isophoric arrays massively thinned phased arrays with well controlled sidelobes traditional filled phased arrays have an element placed in every location of a uniform lattice with half wavelength spacing between the lattice points massively thinned arrays have fewer than half the elements of their filled counterparts such drastic thinning is normally accompanied by loss of sidelobe control this paper describes a class of massively thinned linear and planar arrays that show well behaved sidelobes in spite of the thinning the term isophoric is derived from greek roots to denote uniform weight in isophoric arrays element placement based on difference sets forces uniformly weighted spatial coverage this constraint forces the array power pattern to pass through v uniformly spaced equal and constant values that are less than 1 k times the main beam peak where v is the aperture size in half wavelengths and k is the number of elements in the array the net result is reduced peak sidelobes especially when compared to cut and try random placement approaches an isophoric array will exhibit this sidelobe control even when the array has been thinned to the extent that k is approximately the square root of v where more than one beam must be generated at a time isophoric array designs may be used to advantage even within a traditional filled array by interweaving two isophoric subarrays within a filled array and by appropriate cyclic shifting of the element assignments over time two independent antenna power patterns can be generated each with a sidelobe region that is approximately a constant value of 1 2k relative to the main beam where k is the number of elements in the subarray

fractional order legendre collocation method for solving fractional initial value problems in this paper we present a numerical algorithm for solving second order fractional initial value problems this numerical algorithm is based on a fractional legendre collocation spectral method the governing fractional differential equation is converted into a nonlinear system of algebraic equations the error analysis of the proposed numerical algorithm is presented comparisons with other numerical methods shows that the proposed algorithm is more accurate and simpler to implement several examples are discussed to illustrate the efficiency and accuracy of the present scheme

generalized anderson s inequality abstract let h be a separable infinite dimensional complex hilbert space and let b h denote the algebra of operators on h into itself we study the elementary operator b h b h defined by x axb cxd where a and c respectively b and d are nonzero normal commuting operators we prove that i for all n the kernel of and for all b or ii for all n c the von neumann schatten class 1 b such that c if and only if n n n n 0

a new numerical method for solving high order fractional eigenvalue problems the paper presents a new numerical method for solving eigenvalue problems for fractional high order differential equations with variable coefficients it combines two techniques the method of external excitation mee and the backward substitution method bsm the first one is a mathematical model of physical measurements when a mechanical electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies the bsm consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results in particular fractional problems of the fourth sixth and eighth orders are considered the numerical results show that the proposed method is of a high accuracy and is efficient for solving a wide class of eigenvalue problems

optimization of sparse linear arrays using harmony search algorithms a sparse linear array composed of a uniformly spaced core subarray and an extended sparse subarray is synthesized using a harmony search hs and an exploratory harmony search ehs algorithms the optimal solution is searched by changing the amplitudes of all the elements and the positions of the extended elements under a set of practical constraints performance of the ehs the hs a genetic algorithm ga and a particle swarm optimization pso algorithm has also been compared in synthesizing these sparse linear arrays

directions of arrival estimation through bayesian compressive sensing strategies the estimation of the directions of arrival doas of narrow band signals impinging on a linear antenna array is addressed within the bayesian compressive sensing bcs framework unlike several state of the art approaches the voltages at the output of the receiving sensors are directly used to determine the doas of the signals thus avoiding the computation of the correlation matrix towards this end the estimation problem is properly formulated to enforce the sparsity of the solution in the linear relationships between output voltages i e the problem data and the unknown doas customized implementations exploiting the measurements collected at a unique time instant single snapshot and multiple time instants multiple snapshots are presented and discussed the effectiveness of the proposed approaches is assessed through an extensive numerical analysis addressing different scenarios signal configurations and noise conditions comparisons with state of the art methods are reported as well

elementary operators and orthogonality abstract let h be a separable infinite dimensional hilbert space and let b h denote the algebra of operators on h into itself we study the elementary operators b h b h defined by x i 1 k a i xb i and x i 1 k a i xb i we prove that i when 1 then x x s s for all x b h and all s ker ii when i 1 k a i a i 1 i 1 k a i a i 1 i 1 k b i b i 1 and i 1 k b i b i 1 then for s ker ker c p the von neumann schatten p class 1 p x x s p s p and x x s p s p for all x b h where by convention y p if y c p iii let m i i 1 k and n i i 1 k be separately commuting sequences of normal operators and let b h b h be defined by x i 1 k m i xn i if x c 2 and s ker c 2 then x s 2 2 x 2 2 s 2 2

