Proper Orthogonal Group Research Articles

Semiconvexity of invariant functions of rectangular matrices

The paper deals with the semiconvexity properties (i.e., the rank 1 convexity, quasiconvexity, polyconvexity, and convexity) of rotationally invariant functions f of \(m\times n\) matrices. For \(m\neq n\) the invariance with respect to the proper orthogonal group and the invariance with respect to the full orthogonal group coincide. With each invariant f one can associate a fully invariant function \(f_\flat\) of a square matrix of type \(p\times p\) where \(p = \min \{m,n\}.\) It is shown that f has the semi convexity of a given type if and only if \(f_\flat\) has the semiconvexity of that type. Consequently the semiconvex hulls of f can be determined by evaluating the corresponding hulls of \(f_\flat\) and then extending them to \(m\times n\) matrices by rotational invariance.

Monotonicity of rotationally invariant convex and rank 1 convex functions

Let f : Mn×n → R ∪ {∞} be a function on the space Mn×n of n by n matrices that is invariant with respect to the left and right multiplication by proper orthogonal tensors. It is shown that f(A) ≤ f(Ā) if f is convex and the partial sums of the singular values of A, Ā ∈ Mn×n satisfy certain ordering inequalities. The same holds if f is rank 1 convex and the partial products of the singular values satisfy analogous inequalities. The proofs emphasize the roles of the ordered-forces inequalities and the Baker-Ericksen inequalities for invariant convex and rank 1 convex functions. As an application, the evaluation of the convex and lamination convex hulls of fully rotationally invariant sets by Dacorogna and Tanteri is simplified and similar results are given for sets invariant only with respect to the proper orthogonal group.

On dynamics of flexible branched shell structures undergoing large overall motion using finite elements

The general, statically and kinematically exact, six-field theory of branched shell structures, extended to nonlinear problems of shell dynamics involving also large overall motion, is discussed. The generalized Newmark algorithm on the proper orthogonal group SO(3) with Newton iterations is proposed. Numerical simulations of the behavior of an elastically twisted T-shaped shell structure in forced and free rotations with large overall motion are presented.

Rotationally Invariant Rank 1 Convex Functions

Let f be a function on the set M n xn of all n by n real matrices. If f is rotationally invariant with respect to the proper orthogonal group, it has a representation \tilde f through the signed singular values of the matrix argument A∈ M^nxn. Necessary and sufficient conditions are given for the rank 1 convexity of f in terms of \tilde f .

Rank 1 convex hulls of isotropic functions in dimension 2 by 2

Let $f$ be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set $\text{M}^{2\times 2}$ of all $2$ by $2$ matrices. Based on conditions for the rank 1 convexity of $f$ in terms of signed invariants of $\mathbb{A}$ (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A special case in which the procedure terminates after the second step is determined and examples of the actual calculations are given.

Isotropic invariants of traceless symmetric tensors of orders three and four

We determine integrity bases for functions of a traceless symmetric third-order tensor A ijk and for functions of a traceless symmetric fourth-order tensor A ijkl which are invariant under the three-dimensional proper orthogonal group R 3. Integrity bases are also given for the full orthogonal group O 3.

On the dual of a result of Miranda and Thompson

For A and B real matrices with prescribed singular values, Miranda and Thompson characterized max tr( AUBV) when U and V range over SO( n), the real proper orthogonal group. Motivated by this result, we investigate the location of det( A + UBV) for A, B, U, V as previously. The corresponding problem for A and B complex matrices and U and V ranging over the unitary group is also discussed. Analogies are drawn between the determinantal problems and their tracial versions.

Products of half-turns

Let V be a vector space of any dimension and O + (V) the proper orthogonal group of V. We show, if π ε O + (V), then π is a product of half-turns, and we determine the minimal number of half-turns which is needed to express π.

The theory of spinors via involutions and its application to the representations of the Lorentz group

A simple theory of the spinor representations of the complex orthogonal group O(d,C) in the d-dimensional Euclidean space V(d) is presented via a basic lemma on involutional transformations and Cartan’s theorem on O(d,C). The arbitrary gauge factors of the representations are reduced to ± signs by introducing appropriate phase conventions. The concept of an axial involution is introduced. The plane rotations in V(d) are introduced and used to construct the representations of the proper orthogonal group O+(d,C). The Lorentz group is treated as a subgroup of O(4,C). The general expression for the basic 2×2 irreducible representations A(L0) of the proper orthochronous Lorentz group G(L0) is obtained by direct reduction of the 4×4 spinor representation S(L0) by means of the basic lemma on the involutional transformations. It is completely parameterized by the angle and the axis of the spacial rotation and by the velocity of the pure Lorentz transformation. The finite dimensional irreducible representations of the Lorentz group G(L) are discussed. The transformations of electro-magnetic field under G(L) are discussed in the most general form.

On generating functions for the number of invariants of orthogonal tensors

Generating functions for the number of linearly independent invariants of a set of tensors under a given group of transformations are given by the theory of group representations. For the full and proper orthogonal groups these generating functions are in the form of definite integrals. The classical theory of algebraic invariants gives generating functions for the number of invariants of tensors under two-dimensional unimodular transformations, these generating functions being algebraic expressions. Because of a correspondence between the two-dimensional unimodular group and the three-dimensional proper orthogonal group, the corresponding generating functions are equivalent. The main result of this paper is an explicit demonstration of this equivalence. In addition, algebraic generating functions for the three-dimensional full orthogonal group are obtained and the use of the algebraic generating functions illustrated by applying them to a third order symmetric tensor.

On rotational invariants of vectors and second-order tensors

It may be shown that an arbitrary function ƒ(v i, a j) of the vectors v 1/3 . v n and the second-order tensors a 1/3 ., a M which is invariant under the three-dimensional proper orthogonal group is not expressible as a single-valued function of the invariants I k (v i, a j). Given by Chakrabarti and Wainwright[1] for most of the cases considered in [1]. Examples are given.

Holonomy Groups Of Hypersurfaces

The restricted homogeneous holonomy group of an n–dimensional Riemannian manifold is a connected closed subgroup of the proper orthogonal group SO(n) [1]. In this note we shall prove that the restricted homogeneous holonomy group of an n-dimensional compact hypersurface in the Euclidean space is actually the proper orthogonal group SO(n) itself. This gives a necessary (of course, not sufficient) condition for the imbedding of an n-dimensional compact Riemannian manifold into the (n +1)–dimensional Euclidean space.

On Riemannian manifolds of four dimensions

Introduction. It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretic reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no analogue for orthogonal groups in n ( > 4 ) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifolds of four dimensions, because their tangent spaces possess a geometry of this kind. I t is the purpose of this note to give a study of a compact orientable Riemannian manifold of four dimensions at each point of which is attached a three-dimensional spherical space. This necessitates a more careful study of spherical geometry than hitherto given in the literature, except, so far as the writer is aware, in a paper by E. Study [2] . 2 0ur main result consists of two formulas, which express two topological invariants of a compact orientable differentiable manifold of four dimensions as integrals over the manifold of differential invariants constructed from a Riemannian metric previously given on the manifold. These two topological invariants have a linear combination which is the Euler-Poincare characteristic.