generalization and pac learning some new results for the class of generalized single layer networks the ability of connectionist networks to generalize is often cited as one of their most important properties we analyze the generalization ability of the class of generalized single layer networks gslns which includes volterra networks radial basis function networks regularization networks and the modified kanerva model using techniques based on the theory of probably approximately correct pac learning which have previously been used to analyze the generalization ability of feedforward networks of linear threshold elements ltes an introduction to the relevant computational learning theory is included we derive necessary and sufficient conditions on the number of training examples required by a gsln to guarantee a particular generalization performance we compare our results to those given previously for feedforward networks of ltes and show that on the basis of the currently available bounds the sufficient number of training examples for gslns is typically considerably less than for feedforward networks of ltes with the same number of weights we show that the use of self structuring techniques for gslns may reduce the number of training examples sufficient to guarantee good generalization performance and we provide an explanation for the fact that gslns can require a relatively large number of weights

truncation diffraction phenomena of floquet waves radiated from semi infinite phased array antenna in a general focus problem this paper presents uniform formulations of radiation mechanisms analogous to the ray decompositions of diffraction theory from a semi infinite and periodic array of antennas the antenna array is excited to radiate the electromagnetic em fields focused at a relatively arbitrarily selected location the ray decomposition results from a closed form formulation by asymptotically evaluating the radiation integral of floquet modes that are obtained by applying the poisson sum formula to the summation of elemental radiations the analysis is relatively general and will reduce to the case of a conventional phased array antenna that is excited to radiate directive beams focused in the far zone and exhibits consistent phenomena theoretical investigations as well as numerical examinations are presented to demonstrate the radiation mechanisms

refined error bounds for several learning algorithms this article studies the achievable guarantees on the error rates of certain learning algorithms with particular focus on refining logarithmic factors many of the results are based on a general technique for obtaining bounds on the error rates of sample consistent classifiers with monotonic error regions in the realizable case we prove bounds of this type expressed in terms of either the vc dimension or the sample compression size this general technique also enables us to derive several new bounds on the error rates of general sample consistent learning algorithms as well as refined bounds on the label complexity of the cal active learning algorithm additionally we establish a simple necessary and sufficient condition for the existence of a distribution free bound on the error rates of all sample consistent learning rules converging at a rate inversely proportional to the sample size we also study learning in the presence of classification noise deriving a new excess error rate guarantee for general vc classes under tsybakov s noise condition and establishing a simple and general necessary and sufficient condition for the minimax excess risk under bounded noise to converge at a rate inversely proportional to the sample size

computing eigenfunctions on the koch snowflake a new grid and symmetry in this paper we numerically solve the eigenvalue problem au u u 0 on the fractal region defined by the koch snowflake with zero dirichlet or zero neumann boundary conditions the laplacian with boundary conditions is approximated by a large symmetric matrix the eigenvalues and eigenvectors of this matrix are computed by arpack we impose the boundary conditions in a way that gives improved accuracy over the previous computations of lapidus neuberger renka and griffith we extrapolate the results for grid spacing h to the limit h 0 in order to estimate eigenvalues of the laplacian and compare our results to those of lapidus et al we analyze the symmetry of the region to explain the multiplicity two eigenvalues and present a canonical choice of the two eigenfunctions that span each two dimensional eigenspace

efficient calculation of large finite periodic structures based on surface wave analysis the electromagnetic modeling of practical finite periodic structures is a topic of growing interest due to the truncation of infinite periodic structures surface waves will be excited and localized near discontinuous interfaces leading to the edge effect in this work surface waves are numerically extracted and their magnitudes and decay rates are analyzed for different materials and geometries based on the exponential decay of the surface wave a novel method is developed by connecting the solution to the large finite array problem with that to a relatively small one to achieve low complexity and memory consumption the method numerically reconstructs propagating bloch waves and surface waves according to the bloch floquet theorem of periodic structures and translation invariant properties of semi infinite periodic structures respectively numerical examples are shown for near field distributions and far field radiation patterns the results obtained from small finite periodic structures capture the edge effect and agree well with the results by the rigorous element by element approach

generalization error bounds for threshold decision lists in this paper we consider the generalization accuracy of classification methods based on the iterative use of linear classifiers the resulting classifiers which we call threshold decision lists act as follows some points of the data set to be classified are given a particular classification according to a linear threshold function or hyperplane these are then removed from consideration and the procedure is iterated until all points are classified geometrically we can imagine that at each stage points of the same classification are successively chopped off from the data set by a hyperplane we analyse theoretically the generalization properties of data classification techniques that are based on the use of threshold decision lists and on the special subclass of multilevel threshold functions we present bounds on the generalization error in a standard probabilistic learning framework the primary focus in this paper is on obtaining generalization error bounds that depend on the levels of separation or margins achieved by the successive linear classifiers we also improve and extend previously published theoretical bounds on the generalization ability of perceptron decision trees

scale sensitive dimensions uniform convergence and learnability learnability in valiant s pac learning model has been shown to be strongly related to the existence of uniform laws of large numbers these laws define a distribution free convergence property of means to expectations uniformly over classes of random variables classes of real valued functions enjoying such a property are also known as uniform glivenko cantelli classes in this paper we prove through a generalization of sauer s lemma that may be interesting in its own right a new characterization of uniform glivenko cantelli classes our characterization yields dudley gine and zinn s previous characterization as a corollary furthermore it is the first based on a gine and zinn s previous characterization as a corollary furthermore it is the first based on a simple combinatorial quantity generalizing the vapnik chervonenkis dimension we apply this result to obtain the weakest combinatorial condition known to imply pac learnability in the statistical regression or agnostic framework furthermore we find a characterization of learnability in the probabilistic concept model solving an open problem posed by kearns and schapire these results show that the accuracy parameter plays a crucial role in determining the effective complexity of the learner s hypothesis class

finite element eigenvalues for the laplacian over an l shaped domain abstract the standard finite element schemes for computing eigenvalues of the laplacian converge slowly with decreasing mesh size if the domain boundary contains a re entrant corner a superelement is constructed to cover the region surrounding a corner of angle 3 2 inside the superelement the mesh is refined and the trial function constrained to fit the known analytic form of the solution in the neighborhood of the corner singularity being compatible with linear and bilinear elements the superelement is easily embodied into standard finite element programs calculations are made for an l shaped domain with various boundary conditions comparison with other calculations show that the incorporation of the superelement has neutralised the deterioration in convergence which would otherwise have taken place due to the presence of the re entrant corner

