattice quantization and mse distortion the companding approach is applied to obtain a high resolution quantizing scheme for noisy sources

a godunov type scheme for the isentropic model of a fluid flow in a nozzle with variable cross section we present a godunov type scheme for the isentropic model of a fluid flow in a nozzle with variable cross section the model is of nonconservative form making it hard for standard numerical discretizations of the nonconservative term in particular the error for a standard numerical scheme with a usual discretization of the nonconservative term may become larger as the mesh size gets smaller we first re investigate the riemann problem of the model pointing out several interesting properties of the wave curves and establishing specific existence domain for each type of solutions then we incorporate local riemann solutions to build a godunov type scheme for the model the scheme is constructed in subsonic and supersonic regions where the system is strictly hyperbolic tests show that our scheme can capture standing waves so that it is well balanced furthermore tests also show that our godunov type scheme can give a good accuracy for numerical approximations of exact solutions our godunov type scheme can resolve the difficulty of other existing schemes for similar models of fluid flows with nonconservative terms

adaptive wavelet thresholding for image denoising and compression the first part of this paper proposes an adaptive data driven threshold for image denoising via wavelet soft thresholding the threshold is derived in a bayesian framework and the prior used on the wavelet coefficients is the generalized gaussian distribution ggd widely used in image processing applications the proposed threshold is simple and closed form and it is adaptive to each subband because it depends on data driven estimates of the parameters experimental results show that the proposed method called bayesshrink is typically within 5 of the mse of the best soft thresholding benchmark with the image assumed known it also outperforms sureshrink donoho and johnstone 1994 1995 donoho 1995 most of the time the second part of the paper attempts to further validate claims that lossy compression can be used for denoising the bayesshrink threshold can aid in the parameter selection of a coder designed with the intention of denoising and thus achieving simultaneous denoising and compression specifically the zero zone in the quantization step of compression is analogous to the threshold value in the thresholding function the remaining coder design parameters are chosen based on a criterion derived from rissanen s minimum description length mdl principle experiments show that this compression method does indeed remove noise significantly especially for large noise power however it introduces quantization noise and should be used only if bitrate were an additional concern to denoising

a robust numerical method for approximating solutions of a model of two phase flows and its properties the objective of the present paper is to extend our earlier works on simpler systems of balance laws in nonconservative form such as the model of fluid flows in a nozzle with variable cross section to a more complicated system consisting of seven equations which has applications in the modeling of deflagration to detonation transition in granular materials first we transform the system into an equivalent one which can be regarded as a composition of three subsystems then depending on the characterization of each subsystem we propose a convenient numerical treatment of the subsystem separately precisely in the first subsystem of the governing equations in the gas phase stationary waves are used to absorb the nonconservative terms into an underlying numerical scheme in the second subsystem of conservation laws of the mixture we can take a suitable scheme for conservation laws for the third subsystem of the compaction dynamics equation the fact that the velocities remain constant across solid contacts suggests us to employ the technique of engquist osher s scheme then we prove that our method possesses some interesting properties it preserves the positivity of the volume fractions in both phases and in the gas phase our scheme is capable of capturing equilibrium states preserves the positivity of the density and satisfies the numerical minimum entropy principle numerical tests show that our scheme can provide reasonable approximations for data the supersonic regions but the results are not satisfactory in the subsonic region however the scheme is numerically stable and robust

suboptimality of the karhunen loeve transform for transform coding we examine the performance of the karhunen loeve transform klt for transform coding applications the klt has long been viewed as the best available block transform for a system that orthogonally transforms a vector source scalar quantizes the components of the transformed vector using optimal bit allocation and then inverse transforms the vector this paper treats fixed rate and variable rate transform codes of non gaussian sources the fixed rate approach uses an optimal fixed rate scalar quantizer to describe the transform coefficients the variable rate approach uses a uniform scalar quantizer followed by an optimal entropy code and each quantized component is encoded separately earlier work shows that for the variable rate case there exist sources on which the klt is not unique and the optimal quantization and coding stage matched to a worst klt yields performance as much as 1 5 db worse than the optimal quantization and coding stage matched to a best klt in this paper we strengthen that result to show that in both the fixed rate and the variable rate coding frameworks there exist sources for which the performance penalty for using a worst klt can be made arbitrarily large further we demonstrate in both frameworks that there exist sources for which even a best klt gives suboptimal performance finally we show that even for vector sources where the klt yields independent coefficients the klt can be suboptimal for fixed rate coding

on the uniform quantization of a class of sparse sources we consider the uniform scalar quantization of a class of mixed distributed memoryless sources namely sources having a bernoulli generalized gaussian bgg distribution both for low and high resolutions asymptotic expressions of the distortion for a pth order moment error measure and close approximations of the entropy are provided for these sources operational rate distortion functions at high bit rate and their slope factors at low bit rate are derived the dependence of these results on p and the distribution parameters as well as the relation to the shannon optimal rate distortion bound are then discussed the application of these results to transform coding in two simple cases is finally highlighted

supernormal mappings part 1 orthogonally indecomposable modules and applications abstract let v be a finite dimensional vector space over a field and f v v k a regular hermitian form a linear mapping v v with adjoint mapping i e f v w f v w for all v w v is called supernormal or polynomially normal or simply s normal if s holds true for some polynomial s k x if is a unitary transformation or if is self adjoint or if is anti self adjoint then is s normal for s normal mappings a classification of orthogonally indecomposable modules is obtained the classification distinguishes four types and depends on the form f the minimum polynomial of and whether is nilpotent or not we prove a uniqueness statement for orthogonal decompositions into orthogonally indecomposable modules of similar s normal mappings as an application we generalize the fact that in an orthogonal group each element is a product of two involutions in the generalized setting both factors are self adjoint or anti self adjoint and the first one is an involution the last section establishes a decomposition of v into orthogonally indecomposable modules such that each module is also invariant under both given factors part 2 will be devoted to isometric similarity of s normal mappings

