properly embedded minimal planar domains in 1997 collin proved that any properly embedded minimal surface in r 3 with nite topology and more than one end has

generalized partially linear single index models abstract the typical generalized linear model for a regression of a response y on predictors x z has conditional mean function based on a linear combination of x z we generalize these models to have a nonparametric component replacing the linear combination t 0x t 0z by 0 t 0x t 0z where 0 is an unknown function we call these generalized partially linear single index models gplsim the models include the single index models which have 0 0 using local linear methods we propose estimates of the unknown parameters 0 0 and the unknown function 0 and obtain their asymptotic distributions examples illustrate the models and the proposed estimation methodology

local polynomial variance function estimation the conditional variance function in a heteroscedastic nonparametric regression model is estimated by linear smoothing of squared residuals attention is focused on local polynomial smoothers both the mean and variance functions are assumed to be smooth but neither is assumed to be in a parametric family the biasing effect of preliminary estimation of the mean is studied and a degrees of freedom correction of bias is proposed the corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known a proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions the proposal is illustrated with data from the lidar method of measuring atmospheric pollutants and from turbulence model computations

dressing cmc n noids the purpose of this paper is to construct cmc n noids with bubbletons we recall a specific class of dressing matrices which we will refer to as simple factor dressing matrices which in many known cases add bubbletons to a cmc surface our new surfaces are obtained by dressing known n noids with embedded delaunay ends by well chosen simple factor dressing matrices in such a way that the dressed surface is also a cmc n noid

backlund transformations and loop group actions we construct a local action of the group of rational maps from 2 to gl n on local solutions of flows of the zs akns sl n hierarchy we show that the actions of simple elements linear fractional transformations give local backlund transformations and we derive a permutability formula from different factorizations of a quadratic element we prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities however the action of the subgroup of the rational maps that satisfy the u n reality condition on the space of global rapidly decaying solutions of the flows in the u n hierarchy is global and the action of a simple element gives a global backlund transformation the actions of certain elements in the rational loop group on the vacuum give rise to explicit time periodic multisolitons multibreathers we show that this theory generalizes the classical backlund theory of the sine gordon equation the group structures of backlund transformations for various hierarchies are determined by their reality conditions we identify the reality conditions the group structures for the sl n u k n k kdv kupershmidt wilson and gel fand dikii hierarchies the actions of linear fractional transformations that satisfy a reality condition modulo the center of the group of rational maps give backlund and darboux transformations for the hierarchy defined by the reality condition since the factorization cannot always be carried out under this reality condition the action is again local and backlund transformations only generate local solutions for these hierarchies unless singular solutions are allowed 2000 john wiley sons inc

local polynomial kernel regression for generalized linear models and quasi likelihood functions abstract we investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi likelihood contexts in the ordinary regression case local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity one noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators these properties are shown to carry over to generalized linear model and quasi likelihood settings we also derive the asymptotic distributions of the proposed class of estimators that allow for straightforward interpretation and extensions of state of the art bandwidth selection methods

estimation for a partial linear single index model in this paper we study the estimation for a partial linear single index model a two stage estimation procedure is proposed to estimate the link function for the single index and the parameters in the single index as well as the parameters in the linear component of the model asymptotic normality is established for both parametric components for the index a constrained estimating equation leads to an asymptotically more efficient estimator than existing estimators in the sense that it is of a smaller limiting variance the estimator of the nonparametric link function achieves optimal convergence rates and the structural error variance is obtained in addition the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters a simulation study is performed and an application to a real dataset is illustrated the extension to multiple indices is briefly sketched

on the oscillations appearing in numerical solutions of solvable and nonsolvable integral equations for thin wire antennas differences between certain solvable and nonsolvable ill posed integral equations with the same nonsingular kernel are discussed the main results come from constructing a solvable equation in the context of straight thin wire antennas the kernel of this equation is the usual approximate also called reduced kernel while its exact solution is the familiar sinusoidal current numerical solutions to this solvable equation are compared to corresponding numerical solutions of the usual halle n and pocklington equations with the approximate kernel it is known from previous publications that these last two equations are nonsolvable and that their numerical solutions present severe oscillations when the number of basis functions is sufficiently large it is found that the difficulties encountered in the former solvable equation are much less important compared to those of the nonsolvable ones the same conclusion is brought out from other integral equations arising in different contexts thin wire circular loop antenna method of auxiliary sources and straight wire antenna of infinite length we discuss the consistency of our results with picard s theorem the results in this paper supplement previous publications regarding the difficulties of numerically solving thin wire integral equations with the approximate kernel