information theoretic lower bounds on the oracle complexity of stochastic convex optimization relative to the large literature on upper bounds on complexity of convex optimization lesser attention has been paid to the fundamental hardn4516420ess of these problems given the extensive use of convex optimization in machine learning and statistics gaining an understanding of these complexity theoretic issues is important in this paper we study the complexity of stochastic convex optimization in an oracle model of computation we introduce a new notion of discrepancy between functions and use it to reduce problems of stochastic convex optimization to statistical parameter estimation which can be lower bounded using information theoretic methods using this approach we improve upon known results and obtain tight minimax complexity estimates for various function classes

learning by distances a model of learning by distances is presented in this model a concept is a point in a metric space at each step of the learning process the student guesses a hypothesis and receives from the teacher an approximation of its distance to the target a notion of a distance measuring the proximity of a hypothesis to the correct answer is common to many models of learnability by focusing on this fundamental aspect we discover some general and simple tools for the analysis of learnability tasks as a corollary we present new learning algorithms for valiant s pac scenario with any given distribution these algorithms can learn any pac learnable class and in some cases settle for significantly less information than the usual labeled examples insight gained by the new model is applied to show that every class of subsets c that has a finite vc dimension is pac learnable with respect to any fixed distribution previously known results of this nature were subject to complicated measurability constraints

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

machine learning parameter regions abstract how many random points from an identified set a confidence set or a highest posterior density set suffice to describe them this paper argues that taking random draws from a parameter region in order to approximate its shape is a supervised learning problem analogous to sampling pixels of an image to recognize it misclassification error a common criterion in machine learning provides an off the shelf tool to assess the quality of a given approximation we say a parameter region can be learned if there is an algorithm that yields a misclassification error of at most with probability at least 1 regardless of the sampling distribution we show that learning a parameter region is possible if and only if its potential shapes are not too complex moreover the tightest band that contains a d dimensional parameter region is always learnable from the inside in a sense we make precise with at least max 1 ln 1 3 16 d draws but at most min 2 d ln 2 d exp 1 2 d ln 1 these bounds grow linearly in the dimension of the parameter region and are uniform with respect to its true shape we illustrate the usefulness of our results using structural vector autoregressions we show how many orthogonal matrices are necessary sufficient to evaluate the impulse responses identified set and how many shotgun plots to report when conducting joint inference on impulse responses

noise tolerant parallel learning of geometric concepts we present several efficient parallel algorithms for pac learning geometric concepts in a constant dimensional space the algorithms are robust even against malicious classification noise of any rate less than 1 2 we first give an efficient noise tolerant parallel algorithm to pac learn the class of geometric concepts defined by a polynomial number of d 1 dimensional hyperplanes against an arbitrary distribution where each hyperplane has a slope from a set of known slopes we then describe how boosting techniques can be used so that our algorithms dependence on and does not depend ond next we give an efficient noise tolerant parallel algorithm to pac learn the class of geometric concepts defined by a polynomial number of d 1 dimensional hyperplanes of unrestricted slopes against a uniform distribution we then show how to extend our algorithm to learn this class against any unknown product distribution next we define a complexity measure of any setsof d 1 dimensional surfaces that we call thevariantofsand prove that the class of geometric concepts defined by surfaces of polynomial variant can be efficiently learned in parallel under a product distribution even under malicious classification noise furthermore we show that the vc dimension of the class of geometric concepts defined by a single surface of variant one is finally we give an efficient parallel noise tolerant algorithm to pac learn any 2 dimensional geometric concept defined by a setsof 1 dimensional surfaces of polynomial length under a uniform distribution

results on learnability and the vapnik chervonenkis dimension abstract we consider the problem of learning a concept from examples in the distribution free model by valiant an essentially equivalent model if one ignores issues of computational difficulty was studied by vapnik and chervonenkis we introduce the notion of dynamic sampling wherein the number of examples examined may increase with the complexity of the target concept this method is used to establish the learnability of various concept classes with an infinite vapnik chervonenkis dimension we also discuss an important variation on the problem of learning from examples called approximating from examples here we do not assume that the target concept t is a member of the concept class c from which approximations are chosen this problem takes on particular interest when the vc dimension of c is infinite finally we discuss the problem of computing the vc dimension of a finite concept set defined on a finite domain and consider the structure of classes of a fixed small dimension

convexity and logical analysis of data a boolean function is called k convex if for any pair x y of its true points at hamming distance at most k every point between x and y is also true given a set of true points and a set of false points the central question of logical analysis of data is the study of those boolean functions whose values agree with those of the given points in this paper we examine data sets which admit k convex boolean extensions we provide polynomial algorithms for finding a k convex extension if any and for finding the maximum k for which a k convex extension exists we study the problem of uniqueness and provide a polynomial algorithm for checking whether all k convex extensions agree in a point outside the given data set we estimate the number of k convex boolean functions and show that for small k this number is doubly exponential on the other hand we also show that for large k the class of k convex boolean functions is pac learnable

collocation method for fractional quantum mechanics we show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional laplacian using a collocation approach based on little sinc functions which discretizes the schrodinger equation on a uniform grid the different boundary conditions are naturally implemented using sets of functions with the appropriate behavior good convergence properties are observed a comparison with results based on a wentzel kramers brillouin analysis is performed

minimax analysis of active learning this work establishes distribution free upper and lower bounds on the minimax label complexity of active learning with general hypothesis classes under various noise models the results reveal a number of surprising facts in particular under the noise model of tsybakov 2004 the minimax label complexity of active learning with a vc class is always asymptotically smaller than that of passive learning and is typically signi cantly smaller than the best previously published upper bounds in the active learning literature in high noise regimes it turns out that all active learning problems of a given vc dimension have roughly the same minimax label complexity which contrasts with well known results for bounded noise in low noise regimes we find that the label complexity is well characterized by a simple combinatorial complexity measure we call the star number interestingly we find that almost all of the complexity measures previously explored in the active learning literature have worst case values exactly equal to the star number we also propose new active learning strategies that nearly achieve these minimax label complexities

an efficient computation scheme for the free space green s function of a two dimensional semiinfinite phased array a simple scheme is developed to compute the green s function of a periodic semiinfinite array in free space it is based on the spectral representation of the fields radiated by an infinite linear array of dipoles results related to successive linear arrays are added in the space domain this summation can be accelerated tremendously by using an elementary extrapolation technique the resulting formulation converges everywhere in the plane containing the array and the number of terms required to achieve a given precision increases slowly away from this plane

acceleration of fast multipole method for large scale periodic structures with finite sizes using sub entire domain basis functions an acceleration technique to the fast multipole method fmm has been proposed to handle large scale problems of periodic structures in free space with finite sizes based on the accurate sub entire domain basis functions in the proposed algorithm only nine or 27 elements in the whole impedance matrix are required to be computed and stored for a two dimensional or three dimensional periodic structure and the matrix vector multiply can be performed efficiently using the combination of fast fourier transform and fmm the theoretical analysis and numerical results show that both the memory requirement and computational complexity are only of the order of o n with small constants where n is the total number of unknowns

eigenmodes of isospectral drums recently it was proved that there exist nonisometric planar regions that have identical laplace spectra that is one cannot hear the shape of a drum the simplest isospectral regions known are bounded by polygons with reentrant corners while the isospectrality can be proven mathematically analytical techniques are unable to produce the eigenvalues themselves furthermore standard numerical methods for computing the eigenvalues such as adaptive finite elements are highly inefficient physical experiments have been performed to measure the spectra but the accuracy and flexibility of this method are limited we describe an algorithm due to descloux and tolley comput methods appl mech engrg 39 1983 pp 37 53 that blends singular finite elements with domain decomposition and show that with a modification that doubles its accuracy this algorithm can be used to compute efficiently the eigenvalues for polygonal regions we present results accurate to 12 digits for the most famous pair of isospectral drums as well as results for another pair

bound states in open coupled asymmetrical waveguides and quantum wires the behaviour of bound states in asymmetric cross t and l shaped configurations is considered because of the symmetries of the wavefunctions the analysis can be reduced to the case of an electron localized at the intersection of two orthogonal crossed wires of different width for different values of the ratio of the widths we prove the existence and non existence of bound states in each symmetry class our arguments yield that for the even even case the bound state of the cross configuration persists as one of the arms becomes infinitesimally narrow in the case of odd odd states we find that the lowest mode is bounded when the width of the two arms is the same and stays bound up to a critical value of the ratio between the widths in the case of the even odd states we find that the lowest mode is unbound up to a critical value of the ratio of the widths numerical calculations are used to support those results and to determine precisely the critical values of the ratio of the widths

supervised learning and co training co training under the conditional independence assumption is among the models which demonstrate how radically the need for labeled data can be reduced if a huge amount of unlabeled data is available in this paper we explore how much credit for this saving must be assigned solely to the extra assumptions underlying the co training model to this end we compute general almost tight upper and lower bounds on the sample size needed to achieve the success criterion of pac learning in the realizable case within the model of co training under the conditional independence assumption in a purely supervised setting the upper bounds lie significantly below the lower bounds for pac learning without co training thus co training saves labeled data even when not combined with unlabeled data on the other hand the saving is much less radical than the known savings in the semi supervised setting

learning with prior information a notion of learnability is introduced referred to as learnability with prior information w p i this notion is weaker than the standard notion of probably approximately correct pac learnability a property called dispersability is introduced and it is shown that dispersability plays a key role in the study of learnability w p i specifically dispersability of a function class is always a sufficient condition for the function class to be learnable moreover in the case of concept classes dispersability is also a necessary condition for learnability w p i thus in the case of learnability w p i the dispersability property plays a role similar to the finite metric entropy condition in the case of pac learnability with a fixed distribution next the notion of learnability w p i is extended to the distribution free d f situation and it is shown that a property called d f dispersability is always a sufficient condition for d f learnability w p i and is also a necessary condition for d f learnability in the case of concept classes the approach to learning introduced in the paper is believed to be significant in all problems where a nonlinear system has to be designed based on data this includes direct inverse control and system identification

optimality of svm novel proofs and tighter bounds abstract we provide a new proof that the expected error rate of consistent support vector machines matches the minimax rate up to a constant factor in its dependence on the sample size and margin the upper bound was originally established by 1 while the lower bound follows from an argument of 2 together with reasoning about the vc dimension of large margin classifiers our proof differs from the original in that many of our steps concern reasoning about the primal space while the original carried out these steps by reasoning about the dual space our approach provides a unified framework for analyzing both the homogeneous and non homogeneous cases with slightly better results for the former the fact that our analysis explicitly handles the non homogeneous case offers significant improvements in the bounds compared to the usual textbook approach of reducing to the homogeneous case we also extend our proof to provide a new upper bound on the error rate of transductive svm which yields an improved constant factor compared to inductive svm in addition to these bounds on the expected error rate we also provide a simple proof of a margin based pac style bound for support vector machines and an extension of the agnostic pac analysis that explicitly handles the non homogeneous case

the learnability of quantum states traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n but using ideas from computational learning theory we show that one can do exponentially better in a statistical setting in particular to predict the outcomes of most measurements drawn from an arbitrary probability distribution one needs only a number of sample measurements that grows linearly with n this theorem has the conceptual implication that quantum states despite being exponentially long vectors are nevertheless reasonable in a learning theory sense the theorem also has two applications to quantum computing first a new simulation of quantum one way communication protocols and second the use of trusted classical advice to verify untrusted quantum advice

eigenfrequencies of fractal drums a method for the computation of eigenfrequencies and eigenmodes of fractal drums is presented the approach involves first conformally mapping the unit disk to a polygon approximating the fractal and then solving a weighted eigenvalue problem on the unit disk by a spectral collocation method the numerical computation of the complicated conformal mapping was made feasible by the use of the fast multipole method as described in l banjai l n trefethen a multipole method for schwarz christoffel mapping of polygons with thousands of sides siam j sci comput 25 3 2003 1042 1065 the linear system arising from the spectral discretization is large and dense to circumvent this problem we devise a fast method for the inversion of such a system consequently the eigenvalue problem is solved iteratively we obtain eight digits for the first eigenvalue of the koch snowflake and at least five digits for eigenvalues up to the 20th numerical results for two more fractals are shown

pac learning of concept classes through the boundaries of their items abstract we present a new perspective for investigating the probably approximate correct pac learnability of classes of concepts we focus on special sets of points for characterizing the concepts within their class this gives rise to a general notion of boundary of a concept which holds even in discrete spaces and to a special probability measuring technique this technique is applied i to narrow the gap between the minimum and maximum sample sizes necessary to learn even under a more stringent learnability definition and ii to get self explanatory indices of the complexity of the learning task these indices can be roughly estimated during the learning process and appear very useful in the treatment of nonsymbolic procedures e g in the context of neutral networks

asymptotic floquet mode investigation over two dimensional infinite array antennas phased to radiate near zone focused fields this paper extends the previous work in to consider the radiation from an infinite 2 d antenna array for the near field focus applications it employs the poisson sum formula to expand the radiation into floquet modes which may exhibit a class of wave phenomena in terms of geometrical ray fields this analysis automatically reduces to the conventional case of a linearly phased antenna array and results in consistent radiation phenomena it can be considered as a generalization of phased array analysis theoretical investigations as well as numerical illustrations are presented

bounds on the number of examples needed for learning functions we prove general lower bounds on the number of examples needed for learning function classes within different natural learning models which are related to pac learning and coincide with the pac learning model of valiant in the case of 0 1 valued functions the lower bounds are obtained by showing that all nontrivial function classes contain a hard binary valued subproblem although at first glance it seems to be likely that real valued function classes are much harder to learn than their hardest binary valued subproblem we show that these general lower bounds cannot be improved by more than a logarithmic factor this is done by discussing some natural function classes like nondecreasing functions or piecewise smooth functions the function classes that were discussed in m j kearns and r e schapire proc 31st annual symposium on the foundations of computer science ieee computer society press los alamitos ca 1990 pp 382 392 full version j comput system sci 48 1994 pp 464 497 d kimber and p m long proc 5th annual workshop on computational learning theory acm new york 1992 pp 153 160 with certain restrictions concerning their slope

time domain green s function for an infinite sequentially excited periodic planar array of dipoles the present paper is a continuation of previous explorations by the authors aimed at gaining a basic understanding of the time domain td behavior of large periodic phased i e sequentially turned on array antennas and related configurations our systematic investigation of the relevant canonical td dipole excited green s functions has so far included those for infinite and truncated sequentially pulsed line periodic arrays parameterized in terms of radiating propagating and nonradiating evanescent conical td floquet waves fw and truncation induced td fw modulated tip diffractions the present contribution extends these investigations to an infinite periodic sequentially pulsed planar array which generates pulsed plane propagating and evanescent fw starting from the familiar frequency domain fd transformation of the linearly phased element by element summation synthesis into summations of propagating and evanescent fws we access the time domain by fourier inversion the inversion integrals are manipulated in a unified fashion into exact closed forms which are parameterized by the single nondimensional quantity spl eta c v sup p sub u1 where v sup p sub u1 and c are the excitation phase speed along a preferred phasing direction u sub 1 in the array plane and the ambient wave speed respectively the present study deals with the practically relevant rapidly phased propagating case spl eta 1 regime for a future manuscript numerical reference data generated via element by element summation over the fields radiated by the individual dipoles with ultrawide band limited excitation are compared with results obtained much more efficiently by inclusion of a few td fws physical interpretation of the formal td fw solutions is obtained by recourse to asymptotics instantaneous frequencies and wavenumbers and related constructs of special interest is the demonstration that the td fws emerge along equal delay ellipses from the array plane this furnishes a novel and physically appealing interpretation of the planar array td fw phenomenology

bandwidth choice for nonparametric classification it is shown that for kernel based classification with univariate distributions and two populations optimal bandwidth choice has a dichotomous character if the two densities cross at just one point where their curvatures have the same signs then minimum bayes risk is achieved using bandwidths which are an order of magnitude larger than those which minimize pointwise estimation error on the other hand if the curvature signs are different or if there are multiple crossing points then bandwidths of conventional size are generally appropriate the range of different modes of behavior is narrower in multivariate settings there the optimal size of bandwidth is generally the same as that which is appropriate for pointwise density estimation these properties motivate empirical rules for bandwidth choice

spectral distribution of drums with fractal perimeters the weyl berry lapidus conjecture the weyl berry lapidus conjecture on vibrational spectra of drums with fractal perimeters fractal drums is examined in terms of large scale numerical simulations the integrated densities of states of fractal drums with the perimeter of the koch curve are computed by employing a powerful numerical method it is confirmed that the weyl berry lapidus conjecture holds in the frequency regime higher than a crossover frequency mathrm ensuremath omega mathit c related to the length scale characterizing the fractal boundary

density distribution for isospectral circular membranes a special conformal mapping is used to yield the areal density distribution for inhomogeneous circular membranes which are isospectral with each other and with the congruent circular homogeneous membrane for fixed boundary and for free boundary implications of this nonuniqueness for the inverse problem in two dimensions are discussed

on the complexity of learning from drifting distributions abstract we consider two models of on line learning of binary valued functions from drifting distributions due to bartlett we show that if each example is drawn from a joint distribution which changes in total variation distance by at mosto e3 d log 1 e between trials then an algorithm can achieve a probability of a mistake at mosteworse than the best function in a class of vc dimensiond we prove a corresponding necessary condition ofo e3 d finally in the case that a fixed function is to be learned from noise free examples we show that if the distributions on the domain generating the examples change by at mosto e2 d log 1 e then any consistent algorithm learns to within accuracye

a general lower bound on the number of examples needed for learning abstract we prove a lower bound of 1 ln 1 vcdim c on the number of random examples required for distribution free learning of a concept class c where vcdim c is the vapnik chervonenkis dimension and and are the accuracy and confidence parameters this improves the previous best lower bound of 1 ln 1 vcdim c and comes close to the known general upper bound of o 1 ln 1 vcdim c ln 1 for consistent algorithms we show that for many interesting concept classes including k cnf and k dnf our bound is actually tight to within a constant factor

frequency domain green s function for a planar periodic semi infinite phased array i truncated floquet wave formulation this two part sequence deals with the derivation and physical interpretation of a uniform high frequency solution for the field radiated at finite distance by a planar semi infinite phased array of parallel elementary electric dipoles the field obtained by direct summation over the contributions from the individual radiators is restructured into a double series of wavenumber spectral integrals whose asymptotic reduction yields a series encompassing propagating and evanescent floquet waves fws together with corresponding diffracted rays which arise from scattering of the fw at the edge of the array the formal aspects of the solution are treated in the present paper they involve a sequence of manipulations in the complex spectral wavenumber planes that prepare the integrands for subsequent efficient and physically incisive asymptotics based on the method of steepest descent different species of spectral poles define the various species of propagating and evanescent fw their interception by the steepest descent path sdp determines the variety of shadow boundaries for the edge truncated fw the uniform asymptotic reduction of the sdp integrals performed by the van der waerden 1951 procedure and yielding a variety of edge diffracted fields completes the formal treatment

