on the generators of spl zopf sub 4 cyclic codes of length 2 sup e results are presented on the generators of ideals in the ring spl zopf sub 4 x x sup n 1 in particular each ideal cyclic code has a unique distinguished set of generators that characterizes any cyclic code some results about dual codes are also included

complete distances of all negacyclic codes of length 2 s over bbz _ 2 a various kinds of distances of all negacyclic codes of length 2s over zopf2 a are completely determined using our structure theorems of negacyclic codes of length 2 s over zopf2 a we first calculate the hamming distances of all such negacyclic codes which particularly lead to the hamming weight distributions and hamming weight enumerators of several codes these hamming distances are then used to obtain their homogeneous lee and euclidean distances our techniques are extendable to the more general class of constacyclic codes namely the lambda constacyclic codes of length 2s over zopf2 a where lambda is any unit of zopf2 a with the form 4k 1 we establish the hamming homogeneous lee and euclidean distances of all such constacyclic codes

cyclic and negacyclic codes over finite chain rings the structures of cyclic and negacyclic codes of length n and their duals over a finite chain ring r are established when n is not divisible by the characteristic of the residue field r some cases where n is divisible by the characteristic of the residue field r are also considered namely the structure of negacyclic codes of length 2 sup t over spl zopf sub 2 sup m and that of their duals are derived

self normalized large deviations let x x n n geq 1 be a sequence of independent and identically distributed random variables the classical cramer chernoff large deviation states that lim n to infty n 1 ln p sum i 1 n x i n geq x ln rho x if and only if the moment generating function of x is finite in a right neighborhood of zero this paper uses n p 1 p v n p n p 1 p sum i 1 n x i p 1 p p 1 as the normalizing constant to establish a self normalized large deviation without any moment conditions a self normalized moderate deviation that is the asymptotic probability of p s n v n p geq x n for x n o n p 1 p is also found for any x in the domain of attraction of a normal or stable law as a consequence a precise constant in the self normalized law of the iterated logarithm of griffin and kuelbs is obtained applications to the limit distribution of self normalized sums the asymptotic probability of the t statistic as well as to the erdos renyi shepp law of large numbers are also discussed

gray isometries for finite chain rings and a nonlinear ternary 36 3 sup 12 15 code using tensor product constructions for the first order generalized reed muller codes we extend the well established concept of the gray isometry between z sub 4 spl delta sub l and z sub 2 sup 2 spl delta sub h to the context of finite chain rings our approach covers previous results by carlet see ibid vol 44 p 1543 7 1998 constantinescu see probl pered inform vol 33 no 3 p 22 8 1997 and ph d dissertation tech univ munchen munchen germany 1995 nechaev et al see proc ieee int symp information theory and its applications p 31 4 1996 and overlaps with heise et al see proc acct 6 pskov russia p 123 9 1998 and honold et al see proc acct 6 pskov russia p 135 41 1998 applying the gray isometry on z sub 9 we obtain a previously unknown nonlinear ternary 36 3 sup 12 15 code

negacyclic codes over z sub 4 of even length this paper generalizes the results from wolfmann see ibid vol 45 p 2527 2532 nov 1999 and vol 47 p 1773 1779 july 2001 classifying all negacyclic codes over z sub 4 of even length using a transform approach it is then shown which linear binary cyclic codes are images of negacyclic codes under the gray map in the process the concatenated structure of both negacyclic codes and binary repeated root cyclic codes is given

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

lee metric bch codes and their application to constrained and partial response channels shows that each code in a certain class of bch codes over gf p specified by a code length n spl les p sup m 1 and a runlength r spl les p 1 2 of consecutive roots in gf p sup m has minimum lee distance spl ges 2r for the very high rate range these codes approach the sphere packing bound on the minimum lee distance furthermore for a given r the length range of these codes is twice as large as that attainable by berlekamp s 1984 extended negacyclic codes the authors present an efficient decoding procedure based on euclid s algorithm for correcting up to r 1 errors and detecting r errors that is up to the number of lee errors guaranteed by the designed minimum lee distance 2r bounds on the minimum lee distance for r spl ges p 1 2 are provided for the reed solomon case i e when the bch code roots are in gf p the authors present two applications first lee metric bch codes can be used for protecting against bitshift errors and synchronization errors caused by insertion and or deletion of zeros in d k constrained channels second the code construction with its decoding algorithm can be formulated over the integer ring providing an algebraic approach to correcting errors in partial response channels where matched spectral null codes are used

