The Fundamental Gap For Hyperbolic Triangles

In 1983, M. van den Berg made his Fundamental Gap Conjecture about the difference between the first two Dirichlet eigenvalues (the fundamental gap) of any convex domain in the Euclidean plane. Recently, progress has been made in the case where the domains are polygons and, in particular, triangles. We examine the conjecture for triangles in hyperbolic geometry, though we seek an for an upper bound for the fundamental gap rather than a lower bound.

Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

La trigonometrie rectiligne et spherique ou il est Traite de la construction des tables de sinus, tangentes, secantes et logarithmes : de l'usage de ces tables pour la résolution des triangles avec des questions astronomiques, & ces mêmes tables tres - exactement calculées sur un rayon de 10000000 parties /

Mode of access: Internet.

Recherche méthodique et propriétés des triangles rectangles en nombres entiers.

Available on demand as hard copy or computer file from Cornell University Library.

Tables for calculating the cubic contents of excavations and embankments : by an improved method of diagonals and side triangles /

Mode of access: Internet.

Decompositions of complete tripartite graphs into triangles with an edge attached

Let K(r, s, t) denote the complete tripartite graph with partite sets of size r, s and t, where r less than or equal to s less than or equal to t. Let D be the graph consisting of a triangle with an edge attached. We show that K(r, s, t) may be decomposed into copies of D if and only if 4 divides rs + st + rt and t less than or equal to 3rs/(r + s).

Decompositions of complete graphs into triangles and Hamilton cycles

For all odd integers n greater than or equal to 1, let G(n) denote the complete graph of order n, and for all even integers n greater than or equal to 2 let G,, denote the complete graph of order n with the edges of a 1-factor removed. It is shown that for all non-negative integers h and t and all positive integers n, G, can be decomposed into h Hamilton cycles and t triangles if and only if nh + 3t is the number of edges in G(n). (C) 2004 Wiley Periodicals, Inc.

A molecular dynamics model of the atomic structure of dysprosium alumino-phosphate glass

Molecular dynamics (MD) has been used to identify the relative distribution of dysprosium in the phosphate glass DyAl0.30P3.05O9.62. The MD model has been compared directly with experimental data obtained from neutron diffraction to enable a detailed comparison beyond the total structure factor level. The MD simulation gives Dy ... Dy correlations at 3.80(5) and 6.40(5) angstrom with relative coordination numbers of 0.8(1) and 7.3(5), thus providing evidence of minority rare-earth clustering within these glasses. The nearest neighbour Dy-O peak occurs at 2.30 angstrom with each Dy atom having on average 5.8 nearest neighbour oxygen atoms. The MD simulation is consistent with the phosphate network model based on interlinked PO4 tetrahedra where the addition of network modifiers Dy3+ depolymerizes the phosphate network through the breakage of P-(O)-P bonds whilst leaving the tetrahedral units intact. The role of aluminium within the network has been taken into explicit account, and A1 is found to be predominantly (78 tetrahedrally coordinated. In fact all four A1 bonds are found to be to P (via an oxygen atom) with negligible amounts of Al-O-Dy bonds present. This provides an important insight into the role of Al additives in improving the mechanical properties of these glasses.

Structure of dysprosium and holmium phosphate glasses by the method of isomorphic substitution in neutron diffraction

The relative distribution of rare-earth ions R3+ (Dy3+ or Ho3+) in the phosphate glass RAl0.30P3.05O9.62 was measured by employing the method of isomorphic substitution in neutron diffraction and, by taking the role of Al into explicit account, a self-consistent model of the glass structure was developed. The glass network is found to be made from corner sharing PO4 tetrahedra in which there are, on average, 2.32(9) terminal oxygen atoms, OT, at 1.50(1) Å and 1.68(9) bridging oxygen atoms, OB, at 1.60(1) Å. The network modifying R3+ ions bind to an average of 6.7(1) OT and are distributed such that 7.9(7) R–R nearest neighbours reside at 5.62(6) Å. The Al3+ ion also has a network modifying role in which it helps to strengthen the glass through the formation of OT–Al–OT linkages. The connectivity of the R-centred coordination polyhedra in (M2O3)x(P2O5)1−x glasses, where M3+ denotes a network modifying cation (R3+ or Al3+), is quantified in terms of a parameter fs. Methods for reducing the clustering of rare-earth ions in these materials are then discussed, based on a reduction of fs via the replacement of R3+ by Al3+ at fixed total modifier content or via a change of x to increase the number of OT available per network modifying M3+ cation.

On the Generation of Heronian Triangles

We describe several algorithms for the generation of integer Heronian triangles with diameter at most n. Two of them have running time O(n^(2+ε)). We enumerate all integer Heronian triangles for n ≤ 600000 and apply the complete list on some related problems.

Energy level decay and excited state absorption processes in dysprosium-doped fluoride glass

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)