error correcting codes in projective space the projective space of order n over the finite field bbfq denoted here as pq n is the set of all subspaces of the vector space bbfqn the projective space can be endowed with the distance function d u v dimu dimv 2 dim u v which turns pq n into a metric space with this an n m d code bbc in projective space is a subset of pq n of size m such that the distance between any two codewords subspaces is at least d koetter and kschischang recently showed that codes in projective space are precisely what is needed for error correction in networks an n m d code can correct t packet errors and packet erasures introduced adversarially anywhere in the network as long as 2t 2 d this motivates our interest in such codes in this paper we investigate certain basic aspects of coding theory in projective space first we present several new bounds on the size of codes in pq n which may be thought of as counterparts of the classical bounds in coding theory due to johnson delsarte and gilbert varshamov some of these are stronger than all the previously known bounds at least for certain code parameters we also present several specific constructions of codes and code families in pq n finally we prove that nontrivial perfect codes in pq n do not exist

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

on the capacity of multiplicative finite field matrix channels this paper deals with the multiplicative finite field matrix channel a discrete memoryless channel whose input and output are matrices over a finite field related by a multiplicative transfer matrix the model considered here assumes that all transfer matrices with the same rank are equiprobable so that the channel is completely characterized by the rank distribution of the transfer matrix this model is seen to be more flexible than previously proposed ones in describing random linear network coding systems subject to link erasures while still being sufficiently simple to allow tractability the model is also conservative in the sense that its capacity provides a lower bound on the capacity of any channel with the same rank distribution a main contribution is to express the channel capacity as the solution of a convex optimization problem which can be easily solved by numerical computation for the special case of constant rank input a closed form expression for the capacity is obtained the behavior of the channel for asymptotically large field size or packet length is studied and it is shown that constant rank input suffices in this case finally it is proved that the well known approach of treating inputs and outputs as subspaces is information lossless even in this more general model

existence of a positive solution to kirchhoff type problems without compactness conditions abstract the existence of a positive solution to a kirchhoff type problem on r n is proved by using variational methods and the new result does not require usual compactness conditions a cut off functional is utilized to obtain the bounded palais smale sequences

on ground states for the kirchhoff type problem with a general critical nonlinearity abstract in this paper we consider the following kirchhoff type problem a b r 3 u 2 u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants and f has a critical growth the aim of this paper is to study the existence of ground state solutions for kirchhoff type equations with a general nonlinearity in the critical growth without the assumption of the monotonicity of the function t f t t 3 moreover we will show that the mountain pass value gives the least energy level and also obtain a mountain pass solution

iterative coding for network coding we consider communication over a noisy network under randomized linear network coding possible error mechanisms include node or link failures byzantine behavior of nodes or an overestimate of the network min cut building on the work of kotter and kschischang we introduce a systematic oblivious random channel model within this model codewords contain a header this is the systematic part the header effectively records the coefficients of the linear encoding functions thus simplifying the decoding task under this constraint errors are modeled as random low rank perturbations of the transmitted codeword we compute the capacity of this channel and we define an error correction scheme based on random sparse graphs and a low complexity decoding algorithm by optimizing over the code degree profile we show that this construction achieves the channel capacity in complexity which is jointly quadratic in the number of coded information bits and sublogarithmic in the error probability

high energy solutions of systems of kirchhoff type equations in rn in the present paper five new critical point theorems on the product spaces are given and three existence theorems for the sequence of high energy solutions for the systems of kirchhoff type equations are studied

efficient encoding of low density parity check codes low density parity check ldpc codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy constrained random code ensembles and iterative decoding algorithms we consider the encoding problem for ldpc codes more generally we consider the encoding problem for codes specified by sparse parity check matrices we show how to exploit the sparseness of the parity check matrix to obtain efficient encoders for the 3 6 regular ldpc code for example the complexity of encoding is essentially quadratic in the block length however we show that the associated coefficient can be made quite small so that encoding codes even of length n spl sime 100000 is still quite practical more importantly we show that optimized codes actually admit linear time encoding

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

infinitely many radial solutions for kirchhoff type problems in rn abstract in this paper we concern with a class of kirchhoff type problems in r n by using the fountain theorem we obtain three existence results of infinitely many radial solutions for the problem

infinitely many positive solutions for kirchhoff type problems abstract this paper is concerned with the existence of infinitely many positive solutions to a class of kirchhoff type problem a b u 2 d x u f x u in and u 0 on where is a smooth bounded domain of r n a b 0 0 and f r r is a caratheodory function satisfying some further conditions we obtain a sequence of a e positive weak solutions to the above problem tending to zero in l with f being more general than that of k perera z zhang nontrivial solutions of kirchhoff type problems via the yang index j differential equations 221 2006 246 255 z zhang k perera sign changing solutions of kirchhoff type problems via invariant sets of descent flow j math anal appl 317 2006 456 463

the nehari manifold for a kirchhoff type problem involving sign changing weight functions abstract this paper examines a class of kirchhoff type equations that involve sign changing weight functions using nehari manifold and fibering map the existence of multiple positive solutions is established

