on mds codes via cauchy matrices a special form of cauchy matrix is used to obtain a tighter bound for the validity region of the maximum distance separable mds conjecture and a new compact characterization of generalized reed solomon codes the latter is further used to obtain constructions and some existence results for long 2k k double circulant mds codes

maximum distance separable 2d convolutional codes maximum distance separable mds block codes and mds 1d convolutional codes are the most robust codes for error correction within the class of block codes of a fixed rate and 1d convolutional codes of a certain rate and degree respectively in this paper we generalize this concept to the class of 2d convolutional codes for that we introduce a natural bound on the distance of a 2d convolutional code of rate k n and degree delta which generalizes the singleton bound for block codes and the generalized singleton bound for 1d convolutional codes then we prove the existence of 2d convolutional codes of rate k n and degree delta that reach such bound when n geq k lfloor delta k rfloor 2 lfloor delta k rfloor 3 2 if k nmid delta or n geq k delta k 1 delta k 2 2 if k mid delta by presenting a concrete constructive procedure

the lp minkowski problem and the minkowski problem in centroaffine geometry abstract the l p minkowski problem introduced by lutwak is solved for p n 1 in the smooth category the relevant monge ampere equation 0 1 is solved for all p 1 the same equation for p 1 is also studied and solved for p n 1 1 when p n 1 the equation is interpreted as a minkowski problem in centroaffine geometry a kazdan warner type obstruction for this problem is obtained

on superregular matrices and mdp convolutional codes superregular matrices are a type of lower triangular toeplitz matrix that arises in the context of constructing convolutional codes having a maximum distance profile these matrices are characterized by the property that the only submatrices having a zero determinant are those whose determinants are trivially zero due to the lower triangular structure in this paper we discuss how superregular matrices may be used to construct codes having a maximum distance profile we also present an upper bound on the minimum size a finite field must have in order that a superregular matrix of a given size can exist over that field this in turn gives an upper bound on the smallest field size over which an mdp n k convolutional code can exist

a fisher kpp type equation with density dependent diffusion and convection travelling wave solutions this paper concerns processes described by a nonlinear partial differential equation that is an extension of the fisher and kpp equations including density dependent diffusion and nonlinear convection the set of wave speeds for which the equation admits a wavefront connecting its stable and unstable equilibrium states is characterized there is a minimal wave speed for this wave speed there is a unique wavefront which can be found explicitly it displays a sharp propagation front for all greater wave speeds there is a unique wavefront which does not possess this property for such waves the asymptotic behaviour as the equilibrium states are approached is determined

a remark on rotationally symmetric solutions to the centroaffine minkowski problem abstract in this paper we study the solvability of the rotationally symmetric centroaffine minkowski problem by delicate blow up analyses we remove a technical condition in the existence result obtained by lu and wang 30

sharp profiles in degenerate and doubly degenerate fisher kpp equations this paper investigates the effects of a degenerate diffusion term in reaction diffusion models ut d u ux x g u with fisher kpp type g both in the case when d 0 0 and when d 0 d 1 0 with d u 0 elsewhere we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c and we show the appearance of a sharp type profile when c c these results solve recent conjectures formulated by sanchez garduno and maini j differential equations 117 1995 281 and satnoianu et al discrete continuous dyn systems series b 1 2000 339

on solvability of two dimensional lp minkowski problem abstract given p 0 and a positive continuous function g with g x t g x for some 0 0 the functional f u 0 t u x 2 u 2 x dx c 0 t g x u p x dx 2 p has a minimizer in the class of positive functions u c1 r for which u x t u x for all x r this minimizer is used to prove the existence of a positive periodic solution y c2 r of two dimensional lp minkowski problem y1 p x y x y x g x where p 0 2

a construction of mds 2d convolutional codes of rate 1 n based on superregular matrices abstract in this paper two dimensional convolutional codes with finite support are considered i e convolutional codes whose codewords have compact support indexed in n 2 and take values in f n where f is a finite field the main goal of this work is to analyze the free distance properties of this type of codes of rate 1 n and degree we first establish an upper bound on the maximum possible distance for these codes we then present particular constructions of two dimensional convolutional codes with finite support of rate 1 n and degree that attain such a bound and therefore have the maximum distance among all two dimensional convolutional codes with finite support with the same rate and degree we call such codes maximum distance separable two dimensional convolutional codes

on the _ minkowski problem a volume normalized formulation of the lp minkowski problem is presented this formulation has the advantage that a solution is possible for all p 1 including the degenerate case where the index p is equal to the dimension of the ambient space a new approach to the lp minkowski problem is presented which solves the volume normalized formulation for even data and all p 1

rotationally symmetric solutions to the l p minkowski problem abstract in this paper we study the l p minkowski problem for p n 1 which corresponds to the critical exponent in the blaschke santalo inequality we first obtain volume estimates for general solutions then establish a priori estimates for rotationally symmetric solutions by using a kazdan warner type obstruction finally we give sufficient conditions for the existence of rotationally symmetric solutions by a blow up analysis we also include an existence result for the l p minkowski problem which corresponds to the super critical case of the blaschke santalo inequality

on the uniqueness of lp minkowski problems the constant p curvature case in r3 abstract we study the c 4 smooth convex bodies k r n 1 satisfying k x u x 1 p where x s n k is the gauss curvature of k u is the support function of k and p is a constant in the case of n 2 either when p 1 0 or when p 0 1 in addition to a pinching condition we show that k must be the unit ball this partially answers a conjecture of lutwak yang and zhang about the uniqueness of the l p minkowski problem in r 3 moreover we give an explicit pinching constant depending only on p when p 0 1

convergence to travelling waves for quasilinear fisher kpp type equations we consider the cauchy problem ut u xx u t x r r u 0 x u0 x x r when the increasing function satisfies that 0 0 and the equation may degenerate at u 0 in the case of 0 0 we consider the case of u0 l r 0 u0 x 1 a e x r and the special case of u u u we prove that the solution approaches the travelling wave solution with speed c 1 spreading either to the right or to the left or to the two travelling waves moving in opposite directions

the lp brunn minkowski inequality for p 1 abstract in this paper we confirm the l p brunn minkowski inequality conjecture for p close to 1 the logarithmic brunn minkowski inequality is also verified for convex bodies close to the unit ball in hausdorff distance

on generator matrices of mds codes corresp it is shown that the family of q ary generalized reed solomon codes is identical to the family of q ary linear codes generated by matrices of the form i a where i is the identity matrix and a is a generalized cauchy matrix using cauchy matrices a construction is shown of maximal triangular arrays over gf q which are constant along diagonals in a hankel matrix fashion and with the property that every square subarray is a nonsingular matrix by taking rectangular subarrays of the described triangles it is possible to construct generator matrices i a of maximum distance separable codes where a is a hankel matrix the parameters of the codes are n k d for 1 leq n leq q 1 1 leq k leq n and d n k 1

the logarithmic minkowski problem for polytopes abstract the logarithmic minkowski problem asks for necessary and sufficient conditions for a finite borel measure on the unit sphere so that it is the cone volume measure of a convex body this problem was solved recently by boroczky lutwak yang and zhang for even measures boroczky et al 2013 8 this paper solves the case of discrete measures whose supports are in general position

variational characterization for the planar dual minkowski problem abstract in this paper we give a variational analysis to the planar dual minkowski problem in the sobolev space with the new variational characterization we can deal with existence results for prescribed not necessarily positive data meanwhile functional inequalities and multiple solutions are also obtained

on the uniqueness and continuity of the dual area measure abstract in this paper the uniqueness of the dual minkowski problem for the dual area measure is established via the dual minkowski inequality and the dual log minkowski inequality for real q n 1 it is proved that the weak convergence of the q th dual area measure implies the convergence of the corresponding convex bodies in the hausdorff metric and that the solution to the dual minkowski problem is continuous with respect to q

strongly mds convolutional codes maximum distance separable mds convolutional codes have the property that their free distance is maximal among all codes of the same rate and the same degree in this paper a class of mds convolutional codes is introduced whose column distances reach the generalized singleton bound at the earliest possible instant such codes are called strongly mds convolutional codes they also have a maximum or near maximum distance profile the extended row distances of these codes will also be discussed briefly

on the 2 dimensional dual minkowski problem abstract we study the 2 dimensional dual minkowski problem which is the following nonlinear problem on unit circle u u g u 1 u 2 u 2 2 k 2 s for any given positive continuous function g with 2 m periodic we prove that it is solvable for all k 1 and m 3 4 5

optimal boundary regularity for a singular monge ampere equation abstract in this paper we study the optimal global regularity for a singular monge ampere type equation which arises from a few geometric problems we find that the global regularity does not depend on the smoothness of domain but it does depend on the convexity of the domain we introduce a type to describe the convexity as a result we show that the more convex is the domain the better is the regularity of the solution in particular the regularity is the best near angular points

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

application of circulant matrices to the construction and decoding of linear codes the fourier transform technique is used to analyze and construct several families of double circulant codes the minimum distance of the resulting codes is lower bounded by 2 square root r and can be decoded easily employing the standard bch decoding algorithm or the majority logic decoder of reed muller codes a decoding procedure for reed solomon codes is presented based on a representation of the parity check matrix by circulant blocks the decoding procedure inherits both the relatively low time complexity of the berlekamp massey algorithm and the hardware simplicity characteristic of blahut s algorithm the procedure makes use of the encoding circuit together with a reduced version of blahut s decoder

a generalized rotationally symmetric case of the centroaffine minkowski problem abstract in this paper the centroaffine minkowski problem a critical case of the l p minkowski problem in the n 1 dimensional euclidean space is studied by its variational structure and the method of blow up analyses we obtain two sufficient conditions for the existence of solutions for a generalized rotationally symmetric case of the problem

on the l p monge ampere equation abstract for the case where 0 p 1 we prove the existence of solutions of the l p monge ampere equation for arbitrary measures and show that the solution is in general not unique

lp minkowski problem with not necessarily positive data abstract the l p minkowski problem in the plane has been the object of a number of recent investigations here we present various existence results for 0 p 2 in the case 0 p 2 a necessary and sufficient condition for the solvability of the problem is given

existence of solutions to the orlicz minkowski problem abstract in this paper the orlicz minkowski problem a generalization of the classical minkowski problem is studied using the variational method we obtain a new existence result of solutions to this problem for general measures

constructions of mds convolutional codes maximum distance separable mds convolutional codes are characterized through the property that the free distance attains the generalized singleton bound the existence of mds convolutional codes was established by two of the authors by using methods from algebraic geometry this correspondence provides an elementary construction of mds convolutional codes for each rate k n and each degree spl delta the construction is based on a well known connection between quasi cyclic codes and convolutional codes

on the lp dual minkowski problem abstract the l p dual curvature measure was introduced by lutwak yang zhang in an attempt to unify the l p brunn minkowski theory and the dual brunn minkowski theory the characterization problem for l p dual curvature measure called the l p dual minkowski problem is a fundamental problem in this unifying theory the l p dual minkowski problem contains the l p minkowski problem and the dual minkowski problem two major problems in modern convex geometry that remain open in general in this paper existence results on the l p dual minkowski problem in the weak sense will be provided moreover existence and uniqueness of the solution in the smooth category will also be demonstrated

geometric measures in the dual brunn minkowski theory and their associated minkowski problems a longstanding question in the dual brunn minkowski theory is what are the dual analogues of federer s curvature measures for convex bodies the answer to this is provided this leads naturally to dual versions of minkowski type problems what are necessary and sufficient conditions for a borel measure to be a dual curvature measure of a convex body sufficient conditions involving measure concentration are established for the existence of solutions to these problems

traveling wave phenomena in some degenerate reaction diffusion equations abstract in this paper we study the existence of travelling wave solutions t w s u x t x ct for the equation formula g u where the reactive part g u is as in the fisher kpp equation and different assumptions are made on the non linear diffusion term d u both functions d and g are defined on the interval 0 1 the existence problem is analysed in the following two cases case 1 d 0 0 d u 0 u 0 1 d and g c 2 0 1 d 0 0 and d 0 0 we prove that if there exists a value of c c for which the equation possesses a travelling wave solution of sharp type it must be unique by using some continuity arguments we show that for 0 c c there are no t w s while for c c the equation has a continuum of t w s of front type the proof of uniqueness uses a monotonicity property of the solutions of a system of ordinary differential equations which is also proved case 2 d 0 d 0 0 d and g c 2 0 1 d 0 0 if in addition we impose d 0 0 with d u 0 u 0 1 we give sufficient conditions on c for the existence of t w s of front type meanwhile if d 0 d u u 0 1 we analyse just one example d u u 2 and g u u 1 u which has oscillatory t w s for 0 c 2 and t w s of front type for c 2 in both the above cases we use higher order terms in the taylor series and the centre manifold theorem in order to get the local behaviour around a non hyperbolic point of codimension one in the phase plane

algebraic aspects of two dimensional convolutional codes two dimensional 2d codes are introduced as linear shift invariant spaces of admissible signals on the discrete plane convolutional and in particular basic codes are characterized both in terms of their internal properties and by means of their input output representations the algebraic structure of the class of all encoders that correspond to a given convolutional code is investigated and the possibility of obtaining 2d decoders free from catastrophic errors as veil as efficient syndrome decoders is considered some aspects of the state space implementation of 2d encoders and decoders via finite memory 2d system are discussed

the two dimensional lp minkowski problem and nonlinear equations with negative exponents abstract in this paper we present various existence results for nonlinear differential equations related to the l p minkowski problem in the plane and the one dimensional conformal curvature problem in particular the supercritical case that is p 2 for the l p minkowski problem with sign changing data is addressed a new geometric sharp inequality is also obtained

on the planar dual minkowski problem abstract in this paper we resolve the planar dual minkowski problem proposed by huang et al 2016 31 for all positive indices without any symmetry assumption more precisely given any q 0 and function f on s 1 bounded by two positive constants we show that there exists a convex body in the plane containing the origin in its interior whose dual curvature measure c q has density f in particular if f is smooth then is also smooth

nonuniqueness of solutions to the l p minkowski problem abstract in this paper we show by example that there is no uniform estimate for the l p minkowski problem as a result we obtain the nonuniqueness of solutions to the problem

the lp minkowski problem for n p 1 abstract chou and wang s existence result for the l p minkowski problem on s n 1 for p n 1 and an absolutely continuous measure is discussed and extended to more general measures in particular we provide an almost optimal sufficient condition for the case p 0 1

a priori estimates and existence of solutions to the prescribed centroaffine curvature problem abstract in this paper we study the prescribed centroaffine curvature problem in the euclidean space r n 1 this problem is equivalent to solving a monge ampere equation on the unit sphere it corresponds to the critical case of the blaschke santalo inequality by approximation from the subcritical case and using an obstruction condition and a blow up analysis we obtain sufficient conditions for the a priori estimates and the existence of solutions up to a lagrange multiplier

the lp minkowski problem for polytopes for 0 p 1 abstract necessary and sufficient conditions are given for the existence of solutions to the discrete l p minkowski problem for the critical case where 0 p 1