on ppm sequences with good autocorrelation properties the problem of designing sequences of q ary pulse position modulation ppm symbols that have good periodic autocorrelation properties is investigated two cases are considered in the first it is assumed that only slot synchronization is present and thus cyclic shifts are one slot at a time in the second ppm symbol synchronization is present in which case cyclic shifts are by one symbol q slots at a time in both cases upper bounds are derived on the maximum peak to sidelobe distance which are shown through a computer search to be nearly tight when symbol synchronization is present the bound reduces to the plotkin bound but it is slightly tighter in general

extrapolation vs projection methods for linear systems of equations it is shown that the four vector extrapolation methods minimal polynomial extrapolation reduced rank extrapolation modified minimal polynomial extrapolation and topological epsilon algorithm when applied to linearly generated vector sequences are krylov subspace methods and are equivalent to some well known conjugate gradient type methods a unified recursive method that includes the conjugate gradient conjugate residual and generalized conjugate gradient methods is developed finally the error analyses for these methods are unified and some known and some new error bounds for them are given

extrapolation methods for vector sequences this is an expository paper that describes and compares five methods for extrapolating to the limit or anti limit of a vector sequence without explicit knowledge of the sequence generator the methods are the minimal polynomial extrapolation mpe studied by cabay and jackson mesina and skelboe the reduced rank extrapolation rre of eddy which we show to be equivalent to mesina s version of mpe the vector and scalar versions of the epsilon algorithm vea sea introduced by wynn and extended by brezinski and gekeler and the topological epsilon algorithm tea of brezinski we cover the derivation and error analysis of iterated versions of the algorithms as applied to both linear and nonlinear problems and we show why these versions tend to converge quadratically we also present samples from extensive numerical testing that has led us to the following conclusions a tea in spite of its role as a theoretical link between the polynomial type and the epsilon type methods has no practical appl

a note on the generalized conjugate gradient method each iterate generated by the generalized conjugate gradient method of concus and golub lecture notes in economics and mathematical systems 134 springer verlag new york 1976 and widlund siam j numer anal 15 1978 pp 801 812 is shown to be the best approximation to the solution from a certain affine subspace although not from the natural afline krylov subspace this property is used to improve the error bounds given by widlund and hageman luk and young siam j numer anal 17 1980 pp 852 873

an algebraic approach to the unequal spaced channel allocation problem in wdm lightwave systems to reduce four wave mixing crosstalk in high capacity long haul repeaterless wavelength division multiplexing wdm lightwave systems the use of unequally spaced channels has been proposed instead of being solved by integer linear programming the unequal spaced channel allocation problem is treated by constructing suitable optical orthogonal codes in optical code division multiple access cdma an algebraic framework and three algorithms on finding the frequency locations of unequally spaced wdm channels are introduced where the constructions are based on generating optical cdma codewords with a predetermined pulse separation and aperiodic autocorrelation sidelobes no greater than one the algorithms potentially provide a fast and simple alternative to solve the problem besides the proposed computer search method

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

frame synchronization for optical multi pulse pulse position modulation we investigate frame synchronization algorithms for direct detection multi pulse pulse position modulation mppm optical communication systems the optimal rules in a maximum likelihood sense are derived approximations to them that are computationally easier to implement are identified and upper bounds on the synchronization probability are obtained it is seen that the high snr rules have a performance nearly equal to that of the ml rules but provide a significant performance improvement over the simple correlation rules and are virtually as simple to implement for a wide range of signal energies we also address the problem of designing periodic mppm synchronization sequences with good autocorrelation properties based on a maximum worst case peak to sidelobe distance criterion upper bounds on the peak to sidelobe distance for such sequences are derived and a few sequences generated by computer simulations for different synchronization sequence lengths are given

recursive interpolation extrapolation and projection abstract the recursive projection algorithm derived in a previous paper is related to several well known methods of numerical analysis such as the conjugate gradient method rosen s method and henrici s it is connected with the general interpolation problem with extrapolation methods with orthogonal projection on a subspace and with fourier expansions several other connections and applications are presented

accelerating numerical simulations of strain adaptive bone remodeling predictions abstract bone modeling and remodeling have been extensively studied in many fields of research the process of adaptive bone remodeling can be described mathematically and simulated in a computer model integrated with the finite element method fem the fem can determine the long term behavior of bone and the impact on bone biomechanics produced by prosthetic models as whole bone remodeling simulations are time consuming the aim of this paper is to combine fe analyses with numerical extrapolation techniques to accelerate bone remodeling simulations two vector extrapolation methods reduce ranked extrapolation rre and minimal polynomial extrapolation mpe are used to reduce the simulation time these extrapolation techniques are illustrated by numerical examples of 2 d and 3 d models based on a strain adaptive bone remodeling theory

