a lower bound on the undetected error probability and strictly optimal codes error detection is a simple technique used in various communication and memory systems to enhance reliability we study the probability that a q ary linear or nonlinear block code of length n and size m fails to detect an error a lower bound on this undetected error probability is derived in terms of q n and m the new bound improves upon other bounds mentioned in the literature even those that hold only for linear codes block codes whose undetected error probability equals the new lower bound are investigated we call these codes strictly optimal codes and give a combinatorial characterization of them we also present necessary and sufficient conditions for their existence in particular we find all values of n and m for which strictly optimal binary codes exist and determine the structure of all of them for example we construct strictly optimal binary coded decimal codes of length four and five and we show that these are the only possible lengths of such codes

the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

adaptive numerical analysis in primal elastoplasticity with hardening the quasi static viscoplastic resp elastoplastic evolution problem with isotropic or kinematic hardening is considered with emphasis on optimal convergence and adapted mesh refining in the spatial discretization within one time step of an implicit time discretization the finite element method leads to a minimisation problem for non smooth convex functions on discrete subspaces for piecewise constant resp affine ansatz functions the stress resp displacement approximations are experimentally and theoretically shown to converge linearly an a posteriori error estimate then justifies an automatic adaptive mesh refining algorithm numerical examples support the superiority of the adapted mesh

a semi smooth newton method for elasto plastic contact problems in this paper we reformulate the frictional contact problem for elasto plastic bodies as a set of unconstrained non smooth equations the equations are semi smooth so that pang s newton method for b differentiable equations can be applied an algorithm based on this method is described in detail an example demonstrating the efficiency of the algorithm is presented

reliable and efficient equilibrated a posteriori finite element error control in elastoplasticity and elastoviscoplasticity with hardening this paper establishes the reliability and efficiency of equilibrated a posteriori estimates for l 2 stress error control of conforming displacement finite element approximations of incremental plasticity and viscoplasticity with hardening explicit expressions for upper bounds of the reliability constant that enters the guaranteed upper error bound illustrate its crucial dependence on the hardening material constants numerical experiments show that adaptive finite element solutions with marking strategy based on the max refinement rule and local equilibrated error estimators lead to optimal empirical convergence rates 2005 elsevier b v all rights reserved

a primal dual active set algorithm for three dimensional contact problems with coulomb friction in this paper efficient algorithms for contact problems with tresca and coulomb friction in three dimensions are presented and analyzed the numerical approximation is based on mortar methods for nonconforming meshes with dual lagrange multipliers using a nonsmooth complementarity function for the three dimensional friction conditions a primal dual active set algorithm is derived the method determines active contact and friction nodes and at the same time resolves the additional nonlinearity originating from sliding nodes no regularization and no penalization are applied and superlinear convergence can be observed locally in combination with a multigrid method it defines a robust and fast strategy for contact problems with tresca or coulomb friction the efficiency and flexibility of the method is illustrated by several numerical examples

deep holes and mds extensions of reed solomon codes we study the problem of classifying deep holes of reed solomon codes we show that this problem is equivalent to the problem of classifying maximum distance separable mds extensions of reed solomon codes by one digit this equivalence allows us to improve recent results on the former problem in particular we classify deep holes of reed solomon codes of dimension greater than half the alphabet size we also give a complete classification of deep holes of reed solomon codes with redundancy three in all dimensions

relative submajorization and its use in quantum resource theories we introduce and study a generalization of majorization called relative submajorization and show that it has many applications to the resource theories of thermodynamics bipartite entanglement and quantum coherence relative majorization is an ordering on pairs of vectors induced by stochastic transformations one pair of vectors relatively majorizes another when there exists a stochastic transformation taking the former to the latter relative submajorization is a weakened version in which a substochastic matrix need only generate a vector pair obeying certain positivity conditions relative to the input pair in the context of resource theories we show that the relative submajorization characterizes both the probability and approximation error that can be obtained when transforming one resource to another also when assisted by additional standard resources such as useful work or maximally entangled states these characterizations have a geometric formulation as the ratios or differences respectively between the lorenz curves associated with the input and output resources making them efficient to compute we also find several interesting bounds on the reversibility of a given transformation in terms of the properties of the forward transformation the main technical tool used to establish these results is the linear programming duality which is used to show that any instance of relative submajorization can be dilated to an instance of strict relative majorization we introduce and study a generalization of majorization called relative submajorization and show that it has many applications to the resource theories of thermodynamics bipartite entanglement and quantum coherence relative majorization is an ordering on pairs of vectors induced by stochastic transformations one pair of vectors relatively majorizes another when there exists a stochastic transformation taking the former to the latter relative submajorization is a weakened version in which a substochastic matrix need only generate a vector pair obeying certain positivity conditions relative to the input pair in the context of resource theories we show that the relative submajorization characterizes both the probability and approximation error that can be obtained when transforming one resource to another also when assisted by additional standard resources such as useful work or maximally entangled states these characterizations have a geometric formulation as the ratios or differences respectively

classification of pseudo cyclic mds codes linear codes are considered a code is characterized by the length n the dimension k and the minimum distance d an n k d code over the finite field gf q is said to be maximum distance separable mds if d n k 1 a krishna and d v sarwate 1990 investigated the existence of pseudo cyclic mds codes over gf q of length n where n divides q 1 or q 1 it is shown that the pseudo cyclic mds codes constructed by krishna and sarwate are generalised reed solomon codes pseudo cyclic codes are studied over gf q where n and q p sup h are not relatively prime it is proven that pseudo cyclic n k mds codes modulo x sup n a over gf q exist if and only if n p furthermore any pseudo cyclic p k code modulo x sup p a over gf q turns out to be mds and generalized reed solomon it is explicitly proven that some classes of pseudo cyclic n k mds codes over gf q are generalized reed solomon codes furthermore pseudo cyclic q 1 4 mds codes over gf q q even are completely classified

