rearrangements of functions let g be measurable non negative functions on r and let tf be their equimeasurable symmetric decreasing rearrangements let f r r r be continuous and suppose that the associated rectangle function defined by f r f a c f b d f a d f b c for r x y r2 a x b c y d is non negative then f g d f tf g d where is lebesgue measure the concept of equimeasurable rearrangement is also defined for functions on a more general class of measure spaces and the inequality holds in the general case if f x y x y where is convex and 0 0 then we obtain tf g d g d in particular if x x p 1 p then we find that the operator s tf is a contraction on lp for 1 p

a fast conformal mapping algorithm with no fft abstract an algorithm is presented for the computation of a conformal mapping discretized on a non uniformly spaced point set useful for the numerical solution of many problems of fluid dynamics most existing iterative techniques both those having a linear and those having a quadratic type of convergence rely on the fast fourier transform fft algorithm for calculating a convolution integral which represents the most time consuming phase of the computation the fft however definitely cannot be applied to a non uniform spacing the algorithm presented in this paper has been made possible by the construction of a calculation method for convolution integrals which despite not using an fft maintains a computation time of the same order as that of the fft the new technique is successfully applied to the problem of conformally mapping a closely spaced cascade of airfoils onto a circle which requires an exceedingly large number of points if it is solved with uniform spacing

network based robust h_infty filtering for uncertain linear systems the problem of network based robust hinfin filtering for uncertain linear systems is investigated different from the design of the traditional filter the effects of the network induced delay and data dropout on the performance of a filtering error system are considered the derived criteria for hinfin performance analysis of the filtering error system and filter design are expressed as a set of linear matrix inequalities which can be solved by using convex optimization method numerical examples show the effectiveness of the design method

numerical conformal mapping based on the generalised conjugation operator an iterative procedure for numerical conformal mapping is presented which imposes no restriction on the boundary complexity the formulation involves two analytically equivalent boundary integral equations established by applying the conjugation operator to the real and the imaginary parts of an analytical function the conventional approach is to use only one and ignore the other equation however the discrete version of the operator using the boundary element method bem leads to two non equivalent sets of linear equations forming an over determined system the generalised conjugation operator is introduced so that both sets of equations can be utilised and their least square solution determined without any additional computational cost a strategy largely responsible for the stability and efficiency of the proposed method numerical tests on various samples including problems with cracked domains suggest global convergence although this cannot be proved theoretically the computational efficiency appears significantly higher than that reported earlier by other investigators

a stable richardson iteration method for complex linear systems the richardson iteration method is conceptually simple as well as easy to program and parallelize this makes the method attractive for the solution of large linear systems of algebraic equations with matrices with complex eigenvalues we change the ordering of the relaxation parameters of a richardson iteration method proposed by eiermann niethammer and varga for the solution of such problems the new method obtained is shown to be stable and to have better convergence properties

matrices of sinc methods this paper gives a brief review of sinc methods with emphasis on the matrices of sinc methods a novel procedure is presented based on sinc convolution for solving a poisson problem over a rectangular region although some of the work of gragg may already be applied to the solution of sinc matrix problems this paper also points to new directions of matrix research

an iterative method for conformal mapping abstract in this paper we discuss an iterative method for the calculation of the boundary values of the conformal mapping of a simply connected region g onto a region h with smooth boundary the method is based on a certain riemann hilbert problem it turns out that this problem is the linearized version of a singular integral equation of the second kind hence the method is a newton method whenever the boundaries of g and h are sufficiently smooth its convergence is locally quadratic if g is the unit circle the solution of the linearized problem can be represented explicitly in terms of integral transforms from this one derives a quadratically convergent newton like numerical method that avoids the numerical solution of systems of linear equations and therefore in comparison with other methods based on integral equations is quite economical in terms of computer time and storage requirements

reverse faber krahn and mahler inequalities for the cheeger constant we prove a reverse faber krahn inequality for the cheeger constant stating that every convex body in 2 has an affine image such that the product between its cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle an analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square we also prove a mahler type inequality for the cheeger constant stating that every axisymmetric convex body in 2 has a linear image such that the product between its cheeger constant and the cheeger constant of its polar body is not larger than the same quantity for the square

