ground state solutions for choquard equations with hardy littlewood sobolev upper critical growth and potential vanishing at infinity abstract in this paper we investigate the following non autonomous choquard equation u v x u i u 2 k x f u 2 u 2 2 u k x f u in r n where n 3 i is the riesz potential of order 0 n 2 n n 2 is the upper critical exponent due to the hardy littlewood sobolev inequality the functions v k c r n r may vanish at infinity and the function f c r r is noncritical and f t 0 t f s d s based on variational methods and the hardy type inequalities using the mountain pass theorem and the nehari manifold approach we prove the existence of ground state solution

nonlinear schrodinger equations without compatibility conditions on the potentials abstract we study the existence of nonnegative solutions and ground states to nonlinear schrodinger equations in r n with radial potentials and super linear or sub linear nonlinearities the potentials satisfy power type estimates at the origin and at infinity but no compatibility condition is required on their growth or decay rates at zero and infinity in this respect our results extend some well known results in the literature and we also believe that they can highlight the role of the sum of lebesgue spaces in studying nonlinear equations with weights

preconditioned conjugate gradient methods for solving electromagnetic problems it is shown that appropriately chosen preconditioners can significantly improve the convergence rate of the conjugate gradient cg algorithm as applied to electromagnetic problems preconditioners are constructed for the problems of scattering from frequency selective surfaces and scattering from infinite cylinders

operator algebras with contractive approximate identities weak compactness and the spectrum abstract we continue our study of operator algebras with contractive approximate identities cais by presenting a couple of interesting examples of operator algebras with cais which in particular answer questions raised in previous papers in this series for example about whether roughly speaking weak compactness of an operator algebra or the lack of it can be seen in the spectra of its elements

existence and concentration of positive solutions for a system of coupled saturable schrodinger equations abstract it is considered a saturable system of coupled schrodinger equations with asymptotically linear nonlinearities the bound states solutions with finite energy are proved to exist and have a concentration behavior under hypotheses involving the coupling and saturation parameters the proof is based on variational methods

revisit to sign changing solutions for the nonlinear schrodinger poisson system in r3 abstract in this paper we investigate the existence of solutions for the nonlinear schrodinger poisson system with zero mass by introducing some new ideas we prove via the constraint variational method and the quantitative deformation lemma that the system has a sign changing solution

reduction of the number of unknowns in large stacked planar structures by a fringe aperture formulation for scattering problems comprising a combination of planar structures the total number of unknowns may be significantly reduced if an aperture formulation is employed rather than a patch formulation the rationale behind using the aperture formulation is based on the recognition that the decay rate of the scattered aperture field is independent of the size of the scatterer therefore any scatterer may be surrounded by an aperture of fixed width over which an integral equation is formulated the area of this aperture is proportional to the perimeter of the scatterer rather than its area and it becomes much smaller compared with the entire scatterer area as the size of the scatterer increases hence the reduction in the number of independent unknowns a truncation criterion for the finite aperture is determined via a numerical study of the aperture field behavior for various angles of incidence in addition the a priori knowledge of the physical optics component is also taken into account reducing the unknown function to an aperture field component that is the outcome of the remaining fringe current only the total current distribution can be subsequently derived from this field by adding the known physical optics field and invoking the inverse of the green s function in the spectral domain this analysis of isolated planar scatterers results in a spectral scattering matrix representation that is subsequently used for cascading of stacked structures

on the singularity of the full spectral green s dyad the full 3 times 3 spectral dyadic green s function exhibits certain properties that its popular 2 times 2 transversal portion does not in particular it is shown that the dyad is identically singular the singularity being a manifestation of the plane wave properties of each spectral component this singularity has an impact on the use of certain computational algorithms which in general would require an inversion of the dyad other plane wave properties being dual to maxwell s equations are also discussed

a lattice orthogonalization technique for integrated square error minimization a recursive integrated square error minimization technique is developed to solve integral equations occurring in electromagnetic scattering problems the underlying idea is a generalization of the concept of the second minimization step recently proposed by van den berg this is implemented by a structured lattice algorithm similar to the lattice algorithms which emerged in the last decade in signal processing theory

existence of solution for a class of quasilinear schrodinger equation in rn with zero mass abstract in this paper we use variational methods to establish a berestycki lions type result for two classes of quasilinear schrodinger equations in r n in the zero mass situation

an iterative method for solving electrostatic problems the method of steepest descent is applied to the solution of electrostatic problems the relation between this method and the rayleigh ritz galerkin s and the method of least squares is outlined also explicit error formulas are given for the rate of convergence for this method it is shown that this method is also suitable for solving singular operator equations in that case this method monotonically converges to the solution with minimum norm finally it is shown that the technique yields as a by product the smallest eigenvalue of the operator in the finite dimensional space in which the problem is solved numerical results are presented only for the electrostatic case to illustrate the validity of this procedure which show excellent agreement with other available data

