a tensor singular values and its symmetric embedding eigenvalues national natural science foundation of china 11201092 11261012 scientific research starting foundation for doctors of guizhou normal university china program of young scientists of jiangxi province 20122bcb23003

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

arithmetic duality theorems for 1 motives over function fields in this paper we obtain a poitou tate exact sequence for finite and flat group schemes over a global function field in addition we extend the duality theorems for 1 motives over number fields obtained by d harari and t szamuely to the function field 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

some upper bounds on zt eigenvalues of tensors abstract in this paper we give upper bounds on zt spectral radius of a tensor a t 1 2 which extend the upper bounds of brauer to tensors moreover an upper bound on the z1 spectral radius is proposed via modulus sum of the entries of certain dimension of a which improves the upper bound given by li et al numerical experiments are given to illustrate the utility of the upper bound

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

retract rational fields abstract let k be an infinite field the notion of retract k rationality was introduced by saltman in the study of noether s problem and other rationality problems we will investigate the retract rationality of a field in this paper theorem 1 let k k l be fields if k is retract k rational and l is retract k rational then l is retract k rational theorem 2 for any finite group g containing an abelian normal subgroup h such that g h is a cyclic group for any complex representation g gl v the fixed field c v g is retract c rational theorem 3 if g is a finite group then all the sylow subgroups of g are cyclic if and only if c m g is retract c rational for all g lattices m for all short exact sequences 0 c m m 0 because the unramified brauer group of a retract c rational field is trivial theorems 2 and 3 generalize previous results of bogomolov and barge respectively see theorems 5 9 and 6 1

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

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

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

the existence and uniqueness of eigenvalues for monotone homogeneous mapping pairs abstract in this paper the concept of eigenvalue is introduced to the monotone homogeneous mapping pairs f g and its existence and uniqueness are established successfully under the boundedness of some orbits of f g in the hilbert semi norm also the collatz wielandt min max type property is obtained for such a class of mapping pairs in particular the nonlinear perron frobenius property for nonnegative tensor pairs a b is obtained without involving the calculation of the tensor inversion by particularizing the mapping g or f several results can emerge as corollaries they lead to further existence results and open problems

essential dimension and canonical dimension of gerbes banded by groups of multiplicative type we study gerbes mathcal x over a field banded by groups of multiplicative type a and prove the formula text ed p mathcal x text cdim p mathcal x text ed p a when a is the kernel of a homomorphism of algebraic tori q rightarrow s with q invertible and s split here p is either a prime or 0 this result is applied to prove new results on the essential dimension of algebraic groups

markov chains with memory tensor formulation and the dynamics of power iteration a markov chain with memory is no different from the conventional markov chain on the product state space such a markovianization however increases the dimensionality exponentially instead markov chain with memory can naturally be represented as a tensor whence the transitions of the state distribution and the memory distribution can be characterized by specially defined tensor products in this context the progression of a markov chain can be interpreted as variants of power like iterations moving toward the limiting probability distributions what is not clear is the makeup of the second dominant eigenvalue that affects the convergence rate of the iteration if the method converges at all casting the power method as a fixed point iteration this paper examines the local behavior of the nonlinear map and identifies the cause of convergence or divergence as an application it is found that there exists an open set of irreducible and aperiodic transition probability tensors where the z eigenvector type power iteration fails to converge

rationality problem for generic tori in simple groups abstract we prove that except for several already known cases the generic torus of a simple adjoint or simply connected group is not stably rational this confirms a conjecture by le bruyn on generic norm tori

