a new example of the lavrentiev phenomenon this paper describes a new example of a calculus of variations problem with the lavrentiev phenomenon a problem with different value functions for different classes of admissible trajectories such problems provide examples of nonuniqueness for hamilton jacobi equations since each value function is a solution the authors describe these solutions and the corresponding optimal trajectories and focus on how a standard proof of optimality is adapted for these examples

a numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity in this paper we describe and analyse a numerical method that detects singular minimizers and avoids the lavrentiev phenomenon for three dimensional problems in nonlinear elasticity this method extends to three dimensions the corresponding one dimensional method of ball and knowles

when is a linear multi modal system disturbance decoupled in this paper we study the question under which conditions a linear multi modal system is disturbance decoupled we establish necessary and sufficient geometric conditions from which the existing results on switched linear systems and conewise linear systems can be recovered as special cases also we apply these conditions to a class of linear complementarity systems in order to obtain a more crisp characterization

invariant subspaces for lpv systems and their applications the aim of this paper is to extend the notion of invariant subspaces known in the geometric control theory of the linear time invariant systems to the linear parameter varying lpv systems by introducing the concept of parameter varying invariant subspaces for lpv systems affine in their parameters algorithms are given to compute many parameter varying subspaces relevant in the solution of state feedback and observer design problems

dead beat functional observers for discrete time lpv systems with unknown inputs this technical note deals with functional observers for discrete time linear parameter varying lpv systems more specifically dead beat observers for reconstructing linear combinations of the state are proposed the system inputs are assumed to be unknown and the observers reduce to functions of the output over a finite number of time steps the existence of such observers is proven by resorting to the notion of inverse system together with the concepts of maximal robust invariant subspaces and nilpotent semigroups a constructive approach to derive the explicit equations of the observer is provided an example illustrates the efficiency and the computational aspects of the method

a necessary and sufficient condition for lower semicontinuity abstract it is well known that w 1 p quasiconvexity is a necessary condition for sequential weak lower semicontinuity of the variational integral i u f u x d x on the sobolev space w 1 p and that it is sufficient too provided that the integrand f satisfies suitable growth conditions related to the exponent p we show that for extended real valued integrands a closely related convexity condition defined in terms of gradient young measures is both necessary and sufficient for lower semicontinuity in the more general and flexible setting of compensated compactness our main results identify the relaxation lower semicontinuous envelope in two related situations

on regulation under sampling the paper deals with linear and nonlinear regulation under sampling it is shown that digital solutions exist under assumptions which are closely related to the existence of robust solutions to the continuous problem approximated solutions are computed starting from the continuous ones

generalized invariant subspaces for infinite dimensional systems in this paper some generalized invariant subspaces for infinite dimensional systems are investigated and then some sufficient conditions for parameter insensitive disturbance rejection problems with state feedback and with measurement output feedback to be solvable are studied

perfect elimination of regulation transients in discrete time lpv systems via internally stabilizable robust controlled invariant subspaces this note introduces a geometric solution to the problem of perfect elimination of regulation transients in discrete time linear systems subject to swift and wide a priori known parameter variations the constructive proof of the conditions for problem solvability requires a preliminary strictly geometric interpretation of the multivariable autonomous regulator problem specifically aimed at discrete time linear systems the novel concept of internal stabilizability of a robust controlled invariant subspace plays a key role in the formulation of those conditions as well as in the synthesis of the control scheme

extremal channels of gallager s e_ 0 under the basic polarization transformations we study the extremality of the binary erasure channel and the binary symmetric channel for gallager s reliability function e0 of binary input discrete memoryless channels evaluated under the uniform input distribution from the aspect of channel polarization in particular we show that amongst all binary discrete memoryless channels of a given e0 value for a fixed 0 the binary erasure channel and the binary symmetric channel are extremal in the evolution of e0 under the one step polarization transformations

regularity of relaxed minimizers of quasiconvex variational integrals with p q growth we consider autonomous integrals f u int omega f du rm d x quad rm for u mathbb r n supset omega to mathbb r n in the multidimensional calculus of variations where the integrand f is a strictly quasiconvex c2 function satisfying the p q growth conditions gamma a p leqq f a leqq gamma 1 a q quad rm for every a in mathbb r nn with exponents 1 p leqq q infty