flows of constant mean curvature tori in the 3 sphere the equivariant case we present a deformation for constant mean curvature tori in the 3 sphere we show that the moduli space of equivariant constant mean curvature tori in the 3 sphere is connected and we classify the minimal the embedded and the alexandrov embedded tori therein

on mean convex alexandrov embedded surfaces in the 3 sphere we investigate non compact mean convex alexandrov embedded surfaces in the round unit 3 sphere and show under which conditions it is possible to continuously deform these preserving mean convex alexandrov embeddedness

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

on the application of numerical methods to hallen s equation the so called hallen integral equation for the current on a finite linear antenna center driven by a delta function generator takes two forms depending on the choice of kernel the two kernels are usually referred to as the exact and the approximate or reduced kernel with the approximate kernel the integral equation has no solution nevertheless the same numerical method is often applied to both forms of the integral equation in this paper the behavior of the numerical solutions thus obtained is investigated and the similarities and differences between the two numerical solutions are discussed the numerical method is galerkin s method with pulse functions we first apply this method to the two corresponding forms of the integral equation for the current on a linear antenna of infinite length in this case the method yields an infinite toeplitz system of algebraic equations in which the width of the pulse basis functions enters as a parameter the infinite system is solved exactly for nonzero pulse width the exact solution is then developed asymptotically for the case where the pulse width is small when the asymptotic expressions for the case of the infinite antenna are used as a guide for the behavior of the solutions of the finite antenna the latter problem is greatly facilitated for the approximate kernel the main results of this paper carry over to a certain numerical method applied to the corresponding equation of the pocklington type

surface wave and superdirectivity aspects of effective current for linear antennas previous papers have defined the concept of effective current for linear antennas as a method of postprocessing the unphysical solutions obtained by applying discretization galerkin methods to the usual integral equations that employ the approximate or reduced kernel such an unphysical solution can be thought of as an auxiliary rapidly oscillating current located on the axis of the linear antenna while the effective current is the magnetic field due to the oscillating current at a distance equal to the antenna radius multiplied by a factor equal to the antenna circumference in the present paper we focus on the rapid decrease of the aforementioned magnetic field and point out similarities to surface waves to the well known phenomenon of superdirectivity and to the method of auxiliary sources mas our study sheds light on the unusual nature of electromagnetic waves generated by currents with very rapid spatial variations subject to the condition that the discretization is fine we treat th

fundamental properties of surface waves in lossless stratified structures this paper is focused on dispersive properties of lossless planar layered structures with media having positive constitutive parameters permittivity and permeability possibly uniaxially anisotropic some of these properties have been derived in the past with reference to specific simple layered structures and are here established with more general proofs valid for arbitrary layered structures with positive parameters as a first step a simple application of the smith chart to the relevant dispersion equation is used to prove that evanescent or plasmonic type waves cannot be supported by layers with positive parameters the main part of the paper is then focused on a generalization of a common graphical solution of the dispersion equation in order to derive some general properties about the behaviour of the wavenumbers of surface waves as a function of frequency the wavenumbers normalized with respect to frequency are shown to be always increasing with frequency and at high frequency they tend to the highest refractive index in the layers moreover two surface waves with the same polarization cannot have the same wavenumber at a given frequency the low frequency behaviours are also briefly addressed the results are derived by means of a suitable application of foster s theorem

generalized functional linear models we propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function a linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function and the expected value of the response is related to this linear predictor via a link function if in addition a variance function is specified this leads to a functional estimating equation which corresponds to maximizing a functional quasi likelihood this general approach includes the special cases of the functional linear model as well as functional poisson regression and functional binomial regression the latter leads to procedures for classification and discrimination of stochastic processes and functional data we also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data using a semiparametric quasi likelihood procedure an essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated karhunen loeve expansion we develop asymptotic inference for the proposed class of generalized regression models in the proposed asymptotic approach the truncation parameter increases with sample size and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics we establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable l 2 metric and is defined through a generalized covariance operator as a consequence we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model the proposed estimation inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study we find that the practical selection of the number of components works well with the aic criterion and this finding is supported by theoretical considerations we include an application to the classification of medflies regarding their remaining longevity status based on the observed initial egg laying curve for each of 534 female medflies

