classification of convolutional codes convolutional codes have the structure of an f z module to study their properties it is desirable to classify them as the points of a certain algebraic variety by considering the correspondence of submodules and the points of certain quotient schemes and the inclusion of these as subvarieties of certain grassmannians one has a one to one correspondence of convolutional codes and the points of these subvarieties this classification of convolutional codes sheds light on their structure and proves to be helpful to give bounds on their free distance and to define convolutional codes with good parameters

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

on maximum typical and generic ranks we show that for several notions of rank including tensor rank waring rank and generalized rank with respect to a projective variety the maximum value of rank is at most twice the generic rank we show that over the real numbers the maximum value of the real rank is at most twice the smallest typical rank which is equal to the complex generic rank

homotopy techniques for tensor decomposition and perfect identifiability let t be a general complex tensor of format n1 nd when the fraction i ni 1 i ni 1 is an integer and a natural inequality called balancedness is satisfied it is expected that t has finitely many minimal decomposition as a sum of decomposable tensors we show how homotopy techniques allow us to find all the decompositions of t starting from a given one in particular this gives a guess true with high probability about the total number of these decompositions this guess matches exactly with all cases previously known and predicts several unknown cases some surprising experiments yielded two new cases of generic identifiability formats 3 4 5 and 2 2 2 3 which have a unique decomposition as the sum of 6 and 4 decomposable tensors respectively we conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds i e the only identifiable cases building on the computational experiments we also use basic tools from algebraic geometry to prove these two new cases are indeed generically identifiable

every convolutional code is a goppa code convolutional goppa codes cgc were defined in appl algebra eng comm comput vol 15 pp 51 61 2004 and ieee trans inf theory vol 52 340 344 2006 in this paper we prove that every convolutional code is a cgc defined over a smooth curve over bbf q z and we give an explicit construction of convolutional codes as cgc over the projective line bbp bbf q z 1 we characterize which convolutional codes are defined by a complete linear system over curves of genus 0 1 and over hyperelliptic curves we apply these results to provide detailed constructions of some linear block codes as goppa codes

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

quasi cyclic unit memory convolutional codes unit memory convolutional codes with generator matrices which are composed of circulant submatrices are introduced this structure facilitates the analysis of efficient search for good codes equivalences among such codes and some of the basic structural properties are discussed in particular catastrophic encoders and minimal encoders are characterized and dual codes treated further various distance measures are discussed and a number of good codes some of which result from efficient computer search and some of which result from known block codes are presented

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

optimal and near optimal encoders for short and moderate length tail biting trellises the results of an extensive search for short and moderate length polynomial convolutional encoders for time invariant tail biting representations of block codes at rates r 1 4 1 3 1 2 and 2 3 are reported the tail biting representations found are typically as good as the best known block codes

on the ideals of secant varieties to certain rational varieties abstract if x p n is a reduced and irreducible projective variety it is interesting to find the equations describing the higher secant varieties of x in this paper we find those equations in the following cases x p n 1 p n t p n is the segre embedding of the product and n is large with respect to the n i theorem 2 4 the x are some unbalanced segre veronese embeddings x is a del pezzo surface

column distances of convolutional codes over mathbb z _ p r maximum distance profile codes over finite non binary fields have been introduced and thoroughly studied in the last decade these codes have the property that their column distances are maximal among all codes of the same rate and degree in this paper we aim at studying this fundamental concept in the context of convolutional codes over a finite ring we extensively use the concept of p encoder to establish the theoretical framework and derive several bounds on the column distances in particular a method for constructing not necessarily free maximum distance profile convolutional codes over mathbb z p r is presented

superregular matrices and applications to convolutional codes abstract the main results of this paper are twofold the first one is a matrix theoretical result we say that a matrix is superregular if all of its minors that are not trivially zero are nonzero given a a b a b superregular matrix over a field we show that if all of its rows are nonzero then any linear combination of its columns with nonzero coefficients has at least a b 1 nonzero entries secondly we make use of this result to construct convolutional codes that attain the maximum possible distance for some fixed parameters of the code namely the rate and the forney indices these results answer some open questions on distances and constructions of convolutional codes posted in the literature 6 9

