improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates the problem of estimating the eigenvalues and eigenvectors of the covariance matrix associated with a multivariate stochastic process is considered the focus is on finite sample size situations whereby the number of observations is limited and comparable in magnitude to the observation dimension using tools from random matrix theory and assuming a certain eigenvalue splitting condition new estimators of the eigenvalues and eigenvectors of the covariance matrix are derived that are shown to be consistent in a more general asymptotic setting than the traditional one indeed these estimators are proven to be consistent not only when the sample size increases without bound for a fixed observation dimension but also when the observation dimension increases to infinity at the same rate as the sample size numerical evaluations indicate that the estimators have an excellent performance in small sample size scenarios where the observation dimension and the sample size are comparable in magnitude

quantitative homogenization in nonlinear elasticity for small loads we study quantitative periodic homogenization of integral functionals in the context of nonlinear elasticity under suitable assumptions on the energy densities in particular frame indifference minimality non degeneracy and smoothness at the identity p geqq d growth from below and regularity of the microstructure we show that in a neighborhood of the set of rotations the multi cell homogenization formula of non convex homogenization reduces to a single cell formula the latter can be expressed with the help of correctors we prove that the homogenized integrand admits a quadratic taylor expansion in an open neighborhood of the rotations a result that can be interpreted as the fact that homogenization and linearization commute close to the rotations moreover for small applied loads we provide an estimate on the homogenization error in terms of a quantitative two scale expansion

channel capacity estimation using free probability theory in many channel measurement applications one needs to estimate some characteristics of the channels based on a limited set of measurements this is mainly due to the highly time varying characteristics of the channel in this paper it will be shown how free probability can be used for channel capacity estimation in mimo systems free probability has already been applied in various application fields such as digital communications nuclear physics and mathematical finance and has been shown to be an invaluable tool for describing the asymptotic behavior of many large dimensional systems in particular using the concept of free deconvolution we provide an asymptotically with respect to the number of observations unbiased capacity estimator for mimo channels impaired with noise called the free probability based estimator another estimator called the gaussian matrix mean based estimator is also introduced by slightly modifying the free probability based estimator this estimator is shown to give unbiased estimation of the moments of the channel matrix for any number of observations also the estimator has this property when we extend to mimo channels with phase off set and frequency drift for which no estimator has been provided so far in the literature it is also shown that both the free probability based and the gaussian matrix mean based estimator are asymptotically unbiased capacity estimators as the number of transmit antennas go to infinity regardless of whether phase off set and frequency drift are present the limitations in the two estimators are also explained simulations are run to assess the performance of the estimators for a low number of antennas and samples to confirm the usefulness of the asymptotic results

uniqueness of equilibrium with sufficiently small strains in finite elasticity the uniqueness of equilibrium for a compressible hyperelastic body subject to dead load boundary conditions is considered it is shown for both the displacement and mixed problems that there cannot be two solutions of the equilibrium equations of finite nonlinear elasticity whose nonlinear strains are uniformly close to each other this result is analogous to the result of john commun pure appl math 25 617 634 1972 who proved that for the displacement problem there is a most one equilibrium solution with uniformly small strains the proof in this manuscript utilizes geometric rigidity a new straightforward extension of the fefferman stein inequality to bounded domains and an appropriate adaptation for elasticity of a result from the calculus of variations specifically it is herein shown that the uniform positivity of the second variation of the energy at an equilibrium solution implies that this mapping is a local minimizer of the energy among deformations whose gradient is sufficiently close in rm bmo cap l 1 to the gradient of the equilibrium solution

on morphological operators based on rank filters rank order morphological operators have been designed to better cope with shape variations and noise than the corresponding operators with plain structuring elements in this paper starting from a brief historical overview of rank order filters and erosions dilations in digital image processing rank order based morphological operators are surveyed we also highlight the relations between these operators and show that many are similar if not equivalent an extensive bibliography is provided

a condition for the holder regularity of local minimizers of a nonlinear elastic energy in two dimensions we prove the local holder continuity of strong local minimizers of the stored energy functional e u int omega lambda nabla u 2 h rm det nabla u rm d x subject to a condition of positive twist the latter turns out to be equivalent to requiring that u maps circles to suitably star shaped sets the convex function h s grows logarithmically as s to 0 linearly as s to infty and satisfies h s infty if s leqq 0 these properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers the main innovation is to prove that if a local minimizer has positive twist a e on a ball then an euler lagrange type inequality holds and a caccioppoli inequality can be derived from it the claimed holder continuity then follows by adapting some well known elliptic regularity theory we also demonstrate the regularizing effect that the term int omega h rm det nabla u rm d x can have by analysing the regularity of local minimizers in the class of shear maps in this setting a more easily verifiable condition than that of positive twist is imposed with the result that local minimizers are holder continuous