a note on the q ary image of a q sup m ary repeated root cyclic code for n q p sup s where p ch f sub q s spl ges 1 v a q sup m ary repeated root cyclic code of length n with generator polynomial g x we give a partial answer about whether the q ary image of v is cyclic or not with respect to a certain basis for f sub q m over f sub q

on repeated root cyclic codes a parity check matrix for a q ary repeated root cyclic code is derived using the hasse derivative then the minimum distance of a q ary repeated root cyclic code is expressed in terms of the minimum distance of a certain simple root cyclic code with the help of this result several binary repeated root cyclic codes of lengths up to n 62 are shown to contain the largest known number of codewords for their given length and minimum distance the relative minimum distance d sub min n of q ary repeated root cyclic codes of rate r or r is proven to tend to zero as the largest multiplicity of a root of the generator g x increases to infinity it is further shown that repeated root cycle codes cannot be asymptotically better than simple root cyclic codes

an exponential nonuniform berry esseen bound for self normalized sums in this paper we shall derive exponential nonuniformberry esseen bounds in the central limit theorem for self normalized sums we show that the size of the error can be reduced considerably by replacing the usual standardization by self normalization in particular we establish the exponential bounds for the probability of the self normalized sums under the condition that the third moment is finite whereas an exponential moment assumption is required for the standardized sums applications to t statistics and the probabilities of moderate deviations of self normalized sums are also discussed

repeated root cyclic codes in the theory of cyclic codes it is common practice to require that n q 1 where n is the word length and f sub q is the alphabet it is shown that the even weight subcodes of the shortened binary hamming codes form a sequence of repeated root cyclic codes that are optimal in nearly all other cases one does not find good cyclic codes by dropping the usual restriction that n and q must be relatively prime this statement is based on an analysis for lengths up to 100 a theorem shows why this was to be expected but it also leads to low complexity decoding methods this is an advantage especially for the codes that are not much worse than corresponding codes of odd length it is demonstrated that a binary cyclic code of length 2n n odd can be obtained from two cyclic codes of length n by the well known mod u mod u v mod construction this leads to an infinite sequence of optimal cyclic codes with distance 4 furthermore it is shown that low complexity decoding methods can be used for these codes the structure theorem generalizes to other characteristics and to other lengths some comparisons of the methods using earlier examples are given

type ii codes even unimodular lattices and invariant rings we study self dual codes over the ring z sub 2k of the integers modulo 2k with relationships to even unimodular lattices modular forms and invariant rings of finite groups we introduce type ii codes over z sub 2k which are closely related to even unimodular lattices as a remarkable class of self dual codes and a generalization of binary type ii codes a construction of even unimodular lattices is given using type ii codes several examples of type ii codes are given in particular the first extremal type ii code over z sub 6 of length 24 is constructed which gives a new construction of the leech lattice the complete and symmetrized weight enumerators in genus g of codes over z sub 2k are introduced and the macwilliams identities for these weight enumerators are given we investigate the groups which fix these weight enumerators of type ii codes over z sub 2k and we give the molien series of the invariant rings of the groups for small cases we show that modular forms are constructed from complete and symmetrized weight enumerators of type ii codes shadow codes over z sub 2k are also introduced

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

a0 stable family of single step methods for semidiscretized parabolic problems a general family of single step methods for use in first order odes is presented this family of methods called ss3 methods embraces generalized trapezoidal methods ss21 methods methods and many other well known methods thus providing a good basis for comparison of these methods corresponding parameter values of ss3 methods for these methods are presented the analysis of ss3 methods is restricted to linear and symmetric systems stability convergence accuracy numerical dissipation and overshoot of the methods are considered the conditions for algorithm parameters to exhibit good characteristics in these respects are given it is shown that the optimal subfamily of ss3 methods coincides with methods

