matrix exponentials su n group elements and real polynomial roots the exponential of an n n matrix can always be expressed as a matrix polynomial of order n 1 in particular a general group element for the fundamental representation of su n can be expressed as a matrix polynomial of order n 1 in a traceless n n hermitian generating matrix with polynomial coefficients consisting of elementary trigonometric functions dependent on n 2 invariants in addition to the group parameter these invariants are just angles determined by the direction of a real n vector whose components are the eigenvalues of the hermitian matrix equivalently the eigenvalues are given by projecting the vertices of an n 1 simplex onto a particular axis passing through the center of the simplex the orientation of the simplex relative to this axis determines the angular invariants and hence the real eigenvalues of the matrix

on approximate zeros and rootfinding algorithms for a complex polynomial on donne des criteres pour qu un nombre complexe soit une racine approchee d un polynome f par la methode de newton ou d euler d ordre k on construit des familles d algorithmes surement convergents

extensions of algebraic groups with finite quotient and nonabelian 2 cohomology abstract for a finite smooth algebraic group f over a field k and a smooth algebraic group g over the separable closure of k we define the notion of f kernel in g and we associate to it a set of nonabelian 2 cohomology we use this to study extensions of f by an arbitrary smooth k group g we show in particular that any such extension comes from an extension of finite k groups when k is perfect and we give explicit bounds on the order of these finite groups when g is linear we prove moreover some finiteness results on these sets

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

the exponential map of gl n a finite expansion of the exponential map for a matrix is presented the method uses the cayley hamilton theorem for writing the higher matrix powers in terms of those for the first n 1 the resulting sums over the corresponding coefficients are rational functions of the eigenvalues of the matrix

a fast parallel algorithm for determining all roots of a polynomial with real roots given a polynomial p z of degree n with m bit integer coefficients and an integer mu the problem of determining all its roots with error less than 2 mu is considered it is shown that this problem is in the class nc if p z has all real roots some very interesting properties of a sturm sequence of a polynomial with distinct real roots are proved and used in the design of a fast parallel algorithm for this problem using newton identities and a novel numerical integration scheme for evaluating a contour integral to high precision this algorithm determines good approximations to the linear factors of p z

exact expansions of arbitrary tensor functions f a and their derivatives a general method is presented for determining the exact expansions of a tensor function f a where a is a second order tensor in n dimensional euclidean space and f a is defined by the power series it is shown that f a can be obtained by differentiating a scalar function of the eigenvalues of a using this method closed form singularity free expressions of arbitrary tensor functions and their first derivatives are deduced in two and three dimensional cases

complexity of bezout s theorem iv probability of success extensions we estimate the probability that a given number of projective newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n 1 complex variables of fixed degrees will find all the roots of the system we also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context further complexity theory is developed

covariate adjustment of event histories estimated from markov chains the additive approach summary markov chain models are frequently used for studying event histories that include transitions between several states an empirical transition matrix for nonhomogeneous markov chains has previously been developed including a detailed statistical theory based on counting processes and martingales in this article we show how to estimate transition probabilities dependent on covariates this technique may e g be used for making estimates of individual prognosis in epidemiological or clinical studies the covariates are included through nonparametric additive models on the transition intensities of the markov chain the additive model allows for estimation of covariate dependent transition intensities and again a detailed theory exists based on counting processes the martingale setting now allows for a very natural combination of the empirical transition matrix and the additive model resulting in estimates that can be expressed as stochastic integrals and hence their properties are easily evaluated two medical examples will be given in the first example we study how the lung cancer mortality of uranium miners depends on smoking and radon exposure in the second example we study how the probability of being in response depends on patient group and prophylactic treatment for leukemia patients who have had a bone marrow transplantation a program in r and s plus that can carry out the analyses described here has been developed and is freely available on the internet

