antenna array pattern synthesis via convex optimization we show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems which can be numerically solved with great efficiency by recently developed interior point methods the synthesis problems involve arrays with arbitrary geometry and element directivity constraints on far and near field patterns over narrow or broad frequency bandwidth and some important robustness constraints we show several numerical simulations for the particular problem of constraining the beampattern level of a simple array for adaptive and broadband arrays

constrained least squares optimization in conformal array antenna synthesis the least mean square error of the constrained least squares optimization method is studied the phases of non zero goal values and the phases of non zero constraint values can be selected by a fast iteration for optimum fit to the goal function amplitude this kind of modified least squares optimization method opens up a fast and robust synthesis method for large conformal antenna arrays constraints can be written also for the element excitation amplitudes and a phase synthesis with predefined amplitudes can be done iterative weight correction technique is used for accurate sidelobe control synthesis examples of two dimensional elliptical and large three dimensional conformal antenna arrays are presented

adaptive radial basis function and hermite function pseudospectral methods for computing eigenvalues of the prolate spheroidal wave equation for very large bandwidth parameter asymptotic approximations show that the lowest modes of the prolate spheroidal wave equation are concentrated with an o 1 c length scale where c is the bandwidth parameter of the prolate differential equation accurate computation of the ground state eigenvalue by the long known legendre galerkin method requires roughly 3 8 c legendre polynomials some studies have therefore applied a grid with 20 000 points in conjunction with high order finite differences to achieve c 10 7 here we show that by adaptively applying either hermite functions or gaussian radial basis functions rbfs it is never necessary to use more than eighty degrees of freedom to calculate the lowest dozen eigenvalues of each symmetry class for small c the eigenmodes are not confined to a small portion of the domain 2 2 in latitude but are global we show that by periodizing the basis functions via imbricate series it is possible to apply hermite and rbfs even in the limit c 0 the legendre method is probably a little more efficient in this limit since the prolate functions tend to legendre polynomials in this limit a sideband truncation restricts the discretization to a small block taken from the large hermite galerkin matrix we show that sideband truncation with blocks as small as 5 5 is a very efficient way to compute high order modes in an appendix we prove a rigorous convergence theorem for the periodized hermite expansion

optimal narrow beam low sidelobe synthesis for arbitrary arrays the classical narrow main beam and low sidelobe synthesis problem is addressed and extended to arbitrary sidelobe envelopes in both one and two dimensional scenarios the proposed approach allows also to handle arbitrary arrays this extended basic synthesis problem is first rewritten as a convex optimization problem this problem is then transformed into either a linear program or a second order cone program that can be solved efficiently by readily available software the convex formulation is not strictly equivalent to the initial synthesis problem but arguments are given that establish that the obtained optimum is either identical or can be made as close as wanted to the desired one finally numerical applications and a comparison with a classical solution are presented to both confirm the optimality of the solution and illustrate the potentialities of the proposed approach

relations between elliptic multiple zeta values and a special derivation algebra we investigate relations between elliptic multiple zeta values and describe a method to derive the number of indecomposable elements of given weight and length our method is based on representing elliptic multiple zeta values as iterated integrals over eisenstein series and exploiting the connection with a special derivation algebra its commutator relations give rise to constraints on the iterated integrals over eisenstein series relevant for elliptic multiple zeta values and thereby allow to count the indecomposable representatives conversely the above connection suggests apparently new relations in the derivation algebra under https tools aei mpg de emzv we provide relations for elliptic multiple zeta values over a wide range of weights and lengths ar x iv 1 50 7 02 25 4v 1 he pth 8 j ul 2 01 5

