Related papers: Closed 1/2-Elasticae in the 2-Sphere

We study critical trajectories in the hyperbolic plane for the $1/2$-Bernoulli's bending energy with length constraint. Critical trajectories with periodic curvature are classified into three different types according to the causal…

We consider resonances in the semi-classical limit, generated by a single closed hyperbolic orbit, for an operator on ${\bf R}^2$. We determine all such resonancess in a domain independent of the semi-classical parameter As an application…

Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved…

We minimize elastic energies on framed curves which penalize both curvature and torsion. We also discuss critical points using the infinite dimensional version of the Lagrange multipliers' method. Finally, some examples arising from the…

In this note, we discuss the possible existence of finite critical trajectories connecting two zeros a(t) and b(t) of a family of quadratic differentials satisfying some properties. We treat the cases of holomorphic and meromorphic…

We introduce a new critical value $c_\infty(L)$ for Tonelli Lagrangians $L$ on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that $c_\infty(L)$ is strictly larger than the Ma\~n\'e critical…

In this note, we show there exist infinitely many trajectories which are bi-normal (i.e. normal at initial and final times) to the xz-plane, in the Spatial Circular Restricted Three-Body Problem, for energies below or slightly above the…

Critical points of a function subject to a constraint can be either detected by restricting the function to the constraint or by looking for critical points of the Lagrange multiplier functional. Although the critical points of the two…

We investigate a limit theorem on traversable length inside semi-cylinder in the 2-dimensional supercritical Bernoulli bond percolation, which gives an extension of Theorem 2 in Grimmett(1981). This type of limit theorems was originally…

We prove the existence of chaotic trajectories for the two body problem on a sphere. The trajectories we construct encounter near-collisions and are similar to the second species solutions of Poincar\'e of the classical 3 body problem. The…

We study planar bicycle dynamics via the rotation number function associated with a closed front track and bicycle length R. We prove that mode-locking plateaus occur only at integer rotation numbers and that the rotation number function is…

We study an interacting particle system in which moving particles activate dormant particles linked by the components of critical bond percolation. Addressing a conjecture from Beckman, Dinan, Durrett, Huo, and Junge for a continuous…

Substantial progress has been made in recent years on the 2D critical percolation scaling limit and its conformal invariance properties. In particular, chordal SLE6 (the Stochastic Loewner Evolution with parameter k=6) was, in the work of…

We study the spectrum of the Laplace-Beltrami operator on ellipsoids. For ellipsoids that are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids…

Bernoulli convolutions are certain measures on the unit interval depending on a parameter $\beta$ between 1 and 2. In spite of their simple definition, they are not yet well understood. We study their two-dimensional density which exists by…

We use global bifurcation techniques to establish the existence of arbitrarily many geometrically distinct nonplanar embedded smooth minimal 2-spheres in sufficiently elongated 3-dimensional ellipsoids of revolution. More precisely, we…

Let $G=(V,E)$ be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on $G$. In recent work, we conjectured that if $G$ is nonamenable then the matrix of critical connection probabilities…

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. For most concentrations of the scatterers the trajectories close…

This paper is a sequel to Chaika and Krishnan [arXiv:1612.00434]. We again consider translation invariant measures on families of nearest-neighbor semi-infinite walks on the integer lattice Z^d. We assume that once walks meet, they…

We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the…