76,829 research outputs found
Topological toric manifolds
We introduce the notion of a topological toric manifold and a topological fan
and show that there is a bijection between omnioriented topological toric
manifolds and complete non-singular topological fans. A topological toric
manifold is a topological analogue of a toric manifold and the family of
topological toric manifolds is much larger than that of toric manifolds. A
topological fan is a combinatorial object generalizing the notion of a
simplicial fan in toric geometry.
Prior to this paper, two topological analogues of a toric manifold have been
introduced. One is a quasitoric manifold and the other is a torus manifold. One
major difference between the previous notions and topological toric manifolds
is that the former support a smooth action of an -torus while the latter
support a smooth action of a \C^*-torus. We also discuss their relation in
details.Comment: 42 pages, 4 figure
BV functions and sets of finite perimeters in sub-Riemannian manifolds
We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms are given. When we consider sub-Riemannian manifolds, our definition coincides with the one given in the more general context of metric measure spaces which are doubling and support a Poincaré inequality. We focus on finite perimeter sets, i.e., sets whose characteristic function is BV, in sub-Riemannian manifolds. Under an assumption on the nilpotent approximation, we prove a blowup theorem, generalizing the one obtained for step-2 Carnot groups
Expressiveness of SETAFs and Support-Free ADFs under 3-valued Semantics
Generalizing the attack structure in argumentation frameworks (AFs) has been
studied in different ways. Most prominently, the binary attack relation of Dung
frameworks has been extended to the notion of collective attacks. The resulting
formalism is often termed SETAFs. Another approach is provided via abstract
dialectical frameworks (ADFs), where acceptance conditions specify the relation
between arguments; restricting these conditions naturally allows for so-called
support-free ADFs. The aim of the paper is to shed light on the relation
between these two different approaches. To this end, we investigate and compare
the expressiveness of SETAFs and support-free ADFs under the lens of 3-valued
semantics. Our results show that it is only the presence of unsatisfiable
acceptance conditions in support-free ADFs that discriminate the two
approaches
Learners' mathematical reasoning when generalizing from number patterns in the general education and training phase.
This study aims to explore GET learners’ mathematical (algebraic) reasoning when
generalizing from number patterns. Data was collected in a former model C school in greater
Johannesburg area by means of a questionnaire based task involving number patterns. The
mathematical reasoning of the grade 9 participants when generalizing from number patterns
was examined within a commognitive framework. According to this perspective, thinking is a
special activity of communication in which a participant of a discourse engages. The
participants’ responses to questions in the questionnaire based task were classified according
to particular aspects of the discourse they used, specifically routines (strategies) and visual
mediators. The participants’ generalization routines were further classified into one of the
three main categories; numeric, figural and pragmatic generalizations. The analysis focused
on how the learners’ derived rules for the nth term and their justifications for their responses.
The results of this study strongly support the notion that students’ algebraic reasoning when
generalizing in number patterns is intertwined with their choices of routines and mediators.
Most learners used recursive routines while a few used explicit routines (classified and
categorized as numeric routines) and number-mediators. Also, most participants found it
easier to informally verbalize their generalizations. However participants’ spoken
justifications of their written and spoken responses often did not match their use of routines
and visual mediators. As such, an awareness and appreciation (by teachers) of students’
diverse use of routines and mediators when generalizing from number patterns could have
direct pedagogical implications in the mathematics classrooms
Hadamard Regularization
Motivated by the problem of the dynamics of point-particles in high
post-Newtonian (e.g. 3PN) approximations of general relativity, we consider a
certain class of functions which are smooth except at some isolated points
around which they admit a power-like singular expansion. We review the concepts
of (i) Hadamard ``partie finie'' of such functions at the location of singular
points, (ii) the partie finie of their divergent integral. We present and
investigate different expressions, useful in applications, for the latter
partie finie. To each singular function, we associate a partie-finie (Pf)
pseudo-function. The multiplication of pseudo-functions is defined by the
ordinary (pointwise) product. We construct a delta-pseudo-function on the class
of singular functions, which reduces to the usual notion of Dirac distribution
when applied on smooth functions with compact support. We introduce and analyse
a new derivative operator acting on pseudo-functions, and generalizing, in this
context, the Schwartz distributional derivative. This operator is uniquely
defined up to an arbitrary numerical constant. Time derivatives and partial
derivatives with respect to the singular points are also investigated. In the
course of the paper, all the formulas needed in the application to the physical
problem are derived.Comment: 50 pages, to appear in Journal of Mathematical Physic
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