the calculation of singular coefficients abstract we consider in this paper the calculation of the singular coefficients associated with the solution of an elliptic partial differential equation near a singular point a re entrant corner or a crack tip etc these are the coefficients in the relevant singular expansion of the solution near the point of singularity they often have physical relevance and it is of interest to be able to calculate them accurately we consider the problem in the context of the global element method this is a variable order finite element method designed to be capable of producing highly accurate solutions for singular problems even in the neighbourhood of the singularity if the values of the singular coefficients are needed these must be extracted from the computed solution we show in this paper that a suitably defined least squares fitting procedure allows the calculation of values for the leading singular coefficients which are as accurate as the underlying solution

optimal far field focusing of uniformly spaced arrays subject to arbitrary upper bounds in nontarget directions a new approach providing new theoretical results to the optimal far field focusing of uniformly spaced arrays subject to a completely arbitrary mask for sidelobe bounds is presented and discussed in both cases of linear or planar arrays with rectangular boundaries it is first shown that the problem can be formulated without any loss to performances on the maximum as a linear programming one guaranteeing a globally optimal solution second a sufficient uniqueness criterion for the solution of the overall problem is also developed which shows how the solution may not be unique as is actually the case when planar arrays are considered in addition further globally effective optimization procedures are proposed for the latter case in order to optimize directivity smoothness of excitations or other performance parameters in the set of equivalent solutions last an extension to planar arrays with a nonrectangular boundary is also given a thorough numerical analysis confirms the effectiveness of the approach proposed and of the solution codes developed

a simple algorithm to achieve desired patterns for arbitrary arrays a simple iterative algorithm which can be used to find array weights that produce array patterns with a given look direction and an arbitrary sidelobe specification is presented the method can be applied to nonuniform array geometries in which the individual elements have arbitrary and differing radiation patterns the method is iterative and uses sequential updating to ensure that peak sidelobe levels in the array meet the specification computation of each successive pattern is based on the solution of a linearly constrained least squares problem the constraints ensure that the magnitude of the sidelobes at the locations of the previous peaks takes on the prespecified values phase values for the sidelobes do not change during this process and problems associated with choosing a specific phase value are therefore avoided experimental evidence suggests that the procedure terminates in remarkably few iterations even for arrays with significant numbers of elements

pattern synthesis for arbitrary arrays using an adaptive array method this paper presents a new pattern synthesis algorithm for arbitrary arrays based on adaptive array theory with this algorithm the designer can efficiently control both mainlobe shaping and sidelobe levels the element weights optimize a weighted l sub 2 norm between desired and achieved patterns the values of the weighting function in the l sub 2 norm interpreted as imaginary jammers as in olen and compton s 1990 method are iterated to minimize exceedance of the desired sidelobe levels and minimize the absolute difference between desired and achieved mainlobe patterns the sidelobe control can be achieved by iteration only on sidelobe peaks in comparison to olen and compton s method the new algorithm provides a great improvement in mainlobe shaping control example simulations including both nonuniform linear and planar arrays are shown to illustrate the effectiveness of this algorithm

the complex wavenumber eigenvalues of laplace s tidal equations for oceans bounded by meridians the complex wavenumber eigenvalues of laplace s tidal equations are determined for an ocean of constant depth bounded by meridians a galerkin method is used to expand the tide height and velocities in series of associated legendre functions a homogeneous system of equations results from the continuity and momentum equations the frequency and depth are fixed so that the meridional wavenumbers are the eigenvalues this gives rise to a generalized eigenvalue problem that must be solved numerically by iteration the eigenvalues are not integers and represent inertia gravity wave solutions at the specified tidal forcing frequency that can be excited by the presence of meridional boundaries those complex eigenvalues represent solutions that decay away from meridional boundaries the eigenvalue spectrum is investigated for the semi diurnal fortnightly and monthly tides one complex wavenumber for the semi diurnal tide explains the amphidromic systems within 20 of a north south coastline the fortnightly and monthly tides have only real wavenumber eigenvalues the basin scale deviation of these tides from equilibrium is attributed to low wavenumber divergent inertia gravity waves

