constructing higher order methods for obtaining the multiple roots of nonlinear equations this paper concentrates on iterative methods for obtaining the multiple roots of nonlinear equations using the computer algebra system mathematica we construct an iterative scheme and discuss the conditions to obtain fourth order methods from it all the presented fourth order methods require one function and two derivative evaluation per iteration and are optimal higher order iterative methods for obtaining multiple roots we present some special methods from the iterative scheme including some known already numerical examples are also given to show their performance

design of signature sequences for overloaded cdma and bounds on the sum capacity with arbitrary symbol alphabets in this paper we explore some of the fundamentals of synchronous code division multiple access cdma as applied to wireless and optical communication systems under very general settings of any size for the user symbols and the signature matrix entries the channel is modeled by real complex additive noise of arbitrary distribution two problems are addressed the first problem concerns whether uniquely detectable overloaded matrices exist in the absence of additive noise under these general settings and if so whether there are any practical optimum detection algorithms the second one is about the bounds for the sum channel capacity when user data and signature matrices employ any real or complex alphabets finite or infinite in response to the first problem we have developed practical maximum likelihood detection algorithms for overloaded cdma systems for a large class of alphabets in response to the second problem a general theorem has been developed in which the sum capacity lower bounds with respect to the number of users spreading gain and signal to noise ratio can be derived to show the power and utility of the main theorem a number of sum capacity bounds for special cases are evaluated an important conclusion of this paper is that the lower and upper bounds of the sum capacity for small medium size cdma systems depend on both the input and the signature symbols this is contrary to the asymptotic results for large scale systems reported in the literature also confirmed in this paper where the signature symbols and statistics disappear for signature matrices and input vectors with i i d entries furthermore upper and asymptotic bounds are derived and compared to other derivations

a third order modification of newton s method for multiple roots in this paper we present a new third order modification of newton s method for multiple roots which is based on existing third order multiple root finding methods numerical examples show that the new method is competitive to other methods for multiple roots

construction of optimal order nonlinear solvers using inverse interpolation abstract there is a vast literature on finding simple roots of nonlinear equations by iterative methods these methods can be classified by order by the information used or by efficiency there are very few optimal methods that is methods of order 2 m requiring m 1 function evaluations per iteration here we give a general way to construct such methods by using inverse interpolation and any optimal two point method the presented optimal multipoint methods are tested on numerical examples and compared to existing methods of the same order of convergence

model free forward screening via cumulative divergence feature screening plays an important role in the analysis of ultrahigh dimensional data due to complicated model structure and high noise level existing screening methods often suffer from model

letter to the editor new higher order methods for solving nonlinear equations with multiple roots for a nonlinear equation f x 0 having a multiple root we consider steffensen s transformation t using the transformation say f q x t qf x for integer q 2 repeatedly we develop higher order iterative methods which require neither derivatives of f x nor the multiplicity of the root it is proved that the convergence order of the proposed iterative method is 1 2 q 2 for any equation having a multiple root of multiplicity m 2 the efficiency of the new method is shown by the results for some numerical examples

code construction for the t user noiseless adder channel a general method for constructing codes for the t user noiseless adder channel is proposed which achieves the largest sum rate known the authors consider a multiple access communication system in this system t statistically independent sources which use binary block codes c sub 1 c sub t of equal length n simultaneously transmit information via one common channel maintaining synchronization with respect to words and bits the output of the adder channel is the t l ary vector which is the component wise arithmetic sum of the transmitted binary vectors the task of the decoder is to uniquely determine the messages of all users

martingale difference correlation and its use in high dimensional variable screening in this article we propose a new metric the so called martingale difference correlation to measure the departure of conditional mean independence between a scalar response variable v and a vector predictor variable u our metric is a natural extension of distance correlation proposed by szekely rizzo and bahirov which is used to measure the dependence between v and u the martingale difference correlation and its empirical counterpart inherit a number of desirable features of distance correlation and sample distance correlation such as algebraic simplicity and elegant theoretical properties we further use martingale difference correlation as a marginal utility to do high dimensional variable screening to screen out variables that do not contribute to conditional mean of the response given the covariates further extension to conditional quantile screening is also described in detail and sure screening properties are rigorously justified both simulation results and real data illustrations demonstr

