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

singular values of a real rectangular tensor real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics in this paper we systematically study properties of singular values of a real rectangular tensor and give an algorithm to find the largest singular value of a nonnegative rectangular tensor numerical results show that the algorithm is efficient

comments on explicit criterion for the positive definiteness of a general quartic form the purpose of this note is to point out that the result in the above paper w h ku ieee trans autom control vol ac 10 no 3 pp 372 373 jul 1965 is incomplete and to give a complete and improved result

z eigenvalue methods for a global polynomial optimization problem as a global polynomial optimization problem the best rank one approximation to higher order tensors has extensive engineering and statistical applications different from traditional optimization solution methods in this paper we propose some z eigenvalue methods for solving this problem we first propose a direct z eigenvalue method for this problem when the dimension is two in multidimensional case by a conventional descent optimization method we may find a local minimizer of this problem then by using orthogonal transformations we convert the underlying supersymmetric tensor to a pseudo canonical form which has the same e eigenvalues and some zero entries based upon these we propose a direct orthogonal transformation z eigenvalue method for this problem in the case of order three and dimension three in the case of order three and higher dimension we propose a heuristic orthogonal transformation z eigenvalue method by improving the local minimum with the lower dimensional z eigenvalue methods and a heuristic cross hill z eigenvalue method by using the two dimensional z eigenvalue method to find more local minimizers numerical experiments show that our methods are efficient and promising

gradient skewness tensors and local illumination detection for images in this paper we propose the definition of d eigenvalue for an arbitrary order tensor related with a second order tensor d and introduce the gradient skewness tensor which involves a three order tensor derived from the skewness statistic of gradient images as we happen to find out that the skewness of oriented gradients can measure the directional characteristic of illumination in an image the local illumination detection problem for an image can be abstracted as solving the largest d eigenvalue of gradient skewness tensors we discuss the properties of d eigenvalues and especially for gradient skewness tensors we provide the calculation method of its d eigenvalues and d characteristic polynomial some numerical experiments show its effective application in illumination detection our method also presents excellent results in a class of image authenticity verification problems which is to distinguish artificial flat objects in a photograph

laplacian graph eigenvectors abstract if g is a graph its laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix the main thrust of the present article is to prove several laplacian eigenvector principles which in certain cases can be used to deduce the effect on the spectrum of contracting adding or deleting edges and or of coalescing vertices one application is the construction of two isospectral graphs on 11 vertices having different degree sequences only one of which is bipartite and only one of which is decomposable

ellipticity and deformations with discontinuous gradients in finite elastostatics loss of ellipticity of the equilibrium equations of finite elastostatics is closely related to the possible emergence of elastostatic shocks i e deformations with discontinuous gradients in certain situations where constitutive response functions are essentially one dimentional such as anti plane shear or bar theories strong ellipticity is closely related to convexity of the elastic potential and invertibility of certain constitutive response functions the present work addresses the analogous issues within the context of three dimensional elastostatics of compressible but not necessarily isotropic hyperelastic materials a certain direction dependent resolution of the deformation gradient is introduced and its existence and uniqueness for a given direction are established the elastic potential is expressed as a function of kinematic variables arising from this resolution strong ellipticity is shown to be equivalent to the positive definiteness of the hessian matrix of this function thus sufficing for its strict convexity the underlying variables are interpretable physically as simple shears and extensions their work conjugates define a traction response mapping it is shown that discontinuous deformation gradients are sustainable if and only if this mapping fails to be invertible this result is explicit in the sense that it characterizes the set of all possible piecewise homogeneous deformations given the elastic potential function

enhancing synchronizability of diffusively coupled dynamical networks a survey in this paper we review the literature on enhancing synchronizability of diffusively coupled dynamical networks with identical nodes the last decade has witnessed intensive investigations on the collective behavior over complex networks and synchronization of dynamical systems is the most common form of collective behavior for many applications it is desired that the synchronizability the ability of networks in synchronizing activity of their individual dynamical units is enhanced there are a number of methods for improving the synchronization properties of dynamical networks through structural perturbation in this paper we survey such methods including adding removing nodes and or edges rewiring the links and graph weighting these methods often try to enhance the synchronizability through minimizing the eigenratio of the laplacian matrix of the connection graph a synchronizability measure based on the master stability function formalism we also assess the performance of the methods by numerical simulations on a number of real world networks as well as those generated through models such as preferential attachment watts strogatz and erdos renyi

a procedure for the positive definiteness of forms of even order a positivity method for forms of even order is reported it is shown that testing for positivity of such forms can be reduced to a test for positive definiteness of parametric matrices special emphasis is placed on quartic and sixth order forms examples are given to illustrate the proposed criterion

complex network synchronizability analysis and control in this paper the investigation is first motivated by showing two examples of simple regular symmetrical graphs which have the same structural parameters such as average distance degree distribution and node betweenness centrality but have very different synchronizabilities for a given network with identical node dynamics it is further shown that two key factors influencing the network synchronizability are the network inner linking matrix and the eigenvalues of the network topological matrix several examples are then provided to show that adding new edges to a network can either increase or decrease the network synchronizability in searching for conditions under which the network synchronizability may be increased by adding edges it is found that for networks with disconnected complementary graphs adding edges never decreases their synchronizability moreover it is found that an unbounded synchronized region is always easier to analyze than a bounded synchronized region therefore to effectively enhance the network synchronizability a design method is finally presented for the inner linking matrix of rank 1 such that the resultant network has an unbounded synchronized region for the case where the synchronous state is an equilibrium point of the network

positive maps and separable matrices a linear map between real symmetric matrix spaces is positive if all positive semidefinite matrices are mapped to positive semidefinite ones a real symmetric matrix is separable if it can be written as a summation of kronecker products of positive semidefinite matrices this paper studies how to check if a linear map is positive or not and how to check if a matrix is separable or not we propose numerical algorithms based on lasserre s type of semidefinite relaxations for solving such questions to check the positivity of a linear map we construct a hierarchy of semidefinite relaxations for minimizing the associated biquadratic form over the unit spheres we show that the positivity can be detected by solving a finite number of such semidefinite relaxations to check the separability of a matrix we construct a hierarchy of semidefinite relaxations if it is not separable we can get a mathematical certificate for that if it is we can get a decomposition for the separability

elasticity m tensors and the strong ellipticity condition abstract in this paper we establish two sufficient conditions for the strong ellipticity of any fourth order elasticity tensor and investigate a class of tensors satisfying the strong ellipticity condition the elasticity m tensor the first sufficient condition is that the strong ellipticity holds if the unfolding matrix of this fourth order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries second an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not besides the elasticity m tensor is defined with respect to the m eigenvalues of elasticity tensors we prove that any nonsingular elasticity m tensor satisfies the strong ellipticity condition by employing a perron frobenius type theorem for m spectral radii of nonnegative elasticity tensors other equivalent definitions of nonsingular elasticity m tensors are also established

a seminorm regularized alternating least squares algorithm for canonical tensor decomposition abstract the regularization method could deal with the swamp effect of alternating least squares als algorithms for tensor decomposition usually the regularization term is a norm of the difference between the solution and the current iterate in this paper we show that the norm could be weakened to a seminorm so the selection of the regularization term could be more flexible to overcome the swamp effect and avoid the drawback that the hessian of the subproblem may get close to singular in the iterative process we propose a seminorm regularized als algorithm for solving the canonical tensor decomposition moreover in the new algorithm we introduce a novel extrapolation in the update of each mode factor which makes an immediate impression on the update of subsequent ones by assuming the boundness of the infinite sequence of iterates generated by the new algorithm we establish the global convergence and the weakly linear convergence rate of the sequence of iterates numerical experiments on synthetic and real world problems illustrate that the new method is efficient and promising

altering synchronizability by adding and deleting edges for scale free networks abstract in this paper we propose two methods for altering the synchronizability of scale free networks 1 adding edges between the max degree nodes and min degree nodes 2 deleting edges between the max degree nodes and max degree nodes after adding and deleting edges we find that the former adding process can weaken synchronizability while the latter deleting process can enhance it the two processes adding and deleting can preserve the scale free structure the study of the average clustering coefficient indicates that it is not the most closely correlated with the synchronizability among the topological features considered our work also suggests that there are some essential relations between the network synchronization and the dynamics of economic systems they can be used to deal with some problems in the real world such as relieving the economic crisis in addition the adding and deleting processes may have potential applications in modifying network structure in view of their low cost

