Related papers: Numerical approximations to extremal metrics on to…
Optical surfaces represented by second-degree polynomials (quadratic or conics) are ubiquitous in optics. We revisit the equations of the conic shapes in the context of grazing incidence optics, gathering together the curves commonly used…
We study deterministic online embeddings of metrics spaces into normed spaces and into trees against an adaptive adversary. Main results include a polynomial lower bound on the (multiplicative) distortion of embedding into Euclidean spaces,…
The indicatrix or curvature ellipse and the characteristic curve of a surface in $\mathbf R^4$ are presented, as well as the projective duality connecting them. The characterisation of points in the surfaces as elliptic, parabolic and…
We report on the computation of invariants, covariants, and contravariants of cubic surfaces. All algorithms are implemented in the computer algebra system magma.
We revisit two NP-hard geometric partitioning problems - convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasi-polynomial time algorithms for these problems with improved…
Sharp Moser-Trudinger type inequalities and their extremal functions play an important role in studying nonlinear PDEs and geometry. We establish a new sharp Moser-Trudinger type inequality in the upper half space in two dimensions and…
Extremal length is an important conformal invariant on Riemann surface. It is closely related to the geometry of Teichmuller metric on Teichmuller space. By identifying extremal length functions with energy of harmonic maps from Riemann…
Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of…
In optimal control problems, there exist different kinds of extremals, that is, curves candidates to be solution: abnormal, normal and strictly abnormal. The key point for this classification is how those extremals depend on the cost…
We compute the critical surface for the existence of invariant tori of a family of Hamiltonian systems with two and three degrees of freedom. We use and compare two methods to compute the critical surfaces: renormalization-group…
We prove a correspondence theorem for singular tropical surfaces in real three space, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we…
We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to…
A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the…
The maximum function, on vectors of real numbers, is not differentiable. Consequently, several differentiable approximations of this function are popular substitutes. We survey three smooth functions which approximate the maximum function…
Matrix data sets are common nowadays like in biomedical imaging where the Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) modality produces data sets of 3D symmetric positive definite matrices anchored at voxel positions capturing the…
This note answers extremal questions like: what is the maximum number of edges in a graph of order n, which belongs to some hereditary property. The same question is answered also for the spectral radius and other similar parameters.
This paper is a survey about the Thurston metric on the Teichm\"uller space. The central issue is the constructions of extremal Lipschitz maps between hyperbolic surfaces. We review several constructions, including the original work of…
This paper focuses on computing the directional extremal boundary of a union of equal-radius circles. We introduce an efficient algorithm that accurately determines this boundary by analyzing the intersections and dominant relationships…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…
We determine the gonality and the Clifford index for curves on a compact smooth toric surface. Moreover, it is shown that their gonality are computed by pencils on the ambient surface. From the geometrical view point, this means that the…