two dimensional synchronization patterns for minimum ambiguity a number of closely related combinatorial problems corresponding to specific assumptions about the type of time frequency sequence which may be appropriate in a particular application are formulated in terms of square or rectangular arrays of dots with appropriate constraints on the two dimensional correlation function the current state of knowledge concerning each of these problems is summarized it is hoped that more general constructions may be found leading to larger families of solutions as well as better computational algorithms for finding individual solutions which may lie outside of the general families

tests of homogeneity for the relative risk in multiply matched case control studies summary we examine several tests of homogeneity of the odds ratio in the analysis of 2 x 2 tables arising from epidemiologic 1 r matched case control studies the t4 and t5 statistics proposed by liang and self 1985 biometrika 72 353 358 are unable to detect obvious inhomogeneity in two numerical examples and in simulation studies the null hypothesis is rejected by the chi square statistic of ejigou and mchugh 1984 bioinetrika 71 408 411 and by a new proposed method whose significance level must be simulated

the methods of vorobyev and lanczos abstract this paper deals with the method of moments of vorobyev which is a particular case of galerkin s method this method is generalized in different ways and some of its properties are studied in detail as an application it is showed that the lanczos method for solving systems of linear equations can be derived from it and that some generalizations could be obtained

development of iterative techniques and extrapolation methods for drazin inverse solution of consistent or inconsistent singular linear systems abstract consider the linear system of equations bx where b is an nxn singular matrix in an earlier work by the author it was shown that iterative techniques coupled with standard vector extrapolation methods can be used to obtain or approximate a solution of this system when it is consistent in the present work we expand on that approach to treat the case in which this system is in general inconsistent starting with richardson s iterative method we develop a family of new iterative techniques and vector extrapolation methods that enable us to obtain or approximate the drazin inverse solution of this system whether the index of b is 1 or greater than 1 we show that the drazin inverse solution can be constructed from a finite number of iterations this number being at most n 2 we also provide detailed convergence analyses of the new iterative techniques and vector extrapolation methods and give their precise rates of convergence

convergence and stability analyses for some vector extrapolation methods on the presence of defective iteration matrices abstract in two previous papers 10 11 convergence and stability results for the following vector extrapolation methods were presented minimal polynomial extrapolation reduced rank extrapolation modified minimal polynomial extrapolation and topological epsilon algorithm the analyses were carried out for vector sequences that include those arising from iterative methods for linear systems of equations having diagonalizable iteration matrices in this paper the analyses of 10 11 are extended to vector sequences that include those arising from iterative methods for linear systems having defective iteration matrices the results are illustrated with numerical examples the analyses above naturally suggest some old and some new extensions of the well known power method enabling one to obtain estimates for several dominant eigenvalues of a general matrix

the design and performance analysis for several new classes of codes for optical synchronous cdma and for arbitrary medium time hopping synchronous cdma communication systems new families of spread spectrum codes are constructed that are applicable to optical synchronous code division multiple access cdma communications as well as to arbitrary medium time hopping synchronous cdma communications proposed constructions are based on the mappings from integer sequences into binary sequences the authors use the concept of number theoretic quadratic congruences and a subset of reed solomon codes similar to the one utilized in the welch costas frequency hop fh patterns the properties of the codes are as good as or better than the properties of existing codes for synchronous cdma communications both the number of code sequences within a single code family and the number of code families with good properties are significantly increased when compared to the known code designs possible applications are presented to evaluate the performance of the proposed codes a new class of hit arrays called cyclical hit arrays is recalled which give insight into the previously unknown properties of the few classes of number theoretic fh patterns cyclical hit arrays and the proposed mappings are used to determine the exact probability distribution functions of random variables that represent interference between users of a time hopping or optical cdma system expressions for the bit error probability in multi user cdma systems are derived as a function of the number of simultaneous cdma system users the length of signature sequences and the threshold of a matched filter detector the performance results are compared with the results for some previously known codes

vector sequence transformations methodology and applications to linear systems in this paper a methodology for the construction of various vector sequence transformations is formulated leading to a unified presentation of the subject and to new results the connections to the general interpolation problem and to projections are discussed various particular cases are examined in more details applications to the solution of systems of linear equations will end the paper and in particular their relation with lanczos method will be studied some numerical examples will be given

