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2, 123--139 Hausdorff dimension is the principal notion of dimension in the context of fractal sets in ! We will examine some of these and their inter-relationship. However, other definitions are in widespread use, for example, packing dimension, upper and lower box-counting dimension, upper and lower Minkowski dimension, ... Key words: Isoperimetric Inequality, sets of finite perimeter.
Several seminars will include presentations by members of the Department of Mathematics on proposed topics for dissertations that they would be interested in supervising.
Additional seminars will cover important aspects of writing a dissertation in mathematics, including: what plagiarism is and how to avoid it, the use of libraries for research, electronic research, general aspects of writing mathematics, and preparing a manuscript using mathematical text processing software (in particular, La Te X). Assessment is primarily based entirely on the Dissertation, three hard copies and one electronic copy of which must be submitted by early September (exact date to be specified later).
We will first give a rigorous proof of (S), and then investigate the family of sets such that equality holds true in Steiner inequality. De Giorgi: Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita.
Maggi: Sets of Finite Perimeter and Geometric Variational Problems: an Introduction to Geometric Measure Theory.
$$ v\bigm|_=u\bigm|_ \qquad \implies \qquad \|\nabla u\|_\leq \|\nabla v\|_.$$!
is absolutely minimizing if and only if it is the solution of the infinity Laplacian, which is the (highly degenerate) Euler-Lagrange equation for the prototypical problem in the calculus of variations in ! Recommended modules: Functional Analysis, Partial Differential Equations References: ! and Mc Mullen P., Arcs with increasing chords, Proc. In such problems, it is of course important to choose an appropriate metric on the space of the probability measures. :-)) are scarce and buried in mathematical biology books and journals. 72(1972), 205--207 Studying the convergence properties of sequences of probability measures comes up in many applications (for example in the study of approximations of probability measures and stochastic inverse problems). This is only the case that the gap penalty is 0, but already the project can be only on the LCS. Eﬃcient algorithms computing optimal alignments and optimality regions (what are those? F., Optimal Lipschitz extensions and the infinity Laplacian, Calc. Miroslav Chlebik Presentation [PDF 309.98KB] Key words: Hausdorff dimension, Lipschitz mappings, rectifiable sets, fractals Recommended modules: Measure and Integration, Functional Analysis References: ! Falconer, K., Fractal geometry: Mathematical Foundations and Applications, John Wiley & Sons Ltd., 1990 if ! consists of two sides of a Reuleaux triangle.) Miroslav Chlebik Presentation [PDF 309.98KB] Key words: curve length, Lipschitz curve, calculus of variations Recommended modules: Functional Analysis, Partial Differential Equations References: ! We also look at convergence properties of some sampling techniques. Several other problems are analysed in the area with similar techniques. This project consists of learning about some of the important metrics on the space of probability measures (for example: Hellinger, Prokhorov and Wasserstein), and studying the relationship between them. (3) The problem is a window to an area of mathematics called “Algebraic Statistics” that can serve as an umbrella to the thesis. At the end of summer term, each student will give a presentation to an audience of two members of staff on the topic of their dissertations.This will be graded and worth 10% of the course grade.For full information on the BSc and MMath Final Year Projects, please see this page. These topics are also offered to students in MSc Mathematics.