89,542 research outputs found
Extremal Lipschitz functions in the deviation inequalities from the mean
We obtain an optimal deviation from the mean upper bound \begin{equation}
D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\
x\in\R\label{abstr} \end{equation} where \F is the class of the integrable,
Lipschitz functions on probability metric (product) spaces. As corollaries we
get exact solutions of \eqref{abstr} for Euclidean unit sphere with
a geodesic distance and a normalized Haar measure, for equipped with a
Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond
graph equipped with uniform measure and Hamming distance. We also prove that in
general probability metric spaces the in \eqref{abstr} is achieved on
a family of distance functions.Comment: 7 page
Dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond
We present recent results on counting and distribution of circles in a given
circle packing invariant under a geometrically finite Kleinian group and
discuss how the dynamics of flows on geometrically finite hyperbolic
manifolds are related. Our results apply to Apollonian circle packings,
Sierpinski curves, Schottky dances, etc.Comment: To appear in the Proceedings of ICM, 201
Apollonian circle packings: Dynamics and Number theory
We give an overview of various counting problems for Apollonian circle
packings, which turn out to be related to problems in dynamics and number
theory for thin groups. This survey article is an expanded version of my
lecture notes prepared for the 13th Takagi lectures given at RIMS, Kyoto in the
fall of 2013.Comment: To appear in Japanese Journal of Mat
Simplicial blowups and discrete normal surfaces in simpcomp
simpcomp is an extension to GAP, the well known system for computational
discrete algebra. It allows the user to work with simplicial complexes. In the
latest version, support for simplicial blowups and discrete normal surfaces was
added, both features unique to simpcomp. Furthermore, new functions for
constructing certain infinite series of triangulations have been implemented
and interfaces to other software packages have been improved to previous
versions.Comment: 10 page
Diameters, distortion and eigenvalues
We study the relation between the diameter, the first positive eigenvalue of
the discrete -Laplacian and the -distortion of a finite graph. We
prove an inequality relating these three quantities and apply it to families of
Cayley and Schreier graphs. We also show that the -distortion of Pascal
graphs, approximating the Sierpinski gasket, is bounded, which allows to obtain
estimates for the convergence to zero of the spectral gap as an application of
the main result.Comment: Final version, to appear in the European Journal of Combinatoric
A Relationship Between Problem Solving Ability and Students' Mathematical Thinking
This research have a purpose to know is there an influence of problem solving abilty to students mathematical thinking, and to know how strong problem solving ability affect students mathematical thinking. This research used descriptive quantitative method, which a population is all of students that taking discrete mathematics courses both in department of Information Systems and department of mathematics education. Based on the results of data analysis showed that there are an influence of problem solving ability to students mathematical thinking either at department of mathematics education or at department of information systems. In this study, it was found that the influence of problem solving ability to students mathematical thinking which take place at mathematics education department is stonger than at information system department. This is because, at mathematics education department, problem-solving activities more often performed in courses than at department of information system. Almost 75% of existing courses in department of mathematics education involve problem solving to the objective of courses, meanwhile, in the department of information systems, there are only 10% of these courses. As a result, mathematics education department student's are better trained in problem solving than information system department students. So, to improve students' mathematical thinking, its would be better, at fisrtly enhance the problem solving ability
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