a chebyshev rational chebyshev spectral method for the helmholtz equation in a sector on the surface of a sphere defeating corner singularities when the boundaries of a domain meet at an angle the solutions to an elliptic partial differential equation will usually be singular at the corner using the example of the helmholtz equation on the surface of a sphere in a domain bounded by meridians we show how corner singularities can be defeated by mapping the corner to infinity by applying a chebyshev series in longitude and a rational chebyshev series in the mercator coordinate y arctanh cos colatitude we obtain an exponential rate of convergence despite the corner singularities

beampattern synthesis via a matrix approach for signal power estimation we present new beampattern synthesis approaches based on semidefinite relaxation sdr for signal power estimation the conventional approaches use weight vectors at the array output for beampattern synthesis which we refer to as the vector approaches va instead of this we use weight matrices at the array output which leads to matrix approaches ma we consider several versions of ma including a data adaptive ma ama as well as several data independent ma designs for all of these ma designs globally optimal solutions can be determined efficiently due to the convex optimization formulations obtained by sdr numerical examples as well as theoretical evidence are presented to show that the optimal weight matrix obtained via sdr has few dominant eigenvalues and often only one when the number of dominant eigenvalues of the optimal weight matrix is equal to one ma reduces to va and the main advantage offered by sdr in this case is to determine the globally optimal solution efficiently moreover we show that the ama allows for strict control of main beam shape and peak sidelobe level while retaining the capability of adaptively nulling strong interferences and jammers numerical examples are also used to demonstrate that better beampattern designs can be achieved via the data independent ma than via its va counterpart

optimal array pattern synthesis using semidefinite programming we present algorithms for the solution of two problems in array pattern synthesis the first is the design of nonuniform arrays with a desired magnitude response the second is that of robust design i e design in the presence of uncertainties constraints such as power limitation can be addressed with both problems the algorithms that we present are based on semidefinite programming for which efficient software is readily available we present examples that illustrate the effectiveness of our approach

a fast 3d poisson solver of arbitrary order accuracy we present a direct solver for the poisson and laplace equations in a 3d rectangular box the method is based on the application of the discrete fourier transform accompanied by a subtraction technique which allows reducing the errors associated with the gibbs phenomenon and achieving any prescribed rate of convergence the algorithm requireso n3logn operations wherenis the number of grid points in each direction we show that our approach allows accurate treatment of singular cases which arise when the boundary function is discontinuous or incompatible with the differential equation

traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians we make several observations about eigenvalue problems using as examples laplace s tidal equations and the differential equation satisfied by the associated legendre functions whatever the discretization only some of the eigenvalues of then dimensional matrix eigenvalue problem will be good approximations to those of the differential equation usuallythen 2 eigenvalues of smallest magnitude for the tidal problem however the good eigenvalues are scattered so our first point is it is important to plot the drift of eigenvalues with changes in resolution we suggest plotting the difference between a low resolution eigenvalue and the nearest high resolution eigenvalue divided by the magnitude of the eigenvalue or the intermodal separation whichever is smaller second as a final safeguard it is important to look at the chebyshev coefficients of the mode we show a numerically computed anti kelvin wave which has little eigenvalue drift but is completely spurious as is obvious from its spectral series third inverting the roles of parameters can drastically modify the spectrum legendre s equation may have either an infinite number of discrete modes or only a handful depending on which parameter is the eigenvalue fourth when the modes are singular but decay to zero at the endpoints as is true of tides a tanh mapping can retrieve the usual exponential accuracy of spectral methods fifth the pseudospectral method is more reliable than deriving a banded galerkin matrix by means of recurrence relations the pseudospectral code is simple to check whereas it is easy to make an untestable mistake with the intricate algebra required for the galerkin method sixth we offer a brief cautionary tale about overlooked modes all these cautions are applicable toallforms of spatial discretization including finite difference and finite element methods however we limit our illustrations to spectral schemes where these difficulties are most easily resolved with a bit of care the pseudospectral method is a very robust and efficient method for solving differential eigenproblems even with singular eigenmodes