on the worst case arithmetic complexity of approximating zeros of systems of polynomials let d 1 cdots d n be positive integers let mathcal p denote the set of systems of polynomials f mathbb c n to mathbb c n that have only finitely many zeros including those at infinity and that satisfy degree f i d i for all i let 0 varepsilon leqq r it is shown for fixed d 1 cdots d n that with respect to a certain model of computation the worst case computational complexity of obtaining varepsilon approximations to at least those zeros xi satisfying xi leqq r for arbitrary f in mathcal p is theta log log r varepsilon that is to say both upper and lower bounds are proved an algorithm for proving the upper bound is introduced the number of operations required by this algorithm is o left n mathcal d 4 log mathcal d log log r varepsilon n 2 mathcal d 4 left begin array 20 c 1 sigma d i n end array right 4 right quad text where mathcal d prod limits i 1

more on rotations as spin matrix polynomials any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j the challenge is to find a convenient form for the coefficients of the matrix polynomial the theory of biorthogonal systems is a useful framework to meet this challenge central factorial numbers play a key role in the theoretical development explicit polynomial coefficients for rotations expressed either as exponentials or as rational cayley transforms are considered here structural features of the results are discussed and compared and large j limits of the coefficients are examined

a copula based mixed poisson model for bivariate recurrent events under event dependent censoring in many chronic disease processes subjects are at risk of two or more types of events we describe a bivariate mixed poisson model in which a copula function is used to model the association between two gamma distributed random effects the resulting model is a bivariate negative binomial process in which each type of event arises from a negative binomial process methods for parameter estimation are described for parametric and semiparametric models based on an em algorithm we also consider the issue of event dependent censoring based on one type of event which arises when one event is sufficiently serious that its occurence may influence the decision of whether to withdraw a patient from a study the asymptotic biases of estimators of rate and mean functions from naive marginal analyses are discussed as well as associated treatment effects because the joint model is fit based on a likelihood consistent estimates are obtained simulation studies are carried out to evaluate the empirical performance of the proposed estimators with independent and event dependent censoring and applications to a trial of breast cancer patients with skeletal metastases and a study of patients with chronic obstructive pulmonary disease illustrate the approach

on the complexity of polynomial zeros the parallel complexity of the simultaneous approximation to all the zeros of a polynomial is investigated by modifying and analyzing an algorithm given by householder it is possible to obtain a priori bounds to the number of iterations sufficient to yield a given accuracy and to the number of digits required in the finite arithmetic more classes of polynomials for which the simultaneous approximation to all the zeros can be carried out in polylogarithmic time are found some cases of polynomials customarily considered hard are easily solved the root finding problem for a polynomial of degree n having zeros z i i 1 cdots n is mathcal nc reduced to finding a polynomial a z such that a z i 1 a z i leq 1 1 n c where c is a constant

estimation of excess risk from case control data using aalen s linear regression model we introduce methods for statistical inference in aalen s non parametric linear regression model of disease incidence aalen 1989 statistics in medicine 8 907 925 from nested case control data these methods provide the basis for estimation of excess risk as a linear function of dose and absolute risk for a given exposure history the methods are illustrated by estimating excess and absolute risks associated with radon exposure and smoking from nested case control samples from the colorado plateau uranium miners cohort language en

generalizations of laguerre s method higher order methods laguerre s method is an efficient and reliable method for finding zeros of polynomials and certain other functions a new derivation and motivation of laguerre s method is given which allows it to be included in a class of methods as general as methods of order three or more based on direct generalized hermite or hyperosculatory interpolation members of this new class share with laguerre s method the property of being globally convergent to zeros of polynomials with only real zeros and have the same order of convergence at simple zeros as the classic methods based on generalized hermite interpolation methods of order 4 and 3 303 are investigated and numerical results indicate that for large 100 times 100 eigenvalue problems the method of order 3 303 is as efficient and reliable as laguerre s method