weak lower semicontinuity of polyconvex integrals multiple integrals with polyconvex integrands are studied on the class of all sense preserving diffeomorphisms from w 1 p r n where is an open subset of r n they are proved to be sequentially weakly lower semicontinuous if 1 p n 1 an example is presented showing that a similar result is not valid if p n 1

invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity in this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture the main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface accordingly we define a functional that measures the area of the created surface this functional has relationships with the theory of cartesian currents we show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient and that the weak limit of one to one almost everywhere deformations is also one to one almost everywhere we then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy taking into account orientation preserving and non interpenetration conditions

partial regularity of strong local minimizers in the multi dimensional calculus of variations let n be a bounded domain and f a given strongly quasiconvex integrand of class c2 satisfying the growth condition f xi le c 1 xi p for some c 0 and 2 p consider the multiple integral i u int omega f nabla u where u w1 p n the main result of the paper is the proof that any strong local minimizer of i is of class c1 loc for any 0 1 on an open set of full n dimensional measure in the case of weak local minimizers we establish the same result under the extra assumption that the oscillations in the gradient of the minimizer are not too large without such an assumption weak local minimizers need not be partially regular as we show by a class of examples we also briefly discuss the question of existence of strong local minimizers for i and connections of our results to extensions of weierstrass sufficiency theorem to the multi dimensional setting

statistical eigen inference from large wishart matrices we consider settings where the observations are drawn from a zero mean multivariate real or complex normal distribution with the population covariance matrix having eigenvalues of arbitrary multiplicity we assume that the eigenvectors of the population covariance matrix are unknown and focus on inferential procedures that are based on the sample eigenvalues alone i e eigen inference results found in the literature establish the asymptotic normality of the fluctuation in the trace of powers of the sample covariance matrix we develop concrete algorithms for analytically computing the limiting quantities and the covariance of the fluctuations we exploit the asymptotic normality of the trace of powers of the sample covariance matrix to develop eigenvalue based procedures for testing and estimation specifically we formulate a simple test of hypotheses for the population eigenvalues and a technique for estimating the population eigenvalues in settings where the cumulative distribution function of the nonrandom population eigenvalues has a staircase structure monte carlo simulations are used to demonstrate the superiority of the proposed methodologies over classical techniques and the robustness of the proposed techniques in high dimensional relatively small sample size settings the improved performance results from the fact that the proposed inference procedures are global in a sense that we describe and exploit global information thereby overcoming the inherent biases that cripple classical inference procedures which are local and rely on local information

regularity properties of deformations with finite energy the purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

performance of linear field reconstruction techniques with noise and uncertain sensor locations we consider a wireless sensor network sampling a bandlimited field described by a limited number of harmonics sensor nodes are irregularly deployed over the area of interest or subject to random displacement in addition sensors measurements are affected by noise our goal is to obtain a high quality reconstruction of the field with the mean square error mse of the estimate as performance metric in particular we analytically derive the performance of several reconstruction estimation techniques based on linear filtering for each technique we obtain the mse as well as its asymptotic expression in the case where the number of field harmonics and the number of sensors grow to infinity while their ratio is kept constant through numerical simulations we show the validity of the asymptotic analysis even for a small number of sensors we provide some novel guidelines for the design of sensor networks when many parameters such as field bandwidth number of sensors reconstruction quality and sensor displacement characteristics to be traded off

two dimensional tensor function representation for all kinds of material symmetry all kinds of physically possible material symmetry in two dimensional space were investigated in a recent work of q s zheng and j p boehler in this paper we establish the complete and irreducible representations with respect to every kind of material symmetry for scalar vector and second order tensor valued functions in two dimensional space of any finite number of vectors and second order tensors these representations allow general invariant forms of physical and constitutive laws of anisotropic materials to be developed in plane problems

synthesis of multivalued multithreshold functions for ccd implementation basic multivalued building blocks constructed using ccd charge coupled devices technology are presented they are used to realize simple threshold functions existing techniques for decomposition of multivalued multithreshold functions into simpler subfunctions are reviewed usage of the ccd technology in implementation of these subfunctions is discussed two new decomposition techniques are proposed their aim is to obtain realizations that are better suited to ccd technology in terms of the maximum number of thresholds per element and the generation of negative weights synthesis techniques based on the proposed decompositions are presented and compared although 4 valued logic functions are stressed the techniques are in principle applicable to functions of any radix

fluid films with curvature elasticity the phenomenology of surfactant fluid film microstructures interspersed in bulk fluids poses significant challenges to continuum theory by using simple models of elastic surfaces chemical physicists have been partially successful in describing the qualitative features of

