some bounds for the spectral radius of nonnegative tensors in this paper we extend the well known column sum bound of the spectral radius for nonnegative matrices to the tensor case an upper bound of the spectral radius for a nonnegative tensor is given via the largest eigenvalue of a symmetric tensor also we show some bounds of spectral radius of nonnegative tensors based on the sum of the entries in the other indices of tensors we demonstrate that our new results improve existing results the other main results of this paper is to provide a sharper ky fan type theorem and a comparison theorem for nonnegative tensors finally we make use of our bounds to give a perturbation bound for the spectral radius of symmetric nonnegative tensors this result is similar to the weyl theorem for the matrix case

z eigenpair bounds for an irreducible nonnegative tensor abstract in this paper we consider the z eigenpair of a tensor in particular an irreducible nonnegative tensor we present some bounds for the eigenvector and z spectral radius the proposed bounds improve some existing ones an example with practical applications is given to show the proposed bound

optimality conditions for simultaneous topology and shape optimization new optimality conditions are derived for a class of shape optimization problems the conditions are established on the boundary by an application of the boundary variations technique and in the interior of an optimal domain by exploiting the topological derivative method an example is provided for which the classical second order sufficient optimality conditions are verified for an optimal simply connected domain however the value of the cost can be improved by the topology variations and therefore the optimal solution can be substantially changed by applying the topology optimization

m tensors and nonsingular m tensors abstract the m matrix is an important concept in matrix theory and has many applications recently this concept has been extended to higher order tensors 18 in this paper we establish some important properties of m tensors and nonsingular m tensors an m tensor is a z tensor we show that a z tensor is a nonsingular m tensor if and only if it is semi positive thus a nonsingular m tensor has all positive diagonal entries an m tensor regarding as the limit of a sequence of nonsingular m tensors has all nonnegative diagonal entries we introduce even order monotone tensors and present their spectral properties in matrix theory a z matrix is a nonsingular m matrix if and only if it is monotone this is no longer true in the case of higher order tensors we show that an even order monotone z tensor is an even order nonsingular m tensor but not vice versa an example of an even order nontrivial monotone z tensor is also given

topological derivative for multi scale linear elasticity models applied to the synthesis of microstructures this paper proposes an algorithm for the synthesis optimization of microstructures based on an exact formula for the topological derivative of the macroscopic elasticity tensor and a level set domain representation the macroscopic elasticity tensor is estimated by a standard multi scale constitutive theory where the strain and stress tensors are volume averages of their microscopic counterparts over a representative volume element the algorithm is of simple computational implementation in particular it does not require artificial algorithmic parameters or strategies this is in sharp contrast with existing microstructural optimization procedures and follows as a natural consequence of the use of the topological derivative concept this concept provides the correct mathematical framework to treat topology changes such as those characterizing microstuctural optimization problems the effectiveness of the proposed methodology is illustrated in a set of finite element based numerical examples copyright 2010 john wiley sons ltd

energy change due to the appearance of cavities in elastic solids abstract the paper presents an overview of the problem of assessing an increment of strain energy due to the appearance of small cavities in elastic solids the following approaches are discussed the compound asymptotic method by mazja et al the eshelby like method used in the classical works on the mechanics of composites the homogenization method and the topological derivative method proposed by sokolowski and ochowski the increment of energy is expressed by a quadratic form with respect to strains referring to the virgin solid all the methods lead to the same formula for the increment of energy it is expressed by a quadratic form with respect to strains referring to the virgin solid this quadratic form turns out to be unconditionally positive definite explicit formulae are derived for an elliptical hole and for a spherical cavity the results derived determine the characteristic function of the bubble method of the optimal shape design of elastic 2d and 3d structures

a new algorithm for topology optimization using a level set method the level set method has been recently introduced in the field of shape optimization enabling a smooth representation of the boundaries on a fixed mesh and therefore leading to fast numerical algorithms however most of these algorithms use a hamilton jacobi equation to connect the evolution of the level set function with the deformation of the contours and consequently they can hardly create new holes in the domain at least in 2d in this work we propose an evolution equation for the level set function based on a generalization of the concept of topological gradient this results in a new algorithm allowing for all kinds of topology changes

the topological asymptotic for pde systems the elasticity case the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in this paper such an expansion is obtained and analyzed in the context of linear elasticity for general functionals and arbitrary shaped holes by using an adaptation of the adjoint method and a domain truncation technique the method is general and can be easily adapted to other linear pdes and other types of boundary conditions

