convergence rates for forward backward dynamical systems associated with strongly monotone inclusions abstract we investigate the convergence rates of the trajectories generated by implicit first and second order dynamical systems associated to the determination of the zeros of the sum of a maximally monotone operator and a monotone and lipschitz continuous one in a real hilbert space we show that these trajectories strongly converge with exponential rate to a zero of the sum provided the latter is strongly monotone we derive from here convergence rates for the trajectories generated by dynamical systems associated to the minimization of the sum of a proper convex and lower semicontinuous function with a smooth convex one provided the objective function fulfills a strong convexity assumption in the particular case of minimizing a smooth and strongly convex function we prove that its values converge along the trajectory to its minimum value with exponential rate too

counting markov types balanced matrices and eulerian graphs the method of types is one of the most popular techniques in information theory and combinatorics two sequences of equal length have the same type if they have identical empirical distributions in this paper we focus on markov types that is sequences generated by a markov source of order one we note that sequences having the same markov type share the same so called balanced frequency matrix that counts the number of distinct pairs of symbols we enumerate the number of markov types for sequences of length over an alphabet of size this turns out to be asymptotically equivalent to estimating the number of the balanced frequency matrices the number of integer solutions of a system of linear diophantine equations and the number of connected eulerian multigraphs for fixed we prove that the number of markov types is asymptotically equal to d m nm2 m m2 m where we give an integral representation for d m for m we conclude that asymptotically the number of types is equivalent to 2m3m 2em2 m2m22m m 2 nm2 m provided that m o n1 4 these findings are derived by analytical techniques ranging from analytic combinatorics to multidimensional generating functions to the saddle point method

the asymptotic redundancy of bayes rules for markov chains we derive the asymptotics of the redundancy of bayes rules for markov chains of fixed order over a finite alphabet extending the work of barron and clarke 1990 on independent and identically distributed i i d sources the asymptotics are derived when the actual source is the class of spl phi mixing sources which strictly includes markov chains these results can be used to derive minimax asymptotic rates of convergence for universal codes when a markov chain of fixed order is used as a model

fisher information and stochastic complexity by taking into account the fisher information and removing an inherent redundancy in earlier two part codes a sharper code length as the stochastic complexity and the associated universal process are derived for a class of parametric processes the main condition required is that the maximum likelihood estimates satisfy the central limit theorem the same code length is also obtained from the so called maximum likelihood code

prox penalization and splitting methods for constrained variational problems this paper is concerned with the study of a class of prox penalization methods for solving variational inequalities of the form ax n c x ni0 where mathcal h is a real hilbert space a mathcal h rightrightarrows mathcal h is a maximal monotone operator and n c is the outward normal cone to a closed convex set c subset mathcal h given psi mathcal h to mathbb r cup infty which acts as a penalization function with respect to the constraint x in c and a penalization parameter beta n we consider a diagonal proximal algorithm of the form x n i lambda n a beta n partial psi 1 x n 1 and an algorithm which alternates proximal steps with respect to a and penalization steps with respect to c and reads as x n i lambda n beta n partial psi 1 i lambda na 1 x n 1 we obtain weak ergodic convergence for a general maximal monotone operator a and weak convergence of the whole sequence x n when a is the subdifferential of a proper lower semicontinuous convex function mixing with passty s idea we can extend the ergodic convergence theorem thus obtaining the convergence of a prox penalization splitting algorithm for constrained variational inequalities governed by the sum of several maximal monotone operators our results are applied to an optimal control problem where the state variable and the control are coupled by an elliptic equation we also establish robustness and stability results that account for numerical approximation errors

asymptotic control and stabilization of nonlinear oscillators with non isolated equilibria let h r be a c1 function on a real hilbert space h and let 0 be a positive damping parameter for any control function e r r which tends to zero as t we study the asymptotic behavior of the trajectories of the damped nonlinear oscillator hbfc x t x t x t e t x t 0 we show that if e t does not tend to zero too rapidly as t then the term e t x t asymptotically acts as a tikhonov regularization which forces the trajectories to converge to a particular equilibrium indeed in the main result of this paper it is established that when is convex and s argmin under the key assumption that e is a slow control i e 0 e t dt then each trajectory of the hbfc system strongly converges as t to the element of minimal norm of the closed convex set s as an application we consider the damped wave equation with neumann boundary condition formula

generalized tunstall codes for sources with memory tunstall codes are variable to fixed length codes that maximize the expected number of source letters per dictionary string for discrete memoryless sources we analyze a generalization of tunstall coding to sources with memory and demonstrate that as the dictionary size increases the number of code letters per source symbol comes arbitrarily close to the minimum among all variable to fixed length codes of the same size we also find the asymptotic relationship between the dictionary size and the average length of a dictionary entry

analytical properties and optimization of time delayed feedback control time delayed feedback control is an efficient method for stabilizing unstable periodic orbits of chaotic systems if the equations governing the system dynamics are known the success of the method can be predicted by a linear stability analysis of the desired orbit unfortunately the usual procedures for evaluating the floquet exponents of such systems are rather intricate we show that the main stability properties of the system controlled by time delayed feedback can be simply derived from a leading floquet exponent defining the system behavior under proportional feedback control optimal parameters of the delayed feedback controller can be evaluated without an explicit integration of delay differential equations the method is valid for low dimensional systems whose unstable periodic orbits are originated from a period doubling bifurcation and is demonstrated for the rossler system and the duffing oscillator

asymptotic analysis of the exponential penalty trajectory in linear programming we consider the linear program min c x ax b and the associated exponential penalty functionfr x c x r exp aix bi r forr close to 0 the unconstrained minimizerx r offr admits an asymptotic expansion of the formx r x rd r wherex is a particular optimal solution of the linear program and the error term r has an exponentially fast decay using duality theory we exhibit an associated dual trajectory r which converges exponentially fast to a particular dual optimal solution these results are completed by an asymptotic analysis whenr tends to the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution

properties of jeffreys mixture for markov sources we discuss the properties of jeffreys mixture for a markov model first we show that a modified jeffreys mixture asymptotically achieves the minimax coding regret for universal data compression where we do not put any restriction on data sequences moreover we give an approximation formula for the prediction probability of jeffreys mixture for a markov model by this formula it is revealed that the prediction probability by jeffreys mixture for the markov model with alphabet 0 1 is not of the form nx s ns where nx s is the number of occurrences of the symbol x following the context s 0 1 and ns n0 s n1 s moreover we propose a method to compute our minimax strategy which is a combination of a monte carlo method and the approximation formula where the former is used for earlier stages in the data while the latter is used for later stages

estimating a generating partition from observed time series symbolic shadowing we propose a deterministic algorithm for approximating a generating partition from a time series using tessellations using data generated by henon and ikeda maps we demonstrate that the proposed method produces partitions that uniquely encode all the periodic points up to some order and provide good estimates of the metric and topological entropies the algorithm gives useful results even with a short noisy time series

delay and redundancy in lossless source coding the penalty incurred by imposing a finite delay constraint in lossless source coding of a memoryless source is investigated it is well known that for the so called block tovariable and variable to variable codes the redundancy decays at best polynomially with the delay which in this case is identified with the block or maximal phrase length respectively in stark contrast it is shown that the redundancy can be made to decay exponentially with the delay constraint the corresponding redundancy delay exponent is shown to be bounded from below by the renyi entropy of order 2 of the source and from above for almost all sources in terms of the minimal source symbol probability and the alphabet size

on the construction of explicit khodak s code and its analysis variable to variable vv codes are very attractive yet not well understood data compression schemes in 1972 khodak claimed to provide upper and lower bounds for the achievable redundancy rate however he did not offer explicit construction of such codes in this paper we first present a constructive and transparent proof of khodak s result showing that for memoryless sources there exists a code with the average redundancy bounded by d 5 3 where d is the average delay e g the average length of a dictionary entry we also describe an algorithm that constructs a vv length code with a small redundancy rate for large d then we discuss several generalizations we prove that the worst case redundancy does not exceed d 4 3 furthermore we provide similar upper bound for markov sources of order 1 finally we consider bounds that are valid for almost all memoryless and markov sources for which the set of exceptional source parameters has zero measure in particular for all memoryless sources outside this exceptional class we prove there exists a vv code with the average redundancy rate bounded by d 1 m 3 epsiv and the worst case redundancy rate bounded by d 1 m 3 epsiv where m is the cardinality of the alphabet we complete our analysis with a lower bound showing that for all vv codes the average and the worst case redundancy rates are at least d 2m 1 epsiv for almost all memoryless sources in the sense that the set of exceptional source parameters has zero measure we prove these results using techniques of diophantine approximations

the performance of universal encoding universal coding theory is surveyed from the viewpoint of the interplay between delay and redundancy the price for universality turns out to be acceptably small

limited data x ray tomography using nonlinear evolution equations a novel approach to the x ray tomography problem with sparse projection data is proposed nonnegativity of the x ray attenuation coefficient is enforced by modelling it as max phi x 0 where phi is a smooth function the function phi is computed as the equilibrium solution of a nonlinear evolution equation analogous to the equations used in level set methods the reconstruction algorithm is applied to a simulated full and limited angle projection data of the shepp logan phantom with sparse angular sampling and b measured limited angle projection data of in vitro dental specimens the results are significantly better than those given by traditional backprojection based approaches and similar in quality but faster to compute compared to the algebraic reconstruction technique

optimal periodic orbits of chaotic systems occur at low period invariant sets embedded in a chaotic attractor can generate time averages that differ from the average generated by typical orbits on the attractor motivated by two different topics namely controlling chaos and riddled basins of attraction we consider the question of which invariant set yields the largest optimal value of an average of a given smooth function of the system state we present numerical evidence and analysis that indicate that the optimal average is typically achieved by a low period unstable periodic orbit embedded in the chaotic attractor in particular our results indicate that if we consider that the function to be optimized depends on a parameter gamma then the lebesgue measure in gamma corresponding to optimal periodic orbits of period ital p or greater decreases exponentially with increasing ital p furthermore the set of parameter values for which optimal orbits are nonperiodic typically has zero lebesgue measure copyright ital 1996 the american physical society

proximal point algorithm controlled by a slowly vanishing term applications to hierarchical minimization let phi 0 mathbb r n to mathbb r cup infty be a closed convex function and phi 1 mathbb r n to mathbb r be a finite convex function that are bounded from below our goal is to build an algorithm which first minimizes the map phi 0 and secondly the map phi 1 over the set s 0 rm argmin phi 0 for that purpose we define the following proximal type algorithm x n 1 x n lambda n in partial eta n phi 0 varepsilon n phi 1 x n 1 leqno mathcal a 1 where lambda n is a positive step sequence eta n is a summable error sequence and varepsilon n is a control sequence tending toward 0 partial eta denotes the eta approximate subdifferential when varepsilon n is a slow control i e sum n 0 infty varepsilon n infty we prove that under adequate conditions the sequence x n defined by mathcal a 1 tends toward an element of s 1 rm argmin s 0 phi 1 more generally given finite convex functions phi 2 ldots phi n mathbb r n to mathbb r let us define the sets s i i in 1 ldots n by the recursive relation s i rm argmin s i 1 phi i we introduce an extension of algorithm mathcal a 1 to minimize hierarchically each function phi i on the set s i 1 for i in 1 ldots n

asymptotic behavior of coupled dynamical systems with multiscale aspects abstract we study the asymptotic behavior as time variable t goes to of nonautonomous dynamical systems involving multiscale features as a benchmark case given h a general hilbert space h r and h r two closed convex functions and a function of t which tends to as t goes to we consider the differential inclusion x t x t t x t 0 this system models the emergence of various collective behaviors in game theory as well as the asymptotic control of coupled systems we show several results ranging from weak ergodic to strong convergence of the trajectories as a key ingredient we assume that for every p belonging to the range of n c 0 t p t c p t d t where is the fenchel conjugate of c is the support function of c argmin and n c x is the normal cone to c at x as a by product we revisit the system x t t x t x t 0 where t tends to zero as t goes to and 0 t d t whose asymptotic behavior can be derived from the preceding one by time rescaling applications are given in game theory optimal control variational problems and pdes