some results about vector extrapolation methods and related fixed point iterations abstract in this work we consider polynomial extrapolation methods for vector sequences we give a unified presentation of these methods and a new recursive algorithm for the implementation of the modified minimal polynomial extrapolation mmpe method properties operation counts and storage requirements of the algorithm will be given the second part of the work is devoted to the application of these methods for solving systems of nonlinear equations a sufficient condition for quadratic convergence is given

a family of implicit chebyshev methods for the numerical integration of second order differential equations abstract a family of implicit methods based on intra step chebyshev interpolation is developed for the solution of initial value problems whose differential equations are of the special second order form y f y x x the general procedure allows stepsizes which are considerably larger than commonly used in conventional methods computation overhead is comparable to that required by high order single or multistep procedures in addition the iterative nature of the method substantially reduces local errors while maintaining a low rate of global error growth

bounds and construction for difference triangle sets the definition of difference triangle sets dtss and is short survey of some of their important applications is given a lower bound on the maximal element in any i j dts is then given and a construction of dtss is described finally tables of the best known lower and upper bounds on optimal dtss are presented

bounds and constructions of disjoint sets of distinct difference sets an i j ddd is a set of i disjoint sets of distinct difference sets each having j elements a number of constructions are given upper and lower bounds on the maximal element in a ddd disjoint distinct difference set are given it is shown that regular ddd sets exist for i or approximately 4j

acquisition sequences in ppm communications corresp in pulse position modulation ppm communications an acquisition sequence is needed to establish frame synchronization for subsequent data transmission this sequence must have ppm structure one pulse per frame and desirable autocorrelation properties for its detection the problem of finding aperiodic ppm acquisition sequences having the minimum possible autocorrelation for all off shifted positions is investigated sequences with the maximum possible number of frames have been found up to frame size 15 examples are given for every frame length up to 88 and powers of two up to 1024 an interesting relation between the ppm sequence problem costas and sonar arrays and golomb s ruler is explored

accelerated non linear finite volume method for diffusion a non linear finite volume method with monotone matrix for the diffusion equation is presented it does not extrapolate the primary variable to neumann boundaries as this was previously done in similar methods this change results in faster convergence computation time is significantly shortened further using the reduced rank extrapolation method rre and imposing an upper limit on the number of linear iterations per non linear step second order accuracy and performance improvement are demonstrated by numerical examples

frame synchronization for optical overlapping pulse position modulation systems the maximum likelihood rules for locating data frames in direct detection optical communication systems utilizing overlapping pulse position modulation oppm are derived under two distinct assumptions of symbol and chip synchronization respectively various approximations to the optimal rules are identified as well as upper and lower bounds on the probability of correct synchronization further the problem of sequence design based on a maximum peak to sidelobe distance criterion is studied and upper bounds on peak to sidelobe distance for sequences of oppm symbols are derived some good sequences are obtained through a computer search

convergence and stability properties of minimal polynomial and reduced rank extrapolation algorithms the minimal polynomial and reduced rank extrapolation algorithms are two acceleration of convergence methods far sequences of vectors in a recent survey these methods were tested and compared with the scalar vector and topological epsilon algorithms and were observed to be more efficient than the latter it was also observed that the two methods have similar convergence properties the purpose of the present work is to analyze the convergence and stability properties of these methods and to show that they are bona fide acceleration methods when applied to a class of vector sequences that includes those sequences obtained from systems of linear equations by using matrix iterative methods

acceleration of convergence of vector sequences a general approach to the construction of convergence acceleration methods for vector sequences is proposed using this approach one can generate some known methods such as the minimal polynomial extrapolation the reduced rank extrapolation and the topological epsilon algorithm and also some new ones some of the new methods are easier to implement than the known methods and are observed to have similar numerical properties the convergence analysis of these new methods is carried out and it is shown that they are especially suitable for accelerating the convergence of vector sequences that are obtained when one solves linear systems of equations iteratively a stability analysis is also given and numerical examples are provided the convergence and stability properties of the topological epsilon algorithm are likewise given

hybrid vector transformations we present in this paper hybrid and semi hybrid vector extrapolation methods for a vector sequence transformation we will define the generalized residual which allows us to introduce a new hybrid vector transformation when solving linear systems this new transformation reduces to the hybrid procedure defined by brezinski and redivo zaglia 1994 which is a generalization of the smoothing technique in the second part of this paper we define a new composite transformation called semi hybrid when applied to linearly generated sequences we will show that the residual polynomial of the semi hybrid transformation is the product of two residual polynomials

