Topological Space of the Chemical Elements and its Properties

We carried out a mathematical study of 72 chemical elements taking advantage of the chemotopological method. We selected 128 properties to define the elements (physico-chemical, geochemical and chemical properties). Then, we looked for correlated properties and we reduced the number of them to 90. In this way we defined each element as a 90-tuple. Afterwards, we applied principal component analysis and cluster analysis (4 similarity functions and 5 grouping methodologies). Then, we calculated a consensus tree for the 20 dendrograms generated by the CA. Afterwards, we extracted the similarity relationships from the consensus tree and built up a basis for a topology on the set of chemical elements. Finally, we calculated some topological properties (closures, derived sets, boundaries, interiors and exteriors) of several subsets of chemical elements. We found that alkali metals, alkaline earth metals and noble gases appear not related to the rest of the elements. Also, we found that the boundary of non-metals are the semimetals with a stair-shape on the periodic table     

Cobalt Ion Incorporation into Periodic Mesoporous Organosilica Materials Synthesized by Co-condensation of 1,2-Bistrimethoxysilylethane with 3-Glycidoxypropyltriethoxysilane

Bifunctional periodic mesoporous organosilica materials with and without cobalt ion incorporation were synthesized by co-condensation of 1,2-bistrimethoxysilylethane (BTME) with 3-glycidoxypropyltriethoxysilane (GPTS) in the presence of cetyltrimethylammonium bromide. Nitrogen gas adsorption on samples with varying ratios of BTME:GPTS revealed that increasing the amount of GPTS affects pore size, surface area and pore volume as well as shapes of the isotherms and hysteresis loops. The hysteresis loops of the Type IV isotherms obtained for GPTS-modified ethane silica materials (without cobalt ion) change from Type H3 to Type H4 with increasing GPTS content. There is a tendency for pore sizes to change from mesopore to micropore when the amount of GPTS is increased. Isotherms of cobalt ion incorporated GPTS-modified ethane silica materials change from Type IV to Type I with increasing GPTS content. The surface area, pore volume and pore diameter decrease with increasing loading of GPTS as well as after cobalt ion incorporation. Thermogravimetric analysis and differential thermal analysis show that the surfactant is removed by solvent extraction. Cobalt ion incorporation is confirmed by powder X-ray diffraction and Raman spectroscopy.     

Analysis of critical points on the potential energy surface

A simple classification scheme is proposed for critical points, based only on rankr and signatures of the (n,n)-matrixG of harmonic force constants. The determination ofr ands, e.g. by the well-known factorizationG=L ^T gL (L: triangular matrix,g: diagonal matrix), has several theoretical as well as practical (computational) advantages over the inspection of eigenvalues ofG, so far used in quantum chemistry. The eigenvalues are sufficient butnot necessary for a classification whereas rank and signature are the only necessary and sufficient prerequisites for solving the task. For the purpose of presenting a working example, by calculating only a 2×2 torque constant matrix, it is shown that the coplanar ethylbenzene is unstable in the CNDO/2 picture.     

The generalized order complementarity problem

Given an ordered Banach Space (E,K) andm functionsf ₁,f ₂,...,f _m:EE, the generalized order complementarity problem associated with {f _i} andK is to findx ₀K such thatf _i(x ₀)K,i=1,...,m, and (x ₀,f ₁(x ₀),...,f _m(x ₀))=0. The problem is shown to be equivalent to several fixed-point problems and equivalent to the order complementarity problem studied by Borwein and Dempster and by Isac. Existence and uniqueness of solutions and least-element theory are shown in the spacesC(, ) andL _p(, ). For general locally convex spaces, least-element theory is derived, existence is proved, and an algorithm for computing a solution is presented. Applications to the mixed lubrication theory of fluid mechanics are described.     

Existence of maximal points with respect to ordered bipreference relations

The maximality of a point with respect to an ordered pair of arbitrary preference relations is introduced. Necessary and sufficient conditions for the existence of maximal points are given.The authors are grateful for the referee's comments which led to an improved presentation of the paper.     

Lipschitzian inverse functions,directional derivatives,and applications inC 1,1 optimization

The paper shows that Thibault's limit sets allow an iff-characterization of local Lipschitzian invertibility in finite dimension. We consider these sets as directional derivatives and extend the calculus in a way that can be used to clarify whether critical points are strongly stable inC ^1,1 optimization problems.Many fruitful discussions with colleagues D. Klatte and K. Tammer as well as with H. Th. Jongen and F. Nozicka have influenced the present investigations in a very constructive manner. For the original papers concerning the sets f(x; u), the author is indebted to Prof. L. Thibault.     

Uniformities for the convergence in law and in probability

The convergence (X _n, Y_n)0 is investigated and characterized for probability metrics which metrize convergence in distribution or in probability. Some related metrics are also considered.     

Rare numbers

Suppose thatX ₁,X ₂,... is a sequence of iid random variables taking values inZ ⁺. Consider the random sequenceA(X)(X ₁,X ₂,...). LetY _n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofn ^–(1–) Y _n for [0, 1].     

Critical Points in a Crystal and Procrystal

The critical points in the model electron density distributions of LiF, NaF, NaCl, and MgO crystals, constructed from accurate X-ray diffraction data, are determined. For LiF and MgO they are compared with those obtained from a Hartree–Fock electron density calculation. Both experiment and theory show the same type of critical points on the bond lines. The topological features in areas between structural units, where the electron density is low and near-uniform, turn out to be model dependent and cannot be established well with the data available. Topological analysis of procrystals (hypothetical systems consisting of spherical atoms or ions placed on the same sites as atoms in real crystal) show that (3, –1) critical points, usually connected with bonding interaction, are observed on interatomic lines in these nonbonded systems as well.     

Properties of the random search in global optimization

From theorems which we prove about the behavior of gaps in a set ofN uniformly random points on the interval [0, 1], we determine properties of the random search procedure in one-dimensional global optimization. In particular, we show that the uniform grid search is better than the random search when the optimum is chosen using the deterministic strategy, that a significant proportion of large gaps are contained in the uniformly random search, and that the error in the determination of the point at which the optimum occurs, assuming that it is unique, will on the average be twice as large using the uniformly random search compared with the uniform grid. In addition, some of the properties of the largest gap are verified numerically, and some extensions to higher dimensions are discussed. The latter show that not all of the conclusions derived concerning the inadequacies of the one-dimensional random search extend to higher dimensions, and thaton average the random search is better than the uniform grid for dimensions greater than 6.This paper is based on work started in the Statistics Department of Princeton University when the first author was visiting as a Research Associate. Part of this research was supported by the Office of Naval Research, Contract No. 0014-67-A-0151-0017, and by the US Army Research Office—Durham, Contract No. DA-31-124-ARO-D-215.2 The authors wish to thank B. Omodei for his careful work in preparing the programs for the results of Figs. 1–2 and Table 1. The computations were performed on the IBM 360/50 of the Australian National University's Computer Centre. Thanks are also due to R. Miles for suggestions regarding the extension of the results to multidimensional regions, and to P. A. P. Moran and R. Brent for suggestions regarding the evaluation of the integral ₀ ¹ ... ^0/1 (x ₁ ² + ... +x _n ^/2 )^1/2 dx ₁ ...dx _n.