It is proved that smooth closed curves of given length minimizing the principal eigenvalue of the Schrödinger operator − d 2 ds + κ exist. Here s denotes the arclength and κ the curvature. These minimizers are automatically planar, analytic, convex curves. The straight segment, traversed back and forth, is the only possible exception that becomes admissible in a more generalized setting. In pro...

We prove a Fourier restriction result for general polynomial curves in Rd . Measuring the Fourier restriction with respect to the affine arclength measure of the curve, we obtain a universal estimate for the class of all polynomial curves of bounded degree. Our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right. Applications ...

In R2, we consider an analytic family of fractional integrals , whose convolution kernel is obtained by taking some transverse derivatives of arclength measure on the parabola (t, t2) multiplied by |t|γ , and doing so in a homogeneous way. We determine the exact range of p, q for which the analytic family maps Lp to Lq . We also resolve a similar issue on the Heisenberg group.

It is often desirable to evaluate parametric spline curves at points based on their arc-length instead of the curveÕs original parameter. Techniques are presented here for computing a reparameterization curve allowing approximate arc-length evaluation. This reparameterization curve is also expressed as a spline, allowing rapid evaluation as a function of arc-length. Using composition methods de...

A framework for generating diierential aane invariant signatures based on the gray level images of planar shapes is introduced. Non trivial invariant signatures and their corresponding arclengths are computed for planar shapes. These signatures are useful for pattern recognition and classiication under partial occlusion. We deal only with implementable signatures, which practically means using ...

A Riemannian-geometry approach for modeling and control of dynamics of object manipulation under holonomic or nonholonomic constraints is presented. First, position/force hybrid control of an endeffector of a multijoint redundant (or nonredundant) robot under a holonomic constraint is reinterpreted in terms of “submersion” in Riemannian geometry. A force control signal constructed in the image ...

We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider weighted sum and tangent-point functional. Based on estimates second derivative latter uniform bi-Lipschitz radius, prove stability result implying decay during evolution as well maintenance arclength parametrization. Finally present some ex...

We establish a new relationship between total curvature of knots and crossing number. If K is a smooth knot in R 3 , R the cross-section radius of a uniform tube neighborhood K, L the arclength of K, and κ the total curvature of K, then crossing number of K < 4 L R κ. The proof generalizes to show that for smooth knots in R 3 , the crossing number, writhe, Möbius Energy, Normal Energy, and Symm...