35,299 research outputs found
Introducing Geometric Algebra to Geometric Computing Software Developers: A Computational Thinking Approach
Designing software systems for Geometric Computing applications can be a
challenging task. Software engineers typically use software abstractions to
hide and manage the high complexity of such systems. Without the presence of a
unifying algebraic system to describe geometric models, the use of software
abstractions alone can result in many design and maintenance problems.
Geometric Algebra (GA) can be a universal abstract algebraic language for
software engineering geometric computing applications. Few sources, however,
provide enough information about GA-based software implementations targeting
the software engineering community. In particular, successfully introducing GA
to software engineers requires quite different approaches from introducing GA
to mathematicians or physicists. This article provides a high-level
introduction to the abstract concepts and algebraic representations behind the
elegant GA mathematical structure. The article focuses on the conceptual and
representational abstraction levels behind GA mathematics with sufficient
references for more details. In addition, the article strongly recommends
applying the methods of Computational Thinking in both introducing GA to
software engineers, and in using GA as a mathematical language for developing
Geometric Computing software systems.Comment: Tutorial, 43 pages, 3 figure
Studying cities to learn about minds: some possible implications of space syntax for spatial cognition
What can we learn of the human mind by examining its products? The city is a case in point. Since the beginning of cities human ideas about them have been dominated by geometric ideas, and the real history of cities has always oscillated between the geometric and the ‘organic’. Set in the context of the suggestion from cognitive neuroscience that we impose more geometric order on the world than it actually possesses, and intriguing question arises: what is the role of the geometric intuition in how we understand cities and how we create them? Here I argue, drawing on space syntax research which has sought to link the detailed spatial morphology of cities to observable functional regularities, that all cities, the organic as well as the geometric, are pervasively ordered by geometric intuition, so that neither the forms of the cities nor their functioning can be understood without insight into their distinctive and pervasive emergent geometrical forms. The city is often said to be the creation of economic and social processes, but here it is argued that these processes operate within an envelope of geometric possibility defined by the human mind in its interaction with spatial laws that govern the relations between objects and spaces in the ambient world
Some aspects of braided geometry: differential calculus, tangent space, gauge theory
A new approach is suggested to quantum differential calculus on certain
quantum varieties.
It consists in replacing quantum de Rham complexes with differentials
satisfying Leibniz rule by those which are in a sense close to Koszul complexes
from \cite{G1}. We also introduce the tangent space on a quantum hyperboloid
equipped with an action on the quantum function space and define the notions of
quantum (pseudo)metric and quantum connection (partially defined) on it. All
objects are considered from the viewpoint of flatness of quantum deformations.
A problem of constructing a flatly deformed quantum gauge theory is discussed
as well.Comment: 19 pages, Late
Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces
This paper contains a thorough study of the trigonometry of the homogeneous
symmetric spaces in the Cayley-Klein-Dickson family of spaces of 'complex
Hermitian' type and rank-one. The complex Hermitian elliptic CP^N and
hyperbolic CH^N spaces, their analogues with indefinite Hermitian metric and
some non-compact symmetric spaces associated to SL(N+1,R) are the generic
members in this family. The method encapsulates trigonometry for this whole
family of spaces into a single "basic trigonometric group equation", and has
'universality' and '(self)-duality' as its distinctive traits. All previously
known results on the trigonometry of CP^N and CH^N follow as particular cases
of our general equations. The physical Quantum Space of States of any quantum
system belongs, as the complex Hermitian space member, to this parametrised
family; hence its trigonometry appears as a rather particular case of the
equations we obtain.Comment: 46 pages, LaTe
On several varieties of cacti and their relations
Motivated by string topology and the arc operad, we introduce the notion of
quasi-operads and consider four (quasi)-operads which are different varieties
of the operad of cacti. These are cacti without local zeros (or spines) and
cacti proper as well as both varieties with fixed constant size one of the
constituting loops. Using the recognition principle of Fiedorowicz, we prove
that spineless cacti are equivalent as operads to the little discs operad. It
turns out that in terms of spineless cacti Cohen's Gerstenhaber structure and
Fiedorowicz' braided operad structure are given by the same explicit chains. We
also prove that spineless cacti and cacti are homotopy equivalent to their
normalized versions as quasi-operads by showing that both types of cacti are
semi-direct products of the quasi-operad of their normalized versions with a
re-scaling operad based on R>0. Furthermore, we introduce the notion of
bi-crossed products of quasi-operads and show that the cacti proper are a
bi-crossed product of the operad of cacti without spines and the operad based
on the monoid given by the circle group S^1. We also prove that this particular
bi-crossed operad product is homotopy equivalent to the semi-direct product of
the spineless cacti with the group S^1. This implies that cacti are equivalent
to the framed little discs operad. These results lead to new CW models for the
little discs and the framed little discs operad.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-13.abs.htm
-monopole Floer homology, higher compositions and connected sums
We study the behavior of -monopole Floer homology under
connected sums. After constructing a (partially defined)
-module structure on the -monopole Floer
chain complex of a three manifold (in the spirit of Baldwin and Bloom's
monopole category), we identify up to quasi-isomorphism the Floer chain complex
of a connected sum with a version of the -tensor product
of the modules of the summands. There is an associated Eilenberg-Moore spectral
sequence converging to the Floer groups of the connected sum whose page
is the of the Floer groups of the summands. We discuss in detail
a simple example, and use this computation to show that the
-monopole Floer homology of has non trivial Massey
productsComment: 43 pages, 14 figures. Comments very welcome
General Spinor Structures on Quantum Spaces
A general theory of quantum spinor structures on quantum spaces is presented,
within the conceptual framework of the formalism of quantum principal bundles.