the resonant cases and the riemann problem for a model of two phase flows abstract the resonant cases in the riemann problem for a model of two phase flows with resonance are investigated which complete the construction of riemann solutions the model is given by a nonconservative hyperbolic system of balance laws the phase decomposition method is developed to construct solutions in each phase in a relatively separated way but still glued to each other via the contact discontinuities composite wave curves can be built and intersections of wave curves in each phase plane help to determine various configurations of riemann solutions in particular solutions containing multiple waves propagating with the same shock speed are constructed

word maps and waring type problems waring s problem asks whether every natural number is a sum of g k fcth powers where g is a suitable function this was solved affirmatively by hubert in 1909 optimizing g k has been a central problem in additive number theory ever since see na for more detail and background recently there has been considerable interest in group theoretic analogues of this phenomenon where the aim is to present group elements as short products of certain special elements these special elements can be powers or commutators or values of a general word w or elements of a special conjugacy class in the group the motivation for this is sometimes topological let g be a topologically finitely generated pro p group using expressions of elements of the commutator subgroup g1 as bounded products of commutators serre showed that g is closed and deduced that every finite index subgroup of g is open a recent deep result of nikolov and segal nsi ns2 shows that this holds for every finitely generated profinite group again the core of the proof is presenting group elements as short products of values of certain words

equations in simple lie algebras abstract we prove that for a given element p x 1 x d of the finitely generated free lie algebra l d the induced map p g d g is dominant for any chevalley algebra g provided that k is of characteristic 2 and p is not an identity in sl 2 k we prove that for the engel monomials x y y y and for their linear combinations this map is surjective onto the set of non central elements of g provided that the ground field k is big enough

bayesian tree structured image modeling using wavelet domain hidden markov models wavelet domain hidden markov models have proven to be useful tools for statistical signal and image processing the hidden markov tree hmt model captures the key features of the joint probability density of the wavelet coefficients of real world data one potential drawback to the hmt framework is the need for computationally expensive iterative training to fit an hmt model to a given data set e g using the expectation maximization algorithm we greatly simplify the hmt model by exploiting the inherent self similarity of real world images the simplified model specifies the hmt parameters with just nine meta parameters independent of the size of the image and the number of wavelet scales we also introduce a bayesian universal hmt uhmt that fixes these nine parameters the uhmt requires no training of any kind while extremely simple we show using a series of image estimation denoising experiments that these new models retain nearly all of the key image structure modeled by the full hmt finally we propose a fast shift invariant hmt estimation algorithm that outperforms other wavelet based estimators in the current literature both visually and in mean square error

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

fast global k means clustering using cluster membership and inequality in this paper we present a fast global k means clustering algorithm by making use of the cluster membership and geometrical information of a data point this algorithm is referred to as mfgkm the algorithm uses a set of inequalities developed in this paper to determine a starting point for the jth cluster center of global k means clustering adopting multiple cluster center selection mcs for mfgkm we also develop another clustering algorithm called mfgkm mcs mcs determines more than one starting point for each step of cluster split while the available fast and modified global k means clustering algorithms select one starting point for each cluster split our proposed method mfgkm can obtain the least distortion while mfgkm mcs may give the least computing time compared to the modified global k means clustering algorithm our method mfgkm can reduce the computing time and number of distance calculations by a factor of 3 78 5 55 and 21 13 31 41 respectively with the average distortion reduction of 5 487 for the statlog data set compared to the fast global k means clustering algorithm our method mfgkm mcs can reduce the computing time by a factor of 5 78 8 70 with the average reduction of distortion of 30 564 using the same data set the performances of our proposed methods are more remarkable when a data set with higher dimension is divided into more clusters

fibers of word maps and some applications every word w in the free group fd defines for each group g a word map also denoted w from gd to g we prove that for all w 1 there exists 0 such that for all finite simple groups g and all g g w 1 g o g d where the implicit constant depends only on w in particular the probability that w g1 gd 1 is at most g for some 0 and all large finite simple groups g this result is then applied in the context of subgroup growth and representation varieties

constructing a godunov type scheme for the model of a general fluid flow in a nozzle with variable cross section a godunov type scheme for the model of a general fluid flow in a nozzle with variable cross section is presented the model has the form of a nonconservative system of balance laws which poses many challenging questions for study first exact riemann solvers in computing form are described in both subsonic and supersonic regions second the computable exact solutions of local riemann problem are incorporated into a godunov type scheme third the scheme is shown to be well balanced in the sense that it can capture exactly stationary waves finally numerical tests for data belong to both subsonic and supersonic regions are presented these tests show that the scheme has a very fine accuracy especially the scheme can give very good approximations to the exact solutions even in the resonant phenomenon where the solution contains three waves of the same speed

a unified approach for encoding clean and noisy sources by means of waveform and autoregressive model vector quantization data compression by vector quantization is considered for sources which have been degraded by noise it is shown that by appropriately modifying the given distortion measure the problem becomes a standard quantization problem for the noisy source and the modified distortion measure for the special case of sources corrupted by statistically independent additive noise the authors provide sufficient conditions on the original distortion measure and probability distributions of the source and the noise for convergence of the generalized lloyd algorithm in designing the quantizers the results are specialized to waveform and autoregressive model vector quantization using the weighted quadratic and the itakura saito distortion measures respectively

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed