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

existence of positive solutions for a system of generalized lidstone problems in this paper we study the existence of positive solutions for the system of generalized lidstone problems for ordinary differential equations 1 mu 2 m f 1 t u u 1 m 1u 2 m 2 v v 1 n 1v 2 n 2 1 nv 2 n f 2 t u u 1 m 1u 2 m 2 v v 1 n 1v 2 n 2 a 0u 2 i 0 b 0u 2 i 1 0 a 1u 2 i 1 b 1u 2 i 1 1 0 i 0 1 m 1 a 0v 2 j 0 b 0v 2 j 1 0 a 1v 2 j 1 b 1v 2 j 1 1 0 j 0 1 n 1 here f 1 f 2 c 0 1 xr m n r r 0 our hypotheses imposed on the nonlinearities f 1 and f 2 are formulated in terms of nonnegative matrices we use fixed point index theory to establish our main results based on a priori estimates of positive solutions

on the existence and nonexistence of positive solutions for nonlinear sturm liouville boundary value problems abstract in this paper the existence and nonexistence results of positive solutions are obtained for sturm liouville boundary value problem p x u q x u f x u x 0 1 a u 0 b p 0 u 0 0 c u 1 d p 1 u 1 0 where p c 1 0 1 q c 0 1 p x 0 q x 0 for x 0 1 f c 0 1 r a b c d 0 are constants and satisfy a b c d 0 the discussion is based on the positivity estimation for the green s function of associated linear boundary value problem and the fixed point index theory in cones

positive solutions of two point boundary value problems for second order differential equations with the nonlinearity dependent on the derivative abstract the existence of positive solutions is established for two point boundary value problems for second order differential equations with the nonlinearity dependent on the derivative l x h x f x x 0 x 1 r 1 1 0 1 0 0 r 2 2 1 2 1 0 where l x p x x q x x conditions are given in terms of the relative behaviors of the quotient f u v u v for u v near 0 and

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

multiplicity results for second order two point boundary value problems with superlinear or sublinear nonlinearities we consider boundary value problems for nonlinear second order differential equations of the form u a t f u 0 t epsilon 0 1 u 0 u 1 0 where a epsilon c 0 1 0 infinity and f r r is continuous and satisfies f s s 0 for s not equal 0 we establish existence and multiplicity results for nodal solutions to the problems if either f 0 0 f infinity infinity or f 0 infinity f 0 0 where f s s approaches f 0 and f infinity as s approaches 0 and infinity respectively we use bifurcation techniques to prove our main results c 2004 elsevier inc all rights reserved

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

positive solutions of sub superlinear sturm liouville problems in this paper we introduce the notion of a strict lower upper solution to nonlinear sturm liouville boundary value problems based on the maximum principles we establish a result of leray schauder degree on the ordered intervals induced by the pairs of strict lower and upper solutions applying the result and the fixed point index theory in cones we obtain the global existence results of positive solutions for sub superlinear sturm liouville problems

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

turbo decoding on the binary erasure channel finite length analysis and turbo stopping sets this paper is devoted to the finite length analysis of turbo decoding over the binary erasure channel bec the performance of iterative belief propagation decoding of low density parity check ldpc codes over the bec can be characterized in terms of stopping sets we describe turbo decoding on the bec which is simpler than turbo decoding on other channels we then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure apply turbo decoding until the transmitted codeword has been recovered or the decoder fails to progress further then the set of erased positions that will remain when the decoder stops is equal to the unique maximum size turbo stopping set which is also a subset of the set of erased positions furthermore we present some improvements of the basic turbo decoding algorithm on the bec the proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results finally we give an expression for the turbo stopping set size enumerating function under the uniform interleaver assumption and an efficient enumeration algorithm of small size turbo stopping sets for a particular interleaver the solution is based on the algorithm proposed by garello et al in 2001 to compute an exhaustive list of all low weight codewords in a turbo code

