a local smoothing estimate for the schrodinger equation we show that the schrodinger operator eit is bounded from to for all a 2n 1 2 1 q 2 q and q2 4 n 1 this is almost sharp with respect to the sobolev index we also show that the schrodinger maximal operator sup0 s0 if and only if it is bounded from to when s 2s0 a corollary is that sup0 3 4

fast multipole method for scattering from an arbitrary pec target above or buried in a lossy half space the fast multipole method fmm was originally developed for perfect electric conductors pecs in free space through exploitation of the spectral properties of the free space green s function in the work reported here the fmm is modified for scattering from an arbitrary three dimensional 3 d pec target above or buried in a lossy half space the near terms in the fmm are handled via the original method of moments mom analysis wherein the half space green s function is evaluated efficiently and rigorously through application of the method of complex images the far fmm interactions which employ a clustering of expansion and testing functions utilize an approximation to the green s function dyadic via real image sources and far field reflection dyadics the half space fmm algorithm is validated through comparison with results computed via a rigorous mom analysis further a detailed comparison is performed on the memory and computational requirements of the mom and fmm algorithms for a target in the vicinity of a half space interface

profile decompositions of fractional schrodinger equations with angularly regular data abstract we study the fractional schrodinger equations in r 1 d d 3 of order d d 1 2 under the angular regularity assumption we prove linear and nonlinear profile decompositions which extend the previous results 9 to data without radial assumption as applications we show blowup phenomena of solutions to mass critical fractional hartree equations

a fast volume integral equation solver for electromagnetic scattering from large inhomogeneous objects in planarly layered media a newly developed iterative method the stabilized biconjugate gradient fast fourier transform bcgs fft method is applied to simulate electromagnetic scattering from large inhomogeneous objects embedded in a planarly layered medium in this fast solver the weak form formulation is applied to obtain a less singular discretization of the volume electric field integral equation several techniques are utilized to speed up the dyadic green s function evaluation to accelerate the operation of the dyadic green s function on an induced current i e the green s operation the green s function is split into convolutional and correlational components so that fft can be applied the cpu time and memory cost of this bcgs fft method is o nlogn and o n respectively where n is the number of unknowns significantly more efficient than the method of moments mom as a result this method is capable of solving large scale electromagnetic scattering problems in a planarly layered background a large scale scattering problem in a layered medium with more than three million unknowns has been solved on a sun ultra 60 workstation with 1 2 gbytes memory

fast algorithm for electromagnetic scattering by buried conducting plates of large size this letter presents a fast algorithm for electromagnetic scattering by buried conducting plates of large size and arbitrary shape using the conjugate gradient cg method combined with the fast fourier transform fft due to the use of fft in handling the cyclic convolutions related to toeplitz matrices the sommerfeld integrals evaluation for the buried scattering problem which is usually time consuming has been reduced to a minimum the memory required for this algorithm is of the order n the number of unknowns and the computational complexity is of order n sub iter nlogn n sub iter is the iteration number n sub iter spl lt n for large problems

guaranteed convergence method of analysis of the scattering by an arbitrarily oriented zero thickness pec disk buried in a lossy half space in this paper a guaranteed convergence method for the accurate and efficient analysis of the electromagnetic scattering by an arbitrarily oriented zero thickness perfectly electrically conducting pec disk buried in a lossy half space is presented an electric field integral equation efie in the vector hankel transform domain is obtained by expanding the unknown surface current density in cylindrical harmonics a convergence of exponential type is achieved by means of galerkin s method with expansion functions reconstructing the expected physical behavior of the bm n th harmonic at the center and the edge of the disk the obtained coefficient matrix can be quickly computed since its elements can be reviewed as the sum of single integrals efficiently evaluable by means of an analytical acceleration technique and double integrals of exponentially decaying functions

periodic orbits and semiclassical form factor in barrier billiards using heuristic arguments based on the trace formulas we analytically calculate the semiclassical two point correlation form factor for a family of rectangular billiards with a barrier of height irrational with respect to the side of the billiard and located at any rational position p q from the side to do this we first obtain the asymptotic density of lengths for each family of periodic orbits by a siegel veech formula the result open image in new window obtained for these pseudo integrable non veech billiards is different but not far from the value of 1 2 expected for semi poisson statistics and from values of open image in new window obtained previously in the case of veech billiards

