permutation decoding and the stopping redundancy hierarchy of cyclic and extended cyclic codes we introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the tradeoff between performance and complexity of iterative decoding for the binary erasure channel we derive lower and upper bounds for the stopping redundancy hierarchy via lovasz s local lemma lll and bonferroni type inequalities and specialize them for codes with cyclic parity check matrices based on the observed properties of parity check matrices with good stopping redundancy characteristics we develop a novel decoding technique termed automorphism group decoding that combines iterative message passing and permutation decoding we also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size simulation results demonstrate that for a large number of algebraic codes the performance of the new decoding method is close to that of maximum likelihood ml decoding

an efficient algorithm to find all small size stopping sets of low density parity check matrices in this work we introduce an efficient algorithm to find all stopping sets of size less than some threshold of a fixed low density parity check ldpc matrix the solution is inspired by the algorithm proposed by rosnes and ytrehus in 2005 to find an exhaustive list of all small size turbo stopping sets in a turbo code the efficiency of the proposed algorithm is demonstrated by several numerical examples for instance we have applied the algorithm to the well known 3 5 regular 155 64 tanner code and found all stopping sets of size at most 18 in about 1 min on a standard desktop computer also we have verified that the minimum stopping set size of the 4896 2474 ramanujan margulis code is indeed 24 and that the corresponding multiplicity is exactly 204 furthermore we have applied the algorithm to the ieee 802 16e ldpc codes and determined the minimum stopping set size and the corresponding multiplicity exactly for these codes finally as an application we present a greedy algorithm to find a small number of redundant parity checks to add to the original parity check matrix in order to remove all stopping sets in the corresponding tanner graph of size less than the minimum distance an extensive case study of the 155 64 tanner code illustrates the usefulness of the algorithm and we present a 110 times 155 redundant parity check matrix for this code with no stopping sets of size less than the minimum distance simulation results of iterative decoding on the binary erasure channel show performance improvements for low to medium erasure probabilities when this redundant parity check matrix is used for decoding

tree based construction of ldpc codes having good pseudocodeword weights we present a tree based construction of low density parity check ldpc codes that have minimum pseudocodeword weight equal to or almost equal to the minimum distance and perform well with iterative decoding the construction involves enumerating a d regular tree for a fixed number of layers and employing a connection algorithm based on permutations or mutually orthogonal latin squares to close the tree methods are presented for degrees d ps and d ps 1 for p a prime one class corresponds to the well known finite geometry and finite generalized quadrangle ldpc codes the other codes presented are new we also present some bounds on pseudocodeword weight for p ary ldpc codes treating these codes as p ary ldpc codes rather than binary ldpc codes improves their rates minimum distances and pseudocodeword weights thereby giving a new importance to the finite geometry ldpc codes where p 2

the trapping redundancy of linear block codes we generalize the notion of the stopping redundancy in order to study the smallest size of a trapping set in tanner graphs of linear block codes in this context we introduce the notion of the trapping redundancy of a code which quantifies the relationship between the number of redundant rows in any parity check matrix of a given code and the size of its smallest trapping set trapping sets with certain parameter sizes are known to cause error floors in the performance curves of iterative belief propagation bp decoders and it is therefore important to identify decoding matrices that avoid such sets bounds on the trapping redundancy are obtained using probabilistic and constructive methods and the analysis covers both general and elementary trapping sets numerical values for these bounds are computed for the 2640 1320 margulis code and the class of projective geometry codes and compared with some new code specific trapping set size estimates

on the stopping redundancy of reed muller codes the stopping redundancy of the code is an important parameter which arises from analyzing the performance of a linear code under iterative decoding on a binary erasure channel in this paper we will consider the stopping redundancy of reed muller codes and related codes let r lscr m be the reed muller code of length 2m and order lscr schwartz and vardy gave a recursive construction of parity check matrices for the reed muller codes and asked whether the number of rows in those parity check matrices is the stopping redundancy of the codes we prove that the stopping redundancy of r m 2 m which is also the extended hamming code of length 2m is 2m 1 and thus show that the recursive bound is tight in this case we prove that the stopping redundancy of the simplex code equals its redundancy several constructions of codes for which the stopping redundancy equals the redundancy are discussed we prove an upper bound on the stopping redundancy of r 1 m this bound is better than the known recursive bound and thus gives a negative answer to the question of schwartz and vardy

luenberger boundary observer synthesis for sturm liouville systems a static luenberger observer of a system with sturm liouville operator is synthesised with the aid of a boundary control formulation to this aim approximate observability detectability and stability of the system is studied and design results are worked out for a typical biochemical case study

