We deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid, the elliptic hyperboloid in R implies that for the cone in R. We also prove a new restriction estimate for any surface in R locally isometric to the plane and of finite type.

We describe a technique for measuring the surface profile of a radio telescope with a fast paraboloidal primary. The technique uses a sensor, at the center of curvature of the primary, consisting of a millimeter-wave source and an array of receivers to measure the field in the caustic. The sensor is mounted on the telescope enclosure and it moves with the telescope, so the measurements can be u...

Spanish architect Antoni Gaudı́ drew on natural precedent to create fantastic organically inspired designs. This paper examines some common motifs in his works, and demonstrates how simple geometric operations can be applied to polyhedral models to achieve similar results. The importance of the partial hyperbolic paraboloid surface (or hypar) is discussed in this context. r 2007 Elsevier Ltd. Al...

An open problem in affine geometry is whether an affine complete locally uniformly convex hypersurface in Euclidean (n + 1)-space is Euclidean complete for n ≥ 2. In this paper we give the affirmative answer. As an application, it follows that an affine complete, affine maximal surface in R3 must be an elliptic paraboloid.

Рассматриваются безвихревые векторные поля на поверхности, заданной уравнением $a=z+\alpha(x,y,t)=0$. Изучаются условия, при выполнении которых линии таких полей располагаются этой поверхности. Получены достаточные условия существования гармонического векторного с такими векторными линиями. Изучена переопределенная система уравнений в частных производных, решение которой обеспечивает получение ...

Today, Newton’s Method is commonly used in machine learning and optimization to locate local minima of cost functions. Because it is such a general tool, the cost functions that is implemented on can range widely in terms of both their dimensionality and their complexity. However, little is understood about how Newton’s Method performs on cost functions with different dimensionalities and compl...