the autocorrelation function of a pseudointegrable system abstract the autocorrelation function of a pseudointegrable system is considered the system consists of billiards on plane surface formed out of three squares arranged in an l shape this system has the important property of being constructed from copies of an integrable subsystem the single square the motion can be decomposed into a continuous and a discrete part the unpredictability in the system being associated with the latter a discrete autocorrelation function is calculated and its decay properties investigated structure found in this autocorrelation function is associated with the continued fraction expansion of the ratio of velocity components for repeating continued fractions such as the golden mean the autocorrelation function exhibits a selfsimilar structure for the general case of a randomly chosen velocity ratio we derive the time dependence of the number of occurences of large autocorrelation values which differs from the behavior in integrable and chaotic systems

on the solutions of a class of dual integral equations occurring in diffraction problems it is shown that there exists a category of two dimensional diffraction problems which can be put into a standard form of dual integral equations these diffraction problems include diffraction of electromagnetic waves by a finite strip a finite slit the diffraction of scalar or vector elastic waves by a rigid strip or crack etc a general method for solving such dual integral equations is given by the artifice of constructing a set of functions of compact support biorthogonal to another given set of functions the sufficient conditions for a given dual integral equations to be solvable in this manner are also determined hence the method forms a complement to the weiner hopf method to illustrate the method solutions are obtained for a bench mark problem the diffraction of light by a finite perfectly conducting strip or equivalently the diffraction of sh waves by a crack comparison with results obtained by others for low intermediate and high frequencies show the utility and accuracy of the method for the entire range of frequencies both the near field and the far field are obtained the latter is shown to correspond to the fraunhoffer diffraction pattern for high frequency it is also shown that for the equivalent crack problem the stress intensity factor sif fluctuates rapidly with changes in the angle of incidence for high frequencies thus making the sif especially sensitive to angle of incidence at high frequencies

a new high efficient analysis of the scattering by a perfectly conducting rectangular plate the analysis of the scattering by a perfectly conducting rectangular plate by means of galerkin s method in the spectral domain with products of chebyshev polynomials of first and second kind multiplied by their orthogonal weights as basis functions is fast convergent even for scatterers size of some wavelengths but leads to the numerical evaluation of infinite double integrals of oscillating and slowly decaying functions the aim of this paper is the introduction of a new analytical technique that allows to write such integrals as combinations of very quickly converging integrals

analytically solvable dynamical systems which are not integrable abstract the central two fold aim of this paper is to expose a class of nonintegrable systems whose solutions can be analytically expressed in terms of computationally tractable algorithms and to explicitly construct a demonstrably meaningful general solution for one member of this class toward this end information theoretic results are reviewed which show that dynamical systems of null metric entropy have general solutions of null algorithmic complexity hence all systems exhibiting null metric entropy including ergodic only and mixing only systems have orbits which can be specified by algorithmically meaningful analytic expressions whose information content is very much less than that of the orbits they describe in consequence we may quite legitimately define the class of systems having null metric entropy ot be algorithmically integrable called a integrable herein after establishing the existence of a integrability this paper turns to the problem of exposing a specific nonintegrable but nonetheless a integrable system whose algorithmically meaningful solution can be explicitly derived the billiard moving within a plane polygon having the shape of a 60 120 rhombus is shown to be a simple example indeed the whole class of billiards moving in polygons whose angles are all rational multiples of have null metric entropy and are therefore a integrable systems moreover all rational billiards appear to have explicitly derivable solutions as is discussed herein and in a recent preprint by bjorn birnir because rational billiards are dense in the set of all plane billiards including chaotic ones the present paper suggests that a integrable billiards may provide a long sought analytic route to chaos finally it is pointed out that null entropied a integrable systems are perhaps the broadest category of systems which can reasonably be termed integrable for systems having positive metric entropy are truly nonintegrable due to the presence of orbits which are their own simplest description and their own fastest computer

