Search

We consider the algebra of invariants of $d$-tuples of $n\times n$ matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic $p$ different from two. It is well-known that this algebra is generated by the coefficients of the characteristic polynomial of all products of generic and transpose generic $n\times n$ matrices. We establish that in case $0, Comment: Version 2: some proofs are more detailed; to appear in International Journal of Algebra and Computation

A new lower bound and a new upper bound for the minimum eigenvalue of an 𝓜-tensor are obtained. It is proved that the new lower and upper bounds improve the corresponding bounds provided by He and Huang (J. Inequal. Appl., 2014, 2014, 114) and Zhao and Sang (J. Inequal. Appl., 2016, 2016, 268). Finally, two numerical examples are given to verify the theoretical results.

In this paper, we consider the eigenvalue problem for a tensor of arbitrary even rank. In this connection, we state definitions and theorems related to the tensors of moduli ℂ2p(Ω) and ℝ2p(Ω), where p is an arbitrary natural number and Ω is a domain of the n-dimensional Riemannian space ℝn. We introduce the notions of minor tensors and extended minor tensors of rank (2ps) and order s, the corresponding notions of cofactor tensors and extended cofactor tensors of rank (2ps) and order (N−s), and also the cofactor tensors and extended cofactor tensors of rank 2p(N−s) and order s for rank-(2p) tensor. We present formulas for calculation of these tensors through their components and prove the Laplace theorem on the expansion of the determinant of a rank-(2p) tensor by using the minor and cofactor tensors. We also obtain formulas for the classical invariants of a rank-(2p) tensor through minor and cofactor tensors and through first invariants of degrees of a rank-(2p) tensor and the inverse formulas. A complete orthonormal system of eigentensors for a rank-(2p) tensor is constructed. Canonical representations for the specific strain energy and determining relations are obtained. A classification of anisotropic linear micropolar media with a symmetry center is proposed. Eigenvalues and eigentensors for tensors of elastic moduli for micropolar isotropic and orthotropic materials are calculated.

To investigate complex physical phenomena, bi-dimensional models are often an interesting option. It allows spatial couplings to be produced while keeping them as simple as possible. For linear physical laws, constitutive equations involve the use of tensor spaces. As a consequence the different types of anisotropy that can be described are encoded in tensor spaces involved in the model. In the present paper, we solve the general problem of computing symmetry classes of constitutive tensors in [Formula: see text] using mathematical tools coming from representation theory. The power of this method is illustrated through the tensor spaces of Mindlin strain-gradient elasticity.

In this work we derive important properties regarding matrix invariants which occur in the theory of differential equations with reflection., Preprint

New iterative codes for identifying 𝓗 -tensor are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor, i.e., an even-degree homogeneous polynomial form are given. Advantages of results obtained are illustrated by numerical examples.

We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie’s finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential invariants, and we expose a general method that allows to compute differential invariants from joint invariants.

A tensor is a multidimensional array. First-order tensors and second-order tensors can be viewed as vectors and matrices, respectively. Tensors of higher order, with the ability to include more information, appear more frequently nowadays in image and signal processing, data mining, biomedical engineering, and so on. With the recent work of Kilmer and Martin, familiar matrix-based factorizations in linear algebra can be extended in a straightforward way to third-order tensors based on their new tensor multiplication and concepts. Our method has an advantage over a popular tensor-based face recognition algorithm called TensorFaces, which is based on the higher-order SVD of an arrangement of the images in a database, in that it does not require a least squares solve for the coefficients. In this paper, we give a brief introduction to the new tensor framework and apply the induced tensor decompositions to the application of facial recognition. In the numerical results, we compare our new approaches with the ...

The number of linearly independent homogeneous invariants of degree k for the n-ary form of order d is calculated. Also, a formula for the Poincare series for the algebra of invariants of the n-ary form is found.

This paper gives a concise outline for an extended, multi-dimensional number system from a component perspective. We briefly motivate such a system, sketch the associated algebra and geometry, and conclude with a connection to tensor products that shows it is reasonable, if not desirable, to add tensors of different orders.