a fredholm integral equation of the second kind for conformal mapping abstract we present a fredholm integral equation of the second kind for the derivative of the boundary correspondence function of the conformal mapping of a jordan region with piecewise twice differentiable boundary onto the unit disk one of our derivations of the equation leads to a more general regularisation procedure for some singular integral equations of the first kind the integral equation can be solved numerically by applying the fourier method the convergence to in l 2 of the solution of the discrete equation so obtained follows by simple theorems examples show that the equation often yields better numerical results than both a related method proposed by henrici and warschawski s equation

rearrangement inequalities for functionals with monotone integrands the inequalities of hardy littlewood and riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement it is known that these inequalities extend to integrands of the form f u1 um where f is supermodular in particular they hold when f has nonnegative mixed second derivatives i jf for all i j this paper concerns the regularity assumptions on f and the equality cases it is shown here that extended hardy littlewood and riesz inequalities are valid for supermodular integrands that are just borel measurable under some nondegeneracy conditions all equality cases are equivalent to radially decreasing functions under transformations that leave the functionals invariant i e measure preserving maps for the hardy littlewood inequality translations for the riesz inequality the proofs rely on monotone changes of variables in the spirit of sklar s theorem

a new method for computing delay margins for stability of linear delay systems this note is concerned with stability properties of linear time invariant delay systems we consider delay systems of retarded type modeled both as a high order scalar differential difference equation and as a set of first order differential difference equations expressed in state space form we provide a computational method that can be used to compute a delay interval such that the system under consideration is stable for all delay values that lie in the computed interval this method requires computing only the eigenvalues and generalized eigenvalues of certain constant matrices and it can be implemented efficiently based on this method we further state a simple necessary and sufficient condition concerning stability independent of delay for each of the two types of the models

the accuracy of numerical conformal mapping methods a survey of examples and results this paper shows how the geometry of the region affects the conditioning and the accuracy of numerical conformal mapping methods for simply connected regions especially fourier series methods both explicit examples of popular test cases and more general estimates are discussed the severe ill conditioning that is known as the crowding phenomenon is discussed and its effect on a conformally transplanted boundary value problem is illustrated remarks on various numerical methods are included

the newton method for solving the theodorsen integral equation the newton method for the solution of the theodorsen integral equation in conformal mapping is studied one step of this method consists of solving a linear integral equation the solution of which is given explicitly as the result of a riemann hilbert problem quadratic convergence of the newton method is established under certain assumptions whereas in other methods a so called condition with 1 is required to hold our method converges also for 1 we will also present a numerical implementation in which the result of one step of the newton method is approximately by a vector in r2n which can be computed with 2n log n o n multiplications in comparison one step of the newton method for the discrete theodorsen equation requires o n3 multiplications

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

a nonoverlapping domain decomposition method for symm s equation for conformal mapping symm s equation is a first kind integral equation for computing conformal maps of simply connected regions the package confpack solves symm s equation by an indirect boundary element method using an accurate corner representation this solution technique is extended here to include nonoverlapping domain decomposition degrees of freedom are introduced on one or more interfaces and different unknowns are used leading to a system of second kind equations the corresponding linear system can be expressed in schur complement form the accurate treatment of corners is preserved in the new formulation the results of serial and parallel matlab implementations of the new algorithm show significant speedups as the number of unknowns grows

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

iterative methods for discrete nonlinear riemann hilbert problems abstract let d denote the complex unit disk t d and let f t r 2 r be a given function the paper presents iterative methods for the computation of functions w u iv which are holomorphic in d and satisfy the boundary relation f t u t v t 0 for all t t it is shown that an appropriate discretization of this boundary value problem can be obtained by replacing the class of holomorphic functions with the set of polynomials of degree not exceeding n and imposing the boundary condition at a grid of 2n uniformly distributed points on the circle t the proposed iterative methods for the solution of the discrete as well as the nondiscrete problems converge quadratically in a neighborhood of the solution error estimates for the iterative solutions are given in the scale of the sobolev spaces wk2

