Related papers: Linear Algebraic Number Theory, Part I: Foundation…
A new totally algebraic formalism based on general, abstract ladder operators has been proposed. This approach heavily grounds in the superoperator formalism of Primas. However it is necessary to introduce many improvements in his…
In a previous paper, we have given an algebraic model to the set of intervals. Here, we apply this model in a linear frame. We define a notion of diagonalization of square matrices whose coefficients are intervals. But in this case, with…
While there are numerous linear algebra teaching tools, they tend to be focused on the basics, and not handle the more advanced aspects. This project aims to fill that gap, focusing specifically on methods like Strassen's fast matrix…
We introduce Wave Arithmetic, a smooth analytical framework in which natural, integer, and rational numbers are represented not as discrete entities, but as integrals of smooth, compactly supported or periodic kernel functions. In this…
Linear algebra algorithms often require some sort of iteration or recursion as is illustrated by standard algorithms for Gaussian elimination, matrix inversion, and transitive closure. A key characteristic shared by these algorithms is that…
An associative $*$-algebra is introduced (containing a $TTR$-algebra as a subalgebra) that implements the form factor axioms, and hence indirectly the Wightman axioms, in the following sense: Each $T$-invariant linear functional over the…
We propose and investigate a bi-infinite matrix approach to the multiplication and composition of formal Laurent series. We generalize the concept of Riordan matrix to this bi-infinite context, obtaining matrices that are not necessarily…
A fundamental theorem of linear algebra asserts that every basis for the vector space $\mathbb{R}^n$ has $n$ elements. In this expository note we present a theorem of W. G. Leavitt describing one way in which this invariant basis number…
We create classical (non-quantum) dynamic data structures supporting queries for recommender systems and least-squares regression that are comparable to their quantum analogues. De-quantizing such algorithms has received a flurry of…
A new method for analyzing high-dimensional categorical data, Linear Latent Structure (LLS) analysis, is presented. LLS models belong to the family of latent structure models, which are mixture distribution models constrained to satisfy the…
Randomized numerical linear algebra - RandNLA, for short - concerns the use of randomization as a resource to develop improved algorithms for large-scale linear algebra computations. The origins of contemporary RandNLA lay in theoretical…
Linear response theory is a tool with which one can study systems that are driven out of equilibrium by external perturbations. This monograph presents a thoroughly modern framework to make linear response theory rigorous for a wide array…
We study algebraic tangles as fundamental components in knot theory, developing a systematic approach to classify and tabulate prime tangles using a novel canonical representation. The canonical representation enables us to distinguish…
In this article, we will use elementary number theory techniques to investigate a sequence of integers defined by a sifting process called the lucky numbers. Ulam introduced lucky numbers as a sieve-based analogue of prime numbers. We…
In this note, we propose a simple-looking but broad conjecture about star-algebras over the field of real numbers. The conjecture enables many matrix decompositions to be represented by star-algebras and star-ideals. This paper is written…
This is the second installment of an exposition of an ACL2 formalization of elementary linear algebra. It extends the results of Part I, which covers the algebra of matrices over a commutative ring, but focuses on aspects of the theory that…
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…
The goal of this paper is to present an algebraic approach to the basic results of the theory of linear recurrence relations. This approach is based on the ideas from the theory of representations of one endomorphisms (a special case of…
Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by $T$, producing an asymptotic estimate as $T \to \infty$. This problem can be…
Ordinary binary multiplication of natural numbers can be generalized in a non-trivial way to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of `3-primality' -- primality with…