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 generalization of serre s condition mathrm f f with applications to the finiteness of unramified cohomology in this paper we introduce a condition mathrm f m on a field k for a positive integer m that generalizes serre s condition f and which still implies the finiteness of the galois cohomology of finite galois modules annihilated by m and algebraic k tori that split over an extension of degree dividing m as well as certain groups of etale and unramified cohomology various examples of fields satisfying mathrm f m including those that do not satisfy f are given

formulas for the unramified brauer group of a principal homogeneous space of a linear algebraic group abstract for a smooth compactification v of a principal homogeneous space e under a connected linear algebraic group g defined over a field k of characteristic zero we present two formulas expressing br v br k in terms of g

eigenvectors of laplacian or signless laplacian of hypergraphs associated with zero eigenvalue abstract let g be a connected m uniform hypergraph in this paper we mainly consider the eigenvectors of the laplacian or signless laplacian tensor of g associated with zero eigenvalue called the first laplacian or signless laplacian eigenvectors of g by means of the incidence matrix of g the number of first laplacian or signless laplacian or h eigenvectors can be obtained explicitly by solving the smith normal form of the incidence matrix over z m or z 2 consequently we prove that the number of first laplacian h eigenvectors is equal to the number of first signless laplacian h eigenvectors when zero is an h eigenvalue of the signless laplacian tensor we establish a connection between first laplacian signless laplacian h eigenvectors and the even odd bipartitions of g

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

hilbert irreducibility above algebraic groups this article concerns hilbert irreducibility for covers of algebraic groups with results which appear to be difficult to treat by existing techniques the present method works by first studying irreducibility above torsion specializations e g over cyclotomic extensions and then descending the field by chebotarev theorem among the results we offer an irreducibility theorem for the fibers above a cyclic dense subgroup of a cover of gmn theorem 1 and of a power en of an elliptic curve without cm theorem 2 this had not been treated before for n 1 as a further application in the function field context we obtain a kind of bertini s theorem for algebraic subgroups of gmn in place of linear spaces theorem 3 along the way we shall prove other results as a general lifting theorem above tori theorem 3 1

a measure for the decentralized assignability of eigenvalues abstract the results of 7 are generalized to produce a continuous measure of how close a mode is to being a decentralized fixed mode 13 implications of this measure on controller design are also explored

decentralized stabilization and pole assignment for general proper systems considered is the problem of finding existence conditions and a controller synthesis procedure using decentralized control for assigning the poles of a linear time invariant proper system described by a state space model c a b d where no assumption is made regarding the structure of d this problem has direct application to the decentralized stabilization problem decentralized robust servomechanism problem etc and is a nontrivial extension to the standard decentralized problem where it is assumed that the direct feedthrough terms either are absent or have a block diagonal structure

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

examples of non rational varieties of adjoint groups let k be a field of characteristic 2 and k a separable closure of k s we say that an algebraic variety xrk is stably k rational if there exist two m n m n w x affine spaces a a and a k rational map a x f a merkurjev 9 k k gave a criterion of stable k rationality for the adjoint classical groups with absolute rank f 3 which covers the case of the variety pso q for any quadratic form qrk of rank f 6 this criterion gives examples of field k and quadratic form q of rank 6 with non trivial signed discriminant such that the variety pso q is not stably k rational the main result of this paper is the following

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

existence of real eigenvalues of real tensors abstract we use the brouwer degree to establish the existence of real eigenpairs of higher order real tensors in various settings also we provide some finer criteria for the existence of real eigenpairs of two dimensional real tensors and give a complete classification of the brouwer degree zero and 2 maps induced by general third order two dimensional real tensors

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

an eigenvalue method for testing positive definiteness of a multivariate form in this paper we present an eigenvalue method for testing positive definiteness of a multivariate form this problem plays an important role in the stability study of nonlinear autonomous systems via lyapunov s direct method in automatic control at first we apply the d andrea dickenstein version of the classical macaulay formulas of the resultant to compute the symmetric hyperdeterminant of an even order supersymmetric tensor by using the supersymmetry property we give detailed computation procedures for the bezoutians and specified ordering of monomials in this approach we then use these formulas to calculate the characteristic polynomial of a fourth order three dimensional supersymmetric tensor and give an eigenvalue method for testing positive definiteness of a quartic form of three variables some numerical results of this method are reported

