on nonautonomous linear systems of differential and difference equations with r symmetric coefficient matrices abstract let c n n i denote the set of continuous n n matrices on an interval i we say that r c n n i is a nontrivial k involution if r p l 0 k 1 l i d l p 1 where e 2 i k d 0 d 1 d k 1 n and p p l 0 k 1 u l with u l c d l d l i we say that a c n n i is r symmetric if r t a t r 1 t a t t i and we show that if a is r symmetric then solving x a t x or x a t x f t reduces to solving k independent d l d l systems 0 l k 1 we consider the asymptotic behavior of the solutions in the case where i t 0 finally we sketch analogous results for linear systems of difference equations

monodromy matrix for linear difference operators with almost constant coefficients a new method for solving the discrete scattering problem for a linear single valued difference operator of arbitrary order with almost constant coefficients is proposed the treatment is concerned with the asymptotic behavior of its eigenfunctions as t the purpose of the paper is to investigate the transition between the asymptotically free states of the underlying system defined in terms of the monodromy operator a rare mathematical tool is used an infinite matrix product it is shown that if the coefficients of the aforementioned operator are suitably behaved the monodromy operator exists in the form of the convergent two sided infinite product of matrices associated with the matrix eigenvalue equation corresponding to the scattering problem the application of the method to the solution of the scattering problem for toda lattice equations is demonstrated however the approach is quite general and should be applicable to other forms of lattice equations

the frechet and limiting subdifferentials of integral functionals on the spaces l1 ω e abstract a new approach to computing the frechet subdifferential and the limiting subdifferential of integral functionals is proposed thanks to this way we obtain formulae for computing the frechet and limiting subdifferentials of the integral functional f u f u d u l 1 e here a is a measured space with an atomless finite complete positive measure e is a separable banach space and f e r under some assumptions it turns out that these subdifferentials coincide with the fenchel subdifferential of f

the proximal normal formula in banach space approximation by proximal normals to the clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis we give simple general versions of such formulae in infinite dimensional banach spaces which satisfy different geometrical conditions our first class of spaces with uniformly g teaux differentiable norm includes the hilbert space case and the formulae is attained through dense subsets our second class of reflexive kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of ekeland and lebourg

clarke coderivatives of efficient point multifunctions in parametric vector optimization abstract the paper is devoted to the study of the clarke circatangent coderivatives of the efficient point multifunction of parametric vector optimization problems in banach spaces we provide inner outer estimates for evaluating the clarke circatangent coderivative of this multifunction in a broad class of conventional vector optimization problems in the presence of geometrical operator and finite and infinite functional constraints examples are given for analyzing and illustrating the obtained results

degrees of freedom of two hop wireless networks everyone gets the entire cake we show that fully connected two hop wireless networks with k sources k relays and k destinations have k degrees of freedom both in the case of time varying channel coefficients and constant channel coefficients in which case the result holds for almost all values of constant channel coefficients our main contribution is a new achievability scheme which we call aligned network diagonalization this scheme allows the data streams transmitted by the sources to undergo a diagonal linear transformation from the sources to the destinations thus being received free of interference by their intended destination in addition we extend our scheme to multihop networks with fully connected hops and multihop networks with mimo nodes for which the degrees of freedom are also fully characterized

cut set bound is loose for gaussian relay networks the cut set bound developed by cover and el gamal in 1979 has since remained the best known upper bound on the capacity of the gaussian relay channel we develop a new upper bound on the capacity of the gaussian primitive relay channel which is tighter than the cut set bound our proof uses gaussian measure concentration to establish geometric relations satisfied with high probability between the n letter random variables associated with a reliable code for communicating over this channel we then translate these geometric relations into new information inequalities that cannot be obtained with classical methods combined with a tensorization argument proposed by courtade and ozgur in 2015 our result also implies that the current capacity approximations for gaussian relay networks which have linear gap to the cut set bound in the number of nodes are order optimal and lead to a lower bound on the pre constant

fatou s lemma and lower epi limits of integral functionals abstract in this work with the introduction in the finite case of a modulus of equi integrability we prove some new extensions of fatou s lemma and some of its consequences in the convergence theory of integral functionals we present the case of a sequence of integral functionals using an analog of ioffe s criterion for a sequence of integrands

subdifferential regularities of perturbed distance functions outside the target set in banach spaces abstract this paper is devoted to study the frechet and proximal regularities at points outside the target set of perturbed distance functions d s j determined by a closed subset s and a lipschitz function j also we provide some important results on the frechet proximal and clarke subdifferentials of d s j at those points in banach spaces