decoupling process for better synchronizability on scale free networks we propose a decoupling process performed in scale free networks to enhance the synchronizability of the network together with preserving the scale free structure simulation results show that the decoupling process can effectively promote the network synchronizability which is measured in terms of eigenratio of the coupling matrix moreover we investigate the correlation between some important structural properties and the collective synchronization and find that the maximum vertex betweenness seems to be the most strongly correlated with the synchronizability among the major structural features considered we explain the effect of the decoupling process from a viewpoint of coupling information transmission our work provides some evidence that the dynamics of synchronization is related to that of information or vehicle traffic because of the low cost in modifying the coupling network the decoupling process may have potential applications

comments on a procedure for the positive definiteness of forms of even order hasan and hasan ieee trans automat control vol 41 p 615 17 1996 give a necessary and sufficient condition for the positive definiteness of a quadratic programming problem the purpose of this paper is to point out that the main result in the above mentioned paper by is erroneous

on the z eigenvalues of the adjacency tensors for uniform hypergraphs abstract the adjacency matrices for graphs are generalized to the adjacency tensors for uniform hypergraphs and some fundamental properties for the adjacency tensor and its z eigenvalues of a uniform hypergraph are obtained in particular some bounds on the smallest and the largest z eigenvalues of the adjacency tensors for uniform hypergraphs are presented

block tensors and symmetric embeddings abstract well known connections exist between the singular value decomposition of a matrix a and the schur decomposition of its symmetric embedding sym a 0 a a t 0 in particular if is a singular value of a then and are eigenvalues of the symmetric embedding the top and bottom halves of sym a s eigenvectors are singular vectors for a power methods applied to a can be related to power methods applied to sym a the rank of sym a is twice the rank of a in this paper we develop similar connections for tensors by building on l h lim s variational approach to tensor singular values and vectors we show how to embed a general order d tensor a into an order d symmetric tensor sym a through the embedding we relate power methods for a s singular values to power methods for sym a s eigenvalues finally we connect the multilinear and outer product rank of a to the multilinear and outer product rank of sym a

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

positivity and nonnegativity conditions of a quartic equation and related problems in this paper explicit conditions for positivity no real roots nonnegativity on positive real axis no positive real roots with odd multiplicity and stability aperiodicity all roots are real and negative and simple of a quartic or biquadratic equation are given the derived conditions from the known solution of the quartic equation are not only complete but simpler than those derived from sturm extended hurwitz inners and hankel methods because of abel s theorem no explicit solution in terms of the roots of an equation higher than quartic exists similar simplification for higher degree polynomial equations may not be possible furthermore explicit conditions for positivity and nonnegativity of equations of higher degree than four are extremely difficult to obtain and may not be possible the results of the paper will hopefully shed some light on a century old problem and thus enhance the engineering application of the derived condition to higher order systems

minimal gersgorin tensor eigenvalue inclusion set and its approximation for a tensor its minimal gersgorin tensor eigenvalue inclusion set is presented by establishing a sufficient and necessary condition for the elements belonging to this set we give a sequence of approximation sets and prove that the limit of this sequence is the minimal gersgorin tensor eigenvalue inclusion set for an irreducible tensor

on new classes of nonnegative symmetric tensors in this paper we introduce three new classes of nonnegative forms or equivalently symmetric tensors and their extensions the newly identified nonnegative symmetric tensors constitute distinctive convex cones in the space of general symmetric tensors order six or above for the special case of quartic forms they collapse into the set of convex quartic homogeneous polynomial functions we discuss the properties and applications of the new classes of nonnegative symmetric tensors in the context of polynomial and tensor optimization numerical experiments for solving certain polynomial optimization models based on the new classes of nonnegative symmetric tensors are presented

inhomogeneous polynomial optimization over a convex set an approximation approach in this paper we consider computational methods for optimizing a multivariate inhomogeneous polynomial function over a general convex set the focus is on the design and analysis of polynomial time approximation algorithms the methods are able to deal with optimization models with polynomial objective functions in any fixed degrees in particular we first study the problem of maximizing an inhomogeneous polynomial over the euclidean ball a polynomial time approximation algorithm is proposed for this problem with an assured relative worst case performance ratio which is dependent only on the dimensions of the model the method and approximation ratio are then generalized to optimize an inhomogeneous polynomial over the intersection of a finite number of co centered ellipsoids furthermore the constraint set is extended to a general convex compact set specifically we propose a polynomial time approximation algorithm with a relative worst case performance ratio for polynomial optimization over some convex compact sets e g a polytope finally numerical results are reported revealing good practical performance of the proposed algorithms for solving some randomly generated instances

topological transition features and synchronizability of a weighted hybrid preferential network abstract in this work we propose a weighted hybrid preferential network model by introducing a hybrid preferential attachment mechanism to a weighted network model proposed by barrat barthelemy and vespignani phys rev e 70 2004 066149 the new model is called hybrid preferential bbv hp bbv model topological properties and synchronizability of the hp bbv model are investigated we find that as the hybrid ratio of determinate to random preferential attachment increases the average clustering coefficient increases the average path length and the assortative coefficient decrease the synchronizability of type i hp bbv network is enhanced but the synchronizability of type ii hp bbv network is weakened

uniqueness of nonnegative tensor approximations we show that for a nonnegative tensor a best nonnegative rank r approximation is almost always unique its best rank one approximation may always be chosen to be a best nonnegative rank one approximation and the set of nonnegative tensors with nonunique best rank one approximations forms an algebraic hypersurface we show that the last part holds true more generally for real tensors and thereby determine a polynomial equation so that a real or nonnegative tensor that does not satisfy this equation is guaranteed to have a unique best rank one approximation we also establish an analogue for real or nonnegative symmetric tensors in addition we prove a singular vector variant of the perron frobenius theorem for positive tensors and apply it to show that a best nonnegative rank r approximation of a positive tensor can never be obtained by deflation as an aside we verify that the euclidean distance ed discriminants of the segre variety and the veronese variety are hypersurfaces and give defining equations of these ed discriminants

maximum block improvement and polynomial optimization in this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem the new approach has the following three main ingredients first we establish a block coordinate descent type search method for nonlinear optimization with the novelty being that we accept only a block update that achieves the maximum improvement hence the name of our new search method maximum block improvement mbi convergence of the sequence produced by the mbi method to a stationary point is proved second we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the mbi approach third we propose a scheme to reach a kkt point of the polynomial optimization provided that a stationary solution for the relaxed tensor problem is available numerical experiments have shown that our new method works very efficiently for a majority of the test instances that we have experimented with the method finds the global optimal solution at a low computational cost

d eigenvalues of diffusion kurtosis tensors diffusion kurtosis imaging dki is a new model in medical engineering where a diffusion kurtosis dk tensor is involved a dk tensor is a fourth order three dimensional fully symmetric tensor in this paper we introduce d eigenvalues for a dk tensor the largest the smallest and the average d eigenvalues of a dk tensor correspond with the largest the smallest and the average apparent kurtosis coefficients akc of a water molecule in the space respectively we present their computational methods and discuss related anisotropy value of akc

on nilpotent lie algebras of small breadth abstract a lie algebra l is said to be of breadth k if the maximal dimension of the images of left multiplication by elements of the algebra is k in this paper we give characterization of finite dimensional nilpotent lie algebras of breadth less than or equal to two furthermore using these characterizations we determined the isomorphism classes of these algebras

positive definiteness of paired symmetric tensors and elasticity tensors abstract in this paper we consider higher order paired symmetric tensors and strongly paired symmetric tensors elasticity tensors and higher order elasticity tensors in solid mechanics are strongly paired symmetric tensors a strongly paired symmetric tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite positive definiteness of elasticity and higher order elasticity tensors is strong ellipticity in solid mechanics which plays an important role in nonlinear elasticity theory we mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional strongly paired symmetric tensors we first show that the concerned strongly paired symmetric tensor is positive definite if and only if its smallest m eigenvalue is positive second we propose several necessary and sufficient conditions under which the concerned strongly paired symmetric tensor is positive definite third we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional strongly paired symmetric tensor can be written as a sum of squares of polynomials and further propose several necessary and or sufficient conditions to judge whether the concerned strongly paired symmetric tensors are positive definite or not fourth by using semidefinite relaxation we propose a sequential semidefinite programming method to compute the smallest m eigenvalue of a fourth order three dimensional strongly paired symmetric tensor by which we can check positive definiteness of the concerned tensor the preliminary numerical results confirm our theoretical findings

gpu based parallel solver via the kantorovich theorem for the nonlinear bernstein polynomial systems this paper proposes a parallel solver for the nonlinear systems in bernstein form based on subdivision and the newton raphson method where the kantorovich theorem is employed to identify the existence of a unique root and guarantee the convergence of the newton raphson iterations since the kantorovich theorem accommodates a singular jacobian at the root the proposed algorithm performs well in a multiple root case moreover the solver is designed and implemented in parallel on graphics processing unit gpu with simd architecture thus efficiency for solving a large number of systems is improved greatly an observation validated by our experimental results