stack filters the median and other rank order operators possess two properties called the threshold decomposition and the stacking properties the first is a limited superposition property which leads to a new architecture for these filters the second is an ordering property which allows an efficient vlsi implementation of the threshold decomposition architecture motivated by the success of rank order filters in a wide variety of applications and by the ease with which they can now be implemented we consider in this paper a new class of filters called stack filters they share the threshold decomposition and stacking properties of rank order filters but are otherwise unconstrained they are shown to form a very large class of easily implemented nonlinear filters which includes the rank order operators as well as all compositions of morphological operators the convergence properties of these filters are investigated using techniques similar to those used to determine root signal behavior of median filters the results obtained include necessary conditions for a stack filter to preserve monotone regions or edges in signals the output distribution for these filters is also found all the stack filters of window width 3 are determined along with their convergence properties among these filters are found two which we have named asymmetric median filters they share all the properties of median filters except that they remove impulses of one sign only that is one removes only positive going edges the other removes only negative going edges while the median filter removes impulses of both signs this investigation of the properties of stack filters thus produces several new useful and easily implemented filters

surface interaction potentials in elasticity the problem of modelling body environment interactions is a rather formidable one neither is there a wealth of inspiring examples nor is the corpus of knowledge in mathematical analysis such as to allow indiscriminate generalization of the few well understood cases

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

a theoretical analysis of the properties of median filters necessary and sufficient conditions for a signal to be invariant under a specific form of median filtering are derived these conditions state that a signal must be locally monotone to pass through a median filter unchanged it is proven that the form of successive median filtering of a signal i e the filtered output is itself again filtered eventually reduces the original signal to an invariant signal called a root signal for a signal of length l samples a maximum of frac 1 2 l 2 repeated filterings produces a root signal

infinitely divisible distributions for rectangular free convolution classification and matricial interpretation in a previous paper benaych georges in related convolution 2006 we defined the rectangular free convolution here we investigate the related notion of infinite divisibility which happens to be closely related the classical infinite divisibility there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of infinitely divisible distributions which preserves limit theorems we give an interpretation of this correspondence in terms of random matrices we construct distributions on sets of complex rectangular matrices which give rise to random matrices with singular laws going from the symmetric classical infinitely divisible distributions to their infinitely divisible correspondents when the dimensions go from one to infinity in a ratio

a non classical mindlin plate model incorporating microstructure surface energy and foundation effects a non classical model for a mindlin plate resting on an elastic foundation is developed in a general form using a modified couple stress theory a surface elasticity theory and a two parameter winkler pasternak foundation model it includes all five kinematic variables possible for a mindlin plate the equations of motion and the complete boundary conditions are obtained simultaneously through a variational formulation based on hamilton s principle and the microstructure surface energy and foundation effects are treated in a unified manner the newly developed model contains one material length scale parameter to describe the microstructure effect three surface elastic constants to account for the surface energy effect and two foundation parameters to capture the foundation effect the current non classical plate model reduces to its classical elasticity based counterpart when the microstructure surface energy and foundation effects are all suppressed in addition the new model includes the mindlin plate models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases recovers the kirchhoff plate model incorporating the microstructure surface energy and foundation effects and degenerates to the timoshenko beam model including the microstructure effect to illustrate the new mindlin plate model the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulae derived

explicit examples of lipschitz one homogeneous solutions of log singular planar elliptic systems abstract we give examples of systems of partial differential equations that admit non trivial lipschitz and one homogeneous solutions in the form u r r g where r are plane polar coordinates and g takes values in r m m 2 the systems are singular in the sense that they arise as the euler lagrange equations of the functionals i u b w x u x d x where d f w x f behaves like 1 x as x 0 and w satisfies an ellipticity condition such solutions cannot exist when x d f w x f 0 as x 0 so the condition is optimal the associated analysis exploits the well known fefferman stein duality fefferman and stein 1972 we also discuss conditions for the uniqueness of these one homogeneous solutions and demonstrate that they are minimizers of certain variational functionals

order configuration functions mathematical characterizations and applications to digital signal and image processing we call a function f in n variables an order configuration function if for any x1 xn such that xi1 xin we have f x1 xn xt where t is determined by the n tuple i1 in corresponding to that ordering equivalently it is a function built as a minimum of maxima or a maximum of minima well known examples are the minimum the maximum the median and more generally rank functions or the composition of rank functions such types of functions are often used in nonlinear processing of digital signals or images for example in the median or separable median filter min max filters rank filters etc in this paper we study the mathematical properties of order configuration functions and of a wider class of functions that we call order subconfiguration functions we give several characterization theorems for them we show through various examples how our concepts can be used in the design of digital signal filters or image transformations based on order configuration functions

a theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity the energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by g kirchhoff in 1850 we show that it arises as a 0 limit of three dimensional nonlinear elasticity theory as the thickness of a plate goes to zero a key ingredient in the proof is a sharp rigidity estimate for maps v v u r n u r n we show that the l 2 distance of rv from a single rotation matrix is bounded by a multiple of the l 2 distance from

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

a measure and orientation preserving homeomorphism with approximate jacobian equal 1 almost everywhere we construct an almost everywhere approximately differentiable orientation and measure preserving homeomorphism of a unit n dimensional cube onto itself whose jacobian is equal to 1 almost everywhere moreover we prove that our homeomorphism can be uniformly approximated by orientation and measure preserving diffeomorphisms

a dual parametric finite element method for cavitation in nonlinear elasticity a dual parametric finite element method is introduced in this paper for the computation of singular minimizers in the 2d cavitation problem in nonlinear elasticity the method overcomes the difficulties such as the mesh entanglement and material interpenetration generally encountered in the finite element approximation of problems with extremely large expansionary deformation numerical experiments show that the method is highly efficient in the computation of cavitation problems numerical experiments are also conducted on discrete problems without the radial symmetry to show the validity of the method to more general settings and the potential of its application to the study of mechanism of cavity nucleation in nonlinear elastic materials

eigen inference for energy estimation of multiple sources in this paper a new method is introduced to blindly estimate the transmit power of multiple signal sources in multiantenna fading channels when the number of sensing devices and the number of available samples are sufficiently large compared to the number of sources recent advances in the field of large dimensional random matrix theory are used that result in a simple and computationally efficient consistent estimator of the power of each source a criterion to determine the minimum number of sensors and the minimum number of samples required to achieve source separation is then introduced simulations are performed that corroborate the theoretical claims and show that the proposed power estimator largely outperforms alternative power inference techniques

the convergence of regularized minimizers for cavitation problems in nonlinear elasticity consider a nonlinearly elastic body which occupies the region omega subset mathbb r m m 2 3 in its reference state and which is held in tension under prescribed boundary displacements on partial omega let bf x 0 in omega be any fixed point in the body it is known from variational arguments that for sufficiently large boundary displacements there may exist discontinuous weak solutions of the equilibrium equations corresponding to a hole forming at bf x 0 in the deformed body this is the phenomenon of cavitation for each epsilon 0 define the regularized domains omega epsilon omega backslash overline b epsilon bf x 0 which contain a preexisting hole of radius epsilon 0 centered on bf x 0 now consider the corresponding mixed displacement traction problem on omega epsilon in which the boundary partial omega is subject to the same boundary displacements and the deformed cavity surface i e the image of partial b epsilon is requi

eigenvalue results for large scale random vandermonde matrices with unit complex entries this paper centers on the limit eigenvalue distribution for random vandermonde matrices with unit magnitude complex entries the phases of the entries are chosen independently and identically distributed from the interval various types of distribution for the phase are considered and we establish the existence of the empirical eigenvalue distribution in the large matrix limit on a wide range of cases the rate of growth of the maximum eigenvalue is examined and shown to be no greater than o logn and no slower than o logn loglogn where n is the dimension of the matrix additional results include the existence of the capacity of the vandermonde channel limit integral for the expected log determinant

quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity in this paper we prove the uniqueness of the solution to certain simple displacement boundary value problems in the nonlinear theory of homogeneous hyperelasticity for a body occupying a star shaped reference configuration rn whose boundary is subjected to an affine deformation i e there exists a constant n n matrix f and a constant n vector b such that x fx b for all x we consider all smooth equilibrium configurations satisfying this boundary condition clearly the homogeneous deformation x fx b for all x in bar omega is one such solution our aim is to prove under suitable hypotheses that it is the only such solution

cartesian currents weak diffeomorphisms and existence theorems in nonlinear elasticity in this paper we introduce some new classes of functions among these a class of weak diffeomorphisms in these classes we prove by direct methods the existence of minimizers for several kinds of variational integrals in particular we prove the existence of one to one orientation preserving maps that minimize suitable energies associated with hyperelastic materials the minimizers are also proved to satisfy equilibrium equations finally radial deformations are discussed in connection with cavitation

existence theorem for a minimum problem with free discontinuity set we study the variational problem where is an open set in n n 2g l q l 1 q o andh n 1 is the n 1 dimensional hausdorff measure

a separable median filter for image noise smoothing this paper investigates some properties of the separable filter resulting from successive applications of a one dimensional median filter on the rows and columns of an image although the output of this separable filter is not identical to the corresponding nonseparable two dimensional median filter with a square window its performance in image noise smoothing is close in particular its effectiveness in smoothing noise and its behavior with edges are characterized and compared with those of the two dimensional median filter it is shown that the separable filter has a much simpler implementation in real time hardware at video rates for example

