titchmarsh s asymptotic formula for periodic eigenvalues and an extension to the p laplacian an asymptotic formula for periodic eigenvalues due to titchmarsh is taken to a higher degree of accuracy using a prufer transformation method not depending on floquet theory with the idea of the rotation number this method is then extended to the p laplacian

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

on the decoding of reed solomon and bch codes over integer residue rings we present a decoding procedure for reed solomon rs and bch codes defined over an integer residue ring pgz sub q where q is a power of a prime p the proposed decoding procedure as for rs and bch codes over fields consists of four major steps 1 calculation of the syndromes 2 calculation of the elementary symmetric functions by a modified berlekamp massey 1968 1969 algorithm for commutative rings 3 calculation of the error location numbers and 4 calculation of the error magnitudes the proposed decoding procedure also applies to the synthesis of a shortest linear feedback shift register lfsr capable of generating a prescribed finite sequence of elements lying in a commutative ring with identity

eigenvalues of the radial p laplacian with a potential on 0 brown and reichel recently established the existence of eigenvalues for the p laplacian on r when the potential q is either i large and positive or ii sufficiently large and negative limit circle case at infinity their methods imposed extra restrictions on q in this paper these restrictions are removed in addition the case where q decays at infinity is also shown to produce negative eigenvalues and a condition is given under which there are only a finite number of such eigenvalues

covariance tapering for likelihood based estimation in large spatial data sets maximum likelihood is an attractive method of estimating covariance parameters in spatial models based on gaussian processes but calculating the likelihood can be computationally infeasible for large data sets requiring o n3 calculations for a data set with n observations this article proposes the method of covariance tapering to approximate the likelihood in this setting in this approach covariance matrixes are tapered or multiplied element wise by a sparse correlation matrix the resulting matrixes can then be manipulated using efficient sparse matrix algorithms we propose two approximations to the gaussian likelihood using tapering one of these approximations simply replaces the model covariance with a tapered version whereas the other is motivated by the theory of unbiased estimating equations focusing on the particular case of the matern class of covariance functions we give conditions under which estimators maximizing the tapering approximations are like the maximum likelihood estimat

approximation numbers singular values this paper introduces a generalisation of the notion of singular value for hilbert space operators to more general banach spaces it is shown that for a simple integral operator of hardy type the singular values are the eigenvalues of a non linear sturm liouville equation and coincide with the approximation numbers of the operator finally asymptotic formulas for the singular numbers are deduced

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

on the existence of sign changing radial solutions to nonlinear p laplacian equations in rn abstract a class of nonlinear equations r n 1 u p 2 u r n 1 w r f u on 0 1 where 1 p is considered we study the existence of sign changing solutions to this problem some sufficient conditions for such solutions with the prescribed number of zeros are developed we generalize the result of naito and tanaka 2008 to the radial solutions of a class of nonlinear p laplacian equations in r n some related issues on the half line will be also discussed

a note on interleaved reed solomon codes over galois rings collaborative decoding of interleaved reed solomon rs codes was first proposed by krachkovsky in 1997 their work has since sparked off a series of papers from various authors on the subject in this paper we study the merits of multisequence shift register synthesis based collaborative decoding of interleaved rs codes formed from rs codes over galois rings in terms of key performance measures such as the probabilty of decoding failure and word error probability in particular we show that when is even and large but not exceeding n 3 n 2 6n 1 2 a q ary interleaved code formed from 2 parallel copies of a length n rs code of rate greater than 1 3 2 n over a galois ring of cardinality q 2 can be on par with its finite field counterpart formed from parallel copies of an rs code of the same length and rate over f q in terms of the word error probabilities of their respective decoders while incurring lower decoding complexity

a matrix based list decoding algorithm for linear codes over integer residue rings abstract in this paper we address the problem of list decoding of linear codes over an integer residue ring z q where q is a power of a prime p the proposed procedure exploits a particular matrix representation of the linear code over z p r called the standard form and the p adic expansion of the to be decoded vector in particular we focus on the erasure channel in which the location of the errors is known this problem then boils down to solving a system of linear equations with coefficients in z p r from the parity check matrix representations of the code we recursively select certain equations that a codeword must satisfy and have coefficients only in the field p r 1 z p r this yields a step by step procedure obtaining a list of the closest codewords to a given received vector with some of its coordinates erased we show that such an algorithm actually computes all possible erased coordinates that is the provided list is minimal

