The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincaré inequality for triangles is derived. The proof relies on symmetry of the Neumann fundamental mode for isosceles triangles with aperture less than π/3. Antisymmetry is proved for apertures greater than π/3.

In this note, by using complex variable functions, we present a new simpler proof of the degeneracy property of the longest-edge n-section of triangles for n P 4. This means that the longest-edge n-section of triangles for n P 4 produces a sequence of triangles with minimum interior angle converging to zero. 2012 Elsevier Inc. All rights reserved.

Sommerville [8] and Davies [2] classified the spherical triangles that can tile the sphere in an edge-to-edge fashion. Relaxing this condition yields other triangles, which tile the sphere but have some tiles intersecting in partial edges. This paper shows that no right triangles in a certain subfamily can tile the sphere, although multilayered tilings are possible.

We present an improved algorithm for finding all solutions to Goehl’s problem A = mP for triangles, i.e., the problem of finding all Heronian triangles whose area (A) is an integer multiple (m) of the perimeter (P ). The new algorithm does not involve elimination of extraneous rational triangles, and is a true extension of Goehl’s original method.

We study the existence of equilateral triangles of given side lengths and with integer coordinates in dimension three. We show that such a triangle exists if and only if their side lengths are of the form √ 2(m2 − mn + n2) for some integers m, n. We also show a similar characterization for the sides of a regular tetrahedron in Z: such a tetrahedron exists if and only if the sides are of the for...

Generalized frames are an extension of frames in Hilbert  spaces and  Hilbert $C^*$-modules. In this paper, the concept ''Similar" for modular $g$-frames is introduced and all of operator duals (ordinary duals) of similar $g$-frames with respect to each other are characterized. Also, an operator dual of a given $g$-frame is studied where $g$-frame is constructed by a primary $g$-frame and an or...

Sommerville [10] and Davies [2] classified the spherical triangles that can tile the sphere in an edge-to-edge fashion. Relaxing this condition yields other triangles, which tile the sphere but have some tiles intersecting in partial edges. This paper determines which right spherical triangles within certain families can tile the sphere.

An acute triangulation is a triangulation whose triangles have all their angles less than π 2 . In this paper we prove that i) every planar pentagon can be triangulated into at most 54 acute triangles, and ii) every double pentagon can be triangulated into at most 76 acute triangles.

We describe a method to subdivide the surface of a sphere into spherical triangles of similar, but not identical, shapes and sizes. The Hierarchical Triangular Mesh (HTM) is a quad-tree that is particularly good at supporting searches at different resolutions, from arc seconds to hemispheres. The subdivision scheme is universal, providing the basis for addressing and for fast lookups. The HTM p...

We show that if a graph G has the property that all subsets of vertices of size n/4 contain the “correct” number of triangles one would expect to find in a random graph G(n, 12 ), then G behaves like a random graph, that is, it is quasi-random in the sense of Chung, Graham, and Wilson [6]. This answers positively an open problem of Simonovits and Sós [10], who showed that in order to deduce tha...