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

on computational complexity and honest polynomial degrees abstract it is shown that no 20 set a with a and a both semilow has minimal honest polynomial hp degree the technique extends to show that the hp degrees and polynomial degrees of low r e sets are dense in particular r e sets of minimal hp degree must have high computational complexity in the sense of blum and marques j symbolic logic 38 1973 579 593 these results also hold for tally degrees and polynomial honest m degrees

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

the p t degrees of the recursive sets lattice embeddings extensions of embeddings and the two quantifier theory abstract ambos spies 1984a showed that the two basic nondistributive lattices can be embedded in rp t the polynomial time turing degrees of the recursive sets we introduce more general techniques to extend his results to show that every recursive lattice can be embedded in rp t in addition to lattice theoretic representation theorems we use the scheme of priority style arguments coupled with looking back techniques presented in shinoda and slaman 1988 1990 we also generalize the density type results of ladner 1975 and many others to settle the full extension of the embedding problem for rp t combined with the logical analysis of sentences with one alternation of quantifiers shore 1978 lerman 1983 these results suffice to decide the full theory of rp t they also give a strong nonhomogeneity result the p time degrees of the sets recursive in and if desired p time above two distinct sets a and b are almost never isomorphic the situation for the p time many one degrees is quite different we decide the extension of the embedding problem differently than for rp t but not the theory

a uniform approach to obtain diagonal sets in complexity classes abstract a uniform method for constructing sets which diagonalize over certain complexity classes while preserving other complexity properties is given we obtain some known results as well as some new ones as corollaries of our main theorem the new results concern the complexity classes p np co np pspace apt almost polynomial time r random polynomial time and the polynomial hierarchy

kolmogorov complexity and degrees of tally sets we show that either epm tally epbtt tally or epm tally ep1 tt tally ep2 tt tally ep3 tt tally where erp tally denotes the class of sets which are equivalent to a tally set under rp reductions furthermore the question of whether or not emp tally ebttp tally is equivalent to the question of whether or not ne predicates can be solved in deterministic exponential time the proofs use the techniques of generalized kolmogorov complexity as corollaries to some of the main results we obtain new results about the kolmogorov complexity of sets in p

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

sublattices of the polynomial time degrees we show that any countable distributive lattice can be embedded in any interval of polynomial time degrees furthermore the embeddings can be chosen to preserve the least or the greatest element this holds for both polynomial time bounded many one and turing reducibilities as well as for all of the common intermediate reducibilities

minimal degrees for polynomial reducibilities the existence of minimal degrees is investigated for several polynomial reducibilities it is shown that no set has minimal degree with respect to polynomial many one or turing reducibility this extends a result of ladner in which only recursive sets are considered a polynomial reducibility h t is defined this reducibility is a strengthening of polynomial turing reducibility and its properties relate to the p np question for this new reducibility a set of minimal degree is constructed under the assumption that p np however the set constructed is nonrecursive and it is shown that no recursive set is of minimal h t degree

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

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

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

a high performance parallel implementation of the certified reduced basis method abstract the certified reduced basis method herein rb method is a popular approach for model reduction of parametrized partial differential equations in this paper we introduce new techniques that are required to efficiently implement the offline construction stage of the rb method on high performance parallel supercomputers this enables us to generate certified rb models for large scale three dimensional problems that can be evaluated on standard workstations and other thin computing resources with speedup of many orders of magnitude compared to the corresponding full order model we use our implementation to perform detailed numerical studies for two computationally expensive model problems a natural convection fluid flow problem and a many parameter heat transfer problem in the heat transfer problem we exploit the computational efficiency of the rb method to perform a detailed study of snapshot selection in the greedy algorithm and we also examine statistics of the output sensitivity derivatives to obtain a global view of the relative importance of the parameters

oracle dependent properties of the lattice of np sets abstract we consider under the assumption p np questions concerning the structure of the lattice of np sets together with the sublattice p we show that two questions which are slightly more complex than the known splitting properties of this lattice cannot be settled by arguments which relativize the two questions which we consider are whether every infinite np set contains an infinite p subset and whether there exists an np simple set we construct several oracles all of which make p np and which in addition make the above mentioned statements either true or false in particular we give a positive answer to the question raised by bennett and gill 1981 whether an oracle b exists making p b np b and such that every infinite set in np b has an infinite subset in p b the constructions of the oracles are finite injury priority arguments

on some natural complete operators abstract an operator is a mapping from integer functions to integer functions it is known from a result by baker gill and solovay 1975 that there exists an operator computable in polynomial time by a nondeterministic oracle turing machine called an np operator but not computable in polynomial time by any deterministic oracle turing machine we investigate several natural operators which share similar properties we use the concept of completeness to give a precise classification of the complexity of these operators for example the question of finding maximum values of polynomial time computable functions can be formulated as an operator complete for the class of np operators

computational complexity of recursively enumerable sets we survey a variety of recent notions and results for classifying the computational complexity of a recursively enumerable r e set these complexity theoretic notions are shown to be equivalent to various recursion theoretic notions and are used to relate the complexity properties of an r e set a to its algebraic structure in the appropriate lattice and to the information it encodes

the reduced basis method for initial value problems this paper deals with the application of the reduced basis method to the initial value problem of ordinary differential equations a global approximation of the true solution curve is defined in terms of a sequence of reduced basis curves error bounds are established and these lead to order estimates for specific choices of reduced basis subspaces the effects of numerical integration are then incorporated into the estimates a numerical study is included

certified reduced basis model validation a frequentistic uncertainty framework abstract we introduce a frequentistic validation framework for assessment acceptance or rejection of the consistency of a proposed parametrized partial differential equation model with respect to noisy experimental data from a physical system our method builds upon the hotelling t2 statistical hypothesis test for bias first introduced by balci and sargent in 1984 and subsequently extended by mcfarland and mahadevan 2008 our approach introduces two new elements a spectral representation of the misfit which reduces the dimensionality and variance of the underlying multivariate gaussian but without introduction of the usual regression assumptions a certified verified reduced basis approximation reduced order model which greatly accelerates computational performance but without any loss of rigor relative to the full finite element discretization we illustrate our approach with examples from heat transfer and acoustics both based on synthetic data we demonstrate that we can efficiently identify possibility regions that characterize parameter uncertainty furthermore in the case that the possibility region is empty we can deduce the presence of unmodeled physics such as cracks or heterogeneities

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

certified reduced basis methods and output bounds for the harmonic maxwell s equations we propose certified reduced basis methods for the efficient and reliable evaluation of a general output that is implicitly connected to a given parameterized input through the harmonic maxwell s equations the truth approximation and the development of the reduced basis through a greedy approach is based on a discontinuous galerkin approximation of the linear partial differential equation the formulation allows the use of different approximation spaces for solving the primal and the dual truth approximation problems to respect the characteristics of both problem types leading to an overall reduction in the off line computational effort the main features of the method are the following i rapid convergence on the entire representative set of parameters ii rigorous a posteriori error estimators for the output and iii a parameter independent off line phase and a computationally very efficient on line phase to enable the rapid solution of many query problems arising in control optimization and design the versatility and performance of this approach is shown through a numerical experiment illustrating the modeling of material variations and problems with resonant behavior

certified reduced basis methods for parametrized parabolic partial differential equations with non affine source terms abstract we present rigorous a posteriori output error bounds for reduced basis approximations of parametrized parabolic partial differential equations with non affine source terms the method employs the empirical interpolation method in order to construct affine coefficient function approximations of the non affine parametrized functions our a posteriori error bounds take both error contributions explicitly into account the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation to this end we employ recently developed rigorous error bounds for the empirical interpolation method and develop error estimation and primal dual formulations to provide rigorous bounds for the error in specific outputs of interest we present an efficient offline online computational procedure for the calculation of the reduced basis approximation and associated error bound the method is thus ideally suited for many query or real time contexts as a specific motivational example we consider a three dimensional mathematical model of a welding process our numerical results show that we obtain efficient and reliable mathematical models which may be gainfully employed in manufacturing and product development