mds convolutional codes over a finite ring in this paper we present an analog of the singleton bound for convolutional codes over finite rings codes meeting that bound are called mds a constructive method for the mds codes restricted to free codes is presented via cyclic codes over finite rings examples of nonfree mds codes are provided

nonconcave penalized likelihood with a diverging number of parameters a class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed by fan and li to simultaneously estimate parameters and select important variables they demonstrated that this class of procedures has an oracle property when the number of parameters is finite however in most model selection problems the number of parameters should be large and grow with the sample size in this paper some asymptotic properties of the nonconcave penalized likelihood are established for situations in which the number of parameters tends to as the sample size increases under regularity conditions we have established an oracle property and the asymptotic normality of the penalized likelihood estimators furthermore the consistency of the sandwich formula of the covariance matrix is demonstrated nonconcave penalized likelihood ratio statistics are discussed and their asymptotic distributions under the null hypothesis are obtained by imposing some mild conditions on the penalty functions the asymptotic results are augmented by a simulation study and the newly developed methodology is illustrated by an analysis of a court case on the sexual discrimination of salary

optimal geostatistical model selection in many fields of science predicting variables of interest over a study region based on noisy data observed at some locations is an important problem two popular methods for the problem are kriging and smoothing splines the former assumes that the underlying process is stochastic whereas the latter assumes it is purely deterministic kriging performs better than smoothing splines in some situations but is outperformed by smoothing splines in others however little is known regarding selecting between kriging and smoothing splines in addition how to perform variable selection in a geostatistical model has not been well studied in this article we propose a general methodology for selecting among arbitrary spatial prediction methods based on approximately unbiased estimation of mean squared prediction errors using a data perturbation technique the proposed method accounts for estimation uncertainty in both kriging and smoothing spline predictors and is shown to be optimal in terms of two mean sq

fixed domain asymptotic properties of tapered maximum likelihood estimators when the spatial sample size is extremely large which occurs in many environmental and ecological studies operations on the large covariance matrix are a numerical challenge covariance tapering is a technique to alleviate the numerical challenges under the assumption that data are collected along a line in a bounded region we investigate how the tapering affects the asymptotic efficiency of the maximum likelihood estimator mle for the microergodic parameter in the matern covariance function by establishing the fixed domain asymptotic distribution of the exact mle and that of the tapered mle our results imply that under some conditions on the taper the tapered mle is asymptotically as efficient as the true mle for the microergodic parameter in the matern model

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

bounds for the positive eigenvalues of the p laplacian with decaying potential an upper bound is obtained for the positive eigenvalues of the p laplacian with decaying potential on 0 the bound is expressed in terms of the potential and is shown to be the best possible of its kind

improved list decoding of generalized reed solomon and alternant codes over galois rings we present a two stage list decoder comprising an errors only guruswami sudan gs decoder and an errors and erasures gs decoder as component decoders in the first and second stage respectively the two stages are coupled via a post processor which selects a codeword from the output list of the first component decoder from which erasure locations are obtained for the second stage when applied to a generalized reed solomon rs code over a galois ring r that maps into a generalized rs code of the same length n and minimum hamming distance d over the corresponding residue field the proposed decoder exploits the presence of zero divisors in r to correct s errors where w lceiln radic n n d 1rceil sleslceiln radic n w n d 1rceil with a probability determined by s w and the ratio of the number of nontrivial zero divisors to the number of units in the code alphabet focusing primarily on alternant codes over zopf 2l an important class of subring subcodes of generalized rs codes over gr 2l a we demonstrate that the gs decoding radius w can be exceeded by a substantial margin with significant probability