on optimum asymptotic multiuser efficiency of randomly spread cdma we extend the result by tse and verdu on the optimum asymptotic multiuser efficiency of randomly spread code division multiple access cdma with binary phase shift keying input random gaussian and random binary antipodal spreading are considered we obtain the optimum asymptotic multiuser efficiency of a k user system with spreading gain n when k and n rightarrow infty and the loading factor k n grows logarithmically with k under some conditions it is shown that the optimum detector in a gaussian randomly spread cdma system has a performance close to the single user system at high signal to noise ratio when k and n rightarrow infty and the loading factor k n is kept less than log 3 k 2 random binary antipodal matrices are also studied and a lower bound for the optimum asymptotic multiuser efficiency is obtained furthermore we investigate the connection between detecting matrices in the coin weighing problem and optimum asymptotic multiuser efficiency we obtain a condition such that for any binary input an n times k random matrix whose entries are chosen randomly from a finite set is a detecting matrix as k and n rightarrow infty

new families of nonlinear third order solvers for finding multiple roots in this paper we present two new families of iterative methods for multiple roots of nonlinear equations one of the families require one function and two derivative evaluation per step and the other family requires two function and one derivative evaluation it is shown that both are third order convergent for multiple roots numerical examples suggest that each family member can be competitive to other third order methods and newton s method for multiple roots in fact the second family is even better than the first

a direct approach to sparse discriminant analysis in ultra high dimensions summary sparse discriminant methods based on independence rules such as the nearest shrunken centroids classifier tibshirani et al 2002 and features annealed independence rules fan fan 2008 have been proposed as computationally attractive tools for feature selection and classification with high dimensional data a fundamental drawback of these rules is that they ignore correlations among features and thus could produce misleading feature selection and inferior classification we propose a new procedure for sparse discriminant analysis motivated by the least squares formulation of linear discriminant analysis to demonstrate our proposal we study the numerical and theoretical properties of discriminant analysis constructed via lasso penalized least squares our theory shows that the method proposed can consistently identify the subset of discriminative features contributing to the bayes rule and at the same time consistently estimate the bayes classification direction even when the dimension can grow faster than any polynomial order of the sample size the theory allows for general dependence among features simulated and real data examples show that lassoed discriminant analysis compares favourably with other popular sparse discriminant proposals

families of third and fourth order methods for multiple roots of nonlinear equations this paper presents two families of higher order iterative methods for solving multiple roots of nonlinear equations one is of order three and the other is of order four the presented iterative families all require two evaluations of the function and one evaluation of its first derivative thus the latter is of optimal order the third order family contains several iterative methods known already and different from the optimal fourth order methods for multiple roots known already the presented fourth order family use the modified newton s method as its first step local convergence analyses and some special cases of the presented families are given we also carry out some numerical examples to show their performance

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

a note on convergence radius of osada s method under holder continuous condition abstract in this paper we revise the proofs of the results obtained in convergence radius of osada s method under holder continuous condition 4 because the remainder of the taylor s expansion used for the obtainment of the local convergence radius is not correct so we perform the complete study in order to modify the equation for getting the local convergence radius the uniqueness radius and the error bounds moreover a dynamical study for the third order osada s method is also developed

the kolmogorov filter for variable screening in high dimensional binary classification variable screening techniques have been proposed to mitigate the impact of high dimensionality in classification problems including t test marginal screening fan fan 2008 and maximum marginal likelihood screening fan song 2010 however these methods rely on strong modelling assumptions that are easily violated in real applications to circumvent the parametric modelling assumptions we propose a new variable screening technique for binary classification based on the kolmogorov smirnov statistic we prove that this so called kolmogorov filter enjoys the sure screening property under much weakened model assumptions we supplement our theoretical study by a simulation study copyright 2013 oxford university press

convergence radius of the modified newton method for multiple zeros under holder continuous derivative abstract in recent years a lot of iterative methods for finding multiple zeros of nonlinear equations have been presented and analyzed however almost all these studies give no information for the convergence radius of the corresponding method in this paper we give an estimate of the convergence radius of the well known modified newton s method for multiple zeros when the involved function satisfies a holder and center holder continuity condition

