Exploring Hausdorff Dimension

We live in a 3-dimensional world. We know 2-dimensional objects as those we can see on paper. But what would an object with a nonnatural number (a number that isn\u27t a whole positive number) dimension look like? These dimensions exist and some unique mathematical sets have such dimensions, like log2/log3 . We call such dimensions and its tool Hausdorff dimension. In my talk, I examine Hausdorff dimension and its famous examples including the Cantor Set and multiple manipulations of it

Functional programming framework for GRworkbench

The software tool GRworkbench is an ongoing project in visual, numerical General Relativity at The Australian National University. Recently, the numerical differential geometric engine of GRworkbench has been rewritten using functional programming techniques. By allowing functions to be directly represented as program variables in C++ code, the functional framework enables the mathematical formalism of Differential Geometry to be more closely reflected in GRworkbench . The powerful technique of `automatic differentiation' has replaced numerical differentiation of the metric components, resulting in more accurate derivatives and an order-of-magnitude performance increase for operations relying on differentiation

Higher Dimensional Image Analysis using Brunn-Minkowski Theorem, Convexity and Mathematical Morphology

The theory of deterministic morphological operators is quite rich and has been used on set and lattice theory. Mathematical Morphology can benefit from the already developed theory in convex analysis. Mathematical Morphology introduced by Serra is a very important tool in image processing and Pattern recognition. The framework of Mathematical Morphology consists in Erosions and Dilations. Fractals are mathematical sets with a high degree of geometrical complexity that can model many natural phenomena. Examples include physical objects such as clouds, mountains, trees and coastlines as well as image intensity signals that emanate from certain type of fractal surfaces. So this article tries to link the relation between combinatorial convexity and Mathematical Morphology. Keywords: Convex bodies, convex polyhedra, homothetics, morphological cover, fractal, dilation, erosion

A Fully Verified Executable LTL Model Checker

International audienceWe present an LTL model checker whose code has been completely verified using the Isabelle theorem prover. The checker consists of over 4000 lines of ML code. The code is produced using recent Isabelle technology called the Refinement Framework, which allows us to split its correctness proof into (1) the proof of an abstract version of the checker, consisting of a few hundred lines of “formalized pseudocode”, and (2) a verified refinement step in which mathematical sets and other abstract structures are replaced by implementations of efficient structures like red-black trees and functional arrays. This leads to a checker that, while still slower than unverified checkers, can already be used as a trusted reference implementation against which advanced implementations can be tested. We report on the structure of the checker, the development process, and some experiments on standard benchmarks

The well-ordering of sets

Thesis (M.A.)--Boston Universit