arXiv research

Paper finds local normal forms for wavefronts in flat coordinates.

Microlocal analysis provides deep insight into singularity structures and is often crucial for solving inverse problems, predominately, in imaging sciences. Of particular importance is the analysis of wavefront sets and the correct extraction of those. In this paper, we introduce the first algorithmic approach to extra…

Abstract: Characterizes frontals and wavefronts with formulas.

We give a classification of generic bifurcations of intersections of wavefronts generated by different points of a hypersurface with or without boundaries.

Study on curvature invariants near singularities of wavefronts.

We introduce the notion of reticular Legendrian unfoldings in order to investigate stabilities of bifurcations of wavefronts generated by a hypersurface germ with a boundary, a corner, or an r-corner in a smooth n dimensional manifold. We define several stabilities of reticular Legendrian unfoldings and prove that they…

We investigate genericities of reticular Lagrangian maps and reticular Legendrian maps in order to give generic classifications of caustics and wavefronts generated by a hypersurface germ without or with a boundary in a smooth manifold.

We extend the notion of reticular Legendrian unfoldings in order to investigate multi-time bifurcations of wavefronts generated by an r-corner. We give a classification list of generic and stable bifurcations with two time parameter and give all generic figures in the plane and the space.

Motivated by the study of wave fronts in anisotropic media, we propose an incidence geometry of anisotropic spheres in a Finsler-Minkowski space. An anisotropic version of the Laguerre functional is considered. In some circumstances, this functional can be used to determine that two wavefronts observed at distinct time…

Recent advances in twistor theory are applied to geometric optics in ${\Bbb{R}}^3$. The general formulae for reflection of a wavefront in a surface are derived and in three special cases explicit descriptions are provided: when the reflecting surface is a plane, when the incoming wave is a plane and when the incoming w…

Conflict sets are loci of intersecting wavefronts emanating from $l$ different surfaces. We show that generically conflict sets are Legendrian: locally they admit the structure of wavefronts. Simple stable singularities for this problem in $\R^n$ occur when $0 \leq n-l \leq 4$. Other related sets, such as kite curves a…

We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destablization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of …

Study of fastest paths in anisotropic media via Finsler geometry.

Study finds rigid submanifolds in spacelike waves under specific conditions.

A new model predicts wildfire spread with wind and slope effects.

A systematic geometric theory for the ultradifferentiable (non-quasianalytic and quasianalytic) wavefront set similar to the well-known theory in the classic smooth and analytic setting is developed. In particular an analogue of Bony's Theorem and the invariance of the ultradifferentiable wavefront set under diffeomorp…

Study null surfaces of pseudo-spherical curves in anti-de Sitter space.

In this paper, we consider solutions of the backward heat equation with Ricci flow on manifolds as a type of infinite dimensional limit of solutions of a wave equation on a larger manifold with an analysis of wavefront set. Specifically, the projection of the solution of the wave equation $ \left(\frac{2t}{N} \cdot \fr…

We study diffeologies on locally convex spaces and their application to smooth multiplication of distributions.

We consider the four-dimensional nonholonomic distribution defined by the 4-potential of the electromagnetic field on the manifold. This distribution has a metric tensor with the Lorentzian signature $(+,-,-,-)$, therefore, the causal structure appears as in the general relativity theory. By means of the Pontryagin's m…

Transforms uniquely determine Higgs fields on real-analytic manifolds.

Let $X$ be a compact manifold with boundary. Suppose that the boundary is fibred, $φ:\pa X\longrightarrow Y,$ and let $x\in\CI(X)$ be a boundary defining function. This data fixes the space of `fibred cusp' vector fields, consisting of those vector fields $V$ on $X$ satisfying $Vx=O(x^2)$ and which are tangent to the f…

We extend to the $n$-dimensional case a recent theorem establishing the validity of the Huygens' envelope principle for wavefronts in Finsler spaces. Our results have direct applications in analogue gravity models, for which the Fermat's principle of least time naturally gives origin to an underlying Finslerian geometr…

In this paper, we present a machine learning approach for estimating the number of incident wavefronts in a direction of arrival scenario. In contrast to previous works, a multilayer neural network with a cross-entropy objective is trained. Furthermore, we investigate an online training procedure that allows an adaptio…

Study of billiards in sub-Finsler geometry, including unusual orbits.

Wave propagation framework using cone structures and observers' vector fields.

Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.

We use Machine Learning (ML) and system identification validation approaches to estimate neural network models of large-scale Deformable Mirrors (DMs) used in Adaptive Optics (AO) systems. To obtain the training, validation, and test data sets, we simulate a realistic large-scale Finite Element (FE) model of a faceplat…

Study Legendrian surfaces using N-graphs and flag moduli.

We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differentiable structures. We construct the optimal fiberwise open subset of its tangent bundle and show that the boundary distance map determines the Finsler function in this set but not in …

Study of evolutoids and involutoids of convex curves in 2D space forms.