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The reduction of products of irreducible Cartesian tensors is formulated generally by means of 3‐j tensors. These are special cases of the invariant mappings discussed in Part II [J. A. R. Coope and R. F. Snider, J. Math. Phys. 11, 993 (1970)]. The 3‐j formalism is first developed for a general group. Then, the 3‐j tensors and spinors for the rotation group are discussed in detail, general formulas in terms of elementary invariant tensors being given. The 6‐j and higher n‐j symbols coincide with the familiar ones. Interrelations between Cartesian and spherical tensor methods are emphasized throughout.

Some general properties of three‐dimensional Cartesian tensors (of any rank) which are invariant under the point groups C∞, C∞ y, C∞ h, and D∞ h are obtained. A method is described for obtaining the relationships between the nonzero components of these invariant tensors and is illustrated by application to tensors of ranks six and seven. It is shown how the corresponding relationships for tensors of ranks lower than six can be obtained from tables in the literature. The results are applied in a treatment of nonlinear susceptibilities of an axially isotropic system.

An orthonormal set of irreducible fifth-rank tensors having the required permutation symmetry is constructed. Various problems not encountered in the analogous problem for tensors of ranks two, three, four and six are discussed.

The conformal tensors of a Riemannian space Vn are classified and analysed, mainly in terms of the ‘conformal number’ χ which is defined as a certain linear combination of the numbers that characterize the transformation modes of a conformal tensor. In the case of zero rest mass fields (5), χ – 1 is just the spin. Conformal concomitants of the metric tensor and its derivatives up to a finite order m are considered and, in particular, those tensors which are polynomials in the 2nd and higher derivatives of the metric tensor (called conformal P-tensors of order m). The algebraic structure of such a tensor is described by a set of m − 1 integers (S2, …, Sm), referred to as the structure of F and defined in an invariant way in terms of the various polynomial degrees of F and its partial derivatives. The structure of F is related to the m transformation modes of F by the formula: . Thus for P-tensors, χ = 0, 2, 3, … and m ≤ χ. The set of structures (S2, …, Sm) of all conformal P-tensors of order less than or equal to m, forms an Abelian monoid m, with addition defined componentwise. In a general V4, it is found that 4 consists of all triplets (S2, S3, S4) ∈ 3 for which S3 ≤ 4S2. Finally, new conformal P-tensors are constructed so that examples can be given corresponding to every structure in 4. One also obtains, for each value of χ, a general formula which expresses any conformal P-tensor with conformal number χ in terms of a standard sequence of conformal P-tensors.

The statistical tensors introduced by Fano are discussed as an operator form of the density matrix. The conditions for a statistical tensor to represent a pure state are derived; a general representation of a pure state in terms of 2 j Poincare vectors is exhibited, and a physical interpretation given. The coupling of statistical tensors is discussed briefly.

The vector notation commonly employed in elementary physics cannot be applied in its usual form to spaces of other than three dimensions. In plane dynamics, for instance, it cannot be used to represent the velocity (– ωx2, ωx1) at the point (x1, x2) due to a rotation ω about the origin, or the (scalar) moment about the origin of the force (F1, F2) acting at (x1, x2). In relativity physics the symbol ⋅ is often used to denote the scalar product of two vectors, it is true, and the tensor aαbβ – aβbα is sometimes denoted by a × b, but there exists no body of rules for the manipulation of these symbols that enables one to dispense with the suffix notation as in the case of vectors in three-dimensional space.

The second-order correlation tensors for electromagnetic fields, described relativistically, are introduced. These correlation tensors characterize the correlations which exist between the electromagnetic field vectors at any two world points. With the help of these tensors, generalized energy-momentum tensors are introduced. It is shown that these correlation tensors satisfy certain differential equations and the generalized energy-momentum tensors satisfy certain conservation laws. For stationary and ergodic electromagnetic fields, the results obtained represent the relativistically covariant form of the known correlation theory. Some new invariants for stochastic electromagnetic field are also introduced.

Numerical values of the physical bounds for the components of single and double statistical tensors for spins 2 are calculated. The symmetry properties of these bounds are derived. Correlations between the values of different tensor components are not considered.

It is shown how non-null tensors can be represented by vectors with reference to an orthogonal system of basic tensors called R-tensors. The transformation of vectors is discussed and some new inequalities are deduced from the vector representation.