approximate convexity and submonotonicity abstract it is shown that a locally lipschitz function is approximately convex if and only if its clarke subdifferential is a submonotone operator consequently in finite dimensions the class of locally lipschitz approximately convex functions coincides with the class of lower c 1 functions directional approximate convexity is introduced and shown to be a natural extension of the class of lower c 1 functions in infinite dimensions the following characterization is established a multivalued operator is maximal cyclically submonotone if and only if it coincides with the clarke subdifferential of a locally lipschitz directionally approximately convex function which is unique up to a constant furthermore it is shown that in asplund spaces every regular function is generically approximately convex

on the optimality of simple schedules for networks with multiple half duplex relays this paper studies networks that consist of n half duplex relays assisting the communication between a source and a destination in isit 12 brahma et al conjectured that in gaussian half duplex diamond networks i e without a direct link between the source and the destination and with n non interfering relays an approximately optimal relay scheduling policy i e achieving the cut set upper bound to within a constant gap uniformly over all channel gains has at most n 1 active states i e at most n 1 out of the 2 n possible relay listen transmit configurations have a strictly positive probability such relay scheduling policies were referred to as simple in itw 13 we conjectured that simple approximately optimal relay scheduling policies exist for any gaussian half duplex multi relay network irrespectively of the topology this paper formally proves this more general version of the conjecture and shows it holds beyond gaussian noise networks in particular for any class of memoryless half duplex n relay networks with independent noises and for which independent inputs are approximately optimal in the cut set upper bound an approximately optimal simple relay scheduling policy exists the key step of the proof is to write the minimum of the submodular cut set function by means of its lovasz extension and use the greedy algorithm for submodular polyhedra to highlight structural properties of the optimal solution this together with the saddle point property of min max problems and the existence of optimal basic feasible solutions for linear programs proves the conjecture as an example for n relay gaussian networks with independent noises where each node is equipped with multiple antennas and where each antenna can be configured to listen or transmit irrespectively of the others the existence of an approximately optimal simple relay scheduling policy with at most n 1 active states irrespectively of the total number of antennas in the system is proved

smoothness of the metric projection onto nonconvex bodies in hilbert spaces abstract based on a fundamental work of r b holmes from 1973 we study differentiability properties of the metric projection onto prox regular sets we show that if the set is a nonconvex body with a c p 1 smooth boundary then the projection is c p smooth near suitable open truncated normal rays which are determined only by the function of prox regularity a local version of the same result is established as well namely when the smoothness of the boundary and the prox regularity of the set are assumed only near a fixed point finally similar results are derived when the prox regular set is itself a c p 1 submanifold

limiting subdifferentials of indefinite integrals abstract we compute the limiting subdifferential f x of the indefinite integral of the form f x a x f t d t where f is an essentially bounded measurable function or a function continuous on an interval containing x r except for possibly x or a step function which has a countable number of steps around x the related problem of computing the aumann integral of the limiting subdifferential mapping f where f is a lipschitz real function defined on an open set u r n is also investigated

relaxation of an optimal control problem involving a perturbed sweeping process we establish first in the setting of infinite dimensional hilbert space a result concerning the existence of solutions for perturbed sweeping processes whose perturbations are lipschitz single valued maps then we use this result to extend to the infinite dimensional setting a relaxation result concerning optimal control problems involving such processes

subdifferentials of a minimum time function in banach spaces in general banach space setting we study the minimum time function determined by a closed convex set k and a closed set s this function is simply the usual minkowski function of k if s is the singleton consisting of the origin in particular we show that various subdifferentials of a minimum time function are representable by virtue of corresponding normal cones of sublevel sets of the function

error exponents of modulo additive noise channels with side information at the transmitter consider the optimum strategy for using channel state side information in transmission over a modulo additive noise channel with state dependent noise where the receiver does not have access to the side information si previous work showed that capacity wise the optimum transmitter shifts each code letter by a prediction of the noise sample based on the si we show that this structure achieves also the random coding error exponent and therefore is optimum at some range of rates below capacity specifically the optimum transmitter predictor minimizes the renyi entropy of the prediction error the renyi order depends on the rate and goes to one corresponding to shannon entropy for rates close to capacity in contrast it is shown that this prediction strategy may not be optimal at low transmission rates

lattices which are good for almost everything we define an ensemble of lattices and show that for asymptotically high dimension most of its members are simultaneously good as sphere packings sphere coverings additive white gaussian noise awgn channel codes and mean squared error mse quantization codes these lattices are generated by applying construction a to a random linear code over a prime field of growing size i e by lifting the code to spl ropf sup n

corrected proof of de buda s theorem lattice channel codes an error in de buda s see ieee j select areas commun vol 7 no 6 p 893 1989 proof on the asymptotic optimality of lattice channel codes is pointed out and corrected using a modification of de buda s approach comments are given on the correct interpretation and the limitations of this result

