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.

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.

Raman spectroscopy of the borate mineral ameghinite NaB3O3(OH)4

The molecular structure of the sodium borate mineral ameghinite NaB3O3(OH)4 has been determined by the use of vibrational spectroscopy. The crystal structure consists of isolated [B3O3(OH)4]- units formed by one tetrahedron and two triangles. H bonds and Na atoms link these polyanions to form a 3-dimensional framework. The Raman spectrum is dominated by an intense band at 1027 cm-1, attributed to BO stretching vibrations of both the trigonal and tetrahedral boron. A series of Raman bands at 1213, 1245 and 1281cm-1 are ascribed to BOH in-plane bending modes. The infrared spectra are characterized by strong overlap of broad multiple bands. An intense Raman band found at 620 cm-1 is attributed to the bending modes of trigonal and tetrahedral boron. Multiple Raman bands in the OH stretching region are observed at 3206, 3249 and 3385 cm-1. Raman spectroscopy coupled with infrared spectroscopy has enabled aspects about the molecular structure of the borate mineral ameghinite to be assessed.

Infrared and Raman spectroscopic characterization of the borate mineral colemanite – CaB3O4(OH)3·H2O – implications for the molecular structure

Colemanite CaB3O4(OH)3 H2O is a secondary borate mineral formed from borax and ulexite in evaporate deposits of alkaline lacustrine sediments. The basic structure of colemanite contains endless chains of interlocking BO2(OH) triangles and BO3(OH) tetrahedrons with the calcium, water and extra hydroxide units interspersed between these chains. The Raman spectra of colemanite is characterized by an intense band at 3605 cm-1 assigned to the stretching vibration of OH units and a series of bands at 3182, 3300, 3389 and 3534 cm-1 assigned to water stretching vibrations. Infrared bands are observed in similar positions. The BO stretching vibrations of the trigonal and tetrahedral boron are characterized by Raman bands at 876, 1065 and 1084 cm-1. The OBO bending mode is defined by the Raman band at 611 cm-1. It is important to characterize the very wide range of borate minerals including colemanite because of the very wide range of applications of boron containing minerals.

Wilds of the Canadian Rockies

A travel article about touring in the Rockies, Alberta. MORAINE Lake, held up by a crown of mountains near the small town of Lake Louise, is opal blue. You'd need a precious stone to cut the surface. And even on a hot morning in August, there are plenty around: diamonds of snow decorate the mountain tops and, in their reflections, slice the lake with white triangles. We are at the start of a track that climbs from the lake to Mt Temple, the tallest mountain in the area and two or three hours' walk away. When the track zigzags back, we glimpse the lake through gaps in the conifer forest...

Adaptive finite element analysis of elliptic problems based on bubble-type local mesh generation

A new mesh adaptivity algorithm that combines a posteriori error estimation with bubble-type local mesh generation (BLMG) strategy for elliptic differential equations is proposed. The size function used in the BLMG is defined on each vertex during the adaptive process based on the obtained error estimator. In order to avoid the excessive coarsening and refining in each iterative step, two factor thresholds are introduced in the size function. The advantages of the BLMG-based adaptive finite element method, compared with other known methods, are given as follows: the refining and coarsening are obtained fluently in the same framework; the local a posteriori error estimation is easy to implement through the adjacency list of the BLMG method; at all levels of refinement, the updated triangles remain very well shaped, even if the mesh size at any particular refinement level varies by several orders of magnitude. Several numerical examples with singularities for the elliptic problems, where the explicit error estimators are used, verify the efficiency of the algorithm. The analysis for the parameters introduced in the size function shows that the algorithm has good flexibility.

Self-Assembled Pd(II) Metallocycles Using an Ambidentate Donor and the Study of Square-Triangle Equilibria

The self-assembly reaction of a cis-blocked 90° square planar metal acceptor with a symmetrical linear flexible linker is expected to yield a [4 + 4] self-assembled square, a [3 + 3] assembled triangle, or a mixture of these.However, if the ligand is a nonsymmetrical ambidentate, it is expected to form a complex mixture comprising several linkage isomeric squares and triangles as a result of different connectivities of the ambidentate linker. We report instead that the reaction of a 90° acceptor cis-(dppf)Pd(OTf)2 [where dppf ) 1,1′-bis(diphenylphosphino)- ferrocene] with an equimolar amount of the ambidentate unsymmetrical ligand Na-isonicotinate unexpectedly yields a mixture of symmetrical triangles and squares in the solution. An analogous reaction using cis-(tmen)Pd(NO3)2 instead of cis-(dppf)Pd(OTf)2 also produced a mixture of symmetrical triangles and squares in the solution. In both cases the square was isolated as the sole product in the solid state, which was characterized by a single crystal structure analysis. The equilibrium between the triangle and the square in the solution is governed by the enthalpic and entropic contributions. The former parameter favors the formation of the square due to less strain in the structure whereas the latter one favors the formation of triangles due to the formation of more triangles from the same number of starting linkers. The effects of temperature and concentration on the equilibria have been studied by NMR techniques. This represents the first report on the study of square-triangle equilibria obtained using a nonsymmetric ambidentate linker. Detail NMR spectroscopy along with the ESI-mass spectrometry unambiguously identified the components in the mixture while the X-ray structure analysis determined the solid-state structure.

