In 1956, Shiffman [Sh] proved that any compact minimal annulus with two convex boundary curves (resp. circles) in parallel planes is foliated by convex planar curves (resp. circles) in the intermediate planes. In 1978, Meeks conjectured that the assumption the minimal surface is an annulus is unnecessary [M]; that is, he conjectured that any compact connected minimal surface with two planar con...

A random polygon is the convex hull of uniformly distributed random points in a convex body K ⊂ R. General upper bounds are established for the variance of the area of a random polygon and also for the variance of its number of vertices. The upper bounds have the same order of magnitude as the known lower bounds on variance for these functionals. The results imply a strong law of large numbers ...

An explicit expression is obtained for the perimeter and area generating function G(y, z) = ∑ n>=2 ∑ m>=1 cn,my z, where cn,m is the number of row-convex polygons with area m and perimeter n. A similar expression is obtained for the area-perimeter generating function for staircase polygons. Both expressions contain q-series.

We study a classical problem in communication and wireless networks called Finding White Space Regions. In this problem, we are given a set of antennas (points) some of which are noisy (black) and the rest are working fine (white). The goal is to find a set of convex hulls with maximum total area that cover all white points and exclude all black points. In other words, these convex hulls make i...

In this paper, the notion of $L$-convex fuzzy sublattices is introduced and their characterizations are given. Furthermore,  the notion of the degree to which an $L$-subset is an $L$-convex fuzzy sublattice is proposed and its some  characterizations are given. Besides, the $L$-convex fuzzy sublattice degrees of the homomorphic image and pre-image of an $L$-subset are studied. Finally, we obtai...

4 Colour connected filters 11 4.1 Colour area morphology scale-spaces . . . . . . . . . . . . . . . . . . . . . . . . 12 4.1.1 Colour Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1.2 Distance metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Convex Colour Sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3 Vecto...

We use geometric methods to calculate a formula for the complex Monge-Ampère measure (ddVK) n, for K Rn ⊂ Cn a convex body and VK its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies K. We apply this to show that two methods for deriving Bernstein-Markov type inequalities, i.e., pointwise estimates of gradients of polynomials, yield the same r...

We present fast approximation algorithms for the problem of dividing a given convex geographic region into smaller sub-regions so as to distribute the workloads of a set of vehicles. Our objective is to partition the region in such a fashion as to ensure that vehicles are capable of communicating with one another under limited communication radii. We consider variations of this problem in which...

Conjugate gradient methods are a class of important methods for solving linear equations and nonlinear optimization. In our work, we propose a new stochastic conjugate gradient algorithm with variance reduction (CGVR) and prove its linear convergence with the Fletcher and Revves method for strongly convex and smooth functions. We experimentally demonstrate that the CGVR algorithm converges fast...

In this paper we study the following problem: how to divide a cake among the children attending a birthday party such that all the children get the same amount of cake and the same amount of icing. This leads us to the study of the following. A perfect k-partitioning of a convex set S is a partitioning of S into k convex pieces such that each piece has the same area and 1 k of the perimeter of ...