dynamical analysis of distributed parameter tubular reactors this paper is dedicated to the dynamical analysis of tubular reactor models namely plug flow and axial dispersion reactors involving sequential reactions for which the kinetics only depends on the concentrations of the reactants involved in the reaction this analysis is performed by describing these models as infinite dimensional state space systems with bounded observation and control operators it is shown that these systems are exponentially stable and that a the plug flow reactor model is observable when the component concentrations are measured at the reactor output and observable with respect to its physical domain when only the product concentration is measured and it is reachable with respect to its physical domain when it is controlled at the reactor input and b any finite set of dominant modes of the axial dispersion reactor model is observable when either all the component concentrations or only the product concentration are measured and reachable by using appropriate sensors and actuators the dynamical properties of related finite dimensional models are also discussed

trajectory analysis of nonisothermal tubular reactor nonlinear models the existence and uniqueness of the state trajectories temperature and reactant concentration are analyzed for nonisothermal plug flow and axial dispersion tubular reactor models it is mainly shown that these trajectories exist on the whole nonnegative real time axis and the set of all physically feasible state values is invariant under the dynamics equations the main nonlinearity in the model originates from the arrhenius type kinetics term in the model equations the analysis essentially uses lipschitz and dissipativity properties of the nonlinear operator involved in the dynamics and the concept of state trajectory positivity

average stopping set weight distributions of redundant random ensembles in this paper redundant random ensembles are defined and their average stopping set ss weight distributions are analyzed a redundant random ensemble consists of a set of binary matrices with linearly dependent rows these linearly dependent rows significantly reduce the number of stopping sets ss of small size upper and lower bounds on the average ss weight distribution of the redundant random ensembles are proved based on a combinatorial argument asymptotic forms of these bounds reveal asymptotic behavior of the average ss weight distributions from these bounds a tradeoff between the number of redundant rows corresponding to decoding complexity of belief propagation on binary erasure channel and the average ss weight distribution corresponding to decoding performance can be derived

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

discontinuous feedback stabilization of minimum phase semilinear infinite dimensional systems with application to chemical tubular reactor this paper develops discontinuous control methods for minimum phase semilinear infinite dimensional systems driven in a hilbert space the control algorithms presented ensure asymptotic stability global or local accordingly as state feedback or output feedback is available as well as robustness of the closed loop system against external disturbances with the a priori known norm bounds the theory is applied to stabilization of chemical processes around prespecified steady state temperature and concentration profiles corresponding to a desired coolant temperature two specific cases a plug flow reactor and an axial dispersion reactor governed by hyperbolic and parabolic partial differential equations of first and second order respectively are under consideration to achieve a regional temperature feedback stabilization around the desired profiles with the region of attraction containing a prescribed set of interest a component concentration observer is constructed and included into the closed loop system so that there is no need for measuring the process component concentration which is normally unavailable in practice performance issues of the discontinuous feedback design are illustrated in a simulation study of the plug flow reactor

stopping sets of algebraic geometry codes stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code let c be an n k linear code over fq with parity check matrix h where the rows of h may be dependent let n 1 2 n denote the set of column indices of h a stopping set s of c with parity check matrix h is a subset of n such that the restriction of h to s does not contain a row of weight 1 the stopping set distribution ti h i 0n enumerates the number of stopping sets with size i of c with parity check matrix h denote h the parity check matrix consisting of all the nonzero codewords in the dual code c in this paper we study stopping sets and stopping set distributions of some residue algebraic geometry ag codes with parity check matrix h first we give two descriptions of stopping sets of residue ag codes for the simplest ag codes i e the generalized reed solomon codes it is easy to determine all the stopping sets then we consider the ag codes from elliptic curves we use the group structure of rational points of elliptic curves to present a complete characterization of stopping sets then the stopping sets the stopping set distribution and the stopping distance of the ag code from an elliptic curve are reduced to the search counting and decision versions of the subset sum problem in the group of rational points of the elliptic curve respectively finally for some special cases we determine the stopping set distributions of the ag codes from elliptic curves

improved upper bounds on stopping redundancy for a linear block code with minimum distance d its stopping redundancy is the minimum number of check nodes in a tanner graph representation of the code such that all nonempty stopping sets have size d or larger we derive new upper bounds on stopping redundancy for all linear codes in general and for maximum distance separable mds codes specifically and show how they improve upon previous results for mds codes the new bounds are found by upper bounding the stopping redundancy by a combinatorial quantity closely related to turan numbers the turan number t v k t is the smallest number of t subsets of a v set such that every k subset of the v set contains at least one of the t subsets asymptotically we show that the stopping redundancy of mds codes with length n and minimum distance d 1 is t n d 1 d 2 1 o n 1 for fixed d and is at most t n d 1 d 2 3 o n 1 for fixed code dimension k n d 1 for d 3 4 we prove that the stopping redundancy of mds codes is equal to t n d 1 d 2 for which exact formulas are known for d 5 we show that the stopping redundancy of mds codes is either t n 4 3 or t n 4 3 1

