Search results for "Geometric algebra"

Geometric Algebra Rotors for Sub-symbolic Coding of Natural Language Sentences

2007

A sub-symbolic encoding methodology for natural language sentences is presented. The procedure is based on the creation of an LSA-inspired semantic space and associates rotation operators derived from Geometric Algebra to word bigrams of the sentence. The operators are subsequently applied to an orthonormal standard basis of the created semantic space according to the order in which words appear in the sentence. The final rotated basis is then coded as a vector and its orthogonal part constitutes the sub-symbolic coding of the sentence. Preliminary experimental results for a classification task, compared with the traditional LSA methodology, show the effectiveness of the approach.

An Introduction to Geometric Algebra and Conics

2016

This chapter introduces the conics and characterizes them from an algebraic perspective. While in depth geometrical aspects of the conics lie outside the scopes of this chapter, this chapter is an opportunity to revisit concepts studied in other chapters such as matrix and determinant and assign a new geometric characterization to them.

GAPP Compiler for Hardware Accelerated Geometric Algebra Computing

2016

Because of the high numeric complexity of Geometric Algebra, its use in engineering applications relies heavily on tools for ecient implementations. In this article, we introduce a new quality of Geometric Algebra Computing solutions based on a new compiler for Geometric Algebra Parallelism Programs (GAPP). These programs are already optimized in a sense that only the really needed computations are left. The GAPP compiler is able to generate two output formats leading to advanced hardware accelerated Geometric Algebra Computing. On one hand, there is the direct generation of HSAIL code, in order to more eciently support the solutions of the broad range of heterogeneous computing architectur…

Remarks on the Historiography of Mathematics

2021

In this paper, I examine aspects of the methodological debate that originated in 2010, when the distinguished historian of mathematics Sabetai Unguru reviewed Roshdi Rashed’s edition of the Arabic translation of Apollonius’ Conics. In his review, Unguru criticized what Rashed calls “l’usage instrumental d’une autre mathématique pour commenter une oeuvre ancienne”. I consider this debate very important and will try to place it within in the discussion of the so-called “geometric algebra” that goes back to the seventies, by tracing the contributions of the main figures who took part in it. Published Online (2021-04-30)Copyright © 2021 by Aldo Brigaglia Article PDF Link: https://jps.library.ut…

On the geometric structure of the class of planar quadratic differential systems

2002

In this work we are interested in the global theory of planar quadratic differential systems and more precisely in the geometry of this whole class. We want to clarify some results and methods such as the isocline method or the role of rotation parameters. To this end, we recall how to associate a pencil of isoclines to each quadratic differential equation. We discuss the parameterization of the space of regular pencils of isoclines by the space of its multiple base points and the equivariant action of the affine group on the fibration of the space of regular quadratic differential equations over the space of regular pencils of isoclines. This fibration is principal, with a projective group…

The γ5-problem and anomalies — A Clifford algebra approach

1990

Abstract It is shown that a strong correspondence between noncyclicity and anomalies exists. This allows, by fundamental properties of Clifford algebras, to build a simple and consistent scheme for treating γ 5 without using ( d −4)-dimensional objects

A Geometric Algebra Based Distributional Model to Encode Sentences Semantics

2013

Word space models are used to encode the semantics of natural language elements by means of high dimensional vectors [23]. Latent Semantic Analysis (LSA) methodology [15] is well known and widely used for its generalization properties. Despite of its good performance in several applications, the model induced by LSA ignores dynamic changes in sentences meaning that depend on the order of the words, because it is based on a bag of words analysis. In this chapter we present a technique that exploits LSA-based semantic spaces and geometric algebra in order to obtain a sub-symbolic encoding of sentences taking into account the words sequence in the sentence. © 2014 Springer-Verlag Berlin Heidel…

Graphic Coprocessors with Native Clifford Algebra Support

2009

A brief introduction to Clifford algebra

2010

Geometric algebra (also known as Clifford algebra) is a powerful mathematical tool that offers a natural and direct way to model geometric objects and their transformations. It is gaining growing attention in different research fields as physics, robotics, CAD/CAM and computer graphics. Clifford algebra makes geometric objects (points, lines and planes) into basic elements of computation and defines few universal operators that are applicable to all types of geometric elements. This paper provides an introduction to Clifford algebra elements and operators.

4D Clifford algebra based on fixed-size representation

2008

Geometric algebra (also known as Clifford algebra) is a powerful mathematical tool that offers a natural and direct way to model geometric objects and their transformations. It is gaining growing attention in different research fields as physics, robotics, CAD/CAM and computer graphics. In particular, 4D geometric algebra implements homogeneous coordinates, which are used to model 3D scenery in most computer graphics applications. The research work on Clifford algebra is actually aimed at finding efficient implementations of the algebra. This paper wants to give a contribution to this research effort by proposing a direct hardware support for geometric algebra operators. The paper introduce…