Flow through a plateau border of cellular foam

Experimentally measured average velocities through plateau borders of stationary cellular foam, when compared with those calculated with the assumption of rigid Plateau Border walls, show that the assumption of rigid walls severely underestimates the velocities. An analysis of the situation wherein plateau border walls have velocities, as decided by the surface viscosity of the system, is presented here. The plateau border is idealized as a pipe of equilateral triangular cross-section with vertices of the triangle having zero velocity. The pertinent form of Navier-Stoke's equations with inhomogeneous boundary conditions and its solution through a procedure of successive approximations is presented in dimensionless form. The solution reduces to the known solution of slow steady flow through a triangular pipe, when surface viscosity is infinite. Results indicate that the assumption of rigid plateau border walls is valid only when value of the inverse of dimensionless surface viscosity is less than 0.044. Beyond that the assumption severely underestimates the flow and the effect of nonrigidity of the wall must be considered.

Efficient algorithms for computing Reeb graphs

The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. We describe an algorithm that constructs the Reeb graph of a Morse function defined on a 3-manifold. Our algorithm maintains connected components of the two dimensional levels sets as a dynamic graph and constructs the Reeb graph in O(nlogn+nlogg(loglogg)3) time, where n is the number of triangles in the tetrahedral mesh representing the 3-manifold and g is the maximum genus over all level sets of the function. We extend this algorithm to construct Reeb graphs of d-manifolds in O(nlogn(loglogn)3) time, where n is the number of triangles in the simplicial complex that represents the d-manifold. Our result is a significant improvement over the previously known O(n2) algorithm. Finally, we present experimental results of our implementation and demonstrate that our algorithm for 3-manifolds performs efficiently in practice.

The influence of planting density on the production of 'Goldfinger' (Musa spp., AAAB) in the subtropics

'Goldfinger', a tetraploid banana produced from the Fundación Hondureña de Investigación Agrícola (FHIA) breeding program, was released to the Australian industry in 1995. It was promoted as an apple-flavoured dessert banana with resistance to Fusarium wilt race 1 and subtropical race 4, as well as resistance to black and yellow Sigatoka (Mycosphaerella fijiensis and M. musicola, respectively). This study was initiated to provide agronomic information to the banana industry, which was under threat from Fusarium wilt, on a new cultivar which could replace 'Williams' (AAA, Cavendish subgroup) or 'Lady Finger' (AAB, Pome subgroup) in those areas affected by Fusarium wilt. Also few studies had reported on the production characteristics of the new tetraploid hybrids, especially from subtropical areas, and therefore two field sites, one a steep-land farm and the other a level, more productive site, were selected for planting density and spatial arrangement treatments. The optimum density in terms of commercial production, taking into account bunch weight, finger size, length of the production cycle, plant height and ease of management, was 1680 plants/ha on the steep-land site where plants were planted in single rows with 2.5 m × 2.5 m spacings. However on the level site a double-row triangular layout with inter-row distances of 4.5 m to allow vehicular access (1724 plants/ha) gave the best results. With this arrangement plants were in an alternate, triangular arrangement along a row and a spacing of 1.5 m between plants at the points of each triangle and between each block of triangles.

A Pulsewidth Modulated Control of Induction Motor Drive Using Multilevel 12-Sided Polygonal Voltage Space Vectors

In this paper, a novel 12-sided polygonal space vector structure is proposed for an induction motor drive. The space vector pattern presented in this paper consists of two 12-sided concentric polygons with the outer polygon having a radius double the inner one. As compared to previously reported 12-sided polygonal space vector structures, this paper subdivides the space vector plane into smaller sized triangles. This helps in reducing the switching frequency of the inverters without deteriorating the output voltage quality. It also reduces the device ratings and dv/dt stress on the devices to half. At the same time, other benefits obtained from the existing 12-sided space vector structure, such as increased linear modulation range and complete elimination of 5th and 7th order harmonics in the phase voltage, are also retained in this paper. The space vector structure is realized by feeding an open-end induction motor with two conventional three-level neutral point clamped (NPC) inverters with asymmetric isolated dc link voltage sources. The neutral point voltage fluctuations in the three-level NPC inverters are eliminated by utilizing the switching state multiplicities for a space vector point. The pulsewidth modulation timings are calculated using sampled reference waveform amplitudes and are explained in detail in this paper. Experimental verification on a laboratory prototype shows that this configuration may be considered suitable for high power drives.

A new method for generation of quasi-periodic structures withn fold axes: Application to five and seven folds

A new geometrical method for generating aperiodic lattices forn-fold non-crystallographic axes is described. The method is based on the self-similarity principle. It makes use of the principles of gnomons to divide the basic triangle of a regular polygon of 2n sides to appropriate isosceles triangles and to generate a minimum set of rhombi required to fill that polygon. The method is applicable to anyn-fold noncrystallographic axis. It is first shown how these regular polygons can be obtained and how these can be used to generate aperiodic structures. In particular, the application of this method to the cases of five-fold and seven-fold axes is discussed. The present method indicates that the recursion rule used by others earlier is a restricted one and that several aperiodic lattices with five fold symmetry could be generated. It is also shown how a limited array of approximately square cells with large dimensions could be detected in a quasi lattice and these are compared with the unit cell dimensions of MnAl6 suggested by Pauling. In addition, the recursion rule for sub-dividing the three basic rhombi of seven-fold structure was obtained and the aperiodic lattice thus generated is also shown.