competition in the unstirred chemostat with periodic input and washout the model of an unstirred chemostat is generalized to that of a chemostat with time dependent input washout rates the novelty of the new model is that time periodicity appears in the boundary conditions the asymptotic dynamics of the competition between two microbial populations is determined in terms of the corresponding period map which is shown to preserve the standard competitive ordering it is shown that the dynamics of competition is similar to that of a chemostat with constant boundary conditions simple criteria for coexistence versus competitive exclusion are presented 1 introduction the chemostat represents a basic model of an open system in microbial ecology in its simplest form it consists of three vessels the rst called the feed bottle contains medium with all of the nutrients needed for growth in sur plus except one which hereafter is simply called the nutrient the contents of the feed bottle are pumped at a constant rate into the second vessel called the culture vessel or bioreactor the culture vessel is charged with one or more populations of microorganisms the contents of the culture vessel are pumped into the remaining vessel called the over ow vessel at a constant rate keeping the volume of the reactor constant the organisms compete for the nutrient in a purely exploitative manner basic assumptions include that the vessel is well mixed and that all other parame ters ph temperature etc are strictly controlled the ow rate is assumed to be sucient to preclude wall growth or the accumulation of metabolic products let s t denote the concentration of the nutrient in the culture vessel and xi t i 1 2 denote the concentration of the competitors lets 0 denote the concentration of the input nutrient and let d denote the dilution rate ow rate volume if growth is assumed to be proportional to consumption then the basic equations take the form

periodic trajectories of distributed parameter biochemical systems with time delay abstract this paper deals with a model of a biochemical reactor system with distributed parameters and with a time delay in the growth response time delay has been introduced in microbial growth systems to explain the time lapse between the consumption of liquid substrate and its conversion to solid biomass we study here the properties of the resulting system of partial functional differential equations we first prove the existence positivity and a compactness property of the system trajectories we then prove the existence of periodic solutions of the system for large values of the delay numerical simulations illustrate the existence of such solutions

asymptotic stability of infinite dimensional semilinear systems application to a nonisothermal reactor the concept of asymptotic stability is studied for a class of infinite dimensional semilinear banach state space distributed parameter systems asymptotic stability criteria are established which are based on the concept of strictly m dissipative operator these theoretical results are applied to a nonisothermal plug flow tubular reactor model which is described by semilinear partial differential equations derived from mass and energy balances in particular it is shown that under suitable conditions on the model parameters some equilibrium profiles are asymptotically stable equilibriums of such model

parameterization of positively stabilizing feedbacks for single input positive systems abstract this paper provides parameterizations of all positively stabilizing feedbacks for a particular class of finite dimensional single input positive systems a method to yield these parameterizations by means of tools of linear programming and a convergence analysis which allows to extend the results to a particular infinite dimensional system described by a parabolic partial differential equation it also provides an academic standard example the pure diffusion equation to support the theoretical results

computing the stopping distance of a tanner graph is np hard two decision problems related to the computation f stopping sets in tanner graphs are shown to be np complete it follows as a consequence that there exists no polynomial time algorithm for computing the stopping distance of a tanner graph unless p np

multiple bases belief propagation decoding of high density cyclic codes we introduce a new method for decoding short and moderate length linear block codes with dense parity check matrix representations of cyclic form this approach is termed multiple bases belief propagation the proposed iterative scheme makes use of the fact that a code has many structurally diverse parity check matrices capable of detecting different error patterns we show that this inherent code property leads to decoding algorithms with significantly better performance when compared to standard belief propagation decoding furthermore we describe how to choose sets of parity check matrices of cyclic form amenable for multiple bases decoding based on analytical studies performed for the binary erasure channel for several cyclic and extended cyclic codes the multiple bases belief propagation decoding performance can be shown to closely follow that of the maximum likelihood decoder

characterization of the critical sobolev space on the optimal singularity at the origin in the present paper we investigate the optimal singularity at the origin for the functions belonging to the critical sobolev space hnp p rn 1 p with this purpose we shall show the weighted gagliardo nirenberg type inequality gn u lq rn dx x s c 1n s 1q 1p q1p u lp rn n s pnq n2pu lp rn 1 n s pnq where c depends only on n and p here 0 s n and p q with some p p determined only by n and p additionally in the case n 2 and nn 1 p we can prove the growth orders for s as s n and for q as q are both optimal gn allows us to prove the trudinger type estimate with the homogeneous weight furthermore it is obvious that gn cannot hold with the weight x n itself however with a help of the logarithmic weight of the type log1 x r x n at the origin we cover this critical weight simultaneously we shall give the minimal exponent r q p p so that the continuous embedding can hold