robust disturbance decoupling problem for parameter dependent families of linear systems abstract we consider the problem of disturbance decoupling by means of a single state feedback for a whole family of systems depending on a parameter p we give a geometric necessary and sufficient condition for the existence of solution to such problem under the hypothesis that the systems of the family are generically injective and the dependence on p is polynomial or respectively the set of possible values for p is finite

computing the maximum robust controlled invariant subspace abstract an algorithm is presented to compute the maximum robust controlled invariant subspace contained in a given subspace it is based on a special computational scheme to derive the intersection of a family of subspaces which is particularly convenient when the system coefficients depend polynomially on the uncertain parameter an example is presented to illustrate the algorithm

a geometric approach to the general autonomous regulator problem in the time delay framework abstract the aim of this paper is to show the applicability of geometric techniques to a regulation problem for linear time delay systems given a plant whose dynamics equations include delays and an exosystem that generates a reference signal the problem we consider consists in finding a feedback regulator which guarantees asymptotic stability of the regulation loop and asymptotic command following of the reference signal for any initial condition of the overall system in the presence of disturbances by associating to the time delay plant a corresponding abstract system with coefficients in a ring it is possible to place our investigation in a finite dimensional algebraic context where intuition and results obtained in the classical case that is without delays may be exploited in particular using tools and methods of the geometric approach to systems with coefficients in a ring sufficient conditions for the solvability of the considered problem are found and a constructive procedure which works under specific hypotheses is given

bounds on information combining when the same data sequence is transmitted over two independent channels or when a data sequence is transmitted twice but independently over the same channel the independent observations can be combined at the receiver side from an information theory point of view the overall mutual information between the data sequence and the received sequences represents a combination of the mutual information of the two channels this concept is termed information combining a lower bound and an upper bound on the combined information is presented and it is proved that these bounds are tight furthermore this principle is extended to the computation of extrinsic information on single code bits for a repetition code and for a single parity check code of length three respectively for illustration of the concept and the bounds on information combining two applications are considered first bounds on the information processing characteristic ipc of a parallel concatenated code are derived from its extrinsic information transfer exit chart second bounds on the exit chart for an outer repetition code and for an outer single parity check code of a serially concatenated coding scheme are computed

a necessary and sufficient condition for parameter insensitive disturbance rejection problem with state feedback in this paper a necessary and sufficient condition for a parameter insensitive disturbance rejection problem with state feedback which was pointed out as an open problem by bhattacharyya to be solvable is proved a constructive algorithm of simultaneously a b invariant subspaces for a finite number of linear systems and a relationship between simultaneously a b invariant subspaces and generalized a b invariant subspaces play an important role to prove the main result

finding nicer mappings with lower or equal energy in nonlinear elasticity abstract in the framework of nonlinear elasticity we consider the model energy f u d u x p h det d u x d x where u r n r n with det d u 0 and h 0 0 is convex moreover h t blows up when t 0 we study the problem fix the mapping u with finite energy f u and find a mapping v with the same boundary values with det d v 0 and energy f v not higher than f u such that every component v of v enjoys a kind of maximum and minimum principle due to the constraint det d v 0 a truncation argument does not work on the contrary the constraint det d v 0 makes the problem easy when p n since it is known that every component is weakly monotone so we can take v u in the present work we address the case 2 p n and we give a positive answer if the blow up of h t is of logarithmic type

extremes of error exponents this paper determines the range of feasible values of standard error exponents for binary input memoryless symmetric channels of fixed capacity c and shows that extremes are attained by the binary symmetric and the binary erasure channel the proof technique also provides analogous extremes for other quantities related to gallager s e0 function such as the cutoff rate the bhattacharyya parameter and the channel dispersion