strong asymptotic convergence of evolution equations governed by maximal monotone operators with tikhonov regularization abstract we consider the tikhonov like dynamics u t a u t e t u t where a is a maximal monotone operator on a hilbert space and the parameter function e t tends to 0 as t with 0 e t d t when a is the subdifferential of a closed proper convex function f we establish strong convergence of u t towards the least norm minimizer of f in the general case we prove strong convergence towards the least norm point in a 1 0 provided that the function e t has bounded variation and provide a counterexample when this property fails

chaos order statistics and unstable periodic orbits we present an interesting connection between order statistics and unstable periodic orbits of chaotic maps this can be used to locate all the unstable periodic points of a given order of one dimensional chaotic maps with continuous invariant density the densities of the ordered variates of the iterates are discontinuous exactly at unstable periodic points of the map this is illustrated using the logistic map where densities corresponding to a small number of iterates have been obtained in closed form this scheme can also be applied to a class of continuous time systems where the successive maxima of the time series behave as if they were generated from a unimodal map we demonstrate this by using the lorenz model

asymptotic behavior of solutions to a class of nonlinear evolution equations on etudie les equations de la chaleur a controle de temperature a l interieur ou sur la frontiere du domaine pour cela on considere un type de problemes variationnels paraboliques avec obstacles

optimal periodic orbits of chaotic systems invariant sets embedded in a chaotic attractor can generate time averages that differ from the average generated by typical orbits on the attractor motivated by two different topics namely controlling chaos and riddled basins of attraction we consider the question of which invariant set yields the largest optimal value of an average of a given smooth function of the system state we present numerical evidence and analysis which indicate that the optimal average is typically achieved by a low period unstable periodic orbit embedded in the chaotic attractor

reconstruction of dynamical and geometrical properties of chaotic attractors from threshold crossing interspike intervals we reconstruct the largest lyapunov exponent and fractal dimension of a chaotic attractor using thresholdcrossing interspike intervals alone we show that in certain cases one may reconstruct from this data a set looking very similar to the initial attractor we also give an explanation of this possibility based on the concept of instantaneous frequency s1063 651x 98 51207 x pacs number s 05 45 1b the problem of extracting information about a dynamical system from experimental data is of great importance for all those who study the systems of natural origin that may be thought of as black boxes the traditional way of obtaining experimental data is to fix the sampling stepdt and store the values of some state variable s t corresponding to the time moments idt the fundamental works 1 state that the delay method applied to such time series often allows one to reconstruct a set being topologically equivalent to the initial system attractor on the base of these works a series of methods have been developed allowing one to evaluate attractor dimension 2 and the largest lyapunov characteristic exponent lce 3 to detect unstable equilibrium points 4 and unstable periodic orbits 5 and finally to create in some cases global models 6 or different predictors 7 another way of getting experimental data which is often preferred by biologists is to generate spike train data 8 one imposes a certain condition on the value of state variable s t and records the interspike intervals isi ti between the time moments for which the condition is fulfilled there are two general methods for recording spike train data the first one is to generate a spike when s t crosses some threshold level u in one direction and measure isi as an alternative one may measure the time intervals between the local maxima of s t fig 1 a the values of ti can be treated as the times of trajectory return to the surface defined by the condition imposed to the initial s t the condition s t 5u means the definition of a plane in the phase space while the condition s t 50 s t being a realization x j t o f a dynamical system dx w dt5f

subset source coding this paper studies the fundamental limits of storage for structured data where statistics and structure are both critical to the application accordingly a framework is proposed for optimal lossless and lossy compression of subsets of the possible realizations of a discrete memoryless source dms for the lossless subset compression problem it turns out that the optimal source code may not index the conventional source typical sequences but rather index certain subset typical sequences consistent with the subset of interest building upon an achievability and a strong converse an analytic expression is given based on the shannon entropy relative entropy and subset entropy which identifies such subset typical sequences for a broad class of subsets of a dms intuitively subset typical sequences belong to those typical sets which highly intersect the subset of interest but are still closest to the source distribution in the sense of relative entropy for the lossy subset compression problem an upper bound is derived on the subset rate distortion function in terms of the subset mutual information optimized over the set of conditional distributions that satisfy the expected distortion constraint with respect to the subset typical distribution and over a set of certain auxiliary subsets by proving a strong converse result this upper bound is shown to be tight for a class of symmetric subsets as shown in our numerical examples more often than not one achieves a gain in the fundamental limits in that the optimal compression rate for the subset in both the lossless and lossy settings can be strictly smaller than the source entropy and the source rate distortion function respectively although exceptions are also possible

diophantine approximation on rational quadrics we compute the hausdorff dimension of sets of very well approximable vectors on rational quadrics we use ubiquitous systems and the geometry of locally symmetric spaces as a byproduct we obtain the hausdorff dimension of the set of rays with a fixed maximal singular direction which move away into one end of a locally symmetric space at linear depth infinitely many times

regularized and inertial algorithms for common fixed points of nonlinear operators this paper deals with a general fixed point iteration for computing a point in some nonempty closed and convex solution set included in the common fixed point set of a sequence of mappings on a real hilbert space the proposed method combines two strategies viscosity approximations regularization and inertial type extrapolation the first strategy is known to ensure the strong convergence of some successive approximation methods while the second one is intended to speed up the convergence process under classical conditions on the operators and the parameters we prove that the sequence of iterates generated by our scheme converges strongly to the element of minimal norm in the solution set this algorithm works for instance for approximating common fixed points of infinite families of demicontractive mappings including the classes of quasi nonexpansive operators and strictly pseudocontractive ones