A general theory for finding the invariants of an arbitrary tensor is described. Group theoretic methods produce formulas which precisely determine the number of invariants for a Cartesian tensor of arbitrary rank defined over a space of dimension 2,3 or 4. Explicit expressions are obtained for simple cases.

The trace algebra Cnd over a field of characteristic 0 is generated by all traces of products of d generic n × n matrices, n, d ≥ 2. Minimal sets of generators of Cnd are known for n = 2 and 3 for any d and for n = 4 and 5 and d = 2. The explicit defining relations between the generators are found for n = 2 and any d and for n = 3, d = 2 only. Defining relations of minimal degree for n = 3 and any d are also known. The minimal degree of the defining relations of any homogeneous minimal generating set of C42 is equal to 12. Starting with the generating set given recently by Drensky and Sadikova, we have determined all relations of degree ≤ 14. For this purpose we have developed further algorithms based on representation theory of the general linear group and easy computer calculations with standard functions of Maple.

In the some previous works we have obtained several curvature tensors in the generalized Finsler space GFN (the space with non-symmetric basic tensor and non-symmetric connection in Rund's sence). In this work we study identities for the mentioned tensors (the antisymmetriy with respect of two indices, the cyclic symmetry, the symmetry with respect of pairs of indices).

In the era of big data, the high-order high-dimensional tensor structure has attracted more and more attention, which further leads to the extensive study on tensor theory, computation and applications. Analogous to the matrix case, the positive semidefinite tensors, which form an important class of tensors, have played an essential role in tensor applications. This paper aims to provide a simple survey of the positive semidefiniteness of tensors and some potential research directions on tensor theory.

We generalize the classical isomorphism between symmetric functions and invariants of a matrix. In particular, we show that the invariants over several matrices are given by the abelianization of the symmetric tensors over the free associative algebra. The main result is proved by finding a characteristic free presentation of the algebra of symmetric tensors over a free algebra.

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.

The interrelations between the theory of linear transformations preserving matrix invariants and different branches of mathematics are surveyed here. The preferences are given for those methods and motivations to study these transformations that arise from general algebra.

We find a minimal system of generators and a homogeneous system of parameters for the algebra of invariants of three matrices of third order over a field of an arbitrary characteristic.

We obtain defining relations of the algebra of invariants of the classical subgroups of GL2(C) acting by simultaneous conjugation on m-tuples of 2×2 complex matrices. The sets of defining relations look uniformly for all m≥2 and are derived by translation of classical results on invariant theory of orthogonal groups in the language of 2×2 matrix invariants, combined with arguments of representation theory of the general linear group GL m (C) and ideas coming from the theory of algebras with polynomial identities.

We study the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$ on $m$-tuples of $n \times n$ matrices with entries in an infinite field $K$. We show that invariants of degree $n^2- n$ define the null cone. Consequently, invariants of degree $\leq n^6$ generate the ring of invariants if $\operatorname{char}(K)=0$. We also prove that for $m \gg 0$, invariants of degree at least $n\lfloor \sqrt{n+1}\rfloor$ are required to define the null cone. We generalize our results to matrix invariants of $m$-tuples of $p\times q$ matrices, and to rings of semi-invariants for quivers. For the proofs, we use new techniques such as the regularity lemma by Ivanyos, Qiao and Subrahmanyam, and the concavity property of the tensor blow-ups of matrix spaces. We will discuss several applications to algebraic complexity theory, such as a deterministic polynomial time algorithm for non-commutative rational identity testing, and the existence of small division-free formulas for non-commutative polynomials., Comment: 16 pages

The principal invariants of the stretch tensors in the polar decomposition of the deformation gradient stand in one-to-one relation to the principal invariants of their squares, the Cauchy—Green deformation tensors. These relations are used to obtain the derivatives of the first set of invariants with respect to the deformation gradient. The latter generate expressions for the stretch and rotation factors in the polar decomposition as explicit functions of the deformation gradient, and lead to new formulae for the stress and elastic moduli for isotropic elastic materials.

It is known from the theory of group representations that, in principle, a tensor of any finite order can be decomposed into a sum of irreducible tensors. This paper develops a simple and effective recursive method to realize such decompositions in both two-and three-dimensional spaces. Particularly, such derived decompositions have mutually orthogonal base elements. Quite a few application examples are given for generic and various physical tensors of orders up to six.

First, by using the formulae of Krupka, the trace decomposition for some particular classes of tensors of types (1, 2) and (1, 3) is obtained. Second, it is proved that the traceless part of a tensor is an almost projective invariant of weight 1. We apply this result to Weyl curvature tensors.

We define and derive some properties of the different multiple view tensors. The multiple view geometry is described using a four-dimensional linear manifold in R 3m , where m denotes the number of images. The Grassman co-ordinates of this manifold build up the components of the different multiple view tensors. All relations between these Grassman co-ordinates can be expressed using the quadratic p-relations. From this formalism it is evident that the multiple view geometry is described by four different kinds of projective invariants; the epipoles, the fundamental matrices, the trifocal tensors and the quadrifocal tensors. We derive all constraint equations on these tensors that can be used to estimate them from corresponding points and/or lines in the images as well as all transfer equations that can be used to transfer features seen in some images to another image. As an application of this formalism we show how a representation of the multiple view geometry can be calculated from different combinations of multiple view tensors and how some tensors can be extracted from others. We also give necessary and sufficient conditions for the tensor components, i.e. the constraints they have to obey in order to build up a correct tensor, as well as for arbitrary combinations of tensors. Finally, the tensorial rank of the different multiple view tensors are considered and calculated.

In this paper we use the term 'physical tensor' to stand for a tensor that belongs to a tensor subspace. Based on the relationship among the characters of rotation representation, some techniques are developed in order to give the numbers of independent deviatoric tensors contained in the irreducible decompositions of physical tensors, even prior to the constructions of the irreducible decompositions. A number of examples are shown and many of them are new results.

For G a classical group of type Bn, Cn or Dn defined over an algebraically closed field (of characteristic other than two if G is a special orthogonal group), we show that the centralizer algebra of G on the space of endomorphisms of the module of traceless tensors Wr is naturally isomorphic to the group algebra of the symmetric group Sr if r[les ]n (for G of type Bn or Cn) or if r

In this paper we give purely geometric derivations of the constraints between point and line correspondences in multiple views of a static field. The analysis is carried out using geometric algebra , a system which provides a useful tool in many computer vision applications. It is shown that with a straightforward geometric interpretation it is simple to derive the degrees of freedom of such tensors and to understand their structure. Given such information, minimal parametrizations of the tensors are possible. Such parametrizations may be useful for estimation of the tensors and subsequent matching of points.

The polynomial invariants of finite groups have been studied for more than a century now and continue to find new applications and generate interesting problems. In this article we will survey some of the recent developments coming primarily from algebraic topology and the rediscovery of old open problems.

Let R R be a prime algebra over a commutative ring K K with unity and let f ( X 1 , … , X n ) f(X_{1}, \ldots , X_{n}) be a multilinear polynomial over K K . Suppose that d d is a nonzero derivation on R R such that for all r 1 , … , r n r_{1}, \ldots , r_{n} in some nonzero ideal I I of R R , [ d ( f ( r 1 , … , r n ) ) , f ( r 1 , … , r n ) ] k = 0 \Big [ d\big ( f(r_{1}, \ldots , r_{n})\big ), f(r_{1}, \ldots , r_{n}) \Big ]_{k} = 0 with k k fixed. Then f ( X 1 , … , X n ) f(X_{1}, \ldots , X_{n}) is central–valued on R R except when char R = 2 R=2 and R R satisfies the standard identity s 4 s_{4} in 4 variables.

Let Q m , n {Q_{m,n}} be the space of m m -tuples of n × n n \times n -matrices modulo the simultaneous conjugation action of P G L n PG{L_n} . Let Q m , n ( τ ) {Q_{m,n}}(\tau ) be the set of points of Q m , n {Q_{m,n}} of representation type τ \tau . We show that for m ≥ n + 1 m \geq n + 1 the group Aut ⁡ ( Q m , n ) \operatorname {Aut}({Q_{m,n}}) of representation type preserving algebraic automorphisms of Q m , n {Q_{m,n}} acts transitively on each Q m , n ( τ ) {Q_{m,n}}(\tau ) . Moreover, the action of Aut ⁡ ( Q m , n ) \operatorname {Aut}({Q_{m,n}}) on the Zariski open subset Q m , n ( 1 , n ) {Q_{m,n}}(1,n) of Q m , n {Q_{m,n}} is s s -transitive for every positive integer s s . We also prove slightly weaker analogues of these results for all m ≥ 3 m \geq 3 .