computation of maximal safe sets for switching systems the problem of determining maximal safe sets and hybrid controllers is computationally intractable because of the mathematical generality of hybrid system models given the practical and theoretical relevance of the problem finding implementable procedures that could at least approximate the maximal safe set is important to this end we begin by restricting our attention to a special class of hybrid systems switching systems we exploit the structural properties of the graph describing the discrete part of a switching system to develop an efficient procedure for the computation of the safe set this procedure requires the computation of a maximal controlled invariant set we then restrict our attention to linear discrete time systems for which there is a wealth of results available in the literature for the determination of maximal controlled invariant sets however even for this class of systems the computation may not converge in a finite number of steps we then propose to compute inner approximations that are controlled invariant and for which a procedure that terminates in a finite number of steps can be obtained a tight bound on the error can be given by comparing the inner approximation with the classical outer approximation of the maximal controlled invariant set our procedure is applied to the idle speed regulation problem in engine control to demonstrate its efficiency

lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands the study of regularity properties for minimizers of problems of the form min u u dt u e iv q r u a a u b b 1 1 has been undertaken for a long time in calculus of variations and a number of results are by now available on this subject when f t s z is a smooth coercive function problem 1 1 was considered by tonelli see 18 19 who proved that every minimizer u t of the problem 1 1 is locally lipschitz on a relatively open set of full measure in a b under some additional assumptions on f it is also possible to prove global regularity theorems see for instance cesari 3 chap 2 sect 61 in recent years much attention has been devoted to problems of the form l l on the one hand in weakening the hypotheses of tonelli s regularity results and on the other in finding counterexamples of nonregular minimizers for coercive functionals with respect to this last subject we recall that it has been shown see ball and mizel l and clarke and vinter 6 that there exist polynomial functions f t s z such that for suitable c 0 and p

extremes of information combining extreme densities for information combining are found for two important channel models the binary input symmetric parallel broadcast channel and the parity constrained input symmetric parallel channels following upper and lower mutual information thresholds are stated for per bit maximum a posteriori probability map decoding and low density parity check ldpc code ensembles

invariant dual cones for hybrid systems the concepts of invariance and duality have been analyzed and exploited in systems science and related fields both in continuous state domain and in graphs theory but we are not aware of a theory developed for hybrid systems i e dynamic systems resulting from the interaction of continuous and discrete state systems in this paper we introduce invariance concepts for hybrid cones controlled reset invariance and conditioned reset invariance we show that a reset conditioned invariant cone and a reset controlled invariant cone are related each other by a duality relation obtained by associating an appropriate hybrid system to the given one a discussion on system theoretic interpretation of these cones is provided

model matching with strong stability in switched linear systems abstract this work deals with model matching by output dynamic feedback in switched linear systems the plant and the model are assumed to be subject to different switching signals a necessary and sufficient condition for achieving model matching with asymptotic stability of both the closed loop compensated system and the compensator for a sufficiently large dwell time is proven such condition is obtained by specializing the structural condition with a suitable redefinition of the robust controlled invariant subspace involved capable of capturing not only the structural aspect of the problem but also the stability aspects the effect of the combined action of nonzero initial states and nonzero inputs is dealt with the solution to a more demanding problem formulation where the dwell time is assumed to be fixed and given is also discussed

necessary and sufficient conditions for asymptotic model matching of switching linear systems this paper investigates the problem of designing a feedback controller to force the response of a given plant to match asymptotically that of a prescribed model in case both the plant and the model are switching linear systems matching has to be achieved with stability of the feedback loop for any initial conditions of the plant the model and the controller in case it is dynamic and for any choice of the switching law solvability of the problem is characterized in terms of necessary and sufficient conditions that in part refer to the geometric structure of the difference system which compares the output of the model and of the plant stability is considered both for slow switching and for arbitrary switching proofs of the solvability conditions provide viable procedures for synthesizing the static or dynamic controllers that achieve the matching respectively when the state of the model is measurable and when it is not weaker sufficient conditions that can be practically checked by simple algorithmic procedures are provided both in the general situation and under slightly restrictive hypotheses

a numerical method for detecting singular minimizers a numerical method for computing minimizers in one dimensional problems of the calculus of variations is described such minimizers may have unbounded derivatives even when the integrand is smooth and regular in such cases because of the lavrentiev phenomenon standard finite element methods may fail to converge to a minimizer the scheme proposed is shown to converge to an absolute minimizer and is tested on an example the effect of quadrature is analyzed the implications for higher dimensional problems and in particular for fracture in nonlinear elasticity are discussed

detection filter design for lpv systems a geometric approach this paper investigates fault detection and isolation of linear parameter varying lpv systems by using parameter varying c a invariant subspace and parameter varying unobservability subspaces the so called detection filter approach formulated as the fundamental problem of residual generation fprg for linear time invariant lti systems is extended for a class of lpv systems the question of stability is addressed in the terms of lyapunov quadratic stability by using linear matrix inequalities the results are applied to the model of a generic small commercial aircraft