complementary sequences for isi channel estimation a merit factor based on the sequence autocorrelation function whose minimization leads to the reduction in the cramer rao lower bound crlb for the variance of two sided intersymbol interference isi channel estimation is introduced pairs of binary pilot symbol sequences a preamble and a postamble for channel estimation are jointly designed to minimize this merit factor given that the number of channel taps is l and the length of a pilot symbol sequence is n l 1 where n spl ges l we distinguish between the case when n is even and the case when it is odd for even n we show that complementary sequences not only minimize the merit factor but also the crlb for a subset of odd n we construct almost complementary periodic sequence pairs that minimize the merit factor the optimal pilot symbol block signaling requires alternating between two in most cases different binary sequences that form the merit minimizing pair

the 2 lagrange multiplier method applied to nonlinear transmission problems for the richards equation in heterogeneous soil with cross points we formulate the 2 lagrange multiplier method for the richards equation in heterogeneous soil this allows a rigorous formulation of a discrete version of the richards equation on subdomain decompositions involving cross points using kirchhoff transformation the individual subdomain problems can be transformed into convex minimization problems and solved efficiently using a monotone multigrid method we discuss and compare weak formulations of the time discrete and fully discretized multidomain problem it is shown that in the case of two subdomains when solving the resulting discrete system with a richardson iteration the new method is equivalent to a parallel version of the nonlinear robin method for the richards equation proposed in h berninger and o sander comput vis sci 13 2010 pp 187 205 we give numerical results for a problem with realistic soil parameters

vector extrapolation methods applications and numerical comparison the present paper is a survey of the most popular vector extrapolation methods such as the reduced rank extrapolation rre the minimal polynomial extrapolation mpe the modified minimal polynomial extrapolation mmpe the vector e algorithm vea and the topological e algorithm tea using projectors we derive a different interpretation of these methods and give some theoretical results the second aim of this work is to give a numerical comparison of the vector extrapolation methods above when they are used for practical large problems such as linear and nonlinear systems of equations

vector extrapolation methods with applications to solution of large systems of equations and to pagerank computations an important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors x n where x n c n with n very large such sequences arise for example in the solution of systems of linear or nonlinear equations by fixed point iterative methods and lim n x n are simply the required solutions in most cases of interest however these sequences converge to their limits extremely slowly one practical way to make the sequences x n converge more quickly is to apply to them vector extrapolation methods in this work we review two polynomial type vector extrapolation methods that have proved to be very efficient convergence accelerators namely the minimal polynomial extrapolation mpe and the reduced rank extrapolation rre we discuss the derivation of these methods describe the most accurate and stable algorithms for their implementation along with the effective modes of usage in solving systems of equations nonlinear as well as linear and present their convergence and stability theory we also discuss their close connection with the method of arnoldi and with gmres two well known krylov subspace methods for linear systems we show that they can be used very effectively to obtain the dominant eigenvectors of large sparse matrices when the corresponding eigenvalues are known and provide the relevant theory as well one such problem is that of computing the pagerank of the google matrix which we discuss in detail in addition we show that a recent extrapolation method of kamvar et al that was proposed for computing the pagerank is very closely related to mpe we present a generalization of the method of kamvar et al along with a very economical algorithm for this generalization we also provide the missing convergence theory for it

the lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems many powerful methods for solving systems of equations can be regarded as projection methods most of the projection methods known for solving linear systems are orthogonal projection methods but little attention has been given to the class of nonorthogonal or oblique projection methods which is particularly attractive for large nonsymmetric systems the purpose of the paper is to present some methods in the general setting of oblique projection methods and to give some theoretical results some experiments comparing the various algorithms are reported

testing the homogeneity of the relative risk under multiple matching various problems with this formulation stimulated the approach presented here i the stratification variable may involve only a subset of the set x of all the matching variables for example in the los angeles endometrial cancer study analysed by breslow day only age in three broad groups is used for stratification with the remaining matching variables constituting x such as marital status being ignored hence a subjective decision as to what subset of x to use and therefore what h to use arises in this method ii for moderate to large h this stratification based method may lead to sparseness of data and consequent inaccuracy in the large sample chi squared theory involved iii finally a test for homogeneity of the relative risk may be needed when data from a multiply matched case control study are available but the composite x is no longer available then breslow day s method is inapplicable we propose to assess such data for the homogeneity of the relative risk without requiring the explicit reintroduction of x the test procedure is based upon a probability model developed for an unconditional maximum likelihood approach to relative risk estimation for this epidemiologic design ejigou mchugh 1981 the condition of low disease incidence will be assumed so that the relative risk can be accurately approximated by the odds ratio cornfield 1956