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

the numerical solution of theodorsen integral equation abstract in this paper a numerical solution of the theodorsen integral equation is studied using an adequate quadrature formula which eliminates the singularity of the integral part of the theodorsen equation we obtain a system of nonlinear algebraical equations this system may be solved using a jacobi type method and the procedure can be easily implemented using a programming language with parallel facilities examples are given using ada eval and parallaxis a convergence result is established 1999 elsevier science ltd all rights reserved

an exact method for the stability analysis of time delayed linear time invariant lti systems a general class of linear time invariant systems with time delay is studied recently they attracted considerable interest in the systems and control community the complexity arises due to the exponential type transcendental terms in their characteristic equation the transcendentality brings infinitely many characteristic roots which are cumbersome to elaborate as evident from the literature a number of methodologies have been suggested with limited ability to assess the stability in the parametric domain of time delay this study offers an exact structured and robust methodology to bring a closure to the question at hand ultimately we present a unique explicit analytical expression in terms of the system parameters which not only reveals the stability regions pockets in the domain of time delay but it also declares the number of unstable characteristic roots at any given pocket the method starts with the determination of all possible purely imaginary resonant characteristic roots for any positive time delay to achieve this a simplifying substitution is used for the transcendental terms in the characteristic equation it is proven that the number of such resonant roots for a given dynamics is finite each one of these roots is created by infinitely many time delays which are periodically distributed an interesting property is also claimed next that the root crossing directions at these locations are invariant with respect to the delay and dependent only on the crossing frequency these two unique findings facilitate a simple and practical stability method which is the highlight of the work

on the nonlinear brascamp lieb inequality we prove a nonlinear variant of the general brascamp lieb inequality our proof consists of running an efficient or tight induction on scales argument which uses the existence of gaussian near extremizers to the underlying linear brascamp lieb inequality lieb s theorem in a fundamental way a key ingredient is an effective version of lieb s theorem which we establish via a careful analysis of near minimizers of weighted sums of exponential functions instances of this inequality are quite prevalent in mathematics and we illustrate this with some applications in harmonic analysis

applications of numerical conformal mapping abstract the application of conformal mapping methods to the solution of free surface flow problems is considered methods of numerical conformal mapping based on fourier series are extended to handle efficiently problems with time dependent boundaries they are shown to be practicable only for moderately distorted geometries extensions of the menikoff zemach method to breaking geometries are presented these latter methods are robust at quite large distortions but degrade prematurely in time dependent problems at amplitudes smaller than achieved by our recent vortex methods

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

volume inequalities of convex bodies from cosine transforms on grassmann manifolds abstract the l p cosine transform on grassmann manifolds naturally induces finite dimensional banach norms whose unit balls are origin symmetric convex bodies in r n reverse isoperimetric type volume inequalities for these bodies are established which extend results from the sphere to grassmann manifolds

on the computation of the conjugate trigonometric rational function and on a related splitting problem we prove several splitting theorems related to the following one every continuous real trigonometric rational function of t with period 2 pi can be written as the real part of a rational function of z e it that is analytic for z leqq 1 as an application we obtain immediately a method for computing the conjugate function of a real trigonometric rational function

on a rational transfer function based approach to cal h _ infty filtering design for time delay linear systems this paper introduces a new procedure for h filter design of time delay linear systems a finite order lti system called comparison system is defined in such a way that its h norm is proven to be strongly related to the one of the time delay system differently from what can be viewed as a common feature of filter design methods available in the literature to date the one presented here treats the filtering design of time delay systems with classical numerical routines based on riccati equation and h theory of lti systems the proposed algorithm is simple efficient and easy to implement illustrative examples are solved and discussed in order to put in evidence the most relevant properties of the theoretical results furthermore a practical application is presented

uber die anzahl der losungen der diskreten theodorsen gleichung discretization of the theodorsen integral equation t yields the discrete theodorsen equation t d a system of 2n nonlinear equations a so called condition may be fulfilled it is known that t has exactly one continuous solution this solution gives the boundary correspondence of the normalized conformal map of the unit disc onto a given domaing it is also known that t d has one and only one solution if 0 and any domaing that fulfills an condition we will construct a domaing 1 in the neighbourhood ofg that fulfills a max 1 k condition such that t d forg 1 has an infinite number of solutions the underlying idea of the construction of those domains allows also to give important new facts about iterative methods for the solution of t d even in the case 1

linear systems with commensurate time delays stability and stabilization independent of delay notions of exponential stability independent of delay and stabilizability independent of delay are developed for the class of delay differential systems of the retarded type with commensurate time delays various criteria for exponential stability independent of delay with a given order are specified in terms of matrices whose entries are functions of a single real parameter and polynomials in one variable whose coefficients are functions of a single real parameter sufficient conditions and a necessary condition based on local stabilizability are given for stabilizability independent of delay using state feedback with commensurate time delays constructive methods for determining a stabilizing feedback are also presented the last part of the paper deals with a standard type of observer and regulator with the requirement that the closed loop system be stable independent of delay