an iterative algorithm based on strong h tensors for identifying positive definiteness of irreducible homogeneous polynomial forms abstract identifying the positive definiteness of an even degree homogeneous polynomial form arises in various applications such as evolutionary game dynamics automatical control magnetic resonance imaging and spectral hypergraph theory however it is difficult to determine whether a given homogeneous polynomial is positive definite or not because the problem is np hard in this paper an iterative algorithm of identifying the positive definiteness of irreducible homogeneous polynomial forms is proposed by identifying strong h tensors with weakly irreducibility the validity of the iterative algorithm is proved theoretically experimental results on multilinear systems high order markov chains and symmetric multi player games are presented to illustrate the applications of the proposed methods

principal invariants and inherent parameters of diffusion kurtosis tensors a diffusion kurtosis dk tensor is a fourth order three dimensional fully symmetric tensor which is used in diffusion kurtosis imaging dki a new model in medical engineering to understand the biological and clinical meaning of the dk tensor we have to measure and calculate some quantities and parameters which are independent from coordinate system choices in this paper we study such quantities and parameters they include the largest the smallest and the average apparent kurtosis coefficients akc values which are invariant from the coordinate system choices and some parameters measured in the inherent coordinate system which is formed by the eigenvector system of the second order diffusion tensor we study their computational formulas and relationships

biquadratic optimization over unit spheres and semidefinite programming relaxations this paper studies the so called biquadratic optimization over unit spheres min x in mathbb r n y in mathbb r m sum 1 leq i k leq n 1 leq j l leq m b ijkl x i y j x k y l subject to x 1 y 1 we show that this problem is np hard and there is no polynomial time algorithm returning a positive relative approximation bound then we present various approximation methods based on semidefinite programming sdp relaxations our theoretical results are as follows for general biquadratic forms we develop a frac 1 2 max m n 2 approximation algorithm under a slightly weaker approximation notion for biquadratic forms that are square free we give a relative approximation bound frac 1 nm when min n m is a constant we present two polynomial time approximation schemes ptass which are based on sum of squares sos relaxation hierarchy and grid sampling of the standard simplex for practical computational purposes we propose the first order sos relaxation a convex quadratic sdp relaxation and a simple minimum eigenvalue method and show their error bounds some illustrative numerical examples are also included

controllability of complex networks via pinning we study the problem of controlling a general complex network toward an assigned synchronous evolution by means of a pinning control strategy we define the pinning controllability of the network in terms of the spectral properties of an extended network topology the roles of the control and coupling gains as well as of the number of pinned nodes are also discussed

nilpotent lie algebras with 2 dimensional commutator ideals abstract we classify all finitely dimensional nilpotent lie k algebras h with 2 dimensional commutator ideals h extending a known result to the case where h is non central and k is an arbitrary field it turns out that while the structure of h depends on the field k if h is central it is independent of k if h is non central and is uniquely determined by the dimension of h in the case where k is algebraically or real closed we also list all nilpotent lie k algebras h with 2 dimensional central commutator ideals h and dim k h 11

synchronization diversity and topology of networks of integrate and fire oscillators we study synchronization dynamics of a population of pulse coupled oscillators in particular we focus our attention on the interplay between topological disorder and synchronization features of networks first we analyze synchronization time t in random networks and find a scaling law which relates t to network connectivity then we compare synchronization time for several other topological configurations characterized by a different degree of randomness the analysis shows that regular lattices perform better than a disordered network this fact can be understood by considering the variability in the number of links between two adjacent neighbors this phenomenon is equivalent to having a nonrandom topology with a distribution of interactions and it can be removed by an adequate local normalization of the couplings

on copositive matrices and strong ellipticity for isotropic elastic materials on etablit les conditions necessaires et suffisantes pour une forte ellipticite des equations d un materiau elastique non lineaire isotrope compressible a l equilibre on utilise un theoreme de representation de matrices copositives du a cottle habetler et lemke 1970