In this paper we pursue the study of spectral categories initiated in [26]. More precisely, we construct the Universal matrix invariant of spectral categories, i.e. a functor U with values in an additive category Add, which inverts the Morita equivalences, satisfies matrix invariance, and is universal with respect to these two properties. For example, the algebraic K-theory and the topological Hochschild and cyclic homologies are matrix invariants, and so they factor uniquely throw U. As an application, we obtain for free non-trivial trace maps from the Grothendieck group to the topological Hochschild homology ones., 24 pages

This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher-order generalisation of a part of Arnold's invariants that are first-order invariants for plane immersed curves. The invariants in this theory are developed using the word theory proposed by Turaev., Comment: 8 pages, The earlier versions contain mistakes at the title and Theorem 5.2 in ver 2 (: Theorem 4.2 in ver 1)

We consider certain invariants of links in 3-manifolds, obtained by a specialization of the Turaev-Viro invariants of 3-manifolds, that we call colored Turaev-Viro invariants. Their construction is based on a presentation of a pair (M,L), where M is a closed oriented 3-manifold and L is an oriented link in M, by a triangulation of M such that each component of L is an edge. We analyze some basic properties of these invariants, including the behavior under connected sums of pairs away and along links. These properties allow us to provide examples of links in the three-sphere having the same HOMFLY polynomial and the same Kauffman polynomial but distinct Turaev-Viro invariants, and similar examples for the Alexander polynomial. We also investigate the relations between the Turaev-Viro invariants of (M,L) and those of the complement of L in M, showing that they are sometimes but not always determined by each other., Comment: 40 pages, 23 figures

Stable rationality of the field of matrix invariants ℂMn×Mn)PGLn is proved forn=5 andn=7. In combination with existing results this shows that ℂ (V)PGLn is stably rational wheneverV is an almost free representation ofPGLn andn divides 420=22·3·5·7.

We generalize the Perron–Frobenius Theorem for nonnegative matrices to the class of nonnegative tensors.

A general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarization-problem simplifies the derivation of the Hashin-Shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method, due to Tartar, Murat, Lurie, and Cherkaev, uses translation operators or, equivalently, quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant non-local projection operator, T1. An important class of bounds, namely trace bounds on the effective tensors of two-component media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the two-components are not well ordered. In particular, they generate the bounds of Walpole on the effective bulk modulus. The variational principles of Gibiansky and Cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by Golden and Papanicolaou. An isomorphism is shown to underlie their variational principles. This isomorphism is used to obtain Dirichlet-type variational principles and bounds for the effective tensors of general non-selfadjoint problems. It is anticipated that these variational principles, which stem from the work of Gibiansky and Cherkaev, will have applications in many fields of science.

Consider the vector space of m-tuples of n by n matrices.The linear group GLn(C) acts on Xm, n by simultaneous conjugation. The corresponding ring of polynomial invariantswill be denoted by C(n, m) and is called the ring of matrix invariants of m-tuples of n by n matrices. C. Procesi has shown in [8] that C(n, m) is generated by traces of products of the corresponding generic matrices and, as such, coincides with the center of the trace ring of m generic n by n matrices R (n, m) which is also the ring of equivariant maps from Xm, n to Mn(ℂ).

In the present survey a method is presented for the construction of invariant tensors (with respect to a given representation) called the “inclusion principle” and the possibility of applying the cited method to various problems of algebra, topology, geometry, and analysis.

A complete decomposition of the space of curvature tensors over a Hermitian vector space into irreducible factors under the action of the unitary group is given. The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for almost Hermitian manifolds are given. Conformal invariants are considered and a general Bochner curvature tensor is introduced and shown to be a conformal invariant. Finally curvature tensors on four-dimensional manifolds are studied in detail.