Related papers: Near-linear algebra

Starting from a description of various generalized function algebras based on sequence spaces, we develop the general framework for considering linear problems with singular coefficients or non linear problems. Therefore, we prove…

We show that every quasi-compact and quasi-separated algebraic stack can be approximated by a noetherian algebraic stack. We give several applications such as eliminating noetherian hypotheses in the theory of good moduli spaces.

We study some examples when there is actually an equality in the linear algebra bound. When the vectors considered span in fact the entire space. We would like to point out that in some cases this provides some interesting extra information…

Linear systems often involve, as a basic building block, solutions of equations of the form \begin{align*} A_Sx_S&+A_Px_P =0\\ A'_Sx_S & =0, \end{align*} where our primary interest might be in the vector variable $x_P.$ Usually, neither…

Neural network models often face challenges when processing very small or very large numbers due to issues such as overflow, underflow, and unstable output variations. To mitigate these problems, we propose using embedding vectors for…

In this paper, we analyze the definition Andr\'e proposed for near-vector spaces to make it more transparent. We also study the class of near-vector spaces over division rings and give a characterization of regularity that gives a new…

Let ${\mathbb F}$ be an arbitrary field. We consider a commutative, non-associative, $4$-dimensional algebra ${\mathfrak M}$ of the rock, the paper and the scissors with unit over ${\mathbb F}$ and we prove that the image over ${\mathfrak…

The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show…

We introduce a notion of planar algebra, the simplest example of which is a vector space of tensors, closed under planar contractions. A planar algebra with suitable positivity properties produces a finite index subfactor of a II_1 factor,…

The main aim of this paper is to generalize the concept of vector space by the hyperstructure. We generalize some definitions such as hypersubspaces, linear combination, Hamel basis, linearly dependence and linearly independence. A few…

We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values of analytic functions, how to prove that the…

In this paper we have found a necessary and sufficient condition for equivalence of two norms on a linear space using the theory of exponential vector space. Exponential vector space is an ordered algebraic structure which can be considered…

Let $\mathbb V$ be an arbitrary linear space and $f:\mathbb V \times \ldots \times \mathbb V \to \mathbb V$ an $n$-linear map. It is proved that, for each choice of a basis ${\mathcal B}$ of $\mathbb V$, the $n$-linear map $f$ induces a…

In this paper we derive some basic results of circuit theory using `Implicit Linear Algebra' (ILA). This approach has the advantage of simplicity and generality. Implicit linear algebra is outlined in [1]. We denote the space of all vectors…

A central problem of linear algebra is solving linear systems. Regarding linear systems as equations over general semirings (V,otimes,oplus,0,1) instead of rings or fields makes traditional approaches impossible. Earlier work shows that the…

We present a new look at description of real finite-dimensional Lie algebras. The basic element turns out to be a pair $(F,v)$ consisting of a linear mapping $F\in End(V)$ and its eigenvector $v$. This pair allows to build a Lie bracket on…

Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent…

We study the subspace of the exterior algebra of a simple complex Lie algebra linearly spanned by the copies of the little adjoint representation or, in the case of the Lie algebra of traceless matrices, by the copies of the n-th symmetric…

Motivated by some recent developments in abstract theories of quadratic forms, we start to develop in this work an expansion of Linear Algebra to multivalued structures (a multialgebraic structure is essentially an algebraic structure but…

This paper is aimed at introducing an algebraic model for physical scales and units of measurement. This goal is achieved by means of the concept of ``positive space'' and its rational powers. Positive spaces are 1-dimensional ``semi-vector…