multiple description coding using pairwise correlating transforms the objective of multiple description coding mdc is to encode a source into multiple bitstreams supporting multiple quality levels of decoding in this paper we only consider the two description case where the requirement is that a high quality reconstruction should be decodable from the two bitstreams together while lower but still acceptable quality reconstructions should be decodable from either of the two individual bitstreams this paper describes techniques for meeting mdc objectives in the framework of standard transform based image coding through the design of pairwise correlating transforms the correlation introduced by the transform helps to reduce the distortion when only a single description is received but it also increases the bit rate beyond that prescribed by the rate distortion function of the source we analyze the relation between the redundancy i e the extra bit rate and the single description distortion using this transform based framework we also describe an image coder that incorporates the pairwise transform and show its redundancy rate distortion performance for real images

computation of the para pseudoinverse for oversampled filter banks forward and backward greville formulas frames and oversampled filter banks have been extensively studied over the past few years due to their increased design freedom and improved error resilience in frame expansions the least square signal reconstruction operator is called the dual frame which can be obtained by choosing the synthesis filter bank as the para pseudoinverse of the analysis bank in this paper we study the computation of the dual frame by exploiting the greville formula which was originally derived in 1960 to compute the pseudoinverse of a matrix when a new row is appended here we first develop the backward greville formula to handle the case of row deletion based on the forward greville formula we then study the computation of para pseudoinverse for extended filter banks and laplacian pyramids through the backward greville formula we investigate the frame based error resilient transmission over erasure channels the necessary and sufficient condition for an oversampled filter bank to be robust to one erasure channel is derived a postfiltering structure is also presented to implement the para pseudoinverse when the transform coefficients in one subband are completely lost

the navier stokes equations in a space of bounded functions we establish a blow up rate of the navier stokes equations subject to the non slip boundary condition for a certain class of domains including bounded and exterior domains

extending the applicability of gauss newton method for convex composite optimization on riemannian manifolds we present a semi local convergence analysis of the gauss newton method for solving convex composite optimization problems in riemannian manifolds using the notion of quasi regularity for an initial point using a combination the l average lipszhitz condition and the center l 0 average lipschitz condition we introduce majorizing sequences for the gauss newton method that are more precise than in earlier studies consequently our semi local convergence analysis for the gauss newton method has the following advantages under the same computational cost weaker sufficient convergence conditions more precise estimates on the distances involved and an at least as precise information on the location of the solution

a unifying local semilocal convergence analysis and applications for two point newton like methods in banach space abstract we provide a local as well as a semilocal convergence analysis for two point newton like methods in a banach space setting under very general lipschitz type conditions our equation contains a frechet differentiable operator f and another operator g whose differentiability is not assumed using more precise majorizing sequences than before we provide sufficient convergence conditions for newton like methods to a locally unique solution of equation f x g x 0 in the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results in the local case a larger radius of convergence is obtained several numerical examples are provided to show that our results compare favorably with earlier ones as a special case we show that the famous newton kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the newton kantorovich theorem

notes on the stability of dynamic economic systems the lyapunov function method is the focus of this paper in proving marginal stability asymptotical or globally asymptotical stability of discrete dynamic systems we show that the slightly relaxed versions of the well known sufficient conditions are also necessary

analyticity of the stokes semigroup in spaces of bounded functions the analyticity of the stokes semigroup with the dirichlet boundary condition is established in spaces of bounded functions when the domain occupied with fluid is bounded or more generally admissible which admits a special estimate for the helmholtz decomposition the proof is based on a blow up argument this is the first proof of the analyticity in spaces of bounded functions which was left open more than thirty years

tensor methods for nonlinear equations a new class of methods for solving systems of nonlinear equations called tensor methods is introduced tensor methods are general purpose methods intended especially for problems where the jacobian matrix at the solution is singular or ill conditioned they base each iteration on a quadratic model of the nonlinear function the standard linear model augmented by a simple second order term the second order term is selected so that the model interpolates function values from several previous iterations as well as the current function value and jacobian the tensor method requires no more function and derivative information per iteration and hardly more storage or arithmetic per iteration than a standard method based on newton s method in extensive computational tests a tensor algorithm is significantly more efficient than a similar algorithm based on the standard linear model both on standard nonsingular test problems and on problems where the jacobian at the solution is singular