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

a direct impedance tomography algorithm for locating small inhomogeneities impedance tomography seeks to recover the electrical conductivity distribution inside a body from measurements of current flows and voltages on its surface in its most general form impedance tomography is quite ill posed but when additional a priori information is admitted the situation changes dramatically in this paper we consider the case where the goal is to find a number of small objects inhomogeneities inside an otherwise known conductor taking advantage of the smallness of the inhomogeneities we can use asymptotic analysis to design a direct i e non iterative reconstruction algorithm for the determination of their locations the viability of this direct approach is documented by numerical examples

brualdi type eigenvalue inclusion sets of tensors abstract by using digraphs of tensors we give brualdi type eigenvalue inclusion sets of tensors we also give some applications of our result to nonsingularity and positive definiteness of tensors

modelling of topological derivatives for contact problems the problem of topology optimization is considered for free boundary problems of thin obstacle types the formulae for the first term of asymptotics for energy functionals are derived the precision of obtained terms is verified numerically the topological differentiability of solutions to variational inequalities is established in particular the so called outer asymptotic expansion for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis such results lead to the topological derivatives of shape functionals for contact problems the topological derivatives are used in numerical methods of simultaneous shape and topology optimization

topological sensitivity analysis for three dimensional linear elasticity problem in this work we use the topological shape sensitivity method to obtain the topological derivative for three dimensional linear elasticity problems adopting the total potential energy as cost function and the equilibrium equation as constraint this method based on classical shape sensitivity analysis leads to a systematic procedure to calculate the topological derivative in particular firstly we present the mechanical model later we perform the shape derivative of the corresponding cost function and finally we calculate the final expression for the topological derivative using the topological shape sensitivity method and results from classical asymptotic analysis around spherical cavities in order to point out the applicability of the topological derivative in the context of topology optimization problems we use this information as a descent direction to solve a three dimensional topology design problem furthermore through this example we also show that the topological derivative together with an appropriate mesh refinement strategy are able to capture high quality shapes even using a very simple topology algorithm

an s type eigenvalue localization set for tensors abstract an s type eigenvalue localization set for a tensor is given by breaking n 1 2 n into disjoint subsets s and its complement it is shown that the new set is tighter than those provided by qi 2005 21 and li et al 2014 13 as applications of the results a checkable sufficient condition for the positive definiteness of tensors and a checkable sufficient condition of the positive semi definiteness of tensors are given

incorporating topological derivatives into level set methods the aim of this paper is to investigate the use of topological derivatives in combination with the level set method for shape reconstruction and optimization problems we propose a new approach generalizing the standard speed method which is obtained by using a source term in the level set equation that depends on the topological derivative of the objective functional the resulting approach can be interpreted as a generalized fixed point iteration for the optimality system with respect to topological and shape variations moreover we apply the new approach for a simple model problem in shape reconstruction where the topological derivative can be computed without additional effort finally we present numerical tests related to this model problem which demonstrate that the new method based on shape and topological derivative successfully reconstructs obstacles in situations where the standard level set approach fails

symmetry results for overdetermined boundary value problems of nonlinear elliptic equations abstract we prove the radial symmetry of the solutions of second order nonlinear elliptic equations for overdetermined dirichlet and neumann boundary value problems in addition a global uniqueness theorem of holmgren type is given for nonlinear elliptic equations

topological sensitivity analysis the so called topological derivative concept has been seen as a powerful framework to obtain the optimal topology for several engineering problems this derivative characterizes the sensitivity of the problem when a small hole is created at each point of the domain however the greatest limitation of this methodology is that when a hole is created it is impossible to build a homeomorphic map between the domains in study because they have not the same topology therefore some specific mathematical framework should be developed in order to obtain the derivatives this work proposes an alternative way to compute the topological derivative based on the shape sensitivity analysis concepts the main feature of this methodology is that all the mathematical procedure already developed in the context of shape sensitivity analysis may be used in the calculus of the topological derivative this idea leads to a more simple and constructive formulation than the ones found in the literature further to point out the straightforward use of the proposed methodology it is applied for solving some design problems in steady state heat conduction