numerical quadratures for singular and hypersingular integrals abstract we present a procedure for the design of high order quadrature rules for the numerical evaluation of singular and hypersingular integrals such integrals are frequently encountered in solution of integral equations of potential theory in two dimensions unlike integrals of both smooth and weakly singular functions hypersingular integrals are pseudo differential operators being limits of certain integrals as a result standard quadrature formulae fail for hypersingular integrals on the other hand such expressions are often encountered in mathematical physics see for example 1 and it is desirable to have simple and efficient quadrature formulae for them the algorithm we present constructs high order quadratures for the evaluation of hypersingular integrals the additional advantage of the scheme is the fact that each of the quadratures it produces can be used simultaneously for the efficient evaluation of hypersingular integrals hilbert transforms and integrals involving both smooth and logarithmically singular functions this results in significantly simplified implementations the performance of the procedure is illustrated with several numerical examples

periodic orbits of an almost integrable system the authors enumerate and classify all the periodic orbits of the pi 3 rhombus billiard the billiard flow for this system resides on a classically invariant surface of genus two after transforming the rhombus billiard trajectories on an exactly equivalent set of trajectories on a barrier billiard with barrier to gap ratio equal to 2 they present a criterion facilitating the complete enumeration of periodic billiard trajectories having enumerated the trajectories they classify them distinctly and provide the underlying number theoretic rationale for the same their analysis involves a polar construction which facilitates the determination of lengths of periodic orbits and the areas of the bands in which they occur it is clear from the analysis that there are no isolated periodic trajectories implying that all the periodic orbits close after an even number of reflections from the boundaries these results are used to study the family counting function f x i e the number of families of periodic orbits of length less than x as a function of x

tm scattering by perfectly conducting polygonal cross section cylinders a new surface current density expansion retaining up to the second order edge behavior in the analysis of scattering by perfectly conducting cylinders with polygonal cross section by means of surface integral operator formulations fast convergence can be achieved by expanding the surface current density on each side with basis functions factorizing the correct behavior of the fields on the wedges usually the factorized edge behavior is chosen to be coincident with the first order behavior prescribed by meixner s theory however it could not be the correct one and consequently the convergence of the method becomes increasingly slow as the theoretical behavior differs from the real one this phenomenon is particularly notable when one or more of the predicted singularities unexpectedly disappear to overcome this problem in this work the analysis of tm scattering is made by introducing a new expansion devised so that only the first two terms are responsible for the reconstruction of the singularities while the remaining part of the expansion factorizes the second order edge behavior actually the proposed expansion outperforms the one introduced by the authors in a previous work factorizing the first order edge behavior even when this is the correct one

band limited functions lp convergence we consider the set bp q functions of lp tl whose fourier spectrum lies in i2 i2 we prove that the prolate spheroidal wave func tions constitute a basis of this space if and only if 4 3 p 4 the result is obtained as a consequence of the analogous problem for the spherical bessel functions the proof rely on a weighted inequality for the hubert transform

nystrom type method in three dimensional electromagnetic diffraction by a finite pec rotationally symmetric surface time harmonic electromagnetic wave diffraction by a perfectly electrically conducting pec finite rotationally symmetric surface located in free space is investigated the problem is split to independent azimuth orders and reduced to the sets of coupled hypersingular and singular integral equations ies for the surface current components these ies are discretized by the nystrom type method of discrete singularities using the interpolation type quadrature formulas from the solutions of corresponding matrix equations the near and the far field patterns are obtained the presented method has guaranteed convergence for arbitrary not axially symmetric primary field

hankel transform domain analysis of open circular microstrip radiating structures a full wave analysis of a circular disk conductor printed on a dielectric substrate backed by a ground plane is presented the method is based on the galerkin s method applied in the hankel transform domain and is numerically quite efficient several numerical examples are compared with available experimental data the agreement is found to be quite good and the validity of the present method is confirmed

endpoint bilinear restriction theorems for the cone and some sharp null form estimates recently wolff 25 obtained a nearly sharp l 2 bilinear restriction theorem for bounded subsets of the cone in general dimension we obtain the endpoint of wolff s estimate and generalize to the case when one of the subsets is large as a consequence we are able to deduce some nearly sharp l p null form estimates

a sobolev estimate for the adjoint restriction operator in this note we consider the adjoint restriction estimate for hypersurface under additional regularity assumption we obtain the optimal h s l q estimates and their mixed norm generalizations as applications we prove some weighted strichartz estimates for the propagator varphi rightarrow e it delta alpha 2 varphi alpha 0