a spectral solution of the sturm liouville equation comparison of classical and nonclassical basis sets abstract this paper considers a spectral method of solution of the sturm liouville equation and the associated schroedinger equation the main objective is to develop a collocation method based on quadrature collocation points generated from nonclassical polynomials the polynomials and associated quadrature points are calculated with gautschi s stieltjes procedure from some specified weight function the particular spectral method used here with nonclassical basis sets is referred to as the quadrature discretization method qdm the qdm and a related weighted qdm are applied to several sturm liouville and schroedinger equations and the results are compared with the traditional spectral methods based on chebyshev and legendre quadrature points the results are also compared with the results of other workers wherever available the qdm was found to give the most rapid convergence relative to other methods for the problems studied

pattern synthesis with multipoint accurate array response control in this paper the problem of pattern synthesis with antenna arrays is addressed and new approaches based on the recently developed accurate array response control textrm a 2 textrm rc algorithm are presented it is shown that the array weight vector obtained by the textrm a 2 textrm rc algorithm to control the normalized response at a single direction in each step belongs to a specific set thus an appropriate weight vector chosen from the intersection of weight vector sets corresponding to the desired responses at multiple directions is capable of simultaneously controlling those responses this results in the so called multipoint accurate array response control textrm ma 2 textrm rc algorithm moreover in order to avoid possible beam axis shift in pattern synthesis a modified textrm ma 2 textrm rc textrm m 2 textrm a 2 textrm rc algorithm is proposed by imposing a derivative constraint on the direction of beam axis representative numerical examples are provided to demonstrate the effectiveness of the proposed textrm ma 2 textrm rc and textrm m 2 textrm a 2 textrm rc algorithms for multipoint responses control and pattern synthesis

a flexible array synthesis method using quadratic programming a highly flexible synthesis method for an arbitrary array is proposed to best approximate a desired array pattern in a minimum mean square error sense the basic idea of the technique is to form a quadratic program with its cost function given by the mean square error between the array response and a properly selected pattern described by a known mathematical function this quadratic program can be a constrained or unconstrained optimization problem depending on the requirements of the desired array pattern in formulating the quadratic program no assumption has been made on the gain phase response or characteristics of the individual array elements therefore one can synthesize an array of arbitrary shape to any appropriate pattern with the characteristic of the array elements taken into consideration as long as one is able to model the array accurately the proposed method is used to synthesize arrays of different shapes linear as well as planar arrays including rectangular and circular planar arrays using a chebyshev polynomial or zero function as a design template to illustrate the feasibility of the proposed method

the asymptotic chebyshev coefficients for functions with logarithmic endpoint singularities mappings and singular basis functions abstract when a function is singular at the ends of its expansion interval its chebyshev coefficients a n converge very poorly we analyze three numerical strategies for coping with such singularities of the form 1 x k log 1 x and in the process make some modest additions to the theory of chebyshev expansions the first two numerical methods are the convergence improving changes of coordinate x sin 2 y and x tanh ly 1 y 2 1 2 we derive the asymptotic chebyshev coefficients in the limit n for both mappings and for the original untransformed chebyshev series for the original function the asymptotic approximation for general k is augmented by the exact chebyshev coefficients for integer k numerical tests show that the sine mapping is excellent for k 1 increasing the rate of convergence to b n o 1 n 4 k 1 although the tanh transformation is guaranteed to be better for sufficiently large n we offer both theoretical and numerical evidence to explain why the sine mapping is usually better in practice sufficiently large n is usually huge instead of mapping one may use a third strategy supplementing the chebyshev polynomials with singular basis functions simple experiments show that this approach is also successful