idempotent and co idempotent stack filters and min max operators viewing the elements of rs as images f with pixels i s of grey scale value f i motivates the study of certain nonlinear operators rs rs for translation invariant called stack filters in the signal processing literature we derive the first necessary and sufficient condition for idempotency which can be tested in polynomial time various related properties can be tested in polynomial time as well and many results still apply when the linear lattice r is replaced by an arbitrary distributive lattice r or when the condition of translation invariance is dropped although the main application is in nonlinear image processing several other fields will be touched upon

rigidity and gamma convergence for solid solid phase transitions with so 2 invariance the singularly perturbed two well problem in the theory of solid solid phase transitions takes the form i e u 1 ew u e 2 u 2 where u r n r n is the deformation and w vanishes for all matrices in k so n a u so n b we focus on the case n 2 and derive by means of gamma convergence a sharp interface limit for i e the proof is based on a rigidity estimate for low energy functions our rigidity argument also gives an optimal two well liouville estimate if u has a small bv norm compared to the diameter of the domain then in the l 1 sense either the distance of u from so 2 a or the one from so 2 b is controlled by the distance of vu from k this implies that the oscillation of vu in weak l1 is controlled by the l1 norm of the distance of u to k

asymptotic moments of some random vandermonde matrices abstract appropriately normalized square random vandermonde matrices based on independent random variables with uniform distribution on the unit circle are studied it is shown that as the matrix sizes increases without bound with respect to the expectation of the trace there is an asymptotic distribution equal to that of a c 0 1 valued r diagonal element

asymptotic behavior of random vandermonde matrices with entries on the unit circle analytical methods for finding moments of random vandermonde matrices with entries on the unit circle are developed vandermonde matrices play an important role in signal processing and wireless applications such as direction of arrival estimation precoding and sparse sampling theory just to name a few within this framework we extend classical freeness results on random matrices with independent and identically distributed i i d entries and show that vandermonde structured matrices can be treated in the same vein with different tools we focus on various types of matrices such as vandermonde matrices with and without uniform phase distributions as well as generalized vandermonde matrices in each case we provide explicit expressions of the moments of the associated gram matrix as well as more advanced models involving the vandermonde matrix comparisons with classical i i d random matrix theory are provided and deconvolution results are discussed we review some applications of the results to the fields of signal processing and wireless communications

finite dimensional statistical inference in this paper we derive the explicit series expansion of the eigenvalue distribution of various models namely the case of noncentral wishart distributions as well as correlated zero mean wishart distributions the tools used extend those of the free probability framework which have been quite successful for high dimensional statistical inference when the size of the matrices tends to infinity also known as free deconvolution this contribution focuses on the finite gaussian case and proposes algorithmic methods to compute the moments cases where asymptotic results fail to apply are also discussed

new integral estimates for deformations in terms of their nonlinear strains if u is a bi lipschitzian deformation of a bounded lipschitz domain in ln n 2 we show that the lp norm p 1 p n of a certain nonlinear strain function e u associated with u dominates the distance in lq q np n p if p n from u to a suitably chosen rigid motion of n this work extends that of f john who proved corresponding estimates for p 1 under the hypothesis that u has uniformly small strain we also obtain a bound for the oscillation of du in l2 these estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u in 3 the integral int limits omega e u 2d 3 is dominated by typical hyperelastic energy functionals proposed in the literature for modeling the behavior of rubber thus the case n 3 p 2 gives the first general bound for the deformations of such materials in terms of the associated nonlinear elastic work

eshelby formalism for nano inhomogeneities the eshelby formalism for inclusion inhomogeneity problems is extended to the nano scale at which surface interface effects become important the interior and exterior eshelby tensors for a spherical inhomogeneous inclusion with the interface stress effect subjected to an arbitrary uniform eigenstrain embedded in an infinite alien matrix and the stress concentration tensors for a spherical inhomogeneity subjected to an arbitrary remote uniform stress field are obtained unlike their counterparts at the macro scale the eshelby and stress concentration tensors are in general not uniform inside the inhomogeneity but are position dependent they have the property of radial transverse isotropy it is also shown that the size dependence of the eshelby tensors and the stress concentration tensors follow very simple scaling laws finally the eshelby formula to calculate the strain energy in the presence of the interface effect is given