integration of subdifferentials of nonconvex functions in nonsmooth analysis and optimization subgradients come in many different flavors e g approximate dini proximal clarke generalized see 2 5 6 12 14 191 these subgradients are important and valuable tools however many questions remain unsolved concerning the exact link between the function and its subgradients for instance can two functions not differing by an additive constant have the same subgradients in this paper we study the fundamental problem of determining functions that can be recovered up to an additive constant from the knowledge of their subgradients this integration problem is not very well understood and very few functions or classes of functions are known to be recoverable from their subgradients in section 4 we show that if the basic constraint qualification holds at r then the composition of a closed i e lowersemicontinuous proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients actually in this case all above mentioned subgradients are the same beside the obvious theoretical interest of this integration problem it is our hope or perhaps our long term goal that once this problem is better understood we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives an example of such an equation that well deserves study is the extended hamilton jacobi equation used in optimal control see clarke 2 let us also mention that a problem similar to the integration problem is the one of determining the set valued mappings that are in fact subgradient set valued mappings uniqueness is not mandatory for a contribution to this problem see janin 8 before we look at some of the known cases where the function can be recovered from its subgradients let us look at some negative examples it is clear that not every function can be recovered up to an additive constant from its subgradients we only need to look at the following two functions 0 x50 0 x10 f x 1 x o g x 2 x o

nested linear lattice codes for structured multiterminal binning network information theory promises high gains over simple point to point communication techniques at the cost of higher complexity however lack of structured coding schemes limited the practical application of these concepts so far one of the basic elements of a network code is the binning scheme wyner 1974 1978 and other researchers proposed various forms of coset codes for efficient binning yet these schemes were applicable only for lossless source or noiseless channel network coding to extend the algebraic binning approach to lossy source or noisy channel network coding previous work proposed the idea of nested codes or more specifically nested parity check codes for the binary case and nested lattices in the continuous case these ideas connect network information theory with the rich areas of linear codes and lattice codes and have strong potential for practical applications we review these developments and explore their tight relation to concepts such as combined shaping and precoding coding for memories with defects and digital watermarking we also propose a few novel applications adhering to a unified approach

a digital interface for gaussian relay and interference networks lifting codes from the discrete superposition model for every gaussian network there exists a corresponding deterministic network called the discrete superposition network we show that this discrete superposition network provides a near optimal digital interface for operating a class consisting of many gaussian networks in the sense that any code for the discrete superposition network can be naturally lifted to a corresponding code for the gaussian network while achieving a rate that is no more than a constant number of bits lesser than the rate it achieves for the discrete superposition network this constant depends only on the number of nodes in the network and not on the channel gains or snr moreover the capacities of the two networks are within a constant of each other again independent of channel gains and snr we show that the class of gaussian networks for which this interface property holds includes relay networks with a single source destination pair interference networks multicast networks and the counterparts of these networks with multiple transmit and receive antennas the code for the gaussian relay network can be obtained from any code for the discrete superposition network simply by pruning it this lifting scheme establishes that the superposition model can indeed potentially serve as a strong surrogate for designing codes for gaussian relay networks we present similar results for the k k gaussian interference network mimo gaussian interference networks mimo gaussian relay networks and multicast networks with the constant gap depending additionally on the number of antennas in case of mimo networks

linear asymptotic equilibrium and uniform exponential and strict stability of linear difference systems following an idea originated by conti for continuous matrix functions an equivalence relation called summable similarity is defined on pairs of n n matrix functions a and b special cases of the results show that if a and b are summably similar and the system a xm amxm m 0 1 2 is uniformly exponentially or strictly stable or has linear asymptotic equilibrium then the system b ym bmym m 0 1 2 has the same property more generally the same conclusion is obtained under weaker conditions relating a and b

estimating tangent and normal cones without calculus we study simple estimates of the tangent and normal cones to a set with a nonsmooth boundary we seek estimates that are robust relative to the base point and that depend only on collections of nearby points known to lie inside and outside of the set

invertibly convergent infinite products of matrices abstract the standard definition of convergence of an infinite product of scalars is extended to the infinite product p n 1 b n of k k matrices that is p is convergent according to the definition given here if and only if there is an integer n such that b n is invertible for n n and p lim n m n n b m is invertible a family of sufficient conditions for this kind of convergence is given along with examples showing that they have nontrivial applications