on the matrix equation xa ax xp abstract we study the matrix equation xa ax x p in m n k for 1 p n it is shown that every matrix solution x is nilpotent and that the generalized eigenspaces of a are x invariant for a being a full jordan block we describe how to compute all matrix solutions combinatorial formulas for a m x l x l a m and ax l are given the case p 2 is a special case of the algebraic riccati equation

on the construction of 1 d mds convolutional goppa codes we show that the free distance as a function on a space parameterizing a family of convolutional codes is a lower semicontinuous function and that therefore the property of being maximum distance separable mds is an open condition for a class of convolutional codes an algorithm is offered to compute the free distance the behavior of the free distance by enlargements of the alphabet and by increasing the length is also studied as an application the algebraic equations characterizing the subfamily of mds codes are explicitly computed for families of 1 d convolutional goppa codes

robust feedback stabilization of linear mimo systems using generalized homogenization a robust nonlinear control is designed for stabilizing linear multi input multi output systems the presented control law homogenizes a linear system without its transformation to a canonical form with a specified degree and stabilizes it in a finite time or with a fixed time attraction to any compact set containing the origin if the degree of homogeneity is negative positive the tuning procedure is formalized in an linear matrix inequalities lmi form performance of the approach is illustrated by numerical and experimental examples

algorithmic construction of trellis codes an algorithmic approach is proposed whereby long convolutional codes of rate r sub c k n can easily be constructed for any chosen signal constellation in signal space these algorithms are iterative and in each step a number of candidate codes are found which locally maximize the distance hamming or euclidean between the codewords the result is not necessarily a free distance optimizing code however since the construction complexity can be chosen optimal codes are quite frequently found the codes ae constructed such that a rapid growth of the column distance is achieved a method of combining two codes into a single code of twice the constraint length is also presented

multidimensional quasi cyclic and convolutional codes we introduce multidimensional analogues of quasi cyclic qc codes and study their algebraic structure we demonstrate a concatenated structure for multidimensional qc codes and use this to prove that this class of codes is asymptotically good we also relate the new family of codes to convolutional codes it is known that the minimum distance of qc codes provides a natural lower bound on the free distance of convolutional codes we show that the same relation also holds between certain rank one 2 d convolutional codes and the related multidimensional qc codes we provide examples which show that our bound is sharp in some cases we also present some optimal 2 d qc codes along the way we provide a condition on the encoders of rank one convolutional codes which are equivalent to noncatastrophicity for 1 d convolutional codes in the n text d case n 1 our condition is sufficient for the noncatastrophicity of the encoder

a class of one dimensional mds convolutional codes a class of one dimensional convolutional codes will be presented they are all mds codes i e have the largest distance among all one dimensional codes of the same length n and overall constraint length furthermore their extended row distances are computed and they increase with slope at least n in certain cases of the algebraic parameters we will also derive parity check matrices of vandermonde type for these codes finally cyclicity in the convolutional sense of 8 will be discussed for our class of codes it will turn out that they are cyclic if and only if the field element used in the generator matrix has order n this can be regarded as a generalization of the block code case

construction of partial unit memory mds convolutional codes maximum distance separable mds convolutional codes form an optimal family of convolutional codes the study of which is of great importance there are very few general algebraic constructions of mds convolutional codes in this paper we construct a large family of partial unit memory mds convolutional codes over mathbb f q with flexible parameters compared with the previous work the field size q required to define these codes is much smaller the construction also leads to many new strongly mds convolutional codes an important subclass of mds convolutional codes some examples are presented at the end of this paper