optimal polarization synthesis of arbitrary arrays with focused power pattern the joint synthesis of the spatial power pattern and polarization of arbitrary arrays is addressed specifically the proposed approach gives the solution to a frequently encountered problem namely the array design i e the determination of the radiating element weightings to achieve a pattern that is arbitrarily upper bounded while its polarization is optimized in a given angular region any state of polarization elliptical circular and linear can be synthesized and there is no restriction regarding the array geometry and element patterns the synthesis problem is rewritten as a convex optimization problem that is efficiently solved using readily available software this ensures the optimality of the proposed solution various numerical results are presented to validate the proposed method and illustrate its potentialities the synthesis of a sequentially rotated array is first addressed then a linear array of equispaced randomly oriented dipoles is considered finally a conformal and a planar array of patches where the mutual coupling effects are considered are synthesized to radiate a linear and a circular polarization

a new approach to the optimization of robust antenna array processors the authors present a method for solving a class of optimization problems with nonsmooth constraints in particular they apply the method to the design of narrowband minimum power antenna array processors which are robust in the presence of errors such as array element placement look direction misalignment and frequency offset they first show that the constrained minimum power problem has a unique global minimum provided that the constraint set is nonempty then it is shown how the design problem derived directly from considerations of the sensitivity of the antenna array processor to errors can be transformed into a quadratic programming problem with linear inequality constraints which can be solved efficiently by the standard active set strategy they also present numerical results for two types of nonsmooth constraints developed to provided robustness these results confirm the effectiveness of the method

a numerical pattern synthesis algorithm for arrays a numerical technique for pattern synthesis in arrays is presented for a given set of elements the technique allows one to find a set of array coefficients that steer the main beam in a given direction and yield sidelobes meeting a specified criterion if such a set of array coefficients exists if the pattern specifications cannot be met with the given elements the algorithm finds the best attainable pattern the advantage of this technique is that it can be used with an arbitrary set of array elements different elements in the array can have different element patterns and the array can have arbitrary nonuniform spacing between elements the synthesis technique is based on adaptive array theory the given array elements are assumed to be used as the elements of an adaptive array the main beam is pointed in the proper direction by choosing the steering vector for that direction and the sidelobes are controlled by introducing a large number of interfering signals at many angles throughout the sidelobe region the algorithm iterates on the interference powers until a suitable pattern is obtained

synthesis of minimax sidelobes for arbitrary arrays this paper presents a synthesis algorithm to minimize the maximum sidelobe level for an arbitrary array similar to dolph chebyshev patterns the minimax array patterns for arbitrary arrays have uniform sidelobes

the complex wavenumber eigenvalues of laplace s tidal equations for oceans bounded by meridians corrigendum an error in the article o connor 1995 was brought to my attention by charles sozou this changes the values of the eigenvalues calculated in table 1 and the geophysical conclusions are modified the introduction and 2 on laplace s tidal equations are correct the mistake occured in 3 when the legendre function expansion was substituted into the set of laplace s tidal equations the term a in the expansion for the tide height in equation 12 must be zero otherwise equation 7 cannot be satisfied i did not see this at the time because i neglected to include the a term when substituting the expansions 12 into equation 7 this is why no a term appears in equation 18 the corrections are to omit the a terms in equations 12 and 17

a rational spectral method for the kdv equation on the half line a petrov galerkin method using orthogonal rational functions is proposed for the korteweg de vries kdv equation on the half line with initial boundary values the nonlinear term and the right hand side term are treated by chebyshev rational interpolation explicitly and the linear terms are computed with the galerkin method implicitly such an approach is applicable using fast algorithms numerical results are presented for problems with both exponentially and algebraically decaying solutions respectively highlighting the performance of the proposed method

a fast poisson solver of arbitrary order accuracy in rectangular regions in this paper we propose a direct method for the solution of the poisson equation in rectangular regions it has an arbitrary order accuracy and low cpu requirements which makes it practical for treating large scale problems the method is based on a pseudospectral fourier approximation and a polynomial subtraction technique fast convergence of the fourier series is achieved by removing the discontinuities at the corner points using polynomial subtraction functions these functions have the same discontinuities at the corner points as the sought solution in addition to this they satisfy the laplace equation so that the subtraction procedure does not generate nonperiodic nonhomogeneous terms the solution of a boundary value problem is obtained in a series form in o n log n floating point operations where n2 is the number of grid nodes evaluating the solution at all n2 interior points requires o n2 log n operations