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

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

fast imaging of mean axial and radial diffusion kurtosis diffusion kurtosis imaging dki is being increasingly reported to provide sensitive biomarkers of subtle changes in tissue microstructure however dki also imposes larger data requirements than diffusion tensor imaging dti hence the widespread adaptation and exploration of dki would benefit from more efficient acquisition and computational methods to meet this demand we recently developed a method capable of estimating mean kurtosis with only 13 diffusion weighted images this approach was later shown to provide very accurate mean kurtosis estimates and to be more efficient in terms of contrast to noise per unit time however insofar the computation of two other critical dki parameters radial and axial kurtosis has required the estimation of all 22 variables parameterizing the full dki signal expression here we present two strategies for estimating all of dki s principal parameters mean kurtosis radial kurtosis and axial kurtosis using only 19 diffusion weighted images compared to the current state of the art acquisitions typically requiring about 60 images the first approach is based on axially symmetric diffusion and kurtosis tensors presented here for the first time and referred to as axially symmetric dki the second approach is applicable in tissues with a priori known principal diffusion direction and does not require fitting of any kind the approaches are evaluated in human brain in vivo as well as in fixed rat spinal cord and are demonstrated to provide metrics in good agreement with their full dki counterparts estimated with nonlinear least squares for small data sets and in white matter axially symmetric dki provides more accurate and robust estimates than unconstrained dki the significant acceleration achieved further paves the way to routine application of the technique

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

on p 2 theories of hp t degrees of low sets abstract if i is a set of 0 2 low sets which is closed under recursive join and downward closed under t then it is shown that the 2 theory of the honest polynomial time turing degrees of the sets in i is the same as the 2 theory of the polynomial time turing degrees of the recursive sets and so that it is decidable

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

an hp certified reduced basis method for parametrized elliptic partial differential equations we present a new hp parameter multidomain certified reduced basis rb method for rapid and reliable online evaluation of functional outputs associated with parametrized elliptic partial differential equations we propose and provide theoretical justification for a new procedure for adaptive partition h refinement of the parameter domain into smaller parameter subdomains we pursue a hierarchical splitting of the parameter sub domains based on proximity to judiciously chosen parameter anchor points within each subdomain subsequently we construct individual standard rb approximation spaces p refinement over each subdomain greedy parameter sampling procedures and a posteriori error estimation play important roles in both the h type and p type stages of the new algorithm we present illustrative numerical results for a convection diffusion problem the new hp approach is considerably faster respectively more costly than the standard p type reduced basis method in the online respectively offline stage

multiscale simulations for suspensions of rod like molecules we study the doi model for suspensions of rod like molecules the doi model couples a microscopic fokker planck type equation smoluchowski equation to the macroscopic stokes equation the smoluchowski equation describes the evolution of the distribution of the rod orientations it comes as a drift diffusion equation on the sphere at every point in physical space for sufficiently high macroscopic shear rates high deborah numbers the solution of the coupled system develops internal layers in the macroscopic strain rate the spurt phenomenon in the high deborah numbers regime the drift term in the smoluchowski equation is dominant we thus introduce a finite volume type discretization of the microscopic smoluchowski equation which is motivated by transport dominated pdes we carry out direct numerical simulations of the spurt phenomenon both in the dilute and concentrated regimes below the isotropic nematic transition the solution structure is identical to the one described by purely macroscopic models jso model for higher concentrations we observe the formation of microstructure coming from a position dependent tumbling rate we also investigate the 2 d stability of the spurted solution

local convergence analysis of inverse iteration algorithm for computing the h spectral radius of a nonnegative weakly irreducible tensor abstract in this paper we present an inverse iteration algorithm to find the h spectral radius and the associated positive eigenvector of a nonnegative weakly irreducible tensor which always preserve the positivity of approximate eigenvectors the local quadratical convergence of the proposed algorithm is established based on a basic result of the newton s method for solving nonlinear equations some numerical examples illustrate the efficiency of the proposed algorithm