The construction of independentSU (3) tensors out of octets of fields is considered by investigating numerically invariantSU (3) tensors. A method of obtaining independent sets of these to any rank is discussed and also independent sets are explicitly displayed up to fifth rank. It is shown that this approach allows us to obtain relations among the invariant tensors, and useful new identities involving thed ijk andf ijk tensors are exhibited.

It is known from the theory of group representations that a general orthogonal tensor in three dimensions can be expressed in terms of traceless symmetric tensors and isotropic tensors. This note describes an explicit method of effecting this decomposition for a tensor of arbitrary order. The method employs only elementary algebra and makes no use of group theory.

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with equality if and only if A has a zero row or A is diagonal. We are as yet unable to prove this but the subsequent inequality (3) is a step in this direction. The first purpose of this note is to exhibit (1) as a case of the Pythagorean Theorem in a suitable symmetry class of tensors. Of course, many proofs of (1) are extant and our purpose in reproving it here is to exhibit a technique that is proving itself useful for examining a wide variety of matrix functions. We then show by a similar approach that

This paper may be regarded as a sequel to (1), where the initial value or Cauchy problem for harmonic tensors on a normal hyperbolic Riemann space was treated. The mixed problems to be studied here involve boundary conditions on a timelike boundary surface in addition to the Cauchy data on a spacelike initial manifold. The components of a harmonic tensor satisfy a system of wave equations with similar principal part, and we assign two initial conditions and one boundary condition for each component.

A pair of quadratic differential forms in n dimensions determine at least partly at each point of space, a system of n invariants and a set of n orthogonal unit vectors, with a one to one correspondence between the two sets. It is found very convenient to use these invariants and vectors as auxiliary quantities in terms of which tensors associated with the two forms can be expressed. In order that a tensor, which is a rational function 1 of these auxiliary quantities and their higher derivatives, be also a rational function of the components of the original two forms and their higher derivatives, it is necessary and sufficient that this tensor be invariant under a certain group of transformations on the auxiliary quantities. The Galois theory of groups is applied and a generalization to tensors is obtained of the theorem that every rational symmetric function of the roots of an algebraic equation is a rational function of the coefficients of the equation. The tensors that are rational functions of the components of the original two forms are considered in detail. Certain systems of fundamental tensors, symmetric relative to the auxiliary quantities, are introduced, such that every rational tensor function of the auxiliary quantities and symmetric relative to these quantities, can be expressed uniquely in terms of these fundamental tensors. Only tensors of even order are rational functions of the two forms. If one form is invariantively related to the other, the theory here developed becomes a theory for a single form.

The polynomial invariants of a certain classical linear group of order 336 arise naturally in studying error-correcting codes over GF(7). An incomplete description of these invariants was given by Maschke in 1893. With the aid of the Poincaré series for this group, found by Edge in 1947, we complete Maschke's work by giving a unique representation for the invariants in terms of 12 basic invariants. A conjecture is made concerning the relationship between the Poincaré series and the degrees of the basic invariants for any linear group. A partial answer to this conjecture, due to E. C. Dade, is given.

Following the method of Bhagavantam and Pantulu8 the number of independent components of tensors up to 4th rank is derived for the magnetic classes. This method is extended by Lyubarskii1 to cases of tensors having intrinsic symmetry. The results cover all cases of tensors lacking intrinsic symmetry as well as of tensors having the symmetries of Table II, and are in agreement with previous results.5, 8

Operations with tensors, or multiway arrays, have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-1 outer products using either the CANDECOMP/PARAFAC (CP) or the Tucker models, or some variation thereof. Such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array. The specifics of the approximate representation (such as how many terms to use in the sum, orthogonality constraints, etc.) depend on the application. In this paper, we explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem. Reminiscent of matrix factorizations, we present a new factorization of a tensor as a product of tensors. To derive the new factorization, we define a closed multiplication operation between tensors. A major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications. Specifically, this new multiplication allows us to introduce concepts such as tensor transpose, inverse, and identity, which lead to the notion of an orthogonal tensor. The multiplication also gives rise to a linear operator, and the null space of the resulting operator is identified. We extend the concept of outer products of vectors to outer products of matrices. All derivations are presented for third-order tensors. However, they can be easily extended to the order-p ( p > 3 ) case. We conclude with an application in image deblurring.

It is shown that the full symmetry class of tensors associated with the irreducible character [2,1 n −2 ] of S n does not have an orthogonal basis consisting of decomposable symmetrized tensors.

