Related papers: Integral geometry and unique continuation principl…
We informally review the construction of spacetime geometries with multifractal and, more generally, multiscale properties. Based on fractional calculus, these continuous spacetimes have their dimension changing with the scale; they display…
Numerical continuation techniques are powerful tools that have been extensively used to identify particular solutions of nonlinear dynamical systems and enable trajectory design in chaotic astrodynamics problems such as the Circular…
It is proposed that the mathematical formalism that is most appropriate for the study of spatially non-integrable cosmological models is the transverse geometry of a one-dimensional foliation (congruence) defined by a physical observer. By…
The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi--Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries…
Recently, classical results on completeness of trajectories of Hamiltonian systems obtained at the beginning of the seventies, have been revisited, improved and applied to Lorentzian Geometry. Our aim here is threefold: to give explicit…
Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic…
This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic…
An attempt is made of giving a self-contained introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex manifolds, spinor and twistor methods, heaven…
In this work we present the general differential geometry of a background in which the space-time has both torsion and curvature with internal symmetries being described by gauge fields, and that is equipped to couple spinorial matter…
This is the second part of a two--part series investigating bifurcation phenomena in autonomous Lagrangian systems and geodesic flows on Finsler and Riemannian manifolds. Building upon the abstract bifurcation theorems established in…
In this paper, we are interested in an inverse problem for the active scalar equations with fractional dissipation on the torus. We perform a second order linearization to relate our model to the linear fractional diffusion equation. Our…
In this paper we adopt the pullback approach to global Finsler geometry. We investigate horizontally recurrent Finsler connections. We prove that for each scalar ($\pi$)1-form $A$, there exists a unique horizontally recurrent Finsler…
In continuation of Part I, we study translative integral formulas for certain translation invariant functionals, which are defined on general convex bodies. Again, we consider local extensions and use these to show that the translative…
We consider an inverse variational problem for the lines of constant curvature in (pseudo-)Euclidean two-, three-, and four-dimensional spaces. The accumulated results are physically meaningful in the case of relativistic mechanics of…
Recent links between Finsler Geometry and the geometry of spacetimes are briefly revisited, and prospective ideas and results are explained. Special attention is paid to geometric problems with a direct motivation in Relativity and other…
A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated differential form formalism developed in math.DG/0605113 and an application to general relativity is given.
We study the spectral theory and inverse problem on asymptotically hyperbolic manifolds. The main subjects are as follows: (1)Location of the essential spectrum. (2)Absence of eigenvalues embedded in the continuous spectrum. (3)Limiting…
Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations…
We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or…
We propose to model the topology of three-dimensional (3D) continua by Yin sets, regular open semianalytic sets with bounded boundary. Our model differs from manifold-based models in that singular points of a 3D continuum, i.e., boundary…