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

a superlinearly convergent sequential quadratically constrained quadratic programming algorithm for degenerate nonlinear programming we present an algorithm that achieves superlinear convergence for nonlinear programs satisfying the mangasarian fromovitz constraint qualification and the quadratic growth condition this convergence result is obtained despite the potential lack of a locally convex augmented lagrangian the algorithm solves a succession of subproblems that have quadratic objectives and quadratic constraints both possibly nonconvex by the use of a trust region constraint we guarantee that any stationary point of the subproblem induces superlinear convergence which avoids the problem of computing a global minimum we compare this algorithm with sequential quadratic programming algorithms on several degenerate nonlinear programs

the cayley hilbert metric and positive operators abstract the cayley hilbert metric is defined for a real banach space containing a closed cone by restricting the domain of a particular type of positive nonlinear operator the banach contraction mapping theorem is used to prove the existence of a unique fixed point of the operator with explicit upper and lower bounds applications to quasilinear elliptic partial differential equations and to matrix theory are considered

a theoretically supported scalable tfeti algorithm for the solution of multibody 3d contact problems with friction abstract a total feti tfeti based domain decomposition algorithm with preconditioning by a natural coarse grid defined by the rigid body motions of subdomains is adapted to the solution of multibody contact problems with friction in 3d and proved to be scalable for tresca friction the analysis admits floating bodies the algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition and the discretization parameters is bounded the analysis is based on the classical results by farhat et al on scalability of feti for linear problems and on our development of optimal quadratic programming algorithms the algorithm preserves parallel scalability of the classical feti method both theoretical results and numerical experiments indicate that our algorithm is effective the effectiveness of the method in the inner loop of the fixed point iterations for the solution of the contact problems with coulomb s friction is illustrated on the analysis of a mine support

structure of states which satisfy strong subadditivity of quantum entropy with equality we give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von neumann entropy by combining a result of petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by koashi and imoto we show that such states will have the form of a so called short quantum markov chain which in turn implies that two of the systems are independent conditioned on the third in a physically meaningful sense this characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the holevo bound on accessible information

box constrained quadratic programming with proportioning and projections two new closely related concepts are introduced that depend on a positive constant gamma an iteration is proportional if the norm of violation of the kuhn tucker conditions at active variables does not excessively exceed the norm of the part of the gradient that corresponds to free variables while a progressive direction determines a descent direction that enables the released variables to move far enough from the boundary in a step called proportioning an algorithm that uses the conjugate gradient method to explore the face of the region defined by the current iterate until a disproportional iteration is generated is proposed it then changes the face by means of the progressive direction it is proved that for strictly convex problems the proportioning is a spacer iteration so that the algorithm converges to the solution if the solution is nondegenerate then the algorithm finds the solution in a finite number of steps moreover a simple lower bound on gamma is given to ensure finite termination even for problems with degenerate solutions the theory covers a class of algorithms allowing many constraints to be added or dropped at a time and accepting approximate solutions of auxiliary problems preliminary numerical results are promising

the entropy power inequality for quantum systems when two independent analog signals x and y are added together giving z x y the entropy of z h z is not a simple function of the entropies h x and h y but rather depends on the details of x and y s distributions nevertheless the entropy power inequality epi which states that e2h z e2h x e2h y gives a very tight restriction on the entropy of z this inequality has found many applications in information theory and statistics the quantum analogue of adding two random variables is the combination of two independent bosonic modes at a beam splitter the purpose of this paper is to give a detailed outline of the proof of two separate generalizations of the epi to the quantum regime our proofs are similar in spirit to the standard classical proofs of the epi but some new quantities and ideas are needed in the quantum setting in particular we find a new quantum de bruijin identity relating entropy production under diffusion to a divergence based quantum fisher information furthermore this fisher information exhibits certain convexity properties in the context of beam splitters