an upper bound on the ratio of the probabilities of subgroups and cosets corresp through a counterexample redinbo s upper bound on the ratio of the probability of a group and that of its coset is shown to be incorrect the given example is also shown to provide a new upper bound

tight asymptotic bounds on local hypothesis testing between a pure bipartite state and the white noise state we consider asymptotic hypothesis testing or state discrimination with asymmetric treatment of errors between an arbitrary fixed bipartite pure state psi rangle and the white noise state the completely mixed state under one way local operations and classical communications locc two way locc and separable positive operator valued measures povms as a result we derive the hoeffding bounds under two way locc povms and separable povms further we derive stein s lemma type of optimal error exponents under one way locc two way locc and separable povms up to the third order which clarifies the difference between one way and two way locc povm our results clarify the relationship between the entanglement of renyi entropy and the hypothesis testing under locc since the entanglement of renyi entropy appears in the formula of both the hoeffding bounds and stein s lemma type of error exponents this paper gives a very rare example in which the optimal performance under the infinite round two way locc is also equal to that under separable operations and can be attained with two round communication but not with the one way locc

bayesian m ary hypothesis testing the meta converse and verdu han bounds are tight two alternative exact characterizations of the minimum error probability of bayesian m ary hypothesis testing are derived the first expression corresponds to the error probability of an induced binary hypothesis test and implies the tightness of the meta converse bound by polyanskiy et al the second expression is a function of an information spectrum measure and implies the tightness of a generalized verdu han lower bound the formulas characterize the minimum error probability of several problems in information theory and help to identify the steps where existing converse bounds are loose

applications of ldpc codes to the wiretap channel with the advent of quantum key distribution qkd systems perfect i e information theoretic security can now be achieved for distribution of a cryptographic key qkd systems and similar protocols use classical error correcting codes for both error correction for the honest parties to correct errors and privacy amplification to make an eavesdropper fully ignorant from a coding perspective a good model that corresponds to such a setting is the wire tap channel introduced by wyner in 1975 in this correspondence we study fundamental limits and coding methods for wire tap channels we provide an alternative view of the proof for secrecy capacity of wire tap channels and show how capacity achieving codes can be used to achieve the secrecy capacity for any wiretap channel we also consider binary erasure channel and binary symmetric channel special cases for the wiretap channel and propose specific practical codes in some cases our designs achieve the secrecy capacity and in others the codes provide security at rates below secrecy capacity for the special case of a noiseless main channel and binary erasure channel we consider encoder and decoder design for codes achieving secrecy on the wiretap channel we show that it is possible to construct linear time decodable secrecy codes based on low density parity check ldpc codes that achieve secrecy

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

algorithmic aspects of optimal channel coding a central question in information theory is to determine the maximum success probability that can be achieved in sending a fixed number of messages over a noisy channel this was first studied in the pioneering work of shannon who established a simple expression characterizing this quantity in the limit of multiple independent uses of the channel here we consider the general setting with only one use of the channel we observe that the maximum success probability can be expressed as the maximum value of a submodular function using this connection we establish the following results 1 there is a simple greedy polynomial time algorithm that computes a code achieving a 1 e 1 approximation of the maximum success probability the factor 1 e 1 can be improved arbitrarily close to 1 at the cost of slightly reducing the number of messages to be sent moreover it is np hard to obtain an approximation ratio strictly better than 1 e 1 for the problem of computing the maximum success probability 2 shared quantum entanglement between the sender and the receiver can increase the success probability by a factor of at most 1 1 e 1 in addition this factor is tight if one allows an arbitrary non signaling box between the sender and the receiver 3 we give tight bounds on the one shot performance of the meta converse of polyanskiy poor verdu

several path following methods for a class of gradient constrained variational inequalities path following splitting and semismooth newton methods for solving a class of problems related to elasto plastic material deformations are proposed analyzed and tested numerically while the splitting techniques result in alternating minimization schemes which are typically linearly convergent the proposed moreau yosida regularization based semismooth newton technique and an associated lifting step yield local superlinear convergence in function space the lifting step accounts for the fact that the operator associated with the linear system in the newton iteration need not be boundedly invertible uniformly along the iterates for devising an efficient update strategy for the path following parameter regularity properties of the path are studied and utilized within an inexact path following scheme for all approaches the paper ends by a report on numerical tests of the different approaches

minimum energy to send k bits through the gaussian channel with and without feedback the minimum achievable energy per bit over memoryless gaussian channels has been previously addressed in the limit when the number of information bits goes to infinity in which case it is known that the availability of noiseless feedback does not lower the minimum energy per bit which is 1 59 db below the noise level this paper analyzes the behavior of the minimum energy per bit for memoryless gaussian channels as a function of k the number of information bits it is demonstrated that in this nonasymptotic regime noiseless feedback leads to significantly better energy efficiency in particular without feedback achieving energy per bit of 1 57 db requires coding over at least k 106 information bits while we construct a feedback scheme that transmits a single information bit with energy 1 59 db and zero error we also show that unless k is very small approaching the minimal energy per bit does not require using the feedback link except to signal that transmission should stop