an iterative algorithm for solving a pair of matrix equations ayb e cyd f over generalized centro symmetric matrices a matrix email protected r n x n is said to be a symmetric orthogonal matrix if p p t p 1 a matrix email protected r n x n is said to be generalized centro symmetric generalized central anti symmetric with respect to p if a pap a pap the generalized centro symmetric matrices have wide applications in information theory linear estimate theory and numerical analysis in this paper we propose a new iterative algorithm to compute a generalized centro symmetric solution of the linear matrix equations ayb e cyd f we show when the matrix equations are consistent over generalized centro symmetric matrix y for any initial generalized centro symmetric matrix y 1 the sequence y k generated by the introduced algorithm converges to a generalized centro symmetric solution of matrix equations ayb e cyd f the least frobenius norm generalized centro symmetric solution can be derived when a special initial generalized centro symmetric matrix is chosen furthermore the optimal approximation generalized centro symmetric solution to a given generalized centro symmetric matrix can be derived several numerical examples are given to show the efficiency of the presented method

vector bundles give equations of cactus varieties abstract we prove that vector bundles give equations of cactus varieties we derive from it that equations coming from vector bundles are not enough to define secant varieties of veronese varieties in general

convolutional codes with maximum distance profile abstract maximum distance profile codes are characterized by the property that two trajectories which start at the same state and proceed to a different state will have the maximum possible minimum distance from each other relative to any other convolutional code of the same rate and degree in this paper we use methods from systems theory to characterize maximum distance profile codes algebraically the main result shows that maximum distance profile codes form a generic set inside the variety which parametrizes the set of convolutional codes of a fixed rate and a fixed degree

a new class of superregular matrices and mdp convolutional codes abstract this paper deals with the problem of constructing superregular matrices that lead to mdp convolutional codes these matrices are a type of lower block triangular toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular we present a new class of matrices that are superregular over a sufficiently large finite field f such construction works for any given choice of characteristic of the field f and code parameters n k such that n k we also discuss the size of f needed so that the proposed matrices are superregular

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

convolutional codes with maximum column sum rank for network streaming the column hamming distance of a convolutional code determines the error correction capability when streaming over a class of packet erasure channels we introduce a metric known as the column sum rank that parallels the column hamming distance when streaming over a network with link failures we prove the rank analogues of several known column hamming distance properties and introduce a new family of convolutional codes that maximize the column sum rank up to the code memory our construction involves finding a class of super regular matrices that preserve this property after multiplication with non singular block diagonal matrices in the ground field

on negacyclic mds convolutional codes new families of classical and quantum optimal negacyclic convolutional codes are constructed in this paper this optimality is in the sense that they attain the classical quantum generalized singleton bound the constructions presented in this paper are performed algebraically and not by computational search

typical rank and indscal dimensionality for symmetric three way arrays of order i 2 2 or i 3 3 a peculiar property of three way arrays is that the rank they typically have does not necessarily coincide with the maximum possible rank given their order typical tensorial rank has much been studied over algebraically closed fields however very few results have been found pertaining to the typical rank of three way arrays over the real field these results refer to arrays sampled randomly from continuous distributions arrays that consist of symmetric slices do not fit into this sampling scheme the present paper offers typical rank results over the real field for arrays containing symmetric slices of order 2 2a nd 3 3 symmetric arrays often appear to have lower typical ranks than their asymmetric counterparts this paper also examines whether or not the rank of a symmetric array coincides with the smallest number of dimensions that allow a perfect fit of indscal for all cases considered this is indeed true thus a full indscal solution may require fewer dimensions for a symmetric array than a full cp decomposition applied to an asymmetric array of the same size the reverse situation does not seem to arise next we examine in which cases cp solutions inevitably are indscal solutions finally the rank reducing impact of double standardizing the slices is discussed

