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The mathematical theory underlying Hamiltonian mechanics is called symplectic geometry. So symplectic geometry arose from the roots of mechanics and is seen as one of the most valuable links between physics and mathematics today. Symplectic…
The evolution of 3D visual content calls for innovative methods for modelling shapes based on their intended usage, function and role in a complex scenario. Even if different attempts have been done in this direction, shape modelling still…
This is an attempt to present axioms for Euclidean geometry, aiming at the following goals: to work with geometric notions (thus not merely identify points with pairs of numbers, giving a special status to a particular coordinate system);…
In this paper we will present an ongoing project which aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics. We will argue that this approach provides a geometric semantics for such…
Contact Geometry is an odd dimensional analogue of Symplectic Geometry. This vague idea can actually be formalized in a rather precise way by means of a Symplectic-to-Contact Dictionary. The aim of this review paper is discussing the basic…
The purpose of this paper is to present projective geometry in a synthetic, visual and intuitive style through the central notion of harmonicity which leads to harmonic curves. This presentation includes new results, unpublished proofs of…
We study the geometry of surfaces in $\mathbb R^5$ by relating it to the geometry of regular and singular surfaces in $\mathbb R^4$ obtained by orthogonal projections. In particular, we obtain relations between asymptotic directions, which…
Geometric graphs are a special kind of graph with geometric features, which are vital to model many scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries of translations, rotations, and reflections,…
Text documents are complex high dimensional objects. To effectively visualize such data it is important to reduce its dimensionality and visualize the low dimensional embedding as a 2-D or 3-D scatter plot. In this paper we explore…
This paper lays the foundations for a unified framework for numerically and computationally applying methods drawn from a range of currently distinct geometrical approaches to statistical modelling. In so doing, it extends information…
Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the…
Any procedure applied to data, and any quantity derived from data, is required to respect the nature and symmetries of the data. This axiom applies to refinement procedures and multiresolution transforms as well as to more basic operations…
Soft interfaces can mediate interactions between particles bound to them. The force transmitted through the surface geometry on a particle may be expressed as a closed line integral of the surface stress tensor around that particle. This…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…
The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature.…
In this article, we present a fresh perspective on language, combining ideas from various sources, but mixed in a new synthesis. As in the minimalist program, the question is whether we can formulate an elegant formalism, a universal…
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the…
In this review the foundations of Geometric Quantization are explained and discussed. In particular, we want to clarify the mathematical aspects related to the geometrical structures involved in this theory: complex line bundles, hermitian…
This paper points out the usefulness of the concept of derivation along a map in many problems in Geometry and Physics. In particular it will be shown that this approach allows us to translate the usual concepts arising in Geometrical…
Liquid crystals generally support orientational singularities of the director field known as topological defects. These latter modifiy transport properties in their vicinity as if the geometry was non-Euclidean. We present a state of the…