asynchronous communication exact synchronization universality and dispersion recently tchamkerten and coworkers proposed a novel variation of the problem of joint synchronization and error correction this paper considers a strengthened formulation that requires the decoder to estimate both the message and the location of the codeword exactly such a scheme allows for transmitting data bits in the synchronization phase of the communication thereby improving bandwidth and energy efficiencies it is shown that the capacity region remains unchanged under the exact synchronization requirement furthermore asynchronous capacity can be achieved by universal channel independent codes comparisons with earlier results on another delay compensated definition of rate are made the finite blocklength regime is investigated and it is demonstrated that even for moderate blocklengths it is possible to construct capacity achieving codes that tolerate exponential level of asynchronism and experience only a rather small loss in rate compared to the perfectly synchronized setting in particular the channel dispersion does not suffer any degradation due to asynchronism for the binary symmetric channel a translation coset of a good linear code is shown to achieve the capacity synchronization tradeoff

on mds extensions of generalized reed solomon codes an n k d linear code over f gf q is said to be em maximum distance separable mds if d n k 1 it is shown that an n k n k 1 generalized reed solomon code such that 2 leq k leq n lfloor q 1 2 rfloor k neq 3 rm if q is even can be extended by one digit while preserving the mds property if and only if the resulting extended code is also a generalized reed solomon code it follows that a generalized reed solomon code with k in the above range can be em uniquely extended to a maximal mds code of length q 1 and that generalized reed solomon codes of length q 1 and dimension 2 leq k leq lfloor q 2 rfloor 2 k neq 3 rm if q is even do not have mds extensions hence in cases where the q 1 k mds code is essentially unique n k mds codes with n q 1 do not exist

comparison of channels criteria for domination by a symmetric channel this paper studies the basic question of whether a given channel v can be dominated in the precise sense of being more noisy by a q ary symmetric channel the concept of less noisy relation between channels originated in network information theory broadcast channels and is defined in terms of mutual information or kullback leibler divergence we provide an equivalent characterization in terms of chi 2 divergence furthermore we develop a simple criterion for domination by a q ary symmetric channel in terms of the minimum entry of the stochastic matrix defining the channel v the criterion is strengthened for the special case of additive noise channels over finite abelian groups finally it is shown that domination by a symmetric channel implies via comparison of dirichlet forms a logarithmic sobolev inequality for the original channel

weak flip codes and their optimality on the binary erasure channel this paper investigates fundamental properties of nonlinear binary codes by looking at the codebook matrix not row wise codewords but column wise the family of weak flip codes is presented and shown to contain many beautiful properties in particular the subfamily fair weak flip codes which goes back to shannon et al and which was shown to achieve the error exponent with a fixed number of codewords mathsf m can be seen as a generalization of linear codes to an arbitrary number of codewords the fair weak flip codes are related to binary nonlinear hadamard codes based on the column wise approach to the codebook matrix the r wise hamming distance is introduced as a generalization to the well known and widely used pairwise hamming distance it is shown that the minimum r wise hamming distance satisfies a generalized r wise plotkin bound the r wise hamming distance structure of the nonlinear fair weak flip codes is analyzed and shown to be superior to many codes in particular it is proven that the fair weak flip codes achieve the r wise plotkin bound with equality for all r in the second part of this paper these insights are applied to a binary erasure channel with an arbitrary erasure probability 0 delta r wise hamming distance structure of the code for a number of codewords mathsf m satisfying mathsf m le 4 and an arbitrary finite blocklength n the globally optimal codes in the sense of minimizing the average error probability are found for mathsf m 5 or mathsf m 6 and an arbitrary finite blocklength n the optimal codes are conjectured for larger mathsf m observations regarding the optimal design are presented e g that good codes have a large r wise hamming distance structure for all r numerical results validate our code design criteria and show the superiority of our best found nonlinear weak flip codes compared with the best linear codes

quasi static small strain plasticity in the limit of vanishing hardening and its numerical approximation the quasi static rate independent evolution of the prager ziegler type model of plasticity with hardening is shown to converge to the rate independent evolution of the prandtl reuss elastic perfectly plastic model based on the concept of energetic solutions we study the convergence of the solutions in the limit for hardening coefficients converging to 0 by using the abstract method of gamma convergence for rate independent systems an unconditionally convergent numerical scheme is devised and two and three dimensional numerical experiments are presented a two sided energy inequality is a posteriori verified to document experimental convergence rates

a nonsmooth multiscale method for solving frictional two body contact problems in 2d and 3d with multigrid efficiency we present a nonsmooth multiscale method for the numerical solution of frictional contact problems in 2d and 3d the computational effort is comparable to that of solving linear problems our method does not require any regularization neither of the nonpenetration condition nor of the friction law and can be applied to contact problems with tresca friction as well as to contact problems with coulomb friction for the case of tresca friction the global convergence of the method is shown for the more complicated case of coulomb friction we develop a nonsmooth multiscale method which can be directly applied to the corresponding variational quasi inequality no outer iteration is required moreover our multiscale approach is general in the sense that it can be used in the context of geometric as well as algebraic multigrid methods nonconforming domain decomposition techniques or mortar methods are employed in order to enforce the transmission conditions at the interface between different bodies with nonmatching meshes numerical examples illustrate the high robustness and efficiency of our method

