arXiv research

Archimedes knew that the area between a parabola and any chord $AB$ on the parabola is four thirds of the area of triangle $ΔABP$ where P is the point on the parabola at which the tangent is parallel to $AB$. We consider whether this property (and similar ones) characterizes parabolas. We present five conditions which …

Archimedes showed that the area between a parabola and any chord $AB$ on the parabola is four thirds of the area of triangle $ΔABP$, where P is the point on the parabola at which the tangent is parallel to the chord $AB$. Recently, this property of parabolas was proved to be a characteristic property of parabolas. With…

Study of parabolas in Funk metric on unit disk.

Archimedes determined the center of gravity of a parabolic section as follows. For a parabolic section between a parabola and any chord $AB$ on the parabola, let us denote by $P$ the point on the parabola where the tangent is parallel to $AB$ and by $V$ the point where the line through $P$ parallel to the axis of the p…

A flat complete causal Lorentzian manifold is called {\it strictly causal} if the past and the future of each its point are closed near this point. We consider strictly causal manifolds with unipotent holonomy groups and assign to a manifold of this type four nonnegative integers (a signature) and a parabola in the con…

It is well known that the area $U$ of the triangle formed by three tangents to a parabola $X$ is half of the area $T$ of the triangle formed by joining their points of contact. In this article, we consider whether this property and similar ones characterizes parabolas. As a result, we present three conditions which are…

It is well known that the area $U$ of the triangle formed by three tangents to a parabola $X$ is half of the area $T$ of the triangle formed by joining their points of contact. In this article, we study some properties of $U$ and $T$ for strictly convex plane curves. As a result, we establish a characterization for par…

At each point in an immersed surface in $\mathbb R^4$ there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. More recently, at the singular point of a corank 1 singular surface in $\mathbb R^3$, a curvature parabola in the normal plane which codifies all the …

New curvature defined for corank 1 singular surfaces in 3D.

We establish a link between Archimedes' method of integration for calculating areas, volumes and centers of mass of segments of parabolas and quadrics of revolution by factorization via the moments of a balance and an integration technique for a particular integrable system, namely Bianchi's Bäcklund transformation for…

A method is proposed to construct spiral curves by inversion of a spiral arc of parabola. The resulting curve is rational of 4-th order. Proper selection of the parabolic arc and parameters of inversion allows to match a wide range of boundary conditions, namely, tangents and curvatures at the endpoints, including thos…

We consider smooth 1-parameter families of plane curves tangent to a semicubic parabola, when the curvature radius of their curves at the tangency point vanishes at the cusp point. We find the $\A$-normal form of these families, their envelopes and local patterns near the cusp. We obtain a new codimension 2 singularity…

The paper classifies surfaces with isotropic circles through each point.

The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains Ω of R^4. When Ω is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which Ω is the complement of a parabola is st…

Alternative finance models from physics for non-equilibrium systems.

In stochastic gradient descent, especially for neural network training, there are currently dominating first order methods: not modeling local distance to minimum. This information required for optimal step size is provided by second order methods, however, they have many difficulties, starting with full Hessian having…

We study the geometry of surfaces in $\mathbb{R}^{4}$ with corank $1$ singularities. For such surfaces the singularities are isolated and at each point we define the curvature parabola in the normal space. This curve codifies all the second order information of the surface. Also, using this curve we define asymptotic a…

The cone projection $f_R(z) = z/(1 + |z|/R)$ maps lines to conic arcs with specific properties.

New method constructs translationally equivariant hyperbolic affine spheres.

The study examines spectral properties of the Laplacian on forms for open Riemannian manifolds.

Unified description of aesthetic curves through self-affinities.

We construct simply connected, complete, non-$CMC$ biconservative surfaces in the $3$-dimensional hyperbolic space $\mathbb{H}^3$ in an intrinsic and extrinsic way. We obtain three families of such surfaces, and, for each surface, the set of points where the gradient of the mean curvature function does not vanish is de…

This paper studies the continuous time mean-variance portfolio selection problem with one kind of non-linear wealth dynamics. To deal the expectation constraint, an auxiliary stochastic control problem is firstly solved by two new generalized stochastic Riccati equations from which a candidate portfolio in feedback for…

The Strong Cosmic Censorship conjecture states that for generic initial data to Einstein's field equations, the maximal globally hyperbolic development is inextendible. We prove this conjecture in the class of orthogonal Bianchi class B perfect fluids and vacuum spacetimes, by showing that unboundedness of certain curv…

Solitons are special polygon midpoints under affine transformations.

Using the complex parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$, we transform minimal surfaces in Euclidean space ${\mathbb{R}}^{3} \subset {\mathbb{R}}^{4}$ to a family of degenerate minimal surfaces in Euclidean space ${\mathbb{R}}^{4}$. Applying our deformation to holomorphic null curves in ${…

Paper classifies conic submanifolds in control systems.

Introduces a new geometry based on difference angles, showing unique properties.

The paper generalizes a mean value theorem for solutions of the ultrahyperbolic equation.

Chinchilla Approach 2 biases neural scaling law estimates, leading to unnecessary compute costs.

A cone projection maps complex plane arcs to conic sections with fixed focus and directrix.

New symmetries discovered in Kepler's orbit family.

The paper studies the convex hull of random points in a triangle, focusing on the asymptotic behavior and phase transitions.