on the solution of convex qpqc problems with elliptic and other separable constraints with strong curvature the paper deals with an effective implementation of some algorithms for the solution of convex qpqc problems with elliptic and other separable constraints with strong curvature here we discuss robust quantitative refinement of the karush kuhn tucker conditions extend existing results on the decrease of the cost function along the projected gradient path to separable constraints with elliptic components and plug them into the existing algorithms for the solution of the qpqc problems with r linear rate of convergence in the bounds on the spectrum the results are then extended to the problems with separable inequality and linear equality constraints the performance of the algorithms is demonstrated on the solution of a problem of two cantilever beams in mutual contact with orthotropic tresca and coulomb friction discretized by up to one and half million nodal variables

augmented lagrangians with adaptive precision control for quadratic programming with simple bounds and equality constraints in this paper we discuss a specialization of the augmented lagrangian type algorithm of conn gould and toint to the solution of strictly convex quadratic programming problems with simple bounds and equality constraints the new feature of the presented algorithm is the adaptive precision control of the solution of auxiliary problems in the inner loop of the basic algorithm which yields a rate of convergence that does not have any term that accounts for inexact solution of auxiliary problems moreover boundedness of the penalty parameter is achieved for the precision control used numerical experiments illustrate the efficiency of the presented algorithm and encourage its usage

brunn minkowski and zhang inequalities for convolution bodies abstract a quantitative version of minkowski sum extending the definition of convolution of convex bodies is studied to obtain extensions of the brunn minkowski and zhang inequalities as well as other interesting properties on convex geometry involving convolution bodies or polar projection bodies the extension of this new version to more than two sets is also given

the conditional entropy power inequality for quantum additive noise channels we prove the quantum conditional entropy power inequality for quantum additive noise channels this inequality lower bounds the quantum conditional entropy of the output of an additive noise channel in terms of the quantum conditional entropies of the input state and the noise when they are conditionally independent given the memory we also show that this conditional entropy power inequality is optimal in the sense that we can achieve equality asymptotically by choosing a suitable sequence of gaussian input states we apply the conditional entropy power inequality to find an array of information theoretic inequalities for conditional entropies which are the analogs of inequalities which have already been established in the unconditioned setting furthermore we give a simple proof of the convergence rate of the quantum ornstein uhlenbeck semigroup based on entropy power inequalities we prove the quantum conditional entropy power inequality for quantum additive noise channels this inequality lower bounds the quantum conditional entropy of the output of an additive noise channel in terms of the quantum conditional entropies of the input state and the noise when they are conditionally independent given the memory we also show that this conditional entropy power inequality is optimal in the sense that we can achieve equality asymptotically by choosing a suitable sequence of gaussian input states we apply the conditional entropy power inequality to find an array of information theoretic inequalities for conditional entropies which are the analogs of inequalities which have already been established in the unconditioned setting furthermore we give a simple proof of the convergence rate of the quantum ornstein uhlenbeck semigroup based on entropy power inequalities

corrections to the entropy power inequality for quantum systems we correct an intermediate step in the derivation of our main statements in 1 specifically inequality 63 in the mentioned paper intended to give an upper bound on the entropy of certain gaussian states is incorrect in that paper we used inequality 63 to derive the asymptotic large time scaling of the entropy under the quantum version of the heat equation we provide alternative derivations of this result which sidestep the bound 63 the main conclusions of the paper therefore remain unaffected we thank giacomo de palma andrea mari and vittorio giovannetti for pointing out this issue and simultaneously providing us with a resolution in a very detailed communication we also thank them for their permission to include their discussion in this erratum we note that one of the alternative derivations of the asymptotic scaling presented here has previously appeared in their publication