subsmooth sets functional characterizations and related concepts prox regularity of a set poliquin rockafellar thibault 2000 or its global version proximal smoothness clarke stern wolenski 1995 plays an important role in variational analysis not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist c or the local uniqueness of the projection mapping but also because in the case where c is the epigraph of a locally lipschitz function it is equivalent to the weak convexity lower c 2 property of the function in this paper we provide an adapted geometrical concept called sub smoothness which permits an epigraphic characterization of the approximate convex functions or lower c 1 property subsmooth sets turn out to be naturally situated between the classes of prox regular and of nearly radial sets this latter class has been recently introduced by lewis in 2002 we hereby relate it to the mifflin semismooth functions

prox regular sets and epigraphs in uniformly convex banach spaces various regularities and other properties we continue the study of prox regular sets that we began in a previous work in the setting of uniformly convex banach spaces endowed with a norm both uniformly smooth and uniformly convex e g l p w m p spaces we prove normal and tangential regularity properties for these sets and in particular the equality between mordukhovich and proximal normal cones we also compare in this setting the proximal normal cone with different holderian normal cones depending on the power types s q of moduli of smoothness and convexity of the norm in the case of sets that are epigraphs of functions we show that j primal lower regular functions have prox regular epigraphs and we compare these functions with poliquin s primal lower nice functions depending on the power types s q of the moduli the preservation of prox regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption a conical derivative formula for the metric projection mapping of prox regular sets is also established among other results of the paper it is proved that the attouch wets convergence preserves the uniform r prox regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets

continuity of the fenchel correspondence and continuity of polarities abstract the polar cone of the limit inferior of a sequence of cones of a banach space is shown to be the cesari s limit superior of the sequence of polar cones in the bounded weak topology in a similar way the lower semicontinuity of a not necessarily countable family of closed convex cones is characterized in terms of castaing s notion of pseudo upper semicontinuity more general results are given within the framework of convergence spaces and the pseudo upper semicontinuity of a closed valued multifunction is characterized as closedness with respect to a new convergence applications to the young fenchel correspondence are pointed out

network simplification in half duplex building on submodularity this paper explores the network simplification problem in the context of gaussian half duplex diamond networks specifically given an n relay diamond network this problem seeks to derive fundamental guarantees on the capacity of the best k relay subnetwork as a function of the full network capacity simplification guarantees are presented in terms of a particular approximate capacity termed independent gaussian ig approximate capacity that characterizes the network capacity to within an additive gap which is independent of the channel coefficients and operating snr the main focus of this work is when k n 1 relays are selected out of n relays in a diamond network first a simple algorithm is proposed which selects all relays except the one with the minimum ig approximate half duplex capacity it is shown that the selected n 1 relay subnetwork has an ig approximate half duplex capacity that is at least 1 2 of the ig approximate half duplex capacity of the full network and that for the proposed algorithm this guarantee is tight furthermore this work proves the following tight fundamental guarantee there always exists a subnetwork of k n 1 relays that have an ig approximate half duplex capacity that is at least equal to n 1 n of the ig approximate half duplex capacity of the full network finally these results are extended to derive lower bounds on the fraction guarantee when k in 1 n relays are selected the key steps in the proofs lie in the derivation of properties of submodular functions which provide a combinatorial handle on the network simplification problem for gaussian half duplex diamond networks

prox regular functions in hilbert spaces this paper studies the prox regularity concept for functions in the general context of hilbert space in particular a subdifferential characterization is established as well as several other properties it is also shown that the moreau envelopes of such functions are continuously differentiable

miniversal deformations of matrices under congruence and reducing transformations arnold 1971 1 constructed a miniversal deformation of a square complex matrix under similarity that is a simple normal form to which not only a given square matrix a but all matrices b close to it can be reduced by similarity transformations that smoothly depend on the entries of b we give miniversal deformations of matrices of sesquilinear forms that is of square complex matrices under congruence and construct an analytic reducing transformation to a miniversal deformation analogous results for matrices under congruence were obtained by dmytryshyn futorny and sergeichuk 2012 11

the differentiability of real functions on normed linear space using generalized subgradients abstract the modification of the clarke generalized subdifferential due to michel and penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space the g teaux differentiability of any real function can be deduced from the g teaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative for any distance function on a space with uniformly g teaux differentiable norm the clarke and michel penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for g teaux differentiability however on a banach space with rotund dual the frechet differentiability of a distance function implies that it is a convex function a mean value theorem for the modified generalized subdifferential has implications for g teaux differentiability

infimal convolution and optimal time control problem iii minimal time projection set we continue the investigation of a general minimal time problem with a convex constant dynamics and a lower semicontinuous extended real valued target function defined on a banach space in this paper we obtain an explicit description of the minimal time projection set in terms of the legendre fenchel conjugate function and provide some sufficient conditions for this set to be nonempty the single valuedness and lipschitz property of the minimal time projection are also investigated