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

collocation on uniform grids in this paper we derive four sets of sinc like functions defined on a finite interval and obeying different boundary conditions the functions in each set are orthogonal and their nodes are uniformly distributed on the interval we have applied each set to solve a large class of eigenvalue equations with different boundary conditions both on finite intervals and on the real line showing that precise numerical results can be obtained efficiently and rapidly a comparison with results available in the literature is also performed

sample efficient strategies for learning in the presence of noise in this paper we prove various results about pac learning in the presence of malicious noise our main interest is the sample size behavior of learning algorithms we prove the first nontrivial sample complexity lower bound in this model by showing that order of e 2 d up to logarithmic factors examples are necessary for pac learning any target class of 0 1 valued functions of vc dimension d where e is the desired accuracy and e e 1 e the malicious noise rate it is well known that any nontrivial target class cannot be pac learned with accuracy e and malicious noise rate e e 1 e this irrespective to sample complexity we also show that this result cannot be significantly improved in general by presenting efficient learning algorithms for the class of all subsets of d elements and the class of unions of at most d intervals on the real line this is especialy interesting as we can also show that the popular minimum disagreement strategy needs samples of size d e 2 hence is not optimal with respect to sample size we then discuss the use of randomized hypotheses for these the bound e 1 e on the noise rate is no longer true and is replaced by 2e 1 2e in fact we present a generic algorithm using randomized hypotheses that can tolerate noise rates slightly larger than e 1 e while using samples of size d e as in the noise free case again one observes a quadratic powerlaw in this case d e 2 2e 1 2e e as goes to zero we show upper and lower bounds of this order

a truncated floquet wave diffraction method for the full wave analysis of large phased arrays ii generalization to 3 d cases for pt i see ibid vol 48 no 3 p 594 600 2000 this paper deals with the generalization to three dimensional 3 d arrays of the truncated floquet wave tfw diffraction method for the full wave analysis of large arrays this generalization potentially includes arrays consisting of microstrip excited slots cavity backed apertures and patches the formulation is carried out first by deriving an appropriate fringe integral equation ie and next by defining entire domain basis functions in terms of global array functions shaped as tfw diffracted rays whose analytical expression is derived on the basis of prototype canonical problems the efficiency and the accuracy of this method is demonstrated by comparison with the results of an element by element full wave approach for a rectangular slot array

shaped beam synthesis of real antenna arrays via finite element method floquet modal analysis and convex programming a shaped beam synthesis methodology for planar arrays with control of the ripple amplitude in the shaped region and constraints on the sidelobe and cross polar levels is proposed in this paper a previously developed synthesis formulation for designing finite impulse response fir filters by transforming a nonconvex synthesis problem to a convex optimization scheme enforcing conjugate symmetric excitation weights is extended to real and coupled radiating elements of complex geometry taking into account mutual coupling effects the optimization procedure is integrated with a finite array approach simulating different array environments this approach is based on the infinite array model through a floquet modal and general scattering matrix gsm based analysis where the periodic radiating element is characterized from a hybrid and full wave analysis procedure combining the fem modal analysis and domain decomposition technique numerical results of different synthesized beam patterns are presented for arrays of open ended apertures and microstrip antennas

improved lower bounds for learning from noisy examples this paper presents a general information theoretic approach for obtaining lower bounds on the number of examples required for probably approximately correct pac learning in the presence of noise this approach deals directly with the fundamental information quantities avoiding a bayesian analysis the technique is applied to several different models illustrating its generality and power the resulting bounds add logarithmic factors to or improve the constants in previously known lower bounds

edge effects in axial arrays of circular rings a boundary value problem is solved for the current induced by any source in particular for an incident plane wave on any number of identical rings that are arranged to form an axisymmetric linear array the net aperture distribution along the finite array is approximated by a combination of three physically identifiable components a term from the related infinite array plus contributions from two independent edge effect waves that originate from each of the two ends however if individually resonant loops are closer than a critical spacing the entire structure exhibits a wild resonance and the simple approximation becomes invalid

asymptotic high frequency green s function for a large rectangular planar periodic phased array of dipoles with weakly tapered excitation in two dimensions this paper deals with the derivation and physical interpretation of a uniform high frequency representation of the green s function for a large planar rectangular phased array of dipoles with weakly varying excitation thereby our earlier published results valid for equiamplitude excitation and those for tapered illumination in one dimension are extended to tapering along both dimensions including dipole amplitudes tending to zero at the edges as previously the field obtained by direct summation over the contributions from the individual radiators is restructured into a double spectral integral whose high frequency asymptotic reduction yields a series of propagating and evanescent floquet waves fws together with corresponding fw modulated diffracted fields which arise from fw scattering at the array edges and vertexes to accommodate the weak amplitude tapering new generalized periodicity modulated edge and vertex slope transition functions are introduced accompanied by a systematic procedure for their numerical evaluation special attention is given to the analysis and physical interpretation of the complex vertex diffracted ray fields a sample calculation is included to demonstrate the accuracy of the asymptotic algorithm the resulting array green s function forms the basic building block for the full wave analysis of planar weakly amplitude tapered phased array antennas and for the description of electromagnetic radiation and scattering from weakly amplitude tapered rectangular periodic structures

on the additive properties of the fat shattering dimension the properties of the vc dimension under various compositions are well understood but this is much less the case for classes of continuous functions in this brief we show that a commonly used scale sensitive dimension v y is much less well behaved under minkowski summation than its vc cousin while the fat shattering dimension retains some compositional similarity to the vc dimension as an application we analyze the fat shattering dimension of trigonometric functions and series

floquet mode phenomena of infinite phased array antennas in near field focus applications this paper presents a theoretical investigation on the radiation of an infinite one dimensional antenna array which is phased to radiate electromagnetic fields focused in the near zone of the array aperture in particular the wave phenomena in each of the floquet modes obtained by applying the poisson sum formula to the array excitations are investigated the analysis reduces to the case of a conventional phased antenna array when the radiation is focused in the far zone and exhibits consistent phenomena theoretical investigations as well as numerical illustrations are presented

time harmonic and transient scattering by finite periodic flat strip arrays hybrid ray floquet mode mom algorithm a frequency and short pulse sp time domain td study of two dimensional plane wave scattering from a finite periodic array of thin flat coplanar perfectly conducting strips is presented rigorous analytical numerical reference solutions are established by spatial spectral wave number decomposition and the method of moments mom followed by frequency inversion the analytical portion is approximated so as to yield via high frequency asymptotics for a sufficiently large number of strips a hybrid ray floquet mode mom algorithm that not only explains the phenomena in physical terms but is also numerically efficient and reasonably accurate when compared with the reference solution of special interest are the td floquet modes with their space time dependent frequencies and wave numbers by superposition they can synthesize the highly resolved pulse train return under sp conditions attention is given also to direct sp td synthesis and to processing options for sp td data

