Cover Page
The handle
http://hdl.handle.net/1887/137985
holds various files of this Leiden University
dissertation.
Author:
Berghout, S.
Stellingen
behorende bij het proefschrift Gibbs processes and Applications
van Steven Berghout
1. Translation-invariant Gibbs measures on the integers and g-measures have
remark-ably similar definitions. [Chapter 2]
2. One can find rather elegant necessary and sufficient conditions for a g-measure to be Gibbs. However, the question which Gibbs measures are g-measures remains
open. [Chapter 2]
3. The strengths and weaknesses of one-sided models seem to translate directly into strengths and weaknesses of the resulting two-sided models. [Chapter 3]
4. The regularity of a factor of a Markov chain depends on the topological structure of the support of the Markov chain, as well as on its transition probabilities.
[Chapter 4]
5. Uniform convergence of finite dimensional conditional probabilities is a sufficient condition for the factor measure to be regular. Existence of a continuous measure disintegration is a local convergence condition that gives uniformity “for free”.
[Chapters 4 & 5]
6. One may try to determine the necessity of the existence of a continuous measure disintegration for regularity of the factor of a Gibbs measure by characterizing points that are not Tjur points. This, however, quickly leads to the same difficulties found in a more traditional approach. [Chapter 5]
7. A ‘natural space’ of potentials introduced by Walters is a fantastic source of exam-ples with incredibly rich probabilistic properties; yet, this class is not widely used in the literature.
P. Walters, Ergod. Th. & Dynam. Sys. 27, 1323–1348 (2007).
2
8. The relation between mathematicians and physicists is often described as that of cats and dogs. Teaching physics and math students to recognise and appreciate strengths of another field would be highly beneficial for the society at large.
Ya.G. Sinai, Bull. Am. Math. Soc. 43, no 4, pp. 563–565 (2006).
9. Combining ideas, techniques, and results from different, even distant, areas of math-ematics is time consuming but almost certainly highly rewarding. A truly multidis-ciplinary research, on the other hand, caries significantly higher risks. Nevertheless, it is often viewed as a ‘potential pot of gold’, and researchers are ‘encouraged to follow the rainbow’. Senior professors join in, seeking Enlightenment in their wan-ing years, and young professors jump on board enthusiastically, hopwan-ing to set the world on fire.
L.A. Baker, Urban Ecosyst. 9, pp. 45–47, (2006).
10. Equivalent mathematical formulations of the same problem do not necessarily give rise to equally efficient algorithmic solutions. A journey from a rigorous mathe-matical theorem to an effective practical algorithm is often long and laborious.
11. Use of ‘Folklore Theorems’ in proofs does not necessarily affect an experienced researcher, but might be a source of great uncertainty to a graduate student.
12. With each transition from academia to industry and back, one comes to an even deeper appreciation of cultural differences, strengths and weaknesses of these com-munities.