third order iterative methods with applications to hammerstein equations a unified approach the geometrical interpretation of a family of higher order iterative methods for solving nonlinear scalar equations was presented in s amat s busquier j m gutierrez geometric constructions of iterative functions to solve nonlinear equations j comput appl math 157 1 2003 197 205 this family includes as particular cases some of the most famous third order iterative methods chebyshev methods halley methods super halley methods c methods and newton type two step methods the aim of the present paper is to analyze the convergence of this family for equations defined between two banach spaces by using a technique developed in j a ezquerro m a hernandez halley s method for operators with unbounded second derivative appl numer math 57 3 2007 354 360 this technique allows us to obtain a general semilocal convergence result for these methods where the usual conditions on the second derivative are relaxed on the other hand the main practical difficulty related to the classical third order iterative methods is the evaluation of bilinear operators typically second order frechet derivatives however in some cases the second derivative is easy to evaluate a clear example is provided by the approximation of hammerstein equations where it is diagonal by blocks we finish the paper by applying our methods to some nonlinear integral equations of this type

optimal subband filter banks for multiple description coding multiple description coding refers to the encoding of an information source into multiple bit streams in a manner that each bit stream independently represents a coarse description of the source while multiple descriptions jointly convey a higher fidelity source representation thus multiple description coding allows enhanced resilience to loss of individual descriptions and can be viewed as a form of source based diversity coding this paper addresses the design of subband filter banks to optimize the rate distortion performance of multiple description subband coding the problem is formulated in a lagrangian optimization framework and is solved directly in the frequency domain to produce the optimal filter spectral responses by varying the lagrangian parameter our design obtains all points on the redundancy distortion curve for gaussian wide sense stationary input processes we consider the cases of both orthonormal and more general biorthogonal filter banks we also analyze the connections between these two solutions and quantify the gap in their optimal performance application of the proposed methods to the popular first order autoregressive ar 1 model yields very interesting and insightful results in the two extreme cases of maximum and minimum redundancy our solutions degenerate to previously known results finally comparisons with popularly deployed filter banks reveal large gaps from the theoretical performance bound

existence of weak solutions for the navier stokes equations with initial data in lp addendum the existence of weak solutions for the navier stokes equations for the infinite cylinder with initial data in l p is considered in this paper we study the case of initial data in l p r n 2 n is bridged the existence theorem gives a new method of constructing global solutions the cases p n are treated at the end of the paper

a fast chebyshev s method for quadratic equations a multiresolution transform corresponding to interpolatory technique is used for fast application of chebyshev s method in designing this algorithm we apply data compression to the linear and the bilinear forms that appear on the method a convergence theorem is performed finally some numerical results are presented

analysis and modification of newton s method for algebraic riccati equations when newton s method is applied to find the maximal symmetric solution of an algebraic riccati equation convergence can be guaranteed under moderate conditions in particular the initial guess need not be close to the solution the convergence is quadratic if the frechet derivative is invertible at the solution in this paper we examine the behaviour of the newton iteration when the derivative is not invertible at the solution we find that a simple modification can improve the performance of the newton iteration dramatically

a necessary condition of potential blowup for the navier stokes system in half space assuming that t is a potential blow up time for the navier stokes system in mathbb r 3 we show that the norm of the velocity field in the lorenz space l 3 q with q infty goes to infty as time t approaches t

the rate distortion region for multiple descriptions without excess rate during recent years there has been strong interest in a certain source coding problem which some authors call the problem of multiple descriptions old and new wringing techniques enable us to establish a single letter characterization of the rate distortion region in the case of no excess rate for the joint description

on global weak solutions to the cauchy problem for the navier stokes equations with large l3 initial data abstract the aim of the note is to discuss different definitions of solutions to the cauchy problem for the navier stokes equations with the initial data belonging to the lebesgue space l 3 r 3

multiple description coding for stationary gaussian sources we consider the problem of multiple description coding for stationary gaussian sources under the squared error distortion measure the rate region is characterized for the 2 description case it is shown that each supporting line of the rate region is achievable with a transform lattice quantization scheme we show the optimal coding scheme has a natural spectral domain coding interpretation which yields a reverse water filling solution with a frequency dependent water level instead of the flat water level as in the conventional single description case

a semilocal convergence analysis for directional newton methods a semilocal convergence analysis for directional newton methods in n variables is provided in this study using weaker hypotheses than in the elegant related work by y levin and a ben israel and introducing the center lipschitz condition we provide under the same computational cost as in levin and ben israel a semilocal convergence analysis with the following advantages weaker convergence conditions larger convergence domain finer error estimates on the distances involved and an at least as precise information on the location of the zero of the function a numerical example where our results apply to solve an equation but not the ones in levin and ben israel is also provided in this study