an optimal algorithm and superrelaxation for minimization of a quadratic function subject to separable convex constraints with applications we propose a modification of our mpgp algorithm for the solution of bound constrained quadratic programming problems so that it can be used for minimization of a strictly convex quadratic function subject to separable convex constraints our active set based algorithm explores the faces by conjugate gradients and changes the active sets and active variables by gradient projections possibly with the superrelaxation steplength the error estimate in terms of extreme eigenvalues guarantees that if a class of minimization problems has the spectrum of the hessian matrix in a given positive interval then the algorithm can find and recognize an approximate solution of any particular problem in a number of iterations that is uniformly bounded we also show how to use the algorithm for the solution of separable and equality constraints the power of our algorithm and its optimality are demonstrated on the solution of a problem of two cantilever beams in mutual contact with tresca friction discretized by almost twelve millions nodal variables

geometric inequalities from phase space translations we establish a quantum version of the classical isoperimetric inequality relating the fisher information and the entropy power of a quantum state the key tool is a fisher information inequality for a state which results from a certain convolution operation the latter maps a classical probability distribution on phase space and a quantum state to a quantum state we show that this inequality also gives rise to several related inequalities whose counterparts are well known in the classical setting in particular it implies an entropy power inequality for the mentioned convolution operation as well as the isoperimetric inequality and establishes concavity of the entropy power along trajectories of the quantum heat diffusion semigroup as an application we derive a log sobolev inequality for the quantum ornstein uhlenbeck semigroup and argue that it implies fast convergence towards the fixed point for a large class of initial states

mismatched estimation and relative entropy a random variable with distribution p is observed in gaussian noise and is estimated by a mismatched minimum mean square estimator that assumes that the distribution is q instead of p this paper shows that the integral over all signal to noise ratios snrs of the excess mean square estimation error incurred by the mismatched estimator is twice the relative entropy d p q in nats this representation of relative entropy can be generalized to nonreal valued random variables and can be particularized to give new general representations of mutual information in terms of conditional means inspired by the new representation we also propose a definition of free relative entropy which fills a gap in and is consistent with the literature on free probability

asymptotics of controlled finite memory filters in the paper partially observed controlled problem with the memory consisting of the past k observations is considered controlled finite memory filters are approximated it is shown that average cost per unit time cost functional is characterized by the value of an approximate filter and is continuous with respect to control functions

inexact semimonotonic augmented lagrangians with optimal feasibility convergence for convex bound and equality constrained quadratic programming a variant of the augmented lagrangian type algorithm for strictly convex quadratic programming problems with bounds and equality constraints is considered the algorithm exploits the adaptive precision control in the solution of auxiliary bound constraint problems in the inner loop while the lagrange multipliers for the equality constraints are updated in the outer loop the update rule for the penalty parameter is introduced that depends on the increase of the augmented lagrangian global convergence is proved and an explicit bound on the penalty parameter is given a qualitatively new feature of our algorithm is a bound on the feasibility error that is independent of conditioning of the constraints when applied to the class of problems with the spectrum of the hessian matrix in a given interval the algorithm returns the solution in o 1 matrix vector multiplications the results are valid even for linearly dependent constraints theoretical results are illustrated by numerical experiments including the solution of an elliptic variational inequality

on the linear convergence of descent methods for convex essentially smooth minimization consider the problem of minimizing over a polyhedral set the composition of an affine mapping with a strictly convex essentially smooth function a general result on the linear convergence of descent methods for solving this problem is presented by applying this result the linear convergence of both the gradient projection algorithm of goldstein and levitin and polyak and a matrix splitting algorithm using regular splitting is established the results do not require that the cost function be strongly convex or that the optimal solution set be bounded the key to the analysis lies in a new error bound for estimating the distance from a feasible point to the optimal solution set

reverse brunn minkowski and reverse entropy power inequalities for convex measures we develop a reverse entropy power inequality for convex measures which may be seen as an anegeometric inverse of the entropy power inequality of shannon and stam the specialization of this inequality to log concave measures may be seen as a version of milman s reverse brunn minkowski inequality the proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies and on a study of eective supports of convex measures both of which are of independent interest as well as on milman s deep technology of m ellipsoids and on certain information theoretic inequalities as a by product we also give a

