Related papers: Proofs of Vector Identities Using Tensors
The main aim of the present paper is to represent an exact and simple proof for FLT by using properties of the algebra identities and linear algebra.
In this work we will summarise the recent progress made in constructing consistent theories for a massive vector field with derivative self-interactions. The construction is such that only the three desired polarisations of the Proca field…
The concept of positively and negatively compatible null vectors arises in the study of Clifford geometric algebras with a Lorentz-Minkowski metric. In previous works, the basic properties of such algebras have been set down in terms of a…
Within the field of numerical multilinear algebra, block tensors are increasingly important. Accordingly, it is appropriate to develop an infrastructure that supports reasoning about block tensor computation. In this paper we establish…
Cycloids are particular Petri nets for modelling processes of actions or events. They belong to the fundaments of Petri's general systems theory and have very different interpretations, ranging from Einstein's relativity theory and…
Two approaches are developed to exploit, for simple complex or compact real Lie algebras g, the information that stems from the characteristic equations of representation matrices and Casimir operators. These approaches are selected so as…
We show that the covariant Raychaudhuri identity describing kinematic characteristics of space-time admits a representation involving a geometrical scalar $\xi$ which, depending on circumstances, might be related to, e.g., relativistic…
We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms…
These are the notes for a course on representations of quivers for second year students in Paderborn in summer 2007. My aim was to provide a basic introduction without using any advanced methods. It turns out that a good knowledge of linear…
We study the exponential Hilbert series (both coarsely- and finely-graded) of the Stanley-Reisner ring of an abstract simplicial complex, $\Delta$, and we introduce the $e$-vector of $\Delta$, which relates to the coefficients of the…
We prove various results in infinite-dimensional differential calculus which relate differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: 1. in the…
In this paper, we describe a class of elements in the ring of $\mathrm{SL}(V)$-invariant polynomial functions on the space of configurations of vectors and linear forms of a 3-dimensional vector space $V.$ These elements are related to one…
A proof is given of the vector identity proposed by Gubarev, Stodolsky and Zakarov that relates the volume integral of the square of a 3-vector field to non-local integrals of the curl and divergence of the field. The identity is applied to…
Vertex algebras in higher dimensions provide an algebraic framework for investigating axiomatic quantum field theory with global conformal invariance. We develop further the theory of such vertex algebras by introducing formal calculus…
We show that the counting of observables and correlators for a 3-index tensor model are organized by the structure of a family of permutation centralizer algebras. These algebras are shown to be semi-simple and their Wedderburn-Artin…
We introduce an algorithm to decide isomorphism between tensors. The algorithm uses the Lie algebra of derivations of a tensor to compress the space in which the search takes place to a so-called densor space. To make the method practicable…
VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to…
In this paper we study the Frobenius characters of the invariant subspaces of the tensor powers of a representation V. The main result is a formula for these characters for a polynomial functor of V involving the characters for V. The main…
We study real triality structures through their intrinsic tensor algebra. Starting from a single triality symbol, we construct the associated Lie algebra of two-triality operators, prove the Jacobi identity, and identify the resulting…
We prove the conjecture that higher Verlinde categories are geometrically reductive. This is one of the two properties required in order for recent results on algebraic geometry in tensor categories to apply to these categories. We also…