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

lower bounds for the heights of solutions of linear equations let k be a number eld of degree d we de ne as usual the eld height hk of a number k by hk v m k max 1 dv v here the v adic valuations v are normalized such that on q they coincide with either the standard absolute value or with the p adic absolute value p having p p p 1 moreover dv denotes the local degree the absolute height of then is de ned by h hk lehmer s conjecture says the following let be a nonzero algebraic number of degree d then if is not a root of unity we have

a generalization of the schur siegel smyth trace problem abstract let be a totally positive algebraic integer and define its absolute trace to be tr deg the trace of divided by the degree of elementary considerations show that the absolute trace is always at least one while it is plausible that for any 0 the absolute trace is at least 2 with only finitely many exceptions this is known as the schur siegel smyth trace problem our aim in this paper is to show that the schur siegel smyth trace problem can be considered as a special case of a more general problem

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

improved dynamic programming methods for optimal control of lumped parameter stochastic systems new dynamic programming methods are developed to solve stochastic control problems with a larger number of state variables than previously possible these methods apply accurate interpolation to numerical approximation of continuous cost to go functions greatly reducing the number of discrete states that must be evaluated by efficiently incorporating information on first and second derivatives the approximation reduces computational effort by several orders of magnitude over traditional methods consequently it is practical to apply dynamic programming to complex stochastic problems with a larger number of state variables than traditionally possible results are presented for hypothetical reservoir control problems with up to seven state variables and two random inputs

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

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

on the spectrum of the zhang zagier height from recent work of zhang and of zagier we know that their height n is bounded away from 1 for every algebraic number a different from 0 1 1 2 3 2 the study of the related spectrum is especially interesting for it is linked to lehmer s problem and to a conjecture of bogomolov after recalling some definitions we show an improvement of the so called zhang zagier inequality to achieve this we need some algebraic numbers of small height so in the third section we describe an algorithm able to find them and we give an algebraic number with height 1 2875274z discovered in this way this search up to degree 64 suggests that the spectrum of n may have a limit point less than 1 292 we prove this fact in the fourth part

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

sums of heights of algebraic numbers for at x f x tg x we consider the set at 0 h t q the polynomials f x g x are in z x with only mild restrictions and h is the weil height of we show that this set is dense in d for some effectively computable limit point d

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

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

the smallest perron numbers a perron number is a real algebraic integer of degree d 2 whose conjugates are i such that max 2 i d i in this paper we compute the smallest perron numbers of degree d 24 and verify that they all satisfy the lind boyd conjecture moreover the smallest perron numbers of degree 17 and 23 give the smallest house for these degrees the computations use a family of explicit auxiliary functions these functions depend on generalizations of the integer transfinite diameter of some compact sets in c

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

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

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

flood frequency analysis with systematic and historical or paleoflood data based on the two parameter general extreme value models historical and paleoflood data have become an important source of information for flood frequency analysis a number of studies have been proposed in the literature regarding the value of historical and paleoflood information for estimating flood quantiles these studies have been generally based on computer simulation experiments in this paper the value of using systematic and historical paleoflood data relative to using systematic records alone is examined analytically by comparing the asymptotic variances of flood quantiles assuming a two parameter general extreme value marginal distribution type 1 and type 2 censored data and maximum likelihood estimation method the results of this study indicate that the value of historical and paleoflood data for estimating flood quantiles can be small or large depending on only three factors the relative magnitudes of the length of the systematic record n and the length of the historical period m the return period t of the flood quantile of interest and the return period h of the threshold level of perception for instance for n 50 m 50 and t 500 the statistical gain for type 2 censoring becomes significantly larger than for type 1 censoring as h becomes greater than 100 years in addition computer experiments have shown that the results regarding the statistical gain based on asymptotic considerations are valid for the usual sample sizes

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

orbits of algebraic numbers with low heights we prove that the two smallest values of h h 1 1 h 1 1 are 0 and 0 4218 for any algebraic integer

algorithm developments for discrete adjoint methods this paper presents a number of algorithm developments for adjoint methods using the discrete approach in which the discretisation of the non linear equations is linearised and the resulting matrix is then transposed with a new iterative procedure for solving the adjoint equations exact numerical equivalence is maintained between the linear and adjoint discretisations the incorporation of strong boundary conditions within the discrete approach is discussed as well as a new application of adjoint methods to linear unsteady flow in turbomachinery

two new points in the spectrum of the absolute mahler measure of totally positive algebraic integers for totally positive algebraic integers 0 1 of degree d we consider the set l of values of m 1 d where m is the mahler measure of c j smyth has found the four smallest values of l and conjectured that the fifth point is 2 cos 2 60 2 we prove that this is so and moreover we give the sixth point of l

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

robust design of structures using optimization methods the robust design of structures with stochastic parameters is studied using optimization techniques the first two statistical moments of the stochastic parameters including design variables are considered in conjunction with the second order perturbation method for the approximation of mean value and variance of the structural response in this framework the sensitivities of the mean values and variances of the structural performance function with respect to the design variables are calculated for use in the optimization task the robust design of structures is formulated as a multi criteria optimization problem in which both the expected value and the standard deviation of the objective function are to be minimized the robustness of the feasibility is also taken into account by involving the variability of the structural response in the constraints the two criteria optimization problem is converted into a scalar one and is then solved by a gradient based optimization algorithm to demonstrate the applicability of the presented method numerical examples are given involving static and dynamic response

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

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

flood frequency analysis with historical and paleoflood information an investigation is made of flood quantile estimators which can employ historical and paleoflood information in flood frequency analyses two categories of historical information are considered censored data where the magnitudes of historical flood peaks are known and binomial data where only threshold exceedance information is available a monte carlo study employing the two parameter lognormal distribution shows that maximum likelihood estimators mles can extract the equivalent of an additional 10 30 years of gage record from a 50 year period of historical observation the mle routines are shown to be substantially better than an adjusted moment estimator similar to the one recommended in bulletin 17b of the united states water resources council hydrology committee 1982 the mle methods performed well even when floods were drawn from other than the assumed lognormal distribution

robust flood frequency models the concept of a robust model is briefly explored in the context of flood frequency analysis two necessary properties of a robust model are advanced namely resistance and efficiency strategies for seeking more robust models are discussed because of its versatility the five parameter wakeby distribution can credibly be considered a parent flood distribution four regionalized wakeby parents are employed in simulation studies to search for robust models these parents were shown by houghton to be representative of u s flood experience in the sense that certain raw flood data characteristics could be reproduced a limited range of sampling experiments were undertaken the results suggest that of the site specific estimators considered the two parameter log normal maximum likelihood estimator is most resistant with gumbel estimators employing either maximum likelihood or probability weighted moments displaying comparable resistance several estimators which utilize regional flood information were compared included were empirical bayes estimators which are structurally similar to james stein rules and regionalized estimators based on the flood index method these estimators exhibited substantial improvements in aggregate risk performance over their site specific counterparts particularly for short record lengths regionalized estimators appear to be preferable for short record lengths while estimators which combine both site and regional flood information are preferable for longer record lengths when such estimation procedures are considered other distributional models such as log pearson type iii and wakeby become practical alternatives to the two parameter log normal model

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

censored data correlation and principal component dynamic programming a principal component stochastic dynamic programming algorithm for stochastic multireservoir hydropower system operation optimization was proposed by saad and turgeon 1988 the dimension of the state space of the system is reduced by using only the major principal components of the system s state as determined by a principal component analysis of the results of a deterministic optimization a significant refinement of the procedure is to employ censored data principal component analysis the more sophisticated statistical analysis provides a better model of the full dynamic range of the reservoir system s state explains more observed variability of the reservoir volumes and provides more accurate estimates of the statistical description s parameters in an example presented of a four reservoir system the first component of the censored data principal component analysis for a given month explained all but 7 of the uncensored system s variability whereas the standard analysis for the same month left 12 of the observed system s variability unexplained

a density matching approach for optimization under uncertainty abstract modern computers enable methods for design optimization that account for uncertainty in the system so called optimization under uncertainty ouu we propose a metric for ouu that measures the distance between a designer specified probability density function of the system response the target and the system response s density function at a given design we study an ouu formulation that minimizes this distance metric over all designs we discretize the objective function with numerical quadrature and we approximate the response density function with a gaussian kernel density estimate we offer heuristics for addressing issues that arise in this formulation and we apply the approach to a cfd based airfoil shape optimization problem we qualitatively compare the density matching approach to a multi objective robust design optimization to gain insight into the method

l moment diagrams for censored observations observed data sets containing values above or below the analytical threshold of measuring equipment are referred to as censored such data are frequently encountered in quality and quantity monitoring applications of water soil and air samples most of the previous literature on the statistical analysis of censored data relates to the problems of moment parameter and quantile estimation methods such estimation methods usually assume an underlying probability distribution few goodness of fit methods exist for censored data we introduce l moment diagrams for the evaluation of the goodness of fit of alternative distributional hypotheses for left censored data experiments with artificial censored data sets document the conditions under which l moment diagrams should be useful our approach like hosking s 1995 approach for right censoring derived l moment diagrams for left censored observations from partial probability weighted moments

intersecting curves and algebraic subgroups conjectures and more results this paper solves in the affirmative up to dimension n 5 a question raised in an earlier paper by the authors the equivalence of the problem with a conjecture of shou wu zhang is proved in the appendix

efficient global reliability analysis for nonlinear implicit performance functions many engineering applications are characterized by implicit response functions that are expensive to evaluate and sometimes nonlinear in their behavior making reliability analysis difficult this paper develops an efficient reliability analysis method that accurately characterizes the limit state throughout the random variable space the method begins with a gaussian process model built from a very small number of samples and then adaptively chooses where to generate subsequent samples to ensure that the model is accurate in the vicinity of the limit state the resulting gaussian process model is then sampled using multimodal adaptive importance sampling to calculate the probability of exceeding or failing to exceed the response level of interest by locating multiple points on or near the limit state more complex and nonlinear limit states can be modeled leading to more accurate probability integration by concentrating the samples in the area where accuracy is important i e in the vicinity of the limit state only a small number of true function evaluations are required to build a quality surrogate model the resulting method is both accurate for any arbitrarily shaped limit state and computationally efficient even for expensive response functions this new method is applied to a collection of example problems including one that analyzes the reliability of a microelectromechanical system device that current available methods have difficulty solving either accurately or efficiently

canonical heights transfinite diameters and polynomial dynamics let z be a polynomial of degree at least 2 with coefficients in a number field k iterating gives rise to a dynamical system and a corresponding canonical height function as defined by call and silverman we prove a simple product formula relating the transfinite diameters of the filled julia sets of over various completions of k and we apply this formula to give a generalization of bilu s equidistribution theorem for sequences of points whose canonical heights tend to zero

plotting positions of annual flood extremes considering extraordinary values a generalized plotting position formula for estimation of the recurrence interval or its reciprocal the exceedance probability of hydrologic events in an incomplete data sample containing both extraordinarily large and small historical values is derived the derivation is based on order statistics analysis and is independent of the probability distribution of the hydrologic variable it is shown that the weibull formula is a special case of the generalized formula

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

applied smoothing techniques for data analysis the kernel approach with s plus illustrations 1 density estimation for exploring data 2 density estimation for inference 3 nonparametric regression for exploring data 4 inference with nonparametric regression 5 checking parametric regression models 6 comparing regression curves and surfaces 7 time series data 8 an introduction to semiparametric and additive models references

tree based reinforcement learning for optimal water reservoir operation 1 although being one of the most popular and extensively studied approaches to design water reservoir operations stochastic dynamic programming is plagued by a dual curse that makes it unsuitable to cope with large water systems the computational requirement grows exponentially with the number of state variables considered curse of dimensionality and an explicit model must be available to describe every system transition and the associated rewards costs curse of modeling a variety of simplifications and approximations have been devised in the past which in many cases make the resulting operating policies inefficient and of scarce relevance in practical contexts in this paper a reinforcement learning approach called fitted q iteration is presented it combines the principle of continuous approximation of the value functions with a process of learning off line from experience to design daily cyclostationary operating policies the continuous approximation performed via tree based regression makes it possible to mitigate the curse of dimensionality by adopting a very coarse discretization grid with respect to the dense grid required to design an equally performing policy via stochastic dynamic programming the learning experience in the form of a data set generated combining historical observations and model simulations allows us to overcome the curse of modeling lake como water system italy is used as study site to infer general guidelines on the appropriate setting for the algorithm parameters and to demonstrate the advantages of the approach in terms of accuracy and computational effectiveness compared to traditional stochastic dynamic programming

uncorrelated measurement error in flood frequency inference when measurement error me is present the true value of the annual maximum discharge and of the censoring threshold discharge is unknown the impact of using estimated rather than true discharges on the inference of flood quantiles is considered the likelihood function is formulated for data which consist of annual maximum discharges systematic data and occurrences of floods above some threshold discharge binomial censored data the formulation makes explicit allowance for the corruption of discharges by statistically independent me this formulation can be used to get maximum likelihood estimates or perform a bayesian analysis a limited sampling experiment was conducted to assess the performance of two parameter lognormal maximum likelihood quantile estimators in the presence of simple uncorreiated discontinuous me it was found that me reduces the information content of systematic and binomial censored data with the reduction being greatest for distributions with low coefficients of variation when only using systematic data it was found that the performance of estimators making allowance for me was at best only modestly better than that of estimators ignoring me the principal finding however was that the performance of maximum likelihood estimators using binomial censored data but ignoring me can be substantially worse than that of estimators making explicit allowance for me in fact the loss in performance can be so great that in some cases it may be preferable to ignore the binomial censored data finally it is shown that a bayesian analysis of flood data affected by me provides insights not offered by maximum likelihood estimation when the threshold discharge is in error the posterior distribution of the lognormal parameters can become bimodal as the length of the binomial censored record increases indicating an inconsistency between the information in the systematic and binomial censored data