towards a unified field theory for classical electrodynamics in this paper we introduce a model which describes the relation of matter and the electromagnetic field from a unitarian standpoint in the spirit of ideas of born and infeld in this model based on a semilinear perturbation of maxwell equations the particles are finite energy solitary waves due to the presence of the nonlinearity in this respect the matter and the electromagnetic field have the same nature finite energy means that particles have finite mass and this makes electrodynamics consistent with the special relativity we analyze the invariants of the motion of the semilinear maxwell equations sme and their static solutions in the magnetostatic case i e when the electric field e 0 and the magnetic field h does not depend on time sme are reduced to the semilinear equation open image in new window where denotes the curloperator f is the gradient of a strictly convex smooth function f r3 r and a r3 r3 is the gauge potential related to the magnetic field h h a due to the presence of the curl operator 1 is a strongly degenerate elliptic equation moreover physical considerations impel f to be flat at zero f 0 0 and this fact leads us to study the problem in a functional setting related to the orlicz space lp lq the existence of a nontrivial finite energy solution of 1 is proved under suitable growth conditions on f the proof is carried out by using a suitable variational framework related to the hodge splitting of the vector field a

the discrete fourier transform method of solving differential integral equations in scattering theory an accurate and efficient numerical method is presented for solving many differential integral equations arising from electromagnetic scattering theory it uses the discrete fourier transform technique to treat both the derivatives and the convolution integrals which often appear in these equations as a consequence this method is extremely simple to implement uses less computer memory than comparable methods and yields accurate predictions the differential integral equation is recast into a periodic form conducive to application of the discrete fourier convolution theorem the differential operators are approximated by appropriate finite difference and discrete convolution operators all these quantities are computed by using the fast fourier transform an approximate solution is obtained by using the conjugate gradient method results are compared to experimental data or analytical solutions for a 3 lambda 3 lambda metal plate where lambda is the wavelength a homogeneous and a layered infinite circular dielectric cylinder and a dielectric sphere the accuracy of the method is further illustrated by comparing predictions with independent measurements by r a ross 1966 on a 2 lambda 1 lambda metal plate at grazing incidence in all cases agreement is excellent

excitation of complex modes of periodic structures using inhomogeneous plane wave scattering in fast and slow wave regions periodic metamaterial unit cells are analyzed by using inhomogeneous plane wave scattering the complex modes of the unit cell are derived using either the poles or the zeroes of the complex reflection coefficient of the fundamental floquet mode the required reflection coefficients are computed by a doubly periodic finite element boundary integral technique the search for the eigenvalues is accelerated by the nonlinear nelder mead method usually pairs of eigenvalues are observed which can be both real or conjugate complex also the eigenvalues correspond either to proper or improper modes by selecting the proper or the improper modes during field computation the other type of modes is removed from the observed response and instead zeroes are found in the reflection coefficient this can explain the absorption behavior of single sided open periodic leaky structures by investigating their reflection coefficient with the proposed inhomogeneous plane wave excitation approach the obtained results for various single sided open periodic unit cells are in good agreement with the results of other methods where in the presented excitation based method there is no need to solve any dispersion equation in the complex plane

on the analysis of frequency selective surfaces using subdomain basis functions the problem of scattering from frequency selective surfaces fsss has been investigated by expanding the unknown current distribution with three different sets of basis functions namely the roof top surface patch and triangular patch the boundary condition on the total electric field on the fss due to this current distribution is tested either by a line integral or by the galerkin procedure this results in an operator equation that can be solved either by a direct matrix inversion method or by an iterative procedure namely the conjugate gradient method cgm the performance of each of these basis and testing functions is evaluated it is found that the roof top and the surface patch basis functions in conjunction with the galerkin testing are superior in computational efficiency to other combinations of basis and testing functions that have been studied comparison of the cpu times on a cray x mp 48 supercomputer in solving the operator equation by the direct matrix inversion method and the cgm is provided frequency responses of free standing periodic arrays of conducting and resistive plates are also presented

a characterization of operator algebras abstract an operator algebra is a uniformly closed algebra of bounded operators on a hilbert space in this paper we give a characterization of unital operator algebras in terms of their matricial norm structure more precisely if a is an l matricially normed space and also an algebra with a completely contractive multiplication and an identity of norm 1 then there is a completely isometric isomorphism of a onto a unital operator algebra indeed the multiplication on a need not be assumed to be associative for this conclusion to follow examples are given to show that the condition on the identity is necessary it follows from the above that the quotient of an operator algebra by a closed two sided ideal with the natural matricial structure is again an operator algebra up to complete isometric isomorphism

a modified gradient method for two dimensional problems in tomography abstract a method for reconstructing the complex index of refraction of a bounded two dimensional inhomogeneous object of known geometric configuration from measured scattered field data is presented this work is an extension of recent results on the direct scattering problem wherein the governing domain integral equation was solved iteratively by a successive over relaxation technique the relaxation parameter was chosen to minimize the residual error at each step convergence of this process was established for indices of refraction much larger than required for convergence of the born approximation for the inverse problem the same technique is applied except in this case both the index of refraction and the field are unknown iterative solutions for both unknowns are postulated with two relaxation parameters at each step they are determined by simultaneously minimizing the residual errors in satisfying the domain integral equation and matching the measured data this procedure retains the nonlinear relation between the two unknowns numerical results are presented for a number of representative two dimensional objects the algorithm is shown to be effective in cases where the iterative solution of the direct problem is rapidly convergent

nonlinear elliptic equations with subhomogeneous potentials abstract we prove the existence of nonnegative symmetric solutions to the semilinear elliptic equation u v y 1 y k u g u in r n where x z y 1 y k r n 0 r n 1 r n k r n with n 3 k 1 n 0 0 and n i 2 for i 0 the nonlinearity g and the potential v are respectively a continuous function not necessarily superlinear at infinity and a positive measurable function not necessarily homogeneous but satisfying a subhomogeneity condition which implies vanishing at infinity and singularity at least at the origin this also yields the existence of nonrotating solitary waves and vortices with a critical frequency for nonlinear schrodinger and klein gordon equations with singular cylindrical potentials

