arXiv research

Research on random matrices and machine learning consistency.

Approximates smooth surfaces using Laguerre geometry meshes.

Approximates surfaces using Laguerre geometry with spherical faces.

Laguerre geometry of surfaces in $\R^3$ is given in the book of Blaschke [1], and have been studied by E.Musso and L.Nicolodi [5], [6], [7], B. Palmer [8] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in $\R^n$. For any umbilical free hypersurface $x: M\to\R^n$ with non-zero …

A Laguerre minimal surface is an immersed surface in the Euclidean space being an extremal of the functional \int (H^2/K - 1) dA. In the present paper, we prove that the only ruled Laguerre minimal surfaces are up to isometry the surfaces R(u,v) = (Au, Bu, Cu + D cos 2u) + v (sin u, cos u, 0), where A, B, C, D are fixe…

The study of Laguerre isotropic hypersurfaces with rigidity and isoparametric properties.

In this paper we show that a Dupin hypersurface with constant Möbius curvatures is Möbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat L…

This paper connects Laguerre minimal surfaces to Weierstrass representations.

This exposition gives an introduction to the theory of surfaces in Laguerre geometry and surveys some results, mostly obtained by the authors, about three important classes of surfaces in Laguerre geometry, namely L-isothermic, L-minimal, and generalized L-minimal surfaces. The quadric model of Lie sphere geometry is a…

Spinor representation in isotropic space via Laguerre geometry.

A Laguerre geometric local characterization is given of L-minimal surfaces and Laguerre deformations (T-transforms) of L-minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential. This is used to prove that, via their Laguerre Gauss maps, the T-transforms of L-minimal isotherm…

The paper introduces new methods for Asian option pricing using Laguerre quadrature.

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…

A method is presented to integrate explicitly a class of checkerboard IC-nets.

We use the isotropic projection of Laguerre geometry in order to establish a correspondence between plane curves and null curves in the Minkowski $3$-space. We describe the geometry of null curves (Cartan frame, pseudo-arc parameter, pseudo-torsion, pairs of associated curves) in terms of the curvature of the correspon…

The paper approximates CARMA models for option pricing.

In a recent significant advance, using Laguerre series, the valuation of Asian options has been reduced by Dufresne to computing the negative moments of Yor's accumulation processes. For these he has given functional recursion rules whose probabilistic structure has been the object of intensive recent studies of Yor an…

We develop Fourier methods to expand translation-invariant kernels.

New representations for discrete surfaces derived from dual transforms.

We give an elaborated treatment of discrete isothermic surfaces and their analogs in different geometries (projective, Möbius, Laguerre, Lie). We find the core of the theory to be a novel projective characterization of discrete isothermic nets as Moutard nets. The latter belong to projective geometry and are nets with …

Researchers found a new exact solution for pricing Aunt Michaela options using modified Black-Scholes equation.

Random matrix ensembles yield uniform distributions on manifolds.

We study an analogue of the classical Bianchi-Darboux transformation for L-isothermic surfaces in Laguerre geometry, the Bianchi-Darboux transformation. We show how to construct the Bianchi-Darboux transforms of an L-isothermic surface by solving an integrable linear differential system. We then establish a permutabili…

Recalls and refines the concept of algebraically rectifiable curves.

New methods improve neural connectivity analysis at submillisecond timescales.

We derive the Do and Norbury recursion formula for the one-loop mean of an irregular spectral curve from a variant of replica method by Brezín and Hikami. We express this recursion in special times in which all terms $W_1^{(g)}$ of the genus expansion of the one-loop mean are polynomials. We find a generalization of th…

The study constructs non-planar nets and webs on quadrics and Minkowski space.

Using Maple, we compute some analytical solutions of a modified Black-Scholes equation, recently proposed, in the case of the European put option. We show that the modified Black-Scholes equation with the European put option is exactly solvable in terms of associated Laguerre polynomials. We make some numerical experim…

Using the gauge theoretic approach for Lie applicable surfaces, we characterise certain subclasses of surfaces in terms of polynomial conserved quantities. These include isothermic and Guichard surfaces of conformal geometry and $L$-isothermic surfaces of Laguerre geometry. In this setting one can see that the well kno…

We propose a method for pricing American options whose pay-off depends on the moving average of the underlying asset price. The method uses a finite dimensional approximation of the infinite-dimensional dynamics of the moving average process based on a truncated Laguerre series expansion. The resulting problem is a fin…

Study uses orthogonal polynomials to solve option pricing equations.

In the spirit of Klein's Erlangen Program, we investigate the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix which constitute associated Plücker coordinates. Particular emphasis is put on the restriction to Lie circle g…

New parametrization handles sextactic points on closed curves.

We consider congruences of straight lines in a plane with the combinatorics of the square grid, with all elementary quadrilaterals possessing an incircle. It is shown that all the vertices of such nets (we call them incircular or IC-nets) lie on confocal conics. Our main new results are on checkerboard IC-nets in the p…

The article describes how decorations on hyperbolic surfaces lead to unique tessellations and decompositions.

We simplify matrix computations for block matrices, especially useful for covariance and correlation matrices.

Minimal submanifolds in matrix spaces proven for specific ranks.

Study differential properties of symmetric real matrices with a specific metric.

Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.

Study on random matrices in deep neural networks using Gaussian data.

Study on random matrices in deep neural networks with IID entries.

Financial markets analyzed by reducing correlation matrix complexity.

Study on estimating Monge matrices with statistical methods.

Study isotropy groups for complex orthogonal and skew-symmetric matrices.

We introduce Clique Matrices as an alternative representation of undirected graphs, being a generalisation of the incidence matrix representation. Here we use clique matrices to decompose a graph into a set of possibly overlapping clusters, de ned as well-connected subsets of vertices. The decomposition is based on a s…

Study of strictly accretive matrices using Finsler geometry.

New discretizations of principal curvature lines discovered.

Minimal spectral radii found for specific matrix types.