Search results for "Tangent"
showing 10 items of 123 documents
On Limiting Fréchet ε-Subdifferentials
1998
This paper presents an e-sub differential calculus for nonconvex and nonsmooth functions. We extend the previous work by Jofre et all to the case where the functions are lower semicontinuous instead of locally Lipschitz.
1980
A phenomenological theory of the Phase Distribution Chromatography (PDC)-separation effect is outlined and a theoretical equation for the measured PDC-calibration curves is given. Assuming a reversible-thermodynamical equilibrium in the polystyrene-PDC-column, only a relatively small part of the measured PDC-calibration curves could be explained: namely those running below their tangents. In order to explain the whole sigmoidal shape of the experimental curves, a theory of steady state in the system sol/gel was developed assuming deformation of the polymer coil near the gel front due to the stress related to the velocity gradient. The resulting dynamical flow-equilibrium differs highly from…
Constant angle surfaces in 4-dimensional Minkowski space
2019
Abstract We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant complex angle with respect to a fixed spacelike plane. This notion is the natural Lorentzian analogue of the notion of constant angle surfaces in 4-dimensional Euclidean space. We prove that these surfaces have vanishing Gauss and normal curvatures, obtain representation formulas for the constant angle surfaces with regular Gauss maps and construct constant angle surfaces using PDE’s methods. We then describe their invariants of second order and show…
Orientation of a Surface
2012
We know from Chap. 4 that in order to evaluate the flux of a vector field across a regular surface S, we need to choose a unit normal vector at each point of S in such a way that the resulting vector field is continuous. For instance, if we submerge a permeable sphere into a fluid and we select the field of unit normal outward vectors on the sphere, then the flux of the velocity field of the fluid across the sphere gives the amount of fluid leaving the sphere per unit time. However, if we select the field of unit normal inward vectors on the sphere, then the flux of the velocity field of the fluid across the sphere gives the amount of fluid entering the sphere per unit time (which is the ne…
Surfaces with Boundary
2012
One of the objectives of this book is to obtain a rigorous proof of a version of Green’s formula for compact subsets of \(\mathbb{R}^2\) whose topological boundary is a regular curve of class C 2. These sets are typical examples of what we will call regular 2-surfaces with boundary in \(\mathbb{R}^2\). The analogous three-dimensional example would consist of a compact set of \(\mathbb{R}^3\) whose topological boundary is a regular surface of class C 2. The following example is perhaps instructive.
Particle Filtering for Tracking in 360 Degrees Videos Using Virtual PTZ Cameras
2019
360 degrees cameras are devices able to record spherical images of the environment. Such images can be used to generate views of the scene by projecting the spherical surface onto planes tangent to the sphere. Each of these views can be considered as the output of a virtual PTZ (vPTZ) camera with specific pan, tilt and zoom parameters. This paper proposes to formulate the visual tracking problem as the one of selecting, at each time, the vPTZ camera to foveate on the target from the unlimited set of simultaneously generated vPTZ camera views. Assuming that the selected vPTZ camera is a stochastic variable, the paper proposes to model the posterior distribution of the underlying stochastic p…
Spacelike energy of timelike unit vector fields on a Lorentzian manifold
2004
On a Lorentzian manifold, we define a new functional on the space of unit timelike vector fields given by the L2 norm of the restriction of the covariant derivative of the vector field to its orthogonal complement. This spacelike energy is related with the energy of the vector field as a map on the tangent bundle endowed with the Kaluza–Klein metric, but it is more adapted to the situation. We compute the first and second variation of the functional and we exhibit several examples of critical points on cosmological models as generalized Robertson–Walker spaces and Godel universe, on Einstein and contact manifolds and on Lorentzian Berger’s spheres. For these critical points we have also stu…
A historical account on characterizations ofC1-manifolds in Euclidean spaces by tangent cones
2014
Abstract A historical account on characterizations of C 1 -manifolds in Euclidean spaces by tangent cones is provided. Old characterizations of smooth manifold (by tangent cones), due to Valiron (1926, 1927) and Severi (1929, 1934) are recovered; modern characterizations, due to Gluck (1966, 1968) and Tierno (1997) are restated. All these results are consequences of the Four-cones coincidence theorem due to [1] .
Classification and stable classification of manifolds: some examples
1984
Kähler manifolds with split tangent bundle
2006
( Varietes kahleriennes a fibre tangent scinde). - On etudie dans cet article les varietes kahleriennes compactes dont le fibre tangent se decompose en somme directe de sous-fibres. En particulier, on montre que si le fibre tangent se decompose en somme directe de sous-fibres en droites, alors la variete est uniformisee par un produit de courbes. Les methodes sont issues de la theorie des feuilletages de (co)dimension 1.