action of a scattering map on weighted sobolev spaces in the plane abstract we consider a scattering map that arises in the approach to the scattering theory for the davey stewartson ii equation and show that the map is an invertible map between certain weighted l 2 sobolev spaces

brief paper robust h filtering for a class of uncertain linear systems with time varying delay this paper deals with the problem of delay dependent robust h filtering for uncertain linear systems with time varying delay two cases of time varying delays are fully considered one is the time varying delay being continuous uniformly bounded while the other is the time varying delay being differentiable uniformly bounded with delay derivative bounded by a constant a stable linear filter is designed to ensure that the filtering error system is asymptotically stable with a prescribed level of h noise attenuation based on a new integral inequality delay dependent sufficient conditions for the existence of such a filter are established in terms of linear matrix inequalities lmis through deriving these conditions neither model transformation nor bounding technique for cross terms is employed moreover the relationship between the sufficient conditions for the two cases of time varying delay is disclosed finally two examples are also given to illustrate the effectiveness of the proposed methodology

multimodal method and conformal mapping for the scattering by a rough surface the scattering of a wave from a periodic rough surface in two dimensions is considered the proposed method is based on the use of a conformal mapping and of the multimodal admittance method the use of the conformal mappings induces a spatially varying refractive index and transforms the boundary condition on the rough surface into a boundary condition on a flat surface then the multimodal admittance method reduces this problem to a riccati equation for the modal admittance matrix which is solved numerically with a magnus mobius scheme the method is shown to converge exponentially with the number of fourier modes the method also allows to find geometries having trapped modes or quasi trapped modes at a given frequency by looking at singularities or quasi singularities of this riccati equation besides a simple perturbation expansion based on a small roughness approximation is developed

mean width inequalities for isotropic measures in this paper we establish barthe s mean width inequalities for continuous isotropic measures by a direct approach rather than using the brascamp lieb inequality the following results are obtained among the convex hulls of the support of isotropic mea sures on s n 1 the regular simplex inscribed in the euclidean unit ball has maximal norm in the dual situation there is a reverse result for their polar bodies moreover the case of even isotropic measures is also investigated

global boundary flattening transforms for acoustic propagation under rough sea surfaces this paper introduces a conformal transform of an acoustic domain under a one dimensional rough sea surface onto a domain with a flat top this non perturbative transform can include many hundreds of wavelengths of the surface variation the resulting two dimensional flat topped domain allows direct application of any existing acoustic propagation model of the helmholtz or wave equation using transformed sound speeds such a transform model combination applies where the surface particle velocity is much slower than sound speed such that the boundary motion can be neglected once the acoustic field is computed the bijective one to one and onto mapping permits the field interpolation in terms of the original coordinates the bergstrom method for inverse riemann maps determines the transform by iterated solution of an integral equation for a surface matching term rough sea surface forward scatter test cases provide verification of the method using a particular parabolic equation model of the helmholt

discretized versions of newton type iterative methods for conformal mapping abstract it is shown that for nearly circular regions discretized versions of the newton type methods of wegmann 1978 and hubner 1986 converge locally to fixed points convergence is linear the rates can be determined approximately for several standard regions the dominant operator acts only on a subspace of high order harmonics therefore under conditions of smoothness and or symmetry convergence can be much faster the fixed points satisfy a discrete version of lavrentev s variational principle therefore the resulting approximations for conformal mapping are essentially as accurate as wittich s approximation for the conjugation operation

shadows of convex bodies it is proved that if c is a convex body in rn then c has an affine image c of nonzero volume so that if p is any 1 codimensional orthogonal projection pca 1 cl n l n it is also shown that there is a pathological body k all of whose orthogonal projections have volume about vx times as large as iki n 1 n 0 introduction the problems discussed in this paper concern the areas of shadows orthogonal projections of convex bodies and to a lesser extent the surface areas of such bodies if c is a convex body in irn and 0 a unit vector p c will denote the orthogonal projection of c onto the 1 codimensional space perpendicular to 0 volumes and areas of convex bodies and their surfaces will be denoted with i 1 the relationship between shadows and surface areas of convex bodies is expressed in cauchy s well known formula for each n e n let vn be the volume of the n dimensional euclidean unit ball and let a an be the rotationally invariant probability on the unit sphere sn cauchy s formula states that if c is a convex body in rn then its surface area is ac i s p6ci dcr o vni jn i the classical isoperimetric inequality in rn states that any body has surface area at least as large as a euclidean ball of the same volume the first section of this paper is devoted to the proof of a local isoperimetric inequality showing that all bodies have large shadows rather than merely large surface area or average shadow the principal motivation for this result is its relationship to a conjecture of vaaler and the important problems surrounding it this theorem and its connection with vaaler s conjecture are described at the beginning of 1 an important role is played in the theory of convex bodies by the so called projection body of a convex body it is easily seen by considering polytopes received by the editors september 12 1989 1980 mathematics subject classification 1985 revision primary 52a20 52a40 the author was supported in part by nsf dms 8807243 1991 american mathematical society 0002 9947 91 1 00 25 per page