optimal smoothing in single index models single index models generalize linear regression they have applications to a variety of fields such as discrete choice analysis in econometrics and dose response models in biometrics where high dimensional regression models are often employed single index models are similar to the first step of projection pursuit regression a dimension reduction method in both cases the orientation vector can be estimated root n consistently even if the unknown univariate function or nonparametric link function is assumed to come from a large smoothness class however as we show in the present paper the similarities end there in particular the amount of smoothing necessary for root n consistent orientation estimation is very different in the two cases we suggest a simple empirical rule for selecting the bandwidth appropriate to single index models this rule is studied in a small simulation study and an application in binary response models

embedded minimal tori in s3 and the lawson conjecture we show that any embedded minimal torus in s 3 is congruent to the clifford torus this answers a question posed by h b lawson jr in 1970

estimation of heteroscedasticity in regression analysis consider the regression model y i g t i varepsilon i 1 leq i leq n with nonrandom design variables t i and measurements y i for the unknown regression function g cdot we assume that the data are heteroscedastic i e e varepsilon 2 i sigma 2 i not equiv operatorname const and investigate how to estimate sigma 2 i if sigma 2 i sigma 2 t i with a smooth function sigma 2 cdot initial estimators tilde sigma 2 i can be improved by kernel smoothers and the resulting class of estimators is shown to be uniformly consistent these estimates can be used to improve the estimation of the regression function g itself in parametric and nonparametric models further applications are suggested

quasi likelihood regression with unknown link and variance functions abstract we consider the multiple regression model e y g x t var y sgrave 2 with predictors x link function g and variance function sgrave 2 the aim is to reduce the assumptions in a fully parametric generalized linear model or a quasi likelihood model by allowing the link and the variance functions to be unknown but smooth these functions are then estimated nonparametrically and the estimates are substituted into the quasi likelihood function we propose a three stage approach to identify this semiparametric model by estimating the link function the variance function and the vector of regression coefficients in the linear predictor of the model consistency results for the link and the variance function estimators as well as the asymptotic limiting distribution of the regression coefficients are obtained we show that the resulting parameter estimates are asymptotically efficient as compared to the quasi likelihood parameter estimates obtained for the case where link an

simple factor dressing and the lopez ros deformation of minimal surfaces in euclidean 3 space the aim of this paper is to investigate a new link between integrable systems and minimal surface theory the dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps since a minimal surface in 3 space is a willmore surface its conformal gauss map is harmonic and a dressing on the conformal gauss map can be defined we study the induced transformation on minimal surfaces in the simplest case the simple factor dressing and show that the well known lopez ros deformation of minimal surfaces is a special case of this transformation we express the simple factor dressing and the lopez ros deformation explicitly in terms of the minimal surface and its conjugate surface in particular we can control periods and end behaviour of the simple factor dressing this allows to construct new examples of doubly periodic minimal surfaces arising as simple factor dressings of scherk s first surface

regularity theory and superalgebraic solvers for wire antenna problems we consider the problem of evaluating the current distribution j z that is induced on a straight wire antenna by a time harmonic incident electromagnetic field the scope of this paper is twofold one of its main contributions is a regularity proof for a straight wire occupying the interval 1 1 in particular for a smooth time harmonic incident field this theorem implies that j z i z sqrt 1 z 2 where i z is an infinitely differentiable function the previous state of the art in this regard placed i in the sobolev space w 1 p p 1 the second focus of this work is on numerics we present three superalgebraically convergent algorithms for the solution of wire problems two based on hallen s integral equation and one based on the pocklington integrodifferential equation both our proof and our algorithms are based on two main elements 1 a new decomposition of the kernel of the form g z f 1 z ln z f 2 z where f 1 z and f 2 z are analytic functions on the real line and 2 removal of the end point square root singularities by means of a coordinate transformation the hallen and pocklington based algorithms we propose converge superalgebraically faster than mathcal o n m and mathcal o m m for any positive integer m where n and m are the numbers of unknowns and the number of integration points required for construction of the discretized operator respectively in previous studies at most the leading order contribution to the logarithmic singular term was extracted from the kernel and treated analytically the higher order singular derivatives were left untreated and the resulting integration methods for the kernel exhibit mathcal o m 3 convergence at best a rather comprehensive set of tests we consider shows that in many cases to achieve a given accuracy the numbers n of unknowns required by our codes are up to a factor of five times smaller than those required by the best solvers previously available the required number m of integration points in turn can be several orders of magnitude smaller than those required in previous methods in particular four digit solutions were found in computational times of the order of four seconds and in most cases of the order of a fraction of a second on a contemporary personal computer much higher accuracies result in very small additional computing times

