Related papers: Generic separating sets for 3D elasticity tensors
Consider the representations of an algebraic group G. In general, polynomial invariant functions may fail to separate orbits. The invariant subring may not be finitely generated, or the number and complexity of the generators may grow…
Invariant tensors play an important role in gauge theories, for example, in dualities of N=1 gauge theories. However, for theories with fields in representations larger than the fundamental, the full set of invariant tensors is often…
We study the properties of linear and non-linear determining functionals for dissipative dynamical systems generated by PDEs. The main attention is payed to the lower bounds for the number of such functionals. In contradiction to the common…
This paper introduces the concept of a generating set for stochastic matrices -- a subset of matrices whose repeated composition generates the entire set. Understanding such generating sets requires specifying the "indivisible elements" and…
We present an efficient method for finding the independent invariant tensors of a gauge theory. Our method uses a theorem relating invariant tensors and D-flat directions in field space. We apply our method to several examples-- SO(3) with…
The analysis and visualization of tensor fields is a very challenging task. Besides the cases of zeroth- and first-order tensors, most techniques focus on symmetric second-order tensors. Only a few works concern totally symmetric tensors of…
Motivated by the problems raised by B\"{u}rgisser and Ikenmeyer, we discuss two classes of minimal generic fundamental invariants for tensors of order 3. The first one is defined on $\otimes^3 \mathbb{C}^m$, where $m=n^2-1$. We study its…
We present stable mixed finite elements for planar linear elasticity on general quadrilateral meshes. The symmetry of the stress tensor is imposed weakly and so there are three primary variables, the stress tensor, the displacement vector…
We generalize the concept of the symmetric hyperdeterminants for symmetric tensors to the E-determinants for general tensors. We show that the E-determinant inherits many properties of the determinant of a matrix. These properties include:…
The identifiability of parameters in a probabilistic model is a crucial notion in statistical inference. We prove that a general tensor of rank 8 in C^3\otimes C^6\otimes C^6 has at least 6 decompositions as sum of simple tensors, so it is…
The lowest order nonconforming rectangular element in three dimensions involves 54 degrees of freedom for the stress and 12 degrees of freedom for the displacement. With a modest increase in the number of degrees of freedom (24 for the…
We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined be equations of degree 2. This gives a new…
We outline the construction of differential invariants for higher--rank tensors.
We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger--Reissner mixed…
We study the quadratic invariants of the elasticity tensor in the framework of its unique irreducible decomposition. The key point is that this decomposition generates the direct sum reduction of the elasticity tensor space. The…
Recently, many structured tensors are defined and their properties are discussed in the literature. In this paper, we introduce a new class of structured tensors, called exceptionally regular tensor, which is relevant to the tensor…
We study the Poincar\'e series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of $2\times2$ matrices; a denominator, which we…
Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$…
The integrability has been playing an essential role in the field of differential equations. This property may better help us obtain the topological structure and even the global dynamics for the considered system. A system is called…
A method for the homogenization of 3-D periodic elastic composites is presented. It allows for the evaluation of the averaged overall frequency dependent dynamic material constitutive tensors relating the averaged dynamic field variable…