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

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

on location observability notions for switching systems abstract the focus of this paper is on the analysis of initial discrete state distinguishability notions for switching systems in a discrete time setting moreover the relationship between initial discrete state distinguishability and the problem of reconstructing the current discrete state is addressed

on universal properties of capacity approaching ldpc code ensembles this paper is focused on the derivation of some universal properties of capacity approaching low density parity check ldpc code ensembles whose transmission takes place over memoryless binary input output symmetric mbios channels properties of the degree distributions graphical complexity and the number of fundamental cycles in the bipartite graphs are considered via the derivation of information theoretic bounds these bounds are expressed in terms of the target block bit error probability and the gap in rate to capacity most of the bounds are general for any decoding algorithm and some others are proved under belief propagation bp decoding proving these bounds under a certain decoding algorithm validates them automatically also under any suboptimal decoding algorithm a proper modification of these bounds makes them universal for the set of all mbios channels which exhibit a given capacity bounds on the degree distributions and graphical complexity apply to finite length ldpc codes and to the asymptotic case of an infinite block length the bounds are compared with capacity approaching ldpc code ensembles under bp decoding and they are shown to be informative and are easy to calculate finally some interesting open problems are considered

regularity properties of solutions to the basic problem in the calculus of variations this paper concerns the basic problem in the calculus of variations minimize a functional j defined by j x j l t x t x t dt a over a class of arcs x whose values at a and b have been specified existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs conditions which must be strengthened in order that the standard necessary conditions apply the question arises what necessary conditions hold merely under hypotheses of existence theory say the classical tonelli conditions it is shown that given a solution x there exists a relatively open subset q of a b of full measure on which x is locally lipschitz and satisfies a form of the euler lagrange equation the main theorem of which this is a corollary can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions three such classes are identified and results of bernstein tonelli and morrey are extended one of these classes is of a novel nature and its study implies the new result that when l is independent of t the solution has essentially bounded derivative

generalized controlled and conditioned invariances for linear ω periodic discrete time systems abstract in this paper the three notions of generalized controlled a b invariance generalized conditioned c a invariance and generalized a b c invariance for uncertain linear periodic discrete time systems are studied and then the parameter insensitive disturbance rejection problems with static output feedback and or with state feedback are formulated and their solvability conditions are presented

generalized partial orders for polar code bit channels we study partial orders pos for the synthesized bit channels of polar codes first we give an alternative proof of an existing po for bit channels with the same hamming weight and use the underlying idea to extend the bit channel ordering to some additional cases in particular the bit channel ordering for a given code block length is used to generate additional bit channel ordering relationships for larger block lengths generalizing previously known pos next we consider pos especially for the binary erasure channel bec we identify a symmetry property of the bhattacharyya parameters of complementary bit channel pairs on the bec and provide a condition for the alignment of polarized sets of bit channels for the bec and general binary input memoryless symmetric bms channels numerical examples and further properties about the pos for the bit channels with different hamming weights are provided to illustrate the new pos the bit channels with universal ordering positions which are independent of the channel erasure probability are verified for all of the code block lengths finally we show the threshold behavior of the bhattacharyya parameters of some bit channels by approximating the threshold values the corresponding value for a bit channel can be used to determine whether it is good or bad when the underlying channel is known

noncooperative disturbance decoupling a theory for disturbance decoupling problems has been well developed in the area of geometric control theory the aim of the present study is to introduce disturbance decoupling problems in a dynamic game context a noncooperative version of the disturbance decoupling problem is introduced and two different information structures are studied easily verifiable necessary and sufficient conditions for both problem types are obtained