a new class of alternating proximal minimization algorithms with costs to move given two objective functions f mathcal x mapsto r cup infty and g mathcal y mapsto r cup infty on abstract spaces mathcal x and mathcal y and a coupling function c mathcal x times mathcal y mapsto r we introduce and study alternative minimization algorithms of the following type x 0 y 0 in mathcal x times mathcal y mbox given x n y n rightarrow x n 1 y n rightarrow x n 1 y n 1 mbox as follows left begin array l x n 1 in mbox argmin f xi beta n c xi y n alpha n h x n xi xi in mathcal x y n 1 in mbox argmin g eta mu n c x n 1 eta nu n k y n eta eta in mathcal y end array right their most original feature is the introduction of the terms h mathcal x times mathcal x mapsto r and k mathcal y times mathcal y mapsto r which are costs to change or to move distance like functions relative entropies accounting for various inertial friction or anchoring effects these algorithms are studied in a general abstract framework the introduction of the costs to change h and k leads to proximal minimizations with corresponding dissipative effects as a result the algorithms enjoy nice convergent properties coefficients alpha n beta n mu n nu n are nonnegative parameters when taking alpha n nu n 0 and quadratic costs on a hilbert space one recovers the classical alternating minimization algorithm which itself is a natural extension of the alternating projection algorithm of von neumann a number of new significant results hold in general metric spaces we pay particular attention to the following cases 1 mathcal x d mathcal x and mathcal y d mathcal y are complete metric spaces and h geq d mathcal x k geq d mathcal y high local costs to move the algorithms then provide sequences that converge to nash equilibria 2 mathcal x mathcal y mathcal h is a hilbert space the costs to change are quadratic low local costs to move and the functions f g mathcal h mapsto r cup infty are closed convex proper then some of the classical convergence theorems for alternating convex minimization algorithms including those of acker and prestel are properly extended with original proofs

variable length to variable length source coding a greedy step by step algorithm some results on variable length to variable length source coding are presented a sufficient criterion for the asymptotic optimality of the generic v v code is given this also allows the study of the tunstall huffman scheme performance then a nonoptimal greedy approach to prefix codes is described based on the informational divergence pseudometric and on the gallager algorithm minimization of the current rate associated with the subsequent extensions of the source probability distribution although optimality is not always reached this technique can be usefully employed to improve the tunstall huffman concatenation

asymptotic control and stabilization of nonlinear oscillators with non isolated equilibria a note from l 1 to non l 1 in h attouch m o czarnecki asymptotic control and stabilization of non linear oscillators with non isolated equilibria j differential equations 179 2002 278 310 we exhibited a sharp condition ensuring the efficiency of the tikhonov like control term in the hbfc system precisely let h r be a c1 function on a real hilbert space h let 0 be a positive damping parameter and let r r be a control function which decreases to zero as t in order to select particular equilibria in the important case where has non isolated equilibria we introduced in h attouch m o czarnecki j differential equations 179 2002 278 310 the following damped nonlinear oscillator and studied its asymptotic behavior hbfc x t x t x t t x t 0 we established that when is convex and s argmin under the key assumption that is a slow control i e 0 t dt then each trajectory of the hbfc system strongly converges when t to the element of minimal norm of the closed convex set s the condition on the control term is sharp indeed when 0 t dt the trajectory weakly converges but it may not strongly converge and we have no information a priori on the weak limit in this note we give an answer to the following question when does an l1 control term becomes or behave non l1 precisely take a control term l1 let x be the solution of the corresponding hbfc system take n to be a non increasing truncation of n t t for t 0 n let x be the solution of the corresponding hbfc system we show that limn xn x 0 in particular the weak limits of the trajectories xn strongly converge when n to the strong limit of the trajectory x in other words there is no loss of the information gained by the slow behavior for t n

asymptotic behavior of nonautonomous monotone and subgradient evolution equations in a hilbert setting h we study the asymptotic behavior of the trajectories of nonautonomous evolution equations dot x t a t x t ni 0 where for each t geq 0 a t h rightrightarrows h denotes a maximal monotone operator we provide general conditions guaranteeing the weak ergodic convergence of each trajectory x cdot to a zero of a limit maximal monotone operator a infty as the time variable t tends to infty the crucial point is to use the brezis haraux function or equivalently the fitzpatrick function to express at which rate the excess of mathrm gph a infty over mathrm gph a t tends to zero this approach gives a sharp and unifying view of this subject in the case of operators a t partial phi t which are subdifferentials of proper closed convex functions phi t we show convergence results for the trajectories then we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions illustrations are given in the field of coupled systems and partial differential equations

invariant measurement of strange sets in terms of cycles we argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible but is also a theoretically optimal measurement of the invariant properties of a dynamical system

weak pullback attractors of setvalued processes weak pullback attractors are defined for nonautonomous setvalued processes and their existence and upper semicontinuous convergence under perturbation is established unlike strong pullback attractors invariance and pullback attraction here are required only for at least one trajectory rather than all trajectories at each starting point the concept is useful in for example continuous time control systems and is related to that of viability

precise minimax redundancy and regret recent years have seen a resurgence of interest in redundancy of lossless coding the redundancy regret of universal fixed to variable length coding for a class of sources determines by how much the actual code length exceeds the optimal ideal over the class code length in a minimax scenario one finds the best code for the worst source either in the worst case called also maximal minimax or on average we first study the worst case minimax redundancy over a class of stationary ergodic sources and replace shtarkov s bound by an exact formula among others we prove that a generalized shannon code minimizes the worst case redundancy derive asymptotically its redundancy and establish some general properties this allows us to obtain precise redundancy for memoryless markov and renewal sources for example we present the exact constant of the redundancy for memoryless and markov sources by showing that the integer nature of coding contributes log logm m 1 logm o 1 where m is the size of the alphabet then we deal with the average minimax redundancy and regret our approach here is orthogonal to most recent research in this area since we aspire to show that asymptotically the average minimax redundancy is equivalent to the worst case minimax redundancy for some classes of sources after formulating some general bounds relating these two redundancies we prove our assertion for memoryless and markov sources nevertheless we provide evidence that maximal redundancy of renewal processes does not have the same leading term as the average minimax redundancy however our general results show that maximal and average regrets are asymptotically equivalent