diagonalization uniformity and fixed point theorems abstract we derive new fixed point theorems for subrecursive classes together with a theorem on the uniformity of certain reductions from a general formulation of the technique of delayed diagonalization this formulation extends the main theorem of u schoning theoret comput sci 18 1982 95 103 to cases which involve infinitely many diagonal classes c k and which allow each c k to contain uncountably many members the main technical work ties the familiar concept of a witness function directly to the often studied cantor set topology on languages and provides a delay construction which refines those due to schoning s breidtbart and d schmidt our a e fixed point theorems do not require that the programming system for the subrecursive class in question be well behaved we compare them to results which do the other theorem is similar to the uniform boundedness theorem of classical analysis and extends work of j grollmann and a selman

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

interpolation method for adapting reduced order models and application to aeroelasticity reduced order models are usually thought of as computationally inexpensive mathematical representations that offer the potential for near real time analysis although most reduced order models can operate in near real time their construction can be computationally expensive as it requires accumulating a large number of system responses to input excitations furthermore reduced order models usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation together these two issues underline the need for a fast and robust method for adapting precomputed reduced order models to new sets of physical or modeling parameters to this effect this paper presents an interpolation method based on the grassmann manifold and its tangent space at a point that is applicable to structural aerodynamic aeroelastic and many other reduced order models based on projection schemes this method is illustrated here with the adaptation of computational fluid dynamics based aeroelastic reduced order models of complete fighter configurations to new values of the freestream mach number good correlations with results obtained from direct reduced order model reconstruction high fidelity nonlinear and linear simulations are reported thereby highlighting the potential of the proposed reduced order model adaptation method for near real time aeroelastic predictions using precomputed reduced order model databases

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

certified reduced basis methods for parametrized saddle point problems we present reduced basis approximations and associated rigorous a posteriori error bounds for parametrized saddle point problems first we develop new a posteriori error estimates that unlike earlier approaches provide upper bounds for the errors in the approximations of the primal variable and the lagrange multiplier separately the proposed method is an application of brezzi s theory for saddle point problems to the reduced basis context and exhibits significant advantages over existing methods second based on an analysis of brezzi s theory we compare several options for the reduced basis approximation space from the perspective of approximation stability and computational cost finally we introduce a new adaptive sampling procedure for saddle point problems constructing approximation spaces that are stable and compared to earlier approaches computationally much more efficient the method is applied to a stokes flow problem in a two dimensional channel with a parametrized rectangular obstacle

pseudo spectra theory of tensors and tensor polynomial eigenvalue problems abstract this paper is devoted to the extension of the pseudo spectra theory from matrices to tensors based on the definition of an eigenpair of real symmetric tensors and results on the pseudo spectrum of square matrices we first introduce the pseudo spectrum of a complex tensor and investigate its fundamental properties such as its computational interpretations and the link with the stability of its related homogeneous dynamical system we then extend the pseudo spectrum to the setting of tensor polynomial eigenvalue problems we further derive basic structure of the pseudo spectrum for tensor polynomial eigenvalue problems including the symmetry boundedness and number of connected components under suitable mild assumptions finally we discuss the implication of the pseudo spectrum for computing the backward errors and the distance from a regular tensor polynomial to the nearest irregular tensor polynomial

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

a natural norm successive constraint method for inf sup lower bounds we present a new approach for the construction of lower bounds for the inf sup stability constants required in a posteriori error analysis of reduced basis approximations to anely parametrized partial dierential equations we combine the linearized inf sup statement of the natural norm approach with the approximation procedure of the successive constraint method scm the former natural norm provides an economical parameter expansion and local concavity in parameter a small er optimization problem which enjoys intrinsic lower bound properties the latter scm provides a systematic optimization framework a linear program lp relaxation which readily incorporates continuity and stability constraints the natural norm scm requires a parameter domain decomposition we propose a greedy algorithm for selection of the scm control points as well as adaptive construction of the optimal subdomains the ecacy of the natural norm scm is illustrated through numerical results for two types of non coercive problems the helmholtz equation for acoustics elasticity and electromagnetics and the convection diusi on equation