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 optimal constacyclic codes abstract in this paper we construct new families of maximum distance separable mds convolutional codes derived from classical constacyclic codes additionally we also construct new families of almost mds block and convolutional codes most of the new convolutional codes constructed here are unit memory and have degree 2 moreover based on a different method we construct families of asymmetric quantum codes known in the literature all these results are mainly derived from the investigation of suitable properties on cyclotomic cosets of such codes

bounds on distances and error exponents of unit memory codes binary unit memory codes originally introduced by lee are investigated a few examples of constant unit memory codes are given and bounds on the distance profile and the free distances are discussed for time varying codes asymptotic lower bounds on the distance profile and the free distance are given the error probability for the codes used on a memoryless binary input output symmetric channel is asymptotically upper bounded the asymptotic results for the free distance and the error probability which are in some respects better than for conventional convolutional codes are interpreted by forney s inverse concatenation construction

the existence of strongly mds convolutional codes it is known that maximum distance separable and maximum distance profile convolutional codes exist over large enough finite fields of any characteristic for all parameters n k delta it has been conjectured that the same is true for convolutional codes that are strongly maximum distance separable using methods from linear systems theory we resolve this conjecture by showing that over a large enough finite field of any characteristic codes which are simultaneously maximum distance profile and strongly maximum distance separable exist for all parameters n k delta

algebraic lower bounds on the free distance of convolutional codes a new module structure for convolutional codes is introduced and used to establish further links with quasi cyclic and cyclic codes the set of finite weight codewords of an n k convolutional code over fq is shown to be isomorphic to an fq x submodule of fq n x where fq n x is the ring of polynomials in indeterminate x over fq n an extension field of fq such a module can then be associated with a quasi cyclic code of index n and block length nl viewed as an fq x submodule of fq n x langxl 1rang for any positive integer l using this new module approach algebraic lower bounds on the free distance of a convolutional code are derived which can be read directly from the choice of polynomial generators links between convolutional codes and cyclic codes over the field extension fq n are also developed and bose chaudhuri hocquenghem bch type results are easily established in this setting techniques to find the optimal choice of the parameter l are outlined

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

generalized column distances the notion of generalized hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography in this paper we extend this concept to the context of convolutional codes in particular we focus on column distances and introduce the novel notion of generalized column distances gcd we first show that the hierarchy of gcd is strictly increasing we then provide characterizations of such distances in terms of the truncated parity check matrix of the code that will allow us to determine their values finally the case in which the parity check matrix is in systematic form is treated

inverse system of a symbolic power ii the waring problem for forms abstract the classical waring problem for forms is to determine the smallest length s of an additive decomposition of a general degree d homogeneous polynomial or form f in r variables as sum of s d th powers of linear forms we show that its solution is implied by a result of j alexander and a hirschowitz concerning the hilbert functions of the ideal of functions vanishing to order two at a generic set of s points in p r 1 using macaulay s inverse systems we show that the alexander hirschowitz result is equivalent to determining the number of linear syzygies of s homogeneous forms in r variables that are d th powers of a given set of general linear forms we also determine the dimension of the family of degree d forms that have additive decompositions of length s we then study several notions of length for forms f having to do with the kind of length s zero dimensional schemes z in p r 1 whose defining ideal i z annihilates the inverse system of f when z is to consist of distinct points we obtain the above length of additive decomposition of f when z is smoothable we obtain the smoothable length of f when z is arbitrary we obtain a scheme length of f all these lengths are at least as large as the dimension of the vector space of all order i partial derivates of f for each i the above mentioned length functions are distinct using results about the existence of nonsmoothable gorenstein point singularities in codimension 4 we show that when r 5 there are forms f of scheme length s which are not in the closure of the family of forms having additive decompositions of length s finally we propose a new set of waring problems for forms using these lengths

a note on the matrix equation xa ax xp abstract another more elementary proof is given of proposition 2 3 in a recent paper by dietrich burde this proposition says that n n matrices a and x over an algebraically closed field k of characteristic zero satisfying the equality in the title of this paper can be simultaneously triangularized the proposed proof is based on the shemesh criterion for two matrices to have a common eigenvector

on behaviors and convolutional codes it is well known that a convolutional code is essentially a linear system defined over a finite field in this paper we elaborate on this connection we define a convolutional code as the dual of a complete linear behavior in the sense of willems 1979 using ideas from systems theory we describe a set of generalized first order descriptions for convolutional codes as an application of these ideas we present a new algebraic construction for convolutional codes