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

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

one step sparse estimates in nonconcave penalized likelihood models fan and li propose a family of variable selection methods via penalized likelihood using concave penalty functions the nonconcave penalized likelihood estimators enjoy the oracle properties but maximizing the penalized likelihood function is computationally challenging because the objective function is nondifferentiable and nonconcave in this article we propose a new unified algorithm based on the local linear approximation lla for maximizing the penalized likelihood for a broad class of concave penalty functions convergence and other theoretical properties of the lla algorithm are established a distinguished feature of the lla algorithm is that at each lla step the lla estimator can naturally adopt a sparse representation thus we suggest using the one step lla estimator from the lla algorithm as the final estimates statistically we show that if the regularization parameter is appropriately chosen the one step lla estimates enjoy the oracle properties with good initial estimators computationally the one step lla estimation methods dramatically reduce the computational cost in maximizing the nonconcave penalized likelihood we conduct some monte carlo simulation to assess the finite sample performance of the one step sparse estimation methods the results are very encouraging

computing eigenvalues and fucik spectrum of the radially symmetric p laplacian eigenvalue problems for the radially symmetric p laplacian are discussed we present algorithms which compute a given number of eigenvalues and fucik curves together with the corresponding eigenfunctions the second order p laplacian equation is transformed into a first order system by a generalized prufer transformation to the first order system we apply shooting algorithms newton s method and in case of the fucik curves a predictor corrector method our approach requires analytical and numerical treatment of generalized sine functions singular as well as regular problems are treated and a detailed error analysis for the approximation of singular problems by regular ones are given numerical results are presented

the second eigenfunction of the p laplacian on the disk is not radial abstract in this paper we prove that the second eigenfunction of the p laplacian p 1 on the disk is not radial our proof is a combination of asymptotic analysis for p and the application of interval arithmetic

on b symbol distances of repeated root constacyclic codes let p be a prime s be a positive integer and let b be an integer satisfying 2 leq b in this paper we obtain b symbol distances of all repeated root constacyclic codes of length p s over finite fields using this result we determine b symbol distances of all repeated root constacyclic codes of length p s over finite commutative chain rings we also list all mds b symbol repeated root constacyclic codes of length p s over finite fields and all mds b symbol repeated root constacyclic codes of length p s over finite commutative chain rings in general

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

decoding ldpc codes over integer residue rings this correspondence presents a multistage decoding approach for a free zopfq submodule of zopfq n of rank k defined by a sparse n k timesn parity check matrix overzopf q where q pm p 2 and m 1 the proposed method involves the repeated application of belief propagation decoding to exploit the natural ring epimorphism zopfqrarrzopfp l r rarr sigma i 0 l 1r i pi with kernel p lzopfq for each l 1lesllesm where sigma i 0 m 1r i pi is the p adic expansion of r computer simulations for codes of rate half and moderate length on an additive white gaussian noise awgn channel with various modulation schemes show that such a decoding strategy offers an additional coding gain of between 0 07 0 1 db over a single stage decoding approach

cyclic codes over z sub 4 locator polynomials and newton s identities certain nonlinear binary codes contain more codewords than any comparable linear code presently known these include the kerdock 1972 and preparata 1968 codes that can be very simply constructed as binary images under the gray map of linear codes over z sub 4 that are defined by means of parity checks involving galois rings this paper describes how fourier transforms on galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes these methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of z sub 4 linear codes with length 2 sup m m odd and size 2 2 sup m 1 5m 2 the gray image of the code of length 32 is the best 64 2 sup 37 code that is presently known this paper also determines the exact minimum lee distance of the linear codes over z sub 4 that are obtained from the extended binary two and three error correcting bch codes by hensel lifting the gray image of the hensel lift of the three error correcting bch code of length 32 is the best 64 2 sup 32 code that is presently known this code also determines an extremal 32 dimensional even unimodular lattice

variable selection for binary spatial regression penalized quasi likelihood approach we consider the problem of selecting covariates in a spatial regression model when the response is binary penalized likelihood based approach is proved to be effective for both variable selection and estimation simultaneously in the context of a spatially dependent binary variable an uniquely interpretable likelihood is not available rather a quasi likelihood might be more suitable we develop a penalized quasi likelihood with spatial dependence for simultaneous variable selection and parameter estimation along with an efficient computational algorithm the theoretical properties including asymptotic normality and consistency are studied under increasing domain asymptotics framework an extensive simulation study is conducted to validate the methodology real data examples are provided for illustration and applicability although theoretical justification has not been made we also investigate empirical performance of the proposed penalized quasi likelihood approach for spatial count data to explore suitability of this method to a general exponential family of distributions

