Related papers: Curves of Minimax Spirality
In this paper, we study when we might expect the optimization curve induced by gradient descent to be \emph{convex} -- precluding, for example, an initial plateau followed by a sharp decrease, making it difficult to decide when optimization…
In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature…
This paper deals with finding surfaces in $\mathbb{R}^3$ which are as close as possible to being flat and span a given contour such that the contour is a geodesic on the sought surface. We look for a surface which minimizes the total…
The classical Euler's problem on optimal configurations of elastic rod in the plane with fixed endpoints and tangents at the endpoints is considered. The global structure of the exponential mapping that parameterises extremal trajectories…
The solutions to surface evolution problems like mean curvature flow can be expressed as value functions of suitable stochastic control problems, obtained as limit of a family of regularised control problems. The control-theoretical…
We study the geometric and combinatorial effect of smoothing an intersection point in a collection of arcs or curves on a surface. We prove that all taut arcs with fixed endpoints and all taut 1-manifolds with at least two non-disjoint…
We find conditions under which Almgren-Pitts min-max for the prescribed geodesic curvature functional in a closed oriented Riemannian surface produces a closed embedded curve of constant curvature. In particular, we find a closed embedded…
In this article, we consider the motion planning of a rigid object on the unit sphere with a unit speed. The motion of the object is constrained by the maximum absolute value, $U_{max}$ of geodesic curvature of its path; this constrains the…
Motivated by various applications, this article develops the notion of boundary control for Maxwell's equations in the frequency domain. Surface curl is shown to be the appropriate regularization in order for the optimal control problem to…
We systematically explore a class of constrained optimization problems with linear objective function and constraints that are linear combinations of logarithms of the optimization variables. Such problems can be viewed as a generalization…
We introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by $K>0$ have diameter bounded by $\mbox{diam}(G) \leq…
We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we determine the long-term behavior of open…
We show that transcendental curves in $\mathbb R^n$ (not necessarily compact) have few rational points of bounded height provided that the curves are well behaved with respect to algebraic sets in a certain sense and can be parametrized by…
A new isoperimetric estimate is proved for embedded closed curves evolving by curve shortening flow, normalized to have total length $2\pi$. The estimate bounds the length of any chord from below in terms of the arc length between its…
We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…
A circular arc graph is the intersection graph of a collection of connected arcs on the circle. We solve a Tur'an-type problem for circular arc graphs: for n arcs, if m and M are the minimum and maximum number of arcs that contain a common…
The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2\pi. This is an extremal length problem in…
The aim of this paper is to provide a direct link between maximizing curves that occur in the construction of smooth algebraic surfaces having the maximal possible Picard numbers and reduced free plane curves with simple singularities. We…
In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $\int \sqrt{1+K_\gamma^2} ds$, depending both on length and curvature $K$. We fix starting and ending points…
A number of discrete and continuous optimization problems in machine learning are related to convex minimization problems under submodular constraints. In this paper, we deal with a submodular function with a directed graph structure, and…