polar codes for broadcast channels polar codes are introduced for discrete memoryless broadcast channels for m user deterministic broadcast channels polarization is applied to map uniformly random message bits from m independent messages to one codeword while satisfying broadcast constraints the polarization based codes achieve rates on the boundary of the private message capacity region for two user noisy broadcast channels polar implementations are presented for two information theoretic schemes 1 cover s superposition codes and 2 marton s codes due to the structure of polarization constraints on the auxiliary and channel input distributions are identified to ensure proper alignment of polarization indices in the multiuser setting the codes achieve rates on the capacity boundary of a few classes of broadcast channels e g binary input stochastically degraded the complexity of encoding and decoding is o n log n where n is the block length in addition polar code sequences obtain a stretched exponential decay of o 2 n beta of the average block error probability where 0 reproducible experiments for finite block lengths n 512 1024 2048 corroborate the theory

an existence result for non smooth vibro impact problems we are interested in mechanical systems with a finite number of degrees of freedom submitted to frictionless unilateral constraints we consider the case of a convex non smooth set of admissible positions given by k q mathbb r d phi alpha q geq 0 1 leq alpha leq v v geq 1 and we assume inelastic shocks at impacts we propose a time discretization of the measure differential inclusion which describes the dynamics and we prove the convergence of the approximate solutions to a limit motion which satisfies the constraints moreover if the geometric properties ensuring continuity on data hold at the limit we show that the transmission of velocities at impacts follows the inelastic shocks rule

list decoding of polar codes we describe a successive cancellation list decoder for polar codes which is a generalization of the classic successive cancellation decoder of arikan in the proposed list decoder l decoding paths are considered concurrently at each decoding stage where l is an integer parameter at the end of the decoding process the most likely among the l paths is selected as the single codeword at the decoder output simulations show that the resulting performance is very close to that of maximum likelihood decoding even for moderate values of l alternatively if a genie is allowed to pick the transmitted codeword from the list the results are comparable with the performance of current state of the art ldpc codes we show that such a genie can be easily implemented using simple crc precoding the specific list decoding algorithm that achieves this performance doubles the number of decoding paths for each information bit and then uses a pruning procedure to discard all but the l most likely paths however straightforward implementation of this algorithm requires omega l n 2 time which is in stark contrast with the o n log n complexity of the original successive cancellation decoder in this paper we utilize the structure of polar codes along with certain algorithmic transformations in order to overcome this problem we devise an efficient numerically stable implementation of the proposed list decoder that takes only o l n log n time and o l n space

spatio temporal bifurcations in plasma drift waves abstract experimental data from an experiment on drift waves in plasma is presented the experiment provides a space time diagnostic and has a control parameter that permits the study of the transition from a stable plasma to a turbulent plasma the biorthogonal decomposition is used to analyse the data we introduce the notion of complex modulation for two dimensional systems we decompose the real physical system into complex modulated monochromatic travelling waves and give a simple model describing the speed doubling observed in the data as the control parameter increases

spatiotemporal chaos in nonintegrable three wave interactions abstract the nonliear three wave interaction is ubiquitous having many physical manifestations the conservative equations are integrable by inverse scattering transforms and can possess soliton solutions two nonintegrable three wave models were numerically simulated spatiotemporal chaos where coherent structures evolve and interact chaotically was observed in the long time large system limit correlation functions were measured the behaviour was analysed by using perturbation theory around the inverse scattering transform solutions of the nonperturbed integrable equations saturation amplitudes correlation times and lengths were estimated

time stepping approximation of rigid body dynamics with perfect unilateral constraints i the inelastic impact case we consider a discrete mechanical system with a non trivial mass matrix subjected to perfect unilateral constraints described by the geometrical inequalities f alpha q geqq 0 alpha in 1 dots nu nu geqq 1 we assume that the transmission of the velocities at impact is governed by newton s law with a coefficient of restitution e 0 so that the impact is inelastic we propose a time discretization of the second order differential inclusion describing the dynamics which generalizes the scheme proposed in paoli j differ equ 211 247 281 2005 and for any admissible data we prove the convergence of approximate motions to a solution of the initial value problem

polar codes for the two user multiple access channel in this paper polar codes for the m user multiple access channel mac with binary inputs are constructed it is shown that arikan s polarization technique applied individually to each user transforms independent uses of an m user binary input mac into successive uses of extremal macs this transformation has a number of desirable properties 1 the uniform sum rate of the original mac is preserved 2 the extremal macs have uniform rate regions that are not only polymatroids but matroids and thus 3 their uniform sum rate can be reached by each user transmitting either uncoded or fixed bits in this sense they are easy to communicate over a polar code can then be constructed with an encoding and decoding complexity of o n log n where n is the block length a block error probability of o exp n1 2 e and capable of achieving the uniform sum rate of any binary input mac with arbitrary many users applications of this polar code construction to channels with a finite field input alphabet and to the additive white gaussian noise channel are also discussed

