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

extended immersed boundary method using fem and rkpm abstract the proposed extended immersed boundary method eibm has several distinct features in comparison with the immersed boundary ib method in the ib method the interaction between the immersed elastic boundary and the surrounding viscous fluid is replaced with an equivalent body force distribution within the fluid domain the key ingredient of the ib method is the enforcement of the energy input from the equivalent body force into the fluid domain to be the same at all time as that of the elastic forces within the ib in eibm instead of the volumeless immersed elastic boundary we consider the submerged elastic solid which occupies a finite volume within the fluid domain in particular we replace the kinematic and dynamic matching of the fluid solid interface and the effect of the submerged solid with a judiciously chosen collection of equivalent nodal forces calculated in the context of finite element formulations the employment of the finite element representation enables a realistic stress analysis for the submerged solid subject to large deformations finally in eibm we also replace the discretized delta function commonly used in the ib method with the kernel functions in meshless reproducing kernel particle method rkpm with this improvement using the same finite support domain we could improve the discretized delta function from c 1 to c n continuity where n is chosen according to the required smoothness and resolution of such functions and more importantly enables the use of non uniform meshing in the fluid domain these extensions may eventually open doors to the modelling of complex biological fluid involving molecular cellular and flexible vessel flow interactions

hex splines a novel spline family for hexagonal lattices this paper proposes a new family of bivariate nonseparable splines called hex splines especially designed for hexagonal lattices the starting point of the construction is the indicator function of the voronoi cell which is used to define in a natural way the first order hex spline higher order hex splines are obtained by successive convolutions a mathematical analysis of this new bivariate spline family is presented in particular we derive a closed form for a hex spline of arbitrary order we also discuss important properties such as their fourier transform and the fact they form a riesz basis we also highlight the approximation order for conventional rectangular lattices hex splines revert to classical separable tensor product b splines finally some prototypical applications and experimental results demonstrate the usefulness of hex splines for handling hexagonally sampled data

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

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

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

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

a discontinuous hp finite element method for diffusion problems 1 d analysis abstract this paper presents the mathematical analysis of a new variant of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems not requiring auxiliary variables such as those used in mixed methods the focus of this study is on a class of linear second order boundary value problems for which we prove stability and a priori error estimates in both the finite and infinite dimensional spaces

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

geometric transformations on the hexagonal grid the hexagonal grid has long been known to be superior to the more traditional rectangular grid system in many aspects in image processing and machine vision related fields however systematic developments of the mathematical backgrounds for the hexagonal grid are conspicuously lacking the purpose of this paper is to study geometric transformations on the hexagonal grid formulations of the transformation matrices are carried out in a symmetrical hexagonal coordinate frame a trio of new trigonometric functions are defined in this paper to facilitate the rotation transformations a fast algorithm for rounding an arbitrary point to the nearest hexagonal grid point is also presented

efficient algorithms for computing the 2 d hexagonal fourier transforms in this paper representations of the two dimensional 2 d signals are presented that reduce the computation of 2 d discrete hexagonal fourier transforms 2 d dhfts the representations are based on the concept of the covering that reveals the mathematical structure of the transforms specifically a set of unitary paired transforms is derived that splits the 2 d dhft into a number of smaller one dimensional 1 d dfts examples for the 8 spl times 4 and 16 spl times 8 hexagonal lattices are described in detail the number of multiplications required for computing the 8 spl times 4 and 16 spl times 8 point dhfts are equal 20 and 136 respectively in the general n spl ges 8 case the number of multiplications required to compute the 2n spl times n point dhft by the paired transforms equals n sup 2 log n 1 n

design and evaluation of more accurate gradient operators on hexagonal lattices digital two dimensional images are usually sampled on square lattices while the receptors of the human eye are following a hexagonal structure it is the main motivation for adopting hexagonal lattices the fundamental operation in many image processing algorithms is to extract the gradient information as such various gradient operators have been proposed for square lattices and have been thoroughly optimized accurate gradient operators for hexagonal lattices have however not been researched well enough while the distance between neighbor pixels is constant we therefore derive consistent gradient operators on hexagonal lattices and compare them with the existing optimized filters on square lattices the results show that the derived filters on hexagonal lattices achieve a better signal to noise ratio than those on square lattices results on artificial images also show that the derived filters on hexagonal lattices outperform the square ones with respect to accuracy of gradient intensity and orientation detection