transform coding with integer to integer transforms a new interpretation of transform coding is developed that downplays quantization and emphasizes entropy coding allowing a comparison of entropy coding methods with different memory requirements with conventional transform coding based on computing karhunen loeve transform coefficients and then quantizing them vector entropy coding can be replaced by scalar entropy coding without an increase in rate thus the transform coding advantage is a reduction in memory requirements for entropy coding this paper develops a transform coding technique where the source samples are first scalar quantized and then transformed with an integer to integer approximation to a nonorthogonal linear transform among the possible advantages is to reduce the memory requirement further than conventional transform coding by using a single common scalar entropy codebook for all components the analysis shows that for high rate coding of a gaussian source this reduction in memory requirements comes without any degradation of rate distortion performance

multiple description coding with many channels an achievable region for the l channel multiple description coding problem is presented this region generalizes two channel results of el gamal and cover 1982 and of zhang and berger 1987 it further generalizes three channel results of gray and wyner 1974 and of zhang and berger a source that is successively refinable on chains is shown to be successively refinable on trees a new outer bound on the rate distortion rd region for memoryless gaussian sources with mean squared error distortion is also derived the achievable region meets this outer bound for certain symmetric cases

colored gaussian multiple descriptions spectral and time domain forms it is well known that shannon s rate distortion function rdf in the colored quadratic gaussian qg case can be parametrized via a single lagrangian variable the water level in the reverse water filling solution in this paper we show that the symmetric colored qg multiple description md rdf in the case of two descriptions can be parametrized in the spectral domain via two lagrangian variables which control the tradeoff between the side distortion the central distortion and the coding rate this spectral domain analysis is complemented by a time domain scheme design approach we show that the symmetric colored qg md rdf can be achieved by combining ideas of delta sigma modulation and differential pulse code modulation in particular two source prediction loops one for each description are embedded within a common noise shaping loop whose parameters are explicitly found from the spectral domain characterization

modification of the kantorovich assumptions for semilocal convergence of the chebyshev method this study obtains two semilocal convergence results for the well known chebyshev method which is a third order iterative process the hypotheses required are modifications to the normal kantorovich ones the results obtained are applied to the reduction of nonlinear integral equations of the fredholm type and first kind

sublinear convergence of the chord method at singular points we give sufficient conditions for convergence of the chord method for a class of singular problems the rate of convergence is sublinear

principal component filter banks for optimal multiresolution analysis an important issue in multiresolution analysis is that of optimal basis selection an optimal p band perfect reconstruction filter bank prfb is derived in this paper which minimizes the approximation error in the mean square sense between the original signal and its low resolution version the resulting prfb decomposes the input signal into uncorrelated low resolution principal components with decreasing variance optimality issues are further analyzed in the special case of stationary and cyclostationary processes by exploiting the connection between discrete time filter banks and continuous wavelets an optimal multiresolution decomposition of l sub 2 r is obtained analogous results are also derived for deterministic signals some illustrative examples and simulations are presented

on smoothness of l3 solutions to the navier stokes equations up to boundary we show that l3 solutions to the three dimensional navier stokes equations near a flat part of the boundary are smooth

efficient reconstruction from frame based multiple descriptions redundant bases or frames are a promising technique for multiple description signal coding frames can provide sufficient redundancy for recovery in case of coefficient losses due for instance to packet transmission errors in this paper we consider the problem of dual frame computation in the context of finite impulse response fir analysis and synthesis filterbanks in particular we propose an algorithm for dual frame computation which recovers the missing frame coefficients followed by synthesis with the original filterbank it is shown that the algorithm operates locally and allows for reduced complexity and low delay reconstruction of the original signal the case of excessive losses and incomplete bases is also considered experiments are provided to evaluate the algorithm performance and test its applicability in practical contexts

