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In this paper, we study a category of restricted modules for the Ovsienko-Roger algebra, which is an extension of the Virasoro algebra of its tensor density module of degree one. We construct and characterize simple modules in this category and give natural free field realizations of certain restricted modules using the Weyl vertex algebra.

In seismology, linear inversion techniques were proposed to determine the moment tensor component in both time and frequency domains (Stump & Johnson 1977) and (Kanamori & Given 1981). Although all components of the moment tensor must be determined, the moment tensor inversion with constraints has been normally applied to obtaining stable solutions in seismology (Dziewonski & Woodhouse 1981). This is partly because a fault motion of an earthquake is primarily associated with shear motion, corresponding to off-diagonal components in the moment tensor. One application of the moment tensor inversion with constraints is found in rock mechanics (Dai, Labuz et al. 2000).

In this paper, we determine the first cohomology groups of centerless N = 1 super Virasoro algebras with coefficients in their tensor density modules. As applications, we classify all central exten...

This paper presents Sparseloop, the first infrastructure that implements an analytical design space exploration methodology for sparse tensor accelerators. Sparseloop comprehends a wide set of architecture specifications including various sparse optimization features such as compressed tensor storage. Using these specifications, Sparseloop can calculate a design's energy efficiency while accounting for both optimization savings and metadata overhead at each storage and compute level of the architecture using stochastic tensor density models. We validate Sparseloop on a well-known accelerator design and achieve ∼99% accuracy in terms of runtime activities (e.g., compressed memory accesses). We also present a case study that highlights the key factors (e.g., uncompressed traffic, data density) that affect sparse optimization features' impact on energy efficiency. Tool available at: https://github.com/NVlabs/timeloop.

In this work, with the help of the quantum hydrodynamic formalism, the gravitational equation associated with the classical Dirac field is derived. The hydrodynamic representation of the Dirac equation described by the evolution of four mass densities, subject to the theory-defined quantum potential, has been generalized to the curved space-time in the covariant form. Thence, the metric of space-time has been defined by imposing the minimum action principle. The derived gravity shows the spontaneous emergence of the &ldquo, cosmological&rdquo, gravity tensor (CGT), a generalization of the classical cosmological constant (CC), as a part of the energy-impulse tensor density (EITD). Even if the classical cosmological constant is set to zero, the CGT is non-zero, allowing a stable quantum vacuum (out of the collapsed branched polymer phase). The theory shows that in the classical macroscopic limit, the general relativity equation is recovered. In the perturbative approach, the CGT leads to a second-order correction to Newtonian gravity that takes contribution from the space where the mass is localized (and the space-time is curvilinear), while it tends to zero as the space-time approaches the flat vacuum, leading, as a means, to an overall cosmological constant that may possibly be compatible with the astronomical observations. The Dirac field gravity shows analogies with the modified Brans&ndash, Dicke gravity, where each spinor term brings an effective gravity constant G divided by its field squared. The work shows that in order to obtain the classical minimum action principle and the general relativity limit of the macroscopic classical scale, quantum decoherence is necessary.

Using the Poisson bracket method, we derive continuum equations for a fluid of deformable particles in two dimensions. Particle shape is quantified in terms of two continuum fields: an anisotropy density field that captures the deformations of individual particles from regular shapes and a shape tensor density field that quantifies both particle elongation and nematic alignment of elongated shapes. We explicitly consider the example of a dense biological tissue as described by the Vertex model energy, where cell shape has been proposed as a structural order parameter for a liquid-solid transition. The hydrodynamic model of biological tissue proposed here captures the coupling of cell shape to flow, and provides a starting point for modeling the rheology of dense tissue., Comment: 8 pages, 1 figure

A decomposition of the magnetotelluric tensor is described in terms of quantities which are invariant to the rotation of observing axes, and which also are distinct measures of the 1D, 2D or 3D characteristics of the tensor and so may be useful in dimensionality analysis. When the in-phase and quadrature parts of the tensor are analysed separately there are two invariants which gauge 1D structure, two invariants which gauge 2D structure, and three invariants which gauge 3D structure. A matrix method similar to singular value decomposition is used to determine many of the invariants, and their display is then possible on Mohr diagrams. A particular set of invariants proposed some seventeen years ago is revised to yield an improved set. Several possibilities for the seventh invariant are canvassed, and illustrated by examples from field data. Low values of Δβ, the invariant now preferred for ‘the 7th’, may indicate a particular simplification of otherwise complicated three-dimensional structure.