truncated floquet wave full wave analysis of large phased arrays of open ended waveguides with a nonuniform amplitude excitation recently an efficient hybrid asymptotic method of moments mom approach has been proposed for the analysis of large periodic planar arrays of elements excited with equal amplitude and linear phase the aforementioned method which is based on a floquet wave diffraction representation of the array green s function agf is here extended to treat arrays with tapered amplitude excitation to this end the asymptotic agf is refined by introducing additional slope diffraction contributions an appropriate fringe integral equation solved via a mom scheme provides the effects of array truncation in addition to the infinite array solution the dimension of the corresponding linear algebraic system is independent of the number of elements of the array numerical results are provided to prove the accuracy and the efficiency of this method with respect to an ordinary element by element mom

efficient algorithms for learning functions with bounded variation we show that the class fbv of 0 1 valued functions with total variation at most 1 can be agnostically learned with respect to the absolute loss in polynomial time from o 1 e2 log 1 examples matching a known lower bound to within a constant factor we establish a bound of o 1 m on the expected error of a polynomial time algorithm for learning fbv in the prediction model also matching a known lower bound to within a constant factor applying a known algorithm transformation to our prediction algorithm we obtain a polynomial time pac learning algorithm for fbv with a sample complexity bound of o 1 e log 1 this also matches a known lower bound to within a constant factor

a hybrid spectral spatial method to evaluate the active green s function of large planar rectangular arrays a combined asymptotic numerical algorithm the article illustrates a formulation to evaluate the array green s function agf of large finite planar phased array for observation points on the array plane possibly close to the array contour the procedure is based on the agf representation in terms of a double spectral integral whose integration paths are properly deformed to have an exponential attenuation of the integrand the diffraction integral is evaluated numerically for point close to the array edges while an asymptotic treatment is proposed far from the edges this latter comprises higher order contributions thanks to the convergence properties the final algorithm is numerically accurate stable and more efficient with respect to the individual element summation for large arrays it also constitutes the basic step for the efficient evaluation of the agf of a multilayer environment

a truncated floquet wave diffraction method for the full wave analysis of large phased arrays i basic principles and 2 d cases this two part sequence deals with the formulation of an efficient method for the full wave analysis of large phased array antennas this is based on the method of moments mom solution of a fringe integral equation ie in which the unknown function is the difference between the exact solution of the finite array and that of the associated infinite array the unknown currents can be interpreted as produced by the field diffracted at the array edge which is excited by the floquet waves fws pertinent to the infinite configuration following this physical interpretation the unknown in the ie is efficiently represented by a very small number of basis functions with domain on the entire array aperture in order to illustrate the basic concepts the first part of this sequence deals with the two dimensional example of a linearly phased slit array it is shown that the dominant phenomenon fur describing the current perturbation with respect to the infinite array is accurately represented in most cases by only three diffracted ray shaped unknown functions this also permits a simple interpretation of the element by element current oscillation which was described by other authors

collocation approach to the helmholtz eigenvalue problem on multiply connected domains we present a collocation approach to the numerical solution of the helmholtz eigenvalue problem on multiply connected domains of arbitrary shape in two dimensions a suitable representation of the helmholtz equation on an uniform grid is obtained and the problem is converted to the calculation of the first few eigenvalues and eigenvectors of a large sparse matrix the accuracy of this approach is tested by performing a comparison with results available in the literature the flexibility and simplicity of the method make easy the application to arbitrary geometries

high order eigenvalues for the helmholtz equation in complicated non tensor domains through richardson extrapolation of second order finite differences we apply second order finite differences to calculate the lowest eigenvalues of the helmholtz equation for complicated non tensor domains in the plane using different grids which sample exactly the border of the domain we show that the results obtained applying richardson and pade richardson extrapolations to a set of finite difference eigenvalues corresponding to different grids allow us to obtain extremely precise values when possible we have assessed the precision of our extrapolations comparing them with the highly precise results obtained using the method of particular solutions our empirical findings suggest an asymptotic nature of the fd series in all the cases studied we are able to report numerical results which are more precise than those available in the literature

an eigencurrent approach for the analysis of finite antenna arrays an accurate description of typical finite array behavior such as edge effects and array resonances is essential in the design of various types of antennas the analysis approach proposed in this paper is essentially based on the concept of eigencurrents and is capable of describing finite array behavior in the approach numerical simulation is carried out first by computing element eigencurrents from chosen expansion functions and second by expanding a limited set of array eigencurrents in terms of element eigencurrents that contribute to the mutual coupling in the array both types of eigencurrents are eigenfunctions of an impedance operator that relates the current to the excitation field highlighting both mathematical and physical features we describe the basic concepts of the approach in particular the relation between eigenvalues and mutual coupling we illustrate these features for uniform linear arrays of loops and dipoles and demonstrate that the approach provides significant improvements in terms of computation time and memory use

variational collocation on finite intervals in this paper we study a set of functions defined on an interval of finite width which are orthogonal and which reduce to the sinc functions when the appropriate limit is taken we show that these functions can be used within a variational approach to obtain accurate results for a variety of problems we have applied them to the interpolation of functions on finite domains and to the solution of the schrodinger equation and we have compared the performance of the present approach with others

computing eigenmodes ofelliptic operators using radial basis functions radial basis function rbf approximations have been successfully used to solve boundary value problems numerically we show that rbfs can also be used to compute eigenmodes of elliptic operators particular attention is given to the laplacian operator in two dimensions including techniques to avoid degradation of the solution near the boundaries for regions with corner singularities special functions must be added to the basis to maintain good convergence numerical results compare favorably to basic finite element methods

array pattern synthesis of real antennas using the infinite array approach and linear programming this work proposes an efficient optimization methodology of antenna array excitations accounting for the effects of mutual couplings and real radiating patterns of array elements with complex geometry the method is formulated as a linear programming problem providing multibeam patterns with specified beamwidths constraints on sidelobe and cross polar levels and null pointing directions a matrix formulation is used to integrate the optimization procedure and a finite array analysis based on the infinite array approach in which the periodic radiating element is characterized by a full wave and hybrid procedure combining the domain decomposition technique the finite element method and modal analysis numerical examples of arrays of open ended waveguides microstrip antennas and dielectric resonator antennas are presented