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

the product of the eigenvalues of a symmetric tensor abstract we study e eigenvalues of a symmetric tensor f of degree d on a finite dimensional euclidean vector space v and their relation with the e characteristic polynomial of f we show that the leading coefficient of the e characteristic polynomial of f when it has maximum degree is the d 2 th power respectively the d 2 2 th power when d is odd respectively when d is even of the q discriminant where q is the d th veronese embedding of the isotropic quadric q p v this fact together with a known formula for the constant term of the e characteristic polynomial of f leads to a closed formula for the product of the e eigenvalues of f which generalizes the fact that the determinant of a symmetric matrix is equal to the product of its eigenvalues

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

on algebraic properties of general proper decentralized abstract the new concepts of the decentralized output feedback variable polynomial the decentralized output feedback cycle index of general proper systems and the geometric multiplicities of decentralized fixed modes are introduced their computational methods and some algebraic properties are presented it is shown that the decentralized output feedback cycle index of a general proper system is equal to one when the system has no fixed modes or equal to the maximum of the geometric multiplicities of its decentralized fixed modes it is also shown that almost all decentralized output feedback can be used to make the zeros of the decentralized variable polynomial distinct and disjoint from any given finite set of points on the complex plane

distribution of the eigenvalues of a random system of homogeneous polynomials abstract let f f 1 f n be a system of n complex homogeneous polynomials in n variables of degree d we call c an eigenvalue of f if there exists v c n 0 with f v v generalizing the case of eigenvalues of matrices d 1 we derive the distribution of when the f i are independently chosen at random according to the unitary invariant weyl distribution and determine the limit distribution for n

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

the unramified brauer group of norm one tori let k be a number field and k k galois we transform the construction of the unramified brauer group of the norm one torus rk k1 gm into the construction of a special abelian extension over k if k q and k q biquadratic we explicitly construct the unramified brauer group of rk q1 gm

algebraic properties of decentralized control systems the new concepts of the decentralized output feedback cycle index the geometric multiplicities of decentralized fixed modes dfm and the decentralized output feedback variable polynomial dvp of a decentralized control system dcs are introduced their computational methods and some properties are given it is shown that the decentralized output feedback cycle index is equal to one when the dcs has no decentralized fixed modes dfm or equal to the maximum of the geometric multiplicities of dfm otherwise almost every decentralized output feedback law can be used to make the zeros of the dvp distinct it is also shown that almost any decentralized output feedback law will make the zeros of the dvp disjoint from any given finite set of points on the complex plane it is then shown that if a dcs has no dfm then it can be made both controllable and observable with respect to any non trivial input and output by applying almost any decentralized output feedback law

continuation methods for computing z h eigenpairs of nonnegative tensors abstract in this paper a homotopy continuation method for the computation of nonnegative z h eigenpairs of a nonnegative tensor is presented we show that the homotopy continuation method is guaranteed to compute a nonnegative eigenpair additionally using degree analysis we show that the number of positive z eigenpairs of an irreducible nonnegative tensor is odd a novel homotopy continuation method is proposed to compute an odd number of positive z eigenpairs and some numerical results are presented

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

convergence of an algorithm for the largest singular value of a nonnegative rectangular tensor abstract in this paper we present an iterative algorithm for computing the largest singular value of a nonnegative rectangular tensor we establish the convergence of this algorithm for any irreducible nonnegative rectangular tensor