testing for interaction in multiply matched case control studies we consider the test of homogeneity of relative risks or odds ratios for multiplymatched case control studies suggested by ejigou mchugh 1984 it is shown that this test may be simply derived as a special case of pearsonian chi squared tests with an estimated parameter cramer 1946 ch 30 consider j independent pairs of variates whose components are a single zero one variate for the case and a matched binomial variate of sample size r for the control let the associated probabilities of a one say exposure be plj and p2j respectively we wish to test the hypothesis of a constant odds ratio that is

efficient implementation of minimal polynomial and reduced rank extrapolation methods abstract the minimal polynomial extrapolation mpe and reduced rank extrapolation rre are two very effective techniques that have been used in accelerating the convergence of vector sequences such as those that are obtained from iterative solution of linear and nonlinear systems of equations their definitions involve some linear least squares problems and this causes difficulties in their numerical implementation in this work timewise efficient and numerically stable implementations for mpe and rre are developed a computer program written in fortran 77 is also appended and applied to some model problems among them a hypersonic flow problem involving chemical reactions

on the nonperiodic cyclic equivalence classes of reed solomon codes picking up exactly one member from each of the nonperiodic cyclic equivalence classes of an n k 1 reed solomon code e over gf q gives a code e which has bounded hamming correlation values and the self synchronizing property the exact size of e is shown to be 1 n sigma sub d mod n mu d q sup 1 k d where mu d is the mobius function x is the integer part of x and the summation is over all the divisors d of n q 1 a construction for a subset v of e is given to prove that mod e mod or mod v mod q sup k 1 q sup k 1 n q 1 where n is the number of integers from 1 to k which are relatively prime to q 1 a necessary and sufficient condition for mod e mod mod v mod is proved and some special cases are presented with examples for all possible values of q 2 a number b q is determined such that mod e mod mod v mod for 1 mod v mod for k b q

on conjugate gradient type methods and polynomial preconditioners for a class of complex non hermitian matrices we consider conjugate gradient type methods for the solution of large linear systemsa x b with complex coefficient matrices of the typea t i i wheret is hermitian and a real scalar three different conjugate gradient type approaches with iterates defined by a minimal residual property a galerkin type condition and an euclidean error minimization respectively are investigated in particular we propose numerically stable implementations based on the ideas behind paige and saunder s symmlq and minres for real symmetric matrices and derive error bounds for all three methods it is shown how the special shift structure ofa can be preserved by using polynomial preconditioning and results on the optimal choice of the polynomial preconditioner are given also we report on some numerical experiments for matrices arising from finite difference approximations to the complex helmholtz equation

two dimensional patterns with distinct differences constructions bounds and maximal anticodes a two dimensional 2 d grid with dots is called a configuration with distinct differences if any two lines which connect two dots are distinct either in their length or in their slope these configurations are known to have many applications such as radar sonar physical alignment and time position synchronization rather than restricting dots to lie in a square or rectangle as previously studied we restrict the maximum distance between dots of the configuration the motivation for this is a new application of such configurations to key distribution in wireless sensor networks we consider configurations in the hexagonal grid as well as in the traditional square grid with distances measured both in the euclidean metric and in the manhattan or hexagonal metrics we note that these configurations are confined inside maximal anticodes in the corresponding grid we classify maximal anticodes for each diameter in each grid we present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes our bounds settle in the negative a question of golomb and taylor on the existence of honeycomb arrays of arbitrarily large size we present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various 2 d shapes such as anticodes by considering periodic configurations with distinct differences in the square grid

analysis of some vector extrapolation methods for solving systems of linear equations in this paper we consider some vector extrapolation methods for solving nonsymmetric systems of linear equations when applied to sequences generated linearly these methods namely the minimal polynomial extrapolation mpe and the reduced rank extrapolation rre are krylov subspaces methods and are respectively equivalent to the method of arnoldi and to the gcr and gmres by considering the geometrical aspect of these methods we derive new expressions for their residual norms and give a relationship between them this allows us to compare the two methods using this new approach we will show that for nonsingular skew symmetric matrices the gmres stagnates every two iterations and the restarted version gmres m m ge 2 is always convergent finally the incomplete forms are considered and some convergence results are given

construction and performance analysis of a new family of optical orthogonal codes for cdma fiber optic networks a construction of a new family of optical orthogonal codes oocs for use in code division multiple access cdma fiber optic networks is given previous families of oocs are reviewed and basic requirements for successful implementation of the fiber optic cdma system given the new code construction is based on costas 1970 arrays and the codes are proved to be spl omega 2p 3 spl omega 1 1 p a prime and spl omega q spl les p 1 where q is the maximum number of users oocs the probability of error using the gaussian approximation in the multiuser system as a performance measure is calculated and compared with that of existing oocs the size of the code family is compared against the johnson bound