asymptotic control of pairs of oscillators coupled by a repulsion with nonisolated equilibria ii the singular case let phi h rightarrow mathbb r be a mathcal c 1 function on a real hilbert space h and let gamma 0 be a positive damping parameter for any singular repulsive potential v h setminus 0 to mathbb r i e such that lim z to 0 v z infty and any control function varepsilon mathbb r to mathbb r setminus 0 which tends to zero as t to infty we study the asymptotic behavior of the trajectories of the coupled dissipative system of nonlinear oscillators left begin array l ddot x gamma dot x nabla phi x varepsilon t nabla v x y 0 vspace 1mm ddot y gamma dot y nabla phi y varepsilon t nabla v x y 0 end array right leqno rm hbfc 2 sing this system is the singular version of the regular rm hbfc 2 reg system studied in a cabot and m o czarnecki siam j control optim 41 2002 pp 1254 1280 where the potential v is defined on the whole space h the purpose of this paper is to obtain whenever possible the same existence and convergence results in the singular case as in the regular case considered by a cabot and m o czarnecki this study is mainly motivated by a better convergence behavior of rm hbfc 2 sing with the same sharp condition on the control varepsilon exhibited in h attouch and m o czarnecki j differential equations 179 2002 pp 278 310 and by a cabot and m o czarnecki precisely when h mathbb r and if varepsilon is a slow control i e int 0 infty varepsilon t dt infty then the trajectories x and y converge to extremal points of the set s lambda in mathbb r nabla phi lambda 0 of the equilibria of phi the awkward case in a cabot and m o czarnecki where the trajectories may have the same limit disappears of importance from a physical point of view we thus can consider actual for example electromagnetic repulsive potentials

approaching the solving of constrained variational inequalities via penalty term based dynamical systems abstract we investigate the existence and uniqueness of locally absolutely continuous trajectories of a penalty term based dynamical system associated to a constrained variational inequality expressed as a monotone inclusion problem relying on lyapunov analysis and on the ergodic continuous version of the celebrated opial lemma we prove weak ergodic convergence of the orbits to a solution of the constrained variational inequality under investigation if one of the operators involved satisfies stronger monotonicity properties then strong convergence of the trajectories can be shown

control of nmr laser chaos in high dimensional embedding space the unstable periodic orbits of a chaotic nmr laser have been identified and classified according to their stability properties the dynamical control method by ott grebogi and yorke ogy phys rev lett 64 1196 1990 has been successfully applied in a six dimensional embedding space as required for a proper reconstruction of the attractors an improved method has then been implemented in order to circumvent the occurrence of several possibly complex unstable eigenvalues in the standard ogy procedure and has shown superior performance finally an extension to several control stations per poincar e time is discussed

asymptotic behaviour of quasi autonomous dissipative systems in hilbert spaces abstract by suitably modifying the methods used by b djafari rouhani ergodic theorems for non expansive sequences in hilbert spaces and related problems part i pp 1 76 thesis yale university and a new proof of the weak convergence theorems for non expansive sequences and curves in hilbert spaces preprint 1984 and by b djafari rouhani and s kakutani ergodic theorems for non expansive non linear operators in a hilbert space preprint 1984 we study the asymptotic behaviour of the quasi autonomous dissipative system du dt au where a is a monotone operator in a hilbert space h and l 1 0 h

asymptotic average redundancy of huffman and other block codes we study asymptotically the redundancy of huffman and other codes it has been known from the inception of the huffman 1952 code that in the worst case its redundancy defined as the excess of the code length over the optimal ideal code length is not more than one however to the best of our knowledge no precise asymptotic results have been reported in literature thus far we consider here a memoryless binary source generating a sequence of length n distributed as binomial n p with p being the probability of emitting 0 based on the results of stubley 1994 we prove that for p 1 2 1 m 1 2 sup 1 m 2 sup m o spl rho sup n spl alpha n m rational where spl alpha log sub 2 1 p p and spl beta log sub 2 1 p spl rho x x is the fractional part of x the appearance of the fractal like function explains the erratic behavior of the huffman redundancy and its resistance to succumb to a precise analysis as a side result we prove that the average redundancy of the shannon block code is as n spl rarr spl infin r sub n sup s 1 2 o 1 spl alpha irrational 1 2 1 m 1 2 o spl rho sup n spl alpha n m rational where spl rho 1 finally we derive the redundancy of the golomb 1966 code for the geometric distribution which can be viewed as a special case of the huffman and shannon codes golomb s code redundancy has only oscillating behavior i e there is not convergent mode these findings are obtained by analytic methods such as theory of distribution of sequences modulo 1 and fourier series

a note on simultaneous diophantine approximation on planar curves let mathcal s n psi 1 dots psi n denote the set of simultaneously psi 1 dots psi n approximable points in mathbb r n and mathcal s n psi denote the set of multiplicatively approximable points in mathbb r n let mathcal m be a manifold in mathbb r n the aim is to develop a metric theory for the sets mathcal m cap mathcal s n psi 1 dots psi n and mathcal m cap mathcal s n psi analogous to the classical theory in which mathcal m is simply mathbb r n in this note we mainly restrict our attention to the case that mathcal m is a planar curve mathcal c a complete hausdorff dimension theory is established for the sets mathcal c cap mathcal s 2 psi 1 psi 2 and mathcal c cap mathcal s 2 psi a divergent khintchine type result is obtained for mathcal c cap mathcal s 2 psi 1 psi 2 i e if a certain sum diverges then the one dimensional lebesgue measure on mathcal c of mathcal c cap mathcal s 2 psi 1 psi 2 is full furthermore in the case that mathcal c is a rational quadric the convergent khintchine type result is obtained for both types of approximation our results for mathcal c cap mathcal s 2 psi 1 psi 2 naturally generalize the dimension and lebesgue measure statements of beresnevich et al mem ams 179 846 1 91 2006 moreover within the multiplicative framework our results for mathcal c cap mathcal s 2 psi constitute the first of their type

extremal manifolds and hausdorff dimension the recent proof by d y kleinbock and g a margulis 11 of sprind uk s conjecture for smooth nondegenerate manifolds m means that the set lv m of v approximable points this and other terminology is explained below on m is of zero induced lebesgue measure this raises the question of its hausdorff dimension

