Related papers: Fingerprints, lemniscates and quadratic differenti…
We study Yamabe-type equations on the product of two spheres $(S^n \times S^n, G_\delta)$, where $G_\delta$ is a family of Riemannian metrics parametrized by $\delta > 0$. Using bifurcation theory and isoparametric functions, we establish…
For a non-constant complex rational function $P$, the lemniscate of $P$ is defined as the set of points $z\in \mathbb C$ such that $\vert P(z)\vert =1$. The lemniscate of $P$ coincides with the set of real points of the algebraic curve…
Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic.…
The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the…
Jordan analytic curves which are invariant under rational functions are studied
A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical calculus introduced and developed in that paper to include…
Using fiber bundle theory and conformal mappings, we continuously select a point from the interior of Jordan domains in Riemannian surfaces. This selection can be made equivariant under isometries, and take on prescribed values such as the…
Geometric frameworks for analyzing curves are common in applications as they focus on invariant features and provide visually satisfying solutions to standard problems such as computing invariant distances, averaging curves, or registering…
In this paper we introduce a graded bracket of forms on multicontact manifolds. This bracket satisfies a graded Jacobi identity as well as two different versions of the Leibniz rule, one of them being a weak Leibniz rule, extending the…
We derive the equations of chains for path geometries on surfaces by solving the equivalence problem of a related structure: sub-Riemannian geometry of signature $(1,1)$ on a contact 3-manifold. This approach is significantly simpler than…
A new approach is suggested to quantum differential calculus on certain quantum varieties. It consists in replacing quantum de Rham complexes with differentials satisfying Leibniz rule by those which are in a sense close to Koszul complexes…
Many innovative applications require establishing correspondences among 3D geometric objects. However, the countless possible deformations of smooth surfaces make shape matching a challenging task. Finding an embedding to represent the…
A new approach to the analytic theory of difference equations with rational and elliptic coefficients is proposed. It is based on the construction of canonical meromorphic solutions which are analytical along "thick paths". The concept of…
The aim of this article is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of the high-dimensional Poisson-Neumann partial differential equations (PDEs) with Neumann boundary conditions.…
The pattern avoidance problem seeks to construct a set with large fractal dimension that avoids a prescribed pattern, such as three term arithmetic progressions, or more general patterns, such as finding a set whose Cartesian product avoids…
We derive estimates of the Hessian of two smooth functions defined on Grassmannian manifold. Based on it, we can derive curvature estimates for minimal submanifolds in Euclidean space via Gauss map. In this way, the result for Bernstein…
We introduce a framework on dual complexes for studying Arnold-type invariants of immersed curves and immersed surfaces via local finite-difference structures associated with Alexander numberings. For generic immersed plane curves and…
We generalize a previous result by Fabricius-Bjerre from curves in $\mathbb R^2$ to curves in $\mathbb R P^2$. Applied to the case of real algebraic curves, this recovers the signed count of bitangents of quartics introduced by Larson-Vogt…
One of the most challenging problems in fingerprint recognition continues to be establishing the identity of a suspect associated with partial and smudgy fingerprints left at a crime scene (i.e., latent prints or fingermarks). Despite the…
As an advanced research topic in forensics science, automatic shoe-print identification has been extensively studied in the last two decades, since shoe marks are the clues most frequently left in a crime scene. Hence, these impressions…