exact matrix expansions for group elements of su n some exact closed form expressions are presented for exponentials of matrices of su n in the fundamental representation as a consequence the same formulas apply to embedded subalgebras a su n in higher dimensional representations explicit formulas are given for n 5 as well as applications to skyrmions in the callan klebanov bound state approach

deterministic improvement of complex polynomial factorization based on the properties of the associated resultant abstract we show an improvement of the known best deterministic approach to factorization of a univariate polynomial p x over the complex field previously such an improvement was achieved by means of a random perturbation of the coefficients of the input polynomial we now apply some new techniques which lead to a similar result without any randomization the approach is unusual in exploiting some properties of the resultant of p x and p x which is completely distinct from the techniques usually applied in order to compute complex polynomial factorization

computing matrix eigenvalues and polynomial zeros where the output is real surprisingly simple corollaries from the courant fischer minimax characterization theorem enable us to devise a very effective algorithm for the evaluation of a set s interleaving the set e of the eigenvalues of an n x n real symmetric tridiagonal rst matrix tn as well as a point that splits e into two subsets of comparable cardinalities furthermore we extend this algorithm so as to approximate all the n eigenvalues of tn at nearly optimal sequential and parallel cost that is at the cost of staying within polylogarithmic factors from the straightforward lower bounds the resulting improvement of the known processor bound in nc algorithms for the rst eigenproblem is roughly by factor n our approach extends the previous works m ben or and p tiwari j complexity 6 1990 pp 417 442 and m ben or et al siam j comput 17 1988 pp 1081 1092 for the approximation of the zeros of a polynomial having only real zeros and our algorithm leads to an alternative and simplified derivation of the known record parallel and sequential complexity estimates for the latter problem

grothendieck s theorem on non abelian h 2 and local global principles a theorem of grothendieck asserts that over a perfect field k of cohomological dimension one all non abelian h 2 cohomology sets of algebraic groups are trivial the purpose of this paper is to establish a formally real generalization of this theorem the generalization to the context of perfect fields of virtual cohomological dimension one takes the form of a local global principle for the h 2 sets with respect to the orderings of the field this principle asserts in particular that an element in h 2 is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of k our techniques provide a new proof of grothendieck s original theorem an application to homogeneous spaces over k is also given

nonparametric estimation of stage occupation probabilities in a multistage model with current status data multistage models are used to describe individuals or experimental units moving through a succession of stages corresponding to distinct states e g healthy diseased diseased with complications dead the resulting data can be considered to be a form of multivariate survival data containing information about the transition times and the stages occupied traditional survival analysis is the simplest example of a multistage model where individuals begin in an initial stage say alive and move irreversibly to a second stage death in this article we consider general multistage models with a directed tree structure progressive models in which individuals traverse through stages in a possibly non markovian manner we construct nonparametric estimators of stage occupation probabilities and marginal cumulative transition hazards empirical calculations of these quantities are not possible due to the lack of complete data we consider current status information which represents a more severe form of censoring than the commonly used right censoring asymptotic validity of our estimators can be justified using consistency results for nonparametric regression estimators finite sample behavior of our estimators is studied by simulation in which we show that our estimators based on these limited data compare well with those based on complete data we also apply our method to a real life data set arising from a cardiovascular diseases study in taiwan

a globally convergent method for simultaneously finding polynomial roots a new method for the simultaneous approximation of all the roots of a polynomial is given the method converges for almost every initial approximation the set of the exceptional starting points being a closed set of measure zero at least if all the polynomial roots are real and simple the method exhibits quadratic convergence not only to simple but also to multiple roots

a linear regression model for the analysis of life times a linear model is suggested for the influence of covariates on the intensity function this approach is less vulnerable than the cox model to problems of inconsistency when covariates are deleted or the precision of covariate measurements is changed a method of non parametric estimation of regression functions is presented this results in plots that may give information on the change over time in the influence of covariates a test method and two goodness of fit plots are also given the approach is illustrated by simulation as well as by data from a clinical trial of treatment of carcinoma of the oropharynx