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

injectivity and self contact in nonlinear elasticity let be a bounded open connected subset of 3 with a sufficiently smooth boundary the additional condition det dx vol is imposed on the admissible deformations of a hyperelastic body whose reference configuration is we show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate we also extend j ball s theorems on existence to this case by establishing the existence of a minimizer of the energy in the space w1 p 3 p 3 that is injective almost everywhere

a new b spline collocation method for singular integro differential equations of higher orders abstract two numerical schemes based on b spline collocation method are presented for approximate solution of the systems of singular integro differential equations of higher orders appearing in fracture problems with steigmann ogden surface energy it is shown that the resulting systems of linear equations are well conditioned and the numerical examples are in good agreement with known solutions although this paper is concerned with the numerical solution to the specific equations the proposed method can be easily generalized to a large class of singular integro differential equations

on irregular weak solutions of the energy momentum equations irregular mappings that are weak solutions of the energy momentum equations are presented one example is discontinuous at a countable number of points while the other is c 1 but not c 2 these mappings are not solutions of the usual euler lagrange equations

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

a rigid stamp indentation into a semiplane with a curvature dependent surface tension on the boundary it has been shown that taking into account surface mechanics is extremely important for accurate modeling of many physical phenomena such as those arising in nanoscience fracture propagation and contact mechanics this paper is dedicated to a contact problem of a rigid stamp indentation into an elastic isotropic semiplane with curvature dependent surface tension acting on the boundary of the semiplane cases of both frictionless and adhesive contact of the stamp with the boundary of the semiplane are considered using the method of integral transforms each problem is reduced to a system of singular integro differential equations which is further reduced to one or two weakly singular integral equations it has been shown that the introduction of the curvature dependent surface tension eliminates the classical singularities of the order 1 2 of the stresses and strains at the end points of the contact interval the numerical solution of the problem is obtained by approximation of unknown functions with taylor polynomials

reconstruction of multidimensional signals from irregular noisy samples we focus on a multidimensional field with uncorrelated spectrum and study the quality of the reconstructed signal when the field samples are irregularly spaced and affected by independent and identically distributed noise more specifically we apply linear reconstruction techniques and take the mean square error mse of the field estimate as a metric to evaluate the signal reconstruction quality we find that the mse analysis could be carried out by using the closed form expression of the eigenvalue distribution of the matrix representing the sampling system unfortunately such distribution is still unknown thus we first derive a closed form expression of the distribution moments and we find that the eigenvalue distribution tends to the marcenko pastur distribution as the field dimension goes to infinity finally by using our approach we derive a tight approximation to the mse of the reconstructed field

elastic surface substrate interactions a theory for three dimensional finite deformations of elastic solids with conforming elastic films attached to their bounding surfaces is described the gurtin murdoch theory incorporating elastic resistance of the film to strain is generalized to account for the effects of intrinsic flexural resistance this modification yields a model that can be used to describe equilibrium deformations in the presence of compressive surface stress fields an associated variational theory is given and material symmetry considerations are discussed the theory is illustrated by examples

mappings of finite distortion hausdorff measure of zero sets we prove that a for a mapping f of finite distortion k in l p n p the n p hausdorff measure of any point preimage is zero provided j f is integrable df in l s with s p and the multiplicity function of f is essentially bounded as a consequence for p n 1 we obtain that the mapping is then open and discrete

a meshing strategy for a quadratic iso parametric fem in cavitation computation in nonlinear elasticity abstract the approximation properties of a quadratic iso parametric finite element method for a typical cavitation problem in nonlinear elasticity in two dimensions are analyzed more precisely 1 the finite element interpolation errors are established in terms of the mesh parameters 2 a mesh distribution strategy based on an error equi distribution principle is given 3 the convergence of finite element cavity solutions is proved numerical experiments show that in fact the optimal convergence rate can be achieved by the numerical cavity solutions

uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity introduction 97 constitutive assumptions 99 0 the stored energy function and weak equilibrium solutions 101 1 properties of radial equilibrium solutions 105 2 uniqueness of solutions to boundary value problems for shells 111 3 uniqueness of cavitating solutions 117 4 existence of radial equilibria 121 5 asymptotic behaviour of equilibria for shells 128 6 concluding remarks 132 7 appendix 133 references 135

on the optimal stacking of information plus noise matrices observations of the form d x where d is a matrix representing information and x is a random matrix representing noise can be grouped into a compound observation matrix on the same information noise form there are many ways the observations can be stacked into such a matrix for instance vertically horizontally or quadratically an unbiased estimator for the spectrum of d can be formulated for each stacking scenario in the case of gaussian noise we compare these spectrum estimators for the different stacking scenarios and show that all kinds of stacking actually decrease the variance of the corresponding spectrum estimators when compared to just taking an average of the observations and find which stacking is optimal in this sense when the number of observations grow however it is shown that the difference between the estimators is marginal with only the cases of vertical and horizontal stackings having a higher variance asymptotically