lossy joint source channel coding in the finite blocklength regime this paper finds new tight finite blocklength bounds for the best achievable lossy joint source channel code rate and demonstrates that joint source channel code design brings considerable performance advantage over a separate one in the nonasymptotic regime a joint source channel code maps a block of k source symbols onto a length n channel codeword and the fidelity of reproduction at the receiver end is measured by the probability e that the distortion exceeds a given threshold d for memoryless sources and channels it is demonstrated that the parameters of the best joint source channel code must satisfy nc kr d nv k v d q 1 e where c and v are the channel capacity and channel dispersion respectively r d and v d are the source rate distortion and rate dispersion functions and q is the standard gaussian complementary cumulative distribution function symbol by symbol uncoded transmission is known to achieve the shannon limit when the source and channel satisfy a certain probabilistic matching condition in this paper we show that even when this condition is not satisfied symbol by symbol transmission is in some cases the best known strategy in the nonasymptotic regime

channel coding rate in the finite blocklength regime this paper investigates the maximal channel coding rate achievable at a given blocklength and error probability for general classes of channels new achievability and converse bounds are given which are tighter than existing bounds for wide ranges of parameters of interest and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100 it is also shown analytically that the maximal rate achievable with error probability isclosely approximated by c v n q 1 where c is the capacity v is a characteristic of the channel referred to as channel dispersion and q is the complementary gaussian cumulative distribution function

lower bounds for quantum parameter estimation the laws of quantum mechanics place fundamental limits on the accuracy of measurements and therefore on the estimation of unknown parameters of a quantum system in this paper we prove lower bounds on the size of confidence regions reported by any region estimator for a given ensemble of probe states and probability of success our bounds are derived from a previously unnoticed connection between the size of confidence regions and the error probabilities of a corresponding binary hypothesis test in group covariant scenarios we find that there is an ultimate bound for any estimation scheme which depends only on the representation theoretic data of the probe system and we evaluate its asymptotics in the limit of many systems establishing a general heisenberg limit for region estimation we apply our results to several examples in particular to phase estimation where our bounds allow us to recover the well known heisenberg and shot noise scaling

on the density of classes of closed convex sets with pointwise constraints in sobolev spaces abstract for a banach space x of r m valued functions on a lipschitz domain let k x x be a closed convex set arising from pointwise constraints on the value of the function its gradient or its divergence respectively the main result of the paper establishes under certain conditions the density of k x 0 in k x 1 where x 0 is densely and continuously embedded in x 1 the proof is constructive utilizes the theory of mollifiers and can be applied to sobolev spaces such as h 0 div and w 0 1 p in particular it is also shown that such a density result cannot be expected in general

quasi static multiple antenna fading channels at finite blocklength this paper investigates the maximal achievable rate for a given blocklength and error probability over quasi static multiple input multiple output fading channels with and without channel state information at the transmitter and or the receiver the principal finding is that outage capacity despite being an asymptotic quantity is a sharp proxy for the finite blocklength fundamental limits of slow fading channels specifically the channel dispersion is shown to be zero regardless of whether the fading realizations are available at both transmitter and receiver at only one of them or at neither of them these results follow from analytically tractable converse and achievability bounds numerical evaluation of these bounds verifies that zero dispersion may indeed imply fast convergence to the outage capacity as the blocklength increases in the example of a particular 1 times 2 single input multiple output rician fading channel the blocklength required to achieve 90 of capacity is about an order of magnitude smaller compared with the blocklength required for an awgn channel with the same capacity for this specific scenario the coding decoding schemes adopted in the lte advanced standard are benchmarked against the finite blocklength achievability and converse bounds

the error probability of generalized perfect codes via the meta converse we introduce a definition of perfect and quasi perfect codes for discrete symmetric channels based on the packing and covering properties of generalized spheres whose shape is tilted using an auxiliary probability measure this notion generalizes previous definitions of perfect and quasi perfect codes and encompasses maximum distance separable codes the error probability of these codes whenever they exist is shown to coincide with the estimate provided by the meta converse lower bound we illustrate how the proposed definition naturally extends to cover almost lossless source channel coding and lossy compression

numerical analysis of time depending primal elastoplasticity with hardening the quasi static elastoplastic evolution problem with combined isotropic and kinematic hardening is considered with emphasis on optimal convergence of the lowest order scheme in each time step of a generalized midpoint scheme such as the implicit euler or the crank nicolson scheme the spatial discretization consists of minimizing a convex but nonsmooth function on a subspace of continuous piecewise linear resp piecewise constant trial functions an a priori error estimate is established for the fully discrete method which for smooth data and a smooth exact solution proves linear convergence as the mesh size tends to zero strong convergence of the time derivatives is established under mild conditions on the mesh and time step sizes numerical experiments confirm our theoretical predictions on the improved spatial convergence and indicate that the crank nicolson scheme is not always superior over the implicit euler scheme in practice

path following methods for a class of constrained minimization problems in function space path following methods for primal dual active set strategies requiring a regularization parameter are introduced existence of a primal dual path and its differentiability properties are analyzed monotonicity and convexity of the primal dual path value function are investigated both feasible and infeasible approximations are considered numerical path following strategies are developed and their efficiency is demonstrated by means of examples

path following and augmented lagrangian methods for contact problems in linear elasticity a certain regularization technique for contact problems leads to a family of problems that can be solved efficiently using infinite dimensional semismooth newton methods or in this case equivalently primal dual active set strategies we present two procedures that use a sequence of regularized problems to obtain the solution of the original contact problem first order augmented lagrangian and path following methods the first strategy is based on a multiplier update while path following with respect to the regularization parameter uses theoretical results about the path value function to increase the regularization parameter appropriately comprehensive numerical tests investigate the performance of the proposed strategies for both a 2d as well as a 3d contact problem