scattering from a periodic array of free standing arbitrarily shaped perfectly conducting or resistive patches a fast fourier transform fft based iterative approach for computing the fields scattered by an infinite array of free standing patches is presented the method is capable of handling patches that are lossy and have arbitrary shape it is useful for analyzing configurations that may not have been analyzed previously though a rectangular fft is used the formulation allows the study of the common triangular array periodicities results for various geometries are presented and are compared with existing results

iterative computational techniques in scattering based upon the integrated square error criterion an iterative technique is developed to rigorously compute the electromagnetic time and frequency domain scattering problems the method is based upon a wave function expansion technique this also includes the integral representation techniques in which the electromagnetic field equations and causality conditions are satisfied analytically while the boundary conditions or the constitutive relations have to be satisfied in a computational manner the latter is accomplished by an iterative minimization of the integrated square error for the solution of an integral equation it is shown how to obtain optimum convergence some numerical results pertaining to a number of representative problems illustrate the numerical advantages and disadvantages of the iterative method

convergence criteria for the extended matrix version of the spectral iteration technique em scattering an extended matrix version of the spectral iteration technique sit has been reported recently where it was shown to offer the possibility of significantly improving the convergence rate for homogeneous dielectric or conducting scatterers by introducing a taper it is shown that there are good grounds for introducing such a taper and that it can sometimes significantly improve the convergence of the method for example if a 64 64 grid size allows with a taper convergence in five iterations whereas a 42 42 grid size requires 23 iterations a noticeable improvement in total computer time is effected also there are cases where a taper must be introduced if the sit method is to converge at all without the application of a full orthogonalization procedure the technique requires a correct choice of arbitrary values in the extended matrix and guidelines are presented for such a choice

a concise conjugate gradient computation of plate problems with many excitations one of the major difficulties in the application of the conjugate gradient algorithm for the analysis of electromagnetic scattering problems is the necessity to carry out the calculation separately for each incident wave in the approach suggested rather than handling the incident waves directly a class of possible excitations is represented by a set of strip type basis functions for these functions convergence is predictable and rapid because the majority of the strips are located away from the edges of the scatterer this choice also facilitates the use of the physical optics approximation as a good initial guess once the solutions for all the unit basis functions over the body are known they can be combined to synthesize the solution for any excitation using the weighting coefficients associated with the expansion of the incident field numerical examples are given and they demonstrate the substantial savings achieved by adopting this approach for the analysis of multiple excitations

a generalization of van den berg s integral square error iterative computational technique for scattering van den berg shows a way of using spectral iteration si within a scheme that shows good convergence his method can be reinterpreted as a global expansion of the field quantities in terms of a set of basis functions it is shown here that the orthogonalization of these basis functions can considerably improve convergence rates this involves extra storage requirements but only a negligible decrease in speed of computation

a filtered convolution method for the computation of acoustic wave fields in very large spatiotemporal domains the full wave computation of transient acoustic fields with sizes in the order of 100x100x100 wavelengths by 100 periods requires a numerical method that is extremely efficient in terms of storage and computation iterative integral equation methods offer a good performance on these points provided that the recurring spatiotemporal convolutions are computed with a coarse sampling and relatively few computational operations this paper describes a method for the numerical evaluation of very large scale four dimensional convolutions that employs a fast fourier transformation and that uses a sampling rate close to or at the limit of two points per wavelength and per period to achieve this the functions involved are systematically filtered windowed and zero padded with respect to all relevant coordinates prior to sampling the method is developed in the context of the neumann iterative solution of the acoustic contrast source problem for an inhomogeneous medium the implementation of the method on a parallel computer is discussed the obtained numerical results have a relative root mean square error of a few percent when sampling at two points per wavelength and per period further the results prove that the method enables the computation of transient fields in the order of the indicated size

application of conjugate gradient method for static problems involving conductors of arbitrary shape in this paper two implementations of the conjugate gradient method cgm for the solution of problems in electrostatics involving conductors of arbitrary shapes have been discussed the first method uses a least squares approximation for the computation of the pertinent integral operator and is referred to as lsd a second implementation referred to as point matching discretisation pmd effects considerable saving in the computer time since it uses the midpoint rule for the integral arising in lsd both these techniques require o n storage where n is the number of patches used to model the conductor further a matrix interpretation of the present formulation has been derived this has facilitated the comparison of the techniques described in this paper with the well known method of moments mom formulation and has led to better understanding of the convergence of the results using illustrative examples of canonical square and circular discs and arbitrary shape a pyramid mounted on a cube the convergence of and the computer time for both the implementations have been investigated it has been shown that both the techniques yielded monotonically convergent results for all the examples considered and that the lsd offers better estimate of the capacitance than pmd with lower number of patches

existence of ground state sign changing solutions for a class of generalized quasilinear schrodinger maxwell system in r3 abstract in this paper we study the existence of ground state sign changing solutions for the following generalized quasilinear schrodinger maxwell system d i v g 2 u u g u g u u 2 v x u g u g u k x f u x r 3 g 2 u x r 3 where g c 1 r r v x and k x are positive continuous functions and is a positive parameter by making a change of variable as u g 1 v and g u 0 u g t d t we obtain one ground state sign changing solution v g 1 u by using some new analytical skills and non nehari manifold method furthermore the energy of v g 1 u is strictly larger than twice that of the ground state solutions of nehari type we also establish the convergence property of v g 1 u as the parameter 0 our results improve and generalize some results obtained by chen and tang 2016 zhu et al 2016

