27,167 research outputs found
Conventionalism in Reidâs âGeometry of Visiblesâ
The role of conventions in the formulation of Thomas Reidâs theory of the geometry of vision, which he calls the âgeometry of visiblesâ, is the subject of this investigation. In particular, we will examine the work of N. Daniels and R. Angell who have alleged that, respectively, Reidâs âgeometry of visiblesâ and the geometry of the visual field are non-Euclidean. As will be demonstrated, however, the construction of any geometry of vision is subject to a choice of conventions regarding the construction and assignment of its various properties, especially metric properties, and this fact undermines the claim for a unique non-Euclidean status for the geometry of vision. Finally, a suggestion is offered for trying to reconcile Reidâs direct realist theory of perception with his geometry of visibles
Harmonic fields on the extended projective disc and a problem in optics
The Hodge equations for 1-forms are studied on Beltrami's projective disc
model for hyperbolic space. Ideal points lying beyond projective infinity arise
naturally in both the geometric and analytic arguments. An existence theorem
for weakly harmonic 1-fields, changing type on the unit circle, is derived
under Dirichlet conditions imposed on the non-characteristic portion of the
boundary. A similar system arises in the analysis of wave motion near a
caustic. A class of elliptic-hyperbolic boundary-value problems is formulated
for those equations as well. For both classes of boundary-value problems, an
arbitrarily small lower-order perturbation of the equations is shown to yield
solutions which are strong in the sense of Friedrichs.Comment: 30 pages; Section 3.3 has been revise
Geometric inequalities for black holes
It is well known that the three parameters that characterize the Kerr black
hole (mass, angular momentum and horizon area) satisfy several important
inequalities. Remarkably, some of these inequalities remain valid also for
dynamical black holes. This kind of inequalities play an important role in the
characterization of the gravitational collapse. They are closed related with
the cosmic censorship conjecture. In this article recent results in this
subject are reviewed.Comment: Invited review article for General Relativity and Gravitation. Based
on my plenary lecture at GR20 and the longer review article Classical and
Quantum Gravity, 29(7):073001, 2012, arXiv:1111.3615. 27 pages. 14 figures.
Minor change
Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere and the hyperbolic plane
The Kepler problem is a dynamical system that is well defined not only on the
Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the
theory of central potentials on spaces of constant curvature is studied. All
the mathematical expressions are presented using the curvature \k as a
parameter, in such a way that they reduce to the appropriate property for the
system on the sphere , or on the hyperbolic plane , when
particularized for \k>0, or \k<0, respectively; in addition, the Euclidean
case arises as the particular case \k=0. In the second part we study the main
properties of the Kepler problem on spaces with curvature, we solve the
equations and we obtain the explicit expressions of the orbits by using two
different methods: first by direct integration and second by obtaining the
\k-dependent version of the Binet's equation. The final part of the article,
that has a more geometric character, is devoted to the study of the theory of
conics on spaces of constant curvature.Comment: 37 pages, 7 figure
Common Visual Representations as a Source for Misconceptions of Preservice Teachers in a Geometry Connection Course
In this paper, we demonstrate how atypical visual representations of a triangle, square or a parallelogram may hinder studentsâ understanding of a median and altitude. We analyze responses and reasoning given by 16 preservice middle school teachers in a Geometry Connection class. Particularly, the data were garnered from three specific questions posed on a cumulative final exam, which focused on computing and comparing areas of parallelograms, and triangles represented by atypical images. We use the notions of concept image and concept definition as our theoretical framework for an analysis of the studentsâ responses. Our findings have implication on how typical images can impact studentsâ cognitive process and their concept image. We provide a number of suggestions that can foster conceptualization of the notions of median and altitude in a triangle that can be realized in an enacted lesson
Asymptotics for Hermite-Pade rational approximants for two analytic functions with separated pairs of branch points (case of genus 0)
We investigate the asymptotic behavior for type II Hermite-Pade approximation
to two functions, where each function has two branch points and the pairs of
branch points are separated. We give a classification of the cases such that
the limiting counting measures for the poles of the Hermite-Pade approximants
are described by an algebraic function of order 3 and genus 0. This situation
gives rise to a vector-potential equilibrium problem for three measures and the
poles of the common denominator are asymptotically distributed like one of
these measures. We also work out the strong asymptotics for the corresponding
Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that
characterizes this Hermite-Pade approximation problem.Comment: 102 pages, 31 figure
Symmetries and invariances in classical physics
Symmetry, intended as invariance with respect to a transformation (more precisely, with respect to a transformation group), has acquired more and more importance in modern physics. This Chapter explores in 8 Sections the meaning, application and interpretation of symmetry in classical physics. This is done both in general, and with attention to specific topics. The general topics include illustration of the distinctions between symmetries of objects and of laws, and between symmetry principles and symmetry arguments (such as Curie's principle), and reviewing the meaning and various types of symmetry that may be found in classical physics, along with different interpretative strategies that may be adopted. Specific topics discussed include the historical path by which group theory entered classical physics, transformation theory in classical mechanics, the relativity principle in Einstein's Special Theory of Relativity, general covariance in his General Theory of Relativity, and Noether's theorems. In bringing these diverse materials together in a single Chapter, we display the pervasive and powerful influence of symmetry in classical physics, and offer a possible framework for the further philosophical investigation of this topic
- âŚ