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

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

an embedded mesh method using piecewise constant multipliers with stabilization mathematical and numerical aspects summary an embedded mesh method using piecewise constant multipliers originally proposed by puso et al cmame 2012 is analyzed here to determine effects of the pressure stabilization term and small cut cells the approach is implemented for transient dynamics using the central difference scheme for the time discretization it is shown that the resulting equations of motion are a stable linear system with a condition number independent of mesh size next it is shown that the constraints and the stabilization terms can be recast as non proportional damping such that the time integration of the scheme is provably stable with a critical time step computed from the undamped equations of motion effects of small cuts are discussed throughout the presentation a mesh study is conducted to evaluate the effects of the stabilization on the discretization error and conditioning and is used to recommend an optimal value for stabilization scaling parameter several nonlinear problems are also analyzed and compared with comparable conforming mesh results finally several demanding problems highlighting the robustness of the proposed approach are shown copyright 2014 john wiley sons ltd

the meshless finite element method a meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n and the condition that the shape functions depend only on the node positions furthermore the method proposed also shares several of the advantages of the finite element method such as a the simplicity of the shape functions in a large part of the domain b c0 continuity between elements which allows the treatment of material discontinuities and c ease of introduction of the boundary conditions copyright 2003 john wiley sons ltd

nitsche s method combined with space time finite elements for ale fluid structure interaction problems we propose a weak method for handling the fluid structure interface in finite element fluid structure interaction based on nitsche s method abh math univ hamburg 36 1971 9 we assume transient incompressible newtonian flow and for the structure undamped linear elasticity for the time discretization we use the time continuous energy conserving galerkin method for the structure and for the fluid we employ the time discontinuous galerkin method this means that the velocity becomes piecewise constant on each timestep for the fluid matching the time derivative of the displacements in the solid which is also piecewise constant over a time step we formulate the method and report some numerical examples using space time oriented elements for the fluid in order to mimic lagrangian or ale type simulations

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

immersed finite element method abstract in this paper the immersed finite element method ifem is proposed for the solution of complex fluid and deformable structure interaction problems encountered in many physical models in ifem a lagrangian solid mesh moves on top of a background eulerian fluid mesh which spans over the entire computational domain hence the mesh generation is greatly simplified moreover both fluid and solid domains are modeled with the finite element methods and the continuity between the fluid and solid sub domains are enforced via the interpolation of the velocities and the distribution of the forces with the reproducing kernel particle method rkpm delta function in comparison with the immersed boundary ib method the higher ordered rkpm delta function enables the fluid domain to have nonuniform spatial meshes with arbitrary geometries and boundary conditions the use of such kernel functions may eventually open doors to multi scale and multi resolution modelings of complex fluid structure interaction problems rigid and deformable spheres dropping in channels are simulated to demonstrate the unique capabilities of the proposed method the results compare well with the experimental data to the authors knowledge these are the first solutions that deal with particulate flows with very flexible solids

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

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

an enriched finite element method and level sets for axisymmetric two phase flow with surface tension a finite element method for axisymmetric two phase flow problems is presented the method uses an enriched finite element formulation in which the interface can move arbitrarily through the mesh without remeshing the enrichment is implemented by the extended finite element method x fem which models the discontinuity in the velocity gradient at the interface by a local partition of unity it provides an accurate representation of the velocity field at interfaces on an eulerian grid that is not conformal to the weak discontinuity the interface is represented by a level set which is also used in the construction of the element enrichment surface tension effects are considered and the interface curvature is computed from the level set field the method is demonstrated by several examples copyright 2003 john wiley sons ltd