We prove new versions of A r λ (Ω) -weighted imbedding inequalities for A -harmonic tensors locally and globally. The results are used to estimate the integrals of a homotopy operator T from the Banach space L s (Ω,∧ l ) to the Sobolev space W 1,s (Ω,∧ l−1 ) , l=1,2,…,n , and Sobolev–Poincare imbedding inequality with A r λ (Ω) weight.

An extension of Polya's Theorem inventories the sequence set corresponding to an induced basis in a higher degree symmetry class of tensors.

A recently proposed tensor-tensor multiplication (M.E. Kilmer, C.D. Martin, L. Perrone, A Third-Order Generalization of the Matrix SVD as a Product of Third-Order Tensors , Tech. Rep. TR-2008-4, Tufts University, October 2008) opens up new avenues to understanding the action of n × n × n tensors on a space of n × n matrices. In particular it emphasizes the need to understand the space of objects upon which tensors act. This paper defines a free module and shows that every linear transformation on that module can be represented by tensor multiplication. In addition, it presents a generalization of ideas of eigenvalue and eigenvector to the space of n × n × n tensors.

Here we use invariant theory to describe the Brauer–Severi scheme of the fibers of trace rings of generic matrices with an algebraically closed base field of characteristic zero when the trace ring is viewed as a sheaf of algebras over the variety of matrix invariants. Using this approach, we first prove that the Brauer–Severi scheme of a trace ring is isomorphic to Proj Q , for a graded ring Q whose generators we describe in the first section. This description also has a relevant interpretation over base fields of arbitrary characteristic. In the second section of this paper we show that the Brauer–Severi scheme of the fiber of a trace ring over a point that is not too degenerate will have smooth irreducible components meeting transversally and describe these irreducible components as Brauer–Severi schemes of certain algebras.

The trace algebra C n d is generated by all traces of products of d generic n × n matrices. Minimal generating sets of C n d and their defining relations are known for n 3 and n = 3 , d = 2 . This paper states a minimal generating set and their defining relations for n = d = 3 . Furthermore the computations yield a description of C 33 as a free module over the ring generated by a homogeneous system of parameters.

We suggest a different point of view on some aspects of classical invariant theory. A tensor is regarded as a multiply subscripted array of numbers which represents a multilinear form in a natural way. Two tensors are equivalent if they represent the same symmetric multilinear form. Canonical forms for symmetric 2×2×2 tensors are derived from this point of view and can be regarded as a generalization of Sylvester's law of inertia. In addition, some problems concerning the classification of cubic forms and cubic functionals under the action of various groups are suggested.

In this paper, a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors. What is more, we prove this method has an explicit linear convergence rate for indirectly positive tensors. Interestingly, the algorithm is exactly the NQZ method (proposed by Ng, Qi and Zhou in Finding the largest eigenvalue of a non-negative tensor SIAM J Matrix Anal Appl 31:1090–1099, 2009) by taking a specific parameter. Furthermore, we give a modified NQZ method, which has an explicit linear convergence rate for nonnegative tensors and has an error bound for nonnegative tensors with a positive Perron vector. Besides, we promote an inexact power-type algorithm. Finally, some numerical results are reported.

A necessary and sufficient condition for the existence of the orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with dicyclic group and dihedral group were studied by M. R. Darafsheh and M. R. Pournaki (in press, Linear and Multilinear Algebra) and R. R. Holmes and T. Y. Tam (1992, Linear and Multilinear Algebra32, 21–31). These authors used a certain permutation structure of these groups to prove the necessary condition. In this article we show that the necessary condition found in these previous works is independent of the permutation structures of these groups.

In solid mechanics of isotropic and anisotropic materials representing scalar-valued tensor functions or symmetric second-order tensor-valued tensor functions is of major concern, For instance, the plastic potential is scalar-valued, whereas constitutive equations are tensor-valued. In this paper scalar-valued functions have been represented. Anisotropic effects have been characterized by material tensors of rank two and four. For instance, the effect of damage or the behaviour of oriented solids have been characterized by second-order tensors, and with respect to more general cases inborn anisotropy has been described by using a rank four. In representing scalar-valued tensor functions, a set of irreducible invariant involving the above mentioned tensor variables has been constructed. The central problem is: to find an integrity basis for the argument tensors. Together with the invariants of the single argument tensors the system of simultaneous or joint invariants is considered. Such invariants are given by traces of outer products formed from different argument tensors considered. In finding irreducible invariants of a fourth-order tensor the characteristic polynomial for a fourth-order tensor is derived by the definition of the eigenvektor . Furthermore, Hamilton-Cayley's theorem is applied to a fourth-order tensor. Thus irreducible invariants of a fourth-order tensor can be expressed through sums of principal minors.