towards better mr characterization of neural tissues using directional diffusion kurtosis analysis mr diffusion kurtosis imaging dki was proposed recently to study the deviation of water diffusion from gaussian distribution mean kurtosis the directionally averaged kurtosis has been shown to be useful in assessing pathophysiological changes thus yielding another dimension of information to characterize water diffusion in biological tissues in this study orthogonal transformation of the 4th order diffusion kurtosis tensor was introduced to compute the diffusion kurtoses along the three eigenvector directions of the 2nd order diffusion tensor such axial k and radial k upper left and right quadrants kurtoses measured the kurtoses along the directions parallel and perpendicular respectively to the principal diffusion direction dki experiments were performed in normal adult n 7 and formalin fixed rat brains n 5 dki estimates were documented for various white matter wm and gray matter gm tissues and compared with the conventional diffusion tensor estimates the results showed that kurtosis estimates revealed different information for tissue characterization for example k and k upper left and right quadrants under formalin fixation condition exhibited large and moderate increases in wm while they showed little change in gm despite the overall dramatic decrease of axial and radial diffusivities in both wm and gm these findings indicate that directional kurtosis analysis can provide additional microstructural information in characterizing neural tissues

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

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

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

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

minimal pairs for p abstract we prove a general minimal pair theorem which yields as corollaries many results about minimal pairs for p obtained previously by various authors furthermore it yields the new counterintuitive result that there exist arbitrarily complex minimal pairs we also investigate the question of whether minimal pairs in np are low

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

exploring the 3d geometry of the diffusion kurtosis tensor impact on the development of robust tractography procedures and novel biomarkers diffusion kurtosis imaging dki is a diffusion weighted technique which overcomes limitations of the commonly used diffusion tensor imaging approach this technique models non gaussian behaviour of water diffusion by the diffusion kurtosis tensor kt which can be used to provide indices of tissue heterogeneity and a better characterisation of the spatial architecture of tissue microstructure in this study the geometry of the kt is elucidated using synthetic data generated from multi compartmental models where diffusion heterogeneity between intra and extra cellular media is taken into account as well as the sensitivity of the results to each model parameter and to synthetic noise furthermore based on the assumption that the maxima of the kt are distributed perpendicularly to the direction of well aligned fibres a novel algorithm for estimating fibre direction directly from the kt is proposed and compared to the fibre directions extracted from dki based orientation distribution function odf estimates previously proposed in the literature synthetic data results showed that for fibres crossing at high intersection angles direction estimates extracted directly from the kt have smaller errors than the dki based odf estimation approaches dki odf nevertheless the proposed method showed smaller angular resolution and lower stability to changes of the simulation parameters on real data tractography performed on these kt fibre estimates suggests a higher sensitivity than the dki based odf in resolving lateral corpus callosum fibres reaching the pre central cortex when diffusion acquisition is performed with five b values using faster acquisition schemes kt based tractography did not show improved performance over the dki odf procedures nevertheless it is shown that direct kt fibre estimates are more adequate for computing a generalised version of radial kurtosis maps

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

a certified reduced basis method for the fokker planck equation of dilute polymeric fluids fene dumbbells in extensional flow in this paper we present a reduced basis method for the parametrized fokker planck equation associated with evolution of finitely extensible nonlinear elastic fene dumbbells in a newtonian solvent for a prescribed extensional macroscale flow we apply a proper orthogonal decomposition pod greedy sampling procedure for the stable identification of optimal reduced basis spaces and we develop a rigorous finite time a posteriori bound for the error in the reduced basis prediction of the two outputs of interest the optical anisotropy and the first normal stress difference we present numerical results for stress conformation hysteresis as a function of weissenberg number and final time that demonstrate the rapid convergence of the reduced basis approximation and the effectiveness of the a posteriori error bounds

nonnegative polynomial optimization over unit spheres and convex programming relaxations we consider approximation algorithms for nonnegative polynomial optimization over unit spheres such optimization models have wide applications e g in signal and image processing high order statistics and computer vision since polynomial functions are nonconvex the problems under consideration are all np hard in this paper based on convex polynomial optimization relaxations we propose polynomial time approximation algorithms with new approximation bounds numerical results are reported to show the effectiveness of the proposed approximation algorithms

