Topics in multidimensional persistence

A multiparameter persistence module is a representation of the lattice quiver Nd, where maps along all squares commute. When d = 1, Gabriel’s theorem applies and these modules admit interval decompositions, allowing us to classify one dimen- sional persistence modules through their associated barcode, a combinatorial invariant first introduced in by Carlsson and Zomorodian. When d > 1, no such classification is possible. In this thesis we study these higher dimensional persistence modules, seeking to over- come their lack of simple classification by defining discrete invariants with as much discriminative power as possible. The thesis is composed of three parts. First, we give an in depth analysis of barcode bases. These are bases of one-dimensional persistence modules that realise the interval decom- position given by Gabriel’s theorem. We present a novel algorithm that computes these barcode bases, and give theoretical results that characterise the set of barcode bases of a given persistence module. This allows for a decomposition results of certain types of ladder persistence modules. We generalise all these results to zigzag persistence. Second, we consider Harder-Narasimhan filtrations for quiver representations and define the skyscraper invariant, a novel discrete invariant for multidimensional persistence which is finer than the rank invariant. We further show the skyscraper invariant can be refined to create a complete invariant on certain families of ladder persistence modules. Finally, we discuss computation methods for the skyscraper invariant. We exhibit an algorithm that computes the skyscraper invariant for ladder persistence modules. This is done by leveraging the decomposition result for ladder persistence modules from the first chapter. In doing so, we introduce the ladder invariant, which is computable and more discriminative than the rank invariant. It coincides with the skyscraper invariant on ladder persistence modules and is non-comparable to the skyscraper invariant in general. Algorithms from the first chapter are given as pseudo-code. These were later implemented as a python package and we give an overview of this package in the appendix

Microspore embryogenesis in barley: anther pre-treatment stimulates plant defence gene expression

Microspore embryogenesis (ME) is a process in which the gametophytic pollen programme of the microspore is reorientated towards a new embryo sporophytic programme. This process requires a stress treatment, usually performed in the anther or isolated microspores for several days. Despite the universal use of stress to induce ME, very few studies have addressed the physiological processes that occur in the anther during this step. To further understand the processes triggered by stress treatment, we followed the response of anthers by measuring the expression of stress-related genes in two barley (Hordeum vulgare L.) cultivars differing in their ME response. Genes encoding enzymes involved in oxidative stress (glutathione-S-transferase, GST; oxalate oxidase, OxO), in the synthesis of jasmonic acid (13-lipoxygenase, Lox; allene oxide cyclase, AOC; allene oxide synthase, AOS) and in the phenylpropanoid pathway (phenylalanine ammonia lyase, PAL), as well as those encoding PR proteins (Barwin, chitinase 2b, Chit 2b; glucanase, Gluc; basic pathogenesis-related protein 1, PR1; pathogenesis-related protein 10, PR10) were up-regulated in whole anthers upon stress treatment, indicating that anther perceives stress and reacts by triggering general plant defence mechanisms. In particular, both OxO and Chit 2b genes are good markers of anther reactivity owing to their high level of induction during the stress treatment. The effect of copper sulphate appeared to limit the expression of defence-related genes, which may be correlated with its positive effect on the yield of microspor

Baume-les-Dames – Bois de Framont, site carrier

Les découvertes effectuées sur le site carrier confirment la présence de deux corps de métier : les carriers pour l’extraction de la pierre et les tailleurs de pierre pour le dégrossissage des gros blocs, afin d’alléger au maximum le poids à descendre en la cité. La présence d’outils de précision, tels que compas de report et de marquage, petits burins et outils de tracé, renforcent l’hypothèse de petits blocs sculptés achevés sur le lieu, comme la clé de voûte. Après décapage de la couche ré..

Baume-les-Dames – Ancienne tuilerie

Le site de la tuilerie des Sombeveaux a été localisé au nord – nord-ouest à la limite de la commune de Baume-les-Dames, à 200 m du tracé de la voie romaine à hauteur de Fontenotte. En 1653 (AC), des tuiliers français étaient autorisés à s’installer en ces lieux. Dans le bois de feuillus, les découvertes ont révélé les carrières, les gisements de marne et de terre grasse et, enfouis sous un gros humus, une fontaine, les structures d’un premier bâtiment composé de deux salles, un amas important..

A Bijective Census of Nonseparable Planar Maps

AbstractBijections are obtained between nonseparable planar maps and two different kinds of trees: description trees and skew ternary trees. A combinatorial relation between the latter and ternary trees allows bijective enumeration and random generation of nonseparable planar maps. The involved bijections take account of the usual combinatorial parameters and give a bijective proof of formulae established by Brown and Tutte. These results, combined with a bijection due to Goulden and West, give a purely combinatorial enumeration of two-stack-sortable permutations

Commune de Baume-les-Dames

Des prospections ont eu lieu sur l’ensemble de la commune de Baume-les-Dames et en particulier aux alentours de la carrière des Combes (époques moderne et contemporaine). Ce site est constitué de deux exploitations accolées où plusieurs fronts de taille sont visibles. Deux grandes structures en pierres sèches appareillées sont liées au travail d’extraction de la pierre. L’avancement des travaux de défrichage, le réappareillage de nombreuses structures, la mise au jour d’éléments d’architectur..