results on parity check matrices with optimal stopping and or dead end set enumerators the performance of iterative decoding techniques for linear block codes correcting erasures depends very much on the sizes of the stopping sets associated with the underlying tanner graph or equivalently the parity check matrix representing the code in this correspondence we introduce the notion of dead end sets to explicitly demonstrate this dependency the choice of the parity check matrix entails a tradeoff between performance and complexity we give bounds on the complexity of iterative decoders achieving optimal performance in terms of the sizes of the underlying parity check matrices further we fully characterize codes for which the optimal stopping set enumerator equals the weight enumerator

parity check matrices separating erasures from errors most decoding algorithms of linear codes in general are designed to correct or detect errors however many channels cause erasures in addition to errors in principle decoding over such channels can be accomplished by deleting the erased symbols and decoding the resulting vector with respect to a punctured code for any given linear code and any given maximum number of correctable erasures parity check matrices are introduced that yield parity check equations which do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures these matrices allow for the separation of erasures from errors to facilitate decoding several constructions of such separating parity check matrices are presented to reduce decoding complexity separating parity check matrices with small number of rows are preferred the minimum number of rows in a parity check matrix separating a given maximum number of erasures is called the separating redundancy upper and lower bounds on the separating redundancies are derived in particular it is shown that the separating redundancies tend to grow linearly with the number of rows in full rank parity check matrices of codes the separating redundancies of some classes of codes are determined for some maximum numbers of erasures

lq optimal control of infinite dimensional systems by spectral factorization abstract we consider the standard lq optimal control of an exponentially stabilizable and detectable infinite dimensional semigroup state space system with bounded sensing and control it is reported that the reachable restriction of the lq optimal state feedback operator can be identified 1 by solving a spectral factorization problem delivering a bistable spectral factor with entries in the distributed proper stable transfer function algebra and 2 by obtaining any constant solution of an operator diophantine equation over the properties of the restricted solution feedback operator are discussed it is shown in particular that the lq optimal state feedback operator and its reachable restriction coincide whenever the unreachable state components are unobservable these theoretical results are applied to a simple model of heat diffusion with collocated sensor and actuator leading to an approximation procedure converging exponentially fast to the lq optimal state feedback operator this procedure is based on approximate spectral factorization by symmetric extraction and on simple residue calculus

optimal linear quadratic control of coupled parabolic hyperbolic pdes abstractthis paper focuses on the optimal control design for a system of coupled parabolic hypebolic partial differential equations by using the infinite dimensional state space description and the corresponding operator riccati equation some dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the linear quadratic lq optimal control problem a state lq feedback operator is computed by solving the operator riccati equation which is converted into a set of algebraic and differential riccati equations thanks to the eigenvalues and the eigenvectors of the parabolic operator the results are applied to a non isothermal packed bed catalytic reactor the lq optimal controller designed in the early portion of the paper is implemented for the original nonlinear model numerical simulations are performed to show the controller performances

finite dimensional regulators for a class of regular hyperbolic pde systems abstractin this paper the output regulation problem is addressed for a class of linear hyperbolic infinite dimensional systems with spatially varying coefficients modelling a large class of convection dominated transport reaction systems in particular distributed parameter systems with bounded input and unbounded output operators are considered first we demonstrate a general conclusion about the exponential stability of the considered system by relating the stability to the solution of an associated differential equation based on the assumption that the hyperbolic system satisfies the exponential stability conditions the main manuscript contribution is the development of two novel finite dimensional regulators output and error feedback regulators such that the controlled output of the plant tracks a reference signal generated by a known signal process exosystem in order to guarantee the feasibility of the proposed regulators the solvability of the corresponding sylvester equations is discusse

single exclusion number and the stopping redundancy of mds codes for a linear block code c its stopping redundancy is defined as the smallest number of check nodes in a tanner graph for c such that there exist no stopping sets of size smaller than the minimum distance of c schwartz and vardy conjectured that the stopping redundancy of a maximum distance separable mds code should only depend on its length and minimum distance we define the n t single exclusion number s n t as the smallest number of i subsets of an n set such that for each i subset of the n set i 1 t 1 there exists a i subset that contains all but one element of the i subset new upper bounds on the single exclusion number are obtained via probabilistic methods recurrent inequalities as well as explicit constructions the new bounds are used to better understand the stopping redundancy of mds codes in particular it is shown that for n k n d 1 d mds codes as n rarr infin the stopping redundancy is asymptotic to s n d 2 if d o radic n or if k o radic n k rarr infin thus giving partial confirmation of the schwartz vardy conjecture in the asymptotic sense

finite length analysis of low density parity check codes on the binary erasure channel in this paper we are concerned with the finite length analysis of low density parity check ldpc codes when used over the binary erasure channel bec the main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of ldpc codes when decoded iteratively we also give expressions for upper bounds on the average bit and block erasure probability for regular ldpc ensembles and the standard random ensemble under maximum likelihood ml decoding finally we present what we consider to be the most important open problems in this area