higher integrability for constrained minimizers of integral functionals with p q growth in low dimension abstract we prove higher summability for the gradient of minimizers of strongly convex integral functionals of the calculus of variations f x d u x d x u r n s n 1 with growth conditions of p q type p f x c q 1 p q in low dimension our procedure is set in the framework of fractional sobolev spaces and renders the desired regularity as the result of an approximation technique relying on estimates obtained through a careful use of difference quotients

a geometric approach for structured systems application to disturbance decoupling structured systems described by state space models are considered for these systems the entries of the state space model matrices are supposed to be either fixed zeros or free independent parameters for such systems one can study structural properties i e properties that are valid for almost any value of the parameters we revisit the classical geometric theory in the context of structured systems in particular the well known notions of a b invariant and c a invariant subspaces are redefined and analysed in this context we characterize fixed subspaces included or containing these invariant subspaces and show that they play a key role in solving control problems as an application of this geometric approach for structured systems we propose new solutions to the disturbance decoupling problems by state feedback and by output measurement feedback the solvability conditions are easily checked on the associated graph of the system

interpretation of lavrentiev phenomenon by relaxation the higher order case we consider integral functionals of the calculus of variations of the form f u fj f x u u u n dx defined for u e wn 0 1 and we show that the relaxed functional f with respect to the weak woc 1 0 1 convergence can be written as i f u j xu u n dx l u where the additional term l u the lavrentiev gap is explicitly identified in terms of f

performance bounds of concatenated polar coding schemes a concatenated coding scheme over binary memoryless symmetric bms channels using a polarization transformation followed by outer sub codes is analyzed achievable error exponents and upper bounds on the error rate are derived the first bound is obtained using outer codes which are typical random linear codes as a byproduct of this bound it determines the rates of the outer codes a lower bound on the error exponent that holds for all bms channels with a given capacity is then derived improved bounds and approximations for finite blocklength codes using channel dispersions normal approximation as well as converse and approximate converse results are also obtained the bounds are compared with actual simulation results from the literature for the cases considered when transmitting over the binary input additive white gaussian noise channel there was only a small gap between the channel dispersion based approximation and the actual error rate of concatenated bch polar codes

on invariant sets for constrained discrete time linear systems with disturbances and parametric uncertainties we consider discrete time linear systems with disturbances and parametric uncertainties in the matrix a with constraints on both control and state variables we give the explicit expression of maximal sets of disturbances and or parameters such that it is possible to control the state trajectory of the system to the given constraining set the case of static linear state feedback control is analyzed

interpretation of the lavrentiev phenomenon by relaxation we consider functionals of the calculus of variations of the form f u 01 f x u u dx defined for u w1 0 1 and we show that the relaxed functional f with respect to weak w1 1 0 1 convergence can be written as f u 01 f x u u dx l u where the additional term l u called the lavrentiev term is explicitly identified in terms of f

the output regulation problem with stability for linear switching systems a geometric approach abstract this work presents a solution for the output regulation problem with quadratic stability under arbitrary switching in linear switching systems the extension to other stability requirements like asymptotic stability in particular is also considered and it is shown to affect only few specific features of the proposed solution the main reason is that the geometric approach which is at the basis of the developed methodology establishes a neat separation between the structural aspects and the stability aspects of the problem for the same reason continuous time systems and discrete time systems are given a unified treatment as far as the structural issues are concerned while different technicalities characterize the discussion of the stability issues

disturbance decoupling with closed loop modes stability in switched linear systems this work deals with the problem of rendering the output of a switched linear system insensitive to a disturbance input with the requirement that the dynamics of the closed loop system be exponentially stable for all switching signals with a sufficiently large dwell time a necessary and sufficient condition for the problem to have a solution is shown a complete computational procedure to check the solvability condition and to synthesize the compensation scheme is provided a worked out example is presented with the aim of showing the effectiveness of the devised method

on the robust controlled invariant abstract the robust controlled invariant is defined as a tool to characterize systems subject to parameter changes and to set up a geometric approach to robustness in the large in multi variable control problems as a typical application the robust disturbance decoupling problem an extended version of the well known disturbance decoupling is presented and a necessary and sufficient condition for it to have a solution is derived

