on the measure of totally real algebraic integers ii for a certain measure s2 a defined for totally real algebraic integers a 0 1 we find the four smallest values of q a the methods used involve linear programming and the results are verified using interval arithmetic

network tomography estimating source destination traffic intensities from link data abstract the problem of estimating the node to node traffic intensity from repeated measurements of traffic on the links of a network is formulated and discussed under poisson assumptions and two types of traffic routing regimens deterministic a fixed known path between each directed pair of nodes and markovian a random path between each directed pair of nodes determined according to a known markov chain fixed for that pair maximum likelihood estimation and related approximations are discussed and computational difficulties are pointed out a detailed methodology is presented for estimates based on the method of moments the estimates are derived algorithmically taking advantage of the fact that the first and second moment equations give rise to a linear inverse problem with positivity restrictions that can be approached by an em algorithm resulting in a particularly simple solution to a hard problem a small simulation study is carried out

distribution of algebraic numbers schur studied limits of the arithmetic means an of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk if the leading coefficients are bounded schur proved that lim supn an 1 e 2 we show that an 0 and estimate the rate of convergence by generalizing the erd s turan theorem on the distribution of zeros as an application we show that integer polynomials have some unexpected restrictions of growth on the unit disk schur also studied problems on means of algebraic numbers on the real line when all conjugate algebraic numbers are positive the problem of finding the sharp lower bound for lim infn an was developed further by siegel and others we provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line potential theoretic methods allow us to consider distribution of algebraic numbers in or near general sets in the complex plane we introduce the generalized mahler measure and use it to characterize asymptotic equidistribution of algebraic numbers in arbitrary compact sets of capacity one the quantitative aspects of this equidistribution are also analyzed in terms of the generalized mahler measure 1 schur s problems on means of algebraic numbers let e be a subset of the complex plane c consider the set of polynomials zn e of the exact degree n with integer coefficients and all zeros in e we denote the subset of zn e with simple zeros by zn e given m 0 we write pn anz zn e m if an m and pn zn e respectively pn zn e m if an m and pn zn e schur 45 4 8 studied the limit behavior of the arithmetic means of zeros for polynomials from zn e m as n where m 0 is an arbitrary fixed number his results may be summarized in the following statements let r 0 where r is the real line theorem a schur 45 satz ix given a polynomial pn z an n k 1 z k n define the arithmetic mean of squares of its zeros by sn n k 1 2 k n n if pn zn r m is any sequence of polynomials with degrees n then lim inf n sn e 1 6487 1 1 theorem b schur 45 satz xi for a polynomial pn z an n k 1 z k n define the arithmetic mean of its zeros by an n k 1 k n n if pn zn r m is any sequence of polynomials with degrees n then lim inf n an e 1 6487 1 2 1991 mathematics subject classification primary 11c08 secondary 11r04 26c10 30c15

generalized monte carlo significance tests summary simple monte carlo significance testing has many applications particularly in the preliminary analysis of spatial data the method requires the value of the test statistic to be ranked among a random sample of values generated according to the null hypothesis however there are situations in which a sample of values can only be conveniently generated using a markov chain initiated by the observed data so that independence is violated this paper describes two methods that overcome the problem of dependence and allow exact tests to be carried out the methods are applied to the rasch model to the finite lattice ising model and to the testing of association between spatial processes power is discussed in a simple case

on the absolute mahler measure of polynomials having all zeros in a sector let be an algebraic integer of degree d not 0 or a root of unity all of whose conjugates i are confined to a sector arg z we compute the greatest lower bound c of the absolute mahler measure i 1 d max 1 i 1 d of for belonging to nine subintervals of 0 2 3 in particular we show that c 2 1 12933793 from which it follows that any integer 1 and e i 3 all of whose conjugates have positive real part has absolute mahler measure at least c 2 this value is achieved for satisfying 1 0 2 where 0 1 3247 is the smallest pisot number the real root of 0 3 0 1

a fast procedure for model search in multidimensional contingency tables summary a procedure to select the simplest acceptable models for a multidimensional contingency table is proposed it is based on two rules first that if a model is accepted then all models that include it are considered to be accepted and secondly that if a model is rejected then all its submodels are considered to be rejected two versions are described one for the class of graphical models and the other for the class of hierarchical log linear models application of both versions to a six dimensional table is illustrated the procedure can be regarded as an alternative to fitting all possible models made computationally feasible by application of the two rules it is a generalization of the procedure proposed by havr anek 1984 but is in many cases considerably faster