positive periodic solutions for nonlinear difference equations with periodic coefficients abstract we prove the existence of positive solutions of second order nonlinear difference equations on a finite discrete segment with periodic boundary conditions obtained results yield positive periodic solutions of the equation on the whole discrete axis provided that the coefficients are periodic

a global bifurcation phenomenon for second order singular boundary value problems we establish an existence result of unbounded continuum of solutions from the trivial branch for a second order singular boundary value problem as an application we prove several existence and multiplicity results of positive solutions for certain singular boundary value problems

existence of positive solutions for second order boundary value problem with one parameter abstract in the case of not requiring f t u to be nonnegative by transforming the boundary value problem into the integral equation system and applying the fixed point index theory the author studies the following second order boundary value problem with one parameter u u f t u 0 t 1 u 0 u 1 0 and obtains the results on the existence of its positive solution

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

abstract existence theorems of positive solutions for nonlinear boundary value problems let rn be a bounded domain h l2 l d l h h be a unbounded linear operator and f r r be a continuous function this paper deals with the existence of positive solutions of the operator equation lu f x u u d l which is the abstract form of differential equations boundary value problems the essential conditions on l and f guaranteeing the existence of positive solution are characterized the discussion is based on the krasnoselskii s fixed point theorem of cone mapping and the fixed point index theory in cones the abstract results are applied to some differential equations boundary value problems and some known results are extended and improved

existence and nonexistence results for positive solutions of an integral boundary value problem abstract this paper deals with the existence and nonexistence of positive solutions for the integral boundary value problem a u b u g t f t u cos 0 u 0 sin 0 u 0 0 1 u d cos 1 u 1 sin 1 u 1 0 1 u d we provide sufficient conditions for the existence of either none or one or more positive solutions of the above problem the main tool used in the proofs of existence results is a fixed point theorem in a cone due to krasnoselskii and zabreiko the results presented substantially unify and extend the existing results in the literature

a theoretical analysis of the properties of median filters necessary and sufficient conditions for a signal to be invariant under a specific form of median filtering are derived these conditions state that a signal must be locally monotone to pass through a median filter unchanged it is proven that the form of successive median filtering of a signal i e the filtered output is itself again filtered eventually reduces the original signal to an invariant signal called a root signal for a signal of length l samples a maximum of frac 1 2 l 2 repeated filterings produces a root signal

some statistical properties of median filters some statistical properties of median filters are analyzed it is shown that median filters can remove impulsive plus gaussian white noise better than hanning filters when the amplitude of the impulses is large or the energy of the gaussian noise is relatively low it is also shown that unlike linear filters median filters preserve sharp changes in signals when the noise energy is not too high

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

multiple positive solutions of dirichlet boundary value problems for second order impulsive differential equations abstract this paper is devoted to study the existence of multiple positive solutions for the second order dirichlet boundary value problem with impulse effects the main results here is the generalization of liu and li l liu f y li multiple positive solution of nonlinear two point boundary value problems j math anal appl 203 1996 610 625 for ordinary differential equations existence is established via the theory of fixed point index in cones

nontrivial solutions of singular superlinear m point boundary value problems abstract the singular superlinear m point boundary value problems l x h x f x 0 x 1 0 0 1 i 1 m 2 a i i are considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators where l x p x x q x x and 0

solutions of periodic boundary value problems for second order nonlinear differential equations via variational methods abstract by introducing a variational framework for a class of second order nonlinear differential equations with non separated periodic boundary value conditions some results on the existence of non trivial positive and negative solutions of the problems are obtained some results by atici guseinov graef kong etc obtained by topological degree methods are extended the resonant case of the problems where the nonlinearities are unbounded and satisfy ahmad lazer paul type conditions is also considered

multiplicity of solutions to two point boundary value problems for second order impulsive differential equations abstract in this paper we obtain some sufficient condition for the existence of at least three solutions to two point boundary value problems for second order impulsive differential equations by using leray schauder degree theory we assume the existence of two pairs of strict lower and upper solutions and the nonlinear term f satisfies a nagumo s growth condition with respect to the first order derivative