new results in binary multiple descriptions an encoder whose input is a binary equiprobable memoryless source produces one output of rate r 1 and another of rate r 2 let d 1 d 2 and d 0 respectively denote the average error frequencies with which the source data can be reproduced on the basis of the encoder output of rate r l only the encoder output of rate r 2 only and both encoder outputs the two descriptions problem is to determine the region r of all quintuples r 1 r 2 d 1 d 2 d 0 that are achievable in thc usual shannon sense let r d 1 d log 2 d 1 d log 2 1 d denote the error frequency rate distortion function of the source the no excess rate case prevails when r 1 r 2 r d 0 and the excess rate case when r 1 r 2 r d 0 denote the section of r at r 1 r 2 d 0 by d r 1 r 2 d 0 d 1 d 2 r 1 r 2 d 1 d 2 d 0 in r in the no excess rate case we show that a portion of the boundary of d r 1 r 2 d 0 coincides with the curve frac 1 2 d 1 2d 0 frac 1 2 d 2 2d 0 frac 1 2 1 2d 0 2 this curve is an extension of witsenhausen s hyperbola bound to the case d 0 0 it follows that the projection of r onto the d 1 d 2 plane at fixed d 0 consists of all d 1 geq d 0 and d 2 geq d 0 that lie on or above this hyperbola in the excess rate case we show by counterexample that the achievable region of el gamal and cover is not tight

semilocal convergence of the secant method under mild convergence conditions of differentiability abstract in this work we obtain a semilocal convergence result for the secant method in banach spaces under mild convergence conditions we consider a condition for divided differences which generalizes those usual ones i e lipschitz continuous and holder continuous conditions also we obtain a result for uniqueness of solutions

design of entropy constrained multiple description scalar quantizers the problem of entropy constrained multiple description scalar quantizer design is posed as an optimization problem necessary conditions for optimality are derived and an iterative design algorithm is presented performance results are presented for a gaussian source along with comparisons to the multiple description rate distortion bound and a reference system

multiple description wavelet based image coding we consider the problem of coding images for transmission over error prone channels the impairments we target are transient channel shutdowns as would occur in a packet network when a packet is lost or in a wireless system during a deep fade when data is delivered it is assumed to be error free but some of the data may never reach the receiver the proposed algorithms are based on a combination of multiple description scalar quantizers with techniques successfully applied to the construction of some of the most efficient subband coders a given image is encoded into multiple independent packets of roughly equal length when packets are lost the quality of the approximation computed at the receiver depends only on the number of packets received but does not depend on exactly which packets are actually received when compared with previously reported results on the performance of robust image coders based on multiple descriptions on standard test images our coders attain similar psnr values using typically about 50 60 of the bit rate required by these other state of the art coders while at the same time providing significantly more freedom in the mechanism for allocation of redundancy among descriptions

a shamanskii like acceleration scheme for nonlinear equations at singular roots on applique une variante de la methode de shamanskii pour realiser une methode a convergence superlineaire de resolution d une classe d equations non lineaires ayant une derivee de frechet singuliere a la racine

generalized multiple description coding with correlating transforms multiple description md coding is source coding in which several descriptions of the source are produced such that various reconstruction qualities are obtained from different subsets of the descriptions unlike multiresolution or layered source coding there is no hierarchy of descriptions thus md coding is suitable for packet erasure channels or networks without priority provisions generalizing work by orchard wang vaishampayan and reibman see proc ieee int conf image processing vol i santa barbara ca p 608 11 1997 a transform based approach is developed for producing m descriptions of an n tuple source m spl les n the descriptions are sets of transform coefficients and the transform coefficients of different descriptions are correlated so that missing coefficients can be estimated several transform optimization results are presented for memoryless gaussian sources including a complete solution of the n 2 m 2 case with arbitrary weighting of the descriptions the technique is effective only when independent components of the source have differing variances numerical studies show that this method performs well at low redundancies as compared to uniform md scalar quantization

asymptotic performance of multiple description transform codes the multiple description transform coder is introduced for sources with memory and an asymptotic analysis is presented for the squared error distortion for stationary gaussian sources the optimal transform and the optimal bit allocation for the multiple description coder are identical to those for the single description coder

localised necessary conditions for singularity formation in the navier stokes equations with curved boundary abstract we generalize two results in the navier stokes regularity theory whose proofs rely on zooming in on a presumed singularity to the local setting near a curved portion of the boundary suppose that u is a boundary suitable weak solution with singularity z x t where x then under weak background assumptions the l 3 norm of u tends to infinity in every ball centered at x lim t t u t l 3 b x r r 0 additionally u generates a non trivial mild bounded ancient solution in r 3 or r 3 through a rescaling procedure that zooms in on the singularity our proofs rely on a truncation procedure for boundary suitable weak solutions the former result is based on energy estimates for l 3 initial data and a liouville theorem for the latter result we apply perturbation theory for l initial data based on linear estimates due to k abe and y giga

