Extensions of the Fill–Fishkind formula and the infimum – parallel sum relation

Recently, in (Arias ML, Corach G, Maestripieri A. Range additivity, shorted operator and the Sherman–Morrison–Woodburry formula, Linear Algebra Appl. 2015;467), authors gave a generalization of a formula by Fill and Fishkind, regarding the Moore–Penrose inverse of the sum of two matrices, in the setting of arbitrary Hilbert spaces. We consider this formula under some weaker assumptions and derive certain conclusions generalizing the mentioned result. We also extend a formula connecting the infimum of two orthogonal projections and their parallel sum to a formula connecting the star-infimum and the parallel sum of operators which need not be positive, using the concept of parallel sum that was introduced in (Antezana J, Corach G, Stojanoff D. Bilateral shorted operators and parallel sums. Linear Algebra Appl. 2006;414).     

Barzilai–Borwein method with variable sample size for stochastic linear complementarity problems

A smoothing method for solving stochastic linear complementarity problems is proposed. The expected residual minimization reformulation of the problem is considered, and it is approximated by the sample average approximation (SAA). The proposed method is based on sequential solving of a sequence of smoothing problems where each of the smoothing problems is defined with its own sample average approximation. A nonmonotone line search with a variant of the Barzilai–Borwein (BB) gradient direction is used for solving each of the smoothing problems. The BB search direction is efficient and low cost, particularly suitable for nonmonotone line search procedure. The variable sample size scheme allows the sample size to vary across the iterations and the method tends to use smaller sample size far away from the solution. The key point of this strategy is a good balance between the variable sample size strategy, the smoothing sequence and nonmonotonicity. Eventually, the maximal sample size is used and the SAA problem is solved. Presented numerical results indicate that the proposed strategy reduces the overall computational cost.     

Detection of protein deposits using NIR spectroscopy

The aim of this study was to determine if it is possible to distinguish between the groups of spoiled and unspoiled soft contact lenses using near-infrared spectroscopy and new analytical approach – Aquaphotomics. Using the principal component analysis, it was established that the absorbance spectra of worn and new contact lenses are differed at water absorption band related to hydration of proteins. Detection of proteins thus was performed indirectly by using vibrations of water molecules. This exploratory study showed that near-infrared spectroscopy and Aquaphotomics have potential for non-invasive, chemical-free detection of protein deposits on hydrated soft contact lenses.     

Asymptotic Behavior of the Fourth Painlevé Transcendents in the Space of Initial Values

We study the asymptotic behavior of solutions of the fourth Painlevé equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalization of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any solution that is not rational has an infinite number of poles and infinite number of zeros.     

Abelian actions with globally hypoelliptic leafwise Laplacian and rigidity

In this paper, we prove several results concerning smooth R^k actions on a smooth compact manifold with the property that their leafwise Laplacian is globally hypoelliptic. Such actions are necessarily uniquely ergodic and minimal, and their cohomology is often finite dimensional, even trivial. Further, we consider a class of examples of R² actions on two-step nilmanifolds that have globally hypoelliptic leafwise Laplacian, and we show transversal local rigidity under certain Diophantine conditions.     

Highly symmetric hypertopes

We study incidence geometries that are thin and residually connected. These geometries generalise abstract polytopes. In this generalised setting, guided by the ideas from the polytope theory, we introduce the concept of chirality, a property of orderly asymmetry occurring frequently in nature as a natural phenomenon. The main result in this paper is that automorphism groups of regular and chiral thin residually connected geometries need to be C-groups in the regular case and \({C^+}\)-groups in the chiral case.     

Hyperstrong neutral organic bases: phosphazeno azacalix[3](2,6)pyridines

The gas-phase basicities and pKa values in acetonitrile of tris(phosphazeno)-substituted azacalix[3](2,6)pyridine and some related compounds are examined by DFT computational method. It is shown that the hexakis(phosphazeno) derivative of azacalix[3](2,6)pyridine is a hyperstrong neutral base as evidenced by the absolute proton affinity (APA) of 314.6 kcal mol(-1) and pKa(MeCN) of 37.3 units. It is a consequence of the very strong bifurcated hydrogen bond (32 kcal mol(-1)) and substantial cationic resonance effect.     

Silver(I) mediated folding of a molecular basket

We have investigated Ag(I) mediated folding of a tridentate compound, containing three pyridine flaps tethered to a semirigid scaffold, into a molecular basket, using both experimental and theoretical methods. The basket formation has been shown to be highly favorable in organic media (Delta G degrees = -7.2 kcal/mol), with the assembly process allowing for another ligand to bind preferentially on the outer side.     

Hantzsch synthesis of 2,6-dimethyl-3,5-dimethoxycarbonyl-4-(o-methoxyphenyl)-1,4-dihydropyridine; a novel cyclisation leading to an unusual formation of 1-amino-2-methoxy-carbonyl-3,5-bis(o-methoxyphenyl)-4-oxa-cyclohexan-1-ene

Hantzsch condensation of two equivalents of methyl-3-aminocrotonate with (m- and p)-methoxybenzaldehyde afforded the expected products 2,6-dimethyl-3,5-dimethoxycarbonyl-4-(m-methoxyphenyl)-1,4-dihydropyridine and 2,6-dimethyl-3,5-dimethoxycarbonyl-4-(p-methoxyphenyl)-1,4-dihydropyridine, whereas o-methoxy-benzaldehyde produced mainly 1-amino-2-methoxycarbonyl-3,5-bis(o-methoxy-phenyl)-4-oxa-cyclohexan-1-ene. The structure of the product, not previously reported in the literature, was determined by 1D and 2D NMR spectra and its MS fragmentation. This is the first example of cyclisation leading to a substituted pyran rather than 1,4-DHP under typical Hantzsch reaction conditions. A plausible mechanism for its formation is postulated.