beta beta bounds finite blocklength analog of the golden formula it is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution a relation sometimes referred to as the golden formula this paper is concerned with a finite blocklength extension of this relation this extension consists of two elements 1 a finite blocklength channel coding converse bound by polyanskiy and verdu which involves the ratio of two neyman pearson beta functions beta beta converse bound and 2 a novel beta beta channel coding achievability bound expressed again as the ratio of two neyman pearson beta functions to demonstrate the usefulness of this finite blocklength extension of the golden formula the beta beta achievability and converse bounds are used to obtain a finite blocklength extension of verdu s wideband slope approximation the proof parallels the derivation of the latter with the beta beta bounds used in place of the golden formula the beta beta achievability bound is also shown to be useful in cases where the capacity achieving output distribution is not a product distribution due to e g a cost constraint or structural constraints on the codebook such as orthogonality or constant composition as an example the bound is used to characterize the channel dispersion of the additive exponential noise channel and to obtain a finite blocklength achievability bound the tightest to date for multiple input multiple output rayleigh fading channels with perfect channel state information at the receiver

on the superlinear convergence in computational elasto plasticity we consider the convergence properties of return algorithms for a large class of rate independent plasticity models based on recent results for semismooth functions we can analyze these algorithms in the context of semismooth newton methods guaranteeing local superlinear convergence this recovers results for classical models but also extends to general hardening laws multi yield plasticity and to several non associated models the superlinear convergence is also numerically shown for a large scale parallel simulation of drucker prager elasto plasticity and an example for the modified cam clay model

refinement of the sphere packing bound asymmetric channels we provide a refinement of the sphere packing bound for constant composition codes over asymmetric discrete memoryless channels that improves the subexponential factor in front of the exponent the order of our subexponential factor is n 0 5 1 e r for any 0 where r is the left derivative of the sphere packing exponent at rate r and n is the blocklength

semi smooth newton methods for mixed fem discretizations of higher order for frictional elasto plastic two body contact problems in this article a semi smooth newton method for frictional two body contact problems and a solution algorithm for the resulting sequence of linear systems are presented it is based on a mixed variational formulation of the problem and a discretization by finite elements of higher order general friction laws depending on the normal stresses and elasto plastic material behavior with linear isotropic hardening are considered numerical results show the efficiency of the presented algorithm

solution of one time step problems in elastoplasticity by a slant newton method we discuss a solution algorithm for quasi static elastoplastic problems with hardening such problems can be described by a time dependent variational inequality where the displacement and the plastic strain fields serve as primal variables after discretization in time one variational inequality of the second kind is obtained per time step and can be reformulated as a minimization problem with a convex energy functional which depends smoothly on the displacement and nonsmoothly on the plastic strain there exists an explicit formula for minimizing the energy functional with respect to the plastic strain for a given displacement by substitution the energy functional can be written as a functional depending only on the displacement the theorem of moreau from convex analysis states that the energy functional is differentiable with an explicitly computable first derivative the second derivative of the energy functional does not exist due to the lack of smoothness of the plastic strain across the elastoplastic interface which separates the continuum in elastically and plastically deformed parts a newton like method exploiting slanting functions of the energy functional s first derivative instead of the nonexistent second derivative is applied such a method is called a slant newton method for short the local superlinear convergence of the algorithm in the discrete case is shown and sufficient regularity assumptions are formulated which would guarantee the local superlinear convergence also in the continuous case

unequal message protection asymptotic and non asymptotic tradeoffs we study a form of unequal error protection that we term unequal message protection ump the message set of a ump code is a union of m disjoint message classes each class has its own error protection requirement with some classes needing better error protection than others we analyze the tradeoff between rates of message classes and the levels of error protection our analysis reveals new tradeoffs which were not captured by prior works on ump codes to obtain our results we generalize finite block length achievability and converse bounds due to polyanskiy poor verdu we evaluate our bounds for the binary symmetric and binary erasure channels and analyze the asymptotic characteristic of the bounds in the fixed error and moderate deviations regimes in addition we consider two questions related to the practical construction of ump codes first we study a header construction that prefixes the message class into a header followed by data protection using a standard homogeneous classical code we show that in general this construction is not optimal at finite block lengths we further demonstrate that our main ump achievability bound can be obtained using coset codes which suggests a path to implementation of tractable ump codes

a sequential minimization technique for elliptic quasi variational inequalities with gradient constraints a class of nonlinear elliptic quasi variational inequality qvi problems with gradient constraints in function space is considered problems of this type arise for instance in the mathematical description of the magnetization of superconductors in problems in elastoplasticity or in electrostatics as well as in game theory the paper addresses the iterative solution of the qvis by a sequential minimization technique relying on the repeated solution of variational inequality type problems a monotone operator theoretic approach is developed which does not resort to mosco convergence as is often done in connection with existence analysis for qvis for the numerical solution of the qvis a penalty approach combined with a semismooth newton iteration is proposed the paper ends with a report on numerical tests involving the p laplace operator and various types of nonlinear constraints

a convergent adaptive finite element method for the primal problem of elastoplasticity summary the boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional this paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and up to higher order terms the r linear convergence of the stresses with respect to the number of loops applications include several plasticity models linear isotropic kinematic hardening linear kinematic hardening and multisurface plasticity as model for nonlinear hardening laws for perfect plasticity the adaptive algorithm yields strong convergence of the stresses numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule copyright q 2006 john wiley sons ltd