on estimates for the stokes flow in a space of bounded functions in this paper we study regularizing effects of the composition operator s t p for the stokes semigroup s t and the helmholtz projection p in a space of bounded functions we establish new a priori l estimates of the operator s t p for a certain class of domains including bounded and exterior domains they imply unique existence of mild solutions of the navier stokes equations in a space of bounded functions

extending the applicability of newton s method on lie groups we use newton s method to approximate a zero of a mapping from a lie group into its lie algebra under the same computational cost as before we show the semilocal convergence of newton s method with the following advantages over earlier works 55 weaker sufficient convergence conditions tighter error bounds on the distances involved and at least as precise information on the location of the solution numerical examples are also given for solving equations in cases not covered before

convergence rates for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root the convergence of the newton iterates to the root is linear rather than quadratic in this paper we give a detailed analysis of the linear convergence rates for several types of singular problems for some of these problems we describe modifications of newton s method which will restore quadratic convergence

newton like methods for the computation of fixed points abstract the celebrated banach fixed point theorem provides conditions which assure that the method of successive substitution is convergent the convergence however may take place very slowly so that it may be desirable to use a newton like method for the computation of the fixed point if newton s method itself is applied one ignores the additional information that the problem arises from a frixed point problem with a contraction mapping in the present note some variants of newton s method are discussed which make use of this contraction information it turns out that the convergence of newton s method can be accelerated without any relevant additional computational labour

a new acceleration method for newton s method at singular points when newton s method is used to find a root of a map from a banach space into itself and the derivative is singular at that root convergence of the newton sequence is in general linear in this paper we give a modification of newton s method that under certain conditions converges superlinearly our method is applicable under more general conditions than other techniques in particular it may be used for certain quadratic problems and problems in which the dimension of the null space of the derivative is larger than one

convergence acceleration for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root linear convergence of the newton sequence to the root is the best

multiple description coding with prediction compensation a new multiple description coding paradigm is proposed by combining the time domain lapped transform block level source splitting linear prediction and prediction residual encoding the method provides effective redundancy control and fully utilizes the source correlation the joint optimization of all system components and the asymptotic performance analysis are presented image coding results demonstrate the superior performance of the proposed method especially at low redundancies

on solving nonlinear equations with simple singularities or nearly singular solutions the paper begins with a review of the convergence theory for newton s method near simple and other regular singularities followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy then several ways to accelerate the slow convergence of newton s method are described in some detail these are richardson extrapolation overrelaxation bordering by a singularity condition and appending the usual linear model by a quadratic term the last two approaches require limited second derivative information which can be obtained analytically by differencing or through secant updating numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy

achievable rates for multiple descriptions consider a sequence of independent identically distributed i i d random variables x l x 2 cdots x n and a distortion measure d x i x i on the estimates x i of x i two descriptions i x in 1 2 cdots 2 nr 1 and j x in 1 2 cdots 2 nr 2 are given of the sequence x x 1 x 2 cdots x n from these two descriptions three estimates i x x2 j x and hat x o i x j x are formed with resulting expected distortions e frac 1 n sum n k 1 d x k hat x mk d m m 0 1 2 we find that the distortion constraints d 0 d 1 d 2 are achievable if there exists a probability mass distribution p x p hat x 1 hat x 2 hat x 0 x with ed x hat x m leq d m such that r 1 i x hat x 1 r 2 i x hat x 2 where i cdot denotes shannon mutual information these rates are shown to be optimal for deterministic distortion measures

a class of trust region methods for parallel optimization we present a new class of optimization methods that incorporates a parallel direct search pds method within a trust region newton framework this approach combines the inherent parallelism of pds with the rapid and robust convergence properties of newton methods numerical tests have yielded favorable results for both standard test problems and engineering applications in addition the new method appears to be more robust in the presence of noisy functions which are inherent in many engineering simulations

a two step steffensen s method under modified convergence conditions a simplification of a third order iterative method is proposed the main advantage of this method is that it does not need to evaluate neither any frechet derivative nor any bilinear operator a semilocal convergence theorem in banach spaces under modified kantorovich conditions is analyzed a local convergence analysis is also performed finally some numerical results are presented

discretization of implicit odes for singular root finding problems this paper addresses the use of dynamical system theory to tackle singular root finding problems the use of continuous time methods leads to implicit differential systems when applied to singular nonlinear equations the analysis is based on a taxonomy of singularities and uses previous stability results proved in the context of quasilinear implicit odes the proposed approach provides a framework for the systematic formulation of quadratically convergent iterations to singular roots the scope of the work includes also the introduction of discrete time analysis techniques for singular problems which are based on continuous time stability and numerical stability some numerical experiments illustrate the applicability of the proposed techniques

on iterative methods for the quadratic matrix equation with m matrix in this paper we consider newton s method and bernoulli s method for a quadratic matrix equation arising from an overdamped vibrating system by introducing m matrix to this equation we provide a sufficient condition for the existence of the primary solution moreover we show that newton s method and bernoulli s method with an initial zero matrix converge to the primary solvent under the proposed sufficient condition