We prove that the rings of invariants of 2×2 matrices over an infinite field are Cohen–Macaulay. This result generalizes the similar theorem of Mehta and Ramadas in odd characteristics. Our approach is more elementary and it uses only some standard facts from the theory of modules with good filtrations and the theory of determinantal rings.

LL*L = L. (2) If A = A* is a Hermitian matrix, then, setting C = B* and selecting a Hermitian solution L +, we determine also the relationship among the signatures of these matrices. We recall that the rank and the signature (sgn) of a Hermitian n x n matrix A are invariants of the corresponding Hermitian form in the Sylvester inertia law and are defined by rankA = n - ~(A) = p(A) + q(A), sgnA = p(A) - q(A), where p(A), q(A), and ~(A) are, respectively, the number of positive, negative, and zero eigenvalues of A, taking into account their multiplicities [3]. THEOREM. If the dimensions of the matrices A, B, and C admit the product CAB, then for any semiinverse matrix (CAB) + one has the equality

Other investigators have extended the complementary energy theorem (Castigliano’s theorem) to cover the finite deformation of elastic systems with a finite number of degrees of freedom (structures) and they then have indicated that the extension of the theorem to cover the finite deformation of an elastic continuum involved certain unstated difficulties. The present paper shows that when the strain tensor and Trefftz stress tensor, the usual choice of conjugate deformation and stress tensors, are chosen to characterize the finite deformation of an elastic continuum, one cannot establish a strict complementary energy theorem. It is then shown that a strict complementary energy theorem for the finite deformation of an elastic continuum can be established if what Fritz John calls the Lagrange strain and Lagrange stress tensors are used as the conjugate deformation and stress tensors characterizing the deformation.

The problem of generating a complete set of linearly independent nth-order tensors which are invariant under a crystallographic group is considered. A number of methods for the solution of this problem such as the use of tensor bases, the addition of tensors of lower symmetry, and the method of polynomial invariants are discussed. The limitations of these methods are outlined.

Lagrangian field theories of geometric objects are the most natural framework for investigating the notion of general covariance. We discuss here geometric theories of interacting fields, depending on Lagrangians of arbitrary order, and we give general definitions of energy flow, partial energy flows, energy‐momentum tensors, and stress tensors. We also investigate the role which energy‐momentum tensors and stress tensors play in formulating the natural conservation laws associated with the second theorem of Noether. Examples of application may be found elsewhere.

In this chapter polynomial interpolation is considered and extended to tensor-valued functions of one and two argument tensors. Let xα(α = 1,2,...,n) be distinct points and yα corresponding values. The polynomial of degree n-1 (1) is called “LAGRANGE interpolation formula”, where the polynomials (2) are introduced. It is clear that Lα(xβ) is equal to one for α = β and equal to zero for α ≢ β. The remainder in (1) is given by (3) where min (x, x1,..., xn) < ξ < max (x, xl,..., xn).

This chapter presents the calculation of the fourth order moments for the scalar wave operator. The moments are trace-free, symmetric, and conformally invariant tensors with conformal weight -1 that depends polynomially on the metric tensors, the Riemann curvature tensor, and its covariant derivatives. There are only three linearly independent tensors with these properties and it is possible to determine them explicitly. In a four-dimensional Riemannian or pseudo-Riemannian space, the set of real conformally invariant polynomial covariant 4-tensors of weight -1, which are symmetric and trace-free, is given by the set of linear combinations. It is necessary to determine the coefficients α (υ). The moments and the Wunsch tensors are computed for suitable chosen metrics. The coefficients α(υ) can be determined by a simple comparison. In general relativity, the manifolds with vanishing Ricci tensor describe empty space-times and Maxwell's equations rule the propagation of electro-magnetic waves.

Some of the circumstances in which invariance problems arise in continuum mechanics are described elsewhere in this book. In Chapters 8 and 9 we consider the purely algebraic problem of determining systems of polynomial invariants and tensor polynomial functions for a given set of vectors and tensors, for some of the transformation groups which are of importance in continuum mechanics.

Explicit Brauer Induction is a canonical form for Brauer’s induction theorem. It is designed for use in the construction of invariants of representations from invariants of one-dimensional characters. This paper gives a number of further applications including some new ‘change of field’ maps between representation rings, the behaviour of the canonical form wwith respect to Adams operations and a description of a refinement of Explicit Brauer Induction to produce canonical ‘monomial resolutions’ of representations of finite groups.