the closure of the range of an elementary operator abstract let b h denote the algebra of operators on a hilbert space h and let b b h be the elementary operator defined by x axb cxd a necessary condition for 1 0 b h b h is that 0 is an isolated point of the spectrum of we prove a sufficient condition for 1 0 b h b h applied to the case in which the hyponormal a b and normal c d satisfy certain conditions it is seen that the condition 0 is isolated in the set s a b c and d is sufficient for 1 0 b h b h

fractional sturm liouville eigen problems we first consider a regular fractional sturm liouville problem of two kinds rfslp i and rfslp ii of order 0 2 the corresponding fractional differential operators in these problems are both of riemann liouville and caputo type of the same fractional order 2 0 1 we obtain the analytical eigensolutions to rfslp i ii as non polynomial functions which we define as jacobi poly fractonomials these eigenfunctions are orthogonal with respect to the weight function associated with rfslp i ii subsequently we extend the fractional operators to a new family of singular fractional sturm liouville problems of two kinds sfslp i and sfslp ii we show that the primary regular boundary value problems rfslp i ii are indeed asymptotic cases for the singular counterparts sfslp i ii furthermore we prove that the eigenvalues of the singular problems are real valued and the corresponding eigenfunctions are orthogonal in addition we obtain the eigen solutions to sfslp i ii analytically also as non polynomial functions hence completing the whole family of the jacobi poly fractonomials in numerical examples we employ the new poly fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results

the filter diagonalization method in antenna array optimization for pattern synthesis pattern synthesis of nonuniform antenna arrays has drawn significant attention because of its wide applications with the aim of reducing the number of elements in linear and planar arrays this paper introduces a novel non iterative method based on the filter diagonalization method fdm which was originally applied in the problem of identifying and quantifying chemical molecules with nuclear magnetic resonance nmr in quantum mechanical formalism the proposed method samples the data set from the desired discrete pattern and associates the sample data with a time autocorrelation function of a fictitious dynamical system which is described by an effective hamiltonian operator that contains the array element information the hamiltonian operator can be decomposed by a set of orthonormal eigenvectors therefore the original pattern synthesis is converted into solving the general eigenvalue decomposition with krylov bases the number of nonuniform array elements depends on the number of the krylov bases and the sample data the proposed method can obtain an optimized antenna array to reconstruct the desired radiation pattern with a high accuracy numerical examples show that proposed fdm pattern synthesis can use less prior knowledge to achieve the desired pattern with highly sparse antenna arrays

synthesis of modular contiguously clustered linear arrays through a sparseness regularized solver an innovative methodology for the design of modular and contiguously clustered linear arrays is proposed the approach formulates the subarraying problem as a pattern matching one in which elementary modules taken from a user defined dictionary are combined to synthesize the final antenna layout by recasting such a synthesis problem within the compressive sensing framework an efficient solver is used to determine the array architecture starting from the a priori specification of the features e g the length of the admissible subarrays a set of representative numerical experiments is discussed to assess the advantages and drawbacks of the proposed algorithm also in comparison with some state of the art methods

a stochastic mathematical framework for the analysis of spherically bound random arrays random arrays bound by a spherical volume are rigorously analyzed through a stochastic approach for uniform and gaussian element distributions the full mathematical derivation and process to determine the array characteristics are discussed and a key result is the form solution which describes both the sidelobe behavior and main beam characteristics it is also shown that the proposed approach shares many conceptual similarities to the probabilistic approach applied to circular random arrays and comparisons are made the 3 db beamwidth directivity grating lobes and sidelobe level are examined analytically and in numerical simulation for a conceptual system of isotropic wide band radiators with a 100 1 bandwidth

synthesis of two dimensional antenna array using independent compression regions the synthesis of two dimensional antenna arrays that allows to optimize the number of its elements is of great interest for long term evolution lte advanced wireless systems in this communication we present a novel synthesis approach which focuses on reducing the number of antenna elements using independent compression regions icr we base our approach on the non iterative matrix pencil method mpm and the iterative bayesian compressive sampling bcs method both of these techniques have been previously used for the compression of linear arrays and now we apply them in the context of two dimensional arrays our compression method is based on the idea of segmenting the array in independent regions a set of numerical experiments small and large 2d arrays shaped beam antenna patterns are presented to assess the performance and potentialities of icr design approach

generalized analytical technique for the synthesis of unequally spaced arrays with linear planar cylindrical or spherical geometry an effective method for optimizing the performance of a fixed current distribution uniformly spaced antenna array has been to adjust its element positions to provide performance improvement in comparison with the default uniform structure this approach yields performance improvements such as smaller sidelobe levels or beamwidth values additionally it provides practical advantages such as reductions in size weight and number of antenna elements the objective of this paper is to describe a unified mathematical approach to nonlinear optimization of multidimensional array geometries the approach utilizes a class of limiting properties of sinusoidal bessel or legendre functions that are dictated by the array geometry addressed the efficacy of the method is demonstrated by its generalized application to synthesis of rectangular cylindrical and spherical arrays the unified mathematical approach presented below is a synthesis technique founded on the mathematical transformation of the desired field pattern followed by the application of limiting forms of the transformation and resulting in the development of a closed form expression for the element positions the method offers the following advantages over previous techniques such as direct nonlinear optimization or genetic algorithms first it is not an iterative searching algorithm and provides element spacing values directly in a single run of the algorithm thereby saving valuable cpu time and memory storage second it permits the array designer to place practical constraints on the array geometry e g the minimum maximum spacing between adjacent elements

roast rapid orthogonal approximate slepian transform in this paper we provide a rapid orthogonal approximate slepian transform roast for the discrete vector that one obtains when collecting a finite set of uniform samples from a baseband analog signal the roast offers an orthogonal projection which is an approximation to the orthogonal projection onto the leading discrete prolate spheroidal sequence dpss vectors also known as slepian basis vectors as such the roast is guaranteed to accurately and compactly represent not only oversampled bandlimited signals but also the leading dpss vectors themselves moreover the subspace angle between the roast subspace and the corresponding dpss subspace can be made arbitrarily small the complexity of computing the representation of a signal using the roast is comparable to the fast fourier tranform which is much less than the complexity of using the dpss basis vectors we also give non asymptotic results to guarantee that the proposed basis not only provides a very high degree of approximation accuracy in a mean squared error sense for bandlimited sample vectors but also that it can provide high quality approximations of all sampled sinusoids within the band of interest

microwave imaging within the first order born approximation by means of the contrast field bayesian compressive sensing a new approach based the contrast field cf formulation of the microwave imaging problem that exploits the bayesian compressive sampling bcs paradigm is proposed for the reconstruction of sparse distributions of weak scatterers towards this end the original inverse scattering problem is recast to a probabilistic sparseness constrained optimization by introducing suitable hierarchical priors as sparsity constraints a fast relevance vector machine rvm is then employed to reconstruct the scatterers as well as to estimate the confidence level of the inversion representative numerical results are presented to illustrate the method as well as to assess its potentialities and limitations in terms of inversion accuracy computational efficiency and robustness comparisons with state of the art deterministic and stochastic reconstruction methodologies still within the born approximation ba are discussed as well

reducing the number of elements in a linear antenna array by the matrix pencil method the synthesis of a nonuniform antenna array with as few elements as possible has considerable practical applications this paper introduces a new non iterative method for linear array synthesis based on the matrix pencil method mpm the method can synthesize a nonuniform linear array with a reduced number of elements and can be also used to reduce the number of elements for linear arrays designed by other synthesis techniques in the proposed method the desired radiation pattern is first sampled to form a discrete pattern data set then we organize the discrete data set in a form of hankel matrix and perform the singular value decomposition svd of the matrix by discarding the non principal singular values we obtain an optimal lower rank approximation of the hankel matrix the lower rank matrix actually corresponds to fewer antenna elements the matrix pencil method is then utilized to reconstruct the excitation and location distributions from the approximated matrix numerical examples show the effectiveness and advantages of the proposed synthesis method