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

boundary partial regularity for the high dimensional navier stokes equations abstract we consider suitable weak solutions of the incompressible navier stokes equations in two cases the 4d time dependent case and the 6d stationary case we prove that up to the boundary the two dimensional hausdorff measure of the set of singular points is equal to zero in both cases

geometrically isolated nonisolated solutions and their approximation a solution x x 0 of f x 0 is said to be isolated if the frechet derivative f x 0 is nonsingular it is said to be geometrically isolated if no other solution is in x x 0 leqq rho for some rho 0 isolated solutions are always geometrically isolated sufficient conditions are obtained to insure that a nonisolated solution is also geometrically isolated we then study the application of approximation methods in the general form f h x h 0 to approximate nonisolated solutions which are geometrically isolated under strong consistency conditions the results are somewhat negative the approximations may have an even number including zero or an odd number of roots near x 0 depending upon the multiplicity cf x 0 as a root if the accuracy is o h p and the multiplicity is n then the approximations have error o h p n the relation of these results to limit and bifurcation points is discussed briefly

filter bank frame expansions with erasures we study frames for robust transmission over the internet in our previous work we used quantized finite dimensional frames to achieve resilience to packet losses here we allow the input to be a sequence in l sub 2 z and focus on a filter bank implementation of the system we present results in parallel r sup n or c sup n versus l sub 2 z and show that uniform tight frames as well as newly introduced strongly uniform tight frames provide the best performance

multiple description vector quantization with lattice codebooks design and analysis the problem of designing a multiple description vector quantizer with lattice codebook spl lambda is considered a general solution is given to a labeling problem which plays a crucial role in the design of such quantizers numerical performance results are obtained for quantizers based on the lattices a sub 2 and z sup i i 1 2 4 8 that make use of this labeling algorithm the high rate squared error distortions for this family of l dimensional vector quantizers are then analyzed for a memoryless source with probability density function pdf p and differential entropy h p spl infin for any a spl epsiv 0 1 and rate pair r r it is shown that the two channel distortion d sub o and the channel 1 or channel 2 distortion d sub s satisfy lim sub r spl rarr spl infin d sub o 2 sup 2r 1 a 1 4 g spl lambda 2 sup 2h p and lim sub r spl rarr spl infin d sub s 2 sup 2r 1 a g s sub l 2 sup 2h p where g spl lambda is the normalized second moment of a voronoi cell of the lattice spl lambda and g s sub l is the normalized second moment of a sphere in l dimensions

asymptotic analysis of multiple description quantizers a high rate analysis of multiple description quantizers is presented for rth power distortions and general source densities both fixed length and variable length encoding of the quantizer indices are considered optimal companding functions are shown to be the same as for single channel quantizers as compared to the bound of ozarow 1980 a gap of 8 69 db and 3 07 db exists between the entropy constrained and level constrained cases respectively for a memoryless gaussian source and r 2

approximate methods for the solution of the chandrasekhar h equation we consider two methods of approximate solution to matrix valued analogs of the chandrasekhar h equation we give conditions under which they converge the first method is a generalization of approximation of the integral by a quadrature the second is newton s method

chebyshev halley s method on riemannian manifolds abstract this paper is concerned with the problem of finding singular points of vector fields on riemannian manifolds we using non trivial techniques and results of differential geometry will exhibit a new method defined on complete riemannian manifolds which generalizes the important newton and chebyshev halley s methods moreover a characterization of the convergence under kantorovich type conditions and error estimates are also given in this study using the method introduced in this paper we give an algorithm which will allow us to find singular points of a vector field on the two dimensional sphere s 2 finally in order to illustrate our method and its relevance we develop an example which allows us to find a singularity of a vector field on s 2 such a singularity is not possible to find it using the classical numerical methods on r 3

