We prove that among all triangles of given diameter, the equilateral triangle minimizes the sum of the first n eigenvalues of the Neumann Laplacian, when n 3 . The result fails for n = 2 , because the second eigenvalue is known to be minimal for the degenerate acute isosceles triangle (rather than for the equilateral) while the first eigenvalue is 0 for every triangle. We show the third eigenva...

A subset of the finite dimensional hypercube is said to be equilateral if the distance of any two distinct points equals a fixed value. The equilateral dimension of the hypercube is defined as the maximal size of its equilateral subsets. We study asymptotic bounds on the latter quantity considered as a function of two variables, namely dimension and distance. lorenz minder, thomas sauerwald and...

Figured numbers are sequences of natural that, when represented by dots, form geometric configurations. The immediate sequence figured presents in its configuration the equilateral triangles, being basis for finding several others, as long a relationship between them is discovered. This article shows construction polygonal, stellar and three-dimensional that can be constructed using GeoGebra so...

It is shown that an equilateral convex pentagon tiles the plane if and only if it has two angles adding to 180 o or it is the unique equilateral convex pentagon with Although the area of mathematical tilings has been of interest for a long time there is still much to be discovered. We do not even know which convex polygons tile the plane. Furthermore, for those polygons which do tile, new tilin...

Further morphological observations on the particulate components decorating the lumenal surfaces of membranes of the endocytic complex of the epithelial cells of the suckling rat ileum are presented. The particles each measure approximately 7.5 nm across and give the appearance of the capital letter H in frontal view. They consist of the enzyme n-acetyl-beta-glucosaminidase (NAG). They are arra...

The study of spherical dihedral f-tilings by equilateral and isosceles triangles was introduced in [3]. Taking as prototiles equilateral and scalene triangles, we are faced with three possible ways of adjacency. In [4] and [5] two of these possibilities were studied. Here, we complete this study, describing the f-tilings related to the remaining case of adjacency, including their symmetry group...

The present work deals with heat transfer characteristics as well as fluid flow patterns in laminar flow regime for a circular cylinder with six control rods arranged in equilateral triangular geometries. The computations have been carried out by a finite volume approach using the overset grid method. The unsteady flow at Re= 200 and Pr= 0.7 and 7.0 was examined. The effect of the control rods ...

In the 3/4-filled band system θ-(BEDT-TTF)2X with a two-dimensional triangular lattice, charge ordering (CO) often occurs due to strong inter-site Coulomb repulsion. However, geometrical frustration of lattice can prohibit long-range CO, resulting in charge-glass state which configurations are randomly distributed. Here, we investigate states orthorhombic and monoclinic θ-type BEDT-TTF salts by...

The classical Ramsey theorem, states that every graph contains either a large clique or a large independent set. Here we investigate similar dichotomic phenomena in the context of finite metric spaces. Namely, we prove statements of the form ”Every finite metric space contains a large subspace that is nearly equilateral or far from being equilateral”. We consider two distinct interpretations fo...