Related papers: A Formalization of Elementary Linear Algebra: Part…
We formalize some basic properties of Fourier series in the logic of ACL2(r), which is a variant of ACL2 that supports reasoning about the real and complex numbers by way of non-standard analysis. More specifically, we extend a framework…
We describe the role of algebraic extensions in the theory of commutative, unital normed algebras, with special attention to uniform algebras. We shall also compare these constructions and show how they are related to each other.
We introduce a new framework called linear algebraic number theory (LANT) that reformulates the number-theoretic problem as a regression model and solves it using matrix algebra. This framework restricts all computations to log space,…
Interested in formalizing the generation of fast running code for linear algebra applications, the authors show how an index-free, calculational approach to matrix algebra can be developed by regarding matrices as morphisms of a category…
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…
Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodromy, the homological finiteness properties of the fiber, and the formality properties of the total space. In the process, we prove a more…
In this text, we are concerned with ring epimorphisms, and more specifically universal localisations, from path algebras to matrix algebras. We are mainly focused on constructing ring epimorphisms and universal localisations by extending…
In the paper, we consider the class of so-called endo-commutative algebras. From the identity imposed to specify this class, one can easily see that the product in this class preserves the square of elements. We give a complete…
In this paper we give a small review of some recent results of elementary equivalence of linear and algebraic groups and our last new results of elementary equivalence of categories of modules, endomorphism rings of modules, lattices of…
Elementary Algebraic Geometry can be described as study of zeros of polynomials with integer degrees, this idea can be naturally carried over to `polynomials' with rational degree. This paper explores affine varieties, tangent space and…
We study a general class of recurrence relations that appear in the application of a matrix diagonalization procedure. We find general closed formula and determine analytical properties of the solutions. We finally apply these findings in…
For any finite dimensional basic associative algebra, we study the presentation spaces and their relation with the representation spaces. We prove two propositions about a general presentation, one on its subrepresentations and the other on…
This document presents the solutions to the exercises in the book "Albert algebras over commutative rings" published by Cambridge University Press, 2024, as well as errata and addenda. The addenda include proofs, in the style of the book,…
Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras $A_2$, $B_2$, $C_2$, $G_2$ the complete sets…
An algebraic investigation on bicomplex numbers is carried out here. Particularly matrices and linear maps defined on them are discussed. A new kind of cartesian product, referred to as an idempotent product, is introduced and studied. The…
A double algebra is a linear space $V$ equipped with linear map $V\otimes V\to V\otimes V$. Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
We take matrix decompositions that are usually applied to matrices over the real numbers or complex numbers, and extend them to matrices over an algebra called the double numbers. In doing so, we unify some matrix decompositions: For…
We generalize the Umbral Calculus of G-C. Rota by studying not only sequences of polynomials and inverse power series, or even the logarithms studied in, but instead we study sequences of formal expressions involving the iterated logarithms…
In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame…