Related papers: n-Linear Algebra of type I and its applications
We present an algebraic characterization of the complexity classes Logspace and NLogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is inspired from proof…
We describe a class calculus that is expressive enough to describe and improve its own learning process. It can design and debug programs that satisfy given input/output constraints, based on its ontology of previously learned programs. It…
Let $n>1$ be an integer. The algebras of the title, which we abbreviate as algebras of type $n$, are infinite-dimensional graded Lie algebras $L= \bigoplus_{i=1}^{\infty}L_i$, which are generated by an element of degree $1$ and an element…
This paper reviews the properties and applications of certain n-ary generalizations of Lie algebras in a self-contained and unified way. These generalizations are algebraic structures in which the two entries Lie bracket has been replaced…
Linear algebra is a major field of numerical computation and is widely applied. Most linear algebra libraries (in most programming languages) do not statically guarantee consistency of the dimensions of vectors and matrices, causing runtime…
Bases, mappings, projections and metrics, natural for Neural network training, are introduced. Graph-theoretical interpretation is offered. Non-Gaussianity naturally emerges, even in relatively simple datasets. Training statistics,…
A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the…
In this talk I will introduces two spaces: the first space is the usual n-dimensional vector space with the unusual feature that n is non-integer, the second space is composed by the linear matrices acting on the previous space (physicists…
This survey is meant to provide an introduction to the fundamental theorem of linear algebra and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to linear algebra. Specifically, we…
In this paper we study the tensor powers of the standard representation of the quantum super-algebra $U_q(sl(2|1)$, focusing on the rings of its algebra endomorphisms, called centraliser algebras and denoted by $LG_n$. Their dimensions were…
We construct a class of positive linear maps on matrix algebras. We find conditions when these maps are atomic, decomposable and completely positive. We obtain a large class of atomic positive linear maps. As applications in quantum…
Classical algebraic structures require exact satisfaction of their defining axioms. We propose similarity algebra, a framework extending algebraic and Lie structures to settings where operations satisfy quantitative bounds up to a tolerance…
Despite the wide variety of input types in machine learning, this diversity is often not fully reflected in their representations or model architectures, leading to inefficiencies throughout a model's lifecycle. This paper introduces an…
An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…
A null vector is an algebraic quantity with square equal to zero. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by N. The rules of addition and multiplication in N…
In this book i treat linear algebra over division ring. A system of linear equations over a division ring has properties similar to properties of a system of linear equations over a field. However, noncommutativity of a product creates a…
In this work $n$-dimensional filiform Leibniz algebras admitting a gradation of length $(n-1)$ are classified. Derivations of such algebras are also described.
This document is meant as a pedagogical introduction to the modern language used to talk about quantum theory, especially in the field of quantum information. It assumes that the reader has taken a first traditional course on quantum…
Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…