Manifold of nondegenerate affinor fields

We consider the quotient set of the set of nondegenerate affinor fields with respect to the action of the group of nowhere vanishing functions. This set is endowed with a structure of infinite-dimensional Lie group. On this Lie group, we construct an object of linear connection with respect to which all left-invariant vector fields are covariantly constant (the Cartan connection).     

Three-webs with covariantly constant curvature and torsion tensors

In this paper, we study a special class of multidimensional 3-webs with covariantly constant curvature and torsion tensors. In the first part, we prove that 3-webs of the class belong to G-webs, i.e., there is a subfamily of adapted frames whose components of curvature and torsion tensors are constant. The structure of the homogeneous space G/H carrying the 3-web is described. Structure equations of the G-group are found. In the second part, we find structure equations of the W ^??-web and finite equations of some special web classes.     

On Geodesic Curves on Quotient Manifold of Nondegenerate Affinor Fields

We consider the quotient manifold of the manifold of nondegenerate affinor fields on a compact manifold with respect to the action of the group of nowhere vanishing functions. This manifold is endowed with a structure of infinite-dimensional Lie group. On this Lie group, we construct an object of linear connection with respect to which all left-invariant vector fields are covariantly constant (the Cartan connection). We also find the geodesics of the Cartan connection.     

On one class of three-webs with covariantly constant curvature and torsion tensors

A special class of multidimensional three-webs \( {W^\nabla } \) with covariantly constant curvature and torsion tensors is considered, the curvature tensor having the minimal rank. It is proved that there is a subfamily of adapted frames of the web \( {W^\nabla } \) whose torsion tensor components are constant and whose curvature tensor has a unique nonzero component. The structure equations of the webs of this class are found and some of their properties are described.     

Deformations of open stein manifolds

We consider the deformation of the complex structure on an open Stein manifold. We show that a tame, compactly supported deformation of a Stein manifold is trivial. The remainder of our results are for deformations of the standard complex structure on Cⁿ. A deformation of Cⁿ which tends to a constant deformation faster that r^-3 is trivial. Harmonic deformation tensors (w.r.t to the standard Euclidean metric) which are regular at infinity are constant.     

Maximally movable Riemannian spaces with torsion

We prove that, among all Riemannian spaces of constant curvature, only three-dimensional spaces have torsion which is invariant under the group of motions. The torsion tensor in these spaces is covariantly constant and determines the torsion form. The ratio of the integral of this form over a bounded domain to its volume is a constant determining the torsion of the space. We introduce the notions of volume torsion and scalar torsion.     

Generalized Singular Value Decompositions for Tensors and Their Applications

In this paper, we consider the generalized singular value decompositions for two tensors via the T-product. We investigate and discuss in detail the structures of the quotient singular value decomposition (T-QSVD) and product singular value decomposition (T-PSVD) for two tensors. The algorithms are presented with numerical examples illustrating the results. For applications, we consider color image watermarking processing with T-QSVD and T-PSVD. There are two advantages to T-QSVD and T-PSVD approaches on color watermark processing: two color watermarks can be processed simultaneously and only one key needs to be saved.     

Plane anisotropic rari‐constant materials

We consider here the existence of rari‐constant anisotropic layers and show that actually there are two distinct classes of such materials, mutually exclusive. Also, we show that the correct condition for establishing that a material is of the rari‐constant type is that the number of independent linear tensor invariants of the elastic tensors must reduce to one. We characterize these materials and show that they can be designed by using some basic rules of homogenization. Copyright © 2015 John Wiley & Sons, Ltd.     

Estimations on upper and lower bounds of solutions to a class of tensor complementarity problems

We introduce a class of structured tensors, called generalized row strictly diagonally dominant tensors, and discuss some relationships between it and several classes of structured tensors, including nonnegative tensors, Btensors, and strictly copositive tensors. In particular, we give estimations on upper and lower bounds of solutions to the tensor complementarity problem (TCP) when the involved tensor is a generalized row strictly diagonally dominant tensor with all positive diagonal entries. The main advantage of the results obtained in this paper is that both bounds we obtained depend only on the tensor and constant vector involved in the TCP;and hence, they are very easy to calculate.     

On Holomorphic Hypercomplex Connections

The main purpose of this paper is to study some properties of integrable commutative hypercomplex structures endowed with a holomorphic torsion-free affine connection whose curvature tensor satisfies the purity condition with respect to the covariantly constant structure affinors.     

Deformations of Euclidean supersymmetries

We consider quantum supergroups that arise in nonanticommutative deformations of the N=(1/2, 1/2) and N=(1, 1) four-dimensional Euclidean supersymmetric theories. Twist operators in the corresponding superspaces and deformed superfield algebras contain left spinor generators. We show that nonanticommutative *-products of superfields transform covariantly under the deformed supersymmetries. This covariance guarantees the invariance of the deformed superfield actions of models involving *-products of superfields. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 270–289, May, 2006.     

Invariant tensors and partial differential equations

We consider tensors with coefficients in a commutative differential algebra A. Using the Lie derivative, we introduce the notion of a tensor invariant under a derivation on an ideal of A. Each system of partial differential equations generates an ideal in some differential algebra. This makes it possible to study invariant tensors on such an ideal. As examples we consider the equations of gas dynamics and magnetohydrodynamics.     

A right triangulated version of Gentle-Todorov’s theorem

Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We prove a right triangulated version of Gentle-Todorov’s theorem by introducing the notion of right homotopy cartesian square.     

Two integrable systems with integrals of motion of degree four

We discuss the possibility of using second-order Killing tensors to construct Liouville-integrable Hamiltonian systems that are not Nijenhuis integrable. As an example, we consider two Killing tensors with a nonzero Haantjes torsion that satisfy weaker geometric conditions and also three-dimensional systems corresponding to them that are integrable in Euclidean space and have two quadratic integrals of motion and one fourth-order integral in momenta.     

Bol three-webs with covariantly constant curvature tensor

We find complete systemof tensor relations characterizing the class ofmultidimensional middle Bol three-webs with covariantly constant curvature tensor, and ascertain the algebraic sense of these relations. We prove the existence of such webs and lay the foundation of their classification in terms of torsion tensor rank. We also show that 6-dimensional non-group webs of such type are the known flexible webs E₁ and E₂.     

Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results.     

Covariance in linearized elasticity

In this paper we covariantly obtain all the governing equations of linearized elasticity. Our motivation is to see if one can make a connection between invariance (covariance) properties of the (global) balance of energy in nonlinear elasticity and those of its counterpart in linear elasticity. We start by proving a Green-Naghdi-Rivilin theorem for linearized elasticity. We do this by first linearizing energy balance about a given reference motion and then by postulating its invariance under isometries of the Euclidean ambient space. We also investigate the possibility of covariantly deriving a linearized elasticity theory, without any reference to the local governing equations, e.g. local balance of linear momentum. In particular, we study the consequences of linearizing covariant energy balance and covariance of linearized energy balance. We show that in both cases, covariance gives all the field equations of linearized elasticity.      

Paraholomorphic B-manifold and its properties

This paper is concerned with problem of the geometry of B-manifolds. We give some properties of Riemannian curvature tensors of paraholomorphic B-manifolds. Finally, we consider some examples of paraholomorphic B-manifolds.     

On structure-preserving connections

In this paper we find the formula of connections under which an almost complex structure is covariantly constant. These types of connections on anti-Kähler–Codazzi manifolds are described. Also, twin metric-preserving connections are analyzed for quasi-Kähler manifolds. Finally, anti-Hermitian Chern connections are investigated.