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

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

decidability and invariant classes for degree structures we present a decision procedure for the v3 theory of o o the turing degrees below o the two main ingredients are a new extension of embeddings result and a strengthening of the initial segments results below o of lell first given any finite subuppersemilattice u of 0 0 l with top element o and an isomorphism type v of a poset extending u consistently with its structure as an usl such that v and u have the same top element and v is an end extension of u 0 we construct an extension of u inside 0 0 isomorphic to v second we obtain an initial segment w of 0 0 which is isomorphic to u o such that w u o is a subusl of 0 the decision procedure follows easily from these results as a corollary to the v3 decision procedure for q we show that no degree a 0 is definable by any 3 formula of degree theory as a start on restricting the formulas which could possibly define the various jump classes we classify the generalized jump classes which are invariant for any v or 3 formula the analysis again uses the decision procedure for the v3 theory of 0 a similar analysis is carried out for the high low hierarchy using the decision procedure for the 3 theory of o o a jump class is a invariant if a a holds for every a in w introduction this paper presents some new results dealing with decidability and definability within th o the elementary theory of the poset of degrees of unsolvability in the area of decidability we show that v3 n th o 0 the y3theory of the degrees below 0 is decidable lachlan la has shown that th o is undecidable and epstein e and lerman lel have obtained a similar result for th o 0 schmerl see lel has pulled both of these undecidability results down to the y3v level in the other direction results of kleene and post kp can be used to show that the 3 theory of each of these structures is decidable and shore shl and lerman lel have shown that the v3 theory of is decidable we present a decision procedure for the v3 theory of 0 0 o we first prove an extension of embeddings theorem which allows us to start with a finite subuppersemilattice u of o 0 with top element o and an isomorphism type v of a poset extending u consistently with its structure as an usl such that v and u have the same top element and v is an end extension of u o and then to received by the editors april 15 1987 and in revised form august 10 1987 1980 mathematics subject classification 1985 revision primary 03d30 secondary 03d55

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

an always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor in this paper we propose an iterative method to calculate the largest eigenvalue of a nonnegative tensor we prove this method converges for any irreducible nonnegative tensor we also apply this method to study the positive definiteness of a multivariate form

uniformly hard languages ladner j assoc comput mach 22 1975 155 showed that there are no minimal recursive sets under polynomial time reductions given any recursive set a ladner constructs a set b such that b strictly reduces to a but b does not lie in p the set b does have very long sequences of input lengths of easily computable instances we examine whether ladner s results hold if we restrict ourselves to uniformly hard languages which have no long sequences of easily computable instances under a hard to disprove assumption we show that there exists a minimal recursive uniformly hard set under honest many one polynomial time reductions

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

an h tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms abstract identifying the positive definiteness of an even order homogeneous multivariate form is an important task due to its wide applications in such as medical imaging and the stability analysis of nonlinear autonomous systems via lyapunov s direct method in automatic control and multivariate network realizability analysis in this paper based on the equivalence of the positive definiteness of the form to that of the underlying tensor and the links between the positive definiteness of a tensor with strong h tensor we propose an h tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms the validity of the iterative scheme is guaranteed theoretically and the given numerical experiments show the efficiency of the scheme

honest polynomial degrees and p np abstract we present a relatively simple proof of a result from homer 1986 showing that if nonrecursive sets cannot be minimal for honest polynomial time turing reducibility h t minimal then p np as a corollary to our proof we strengthen homer s result by showing without assuming that p np that there are h t minimal sets for all tally sets we also consider the converse of homer s result providing some evidence that the nonexistence of h t minimal sets may folow from p np in an interesting way finally we consider structural and or computability properties of sets that cannot be h t minimal