numerical analysis of singularities in two dimensions part 1 computation of eigenpairs a numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second order elliptic partial differential equations in two dimensions in the vicinity of singular points the singularities may be caused by re entrant corners and abrupt changes in boundary conditions or material properties such singularities are of great interest from the point of view of failure initiation the eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes for this reason failure theories directly or indirectly involve the eigenpairs and their amplitudes this paper addresses the problem of determining the eigenpairs numerically on the basis of the steklov formulation numerical results are presented for several cases importantly the method is applicable to three dimensional cases

pseudospectral methods on a semi infinite interval with application to the hydrogen atom a comparison of the mapped fourier sine method with laguerre series and rational chebyshev expansions the fourier sine with mapping pseudospectral algorithm of fattal et al phys rev e 53 1996 1217 has been applied in several quantum physics problems here we compare it with pseudospectral methods using laguerre functions and rational chebyshev functions we show that laguerre and chebyshev expansions are better suited for solving problems in the interval r 0 for example the coulomb schrodinger equation than the fourier sinemapping scheme all three methods give similar accuracy for the hydrogen atom when the scaling parameter l is optimum but the laguerre and chebyshev methods are less sensitive to variations in l we introduce a new variant of rational chebyshev functions which has a more uniform spacing of grid points for large r and gives somewhat better results than the rational chebyshev functions of boyd j comp phys 70 1987 63

synthesis of planar arrays with elements in concentric rings the circular symmetry of concentric ring arrays becomes an advantage for situations where the radiation patterns are symmetric in the azimuthal direction for planar arrays with elements in an equispaced grid the fast fourier transform can be applied in the calculations involved between the array factor and the source distribution however for circular arrays the elements are in a non equispaced grid making the application of fast computation algorithms difficult therefore two techniques are proposed to synthesize circular arrays the first one provides the control of the pattern by sampling the array factor in this case the fast fourier transform can be applied the windows technique is also utilized to control the sidelobe levels and the ripple structure of the shaped beam the second technique considers the symmetric nature of concentric circular arrays and the control of the array factor is performed imposing some specific points of the pattern

summary of sinc numerical methods sinc approximation methods excel for problems whose solutions may have singularities or infinite domains or boundary layers this article summarizes results obtained to date on sinc numerical methods of computation sinc methods provide procedures for function approximation over bounded or unbounded regions encompassing interpolation approximation of derivatives approximate definite and indefinite integration solving initial value ordinary differential equation problems approximation and inversion of fourier and laplace transforms approximation of hilbert transforms and approximation of indefinite convolutions the approximate solution of partial differential equations and the approximate solution of integral equations methods for constructing conformal maps and methods for analytic continuation indeed sinc are ubiquitous for approximating every operation of calculus

an effective approach for the optimal focusing of array fields subject to arbitrary upper bounds a new approach to the optimal focusing of array fields subject to arbitrary upper bounds is presented the approach formulates the problem as the minimization of a linear function in a convex set unlike other approaches this one guarantees the achievement of the global optimum by using local optimization techniques and can moreover deal with any convex constraint on the unknowns such as near field constraints optimization is performed by two ad hoc developed solution algorithms which exploit the geometrical characteristics of the problem at hand thus leading to extremely effective and computationally efficient numerical codes an extensive numerical analysis has been performed in all cases of linear planar and circular arc arrays the enhanced performance of the proposed technique with respect to the solution algorithms available in the literature fully confirms the effectiveness of the approach