on sign changing solutions for nonlinear operator equations abstract in this paper the existence of sign changing solutions for nonlinear operator equations is discussed by using the topological degree and fixed point index theory the main theorems are some new three solution theorems which are different from the famous amann s and leggett williams three solution theorems as well as the results in f li g han generalization for amann s and leggett williams three solution theorems and applications j math anal appl 298 2004 638 654 these three solutions are all nonzero one of them is positive another is negative and the third one is a sign changing solution furthermore the theoretical results are successfully applied to both integral and differential equations

positive solutions of two point boundary value problems for systems of nonlinear second order singular and impulsive differential equations abstract this paper is devoted to the study of multiple and single positive solutions of two point boundary value problems for nonlinear second order singular and impulsive differential systems by constructing a cone k 1 k 2 which is the cartesian product of two cones in the space c 0 1 and computing the fixed point index in the k 1 k 2 we establish the conditions for the existence of at least one or at least two positive solutions to two point boundary value problems for systems of nonlinear second order singular and impulsive differential equations

multiple positive solutions of some boundary value problems abstract we study the existence of multiple positive solutions of the equations u t u subject to linear boundary conditions we show that there are at least two positive solutions if t u is superlinear at one end point zero or infinity and sublinear at the other applications of these results are provided to yield multiple positive solutions of some elliptic boundary value problems on an annulus

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

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

eigenvalue criteria for existence of multiple positive solutions to boundary value problems of second order delay differential equations abstract criteria are established for existence of multiple positive solutions to the boundary value problems expressed by second order delay differential equations of the form p t y t q t y t f t y t y t a t 0 1 y t y t t t 0 y t y t t t 1 1 a where 0 a 1 4 are suitably small p c 1 0 1 p 0 q c 0 1 and q 0

on linear and nonlinear fourth order eigenvalue problems with indefinite weight abstract we determine the principal eigenvalues of the linear indefinite weight problem u x r g x u x x 0 1 u 0 u 1 u 0 u 1 0 moreover we investigate the existence of positive solutions for the corresponding nonlinear indefinite weight problem where g 0 1 r is a continuous function which attains both positive and negative values f c r r and r is a parameter

positive solutions of second order two delay differential systems with twin parameter abstract the paper is concerned with the existence of positive solutions for a second order two delay differential systems with twin parameter firstly we present some sufficient conditions with the twin parameter belonging to an open rectangular region eigenvalue region to ensure the existence of positive solutions for the differential systems with singularity secondly we deal with the superlinear semipositone problem of the differential systems and establish the existence of positive solutions for the systems the proofs are based on a fixed point theorem in cones

existence of positive solutions forboundary value problems of second order functional difference equations abstract we prove the existence of positive solutions for boundary value problems of second orderfunctional difference equations the result here is the generalization of a corresponding result for ordinary difference equations

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

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

a generalization of median filtering using linear combinations of order statistics we consider a class of nonlinear filters whose output is given by a linear combination of the order statistics of the input sequence assuming a constant signal in white noise the coefficients in the linear combination are chosen to minimize the output mse for several noise distributions it is shown that the optimal order statistic filter osf tends toward the median filter as the noise becomes more impulsive the optimal osf is applied to an actual noisy image and is shown to perform well combining properties of both the averaging and median filters a more general design scheme for applications involving nonconstant signals is also given

decimated rank order filtering rank order filters are used for a wide range of applications they are noted for their ability to remove impulsive noise and to preserve sharp changes in the signal however there is a tradeoff between noise removing capability and edge retention when impulsive noise and sharp edges in the signal coincide edges are no longer preserved another tradeoff is noise removal versus computational cost the increase in the number of operations is proportional to the noise removal capability the authors introduce a new filter the decimated rank order filter that achieves large amount of noise removal with much less detail and edge smoothing as compared with the standard rank order filter furthermore this is possible at a reduced computational burden the decimated median filter is a particular example of this class the authors investigate its deterministic and statistical properties and show that it compares favorably with the standard median filter in terms of impulsive noise removal especially in the neighborhood of edges they extend the structure of decimated rank order filters they find all rank orders of the signal to obtain the decimated sorter and show that the decimated sorter is the bitonic sorter used in switching networks they point out the connection between their research and concepts in sorting and switching structures