identification of small inhomogeneities of extreme conductivity by boundary measurements a theorem on continuous dependence we consider an electrostatic problem for a conductor consisting of finitely many small inhomogeneities of extreme conductivity embedded in a spatially varying reference medium firstly we establish an asymptotic formula for the voltage potential in terms of the reference voltage potential the location of the inhomogeneities and their geometry secondly we use this representation formula to prove a lipschitz continuous dependence estimate for the corresponding inverse problem this estimate bounds the difference in the location and in certain geometric properties of two sets of inhomogeneities by the difference in the boundary voltage potentials corresponding to a fixed current distribution

a new reconstruction method for the inverse potential problem abstract the inverse potential problem consists in reconstructing an unknown measure with support in a geometrical domain from a single boundary measurement in order to deal with this severely ill posed inverse problem we rewrite it as an optimization problem where a kohn vogelius type functional measuring the misfit between the solutions of two auxiliary problems is minimized one auxiliary problem contains information on the boundary measurement while the other one corresponds to the boundary excitation the solutions of the auxiliary problems coincide once the inverse problem is solved in order to minimize the kohn vogelius criterion its total variation with respect to a set of ball shaped perturbations on the measure is explicitly evaluated then a new method for solving the inverse potential problem based on the expression obtained is devised finally some numerical results are presented in order to show the effectiveness of the proposed reconstruction algorithm

some new inequalities for the minimum h eigenvalue of nonsingular m tensors abstract this paper is devoted to deriving some new upper and lower bounds for the minimum h eigenvalue of irreducible nonsingular m tensors we demonstrate that the proposed bounds improve some existing ones and numerical examples are given to illustrate this fact additionally we extend these bounds to general nonsingular m tensors the other main results in this paper is to provide sharper ky fan theorem and comparison theorem for nonsingular m tensors which are better than the corresponding ones obtained by he and huang 2014 12

on the topological derivative in shape optimization in this paper the topological derivative for an arbitrary shape functional is defined examples are provided for elliptic equations and the elasticity system in the plane the topological derivative can be used for solving shape optimization problems in structural mechanics

eigenvalues and invariants of tensors abstract a tensor is represented by a supermatrix under a co ordinate system in this paper we define e eigenvalues and e eigenvectors for tensors and supermatrices by the resultant theory we define the e characteristic polynomial of a tensor an e eigenvalue of a tensor is a root of the e characteristic polynomial in the regular case a complex number is an e eigenvalue if and only if it is a root of the e characteristic polynomial we convert the e characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor i e they are invariant under co ordinate system changes we call them principal invariants of that tensor the maximum number of principal invariants of mth order n dimensional tensors is a function of m and n we denote it by d m n and show that d 1 n 1 d 2 n n d m 2 m for m 3 and d m n m n 1 m for m n 3 we also define the rank of a tensor all real eigenvectors associated with nonzero e eigenvalues are in a subspace with dimension equal to its rank

a new eigenvalue inclusion set for tensors and its applications abstract a new tensor eigenvalue inclusion set is given and proved to be tighter than those in l q qi 2005 18 and c q li y t li x kong 2014 12 in addition we study the eigenvalues lying on the boundary of the eigenvalue inclusion set provided by qi 2005 for weakly irreducible tensors as applications we give new bounds for the spectral radius of nonnegative tensors some sufficient conditions for a tensor to be a strong m tensor and some inequalities to identify the positive definiteness for an even order real supersymmetric tensor

the shape and topological optimizations connection abstract in 1973 jean cea introduced a topological shape optimization algorithm based on a fixed point method the recent topological gradient concept provides a mathematical justification of cea s powerful method the aim of the topological gradient is to compute the sensitivity of a cost function when we drill a little hole in the domain some optimality conditions are derived when classical shape gradient and topological gradient are used simultaneously and numerical results are shown both in mechanics and electromagnetics