quasistatic evolution problems for linearly elastic perfectly plastic materials the problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate independent processes existence of solutions is proved through the use of incremental variational problems in spaces of functions with bounded deformation this approach provides a new approximation result for the solutions of the quasistatic evolution problem which are shown to be absolutely continuous in time four equivalent formulations of the problem in rate form are derived a strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain

performance bounds for nonbinary linear block codes over memoryless symmetric channels the performance of nonbinary linear block codes is studied in this paper via the derivation of new upper bounds on the block error probability under maximum likelihood ml decoding the transmission of these codes is assumed to take place over a memoryless and symmetric channel the new bounds which are based on the gallager bounds and their variations are applied to the gallager ensembles of nonbinary and regular low density parity check ldpc codes these upper bounds are also compared with sphere packing lower bounds this study indicates that the new upper bounds are useful for the performance evaluation of coded communication systems which incorporate nonbinary coding techniques

a tight upper bound for the third order asymptotics for most discrete memoryless channels this paper shows that the logarithm of the e error capacity average error probability for n uses of a discrete memoryless channel dmc is upper bounded by the normal approximation plus a third order term that does not exceed 1 2 logn o 1 if the e dispersion of the channel is positive this matches a lower bound by y polyanskiy 2010 for dmcs with positive reverse dispersion if the e dispersion vanishes the logarithm of the e error capacity is upper bounded by n times the capacity plus a constant term except for a small class of dmcs and e 1 2

duality for the group coset probability ratio corresp from the macwilliams and assmus mattson identifies it is shown that sullivan s lower bound on the ratio of the probability of a group and a coset has a dual relation with the upper bound

an interior point algorithm for elastoplasticity the problem of small deformation rate independent elastoplasticity is treated using convex programming theory and algorithms a finite step variational formulation is first derived after which the relevant potential is discretized in space and subsequently viewed as the lagrangian associated with a convex mathematical program next an algorithm based on the classical primal dual interior point method is developed several key modifications to the conventional implementation of this algorithm are made to fully exploit the nature of the common elastoplastic boundary value problem the resulting method is compared to state of the art elastoplastic procedures for which both similarities and differences are found finally a number of examples are solved demonstrating the capabilities of the algorithm when applied to standard perfect plasticity hardening multisurface plasticity and problems involving softening copyright 2006 john wiley sons ltd

a duality based path following semismooth newton method for elasto plastic contact problems a fenchel dualization scheme for the one step time discretized contact problem of quasi static elasto plasticity with combined kinematic isotropic hardening is considered the associated path is induced by a coupled moreau yosida tikhonov regularization of the dual problem the sequence of solutions to the regularized problems is shown to converge strongly to the optimal displacement stress strain triple of the original elasto plastic contact problem in the space continuous setting this property relies on the density of the intersection of certain convex sets which is shown as well it is also argued that the mappings associated with the resulting problems are newton or slantly differentiable consequently each regularized subsystem can be solved mesh independently at a local superlinear rate of convergence for efficiency purposes an inexact path following approach is proposed and a numerical validation of the theoretical results is given

one and two level schwarz methods for variational inequalities of the second kind and their application to frictional contact we present and analyze subspace correction methods for the solution of variational inequalities of the second kind and apply these theoretical results to non smooth contact problems in linear elasticity with tresca and non local coulomb friction we introduce these methods in a reflexive banach space prove that they are globally convergent and give error estimates in the context of finite element discretizations where our methods turn out to be one and two level schwarz methods we specify their convergence rate and its dependence on the discretization parameters and conclude that our methods converge optimally transferring this results to frictional contact problems we thus can overcome the mesh dependence of some fixed point schemes which are commonly employed for contact problems with coulomb friction

on subsets of the normal rational curve a normal rational curve of the k 1 dimensional projective space over mathbb f q is an arc of size q 1 since any k points of the curve span the whole space in this paper we will prove that if q is odd then a subset of size 3k 6 of a normal rational curve cannot be extended to an arc of size q 2 in fact we prove something slightly stronger suppose that q is odd and e is a 2k 3 subset of an arc g of size 3k 6 if g projects to a subset of a conic from every k 3 subset of e then g cannot be extended to an arc of size q 2 stated in terms of error correcting codes we prove that a k dimensional linear maximum distance separable code of length 3k 6 over a field mathbb f q of odd characteristic which can be extended to a reed solomon code of length q 1 cannot be extended to a linear maximum distance separable code of length q 2

a note on the poor verdu upper bound for the channel reliability function in an earlier work poor and verdu 1995 established an upper bound for the reliability function of arbitrary single user discrete time channels with memory they also conjectured that their bound is tight for all coding rates we demonstrate via a counterexample involving memoryless binary erasure channels becs that the poor verdu upper bound is not tight at low rates we conclude by examining possible improvements to this bound

feedback in the non asymptotic regime without feedback the backoff from capacity due to non asymptotic blocklength can be quite substantial for blocklengths and error probabilities of interest in many practical applications in this paper novel achievability bounds are used to demonstrate that in the non asymptotic regime the maximal achievable rate improves dramatically thanks to variable length coding and feedback for example for the binary symmetric channel with capacity 1 2 the blocklength required to achieve 90 of the capacity is smaller than 200 compared to at least 3100 for the best fixed blocklength code even with noiseless feedback virtually all the advantages of noiseless feedback are shown to be achievable even if the feedback link is used only to send a single signal informing the encoder to terminate the transmission stop feedback it is demonstrated that the non asymptotic behavior of the fundamental limit depends crucially on the particular model chosen for the end of packet control signal fixed blocklength codes and related questions concerning communicating with a guaranteed delay are discussed in which situation feedback is demonstrated to be almost useless even non asymptotically