the disturbance decoupling problem with stability for switching dynamical systems abstract the disturbance decoupling problem with stability is dealt with by means of the geometric approach for switching systems the existence of feedbacks which decouple the disturbance and at the same time assure stability is difficult to characterize since the action of the feedback couples with that of the switching law under suitable conditions it is shown that the above requirement can be dealt with in separate ways and this allows us to state a checkable necessary condition and on that basis also a sufficient condition for solvability of the problem

brief a polytopic game we consider a discrete time linear dynamic game in state space form where the actions of two players are confined to belong to prescribed convex polytopes at all times each player has a home set a polyhedral domain in state space and should devise a cooperative or a non cooperative strategy contingent on the known initial state to reach her own home set in a finite number of steps and stay there irrespective of the action taken by the opponent players act independently on the basis of the state at each time using feedback strategies purpose of the analysis is to classify the set of initial states into winning sets and draw sets for each player and prescribe strategies able to implement the kind of invariance home sets should enjoy in the paper necessary and sufficient conditions for the existence of invariant sets with the above properties are derived these conditions and the relative strategies are characterized in terms of linear programming problems that provide a straightforward algorithmic solution to the game

singular minimizers for regular one dimensional problems in the calculus of variations i i f x u x u x dx ja where a b is a finite interval denotes d dx f f x u p is c 0 and fpp 0 regularity we consider the problem of minimizing in the set a of absolutely continuous functions u a b r satisfying u a a u b 3 where a and 5 are given constants as is well known if a minimizer u of i in a is lipschitz continuous then u is smooth and satisfies the euler lagrange equation

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

disturbance decoupling with quadratic stability for switched linear systems in this paper disturbance decoupling problems without stability and with quadratic stability for switched linear systems are formulated in the framework of the so called geometric approach firstly necessary and sufficient conditions for the problem without stability to be solvable are given secondly sufficient conditions for the problem with quadratic stability to be solvable are given further for switched linear systems composed of two subsystems necessary and sufficient conditions for the problem with quadratic stability to be solvable are also investigated finally an illustrative example is shown

on the lavrentiev phenomenon for multiple integral scalar variational problems abstract we prove the non occurrence of lavrentiev gaps between lipschitz and sobolev functions for functionals of the form i u f u u u when r n r is lipschitz and belongs to a wide class of open bounded sets in r n containing lipschitz domains the lagrangian f is assumed to be either convex in both variables or a sum of functions f s a s g b s with g convex and s a s g 0 b s satisfying a non oscillatory condition at infinity we thus derive the non occurrence of the lavrentiev phenomenon for unnecessarily convex functionals of the gradient no growth conditions are assumed

random coding techniques for nonrandom codes this work provides techniques to apply the channel coding theorem and the resulting error exponent which was originally derived for totally random block code ensembles to ensembles of codes with less restrictive randomness demands as an example the random coding technique can even be applied for an ensemble that contains a single code for a specific linear code we get an upper bound for the error probability which equals gallager s 1968 random coding bound up to a factor determined by the maximum ratio between the weight distribution of the code and the expected random weight distribution

channel polarization a method for constructing capacity achieving codes for symmetric binary input memoryless channels a method is proposed called channel polarization to construct code sequences that achieve the symmetric capacity i w of any given binary input discrete memoryless channel b dmc w the symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability channel polarization refers to the fact that it is possible to synthesize out of n independent copies of a given b dmc w a second set of n binary input channels wn i 1 les i les n such that as n becomes large the fraction of indices i for which i wn i is near 1 approaches i w and the fraction for which i wn i is near 0 approaches 1 i w the polarized channels wn i are well conditioned for channel coding one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining codes constructed on the basis of this idea are called polar codes the paper proves that given any b dmc w with i w 0 and any target rate r i w there exists a sequence of polar codes cfrn nges1 such that cfrn has block length n 2n rate ges r and probability of block error under successive cancellation decoding bounded as pe n r les o n 1 4 independently of the code rate this performance is achievable by encoders and decoders with complexity o n logn for each