Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Supervised and weakly supervised counting-by-segmentation: the fluorescent microscopy use case

This thesis focuses on automating the time-consuming task of manually counting activated neurons in fluorescent microscopy images, which is used to study the mechanisms underlying torpor. The traditional method of manual annotation can introduce bias and delay the outcome of experiments, so the author investigates a deep-learning-based procedure to automatize this task. The author explores two of the main convolutional-neural-network (CNNs) state-of-the-art architectures: UNet and ResUnet family model, and uses a counting-by-segmentation strategy to provide a justification of the objects considered during the counting process. The author also explores a weakly-supervised learning strategy that exploits only dot annotations. The author quantifies the advantages in terms of data reduction and counting performance boost obtainable with a transfer-learning approach and, specifically, a fine-tuning procedure. The author released the dataset used for the supervised use case and all the pre-training models, and designed a web application to share both the counting process pipeline developed in this work and the models pre-trained on the dataset analyzed in this work.