on integer chebyshev polynomials we are concerned with the problem of minimizing the supremum norm on o 1 of a nonzero polynomial of degree at most n with integer coefficients we use the structure of such polynomials to derive an efficient algorithm for computing them we give a table of these polynomials for degree up to 75 and use a value from this table to answer an open problem due to p borwein and t erdelyi and improve a lower bound due to flammang et al

tiles in quasicrystals with quartic irrationality in 2003 pelantova and twarock did research into the number of and types of tiles found in 1 dimensional cut and project quasicrystals associated with 7 order symmetry in this paper we extend this to symmetries of order 9 degree 3 as well as orders 15 16 20 and 24 degree 4 some discussion of the next case order 11 degree 5 is given

on the linear independence measure of logarithms of rational numbers in this paper we give a general theorem on the linear independence measure of logarithms of rational numbers and in particular the linear independence measure of 1 log 2 log 3 log 5 and of 1 log 2 log 3 log 5 log 7 we also give a method to search for polynomials of smallest norm on a real interval a b which may be suitable for computing or improving the linear independence measure of logarithms of rational numbers

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

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

algebraic algorithms for sampling from conditional distributions we construct markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic examples include contingency tables logistic regression and spectral analysis of permutation data the algorithms involve computations in polynomial rings using grobner bases

littlewood type problems on 0 1 we consider the problem of minimizing the uniform norm on 0 1 over non zero polynomials p of the form p x sum j 0 n a jx j quad text with a j le 1 a j in bbb c where the modulus of the first non zero coefficient is at least delta 0 essentially sharp bounds are given for this problem an interesting related result states that there are absolute constants c 1 0 and c 2 0 such that exp c 1 sqrt n le inf 0 ne p in cal f n p 0 1 le exp c 2 sqrt n for every n ge 2 where cal f n denotes the set of polynomials of degree at most n with coefficients from 1 0 1 this chebyshev type problem is closely related to the question of how many zeros a polynomial from the above classes can have at 1 we also give essentially sharp bounds for this problem em inter alia we prove that there is an absolute constant c 0 such that every polynomial p of the form p x sum j 0 n a j x j quad text with a j le 1 a 0 a n 1 a j in bbb c has at most c sqrt n real zeros this improves the old bound c sqrt n log n given by schur in 1933 as well as more recent related bounds of bombieri and vaaler and up to the constant c this is the best possible result all the analysis rests critically on the key estimate stating that there are absolute constants c 1 0 and c 2 0 such that f 0 c 1 a le exp c 2 a f 1 a 1 for every f in cal s and a in 0 1 where cal s denotes the collection of all analytic functions f on the open unit disk d z in bbb c z f z le frac 1 1 z quad text for z in d

the integer chebyshev problem the main problem of this chapter is to find a polynomial in z n of minimal supremum norm on an interval this is p1 and it is of a slightly different flavour than most of the other problems in this book in that there is no restriction on the size of the coefficients we now state p1 with greater precision

group decision making models for sequential tasks the sequential probability ratio test sprt and related drift diffusion model ddm are optimal for choosing between two hypotheses using the minimal average number of samples and relevant for modeling the decision making process in human observers this work extends these models to group decision making previous works have focused almost exclusively on group accuracy here we explicitly address group decision time first we derive explicit solutions for the error rate and probability distribution function of decision times for a group of independent possibly nonidentical decision makers using one of three simple rules race majority total and majority first we illustrate our solutions with a group of n i i d decision makers who each make an individual decision using the sprt based ddm then compare the performance of each group rule under different constraints we then generalize these group rules to the eta total and eta first schemes to demonstrate the flexibility and power of our approach in characterizing the performance of a group given the performance of its individual members

the trace of totally positive algebraic integers for all totally positive algebraic numbers except a finite number of explicit exceptions the following inequality holds 1 d 1 d max 1 780022 1 66 1 where d is the degree of a and 0 1 d its conjugates this improves previous results of smyth flammang and rhin

algebraic integers whose conjugates all lie in an ellipse we find all 15909 algebraic integers whose conjugates all lie in an ellipse with two of them nonreal while the others lie in the real interval 1 2 this problem has applications to finding certain subgroups of sl 2 c we use explicit auxiliary functions related to the generalized integer transfinite diameter of compact subsets of c this gives good bounds for the coefficients of the minimal polynomial of a

two stage wilcoxon detectors using conditional tests the idea of the wilcoxon detector using a conditional test is extended to two stage detectors which constitute a class of sequential detectors three different two stage detectors are considered using the wilcoxon statistic with conditional tests the three detectors differ in the usage of the samples of the first stage for computing the test statistic in the second stage in computing the test statistic in the second stage the first detector disregards all the samples of the first stage the second detector uses the value of the test statistic of the first stage and the third detector uses all the samples of the first stage the asymptotic relative efficiency improves from the first to the second detector comparisons of these detectors with fixed sample size fss detectors show that they are better than both the fss linear detector and the fss conditional wilcoxon detector under the same false alarm and detection probabilities

