相关论文: Geometric Algebras and Extensors

We argue for more widespread use of manifold-like polyfolds (M-polyfolds) as differential geometric objects. M-polyfolds possess a distinct advantage over differentiable manifolds, enabling a smooth and local change of dimension. To…

Geometric models have emerged as an important tool in the representation theory of algebras. Surface models associated to gentle algebras have been particularly fruitful in advancing our understanding of their module and derived categories.…

In this expository paper we want to give a brief introduction, with few key references for further reading, to the inner functioning of the new and successfull algorithms of Deep Learning and Geometric Deep Learning with a focus on Graph…

This paper shows how beginning with Justus Grassmann's work, Hermann Grassmann was influenced in his mathematical thinking by crystallography. H. Grassmann's Ausdehnungslehre in turn had a decisive influence on W.K. Clifford in the genesis…

Let $A$ be a unital associative algebra over a field $k$. All unital associative algebras containing $A$ as a subalgebra of a given codimension $\mathfrak{c}$ are described and classified. For a fixed vector space $V$ of dimension…

The discussion of how to apply geometric algebra to euclidean $n$-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes…

Formalism based on GA is an alternative to distributed representation models developed so far --- Smolensky's tensor product, Holographic Reduced Representations (HRR) and Binary Spatter Code (BSC). Convolutions are replaced by geometric…

In recent years, the intersection of algebra, geometry, and combinatorics with particle physics and cosmology has led to significant advances. Central to this progress is the twofold formulation of the study of particle interactions and…

This paper surveys the application of geometric algebra to the physics of electrons. It first appeared in 1996 and is reproduced here with only minor modifications. Subjects covered include non-relativistic and relativistic spinors, the…

$G$-deformability of maps into projective space is characterised by the existence of certain Lie algebra valued 1-forms. This characterisation gives a unified way to obtain well known results regarding deformability in different geometries.

Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and also on the choice of the contravariant components, it was shown that a wide variety of third,…

The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some…

This paper establishes the basis of the quaternionic differential geometry ($\mathbbm H$DG) initiated in a previous article. The usual concepts of curves and surfaces are generalized to quaternionic constraints, as well as the curvature and…

This paper presents geometrical foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theories based on the representation theory of SL(2,R) group. We describe here geometries of…

An alternative, pedagogically simpler derivation of the allowed physical wave fronts of a propagating electromagnetic signal is presented using geometric algebra. Maxwell's equations can be expressed in a single multivector equation using…

In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to…

We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised…

In this paper we use Euler-Seidel matrices method to find out some properties of exponential and geometric polynomials and numbers. Some known results are reproved and some new results are obtained.

By "parallelogram geometry" we mean the elementary, "commutative", geometry corresponding to vector addition, and by "trapezoid geometry" a certain "non-commutative deformation" of the former. This text presents an elementary approach via…

The weighted triangulation algebras associated to triangulation quivers and their socle deformations were recently introduced and studied in [15]-[20] and [2]. These algebras, based on surface triangulations and originated from the theory…