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

negacyclic and cyclic codes over z sub 4 the negashift spl nu of z sub 4 sup n is defined as the permutation of z sub 4 sup n such that spl nu a sub 0 a sub 1 spl middot spl middot spl middot a sub i spl middot spl middot spl middot a sub n 1 a sub n 1 a sub 0 spl middot spl middot spl middot a sub i spl middot spl middot spl middot a sub n 2 and a negacyclic code of length n over z sub 4 is defined as a subset c of z sub 4 sup n such that spl nu c c we prove that the gray image of a linear negacyclic code over z sub 4 of length n is a binary distance invariant not necessary linear cyclic code we also prove that if n is odd then every binary code which is the gray image of a linear cyclic code over z sub 4 of length n is equivalent to a not necessary linear cyclic code and this equivalence is explicitely described this last result explains and generalizes the existence already known of versions of kerdock preparata and others codes as doubly extended cyclic codes furthermore we introduce a family of binary linear cyclic codes which are gray images of z sub 4 linear negacyclic codes

quaternary quadratic residue codes and unimodular lattices we construct new self dual and isodual codes over the integers module 4 the binary images of these codes under the gray map are nonlinear but formally self dual the construction involves hensel lifting of binary cyclic codes quaternary quadratic residue codes are obtained by hensel lifting of the classical binary quadratic residue codes repeated hensel lifting produces a universal code defined over the 2 adic integers we investigate the connections between this universal code and the codes defined over z sub 4 the composition of the automorphism group and the structure of idempotents over z sub 4 we also derive a square root bound on the minimum lee weight and explore the connections with the finite fourier transform certain self dual codes over z sub 4 are shown to determine even unimodular lattices including the extended quadratic residue code of length q 1 where q spl equiv 1 mod8 is a prime power when q 23 the quaternary golay code determines the leech lattice in this way this is perhaps the simplest construction for this remarkable lattice that is known

nonlinear dynamics and loop formation in kirchhoff rods with implications to the mechanics of dna and cables the paper contributes a general dynamical formulation and numerical solution procedure for studying nonlinear and three dimensional dynamics of kirchhoff rods target applications include the dynamic formation of dna loops and supercoils as well as loops hockles in marine cables the formulation accommodates non homogeneous and non isotropic inextensible rods both with and without coupling of tension and torsion the utility of this formulation is illustrated by studying the dynamics and quasi static response of a clamped clamped rod subject to compression and or twist for slow loading rates the computed quasi static responses converge to published equilibrium solutions for a benchmark problem as loading rates increase new behaviors are observed including hysteresis in the neighborhood of equilibrium bifurcations the addition of chirality in the form of tension torsion coupling has a pronounced influence on the computed looped geometries this finding has implications for dna loops formed by dna protein binding as well as loop formation in helically wound wire and synthetic cables

asymptotic distribution of the largest off diagonal entry of correlation matrices suppose that we have observations from a dimensional population we are interested in testing that the variates of the population are independent under the situation where goes to infinity as a test statistic is chosen to be where is the sample correlation coefficient between the th coordinate and the th coordinate of the population under an independent hypothesis we prove that the asymptotic distribution of is an extreme distribution of type by using the chen stein poisson approximation method and the moderate deviations for sample correlation coefficients as a statistically more relevant result a limit distribution for where is spearman s rank correlation coefficient between the th coordinate and the th coordinate of the population is derived

constacyclic codes of length 2 s over galois extension rings of bbf _ 2 u bbf _2 we study all constacyclic codes of length 2s over gr rfr m the galois extension ring of dimension m of the ring rfr f2 uf2 the units of the ring gr rfr m are of the forms alpha and alpha ubeta where alpha beta are nonzero elements of f2m which correspond to 2 m 2m 1 such constacyclic codes first the structure and hamming distances of 1 ugamma constacyclic codes are established we then classify all cyclic codes of length 2s over gr rfr m and obtain a formula for the number of those cyclic codes as well as the number of codewords in each code finally one to one correspondences between cyclic and alpha constacyclic codes as well as 1 ugamma constacyclic and alpha ubeta constacyclic codes are provided via ring isomorphisms that allow us to carry over the results about cyclic and 1 ugamma constacyclic accordingly to all constacyclic codes of length 2s over gr rfr m

