The (negative) Torricelli triangle T1(ABC) of a nondegenerate (positively oriented) triangle ABC is defined to be the triangle A1B1C1, where ABC1, BCA1, and CAB1 are the equilateral triangles drawn outwardly on the sides of ABC. It is known that not every triangle is the Torricelli triangle of some initial triangle, and triangles that are not Torricelli triangles are characterized in [28]. In t...

This paper is concerned with the following elliptic system:$$ left{ begin{array}{ll} -triangle u + b(x)nabla u + V(x)u=g(x, v), -triangle v - b(x)nabla v + V(x)v=f(x, u), end{array} right. $$ for $x in {R}^{N}$, where $V $, $b$ and $W$ are 1-periodic in $x$, and $f(x,t)$, $g(x,t)$ are super-quadratic. In this paper, we give a new technique to show the boundedness of Cerami sequences and estab...

In the field of view-dependant continuous level of detail of triangle-meshes it is often necessary to extract the current LOD triangle by triangle. Thus, triangle strips are only of very limited use, or only usable with a high effort. In this work a method is proposed that allows a stepwise reduction of a great, fine stepped LOD hierarchy by merging nodes. The result of this process is a reduce...

We present a novel framework for the efficient simulation and animation of discrete thin shells. Our method takes a point sampled surface as input and performs all necessary computations without intermediate triangulation. We discretize the thin shell functional using so-called fibers. Such fibers are locally embedded parametric curves crisscrossing individual point samples. In combination, the...

We determine the possible eigenvalues of elliptic matrices A,B, C in PU(2, 1) satisfying ABC = 1. This is done by describing geometrically the image of a certain momentum map for the (non-compact) group action of PU(2, 1) by conjugation on C1 × C2 where C1 and C2 are fixed elliptic conjugacy classes in PU(2, 1). Surprisingly, this image is not always convex, rather it is the union of one, two o...

We report on the initial measurements of the angle γ and the sum of angles 2β + γ of the Unitarity Triangle. When compared with indirect information on the value of γ from other measurements of CKM parameters, the measurement of these angles will provide a precise test of Standard Model predictions, as statistics increase. There are several methods for directly measuring γ and 2β + γ. We report...

We show that a natural hybridation construction of lattices in SU(n, 1) to produce (non-arithmetic) lattices in SU(n + 1, 1) fails when n = 1 for most triangle groups in SU(1, 1).

We study Euler’s problem of determination of a triangle from its circumcenter, orthocenter, and incenter as a problem of geometric construction. While it cannot be solved using ruler and compass, we construct the vertices by intersecting a circle with a rectangular hyperbola, both easily constructed from the given triangle centers.

A non-contradictible axiomatic theory is constructed under the local reversibility of the metric triangle inequality. The obtained notion includes the metric spaces as particular cases and the generated metric topology is T$_{1}$-separated and generally, non-Hausdorff.

We construct a Teichmüller curve uniformized by the Fuchsian triangle group ∆(m, n, ∞) for every m < n ≤ ∞. Our construction includes the Teichmüller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. We interprete some of the so-called Lyapunov exponents of the Kontsevich–Zorich cocycle as normalize...