ground state solutions for a class of choquard equations with potential vanishing at infinity abstract this paper is dedicated to studying the following nonlinear choquard equation u v x u r n q y f u y x y d y q x f u u d 1 2 r n where n 3 0 n v c r n 0 q c r n 0 f c r r and f t 0 t f s d s by combining the non nehari manifold approach with some new inequalities we prove that the above equation has a ground state solution in the case when v x 0 as x where the strict monotonicity condition on f is not required moreover by using perturbation method we obtain the existence of a least energy solution in the zero mass case i e v 0 where f satisfies the condition that f t 0 0 for some t 0 r instead of the usual ambrosetti rabinowitz type condition these results extend the ones in alves figueiredo and yang 2015 1 and some related literature

existence of bounded variation solutions for a 1 laplacian problem with vanishing potentials abstract in this work it is studied a quasilinear elliptic problem in the whole space r n involving the 1 laplacian operator with potentials which can vanish at infinity the euler lagrange functional is defined in a space whose definition resembles b v r n it is proved the existence of a nonnegative nontrivial bounded variation solution and the proof relies on a version of the mountain pass theorem without the palais smale condition to lipschitz continuous functionals

existence of a least energy nodal solution for a schrodinger kirchhoff equation with potential vanishing at infinity in this paper we study a class of nonlocal schrodinger kirchhoff problems involving only continuous functions using a minimization argument and a quantitative deformation lemma we find a least energy nodal or sign changing solution to this problem moreover when the problem presents symmetry we show that it has infinitely many nontrivial solutions

sign changing solutions for a fractional kirchhoff equation abstract using a minimization argument and a quantitative deformation lemma we establish the existence of least energy sign changing solutions for the following nonlinear kirchhoff problem a b u 2 s u v x u k x f u in r 3 where a b 0 are constants s 0 1 s is the fractional laplacian v k are continuous positive functions allowed for vanishing behavior at infinity and f is a continuous function satisfying suitable growth assumptions moreover when the nonlinearity f is odd we obtain the existence of infinitely many nontrivial weak solutions not necessarily nodals

a technique for organizing large moment calculations for use with iterative solution methods through a specific choice of evenly spaced basis and testing functions one can construct a moment matrix formulation wherein significant redundancy is embodied in the matrix a scheme that exploits this redundancy to effect substantial computer storage reduction in an iterative field solution such as the conjugate gradient method is described the expense incurred is a modest increase in computation time the redundancy results from translational similarity features of specific classes of geometries and leads to matrix equations that may be interpreted as discrete convolutions for illustration the analysis here is carried out on a planar scatterer but the strategy can be applied to spherical or cylindrical scatterers

a weak form of the conjugate gradient fft method for plate problems a number of electromagnetic field problems for planar structures can be formulated in terms of a hypersingular integral equation in which a grad div operator acts on a vector potential the vector potential is a convolution of the free space green s function and some surface current density over the domain of interest a weak form of this integral equation is obtained by testing it with subdomain basis functions defined over the plate domain only as a next step the vector potential is expanded in a sequence of subdomain basis functions and the grad div operator is integrated analytically over the plate domain only for the problem of electromagnetic scattering by a plate the method shows excellent numerical performance the numerical difficulties encountered in some previous conjugate gradient fast fourier transform cgfft methods have been eliminated

array decomposition method for the accurate analysis of finite arrays presented in this paper is a fast method to accurately model finite arrays of arbitrary three dimensional elements the proposed technique referred to as the array decomposition method adm exploits the repeating features of finite arrays and the free space green s function to assemble a nonsymmetric block toeplitz matrix system the toeplitz property is used to significantly reduce storage requirements and allows the fast fourier transform fft to be applied in accelerating the matrix vector product operations of the iterative solution process each element of the array is modeled using the finite element boundary integral fe bi technique for rigorous analysis consequently we demonstrate that the complete lu decomposition of the matrix system from a single array element can be used as a highly effective block diagonal preconditioner on the larger array matrix system this rigorous method is compared to the standard fe bi technique for several tapered slot antenna tsa arrays and is demonstrated to generate the same accuracy with a fraction of the storage and solution time

infinitely many solutions for quasilinear schrodinger equation with critical exponential growth in rn abstract this paper shows the existence of infinitely many solutions for the quasilinear equations of the form 0 1 n u v x u n 2 u n u 2 u k x u q 2 u h u x r n where n u is the n laplacian operator n 3 0 1 q n k l r n n n q and h is an odd continuous function having critical exponential growth the potential function v x c r n and 0 inf x r n v x sup x r n v x using mountain pass theorem and some special techniques we demonstrate that there exists 0 0 such that problem 0 1 admits infinitely many high energy solutions provided that 0 0

removal of surface related wave phenomena the marine case removal of the effects of the free surface from seismic reflection data is an essential preprocessing step before prestack migration the problem can be formulated by means of rayleigh s reciprocity theorem which leads to an integral equation of the second kind for the desired pressure field that does not include these free surface effects this integral equation can be solved numerically both in the spatial domain and in the double radon domain solving the integral equation in the double radon domain has the advantage of reducing the computation time significantly since the kernel of the integral equation becomes dominant diagonally two methods are proposed to solve the integral equation direct matrix inversion and a recursive subtraction of the free surface multiples using a neumann series both methods have been developed and tested on a synthetic data set which was computed with the help of an independent forward modeling scheme