eliminating unphysical oscillations arising in galerkin solutions to classical integral equations of antenna theory an asymptotic study previous works have discussed in detail the difficulties occurring when one applies numerical methods to hallen s and pocklington s integral equations for the current distribution along a linear antenna when the so called approximate kernel is used the main difficulty is the appearance of unphysical oscillations near the driving point and or near the ends of the antenna another work has proposed an easy to apply possible remedy to overcome these unnatural oscillations the basic idea is to define a new current from the near magnetic field produced by the original oscillating current in the present paper for the case of an antenna center driven by a delta function generator we place this remedy on a much firmer basis by means of an asymptotic study for the case of the linear antenna of infinite length we provide specific connections of our study to actual finite length antennas

embedded constant mean curvature tori in the three sphere we prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric and use this to give a complete classification of such surfaces for any given value of the mean curvature

on the use of nonsingular kernels in certain integral equations for thin wire circular loop antennas fundamental properties of certain popular integral equations for thin wire circular loop antennas with nonsingular kernels are studied the cornerstone of our study is a large asymptotic formula for the n th fourier coefficient of the nonsingular kernel four different methods of driving the circular loop are considered namely the delta function generator the finite gap generator a certain type of frill generator and the case of plane wave incidence the excitation model is crucial to our discussions since it determines the behavior of the solution convergence also discussed are associated difficulties that may arise when moment methods are applied to the aforementioned equations as well as a simple improvement to the fourier series method for determining the current in the case of the frill generator the main results herein closely parallel recent results for the case of the straight antenna and similarities and differences between the straight and circular cases are discussed

semiparametric analysis of heterogeneous data using varying scale generalized linear models this article describes a class of heteroscedastic generalized linear regression models in which a subset of the regression parameters are rescaled nonparametrically and develops efficient semiparametric inferences for the parametric components of the models such models provide a means to adapt for heterogeneity in the data due to varying exposures varying levels of aggregation and so on the class of models considered includes generalized partially linear models and nonparametrically scaled link function models as special cases we present an algorithm to estimate the scale function nonparametrically and obtain asymptotic distribution theory for regression parameter estimates in particular we establish that the asymptotic covariance of the semiparametric estimator for the parametric part of the model achieves the semiparametric lower bound we also describe bootstrap based goodness of scale test we illustrate the methodology with simulations published data and data from collaborative research on ultrasound safety

on two types of convergence in the method of auxiliary sources the method of auxiliary sources mas is often applied to problems involving an externally illuminated smooth perfect electric conductor pec one seeks to approximately satisfy the boundary condition on the pec surface using auxiliary sources located inside the pec surface usually the underlying auxiliary surface is also smooth and closed the currents on the auxiliary sources mas currents are the initial unknowns once they have been found one can easily determine the field due to them mas field at all points external to the auxiliary surface and in particular at all points external to the pec scatterer we show that in the limit n it is possible to have a convergent mas field together with divergent mas currents and that this phenomenon is accompanied by an abrupt behavior of the limiting value of the mas field we show this possibility through an analytical study of a two dimensional scattering problem involving a circular cylinder in which mas fields and currents can be determined explicitly the analytical study proceeds from first principles it involves fundamental electromagnetics and relatively simple mathematical manipulations numerical results supplement the analytical study and demonstrate the nature of the aforementioned divergence our study sheds light on other aspects of mas in particular it establishes interesting similarities and differences between mas and its continuous version and reveals many similarities between mas currents and numerical solutions of hallen s integral equation with the approximate kernel

