Related papers: Typing linear algebra: A biproduct-oriented approa…
Despite a growing body of work at the intersection of deep learning and formal languages, there has been relatively little systematic exploration of transformer models for reasoning about typed lambda calculi. This is an interesting area of…
This paper outlines our ideas on how to teach linear algebra in a mechanized mathematical environment, and discusses some of our reasons for thinking that this is a better way to teach linear algebra than the ``old fashioned way''. We…
Given a full column rank matrix $A \in \mathbb{R}^{m\times n}$ ($m\geq n$), we consider a special class of linear systems of the form $A^\top Ax=A^\top b+c$ with $x, c \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. The occurrence of $c$ in…
Graded rings provide a natural algebraic framework for encoding symmetry via decompositions into homogeneous components indexed by a group, together with multiplication rules reflecting the group operation. Among graded rings, strongly…
We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the Lambda-calculus…
Linear type systems have a long and storied history, but not a clear path forward to integrate with existing languages such as OCaml or Haskell. In this paper, we study a linear type system designed with two crucial properties in mind:…
Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive…
In this article, a new approach based on linear algebra is adopted to study a hybrid Sheffer polynomial sequences. The recurrence relations and differential equation for these polynomials are derived by using the properties and…
We propose a hierarchical architecture for efficiently computing high-quality solutions to structured mixed-integer programs (MIPs). To reduce computational effort, our approach decouples the original problem into a higher level problem and…
Iteration method is commonly used in solving linear systems of equations. We present quantum algorithms for the relaxed row and column iteration methods by constructing unitary matrices in the iterative processes, which generalize row and…
Multiplication of a sparse matrix with another (dense or sparse) matrix is a fundamental operation that captures the computational patterns of many data science applications, including but not limited to graph algorithms, sparsely connected…
A Bialgebra is a module over a ring that is both an associative algebra and a co-associative coalgebra with the product and coproduct additionally satisfying an appropriate commutative relationship. One application of Bialgebras is in the…
In this paper we generalize the well-known construction of shuffle product algebras by using mixable shuffles, and prove that any free Baxter algebra is isomorphic to a mixable shuffle product algebra. This gives an explicit construction of…
Tensor decompositions have become essential tools for feature extraction and compression of multiway data. Recent advances in tensor operators have enabled desirable properties of standard matrix algebra to be retained for multilinear…
Many tasks require finding groups of elements in a matrix of numbers, symbols or class likelihoods. One approach is to use efficient bi- or tri-linear factorization techniques including PCA, ICA, sparse matrix factorization and plaid…
In this note we develop a coalgebraic approach to the study of solutions of linear difference equations over modules and rings. Some known results about linearly recursive sequences over base fields are generalized to linearly (bi)recursive…
Studies on time and memory costs of products in geometric algebra have been limited to cases where multivectors with multiple grades have only non-zero elements. This allows to design efficient algorithms for a generic purpose; however, it…
The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief economic…
The main purpose of this book is to propose an introduction to the modern tools of algebraic complexity. To remain as simple as possible while providing meaningful examples, we chose to focus on effective linear algebra; this is certainly…
We recall the concept of Baxterisation of an R-matrix, or of a monodromy matrix, which corresponds to build, from one point in the $ R$-matrix parameter space, the algebraic variety where the spectral parameter(s) live. We show that the…