the minimal polynomial of a sequence obtained from the componentwise linear transformation of a linear recurring sequence let s s1 s2 sm be a linear recurring sequence with terms in gf qn and t be a linear transformation of gf qn over gf q denote t s t s1 t s2 t sm in this paper we first present counter examples to show that the main result in a m youssef g gong on linear complexity of sequences over gf 2n theoret comput sci 352 2006 288 292 is not correct in general since lemma 3 in that paper is incorrect then we determine the minimal polynomial of t s if the canonical factorization of the minimal polynomial of s without multiple roots is known and thus present the solution to the main problem which was considered in the above paper but incorrectly solved additionally as a special case we determine the minimal polynomial of t s if the minimal polynomial of s is primitive finally we give an upper bound on the linear complexity of t s when t exhausts all possible linear transformations of gf qn over gf q this bound is tight in some cases

the ratio of uniforms approach for generating discrete random variates abstract the most efficient algorithms for sampling from classical discrete distributions are based upon the acceptance rejection principle they are complicated and not easy to understand by adapting the ratio of uniforms method to unimodal discrete distributions sampling procedures can be established which are both simple and fast algorithms for the hypergeometric distribution are developed and compared with competing methods

the exact approximation method for generating random variables in a computer abstract a suitably chosen approximation to the inverse of a probability distribution can lead to exact and very fast methods for generating random variables if the approximation is made exact by adjusting the argument of the approximating function this article describes the basic method and extensions of it it gives four examples of which two are methods for generating gamma and t variates that while meant to illustrate the basic method show promise of being faster than the best current methods

solving homogeneous linear equations over gf 2 via block wiedemann algorithm we propose a method of solving large sparse systems of homogeneous linear equations over g f 2 the field with two elements we modify an algorithm due to wiedemann a block version of the algorithm allows us to perform 32 matrix vector operations for the cost of one the resulting algorithm is competitive with structured gaussian elimination in terms of time and has much lower space requirements it may be useful in the last stage of integer factorization we address here the problem of solving a large sparse system of homogeneous linear equations over gf 2 the field with two elements one important application which motivates the present work arises in integer factorization during the last stage of most integer factorization algorithms we are presented with a large sparse integer matrix and are asked to find linear combinations of the columns of this matrix which vanish modulo 2 for example 7 the matrix may have 100 000 columns with an average of 15 nonzero entries per column for this application we would like to obtain several solutions because a given solution will lead to a nontrivial factorization with probability 112 with n independent solutions our probability of finding a factorization rises to 1 2 structured gaussian elimination can be used 7 but as problems get larger it may become infeasible to store the matrices obtained in the intermediate stages of gaussian elimination the wiedemann algorithm 9 71 has smaller storage requirements one need only store a few vectors and an encoding of a sparse matrix not a dense matrix as occurs in gaussian elimination after fillin and it may have fewer computational steps since one takes advantage of the sparseness of the matrix but its efficiency is hampered by the fact that the algorithm acts on only one bit at a time in the present paper we work with blocks of vectors at a single time by treating 32 vectors at a time on a machine with 32 bit words we can perform 32 matrix vector products at once thus considerably decreasing the cost of indexing this can be viewed as a block wiedemann algorithm the main technical difficulty is in obtaining the correct generalization of the berlekamp massey algorithm to a block version namely a multidimensional version of the extended euclidean algorithm received by the editor november 20 1991 and in revised form july 24 1992 1991 mathematics subject classijication primary 15a33 11y05 11 04 15 04 1994 amencan mathernat cal soc ety 0025 571 8 94 1 oo 25 per page

quadriphase sequences for spread spectrum multiple access communication this paper studies constructions for quadriphase sequences that are suitable for use as signature sequences in quadriphase spread spectrum multiple access communications quadriphase sequences are closely related to biphase i e binary sequences and many of the properties of the former can be expressed in terms of properties of the latter methods of construction are presented which obtain sets of quadriphase sequences from sets of biphase sequences methods of construction of quadriphase sequences are provided based upon the properties of the multiplicative characters of the finite field gf q where q equiv 1 mod 4 these methods use codewords from low rate reed solomon codes which are mapped onto the fourth roots of unity 2q 2 sequences of period q 1 are constructed for which the maximum magnitudes of the periodic cross correlation and the periodic out of phase autocorrelation are bounded by 3 sqrt q 5

shift register synthesis modulo m the berlekamp algorithm takes a sequence of elements from a field and finds the shortest linear recurrence or linear feedback shift register that can generate the sequence we present an algorithm which generalizes berlekamp s to the case when the elements of the sequence are integers modulo m where m is an arbitrary but known integer details will be published elsewhere

theory and applications of q ary interleaved sequences a new class of q ary sequences called interleaved sequence is introduced their periods shift equivalence linear spans and autocorrelation functions are derived the interleaved sequences include a large number of popular sequences such as multiplexed sequences clock controlled sequences kasami 1966 sequences gmw sequences geometric sequences and no 1989 sequences a special class of the interleaved sequences is constructed by mapping gf q sup m sequences into gf q sequences in terms of different bases of gf q sup m over gf q as an application of the theory of interleaved sequences some new families of binary pseudo random sequences are constructed which have large linear spans optimal periodic cross autocorrelation functions balance and the rapidly hopped properties a complete comparison of the new family of sequences with the gold sequence family the kasami small and large set sequence families the bent function sequence family and the no sequence family is discussed this shows that the new sequence families have important advantages for use in spread spectrum multiple access communication systems

new array codes for multiple phased burst correction an optimal family of array codes over gf q for correcting multiple phased burst errors and erasures where each phased burst corresponds to an erroneous or erased column in a code array is introduced as for erasures these array codes have an efficient decoding algorithm which avoids multiplications or divisions over extension fields replacing these operations with cyclic shifts of vectors over gf q the erasure decoding algorithm can be adapted easily to handle single column errors as well the codes are characterized geometrically by means of parity constraints along certain diagonal lines in each code array thus generalizing a previously known construction for the special case of two erasures algebraically they can be interpreted as reed solomon codes when q is primitive in gf q the resulting codes become conventional reed solomon codes of length p over gf q sup p 1 in which case the new erasure decoding technique can be incorporated into the berlekamp massey algorithm yielding a faster way to compute the values of any prescribed number of errors

non uniform random variate generation by the vertical strip method abstract in this paper a method called the vertical strip vs method is proposed for generating non uniform random variates with a given density it can be considered as an improvement of the grid method as the vs method avoids setting up a directory to store information on big rectangles unlike the horizontal strip method that is based on the riemann integral the vs method is based on the lebesgue integral and can be applied to unbounded densities or densities with infinite support applications of the vs method for generating random variates which follow the exponential distribution and normal density are also given

the random generation of directed animals abstract in this paper we propose an algorithm to randomly generate a directed animal directed animals are well known combinatorial objects and have been widely used for modelling the physical phenomenon of percolation the algorithm consists of three steps and we prove that each of them is performed in linear time finally we report the results of our experiments made by means of appropriate computer programs in order to give empirical evidence that our algorithm really works

relative quantification based on logistic models for individual polymerase chain reactions the quantitative real time reverse transcription polymerase chain reaction rt pcr technology measures molecular variations in specific biomarkers relative quantification determines the target expression relative to an external standard or reference sample and should be adjusted for the pcr efficiencies actually achieved more accurate methods of estimating pcr efficiency require a number of serial dilutions of the target sample which is not generally feasible for clinical specimens alternatively the efficiency of a single reaction may be estimated by considering kinetic data from this reaction the current methods of estimating individual reaction efficiency require finding its exponential phase which may affect the accuracy and precision of efficiency estimates thus a model adequately representing all available kinetic rt pcr data is preferable but no such model is currently in use for relative quantification in this work we use a logistic model for all kinetic data from each rt pcr and propose a new method of efficiency adjusted relative quantification based on the estimates from the fitted logistic models this method allows incorporating multiple replicates and possibly multiple reference housekeeping genes for estimating relative expression and corresponding confidence interval real kinetic rt pcr data are used to compare the proposed and standard methods the methods are applied to the clinical data from the ongoing study of guanylyl cyclase c as a biomarker for colorectal cancer

random generation of trees and other combinatorial objects abstract in this paper we present a general method for the random generation of some classes of combinatorial objects our basic idea is to translate eco method enumerating combinatorial objects from a method for the enumeration of combinatorial objects into a random generation method the algorithms we illustrate are based on the concepts of succession rule and generating tree the former is a law that predicts the combinatorial object class growth according to a given parameter the generating tree related to a given succession rule is a particular labelled plane tree that represents the rule in an extensive way each node of a generating tree can also be seen as a particular combinatorial object and so a random path in the generating tree coincides with the random generation of that combinatorial object the generation is uniform if we take the probability of each branch to be selected into account when the path is generated we also give the formulae evaluating complexity finally we take the class of m ary trees into consideration in order to illustrate our general method in this case the average time complexity of the generating algorithm can be estimated as o mn

sequential importance sampling for multiway tables we describe an algorithm for the sequential sampling of entries in multiway contingency tables with given constraints the algorithm can be used for computations in exact conditional inference to justify the algorithm a theory relates sampling values at each step to properties of the associated toric ideal using computational commutative algebra in particular the property of interval cell counts at each step is related to exponents on lead indeterminates of a lexicographic grobner basis also the approximation of integer programming by linear programming for sampling is related to initial terms of a toric ideal we apply the algorithm to examples of contingency tables which appear in the social and medical sciences the numerical results demonstrate that the theory is applicable and that the algorithm performs well

the equivalence of weak strong and complete convergence in l_1 for kernel density estimates theorem 1 let k be a nonnegative borel measurable function on rd with f k x dx 1 then the following conditions are equivalent i j 0 in probability as n oo some f ii jn o 0 in probability as n s oo all f iii jn s 0 almost surely as n oo all f iv jn 0 exponentially as n oc i e for all e 0 there exist r no 0 such that p jne no all f v lim0h 0 and limnnhd oo also v implies iv when k is merely absolutely integrable and f k x dx 1 0

compression mappings on primitive sequences over z p sup e let z p sup e be the integer residue ring with odd prime p spl ges 5 and integer e spl ges 2 for a sequence a over z p sup e there is a unique p adic expansion a a sub 0 a spl middot p a sub e 1 spl middot p sup e 1 where each a sub i is a sequence over 0 1 p 1 and can be regarded as a sequence over the finite field gf p naturally let f x be a primitive polynomial over z p sup e and g f x p sup e the set of all primitive sequences generated by f x over z p sup e set spl phi sub e 1 x sub 0 x sub e 1 x sub e 1 sup k spl eta sub e 2 1 x sub 0 x sub 1 x sub e 2 spl psi sub e 1 x sub 0 x sub e 1 x sub e 1 sup k spl eta sub e 2 2 x sub 0 x sub 1 x sub e 2 where spl eta sub e 2 1 and spl eta sub e 2 2 are arbitrary functions of e 1 variables over gf p and 2 spl les k spl les p 1 then the compression mapping spl phi sub e 1 g f x p sup e spl rarr gf p sup spl infin a spl rarr spl phi sub e 1 a sub 0 a sub e 1 is injective that is a b if and only if spl phi sub e 1 a sub 0 a sub e 1 spl phi sub e 1 b sub 0 b sub e 1 for a b spl isin g f x p sup e furthermore if f x is a strongly primitive polynomial over z p sup e then spl phi sub e 1 a sub 0 a sub e 1 spl psi sub e 1 b sub 0 b sub e 1 if and only if a b and spl phi sub e 1 x sub 0 x sub e 1 spl psi sub e 1 x sub 0 x sub e 1 for a b spl isin g f x p sup e

patchwork rejection algorithms abstract new methods for implementing rejection algorithms are suggested the objectives are to avoid the generation of unwanted envelope variates to use fewer pseudorandom uniform variates and to bypass as far as possible the need to decide whether to accept or to reject considerably fewer than two uniforms on average per attempted output of a target random variable can be achieved together with an increased acceptance rate the generation of tail variates from a normal distribution is used as illustration

monte carlo algorithms for hardy weinberg proportions the hardy weinberg law is among the most important principles in the study of biological systems given its importance many tests have been devised to determine whether a finite population follows hardy weinberg proportions because asymptotic tests can fail guo and thompson developed an exact test unfortunately the monte carlo method they proposed to evaluate their test has a running time that grows linearly in the size of the population n here we propose a new algorithm whose expected running time is linear in the size of the table produced and completely independent of n in practice this new algorithm can be considerably faster than the original method

heteroscedastic nonlinear regression several parameter estimation methods for dealing with heteroscedasticity in nonlinear regression are described these include variations on ordinary weighted iteratively reweighted extended and generalized least squares some of these variations are new and one of them in particular modified extended iteratively reweighted least squares meirls allows parameters of an assumed heteroscedastic variance model to be estimated with an adjustment for bias due to estimation of the regression parameters the context of the discussion is primarily that of pharmacokinetic type data although an example is given involving chemical reaction data using simulated data from 21 heteroscedastic pharmacokinetic type models some of the methods are compared in terms of mean absolute error and 95 confidence interval coverage from these comparisons meirls and the variations on generalized least squares emerge as the methods of choice

sudoku based space filling designs sudoku is played by millions of people across the globe it has simple rules and is very addictive the game board is a nine by nine grid of numbers from one to nine several entries within the grid are provided and the remaining entries must be filled in subject to no row column or three by three subsquare containing duplicate numbers by exploiting these three types of uniformity we propose an approach to constructing a new type of design called a sudoku based space filling design such a design can be divided into groups of subdesigns so that the complete design and each subdesign achieve maximum uniformity in univariate and bivariate margins examples are given illustrating the proposed construction method applications of such designs include computer experiments with qualitative and quantitative factors linking parameters in engineering and crossvalidation copyright 2011 oxford university press

generating random binary trees a survey this paper surveys algorithms for generating unbiased random binary trees there exist several linear time algorithms the best algorithms use only integers of size o n to generate binary trees on n nodes

on the linear complexity of some new q ary sequences some new q ary sequences with period q 3 e k 1 q p m p an odd prime m e k integers are first constructed and then inspired by antweiler s method their linear complexity is examined the exact value of linear complexity k 6e w is determined when email protected i 1 wp e i furthermore an upper bound of the linear complexity is given for the other values of r our results show that this sequence has larger linear span than gmw sequence with the same parameters finally the results of a maple program are included to illustrate the validity of the results

a general approach for two stage analysis of multilevel clustered non gaussian data in this article we propose a two stage approach to modeling multilevel clustered non gaussian data with sufficiently large numbers of continuous measures per cluster such data are common in biological and medical studies utilizing monitoring or image processing equipment we consider a general class of hierarchical models that generalizes the model in the global two stage gts method for nonlinear mixed effects models by using any square root n consistent and asymptotically normal estimators from stage 1 as pseudodata in the stage 2 model and by extending the stage 2 model to accommodate random effects from multiple levels of clustering the second stage model is a standard linear mixed effects model with normal random effects but the cluster specific distributions conditional on random effects can be non gaussian this methodology provides a flexible framework for modeling not only a location parameter but also other characteristics of conditional distributions that may be of specific interest for estimation of the population parameters we propose a conditional restricted maximum likelihood creml approach and establish the asymptotic properties of the creml estimators the proposed general approach is illustrated using quartiles as cluster specific parameters estimated in the first stage and applied to the data example from a collagen fibril development study we demonstrate using simulations that in samples with small numbers of independent clusters the creml estimators may perform better than conditional maximum likelihood estimators which are a direct extension of the estimators from the gts method

efficient inference for random coefficient growth curve models with unbalanced data growth and dose response curve studies often result in incomplete or unbalanced data random effects models together with a variety of computer intensive iterative techniques have been suggested for the analysis of such data this paper is concerned with a noniterative method for estimating and comparing location parameters in random coefficient growth curve models consistent and asymptotically efficient estimators of the location parameters are obtained using estimated generalized least squares two criteria for testing multivariate general linear hypotheses are introduced and their asymptotic properties are investigated the results are applied to clinical data obtained on the blood ultrafiltration performance of hemodialyzers used in the treatment of patients with end stage renal disease

nonlinear mixed effects models for repeated measures data we propose a general nonlinear mixed effects model for repeated measures data and define estimators for its parameters the proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood or restricted maximum likelihood estimators for linear mixed effects models we implement newton raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed

uniform random generation of strings in a context free language let s be the set of all strings of length n generated by a given context free grammar a uniform random generator is one which produces strings from s with equal probability in generating these strings care must be taken in choosing the disjuncts that form the right hand side of a grammar rule so that the produced string will have the specified length uniform random generators have applications in studying the complexity of parsers in estimating the average efficiency of theorem provers for the propositional calculus in establishing a measure of ambiguity of a grammar etc two methods are presented for generating uniform random strings in an unambiguous context free language the first method will generate a random string of length n in linear time but must use a precomputed table of size o n r 1 where r is the number of nonterminals in the grammar used to specify the language the second method precomputes part of the table and calculates the other entries as they are called for it requi

on the hamming distance of linear codes over a finite chain ring let r be a finite chain ring e g a galois ring k its residue field and c a linear code over r we prove that d c the hamming distance of c is d c spl alpha where c spl alpha is a submodule quotient spl alpha is a certain element of r and denotes the canonical projection to k these two codes also have the same set of minimal codeword supports we explicitly construct a generator matrix polynomial of c spl alpha from the generator matrix polynomials of c we show that in general d c spl les d c with equality for free codes i e for free r submodules of r sup n and in particular for hensel lifts of cyclic codes over k most of the codes over rings described in the literature fall into this class we characterize minimum distance separable mds codes over r and prove several analogs of properties of mds codes over finite fields we compute the hamming weight enumerator of a free mds code over r

simulating size constrained galton watson trees we discuss various methods for generating random galton watson trees conditional on their sizes being equal to n a linear expected time algorithm is proposed

hamming metric decoding of alternant codes over galois rings the standard decoding procedure for alternant codes over fields centers on solving a key equation which relates an error locator polynomial and an error evaluator polynomial by a syndrome sequence we extend this technique to decode alternant codes over galois rings we consider the module m a b as spl equiv b mod x sup r of all solutions to the key equation where s is the syndrome polynomial and r is the number of rows in a parity check matrix for the code in decoding we seek a particular solution spl sigma spl omega spl isin m which we prove can be found in a grobner basis for m we present an iterative algorithm which generates a grobner basis modulo x sup k 1 from a given basis modulo x sup k at the rth step a grobner basis for m is found and the required solution recovered

multisequence shift register synthesis over commutative rings with identity with applications to decoding cyclic codes over integer residue rings we present a new algorithm for solving the multisequence shift register synthesis problem over a commutative ring r with identity given a finite set of r sequences each of length l the complexity of our algorithm in terms of r multiplications is o l sup 2 as l spl rarr spl infin an important application of this algorithm is in the decoding of cyclic codes over z sub q up to the hartmann tzeng bound where q is a prime power characterization of the set of monic characteristic polynomials of a prescribed set of multiple syndrome sequences leads to an efficient decoding procedure which we further extend to decode cyclic codes over z sub m where m is a product of prime powers

some simple methods for estimating intraindividual variability in nonlinear mixed effects models methods are proposed to improve both population and individual inference within the nonlinear random coefficient regression framework by incorporating possible heterogeneity in the intraindividual variance structure these methods extend existing variance function estimation techniques by pooling information across individuals the methods are appropriate when it is reasonable to assume that there exists a common intraindividual variance structure

efficient bit parallel gf 2 m multiplier for a large class of irreducible pentanomials this work studies efficient bit parallel multiplication in gf 2m for irreducible pentanomials based on the so called shifted polynomial bases spbs we derive a closed expression of the reduced spb product for a class of polynomials xm xk s xk s 1 hellip xk 1 1 with ks k1 les m 1 2 then we apply the above formulation to the case of pentanomials the resulting multiplier outperforms or is as efficient as the best proposals in the technical literature but it is suitable for a much larger class of pentanomials than those studied so far unlike previous works this property enables the choice of pentanomials optimizing different field operations for example inversion yet preserving an optimal implementation of field multiplication as discussed and quantitatively proved in the last part of the paper

testing hardy weinberg equilibrium an objective bayesian analysis we analyze the general multiallelic hardy weinberg equilibrium problem from an objective bayesian testing standpoint we argue that for small or moderate sample sizes the answer is rather sensitive to the prior chosen and this suggests to carry out a sensitivity analysis with respect to the prior this goal is achieved through the identification of a class of priors specifically designed for this testing problem in this paper we consider the class of intrinsic priors under the full model indexed by a tuning quantity the training sample size these priors are objective satisfy savage s continuity condition and have proved to behave extremely well for many statistical testing problems we compute the posterior probability of the hardy weinberg equilibrium model for the class of intrinsic priors assess robustness over the range of plausible answers as well as stability of the decision in favor of either hypothesis copyright 2010 john wiley sons ltd

a survey of combinatorial gray codes the term combinatorial gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified small way this notion generalizes the classical binary reflected gray code scheme for listing n bit binary numbers so that successive numbers differ in exactly one bit position as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families including permutations and combinations the area of combinatorial gray codes was popularized by herbert wilf in his invited address at the siam conference on discrete mathematics in 1988 and his subsequent siam monograph combinatorial algorithms an update 1989 in which he posed some open problems and variations on the theme this resulted in much recent activity in the area and most of the problems posed by wilf are now solved in this paper we survey the area of combinatorial gray codes describe recent results variations and trends and highlight some open problems

further result of compressing maps on primitive sequences modulo odd prime powers let be the integer residue ring with odd prime and integer for a sequence over there is an unique adic expansion where each is a sequence over and can be regarded as a sequence over the prime field gf naturally let be a strongly primitive polynomial over and the set of all primitive sequences generated by over suppose that gf gf it is shown that any function in is an injective map from to gf and the derived sequences of different functions are also different that is if and only if and for and these injective functions in can be considered as good candidates for the keys of a stream cipher

synthesis and uniqueness of m sequences over gf q sup n as n phase sequences over gf q the construction of m sequences over gf q sup n from known m sequences over gf q is discussed an algorithm is given which generates m sequences over gf q sup n from m sequences over gf q where n phase shifts of known m sequences over gf q are determined by the iterative polynomials it is shown that n shift distinct m sequences over gf q sup n generated by the same m sequences over gf q are unique when the elements of gf q sup n are represented by n tuples of the elements from gf q

performing the exact test of hardy weinberg proportion for multiple alleles summary the hardy weinberg law plays an important role in the field of population genetics and often serves as a basis for genetic inference because of its importance much attention has been devoted to tests of hardy weinberg proportions hwp over the decades it has long been recognized that largesample goodness of fit tests can sometimes lead to spurious results when the sample size and or some genotypic frequencies are small although a complete enumeration algorithm for the exact test has been proposed it is not of practical use for loci with more than a few alleles due to the amount of computation required we propose two algorithms to estimate the significance level for a test of hwp the algorithms are easily applicable to loci with multiple alleles both are remarkably simple and computationally fast relative efficiency and merits of the two algorithms are compared guidelines regarding their usage are given numerical examples are given to illustrate the practicality of the algorithms

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

a new class of nonlinear pn sequences over gf q sup n a new class of pseudo random sequences called cascaded gmw type permutation polynomial cgpp sequences over gp q sup n is constructed using a system of n orthogonal cascaded gmw functions in gf q their statistical properties are given classification evaluation and implementation of the cgpp sequences are derived a sequence generated by mapping elements of a cgpp sequence into gf q is also presented

relationships between m sequences over gf q and gf q sup m it is shown that m sequences over gf q sup m of length q sup nm 1 corresponding to primitive polynomials in gf q sup m x of degree n can be generated from known m sequences over gf q of length q sup nm 1 obtained from primitive polynomials in gf q x of degree mn a procedure for generating the m sequences over gf q sup 2 from m sequences over gf q was given which enables the generation of m sequences over gf p sup 2n in addition it was shown that all of the primitive polynomials in gf q sup m x can be obtained from a complete set of the primitive polynomials in gf q x

on the phase sequence set of slm ofdm scheme for a crest factor reduction the crest factor distribution of orthogonal frequency division multiplexing ofdm symbol sequences is evaluated and it is shown that ofdm symbol sequences with a short period are expected to have a high crest factor the crest factor relationship between two input symbol sequences hamming distance d apart is also derived using these two results two criteria are proposed for a phase sequence set of the selected mapping slm ofdm scheme and suggest the rows of the cyclic hadamard matrix constructed from an m sequence as a near optimal phase sequence set of the slm ofdm scheme