the borell ehrhard game a precise description of the convexity of gaussian measures is provided by sharp brunn minkowski type inequalities due to ehrhard and borell we show that these are manifestations of a game theoretic mechanism a minimax variational principle for brownian motion as an application we obtain a gaussian improvement of barthe s reverse brascamp lieb inequality

on fornberg s numerical method for conformal mapping it is shown that fornberg s method 1 is a discretized version of an iterative process which is essentially the same we introduced and studied in 7 8 the main difference consists in the determination of the shift u k in each iterative step while we give an explicit representation of u k fornberg determines u k from a linear equation which involves a certain integral operator r we prove some properties and estimates for this operator which explain several experimental results reported in 1 especially the fact that the conjugate gradient method is so efficient the results of 8 concerning the convergence can be applied also for fornberg s method finally we give estimates for the numerical error

stationary and almost stationary iterative k l step methods for linear and nonlinear systems of equations we first discuss the solution of a fixed point equationx x of a frechet differentiable self mapping by iterative methods of the general form x m left phi left sum limits i 0 m 1 x i gamma i m 1 right sum limits j 0 m 1 x j beta j m right frac 1 beta m m m 1 2 ldots defined by two infinite nonsingular upper triangular matricesb j m andc j m with column sums 1 we show that two such methods defined byb c and tilde b tilde c respectively are in a certain sense equivalent if and only if c 1 b tilde c 1 tilde b in particular hat b c 1 and the unit matrix replacingc define an equivalent so called semiiterative method we introduce k l step methods as those whereb andc have upper bandwidthk andl 1 respectively they require storing max k l previous iterates only for stationary methodsb andc have toeplitz structure except for their first row which is chosen such that the column sum condition holds an euler method which may require to store all iterates is equivalent to a stationary k l step method if and only if the underlying conformal mapg is a rational function of the formg w w 0 k k v 0 v l 1 w 1 by choosingg withg 1 1 such that for some onto the exterior of some continuums known to contain the eigen values of the frechet derivative of one obtains a feasible procedure for designing a locally converging stationary k l step method custom made for a set of problems in the case x tx d with a linear operatort where one wants to solve a linear system of equations we show that the residual polynomials of a stationary semiiterative method are generalized faber polynomials with respect to a particular weight function using another weight function leads to what we call almost stationary methods the classical chebyshev iteration is an example of such a method we define equivalent almost stationary k l step methods and give a corresponding convergence result

on some relations among numerical conformal mapping methods abstract gaier and gutknecht have shown that many numerical methods for producing the conformal map from the unit disk to a simply connected region share a common theoretical basis as solutions of nonlinear integral equations arising from the hilbert transform of a function of the boundary correspondence we give a brief presentation of this classification and extend it somewhat to include some equations for the inverse correspondence such as those of menikoff and zemach noble and schwarz christoffel the use of explicit maps and the method of bisshopp are also brought into this framework an example illustrating the use of explicit maps is given

numerical conformal mapping methods based on function conjugation abstract a unifying treatment of methods for computing conformal maps from the unit disk onto a jordan region is presented integral and integro differential equations involving the conjugation operator for the boundary correspondence function are first derived using an arbitrary auxiliary function having certain properties then various iteration methods for solving these equations are described in this generality so that the basic ideas become manifest specific methods are then treated as examples of the general theory among them are in particular the successive conjugation methods of theodorsen melentiev and kulisch timman and friberg the projection method of bergstrom and the newton methods of vertgeim wegmann and hubner which make use of the easy construction of the solutions of riemann hilbert problems many of these methods are treated in greater generality than in the literature the connections with the methods of fornberg menikoff zemach chakravarthy anderson and challis burley are also outlined

on a reverse form of the brascamp lieb inequality we prove a reverse form of the multidimensional brascamp lieb inequality our method also gives a new way to derive the brascamp lieb inequality and is rather convenient for the study of equality cases

network based h_ infty filtering for discrete time systems this correspondence is concerned with network based h filtering for discrete time systems the output signals of the system under consideration are transmitted to the filter through a constraint communication network which usually leads to network induced delays and packet dropouts by introducing a logic data packet processor to choose the newest data signal from the network to actuate the filter network induced delays and packet dropouts are modeled as a markov chain taking values in a finite set as a result the filter to be designed is modeled as a markov jumping linear filter by introducing some slack matrix variables in terms of prob ability identity a less conservative bounded real lemma brl is derived to ensure that the filtering error system is stochastically stable with a pre scribed h level based on this brl suitable h filters are designed by employing a cone complementary approach a practical example on the leslie model about some certain pest s structured population dynamics is given to show the effectiveness of the proposed approach

a formula for optimal integration in h 2 abstract the weights j of the optimal integration formula q j jf zj in h2 for given integration points zj are the exact integrals of the cardinal functions in the corresponding formula for optimal evaluation by writing these cardinal functions as sums of their principal values we very easily obtain a closed formula for the weights in the case of real zj s this formula makes explicit a series formula of wilf we compare numerically the accuracy of the optimal formula with that of some well known integration formulae for points equidistant on a circle of radius r the formula allows an alternate derivation of a formula obtained by golomb we give also the barycentric formula for optimal evaluation with these points as well as an experimentally stable sequence of radii r for integrating with an increasing number of points

solving theodorsen s integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods we investigate several iterative methods for the numerical solution of theodorsen s integral equation the discretization of which is either based on trigonometric polynomials or function families with known attenuation factors all our methods require simultaneous evaluations of a conjugate periodic function at each step and allow us to apply the fast fourier transform for this in particular we discuss the nonlinear jor iteration the nonlinear sor iteration a nonlinear second order euler iteration the nonlinear chebyshev semi iterative method and its cyclic variant under special symmetry conditions for the region to be mapped onto we establish local convergence in the case of discretization by trigonometric interpolation and give simple formulas for the optimal parameters e g the underrelaxation factor and the asymptotic convergence factor weaker related results for the general non symmetric case are presented too practically our methods extend the range of application of theodorsen s method and improve its effectiveness strikingly

complex variable and regularization methods of inversion of the laplace transform in this paper three methods are derived for approximating f given its laplace transform g on 0 oo i e fo f t exp st dt g s assuming that g e l 0 oo the first method is based on a sinc like rational approximation of 9 the second on a sinc solution of the integral equation fo0 f t exp st dt g s via standard regularization and the third method is based on first converting fo0 f t exp st dt g s to a convolution integral over r and then finding a sinc approximation to f via the application of a special regularization procedure to solve the fourier transform problem we also obtain bounds on the error of approximation which depend on both the method of approximation and the regularization parameter

an iterative method for solving nonlinear riemann hilbert problems abstract a geometrically motivated linearization is used to construct an iterative method for solving nonlinear boundary value problems for holomorphic functions the nondiscrete method is quadratically convergent results of some test calculations are reported

numerical algorithms based on analytic function values at roots of unity let f z be an analytic or meromorphic function in the closed unit disk sampled at the n th roots of unity based on these data how can we approximately evaluate f z or f m z at a point z in the disk how can we calculate the zeros or poles of f in the disk these questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations partial differential equations and large scale linear algebra we analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of cauchy integrals we then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension finally we highlight the power of rational in comparison with polynomial approximations for some of these problems

an iterative method for the conformal mapping of doubly connected regions an iterative method is described for the determination of the conformal mapping of a circular annulus rq z q z 1 onto a doubly connected region g with smooth boundary curves the method is analogous to the method of 7 for simply connected regions it requires in each step the solution of a riemann hilbert problem on rq this problem can be solved explicitly in terms of a generalized conjugation operator k a degeneracy of this problem leads in a natural way to an adjustment of the parameter q in each step if the boundary curves of g admit parametrizations by functions with second derivatives which are holder continuous with exponent 0 1 then the method converges locally in the sobolev space w of 2 periodic absolutely continuous functions with square integrable derivatives the order of convergence is at least 1 the convergence is quadratic if the have lipschitz continuous second derivatives for the numerical implementation of the method the operator k can be approximated by a wittich type method which can be performed very effectively using fft a calculation with n 2m grid points on the computer requires storage of the order o n and computing time o mn the results of some test calculations are reported and compared with results of calculations with the method of theodorsen garrick