relaxed polar codes polar codes are the latest breakthrough in coding theory as they are the first family of codes with explicit construction that provably achieve the symmetric capacity of binary input discrete memoryless channels polar encoding and successive cancellation decoding have the complexities of n log n for code length n although the complexity bound of n log n is asymptotically favorable we report in this work methods to further reduce the encoding and decoding complexities of polar coding the crux is to relax the polarization of certain bit channels without performance degradation we consider schemes for relaxing the polarization of both very good and very bad bit channels in the process of channel polarization relaxed polar codes are proved to preserve the capacity achieving property of polar codes analytical bounds on the asymptotic and finite length complexity reduction attainable by relaxed polarization are derived for binary erasure channels we show that the computation complexity can be reduced by a factor of six while preserving the rate and error performance we also show that relaxed polar codes can be decoded with significantly reduced latency for additive white gaussian noise channels with medium code lengths we show that relaxed polar codes can have lower error probabilities than conventional polar codes while having reduced encoding and decoding computation complexities

performance limits and practical decoding of interleaved reed solomon polar concatenated codes a scheme for concatenating the recently invented polar codes with non binary mds codes as reed solomon codes is considered by concatenating binary polar codes with interleaved reed solomon codes we prove that the proposed concatenation scheme captures the capacity achieving property of polar codes while having a significantly better error decay rate we show that for any e 0 and total frame length n the parameters of the scheme can be set such that the frame error probability is less than 2 n 1 e while the scheme is still capacity achieving this improves upon 2 n 0 5 e the frame error probability of arikan s polar codes the proposed concatenated polar codes and arikan s polar codes are also compared for transmission over channels with erasure bursts we provide a sufficient condition on the length of erasure burst which guarantees failure of the polar decoder on the other hand it is shown that the parameters of the concatenated polar code can be set in such a way that the capacity achieving properties of polar codes are preserved we also propose decoding algorithms for concatenated polar codes which significantly improve the error rate performance at finite block lengths while preserving the low decoding complexity

spatiotemporal analysis of complex signals theory and applications we present a space time description of regular and complex phenomena which consists of a decomposition of a spatiotemporal signal into orthogonal temporal modes that we call chronos and orthogonal spatial modes that we call topos this permits the introduction of several characteristics of the signal three characteristic energies and entropies one temporal one spatial and one global and a characteristic dimension although the technique is general we concentrate on its applications to hydrodynamic problems specifically the transition to turbulence we consider two cases of application a coupled map lattice as a dynamical system model for spatiotemporal complexity and the open flow instability on a rotating disk in the latter we show a direct relation between the global entropy and the different instabilities that the flow undergoes as reynolds number increases

dynamical systems with active singularities of elastic type a modeling and controller synthesis framework a new class of systems characterized by admitting impulsive control action within their singular motion phases such as dimension changes state discontinuities and other irregularities is introduced this class termed dynamical systems with active or controlled singularities encompasses various applications with sensing and or actuation ultra fast in comparison with the natural system time scale including mechanisms with impact induced motion power and sensor networks under faults fast positioning devices smart skins and switching electronic circuits the present work focuses on systems with impact type singularities induced through system interaction with controlled or active state constraints the latter are assumed to be characterized by parameter dependent elastic type constraint violation that vanishes in the limit as the parameter value tends to infinity a physically well justified representation of this class of systems is proposed but found to be not well suited for controller synthesis in the singular phase to address this problem two equations are derived one describing controlled rigid impacts in auxiliary stretched time and compatible with the regular control design techniques termed controlled infinitesimal dynamics equation and another incorporating controlled impact dynamics in the infinitesimal form in terms of the shift operator along the trajectories of the first equation termed the limit model the latter is demonstrated to generate a unique isolated discontinuous system motion i e to provide a tight and well behaved description of the collision with rigid constraint it is then shown that the corresponding paths generated by the original physically based and the limit representations can be made arbitrarily close to each other uniformly except possibly in the vicinities of the jump points by the appropriate choice of the value of the constraint violation parameter this is shown to permit enforcing for sufficiently large values of the parameter the desired limit system behavior onto the original system by simply taking the control signals found through the limit representation time rescaling them and inserting the resulting signals directly into the original system these features show that the procedure developed in fact provides a well posed controller synthesis framework for the class of systems considered using this framework a ball racket rotationally controlled impact representation is developed and on its basis a singular phase control signal in the form of racket rotation velocity is designed demonstrating that the soft racket provides the specified bounce off angle increment under a noticeably lower racket rotation velocity and a longer rotation phase than the hard racket