an improved global best harmony search algorithm this paper introduces an improved global best harmony search ighs algorithm the proposed modifications effectively combines a novel improvisation scheme with a previously developed mechanism for updating the pitch adjustment rate par and the distance bandwidth bw the aim of this modification is to efficiently investigate the search space by going through the stages of exploration and exploitation the proposed algorithm is compared against seven previous modifications to hs using rigorous statistical tests when applied to the cec05 benchmark functions showing a superior performance on most of the tested functions

existence and concentration result for the kirchhoff type equations with general nonlinearities in this paper we study the existence and concentration behaviors of positive solutions to the kirchhoff type equations varepsilon 2 m left varepsilon 2 n int mathbf r n nabla u 2 mathrm d x right delta u v x u f u quad rm in mathbf r n quad u in h 1 mathbf r n n geqq 1 where m and v are continuous functions under suitable conditions on m and general conditions on f we construct a family of positive solutions u varepsilon varepsilon in 0 tilde varepsilon which concentrates at a local minimum of v after extracting a subsequence ek

coding for errors and erasures in random network coding the problem of error control in random linear network coding is considered a ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic motivated by the property that linear network coding is vector space preserving information transmission is modeled as the injection into the network of a basis for a vector space v and the collection by the receiver of a basis for a vector space u a metric on the projective geometry associated with the packet space is introduced and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space v capu is sufficiently large if the dimension of each codeword is restricted to a fixed integer the code forms a subset of a finite field grassmannian or equivalently a subset of the vertices of the corresponding grassmann graph sphere packing and sphere covering bounds as well as a generalization of the singleton bound are provided for such codes finally a reed solomon like code construction related to gabidulin s construction of maximum rank distance codes is described and a sudan style ldquolist 1rdquo minimum distance decoding algorithm is provided

existence and multiplicity of non trivial solutions for schrodinger kirchhoff type equations with radial potential abstract in the present paper the following schrodinger kirchhoff type problem p a b r n u 2 d x u v x u q x f u in r n u x 0 as x is studied four existence theorems of nontrivial solutions and a sequence of high energy solutions for problem p are obtained by the mountain pass theorem and symmetric mountain pass theorem

communication over finite field matrix channels this paper is motivated by the problem of error control in network coding when errors are introduced in a random fashion rather than chosen by an adversary an additive multiplicative matrix channel is considered as a model for random network coding the model assumes that n packets of length m are transmitted over the network and up to t erroneous packets are randomly chosen and injected into the network upper and lower bounds on capacity are obtained for any channel parameters and asymptotic expressions are provided in the limit of large field or matrix size a simple coding scheme is presented that achieves capacity in both limiting cases the scheme has decoding complexity o n 2 m and a probability of error that decreases exponentially both in the packet length and in the field size in bits extensions of these results for coherent network coding are also presented

existence results of positive solutions of kirchhoff type problems abstract in the present paper we use variational methods to prove two existence results of positive solutions of the following kirchhoff type problems a b u 2 u f x u in u 0 on one deals with the asymptotic behaviors of f near zero and infinity and the other deals with 4 superlinear of f at infinity

finite field matrix channels for network coding in 2010 silva et al studied certain classes of finite field matrix channels in order to model random linear network coding where exactly t random errors are introduced in this paper we consider a generalization of these matrix channels where the number of errors is not required to be constant indeed the number of errors may follow any distribution we show that a capacity achieving input distribution can always be taken to have a very restricted form the distribution should be uniform given the rank of the input matrix this result complements and is inspired by a paper of nobrega et al which establishes a similar result for a class of matrix channels that model network coding with link erasures our result shows that the capacity of our channels can be expressed as maximization over probability distributions on the set of possible ranks of input matrices a set of linear rather than exponential size

a rank metric approach to error control in random network coding the problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of rotter and kschischang a large class of constant dimension subspace codes is investigated it is shown that codes in this class can be easily constructed from rank metric codes while preserving their distance properties moreover it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank metric codes where partial information about the error is available this partial information may be in the form of erasures knowledge of an error location but not its value and deviations knowledge of an error value but not its location taking erasures and deviations into account when they occur strictly increases the error correction capability of a code if mu erasures and delta deviations occur then errors of rank t can always be corrected provided that 2t les d 1 mu delta where d is the minimum rank distance of the code for gabidulin codes an important family of maximum rank distance codes an efficient decoding algorithm is proposed that can properly exploit erasures and deviations in a network coding application where n packets of length m over f q are transmitted the complexity of the decoding algorithm is given by o dm operations in an extension field f qn

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

new existence and multiplicity of nontrivial solutions for nonlocal elliptic kirchhoff type problems abstract in the present paper we study the existence and multiplicity of nontrivial solutions for nonlocal elliptic kirchhoff type problems a b u 2 d x u f x u in u 0 on where a b 0 are real numbers some new existence and multiplicity results of nontrivial solutions are obtained by means of the mountain pass theorem and a cut off technique one known result is improved

existence and multiplicity of solutions for kirchhoff type equations abstract in this paper the existence and multiplicity of weak solutions are obtained for a class of kirchhoff type problems with dirichlet boundary value conditions by using the mountain pass theorem the local linking theorem the fountain theorem and the symmetric mountain pass lemma in critical point theory