Angular Gaussian and Cauchy estimation

This paper proposes a unified treatment of maximum likelihood estimates of angular Gaussian and multivariate Cauchy distributions in both the real and the complex case. The complex case is relevant in shape analysis. We describe in full generality the set of maxima of the corresponding log-likelihood functions with respect to an arbitrary probability measure. Our tools are the convexity of log-likelihood functions and their behaviour at infinity

Barycentre sur le bord de SL(3, R)/SO(3, R)

1995 wiederholten Besson, Courtois und Gallot [BCG] den Beweis eines bekannten Satzes, nämlich, dass jedes positive Mass µ auf dem Rand M(∞) eines lokal symmetrischen Raumes M vom Rang 1 einen Schwerpunkt p innerhalb des Raumes besitzt, der als Lösung p der Gleichung: beschrieben werden kann, wobei Bθ die Busemann Funktion für θ ∈ M(∞) ist. Wenn der Rang höher ist, ist es klar, dass der Schwerpunkt nicht immer existieren kann, weil er zum Beispiel auf euklidischen Untermannigfaltigkeiten, der sogenannten Flachs, nicht existiert. Nach der präzisen Definition des Schwerpunktes im Kapitel 3 folgt im Kapitel 4 die Definition des Cauchy Randes C, als einen Teil des üblichen Randes von SL(3,R)/SO(3,R), auf dem wir einen Schwerpunkt erwarten können. Im Kapitel 4 wird die Existenz und Eindeutigkeit des Schwerpunktes für ein diskretes Mass, das heisst für eine endliche Menge von Punkten auf dem Cauchy Rand von SL(3,R)/SO(3,R) bewiesen. Dazu muss noch eine schwache zusätzliche Bedingung hinzugefügt werden: Wenn im Fall von Rang 1 symmetrischen Räumen einen diskreten Mass betrachtet wird, so muss verlangt werden, dass mindestens drei Punkte vorhanden sind. Diese Bedingung ist auf dem Fall höheren Ranges zu verallgemeinern. Punkte die diese Bedingung erfüllen werden dann gut verteilte Punkte genannt. Im Kapitel 6 wird ein Algorithmus beschrieben, um diesen Schwerpunkt zu rechnen. Ein Programm wird in den Fällen M = H2 und M = SL(3,R)/SO(3,R) gegeben, und ein Paar Beispiele vorgestellt. Eine mögliche Beziehung zur Statistik wird im Kapitel 7 vorgeschlagen; es wird eine Beziehung zwischen dem Schwerpunkt einer Familie von Punkten in R bzw. R2 und ihrem Schätzer in der Familie der Cauchy Verteilungen aufgezeigt.In 1995, Besson, Courtois and Gallot [BCG] repeated the proof of a theorem that stated that any positive measure µ on the boundary M(∞) of a locally symmetric space M with rank 1 admits a unique center of mass as the solution of the equation: where Bθ is the Busemann function for θ ∈ M(∞). In the higher rank case it is obvious that the center of mass does not exist anymore. As an example we know that the center of mass on an Euclidean manifold, called a flat, does not exist. After the precise definition of the center of mass in chapter 3 the Cauchy boundary C is defined in chapter 4. The Cauchy boundary is a part of the usual boundary of SL(3,R)/SO(3,R) on which we can try to prove the existence of a center of mass: The existence and uniqueness of the center of mass for a discrete measure, that means for points, on the Cauchy boundary of SL(3,R)/SO(3,R) is given in chapter 5. We must add a slight restriction: If we consider a discrete measure on a rank 1 symmetric space we must have at least three points. In the higher rank case we generalize somewhat this condition. Points that satisfy this condition are called well-spread points. In chapter 6 there is an algorithm to compute that center of mass. The code is given for the cases where M = H2 and M = SL(3,R)/SO(3,R), as well as a list of examples. Some possible relation to statistics is given in chapter 7. More precisely we relate the center of mass of a family of points in R resp. R2 to an estimator in the family of Cauchy densities

Angular Gaussian and Cauchy estimation

This paper proposes a unified treatment of maximum likelihood estimates of angular Gaussian and multivariate Cauchy distributions in both the real and the complex case. The complex case is relevant in shape analysis. We describe in full generality the set of maxima of the corresponding log-likelihood functions with respect to an arbitrary probability measure. Our tools are the convexity of log-likelihood functions and their behaviour at infinity.Multivariate Cauchy Angular Gaussian Directional and shape analysis Maximum likelihood Differential geometry Equivariance Geodesics Symmetric spaces

Timing of the floral induction in spinach

A general outline of the cellular and metabolic events linked to floral induction in spinach, a LDP, will be presented. A special emphasis will be put on the chronology of these events in the meristem, in the leaves and in the petioles. The vegetative and the floral states were first characterized by time series studies including electron microscopy, cytochemical and biochemical techniques. The development of the induced state was then kinetically followed. Various inhibitors and promoters of the flowering process have also been used. According to this experimental protocol, it was shown that the first sign of floral evocation in the meristem is an activation of the pentose phosphate pathway (PPP) which occurs three hours after the critical photoperiod has been reached. The first ultrastructural events (budding and fragmentation of the plastids) are seen only six hours later. In the leaves, newly synthesized proteins, which are specific for the induced state, are detectable nine hours after the critical photoperiod. Earlier physiological markers have been identified. Among them, the R-FR photocontrol of a peroxidase activity is switched to its specific floral response after three hours, while changes in energy metabolism (adenine and pyridine nucleotides ratios) already occur after one hour. In the petioles, the glucose level is immediately and steadily increased and reaches its maximum value when the PPP is activated in the meristem. Other results have shown that the signal transmitted from the leaves to the meristem can be slowed down by a LiCl pretreatment, suggesting that membrane permeability changes could be implied in floral induction. It can, therefore, be proposed that most of the earliest events of the transition to the floral state are rapid biophysical and biochemical modifications of the structures supporting vegetative growth. Changes in hormonal balances, though still not studied, are also considered as part of these early events which finally result in gene activation, the ultimate event of flower induction

Flowering and leaf-shoot interactions

Three aspects of the flowering process are presented and discussed: the activity changes in the shoot apex, and the role of leaves from the point of view of energy transduction and signal production. A few markers are proposed to detect the early transition from the vegetative to the floral state. Floral induction seems to be dependent on changes in membrane properties and on communication between the different parts of the plant. An hypothesis is proposed which suggests that the transformation of plasma membranes is one of the components of the basic mechanism of floral induction in the leaves

Le mécanisme de l'induction florale

The mechanism of flowering induction.- We present different aspects of the flowering process: the activity of shoot apex, the role of energy and signais in leaves (macro-functions). A few number of markers are proposed to detect the transition from the vegetative to the floral state. The flowering seems to be essentially depending on membranes properties. At the moment of induction the plasmalemma is modified