properties of monotonic effects on directed acyclic graphs various relationships are shown hold between monotonic effects and weak monotonic effects and the monotonicity of certain conditional expectations counterexamples are provided to show that the results do not hold under less restrictive conditions monotonic effects are furthermore used to relate signed edges on a causal directed acyclic graph to qualitative effect modification the theory is applied to an example concerning the direct effect of smoking on cardiovascular disease controlling for hypercholesterolemia monotonicity assumptions are used to construct a test for whether there is a variable that confounds the relationship between the mediator hypercholesterolemia and the outcome cardiovascular disease

product rules on the unit circle with uniformly distributed nodes error bounds for analytic functions this paper deals with the numerical calculation of integrals over the unit circle in the complex plane by means of quadrature formulas with uniformly distributed nodes on such region several properties of the coefficients as well as error bounds for analytic functions are established

fundamental concepts of qualitative probabilistic networks abstract graphical representations for probabilistic relationships have recently received considerable attention in ai qualitative probabilistic networks abstract from the usual numeric representations by encoding only qualitative relationships which are inequality constraints on the joint probability distribution over the variables although these constraints are insufficient to determine probabilities uniquely they are designed to justify the deduction of a class of relative likelihood conclusions that imply useful decision making properties two types of qualitative relationship are defined each a probabilistic form of monotonicity constraint over a group of variables qualitative influences describe the direction of the relationship between two variables qualitative synergies describe interactions among influences the probabilistic definitions chosen justify sound and efficient inference procedures based on graphical manipulations of the network these procedures answer queries about qualitative relationships among variables separated in the network and determine structural properties of optimal assignments to decision variables

on the convergence of quadrature formulas for complex weight functions abstract we consider quadrature formulas for complex weights and prove their convergence for a large class of functions

assessing the sensitivity of regression results to unmeasured confounders in observational studies this paper presents a general approach for assessing the sensitivity of the point and interval estimates of the primary exposure effect in an observational study to the residual confounding effects of unmeasured variable after adjusting for measured covariates the proposed method assumes that the true exposure effect can be represented in a regression model that includes the exposure indicator as well as the measured and unmeasured confounders one can use the corresponding reduced model that omits the unmeasured confounder to make statistical inferences about the true exposure effect by specifying the distributions of the unmeasured confounder in the exposed and unexposed groups along with the effects of the unmeasured confounder on the outcome variable under certain conditions there exists a simple algebraic relationship between the true exposure effect in the full model and the apparent exposure effect in the reduced model one can then estimate the true exposure effect by making a simple adjustment to the point and interval estimates of the apparent exposure effect obtained from standard software or published reports the proposed method handles both binary response and censored survival time data accommodates any study design and allows the unmeasured confounder to be discrete or normally distributed we describe applications on two major medical studies

causal diagrams for empirical research summary the primary aim of this paper is to show how graphical models can be used as a mathematical language for integrating statistical and subject matter information in particular the paper develops a principled nonparametric framework for causal inference in which diagrams are queried to determine if the assumptions available are sufficient for identifying causal effects from nonexperimental data if so the diagrams can be queried to produce mathematical expressions for causal effects in terms of observed distributions otherwise the diagrams can be queried to suggest additional observations or auxiliary experiments from which the desired inferences can be obtained

directed acyclic graphs with edge specific bounds we give a definition of a bounded edge within the causal directed acyclic graph framework a bounded edge generalizes the notion of a signed edge and is defined in terms of bounds on a ratio of survivor probabilities we derive rules concerning the propagation of bounds bounds on causal effects in the presence of unmeasured confounding are also derived using bounds related to specific edges on a graph we illustrate the theory developed by an example concerning estimating the effect of antihistamine treatment on asthma in the presence of unmeasured confounding

rational interpolation i least square convergence abstract given a positive bounded borel measure on the interval 1 1 we provide convergence results in l 2 norm to a function f of its sequence of interpolating rational functions at the nodes of rational gauss type quadrature formulas associated with the measure for this we use the connection between rational gauss type quadrature formulas on 1 1 and rational szeg quadrature formulas associated with a positive symmetric borel measure on the complex unit circle

properties of interpolatory product integration rules this paper is concerned with the approximate evaluation of int 1 1 k x f x d x where k is lebesgue integrable and f is at least riemann integrable and preferably smooth the integral is approximated by a rule of the form sum nolimits i 1 n w ni k f x ni where the points x ni are chosen in some prescribed way and the weights w ni k are such that the rule is exact if f is any polynomial of degree n for suitable choices of the points and suitable functions k the rule is shown to converge to the exact integral and the companion rule sum w ni k f x ni to int k x f x dx for all riemann integrable functions f the companion property ensures for example that the weights have the asymptotic positivity property if k is positive error bounds are given that guarantee rapid convergence of the rule if f is smooth finally by establishing a connection with mean convergence of lagrangian interpolation some negative theorems are proved in pa

signed directed acyclic graphs for causal inference formal rules governing signed edges on causal directed acyclic graphs are described in this paper and it is shown how these rules can be useful in reasoning about causality specifically the notions of a monotonic effect a weak monotonic effect and a signed edge are introduced results are developed relating these monotonic effects and signed edges to the sign of the causal effect of an intervention in the presence of intermediate variables the incorporation of signed edges into the directed acyclic graph causal framework furthermore allows for the development of rules governing the relationship between monotonic effects and the sign of the covariance between two variables it is shown that when certain assumptions about monotonic effects can be made then these results can be used to draw conclusions about the presence of causal effects even when data is missing on confounding variables

foreword to the proceedings of the orthoquad 2014 conference orthoquad 2014 was an international symposium on orthogonality quadrature and related topics held on january 20 24 2014 in puerto de la cruz tenerife spain it was held in memory of professor pablo gonzalez vera 1955 2012

sequences of orthogonal laurent polynomials bi orthogonality and quadrature formulas on the unit circle in this paper the construction of orthogonal bases in the space of laurent polynomials on the unit circle is considered as an application a connection with the so called bi orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on laurent polynomials are studied

rational quadrature formulae on the unit circle with arbitrary poles interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered the poles are prescribed under the only restriction of not lying on the unit circle a computable upper bound of the error is obtained which is valid for any choice of poles arbitrary weight functions and any degree of exactness provided that the integrand is analytic on a neighborhood of the unit circle a number of numerical examples are given which show the advantages of using such rules as well as the sharpness of the error bound also a comparison is made with other error bounds appearing in the literature

on a completely integrable nonlinear hyperbolic variational equation abstract we show that the nonlinear partial differential equation u t uu x xx 1 2 u x 2 x is a completely integrable bi variational bi hamiltonian system the corresponding equation for w u xx belongs to the harry dym hierarchy this equation arises in two different physical contexts in two nonequivalent variational forms it describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field and it is the high frequency limit of the camassa holm equation which is an integrable model equation for shallow water waves using the bi hamiltonian structure we derive a recursion operator a lax pair and an infinite family commuting hamiltonian flows together with the associated conservation laws we also give the transformation to action angle coordinates smooth solutions of the partial differential equation break down because their derivative blows up in finite time nevertheless the hamiltonian structure and complete integrability appear to remain valid globally in time even after smooth solutions break down we show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions we compute the bi hamiltonian structure on this invariant manifold which is obtained by restricting the bi hamiltonian structure of the partial differential equation

periodic solitons of an equation for short capillary gravity waves we consider the periodic solitons of an equation for short capillary gravity waves although their profiles cannot be given a description in terms of a mathematical expression in closed form we show that they are transformed by a natural change of variables to the well known cnoidal periodic solitons of the kdv equation

rational interpolation ii quadrature and convergence abstract consider an n th rational interpolatory quadrature rule j n f j 1 n j f x j to approximate integrals of the form j f 1 1 f x d x where is a possibly complex bounded measure with infinite support on the interval 1 1 first we discuss the connection of j n f with certain rational interpolatory quadratures on the complex unit circle to approximate integrals of the form f e i d c where c is a possibly complex bounded measure with infinite support on next we provide conditions to ensure the convergence of j n f to j f for n tending to infinity finally an upper bound for the error on the n th approximation and an estimate for the rate of convergence is provided

a connection between quadrature formulas on the unit circle and the interval 1 1 abstract we establish a relation between gauss quadrature formulas on the interval 1 1 that approximate integrals of the form i f 1 1 f x x d x and szeg quadrature formulas on the unit circle of the complex plane that approximate integrals of the form i f f e i d the weight x is positive on 1 1 while the weight is positive on it is shown that if cos sin then there is an intimate relation between the gauss and szeg quadrature formulas moreover as a side result we also obtain an easy derivation for relations between orthogonal polynomials with respect to x and orthogonal szeg polynomials with respect to inclusion of gauss lobatto and gauss radau formulas is natural

positive interpolatory quadrature formulas and para orthogonal polynomials we establish a relation between quadrature formulas on the interval 1 1 that approximate integrals of the form j f 11 f x x dx and szego quadrature formulas on the unit circle that approximate integrals of the form i f f ei d the functions x and are assumed to be weight functions on 1 1 and respectively and are related by cos sin it is well known that the nodes of szego formulas are the zeros of the so called para orthogonal polynomials bn z n z n z 1 n z and n z being the orthogonal and reciprocal polynomials respectively with respect to the weight function furthermore for 1 we have recently obtained gauss type quadrature formulas on 1 1 see bultheel et al j comput appl math 132 1 2000 1 in this paper making use of the para orthogonal polynomials with 1 a one parameter family of interpolatory quadrature formulas with positive coefficients for j f is obtained along with error expressions for analytic integrands finally some illustrative numerical examples are also included

szegő type quadrature formulas abstract we consider the approximation of integrals with respect to measures supported on the unit circle by means of positive quadrature formulas with maximal domain of exactness and up to three preassigned nodes the quadrature formulas are exact in a new nested sequence of subspaces of laurent polynomials that allows us to extend analogous properties to the usual quadrature formulas on the real line estimations for the error of these quadrature formulas depending on the regularity of the integrand are obtained

error bounds for interpolatory quadrature rules on the unit circle for the construction of an interpolatory integration rule on the unit circle t with n nodes by means of the laurent polynomials as basis functions for the approximation we have at our disposal two nonnegative integers p n and q n p n q n n 1 which determine the subspace of basis functions the quadrature rule will integrate correctly any function from this subspace in this paper upper bounds for the remainder term of interpolatory integration rules on t are obtained these bounds apply to analytic functions up to a finite number of isolated poles outside t in addition if the integrand function has no poles in the closed unit disc or is a rational function with poles outside t we propose a simple rule to determine the value of p n and hence q n in order to minimize the quadrature error term several numerical examples are given to illustrate the theoretical results

convergent difference schemes for the hunter saxton equation we propose and analyze several finite difference schemes for the hunter saxton equation hs u t uu x 1 2 x 0 u x 2 d x x 0 t 0 this equation has been suggested as a simple model for nematic liquid crystals we prove that the numerical approximations converge to the unique dissipative solution of hs as identified by zhang and zheng a main aspect of the analysis in addition to the derivation of several a priori estimates that yield some basic convergence results is to prove strong convergence of the discrete spatial derivative of the numerical approximations of u which is achieved by analyzing various renormalizations in the sense of diperna and lions of the numerical schemes finally we demonstrate through several numerical examples the proposed schemes as well as some other schemes for which we have no rigorous convergence results

rational approximants associated with measures supported on the unit circle and the real line abstract as a continuation of the well known connection between the theory of orthogonal polynomials on the unit circle and the interval 1 1 in this paper properties concerning error and convergence of certain rational approximants associated with the measures d t and d d cos supported on 1 1 and the unit circle respectively are deduced numerical illustrations are also given

rational szego quadratures associated with chebyshev weight functions in this paper we characterize rational szeg o quadrature formulas associated with chebyshev weight functions by giving explicit expressions for the corresponding para orthogonal rational functions and weights in the quadratures as an application we give characterizations for szeg o quadrature formulas associated with rational modifications of chebyshev weight functions some numerical experiments are finally presented

orthogonality interpolation and quadratures on the unit circle and the interval 1 1 given a weight function s x on 1 1 or more generally a positive borel measure the erdos turan theorem assures the convergence in l 2 s norm to a function f of its sequence of interpolating polynomials at the zeros of the orthogonal polynomials or equivalently at the nodes of the gauss christoffel quadrature formulas associated with s in this paper we will extend this result to the nodes of the gauss radau and gauss lobatto quadrature formulas by passing to the unit circle and taking advantage of the results on interpolation by means of laurent polynomials at the zeros of certain para orthogonal polynomials with respect to the weight function w q s cos q sin q on p p as a consequence an application to the construction of certain product integration rules on finite intervals of the real line will be given several numerical experiments are finally carried out

error bounds for rational quadrature formulae of analytic functions we study the error of rational quadrature rules when functions which are analytic on a neighborhood of the set of integration are considered a computable upper bound of the error is presented which is valid for a broad range of rational quadrature formulae and a comparison is made with the exact error for a number of numerical examples

quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity in this paper we investigate the szego radau and szego lobatto quadrature formulas on the unit circle these are n m point formulas for which m nodes are fixed in advance with m 1 and m 2 respectively and which have a maximal domain of validity in the space of laurent polynomials this means that the free parameters free nodes and positive weights are chosen such that the quadrature formula is exact for all powers z j email protected email protected p with p p n m as large as possible