on universal types the universal type class of a sequence xn is defined in analogy to the notion underlying the classical method of types two sequences of the same length are said to be of the same universal lz type if and only if they yield the same dictionary or equivalently parsing tree in the incremental parsing of ziv and lempel 1978 it is shown that for any finite order k the variational distance between the kth order empirical probability distributions of two sequences of the same universal type vanishes as the sequence length tends to infinity consequently for any k and any kth order probability assignment the difference between the normalized logarithms of the probabilities assigned to two sequences of the same universal type also vanishes asymptotically the size of a universal type class is studied and it is shown that its asymptotic behavior parallels that of the conventional counterpart with the lz78 code length playing the role of the empirical entropy the number of universal types for sequences of length n is estimated and shown to be of the form exp 1 o 1 gamman logn for a well characterized constant gamma algorithms for enumerating the sequences in a universal type class and for drawing a sequence from the class with uniform probability are described as an application the problem of universal simulation of individual sequences is considered a sequence drawn with uniform probability from the universal type class of xn is an optimal simulation of xn in a well defined mathematical sense

markov types and minimax redundancy for markov sources redundancy of universal codes for a class of sources determines by how much the actual code length exceeds the optimal code length in the minimax scenario one designs the best code for the worst source within the class such minimax redundancy comes in two flavors average minimax or worst case minimax we study the worst case minimax redundancy of universal block codes for markovian sources of any order we prove that the maximal minimax redundancy for markov sources of order r is asymptotically equal to 1 2m sup r m 1 log sub 2 n log sub 2 a sub m sup r lnlnm sup 1 m 1 lnm o 1 where n is the length of a source sequence m is the size of the alphabet and a sub m sup r is an explicit constant e g we find that for a binary alphabet m 2 and markov of order r 1 the constant a sub 2 sup 1 16 spl middot g spl ap 14 655449504 where g is the catalan number unlike previous attempts we view the redundancy problem as an asymptotic evaluation of certain sums over a set of matrices representing markov types the enumeration of markov types is accomplished by reducing it to counting eulerian paths in a multigraph in particular we propose exact and asymptotic formulas for the number of strings of a given markov type all of these findings are obtained by analytic and combinatorial tools of analysis of algorithms

type classes of context trees it is well known that a tree model does not always admit a finite state machine fsm representation with the same minimal number of parameters therefore known characterizations of type classes for fsms do not apply in general to tree models in this paper the type class of a sequence with respect to a given context tree is studied an exact formula is derived for the size of the class extending whittle s formula for type classes with respect to fsms the derivation is more intricate than in the fsm case since some basic properties of fsm types do not hold in general for tree types the derivation also yields an efficient enumeration of the tree type class a formula for the number of type classes with respect to is also derived the formula is asymptotically tight up to a multiplicative constant and also extends the corresponding result for fsms the asymptotic behavior of the number of type classes and of the size of a class is expressed in terms of the so called minimal canonical extension of t a tree that is generally larger than but smaller than its fsm closure

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

alternating proximal algorithms for linearly constrained variational inequalities application to domain decomposition for pde s abstract let x y z be real hilbert spaces let f x r g y r be closed convex functions and let a x z b y z be linear continuous operators let us consider the constrained minimization problem p min f x g y a x b y given a sequence n which tends toward 0 as n we study the following alternating proximal algorithm a x n 1 argmin n 1 f 1 2 a b y n z 2 2 x n x 2 x y n 1 argmin n 1 g 1 2 a x n 1 b z 2 2 y n y 2 y where and are positive parameters it is shown that if the sequence n tends moderately slowly toward 0 then the iterates of a weakly converge toward a solution of p the study is extended to the setting of maximal monotone operators for which a general ergodic convergence result is obtained applications are given in the area of domain decomposition for pde s

extracting unstable periodic orbits from chaotic time series data a general nonlinear method to extract unstable periodic orbits from chaotic time series is proposed by utilizing the estimated local dynamics along a trajectory we devise a transformation of the time series data such that the transformed data are concentrated on the periodic orbits thus one can extract unstable periodic orbits from a chaotic time series by simply looking for peaks in a finite grid approximation of the distribution function of the transformed data our method is demonstrated using data from both numerical and experimental examples including neuronal ensemble data from mammalian brain slices the statistical significance of the results in the presence of noise is assessed using surrogate data s1063 651x 97 15205 9

diophantine approximation on planar curves and the distribution of rational points let c be a nondegenerate planar curve and for a real positive decreasing function let c denote the set of simultaneously approximable points lying on c we show that c is of khintchine type for divergence i e if a certain sum diverges then the one dimensional lebesgue measure on c of c is full we also obtain the hausdorff measure analogue of the divergent khintchine type result in the case that c is a rational quadric the convergence counterparts of the divergent results are also obtained furthermore for functions with lower order in a critical range we determine a general exact formula for the hausdorff dimension of c these results constitute the first precise and general results in the theory of simultaneous diophantine approximation on manifolds

inhomogeneous diophantine approximation on curves and hausdorff dimension abstract the goal of this paper is to develop a coherent theory for inhomogeneous diophantine approximation on curves in r n akin to the well established homogeneous theory more specifically the measure theoretic results obtained generalize the fundamental homogeneous theorems of r c baker 1978 2 dodson dickinson 2000 18 and beresnevich bernik kleinbock margulis 2002 8 in the case of planar curves the complete hausdorff dimension theory is developed

tunstall code khodak variations and random walks a variable to fixed length encoder partitions the source string into variable length phrases that belong to a given and fixed dictionary tunstall and independently khodak designed variable to fixed length codes for memoryless sources that are optimal under certain constraints in this paper we study the tunstall and khodak codes using variety of techniques ranging from stopping times for sums of independent random variables to tauberian theorems and mellin transform after proposing an algebraic characterization of the tunstall and khodak codes we present new results on the variance and a central limit theorem for dictionary phrase lengths this analysis also provides a new argument for obtaining asymptotic results about the mean dictionary phrase length and average redundancy rates

a convergence result for nonautonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming abstract we present a new result on the asymptotic behavior of nonautonomous subgradient evolution equations of the form u t t u t where t t 0 is a family of closed proper convex functions the result is used to study the flow generated by the family t x f x r t where f x r ctx r exp aix bi r is the exponential penalty approximation of the linear program min ctx ax b and r t is a positive function tending to 0 when t we prove that the trajectory u t converges to an optimal solution u of the linear program and we give conditions for the convergence of an associated dual trajectory t toward an optimal solution of the dual program

