In this note we prove only the important special case (1.1), omitting the parts of the argument required for other cases. In addition, we present an alternate proof, dating from 2002, of the lower bound implicit in (1.1). This proof avoids the use of results about uniform order statistics required in [3], and instead utilizes the cycle lemma from combinatorics. Although shorter and technically ...

Neural stimulus selectivity is thought to be optimized for the representation of real-world stimuli. Neural coding properties, therefore, may adapt to different environments. Here, we address the question if tuning curves depend on the statistics of visual stimuli. This is done by studying the directional tuning of macaque area MT neurons exposed to dynamic motion stimuli of two different direc...

In this paper, we introduce a novel approach for active contours with free endpoints. A scheme is presented for image segmentation and restoration based on a discrete version of the Mumford-Shah functional where the contours can be both closed and open curves. Additional to a flow of the curves in normal direction, evolution laws for the tangential flow of the endpoints are derived. Using a par...

We use rational parametrizations of certain cubic surfaces and an explicit formula for descent via 3-isogeny to construct the first examples of elliptic curves Ek : x 3 + y = k of ranks 8, 9, 10, and 11 over Q. As a corollary we produce examples of elliptic curves over Q with a rational 3-torsion point and rank as high as 11. We also discuss the problem of finding the minimal curve Ek of a give...

In this paper we investigate an analogue for curves of the famous Kakeya conjecture about straight lines. The simplest version of the latter asks whether a set in R that includes a unit line segment in every direction must necessarily have dimension n. The analogue we have in mind replaces the line segments by curved arcs from a specified family. (This is a quite different problem from that con...

The geodesics for a sub-Riemannian metric on a threedimensional contact manifold M form a 1-parameter family of curves along each contact direction. However, a collection of such contact curves on M , locally equivalent to the solutions of a fourth-order ODE, are the geodesics of a sub-Riemannian metric only if a sequence of invariants vanish. The first of these, which was first identified by F...

We analyze characteristic curves of vector fields and report on locations where they have cusps in their spatial projection, i.e., isolated points on the curve with abruptly turning tangent direction. Cusps appear in places where a projection of the corresponding tangent curve vector field exhibits critical points. We show that such cusps are only possible for streak and path lines, whereas the...

In this paper, we introduce a novel approach for active contours with free endpoints. A scheme is presented for image segmentation and restoration based on a discrete version of the Mumford-Shah functional where the contours can be both closed and open curves. Additional to a flow of the curves in normal direction, evolution laws for the tangential flow of the endpoints are derived. Using a par...

This paper extends previous research in that sense that for orthogonal projections of rigid smooth (true-3D) curves moving totally free it reduces the number of required traceable points to two only (the best results known so far to the author are 3 points from free motion and 2 for motion restricted to rotation around a fixed direction and and 2 for motion restricted to influence of a homogene...

We consider a curve reconstruction problem from unorganized point clouds with noise. In general, the result of curve reconstruction depends on how to select and order the representative points to resemble the shape of the clouds. We exploit a natural distance based on a property of onedimensional Brownian motion to order sample points, which simultaneously reflect smoothness and nearness of poi...