dynamic response of intrinsic continua for use in biological and molecular modeling explicit finite element formulation an intrinsic beam formulation has recently appeared aiaa j 2003 41 6 1131 1137 which presents the three dimensional equations of motion governing spatial and temporal changes in a one dimensional continua s curvature strain rectilinear velocity and angular velocity the formulation would suggest several computational advantages over more traditional displacement based continua approaches low order interpolation functions can describe generally curved and twisted continua configurations inter element displacements slopes strains and curvatures can be matched and finite rotational variables and their complexities are absent here we present a completed intrinsic continua finite element development and critical analysis which follows from an earlier preliminary treatment as applied to carbon nanotubes int j solids struct 2007 44 874 894 modeling of nodal displacements and rotations are included explicit time stepping with desired high frequency damping is accomplished using an implementation of the generalized method zero energy modes inherent in the formulation are also identified and rectified finally we document very good agreement between results predicted with the intrinsic continua finite element simulator and results generated using more traditional finite element simulations copyright 2009 john wiley sons ltd

self normalized cramer type large deviations for independent random variables let x 1 x 2 be independent random variables with zero means and finite variances it is well known that a finite exponential moment assumption is necessary for a cramer type large deviation result for the standardized partial sums in this paper we show that a cramer type large deviation theorem holds for self normalized sums only under a finite 2 th moment 0 1 in particular we show p s n v n x 1 x 1 o 1 1 x 2 d 2 n for 0 x d n where d n n i 1 ex 2 i 1 2 n i 1 e x i 2 1 2 and v n n i 1 x 2 i 1 2 applications to the studentized bootstrap and to the self normalized law of the iterated logarithm are discussed

negacyclic codes of length 2 sup s over galois rings codes over the ring of integers modulo 4 have been studied by many researchers negacyclic codes such that the length n of the code is odd have been characterized over the alphabet zopf4 and furthermore have been generalized to the case of the alphabet being a finite commutative chain ring in this paper we investigate negacyclic codes of length 2s over galois rings the structure of negacyclic codes of length 2s over the galois rings gr 2a m as well as that of their duals are completely obtained the hamming distances of negacyclic codes over gr 2a m in general and over zopf2 a in particular are studied among other more general results the hamming distances of all negacyclic codes over zopf2 a of length 4 8 and 16 are given the weight distributions of such negacyclic codes are also discussed

non linear unstable response of hanging chains abstract a chain hanging under its own weight is the canonical system to study the transition from taut to low tension chain behaviour the three dimensional non linear dynamics of a hanging chain driven harmonically from the top demonstrate rich dynamic behaviour as a function of frequency and forcing level the response may also result in catastrophic collapse at the lower end the driving mechanism is the loss of tension near the lower end when the response is dominated by large impulse like tension peaks the subsequent behaviour is that of an impulsively loaded cable characteristic of low tension lines

a linear construction for certain kerdock and preparata codes the nordstrom robinson kerdock and slightly modified preparata codes are shown to be linear over z4 the integers mod 4 the kerdock and preparata codes are duals over z4 and the nordstrom robinson code is self dual all these codes are just extended cyclic codes over z4 this provides a simple definition for these codes and explains why their hamming weight distributions are dual to each other firstand second order reed muller codes are also linear codes over z4 but hamming codes in general are not nor is the golay code

application of the generalized α method to the time integration of the cable dynamics equations abstract the box method is a popular finite difference scheme for solving the cable dynamics equations in ocean engineering applications in this paper we analyze the shortcomings of the temporal integration defined by the box method and propose a new procedure that retains the box method s spatial discretization but employs the generalized method for temporal discretization the stability and accuracy of the generalized method as applied to the cable dynamics equations are developed this analysis procedure demonstrates the formulation of the semi discrete governing equations using spatial finite differences and the applicability of amplification matrix stability analysis to finite difference problems an example is presented that demonstrates the robustness of the new approach

poisson convergence for dissociated statistics summary a poisson limit theorem is derived for the number of large values observed among comparisons of independent but not necessarily identically distributed random variables the comparisons made need not be the same and may depend on the two variables being compared an application to the assessment of large numbers of correlation coefficients is given