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

discrete spline methods for solving two point fractional bagley torvik equation a new discrete spline method is developed to solve the two point fractional bagley torvik equation the method is based on discrete spline function and a nonstandard grunwald letnikov difference nsgd and the weighted and shifted grunwald letnikov difference wsgd operators to approximate the fractional derivative bounds for grunwald letnikov weights are considered convergence analysis is discussed and a class of second order and third order methods are obtained illustrative examples are presented to validate the practical usefulness of the methods

bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

best constants in martingale version of rosenthal s inequality the following generalization of rosenthal s inequality was proved by burkholder a 1 p s f p d ast p leq f ast p leq b p s f p d ast p for all martingales f n it is known that a p grows like sqrt p as p rightarrow infty in this paper we prove that the growth rate of b p as p rightarrow infty is p ln p

pattern synthesis of narrowband conformal arrays using iterative second order cone programming a new design method is proposed for the power or shaped beam pattern synthesis problem of narrowband conformal arrays where only the magnitude response is specified the proposed method iteratively linearizes the non convex power pattern function to obtain a convex subproblem in the design variables which can be solved optimally using second order cone programming socp in addition a wide variety of magnitude constraints such as non convex lower bound magnitude constraints can be incorporated an efficient technique for determining a reasonably good initial guess to the problem is also proposed to further improve the reliability of the method computer simulations show that the initial guesses so obtained converge to satisfactory solutions while satisfying various prescribed magnitude constraints design results show that the performance of the proposed method is comparable to the optimal solution previously obtained for uniform linear arrays with isotropic elements moreover we show by means of examples that the proposed method is also applicable to general non convex power pattern synthesis problems involving arbitrary array geometries arbitrary polarization characteristics and mutual coupling effect

universality properties of gelfand tsetlin patterns a standard gelfand tsetlin pattern of depth n is a configuration of particles in 1 ldots n times mathbb r for each r in 1 ldots n r times mathbb r is referred to as the rth level of the pattern a standard gelfand tsetlin pattern has exactly r particles on each level r and particles on adjacent levels satisfy an interlacing constraint probability distributions on the set of gelfand tsetlin patterns of depth n arise naturally as distributions of eigenvalue minor processes of random hermitian matrices of size n we consider such probability spaces when the distribution of the matrix is unitarily invariant prove a determinantal structure for a broad subclass and calculate the correlation kernel in particular we consider the case where the eigenvalues of the random matrix are fixed this corresponds to choosing uniformly from the set of gelfand tsetlin patterns whose nth level is fixed at the eigenvalues of the matrix fixing q n in 1 ldots n and letting n under the assumption that frac q n n to alpha in 0 1 and the empirical distribution of the particles on the nth level converges weakly the asymptotic behaviour of particles on level q n is relevant to free probability theory saddle point analysis is used to identify the set in which these particles behave asymptotically like a determinantal random point field with the sine kernel

a bayesian approach to robust adaptive beamforming an adaptive beamformer that is robust to uncertainty in source direction of arrival doa is derived using a bayesian approach the doa is assumed to be a discrete random variable with a known a priori probability density function pdf that reflects the level of uncertainty in the source doa the resulting beamformer is a weighted sum of minimum variance distortionless response mvdr beamformers pointed at a set of candidate doas where the relative contribution of each mvdr beamformer is determined from the a posteriori pdf of the doa conditioned on previously observed data a simple approximation to the a posteriori pdf results in a straightforward implementation performance of the approximate bayesian beamformer is compared with linearly constrained minimum variance lcmv beamformers and data driven approaches that attempt to estimate signal characteristics or the steering vector from the data

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

product of random projections jacobi ensembles and universality problems arising from free probability we consider the product of two independent randomly rotated projectors the square of its radial part turns out to be distributed as a jacobi ensemble we study its global and local properties in the large dimension scaling relevant to free probability theory we establish asymptotics for one point and two point correlation functions as well as properties of largest and smallest eigenvalues

a fractional differential equation for a mems viscometer used in the oil industry a mathematical model is developed for a micro electro mechanical system mems instrument that has been designed primarily to measure the viscosity of fluids that are encountered during oil well exploration it is shown that in one mode of operation the displacement of the device satisfies a fractional differential equation fde the theory of fdes is used to solve the governing equation in closed form and numerical solutions are also determined using a simple but efficient central difference scheme it is shown how knowledge of the exact and numerical solutions enables the design of the device to be optimised it is also shown that the numerical scheme may be extended to encompass the case of a nonlinear spring where the resulting fde is nonlinear

a random matrix model from two dimensional yang mills theory we use fat graphs to give a unified treatment of various asymptotic freeness results in particular a random matrix model from two dimensional yang mills theory on the plane is presented

achieving the welch bound with difference sets consider a codebook containing n unit norm complex vectors in a k dimensional space in a number of applications the codebook that minimizes the maximal cross correlation amplitude i sub max is often desirable relying on tools from combinatorial number theory we construct analytically optimal codebooks meeting in certain cases the welch lower bound when analytical constructions are not available we develop an efficient numerical search method based on a generalized lloyd algorithm which leads to considerable improvement on the achieved i sub max over existing alternatives we also derive a composite lower bound on the minimum achievable i sub max that is effective for any codebook size n

optimal bounds for convergence of expected spectral distributions to the semi circular law let mathbf x x jk j k 1 n denote a hermitian random matrix with entries x jk which are independent for 1 le j le k le n we consider the rate of convergence of the empirical spectral distribution function of the matrix mathbf x to the semi circular law assuming that mathbf e x jk 0 mathbf e x jk 2 1 and that begin aligned sup n ge 1 sup 1 le j k le n mathbf e x jk 4 mu 4 infty end aligned and begin aligned sup 1 le j k le n x jk le d 0n frac 1 4 end aligned by means of a recursion argument it is shown that the kolmogorov distance between the expected spectral distribution of the wigner matrix mathbf w frac 1 sqrt n mathbf x and the semicircular law is of order o n 1

linear array thinning using iterative fft techniques a new approach for the synthesis of thinned uniformly spaced linear arrays featuring a minimum sidelobe level is presented the method is based on the iterative fourier technique to derive element excitations from the prescribed array factor using successive forward and backward fourier transforms array thinning is accomplished by setting the amplitudes of the largest element excitations to unity and the others to zero during each iteration cycle the number of turned on elements depends on the array filling factor and the total count of element positions the effectiveness of the new method will be demonstrated for various 200 element linear arrays and compared with published results

bulk universality for generalized wigner matrices with few moments in this paper we consider n times n real generalized wigner matrices whose entries are only assumed to have finite 2 varepsilon th moment for some fixed but arbitrarily small varepsilon 0 we show that the stieltjes transforms m n z of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale when eta mathfrak i z is almost of order n 1 as a consequence we establish bulk universality for local spectral statistics of these matrices at fixed energy levels both in terms of eigenvalue gap distributions and correlation functions meaning that these statistics converge to those of the gaussian orthogonal ensemble in the large n limit

cumulants for random matrices as convolutions on the symmetric group in this paper we show that free cumulants can be naturally seen as the limiting value of cumulants of matrices we define these objects as functions on the symmetric group by some convolution relations involving the generalized moments we state that some characteristic properties of the free cumulants already hold for these cumulants

a novel adaptive beamformer based on semidefinite programming sdp with magnitude response constraints a novel robust adaptive beamformer formulated as a semidefinite programming sdp problem is proposed in this paper with new constraints on the magnitude response the beamwidth and response ripple of the robust response region can be well controlled moreover only a small part of these inequality constraints on the magnitude response are active during optimization so that few degrees of freedom dofs of the adaptive beamformer are consumed consequently the resultant beamformer has significant improvement on signal to interference plus noise ratio sinr an important problem in the proposed beamformer is how to generate the array weight vector from the optimal semidefinite matrix in this paper a method utilizing the extended spectral factorization method is proposed to solve this problem simple implementation flexible performance control as well as significant sinr enhancement support the practicability of the proposed method

on haar wavelet operational matrix of general order and its application for the numerical solution of fractional bagley torvik equation abstract in the present analysis the motion of an immersed plate in a newtonian fluid described by torvik and bagley s fractional differential equation 1 has been considered this bagley torvik equation has been solved by operational matrix of haar wavelet method the obtained result is compared with analytical solution suggested by podlubny 2 haar wavelet method is used because its computation is simple as it converts the problem into algebraic matrix equation

restricted isometry of fourier matrices and list decodability of random linear codes we prove that a random linear code over mathbb f q with probability arbitrarily close to 1 is list decodable at radius 1 1 q epsilon with list size l o 1 epsilon 2 and rate r omega q epsilon 2 log 3 1 epsilon up to the polylogarithmic factor in 1 epsilon and constant factors depending on q this matches the lower bound l omega q 1 epsilon 2 for the list size and upper bound r o q epsilon 2 for the rate previously only existence and not abundance of such codes was known for the special case q 2 guruswami et al 2002 in order to obtain our result we employ a relaxed version of the well known johnson bound on list decoding that translates the average hamming distance between codewords to list decoding guarantees we furthermore prove that the desired average distance guarantees hold for a code provided that a natural complex matrix encoding the codewords satisfies the restricted isometry property with respect to the euclidean norm for the case of random binary

robust minimum variance beamforming this paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold in our method uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction we choose weights that minimize the total weighted power output of the array subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid the robust weight selection process can be cast as a second order cone program that can be solved efficiently using lagrange multiplier techniques if the ellipsoid reduces to a single point the method coincides with capon s method we describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response we form separate uncertainty ellipsoids for each component in the signal path e g antenna electronics and then determine an aggregate uncertainty ellipsoid from these we give new results for modeling the element wise products of ellipsoids we demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples

the grunwald letnikov method for fractional differential equations this paper is devoted to the numerical treatment of fractional differential equations based on the grunwald letnikov definition of fractional derivatives finite difference schemes for the approximation of the solution are discussed the main properties of these explicit and implicit methods concerning the stability the convergence and the error behavior are studied related to linear test equations the asymptotic stability and the absolute stability of these methods are proved error representations and estimates for the truncation propagation and global error are derived numerical experiments are given

exponentially fitted methods for solving time fractional nonlinear reaction diffusion equation abstract in this article we will develop a new numerical scheme with the second order in time and a class of fourth order or sixth order in space based on the exponential fitting techniques to approximate the nonlinear time fractional reaction diffusion equation with fixed order and distributed order derivatives these techniques depend on a parameter which will be used to annihilate the error and increase the order of accuracy the proposed methods are proved to be unconditionally stable and convergent by fourier analysis also the theoretical results and the effectiveness of the numerical scheme are confirmed by numerical test problems and a comparison with other methods is presented

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

multivariate analysis and jacobi ensembles largest eigenvalue tracy widom limits and rates of convergence abstract let a and b be independent central wishart matrices in p variables with common covariance and having m and n degrees of freedom respectively the distribution of the largest eigenvalue of a b 1 b has numerous applications in multivariate statistics but is difficult to calculate exactly suppose that m and n grow in proportion to p we show that after centering and scaling the distribution is approximated to second order o p 2 3 by the tracy widom law the results are obtained for both complex and then real valued data by using methods of random matrix theory to study the largest eigenvalue of the jacobi unitary and orthogonal ensembles asymptotic approximations of jacobi polynomials near the largest zero play a central role

spectral distribution of random matrices from binary linear block codes let c be a binary linear block code of length n dimension k and minimum hamming distance d over gf 2 n let d denote the minimum hamming distance of the dual code of c over gf 2 n let e gf 2 n 1 1 n be the component wise mapping e vi 1 vi for v v1 v2 vn gf 2 n finally for p n let mmb c be a p n random matrix whose rows are obtained by mapping a uniformly drawn set of size p of the codewords of c under e it is shown that for d large enough and y p n 0 1 fixed as n the empirical spectral distribution of the gram matrix of 1 n mmb c resembles that of a random i i d rademacher matrix i e the marchenko pastur distribution moreover an explicit asymptotic uniform bound on the distance of the empirical spectral distribution of the gram matrix of 1 n mmb c to the marchenko pastur distribution as a function of y and d is presented

the ppt square conjecture holds generically for some classes of independent states let be a random pure state on where is a random unit vector uniformly distributed on the sphere in let be a random induced state whose distribution is and let be a random induced state following the same distribution independent of let be a random state induced by the entanglement swapping of and we show that the empirical spectrum of converges almost surely to the marcenko pastur law with parameter c 2 as and as an application we prove that the state is separable generically if are ppt entangled

the spectral edge of unitary brownian motion the brownian motion u n t t ge 0 on the unitary group converges as a process to the free unitary brownian motion u t t ge 0 as n rightarrow infty in this paper we prove that it converges strongly as a process not only in distribution but also in operator norm in particular for a fixed time t 0 we prove that the unitary brownian motion has a spectral edge there are no outlier eigenvalues in the limit we also prove an extension theorem any strongly convergent collection of random matrix ensembles independent from a unitary brownian motion also converge strongly jointly with the brownian motion we give an application of this strong convergence to the jacobi process

a fourth order approximation of fractional derivatives with its applications a new fourth order difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted grunwald formulae combining the compact technique the properties of proposed fractional difference quotient operator are presented and proved then the new approximation formula is applied to solve the space fractional diffusion equations by the energy method the proposed quasi compact difference scheme is proved to be unconditionally stable and convergent in l2 norm for both 1d and 2d cases several numerical examples are given to confirm the theoretical results

robust adaptive beamforming for general rank signal models the performance of adaptive beamforming methods is known to degrade severely in the presence of even small mismatches between the actual and presumed array responses to the desired signal such mismatches may frequently occur in practical situations because of violation of underlying assumptions on the environment sources or sensor array this is especially true when the desired signal components are present in the beamformer training data snapshots because in this case the adaptive array performance is very sensitive to array and model imperfections the similar phenomenon of performance degradation can occur even when the array response to the desired signal is known exactly but the training sample size is small we propose a new powerful approach to robust adaptive beamforming in the presence of unknown arbitrary type mismatches of the desired signal array response our approach is developed for the most general case of an arbitrary dimension of the desired signal subspace and is applicable to both the rank one point source and higher rank scattered source fluctuating wavefront desired signal models the proposed robust adaptive beamformers are based on explicit modeling of uncertainties in the desired signal array response and data covariance matrix as well as worst case performance optimization simple closed form solutions to the considered robust adaptive beamforming problems are derived our new beamformers have a computational complexity comparable with that of the traditional adaptive beamforming algorithms while at the same time offer a significantly improved robustness and faster convergence rates

rate of convergence to the semi circular law a stochastic bound of order op n 1 2 for the kolmogorov distance between the spectral distribution function of an n n matrix from wigner ensemble and the distribution function of the semi circular law is obtained the result holds assuming that the twelfth moment of the entries of the matrix is uniformly bounded

exact and discretized stability of the bagley torvik equation abstract the paper discusses stability and asymptotic properties of a two term linear fractional differential equation involving the bagley torvik equation as the particular case these properties are analysed for the exact as well as numerical solutions obtained from the grunwald letnikov discretization of the studied differential equation as the main results precise descriptions of the exact and discretized stability regions are presented including the decay rate of the solutions these results enable us among others to observe similarities and distinctions between the asymptotic behaviour of the classical and fractional damping models