multiple description source coding with no excess marginal rate multiple description source coding concerns situations in which the transmission of the source information is distributed over two data streams at rates r sub 1 and r sub 2 respectively when both data streams are received the decoder uses the combined data at rate r sub 1 r sub 2 to reconstruct the source information with average distortion d sub 0 if a communication breakdown prevents one of the data streams from reaching the receiver the decoder has to base its reconstruction solely on the available data at rate either r sub 1 or r sub 2 this results in a higher distortion of either d sub 1 or d sub 2 respectively the region spl rscr of all achievable quintuples r sub 1 r sub 2 d sub 0 d sub 1 d sub 2 has been determined in the so called no excess rate sum case defined by imposing the requirement r sub 1 r sub 2 r d sub 0 where r spl middot is the rate distortion function of the source the case with excess rate sum characterized by r sub 1 r sub 2 r d sub 0 is challenging we study in this paper a special case of it in which the requirements r sub t r d sub t t 1 2 are imposed we refer to this as the no excess marginal rate case the lower and upper bounds on d sub 0 we obtain are separated by only a tiny gap when evaluated for a binary equiprobable source and the hamming distortion measure

a uniparametric family of iterative processes for solving nondifferentiable equations in this work we study a class of secant like iterations for solving nonlinear equations in banach spaces we consider a condition for divided differences which generalizes the usual ones i e lipschitz and holder continuous conditions a semilocal convergence result is obtained for nondifferentiable operators for that we use a technique based on a new system of recurrence relations to obtain domains of existence and uniqueness of the solution finally we apply our results to the numerical solution of several examples

newton s method for singular problems when the dimension of the null space is 1 a theorem is proved concerning the convergence of newton s method when the dimension of the null space of the jacobian matrix is 1 it improves on previous results of decker and kelley siam j numer anal 17 1980 pp 66 70 and reddien comput math appl 5 1980 pp 79 86

analysis of newton s method at irregular singularities recent work on newton s method at singularities of the jacobian has established linear convergence under certain regularity assumptions here newton s method is analyzed in the neighborhood of irregular singularities which include all minimizers at which the hessian has a one dimensional null space depending on a parameter specific to the underlying optimization problem or system of simultaneous equations newton s method is found either to converge with a limiting ratio of about frac 2 3 or to diverge from arbitrarily close starting points or to behave in a certain sense chaotically

continuous alignment of vorticity direction prevents the blow up of the navier stokes flow under the no slip boundary condition abstract this paper is concerned with a regularity criterion based on vorticity direction for navier stokes equations in a three dimensional bounded domain under the no slip boundary condition it asserts that if the vorticity direction is uniformly continuous in space uniformly in time there is no type i blow up a similar result has been proved for a half space by y maekawa and the first and the last authors 2014 the result of this paper is its natural but non trivial extension based on l theory of the stokes and the navier stokes equations recently developed by k abe and the first author

stability of singular equilibria in quasilinear implicit differential equations abstract this paper addresses stability properties of singular equilibria arising in quasilinear implicit odes under certain assumptions local dynamics near a singular point may be described through a continuous or directionally continuous vector field this fact motivates a classification of geometric singularities into weak and strong ones stability in the weak case is analyzed through certain linear matrix equations a singular version of the lyapunov equation being especially relevant in the study weak stable singularities include singular zeros having a spherical domain of attraction which contains other singular points regarding strong equilibria stability is proved via a lyapunov schmidt approach under additional hypotheses the results are shown to be relevant in singular root finding problems

a modification of newton method with third order convergence in this paper we present a new modification of newton method for solving non linear equations analysis of convergence shows that the new method is cubically convergent per iteration the new method requires two evaluations of the function and one evaluation of its first derivative thus the new method is preferable if the computational costs of the first derivative are equal or more than those of the function itself its practical utility is demonstrated by numerical examples

minimal initial data for potential navier stokes singularities abstract assuming some initial data u 0 h 1 2 r 3 lead to a singularity for the 3d navier stokes equations we show that there are also initial data with the minimal h 1 2 norm which will produce a singularity

filter banks for prediction compensated multiple description coding in this paper a prediction compensated multiple description md coding framework for two band filter banks is proposed in which the coefficients in each subband are split into two descriptions each description also includes the prediction residuals of the data in the other description the designs of the optimal orthogonal and biorthogonal filter banks are formulated in a unified framework and both one level and multiple level decompositions are analyzed contrary to the existing md filter banks in the literature the optimal filter banks in the proposed scheme are quite similar to those in single description coding therefore the method can be applied to systems with single description optimized filter banks and still attain near optimal performance image coding results show that this method achieves better performance and lower complexity than the latest jpeg 2000 based mdc