positive solutions of nonlinear singular two point boundary value problems for second order impulsive differential equations in this paper we study the positive solutions of nonlinear singular two point boundary value problems for second order impulsive differential equations the existence of one or two positive solutions are established by using the fixed point index theorem in cone

existence of solutions to a class of nonlinear second order two point boundary value problems in this paper the existence and multiplicity results of solutions are obtained for the second order two point boundary value problem u t f t u t for all t 0 1 subject to u 0 u 1 0 where f is continuous the monotone operator theory and critical point theory are employed to discuss this problem respectively in argument quadratic root operator and its properties play an important role

a new class of detail preserving filters for image processing a new class of median type filters for image processing is proposed in the filters linear fir substructures are used in conjunction with the median operation the root signals and noise attenuation properties of the fir median hybrid filters are analyzed and compared to representative edge preserving filtering operations the concept of multilevel median operation is introduced to improve the detail preserving property of conventional median and the fir median hybrid filters in the multilevel filters there exists a tradeoff between noise attenuation and detail preservation the analysis and examples indicate that fir median hybrid filters preserve details better and are computationally much more efficient than the conventional median and the k nearest neighbor averaging filters

on the existence of positive solutions for nonlinear differential equations with periodic boundary conditions we prove the existence of positive solutions of second order nonlinear differential equations on a finite interval with periodic boundary conditions and give upper and lower bounds for these positive solutions obtained results yield positive periodic solutions of the equation on the whole real axis provided that the coefficients are periodic

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

a separable median filter for image noise smoothing this paper investigates some properties of the separable filter resulting from successive applications of a one dimensional median filter on the rows and columns of an image although the output of this separable filter is not identical to the corresponding nonseparable two dimensional median filter with a square window its performance in image noise smoothing is close in particular its effectiveness in smoothing noise and its behavior with edges are characterized and compared with those of the two dimensional median filter it is shown that the separable filter has a much simpler implementation in real time hardware at video rates for example

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

positive solutions to some equations with homogeneous operator abstract in this paper we discuss the positive solutions to the equation u u a a u b u u 0 where a is a positive linear completely continuous operator b is an homogeneous operator defined on a cone in a real banach space and u a b u by using the fixed point index theory when u 0 is sufficiently small the spectral radius r a 1 and 1 where sgn b we obtain a positive solution to the above equation under some appropriate conditions the new results generalize the previous research about the homogeneous operator equation as an application by using our main theorem we can obtain a symmetrical positive solution to the one dimensional kirchhoff equation

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

existence of positive solutions of a fourth order boundary value problem abstract we consider the fourth order boundary value problem u f t u u 0 t 1 u 0 u 1 u 0 u 1 0 where f t u p au b p u p near 0 0 and f t u p cu dp u p near we give conditions on the constants a b c d that guarantee the existence of positive solutions the proof of our main result is based upon global bifurcation techniques

fir median hybrid filters this paper introduces a new generalized class of median type filters which we call fir median hybrid fmh filters the input signal x n is filtered with m linear phase fir filters and the output of the fmh filter is the median of the outputs of the fir filters the statistical properties of the fmh filters are analyzed for input signals with gaussian double exponential and uniform density functions it is shown that fmh filters have essentially the same statistical properties and same type of root signals as the standard median sm filters fmh filters preserve corner points and ramps and attenuate impulsive type noise components effectively an interesting subclass of the fmh filters requiring only one scaling multiplier two additions and three compare swap operations irrespective of the filter length is introduced in the examples speed improvement by a factor of 450 is obtained over an otherwise approximately equivalent sm filter

on the number of solutions of two classes of sturm liouville boundary value problems abstract we employ global bifurcation theory of p h rabinowitz together with sturm comparison theorem to establish multiplicity results for some classes of sturm liouville boundary value problems

