Related papers: Logarithms Over a Real Associative Algebra
Holomorphic functions of exponential type on a complex Lie group $G$ (introduced by Akbarov) form a locally convex algebra, which is denoted by $\cO_{exp}(G)$. Our aim is to describe the structure of $\cO_{exp}(G)$ in the case when $G$ is…
Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none…
We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.
We give a rigorous formulation of the intuitive idea that a differentiable map should be thesame thing as a locally, or infinitesimally, linear map: just as a linear map respects the operations of addition and multiplication by scalars ina…
We give an introduction to logic tailored for algebraists, explaining how proofs in linear logic can be viewed as algorithms for constructing morphisms in symmetric closed monoidal categories with additional structure. This is made explicit…
In the paper I considered algebra of polynomials over associative D-algebra with unit. Using the tensor notation allows to simplify the representation of polynomial. I considered questions related to divisibility of polynomial of any power…
The notion of associativity (which differs from the straightforward generalization of the usual associativity given by the move of parentheses in the relevant expression) for operations of high arity is introduced. It is proved that the…
We interpret augmented racks as a certain kind of multiplicative graphs and show that this point of view is natural for defining rack homology. We also define the analogue of the group algebra for these objects; in particular, we see how…
We prove recursive formulas involving sums of divisors and sums of triangular numbers and give a variety of identities relating arithmetic functions to divisor functions providing inductive identities for such arithmetic functions.
We investigate the representation and complete representation classes for algebras of partial functions with the signature of relative complement and domain restriction. We provide and prove the correctness of a finite equational…
We investigate the expressive power of regular expressions for languages of countable words and establish their expressive equivalence with logical and algebraic characterizations. Our goal is to extend the classical theory of regular…
We prove some interesting multiplicative relations which hold between the coefficients of the logarithmic derivatives obtained in a few simple ways from $\mathbb{F}_q$-linear formal power series. Since the logarithmic derivatives connect…
Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an…
In a recent work Foulis and Pulmannov\' a \cite{Foulis2012} studied the logical connectives in lattice effect algebras. In this paper we extend their study and investigate further the logical calculus for which the lattice effect algebras…
We define several versions of the cohomology ring of an associative algebra. These ring structures unify some well known operations from homological algebra and differential geometry. They have some formal resemblance with the quantum…
A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of…
Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proofs, and topological proofs. Algebraic proofs make use of the…
To any directed graph we associate an algebra with edges of the graph as generators and with relations defined by all pairs of directed paths with the same origin and terminus. Such algebras are related to factorizations of polynomials over…
We show that the iterative logarithm of each non-linear entire function is differentially transcendental over the ring of entire functions, and we give a sufficient criterion for such an iterative logarithm to be differentially…
We introduce and study the notion of a logarithmic vertex algebra, which is a vertex algebra with logarithmic singularities in the operator product expansion of quantum fields; thus providing a rigorous formulation of the algebraic…