scattering by polygonal cross section dielectric cylinders at oblique incidence we analyze the scattering by polygonal cross section lossless dielectric cylinders illuminated by an obliquely incident plane wave the problem is formulated in terms of a system of surface integral equations opportunely devised so as to be valid for objects with edges and incident angles including the total reflection limit angle by means of galerkin s method in the spectral domain with analytically fourier transformable expansion functions factorizing the correct edge behaviors and continuity conditions of the unknowns convergence of exponential type is achieved and the coefficients of the scattering matrix are reduced to single integrals that can be efficiently evaluated numerical results for both near field and far field parameters are presented showing the quick convergence of the method even when applied to composite cylindrical objects

full wave analysis and optimization of a tara like shield assisted paraboloidal reflector antenna using a nystrom type method using the recently developed method based on the rigorous theory of singular and hypersingular integral equations ies and nystrom type discretization a paraboloidal reflector assisted with a conical shield is investigated to decrease the computation time we propose a new method of calculation of the modal green s functions in the quasi optical range the far and near fields of the studied reflector are analyzed in the transmission and reception cases and reveal fine features that escape asymptotic analysis elementary numerical optimization of the shield assisted paraboloidal antenna is performed

radial weights and mixed norm inequalities for the disc multiplier in this paper we show that the l2 weighted inequalities with radial weights of the multiplier for the ball are controlled by the maximal function defined by averages on arbitrary rectangles this solves the case of radial weights of a problem proposed by e stein in 1978 it also gives an elegant and simple proof of the boundedness of the multiplier for the ball on the spaces of mixed norm lradp lang2 for 2n n 1 p 2n n 1 obtained by a cordoba and by g mockenhaupt additional work is presented in order to extend these mixed norm estimates to the end points 2n n 1 and 2n n 1 in the sharp form of restricted weak type inequalities

endpoint strichartz estimates and global solutions for the nonlinear dirac equation abstract we prove endpoint strichartz estimates for the klein gordon and wave equations in mixed norms on the polar coordinates in three spatial dimensions as an application global wellposedness of the nonlinear dirac equation is shown for small data in the energy class with some regularity assumption for the angular variable

non universal spectral rigidity of quantum pseudo integrable billiards we obtain the dyson mehta statistic for pseudo integrable billiards and show that it is non universal with a universal trend also that this trend is similar to the one for integrable billiards we present a formula based on exact semiclassical calculations and the proliferation law of periodic orbits which gives rigidity for the entire range of l to consolidate our theory we discuss several examples finding complete agreement with the numerical results and also the underlying fundamental reasons for the non universality

numerical evaluation of hypersingular integrals abstract boundary integral equations which employ integrals which exist only if defined in the cauchy principal value sense or as the hadamard finite part are currently used with success to solve many two and three dimensional problems of applied mechanics we will recall definitions and main properties of these integrals examine some numerical approaches for their evaluation and present several new results

remarks on the gauss quadrature rule for a particular class of finite part integrals the method of finite part hadamard type or hypersingular integrals is already a standard approach in applied numerical methods in engineering and mainly in acoustics fluid dynamics elasticity and fracture mechanics here we study the convergence of the non classical possessing a negative node outside the integration interval as well as a negative weight corresponding to this node gauss legendre quadrature rule for finite part integrals on the interval 0 1 with the weight function 1 y and we prove its validity for integrands possessing a continuous first derivative the case of integrands possessing a higher order derivative is also considered and the rate of convergence is established these results are based on the derivation of a bound for the negative node in the above quadrature rule and further on the theory of convergence of the classical gauss quadrature rule numerical results verifying the results of this paper are also displayed the present results can be generalized to the gauss laguerre similarly non classical quadrature rule for finite part integrals on the interval 0 as well as to the two special but important in engineering applications non classical gaussian quadrature rules for cauchy type principal value integrals and mangler type finite part integrals have been proposed by tsamasphyros and dimou

method of analytical regularization in the analysis of axially symmetric excitation of imperfect circular disk antennas abstract this work presents accurate numerical method and computational results in the modeling of the axisymmetric radiation of resistive and thin dielectric circular disk on substrate antennas to simulate resistive and thin dielectric disk features we use two side generalized boundary conditions gbc the analysis is based on the singular integral equations ies and the method of analytical regularization mar exploiting a galerkin method gm with judicious basis functions which convert ies into the fredholm second kind infinite matrix equations unlike previously known numerical techniques this guarantees convergence and enables us to compute the solution with controlled accuracy even in sharp resonances numerical results demonstrate the effect of disk losses and transparency on the fundamental radiation phenomena and help improve antenna characteristics such as radiation efficiency