approximation of haar distributed matrices and limiting distributions of eigenvalues of jacobi ensembles we develop a tool to approximate the entries of a large dimensional complex jacobi ensemble with independent complex gaussian random variables based on this and the author s earlier work in this direction we obtain the tracy widom law of the largest singular values of the jacobi emsemble moreover the circular law the marchenko pastur law the central limit theorem and the laws of large numbers for the spectral norms are also obtained

random subsets of structured deterministic frames have manova spectra abstract we draw a random subset of k k rows from a frame with n n rows vectors and m m columns dimensions where k k and m m are proportional to n n for a variety of important deterministic equiangular tight frames etfs and tight non etfs we consider the distribution of singular values of the k k subset matrix we observe that for large n n they can be precisely described by a known probability distribution wachter s manova multivariate anova spectral distribution a phenomenon that was previously known only for two types of random frames in terms of convergence to this limit the k k subset matrix from all of these frames is shown to be empirically indistinguishable from the classical manova jacobi random matrix ensemble thus empirically the manova ensemble offers a universal description of the spectra of randomly selected k k subframes even those taken from deterministic frames the same universality phenomena is shown to hold for notable random frames as well this description enables exact calculations of properties of solutions for systems of linear equations based on a random choice of k k frame vectors of n n possible vectors and has a variety of implications for erasure coding compressed sensing and sparse recovery when the aspect ratio m n m n is small the manova spectrum tends to the well known marcenko pastur distribution of the singular values of a gaussian matrix in agreement with previous work on highly redundant frames our results are empirical but they are exhaustive precise and fully reproducible

robust beamforming with magnitude response constraints using iterative second order cone programming the problem of robust beamforming for antenna arrays with arbitrary geometry and magnitude response constraints is one of considerable importance due to the presence of the non convex magnitude response constraints conventional convex optimization techniques cannot be applied directly a new approach based on iteratively linearizing the non convex constraints is then proposed to reformulate the non convex problem to a series of convex subproblems each of which can be optimally solved using second order cone programming socp moreover in order to obtain a more robust beamformer against array imperfections the proposed method is further extended by optimizing its worst case performance using again socp different from some conventional methods which are restricted to linear arrays the proposed method is applicable to arbitrary array geometries since the weight vector rather than its autocorrelation sequence is used as the variable simulation results show that the performance of the proposed method is comparable to the optimal solution previously proposed for uniform linear arrays and it also gives satisfactory results under different array specifications and geometries tested

robust adaptive beamformers based on worst case optimization and constraints on magnitude response in this paper novel robust adaptive beamformers are proposed with constraints on array magnitude response with the transformation from the array output power and the magnitude response to linear functions of the autocorrelation sequence of the array weight the optimization of an adaptive beamformer which is often described as a quadratic optimization problem in conventional beamforming methods is then reformulated as a linear programming lp problem unlike conventional robust beamformers the proposed method is able to flexibly control the robust response region with specified beamwidth and response ripple in practice an array has many imperfections besides steering direction error in order to make the adaptive beamformer robust against all kinds of imperfections worst case optimization is exploited to reconstruct the robust beamformer by minimizing array output power with the existence of the worst case array imperfections the robust beamforming can be expressed as a second order cone programming socp problem the resultant beamformer possesses superior robustness against arbitrary array imperfections with the proposed methods a large robust response region and a high signal to interference plus noise ratio sinr enhancement can be achieved readily simple implementation flexible performance control as well as significant sinr enhancement support the practicability of the proposed methods

universality of the local spacing distribution in certain ensembles of hermitian wigner matrices consider an n n hermitian random matrix with independent entries not necessarily gaussian a so called wigner matrix it has been conjectured that the local spacing distribution i e the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is in the limit as n the same as that of hermitian random matrices from gue we prove this conjecture for a certain subclass of hermitian wigner matrices