Boolean products of lattices

A study is made of Boolean product representations of bounded lattices over the Stone space of their centres. Special emphasis is placed on relating topological properties such as clopen or regular open equalizers to their equivalent lattice theoretic counterparts. Results are also presented connecting various properties of a lattice with properties of its individual stalks.Research supported by the Natural Sciences and Engineering Research Council of Canada.Research supported by ONR Grant N00014-90-J-1008.     

The homology of the cyclic coloring complex of simple graphs

Let G be a simple graph on n vertices, and let χG(λ) denote the chromatic polynomial of G. In this paper, we define the cyclic coloring complex, Δ(G), and determine the dimensions of its homology groups for simple graphs. In particular, we show that if G has r connected components, the dimension of (n−3)rd homology group of Δ(G) is equal to (n−(r+1)) plus , where is the rth derivative of χG(λ). We also define a complex ΔC(G), whose r-faces consist of all ordered set partitions [B₁,…,Br+2] where none of the Bi contain an edge of G and where 1∈B₁. We compute the dimensions of the homology groups of this complex, and as a result, obtain the dimensions of the multilinear parts of the cyclic homology groups of C[x₁,…,xn]/{xixj|ij is an edge of G}. We show that when G is a connected graph, the homology of ΔC(G) has nonzero homology only in dimension n−2, and the dimension of this homology group is . In this case, we provide a bijection between a set of homology representatives of ΔC(G) and the acyclic orientations of G with a unique source at v, a vertex of G.     

Examination of Pt(111)/Ru and Pt(111)/Os surfaces: STM imaging and methanol oxidation activity

Ruthenium and osmium were deposited in submonolayer amounts on Pt(111) single crystal surfaces using the previously reported ‘spontaneous deposition’ procedure [Chrzanowski et al., Langmuir, 13 (1997) 5974]. Such surfaces were first explored using ex situ scanning tunneling microscopy (STM) to image the deposition characteristics of ruthenium and osmium islands on Pt(111). It was found that, using the spontaneous deposition procedure, a maximum coverage of 0.20 ML ruthenium is formed on the surface after 120 s of exposure to a RuCl₃ solution in 0.1 M HClO₄. A homogeneous deposition on the Pt(111) surface was found, with no observed preferential deposition on step edges or surface defect sites. In contrast, in the spontaneous deposition of osmium, osmium clusters form preferentially at, though not limited to, surface defect sites and step edges. Osmium island deposition occurs at a greater rate than ruthenium on Pt(111), and possible explanations are presented. Methanol activity on the Pt(111)/Ru and Pt(111)/Os surfaces is also studied, using the coverage values determined to yield the highest activity for methanol electro-oxidation (0.20 ML coverage for Ru and 0.15 ML for Os). At potentials more negative than 0.40 V vs. RHE, the Pt(111)/Ru surface yields a higher surface activity than Pt(111)/Os. However, at potentials more positive than 0.04 V, Pt(111)/Os exhibits demonstrably higher surface activity. The relevance of this data is discussed and future avenues of interest are indicated.     

A new PEC algorithm for the numerical solution of ordinary differential equations

Significant time reduction in obtaining numerical solutions of ordinary differential equations for which function evaluations are time consuming can be obtained with PEC methods as compared to PECE methods. In this report we present two PEC methods: a fourth-order algorithm for which stability characteristics and numerical examples are presented, and a second-order algorithm which is just mentioned. It is believed that PEC methods represent a useful addition to the library of solution techniques.     

The Hodge structure of the coloring complex of a hypergraph

Let G be a simple graph with n vertices. The coloring complex Δ(G) was defined by Steingrímsson, and the homology of Δ(G) was shown to be nonzero only in dimension n−3 by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group H_n−3(Δ(G)) where the dimension of the jth component in the decomposition, , equals the absolute value of the coefficient of λ^j in the chromatic polynomial of G, χ_G(λ).Let H be a hypergraph with n vertices. In this paper, we define the coloring complex of a hypergraph, Δ(H), and show that the coefficient of λ^j in χ_H(λ) gives the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of Δ(H). We also examine conditions on a hypergraph, H, for which its Hodge subcomplexes are Cohen–Macaulay, and thus where the absolute value of the coefficient of λ^j in χ_H(λ) equals the dimension of the jth Hodge piece of the Hodge decomposition of Δ(H). We also note that the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of the intersection of coloring complexes is given by the coefficient of jth term in the associated chromatic polynomial.     

The coloring complex and cyclic coloring complex of a complete k-uniform hypergraph

In this paper, we study the homology of the coloring complex and the cyclic coloring complex of a complete k-uniform hypergraph. We show that the coloring complex of a complete k-uniform hypergraph is shellable, and we determine the rank of its unique nontrivial homology group in terms of its chromatic polynomial. We also show that the dimension of the $(n?k?1)\text{st}$ homology group of the cyclic coloring complex of a complete k-uniform hypergraph is given by a binomial coefficient. Further, we discuss a complex whose r-faces consist of all ordered set partitions $[B_1,\dots ,B_{r+2}]$ where none of the $B_i$ contain a hyperedge of the complete k-uniform hypergraph H and where $1\in B_1$. It is shown that the dimensions of the homology groups of this complex are given by binomial coefficients. As a consequence, this result gives the dimensions of the multilinear parts of the cyclic homology groups of $\mathbb{C}[x_1,\dots ,x_n]/\{x_{i_1}\dots x_{i_k}|i_1\dots i_k\text{is a hyperedge of}H\}$.     

Density embedded VB/MM: a hybrid ab initio VB/MM with electrostatic embedding

A hybrid QM/MM method that combines ab initio valence-bond (VB) with molecular mechanics (MM) is presented. The method utilizes the ab initio VB approach to describe the reactive fragments and MM to describe the environment thus allows VB calculations of reactions in large biological systems. The method, termed density embedded VB/MM (DE-VB/MM), is an extension of the recently developed VB/MM method. It involves calculation of the electrostatic interaction between the reactive fragments and their environment using the electrostatic embedding scheme. Namely, the electrostatic interactions are represented as one-electron integrals in the ab initio VB Hamiltonian, hence taking into account the wave function polarization of the reactive fragments due to the environment. Moreover, the assumptions that were utilized in an earlier version of the method, VB/MM, to formulate the electrostatic interactions effect on the off-diagonal matrix elements are no longer required in the DE-VB/MM methodology. Using DE-VB/MM, one can calculate, in addition to the adiabatic ground state reaction profile, the energy of the diabatic VB configurations as well as the VB state correlation diagram for the reaction. The abilities of the method are exemplified on the identity SN2 reaction of a chloride anion with methyl chloride in aqueous solution. Both the VB configurations diagram and the state correlation diagram are presented. The results are shown to be in very good agreement with both experimental and other computational data, suggesting that DE-VB/MM is a proper method for application to different reactivity problems in biological systems.