sequential monte carlo methods for statistical analysis of tables we describe a sequential importance sampling sis procedure for analyzing two way zero one or contingency tables with fixed marginal sums an essential feature of the new method is that it samples the columns of the table progressively according to certain special distributions our method produces monte carlo samples that are remarkably close to the uniform distribution enabling one to approximate closely the null distributions of various test statistics about these tables our method compares favorably with other existing monte carlo based algorithms and sometimes is a few orders of magnitude more efficient in particular compared with markov chain monte carlo mcmc based approaches our importance sampling method not only is more efficient in terms of absolute running time and frees one from pondering over the mixing issue but also provides an easy and accurate estimate of the total number of tables with fixed marginal sums which is far more difficult for an mcmc method to achieve

sequential imputations and bayesian missing data problems abstract for missing data problems tanner and wong have described a data augmentation procedure that approximates the actual posterior distribution of the parameter vector by a mixture of complete data posteriors their method of constructing the complete data sets is closely related to the gibbs sampler both required iterations and similar to the em algorithm convergence can be slow we introduce in this article an alternative procedure that involves imputing the missing data sequentially and computing appropriate importance sampling weights in many applications this new procedure works very well without the need for iterations sensitivity analysis influence analysis and updating with new data can be performed cheaply bayesian prediction and model selection can also be incorporated examples taken from a wide range of applications are used for illustration

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

trace of totally positive algebraic integers and integer transfinite diameter the diameter of a totally real algebraic integer of degree d with conjugates 1 2 d is diam d 1 for all positive integers k and n diam 2 cos 2k n is less than 4 r m robinson has computed modulo integer translations all the other totally real algebraic integers with diam 4 for d 8 we have done the computations for all d 15 we use a large family of explicit auxiliary functions related to generalized integer transfinite diameter of real intervals they give good bounds for the coefficients of the minimal polynomial of for d 15 we prove a recent conjecture of capparelli et al mathematics subject classification 2000 primary 11c08 11y40

sampling for conditional inference on case control data summary the problem of exact conditional inference for discrete multivariate case control data has two forms the first is grouped case control data where monte carlo computations can be done using the importance sampling method of booth and butler 1999 biometrika86 321 332 or a proposed alternative sequential importance sampling method the second form is matched case control data for this analysis we propose a new exact sampling method based on the conditional poisson distribution for conditional testing with one binary and one integral ordered covariate this method makes computations on data sets with large numbers of matched sets fast and accurate we provide detailed derivation of the constraints and conditional distributions for conditional inference on grouped and matched data the methods are illustrated on several new and old data sets

testing for independence in a two way table new interpretations of the chi square statistic the classical chi square test for independence in a two way contingency table often rejects the independence hypothesis at an extremely small significance level particularly when the sample size is large this paper proposes some alternative distributions to independence to help interpret the chi 2 statistic in such situations the uniform alternative in which every possible contingency table of the given dimension and sample size receives equal probability leads to the volume test as originally suggested in a regression context by h hotelling exponential family theory is used to generate a class of intermediate alternatives between independence and uniformity leading to a random effects model for contingency tables

integer transfinite diameter and polynomials with small mahler measure in this work we show how suitable generalizations of the integer transfinite diameter of some compact sets in c give very good bounds for coefficients of polynomials with small mahler measure by this way we give the list of all monic irreducible primitive polynomials of z x of degree at most 36 with mahler measure less than 1 324 and of degree 38 and 40 with mahler measure less than 1 31

connecting the latent multinomial link et al 2010 biometrics 66 178 185 define a general framework for analyzing capture recapture data with potential misidentifications in this framework the observed vector of counts y is considered as a linear function of a vector of latent counts x such that y ax with x assumed to follow a multinomial distribution conditional on the model parameters bayesian methods are then applied by sampling from the joint posterior distribution of both x and in particular link et al 2010 propose a metropolis hastings algorithm to sample from the full conditional distribution of x where new proposals are generated by sequentially adding elements from a basis of the null space kernel of a we consider this algorithm and show that using elements from a simple basis for the kernel of a may not produce an irreducible markov chain instead we require a markov basis as defined by diaconis and sturmfels 1998 the annals of statistics 26 363 397 we illustrate the importance of markov bases with three capture recapture examples we prove that a specific lattice basis is a markov basis for a class of models including the original model considered by link et al 2010 and confirm that the specific basis used in their example with two sampling occasions is a markov basis the constructive nature of our proof provides an immediate method to obtain a markov basis for any model in this class