further results on the non parametric linear regression model in survival analysis this paper gives further developments of a non parametric linear regression model in survival analysis three subjects are studied first martingale residuals originally developed for the cox model are introduced for our linear model their theory is developed and they are shown to be useful for judging goodness of fit the second focus of the paper is on the use of bootstrap replications to judge which features of the cumulative regression plots are likely to reflect real phenomena and not merely random variation in particular this is applied to judging whether the effect of a covariate disappears over time a problem for which no formal test exists the third subject is density type or kernel estimation of the regression functions themselves this might give more direct information than the cumulative plots the approaches are illustrated by data from a clinical trial of carcinoma of the oropharynx and by survival times of grafts in renal patients

direct modeling of regression effects for transition probabilities in the progressive illness death model in this work we present direct regression analysis for the transition probabilities in the possibly non markov progressive illness death model the method is based on binomial regression where the response is the indicator of the occupancy for the given state along time randomly weighted score equations that are able to remove the bias due to censoring are introduced by solving these equations one can estimate the possibly time varying regression coefficients which have an immediate interpretation as covariate effects on the transition probabilities the performance of the proposed estimator is investigated through simulations we apply the method to data from the registry of systematic lupus erythematosus relesser a multicenter registry created by the spanish society of rheumatology specifically we investigate the effect of age at lupus diagnosis sex and ethnicity on the probability of damage and death along time copyright 2017 john wiley sons ltd

on taylor s formula for the resolvent of a complex matrix the resolvent r l a of a complex rxr matrix a is an analytic function in any domain with empty intersection with the spectrum s a of a the well known taylor expansion of r l a in a neighborhood of any given l email protected email protected a is modified taking into account that only the first powers of r l 0 a are linearly independent the main tool in this framework is given by the multivariable polynomials f k n v 1 v 2 v r n 1 0 1 k 1 2 email protected r depending on the invariants v 1 v 2 v r of r l a m denotes the degree of the minimal polynomial these functions are used in order to represent the coefficients of the subsequent powers of r l 0 a as a linear combination of the first m of them

on rotations as spin matrix polynomials recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods structural features of the results are then examined in the framework of biorthogonal systems to obtain an alternate derivation the central factorial numbers play key roles in both derivations

additive hazards markov regression models illustrated with bone marrow transplant data summary when there are covariate effects to be considered multi state survival analysis is dominated either by parametric markov regression models or by semiparametric markov regression models using cox s 1972 proportional hazards models for transition intensities between the states the purpose of this research work is to study alternatives to cox s model in a general finite state markov process setting we shall look at two alternative models aalen s 1989 nonparametric additive hazards model and lin ying s 1994 semiparametric additive hazards model the former allows the effects of covariates to vary freely over time while the latter assumes that the regression coefficients are constant over time with the basic tools of the product integral and the functional delta method we present an estimator of the transition probability matrix and develop the large sample theory for the estimator under each of these two models data on 1459 hla identical sibling transplants for acute leukaemia from the international bone marrow transplant registry serve as illustration

assessing cumulative incidence functions under the semiparametric additive risk model in analyzing competing risks data a quantity of considerable interest is the cumulative incidence function often the effect of covariates on the cumulative incidence function is modeled via the proportional hazards model for the cause specific hazard function as the proportionality assumption may be too restrictive in practice we consider an alternative more flexible semiparametric additive hazards model of biometrika 1994 81 501 514 for the cause specific hazard this model specifies the effect of covariates on the cause specific hazard to be additive as well as allows the effect of some covariates to be fixed and that of others to be time varying we present an approach for constructing confidence intervals as well as confidence bands for the cause specific cumulative incidence function of subjects with given values of the covariates furthermore we also present an approach for constructing confidence intervals and confidence bands for comparing two cumulative incidence functions given values of the covariates the finite sample property of the proposed estimators is investigated through simulations we conclude our paper with an analysis of the well known malignant melanoma data using our method