The authors consider a generic configuration of regions, consisting of a collection of distinct compact regions \{ \Omega_i\} in \mathbb{R}^{n+1} which may be either regions with smooth boundaries disjoint from the others or regions which meet on their piecewise smooth boundaries \mathcal{B}_i in a generic way. They introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions' individual shapes and geometric properties as well as the "positional geometry" of the collection. The linking structure extends in a minimal way the individual "skeletal.

The authors consider a generic configuration of regions, consisting of a collection of distinct compact regions in which may be either regions with smooth boundaries disjoint from the others or regions which meet on their piecewise smooth boundaries in a generic way. They introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions' individual shapes and geometric properties as well as the “positional geometry” of the collection. The linking structure extends in a minimal way the individual “skeletal structures” on each of the regions. This allows the authors to significantly extend the mathematical methods introduced for single regions to the configuration of regions.

Based on the second Women in Shape (WiSH) workshop held in Sirince, Turkey in June 2016, these proceedings offer the latest research on shape modeling and analysis and their applications. The 10 peer-reviewed articles in this volume cover a broad range of topics, including shape representation, shape complexity, and characterization in solving image-processing problems. While the first six chapters establish understanding in the theoretical topics, the remaining chapters discuss important applications such as image segmentation, registration, image deblurring, and shape patterns in digital fabrication. The authors in this volume are members of the WiSH network and their colleagues, and most were involved in the research groups formed at the workshop. This volume sheds light on a variety of shape analysis methods and their applications, and researchers and graduate students will find it to be an invaluable resource for further research in the area.

This volume consists of ten articles which provide an in-depth and reader-friendly survey of some of the foundational aspects of singularity theory. Authored by world experts, the various contributions deal with both classical material and modern developments, covering a wide range of topics which are linked to each other in fundamental ways. Singularities are ubiquitous in mathematics and science in general. Singularity theory interacts energetically with the rest of mathematics, acting as a crucible where different types of mathematical problems interact, surprising connections are born and simple questions lead to ideas which resonate in other parts of the subject. This is the first volume in a series which aims to provide an accessible account of the state-of-the-art of the subject, its frontiers, and its interactions with other areas of research. The book is addressed to graduate students and newcomers to the theory, as well as to specialists who can use it as a guidebook.

In this article the authors study Hamiltonian flows associated to smooth functions R R restricted to energy levels close to critical levels. They assume the existence of a saddle-center equilibrium point in the zero energy level . The Hamiltonian function near is assumed to satisfy Moser's normal form and is assumed to lie in a strictly convex singular subset of . Then for all small, the energy level contains a subset near , diffeomorphic to the closed -ball, which admits a system of transversal sections , called a foliation. is a singular foliation of and contains two periodic orbits and as binding orbits. is the Lyapunoff orbit lying in the center manifold of , has Conley-Zehnder index and spans two rigid planes in . has Conley-Zehnder index and spans a one parameter family of planes in . A rigid cylinder connecting to completes . All regular leaves are transverse to the Hamiltonian vector field. The existence of a homoclinic orbit to in follows from this foliation.

This article is concerned with the maximal accretive realizations of geometric Kramers-Fokker-Planck operators on manifolds with boundaries. A general class of boundary conditions is introduced which ensures the maximal accretivity and some global subelliptic estimates. Those estimates imply nice spectral properties as well as exponential decay properties for the associated semigroup. Admissible boundary conditions cover a wide range of applications for the usual scalar Kramer-Fokker-Planck equation or Bismut's hypoelliptic laplacian.