further results on coding for t user multiple access channels corresp coding schemes for the noiseless t user binary adder multiple access channel are discussed a class of uniquely decodable codes of arbitrary length satisfying a specific recursive relationship and which are asymptotically good is explicitly constructed

how an erdos renyi type search approach gives an explicit code construction of rate 1 for random access with multiplicity feedback pippenger 1981 showed in a probabilistic way that the capacity of a collision channel with multiplicity feedback is one in this correspondence using an erdos renyi type search strategy we settle a long standing open problem by giving a constructive proof of this result moreover we prove that two different capacity definitions are equivalent thereby solving a problem posed by tsybakov 1985

a family of k 1 ary signature codes for noisy multiple access adder channel a coding scheme of k 1 ary error correcting signature codes for a noisy multiple access adder channel is proposed given a signature matrix a and a difference matrix d d d a priori a larger signature matrix is obtained by replacing each element in the hadamard matrix with a or d or d depending on the values of the elements and their locations in the hadamard matrix the set of rows in the proposed matrix gives an error correcting signature code introducing a difference matrix makes it possible to construct an error correcting signature code whose sum rate is increased with an increase in the order of the hadamard matrix either binary or non binary signature codes are constructed when the pairs of matrices a and d are given

high order nonlinear solver for multiple roots a method of order four for finding multiple zeros of nonlinear functions is developed the method is based on jarratt s fifth order method for simple roots and it requires one evaluation of the function and three evaluations of the derivative the informational efficiency of the method is the same as previously developed schemes of lower order for the special case of double root we found a family of fourth order methods requiring one less derivative thus this family is more efficient than all others all these methods require the knowledge of the multiplicity

generalized t user codes for multiple access channels corresp large classes of t user codes for the binary adder channel are shown to be easily constructed by generalizing some previous results in two basic ways the first is an iterative procedure the second is by introducing an equivalence relation on such codes decoding algorithms are presented in both cases the number of codes in the equivalence class of some previously defined codes is determined exactly

convergence radius of halley s method for multiple roots under center holder continuous condition recently a new treatment based on taylor s expansion to give the estimate of the convergence radius of iterative method for multiple roots has been presented it has been successfully applied to enlarge the convergence radius of the modified newton s method and osada s method for multiple roots this paper re investigates the convergence radius of halley s method under the condition that the derivative f m 1 of function f satisfies the center holder continuous condition we show that our result can be obtained under much weaker condition and has a wider range of application than that given by bi et al 2011 21

recursive constructions of detecting matrices for multiuser coding a unifying approach detecting matrices are a class of combinatorial objects originated from the coin weighing problem of soderberg and shapiro in the early 1960s in this paper various known recursive construction techniques for binary bipolar and ternary detecting matrices are reexamined in a unifying framework new general recursive constructions of detecting matrices which include previous recursive constructions as special cases are derived such matrices find applications in multiuser coding since they are equivalent to a certain class of uniquely decodable multiuser codes for the binary adder channel interestingly it is found that among the three kinds of detecting matrices ternary detecting matrices are of fundamental significance from the combinatorial theoretic as well as from the multiuser coding application point of view

model free feature screening for ultrahigh dimensional data with the recent explosion of scientific data of unprecedented size and complexity feature ranking and screening are playing an increasingly important role in many scientific studies in this article we propose a novel feature screening procedure under a unified model framework which covers a wide variety of commonly used parametric and semiparametric models the new method does not require imposing a specific model structure on regression functions and thus is particularly appealing to ultrahigh dimensional regressions where there are a huge number of candidate predictors but little information about the actual model forms we demonstrate that with the number of predictors growing at an exponential rate of the sample size the proposed procedure possesses consistency in ranking which is both useful in its own right and can lead to consistency in selection the new procedure is computationally efficient and simple and exhibits a competent empirical performance in our intensive simulations and real

convergence radius of osada s method under center holder continuous condition recently a new treatment based on taylor s expansion to give the estimate of the convergence radius of iterative method for multiple roots has been presented it has been successfully applied to enlarge the estimate of the convergence radius of the modified newton s method for multiple roots using the similarly treatment this paper investigates the convergence radius of the osada s method under the condition that the derivative f m 1 of function f satisfies the center holder continuous condition by some examples we show the treatment is simpler and efficient once again the uniqueness ball of solution is also discussed