efficient root finding algorithm with application to list decoding of algebraic geometric codes a list decoding for an error correcting code is a decoding algorithm that generates a list of codewords within a hamming distance t from the received vector where t can be greater than the error correction bound in previous work by m shokrollahi and h wasserman see ibid vol 45 p 432 7 march 1999 a list decoding procedure for reed solomon codes was generalized to algebraic geometric codes recent work by v guruswami and m sudan see ibid vol 45 p 1757 67 sept 1999 gives improved list decodings for reed solomon codes and algebraic geometric codes that work for all rates and have many applications however these list decoding algorithms are rather complicated r roth and g ruckenstein see ibid vol 46 p 246 57 jan 2000 proposed an efficient implementation of the list decoding of reed solomon codes in this correspondence extending roth and ruckenstein s fast algorithm for finding roots of univariate polynomials over polynomial rings i e the reconstruct algorithm we present an efficient algorithm for finding the roots of univariate polynomials over function fields based on the extended algorithm we give an efficient list decoding algorithm for algebraic geometric codes

a semilinear parabolic system arising in the theory of superconductivity arises in the theory of superconductivity of liquids proposed by abrahams and tsuneto in 11 system 1 has also been briefly treated as an example by chueh conley and smaller in 61 throughout this paper we shall assume that d is a bounded domain in r n 1 2 or 3 with smooth boundary x a denotes the laplacian all the results of the paper which hold for n 1 also hold when a is replaced by any second order symmetric elliptic differential expression with positive spectrum because of the very symmetric nature of l it is reasonable to expect solutions of 1 to have nice properties we show in this paper that this is in fact the case in section 2 we give a complete description of steady state solutions of 1 satisfying dirichlet or neumann boundary conditions we show that often all steady states are of the form u u t au for some constant a where u is the solution of a single differential equation it is possible using standard techniques for scalar nonlinear boundary value problems to obtain lots of information about the solutions of this single equation and so about solutions of 1 in 6 1 it is shown that all solutions of 1 are attracted towards the set u v u v 1 in section 3 we show that 1 has a lyapunov function and use this fact and the standard theory of w limit sets to prove that any solution of 1 tends as t 03 towards the family of all steady state solutions of 1 often in the case of scalar equations it can be shown that

fast decoding of algebraic geometric codes up to the designed minimum distance we present a decoding algorithm for algebraic geometric codes from regular plane curves in particular the hermitian curve which corrects all error patterns of weight less than d 2 with low complexity the algorithm is based on the majority scheme of feng and rao 1993 and uses a modified version of sakata s 1988 generalization of the berlekamp massey algorithm

non constant positive steady states of a predator prey system with non monotonic functional response and diffusion this paper deals with non constant positive steady state solutions of a predator prey system with non monotonic functional response also called holling type iv interaction terms and diffusion under the homogeneous neumann boundary condition we first establish positive upper and lower bounds for such solutions and then study their non existence global existence and bifurcation

the stability of spike solutions to the one dimensional gierer meinhardt model abstract the stability properties of an n spike equilibrium solution to a simplified form of the gierer meinhardt activator inhibitor model in a one dimensional domain is studied asymptotically in the limit of small activator diffusivity e the equilibrium solution consists of a sequence of spikes of equal height the two classes of eigenvalues that must be considered are the o 1 eigenvalues and the o e2 eigenvalues which are referred to as the large and small eigenvalues respectively the spike pattern is stable when the parameters in the gierer meinhardt model are such that both sets of eigenvalues lie in the left half plane for a certain range of these parameters and for n 2 and e 0 it is shown the o 1 eigenvalues are in the left half plane only when d d n where d n is another critical value of d which satisfies d n n thus when n 2 and e 1 the spike pattern is stable only when d n an explicit formula for d n is given for the special case n 1 it is shown that a one spike equilibrium solution is stable when d d1 e an asymptotic formula for d1 e is given finally the dynamics of a one spike solution is studied by deriving a differential equation for the trajectory of the center of the spike

approximately certifying the restricted isometry property is hard a matrix is said to possess the restricted isometry property rip if it acts as an approximate isometry when restricted to sparse vectors previous work has shown it to be np hard to determine whether a matrix possess this property but only in a narrow range of parameters in this paper we show that it is np hard to make this determination for any accuracy parameter even when we restrict ourselves to instances which are either rip or far from being rip this result implies that it is np hard to approximate the range of parameters for which a matrix possesses the rip with accuracy better than some constant ours is the first work to prove such a claim without any additional assumptions

stripes or spots nonlinear effects in bifurcation of reaction diffusion equations on the square bifurcation to spatial patterns in a two dimensional reaction diffusion medium is considered the selection of stripes versus spots is shown to depend on the nonlinear terms and cannot be discerned from the linearized model the absence of quadratic terms leads to stripes but in most common models quadratic terms will lead to spot patterns examples that include neural nets and more general pattern formation equations are considered

certifying the restricted isometry property is hard this paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property rip we demonstrate that testing whether a matrix satisfies rip is np hard as a consequence of our result it is impossible to efficiently test for rip provided p np