synthesis of non rational controllers for linear delay systems this paper addresses the control of linear delay systems using non rational controllers the structure of the controller is chosen so as to copy the structure of the plant reproducing the delays in the state and in the output the resulting stabilization and performance design problems are entirely expressed as linear matrix inequalities although the design inequalities are based on delay independent stability conditions the overall design is delay dependent in the sense that the controller makes use of the delay parameter of the plant this parameter is assumed to be constant yet arbitrary using non rational controllers we overcome the main difficulty faced when designing rational controllers for linear delay systems which is to incorporate in the design problem the matrix multiplier used to prove stability with respect to the delayed part of the system we illustrate the paper with several examples and provide extensive comparisons with existent results

near optimality of the sinc approximation near optimality of the sinc approximation is established in a variety of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity

optimal linear filtering under parameter uncertainty this paper addresses the problem of designing a guaranteed minimum error variance robust filter for convex bounded parameter uncertainty in the state output and input matrices the design procedure is valid for linear filters that are obtained from the minimization of an upper bound of the error variance holding for all admissible parameter uncertainty the results provided generalize the ones available in the literature to date in several directions first all system matrices can be corrupted by parameter uncertainty and the admissible uncertainty may be structured assuming the order of the uncertain system is known the optimal robust linear filter is proved to be of the same order as the order of the system in the present case of convex bounded parameter uncertainty the basic numerical design tools are linear matrix inequality lmi solvers instead of the riccati equation solvers used for the design of robust filters available in the literature to date the paper that contains the most important and very recent results on robust filtering is used for comparison purposes in particular it is shown that under the same assumptions our results are generally better as far as the minimization of a guaranteed error variance is considered some numerical examples illustrate the theoretical results

gaussian kernels have only gaussian maximizers a gaussian integral kernel g x y on rn x rn is the exponential of a quadratic form in x and y the fourier transform kernel is an example the problem addressed here is to find the sharp bound of g as an operator from l p rp to l q rn and to prove that the l p rn functions that saturate the bound are necessarily gaussians this is accomplished generally for 1 q in some special cases besides greatly extending previous results in this area the proof technique is also essentially different from earlier ones a corollary of these results is a fully multidimensional multilinear generalization of young s inequality

convergence proofs and error estimates for an iterative method for conformal mapping the iterative method as introduced in 8 and 9 for the determination of the conformal mapping of the unit disc onto a domaing is here described explicitly in terms of the operatork which assigns to a periodic functionu its periodic conjugate functionk u it is shown that whenever the boundary curve ofg is parametrized by a function with lipschitz continuous derivative dot eta ne 0 then the method converges locally in the sobolev spacew of 2 periodic absolutely continuous functions with square integrable derivative if is in a holder classc 2 the order of convergence is at least 1 if is inc l 1 withl 1 0 1 then the iteration converges inc l for analytic boundary curves the convergence takes place in a space of analytic functions for the numerical implementation of the method the operatork can be approximated by wittich s method which can be applied very effectively using fast fourier transform the sobolev norm of the numerical error can be estimated in terms of the numbern of grid points it iso n 1 l if is inc l 1 ando exp n 2 if is an analytic curve the number in the latter formula is bounded by logr wherer is the radius of the largest circle into which can be extended analytically such that z 0 for z r the results of some test calculations are reported

the evaluation of the conjugate function of a periodic spline on a uniform mesh abstract we present an o n log n algorithm for the simultaneous evaluation at all n points of a uniform mesh of a function conjugate to a periodic spline the algorithm is based on the attenuation factor theory and makes use of the fft it can be applied to other function classes with known attenuation factors also for the approximation by functions conjugate to splines of odd degree we state several optimality criterions and error bounds

invariances in variance estimates we provide variants and improvements of the brascamp lieb variance inequality which takes into account the invariance properties of the underlying measure this is applied to spectral gap estimates for log concave measures with many symmetries and to non interacting conservative spin systems