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

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

on methods for stabilizing constraints over enriched interfaces in elasticity summary enriched finite element approaches such as the extended finite element method provide a framework for constructing approximations to solutions of non smooth problems internal features such as boundaries are represented in such methods by using discontinuous enrichment of the standard finite element basis within such frameworks however imposition of interface constraints and or constitutive relations can cause unexpected difficulties depending upon how relevant fields are interpolated on un gridded interfaces this work address the stabilized treatment of constraints in an enriched finite element context both the lagrange multiplier and penalty enforcement of tied constraints for an arbitrary boundary represented in an enriched finite element context can lead to instabilities and artificial oscillations in the traction fields we demonstrate two alternative variational methods that can be used to enforce the constraints in a stable manner in a bubble stabilized approach fine scale degrees of freedom are added over elements supporting the interface the variational form can be shown to have a similar form to a second approach we consider nitsche s method with the exception that the stabilization terms follow directly from the bubble functions in this work we examine alternative variational methods for enforcing a tied constraint on an enriched interface in the context of two dimensional elasticity we examine several benchmark problems in elasticity and show that only nitsche s method and the bubble stabilization approach produce stable traction fields over internal boundaries we also demonstrate a novel difference between the penalty method and nitsche s method in that the latter passes the patch test exactly regardless of the stabilization parameter s magnitude results for more complicated geometries and triple interface junctions are also presented copyright 2008 john wiley sons ltd

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

subdivision stabilised immersed b spline finite elements for moving boundary flows an immersed finite element method is presented to compute flows with complex moving boundaries on a fixed cartesian grid the viscous incompressible fluid flow equations are discretized with b spline basis functions the two scale relation for b splines is used to implement an elegant and efficient technique to satisfy the lbb condition on non grid aligned fluid domains and at moving boundaries the boundary conditions are enforced with a consistent penalty method as originally proposed by nitsche in addition a special extrapolation technique is employed to prevent the loss of numerical stability in presence of arbitrarily small cut cells the versatility and accuracy of the proposed approach is demonstrated by means of convergence studies and comparisons with previous experimental and computational investigations

on the determination of functions from their integral values along certain manifolds when one integrates a function of two variables x y a point function f p in the plane subject to suitable regularity conditions along an arbitrary straight line g then one obtains in the integral values f g a line function in part a of the present paper the problem which is solved is the inversion of this linear functional transformation that is the following questions are answered can every line function satisfying suitable regularity conditions be regarded as constructed in this way if so is f uniquely known from f and how can f be calculated in part b a solution of the dual problem of calculating a line function f g from its point mean values f p is solved in a certain sense finally in part c certain generalizations are discussed prompted by consideration of non euclidean manifolds as well as higher dimensional spaces the treatment of these problems themselves of interest gains enhanced importance through the numerous relationships that exist between this topic and the theory of logarithmic and newtonian potentials these are mentioned at appropriate places in the text

a unified treatment of cooley tukey algorithms for the evaluation of the multidimensional dft in this paper the cooley tukey fast fourier transform fft algorithm is generalized to the multidimensional case in a natural way which allows for the evaluation of discrete fourier transforms of rectangularly or hexagonally sampled signals or of signals which are sampled on an arbitrary periodic grid in either the spatial or fourier domain this general algorithm incorporates both the traditional rectangular row column and vector radix algorithms as special cases this fft algorithm is shown to result from the factorization of an integer matrix for each factorization of that matrix a different algorithm can be developed this paper presents the general algorithm discusses its computational efficiency and relates it to existing multi dimensional fft algorithms

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

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

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

an embedded mesh method in a multiple material ale abstract a new approach for treating the mechanical interactions of overlapping finite element meshes is presented referred to as embedded mesh methods here these overlapping mesh methods typically include a foreground solid mesh and a background euler fluid grid or solid mesh a number of different approaches have been used in previous work to characterize the interactions of the background and foreground meshes at the interface lagrange multipliers are well suited to enforce the continuity constraints but care must be taken such that the resulting formulation is stable several lagrange multiplier techniques are examined in this work and applied to coupling solid meshes and fluid structure interaction type problems in addition details regarding implementation in a two step multi material arbitrary lagrangian eulerian ale code are presented example problems demonstrate convergence and applicability to a range of problems in particular the fluid structure interaction examples focus on blast applications

the discontinuous enrichment method for elastic wave propagation in the medium frequency regime the discontinuous enrichment method dem is specified and developed for the solution of two dimensional elastic wave propagation problems in the frequency domain the classical finite element polynomial approximation of the displacement field is enriched by the superposition of discontinuous pressure and shear wave functions the continuity of the solution across the element interfaces is weakly enforced by suitable lagrange multipliers higher order rectangular dem elements are constructed and benchmarked against the standard higher order polynomial galerkin elements for two dimensional problems in the medium frequency regime in general it is found that for such applications dem can achieve the same accuracy as the p finite element method using a similar computational complexity but with about 10 times fewer degrees of freedom this highlights the potential of this hybrid method for problems where the scale of vibrations is very small compared to the characteristic dimension of the physical medium copyright 2006 john wiley sons ltd

an extended finite element method lagrange multiplier based approach for fluid structure interaction abstract this paper presents a new fixed grid fluid structure interaction scheme that can be applied to the interaction of most general structures with incompressible flow it is based on an extended finite element method xfem based strategy the extended eulerian fluid field and the lagrangian structural field are partitioned and iteratively coupled using lagrange multiplier techniques for non matching grids the approach allows the simulation of the interaction of thin and bulky structures exhibiting large deformations finally qualitative examples and a benchmark computation demonstrate key features and accuracy of the method

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

fast algorithms for the multidimensional discrete fourier transform in this paper the prime factor algorithm for the evaluation of a one dimensional discrete fourier transform is generalized to the evaluation of multidimensional discrete fourier transforms defined on arbitrary periodic sampling lattices it is shown that such an algorithm is equivalent in computational complexity to the evaluation of a rectangular discrete fourier transform as a sidelight to the derivation of the algorithm a chinese remainder theorem is derived for integer lattices

the immersed fictitious element method for fluid structure interaction volumetric consistency compressibility and thin members a weak form and an implementation are given for fluid structure interaction by the immersed fictitious element method for compressible fluids the weak form is applicable to models where the fluid is described by eulerian coordinates while the solid is described by lagrangian coordinates which suits their intrinsic characteristics a unique feature of the method is the treatment of the fictitious fluid by a lagrangian description which simplifies the interface conditions methods for enforcing volumetric consistency between the fluid and solid and treating thin members are given although a compressible viscous fluid is considered here the new developments can be applied to incompressible fluids

dislocations by partition of unity a new finite element method for accurately modelling the displacement and stress fields produced by a dislocation is proposed the methodology is based on a local enrichment of the finite element space by closed form solutions for dislocations in infinite media via local partitions of unity this allows the treatment of both arbitrary boundary conditions and interfaces between materials the method can readily be extended to arrays of dislocations 3d problems large strains and non linear constitutive models results are given for an edge dislocation in a hollow cylinder and in an infinite medium for the cases of a glide plane intersecting a rigid obstacle and an interface between two materials copyright 2005 john wiley sons ltd

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

2 d and 1 d multipaired transforms frequency time type wavelets a concept of multipaired unitary transforms is introduced these kinds of transforms reveal the mathematical structure of fourier transforms and can be considered intermediate unitary transforms when transferring processed data from the original real space of signals to the complex or frequency space of their images considering paired transforms we analyze simultaneously the splitting of the multidimensional fourier transform as well as the presentation of the processed multidimensional signal in the form of the short one dimensional 1 d signals that determine such splitting the main properties of the orthogonal system of paired functions are described and the matrix decompositions of the fourier and hadamard transforms via the paired transforms are given the multiplicative complexity of the two dimensional 2 d 2 sup r spl times 2 sup r point discrete fourier transform by the paired transforms is 4 sup r 2 r 7 3 8 3 12 r 3 which shows the maximum splitting of the 5 d fourier transform into the number of the short 1 d fourier transforms the 2 d paired transforms are not separable and represent themselves as frequency time type wavelets for which two parameters are united frequency and time the decomposition of the signal is performed in a way that is different from the traditional haar system of functions

discontinuous galerkin methods for non linear elasticity this paper presents the formulation and a partial analysis of a class of discontinuous galerkin methods for quasistatic non linear elasticity problems these methods are endowed with several salient features the equations that define the numerical scheme are the euler lagrange equations of a one field variational principle a trait that provides an elegant and simple derivation of the method in consonance with general discontinuous galerkin formulations it is possible within this framework to choose different numerical fluxes numerical evidence suggests the absence of locking at near incompressible conditions in the finite deformations regime when piecewise linear elements are adopted finally a conceivable surprising characteristic is that as demonstrated with numerical examples these methods provide a given accuracy level for a comparable and often lower computational cost than conforming formulations stabilization is occasionally needed for discontinuous galerkin methods in linear elliptic problems in this paper we propose a sufficient condition for the stability of each linearized non linear elastic problem that naturally includes material and geometric parameters the latter needed to account for buckling we then prove that when a similar condition is satisfied by the discrete problem the method provides stable linearized deformed configurations upon the addition of a standard stabilization term we conclude by discussing the complexity of the implementation and propose a computationally efficient approach that avoids looping over both elements and element faces several numerical examples are then presented in two and three dimensions that illustrate the performance of a selected discontinuous galerkin method within the class copyright 2006 john wiley sons ltd

a local space time discontinuous finite element method we present a locally enriched space time finite element method for solving hyperbolic problems with discontinuities the space time formulation is coupled to a standard semidiscrete finite element method the discontinuities are captured by a space time version of the extended finite element method which treats arbitrary moving discontinuities since the discontinuities are local the enriched space time method is only needed around the discontinuities which provides significant computational savings the coupling is implemented through a weak enforcement of the continuity of the flux between the space time and semidiscrete domains in a manner similar to discontinuous galerkin methods the method is illustrated through one dimensional problems it displays remarkable accuracy in capturing moving discontinuities both in their amplitude and speed

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

fluid structure interaction by the discontinuous galerkin method for large deformations a discontinuous galerkin method for large deformation fluid structure interaction problems is developed the fluid structure interface can be arbitrarily aligned relative to the fluid grid an eulerian description is used for the fluid with a lagrangian description of the solid results are presented for several examples that show excellent agreement with immersed fictitious element methods copyright 2008 john wiley sons ltd

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

an eulerian lagrangian method for fluid structure interaction based on level sets abstract we describe a method for fluid structure interaction with the interface and free surfaces defined by level sets the fluid is treated by an eulerian mesh whereas the solid or structure by a lagrangian mesh the level set method provides a compact framework for writing the weak form and for developing and computing the discretized equations lagrange multiplier methods and penalty methods are used to couple the fluid and structure results are presented for a few compressible fluid problems representative of high pressure waves

a general discontinuous galerkin method for finite hyperelasticity formulation and numerical applications a discontinuous galerkin formulation of the boundary value problem of finite deformation elasticity is presented the primary purpose is to establish a discontinuous galerkin framework for large deformations of solids in the context of statics and simple material behaviour with a view toward further developments involving behaviour or models where the dg concept can show its superiority compared to the continuous formulation the method is based on a general hu washizu de veubeke functional allowing for displacement and stress discontinuities in the domain interior it is shown that this approach naturally leads to the formulation of average stress fluxes at interelement boundaries in a finite element implementation the consistency and linearized stability of the method in the non linear range as well as its convergence rate are proven an implementation in three dimensions is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward manner in order to demonstrate the versatility accuracy and robustness of the method examples of application and convergence studies in three dimensions are provided copyright 2006 john wiley sons ltd

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

a symmetric positive definite formulation for monolithic fluid structure interaction in this paper we consider a strongly coupled monolithic fluid structure interaction framework for incompressible flow as opposed to a loosely coupled partitioned method this requires solving a single linear system that combines the unknown velocities of the structure with the unknown pressures of the fluid in our previous work we were able to obtain a symmetric formulation of this coupled system however it was also indefinite making it more difficult to solve in fact in practice there have been cases where we have been unable to invert the system in this paper we take a novel approach that consists of factoring the damping matrix of deformable structures and show that this can be used to obtain a symmetric positive definite system at least to the extent that the uncoupled systems were symmetric positive definite we use a traditional mac grid discretization of the fluid and a fully lagrangian discretization of the structures for the sake of exposition noting that our procedure can be generalized to other scenarios for the special case of rigid bodies where there are no internal damping forces we exactly recover the system of batty et al 2007 4