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

numerical solution of singular poisson problems via the sinc galerkin method the galerkin method is used with sinc functions for the basis elements to approximate the solution of the poisson problem particular attention is paid to singular problems a first or higher order partial derivative of the solution is undefined on the boundary the linear system is derived and a spectral decomposition technique of solution is used explicit parameter selections that enhance the performance of the method are given the method is thoroughly tested with the numerical results presented

beampattern synthesis for linear and planar arrays with antenna selection by convex optimization a convex optimization based beampattern synthesis method with antenna selection is proposed for linear and planar arrays conjugate symmetric beamforming weights are used so that the upper and non convex lower bound constraints on the beampattern can be convex thus a mainlobe of an arbitrary beamwidth and response ripple can be obtained this method can achieve completely arbitrary sidelobe levels by minimizing a re weighted objective function based on the magnitudes of the elements in the beamforming weight vector iteratively the proposed method selects certain antennas in an array to satisfy the prescribed beampattern specifications precisely interestingly a sparse array with fewer antennas compared to other methods is produced this method can design non uniformly spaced arrays with inter element spacings larger than one half wavelength without the appearance of grating lobes in the resulting beampattern simulations are shown using arrays of up to a few hundred antennas to illustrate the practicality of the proposed method

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

prolate spheroidal wavefunctions as an alternative to chebyshev and legendre polynomials for spectral element and pseudospectral algorithms prolate spheroidal functions of order zero are generalizations of legendre polynomials which when the bandwidth parameter c 0 oscillate more uniformly on x 1 1 than either chebyshev or legendre polynomials this suggests that compared to these polynomials prolate functions give more uniform spatial resolution further when used as the spatial discretization for time dependent partial differential equations in combination with explicit time marching prolate functions allow a longer stable timestep than legendre polynomials we show that these advantages are real and further that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis however improvements in spatial resolution are at most a factor of 2 approached slowly as n the timestep can be lengthened by several times but not by a factor that grows rapidly with n the prolate basis is not likely to radically expand the range of problems that can be done on a workstation however for production runs on the bleeding edge edge of arithmurgy such as numerical weather prediction the rewards for switching to a prolate basis may be considerable

motivic multiple zeta values and superstring amplitudes the structure of tree level open and closed superstring amplitudes is analyzed for the open superstring amplitude we find a striking and elegant form which allows one to disentangle its expansion into several contributions accounting for different classes of multiple zeta values this form is bolstered by the decomposition of motivic multiple zeta values i e the latter encapsulate the expansion of the superstring amplitude moreover a morphism induced by the coproduct maps the expansion onto a non commutative hopf algebra this map represents a generalization of the symbol of a transcendental function in terms of elements of this hopf algebra the expansion assumes a very simple and symmetric form which carries all the relevant information equipped with these results we can also cast the closed superstring amplitude into a very elegant form

fast pencil beam pattern synthesis of large unequally spaced antenna arrays reducing the computational cost of large array pattern synthesis is attractive in many applications in this paper a fast pencil beam pattern synthesis method for large nonuniform antenna arrays is proposed this method is based on an interpolation in a least square sense and iterative fast fourier transform fft i e interpolate the nonuniform distributed elements into uniform virtual elements and then apply fft to synthesis the uniform array thanks to the efficiency of fft the proposed method is much faster and can handle much larger arrays than the existing methods to guarantee the interpolation accuracy the choice of the interpolation parameters is discussed both linear and planar nonuniform array examples are shown to validate the advantages of the proposed method

harmonic sums and polylogarithms generated by cyclotomic polynomials the computation of feynman integrals in massive higher order perturbative calculations in renormalizable quantum field theories requires extensions of multiply nested harmonic sums which can be generated as real representations by mellin transforms of poincare iterated integrals including denominators of higher cyclotomic polynomials we derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations the analytic continuation of cyclotomic harmonic sums to complex values of n is performed using analytic representations we also consider special values of the cyclotomic harmonic polylogarithms at argument x 1 resp for the cyclotomic harmonic sums at n which are related to colored multiple zeta values deriving various of their relations based on the stuffle and shuffle alge bras and three multiple ar gument relations we also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums basis representations are derived for weight w 1 2 sums up to cyclotomy l 20