inequalities for quantum entropy a review with conditions for equality this article presents self contained proofs of the strong subadditivity inequality for von neumann s quantum entropy s and some related inequalities for the quantum relative entropy most notably its convexity and its monotonicity under stochastic maps moreover the approach presented here which is based on klein s inequality and lieb s theorem that the function a tr ek log a is concave allows one to obtain conditions for equality in the case of strong subadditivity which states that s 123 s 2 s 12 s 23 where the subscripts denote subsystems of a composite system equality holds if and only if log 123 log 12 log 2 log 23 using the fact that the holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy we show that equality can be attained for that bound only when the states in the ensemble commute the article concludes with an appendix giving a short description of epstein s elegant proof of lieb s the

gaussian optimizers for entropic inequalities in quantum information we survey the state of the art for the proof of the quantum gaussian optimizer conjectures of quantum information theory these fundamental conjectures state that quantum gaussian input states are the solution to several optimization problems involving quantum gaussian channels these problems are the quantum counterpart of three fundamental results of functional analysis and probability the entropy power inequality the sharp young s inequality for convolutions and the theorem gaussian kernels have only gaussian maximizers quantum gaussian channels play a key role in quantum communication theory they are the quantum counterpart of gaussian integral kernels and provide the mathematical model for the propagation of electromagnetic waves in the quantum regime the quantum gaussian optimizer conjectures are needed to determine the maximum communication rates over optical fibers and free space the restriction of the quantum limited gaussian attenuator to input states diagonal in the fock basis coincide

a solution of gaussian optimizer conjecture for quantum channels the long standing conjectures of the optimality of gaussian inputs and additivity are solved for a broad class of gauge covariant or contravariant bosonic gaussian channels which includes in particular thermal additive classical noise and amplifier channels restricting to the class of states with finite second moments we show that the vacuum is the input state which minimizes the entropy at the output of such channels this allows us to show also that the classical capacity of these channels under the input energy constraint is additive and is achieved by gaussian encodings

scalable tfeti algorithm for the solution of multibody contact problems of elasticity a total feti tfeti based domain decomposition algorithm with preconditioning by a natural coarse grid of the rigid body motions is adapted to the solution of multibody contact problems of elasticity in 2d and 3d and proved to be scalable the algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition parameter and the discretization parameter is bounded the analysis is based on the classical results by farhat mandel and roux on scalability of feti with a natural coarse grid for linear problems and on our development of optimal quadratic programming algorithms for bound and equality constrained problems the algorithm preserves parallel scalability of the classical feti method both theoretical results and numerical experiments indicate a high efficiency of our algorithm in addition its performance is illustrated on a real world problem of analysis of the ball bearing copyright 2009 john wiley sons ltd

minimization of a large scale quadratic function subject to a spherical constraint an important problem in linear algebra and optimization is the trust region subproblem minimize a quadratic function subject to an ellipsoidal or spherical constraint this basic problem has several important large scale applications including seismic inversion and forcing convergence in optimization methods existing methods to solve the trust region subproblem require matrix factorizations which are not feasible in the large scale setting this paper presents an algorithm for solving the large scale trust region subproblem that requires a fixed size limited storage proportional to the order of the quadratic and that relies only on matrix vector products the algorithm recasts the trust region subproblem in terms of a parameterized eigenvalue problem and adjusts the parameter with a superlinearly convergent iteration to find the optimal solution from the eigenvector of the parameterized problem only the smallest eigenvalue and corresponding eigenvector of the parameterized problem needs to be computed the implicitly restarted lanczos method is well suited to this subproblem

