arXiv research

Study on Jones polynomials and their roots in the unit circle and complex plane.

Develops diffusion models for time-varying correlation on the circle.

Classifies surfaces with great and small circles through each point.

Method detects intersections between ellipses for Borromean linking.

Harmonic and minimal great circle fibrations have special Gauss maps.

The paper studies the geometry of probability measures on the unit circle.

Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.

We give parameterizations of homeomorphisms, quasisymmetric maps and symmetric maps of the unit circle in terms of shear coordinates for the Farey tesselation.

Study infinite circle patterns in the Weil-Petersson class using discrete harmonic functions.

A new proof shows how to characterize maps using simple geometry.

Classifies hexagonal circular 3-webs with cubic polar curves.

Polynomials' roots count tied to surface umbilics.

By a theorem of A'Campo, the eigenvalues of certain Coxeter transformations are positive real or lie on the unit circle. By optimally bounding the signature of tree-like positive Hopf plumbings from below by the genus, we prove that at least two thirds of them lie on the unit circle. In contrast, we show that for divid…

This work introduces a geometric approach to probability representation and option pricing.

Deep belief networks can approximate any multivariate density with binary hidden units.

We prove that the set of smooth, $π$-periodic, positive functions on the unit circle for which the $L_{-2}$ Minkowski problem is solvable is dense in the set of all smooth, $π$-periodic, positive functions on the unit circle with respect to the $L^{\infty}$ norm. Furthermore, we obtain a necessary condition on the solv…

Bayesian model predicts circular data with fast Gibbs sampling.

Given a matrix $A\in SL(N,\Z)$, form the semidirect product $G=\Z^N\rtimes_A \Z$ where the $\Z$ factor acts on $\Z^N$ by $A$. Such a $G$ arises naturally as the fundamental group of an $N$-dimensional torus bundle which fibers over the circle. In this paper we prove that if $A$ has distinct eigenvalues not lying on the…

We give a short proof of the fact that bounded earthquakes of the unit disk induce quasisymmetric maps of the unit circle. By a similar method, we show that symmetric maps are induced by bounded earthquakes with asymptotically trivial measures.

Eta invariant computed for circle bundles over Fano manifolds.

In this paper we investigate free boundary minimal surfaces in the unit ball in Euclidean 3-space, and by using holomorphic techniques we prove that intersection curves of free boundary minimal surfaces with the unit sphere are all circles.

Asymptotics for equidistribution of circles on hyperbolic surfaces.

Proves stability of cone-volume measure with nearly constant density.

Given any smooth fibration of the unit 3-sphere by great circles, we show that the distribution of 2-planes orthogonal to the great circle fibres is a tight contact structure, a fact well known in the special case of the Hopf fibrations. The proof expresses hypothesis and conclusion as differential inequalities involvi…

The complex wave representation (CWR) converts unsigned 2D distance transforms into their corresponding wave functions. Here, the distance transform S(X) appears as the phase of the wave function φ(X)---specifically, φ(X)=exp(iS(X)/τwhere τis a free parameter. In this work, we prove a novel result using the higher-orde…

The study explores continuous noncrossing partitions and their relation to weighted circular factorizations.

Let $Σ$ be a hypersurface in an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We study the isometric extension problem for isometric immersions $f:Σ\to\mathbb R^n$, where $\mathbb R^n$ is equipped with the Euclidean standard metric. We prove a general curvature obstruction to the existence of merely differen…

New inequality on sphere generalizes circle inequality.

We classify all Kahler metrics in an open subset of $C^2$ whose real geodesics are circles. All such metrics are equivalent (via complex projective transformations) to Fubini metrics (i.e. to Fubini-Study metric on $CP^2$ restricted to an affine chart, to the complex hyperbolic metric in the unit ball model or to the E…

Study on curve diffusion flows with scale-critical curvature term.

Let $Φ\colon \sbat \times M \to M$ be a smooth action of the unit circle $ \sbat$ on a manifold $M$. In this work, we compute the minimal model of $M$ in terms of the orbit space $B$ and the fixed point set $F\subset B$, as a dg-module over the Sullivan's minimal model of $B$.

Constructs minimal surfaces near the boundary of a ball.

We deal with minimal surfaces in the unit sphere $S^3$, which are one-parameter families of circles. Minimal surfaces in $\R^3$ foliated by circles were first investigated by Riemann, and a hundred years later Lawson constructed examples of such surfaces in $S^3$. We prove that in $S^3$ there are only two types of mini…

We determine for which complex numbers on the unit circle the Levine-Tristram signature and the nullity give rise to link concordance invariants.

An embedding of the group $\Diff(S^{1})$ of orientation preserving diffeomorphims of the unit circle $S^1$ into an infinite-dimensional symplectic group, $\Sp(\infty)$, is studied. The authors prove that this embedding is not surjective. A Brownian motion is constructed on $\Sp(\infty)$. This study is motivated by rece…

Knots in circle bundles are uniquely identified by their complements.

We investigate the classification of topological quandles on some simple manifolds. Precisely we classify all Alexander quandle structures, up to isomorphism, on the real line and the unit circle. For the closed unit interval $[0, 1]$, we conjecture that there exists only one topological quandle structure on it, i.e. t…

Researchers create a teapot model for Mandelbrot set, proving connectedness.

Maps asymptotically embed conic transforms from circle bundles.

In this paper, it is shown that any surface automorphism of positive mapping-class entropy possesses a virtual homological eigenvalue which lies outside the unit circle of the complex plane.

We consider the isoperimetric problem in planar sectors with density $r^{p}$, and with density $a>1$ inside the unit disk and $1$ outside. We characterize solutions as a function of sector angle. We also solve the isoperimetric problem in $\mathbb{R}^{n}$ with density $r^{p},\; p<0$.

The signature function of a knot is an integer-valued step function on the unit circle in the complex plane. Necessary and sufficient conditions for a function to be the signature function of a knot are presented.

The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume, whose fun…

We compute the Szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kaehler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for w…

Extends Fatou theorem to bounded harmonic maps.

It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if $T$ is a Riemannian 2-torus with boundary in $\mathbb R ^n$, such that the boundary curve is a standard unit circle, then the length o…

The signature function of a knot is a locally constant integer valued function with domain the unit circle. The jumps (i.e., the discontinuities) of the signature function can occur only at the roots of the Alexander polynomial on the unit circle. The latter are important in deforming U(1) representations of knot group…

I propose a frequency domain adaptation of the Expectation Maximization (EM) algorithm to group a family of time series in classes of similar dynamic structure. It does this by viewing the magnitude of the discrete Fourier transform (DFT) of each signal (or power spectrum) as a probability density/mass function (pdf/pm…