nonparametric quasi likelihood the quasi likelihood function proposed by wedderburn broadened the scope of generalized linear models by specifying the variance function instead of the entire distribution however complete specification of variance functions in the quasi likelihood approach may not be realistic we define a nonparametric quasi likelihood by replacing the specified variance function in the conventional quasi likelihood with a nonparametric variance function estimate this nonparametric variance function estimate is based on squared residuals from an initial model fit the rate of convergence of the nonparametric variance function estimator is derived it is shown that the asymptotic limiting distribution of the vector of regression parameter estimates is the same as for the quasi likelihood estimates obtained under correct specification of the variance function thus establishing the asymptotic efficiency of the nonparametric quasi likelihood estimates we propose bandwidth selection strategies based on deviance and pearson s chi square statistic it is demonstrated in simulations that for finite samples the proposed nonparametric quasi likelihood method can improve upon extended quasi likelihood or pseudo likelihood methods where the variance function is assumed to fall into a parametric class with unknown parameters we illustrate the proposed methods with applications to dental data and cherry tree data

reply to comments on fundamentals of thin wire integral equations with the finite gap generator in papers 1 2 the authors extend the so called effective current method to the hallen s and pocklington s equations with the finite gap generator fgg the indicated method considered earlier in 3 with a reference to 4 is associated with application of the method of auxiliary sources to the analysis of cylindrical dipole antennas 5 7 the purpose of these comments is to draw the authors and readers attention to the existence ofpaper 8 where the pocklington s equation with the fgg is solved for the axial current considered there to be auxiliary the equation is solved by the galerkin method using the piecewise sinusoidal basis and testing functions the auxiliary axial current is used further for calculation of the surface current which is referred to as the effective current in 1 4 and the surface current itself is used for calculation of the input impedance so in some respects papers 1 7 just repeat the main idea and technique of paper 8 although of course papers 1 7 also contain a lot of their own new interesting and useful analytical and numerical results

on two families of transformations to additivity for binary response data summary two families of power transformations for probabilities are introduced to model symmetric and asymmetric departures from the logistic model the aim is to provide in a comprehensive way a representation of alternative scales in terms of which to carry out the analysis of binary response data maximum likelihood methods are employed to estimate an appropriate transformation using glim tests for detection of symmetric and asymmetric departures are proposed

two dimensional highly directive currents on large circular loops properties of idealized two dimensional current distributions on circular loops are investigated analytically via the solution of a constrained optimization problem the directivity in the far field is maximized under a fixed c n t where n is the integral of the square of the current magnitude and t is the total radiated power c enters the ensuing fourier series for the current implicitly through a lagrange multiplier for non negative and large electrical radius ka the directivity and the current are evaluated approximately via combined use of the poisson summation formula and the mellin transform technique as a result a geometrical ray representation for the current is derived for the case of directivities that are slightly larger than that of the uniform distribution the analysis indicates certain advantages of large radiating structures for moderate values of the constraint c in the limit c of oseen s einstein needle radiation an asymptotic formula for the directivity is obtained pos

adapting for the missing link we consider the fitting of generalized linear models in wich the link function is assumed to be unknown and propose the following computational method first estimate regression coefficients using the canonical link then estimate the link via a kernel smoother treating the direction in the predictor space determined by the regression coefficients as known then reestimate the direction using the estimated link and alternate between these two steps we show that under fairly general conditions n 1 2 consistent estimates of the direction are obtained a small monte carlo study is presented

highly directive current distributions general theory a theoretical scheme for studying the properties of localized monochromatic and highly directive classical current distributions in two and three dimensions is formulated and analyzed for continuous current distributions it is shown that maximizing the directivity d in the far field while constraining c5n t where n is the integral of the square of the magnitude of the current density and t is proportional to the total radiated power leads to a fredholm integral equation of the second kind for the optimum current this equation is a useful analytical tool for studying currents that produce optimum directivities above the directivity of a uniform distribution various consequences of the present formulation are examined analytically for essentially arbitrary geometries of the current carrying region in particular certain properties of the optimum directivity are derived and differences between the continuous and discrete cases are pointed out when c the directivity tends to infinity monotonically in accord with oseen s einstein needle radiation s1063 651x 98 02108 4