arXiv research

Study examines Hilbert area of inscribed polygons in projective geometry.

Generalized Cauchy's surface area formula to arbitrary submanifolds in R^n.

Formula for Heisenberg group surface areas derived.

The study connects polygon areas and projective structures in 3D space.

Entropy study of geodesic flow on convex projective surfaces.

The paper proves rigidity theorems for area widths of Riemannian manifolds.

Cost overruns in transport infrastructure projects know no geographical limits, overruns are a global phenomenon. Nevertheless, the size of cost overruns varies with location. In the Netherlands, cost overruns appear to be smaller compared to the rest of the world. This paper tests whether Dutch projects perform signif…

We consider a projection from the center of the unit sphere to a tangent space of it, the central projection, and study two area minimizing problems of the image of a closed subset in the sphere. One of the problems is the uniqueness of the tangent plane that minimizes the area for an arbitrary fixed subset. The other …

Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.

We prove an isoperimetric inequality for the second non-zero eigenvalue of the Laplace-Beltrami operator on the real projective plane. For a metric of the unit area this eigenvalue is not greater than 20π. This value is attained in the limit by a sequence of metrics of area one on the projective plane. The limiting met…

A new systolic inequality with a remainder for the real projective plane.

Minimal surfaces in a Riemannian manifold $M^n$ are surfaces which are stationary for area: the first variation of area vanishes. In this paper we focus on surfaces of the topological type of the real projective plane $\R P^2$. We show that a minimal surface $f:\R P^2\to M^3$ which has the smallest area, among those ma…

We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer--Mrowka proof of the Thom conjecture.

The area of a convex projective surface of genus $g\ge 2$ is at least $(g-1)π^2/2+\|τ\|^2/8$ where $τ=(\log t_i)$ is the vector of triangle invariants of Bonahon-Dreyer and $t_i$ are the Fock-Goncharov triangle coordinates.

We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.

We characterize the first min-max width of real projective spaces of any dimension. The width is the minimum area over the Clifford hypersurfaces. We also compute the Morse index of the Clifford hypersurfaces in the complex and quaternionic projective spaces.

Minimal surfaces in a ball have limited area.

In 3-d the average projected area of a convex solid is 1/4 the surface area, as Cauchy showed in the 19th century. In general, the ratio in n dimensions may be obtained from Cauchy's surface area formula, which is in turn a special case of Kubota's theorem. However, while these latter results are well-known to those wo…

The coarea formula is proven for Heisenberg group maps, addressing open questions.

We consider the configuration space of planar $n$-gons with fixed perimeter, which is diffeomorphic to the complex projective space $\mathbb{C}P^{n-2}$. The oriented area function has the minimal number of critical points on the configuration space. We describe its critical points (these are regular stars) and compute …

Optimizes energy of mappings from complex projective spaces.

The paper explores projective structures on curves and their applications in conformal geometry.

Adapts stereographic projection for ellipsoid and elliptic paraboloid.

We present an algorithm for converting an indoor spherical panorama into a photograph with a simulated overhead view. The resulting image will have an extremely wide field of view covering up to 4π steradians of the spherical panorama. We argue that our method complements the stereographic projection commonly used in t…

For the moduli space of unmarked convex $\mathbb{RP}^2$ structures on the surface $S_{g,m}$ with negative Euler characteristic, we investigate the subsets of the moduli space defined by the notions like boundedness of projective invariants, area, Gromov hyperbolicity constant, quasisymmetricity constant etc. These subs…

A formula for triangle area in Deep Sets form.

We introduce a natural stratification of the space of projective classes of measured laminations on a complete hyperbolic surface of finite area. We prove a rigidity result, namely, the group of self-homeomorphisms of the space of projective measured laminations that preserve such a stratification is in general identif…

This work is the first part of a project dealing with an in-depth study of effective techniques used in econometrics in order to make accurate forecasts in the concrete framework of one of the major economies of the most productive Italian area, namely the province of Verona. In particular, we develop an approach mainl…

Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will focus on their algebraic-geometric aspects, explaining how the representation the…

Studies projective geometry and partial differential equations prolongation.

In the present paper we use twistor theory in order to solve two problems related to harmonic maps from surfaces to Euclidean spheres $\mathbb{S}^n$. First, we propose a new approach to isoperimetric inequalities based on energy index. Using this approach we show that for any positive $k$, the $k$-th non-zero eigenvalu…

We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called {\it horizontal} area functional associated to the canonical {\it horizontal} area form. We derive the intrinsic equation in the general case…

Modified neural network enhances unsupervised anomaly detection.

From a fresh data science perspective, this thesis discusses the prediction of coronary artery disease based on genetic variations at the DNA base pair level, called Single-Nucleotide Polymorphisms (SNPs), collected from the Ontario Heart Genomics Study (OHGS). First, the thesis explains two commonly used supervised le…

The pure braid group cannot be realized as area-preserving homeomorphisms.

Nielsen realization problem for the mapping class group $\text{Mod}(S_g)$ asks whether the natural projection $p_g: \text{Homeo}_+(S_g)\to \text{Mod}(S_g)$ has a section. While all the previous results use torsion elements in an essential way, in this paper, we focus on the much more difficult problem of realization of…

New quantity helps map homotopy classes in complex spaces.

Let (M,g) be a compact Riemannian manifold of dimension 3, and let \mathscr{F} denote the collection of all embedded surfaces homeomorphic to \mathbb{RP}^2. We study the infimum of the areas of all surfaces in \mathscr{F}. This quantity is related to the systole of (M,g). It makes sense whenever \mathscr{F} is non-empt…

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be reg…

The paper studies projectively equivalent para-Kaehler metrics in 4D.

Study uniformly differentiable graphs in Carnot groups, proving area formulas.

The abstract discusses a new causal structure on manifolds using paths and points.

New examples of non-bumpy metrics on spheres and projective spaces with multiplicity.

Collision avoidance is a critical task in many applications, such as ADAS (advanced driver-assistance systems), industrial automation and robotics. In an industrial automation setting, certain areas should be off limits to an automated vehicle for protection of people and high-valued assets. These areas can be quaranti…

Paper extends RPD for better handling multiple modalities and non-convexity.

Study on Santaló point for convex bodies in normed spaces.

Study bounds on cusp volumes of alternating knots on surfaces.

New pseudometrics defined on knot spaces based on curve thickness and length.