optimal lq feedback regulation of a nonisothermal plug flow reactor model by spectral factorization the linear quadratic lq optimal temperature and reactant concentration regulation problem is studied for a partial differential equation model of a nonisothermal plug flow tubular reactor by using a nonlinear infinite dimensional hilbert state space description first the dynamical properties of the linearized model around a constant temperature equilibrium profile along the reactor are studied it is shown that it is exponentially stable and approximately reachable next the general concept of lq feedback is introduced it turns out that any lq feedback is optimal from the input output viewpoint and stabilizing for the plug flow reactor linearized model a state lq feedback operator is computed via the solution of a matrix riccati differential equation mrde in the space variable thanks to the reachability property the computed lq feedback is actually the optimal one then the latter is applied to the nonlinear model and the resulting closed loop system dynamical performances are analyzed a criterion is given which guarantees that the constant temperature equilibrium profile is an asymptotically stable equilibrium of the closed loop system moreover under the same criterion it is shown that the control law designed previously is optimal along the nonlinear closed loop model with respect to some cost criterion the results are illustrated by some numerical simulations

a sliding mode observer for a linear reaction convection diffusion equation with disturbances abstract a sliding mode observer is designed and analyzed for a linear reaction convection diffusion equation subject to external disturbances based on a discontinuous input the proposed observer ensures both state estimation and disturbance rejection using only one boundary measurement an appropriate lyapunov function is used to prove the asymptotic convergence of the observation error the performance of the proposed observer is illustrated by some numerical simulation results

feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations general second order parabolic and hyperbolic equations on a bounded domain are considered the input is applied in the neumann or mixed boundary condition and is expressed as a finite dimensional feedback in the parabolic case the feedback acts in particular on the dirichlet trace of the solution here it is shown that the resulting closed loop system defines a feedback c0 semigroup on l2 in fact on h32 2 0 that is both analytic and compact for positive times and whose generator has compact resolvent in the hyperbolic case the feedback acts on the position vector only or on its dirichlet trace in a special case here a similar result is established regarding the existence of a feedback c0 cosine operator moreover an example is given which hints that the class of prescribed feedbacks acting on the dirichlet trace cannot be substantially enlarged functional analytic techniques are employed in particular perturbation theory however perturbation theory for the original variable fails on l2 the space in which the final result is sought therefore our approach employs perturbation theory after a suitable continuous extension on the larger space h12 2

on the pseudocodeword redundancy of binary linear codes for a binary linear code the pseudocodeword redundancy with respect to the additive white gaussian noise channel the binary symmetric channel or the max fractional weight is defined to be the smallest number of rows in a parity check matrix such that the corresponding minimum pseudoweight is equal to the minimum hamming distance of the code it is shown that most codes do not have a finite pseudocodeword redundancy also upper bounds on the pseudocodeword redundancy for some families of codes including codes based on designs are provided the pseudocodeword redundancies for all codes of small length at most 9 are computed furthermore comprehensive results are provided on the cases of cyclic codes of length at most 250 for which the eigenvalue bound of vontobel and koetter is sharp

on generic erasure correcting sets and related problems motivated by iterative decoding techniques for the binary erasure channel hollmann and tolhuizen introduced and studied the notion of generic erasure correcting sets for linear codes a generic r s erasure correcting set generates for all codes of codimension r a parity check matrix that allows iterative decoding of all correctable erasure patterns of size s or less the problem is to derive bounds on the minimum size f r s of generic erasure correcting sets and to find constructions for such sets in this paper we continue the study of these sets we derive better lower and upper bounds hollmann and tolhuizen also introduced the stronger notion of r s sets and derived bounds for their minimum size g r s here also we improve these bounds we observe that these two conceps are closely related to so called s wise intersecting codes an area in which g r s has been studied primarily with respect to ratewise performance we derive connections finally we observed that hypergraph covering can be used for both problems to derive good upper bounds

brief paper lq control design of a class of hyperbolic pde systems application to fixed bed reactor a general linear controller design method for a class of hyperbolic linear partial differential equation pdes systems is presented this is achieved by using an infinite dimensional hilbert state space description with infinite dimensional distributed input and output a state lq feedback operator is computed via the solution of a matrix riccati differential equation in the space variable the proposed method is applied to a fixed bed chemical reactor control problem where one elementary reaction takes place an optimal controller is designed for linearized fixed bed reactor model it is applied to the original nonlinear model and the resulting closed loop stability is analyzed numerical simulations are performed to show the performance of the designed controller

on the stopping distance and the stopping redundancy of codes it is now well known that the performance of a linear code copf under iterative decoding on a binary erasure channel and other channels is determined by the size of the smallest stopping set in the tanner graph for copf several recent papers refer to this parameter as the stopping distance s of copf this is somewhat of a misnomer since the size of the smallest stopping set in the tanner graph for copf depends on the corresponding choice of a parity check matrix it is easy to see that s les d where d is the minimum hamming distance of copf and we show that it is always possible to choose a parity check matrix for copf with sufficiently many dependent rows such that s d we thus introduce a new parameter the stopping redundancy of copf defined as the minimum number of rows in a parity check matrix h for copf such that the corresponding stopping distance s h attains its largest possible value namely s h d we then derive general bounds on the stopping redundancy of linear codes we also examine several simple ways of constructing codes from other codes and study the effect of these constructions on the stopping redundancy specifically for the family of binary reed muller codes of all orders we prove that their stopping redundancy is at most a constant times their conventional redundancy we show that the stopping redundancies of the binary and ternary extended golay codes are at most 34 and 22 respectively finally we provide upper and lower bounds on the stopping redundancy of mds codes

control of parabolic pdes with time varying spatial domain czochralski crystal growth process this paper considers the optimal control problem for a class of convection diffusion reaction systems modelled by partial differential equations pdes defined on time varying spatial domains the class of pdes is characterised by the presence of a time dependent convective transport term which is associated with the time evolution of the spatial domain boundary the functional analytic description of the pde yields the representation of the initial and boundary value problem as a nonautonomous parabolic evolution equation on an appropriately defined infinite dimensional function space the properties of the time varying evolution operator to guarantee existence and well posedness of the initial and boundary value problem are demonstrated which serves as the basis for the optimal control problem synthesis an industrial application of the crystal temperature regulation problem for the czochralski crystal growth process is considered and numerical simulation results are provided

on critical cases of sobolev s inequalities abstract we present a new form of the trudinger type inequality which shows an explicit dependence moreover we give an alternative proof of the brezis gallouet wainger inequality

remarks on gagliardo nirenberg type inequality with critical sobolev space and bmo we consider the generalized gagliardo nirenberg inequality in mathbb r n in the homogeneous sobolev space dot h s r mathbb r n with the critical differential order s n r which describes the embedding such as l p mathbb r n cap dot h n r r mathbb r n subset l q mathbb r n for all q with p q n r which may be regarded as a generalization of the brezis gallouet wainger inequality

optimal control design for time varying catalytic reactors a riccati equation based approach the linear quadratic lq optimal control problem is studied for a partial differential equation model of a time varying catalytic reactor first the dynamical properties of the linearised model are studied next an lq control feedback is computed by using the corresponding operator riccati differential equation whose solution can be obtained via a related matrix riccati partial differential equation finally the designed controller is applied to the non linear reactor system and tested numerically

stopping set distributions of some reed muller codes stopping sets and stopping set distribution of a linear code are used to determine the performance of this code under iterative decoding over a binary erasure channel bec let c be a binary n k linear code with parity check matrix h where the rows of h may be dependent a stopping set s of c with parity check matrix h is a subset of column indices of h such that the restriction of h to s does not contain a row of weight one the stopping set distribution ti h i 0n enumerates the number of stopping sets with size i of c with parity check matrix h note that stopping sets and stopping set distribution are related to the parity check matrix h of c let h be the parity check matrix of c which is formed by all the nonzero codewords of its dual code c a parity check matrix h is called bec optimal if ti h ti h i 0 1 n and h has the smallest number of rows in this paper we study stopping sets stopping set distributions and bec optimal parity check matrices of binary linear codes using finite geometry in combinatorics we obtain bec optimal parity check matrices and then determine the stopping set distributions for the simplex codes the hamming codes the first order reed muller codes and the extended hamming codes which are some reed muller codes or their shortening or puncturing versions

brief paper on asymptotic stability of semi linear distributed parameter dissipative systems the concept of asymptotic stability is studied for a class of semi linear distributed parameter dissipative systems with nonlinearity defined on a convex subset of the state space this is achieved by using infinite dimensional banach state space description stability criteria are established which are based on a weaker technical condition of the m dissipativity these theoretical results are applied to a class of transport reaction processes different types of nonlinearities are studied by adapting the criteria given in the early portions of the paper

lq boundary control of a diffusion convection reaction system in this work the boundary control of a distributed parameter system modelled by linear parabolic partial differential equations pdes with spatially varying coefficients is studied an infinite dimensional state space setting is considered and an exact transformation of the boundary actuation is realised to obtain an evolutionary model the evolutionary model which incorporates the spatially varying coefficients of the underlying set of the pdes is used for subsequent linear quadratic regulator synthesis the formulated linear quadratic state feedback controller is applied to a nonlinear model of the reactor and its performance is studied

distributed fuzzy control design of nonlinear hyperbolic pde systems with application to nonisothermal plug flow reactor this paper considers the problem of fuzzy control design for a class of nonlinear distributed parameter systems that is described by first order hyperbolic partial differential equations pdes where the control actuators are continuously distributed in space the goal of this paper is to develop a fuzzy state feedback control design methodology for these systems by employing a combination of pde theory and concepts from takagi sugeno t s fuzzy control first the t s fuzzy hyperbolic pde model is proposed to accurately represent the nonlinear first order hyperbolic pde system subsequently based on the t s fuzzy pde model a lyapunov technique is used to analyze the closed loop exponential stability with a given decay rate then a fuzzy state feedback control design procedure is developed in terms of a set of spatial differential linear matrix inequalities sdlmis from the resulting stability conditions furthermore utilizing the finite difference approximation method with a backward difference for the spatial derivative a recursive algorithm is presented to solve the sdlmis via the existing lmi optimization techniques finally the developed design methodology is successfully applied to the control of a nonisothermal plug flow reactor

asymptotic behaviour of contraction non autonomous semi flows in a banach space the asymptotic behaviour is studied for a class of non autonomous infinite dimensional non linear dissipative systems this is achieved by using the concept of contraction semi flow which is a generalization of contraction non linear semigroup conditions are presented under which the solution of the abstract differential equation converges to the omega limit set the equilibrium profile respectively the general development is applied to semi linear systems with time varying non linearity asymptotic behaviour and stability criteria are established on the basis of the conditions given in the early portion of the paper the theoretical results are applied to a general class of first order hyperbolic time varying semi linear pdes

asymptotic behavior and stability for solutions of a biochemical reactor distributed parameter model this paper deals with the dynamical analysis of a tubular biochemical reactor the existence of nonnegative state trajectories and the invariance of the set of all physically feasible state values under the dynamical equation as well as the convergence of the state trajectories to equilibrium profiles are proved in addition the existence of multiple equilibrium profiles is analyzed it is proved that under physically meaningful conditions the system has two stable and one unstable equilibrium profiles

lq optimal control of hyperbolic pdaes the linear quadratic control synthesis for a set of coupled first order hyperbolic partial differential and algebraic equations is presented by using the infinite dimensional hilbert state space representation of the system and the well known operator riccati equation ore method solving the algebraic equations and substituting them into the partial differential equations pdes results in a model consisting of a set of pure hyperbolic pdes the resulting pde system involves a hyperbolic operator in which the velocity matrix is spatially varying non symmetric and its eigenvalues are not necessarily negative through of the domain the c0 semigroup generation property of such an operator is proven and it is shown that the generated c0 semigroup is exponentially stable and consequently the ore has a unique and non negative solution conversion of the ore into a matrix riccati differential equation allows the use of a numerical scheme to solve the control problem

bounds on the expansion properties of tanner graphs this work focuses on the expansion properties of a tanner graph because they are known to be related to the performance of associated iterative message passing algorithms over various channels by analyzing the eigenvalues and corresponding eigenvectors of the normalized incidence matrix representing a tanner graph lower bounds on these expansion properties are derived specifically for the binary erasure channel these results lead to two lower bounds on stopping distance for any given binary linear code and an upper bound on stopping redundancy for the family of difference set codes type i 2 d projective geometry low density parity check ldpc codes

on the stability of a periodic solution of distributed parameters biochemical system this paper studies the stability of periodic solutions of distributed parameters biochemical system with periodic input sin t we prove that if sin t is periodic then the system has a periodic solution that is input to state stable when small perturbations are acting on the input concentration sin t

entropy numbers of embeddings of weighted besov spaces iii weights of logarithmic type we determine the exact asymptotic order of the entropy numbers of compact embeddings b s 1 p 1 q 1 mathbb r d w 1 hookrightarrow b s 2 p 2 q 2 mathbb r d w 2 of weighted besov spaces in the case where the ratio of the weights w x w 1 x w 2 x is of logarithmic type this complements the known results for weights of polynomial type the estimates are given in terms of the number 1 p 1 p 1 1 p 2 and the function w x we find an interesting new effect if the growth rate at infinity of w x is below a certain critical bound then the entropy numbers depend only on w x and no longer on the parameters of the two besov spaces all results remain valid for triebel lizorkin spaces as well

on the effect of equivalent constraints on a maximizing problem associated with the sobolev type embeddings in mathbb r n in this paper we consider the attainability of a maximizing problem begin aligned d sup vert u vert h 1 n gamma 1 left vert u vert n n alpha vert u vert p p right end aligned where n ge 2 n 0 0 gamma le n and vert u vert h 1 n gamma left vert u vert n gamma vert nabla u vert n gamma right frac 1 gamma the existence of a maximizer for d is closely related to the exponent gamma in fact we show that the value begin aligned alpha alpha inf vert u vert h 1 n gamma 1 left frac 1 vert u vert n n vert u vert p p right end aligned is a threshold in terms of the attainability of d