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

classification of 6 dimensional nilpotent lie algebras over fields of characteristic not 2 abstract nilpotent 6 dimensional lie algebras over any field of characteristic not 2 are classified the proof of this classification is essentially constructive for a given 6 dimensional nilpotent lie algebra l following the steps of the proof it is possible to find a lie algebra m that occurs in the classification and an isomorphism l m in the proof a method due to skjelbred and sund is used along with a method based on grobner bases to find isomorphisms

minimizing polynomials via sum of squares over the gradient ideal a method is proposed for finding the global minimum of a multivariate polynomial via sum of squares sos relaxation over its gradient variety that variety consists of all points where the gradient is zero and it need not be finite a polynomial which is nonnegative on its gradient variety is shown to be sos modulo its gradient ideal provided the gradient ideal is radical or the polynomial is strictly positive on the real gradient variety this opens up the possibility of solving previously intractable polynomial optimization problems the related problem of constrained minimization is also considered and numerical examples are discussed experiments show that our method using the gradient variety outperforms prior sos methods

e cospectral hypergraphs and some hypergraphs determined by their spectra two k uniform hypergraphs are said to be cospectral e cospectral if their adjacency tensors have the same characteristic polynomial e characteristic polynomial a k uniform hypergraph h is said to be determined by its spectrum if there is no non isomorphic k uniform hypergraph cospectral with h in this note we give a method for constructing e cospectral hypergraphs which is similar with godsil mckay switching some hypergraphs are shown to be determined by their spectra

criterions for the positive definiteness of real supersymmetric tensors the positive definiteness of an even degree homogeneous polynomial form f x plays an important role in the stability study of nonlinear autonomous systems via lyapunov s direct method in automatic control and the positive definiteness of f x is equivalent to that of an even order supersymmetric tensor which defines f x in this paper we provide some criterions for identifying the positive definiteness of an even order real supersymmetric tensor moreover an iterative algorithm for identifying the positive definiteness of an even order real supersymmetric tensor is obtained numerical examples are given to verify the corresponding results

the number of eigenvalues of a tensor abstract eigenvectors of tensors as studied recently in numerical multilinear algebra correspond to fixed points of self maps of a projective space we determine the number of eigenvectors and eigenvalues of a generic tensor and we show that the number of normalized eigenvalues of a symmetric tensor is always finite we also examine the characteristic polynomial and how its coefficients are related to discriminants and resultants

enhanced synchronizability via age based coupling in this brief report we study the synchronization of growing scale free networks an asymmetrical age based coupling method is proposed with only one free parameter alpha although the coupling matrix is asymmetric our coupling method could guarantee that all the eigenvalues are non negative reals the eigenratio r will approach 1 in the large limit of alpha

a levenberg marquardt method for solving semi symmetric tensor equations abstract in this paper we propose a levenberg marquardt lm method for solving tensor equations with semi symmetric coefficient tensor and prove its global convergence and local quadratic convergence under the local error bound condition which is weaker than non singularity as application we solve h eigenvalue of real semi symmetric tensor by the lm method at last some numerical examples are provided to illustrate the efficiency and validity of these methods proposed

a modified newton s method for best rank one approximation to tensors in this paper a modified newton s method for the best rank one approximation problem to tensor is proposed we combine the iterative matrix of jacobi gauss newton jgn algorithm or alternating least squares als algorithm with the iterative matrix of grq newton method and present a modified version of grq newton algorithm a line search along the projective direction is employed to obtain the global convergence preliminary numerical experiments and numerical comparison show that our algorithm is efficient

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

the degree of the e characteristic polynomial of an even order tensor the e characteristic polynomial of an even order supersymmetric tensor is a useful tool in determining the positive definiteness of an even degree multivariate form in this paper for an even order tensor we first establish the formula of its e characteristic polynomial by using the classical macaulay formula of resultants then give an upper bound for the degree of that e characteristic polynomial examples illustrate that this bound is attainable in some low order and dimensional cases

approximation algorithms for homogeneous polynomial optimization with quadratic constraints in this paper we consider approximation algorithms for optimizing a generic multi variate homogeneous polynomial function subject to homogeneous quadratic constraints such optimization models have wide applications e g in signal processing magnetic resonance imaging mri data training approximation theory and portfolio selection since polynomial functions are non convex the problems under consideration are all np hard in general in this paper we shall focus on polynomial time approximation algorithms in particular we first study optimization of a multi linear tensor function over the cartesian product of spheres we shall propose approximation algorithms for such problem and derive worst case performance ratios which are shown to be dependent only on the dimensions of the model the methods are then extended to optimize a generic multi variate homogeneous polynomial function with spherical constraint likewise approximation algorithms are proposed with provable approximation performance ratios furthermore the constraint set is relaxed to be an intersection of co centered ellipsoids namely we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin and propose polynomial time approximation algorithms with provable worst case performance ratios numerical results are reported illustrating the effectiveness of the approximation algorithms studied