Related papers: Implicit Linear Algebra and its Applications
A solution of linear systems of equations Ax=b and Ax=0 is a vital part of many computational packages. This paper presents a novel formulation based on the projective extension of the Euclidean space using the outer product (extended…
In this paper, we present and analyze methods for solving a system of linear equations over idempotent semifields. The first method is based on the pseudo-inverse of the system matrix. We then present a specific version of Cramer's rule…
In this article we dwell into the class of so called ill posed Linear Inverse Problems (LIP) in machine learning, which has become almost a classic in recent times. The fundamental task in an LIP is to recover the entire signal / data from…
We explain how recent developments in the fields of realisability models for linear logic -- or geometry of interaction -- and implicit computational complexity can lead to a new approach of implicit computational complexity. This…
Linearization of coupled second order nonlinear ordinary differential equations (SNODEs) is one of the open and challenging problems in the theory of differential equations. In this paper we describe a simple and straightforward method to…
A previous article was devoted to an analysis of the symmetry properties of a class of first-order delay ordinary differential systems (DODSs). Here we concentrate on linear DODSs. They have infinite-dimensional Lie point symmetry groups…
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…
In this paper, some real-world motivated examples are provided illustrating the power of linear algebra tools as the product of matrices, determinants, eigenvalues and eigenvectors. In this sense, some practical applications related to…
The integrability condition called shape invariance is shown to have an underlying algebraic structure and the associated Lie algebras are identified. These shape-invariance algebras transform the parameters of the potentials such as…
A natural construction of the logarithmic extension of the M(2,p) minimal models is presented, which generalises our previous model [0708.0802] of percolation (p=3). Its key aspect is the replacement of the minimal model irreducible modules…
Integer linear programming (ILP) encompasses a very important class of optimization problems that are of great interest to both academia and industry. Several algorithms are available that attempt to explore the solution space of this class…
We consider two classes of computations which admit taking linear combinations of execution runs: probabilistic sampling and generalized animation. We argue that the task of program learning should be more tractable for these architectures…
We study in this paper solutions to several kinds of linear bimatrix equations arising from pole assignment and stability analysis of complex-valued linear systems, which have several potential applications in control theory, particularly,…
The Wilson loop with a wavy line contour is studied using integrable methods. The auxiliary problem is solved and the Lax operator is built to first order in perturbation theory, considering a small perturbation from the straight line.…
This work provides a complete characterization of the solutions of a linear interpolation problem for vector polynomials. The interpolation problem consists in finding n scalar polynomials such that an equation involving a linear…
Given an associative, not necessarily commutative, ring R with identity, a formal matrix calculus is introduced and developed for pairs of matrices over R. This calculus subsumes the theory of homogeneous systems of linear equations with…
A vector space is commonly defined as a set that satisfies several conditions related to addition and scalar multiplication. However, for beginners, it may be hard to immediately grasp the essence of these conditions. There are probably a…
Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute…
There is strong interest in developing mathematical methods that can be used to understand complex neural networks used in image analysis. In this paper, we introduce techniques from Linear Algebra to model neural network layers as maps…
We introduce a family of implicit probabilistic integrators for initial value problems (IVPs), taking as a starting point the multistep Adams-Moulton method. The implicit construction allows for dynamic feedback from the forthcoming…