achieving the uniform rate region of general multiple access channels by polar coding we consider the problem of polar coding for transmission over m user multiple access channels in the proposed scheme all users encode their messages using a polar encoder while a multiuser successive cancellation decoder is deployed at the receiver the encoding is done separately across the users and is independent of the target achievable rate for the code construction the positions of information bits and frozen bits for each of the users are decided jointly this is done by treating the polar transformations across all the m users as a single polar transformation with a certain polarization base we characterize the resolution of achievable rates on the dominant face of the uniform rate region in terms of the number of users m and the length of the polarization base l in particular we prove that for any target rate on the dominant face there exists an achievable rate also on the dominant face within the distance at most frac m 1 sqrt m l from the target rate we then prove that the proposed mac polar coding scheme achieves the whole uniform rate region with fine enough resolution by changing the decoding order in the multiuser successive cancellation decoder as l and the code block length n grow large the encoding and decoding complexities are o n log n and the asymptotic block error probability of o 2 n 0 5 epsilon is guaranteed examples of achievable rates for the 3 user multiple access channel are provided

how to construct polar codes a method for efficiently constructing polar codes is presented and analyzed although polar codes are explicitly defined straightforward construction is intractable since the resulting polar bit channels have an output alphabet that grows exponentially with the code length thus the core problem that needs to be solved is that of faithfully approximating a bit channel with an intractably large alphabet by another channel having a manageable alphabet size we devise two approximation methods which sandwich the original bit channel between a degraded and an upgraded version thereof both approximations can be efficiently computed and turn out to be extremely close in practice we also provide theoretical analysis of our construction algorithms proving that for any fixed e 0 and all sufficiently large code lengths n polar codes whose rate is within e of channel capacity can be constructed in time and space that are both linear in n

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

efficient design and decoding of polar codes polar codes are shown to be instances of both generalized concatenated codes and multilevel codes it is shown that the performance of a polar code can be improved by representing it as a multilevel code and applying the multistage decoding algorithm with maximum likelihood decoding of outer codes additional performance improvement is obtained by replacing polar outer codes with other ones with better error correction performance in some cases this also results in complexity reduction it is shown that gaussian approximation for density evolution enables one to accurately predict the performance of polar codes and concatenated codes based on them

polar codes characterization of exponent bounds and constructions polar codes were recently introduced by arikan they achieve the symmetric capacity of arbitrary binary input discrete memoryless channels under a low complexity successive cancellation decoding scheme the original polar code construction is closely related to the recursive construction of reed muller codes and is based on the 2 2 matrix 1 0 1 1 it was shown by arikan telatar that this construction achieves an error exponent of 1 2 i e that for sufficiently large blocklengths the error probability decays exponentially in the square root of the blocklength it was already mentioned by arikan that in principle larger matrices can be used to construct polar codes in this paper it is first shown that any l l matrix none of whose column permutations is upper triangular polarizes binary input memoryless channels the exponent of a given square matrix is characterized upper and lower bounds on achievable exponents are given using these bounds it is shown that there are no matrices of size smaller than 15 15 with exponents exceeding 1 2 further a general construction based on bch codes which for large i achieves exponents arbitrarily close to 1 is given at size 16 16 this construction yields an exponent greater than 1 2

rate splitting multiple access for discrete memoryless channels it is shown that the encoding decoding problem for any asynchronous m user discrete memoryless multiple access channel can be reduced to corresponding problems for at most 2m 1 single user discrete memoryless channels this result which extends a similar result for gaussian channels reduces the seemingly hard task of finding good multiple access codes to the much better understood task of finding good codes for single user channels as a by product some interesting properties of the capacity region of m user asynchronous discrete memoryless channels are derived

penalty approximation for non smooth constraints in vibroimpact abstract we examine the penalty approximation of the free motion of a material point in an angular domain we choose an over damped penalty approximation and we prove that if the first impact point is not at the vertex then the limit of the approximation exists and is described by moreau s rule for inelastic impacts the proofs rely on validated asymptotics and use some classical tools of the theory of dynamical systems

optimal control problems in hybrid systems with active singularities abstract the optimal control problem for systems with controlled unilateral phase constraints is considered the definition of the generalized solutions is introduced the transformation method for the original optimal control problem within the class of generalized solution to a standard optimal control problem is proposed and the necessary optimality conditions are found

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