asymptotic behavior of gradient like dynamical systems involving inertia and multiscale aspects abstract in a hilbert space h we study the asymptotic behavior as time variable t goes to of nonautonomous gradient like dynamical systems involving inertia and multiscale features given h r and h r two convex differentiable functions a positive damping parameter and t a function of t which tends to zero as t goes to we consider the second order differential equation x t x t x t t x t 0 this system models the emergence of various collective behaviors in game theory as well as the asymptotic control of coupled nonlinear oscillators assuming that t tends to zero moderately slowly as t goes to infinity we show that the trajectories converge weakly in h the limiting equilibria are solutions of the hierarchical minimization problem which consists in minimizing over the set c of minimizers of as key assumptions we suppose that 0 t d t and that for every p belonging to a convex cone c depending on the data and 0 t p c t p d t where is the fenchel conjugate of and c is the support function of c an application is given to coupled oscillators

the gradient and heavy ball with friction dynamical systems the quasiconvex case we consider the gradient system dot x t nabla phi x t 0 and the so called heavy ball with friction dynamical system ddot x t lambda dot x t nabla phi x t 0 as well as an implicit discrete proximal version of it and study the asymptotic behavior of their solutions in the case of a smooth and quasiconvex objective function minimization properties of trajectories are obtained under various additional assumptions we finally show a minimizing property of the heavy ball method which is not shared by the gradient method the genericity of the convergence of each trajectory at least when is a morse function towards local minimum of

types of markov fields and tilings the method of types is one of the most popular techniques in information theory and combinatorics however thus far the method has been mostly applied to 1 d markov processes and it has not been thoroughly studied for general markov fields markov fields over a finite alphabet of size m ge 2 can be viewed as models for multidimensional systems with local interactions the locality of these interactions is represented by a shape mathcal s while its marking by symbols of the underlying alphabet is called a tile two assignments in a markov field have the same type if they have the same empirical distribution i e if they have the same number of tiles of a given type our goal is to study the growth of the number of possible markov field types in either a d dimensional box of lengths n 1 ldots n d or its cyclic counterpart a d dimensional torus we relate this question to the enumeration of nonnegative integer solutions of a large system of diophantine linear equations called the conservation laws we view a markov type as a vector in a d m mathcal s dimensional space and count the number of such vectors satisfying the conservation laws which turns out to be the number of integer points in a certain polytope for the torus this polytope is of dimension mu d 1 mathrm rk mathrm mathrm c where mathrm rk mathrm mathrm c is the number of linearly independent conservation laws mathrm mathrm c this provides an upper bound on the number of types then we construct a matching lower bound leading to the conclusion that the number of types in the torus markov field is theta n mu where n n 1 ldots n d these results are derived by geometric tools including ideas of discrete and convex multidimensional geometry

a dynamic gradient approach to pareto optimization with nonsmooth convex objective functions in a general hilbert framework we consider continuous gradient like dynamical systems for constrained multiobjective optimization involving non smooth convex objective functions our approach is in the line of a previous work where was considered the case of convex di erentiable objective functions based on the yosida regularization of the subdi erential operators involved in the system we obtain the existence of strong global trajectories we prove a descent property for each objective function and the convergence of trajectories to weak pareto minima this approach provides a dynamical endogenous weighting of the objective functions applications are given to cooperative games inverse problems and numerical multiobjective optimization

the minimum description length principle in coding and modeling we review the principles of minimum description length and stochastic complexity as used in data compression and statistical modeling stochastic complexity is formulated as the solution to optimum universal coding problems extending shannon s basic source coding theorem the normalized maximized likelihood mixture and predictive codings are each shown to achieve the stochastic complexity to within asymptotically vanishing terms we assess the performance of the minimum description length criterion both from the vantage point of quality of data compression and accuracy of statistical inference context tree modeling density estimation and model selection in gaussian linear regression serve as examples

average redundancy of the shannon code for markov sources it is known that for memoryless sources the average and maximal redundancy of fixed to variable length codes such as the shannon and huffman codes exhibit two modes of behavior for long blocks it either converges to a limit or it has an oscillatory pattern depending on the irrationality or rationality respectively of certain parameters that depend on the source in this paper we extend these findings concerning the shannon code to the case of a markov source we provide a precise characterization of the convergent versus oscillatory behavior of the shannon code redundancy for a class of irreducible periodic and aperiodic markov sources these findings are obtained by analytic methods such as fourier fejer series analysis and spectral analysis of matrices

second order forward backward dynamical systems for monotone inclusion problems we begin by considering second order dynamical systems of the from ddot x t gamma t dot x t lambda t b x t 0 where b cal h rightarrow cal h is a cocoercive operator defined on a real hilbert space cal h lambda 0 infty rightarrow 0 infty is a relaxation function and gamma 0 infty rightarrow 0 infty is a damping function both depending on time for the generated trajectories we show existence and uniqueness of the generated trajectories as well as their weak asymptotic convergence to a zero of the operator b the framework allows us to address from similar perspectives second order dynamical systems associated with the problem of finding zeros of the sum of a maximally monotone operator and a cocoercive one this captures as a particular case the minimization of the sum of a nonsmooth convex function with a smooth convex one furthermore we prove that when b is the gradient of a smooth convex function the value of the latter converges along the ergo

inhomogeneous theory of dual diophantine approximation on manifolds the theory of inhomogeneous diophantine approximation on manifolds is developed in particular the notion of nice manifolds is introduced and the divergence part of the groshev type theory is established for all such manifolds our results naturally incorporate and generalize the homogeneous measure and dimension theorems for non degenerate manifolds established to date the results have natural applications beyond the standard inhomogeneous theory such as diophantine approximation by algebraic integers