fractional sturm liouville problem in this paper we define some fractional sturm liouville operators fslos and introduce two classes of fractional sturm liouville problems fslps namely regular and singular fslp the operators defined here are different from those defined in the literature in the sense that the operators defined here contain left and right riemann liouville and left and right caputo fractional derivatives for both classes we investigate the eigenvalue and eigenfunction properties of the fslos in the class of regular fslps we discuss two types of fslps as an operator for the class of singular fslps we introduce a fractional legendre equation fle and discuss its solution it is shown that the legendre polynomials resulting from an fle are the same as those obtained from the integer order legendre equation however the eigenvalues of the two equations differ using the legendre integral transform we demonstrate some applications of our results by solving two fractional differential equations one ordinary and the other partial it is our hope that this paper will initiate new research in the area of fslps and many of its variations

linear array thinning exploiting almost difference sets this paper describes a class of linear thinned arrays with predictable and well behaved sidelobes the element placement is based on almost difference sets and the array power pattern is forced to pass through n uniformly spaced values that although neither equal nor constant as for difference sets are a priori known from the knowledge of the aperture size the number of active array elements k and the features of the correlation function such a property allows one to predict the bounds of the confidence range of the peak sidelobe of the admissible arrays obtainable through simple shift operations on a binary sequence the expected peak sidelobe performances turn out to be comparable with those from difference sets even though obtainable in a wider set of array configurations and improved in comparison with cut and try random placements

non local fractional calculus from different viewpoint generated by truncated m derivative abstract the core of this paper is to propose more favored novel non local fractional integral and derivative operators with three parameters by using the iteration manner on truncated m derivative involving truncated mittag leffler function this is related closely to the fact that in a certain sense to iterate a convenient local derivative for real phenomena under investigation is one of the best ways for obtaining more comprehensive fractional integration and differentiation in order to gain a deeper grip we address some significant theorems reflected accomplishedly the features of the standard fractional calculus about these aidful operators encompassing some other types of fractional operators the presented new operators in riemann liouville and caputo sense are utilized to solve sturm liouville problem subjected to the initial conditions as a useful method laplace transform of the aforementioned fractional integral and derivative is introduced to solve anomalous dynamics of various complex systems moreover we carry out comparison analysis by plotting governing sturm liouville equation in order to grasp the advantages of these new fractional operators performing much better on account of the additional parameter

aperiodic array layout optimization by the constraint relaxation approach an optimization procedure for the layout assessment of electrically large but finite planar arrays is presented the synthesis takes into account the desired directivity pattern that is prescribed employing bound constraints moreover the size of the radiators is taken into account which results in a hard nonoverlapping between the elements constraint the latter should not be violated if we want the attained solution not only to obey the far field mask but also to be physically realizable as stated the optimization problem is twofold an antenna design is associated with a packing problem in order to take the constraints on the layout into account and solve the whole problem we propose the constraint relaxation approach which is equipped with a packing algorithm our study is applied to various initial geometries and the resulting arrays appear to comply with the desired pattern and the nonoverlapping constraint several examples for different cases including symmetric arrays and a study on maximally sparse arrays are presented which show the applicability and merit of the method

discrete mittag leffler kernel type fractional difference initial value problems and gronwall s inequality abstract in this article we studied the caputo and riemann liouville type discrete fractional difference initial value problems with discrete mittag leffler kernels the existence and uniqueness of the solution is proved by using banach contraction principle the linear type equations are used to prove new discrete fractional versions of the gronwall s inequality the nabla discrete laplace transform is used to obtain solution representations the proven gronwall s inequality under a new defined lipschitzian is used to prove that small changes in the initial conditions yield small changes in solutions numerical examples are discussed to demonstrate the reliability of the theoretical results

wavelet based compressive imaging of sparse targets the application of the compressive sensing cs paradigm to retrieve non single pixels contrast profiles is discussed by exploiting a wavelet representation to model complex scatterer distributions with sparse vectors of coefficients an efficient bayesian cs bcs strategy is adopted to solve the arising inverse scattering problem a set of representative numerical examples is presented to illustrate the advantages and the limitations of the proposed approach also with respect to comparable state of the art inversion methods

elementary operators finite ascent range closure and compactness abstract given banach space operators a i b i b x 1 i 2 let ab b b x denote the elementary operator ab x a 1 x b 1 a 2 x b 2 then ab has finite ascent 1 for a number of fairly general choices of the operators a i and b i this information is applied to prove some necessary and sufficient conditions for the range of ab to be closed and in deciding conditions on the tuples a 1 a 2 and b 1 b 2 so that ab n x compact for some integer n 1 and operator x implies ab x compact this generalizes some well known results of anderson and foias 4 and yosun 25 also considered is the question what is a necessary and sufficient condition on the tuples a 1 a 2 b 1 b 2 and ab for ab n to be compact for some integer n 1

bayesian compressive sampling for pattern synthesis with maximally sparse non uniform linear arrays a numerically efficient technique based on the bayesian compressive sampling bcs for the design of maximally sparse linear arrays is introduced the method is based on a probabilistic formulation of the array synthesis and it exploits a fast relevance vector machine rvm for the problem solution the proposed approach allows the design of linear arrangements fitting desired power patterns with a reduced number of non uniformly spaced active elements the numerical validation assesses the effectiveness and computational efficiency of the proposed approach as a suitable complement to existing state of the art techniques for the design of sparse arrays

minimum sidelobe levels for linear arrays a method of finding the optimum sidelobe minimizing weights for an arbitrary linear array is derived that holds for any scan direction beamwidth and type of antenna element used optimum linear array positions are then determined via the particle swarm optimization method hence the two together can give a global bound on antenna array performance with regards to sidelobe minimization results are presented for linear arrays of 2 6 elements for different scan angles beamwidths and antenna elements

element positioning for linear arrays using generalized gaussian quadrature we present a novel approach for designing small narrowband linear arrays with unequally spaced elements our approach involves discretising a line source with a desirable beam pattern and relies on the fact that beam patterns are bandlimited we show that given a nonnegative weighting function for a line source one can design a linear array of elements whose beam pattern approximates that of the line source to any desired degree of accuracy we make use of the prolate spheroidal wave functions to analyze line source beam patterns since they are eigenfunctions of the finite fourier transform these functions form a tchebycheff system which allows us to calculate exactly integrals involving these functions via a generalized gaussian quadrature rule this quadrature rule then gives the appropriate set of element positions and weights we give results for several weighting functions and we consider both broadside and electronically scanned linear arrays as an example we show how a broadside array with half wavelength spacing and dolph tchebycheff weights may be improved upon using the method described

variational methods for the fractional sturm liouville problem abstract this article is devoted to the regular fractional sturm liouville eigenvalue problem by applying the methods of fractional variational analysis we prove the existence of a countable set of orthogonal solutions and corresponding eigenvalues moreover we formulate two results showing that the lowest eigenvalue is the minimum value for a certain variational functional

hybrid design of non regular linear arrays with accurate control of the pattern sidelobes the hybridization of an analytical thinning technique based on almost difference sets adss and a convex programming cp strategy is proposed for designing non regular linear arrays over a half wavelength lattice that fit arbitrary upper bounds on the far field pattern the proposed approach combines the efficiency of the ads procedure in defining a low sidelobe thinned arrangement with the effectiveness of the cp approach in deducing the optimal tapering to control the beam shape a set of numerical results is presented to assess features and potentialities of the ads cp method

hybrid bcs deterministic approach for sparse concentric ring isophoric arrays an innovative methodology for the synthesis of sparse concentric ring isophoric arrays matching a user defined reference pattern is proposed the synthesis problem is recast as a two step approach firstly aimed at a determining the sparsest auxiliary layout of concentric current rings matching the reference pattern through a bayesian compressive sensing method and successively b yielding the concentric ring isophoric array by means of a density tapering technique a selected set of numerical experiments is presented to provide some insights about the effectiveness the numerical efficiency and the flexibility of the proposed hybrid method

design of broadband planar arrays based on the optimization of aperiodic tilings antenna arrays based on aperiodic tilings have been shown to exhibit low sidelobe levels and modest bandwidths over which grating lobes are suppressed in addition compared to conventional periodic arrays these arrays are naturally thinned i e mean inter element spacing is greater than the generation of these arrays involves placing array elements at the locations of the vertices of an aperiodic tiling to obtain a realizable design the entire array is then scaled and truncated to achieve a desired minimum element spacing and aperture size this paper demonstrates that it is possible to greatly extend the bandwidth of these arrays by incorporating a simple perturbation scheme into the basic array generation process the implementation of this perturbation scheme is straightforward and it lends itself well to being combined with an optimization technique such as the genetic algorithm it is successfully used to generate arrays that have large bandwidths peak sidelobe level db with no grating lobes of up to a minimum element spacing of moreover the flexibility of this technique will be further demonstrated by introducing a slight variation of the basic scheme that is capable of generating arrays with extremely wide bandwidths an example will be presented for an array design that has a bandwidth corresponding to a minimum element spacing of up to

on the design of maximally sparse beamforming arrays the problem of designing nonuniformly spaced arrays is formulated as one of constrained optimization in which the cost function is chosen to select the array with the minimum number of elements the response of the array is controlled by a set of inequality point response constraints it is shown that a suitable cost function for this problem is the l sub p quasi norm for 0

autocorrelation driven synthesis of antenna arrays the case of ds based planar isophoric thinned arrays a new methodology for the design of isophoric thinned arrays with a priori controlled pattern features is introduced a fully analytical and general i e valid for any lattice and set of weights relationship between the autocorrelation of the array excitations and the power pattern samples is first derived binary 2 d sequences with known autocorrelation properties namely the difference sets ds s are then chosen as a representative benchmark to prove that it is possible to deduce closed form synthesis formulas that a priori guarantee to fit requirements on the sidelobe level sll the directivity the half power beamwidth and the power pattern in user defined directions the selected results from a wide numerical assessment which also includes full wave simulations with realistic radiators are illustrated to validate the reliability and the accuracy of the proposed design equations and the associated performance bounds

peak sidelobe level reduction with a hybrid approach based on gas and difference sets this paper presents an approach for the optimization of the beam pattern produced by massively thinned arrays the method which combines the most attractive features of a genetic algorithm and those of a combinatorial technique namely the difference sets method is aimed at synthesizing massively thinned antenna arrays in order to suitably reduce the peak sidelobe level selected numerical results are presented in order to assess the effectiveness and reliability of the proposed approach

reducing the number of elements in multiple pattern linear arrays by the extended matrix pencil methods previously the matrix pencil method mpm and the forward backward mpm fbmpm were used to effectively reduce the number of antenna elements in the single pattern linear arrays this work extends the mpm and fbmpm based synthesis methods to the synthesis of multiple pattern linear arrays with a smaller number of elements the extended mpm resp the extended fbmpm method organizes all the multiple pattern data into a composite hankel resp composite hankel toeplitz matrix from which the minimum number of elements and the common poles corresponding to element positions can be obtained with similar processing used in the original mpm or fbmpm synthesis method in particular the extended fbmpm inherits the advantage of the original fbmpm that a useful restriction is put on the distribution of poles which makes the element positions obtained much more accurate and robust numerical experiments are conducted to validate the effectiveness and robustness of the proposed methods for the tested cases the element saving is about 20 25 for reconfigurable shaped patterns and can be even more for electrically large linear arrays with scanned pencil beams

on the minimum gap and the angle between two subspaces in this note we give the expressions of the minimum gap and the angle between two closed subspaces of a hilbert space in terms of the properties of the projections on these subspaces

sparsening conformal arrays through a versatile bcs based method sparsening conformal arrangements is carried out through a versatile multi task bayesian compressive sensing mt bcs strategy the problem formulated in a probabilistic fashion as a pattern matching synthesis is that of determining the sparsest excitation set locations and weights fitting a reference pattern subject to user defined geometrical constraints results from a set of representative numerical experiments are presented to illustrate the key features of the proposed approach as well as to assess also through comparisons its potentials in terms of matching accuracy element saving and computational costs

an efficient algorithm for solving higher order fractional sturm liouville eigenvalue problems abstract in this paper we present a simple and efficient computational algorithm for solving eigenvalue problems of high fractional order differential equations with variable coefficients the method of solution is based on utilizing the series solution to convert the governing fractional differential equation into a linear system of algebraic equations then the eigenvalues can be calculated by finding the roots of the corresponding characteristic polynomial notice that this class of eigenvalue problems is very promising to the solution of linear fractional partial differential equations fpde the numerical results demonstrate reliability and efficiency of the proposed algorithm based on our simulations some theoretical conjectures are reported

range kernel orthogonality of the elementary operator x i 1naixbi x abstract let h be a separable infinite dimensional complex hilbert space and let b h denote the algebra of operators on h into itself let a a 1 a 2 a n and b b 1 b 2 b n be n tuples in b h define the elementary operators ab and ab b h b h by ab x i 1 n a i xb i x and ab x i 1 n a i xb i x this note considers the range kernel orthogonality of the restrictions of ab and ab to schatten p classes c p it is proved that a if 1 p s c p and i 1 n a i a i i 1 n a i a i i 1 n b i b i and i 1 n b i b i are all 1 then min ab x s p ab x s p s p for all x c p if and only if ab s 0 ab s b if p 2 and s c 2 then ab x s 2 2 ab x 2 2 s 2 2 and ab x s 2 2 ab x 2 2 s 2 2 if and only if ab s 0 ab s and c if a and b are the n tuples of a such that bb s 0 bb s for some injective s c 1 then the inequality of a holds with p 1 and for all x c 1 if and only if ab s 0 ab s

exponentially accurate spectral and spectral element methods for fractional odes current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy here we present different methods for fractional odes that lead to exponentially fast decay of the error first we develop a petrov galerkin pg spectral method for fractional initial value problems fivps of the form d t 0 u t f t and fractional final value problems ffvps d t t u t g t where 0 1 subject to dirichlet initial final conditions these schemes are developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently developed in 1 specifically we obtain solutions to fivps and ffvps in terms of the new fractional non polynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of second kind fslp ii subsequently we develop a discontinuous spectral method dsm of petrov galerkin sense for the aforementioned fivps and ffvps where the basis functions do not satisfy the initial final conditions finally we extend the dsm scheme to a discontinuous spectral element method dsem for efficient longer time integration and adaptive refinement in these discontinuous schemes we employ the asymptotic eigensolutions to fslp i ii which are of jacobi polynomial forms as basis and test functions our numerical tests confirm the exponential algebraic convergence respectively in p and h refinements for various test cases with integer and fractional order solutions

the fractional sturmliouville problemnumerical approximation and application in fractional diffusion the numerical method of solving the fractional eigenvalue problem is derived in the case when the fractional sturmliouville equation is subjected to the mixed boundary conditions this non integer order differential equation is discretized to the scheme with the symmetric matrix representing the action of the numerically expressed composition of the left and the right caputo derivative the numerical eigenvalues are thus real and the eigenvectors associated to distinct eigenvalues are orthogonal in the respective finite dimensional hilbert space the advantage of the proposed method is the formulation which allows us to construct the approximate eigenfunctions which form an orthonormal function system in the infinite dimensional weighted lebesgue integrable function space the developed numerical method of calculation of the eigenvalues and eigenfunctions is then applied in construction of the approximate solution to the 1d space time fractional diffusion problem in a bounded domain a new numerical method for solving a fractional eigenvalue problem has been proposed the approximate eigenfunctions are orthogonal at each step of the proposed method the method is applied in the approximate solution to fractional diffusion problem

array thinning of coupled antennas based on the orthogonal matching pursuit method and a spherical wave expansion for far field synthesis this work is focused on the array thinning problem in shaped beam far field synthesis the orthogonal matching pursuit algorithm which allows finding sparse solutions in linear systems is combined with a fast full wave analysis method for antenna arrays based on a spherical wave expansion thereby taking mutual coupling between real antennas into account in the thinning process to this aim the original iterative algorithm is modified so that the current residual is obtained for the selected real coupled antennas in each step of the algorithm in this way the remaining nonselected elements are effectively removed instead of turned off results for arrays made up of microstrip patch antennas and dielectric resonator antennas arranged in triangular and rectangular lattices are presented

operators that are orthogonal to the range of a derivation the gateaux differentiability of the schatten p norms and the usual operator norm is used to characterize those operators that are orthogonal to the range of a derivation with respect to these norms

reconfigurable adaptive array beamforming by antenna selection traditional adaptive array beamforming with a fixed array configuration can lead to significant inefficiencies and performance loss under different scenarios as antennas become smaller and cheaper relative to front ends it becomes important to devise a reconfigurable adaptive antenna array raaa strategy to yield high signal to noise and interference ratio using fewer antennas this is achieved by selecting k from n antennas to minimize the spatial correlation coefficient scc between the desired signal and the interference the lower bound of optimum scc is formulated with two relaxation methods to give information about the suitable number of selected antennas k a correlation measurement cm method is proposed to select the optimum subarray with k antennas thereby reducing complexity we carry out performance analysis and show that a 1 k2 suboptimum solution can be guaranteed with arbitrary shaped arrays furthermore a difference of convex sets dcs method is proposed to select the optimum subarray with controlled quiescent pattern in order to reduce the effect of interference during the reconfiguration time the utility of the proposed array reconfiguration for performance improvement without increasing the cost is demonstrated using both simulated and experimental data

synthesis of linear and planar arrays with minimum element selection a new method for the synthesis of linear and planar arrays having prescribed beamwidth and sidelobe levels and a minimum number of elements is proposed in the method the number of elements in an array is minimized while constraining the amplitude response error in the mainlobe region the attenuation in the sidelobe region and the array dimensions an iterative constrained optimization method is used where the amplituderesponse error is linearly approximated at each iteration while concurrently minimizing a re weighted l1 norm of the array coefficients to ensure robustness of the array we constrain a sensitivity parameter namely the white noise gain to be above a prescribed level furthermore the method also provides the additional flexibility of controlling the array dimensions symmetry properties and element positions of the array two variants have been developed in the first variant both the array coefficients and the positions of the elements are optimized in the second variant only the array coefficients are optimized while the elements are fixed at predefined positions experimental comparisons with several state of the art competing methods show that the proposed method provides greater flexibility of controlling the robustness beampattern response error array dimensions and element positions while at the same time the number of elements is less than or equal to that of the competing methods

on the range closure of an elementary operator let a b 2 b h be hilbert space contractions and let 4ab be the elementary operator 4ab x i axb i x a number of conditions which are equivalent to 4ab has closed range are proved

fractional spectral collocation method we develop an exponentially accurate fractional spectral collocation method for solving steady state and time dependent fractional pdes fpdes we first introduce a new family of interpolants called fractional lagrange interpolants which satisfy the kronecker delta property at collocation points we perform such a construction following a spectral theory recently developed in m zayernouri and g e karniadakis j comput phys 47 2013 pp 2108 2131 for fractional sturm liouville eigenproblems subsequently we obtain the corresponding fractional differentiation matrices and we solve a number of linear fodes in addition to linear and nonlinear fpdes to investigate the numerical performance of the fractional collocation method we first examine space fractional advection diffusion problem and generalized space fractional multiterm fodes next we solve fpdes including the time and space fractional advection diffusion equation time and space fractional multiterm fpdes and finally the space

a novel method for solving second order fractional eigenvalue problems the paper presents a new numerical method for solving eigenvalue problems for fractional differential equations it combines two techniques the method of external excitation mee and the backward substitution method bsm the first one is a mathematical model of physical measurements when a mechanical electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies the bsm consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results the numerical results show that the proposed method is of a high accuracy and is efficient for solving of a wide class of eigenvalue problems

an effective method for synthesizing multiple pattern linear arrays with a reduced number of antenna elements an innovative technique based on the enhanced unitary matrix pencil mp method is presented for the design of sparse multiple pattern linear arrays by virtue of the equivalent mp obtained with a unitary transformation the relation between the element positions and the generalized eigenvalues is achieved in this method which contributes to the real solutions of the common element positions for all multiple patterns owing to the utilization of a unitary transformation that can convert a complex matrix to a real one the computational complexity can be significantly reduced since only the real computations are involved in the singular value decomposition and eigenvalue decomposition procedures consequently by only varying the obtained excitation distributions different patterns are generated with a higher matching accuracy representative experiments are provided to validate the effectiveness and advantages of the proposed method

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

complex weight sparse linear array synthesis by bayesian compressive sampling an innovative method for the synthesis of maximally sparse linear arrays matching arbitrary reference patterns is proposed in the framework of sparseness constrained optimization the approach exploits the multi task mt bayesian compressive sensing bcs theory to enable the design of complex non hermitian layouts with arbitrary radiation and geometrical constraints by casting the pattern matching problem into a probabilistic formulation a relevance vector machine rvm technique is used as solution tool the numerical assessment points out the advances of the proposed implementation over the extension to complex patterns of and it gives some indications about the reliability flexibility and numerical efficiency of the mt bcs approach also in comparison with state of the art sparse arrays synthesis methods