exact free vibration analysis for membrane assemblies with general classical boundary conditions abstract this paper proposes exact dynamic stiffness formulations for membranes and their assemblies under any arbitrary classical boundary conditions first by taking exact solutions in one direction satisfying all possible opposite edge supports we can derive exact general solutions of the helmholtz equation for membrane vibration then generic force and displacement boundary conditions in the other direction are expressed in terms of the general solutions finally the dynamic stiffness matrices of rectangular membrane elements are formulated which can be assembled directly and allows applications of arbitrary boundary conditions as an accurate and efficient modal solution technique the wittrick williams ww algorithm is applied onto the global dynamic stiffness matrix of the final structure the most important issue of the ww algorithm j 0 count has been resolved with the analytical expressions derived for all possible cases the proposed dynamic stiffness method dsm is then applied to several examples including individual membranes and their assemblies high accuracy and exactness of the proposed method within the whole frequency range is demonstrated by comparison with the finite element method besides interesting findings have been observed on repeated eigenvalues with distinct mode shapes corresponding to certain aspect ratio and tension ratio where physical and mathematical understanding has been provided

solving the helmholtz equation for membranes of arbitrary shape numerical results i calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on little sinc functions the matrix representation of the pde obtained using this method is explicit and does not require the calculation of integrals to illustrate the virtues of this approach i have considered a large number of examples part of them are taken from the literature and part of them new when possible i have tested the accuracy of these results by comparing them with the exact results when available or with results from the literature in particular in the case of the l shaped membrane the first example discussed in the paper i show that it is possible to extrapolate the results obtained with different grid sizes to obtain highly precise results finally i also show that the present collocation technique can be easily combined with conformal mapping to provide numerical approximations to the energies which quite rapidly converge to the exact results

a new method for studying the vibration of non homogeneous membranes abstract we present a method to solve the helmholtz equation for a non homogeneous membrane with dirichlet boundary conditions at the border of arbitrary two dimensional domains the method uses a collocation approach based on a set of localized functions called little sinc functions which are used to discretize two dimensional regions we have performed extensive numerical tests and we have compared the results obtained with the present method with the ones available from the literature our results show that the present method is very accurate and that its implementation for general problems is straightforward

the one inclusion graph algorithm is near optimal for the prediction model of learning haussler littlestone and warmuth 1994 described a general purpose algorithm for learning according to the prediction model and proved an upper bound on the probability that their algorithm makes a mistake in terms of the number of examples seen and the vapnik chervonenkis vc dimension of the concept class being learned we show that their bound is within a factor of 1 o 1 of the best possible such bound for any algorithm

quantifying inductive bias ai learning algorithms and valiant s learning framework abstract we show that the notion of inductive bias in concept learning can be quantified in a way that directly relates to learning performance in the framework recently introduced by valiant our measure of bias is based on the growth function introduced by vapnik and chervonenkis and on the vapnik chervonenkis dimension we measure some common language biases including restriction to conjunctive concepts conjunctive concepts with internal disjunction k dnf and k cnf concepts we also measure certain types of bias that result from a preference for simpler hypotheses using these bias measurements we analyze the performance of the classical learning algorithm for conjunctive concepts from the perspective of valiant s learning framework we then augment this algorithm with a hypothesis simplification routine that uses a greedy heuristic and show how this improves learning performance on simpler target concepts improved learning algorithms are also developed for conjunctive concepts with internal disjunction k dnf and k cnf concepts we show that all our algorithms are within a logarithmic factor of optimal in terms of the number of examples they require to achieve a given level of learning performance in the valiant framework our results hold for arbitrary attribute based instance spaces defined by either tree structured or linear attributes

reviving the method of particular solutions fox henrici and moler made famous a method of particular solutions for computing eigenvalues and eigenmodes of the laplacian in planar regions such as polygons we explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties the crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant similar methods may be used to improve other mesh free algorithms for a variety of computational problems

numerical investigation of isospectral cavities built from triangles we present computational approaches as alternatives to a recent microwave cavity experiment by s sridhar and a kudrolli phys rev lett 72 2175 1994 on isospectral cavities built from triangles a straightforward proof of isospectrality is given based on the mode matching method our results show that the experiment is accurate to 0 3 for the first 25 states the level statistics resemble those of a gaussian orthogonal ensemble when the integrable part of the spectrum is removed

truncated floquet wave full wave t fw sup 2 analysis of large periodic arrays of rectangular waveguides a new hybrid technique has been recently proposed for the analysis of large periodic arrays this method is based on a high frequency representation of the active green s function of a finite array involving a floquet wave fw expansion with the relevant edge and vertex diffracted ray fields originating at the array border using this representation the unknown current of an appropriate fringe integral equation is expanded in terms of a few basis functions shaped as fw induced diffracted rays this representation drastically reduces the numerical effort and provides a physical insight into the mechanism of the array truncation in this paper this method is applied to the analysis of large arrays of rectangular waveguides opened on an infinite ground plane numerical results are shown to demonstrate the effectiveness and the numerical efficiency of the method

neural nets with superlinear vc dimension it has been known for quite a while that the vapnik chervonenkis dimension vc dimension of a feedforward neural net with linear threshold gates is at most o w log w where w is the total number of weights in the neural net we show in this paper that this bound is in fact asymptotically optimal more precisely we exhibit for any depth d 3 a large class of feedforward neural nets of depth d with w weights that have vc dimension w log w this lower bound holds even if the inputs are restricted to boolean values the proof of this result relies on a new method that allows us to encode more program bits in the weights of a neural net than previously thought possible

a hybrid mom solution of scattering from finite arrays of cylindrical cavities in a ground plane in this paper we propose a novel hybrid space spectral domain method of moments mom calculation of canonical plane wave scattering from finite periodic structures comprised of dissimilar cylindrical cavities in an infinite ground plane covered by a dielectric superstrate we take full advantage of analyzing this particular type of problem comprised of a canonical geometry by employing a set of entire domain basis functions waveguide modes which truly span the hilbert space in which the desired solution resides the convergence behavior of a solution using said functions is quite robust results show that typically only a few basis functions are required to adequately represent the aperture fields with fidelity we have also reformulated the conventional magnetic field integral equation in a hybrid fashion whereby the exterior unbounded half space mom interaction terms have been cast in the spectral domain and the interior cavity interactions cast in the space domain directly the spectral representation allows us to trivially include a superstrate layer in the geometry and additionally leads to an efficient evaluation of the interaction integrals due to an overall reduction in the spatial dimensionality the interior interactions are readily handled in the space domain where we take advantage of the orthogonality property of waveguide modes most importantly the technique is significantly more accurate than purely numerical methods i e those in which a discretization of the geometry is necessary