a bayesian approach to nonlinear random effects models nonlinear random effects models are considered from the bayesian point of view the method of analysis follows closely that of lindley and smith 1972 journal of the royal statistical society series b 34 1 42 the numerical method is related to the em algorithm

on linear complexity of sequences over gf 2 n in this paper we consider some aspects related to determining the linear complexity of sequences over gf 2n in particular we study the effect of changing the finite field basis on the minimal polynomials and thus on the linear complexity of sequences defined over gf 2n but given in their binary representation let a ai be a sequence over gf 2n then ai can be represented by ai j 0n 1 aij j aij gf 2 where is the root of the irreducible polynomial defining the field consider the sequence b bi whose elements are obtained from the same binary representation of a but assuming a different set of basis say 0 1 n 1 i e bi j 0r 1 aij j we study the relation between the minimal polynomial of a and b

statistical tests for large tree structured data we develop a general statistical framework for the analysis and inference of large tree structured data with a focus on developing asymptotic goodness of fit tests we first propose a consistent statistical model for binary trees from which we develop a class of invariant tests using the model for binary trees we then construct tests for general trees by using the distributional properties of the continuum random tree which arises as the invariant limit for a broad class of models for tree structured data based on conditioned galton watson processes the test statistics for the goodness of fit tests are simple to compute and are asymptotically distributed as 2 and f random variables we illustrate our methods on an important application of detecting tumour heterogeneity in brain cancer we use a novel approach with tree based representations of magnetic resonance images and employ the developed tests to ascertain tumor heterogeneity between two groups of patients

minimal grobner bases and the predictable leading monomial property abstract we focus on grobner bases for modules of univariate polynomial vectors over a ring we identify a useful property the predictable leading monomial plm property that is shared by minimal grobner bases of modules in f x q no matter what positional term order is used the plm property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable degree property row reducedness a terminology introduced by forney in the 70 s because of the presence of zero divisors minimal grobner bases over a finite ring of the type z p r where p is a prime integer and r is an integer 1 do not necessarily have the plm property in this paper we show how to derive from an ordered minimal grobner basis a so called minimal grobner p basis that does have a plm property we demonstrate that minimal grobner p bases lend themselves particularly well to derive minimal realization parametrizations over z p r applications are in coding and sequences over z p r

a calculus for the random generation of labelled combinatorial structures a systematic approach to the random generation of labelled combinatorial objects is presented it applies to structures that are decomposable i e formally specifiable by grammars involving set sequence and cycle constructions a general strategy is developed for solving the random generation problem with two closely related types of methods for structures of size n the boustrophedonic algorithms exhibit a worst case behaviour of the form o n log n the sequential algorithms have worst case o n2 while offering good potential for optimizations in the average case the complexity model is in terms of arithmetic operations and both methods appeal to precomputed numerical table of linear size that can be computed in time o n2 a companion calculus permits systematically to compute the average case cost of the sequential generation algorithm associated to a given specification using optimizations dictated by the cost calculus several random generation algorithms of the sequential type are developed most of them have expected complexity 1 2n log n and are thus only slightly superlinear the approach is exemplified by the random generation of a number of classical combinatorial structures including cayley trees hierarchies the cycle decomposition of permutations binary trees functional graphs surjections and set partitions

on recursiveness and related topics in linear systems a decomposition of matrix formal power series linear systems is developed the interconnections between formal power series realization pade approximation recursiveness berlekamp massey algorithm feedback interconnections matrix linear fractional representations matrix continued fractions and the matrix euclidean algorithm are thereby revealed

bayesian methods for examining hardy weinberg equilibrium testing for hardy weinberg equilibrium is ubiquitous and has traditionally been carried out via frequentist approaches however the discreteness of the sample space means that uniformity of p values under the null cannot be assumed with enumeration of all possible counts conditional on the minor allele count offering a computationally expensive way of p value calibration in addition the interpretation of the subsequent p values and choice of significance threshold depends critically on sample size because equilibrium will always be rejected at conventional levels with large sample sizes we argue for a bayesian approach using both bayes factors and the examination of posterior distributions we describe simple conjugate approaches and methods based on importance sampling monte carlo the former are convenient because they yield closed form expressions for bayes factors which allow their application to a large number of single nucleotide polymorphisms snps in particular in genome wide contexts we also describe straightforward direct sampling methods for examining posterior distributions of parameters of interest for large numbers of alleles at a locus we resort to markov chain monte carlo we discuss a number of possibilities for prior specification and apply the suggested methods to a number of real datasets

primitive polynomials and m sequences over gf q sup m procedures for obtaining primitive polynomials and m sequences over gf q sup m in terms of primitive polynomials and m sequences over gf q are presented using a degree mn primitive polynomial g x in gf g x an m sequence over gf q sup m can be expressed as a vector m sequence whose component m sequences are shifted versions of the m sequence generated by g x the degree n primitive polynomials in gf q sup m x with root alpha q sup i that are factors of g x with root alpha when g x is viewed in gf q sup m x are then developed from the m sequence over gf q sup m expressions for the shifts and corresponding primitive polynomial for the m sequence generated by the uth decimation of the m sequence generated by the polynomials are also given expressions for the shifts and corresponding primitive polynomial factors of g x for different bases expressing gf q sup m over gf q are presented

complex sequences over gf p sup m with a two level autocorrelation function and a large linear span new complex sequences with elements are proposed that have constant absolute values of 1 the periodic autocorrelation functions of these sequences are shown to be two level the sequences are generated by three consecutive mapping processes the number of sequences of fixed length is determined which is larger than the number of binary sequences for cryptographical reasons the large linear span of the sequences is of great interest the linear span of the new sequences is examined and is proven to be larger than the linear span of complex m sequences if the parameters of the mapping processes are appropriately chosen the formula for generating the sequences is generalized to enlarge the linear span of sequences

on secrecy capacity of minimum storage regenerating codes in this paper we revisit the problem of characterizing the secrecy capacity of minimum storage regenerating msr codes under the passive l 1 l 2 eavesdropper model where the eavesdropper has access to data stored on l 1 nodes and the repair data for an additional l 2 nodes we study it from the information theoretic perspective first some general properties of msr codes as well as a simple and generally applicable upper bound on secrecy capacity are given second a new concept of stable msr codes is introduced where the stable property is shown to be closely linked with secrecy capacity finally a comprehensive and explicit result on secrecy capacity in the linear msr scenario is present which generalizes all related works in the literature and also predicts certain results for some unexplored linear msr codes

markov chains for monte carlo tests of genetic equilibrium in multidimensional contingency tables hardy weinberg equilibrium and linkage equilibrium are fundamental concepts in population genetics in practice testing linkage equilibrium in haplotype data is equivalent to testing independence in a large sparse multidimensional contingency table testing hardy weinberg and linkage equilibrium simultaneously on multilocus genotype data introduces the additional complications of missing information and symmetry constraints on marginal probabilities to avoid unreliable large sample approximations for sparse contingency tables one can use exact tests like fisher s classical test that condition on observed marginal totals unfortunately computing p values for exact tests is often infeasible because of the large number of tables consistent with the marginal totals of an observed table we develop here markov chains for sampling from the appropriate conditional distributions for testing genetic equilibrium these chains compare favorably with a parallel independent sampling method that we present for n haplotype observations on j loci the markov chains converge to their stationary distributions in j 1 n ln n 2 0 n steps and can be an efficient tool for estimating p values our theoretical treatment of these results involves strong stationary stopping times order statistics large deviations and the embedding of poisson processes we include some general results on the application of strong stationary times to bounding the precision and bias of sample average estimators

inferring sequences produced by pseudo random number generators in this paper efficient algorithms are given for inferring sequences produced by certain pseudo random number generators the generators considered are all of the form x n s k j l j j x o x l x n l mod m in each case we assume that the functions j are known and polynomial time computable but that the coefficients aj and the modulus m are unknown using this general method specific examples of generators having this form the linear congruential method linear congruences with n terms in the recurrence and quadratic congruences are shown to be cryptographically insecure