Multi-linear functionals or tensors are useful in study and analysis multi-dimensional signal and system. Tensor approximation, which has various applications in signal processing and system theory, can be achieved by generalizing the notion of singular values and singular vectors of matrices to tensor. In this paper, we showed local convergence of a parallelizable numerical method (based on the Jacobi iteration) for obtaining the singular values and singular vectors of a tensor.

We give a sufficient criterion for a lower bound of the cactus rank of a tensor. Then we refine that criterion in order to be able to give an explicit sufficient condition for a non-redundant decomposition of a tensor to be minimal and unique., 14 pages

In the present paper we consider some internal tensor structures. It is proved that if on the space Lm ( V n ) there are tensor fields a j i and a β α defining almost a dual structure, then there exist such lifts of these tensor fields which represent either almost a binary structure, or almost a complex structure on T ( Lm ( V n ) ) . Keywords: Vector fibration, Tangent bundle, Linear connectedness, Lifts, Tensor structures

In this paper, firstly, we introduce the variant tensor splittings, and present some equivalent conditions for a strong M -tensor based on the tensor splitting. Secondly, the existence and uniqueness conditions of the solution for multi-linear systems are given. Thirdly, we propose some tensor splitting algorithms for solving multi-linear systems with coefficient tensor being a strong M -tensor. As an application, a tensor splitting algorithm for solving the multi-linear model of higher order Markov chains is proposed. Numerical examples are given to demonstrate the efficiency of the proposed algorithms.

Transformations of vector or tensor properties from a space-fixed to a molecule-fixed axis system are often required in the study of rotating molecules. Spherical components λμ,ν of a first rank irreducible tensor can be obtained from the direction cosines between the two axis systems, and a second rank tensor with spherical components λ μ , ν ( 2 ) can be built from the direct product λ × λ. It is shown that the treatment of the interaction between molecular rotation and the electric quadrupole of a nucleus is greatly simplified, if the coefficients in the axis-system transformation of the gradient of the electric field of the outer charges at the coupled nucleus are arranged as spherical components λ μ , ν ( 2 ) . Then the reduced matrix elements of the field gradient operators in a symmetric top eigenfunction basis, including their dependence on the molecule-fixed z-angular momentum component k, can be determined from the knowledge of those of λ ( 2 ) . The hyperfine structure Hamiltonian Hq is expressed as the sum of terms characterized each by a value of the molecule-fixed index ν, whose matrix elements obey the rule Δk = ν. Some of these terms may vanish because of molecular symmetry, and the specific cases of linear and symmetric top molecules, orthorhombic molecules, and molecules with symmetry lower than orthorhombic are considered. Each ν-term consists of a contraction of the rotational tensor λ ( 2 ) and the nuclear quadrupole tensor in the space-fixed frame, and its matrix elements in the rotation-nuclear spin coupled representation can be determined by the standard spherical tensor methods.

The teleparallel gravitational energy–momentum tensor density of the Friedmann–Lemaître–Robertson–Walker spacetime is calculated and analyzed: it is decomposed into density, isotropic pressure, non-isotropic pressures, and the heat-flux 4-vector; the antisymmetric part is decomposed into “electric” and “magnetic” components. It is found that the gravitational field obeys a radiation-like equation of state, the antisymmetric part does not contribute to the gravitational energy–momentum; and the total energy density, the non-isotropic pressures and the heat-flux 4-vector vanish for spatially flat universes. Finally, it is verified that the field equations have a well-defined vacuum.

In this paper we prove that the Fremlin tensor product of two f-algebras can be endowed with an f-algebra structure and satisfies an appropriate universal property. In particular, the Riesz...

This paper is devoted to the extension of the ϵ-pseudo-spectra theory from matrices to tensors. Based on the definition of an eigenpair of real symmetric tensors and results on the ϵ-pseudo-spectrum of square matrices, we first introduce the ϵ-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 ϵ-pseudo-spectrum to the setting of tensor polynomial eigenvalue problems. We further derive basic structure of the ϵ-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 ϵ-pseudo-spectrum for computing the backward errors and the distance from a regular tensor polynomial to the nearest irregular tensor polynomial.

