New two stages high order symmetric six-step method with vanished phase–lag and its first,second and third derivatives for the numerical solution of the Schrödinger equation

In the present paper, we obtain and analyze, for the first time in the literature, a new two-stages high order symmetric six-step method. The specific characteristics of the new proposed method are the highest possible algebraic order, the elimination of the phase–lag and its first, second and third derivatives. Additionally, for the new method we give the analysis of the method (both error and stability and interval of periodicity analysis) and the comparison of the effectiveness of the new developed method with the effectiveness of well known methods and very recently produced methods in the literature. The comparison is based on the numerical solution of the Schrödinger equation. The theoretical achievements and the numerical results show the effectiveness of the new developed method in comparison with other well known or recently developed numerical methods.     

An efficient and computational effective method for second order problems

An efficient and computational effective algorithm is introduced, for the first time in the literature, in the present paper. The main properties of the scheme are: (1) the algorithm is a two-step scheme, (2) the algorithm is symmetric one, (3) it is a hight algebraic order scheme (i.e of eight algebraic order), (4) it is a three-stages algorithm, (5) the first layer of the new method is based on an approximation to the point \(x_{n-1}\), (6) the scheme has vanished phase-lag and its first, second and third derivatives, (7) the new proposed algorithm has an interval of periodicity equal to \(\left( 0, 9.8 \right) \). For the present new scheme we study: (1) its construction, (2) its error analysis (3) its stability analysis. Finally, the investigation of the effectiveness of the new algorithm leads to its application to systems of differential equations arising from the Schrödinger equation.     

On the effectiveness of exponential type orbitals with hyperbolic cosine functions in atomic calculations

Unconventional basis functions, constructed from exponential type orbitals (ETOs) with hyperbolic cosine functions, are applied to Roothaan-Hartree-Fock calculations of atoms within the minimal basis sets framework. The most popular ETOs, Slater type orbitals, B functions and \(\psi ^{(\alpha ^*)}\) functions with \(\alpha ^*=2\), and two types of hyperbolic cosine functions, \(\cosh (\beta r)\) and \(\cosh (\beta r+\gamma )\), are used in this work. The performance of the present basis functions is investigated and compared to the conventional double-zeta Slater-type basis set and numerical Hartree-Fock results. The improvement in the atomic energies clearly demonstrates how the accuracy increases when we move from ETO to ETO with hyperbolic cosine basis functions. The resulting improved minimal basis sets can also be useful in molecular calculations.     

Hydrogenation of diacetyl over composite-supported egg-shell noble metal catalysts

Two composite supports with a mixed inorganic–organic structure were synthesized: BTAl and UTAl. Hydrophilic–hydrophobic dual properties of the supports were suitable for preparing egg-shell-supported metal catalysts for selective hydrogenation reactions. The catalysts were characterized by ICP, XRD, OM, TEM, EPMA, XPS and TGA. Their mechanical resistance was assessed. Activity and selectivity were tested with the hydrogenation of 2,3-butanedione (diacetyl) to 3-hydroxy-2-butanoneacetoin (acetoin). The same order of increasing metal particle size was found for the two tested supports: Pt < Ru < Pd. The XPS analysis showed that the metal/composite catalysts reduced in H2 at 503 K had two kinds of active sites: reduced (Me°) and electron-deficient (Me⁺). It was rationalized that the hydrogen bond cleavage was performed on the Me° active sites, while reactant adsorption occurred on the Me⁺ sites. The differences in activity and selectivity between the composite catalysts were attributed to electronic effects on the different metals and to different adsorptive properties of the different polymers. The high selectivity to acetoin was attributed to the preferential adsorption of diacetyl as compared to the adsorption of acetoin. The BTAl catalysts were slightly more active and selective than the UTAl ones. This was attributed to electronic effects caused by remnant organic groups on the composite supports (urethane or biphenyl on UTAl or BTAl, respectively). Pd-BTAl was the most active and selective catalyst, a fact related to electronic effects of both palladium and the support.     

Fluorene-containing tetraphenylethylene molecules as lasing materials

A series of star-shaped oligofluorene molecules, each containing a TPE core, have been specifically designed and produced to show effective aggregation-induced emission (AIE). Each molecule differs either in the number of fluorene units within the arms (e.g., 1 or 4, compounds 4 and 5 ), or the terminal group positioned at the end of each arm (e.g., H, TMS, or TPA, compounds 4 , 6 , and 7 ). Although they are all poor emitters in solution phase they become efficient yellow-green luminogens in the condensed state. Their AIE properties were investigated in THF/H₂O mixtures, with each molecule exhibiting a clear emission enhancement at specific water contents. An all-organic distributed feedback (DFB) laser was fabricated using compound 4 as the gain material and exhibited an average threshold energy fluence of 60 ± 6 μJ/cm² and emission in the green region. Furthermore, piezofluorochromism studies on a thin film of this material displayed a linear dependence of the amplified spontaneous emission (ASE) peak position on applied pressure, indicating potential applications as lasing-based pressure sensors. © 2016 The Authors. Journal of Polymer Science Part A: Polymer Chemistry Published by Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2017 , 55, 734–746     

Optical activity of films based on chitosan of various molecular masses and modifications

The effect of chemical (polysalt → polybase reaction) and physicochemical (heat and vapor processing) modifications of the films of chitosan of various molecular masses and prehistories on the optical activity of the polysaccharide is studied. For both of the chitosan chemical modifications, the following dependence that is nontrivial for high-molecular compounds is established: [α] = f(log\({\bar M_\eta }\)); as the degree of polymerization decreases, the modulus of [α] of the films increases. The X-ray diffraction study shows that the differences in the optical activity of the samples with different \({\bar M_\eta }\) are caused by a restriction in the mobility of the macromolecular sections upon formation of the films of high-molecular chitosan and incompleteness of the relaxation processes of the film systems to the state with energetically favorable conformations. The comparison of humidity and optical and biological activities of the initial and dehydrated chitosan films is performed. The chitosan films in the salt modification feature moderate bactericidal activity, which decreases with an increase in \({\bar M_\eta }\) of the polymer. The thermal processing of the salt modification reduces the antibacterial action; a conversion to the base form is accompanied by the loss of bactericidal activity.     

Study of oriented polyethylene structures using a Raman longwave acoustic mode

This paper concerns the basics of using longwave (low-frequency) Raman spectroscopy in the region of a longitudinal acoustic mode for the study of structural features of polymers. Using ultrahighmolecular- weight polyethylene (UHMW PE)-based reactor powders, gels, and oriented fibers obtained by the gel technology, the length distribution of straight-chain segments and its relation to the morphology of the sample were analyzed. An important role of the low-frequency Raman spectroscopy in creating domestic gel technology for preparing ultrahigh-strength threads from polyethylene was demonstrated.     

On parallel Branch and Bound frameworks for Global Optimization

Branch and Bound (B&B) algorithms are known to exhibit an irregularity of the search tree. Therefore, developing a parallel approach for this kind of algorithms is a challenge. The efficiency of a B&B algorithm depends on the chosen Branching, Bounding, Selection, Rejection, and Termination rules. The question we investigate is how the chosen platform consisting of programming language, used libraries, or skeletons influences programming effort and algorithm performance. Selection rule and data management structures are usually hidden to programmers for frameworks with a high level of abstraction, as well as the load balancing strategy, when the algorithm is run in parallel. We investigate the question by implementing a multidimensional Global Optimization B&B algorithm with the help of three frameworks with a different level of abstraction (from more to less): Bobpp, Threading Building Blocks (TBB), and a customized Pthread implementation. The following has been found. The Bobpp implementation is easy to code, but exhibits the poorest scalability. On the contrast, the TBB and Pthread implementations scale almost linearly on the used platform. The TBB approach shows a slightly better productivity.     

Stable iterative Lagrange principle in convex programming as a tool for solving unstable problems

A convex programming problem in a Hilbert space with an operator equality constraint and a finite number of functional inequality constraints is considered. All constraints involve parameters. The close relation of the instability of this problem and, hence, the instability of the classical Lagrange principle for it to its regularity properties and the subdifferentiability of the value function in the problem is discussed. An iterative nondifferential Lagrange principle with a stopping rule is proved for the indicated problem. The principle is stable with respect to errors in the initial data and covers the normal, regular, and abnormal cases of the problem and the case where the classical Lagrange principle does not hold. The possibility of using the stable sequential Lagrange principle for directly solving unstable optimization problems is discussed. The capabilities of this principle are illustrated by numerically solving the classical ill-posed problem of finding the normal solution of a Fredholm integral equation of the first kind.