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

stationary patterns of strongly coupled prey predator models abstract we study some elliptic systems arising from 3 component predator prey models where cross diffusions are included in such a way that predator chases the prey and the prey runs away from the predator we establish the existence and non existence of non constant positive solutions our results show that the cross diffusions can create the stationary patterns

dynamics of lotka volterra competition systems with large interaction abstract we show that under certain additional hypothesis two population competing species models in bounded domains with diffusion and large interaction have simple dynamics in particular the solutions approach stationary states as t tends to infinity

positive steady state solutions of the noyes field model for belousov zhabotinskii reaction abstract in the paper we investigate the noyes field model for belousov zhabotinskii reaction and study positive steady state solutions of this model with the homogeneous neumann boundary condition we obtain the existence and non existence of non constant positive steady state solutions

strategy and stationary pattern in a three species predator prey model abstract in this paper we study a strongly coupled system of partial differential equations which models the dynamics of a two predator one prey ecosystem in which the prey exercises a defense switching mechanism and the predators collaboratively take advantage of the prey s strategy we demonstrate the emergence of stationary patterns for this system and show that it is due to the cross diffusion that arises naturally in the model as far as the authors are aware this is the first example of stationary patterns in a predator prey system arising solely from the effect of cross diffusion

algebraic soft decision decoding of hermitian codes an algebraic soft decision decoder for hermitian codes is presented we apply koetter and vardy s soft decision decoding framework now well established for reed solmon codes to hermitian codes first we provide an algebraic foundation for soft decision decoding then we present an interpolation algorithm to find the q polynomial that plays a key role in the decoding with some simulation results we compare performances of the algebraic soft decision decoders for hermitian codes and reed solmon codes favorable to the former

iterative soft decision decoding of hermitian codes this paper proposes an iterative soft decision decoding algorithm for one of the most popular algebraic geometric ag codes hermitian codes the algorithm is designed by integrating the two most powerful soft decision decoding algorithms the adaptive belief propagation abp algorithm and the koetter vardy kv list decoding algorithm the abp algorithm performs iterative decoding based on an adapted parity check matrix of a hermitian code to enhance the reliability of the soft received information with the enhanced reliability the kv algorithm performs soft decision list decoding to obtain the original message since the matrix adaptation relies on bit reliabilities regrouping of the unreliable bits is introduced to assist the abp decoding a complexity reducing abp kv decoding approach is proposed based on assessing the soft information provided by the abp algorithm and determining whether the following kv decoding steps should be carried out geometric interpretation of the abp algorithm is presented demonstrating the necessity of performing matrix adaptation our performance analysis shows the proposed iterative decoding algorithm outperforms both the existing decoding approaches for hermitian codes and the abp kv decoding of reed solomon rs codes

qualitative analysis of a ratio dependent predator prey system with diffusion ratio dependent predator prey models are favoured by many animal ecologists recently as they better describe predator prey interactions where predation involves a searching process when densities of prey and predator are spatially homogeneous the so called michaelis menten ratio dependent predator prey system which is an ordinary differential system has been studied by many authors the present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous neumann boundary condition its main purpose is to study qualitative properties of solutions to this reaction diffusion partial differential system in particular we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary differential equation dynamics non constant positive steady states exist for the partial differential equation model this demonstrates that stationary patterns arise as a result of diffusion

soft decision list decoding of hermitian codes this paper proposes the first complete soft decision list decoding algorithm for hermitian codes based on the koetter vardy s reed solomon code decoding algorithm for hermitian codes interpolation processes trivariate polynomials which are defined over the pole basis of a hermitian curve in this paper the interpolated zero condition of a trivariate polynomial with respect to a multiplicity matrix m is redefined followed by a proof of the validity of the soft decision scheme this paper also introduces a new stopping criterion for the algorithm that tranforms the reliability matrix pi to the multiplicity matrix m geometric characterisation of the trivariate monomial decoding region is investigated resulting in an asymptotic optimal performance bound for the soft decision decoder by defining the weighted degree upper bound of the interpolated polynomial two complexity reducing modifications are introduced for the soft decision scheme elimination of unnecessary interpolated polynomials and pre calculation of the coefficients that relate the pole basis monomials to the zero basis functions of a hermitian curve our simulation results and analyses show that soft decision list decoding of hermitian code can outperform koetter vardy decoding of reed solomon code which is defined in a larger finite field but with less decoding complexity

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

bit level soft decision decoding of reed solomon codes a reed solomon decoder that makes use of bit level soft decision information is presented a reed solomon generator matrix that possesses a certain inherent structure in gf 2 is derived this structure allows the code to be represented as a union of cosets each coset being an interleaver of several binary bch codes such partition into cosets provides a clue for efficient bit level soft decision decoding two decoding algorithms are derived in the development of the first algorithm a memoryless channel is assumed making the value of this algorithm more conceptual than practical the second algorithm which is obtained as a modification of the first does account for channel memory and thus accommodates a bursty channel both decoding algorithms are in many cases orders of magnitude more efficient than conventional techniques

decoding algebraic geometric codes up to the designed minimum distance a simple decoding procedure for algebraic geometric codes c sub omega d g is presented this decoding procedure is a generalization of peterson s decoding procedure for the bch codes it can be used to correct any d 1 2 or fewer errors with complexity o n sup 3 where d is the designed minimum distance of the algebraic geometric code and n is the codelength

steiner equiangular tight frames abstract we provide a new method for constructing equiangular tight frames etfs the construction is valid in both the real and complex settings and shows that many of the few previously known examples of etfs are but the first representatives of infinite families of such frames it provides great freedom in terms of the frame s size and redundancy this method also explicitly constructs the frame vectors in their native domain as opposed to implicitly defining them via their gram matrix moreover in this domain the frame vectors are very sparse the construction is extremely simple a tensor like combination of a steiner system and a regular simplex this simplicity permits us to resolve an open question regarding etfs and the restricted isometry property rip we show that the rip behavior of some etfs is unfortunately no better than their coherence indicates

iterative soft input soft output decoding of reed solomon codes by adapting the parity check matrix an iterative algorithm is presented for soft input soft output siso decoding of reed solomon rs codes the proposed iterative algorithm uses the sum product algorithm spa in conjunction with a binary parity check matrix of the rs code the novelty is in reducing a submatrix of the binary parity check matrix that corresponds to less reliable bits to a sparse nature before the spa is applied at each iteration the proposed algorithm can be geometrically interpreted as a two stage gradient descent with an adaptive potential function this adaptive procedure is crucial to the convergence behavior of the gradient descent algorithm and therefore significantly improves the performance simulation results show that the proposed decoding algorithm and its variations provide significant gain over hard decision decoding hdd and compare favorably with other popular soft decision decoding methods

stationary patterns created by cross diffusion for the competitor competitor mutualist model abstract in this paper we consider a competitor competitor mutualist model with cross diffusion we prove some existence and non existence results concerning non constant positive steady states patterns in particular we demonstrate that the cross diffusion can create patterns when the corresponding model without cross diffusion fails

mini maximizers for reaction diffusion systems with skew gradient structure abstract a reaction diffusion system with skew gradient structure is a sort of activator inhibitor system that consists of two gradient systems coupled in a skew symmetric way any steady state of such a system corresponds to a critical point of some functional the aim of this paper is to study the relation between a stability property as a steady state of the reaction diffusion system and a mini maximizing property as a critical point of the functional it is shown that a steady state of the skew gradient system is stable regardless of time constants if and only if it is a mini maximizer of the functional it is also shown that the mini maximizing property is closely related with the diffusion induced instability moreover by using the property that any mini maximizer on a convex domain is spatially homogeneous quite a general instability criterion is obtained for some activator inhibitor systems these results are applied to the diffusive fitzhugh nagumo system and the gierer meinhardt system

soft decision decoding of reed solomon codes this paper presents a maximum likelihood decoding mld and a suboptimum decoding algorithm for reed solomon rs codes the proposed algorithms are based on the algebraic structure of the binary images of rs codes theoretical bounds on the performance are derived and shown to be consistent with simulation results the proposed suboptimum algorithm achieves near mld performance with significantly lower decoding complexity it is also shown that the proposed suboptimum algorithm has better performance compared with generalized minimum distance decoding while the proposed mld algorithm has significantly lower decoding complexity than the well known vardy be ery 1991 mld algorithm

existence and instability of neumann layer solutions for a 3 component lotka volterra model with diffusion abstract we consider a 3 component lotka volterra model with diffusion which describes the dynamics of the population of two competing prey and one predator and we discuss the existence of positive stationary solutions and their stability property to do this the singular perturbation method and the associated singular limit spectral analysis are employed

non constant positive steady states of the sel kov model abstract this paper deals with the reaction diffusion system known as the sel kov model with the homogeneous neumann boundary condition this model has been applied to various problems in chemistry and biology we first give a priori estimates positive upper and lower bounds of positive steady states and then study the non existence bifurcation and global existence of non constant positive steady states as the parameters and are varied