third order iterative methods under kantorovich conditions we study a class of third order iterative methods for nonlinear equations on banach spaces a characterization of the convergence under kantorovich type conditions and optimal estimates of the error are found though in general these methods are not very extended due to their computational costs we will show some examples in which they are competitive and even cheaper than other simpler methods we center our analysis in both analytic and computational aspects

optimal filter banks for multiple description coding analysis and synthesis multiple description md coding is a source coding technique for information transmission over unreliable networks in md coding the coder generates several different descriptions of the same signal and the decoder can produce a useful reconstruction of the source with any received subset of these descriptions in this paper we study the problem of md coding of stationary gaussian sources with memory first we compute an approximate md rate distortion region for these sources which we prove to be asymptotically tight at high rates this region generalizes the md rate distortion region of el gamal and cover 1982 and ozarow 1980 for memoryless gaussian sources then we develop an algorithm for the design of optimal two channel biorthogonal filter banks for md coding of gaussian sources we show that optimal filters are obtained by allocating the redundancy over frequency with a reverse water filling strategy finally we present experimental results which show the effectiveness of our filter banks in the low complexity low rate regime

design of multiple description scalar quantizers the design of scalar quantizers for communication systems that use diversity to overcome channel impairments is considered the design problem is posed as an optimization problem and necessary conditions for optimality are derived a design algorithm a generalization of s p lloyd s 1962 algorithm for quantizer design is developed unlike a single channel scalar quantizer the performance of a multiple description scalar quantizer is dependent on the index assignment the problem of index assignment is addressed good index assignments performance results and sample quantizer designs are presented for a memoryless gaussian source comparisons are made with rate distortion bounds for the multiple description problem

liouville theorems for the navier stokes equations and applications we study bounded ancient solutions of the navier stokes equations these are solutions with bounded velocity defined in rn 1 0 in two space dimensions we prove that such solutions are either constant or of the form u x t b t depending on the exact definition of admissible solutions the general 3 dimensional problem seems to be out of reach of existing techniques but partial results can be obtained in the case of axisymmetric solutions we apply these results to some scenarios of potential singularity formation for axi symmetric solutions and obtain extensions of results in a recent paper by chen strain tsai and yau 4

newton type methods on riemannian manifolds under kantorovich type conditions one of the most studied problems in numerical analysis is the approximation of nonlinear equations using iterative methods in the last years attention has been paid in studying newton s method on manifolds in this paper we generalize this study considering some newton type iterative methods a characterization of the convergence under kantorovich type conditions and optimal estimates of the error are found using normal coordinates the order of convergence is derived the sufficient semilocal convergence criteria are weaker and the majorizing sequences are tighter for the special cases of simplified newton and newton methods than in earlier studies such as argyros 2004 2007 2008 6 8 12 and kantorovich and akilov 1964 32

convergence analysis of a variant of the newton method for solving nonlinear equations the paper presents a convergence analysis of a modified newton method for solving nonlinear systems of equations the convergence results show that this method converges cubically in the nonsingular case and linearly with the rate 3 8 under some sufficient conditions when the jacobian is singular at the root the convergence theory is used to analyze the convergence behavior when the modified newton method is applied to a nonsymmetric algebraic riccati equation arising in transport theory numerical experiment confirms the theoretical results

third order iterative methods without using any frechet derivative a modification of classical third order methods is proposed the main advantage of these methods is they do not need to evaluate any frechet derivative a convergence theorem in banach spaces just assuming the second divided difference is bounded and a punctual condition is analyzed finally some numerical results are presented

on a bilinear operator free third order method on riemannian manifolds we present a semilocal convergence analysis of a bilinear operator free third order method on riemannian manifolds using a combination of generalized lipschitz and center lipschitz conditions we provide a convergence analysis which expands the applicability of the method even in the setting of nonlinear equations considered in earlier studies such as cf 4 9 38

global weak besov solutions of the navier stokes equations and applications we introduce a notion of global weak solution to the navier stokes equations in three dimensions with initial values in the critical homogeneous besov spaces dot b 1 frac 3 p p infty p 3 these solutions satisfy a certain stability property with respect to the weak ast convergence of initial conditions to illustrate this property we provide applications to blow up criteria minimal blow up initial data and forward self similar solutions our proof relies on a new splitting result in homogeneous besov spaces that may be of independent interest