multivariable nyquist criteria root loci and pole placement a geometric viewpoint classical control theory is concerned with the topics in our title in the context of single input single output systems there is now a large and growing literature on the extension of these ideas to the multiinput multioutput case this development has posed certain difficulties some due to the intrinsic nature of the problem and some we would argue due to an inadequate reflection on what the multivariable problem calls for in this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory a rather satisfactory generalization of the nyquist criterion will be described and a clear analog of the asymptotic properties of the root locus will be obtained in the multiparameter case however an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains this calls for a new ingredient a compactification of the space of gains and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement this problem is difficult in the multivariable case but by establishing a correspondence with a classical set of problems in geometry we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field

on the fast solution of a linear system arising in numerical conformal mapping abstract one step of the newton method for the discretized theodorsen equation in conformal mapping requires the solution of a certain 2n 2n system application of the gaussian algorithm costs o n3 arithmetic operations a o we present an algorithm which reduces the problem to the solution of three n n linear toeplitz systems these systems can be solved in o n log2n a o this is also the amount of work required by the whole algorithm

robust stability of time delay systems with an uncertain time delay constant frequency domain methods are described for testing the robust stability of a class of time delay systems with an uncertain time constant both parametric and non parametric uncertainties are considered our technique determines without a sweep of time delay the maximum time delay constant which preserves the robust stability

best constant in sobolev trace inequalities on the half space using a mass transportation method we study optimal sobolev trace inequalities on the half space and prove a conjecture made by escobar in 1988 about the minimizers

explicit nearly optimal linear rational approximation with preassigned poles this paper gives explicit rational functions for interpolating and approximating functions on the intervals 1 1 0 oc and oc oc the rational functions are linear in the functions to be approximated and they have preassigned poles the error of approximation of these rationals is nearly as small as the error of best rational approximation with numerator and denominator polynomials of the same degrees regions of analyticity are described which make it possible to tell a priori the accuracy which we can expect from this type of rational approximation 1 introduction and summary in this paper we attempt to give a constructive affirmative answer to each of the following questions

h sub spl infin memoryless control with an spl alpha stability constraint for time delay systems an lmi approach focuses on the design of h sub spl infin memoryless state feedback controllers satisfying some spl alpha stability constraints on the closed loop poles for a class of linear systems with delayed state subject to disturbance inputs a sufficient condition for feasibility is derived in terms of linear matrix inequalities furthermore we also consider a particular optimization problem the maximal bound allowed on time delay such that the prescribed level on disturbance attenuation and the spl alpha stability constraints are still preserved this problem is converted into a linear matrix inequality lmi optimization problem and we give a suboptimal bound on the delay

numerical methods based on whittaker cardinal or sinc functions this paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis described in this paper are methods of interpolation and approximation of functions and their derivatives quadrature the approximate evaluation of transforms hilbert fourier laplace hankel and mellin and the approximate solution of differential and integral equations the methods have many advantages over classical methods which use polynomials as bases in addition all of the methods converge at an optimal rate if singularities on the boundary of approximation are ignored

numerical conformal mapping onto a rectangle with applications to the solution of laplacian problems abstract let f be the function which maps conformally a simply connected domain onto a rectangle r so that four specified points on are mapped respectively onto the four vertices of r in this paper we consider the problem of approximating the conformal map f and present a survey of the available numerical methods we also illustrate the practical significance of the conformal map by presenting a number of applications involving the solution of laplacian boundary value problems

the flatness theorem for nonsymmetric convex bodies via the local theory of banach spaces let l be a lattice in rn and k a convex body disjoint from l the classical flatness theorem asserts that then w k l the l width of k does not exceed some bound depending only on the dimension n this fact was later found relevant to questions in integer programming kannan and lovasz 1988 showed that under the above assumptions w k l cn2 where c is a universal constant banaszczyk 1996 proved that w k l cn 1 log n if k has a centre of symmetry in the present paper we show that w k l cn3 2 for an arbitrary k it is conjectured that the exponent 3 2 may be replaced by 1 perhaps at the cost of a logarithmic factor we prove that for some naturally arising classes of bodies

bessel transforms and rational extrapolation a numerical method is developed which handles the bessel transform of functions having slow rates of decrease the method replaces the bessel transform by a related damped transform for which the sinc quadrature rule provides an efficient and accurate approximation it is then shown that the value of the original bessel transform can be obtained from the damped transform by extrapolation with the thiele algorithm

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem