outliers in the spectrum of iid matrices with bounded rank perturbations it is known that if one perturbs a large iid random matrix by a bounded rank error then the majority of the eigenvalues will remain distributed according to the circular law however the bounded rank perturbation may also create one or more outlier eigenvalues we show that if the perturbation is small then the outlier eigenvalues are created next to the outlier eigenvalues of the bounded rank perturbation but if the perturbation is large then many more outliers can be created and their law is governed by the zeroes of a random laurent series with gaussian coefficients on the other hand these outliers may be eliminated by enforcing a row sum condition on the final matrix

smallest singular value of random matrices and geometry of random polytopes abstract we study the behaviour of the smallest singular value of a rectangular random matrix i e matrix whose entries are independent random variables satisfying some additional conditions we prove a deviation inequality and show that such a matrix is a good isomorphism on its image then we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes absolutely convex hulls of rows of random matrices all our results hold with high probability that is with probability exponentially in dimension close to 1

orbital hp clouds for solving schrodinger equation in quantum mechanics solving schrodinger equation in quantum mechanics presents a challenging task in numerical methods due to the high order behavior and high dimension characteristics in the wave functions in addition to the highly coupled nature between wave functions this work introduces orbital and polynomial enrichment functions to the partition of unity for solution of schrodinger equation under the framework of hp clouds an intrinsic enrichment of orbital function and extrinsic enrichment of monomial functions are proposed due to the employment of higher order basis functions a higher order stabilized conforming nodal integration is developed the proposed methods are implemented using the density functional theory for solution of schrodinger equation analysis of several single and multi electron nucleus structures demonstrates the effectiveness of the proposed method

asymptotic shape of a random polytope in a convex body abstract let k be an isotropic convex body in r n and let z q k be the l q centroid body of k for every n n consider the random polytope k n conv x 1 x n where x 1 x n are independent random points uniformly distributed in k we prove that a random k n is asymptotically equivalent to z ln n n k in the following sense there exist absolute constants 1 2 0 such that for all 0 1 2 and all n n n one has i k n c z q k for every q 1 ln n n with probability greater than 1 c 1 exp c 2 n 1 n ii for every q 2 ln n n the expected mean width e w k n of k n is bounded by c 3 w z q k as an application we show that the volume radius k n 1 n of a random k n satisfies the bounds c 4 ln 2 n n n k n 1 n c 5 l k ln 2 n n n for all n exp n

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

the circular law for random matrices we consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance we prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries we assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices the results are based on previous work of bai rudelson and the authors extending results to a larger class of sparse matrices

limit of the smallest eigenvalue of a large dimensional sample covariance matrix in this paper the authors show that the smallest if p leq n or the p n 1 th smallest if p n eigenvalue of a sample covariance matrix of the form 1 n xx tends almost surely to the limit 1 sqrt y 2 as n rightarrow infty and p n rightarrow y in 0 infty where x is a p times n matrix with iid entries with mean zero variance 1 and fourth moment finite also as a by product it is shown that the almost sure limit of the largest eigenvalue is 1 sqrt y 2 a known result obtained by yin bai and krishnaiah the present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices

on the approximability and the selection of particle shape functions particle methods also known as meshless or meshfree methods have become popular in approximating solutions of partial differential equations especially in the engineering community these methods do not employ a mesh or use it minimally in the construction of shape functions there is a wide variety of classes of shape functions that can be used in particle methods in this paper we primarily address the issue of selecting a class of shape functions among this wide variety that would yield efficient approximation of the unknown solution we have also made several comments and observations on the order of convergence of the interpolation error when these shape functions are used specifically we have shown that the interpolation error estimate for certain classes of shape functions may not indicate the actual order of convergence of the approximation error

on the finite element solution of the pure neumann problem this paper considers the finite element approximation and algebraic solution of the pure neumann problem our goal is to present a concise variational framework for the finite element solution of the neumann problem that focuses on the interplay between the algebraic and variational problems while many of the results that stem from our analysis are known by some experts they are seldom derived in a rigorous fashion and remain part of numerical folklore as a result this knowledge is not accessible or appreciated by many practitioners both novices and experts in one source our paper contributes a simple yet insightful link between the continuous and algebraic variational forms that will prove useful