superregular lower triangular toeplitz matrices for low delay wireless streaming a matrix is termed superregular if all of its possible submatrices are non singular superregular lower triangular toeplitz matrices are useful for mds convolutional codes and sequential network codes in this paper we present the explicit matrix constructions for superregular lower triangular toeplitz matrices in mathrm gf 2 p k times k k leq 5 for k 5 we provide a greedy algorithm which over sufficiently large fields is guaranteed to find a superregular lower triangular toeplitz matrix we introduce product preserving joint superregularity and extend our explicit matrix constructions to these cases we provide methods for deriving the exact symbol loss probability and delay for any deterministic block code we derive the exact symbol loss probability and delay for codes using a superregular lower triangular matrix and for codes using two product preserving jointly superregular lower triangular matrices we then compare these results with those obtained from both simulations and our practical implementation and for each case we also compare with random based codes furthermore our experiments show a gain in coding throughput above 40 for superregular lower triangular toeplitz matrices over random matrices

a bound for the waring rank of the determinant via syzygies abstract we show that the waring rank of the 3 3 determinant previously known to be between 14 and 18 is at least 15 we use syzygies of the apolar ideal which have not been used in this way before additionally we show that the symmetric cactus rank of the 3 3 permanent is at least 14

how to solve the matrix equation xa ax f x abstract let f be an analytic function defined on a complex domain and a m n c we assume that there exists a unique satisfying f 0 when f 0 and a is non derogatory we completely solve the equation xa ax f x this generalizes burde s results when f 0 we give a method to solve completely the equation xa ax f x we reduce the problem to solving a sequence of sylvester equations solutions of the equation f xa ax x are also given in particular cases

on the rank of a symmetric form abstract we give a lower bound for the degree of a finite apolar subscheme of a complex symmetric form f in terms of the degrees of the generators of the annihilator ideal f in the special case when f is a monomial x 0 d 0 x 1 d 1 x n d n with d 0 d 1 d n 1 d n we deduce that the minimal length of an apolar subscheme of f is d 0 1 d n 1 1 and if d 0 d n then this minimal length coincides with the rank of f

on the algebraic parameters of convolutional codes with cyclic structure in this paper convolutional codes with cyclic structure will be investigated these codes can be understood as left principal ideals in a suitable skew polynomial ring it has been shown in 4 that only certain combinations of the algebraic parameters field size length dimension and forney indices can occur for such cyclic codes we will investigate whether all these combinations can indeed be realized by a suitable cyclic code and if so how to construct such a code a complete characterization and construction will be given for minimal cyclic codes it is derived from a detailed investigation of the units in the skew polynomial ring

on cyclotomic cosets and code constructions abstract new properties of q ary cyclotomic cosets modulo n q m 1 where q 3 is a prime power are investigated in this paper based on these properties the dimension as well as bounds for the designed distance of some families of classical cyclic codes can be computed as an application new families of nonbinary calderbank shor steane css quantum codes as well as new families of convolutional codes are constructed in this work these new css codes have parameters better than the ones available in the literature the convolutional codes constructed here have free distance greater than the ones available in the literature

on the weight distribution of convolutional codes detailed information about the weight distribution of a convolutional code is given by the adjacency matrix of the state diagram associated with a minimal realization of the code we will show that this matrix is an invariant of the code moreover it will be proven that codes with the same adjacency matrix have the same dimension and the same forney indices and finally that for one dimensional binary convolutional codes the adjacency matrix determines the code uniquely up to monomial equivalence

convolutional goppa codes in this correspondence we define convolutional goppa codes over algebraic curves and construct their corresponding dual codes examples over the projective line and over elliptic curves are described obtaining in particular some maximum distance separable mds convolutional codes

