arXiv research

New method studies moving points on curves using rotating frames.

Mathematical formulas for elliptic curve integrals solve anomaly equations.

This paper uses crypto derivatives data to estimate yield curves for cryptocurrencies.

Sub-Riemannian geometry connects bike paths to mathematical curves.

Explains Conway's tangle trick and its mathematical origins.

A spiral unibike track emerges from a mathematical construction.

Consider two families of closed oriented curves in a d-manifold. At each point of intersecction of a curve of one family with a curve of the other family, form a new closed curve by going around the first curve and then going around the second. Typically, an i-dimensional family and a j-dimensional family will produce …

Survey of contact homology for contact manifolds.

Study evaluates different mathematical models for three case studies using statistical fitting.

Research connects physics and math through ceramic art of Riemann surfaces.

Discretization of curves is an ancient topic. Even discretization of curves with an eye toward differential geometry is over a century old. However there is no general theory or methodology in the literature, despite the ubiquitous use of discrete curves in mathematics and science. There are conflicting definitions of …

Developable ruled surfaces generated by curvature axes of curves.

Clarifies when solutions to stochastic PDEs stay near given subsets.

The paper defines and analyzes $f$-biharmonic $θ_{α}$-slant curves in $\mathcal{S}$-space forms.

We prove that if $ρ\neq1/2$ is a rational number between zero and one, then there is no integer $n>1$ such that $n\tan(πρ)=\tan(nπρ).$ This has interpretations both in the theory of bicycle curves and that of mathematical billiards.

This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in $n$-dimensional euclidean space $\EE^n$ to a surface or a space curve as physical models. These Dirac operators are identified with operators of the Frenet-Serret re…

We constructed physically stable sp2 negatively curved cubic carbon structures which reticulate a Schwarz P-like surface. The method for constructing such crystal structures is based on the notion of the standard realization of abstract crystal lattices. In this paper, we expound on the mathematical method to construct…

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

The work deals with the risk assessment theory. An unitary risk algorithm is elaborated. The algorithm is based on parallel curves. The basic curve of risk is a hyperbolic curve, obtained as a multiplication between the probability of occurrence of certain event and its impact. Section 1 contains the problem formulatio…

Study on curve shortening flow in 3D space curves, showing convexity preservation and avoidance principle.

New metrics on curve spaces improve shape analysis.

The topological recursion of Eynard and Orantin governs a variety of problems in enumerative geometry and mathematical physics. The recursion uses the data of a spectral curve to define an infinite family of multidifferentials. It has been conjectured that, under certain conditions, the spectral curve possesses a non-c…

Automatically explores geometric loci of curves using software networking.

Explains Arnold's J+ invariant for curves, using basic math.

This manuscript, titled Differential Geometry of Curves and Surfaces in Three-dimensional Euclidean Space, is intended for undergraduate students of mathematics and other sciences with the need of use of Euclidean differential geometry. We deal with the differential geometry of curves and surfaces in three dimensions, …

New dynamic curves improve cryptocurrency exchange liquidity.

New approach to QFT divergences uses curved momentum space.

Explains a theorem about curves in the plane.

Machine learning classifies complex geometric patterns with high accuracy.

Classifies shapes of yield curves in the Svensson family.

The paper connects hyperbolic spinors to non-null framed curves in Minkowski 3-space.

This paper introduces a new mathematical formulation and numerical approach for the computation of distances and geodesics between immersed planar curves. Our approach combines the general simplifying transform for first-order elastic metrics that was recently introduced by Kurtek and Needham, together with a relaxatio…

In the recent years, Riemannian shape analysis of curves and surfaces has found several applications in medical image analysis. In this paper we present a numerical discretization of second order Sobolev metrics on the space of regular curves in Euclidean space. This class of metrics has several desirable mathematical …

Adaptive pricing framework for perpetual contracts using liquidity curves and oracles.

We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived in a mathematically…

In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface $S$, and fix a number of points $F$ on its boundary. We ask: how many configurations of disjoint arcs are there on $S$ whose boundary is $F$? We find that thi…

Contact Lie systems analyze integral curves of Hamiltonian vector fields.

Submanifolds in Lorentz-Minkowski space are investigated from various mathematical viewpoints and are of interest also in relativity theory. We define the hyperbolic surface and the de Sitter surface of a curve in the spacelike hypersurface M in the Minkowski 4-space. These surfaces are respectively located in the hype…

These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the second cohomology group of the level p congruence subgroup of the mapping class…

We consider the region of closed timelike curves (CTC's) in three-dimensional flat Lorentz spacetimes. The interest in this global geometrical feature goes beyond the purely mathematical. Such spacetimes may be considered lower-dimensional toy models of sourceless Einstein gravity or cosmology. In particular, our inter…

Overparameterized MLR fits hyper-curves, improving model robustness.

We study 1-parameter families of holomorphic curves with Lagrangian boundary in Calabi-Yau 3-folds. We show that the expected codimension one phenomena can be organized to match the HOMFLYPT skein relations from quantum topology. It follows that counting holomorphic curves by the class of their boundaries in the skein …

Capital distribution curve is defined as log-log plot of normalized stock capitalizations ranked in descending order. The curve displays remarkable stability over periods of time. Theory of exchangeable distributions on set partitions, developed for purposes of mathematical genetics and recently applied in non-parametr…

This is an introductory chapter in a series in which we take a systematic study of the Yang-Mills equations on curved space-times. In this first, we provide standard material that consists in writing the proof of the global existence of Yang-Mills fields on arbitrary curved space-times using the Klainerman-Rodnianski p…

Defines a map connecting 3d-index and skein module.

A new mathematical approach to general covariance using stacks and Lie algebras.

The "double descent" risk curve was proposed to qualitatively describe the out-of-sample prediction accuracy of variably-parameterized machine learning models. This article provides a precise mathematical analysis for the shape of this curve in two simple data models with the least squares/least norm predictor. Specifi…

In this paper we give the stable classification of ordered, pointed, oriented multi-component curves on surfaces with minimal crossing number less than or equal to 2 such that any equivalent curve has no simply closed curves in its components. To do this, we use the theory of words and phrases which was introduced by V…