a variational sinc collocation method for strong coupling problems we have devised a variational sinc collocation method vscm which can be used to obtain accurate numerical solutions to many strong coupling problems sinc functions with an optimal grid spacing are used to solve the linear and nonlinear schrodinger equations and a lattice 4 model in 1 1 our results indicate that errors decrease exponentially with the number of grid points and that a limited numerical effort is needed to reach high precision

analysis of large complex structures with the synthetic functions approach an innovative procedure is presented that allows the method of moments mom analysis of large and complex antenna and scattering problems at a reduced memory and cpu cost bounded within the resources provided by a standard 32 bit personal computer the method is based on the separation of the overall geometry into smaller portions called blocks and on the degrees of freedom of the field the blocks need not be electrically unconnected on each block basis functions are generated with support on the entire block that are subsequently used as basis functions for the analysis of the complete structure only a small number of these functions is required to obtain an accurate solution therefore the overall number of unknowns is drastically reduced with consequent impact on storage and solution time these entire domain basis functions are called synthetic functions they are generated from the solution of the electromagnetic problem for the block in isolation under excitation by suitably defined sources the synthetic functions are obtained from the responses to all sources via a procedure based on the singular value decomposition because of the strong reduction of the global number of unknowns one can store the mom matrix and afford a direct solution the method is kernel free and can be implemented on existing mom codes

edge born waves in connected arrays a finite times infinite analytical representation connected arrays constitute one of the most promising options for wideband phased arrays like most phased arrays they are designed using infinite array theory however when finiteness is included edge effects perturb their behavior these effects are more severe when the arrays are designed to operate over very broad frequency ranges since the mutual coupling between the elements facilitates the propagation of edge born waves that can become dominant over large portions of the arrays finite array simulations which would predict these behaviors are computationally unwieldy in this paper we present a green s function based procedure to assess edge effects in finite connected arrays first the electric current distribution on the array is rigorously derived later on the introduction of a few simplifying assumptions allows the derivation of an analytical approximation for the current distribution this latter provides meaningful insights in the induced dominant edge wave mechanism the efficiency of connected arrays as a function of their dimension in terms of the wavelength and of the loading feed impedances is investigated

lower bounds in pattern recognition and learning lower bounds are derived for the performance of any pattern recognition algorithm which using training data selects a discrimination rule from a certain class of rules the bounds involve the vapnik chervonenkis dimension of the class and l the minimal error probability within the class we provide lower bounds when l 0 the usual assumption in valiant s theory of learning and l 0

on the necessity of irrelevant variables this work explores the effects of relevant and irrelevant boolean variables on the accuracy of classifiers the analysis uses the assumption that the variables are conditionally independent given the class and focuses on a natural family of learning algorithms for such sources when the relevant variables have a small advantage over random guessing the main result is that algorithms relying predominately on irrelevant variables have error probabilities that quickly go to 0 in situations where algorithms that limit the use of irrelevant variables have errors bounded below by a positive constant we also show that accurate learning is possible even when there are so few examples that one cannot determine with high confidence whether or not any individual variable is relevant

on the spectrum of the dirichlet laplacian on broken strips we investigate the spectrum of the dirichlet laplacian on broken strips it is shown that there is exactly one eigenvalue below the bottom of the essential spectrum for any angle corresponding to the broken strip

computational limitations on learning from examples the computational complexity of learning boolean concepts from examples is investigated it is shown for various classes of concept representations that these cannot be learned feasibly in a distribution free sense unless r np these classes include a disjunctions of two monomials b boolean threshold functions and c boolean formulas in which each variable occurs at most once relationships between learning of heuristics and finding approximate solutions to np hard optimization problems are given

semi infinite arrays of isotropic point scatterers a unified approach we solve the two dimensional problem of acoustic scattering by a semi infinite pe riodic array of identical isotropic point scatterers i e objects whose size is negligible compared to the incident wavelength and which are assumed to scatter incident waves uniformly in all directions this model is appropriate for scatterers on which dirichlet boundary conditions are applied in the limit as the ratio of wavelength to body size tends to infinity the problem is also relevant to the scattering of an e polarized electromagnetic wave by an array of highly conducting wires the actual geometry of each scatterer is characterized by a single parameter in the equations related to the single body scattering problem and determined from a harmonic boundary value problem using a mixture of analytical and numerical techniques we confirm that a number of phenomena reported for specific geometries are in fact present in the general case such as the presence of shadow boundaries in the far field and the vanishing of the circular wave scattered by the end of the array in certain specific directions we show that the semi infinite array problem is equivalent to that of inverting an infinite toeplitz matrix which in turn can be formulated as a discrete wiener hopf problem numerical results are presented which compare the amplitude of the wave diffracted by the end of the array for scatterers having different shapes

new series for the cosine lemniscate function and the polynomialization of the lemniscate integral we discuss the numerical computation of the cosine lemniscate function and its inverse the lemniscate integral these were previously studied by bernoulli euler gauss abel jacobi and ramanujan we review general elliptic formulas for this special case and provide some novelties we show that a fourier series by ramanujan converges twice as fast as the standard elliptic cosine fourier series specialized to this case the so called imbricate series however converges geometrically fast over the entire complex plane we derive two new expansions the rational plus fourier series converges much faster than ramanujan s for real z each term is asymptotically 12 400 times smaller than its immediate predecessor coslem z 4b q 1 q cos bz 1 q 2 4qcos 2 bz n 1 q n 1 2 1 1 q 2 n 1 1 cos 2n 1 bz where q exp p is the elliptic nome k 1 85 is the complete elliptic integral of the first kind for a modulus m 1 2 and b p k2 the rational imbricate series is uniformly valid over the complex plane but converges twice as fast as the sech imbricate series coslem z 4bq 1 q j q 1 q cos bz 1 q 2 4qcos 2 b z jpi where p 4 2 k is the period in both the real and imaginary directions we devise a new approximation for the lemniscate integral for real argument as the arccosine of a chebyshev series and show that 17 terms yield about 15 digits of accuracy for complex argument we show that the lemniscate integral can be found to near machine precision assumed as sixteen decimal digits by computing the roots of a polynomial of degree thirteen alternatively newton s iteration converges in three iterations with an initialization accurate to four decimal places that is the chosen root of a cubic equation