The authors are the first to present a closed procedure for numerical solution of inverse coefficient problems of heat conduction in anisotropic materials used as heat-shielding ones in rocket and space equipment. The reconstructed components of the thermal-conductivity tensor depend on temperature (are nonlinear). The procedure includes the formation of experimental data, the implicit gradient-descent method, the economical absolutely stable method of numerical solution of parabolic problems containing mixed derivatives, the parametric identification, construction, and numerical solution of the problem for elements of sensitivity matrices, the development of a quadratic residual functional and regularizing functionals, and also the development of algorithms and software systems. The implicit gradient-descent method permits expanding the quadratic functional in a Taylor series with retention of the linear terms for the increments of the sought functions. This substantially improves the exactness and stability of solution of the inverse problems. Software systems are developed with account taken of the errors in experimental data and disregarding them. On the basis of a priori assumptions of the qualitative behavior of the functional dependences of the components of the thermal-conductivity tensor on temperature, regularizing functionals are constructed by means of which one can reconstruct the components of the thermal-conductivity tensor with an error no higher than the error of the experimental data. Results of the numerical solution of the inverse coefficient problems on reconstruction of nonlinear components of the thermal-conductivity tensor have been obtained and are discussed.

The visualization of symmetric second-order tensor fields in two or three dimensions is still a challenging task, particularly if global structures of the data are desired. One approach is tensor field topology which provides structures characterizing the behavior of the eigenvector fields. Another widely used approach is analyzing tensor fields by means of scalar invariants, i.e., quantities invariant with respect to changes of the coordinate system. In this case, the selection of the relevant invariants might be difficult. Thus, we propose an approach which analyzes the complete invariant part of the tensor. We define extremal points for tensor fields in a mathematically rigorous way, which form curves for two-dimensional and surfaces for three-dimensional tensor fields. We propose a way to compute extremal curves or surfaces from a suitable set of two or three invariants, respectively. We also show that commonly used sets of invariants lead to the same extremal points. Consequently, extremal points contain minima and maxima of most invariants used in tensor field analysis and they are linked to the tensor field topology by containing the degenerate points. Moreover, we show that each extremal point is an extremum or a saddle of a certain invariant. The method is demonstrated on synthetic datasets as well as on stress tensor fields from structure simulations.

Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares (SOS) algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors (EEN-tensors), which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a nonnegative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.

In Relativity the sum of 4−vectors in different points does not generally represent a 4−vector. By using this result, it is shown by simple methods that the total energy-momentum of a system of point particles represents a well-defined 4−vector if the particles do not interact. It is proved that this is equivalent to the no-interaction theorem in Classical Physics. This theorem difficulties the study of a system of interacting particles since it is not even possible to define the total energy-momentum nor the reference frame where the system is at rest. This impediment is avoided by adding to the energy-momentum tensor the stress tensor describing the interaction. As an example, this is applied to a system of charged particles. In the process, the equation of motion for a charged particle including the self-force is formally obtained. However, when a thermodynamic system is analyzed from two different reference frames with a relativistic relative velocity, the interaction between the particles and the walls of the volume cannot be described by means of a covariant stress tensor and consequently the proposed technique is not feasible. Despite the above mentioned drawbacks, a covariant theory of the relativistic transformation laws of the thermodynamic quantities is developed.

Low-rankness has been widely exploited for the tensor completion problem. Recent advances have suggested that the tensor nuclear norm often leads to a promising approximation for the tensor rank. It treats the singular values equally to pursue the convexity of the objective function, while the singular values for the practical images have clear physical meanings with different importance and should be treated differently. In this paper, we propose a non-convex logDet function as a smooth approximation for tensor rank instead of the convex tensor nuclear norm and introduce it into the low-rank tensor completion problem. An alternating direction method of multiplier (ADMM)-based method is developed to solve the problem. Experimental results have shown that the proposed method can significantly outperform existing state-of-the-art nuclear norm-based methods for tensor completion.

We investigate how asymptotic properties of real sequences s=(σm) and t=(τn) are passed on to their tensor product s⊗t=(σmτn). The main result concerns sequences of the form xγϱ:=(1+log2l)γlϱwithl=1,2,⋯,ϱ>0,andγ∈R. This problem is closely related to tensor stability of operator ideals on the Hilbert space.

We study the tensor rank of the tensor corresponding to the algebra of n-variate complex polynomials modulo the dth power of each variable. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a counterexample to a conjecture of Rhodes. At the same time we obtain a new lower bound on the tensor rank of tensor powers of the generalised W-state tensor. In addition, we exactly determine the tensor rank of the tensor cube of the three-party W-state tensor, thus answering a question of Chen et al., Comment: To appear in Linear Algebra and its Applications

Certain characterizations of the Friedrichs and the Kre\(\breve{\imath }\)n von-Neumann extensions of the tensor product of two nonnegative linear relations A and B in terms of the Friedrichs and the Kre\(\breve{\imath }\)n-von Neumann extensions of A and B are provided. A characterization of the extremal extensions of the tensor product of A and B is also given.