channels with cost constraints strong converse and dispersion this paper shows the strong converse and the dispersion of memoryless channels with cost constraints and performs a refined analysis of the third order term in the asymptotic expansion of the maximum achievable channel coding rate showing that it is equal to 1 2 log n n in most cases of interest the analysis is based on a nonasymptotic converse bound expressed in terms of the distribution of a random variable termed the mathsf b tilted information density which plays a role similar to that of the mathsf d tilted information in lossy source coding we also analyze the fundamental limits of lossy joint source channel coding over channels with cost constraints

finite blocklength converse bounds for quantum channels we derive upper bounds on the rate of transmission of classical information over quantum channels by block codes with a given blocklength and error probability for both entanglement assisted and unassisted codes in terms of a unifying framework of quantum hypothesis testing with restricted measurements our bounds do not depend on any special property of the channel such as memorylessness and generalize both a classical converse of polyanskiy poor and verdu as well as a quantum converse of renner and wang and have a number of desirable properties in particular our bound on entanglement assisted codes is a semidefinite program and for memoryless channels its large blocklength limit is the well known formula for entanglement assisted capacity due to bennett shor smolin and thapliyal

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

an improved sphere packing bound for finite length codes over symmetric memoryless channels this paper derives an improved sphere packing isp bound for finite length error correcting codes whose transmission takes place over symmetric memoryless channels and the codes are decoded with an arbitrary list decoder we first review classical results i e the 1959 sphere packing sp59 bound of shannon for the gaussian channel and the 1967 sphere packing sp67 bound of shannon et al for discrete memoryless channels an improvement on the sp67 bound as suggested by valembois and fossorier is also discussed these concepts are used for the derivation of a new lower bound on the error probability of list decoding referred to as the isp bound which is uniformly tighter than the sp67 bound and its improved version the isp bound is applicable to symmetric memoryless channels and some of its applications are presented its tightness under maximum likelihood ml decoding is studied by comparing the isp bound to previously reported upper and lower bounds on the ml decoding error probability and also to computer simulations of iteratively decoded turbo like codes this paper also presents a technique which performs the entire calculation of the sp59 bound in the logarithmic domain thus facilitating the exact calculation of this bound for moderate to large block lengths without the need for the asymptotic approximations provided by shannon

saddle point in the minimax converse for channel coding a minimax metaconverse has recently been proposed as a simultaneous generalization of a number of classical results and a tool for the nonasymptotic analysis in this paper it is shown that the order of optimizing the input and output distributions can be interchanged without affecting the bound in the course of the proof a number of auxiliary results of separate interest are obtained in particular it is shown that the optimization problem is convex and can be solved in many cases by the symmetry considerations as a consequence it is demonstrated that in the latter cases the multiletter input distribution in information spectrum verdu han converse bound can be taken to be a memoryless product of single letter ones a tight converse for the binary erasure channel is rederived by computing the optimal nonproduct output distribution for discrete memoryless channels a conjecture of poor and verdu regarding the tightness of the information spectrum bound on the error exponents is resolved in the negative concept of the channel symmetry group is established and relations with the definitions of symmetry by gallager and dobrushin are investigated

a mesh independence result for semismooth newton methods for a class of semismooth operator equations a mesh independence result for generalized newton methods is established the main result states that the continuous and the discrete newton process when initialized properly converge q linearly with the same rate the problem class considered in the paper includes mcp function based reformulations of first order conditions of a class of control constrained optimal control problems for partial differential equations for which a numerical validation of the theoretical results is given

deep holes of projective reed solomon codes projective reed solomon prs codes are reed solomon codes of the maximum possible length q 1 the classification of deep holes received words with maximum possible error distance for prs codes is an important and difficult problem in this paper we use algebraic methods to explicitly construct three classes of deep holes for prs codes we show that these three classes completely classify all deep holes of prs codes with redundancy four previously the deep hole classification was only known for prs codes with redundancy at most three

pde constrained optimization subject to pointwise constraints on the control the state and its derivative a general moreau yosida based framework for minimization problems subject to partial differential equations and pointwise constraints on the control the state and its derivative is considered a range space constraint qualification is used to argue existence of lagrange multipliers and to derive a kkt type system for characterizing first order optimality of the unregularized problem the theoretical framework is then used to develop a semismooth newton algorithm in function space and to prove its locally superlinear convergence when solving the regularized problems further for maintaining the local superlinear convergence in function space it is demonstrated that in some cases it might be necessary to add a lifting step to the newton framework in order to bridge an l 2 l r norm gap with r 2 the paper ends by a report on numerical tests

a minimax approach for the study of systems of variational equations and related galerkin schemes abstract the aim of this work is to analyze the existence of a solution for a quite general variational inequalities system which includes those appearing in the theory of mixed variational equations the so called babuska brezzi theory it is done by means of a minimax inequality which also provides us with a estimate of the norm of the solution then we derive a numerical method of the galerkin type to approximate the solution of such a system its stability follows from the mentioned control of the norm of the solution