on the electromagnetic scattering from infinite rectangular grids with finite conductivity the development of a new numerical technique for the analysis of electromagnetic scattering from a rectangular wire or strip mesh according to this technique the conjugate gradient method cgm and the fast fourier transform fft are used to solved for the aperture fields and the induced current densities the method is efficient because 1 it makes use of the fast fourier transform 2 it converts a set of integrodifferential equations into a set of algebraic equations and 3 it does not require inversion of any matrices unlike the moment method the spectral iteration approach sit and the averaged boundary conditions method this technique can be used for arbitrary size and spacing between adjacent strips the equivalent radius principle is used to determine the reflection coefficient from a wire mesh results examples and comparisons of this method with other methods support the validity of the algorithm

compactness and existence results for the p laplace equation abstract given 1 p n and two measurable functions v r 0 and k r 0 r 0 we define the weighted spaces w u d 1 p r n r n v x u p d x l k q l q r n k x d x and study the compact embeddings of the radial subspace of w into l k q 1 l k q 2 and thus into l k q l k q l k q as a particular case we consider exponents q 1 q 2 q that can be greater or smaller than p our results do not require any compatibility between how the potentials v and k behave at the origin and at infinity and essentially rely on power type estimates of their relative growth not of the potentials separately we then apply these results to the investigation of existence and multiplicity of finite energy solutions to nonlinear p laplace equations of the form p u v x u p 2 u g x u in r n 1 p n where v and g u with u fixed may be vanishing or unbounded at zero or at infinity both the cases of g super and sub p linear in u are studied and in the sub p linear case nonlinearities with g 0 0 are also considered

multiple solutions for a class of quasilinear schrodinger equations in ℝn in this work we study the existence of multiple solutions to a class of quasilinear schrodinger equation pu v x up 2u p u2 u2 2u g u x rn where pu div up 2 u is the p laplacian operator and p 2 n 12 is a parameter the potential v c n is positive and bounded in n g u is odd and continuous using symmetric mountain pass lemma we obtain infinitely many solutions to schrodinger equation

a least square iterative technique for solving time domain scattering problems an alternative method for the numerical solution of time domain integral equations pertaining to three dimensional transient scattering problems is developed and applied to two representative acoustic scattering configurations it is based on the iterative minimization of an integrated square error for the cases considered the method gives more accurate results than the frequently used marching on in time method within comparable computation times

generalized quasilinear asymptotically periodic schrodinger equations with critical growth this paper is concerned with the following quasilinear schrodinger equations div g 2 u u g u g u u 2 v x u f x u g u g u 2 2 g u x r n u x 0 x r n where n 3 2 2 n n 2 g u 0 u g t d t and 0 is a parameter by using a change of variable the quasilinear equation is reduced to a semilinear one whose associated functional is well defined in h 1 r n we establish the existence of positive solutions for this problem by using the mountain pass theorem in combination with the concentration compactness principle under appropriate assumptions on v x and f x u recent results from the literature are improved and extended

a spectral iteration technique for analyzing scattering from arbitrary bodies part ii conducting cylinders with h wave incidence the stacked two dimensional spectral iteration technique sit introduced in an accompanying paper is further developed and extended to the case of an h polarized field incident on an arbitrary conducting cylinder as before the method takes advantage of the simplicity with which the relationships between planar sources and fields are expressed in the spectral domain

ground states of degenerate quasilinear schrodinger equation with vanishing potentials abstract in this paper we study the existence of nontrivial ground state solutions for the following class of p laplacian type equation div a x u v x x p u p 2 u k x x p f u in r n where 1 p n n 3 n p p e 1 d 1 e p p e n p n d p critical hardy sobolev exponent f has a quasicritical growth v and k are nonnegative potentials the function a satisfies a x u c 0 x p h 0 x u p 1 c 0 1 x p h 1 x u p 1 for any r n a e x r n for any two positive functions h 1 l l o c r n h 0 l p p 1 r n with p p

an approach to symmetrization via polarization we prove that the steiner symmetrization of a function can be approximated in l p r n by a sequence of very simple rearrangements which are called polarizations this result is exploited to develop elementary proofs of many inequalities including the isoperimetric inequality in euclidean space in this way we also obtain new symmetry results for solutions of some varia tional problems furthermore we compare the solutions of two boundary value problems one of them having a polarized geometry and we show some point wise inequalities between the solutions this leads to new proofs of well known functional inequalities which compare the solutions of two elliptic or parabolic problems one of them having a steiner symmetrized geometry the method also allows us to investigate the case of equality in the inequalities roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry

an add on method for the analysis of scattering from large planar structures a so called add on procedure is proposed to deal with the data analysis problem resulting from the collection of scattering data from large planar structures the computations involving of the order of a few thousand unknowns is undertaken in a gradual manner by building up the body from small patches which are added sequentially the procedure is based on an initial expansion of the unknown current distribution into subsectional pulse type basic functions each segment of the scatterer carries an unknown amplitude which is the response to an incident wave however rather than forming a matrix equation for these responses they are computed in a gradual manner where the scatterer is built up from these segments as they are added one at a time at the end of each addition of a segment the result for scattering from a partial body is obtained at each stage the problem solved reflects the size of the small addition only and the solution to an actual partial body is obtained an important feature of this method is its ability to utilize a priori known information on a portion of the scatterer as an initial stage for the economic analysis of the entire structure the process takes into account the interactions between all segments of the body the process proves to be very efficient both in terms of computation time and storage requirements as seen in the computed examples on of the order of 1000 to 6000 unknowns