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

differentiability implies continuity in neuronal dynamics recent work has identified nonlinear deterministic structure in neuronal dynamics using periodic orbit theory troublesome in this work were the significant periods of time where no periodic orbits were extracted dynamically dark regions tests for periodic orbit structure typically require that the underlying dynamics are differentiable since continuity of a mathematical function is a necessary but insufficient condition for differentiability regions of observed differentiability should be fully contained within regions of continuity we here verify that this fundamental mathematical principle is reflected in observations from mammalian neuronal activity first we introduce a null jacobian transformation to verify the observation of differentiable dynamics when periodic orbits are extracted second we show that a less restrictive test for deterministic structure requiring only continuity demonstrates widespread nonlinear deterministic structure only partially appreciated with previous approaches 2001 published by elsevier science b v

rational points near manifolds and metric diophantine approximation this work is motivated by problems on simultaneous diophantine approximation on manifolds namely establishing khintchine and jarn k type theorems for submanifolds of r n these problems have attracted a lot of interest since kleinbock and margulis proved a related conjecture of alan baker and v g sprind zuk they have been settled for planar curves but remain open in higher dimensions in this paper khintchine and jarn k type divergence theorems are established for arbitrary analytic nondegenerate manifolds regardless of their dimension the key to establishing these results is the study of the distribution of rational points near manifolds a very attractive topic in its own right here for the rst time we obtain sharp lower bounds for the number of rational points near nondegenerate manifolds in dimensions n 2 and show that they are ubiquitous that is uniformly distributed

minimax redundancy for the class of memoryless sources let x sup n x sub 1 x sub n be a memoryless source with unknown distribution on a finite alphabet of size k we identify the asymptotic minimax coding redundancy for this class of sources and provide a sequence of asymptotically minimax codes equivalently we determine the limiting behavior of the minimax relative entropy min sub qxn max sub pxn d p sub xn spl par q sub xn where the maximum is over all independent and identically distributed i i d source distributions and the minimum is over all joint distributions we show in this paper that the minimax redundancy minus k 1 2 log n 2 spl pi e converges to log spl int spl radic det i spl theta d spl theta log spl gamma 1 2 sup k spl gamma k 2 where i spl theta is the fisher information and the integral is over the whole probability simplex the bayes strategy using jeffreys prior is shown to be asymptotically maximin but not asymptotically minimax in our setting the boundary risk using jeffreys prior is higher than that of interior points we provide a sequence of modifications of jeffreys prior that put some prior mass near the boundaries of the probability simplex to pull down that risk to the asymptotic minimax level in the limit

convergence of damped inertial dynamics governed by regularized maximally monotone operators in a hilbert space setting we study the asymptotic behavior as time t goes to infinity of the trajectories of a second order differential equation governed by the yosida regularization of a maximally monotone operator with time varying positive index lambda t the dissipative and convergence properties are attached to the presence of a viscous damping term with positive coefficient gamma t a suitable tuning of the parameters gamma t and lambda t makes it possible to prove the weak convergence of the trajectories towards zeros of the operator when the operator is the subdifferential of a closed convex proper function we estimate the rate of convergence of the values these results are in line with the recent articles by attouch cabot and attouch peypouquet in this last paper the authors considered the case gamma t frac alpha t which is naturally linked to nesterov s accelerated method we unify and often improve the results already present in the literature

viscosity solutions of minimization problems viscosity methods for minimization problems are revisited from some modern perspectives in variational analysis variational convergences for sequences of functions epi convergence gamma convergence mosco convergence and for sequences of operators graph convergence provide a flexible tool for such questions it is proved in a rather large setting that the solutions of the approximate problems converge to a viscosity solution of the original problem that is a solution that is minimal among all the solutions with respect to some viscosity criteria various examples coming from mathematical programming calculus of variations semicoercive elliptic equations phase transition theory hamilton jacobi equations singular perturbations and optimal control theory are considered

coupling general penalty schemes for convex programming with the steepest descent and the proximal point algorithm we study the asymptotic behavior of the continuous flow generated by coupling the steepest descent method with a general class of penalty schemes for convex programming we establish convergence of the trajectories towards primal optimal solutions as well as convergence of some naturally associated dual paths the results are extended to the sequences generated by an implicit discretization scheme which corresponds to the coupling of an inexact proximal point iteration with the penalty schemes

minimax noiseless universal coding for markov sources upper and lower bounds are presented on the minimax redundancy for markov source noiseless block to variable universal coding the number of states block size and markov order are arbitrary but finite unlike earlier results the upper and lower bounds are absolute that is not merely asymptotic the upper bound is established by combinatorial bounds the lower bound is established by a bound on the average redundancy for an arbitrary distribution on the transition probabilities the bound can be optimized across the choice of distribution

complexity of strings in the class of markov sources shannon s self information of a string is generalized to its complexity relative to the class of finite state machine fsm defined sources unlike an earlier generalization the new one is valid for both short and long strings the definition is justified in part by a theorem stating that asymptotically the mean complexity provides a tight lower bound for the mean length of all so called regular codes this also generalizes shannon s noiseless coding theorem for a large subclass of fsm sources a simple algorithm is described for computing the complexity

flows on homogeneous spaces and diophantine approximation on manifolds we present a new approach to metric diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces this approach yields a new proof of a conjecture of mahler originally settled by v g sprindzuk in 1964 we also prove several related hypotheses of baker and sprindzuk formulated in 1970s the core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices

convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity abstract we prove convergence of global bounded and smooth solutions of the wave equation with linear dissipation and analytic nonlinearity a generalization and examples of applications are given at the end of the paper

asymptotic expansion of penalty gradient flows in linear programming we establish asymptotic expansions for nonautonomous gradient flows of the form dot u t nabla f u t r t where f x r is a penalty approximation of a linear program and the penalty parameter r t tends to 0 as t to infty under appropriate conditions we show that every integral curve satisfies u t u infty r t d 0 dot r t r t w 0 o dot r t r t for suitable vectors u infty d 0 and w 0 we deduce an asymptotic expansion for a related dual trajectory and we show that the primal dual limit point is a pair of strictly complementary optimal solutions for the linear program

primal and dual convergence of a proximal point exponential penalty method for linear programming we consider the diagonal inexact proximal point iteration frac u k u k 1 lambda k in partial varepsilon k f u k r k nu k where f x r ctx r exp a i x b i r is the exponential penalty approximation of the linear program min ctx ax b we prove that under an appropriate choice of the sequences k e k and with some control on the residual k for every r k 0 the sequence u k converges towards an optimal point u of the linear program we also study the convergence of the associated dual sequence i k exp a i u k b i r k towards a dual optimal solution