the third order term in the normal approximation for the awgn channel this paper shows that under the average error probability formalism the third order term in the normal approximation for the additive white gaussian noise channel with a maximal or equal power constraint is at least 1 2 log n o 1 this improves on the lower bound by polyanskiys poor verdu 2010 and matches the upper bound proved by the same authors

operational interpretation of renyi information measures via composite hypothesis testing against product and markov distributions we revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis we introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms in this case we find the threshold rate the optimal error and strong converse exponents at large deviations from the threshold and the second order asymptotics at small deviations from the threshold we apply our results to find the operational interpretations of various renyi information measures in case the alternative hypothesis is comprised of bipartite product distributions we find that the optimal error and strong converse exponents are determined by the variations of renyi mutual information in case the alternative hypothesis consists of tripartite distributions satisfying the markov property we find that the optimal exponents are determined by the variations of renyi conditional mutual information in either case the relevant notion of renyi mutual information depends on the precise choice of the alternative hypothesis as such this paper also strengthens the view that different definitions of renyi mutual information conditional entropy and conditional mutual information are adequate depending on the context in which the measures are used

a linear program for the finite block length converse of polyanskiy poor verdu via nonsignaling codes motivated by recent work on entanglement assisted codes for sending messages over classical channels the larger easily characterized class of nonsignaling codes is defined analyzing the optimal performance of these codes yields an alternative proof of the finite block length converse of polyanskiy poor and verdu and shows that they achieve this converse this provides an explicit formulation of the converse as a linear program which has some useful features for discrete memoryless channels it is shown that nonsignaling codes attain the channel capacity with zero error probability if and only if the dispersion of the channel is zero

optimal ultrasmall block codes for binary discrete memoryless channels optimal block codes in the sense of minimum average error probability using maximum likelihood decoding with a small number of codewords are investigated for the binary asymmetric channel bac including the two special cases of the binary symmetric channel bsc and the z channel zc both with arbitrary cross over probabilities for the zc the optimal code structure for an arbitrary finite blocklength is derived in the cases of two three and four codewords and conjectured in the case of five codewords for the bsc the optimal code structure for an arbitrary finite blocklength is derived in the cases of two and three codewords and conjectured in the case of four codewords for a general bac the best codebooks under the assumption of a threshold decoder are derived for the case of two codewords the derivation of these optimal codes relies on a new approach of constructing and analyzing the codebook matrix not rowwise codewords but columnwise this new tool leads to an elegant definition of interesting code families that is recursive in the blocklength n and admits their exact analysis of error performance this allows for a comparison of the average error probability between all possible codebooks

density of convex intersections and applications in this paper we address density properties of intersections of convex sets in several function spaces using the concept of convergence it is shown in a general framework how these density issues naturally arise from the regularization discretization or dualization of constrained optimization problems and from perturbed variational inequalities a variety of density results and counterexamples for pointwise constraints in sobolev spaces are presented and the corresponding regularity requirements on the upper bound are identified the results are further discussed in the context of finite element discretizations of sets associated with convex constraints finally two applications are provided which include elasto plasticity and image restoration problems

nonlinear complementarity functions for plasticity problems with frictional contact in this paper a unifying framework for the numerical treatment of frictional contact in combination with elastoplasticity with hardening is presented both kinds of problems lead to the mathematical structure of variational inequalities which are discretized by introducing additional variables for the contact stresses or the deviatoric stresses leading to a mixed formulation by defining a family of nonlinear complementarity functions which can be applied to both frictional and plastic yield conditions these inequality constraints are reformulated as equalities combined with the equilibrium equations a system of nonlinear equations is obtained that is solved in terms of a semismooth newton method to obtain a larger domain of convergence and better stability properties the introduction of additional scaling parameters is motivated and their benefit to the convergence of the scheme is demonstrated due to the locality of the contact and plastic constraints the resulting method can be interpreted as a primal dual active set strategy furthermore for linear hardening laws our method is a generalization of the well known radial return algorithm which is obtained for a special suboptimal choice of the plastic scaling parameters the efficiency and the generality of the proposed quasi newton scheme is demonstrated by several numerical examples including the simulation of a three dimensional contact problem with several elastoplastic bodies exponential isotropic hardening and coulomb friction

a lower bound on the probability of error in multihypothesis testing consider two random variables x and y where x is finitely or countably infinitely valued and where y is arbitrary let spl epsiv denote the minimum probability of error incurred in estimating x from y it is shown that spl epsiv spl ges sub 0 spl les spl alpha spl les 1 sup sup 1 spl alpha p spl pi x y spl les spl alpha where spl pi x y denotes the posterior probability of x given y this bound finds information theoretic applications in the proof of converse channel coding theorems it generalizes and strengthens previous lower bounds due to shannon and to verdu and han 1994

on the evaluation of the polyanskiy poor verdu converse bound for finite block length coding in awgn a tight converse bound to the channel coding rate in the finite block length regime and under additive white gaussian noise conditions was recently proposed by polyanskiy poor verdu ppv the bound is a generalization of a number of other classical results and it was also claimed to be equivalent to shannon s 1959 cone packing bound in this paper we investigate methods for a reliable numerical evaluation of the bound which is troublesome even for not too large values of the block length n by compactly expressing the polyanskiy poor and verdu ppv converse bound in terms of non central chi squared distributions and by evaluating those through an integral expression and a corresponding series expansion which exploit a method proposed by temme as a result a robust evaluation method and new insights on the bound s asymptotics as well as new approximate expressions are obtained