asymptotic redundancies for universal quantum coding clarke and barren 1990 1994 1995 have shown that the jeffreys invariant prior of bayesian theory yields the common asymptotic minimax and maximin redundancy of universal data compression in a parametric setting we seek a possible analog of this result for the two level quantum systems we restrict our considerations to prior probability distributions belonging to a certain one parameter family q sub u spl infin u 1 within this setting we are able to compute exact redundancy formulas for which we find the asymptotic limits we compare our quantum asymptotic redundancy formulas to those derived by naively applying the nonquantum counterparts of clarke and barren and find certain common features our results are based on formulas we obtain for the eigenvalues and eigenvectors of 2 sup n spl times 2 sup n bayesian density matrices spl zeta sub n u these matrices are the weighted averages with respect to q sub u of all possible tensor products of n identical 2 spl times 2 density matrices representing the two level quantum systems we propose a form of universal coding for the situation in which the density matrix describing an ensemble of quantum signal states is unknown a sequence of n signals would be projected onto the dominant eigenspaces of spl zeta sub n u

first order asymptotics of matrix integrals a rigorous approach towards the understanding of matrix models we investigate the large n limit of spectral measures of matrices which relate to the gibbs measures of a number of statistical mechanical systems on random graphs these include the ising and potts models on random graphs for most of these models we prove that the spectral measures converge almost surely and describe their limit via solutions to an euler equation for isentropic flow with negative pressure p 3 1 2 3

the conditional entropy power inequality for bosonic quantum systems we prove the conditional entropy power inequality for gaussian quantum systems this fundamental inequality determines the minimum quantum conditional von neumann entropy of the output of the beam splitter or of the squeezing among all the input states where the two inputs are conditionally independent given the memory and have given quantum conditional entropies we also prove that for any couple of values of the quantum conditional entropies of the two inputs the minimum of the quantum conditional entropy of the output given by the conditional entropy power inequality is asymptotically achieved by a suitable sequence of quantum gaussian input states our proof of the conditional entropy power inequality is based on a new stam inequality for the quantum conditional fisher information and on the determination of the universal asymptotic behaviour of the quantum conditional entropy under the heat semigroup evolution the beam splitter and the squeezing are the central elements of quantum optics and can model the attenuation the amplification and the noise of electromagnetic signals this conditional entropy power inequality will have a strong impact in quantum information and quantum cryptography among its many possible applications there is the proof of a new uncertainty relation for the conditional wehrl entropy

local minimizers of quadratic functions on euclidean balls and spheres in this paper a characterization of the local nonglobal minimizes of a quadratic function defined on a euclidean ball or a sphere is given it is proven that there exists at most one local nonglobal minimizes and that the lagrange multiplier that corresponds to this minimizes is the largest solution of a nonlinear scalar equation an algorithm is proposed for computing the local nonglobal minimizes

on the maximization of a concave quadratic function with box constraints a new method for maximizing a concave quadratic function with bounds on the variables is introduced the new algorithm combines conjugate gradients with gradient projection techniques as the algorithm of more and toraldo siam j optimization 1 1991 pp 93 113 and other well known methods do a new strategy for the decision of leaving the current face is introduced that makes it possible to obtain finite convergence even for a singular hessian and in the presence of dual degeneracy numerical experiments are presented

using krylov subspace and spectral methods for solving complementarity problems in many body contact dynamics simulation summary many body dynamics problems are expected to handle millions of unknowns when for instance investigating the three dimensional flow of granular material unfortunately the size of the problems tractable by existing numerical solution techniques is severely limited on convergence grounds this is typically the case when the equations of motion embed a differential variational inequality problem that captures contact and possibly frictional interactions between rigid and or flexible bodies as the size of the physical system increases the speed and or the quality of the numerical solution decreases this paper describes three methods the gradient projected minimum residual method the preconditioned spectral projected gradient with fallback method and the modified proportioning with reduced gradient projection method that demonstrate better scalability than the projected jacobi and gauss seidel methods commonly used to solve contact problems that draw on a differential variational inequality based modeling approach copyright 2013 john wiley sons ltd

