Metal triangles as building blocks in metal cluster chemistry

Many trinuclear metal clusters have structures based on isolated metal triangles with either single bonds (e.g.,M ₃(CO)₁₂ whereM = Fe, Ru, Os) or double bonds (e.g., Re₃ Cl ₁₂ ^3– ) along each edge of the triangle. Individual metal triangles can be joined in the following ways to form more complicated triangulated networks: (1) Bridging an edge of a triangle with a new vertex to give rafts in which adjacent triangles share edges; (2) Bridging a vertex of a triangle with a new edge to give bowties in which adjacent triangles share vertices; (3) Capping a triangular face with a new vertex to give a chain of tetrahedra in which adjacent tetrahedra share faces. Such triangulated metal networks are particularly common in osmium carbonyl chemistry and in mixed osmium/platinum carbonyl derivatives. Platinum triangles of the type Pt₃L₆ are analogous to cyclopropenyl rings and can form sandwiches with one or two mercury atoms in the center such as the mercuric derivative Hg[Pt.₃(µ₂-2,6-Me₂C₆H₃NC)₃] (2,6-Me₂C₆H₃NC)₃]₂ and the mercurous derivative Hg₂[Pt₃(µ₂-CO)₃L₃]₂. Platinum triangles can also be stacked in the absence of filling to give [Pt₃(µ₂-CO)₃(CO)₃] _n ^2– (n=2, 3, 4, 5, 6, 10). Metal triangles also form the faces of metal deltahera of which the octahedron, bicapped square antiprism, and icosahedron are found in globally delocalized transition metal clusters.This article is dedicated to Prof. L. F. Dahl in recognition of his many seminal contributions to metal cluster chemistry.     

Duality Theorem for a Three-Phase Partition Problem

In some nonlinear diffusive phenomena, the systems have three or more stable states. Sternberg and Zeimer established the existence of minimal solutions for the problem of partitioning a certain domain Ω⊂ℝ² into three subdomains having least interfacial area. Ikota and Yanagida investigated stability and instability for stationary curves with one triple junction and for stationary binary-tree type interfaces. In this paper, we introduce a new concept of separation of three convex sets by a triangle, define a dual problem to the three-phase partition problem, and present a duality theorem. The author thanks Professor F. Giannessi for valuable comments, especially on Gale and Klee-type separation theorems. This research was partially supported by Kyushu University 21st Century COE Program (Development of Dynamic Mathematics with High Functionality) and by the Grant-in-Aid for General Scientific Research from the Japan Society for the Promotion of Science 14340037.     

NMR for equilateral triangular geometry under conditions of surface relaxivity—analytical and random walk solution

We consider analytical and numerical solution of NMR relaxation under the condition of surface relaxation in an equilateral triangular geometry. We present an analytical expression for the Green’s function in this geometry. We calculate the transverse magnetic relaxation without magnetic gradients present, single-phase, both analytically and numerically. There is a very good match between the analytical and numerical results. We also show that the magnetic signal from an equilateral triangular geometry is qualitatively different from the known solution: plate, cylinder, and sphere, in the case of a nonuniform initial magnetization. Nonuniform magnetization close to the sharp corners makes the magnetic signal very fast multiexponential. This type of initial configuration fits qualitatively with the experimental results by Song (Phys. Rev. Lett. 85, 3878 (2000)), Song et al. (Nature 406, 178 (2000)), Song (Mag. Reson. Imag. 19, 417 (2001)) and Lisitza and Song (Phys. Rev. B 65, 172406 (2002)). It should also be noted that the solution presented here can be used to describe absorption of a chemical substance in an equilateral triangular geometry (for a stationary fluid).     

An improvement of fixed point algorithms by using a good triangulation

We consider measures for triangulations ofR ⁿ. A new measure is introduced based on the ratio of the length of the sides and the content of the subsimplices of the triangulation. In a subclass of triangulations, which is appropriate for computing fixed points using simplicial subdivisions, the optimal one according to this measure is calculated and some of its properties are given. It is proved that for the average directional density this triangulation is optimal (within the subclass) asn goes to infinity. Furthermore, we compare the measures of the optimal triangulation with those of other triangulations. We also propose a new triangulation of the affine hull of the unit simplex. Finally, we report some computational experience that confirms the theoretical results.     

Geometric Characterizations for Variational Minimization Solutions of the 3-Body Problem

Abstract In this paper, we prove that for any given positive masses the variational minimization solutions of the 3-body problem in R ³ or R ² are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular restricted 3-body problem in R ³ or R ² are also planar equilateral triangle circular solutions. *Partially supported by the NNSF and MCME of China, the Qiu Shi Sci. and Tech. Foundation, and Edu. Comm. of Tianjin City. Associate Member of the ICTP. **Partially supported by the NNSF of China     

Programmable 3D Hexagonal Geometry of DNA Tensegrity Triangles

Non-canonical interactions in DNA remain under-explored in DNA nanotechnology. Recently, many structures with non-canonical motifs have been discovered, notably a hexagonal arrangement of typically rhombohedral DNA tensegrity triangles that forms through non-canonical sticky end interactions. Here, we find a series of mechanisms to program a hexagonal arrangement using: the sticky end sequence; triangle edge torsional stress; and crystallization condition. We showcase cross-talking between Watson–Crick and non-canonical sticky ends in which the ratio between the two dictates segregation by crystal forms or combination into composite crystals. Finally, we develop a method for reconfiguring the long-range geometry of formed crystals from rhombohedral to hexagonal and vice versa. These data demonstrate fine control over non-canonical motifs and their topological self-assembly. This will vastly increase the programmability, functionality, and versatility of rationally designed DNA constructs.     

Auslander-Reiten Triangles, Ziegler Spectra and Gorenstein Rings

We investigate (existence of) Auslander—Reiten triangles in a triangulated category in connection with torsion pairs, existence of Serre functors, representability of homological functors and realizability of injective modules. We also develop an Auslander—Reiten theory in a compactly generated triangulated category and we study the connections with the naturally associated Ziegler spectrum. Our analysis is based on the relative homological theory of purity and Brown's Representability Theorem. Our main interest lies in the structure of Auslander—Reiten triangles in the full subcategory of compact objects. We also study the connections and the interplay between Auslander—Reiten theory, pure-semisimplicity and the finite type property, Grothendieck groups, and we give applications to derived categories of Gorenstein rings.