225,943 research outputs found
A geometry of information, I: Nerves, posets and differential forms
The main theme of this workshop (Dagstuhl seminar 04351) is `Spatial
Representation: Continuous vs. Discrete'. Spatial representation has two
contrasting but interacting aspects (i) representation of spaces' and (ii)
representation by spaces. In this paper, we will examine two aspects that are
common to both interpretations of the theme, namely nerve constructions and
refinement. Representations change, data changes, spaces change. We will
examine the possibility of a `differential geometry' of spatial representations
of both types, and in the sequel give an algebra of differential forms that has
the potential to handle the dynamical aspect of such a geometry. We will
discuss briefly a conjectured class of spaces, generalising the Cantor set
which would seem ideal as a test-bed for the set of tools we are developing.Comment: 28 pages. A version of this paper appears also on the Dagstuhl
seminar portal http://drops.dagstuhl.de/portals/04351
Supersymmetric D-branes on flux backgrounds
Several aspects concerning the physics of D-branes in Type II flux
compactifications preserving minimal N=1 supersymmetry in four dimensions are
considered. It is shown how these vacua are completely characterized in terms
of properly defined generalized calibrations for D-branes and the relation with
Generalized Complex Geometry is discussed. General expressions for
superpotentials and D-terms associated with the N=1 four-dimensional
description of space-time filling D-branes are presented. The massless spectrum
of calibrated D-branes can be characterized in terms of cohomology groups of a
differential complex canonically induced on the D-branes by the underlying
generalized complex structure.Comment: 9 pages; contribution to the proceedings of the RTN project
`Constituents, Fundamental Forces and Symmetries of the Universe' conference
in Napoli, October 9 - 13, 2006; (v2) references adde
The hole argument and beyond: Part I: the story so far
In this two-part paper, we review, and then develop, the assessment of the hole argument for general relativity. This first Part reviews the literature hitherto, focussing on the philosophical aspects. It also introduces two main ideas we will need in Part II: which will propose a framework for making comparisons of non-isomorphic spacetimes.
In Section 1 of this paper, we recall Einstein's original argument. Section 2 recalls the argument's revival by philosophers in the 1980s and 1990s. This includes the first main idea we will need in Part II: namely, that two spacetime points in different possible situations are never strictly identicalâthey are merely counterparts.
In Section 3, we reportâand rebutâmore recent claims to "dissolve" the argument. Our rebuttal is based on the fact that in differential geometry, and its applications in physics such as general relativity, points are in some cases identified, or correspond with each other, between one context and another, by means other than isometry (or isomorphism). We call such a correspondence a threading of points. This is the second main idea we shall use in Part II
Geometric aspects of higher order variational principles on submanifolds
The geometry of jets of submanifolds is studied, with special interest in the
relationship with the calculus of variations. A new intrinsic geometric
formulation of the variational problem on jets of submanifolds is given.
Working examples are provided.Comment: 17 page
On some aspects of the geometry of differential equations in physics
In this review paper, we consider three kinds of systems of differential
equations, which are relevant in physics, control theory and other applications
in engineering and applied mathematics; namely: Hamilton equations, singular
differential equations, and partial differential equations in field theories.
The geometric structures underlying these systems are presented and commented.
The main results concerning these structures are stated and discussed, as well
as their influence on the study of the differential equations with which they
are related. Furthermore, research to be developed in these areas is also
commented.Comment: 21 page
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