a new method for inverse electromagnetic casting problems based on the topological derivative the inverse electromagnetic casting problem consists in looking for a suitable set of electric wires such that the electromagnetic field induced by an alternating current passing through them makes a given mass of liquid metal acquire a predefined shape in this paper we propose a new method for the topology design of such inductors the inverse electromagnetic casting problem is formulated as an optimization problem and topological derivatives are considered in order to locate new wires in the right position several numerical examples are presented showing that the proposed technique is effective to design suitable inductors

some properties of strong h tensors and general h tensors abstract h matrices matrices whose comparison matrix is an m matrix are well studied in matrix theory and have numerous applications e g linear complementarity problems and iterative methods for solving systems of linear equations in this article we establish some properties of their tensor counterparts strong h tensors and general h tensors

numerical simulation of tridimensional electromagnetic shaping of liquid metals we describe a numerical method to compute free surfaces in electromagnetic shaping and levitation of liquid metals we use an energetic variational formulation and optimization techniques to compute a critical point the surfaces are represented by piecewise linear finite elements each step of the algorithm requires solving an elliptic boundary value problem in the exterior of the intermediate surfaces this is done by using an integral representation on these surfaces

symmetry breaking for positive solutions of semilinear elliptic equations in a recent interesting paper gidas ni and nirenberg 2 proved that positive solutions of the dirichlet problem for second order semi linear elliptic equations on balls must themselves be spherically symmetric functions here we consider the bifurcation problem for such solutions specifically we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solution when this happens we say that the symmetry breaks to carry out this program we rely on certain results in 6 where we studied the kernel of the associated linearized operator this enables us to give some necessary conditions for symmetry to break we also find a class of functionsf where these conditions are also sufficient see equation 18 below the problem is of course to show that when zero comes into the spectrum of the linearized operator and our conditions are fulfilled bifurcation into the non radial direction actually occurs this is done by showing first that the bifurcation curve is a smooth manifold near the bifurcation point and then appealing to a theorem of vanderbauwhede 8 which gives sufficient conditions for bifurcation to occur in the presence of symmetries

on eigenvalue problems of real symmetric tensors abstract we use variational methods to give a positive answer to a conjecture posed by liqun qi l qi eigenvalues of a real supersymmetric tensor j symbolic comput 40 2005 1302 1324 regarding the real eigenvalues of certain higher order tensors

symmetric nonnegative tensors and copositive tensors abstract we first prove two new spectral properties for symmetric nonnegative tensors we prove a maximum property for the largest h eigenvalue of a symmetric nonnegative tensor and establish some bounds for this eigenvalue via row sums of that tensor we show that if an eigenvalue of a symmetric nonnegative tensor has a positive h eigenvector then this eigenvalue is the largest h eigenvalue of that tensor we also give a necessary and sufficient condition for this we then introduce copositive tensors this concept extends the concept of copositive matrices symmetric nonnegative tensors and positive semi definite tensors are examples of copositive tensors the diagonal elements of a copositive tensor must be nonnegative we show that if each sum of a diagonal element and all the negative off diagonal elements in the same row of a real symmetric tensor is nonnegative then that tensor is a copositive tensor some further properties of copositive tensors are discussed

some variational principles for z eigenvalues of nonnegative tensors many important spectral properties of nonnegative matrices have recently been successfully extended to higher order nonnegative tensors for example see chang et al 2008 2011 friedland et al in press lim 2005 liu et al 2010 ng et al 2010 qi et al 2007 yang and yang 2010 2 3 9 17 23 24 27 28 however most of these results focus on the h eigenvalues introduced by qi 2005 2007 25 26 the key results of this paper reveal some similarities as well as some crucial differences between z eigenvalues and h eigenvalues of a nonnegative tensor in particular neither the positive z eigenvalue nor the associated positive z eigenvector of an irreducible nonnegative tensor has to be unique in general as demonstrated by example 2 7 furthermore the collatz type min max characterizations of the largest positive z eigenvalue of an irreducible nonnegative tensor is only partially true in general as seen in theorem 4 7 and example 4 8