improved probabilistic bounds on stopping redundancy for a linear code c the stopping redundancy of c is defined as the minimum number of check nodes in a tanner graph t for c such that the size of the smallest stopping set in t is equal to the minimum distance of c han and siegel recently proved an upper bound on the stopping redundancy of general linear codes using probabilistic analysis for most code parameters this bound is the best currently known in this correspondence we present several improvements upon this bound

existence and non existence of maximizers for the moser trudinger type inequalities under inhomogeneous constraints in this paper we study the existence and non existence of maximizers for the moser trudinger type inequalities in mathbb r n of the form begin aligned d n alpha a b sup u in w 1 n mathbb r n vert nabla u vert l n mathbb r n a vert u vert l n mathbb r n b 1 int mathbb r n phi n left alpha u n right dx end aligned here n ge 2 n frac n n 1 a b 0 alpha in 0 alpha n and phi n t e t sum j 0 n 2 frac t j j where alpha n n omega n 1 1 n 1 and omega n 1 denotes the surface area of the unit ball in mathbb r n we show the existence of the threshold alpha alpha a b n in 0 alpha n such that d n alpha a b is not attained if alpha in 0 alpha and is attained if alpha in alpha alpha n we also provide the conditions on a b in order that the inequality alpha alpha n holds

exact multiplicity of solutions and s shaped bifurcation curve for a class of semilinear elliptic equations abstract the set of steady state solutions to a reaction diffusion equation modeling an autocatalytic chemical reaction is completely determined when the reactor has spherical geometry and the spatial dimension is n 1 or 2 for any reaction order or n 3 for subcritical reaction order bifurcation approach and analysis of linearized problems are used to establish exact multiplicity and precise global bifurcation diagram of positive steady states

on parity check collections for iterative erasure decoding that correct all correctable erasure patterns of a given size recently there has been interest in the construction of small parity check sets for iterative decoding of the hamming code with the property that each uncorrectable or stopping set of size three is the support of a codeword and hence uncorrectable anyway here we reformulate and generalize the problem and improve on this construction we show that a parity check collection that corrects all correctable erasure patterns of size m for the hamming code with codimension r provides in fact for all codes of codimension r a corresponding generic parity check collection with this property this leads in a natural way to a necessary and sufficient condition for such generic parity check collections we use this condition to construct a generic parity check collection for codes of codimension r correcting all correctable erasure patterns of size at most m for all r and mlesr thus generalizing the known construction for m 3 then we discuss optimality of our construction and show that it can be improved for mges3 and r large enough finally we discuss some directions for further research

stabilization of a coupled transport diffusion system with boundary input we consider the problem of stabilizing a coupled transport diffusion system with boundary input the system is described by two linear transport diffusion equations and is not asymptotically stable in order to stabilize the system with boundary input sensor influence functions are assumed to be located at interior of the domain first we formulate the system as an evolution equation with unbounded output operators in a hilbert space using variable transformation next we derive a reduced order model with a finite dimensional state variable for the infinite dimensional system then a stabilizing controller is constructed for the reduced order model under an additional assumption it is shown that the finite dimensional controller together with a residual mode filter plays a role of a finite dimensional stabilizing controller for the original infinite dimensional system if the order of the residual mode filter is chosen sufficiently large finally the validity of the design method is demonstrated through a numerical simulation

a three dimensional fuzzy control methodology for a class of distributed parameter systems the traditional fuzzy set is two dimensional 2 d with one dimension for the universe of discourse of the variable and the other for its membership degree this 2 d fuzzy set is not able to handle the spatial information the traditional fuzzy logic controller flc developed from this 2 d fuzzy set should not be able to control the distributed parameter system that has the tempo spatial nature a three dimensional 3 d fuzzy set is defined to be made of a traditional fuzzy set and an extra dimension for spatial information based on concept of the 3 d fuzzy set a new fuzzy control methodology is proposed to control the distributed parameter system similar to the traditional flc it still consists of fuzzification rule inference and defuzzification operations different to the traditional flc it uses multiple sensors to provide 3 d fuzzy inputs and possesses the inference mechanism with 3 d nature that can fuse these inputs into a so called ldquospatial membership function rdquo thus a simple 2 d rule base can still be used for two obvious advantages one is that rules will not increase as sensors increase for the spatial measurement the other is that computation of this 3 d fuzzy inference can be significantly reduced for real world applications using only a few more sensors the proposed flc is able to process the distributed parameter system with little complexity increased from the traditional flc the 3 d flc is successfully applied to a catalytic packed bed reactor and compared with the traditional flc the results demonstrate its effectiveness to the nonlinear unknown distributed parameter process and its potential to a wide range of engineering applications