a justification of nonlinear properly invariant plate theories a single asymptotic derivation of three classical nonlinear plate theories is presented in a setting which preserves the frame invariance properties of three dimensional finite elasticity by a successive scaling of the external loading on the three dimensional body the nonlinear membrane theory the nonlinear inextensional theory and the von karman equations are derived as the leading order terms in the asymptotic expansion of finite elasticity the governing equations of the nonlinear inextensional theory are of particular interest where 1 plane strain kinematics and plane stress constitutive equations are derived simultaneously from the asymptotic analysis 2 the theory can be phrased as a minimization problem over the space of isometric deformations of a surface and 3 the local equilibrium equations are identical to those arising in the one director cosserat shell model furthermore it can be concluded that with a regular single scale asymptotic expansion it is not possible to obtain a system of plate equations in which finite membrane strain and finite bending strain occur simultaneously in the leading order term of an asymptotic analysis

spectrum estimation for large dimensional covariance matrices using random matrix theory estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics the eigenvalues of covariance matrices play a key role in many widely used techniques in particular in principal component analysis pca in many modern data analysis problems statisticians are faced with large datasets where the sample size n is of the same order of magnitude as the number of variables p random matrix theory predicts that in this context the eigenvalues of the sample covariance matrix are not good estimators of the eigenvalues of the population covariance we propose to use a fundamental result in random matrix theory the marcenko pastur equation to better estimate the eigenvalues of large dimensional covariance matrices the marcenko pastur equation holds in very wide generality and under weak assumptions the estimator we obtain can be thought of as shrinking in a nonlinear fashion the eigenvalues of the sample covariance matrix to estimate the population eigenvalues inspired by ideas of random matrix theory we also suggest a change of point of view when thinking about estimation of high dimensional vectors we do not try to estimate directly the vectors but rather a probability measure that describes them we think this is a theoretically more fruitful way to think about these problems our estimator gives fast and good or very good results in extended simulations our algorithmic approach is based on convex optimization we also show that the proposed estimator is consistent

boundary regularity and sufficient conditions for strong local minimizers abstract in this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning c 1 extremals at which the second variation is strictly positive the results are presented in the quasiconvex setting in accordance with the original statement by grabovsky and mengesha 31 the strategy that we follow relies on a decomposition theorem that allows to split a sequence of variations into its oscillating and its concentrating parts as well as on a sufficiency result according to which smooth extremals are spatially local minimizers furthermore we prove partial regularity up to the boundary for strong local minimizers in the non homogeneous case and a full regularity result for lipschitz extremals with gradient of vanishing mean oscillation as a consequence we also establish a sufficiency result for this class of extremals the regularity results are established via a blow up argument

tension field theory a general theory of the tension field is developed for application to the analysis of wrinkling in isotropic elastic membranes undergoing finite deformations the principal contribution is a partial differential equation describing a geometrical property of tension trajectories this is one of a system of two equations which describes the state of stress independently of the deformation this system is strongly elliptic at any stable solution whereas the deformation is described by a system of parabolic type controllable solutions i e those states that can be maintained in any isotropic elastic material by application of edge tractions and lateral pressure alone are obtained the general axisymmetric problem is solved implicitly and the theory is applied to the solution of two representative examples existing small strain theories are shown to correspond to a singular limit of the general theory at which the underlying system changes from elliptic to parabolic type

an existence theory for nonlinear elasticity that allows for cavitation in this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body such cavities have been observed in experiments on elastomers and a mathematical theory for radially symmetric cavities has been developed by ball here the full three dimensional problem is considered and an additional physically motivated energy term that is proportional to the area of the boundary of the deformed body is included the minimizers lie in a subclass of those maps in w1 p 2 p 3 that are one to one almost everywhere and preserve orientation roughly speaking this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body such maps are shown to be much more regular than expected in particular some ideas of sverak are used to show that each map in this subclass has a representative which is continuous outside a set of hausdorff dimension 3 p and that this representative also satisfies lusin s condition n i e it maps lebesgue null sets onto such sets it is also shown that the distributional jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to lebesgue measure and at most a countable number of dirac measures

on edge detection of x ray images using fuzzy sets the effectiveness of the theory of fuzzy sets in detecting different regional boundaries of x ray images is demonstrated the algorithm includes a prior enhancement of the contrast among the regions having small change in gray levels using the contrast intensification int operation along with smoothing in the fuzzy property plane before detecting its edges the property plane is extracted from the spatial domain using s and 1 functions and the fuzzifiers final edge detection is achieved using max or min operator the system performance for different parameter conditions is illustrated by application to an image of a radiograph of the wrist

on the response and symmetry of elastic and hyperelastic membrane points in continuum mechanics constitutive equations for elastic and hyperelastic 3 dim material points are important mathematical models with many applications in this paper we present constitutive equations for elastic and hyperelastic 2 dim membrane points as in the theory of 3 dim material points the central topic for the theory of 2 dim membrane points are conditions of material frame indifference conditions of material symmetry and representations for the response functions and for the stored energy functions