tuning parameter selectors for the smoothly clipped absolute deviation method summary the penalized least squares approach with smoothly clipped absolute deviation penalty has been consistently demonstrated to be an attractive regression shrinkage and selection method it not only automatically and consistently selects the important variables but also produces estimators which are as efficient as the oracle estimator however these attractive features depend on appropriate choice of the tuning parameter we show that the commonly used generalized crossvalidation cannot select the tuning parameter satisfactorily with a nonignorable overfitting effect in the resulting model in addition we propose a bic tuning parameter selector which is shown to be able to identify the true model consistently simulation studies are presented to support theoretical findings and an empirical example is given to illustrate its use in the female labor supply data

list decoding of generalized reed solomon codes over commutative rings we show that the list decoding procedure of guruswami and sudan may be used to decode generalized reed solomon rs codes defined over commutative rings with identity an algorithm for performing the first of the two phases of this decoding procedure is also given in particular we show that for generalized rs codes over chain rings the probability of picking an incorrect codeword from a given list can be reduced by increasing the size of the code alphabet while keeping the length and minimum hamming distance of the code constant

automatic solution of sturm liouville problems using the pruess method abstract we develop algorithms based on coefficient approximation for the automatic solution of regular and singular sturm liouville problems

analytical and numerical results for the fucik spectrum of the laplacian nontrivial solutions of u u vu 0 in rn with zero dirichlet condition on are studied the collection of the pairs v is the so called fucik spectrum f a new variational formulation for parts of f is presented and analyzed based on this formulation a minimization algorithm for the computation of a part of f is developed alternatively an approach using an implicit function argument is discussed analytically and via newton s method also numerically by combining the variational minimization method with newton s method a new bifurcation result is first observed numerically and then proved rigorously by replacing the variational minimization method by the mountain pass algorithm higher curves in f are found numerically several numerical results are discussed including a further example of the previously recorded phenomenon of crossing of fucik curves originating from different eigenvalues

maximum likelihood estimation of models for residual covariance in spatial regression we describe the maximum likelihood method for fitting the linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model containing unknown parameters observations are assumed to be gaussian we give conditions which ensure consistency and asymptotic normality of the estimators our main concern is with the analysis of spatial data and in this context we describe some simulation experiments to assess the small sample behaviour of estimators we also discuss an application of the spectral approximation to the likelihood for processes on a lattice

additive model building for spatial regression summary spatial regression is an important predictive tool in many scientific applications and an additive model provides a flexible regression relationship between predictors and a response variable we develop a regularized variable selection technique for building a spatial additive model we find that the methods developed for independent data do not work well for spatially dependent data this motivates us to propose a spatially weighted error norm with a group lasso type of penalty to select additive components in spatial additive models we establish the selection consistency of the approach proposed where the penalty parameter depends on several factors such as the order of approximation of additive components characteristics of the spatial weight and spatial dependence an extensive simulation study provides a vivid picture of the effects of dependent data structure and choice of a spatial weight on selection results as well as the asymptotic behaviour of the estimators as an illustrative example the method is applied to lung cancer mortality data over the period of 2000 2005 obtained from the surveillance epidemiology and end results programme national cancer institute usa

computing the first eigenpair of the p laplacian in annuli abstract we propose a method for computing the first eigenpair of the dirichlet p laplacian p 1 in the annulus a b x r n a x b n 1 for each t a b we use an inverse iteration method to solve two radial eigenvalue problems one in the annulus a t with the corresponding eigenvalue t and boundary conditions u a 0 u t and the other in the annulus t b with the corresponding eigenvalue t and boundary conditions u t 0 u b next we adjust the parameter t using a matching procedure to make t coincide with t thereby obtaining the first eigenvalue p hence by a simple splicing argument we obtain the positive l normalized radial first eigenfunction u p the matching parameter is the maximum point of u p in order to apply this method we derive estimates for t and t and we prove that these functions are monotone and locally lipschitz continuous moreover we derive upper and lower estimates for the maximum point which we use in the matching procedure and we also present a direct proof that u p converges to the l normalized distance function to the boundary as p we also present some numerical results obtained using this method