Use of exchange maximization to generate starting vectors for self-consistent field calculations on metal cluster/adsorbate systems

Localized molecular orbitals (LMOs) derived from exchange maximization with respect to all atom-centered basis functions in the basis set are shown to generate a good starting electronic field for self-consistent field calculations on extended systems such as metal clusters, for which well-defined chemical bonds are not present. Examples studied are a cluster of 20 Ni atoms and the Pt(97)CO, Ag(43)/H(3)CNON, Ag(91)/H(2)CO, and vinylidene/Ni metal cluster plus adsorbate systems. It is also shown that improved starting vectors can be obtained by remixing a subset of the LMOs with the largest exchange eigenvalues through diagonalization of the Fock matrix computed with a null electronic field. Employing only a subset of the exchange-maximized LMOs in the first iterations, and then gradually expanding the space in which the diagonalizations are carried out in succeeding cycles, is shown to be an effective means of guiding the SCF procedure to the converged full-basis solution.     

On normalized Laplacians,multiplicative degree-Kirchhoff indices,and spanning trees of the linear [n]phenylenes and their dicyclobutadieno derivatives

Let be the molecular graph of the linear [n] phenylene with n hexagons and n − 1 squares, and let be the graph obtained by attaching four-membered rings to the terminal hexagons of . In this article, the normalized Laplacian spectrum of consisting of the eigenvalues of two symmetric tridiagonal matrices of order 3n is determined. An explicit closed-form formula of the multiplicative degree-Kirchhoff index (respectively the number of spanning trees) of is derived. Similarly, explicit closed-form formulas of the multiplicative degree-Kirchhoff index and the number of spanning trees of are obtained. It is interesting to see that the multiplicative degree-Kirchhoff index of (respectively ) is approximately to one half of its Gutman index.     

Boundaries of rotation sets for homeomorphisms of the -torus

We construct a diffeomorphism of the 3-torus whose rotation set is not closed. We prove that the rotation set of a homeomorphism of the -torus contains the extreme points of its closed convex hull. Finally, we show that each pseudo-rotation set is closed for torus homeomorphisms.

     

On Mad and Comedians

A popular robust measure of dispersion of a random variable (rv) X is the median absolute deviation from the median med(|X - med(X)|), MAD for short, which is based on the median med(X) of X. By choosing Y = X, the MAD turns out to be a special case of the comedian med((X - med(X))(Y - med(Y))), which is a robust measure of covariance between rvs X and Y. We investigate the comedian in detail, in particular in the normal case, and establish strong consistency and asymptotic normality of empirical counterparts. This leads to a robust competitor of the coefficient of correlation as an asymptotic level--statistic for testing independence of X and Y. An example shows the weird fact that knowledge of the population med(X) does not necessarily pay (in the sense of asymptotic relative efficiency) when estimating the MAD.     

Percolation and Minimal Spanning Trees

Consider a random set of points in the box [n, –n) ^d , generated either by a Poisson process with density p or by a site percolation process with parameter p. We analyze the empirical distribution function F _n of the lengths of edges in a minimal (Euclidean) spanning tree on . We express the limit of F _n, as n , in terms of the free energies of a family of percolation processes derived from by declaring two points to be adjacent whenever they are closer than a prescribed distance. By exploring the singularities of such free energies, we show that the large-n limits of the moments of F _n are infinitely differentiable functions of p except possibly at values belonging to a certain infinite sequence (p _c(k): k 1) of critical percolation probabilities. It is believed that, in two dimensions, these limiting moments are twice differentiable at these singular values, but not thrice differentiable. This analysis provides a rigorous framework for the numerical experimentation of Dussert, Rasigni, Rasigni, Palmari, and Llebaria, who have proposed novel Monte Carlo methods for estimating the numerical values of critical percolation probabilities.     

Typology of secondary cyanobacterial metabolites from minimum spanning tree analysis

Recently, two main events have spurred a rapid increase in cyanobacteria chemical, toxicological, and ecological research. The first deals with the interest in isolating compounds from these organisms as source of active products with potential therapeutic applications. The second pertains the crucial problem of harmful cyanobacterial blooms in the aquatic environments.

In this context, 594 secondary metabolites belonging to more than 30 genera of cyanobacteria were retrieved from literature. In order to perform their typology, they were first associated with 87 different molecular archetypes and two orphan classes. These 89 groups of molecular structures were then confronted to minimum spanning tree analysis. Attempts were made to graphically derive chemotaxonomical relationships. The interest of QSAR models for estimating the potential pharmacological interest of the cyanobacterial secondary metabolites was also discussed.     

有向网络中具有一个枢纽点的最小支撑树的计算方法

对有向网络中具有一个枢纽点的支撑树的问题和性质进行了研究,给出了在有向网络图中寻找以某一定点为枢纽点的最小支撑树的计算方法,并对算法的复杂性进行了讨论,最后将该算法应用于实际算例的计算．     

几类图的支撑树的个数

推广了计算图的支撑树个数的递归公式,解释了组合计数原理的用法.用组合技巧和常系数线性递归序列的解法,对n步梯、n-棱柱、Mobius n-棱柱及有关图,找到了计算它们的支撑树的个数的若干公式.     

Spanning Forests and the q-State Potts Model in the Limit q →0

We study the q-state Potts model with nearest-neighbor coupling v=e^βJ−1 in the limit q,v → 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of spanning forests; physically, it provides information about the Potts-model phase diagram in the neighborhood of (q,v) = (0,0). We have studied this model on the square and triangular lattices, using a transfer-matrix approach at both real and complex values of w. For both lattices, we have computed the symbolic transfer matrices for cylindrical strips of widths 2≤ L ≤ 10, as well as the limiting curves B of partition-function zeros in the complex w-plane. For real w, we find two distinct phases separated by a transition point w=w₀, where w₀ =−1/4 (resp. w₀=−0.1753 ± 0.0002) for the square (resp. triangular) lattice. For w>w₀ we find a non-critical disordered phase that is compatible with the predicted asymptotic freedom as w → +∞. For w0 our results are compatible with a massless Berker–Kadanoff phase with central charge c=−2 and leading thermal scaling dimension x_T,1 = 2 (marginally irrelevant operator). At w=w₀ we find a “first-order critical point”: the first derivative of the free energy is discontinuous at w₀, while the correlation length diverges as w↓ w₀ (and is infinite at w=w₀). The critical ehavior at w=w₀ seems to be the same for both lattices and it differs from that of the Berker–Kadanoff phase: our results suggest that the central charge is c=−1, the leading thermal scaling dimension is x_T,1=0, and the critical exponents are ν=1/d=1/2 and α=1.     

The robust minimum spanning tree problem: Compact and convex uncertainty

We consider the robust minimum spanning tree problem where edges costs are on a compact and convex subset of Rⁿ. We give the location of the robust deviation scenarios for a tree and characterizations of strictly strong edges and non-weak edges leading to recognition algorithms.