matrix interpretation of the spectral iteration technique electromagnetic scattering a matrix interpretation of the spectral iteration technique is presented to illustrate improvements in accuracy and convergence for both transverse magnetic and transverse electric waves incident on two dimensional homogeneous scatterers producing solutions identical to a method of moments scheme in this scheme it is possible to improve the convergence rate of the technique by the use of nonphysical green s function terms in the extended matrix these terms result in the generation of a nonphysical field outside the scatterer while still maintaining the correct solution of the current although the problem of nonconvergence has not been entirely overcome a particular taper method used in the examples provided shows an improvement over their nontapered counterparts

standing wave solutions for generalized quasilinear schrodinger equations with critical growth abstract we study the existence of standing wave solutions for generalized quasilinear schrodinger equations involving critical sobolev exponents we use the variational methods to obtain a nonexistence theorem and two existence results

a weak form of the conjugate gradient fft method for two dimensional elastodynamics the problem of two dimensional scattering of elastic waves by an elastic inclusion can be formulated in terms of a domain integral equation in which the grad div operator acts on a vector potential the vector potential is the spatial convolution of a green s function with the product of the density and the displacement over the domain of interest a weak form of the integral equation for the unknown displacement is obtained by testing it with rooftop functions this method shows excellent numerical performance

iterative analysis of finite sized planar frequency selective surfaces with rectangular patches or perforations frequency selective surfaces fss and other periodic gratings are often analyzed under the assumption that they are infinite in extent most existing methods for analyzing periodic structures are based on the use of a floquet type representation of the fields in a unit cell whose dimensions are typically comparable to the wavelength in this work a finite truncated version of an infinite periodic structure is dealt with directly without the benefit of the assumption that the structure is periodic this in turn requires the handling of a large number of unknowns and makes it difficult to solve the problem using conventional matrix methods two different iteration approaches to solving the finite fss problem are discussed in the paper both of which employ the spectral domain formulation the first of these employs the spectral iteration technique and the second one uses the conjugate gradient cg iteration algorithm convergence characteristics of both of these methods are investigated and the results are reported

critical fractional p laplacian problems with possibly vanishing potentials abstract we obtain nontrivial solutions of a critical fractional p laplacian equation in the whole space and with possibly vanishing potentials in addition to the usual difficulty of the lack of compactness associated with problems involving critical sobolev exponents the problem is further complicated by the absence of a direct sum decomposition suitable for applying classical linking arguments we overcome this difficulty using a generalized linking construction based on the z 2 cohomological index

solution stability of iterative schemes utilizing the discrete fourier transform em scattering the fredholm integral equation of the first kind can be solved numerically using iterative schemes which minimize the integral square error when the kernal is of the convolution type the discrete fourier transform dft can be used when evaluating the operator equation in the iterative schemes however the dft imposes an artificial periodicity thus changing the nature of the problem the effect of this on the solution has been studied and convergence investigated by comparison with a method of moments solution a proposed method of avoiding the periodicity problem has been studied the effect of introducing losses in the medium surrounding the scatterer has been investigated provided losses are low there is little effect on the solution

existence of solutions for a class of nonlinear schrodinger equations with potential vanishing at infinity abstract in this paper we investigate the existence of positive ground state solution for the following class of elliptic equations u v x u k x f u in r n where n 3 v k are nonnegative continuous functions and f is a continuous function with a quasicritical growth here we prove a hardy type inequality and use it together with variational method to get a ground state solution

convergence of the conjugate gradient method when applied to matrix equations representing electromagnetic scattering problems an iterative procedure based on the conjugate gradient method is used to solve a variety of matrix equations representing electromagnetic scattering problems in an attempt to characterize the typical rate of convergence of that method it is found that this rate depends on the cell density per wavelength used in the discretization the presence of symmetries in the solution and the degree to which mixed cell sizes are used in the models assuming cell densities used in the discretization are in the range of ten per linear wavelength the iterative algorithm typically requires n 4 to n 2 steps to converge to necessary accuracy where n is the order of the matrix under consideration

a spectral iteration technique for analyzing scattering from arbitrary bodies part i cylindrical scatterers with e wave incidence in the past methods for solving electromagnetic scattering problems in the frequency domain have been developed largely for the low frequency moment method and high frequency asymptotic techniques regimes the intermediate frequency range has been analyzed by combinations of these two approaches or by separation of variables when possible this paper is devoted to the development of an independent approach viz the stacked spectral iteration technique which is capable of handling arbitrary scatterers with dimensions ranging from small to moderately large the method takes advantage of the simplicity with which the planar source planar field relationships are expressed in the spectral domain the boundary conditions or constitutive relationships on the other hand are expressed most simply in the spatial domain alternating between the two domains is carried out with the aid of the fast fourier transform fft algorithm the spectral iteration technique sit was applied in the past to thin planar structures which allow the analysis to be carried out on a plane the generalization of the two dimensional formulation to arbitrary three dimensional bodies can be accomplished by sampling the current distribution on the scatterer over a number of parallel planes and using the simple spectral domain interaction relationships between the planes this new approach involves no matrix inversion and is capable of analyzing scatterers whose sizes far exceed those treatable by the moment method in addition to being arbitrarily shaped the scatterer may be conducting dielectric or lossy dielectric thus the sit provides an efficient approach to filling the much needed gap between low and high frequency conventional techniques e g the moment method mom and the geometrical theory of diffraction gtd and to extending the range of applicability to dielectric scatterers with or without loss though the concepts presented herein are applicable to arbitrary three dimensional scatterers the problem of arbitrary cylinders with e polarized excitation is addressed in this paper while the h case is treated in an accompanying work the three dimensional case is to be reported in a future communication which treats the problem of scattering by a lossy inhomogeneons dielectric cylinder of finite length

multiple solutions to a class of generalized quasilinear schrodinger equations with a kirchhoff type perturbation abstract in this paper we consider a class of generalized quasilinear schrodinger equations with a kirchhoff type perturbation under the assumption that the potential may be vanishing at infinity the existence of both the ground state and the ground state sign changing solutions is established furthermore the behavior of these solutions is studied when the perturbation vanishes it is a surprise that we find an interesting phenomenon about the monotonicity for the quotient function as a byproduct

spectral domain iterative analysis of single and double layered microstrip antennas using the conjugate gradient algorithm an analysis for single and double layered microstrip antennas is described the planarity of these structures makes it possible to construct a general spectral representation for any number of layers and to derive the spectral green s dyad in a compact fashion using a transmission line analogy a formulation of the antenna problem in the spectral domain incorporating this dyad is coupled with the conjugate gradient algorithm whose applicability as an efficient way of analyzing a number of microstrip antennas is studied results are quite accurate conclusions pertaining to the applicability of the method including effects of problem parameters on convergence rates are drawn

the conjugate gradient spectral iterative technique for planar structures it is shown that using the spectral iterative technique sit to solve the first kind integral equation is equivalent to the neumann iterative solution of a related second kind integral equation it is thus shown that sit only converges when the norm of the operator in the second kind equation is small enough applying a conjugate gradient technique to the second kind equation results in a convergent iterative scheme some representative numerical results show a superiority in the rate of convergence of the conjugate gradient scheme for the second kind equation cgsit scheme when compared with the convergence of the conjugate scheme for the original first kind equation cg scheme the cgsit scheme combines the advantages of the conjugate gradient method with those of the spectral iterative technique

application of the discrete fourier transform method to thin dielectric structures em scattering the discrete fourier transform method is a numerical technique designed to take advantage of the convolution structure that often appears in the differential integral equations arising from electromagnetic scattering it used the fast fourier transform fft to compute both the derivatives and the convolution integrals as a consequence this method is easy to program uses less computer memory than comparable methods yields accurate predictions and in general offers a better rate of convergence this technique which is particularly suited for solving problems where the scatterer is made of a dielectric material and has a shape which can be approximated accurately by a rectangular grid is applied to thin dielectric slabs with both electric and magnetic properties it is shown that by choosing the conductivity sufficiently large a thin dielectric slab behaves like a metallic plate on the other hand with a suitable choice of conductivity a particular thin dielectric slab will act like a resistive plate

on the quest for positivity in operator algebras abstract we show that in every nonzero operator algebra with a contractive approximate identity or c a i there is a nonzero operator t such that i t 1 in fact there is a c a i consisting of operators t with i 2 t 1 so the numerical range of the elements of our contractive approximate identity is contained in the closed disk center 1 2 and radius 1 2 this is the necessarily weakened form of the result for c algebras where there is always a contractive approximate identity consisting of operators with 0 t 1 the numerical range is contained in the real interval 0 1 so if an operator algebra has a c a i it must have operators with a certain amount of positivity

an improved pulse basis conjugate gradient fft method for the thin conducting plate problem a conjugate gradient fast fourier transform cg fft formulation for the scattering by a thin perfectly conducting plate is presented pulse basis functions and a dirac delta function are used for expansion and testing purposes respectively particular attention is paid to the generation of the terms in the impedance matrix of the resulting matrix equation except for the self coupling terms all the other terms are computed by explicit integrations comparison with two previously reported pulse basis cg fft formulations shows that the present method is more stable when the error tolerance of the solution is reduced when sufficient cell discretizations are used smooth distributions can be obtained which resemble those obtained via rooftop cg fft formulation the numerical results are further validated by comparing the far field radar cross section with an analytical technique for a circular plate

on a class of finite step iterative methods conjugate directions for the solution of an operator equation arising in electromagnetics a class of finite step iterative methods for the solution of linear operator equations is presented specifically the basic principles of the method of conjugate directions are developed gaussian elimination and the method of conjugate gradients are then presented as two special cases with an arbitrary initial guess the method of conjugate gradient always converges to the solution in at most n iterations where n is the number of independent eigenvalues for the operator in the finite dimensional space in which the problem is being solved the conjugate gradient method requires much less storage sim 5n than the conventional matrix methods sim n 2 in the solution of problems of higher complexity also after each iteration the quality of the solution is known in the conjugate gradient method the conjugate gradient method is also superior to the spectral iterative method as the latter does not always converge and it doubles the complexity of a given problem unnecessarily four versions of the conjugate gradient method are presented in detail and numerical results for a thin wire scatterer are given to illustrate various properties of each version

minimization problems in l1 r3 abstract in this paper minimization problems in l1 ir3 are considered these problems arise in astrophysics for the determination of equilibrium configurations of axially symmetric rotating fluids rotating stars under nearly optimal assumptions a minimizer is proved to exist by a direct variational method which uses heavily the symmetry of the problem in order to get some compactness finally by looking directly at the euler equation we give some existence results of solutions of the euler equation even if the infimum is not finite author