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

on the hilbert property and the fundamental group of algebraic varieties in this paper we link the so called hilbert property hp for an algebraic variety over a number field with its fundamental group in a perspective which appears new the notion of hp comes from hilbert s irreducibility theorem and has important implications for instance towards the inverse galois problem we shall observe that the hp is in a sense opposite to the chevalley weil theorem this shall immediately entail the result that the hp can possibly hold only for simply connected varieties in the appropriate sense in turn this leads to new counterexamples to the hp involving enriques surfaces in this view we shall also formulate an alternative related property possibly holding in full generality and some conjectures which unify part of what is known in the topic these predict that for a variety with a zariski dense set of rational points the validity of the hp which is defined arithmetically is indeed of purely topological nature also a consequence of these conjectures would be a positive solution of the inverse galois problem in the paper we shall also prove the hp for a k3 surface related to the above enriques surface providing what appears to be the first example of a variety non rational over mathbb c for which the hp can be proved in an appendix we shall further discuss among other things the hp also for kummer surfaces all of this basically exhausts the study of the hp for surfaces with a zariski dense set of rational points

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

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

universal torsors and values of quadratic polynomials represented by norms let k k be an extension of number fields and let p t be a quadratic polynomial over k let x be the affine variety defined by p t n k k mathbf z we study the hasse principle and weak approximation for x in three cases for k k 4 and p t irreducible over k and split in k we prove the hasse principle and weak approximation for k mathbb q with arbitrary k we show that the brauer manin obstruction to the hasse principle and weak approximation is the only one for k k 4 and p t irreducible over k we determine the brauer group of smooth proper models of x in a case where it is non trivial we exhibit a counterexample to weak approximation

numerical ranges of tensors abstract the main aim of this paper is to generalize matrix numerical ranges to the tensor case based on tensor norms we show that the basic properties of matrix numerical ranges such as compactness and convexity are valid for tensor numerical ranges we make use of convexity property to propose an algorithm for approximating tensor numerical ranges in which tensor eigenvalues are contained also we consider tensor numerical ranges based on inner products however they may not be convex in general

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

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

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

stable rationality of certain pgln quotients stable rationality of the field of matrix invariants m n m n pgl n is proved forn 5 andn 7 in combination with existing results this shows that v pgl n is stably rational wheneverv is an almost free representation ofpgl n andn divides 420 22 3 5 7

the center of the generic g crossed product abstract let g be a finite group and let f be a field of characteristic zero in this paper we construct a generic g crossed product over f using generic graded matrices the center of this generic g crossed product denoted by f g is then the invariant field of a suitable g action on a field of rational functions in several indeterminates the main goal of this paper is to study the extensions f g f given that f contains enough roots of unity and determine how close they are to being purely transcendental in particular we show that f g f is a stably rational extension for g c 2 c 2 n where n is odd and for g n 2 m e 1 1 where g c d n 2 m 1 furthermore we prove that if h k are groups of coprime orders then f h k is stably rationally equivalent to the fraction field of f h f k

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

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

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

arithmetical birational invariants of linear algebraic groups over two dimensional geometric fields let g be a connected linear algebraic group over a geometric field k of cohomological dimension 2 of one of the types which were considered by colliot thelene gille and parimala basing on their results we compute the group of classes of r equivalence g k r the defect of weak approximation a g the first galois cohomology h 1 k g and the tate shafarevich kernel x 1 k g for suitable k in terms of the algebraic fundamental group 1 g we prove that the groups g k r and a g and the set x 1 k g are stably k birational invariants of g

motivic classes of classifying stacks of some semi direct products abstract let k be a field let g be a finite group and let t be a split k torus on which g acts multiplicatively for every m 1 denote by t m the m torsion subgroup of t under a suitable assumption on m we show that the motivic class of b t m g in k 0 stacks k equals that of bg as a consequence we prove that the motivic class of bw is trivial for a large class of complex reflection groups w

structurally fixed modes a notion of structurally fixed modes is introduced in the framework of decentralized control systems to identify the modes that cannot be shifted by decentralized feedback regardless of the numerical values of system parameters in this way a qualitative characterization of decentralized fixed modes is provided which is suitable for delineating decentrally stabilizable systems the characterization leads naturally to the notion of fixed modes with respect to arbitrary feedback structure constraints

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