general imperfect interfaces abstract the objective of this contribution is to develop a thermodynamically consistent theory for general imperfect interfaces and to establish a unified computational framework to model all classes of such interfaces using the finite element method the interface is termed general imperfect in the sense that it allows for a jump of the temperature as well as for a jump of the normal heat flux across the interface conventionally imperfect interfaces with respect to their thermal behavior are restricted to being either highly conducting hc or lowly conducting lc also known as kapitza for a hc interface the temperature is continuous across the interface while the jump of the normal heat flux is admissible on the contrary a lc interface does not allow for a jump of the heat flux across the interface but it does allow for a temperature jump the temperature jump of a lc interface is frequently assumed to be proportional to the average heat flux across the interface in this contribution we prove that this common assumption is indeed an appropriate condition to sufficiently and not necessarily satisfy the second law of thermodynamics while hc and lc interfaces are generally accepted and well established today the general imperfect interfaces remain poorly understood here we propose a thermodynamically consistent theory of general imperfect interfaces and we show that the dissipative structure of the interface suggests firstly to classify such interfaces as semi dissipative sd and fully dissipative fd secondly for a fd interface the interface temperature shall be considered as an independent degree of freedom and a new constitutive equation is obtained to calculate the interface temperature using a new interface material parameter i e the sensitivity furthermore we show how all types of interfaces are derived from a fd general imperfect interface model this finding allows us to establish a unified finite element framework to model all classes of interfaces full details of the novel numerical scheme are provided key features of general imperfect interfaces are then elucidated via a series of three dimensional numerical examples in particular we show that according to the second law the interface temperature may not necessarily be the average of or even between the temperatures across the interface finally we recall since the influence of interfaces on the overall response of a body increases as the scale of the problem decreases this contribution has certain applications to nano composites

interfacial dynamics for stokes flow theoretical and computational aspects of the method of interfacial dynamics for flow at vanishing reynolds number are considered the mathematical formulation relies on the boundary integral representation that expresses the flow in terms of distributions of stokes flow singularities over the interfaces the densities of the distributions are identified with the jump in the hydrodynamic traction due to interfacial in plane and transverse tensions the interfacial velocity or the strength of a hydrodynamic potential the numerical procedure involves describing the interfaces in terms of interfacial marker points that reproduce the evolving shapes of the interfaces by global or local interpolation solving integral equations of the second kind for the interfacial velocity or for the density of a hydrodynamic potential and computing the motion of the marker points while simultaneously updating interfacial fields relevant to the dynamics including the concentration of a surfactant and the position of interfacial point particles at an equilibrium configuration interfaces exhibiting isotropic tension elastic tensions viscous and incompressible behavior are considered the mathematical modeling of the tensions and bending moments developing over interfaces with a membrane like constitution is discussed in the context of the theory of thin shells to facilitate the numerical implementation the coupling of the interfacial mechanics to the hydrodynamics by means of interface force and torque balances is formulated in global cartesian coordinates recent progress in the implementation of boundary element methods is reviewed and areas for further research are identified

convolution operations arising from vandermonde matrices different types of convolution operations involving large vandermonde matrices are considered the convolutions parallel those of large gaussian matrices and additive and multiplicative free convolution and include additive and multiplicative convolution of vandermonde matrices and deterministic diagonal matrices and cases where two independent vandermonde matrices are involved it is also shown that the convergence of any combination of vandermonde matrices is almost sure the convolutions are divided into two types those which depend on the phase distribution of the vandermonde matrices and those which depend only on the spectra of the matrices a general criterion is presented to find which type applies for any given convolution a simulation is presented verifying the results implementations of the presented convolutions are provided and discussed the implementation is based on the technique of fourier motzkin elimination and is quite general as it can be applied to virtually any combination of vandermonde matrices connections with related matrices such as toeplitz and hankel matrices are also discussed

median filtering by threshold decomposition median filters are a special class of ranked order filters used for smoothing signals repeated application of the filter on a quantized signal of finite length ultimately results in a sequence termed a root signal which is invariant to further passes of the median filter in this paper it is shown that median filtering an arbitrary level signal to its root is equivalent to decomposing the signal into binary signals filtering each binary signal to a root with a binary median filter and then reversing the decomposition this equivalence allows problems in the analysis and the implementation of median filters for arbitrary level signals to be reduced to the equivalent problems for binary signals since the effects of median filters on binary signals are well understood this technique is a powerful new tool

energy minimizers in nonlinear elastostatics and the implicit function theorem the purpose of this paper is to give conditions under which the equilibria obtained by two different approaches are the same these two approaches are the calculus of variation and the theory of polyconvex functions and one using the implicit function theorem in infinite dimensional space