skew symmetric matrix pencils codimension counts and the solution of a pair of matrix equations the homogeneous system of matrix equations x t a ax xb b t bx 0 0 where a b is a pair of skew symmetric matrices of the same size is considered we establish the general solution

on the parametrization of conditioned invariant subspaces and observer theory the present paper is an in depth analysis of the set of conditioned invariant subspaces of a given observable pair c a we do this analysis in two different ways one based on polynomial models starting with a characterization obtained in p a fuhrmann linear operators and systems in hilbert space 1981 ieee trans automat control ac 26 1981 284 the other being a state space approach toeplitz operators projections in polynomial and rational models wiener hopf factorizations and factorization indices all appear and are tools in the characterizations we single out an important subclass of conditioned invariant subspaces namely the tight ones which already made an appearance in p a fuhrmann u helmke systems control lett 30 1997 217 a precursor of the present paper of particular importance for the study of the parametrization of the set of conditioned invariant subspaces of an observable pair c a is the structural map that associates with any reachable pair with only the input dimension constrained a uniquely determined conditioned invariant subspace the construction of this map uses polynomial models and the shift realization new objects the partial observability and reachability matrices are introduced which are needed for the state space characterizations kernel and image representations for conditioned invariant subspaces are derived uniqueness of a kernel representation of a conditioned invariant subspace is shown to be equivalent to tightness we pass on to an analysis and derivation of the kronecker hermite canonical form for full column rank rectangular polynomial matrices this extends the work of a e eckberg a characterization of linear systems via polynomial matrices and module theory ph d thesis mit cambridge ma 1974 g d forney siam j control optim 13 1973 493 and d hinrichsen h f munzner and d pratzel wolters systems

the solution to the waring problem for monomials and the sum of coprime monomials abstract in this paper we compute the waring rank of any polynomial of the form f i 1 r m i where the m i are pairwise coprime monomials i e gcd m i m j 1 for i j in particular we determine the waring rank of any monomial as an application we show that certain monomials in three variables give examples of forms of rank higher than the generic form as a further application we produce a sum of power decomposition for any form which is the sum of pairwise coprime monomials

brief paper time scaling symmetry and zeno solutions symmetries are often helpful in reducing the complexity of the analysis of dynamical systems here we discuss a symmetry that combines a transformation in space with a scaling of time examples are given of a number of nonsmooth dynamical systems in which a symmetry of this type occurs the time scaling symmetry can be combined with return mapping analysis to achieve a dimension reduction in addition to the one already obtained from considering the return map the method is applied in a study of zeno behavior in linear systems with relay feedback

on cyclic codes of composite length and the minimum distance in an interesting paper prof c ding provided three constructions of cyclic codes of length being a product of two primes numerical data shows that many codes from these constructions are best cyclic codes of the same length and dimension over the same finite field however not much is known about these codes in this paper we explain some of the numerical data by developing a general method on cyclic codes of composite length and on estimating the minimum distance we also provide a general construction of cyclic codes of composite length which are related to ding s constructions numerical data shows that it produces many best cyclic codes as well finally we point out how these cyclic codes can be used to construct convolutional codes with large free distance

on the algebraic boundaries among typical ranks for real binary forms abstract we describe the algebraic boundaries of the regions of real binary forms with fixed typical rank and of degree at most eight showing that they are dual varieties of suitable coincident root loci

a comparison of different notions of ranks of symmetric tensors abstract we introduce various notions of rank for a high order symmetric tensor taking values over the complex numbers namely rank border rank catalecticant rank generalized rank scheme length border scheme length extension rank and smoothable rank we analyze the stratification induced by these ranks the mutual relations between these stratifications allow us to describe the hierarchy among all the ranks we show that strict inequalities are possible between rank border rank extension rank and catalecticant rank moreover we show that scheme length generalized rank and extension rank coincide

representing elements of the weyl algebra by labeled trees it is shown how arbitrary elements of the weyl algebra can be represented by labeled plane trees using normal ordering several examples are treated and combinatorial aspects are discussed also possible avenues for future research are described