Quantum analogs of all basic objects of the classical theory are constructed
and analyzed. This includes Laplace and Dirac operators, quantum versions of
Clifford and spinor bundles, a Hodge *-operator, appropriate integration
operators, and mutual relations of these objects. We also present a
self-contained formalism of braided Clifford algebras. Quantum phenomena
appearing in the theory are discussed, including a very interesting example of
the Dirac operator associated to a quantum Hopf fibration.Comment: 40 pages, AMSLaTeX; Visit http://www.matem.unam.mx/~micho for the
latest versions of this and related author's paper
Twisted submanifolds of R^n
We propose a general procedure to construct noncommutative deformations of an
embedded submanifold of determined by a set of smooth
equations . We use the framework of Drinfel'd twist deformation of
differential geometry of [Aschieri et al., Class. Quantum Gravity 23 (2006),
1883]; the commutative pointwise product is replaced by a (generally
noncommutative) -product determined by a Drinfel'd twist. The twists we
employ are based on the Lie algebra of vector fields that are tangent
to all the submanifolds that are level sets of the ; the twisted Cartan
calculus is automatically equivariant under twisted tangent infinitesimal
diffeomorphisms. We can consistently project a connection from the twisted
to the twisted if the twist is based on a suitable Lie
subalgebra . If we endow with a metric
then twisting and projecting to the normal and tangent vector fields commute,
and we can project the Levi-Civita connection consistently to the twisted ,
provided the twist is based on the Lie subalgebra
of the Killing vector fields of the metric; a
twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can
be characterized in terms of generators and polynomial relations. We present in
some detail twisted cylinders embedded in twisted Euclidean and
twisted hyperboloids embedded in twisted Minkowski [these are
twisted (anti-)de Sitter spaces ].Comment: Latex file, 48 pages, 1 figure. Slightly adapted version to the new
preprint arXiv:2005.03509, where the present framework is specialized to
quadrics and other algebraic submanifolds of R^n. Several typos correcte
Resonance Gyrons and Quantum Geometry
We describe irreducible representations, coherent states and star-products
for algebras of integrals of motions (symmetries) of two-dimensional resonance
oscillators. We demonstrate how the quantum geometry (quantum K\"ahler form,
metric, quantum Ricci form, quantum reproducing measure) arises in this
problem. We specifically study the distinction between the isotropic resonance
and the general resonance for arbitrary coprime . Quantum
gyron is a dynamical system in the resonance algebra. We derive its Hamiltonian
in irreducible representations and calculate the semiclassical asymptotics of
the gyron spectrum via the quantum geometrical objects.Comment: Latex, 31page
Noncommutative Geometry and Gauge Theories on Discrete Groups
We build and investigate a pure gauge theory on arbitrary discrete groups. A
systematic approach to the construction of the differential calculus is
presented. We discuss the metric properties of the models and introduce the
action functionals for unitary gauge theories. A detailed analysis of two
simple models based on \z_2 and \z_3 follows. Finally we study the method
of combining the discrete and continuous geometry.Comment: LaTeX file, 23 pages, TPJU 7/92 (revised version
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