diffraction of electromagnetic plane wave by a rectangular plate and a rectangular hole in the conducting plate the problems of diffraction of an electromagnetic plane wave by a perfectly conducting rectangular plate and its complementary problem diffraction by a rectangular hole in an infinite conducting plate are rigorously solved using the method of the kobayashi 1931 potential the mathematical formulation involves dual integral equations derived from the potential integrals and the boundary condition on the plane where a plate or hole is located the weighting functions in the potential integrals are determined by applying the properties of the weber schafheitlin s integrals and the solution is obtained in the form of a matrix equation illustrative computations are given for the far diffracted field pattern and the current densities induced on the plate the results of the patterns are compared with the results obtained from physical optics po and the physical theory of diffraction ptd the agreement is fairly good particularly with the ptd solutions

barrier billiards a simple pseudo integrable system a hard line segment barrier lies in a periodic cell in the plane parallel to one edge and half its length this constitutes perhaps the simplest pseudo integrable system and the classical motion of a particle within it admits concise analysis the hitting and missing of the barrier defines a quasi periodic doubly periodic sequence with the period ratio depending on the invariant gradient of the trajectory the consequent motion is expressed in terms of its continued fraction expansion rational gradients correspond to periodic orbits which can be classified in full quadratic irrational gradients lead to trajectories of calculable fractal dimension or power law correlation decay

exponential energy growth in a fermi accelerator an unbounded energy growth of particles bouncing off two dimensional 2d smoothly oscillating polygons is observed notably such billiards have zero lyapunov exponents in the static case for a special 2d polygon geometry a rectangle with a vertically oscillating horizontal bar we show that this energy growth is not only unbounded but also exponential in time for the energy averaged over an ensemble of initial conditions we derive an a priori expression for the rate of the exponential growth as a function of the geometry and the ensemble type we demonstrate numerically that the ensemble averaged energy indeed grows exponentially at a close to the analytically predicted rate namely the process is controllable

thin dielectric sheet simulation by surface integral equation using modified rwg and pulse bases the conventional thin dielectric sheet surface integral equation proposed by harrington and mautz is reviewed due to its simplicity and inadequency for certain problems a new surface integral equation is proposed to fill the gap this new formulation takes both tangential and normal currents into account so that it can handle problems with high and low permittivity and grazing incident waves based on the d field formulation inhomogeneity and surface charge can be treated correctly modelling the geometry by triangles this surface integral equation is discretized and is suitable for the method of moments to solve rao wilson glisson and pulse functions are employed simultaneously for basis and testing functions and so the moment solution has full descriptions of polarization current in the thin dielectric layer several examples are examined by both analytical and other published numerical methods these results show excellent agreement and demonstrate the validity and effectiveness of this new integral equation

on radial weights for the spherical summation operator abstract in this paper we give a sufficient condition for radial weights such that the spherical summation operator is bounded on l 2 r n x dx n 2 when applied to power weights x x x these conditions are also necessary moreover in the general case we shall show that our condition is almost necessary furthermore we give an example which shows that the conjecture in anderson proc amer math soc 83 no 2 1981 269 275 fails to hold

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

scattering by a circular cylinder buried under a slightly rough surface the cylindrical wave approach an analytical numerical technique for the scattering problem of a plane wave by a cylinder buried under a rough surface based on the cylindrical wave approach is presented the rough deviations on the interface are dealt with by means of the small perturbation method reflection and transmission coefficients are evaluated in a first order approximation and fields are the sum of a zeroth order solution relevant to flat surface and first order perturbation fields associated to the surface roughnesses numerical results are obtained through an exact evaluation of the spectral integrals giving results both in near and far field regions for the case of an interface with sinusoidal profile the approach is validated through comparisons with the literature and results showing the effect of geometrical and physical parameters on the scattered field are reported

linear adjoint restriction estimates for paraboloid we prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices this result generalizes the paraboloid restriction estimate in radial case from shao rev mat iberoam 25 2009 1127 1168 as well as the result from miao et al proc ams 140 2012 2091 2102 as an application we show a local smoothing estimate for a solution of the linear schrodinger equation under the assumption that the initial datum has additional angular regularity

quadrature rules for weakly singular strongly singular and hypersingular integrals in boundary integral equation methods we present n point gauss gegenbauer quadrature rules for weakly singular strongly singular and hypersingular integrals that arise in integral equation formulations of potential problems in domains with edges and corners the rules are tailored to weight functions with algebraic endpoint singularities related to the geometrical singularities of the domain each rule has two different expressions involving legendre functions and hypergeometric functions respectively numerical examples amply demonstrate the accuracy and stability of the proposed algorithms application to the solution of a singular integral equation is exemplified

locally corrected multidimensional quadrature rules for singular functions accurate numerical integration of singular functions usually requires either adaptivity or product integration both interfere with fast summation techniques and thus hamper large scale computations this paper presents a method for computing highly accurate quadrature formulas for singular functions which combine well with fast summation methods given the singularity and the n nodes we first construct weights which integrate smooth functions with order k accuracy then we locally correct a small number of weights near the singularity to achieve order k accuracy on singular functions as well the method is highly efficient and runs in o nk 2d n log 2 n time and o k 2d n space we derive precise error bounds and time estimates and confirm them with numerical results which demonstrate the accuracy and efficiency of the method in large scale computations as part of our implementation we also construct a new adaptive multidimensional product gauss quadrature routine with an effective error

exponentially converging nystro spl uml m s methods for systems of singular integral equations with applications to open closed strip or slot loaded 2 d structures this paper concerns the fast highly accurate and exponentially convergent solution of three sets of integral equations pertaining to two dimensional generalized microstrip or microslot structures the analysis relies on nystroumlm s method with due regard to all singular integrals and slowly converging series that appear in the kernels the sophisticated treatment and efficient computation of such series and integrals is of paramount importance entailing the use of advanced techniques as a result of our specialized treatment the discretization procedures developed herein a fully account for both the singular nature of the kernels and the singularities of the solution at the edges b obviate the need for taking inner products with testing functions c appear to converge exponentially versus matrix size and d enable expressing all matrix elements via single finite sums detailed numerical examples and case studies illustrate the simplicity flexibility and remarkable efficiency of the algorithms

generalized gaussian quadrature rules for systems of arbitrary functions a numerical algorithm is presented for the construction of generalized gaussian quadrature rules originally introduced by s karlin and w studden over three decades ago the quadrature rules to be discussed possess most of the desirable properties of the classical gaussian integration formulae such as positivity of the weights rapid convergence mathematical elegance etc the algorithm is applicable to a wide class of functions including smooth functions not necessarily polynomials as well as functions with end point singularities such as those encountered in the solution of integral equations complex analysis potential theory and several other areas the performance of the algorithm is illustrated with several numerical examples

periodic orbits and a correlation function for the semiclassical density of states through the rule that there is one quantum state per planck cell in phase space classical mechanics supplies an average semiclassical density of states for a system with bound motion can it supply more refined averages such as the correlation function for the density of states i shall argue that for the two extreme cases of integrable classical motion and ergodic classical motion information can indeed be obtained on not quite this but a closely related correlation function the results for the two types of motion are markedly different

analysis of a circular microstrip disk antenna with a thick dielectric substrate the problem of a circular microstrip disk excited by a probe is solved using rigorous analysis the disk is assumed to have zero thickness and the current on the probe is taken to be uniform using vector hankel transforms the problem is formulated in terms of vector dual integral equations from which the unknown current can be solved for due to the singular nature of the current distribution arising from probe excitation the direct application of galerkin s basis function expansion method gives a slowly convergent result therefore the singular part of the current is removed since the singularity is known a priori the unknown current to be solved for is then regular and tenable to galerkin s method of analysis it is shown that this analysis agrees with the single mode approximation when the dielectric substrate layer is thin and that it deviates from the single mode approximation when the substrate layer is thick excellent agreement of both the computed real and imaginary parts of the input impedance with experimental data is noted the radiation patterns and the current distributions on the disk are also presented

radial multipliers and restriction to surfaces of the fourier transform in mixed norm spaces in this article we revisit some classical conjectures in harmonic analysis in the setting of mixed norm spaces l rad p l ang 2 left mathbb r n right we produce sharp bounds for the restriction of the fourier transform to compact hypersurfaces of revolution in the mixed norm setting and study an extension of the disc multiplier we also present some results for the discrete restriction conjecture and state an intriguing open problem

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces