Shell Partition and Metric Semispaces: Minkowski Norms, Root Scalar Products, Distances and Cosines of Arbitrary Order

Vector semispaces are studied from a realistic way with the intention to define a natural metric, adapted to their peculiar structure, which reside on the essential positive definiteness of their elements. From this point of view, Minkowski norms allow classifying semispaces in shells, that is: subsets where all the vector elements possess the same norm values. Shell structure appears to be a possible disjoint partition of any semispace and so shells become equivalence classes Then, the unit shell appears to be the core of the semispace homothetic construction as well as the origin of the semispace metrics. Minkowski or root scalar products permit to connect two or more semispace elements and conduct towards generalized definitions of Pth order root distances and cosines. Finally, the unit shell of a given semispace, in company of both Boolean tagged sets, inward matrix products and with the aid of the matrix signatures as well, it is seen as the seed to construct any arbitrary element of the semispace connected vector space. Finite and infinite dimensional vector spaces application examples are provided along the work discussion.     

Improving the accuracy of 1H–19F internuclear distance measurement using 2D 1H–19F HOESY

With the rise in fluorinated pharmaceuticals, it is becoming increasingly important to develop new ¹⁹F NMR-based methods to assist in their analysis. Crucially, obtaining information regarding the conformational dynamics of a molecule in solution can aid the design of strongly binding therapeutics. Herein, we report the development of a 2D ¹H–¹⁹F Heteronuclear Overhauser Spectroscopy (HOESY) experiment to measure ¹H–¹⁹F internuclear distances, with accuracies of ~5% when compared with ¹H–¹⁹F internuclear distances calculated by quantum chemical methods. We demonstrate that correcting for cross-relaxation of ¹H, using the diagonal peaks from the 2D ¹H–¹H Nuclear Overhauser Enhancement Spectroscopy (NOESY), is critical in obtaining accurate values for ¹H–¹⁹F internuclear distances. Finally, we show that by using the proposed method to measure ¹H–¹⁹F internuclear distances, we are able to determine the relative stereochemistry of two fluorinated pharmaceuticals.     

Electronic and structural properties of Ti9XO20 (X=Ti,C, si,Ge, Sn and pb) clusters: A DFT study

The electronic and structural properties of Ti₉XO₂₀ (X=Ti, C, Si, Ge, Sn and Pb) clusters have been obtained in the density functional theory (DFT) framework. The changes in the bond length, binding energy, frontier orbitals, and electronic potential have been fully analyzed when one titanium atom in the (TiO₂)₁₀ cluster is replaced by elements with four valence electrons. When one titanium atom is substituted by one carbon atom, a charge excess among the guest and the surrounding oxygen atoms is generated, which is approximately 1.5 times that of the pristine case, and this structure has been shown to be the most stable among the studied systems. In addition, the Ti₁₀O₂₀–Cd₂ and Ti₉CO₂₀–Cd₂ clusters exhibit HOMO–LUMO gaps that have decreased by 0.58 and 2.12 eV, respectively, with respect to the bare cases.     

Prediction of wood property in Chinese Fir based on visible/near-infrared spectroscopy and least square-support vector machine

A method for the quantification of density of Chinese Fir samples based on visible/near-infrared (vis–NIR) spectrometry and least squares-support vector machine (LS-SVM) was proposed. Sample set partitioning based on joint x–y distances (SPXY) algorithm was used for dividing calibration and prediction samples, it is of value for prediction of property involving complex matrices. A stepwise procedure is employed to select samples according to their differences in both x (instrumental responses) and y (predicted parameter) spaces. For comparison, the models were also constructed by Kennard–Stone method, as well as by using the duplex and random sampling methods for subset partitioning. The results revealed that the SPXY algorithm may be an advantageous alternative to the other three strategies. To validate the reliability of LS-SVM, comparisons were made among other modeling methods such as support vector machine (SVM) and partial least squares (PLS) regression. Satisfactory models were built using LS-SVM, with lower prediction errors and superior performance in relation to SVM and PLS. These results showed possibility of building robust models to quantify the density of Chinese Fir using near-infrared spectroscopy and LS-SVM combined SPXY algorithm as a nonlinear multivariate calibration procedure.     

Ricci curvature and measures

In the last thirty years three a priori very different fields of mathematics, optimal transport theory, Riemannian geometry and probability theory, have come together in a remarkable way, leading to a very substantial improvement of our understanding of what may look like a very specific question, namely the analysis of spaces whose Ricci curvature admits a lower bound. The purpose of these lectures is, starting from the classical context, to present the basics of the three fields that lead to an interesting generalisation of the concepts, and to highlight some of the most striking new developments. This article is based on the 5th Takagi Lectures that the author delivered at the University of Tokyo on October 4 and 5, 2008.     

On the distribution of distances between specified nodes in increasing trees

We study the quantity distance between nodejand nodenin a random tree of sizen chosen from a family of increasing trees. For those subclass of increasing tree families, which can be constructed via a tree evolution process, we give closed formulæ for the probability distribution, the expectation and the variance. Furthermore we derive a distributional decomposition of the random variable considered and we show a central limit theorem of this quantity, for arbitrary labels 1≤j<n and n→∞.Such tree models are of particular interest in applications, e.g., the widely used models of recursive trees, plane-oriented recursive trees and binary increasing trees are special instances and are thus covered by our results.     

Weighted distances in scale‐free preferential attachment models

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a nonnegative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely two universality classes of weight distributions, called the explosive and conservative class. In the explosive class, we show that the typical weighted distance converges in distribution to the sum of two i.i.d. finite random variables. In the conservative class, we prove that the typical weighted distance tends to infinity, and we give an explicit expression for the main growth term, as well as for the hopcount. Under a mild assumption on the weight distribution the fluctuations around the main term are tight.     

Interior proximal methods and central paths for convex second-order cone programming

We make a unified analysis of interior proximal methods of solving convex second-order cone programming problems. These methods use a proximal distance with respect to second-order cones which can be produced with an appropriate closed proper univariate function in three ways. Under some mild conditions, the sequence generated is bounded with each limit point being a solution, and global rates of convergence estimates are obtained in terms of objective values. A class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution. These results are illustrated with some examples. In addition, we also study the central paths associated with these distance-like functions, and for the linear SOCP we discuss their relations with the sequence generated by the interior proximal methods. From this, we obtain improved convergence results for the sequence for the interior proximal methods using a proximal distance continuous at the boundary of second-order cones.