The object of the present paper is to study the symmetric and skew-symmetric properties of a second order parallel tensor on paracontact metric (k, µ)-spaces and almost β-para-Kenmotsu (k, µ)-spaces. In this paper, we prove that if there exists a second order symmetric parallel tensor on a paracontact metric (k, µ)-space M, then either M is locally isometric to a product of a flat n + 1-dimensional manifold and an n-dimensional manifold of constant sectional curvature −4, or the second order parallel tensor is a constant multiple of the associated metric tensor g of M2n+1. If there is a second order parallel tensor on an almost β-para-Kenmotsu (k, µ)-space with k ≠ 0, then it is a constant multiple of the associated metric tensor g of M2n+1.

In this paper, we present a cubically convergent method for finding the largest eigenvalue of a nonnegative irreducible tensor. A cubically convergent method is used to solve an equivalent system of nonlinear equations which is transformed by the tensor eigenvalue problem. Due to particular structure of tensor, Chebyshev’s direction is added to the method with a few extra computation. Two rules are designed such that the descendant property of the search directions is ensured. The global convergence is proved by using the line search technique. Numerical results indicate that the proposed method is competitive and efficient on some test problems.

In this paper, we mainly focus on new inclusion sets for eigenvalues of a tensor. First, we propose new inclusion sets for eigenvalues of a tensor, which are sharper than some existing inclusion sets, and obtain the law of distribution of the number of eigenvalues for a tensor. Second, two new classes of tensors are introduced. Third, some bounds on the spectral radii for nonnegative tensors are given. Fourth, some checkable sufficient conditions for the positive definiteness (positive semidefiniteness) of some classes of even-order real symmetric tensors are obtained.

The aim of this paper is to show some rigidity results for complete Riemannian manifolds with parallel Cotton tensor. In particular, we prove that any compact manifold of dimension $$n\ge 3$$ with parallel Cotton tensor and positive constant scalar curvature is isometric to a finite quotient of $${\mathbb {S}}^n$$ under a pointwise or integral pinching condition. Moreover, a rigidity theorem for stochastically complete manifolds with parallel Cotton tensor is also given. The proofs rely mainly on curvature elliptic estimates and the weak maximum principle.

Interest in higher-order tensors has recently surged in data-intensive fields, with a wide range of applications including image processing, blind source separation, community detection, and feature extraction. A common paradigm in tensor-related algorithms advocates unfolding (or flattening) the tensor into a matrix and applying classical methods developed for matrices. Despite the popularity of such techniques, how the functional properties of a tensor changes upon unfolding is currently not well understood. In contrast to the body of existing work which has focused almost exclusively on matricizations, we here consider all possible unfoldings of an order-$k$ tensor, which are in one-to-one correspondence with the set of partitions of $\{1,\ldots,k\}$. We derive general inequalities between the $l^p$-norms of arbitrary unfoldings defined on the partition lattice. In particular, we demonstrate how the spectral norm ($p=2$) of a tensor is bounded by that of its unfoldings, and obtain an improved upper bound on the ratio of the Frobenius norm to the spectral norm of an arbitrary tensor. For specially-structured tensors satisfying a generalized definition of orthogonal decomposability, we prove that the spectral norm remains invariant under specific subsets of unfolding operations., Comment: 17 pages, 1 figure

In this paper, we generalize algebraic Lyapunov and Stein stability results to tensors: let A be an real matrix and be a positive definite tensor of order m (even) and dimension n. Then, the tensor equationhas a unique positive definite solution if and only if A is positive stable. Similarly, the tensor equationhas a unique positive definite solution if and only if A is Schur stable (that is, all its eigenvalues lie in the open unit disk).

In this paper, a way of generalizing the tensor renormalization group(TRG) is proposed. Mathematically, the connection between patterns of tensor renormalization group and the concept of truncation sequence in polytope geometry is discovered. A theoretical contraction framework is therefore proposed. Furthermore, the canonical polyadic decomposition is introduced to tensor network theory. A lowest order numerical verification of this method on the 3-D Ising model is carried out., 31 pages, 23 figures