ergodic control of partially observed markov chains the ergodic or long run average cost control problem for a partially observed finite state markov chain is studied via the associated fully observed separated control problem for the nonlinear filter dynamic programming equations for the latter are derived leading to existence and characterization of optimal stationary policies

rate distortion theory for gaussian multiterminal source coding systems with several side informations at the decoder in this paper we consider the separate coding problem for l 1 correlated gaussian memoryless sources we deal with the case where l sources work as partial side information at the decoder for the reconstruction of the remaining source the determination problem of the rate distortion region for this system is the so called many help one problem and it has been known as a highly challenging problem for almost 20 years in this paper we give a partial solution to this problem we determine the rate distortion region in the case where the l sources working as partial side information are conditionally independent if the remaining source we wish to reconstruct is given the additive white gaussian noise ceo problem is a special case of this we also discuss the relation of the result previous results of ours

the conditional entropy power inequality for gaussian quantum states we propose a generalization of the quantum entropy power inequality involving conditional entropies for the special case of gaussian states we give a proof based on perturbation theory for symplectic spectra we discuss some implications for entanglement assisted classical communication over additive bosonic noise channels

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

entropy power inequalities for qudits shannon s entropy power inequality epi can be viewed as a statement of concavity of an entropic function of a continuous random variable under a scaled addition rule f a x 1 a y af x 1 a f y a 0 1 here x and y are continuous random variables and the function f is either the differential entropy or the entropy power konig and smith ieee trans inf theory 60 3 1536 1548 2014 and de palma mari and giovannetti nat photonics 8 12 958 964 2014 obtained quantum analogues of these inequalities for continuous variable quantum systems where x and y are replaced by bosonic fields and the addition rule is the action of a beam splitter with transmissivity a on those fields in this paper we similarly establish a class of epi analogues for d level quantum systems i e qudits the underlying addition rule for which these inequalities hold is given by a quantum channel that depends on the parameter a 0 1 and acts like a finite dimensional analogue of a beam splitter with transmissiv

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

a method of analytic centers for quadratically constrained convex quadratic programs an interior point method is developed for maximizing a concave quadratic function under convex quadratic constraints the algorithm constructs a sequence of nested convex sets and finds their approximate centers using a partial newton step given the first convex set and its approximate center the total arithmetic operations required to converge to an approximate solution are of order o sqrt m m n n 2 ln varepsilon where m is the number of constraints n is the number of variables and varepsilon is determined by the desired tolerance of the optimal value and the size of the first convex set a method to initialize the algorithm is also proposed so that the algorithm can start from an arbitrary perhaps infeasible point

discussions around voiculescu s free entropies abstract in this article we set two analogous definitions of the free entropies and introduced by voiculescu invent math 118 1994 411 132 1998 189 we discuss their relations improving the preceding results obtained in cabanal duvillard and guionnet ann probab 2001 to appear where a bound on the microstates entropy was established

convergence rate of an optimization algorithm for minimizing quadratic functions with separable convex constraints a new active set algorithm for minimizing quadratic functions with separable convex constraints is proposed by combining the conjugate gradient method with the projected gradient it generalizes recently developed algorithms of quadratic programming constrained by simple bounds a linear convergence rate in terms of the hessian spectral condition number is proven numerical experiments including the frictional three dimensional 3d contact problems of linear elasticity illustrate the computational performance