a conjugate gradient procedure for the analysis of planar conductors with alternating patch and aperture formulations em scattering the formulation of an electromagnetic scattering problem by an operator equation can be done in different ways leading to different rates of convergence when the conjugate gradient algorithm is used the convergence can tend to slow down at certain stages depending on the formulation used thus whenever the error tends to level off in one formulation it is advisable to switch to another one to this end a procedure that alternates repeatedly between formulations is presented it is shown to accelerate the convergence rate during the earlier stages of the iterative process making it possible to arrive at a prescribed practical error criterion faster a substantial saving in storage allocation is attained

existence of solution for two classes of elliptic problems in r n with zero mass in this paper we show the existence of a positive solution for the following class of elliptic equations u k x f u in rn where f is a continuous function with quasicritical growth and k is a nonnegative function verifying some conditions of two kinds we analyze the problem with k asymptotically periodic k periodic and k belonging to some lr

operator algebras with contractive approximate identities ii abstract we make several contributions to our recent program investigating structural properties of algebras of operators on a hilbert space for example we make substantial contributions to the noncommutative peak interpolation program begun by hay and the first author hay and neal another sample result an operator algebra has a contractive approximate identity iff the linear span of the elements with positive real part is dense we also extend the theory of compact projections to the most general case despite the title our algebras are often allowed to have no approximate identity

standing waves with a critical frequency for nonlinear schrodinger equations this paper is concerned with the existence and qualitative property of standing wave solutions for the nonlinear schrodinger equation with e being a critical frequency in the sense that we show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of v and whose amplitude goes to 0 as moreover depending upon the local behaviour of the potential function v x near the minimum points the limiting profile of the standing wave solutions will be shown to exhibit quite different characteristic features this is in striking contrast with the non critical frequency case which has been extensively studied in recent years

comments with reply on a generalization of van den berg s integral square error computational technique for scattering by a j mackay and a mccowen the commenters note that if the conjugate gradient method were used in the above named work ibid vol ap 35 p 213 20 feb 1987 its use automatically enforces the orthogonality of the authors functions without requiring any additional storage in replying the authors agree that it was perhaps an oversight not to compare van den berg s method with the conjugate gradient method however they note that it was not their aim to provide such a comparison but only to show how improvements could be made to the spectral iteration technique sit

a constrained conjugate gradient method for solving the magnetic field boundary integral equation it is well known that electromagnetic solutions of boundary integral equations for perfectly electrically conducting scatterers are nonunique for those frequencies which correspond to interior resonances of the scatterer in this paper a simple and efficient computational method is developed in which the interior integral representations required to hold on an interior closed surface are used as a sufficient constraint to restore uniqueness we use the interior equations together with the second kind magnetic field integral equation so that the ill posedness of the interior equations does not give a problem we develop a constrained conjugate gradient method that minimizes a cost functional consisting of two terms the first term is the error norm with respect to the magnetic field boundary integral equation while the second term is the error norm with respect to the interior equations over a closed interior surface which is chosen as small as possible some numerical examples show the robustness and efficiency of the pertaining computational procedure

compact imbeddings between variable exponent spaces with unbounded underlying domain abstract in this paper we have established some compact imbedding theorems for some subspaces of w 1 p x u when the underlying domain u is unbounded the domain we consider is mainly of type r n n 2 or r l l 2 where r m is a bounded domain with smooth boundary

survey of numerical methods for solution of large systems of linear equations for electromagnetic field problems many of the popular methods for the solution of large matrix equations are surveyed with the hope of finding an efficient method suitable for both electromagnetic scattering and radiation problems and system identification problems

existence of solutions for a class of elliptic equations in r n with vanishing potentials in this paper we investigate the existence of positive solution for the following class of elliptic equation u v x u f u in rn where f has a subcritical growth and v is a nonnegative potential which can vanish at infinity that is v x 0 as x or shortly v 0

existence of ground state solutions to dirac equations with vanishing potentials at infinity abstract in this work we study the existence of ground state solutions of dirac equations with potentials which are allowed to vanish at infinity the approach is based on minimization of the energy functional over a generalized nehari set some conditions on the potentials are given in order to overcome the lack of compactness

application of the discrete fourier transform method to nonorthogonal dihedral reflectors the discrete fourier transform method dftm is used to calculate the radar cross section rcs or orthogonal or nonorthogonal dihedral reflectors in order to accurately model the mutual interactions between the two plates a four point bivariate lagrange interpolation is incorporated into the dftm the resulting algorithm is simple and efficient results obtained with it compare favorably with measurements and predictions computed using the method of moments

existence of least energy positive negative and nodal solutions for a class of p q problems with potentials vanishing at infinity abstract in this paper we prove the existence of at least three nontrivial solutions for the class of p q problems given by div a u p u p 2 u v x b u p u p 2 u k x f u in r n where n 3 2 p n a b f are c 1 real functions and v k are positive and continuous functions thanks to a hardy type inequality we are able to prove compact embeddings in weighted sobolev spaces and by means of mountain pass theorem we get a positive and a negative ground state weak solution moreover by a minimization argument and a quantitative deformation lemma we find a nodal or sign changing weak solution with two nodal domains or that changes sign exactly once in r n the functions f a and b satisfy suitable growth and monotonicity conditions which allow this result to extend the previous ones to a larger class of p q type problems