invertibility of sparse non hermitian matrices abstract we consider a class of sparse random matrices of the form a n i j i j i j 1 n where i j are i i d centered random variables and i j are i i d bernoulli random variables taking value 1 with probability p n and prove a quantitative estimate on the smallest singular value for p n log n n under a suitable assumption on the spectral norm of the matrices this establishes the invertibility of a large class of sparse matrices for p n n with some 0 1 we deduce that the condition number of a n is of order n with probability tending to one under the optimal moment assumption on i j this in particular extends a conjecture of von neumann about the condition number to sparse random matrices with heavy tailed entries in the case that the random variables i j are i i d sub gaussian we further show that a sparse random matrix is singular with probability at most exp c n p n whenever p n is above the critical threshold p n log n n the results also extend to the case when i j have a non zero mean we further find quantitative estimates on the smallest singular value of the adjacency matrix of a directed erd s reyni graph whenever its edge connectivity probability is above the critical threshold log n n

the littlewood offord problem and invertibility of random matrices abstract we prove two basic conjectures on the distribution of the smallest singular value of random n n matrices with independent entries under minimal moment assumptions we show that the smallest singular value is of order n 1 2 which is optimal for gaussian matrices moreover we give a optimal estimate on the tail probability this comes as a consequence of a new and essentially sharp estimate in the littlewood offord problem for i i d random variables x k and real numbers a k determine the probability p that the sum k a k x k lies near some number v for arbitrary coefficients a k of the same order of magnitude we show that they essentially lie in an arithmetic progression of length 1 p

random matrices tail bounds for gaps between eigenvalues gaps or spacings between consecutive eigenvalues are a central topic in random matrix theory the goal of this paper is to study the tail distribution of these gaps in various random matrix models we give the first repulsion bound for random matrices with discrete entries and the first super polynomial bound on the probability that a random graph has simple spectrum along with several applications

on mds extensions of generalized reed solomon codes an n k d linear code over f gf q is said to be em maximum distance separable mds if d n k 1 it is shown that an n k n k 1 generalized reed solomon code such that 2 leq k leq n lfloor q 1 2 rfloor k neq 3 rm if q is even can be extended by one digit while preserving the mds property if and only if the resulting extended code is also a generalized reed solomon code it follows that a generalized reed solomon code with k in the above range can be em uniquely extended to a maximal mds code of length q 1 and that generalized reed solomon codes of length q 1 and dimension 2 leq k leq lfloor q 2 rfloor 2 k neq 3 rm if q is even do not have mds extensions hence in cases where the q 1 k mds code is essentially unique n k mds codes with n q 1 do not exist

inverse littlewood offord theorems and the condition number of random discrete matrices consider a random sum r v r nvn where 771 rin are independently and identically distributed i i d random signs and vi vn are integers the littlewood offord problem asks to maximize concentration probabilities such as p r i ih in this paper we develop an inverse littlewood offord theory somewhat in the spirit of freiman s inverse theory in additive combinatorics which starts with the hypothesis that a concentration probability is large and concludes that almost all of the v vn are efficiently contained in a generalized arithmetic progression as an application we give a new bound on the magnitude of the

on 0 1 polytopes with many facets there exist n dimensional 0 1 polytopes with as many as cn logn n 4 facets this is our main result it answers a question of komei fukuda and gunter m ziegler c 2001 academic press

random embedding of ell_p n into ell_r n for any 0 n we give an explicit definition of a random operator s ell p n to mathbb r n such that for every 0 0 is a real number depending only on p and r one of the main tools that we develop is a new type of multidimensional esseen inequality for studying small ball probabilities

on the convergence of the extremal eigenvalues of empirical covariance matrices with dependence consider a sample of a centered random vector with unit covariance matrix we show that under certain regularity assumptions and up to a natural scaling the smallest and the largest eigenvalues of the empirical covariance matrix converge when the dimension and the sample size both tend to infinity to the left and right edges of the marchenko pastur distribution the assumptions are related to tails of norms of orthogonal projections they cover isotropic log concave random vectors as well as random vectors with i i d coordinates with almost optimal moment conditions the method is a refinement of the rank one update approach used by srivastava and vershynin to produce non asymptotic quantitative estimates in other words we provide a new proof of the bai and yin theorem using basic tools from probability theory and linear algebra together with a new extension of this theorem to random matrices with dependent entries