bounds on information combining with quantum side information bounds on information combining are entropic inequalities that determine how the information entropy of a set of random variables can change when these are combined in certain prescribed ways such bounds play an important role in classical information theory particularly in coding and shannon theory entropy power inequalities are special instances of them the arguably most elementary kind of information combining is the addition of two binary random variables a cnot gate and the resulting quantities play an important role in belief propagation and polar coding we investigate this problem in the setting where quantum side information is available which has been recognized as a hard setting for entropy power inequalities our main technical result is a non trivial and close to optimal lower bound on the combined entropy which can be seen as an almost optimal quantum mrs gerber s lemma our proof uses three main ingredients 1 a new bound on the concavity of von neumann entropy which is tight in the regime of low pairwise state fidelities 2 the quantitative improvement of strong subadditivity due to fawzi renner in which we manage to handle the minimization over recovery maps and 3 recent duality results on classical quantum channels due to renes et al we furthermore present conjectures on the optimal lower and upper bounds under quantum side information supported by interesting analytical observations and strong numerical evidence we finally apply our bounds to polar coding for binary input classical quantum channels and show the following three results 1 even non stationary channels polarize under the polar transform 2 the blocklength required to approach the symmetric capacity scales at most sub exponentially in the gap to capacity and 3 under the aforementioned lower bound conjecture a blocklength polynomial in the gap suffices

decay of correlations dedicated to micheline ishay i would like to thank p boyland l chierchia v donnay g de martino c gole j l lebowitz m lyubich m rychlik i g schwarz s vaienti and especially g gallavotti for helpful and enlightening discussions particularly warm thanks go to n chernov for carefully reading and finding a mistake in an early version the present paper benefits from several improvements due to his sharp criticism in addition i am indebted to p collet and most of all m wojtkowski for introducing me to the magical world of cones finally i thank j milnor director of the institute for mathematical sciences at stony brook university where i was visiting during part of this work and the italian c n r gnfm for providing travel funds

a free analogue of shannon s problem on monotonicity of entropy abstract we prove a free probability analog of a result of s artstein k ball f barthe a naor solution of shannon s problem on monotonicity of entropy j amer math soc 17 2004 975 982 in particular we prove that if x 1 x 2 are freely independent identically distributed random variables then the function n x 1 x n n is monotone increasing for all n

an algorithm for the numerical realization of 3d contact problems with coulomb friction this contribution deals with the numerical realization of static contact problems with coulomb friction for three dimensional elastic bodies we first introduce auxiliary contact problems with given friction which define a mapping associating with a given slip bound the normal contact stress in the equilibrium state solutions to contact problems with coulomb friction are defined as fixed points of and are computed by using the method of successive approximations the mathematical model of contact problems with given friction leads to a variational inequality of the second kind its discretization is based on the so called reciprocal variational formulation i e the formulation in terms of the normal and tangential components of stresses on the contact boundary unlike the two dimensional case constraints imposed on the tangential components of contact stresses are quadratic the main goal of this contribution is to show how to solve this problem by using existing fast algorithms for simple box constraints numerical experiments for several variants of our algorithm will be shown and compared

volumes of restricted minkowski sums and the free analogue of the entropy power inequality in noncommutative probability theory independence can be based on free products instead of tensor products this yields a highly noncommutative theory free probability theory for an introduction see 9 the analogue of entropy in the free context was introduced by the second named author in 8 here we show that shannon s entropy power inequality 6 1 has an analogue for the free entropy x theorem 2 1

additivity of the classical capacity of entanglement breaking quantum channels we show that for the tensor product of an entanglement breaking quantum channel with an arbitrary quantum channel both the minimum entropy of an output of the channel and the holevo schumacher westmoreland capacity are additive in addition for the tensor product of two arbitrary quantum channels we give a bound involving entanglement of formation for the amount of subadditivity for minimum entropy output or superadditivity for classical capacity that can occur

an optimal algorithm for minimization of quadratic functions with bounded spectrum subject to separable convex inequality and linear equality constraints an in a sense optimal algorithm for minimization of quadratic functions subject to separable convex inequality and linear equality constraints is presented its unique feature is an error bound in terms of bounds on the spectrum of the hessian of the cost function if applied to a class of problems with the spectrum of the hessians in a given positive interval the algorithm can find approximate solutions in a uniformly bounded number of simple iterations such as matrix vector multiplications moreover if the class of problems admits a sparse representation of the hessian it simply follows that the cost of the solution is proportional to the number of unknowns theoretical results are illustrated by numerical experiments