66,786 research outputs found
-triangulations of semialgebraic sets
We show that every semialgebraic set admits a semialgebraic triangulation
such that each closed simplex is differentiable. As an application, we
give a straightforward definition of the integration over a
compact semialgebraic subset of a differential form on an ambient
algebraic manifold, that provides a significant simplification of the theory of
semialgebraic singular chains and integrations. Our results hold over every
(possibly non-archimedian) real closed field.Comment: 12 page
Studentsâ Productive Struggles in Mathematics Learning
Using a predetermined framework on studentsâ productive struggles, the purpose of this study is to explore high school studentsâ productive struggles during the simplification of rational algebraic expressions in a high school mathematics classroom. This study is foregrounded in the anthropological theory of the didactic, and its central notion of a âpraxeologyâ â a praxeology refers to the study of human action, based on the notion that humans engage in purposeful behavior of which the simplification of rational algebraic expressions is an example. The research methodology comprised a lesson study involving a sample of 28 students, and the productive struggle framework was used for data analysis. Findings show that the productive struggle framework is a useful tool that can be used to analyze studentsâ thinking processes during the simplification of rational algebraic expressions. Further research is required on the roles that noticing and questioning can play for mathematics teachers to respond to and effectively support the studentsâ struggles during teaching and learning
Cusps of lattices in rank 1 Lie groups over local fields
Let G be the group of rational points of a semisimple algebraic group of rank
1 over a nonarchimedean local field. We improve upon Lubotzky's analysis of
graphs of groups describing the action of lattices in G on its Bruhat-Tits tree
assuming a condition on unipotents in G. The condition holds for all but a few
types of rank 1 groups. A fairly straightforward simplification of Lubotzky's
definition of a cusp of a lattice is the key step to our results. We take the
opportunity to reprove Lubotzky's part in the analysis from this foundation.Comment: to appear in Geometriae Dedicat
Operators on superspaces and generalizations of the Gelfand-Kolmogorov theorem
(Write-up of a talk at the Bialowieza meeting, July 2007.)
Gelfand and Kolmogorov in 1939 proved that a compact Hausdorff topological
space can be canonically embedded into the infinite-dimensional vector
space , the dual space of the algebra of continuous functions
as an "algebraic variety" specified by an infinite system of quadratic
equations.
Buchstaber and Rees have recently extended this to all symmetric powers
\Sym^n(X) using their notion of the Frobenius -homomorphisms.
We give a simplification and a further extension of this theory, which is
based, rather unexpectedly, on results from super linear lgebra.Comment: LaTeX, 7 pages. Based on a talk at the Bialowieza meeting, July 200
Convexity package for momentum maps on contact manifolds
Let a torus T act effectively on a compact connected cooriented contact
manifold, and let Psi be the natural momentum map on the symplectization. We
prove that, if dim T > 2, the union of the origin with the image of Psi is a
convex polyhedral cone, the non-zero level sets of Psi are connected (while the
zero level set can be disconnected), and the momentum map is open as a map to
its image. This answers a question posed by Eugene Lerman, who proved similar
results when the zero level set is empty. We also analyze examples with dim T
<= 2.Comment: 39 pages. Contains small corrections and a small simplification of
the argument. To appear in Algebraic and Geometric Topology
- âŠ