asymptotics for high dimensional regression m estimates fixed design results we investigate the asymptotic distributions of coordinates of regression m estimates in the moderate p n regime where the number of covariates p grows proportionally with the sample size n under appropriate regularity conditions we establish the coordinate wise asymptotic normality of regression m estimates assuming a fixed design matrix our proof is based on the second order poincare inequality and leave one out analysis some relevant examples are indicated to show that our regularity conditions are satisfied by a broad class of design matrices we also show a counterexample namely an anova type design to emphasize that the technical assumptions are not just artifacts of the proof finally numerical experiments confirm and complement our theoretical results

invertibility of random matrices norm of the inverse let a be an n n matrix whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate we prove that the operator norm of a 1 does not exceed cn3 2 with probability close to 1

mdr codes over z sub k in this correspondence we study maximum distance with respect to rank mdr codes over the ring z sub k we generalize the construction of bose chaudhuri hocquenghem bch and reed solomon codes and apply the generalized chinese remainder theorem to construct codes

poincare inequality in mean value for gaussian polytopes let k n g 1 g n be the absolute convex hull of n independent standard gaussian random points in mathbb r n with n n we prove that for any 1 lipschitz function f mathbb r n rightarrow mathbb r the polytope k n satisfies the following poincare inequality in mean value mathbb e omega int limits k n omega left f x frac 1 textup vol n left k n omega right int limits k n omega f y dy right 2 dx leq frac c n mathbb e omega int limits k n omega x 2dx where c 0 is an absolute constant this poincare inequality is the one suggested by a conjecture of kannan lovasz and simonovits for general convex bodies moreover we prove in mean value that the volume of the polytope k n is concentrated in a subexponential way within a thin euclidean shell with the optimal dependence of the dimension n an important tool of the proofs is a representation of the law of g 1 g n conditioned by the event that the convex hull of g 1 g n is a n 1 face of k n as an application we also get an estimate of the number of n 1 faces of the polytope k n valid for every n n

euclidean embeddings in spaces of finite volume ratio via random matrices let rn be the space rn equipped with a norm whose unit ball has a bounded volume ratio with respect to the euclidean unit ball let be any random n n matrix with n n whose entries are independent random variables satisfying some moment assumptions we show that with high probability is a good isomorphism from the n dimensional euclidean space rn onto its image in rn i e there exist 0 such that for all x rn n x x n x this solves a conjecture of schechtman on random embeddings of 2 into n 1

effect of numerical integration on meshless methods abstract in this paper we present the effect of numerical integration on meshless methods with shape functions that reproduce polynomials of degree k 1 the meshless method was used on a second order neumann problem and we derived an estimate for the energy norm of the error between the exact solution and the approximate solution from the meshless method under the presence of numerical integration this estimate was obtained under the assumption that the numerical integration scheme satisfied a form of green s formula we also indicated how to obtain numerical integration schemes satisfying this property

smooth analysis of the condition number and the least singular value let x be a complex random variable with mean zero and bounded variance let n n be the random matrix of size n whose entries are iid copies of x and let m be a fixed matrix of the same size the goal of this paper is to give a general estimate for the condition number and least singular value of the matrix m n n generalizing an earlier result of spielman and teng for the case when x is gaussian our investigation reveals an interesting fact that the core matrix m does play a role on tail bounds for the least singular value of m n n this does not occur in spielman teng studies when x is gaussian consequently our general estimate involves the norm m in the special case when m is relatively small this estimate is nearly optimal and extends or refines existing results

row products of random matrices abstract let 1 k be d n matrices we define the row product of these matrices as a d k n matrix whose rows are entry wise products of rows of 1 k this construction arises in certain computer science problems we study the question to which extent the spectral and geometric properties of the row product of independent random matrices resemble those properties for a d k n matrix with independent random entries in particular we show that the largest and the smallest singular values of these matrices are of the same order as long as n d k we also consider a problem of privately releasing the summary information about a database and use the previous results to obtain a bound for the minimal amount of noise which has to be added to the released data to avoid a privacy breach

on the geometry of random convex sets between polytopes and zonotopes abstract in this work we study a class of random convex sets that interpolate between polytopes and zonotopes these sets arise from considering a q t h moment q 1 of an average of order statistics of 1 dimensional marginals of a sequence of n n independent random vectors in r n we consider the random model of isotropic log concave distributions as well as the uniform distribution on an l p n sphere 1 p with respect to the cone probability measure and study the geometry of these sets in terms of the support function and mean width we provide asymptotic formulas for the expectation of these geometric functionals which are sharp up to absolute constants our model includes and generalizes the standard one for random polytopes

complete arcs and algebraic curves in pg 2 q a k arc k of pg 2 q is a set of k points no three of which are collinear if q is even then k q 2 if q is odd then k q 1 a k arc is called complete if it is not contained in a k 1 arc b segre considered the following problem what is the smallest k0 q n such that any k arc with q k k0 q is not complete he succeeded in finding two interesting bounds for k0 q using the theorem of hasse weil in the case of a square even q it was proved recently that his bound is the exact value for k0 q in the odd case we slightly improve his bound further we show that the bounds can be obtained without using the hasse weil theorem only applying elementary projective and algebraic geometry finally a short proof is given of segre s theorem stating that for q odd each q arc is contained in a q 1 arc

finding approximately rank one submatrices with the nuclear norm and ell_1 norm we propose a convex optimization formulation with the nuclear norm and ell 1 norm to find a large approximately rank one submatrix of a given nonnegative matrix we develop optimality conditions for the formulation and characterize the properties of the optimal solutions we establish conditions under which the optimal solution of the convex formulation has a specific sparse structure finally we show that under certain hypotheses with high probability the approach can recover the rank one submatrix even when it is corrupted with random noise and inserted as a submatrix into a much larger random noise matrix

smallest singular value of a random rectangular matrix we prove an optimal estimate of the smallest singular value of a random sub gaussian matrix valid for all dimensions for an nn matrix a with inde pendent and identically distributed sub gaussian entries the smallest singular value of a is at least of the order p np n 1 with high probability a sharp

the limit of the smallest singular value of random matrices with i i d entries abstract let a i j 1 i j be i i d real valued random variables with zero mean and unit variance and let an integer sequence n m m 1 satisfy m n m z for some z 0 1 for each m n denote by a m the n m m random matrix a i j 1 i n m 1 j m and let s m a m be its smallest singular value we prove that the sequence n m 1 2 s m a m m 1 converges to 1 z almost surely our result does not require boundedness of any moments of a i j s higher than the 2 nd and resolves a long standing question regarding the weakest moment assumptions on the distribution of the entries sufficient for the convergence to hold

near optimal sample complexity bounds for robust learning of gaussian mixtures via compression schemes we introduce a novel technique for distribution learning based on a notion of sample compression any class of distributions that allows such a compression scheme can be learned with few samples m

on the weights of general mds codes the weight spectra of mds codes of length n and dimension k over arbitrary alphabets are studied for all q ary mds codes of dimension k and length n ne q k 1 containing the zero codeword it is shown that all k weights from n to n k 1 are realized the weight spectrum in the remaining case n q k 1 is also determined additionally we prove that all binary mds codes are equivalent to linear mds codes the proofs are combinatorial and self contained

error analysis of the reproducing kernel particle method interest in meshfree or meshless methods has grown rapidly in recent years in solving boundary value problems bvps arising in mechanics especially in dealing with difficult problems involving large deformation moving discontinuities etc in this paper we provide a theoretical analysis of the reproducing kernel particle method rkpm which belongs to the family of meshfree methods one goal of the paper is to set up a framework for error estimates of rkpm we introduce the concept of a regular family of particle distributions and derive optimal order error estimates for rkp interpolants on a regular family of particle distributions the interpolation error estimates can be used to yield error estimates for rkp solutions of bvps

on delocalization of eigenvectors of random non hermitian matrices we study delocalization of null vectors and eigenvectors of random matrices with i i d entries let a be an n times n random matrix with i i d real subgaussian entries of zero mean and unit variance we show that with probability at least 1 e log 2 n begin aligned min limits i subset n i m vert mathbf v i vert ge frac m 3 2 n 3 2 log cn vert mathbf v vert end aligned for any real eigenvector mathbf v and any m in log c n n where mathbf v i denotes the restriction of mathbf v to i further when the entries of a are complex with i i d real and imaginary parts we show that with probability at least 1 e log 2 n all eigenvectors of a are delocalized in the sense that begin aligned min limits i subset n i m vert mathbf v i vert ge frac m n log cn vert mathbf v vert end aligned for all m in log c n n comparing with related results in the range m in log c n n log c n in the i i d setting and with weaker probability estimates our lower bounds on vert mathbf v i vert strengthen an earlier estimate min nolimits i m vert mathbf v i vert ge c m n 6 vert mathbf v vert obtained in rudelson and vershynin geom funct anal 26 6 1716 1776 2016 and bounds min nolimits i m vert mathbf v i vert ge c m n 2 vert mathbf v vert in the real setting and min nolimits i m vert mathbf v i vert ge c m n 3 2 vert mathbf v vert in the complex setting established in luh and o rourke eigenvector delocalization for non hermitian random matrices and applications arxiv 1810 00489 as the case of real and complex gaussian matrices shows our bounds are optimal up to the polylogarithmic multiples we derive stronger estimates without the polylogarithmic error multiples for null vectors of real n 1 times n random matrices