quantum riemann surfaces i the unit disc we construct a non commutative algebra c mu bar u which is a quantum deformation of the algebra of continuous functions on the closed unit disc bar u c mu bar u is generated by the toeplitz operators on a suitable hilbert space of holomorphic functions onu

subdifferentials of a minimal time function in normed spaces abstract in a general normed vector space we study the minimal time function determined by a differential inclusion where the set valued mapping involved has constant values of a bounded closed convex set u and by a closed target set s we show that proximal and frechet subdifferentials of a minimal time function are representable by virtue of corresponding normal cones of sublevel sets of the function and level or suplevel sets of the support function of u the known results in the literature require the set u to have the origin as an interior point or u be compact in particular if the set u is the unit closed ball the results obtained reduce to the subdifferential of the distance function defined by s

proximal analysis in reflexive smooth banach spaces abstract in the present paper we introduce and study a new proximal normal cone in reflexive banach spaces in terms of a generalized projection operator two new variants of generalized proximal subdifferentials are also introduced in reflexive smooth banach spaces the density theorem for both proximal subdifferentials has been proved in p uniformly convex and q uniformly smooth banach spaces various important properties and applications of our concepts are also proved

convexity in hamilton jacobi theory i dynamics and duality value functions propagated from initial or terminal costs and constraints by way of a differential inclusion or more broadly through a lagrangian that may take on infty are studied in the case where convexity persists in the state argument such value functions themselves taking on infty are shown to satisfy a subgradient form of the hamilton jacobi equation which strongly supports properties of local lipschitz continuity semidifferentiability and clarke regularity an extended method of characteristics is developed which determines them from the hamiltonian dynamics underlying the given lagrangian close relations with a dual value function are revealed

closest point search in lattices in this semitutorial paper a comprehensive survey of closest point search methods for lattices without a regular structure is presented the existing search strategies are described in a unified framework and differences between them are elucidated an efficient closest point search algorithm based on the schnorr euchner 1995 variation of the pohst 1981 method is implemented given an arbitrary point x spl isin spl ropf sup m and a generator matrix for a lattice spl lambda the algorithm computes the point of spl lambda that is closest to x the algorithm is shown to be substantially faster than other known methods by means of a theoretical comparison with the kannan 1983 1987 algorithm and an experimental comparison with the pohst 1981 algorithm and its variants such as the viterbo boutros see ibid vol 45 p 1639 42 1999 decoder modifications of the algorithm are developed to solve a number of related search problems for lattices such as finding a shortest vector determining the kissing number computing the voronoi 1908 relevant vectors and finding a korkine zolotareff 1873 reduced basis

a note on regularity of sets and of distance functions in banach space abstract we show that in every infinite dimensional banach space there is a compact set c and x in c such that c is regular at x in the sense of nonsmooth analysis but such that the associated distance function d c is not regular at x and neither is d c

approximating subdifferentials by random sampling of gradients many interesting real functions on euclidean space are differentiable almost everywhere all lipschitz functions have this property but so for example does the spectral abscissa of a matrix a non lipschitz function in practice the gradient is often easy to compute we investigate to what extent we can approximate the clarke subdifferential of such a function at some point by calculating the convex hull of some gradients sampled at random nearby points

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

a robust gradient sampling algorithm for nonsmooth nonconvex optimization let f be a continuous function on rl n and suppose f is continuously differentiable on an open dense subset such functions arise in many applications and very often minimizers are points at which f is not differentiable of particular interest is the case where f is not convex and perhaps not even locally lipschitz but is a function whose gradient is easily computed where it is defined we present a practical robust algorithm to locally minimize such functions based on gradient sampling no subgradient information is required by the algorithm when f is locally lipschitz and has bounded level sets and the sampling radius eps is fixed we show that with probability 1 the algorithm generates a sequence with a cluster point that is clarke eps stationary furthermore we show that if f has a unique clarke stationary point bar x then the set of all cluster points generated by the algorithm converges to bar x as eps is reduced to zero numerical results are presented demonstrating the robustness of the algorithm and its applicability in a wide variety of contexts including cases where f is not locally lipschitz at minimizers we report approximate local minimizers for functions in the applications literature which have not to our knowledge been obtained previously when the termination criteria of the algorithm are satisfied a precise statement about nearness to clarke eps stationarity is available a matlab implementation of the algorithm is posted at http www cs nyu edu overton papers gradsamp alg

weak regularity of functions and sets in asplund spaces abstract in this paper we study a new concept of weak regularity of functions and sets in asplund spaces we show that this notion includes prox regular functions functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension we establish also that weak regularity is equivalent to mordukhovich regularity in finite dimension finally we give characterizations of the weak regularity of epi lipschitzian sets in terms of their local representations

invertibly convergent infinite products of matrices with applications to difference equations abstract the standard definition of convergence of an infinite product of scalars is extended to the infinite product p m 1 b m of k k matrices that is p is convergent according to the definition given here if and only if there is an integer n such that bm is invertible for m n and p lim n n m n i a m is invertible sufficient conditions for this kind of convergence are given some of the results seem to be new even for infinite products of scalars the results are derived by considering related systems of difference equations and have implications concerning the asymptotic behavior of solutions of these systems

the approximate capacity of half duplex line networks this paper investigates the problem of characterizing the capacity of half duplex hd line networks where a source node communicates to a destination node through a multi hop path of n relays if the relays operate in full duplex fd it is well known that the capacity of the line network equals the minimum among the point to point link capacities in the path in contrast this paper considers a different case where the relays operate in hd in the first part of the paper it is shown that the approximate capacity optimal up to a constant additive gap that only depends on the number of nodes in the network of an hd n relay line network equals half the minimum of the harmonic means of the point to point link capacities of each two consecutive links in the path it is then proved that the n 1 listen transmit states out of the 2 n possible ones sufficient to characterize the approximate capacity can be found in linear time in the second part of the paper it is shown that the problem of finding the path that has the largest hd approximate capacity in a network that can be represented as a graph is np hard however if the number of cycles in the network is polynomial in the number of nodes then a polynomial time algorithm can indeed be designed

efficient capacity computation and power optimization for relay networks the capacity or approximations to capacity of various single source single destination relay network models has been characterized in terms of the cut set upper bound in principle a direct computation of this bound requires evaluating the cut capacity over exponentially many cuts we show that the minimum cut capacity of a relay network under some special assumptions can be cast as a minimization of a submodular function and as a result can be computed efficiently we use this result to show that the capacity or an approximation to the capacity within a constant gap for the gaussian wireless erasure and avestimehr diggavi tse deterministic relay network models can be computed in polynomial time we present some empirical results showing that computing constant gap approximations to the capacity of gaussian relay networks with around 300 nodes can be done in order of minutes for gaussian networks cut set capacities are also functions of the powers assigned to the nodes we consider a family of power optimization problems and show that they can be solved in a polynomial time in particular we show that the minimization of the sum of powers assigned to the nodes subject to a minimum rate constraint measured in terms of cut set bounds can be computed in the polynomial time we propose a heuristic algorithm to solve this problem and measure its performance through simulations on random gaussian networks we observe that in the optimal allocations most of the power is assigned to a small subset of relays which suggests that network simplification may be possible without excessive performance degradation

lattice coding and decoding for multiple antenna ergodic fading channels for ergodic fading a lattice coding and decoding strategy is proposed and its performance is analyzed for the single input single output siso and multiple input multiple output mimo point to point channel as well as the multiple access channel mac with channel state information available only at the receiver csir at the decoder a novel strategy is proposed consisting of a time varying equalization matrix followed by decision regions that depend only on channel statistics not individual realizations our encoder has a similar structure to that of erez and zamir for the siso channel the gap to capacity is bounded by a constant under a wide range of fading distributions for the mimo channel under rayleigh fading the rate achieved is within a gap to capacity that does not depend on the signal to noise ratio snr and diminishes with the number of receive antennas the analysis is extended to the k user mac where similar results hold achieving a small gap to capacity while limiting the use of csir to the equalizer highlights the scope for efficient decoder implementations since decision regions are fixed i e independent of channel realizations

limiting subdifferentials of perturbed distance functions in banach spaces abstract we explore in arbitrary banach spaces the frechet type e subdifferentials and the limiting subdifferentials for the perturbed distance function d s j determined by a closed subset s and a lower semicontinuous function j defined on s in particular upper and lower estimates for the frechet type e subdifferentials and for the limiting subdifferentials are provided in terms of the corresponding subdifferentials of the sum of the associated functions j and s

a close to capacity dirty paper coding scheme the writing on dirty paper channel model offers an information theoretic framework for precoding techniques for canceling arbitrary interference known at the transmitter it indicates that lossless precoding is theoretically possible at any signal to noise ratio snr and thus dirty paper coding may serve as a basic building block in both single user and multiuser communication systems we design an end to end coding realization of a system materializing a significant portion of the promised gains we employ multidimensional quantization based on trellis shaping at the transmitter coset decoding is implemented at the receiver using virtual bits combined with iterative decoding of capacity approaching codes we achieve an improvement of 2db over the best scalar quantization scheme code design is done using the exit chart technique

bv solutions of nonconvex sweeping process differential inclusion with perturbation abstract this paper is devoted to the study of differential inclusions particularly discontinuous perturbed sweeping processes in the infinite dimensional setting on the one hand the sets involved are assumed to be prox regular and to have a variation given by a function which is of bounded variation and right continuous on the other hand the perturbation satisfies a linear growth condition with respect to a fixed compact subset finally the case where the sets move in an absolutely continuous way is recovered as a consequence

extensions of a theorem of wintner on systems with asymptotically constant solutions a theorem of wintner concerning sufficient conditions for a system y a t y to have linear asymptotic equilibrium is extended to a system x a t x f t x the integrability conditions imposed on permit conditional convergence of some of the improper integrals that occur the results improve on wintner s even if 0

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

linear regularity and ϕ regularity of nonconvex sets in this paper we discuss some sufficient conditions for the linear regularity and bounded linear regularity and their variations of finitely many closed not necessarily convex sets in a normed vector space the accompanying necessary conditions are also given in the setting of asplund spaces

on asymptotic summations of almost diagonal difference systems and their applications in this paper we derive some new results of asymptotic summations of left and right almost diagonal difference systems in particular we study a special case where the diagonal elements are constants with magnitude 1 conditions on the perturbation matrices are given under which the system is left and right almost diagonal an application of this result in a discrete adiabatic oscillator equation is discussed we also introduce a result for the case when the diagonal matrix is approaching the identity matrix i an example is given to illustrate that the new result can be applied to some cases that do not satisfy the growth condition

semiregularity and generalized subdifferentials with applications to optimization the michel penot subdifferential of a locally lipschitzian function is the principal part of the clarke subdifferential it coincides with the g derivative at differentiable points a locally lipschitzian function can be determined by its michel penot subdifferential uniquely up to an additive constant though this cannot be done by its clarke subdifferential if the set of abnormal points is not negligible a set valued operator is the michel penot subdifferential of a locally lipschitzian function if and only if it is a seminormal operator satisfying a cyclical condition various calculus rules hold for the michel penot subdifferential equalities hold for these rules at a point under semiregularity which is weaker than regularity for a locally lipschitzian function in a separable banach space semiregularity holds everywhere except for a haar zero set applications in optimization are discussed

softness sleekness and regularity properties in nonsmooth analysis abstract we study continuity properties of tangent and normal cones to a closed subset of a banach space such a study can be seen as a nonsmooth set theoretic analogue of the study of continuously differentiable functions we give particular attention to the case where different concepts coincide a desirable feature such properties are able to offer we identify a wide class of sets and functions for which this property occurs

miniversal deformations of matrices of bilinear forms arnold v i arnold on matrices depending on parameters russian math surveys 26 2 1971 29 43 constructed miniversal deformations of square complex matrices under similarity that is a simp

noncoincidence of approximate and limiting subdifferentials of integral functionals for a locally lipschitz integral functional i f on l 1 t mathbf r n associated with a measurable integrand f the limiting subdifferential and the approximate subdifferential never coincide at a point x 0 where f t cdot is not subdifferentially regular at x 0 t for a e t in t the coincidence of both subdifferentials occurs on a dense set of l 1 t mathbf r n if and only if f t cdot is convex for a e t in t our results allow us to characterize aubin s lipschitz like property as well as the convexity of multivalued mappings between l 1 spaces new necessary optimality conditions for some bolza problems are also obtained

dirac type operators on the quantum solid torus with global boundary conditions abstract we define a noncommutative space we call the quantum solid torus it is an example of a noncommutative manifold with a noncommutative boundary we study quantum dirac type operators subject to atiyah patodi singer like boundary conditions on the quantum solid torus we show that such operators have compact inverse which means that the corresponding boundary value problem is elliptic

achieving 1 2 log 1 snr on the awgn channel with lattice encoding and decoding we address an open question regarding whether a lattice code with lattice decoding as opposed to maximum likelihood ml decoding can achieve the additive white gaussian noise awgn channel capacity we first demonstrate how minimum mean square error mmse scaling along with dithering lattice randomization techniques can transform the power constrained awgn channel into a modulo lattice additive noise channel whose effective noise is reduced by a factor of spl radic 1 snr snr for the resulting channel a uniform input maximizes mutual information which in the limit of large lattice dimension becomes 1 2 log 1 snr i e the full capacity of the original power constrained awgn channel we then show that capacity may also be achieved using nested lattice codes the coarse lattice serving for shaping via the modulo lattice transformation the fine lattice for channel coding we show that such pairs exist for any desired nesting ratio i e for any signal to noise ratio snr furthermore for the modulo lattice additive noise channel lattice decoding is optimal finally we show that the error exponent of the proposed scheme is lower bounded by the poltyrev exponent

a numerical scheme for a class of sweeping processes the aim of this paper is to study a whole class of first order differential inclusions which fit into the framework of perturbed sweeping process by uniformly prox regular sets after obtaining well posedness results we propose a numerical scheme based on a prediction correction algorithm and we prove its convergence finally we apply these results to a problem coming from the modelling of crowd motion

