Related papers: Conjugate points in Euler's elastic problem

J. J. Sylvester's four-point problem asks for the probability that four points chosen uniformly at random in the plane have a triangle as their convex hull. Using a combinatorial classification of points in the plane due to Goodman and…

The bifurcation of an incompressible neo-Hookean thick block with a ratio of thickness to length {eta}, subject to pure bending, is considered. The two incremental equilibrium equations corresponding to a nonlinear pre-buckling state of…

We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the…

The simplest way to generate a lattice of convex sets is to consider an initial set of points and draw segments, triangles, and any convex hull from it, then intersect them to obtain new points, and so forth. The result is an infinite…

We construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable H\"older space and the…

In this paper, we study the relaxation limit of the relaxing Cauchy problem for non-isentropic compressible Euler equations with damping in multi-dimensions. We prove that the velocity of the relaxing equations converges weakly to that of…

We generalize Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an n-dimensional polytope with n+1 rational vertices, we use its description as the intersection of n+1 halfspaces,…

We prove constant-curvature analogues of several results regarding the hot spots conjecture in dimension two. Our main theorem shows that the hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature. We…

Let $L$ be a non-split prime alternating link with $n>0$ crossings. We show that for each fixed $g$, the number of genus-$g$ Seifert surfaces for $L$ is bounded by an explicitly given polynomial in $n$. The result also holds for all…

We first recall several historical oscillating bridges that, in some cases, led to collapses. Some of them are quite recent and show that, nowadays, oscillations in suspension bridges are not yet well understood. Next, we survey some…

Three-dimensional convex bodies can be classified in terms of the number and stability types of critical points on which they can balance at rest on a horizontal plane. For typical bodies these are nondegenerate maxima, minima, and…

We study how electron-electron interactions renormalize tunneling into a Luttinger liquid beyond the lowest order of perturbation in the tunneling amplitude. We find that the conventional fixed point has a finite basin of attraction only in…

We streamline and generalize the recent progress in understanding entanglement between spatial regions in Abelian gauge theories. We provide an unambiguous and explicit prescription for calculating entanglement entropy in a $\mathbb Z_N$…

We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper we show that,…

After reformulate the incompressible Euler-$\alpha$ equations in 3D smooth domain with Drichlet data, we obtain the unique classical solutions to Euler-$\alpha$ equations exist in uniform time interval independent of $\alpha$. We also show…

This paper concerns the number of lattice points in a circle.

In this note, we present a conjecture on intersections of set families, and a rephrasing of the conjecture in terms of principal downsets of Boolean lattices. The conjecture informally states that, whenever we can express the measure of a…

Using an elementary argument, we prove new fixed point theorems for classical elliptic complexes. We obtain new results for conformal relations and coisotropic intersections. We obtain theorems for the average intersections of families of…

A generalization of the Euler-Plateau problem to account for the energy contribution due to twisting of the bounding loop is proposed. Euler-Lagrange equations are derived in a parameterized setting and a bifurcation analysis is performed.…

Exceptional points of a dissipative chain of three coupled oscillators (trimer), which is driven by quadratic photon, are investigated. The exceptional points emerge from the coalescence of both eigenvalues and eigenvectors of the dynamical…