spectrum of markov generators on sparse random graphs we investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph this is the non hermitian random nxn matrix l defined by l j k x j k if k j and l j j sum l j k k j where x j k are i i d random weights under mild assumptions on the law of the weights we establish convergence as n tends to infinity of the empirical spectral distribution of l after centering and rescaling in particular our assumptions include sparse random graphs such as the oriented erd s renyi graph where each edge is present independently with probability p n 0 as long as np n log n 6 the limiting distribution is characterized as an additive gaussian deformation of the standard circular law in free probability terms this coincides with the brown measure of the free sum of the circular element and a normal operator with gaussian spectral measure the density of the limiting distribution is analyzed using a subordination formula furthermore we study the convergence of the invariant measure of l to the uniform distribution and establish estimates on the extremal eigenvalues of l

single pass randomized algorithms for lu decomposition abstract in this paper we present some single pass randomized algorithms to compute lu decomposition these algorithms need only one pass over the original matrix and hence are very suitable for extremely large and high dimensional matrix stored outside of core memory or generated in a streaming fashion rigorous error bounds and complexity of these algorithms are provided numerical experiments show that these single pass algorithms have the similar accuracy and runtime excluding the cost of matrix transfer compared with the state of the art randomized algorithms for lu decomposition

spectral norm of products of random and deterministic matrices we study the spectral norm of matrices w that can be factored as w ba where a is a random matrix with independent mean zero entries and b is a fixed matrix under the 4 e th moment assumption on the entries of a we show that the spectral norm of such an m n matrix w is bounded by sqrt m sqrt n which is sharp in other words in regard to the spectral norm products of random and deterministic matrices behave similarly to random matrices with independent entries this result along with the previous work of rudelson and the author implies that the smallest singular value of a random m n matrix with i i d mean zero entries and bounded 4 e th moment is bounded below by sqrt m sqrt n 1 with high probability

diameters of sections and coverings of convex bodies abstract we study the diameters of sections of convex bodies in r n determined by a random n n matrix either as kernels of or as images of entries of are independent random variables satisfying some boundedness conditions and typical examples are matrices with gaussian or bernoulli random variables we show that if a symmetric convex body k in r n has one well bounded k codimensional section then for any m ck random sections of k of codimension m are also well bounded where c 1 is an absolute constant it is noteworthy that in the gaussian case when determines randomness in sense of the haar measure on the grassmann manifold we can take c 1

poisson polyhedra in high dimensions abstract the zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated as the space dimension tends to infinity the model includes the zero cell of stationary and isotropic poisson hyperplane tessellations as well as the typical cell of a stationary poisson voronoi tessellation as special cases it is shown that asymptotically in the space dimension with overwhelming probability these cells satisfy the hyperplane conjecture if the distance exponent and the intensity parameter are suitably chosen dimension dependent functions also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed in the background are new identities linking the f vector of the zero cell to certain dual intrinsic volumes

the smallest singular value of a shifted d regular random square matrix we derive a lower bound on the smallest singular value of a random d regular matrix that is the adjacency matrix of a random d regular directed graph specifically let c 1 d c n log 2 n and let mathcal m n d be the set of all n times n square matrices with 0 1 entries such that each row and each column of every matrix in mathcal m n d has exactly d ones let m be a random matrix uniformly distributed on mathcal m n d then the smallest singular value s n m of m is greater than n 6 with probability at least 1 c 2 log 2 d sqrt d where c c 1 and c 2 are absolute positive constants independent of any other parameters analogous estimates are obtained for matrices of the form m z mathrm id where mathrm id is the identity matrix and z is a fixed complex number