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

precoded integer forcing universally achieves the mimo capacity to within a constant gap an open loop single user multiple input multiple output mimo communication scheme is considered where a transmitter equipped with multiple antennas encodes the data into independent streams all taken from the same linear code the coded streams are then linearly precoded using the encoding matrix of a perfect linear dispersion space time code at the receiver side integer forcing equalization is applied followed by standard single stream decoding it is shown that this communication architecture achieves the capacity of any gaussian mimo channel up to a gap that depends only on the number of transmit antennas

subdifferentiation of integral functionals we examine how the subdifferentials of nonconvex integral functionals can be deduced from the subdifferentials of the corresponding integrand or at least be estimated with the help of them in fact assuming some regularity properties of the integrands we obtain exact expressions for the subdifferentials of the integral functionals we draw some consequences in terms of duality for such integral functionals extending in this way the early work of rockafellar to the nonconvex case

approximately achieving gaussian relay network capacity with lattice based qmf codes recently a new relaying strategy quantize map and forward qmf scheme has been demonstrated to approximately achieve within an additive constant number of bits the gaussian relay network capacity universally i e for arbitrary topologies channel gains and snrs this was established using gaussian codebooks for transmission and random mappings at the relays in this paper we develop structured lattice codes that implement the qmf strategy the main result of this paper is that such structured lattice codes can approximately achieve the gaussian relay network capacity universally again within an additive constant in addition we establish a similar result for half duplex networks where we demonstrate that one can approximately achieve the capacity using fixed transmit receive tx rx schedules for the relays with no transmit power optimization across the different tx rx states of the network

asymptotic property of linear volterra difference systems by using the resolvent matrix and the comparison principle we investigate the asymptotic behavior of linear volterra difference systems

gaussian half duplex relay networks improved constant gap and connections with the assignment problem this paper considers a gaussian relay network where a source transmits a message to a destination with the help of n half duplex relays the information theoretic cut set upper bound to the capacity is shown to be achieved to within 1 96 n 2 bits by noisy network coding thereby reducing the previously known gap this gap is obtained as a special case of a more general constant gap result for gaussian half duplex multicast networks it is then shown that the generalized degrees of freedom of this network is the solution of a linear program where the coefficients of the linear inequality constraints are proved to be the solution of several linear programs referred as the assignment problem in graph theory for which efficient numerical algorithms exist the optimal schedule that is the optimal value of the 2 n possible transmit receive configuration states for the relays is investigated and known results for diamond networks are extended to general relay networks it is shown for the case of n 2 relays that only n 1 3 out of the 2 n 4 possible states have a strictly positive probability and suffice to characterize the capacity to within a constant gap extensive experimental results show that for a general n relay network with n leq 8 the optimal schedule has at most n 1 states with a strictly positive probability as an extension of a conjecture presented for diamond networks it is conjectured that this result holds for any half duplex relay network and any number of relays finally a network with n 2 relays is studied in detail to illustrate the channel conditions under which selecting the best relay is not optimal and to highlight the nature of the rate gain due to multiple relays

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

approximately convex functions and approximately monotonic operators abstract we present characterizations of some generalized convexity properties of functions with the help of a general subdifferential we stress the case of lower semicontinuous functions we also study the important case of marginal functions and we provide representation results

sweeping process with regular and nonregular sets abstract differential inclusions involving the normal cone to a moving set are investigated a special attention is paid to the sweeping process associated with sets for which no regularity assumption is required

averaging bounds for lattices and linear codes general random coding theorems for lattices are derived from the minkowski hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out a new version of the minkowski hlawka theorem itself is obtained as the limit for p spl rarr spl infin of a simple lemma for linear codes over gf p used with p level amplitude modulation the relation between the combinatorial packing of solid bodies and the information theoretic soft packing with arbitrarily small but positive overlap is illuminated the soft packing results are new when specialized to the additive white gaussian noise channel they reduce to a version of the de buda poltyrev result that spherically shaped lattice codes and a decoder that is unaware of the shaping can achieve the rate 1 2 log sub 2 p n

a fuzzy necessary optimality condition for non lipschitz optimization in asplund spaces in this paper we consider a general optimization problem in the broad class of asplund spaces we derive a new necessary optimality condition in the so called lagrangian fuzzy form without standard lipschitz conditions we also give a chain rule for frechet subdifferentials and subdifferential representations of frechet normals to constrained sets