Related papers: Representing and Implementing Matrices Using Algeb…
Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of…
The ZX-calculus is a graphical language for reasoning about ZX-diagrams, a type of tensor networks that can represent arbitrary linear maps between qubits. Using the ZX-calculus, we can intuitively reason about quantum theory, and optimise…
We propose a sparse algebra for samplet compressed kernel matrices, to enable efficient scattered data analysis. We show the compression of kernel matrices by means of samplets produces optimally sparse matrices in a certain S-format. It…
The representation problem of finite-dimensional Markov matrices in Markov semigroups is revisited, with emphasis on concrete criteria for matrix subclasses of theoretical or practical relevance, such as equal-input, circulant, symmetric or…
Matrix models play an important role in studies of quantum gravity, being candidates for a formulation of M-theory, but are notoriously difficult to solve. In this work, we present a fresh approach by introducing a novel exact model…
We introduce an associative algebra $\M_k(x)$ whose dimension is the $2k$-th Motzkin number. The algebra $\M_k(x)$ has a basis of "Motzkin diagrams," which are analogous to Brauer and Temperley-Lieb diagrams, and it contains the…
Bidiagonal matrices are widespread in numerical linear algebra, not least because of their use in the standard algorithm for computing the singular value decomposition and their appearance as LU factors of tridiagonal matrices. We show that…
The ZX-calculus is a universal graphical language for qubit quantum computation, meaning that every linear map between qubits can be expressed in the ZX-calculus. Furthermore, it is a complete graphical rewrite system: any equation…
We extend the Ax-Katz theorem for a single polynomial from finite fields to the rings Z_m with m composite. This extension not only yields the analogous result, but gives significantly higher divisibility bounds. We conjecture what computer…
This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support…
To a representation of $\O_N$ (the Cuntz algebra with $N$ generators) we associate a projection valued measure and we study the case when this measure has atoms. The main technical tool are the spaces invariant for all the operators…
The various non-linear transformations incurred by the rays in an optical system can be modelled by matrix products up to any desired order of approximation. Mathematica software has been used to find the appropriate matrix coefficients for…
One of the grand challenges of Mathematics instruction is to provide students with problems that are both accessible and have a reasonably elegant solution. Instructors commonly resort to resources like course textbooks, online-learning…
In this work, new closed-form formulas for the matrix exponential are provided. Our method is direct and elementary, it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Moreover,…
Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, linear deformation of matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such…
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…
Multivalued projections are applied to the study of weighted least squares solutions of linear relations equations (or inclusions) and some of its applications. To this end a matrix representation of multivalued projections with respect to…
The ZX-calculus is a graphical language for reasoning about quantum computation that has recently seen an increased usage in a variety of areas such as quantum circuit optimisation, surface codes and lattice surgery, measurement-based…
Optical computing harnesses the speed of light to perform vector-matrix operations efficiently. It leverages interference, a cornerstone of quantum computing algorithms, to enable parallel computations. In this work, we interweave quantum…
With the advent of computers, one needs algebraic structures that can simultaneously work with bulk data. One such algebraic structure, namely, n-linear algebras of type I are introduced in this book and its applications to n-Markov chains…