positive solutions of second order boundary value problems with sign changing nonlinear terms abstract in this paper the existence results of positive solutions are obtained for second order boundary value problem u f t u t 0 1 u 0 u 1 0 where f c 0 1 r whose sign may change the proof is based on the fixed point index theory in cones

detail preserving ranked order based filters for image processing the theoretical analysis of multistage median filters is developed it is shown that multistage median filters are a combination of max median and min median filters since multistage median filters belong to the class of two dimensional stack filters they have threshold decomposition attributes making their theoretical analysis simple statistical threshold decomposition is applied to derive the statistical characteristics of these filters and the results are used to evaluate the performance of these two types of multistage filters finally a quantitative and qualitative comparison of the multistage filters and of other efficient detail preserving filters is presented the comparisons are made using the mean squared error and the mean absolute error criteria

unbounded solutions of the singular boundary value problems for second order differential equations on the half line this paper presents some results on the existence of multiple unbounded solutions for the boundary value problems for second order differential equations on half line and on the uniqueness of unbounded solution

nontrivial solutions of singular sublinear sturm liouville problems abstract the singular superlinear sturm liouville problems l x h x f x 0 x 1 r 1 1 0 1 0 0 r 2 2 1 2 1 0 are considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators where l x p x x q x x and h x is allowed to be singular at both x 0 and x 1 in particular f is not necessary to be nonnegative the existence results of nontrivial solutions and positive solutions are given by means of the topological degree theory

positive solutions of a second order integral boundary value problem in this paper we study the existence of positive solutions of a second order integral boundary value problems for ordinary differential equations our results presented here unify generalize and substantially improve the existing results in the literature moreover it is worthwhile to point out that our method will dispense with constructing a new green function

the effect of median filtering on edge estimation and detection in this paper we consider the effect of median prefiltering on the subsequent estimation and detection of edges in digital images where possible a quantitative statistical comparison is made for a number of filters defined with two dimensional geometries in some cases one dimensional analyses are required to illustrate certain points noise images prefiltered by median filters defined with a variety of windowing geometries are used to support the analysis and it is found that median prefiltering improves the performance of both thresholding and zero crossing based edge detectors

existence of positive solutions of lidstone boundary value problems abstract we show that there exists at least one positive solution for lidstone boundary value problem 1 n u 2 n t f t u t u t u 2 n 1 t 0 t 1 u 2 i 0 u 2 i 1 0 i 0 1 n 1 under some suitable conditions our proof is based upon global bifurcation techniques

existence of solutions of semilinear systems with gradient dependence via eigenvalue criteria abstract in this paper new criteria are established for the existence of positive radial solutions of a semilinear elliptic system depending on the gradient these criteria are determined by some relationships between the upper and lower bounds on suitable strips of r n of the nonlinearities of the system and the principal characteristic values of some associated linear hammerstein integral operators moreover using smoothing tools the totality of the involved cone is established

positive solutions and eigenvalue regions of two delay singular systems with a twin parameter abstract the paper is concerned with a two delay singular differential system with a twin parameter applying fixed point index theory we show the relationship between the asymptotic behaviors of nonlinearities at zero and infinity and the open regions eigenvalue regions of parameters which are correlated with delays such that the system has zero one and two positive solution s

generalization for amann s and leggett williams three solution theorems and applications abstract in this paper some new three solution theorems are obtained moreover the famous amann s and leggett williams three solution theorems in nonlinear functional analysis can be seen as their special cases namely these two famous theorems are united so they are both improved and that some new three solution theorems are applied to ordinary differential equations and some new existence results are obtained

multiple positive solutions of nonlinear two point boundary value problems we improve the results obtained by erbe hu and wang in a recent paper we show that there exist at least two positive solutions of two point boundary value problems under conditions weaker than those used by erbe hu and wang

a three point boundary value problem for nonlinear fourth order differential equations abstract in this paper the authors consider the nonlinear fourth order ordinary differential equation e u t g t f u 0 with the boundary conditions b u 0 u 1 u 0 u p u 1 0 some results on the existence and nonexistence of positive solutions to problem e b are obtained results on the existence of infinitely many positive solutions are also presented examples are included to demonstrate that the results are sharp

edge preserving filtering by combining nonlinear mean and median filters a very general class of nonlinear filters called mapping order statistics filters mosfs is introduced a subclass of the mosfs called l sub p mean median filters l sub p mmfs is discussed the l sub p mmf shows better properties in preserving edges and attenuating positive and negative impulse noise simultaneously than do the nonlinear mean filters in the analysis of the deterministic properties of the l sub p mmf the concept of a monotonic trend region is introduced a necessary condition for a signal to be a root signal is given this necessary condition is also valid for the known finite impulse response median hybrid filter

comparison of algorithms for standard median filtering standard median filtering that is searched repeatedly for a median from a sample set which changes only slightly between the subsequent searches is discussed several well known methods for solving this running median problem are reviewed the asymptotical time complexities of the methods are analyzed and simple variants are proposed which are especially suited for small sample sets a frequent situation although the discussion is restricted to the one dimensional case the ideas are easily extended to higher dimensions

positive solutions of m point boundary value problems the second order m point boundary value problem x h x f x 0 0 x 1 0 0 1 i 1m 2ai i is considered under some conditions concerning the first eigenvalue of the relevant linear operator where i 0 1 with 0 1 2 m 2 1 ai 0 with i 1m 2ai 1 h x is allowed to be singular at x 0 and x 1 the existence of positive solutions and multiple positive solutions is obtained by means of fixed point index theory similarly conclusions hold for some other m point boundary value conditions

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

multiplicity results for second order m point boundary value problem by using fixed point theorems some new results for multiplicity of positive solutions for a class of second order m point boundary value problem are obtained the associated green s function of the problem is also given

on positive solutions of some nonlinear fourth order beam equations abstract the existence uniqueness and multiplicity of positive solutions of the following boundary value problem is considered u 4 t f t u t 0 for 0 where 0 is a constant f 0 1 0 0 is continuous

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

positive solutions of a 2nth order boundary value problem involving all derivatives via the order reduction method this paper is mainly concerned with the existence multiplicity and uniqueness of positive solutions for the 2nth order boundary value problem 1 nu 2 n f t u u 1 i 2 u i 1 n 1u 2 n 1 u 2 i 0 u 2 i 1 1 0 i 0 1 n 1 where n 2 and f c 0 1 xr 2 n r r 0 we first use the method of order reduction to transform the above problem into an equivalent initial value problem for a first order integro differential equation and then use the fixed point index theory to prove the existence multiplicity and uniqueness of positive solutions for the resulting problem based on a priori estimates achieved by developing spectral properties of associated parameterized linear integral operators finally as a byproduct our main results are applied for establishing the existence multiplicity and uniqueness of symmetric positive solutions for the lidstone problem involving all derivatives

median filtering by threshold decomposition median filters are a special class of ranked order filters used for smoothing signals repeated application of the filter on a quantized signal of finite length ultimately results in a sequence termed a root signal which is invariant to further passes of the median filter in this paper it is shown that median filtering an arbitrary level signal to its root is equivalent to decomposing the signal into binary signals filtering each binary signal to a root with a binary median filter and then reversing the decomposition this equivalence allows problems in the analysis and the implementation of median filters for arbitrary level signals to be reduced to the equivalent problems for binary signals since the effects of median filters on binary signals are well understood this technique is a powerful new tool

existence of three positive solutions for some second order boundary value problems in this paper a new fixed point theorem of functional type in a cone is established with using the new fixed point theorem and imposing growth conditions on the nonlinearity the existence of three positive solutions for the boundary value problemx t f t x t x t 0 0 0 is continuous finally an example is given to illustrate the importance of results obtained

global bifurcation and multiple results for sturm liouville problems we consider the nonlinear sturm liouville boundary value problem lu t email protected t f u t 0