a trace bound for positive definite connected integer symmetric matrices abstract let a be a connected integer symmetric matrix i e a a i j m n z for some n a a t and the underlying graph vertices corresponding to rows with vertex i joined to vertex j if a i j 0 is connected we show that if all the eigenvalues of a are strictly positive then tr a 2 n 1 there are two striking corollaries first the analogue of the schur siegel smyth trace problem is solved for characteristic polynomials of connected integer symmetric matrices second we find new examples of totally real separable irreducible monic integer polynomials that are not minimal polynomials of integer symmetric matrices

maximum likelihood estimation in log linear models we study maximum likelihood estimation in log linear models under conditional poisson sampling schemes we derive necessary and sufficient conditions for existence of the maximum likelihood estimator mle of the model parameters and investigate estimability of the natural and mean value parameters under a nonexistent mle our conditions focus on the role of sampling zeros in the observed table we situate our results within the framework of extended exponential families and we exploit the geometric properties of log linear models we propose algorithms for extended maximum likelihood estimation that improve and correct the existing algorithms for log linear model analysis

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

on the absolute mahler measure of polynomials having all zeros in a sector ii kklet be an algebraic integer of degree d not 0 or a root of unity all of whose conjugates i are confined to a sector arg z in the paper on the absolute mahler measure of polynomials having all zeros in a sector g rhin and c smyth compute the greatest lower bound c of the absolute mahler measure d i 1 max 1 i 1 d of for belonging to nine subintervals of 0 2 3 in this paper we improve the result to thirteen subintervals of 0 and extend some existing subintervals

a modified sequential detection procedure a modified version of sequential testing is proposed and applied to discrete time signal detection the resulting detection procedure exhibits simplicity in structure and in analysis and retains most of the optimal features of sequential detection also two simple and efficient truncation schemes are suggested for the proposed detection procedure which is unrealizable without truncation

the monic integer transfinite diameter we study the problem of finding nonconstant monic integer polynomials normalized by their degree with small supremum on an interval i the monic integer transfinite diameter t m i is defined as the infimum of all such supremums we show that if i has length 1 then t m i we make three general conjectures relating to the value of t m i for intervals i of length less than 4 we also conjecture a value for t m 0 b where 0 b 1 we give some partial results as well as computational evidence to support these conjectures we define functions l t and l t which measure properties of the lengths of intervals i with t m i on either side of t upper and lower bounds are given for these functions we also consider the problem of determining t m i when i is a farey interval we prove that a conjecture of borwein pinner and pritsker concerning this value is true for an infinite family of farey intervals

computing totally positive algebraic integers of small trace we construct minimal polynomials of totally positive algebraic integers of small absolute trace by consideration of their reductions modulo auxiliary polynomials many new examples of such polynomials of minimal absolute trace for given degree are found the computations are pushed to degrees that previously were unattainable and one consequence is that the new examples form the majority of all those known as an application we produce a new bound for the schur siegel smyth trace problem

monic integer chebyshev problem we study the problem of minimizing the supremum norm by monic polynomials with integer coefficients let mn z denote the monic polynomials of degree n with integer coefficients a monic integer chebyshev polynomial mn mn z satisfies mn e infpn mn z pn e and the monic integer chebyshev constant is then defined by tm e limn mn e1 n this is the obvious analogue of the more usual integer chebyshev constant that has been much studied we compute tm e for various sets including all finite sets of rationals and make the following conjecture which we prove in many cases conjecture suppose a2 b2 a1 b1 is an interval whose endpoints are consecutive farey fractions this is characterized by a1b2 a2b1 1 then tm a2 b2 a1 b1 max 1 b1 1 b2 this should be contrasted with the nonmonic integer chebyshev constant case where the only intervals for which the constant is exactly computed are intervals of length 4 or greater

sampling schemes for sequential detection with dependent observations several sampling schemes and their corresponding sequential detection procedures in autoregressive noise are presented in this paper two of them use uniform sampling procedures with high and low sampling rates respectively the other two employ groups of samples which are separated by long intergroup delays such that the intergroup correlations are negligible one of the group sampling schemes also employs optimal signaling waveforms to further improve its energy efficiency in all the schemes data sampling and transformation are designed in such a way that wald s sequential probability ratio test sprt can still be implemented the performances of different schemes in terms of average termination time att are derived analytically when all the schemes employ the same sampling interval and under a constant signal amplitude constraint their performances are compared through analytical and numerical methods in addition under a constant power constraint their atts and energy efficiency are compared it is theoretically proved that the scheme using groups of samples with the optimal signaling waveform is the most energy efficient