Picard and weakly Picard operators technique for nonlinear differential equations in Banach spaces

In this paper we study nonlocal Cauchy problems for differential equations in Banach spaces. Using Picard and weakly Picard operators technique and suitable Bielecki norms, some existence, uniqueness and data dependence results are obtained under some mild conditions. As an application, we also discuss a class of impulsive Cauchy problems by adapting the same methods.     

Integrability and ergodicity of classical billiards in a magnetic field

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not trivially integrable can be described by a bouncing map. We compute a general expression for the Jacobian matrix of this map, which allows us to determine stability and bifurcation values of specific periodic orbits. In some cases, the bouncing map is a twist map and admits a generating function. We give a general form for this function which is useful to do perturbative calculations and to classify periodic orbits. We prove that billiards in convex domains with sufficiently smooth boundaries possess invariant tori corresponding to skipping trajectories. Moreover, in strong field we construct adiabatic invariants over exponentially large times. To some extent, these results remain true for a class of nonconvex billiards. On the other hand, we present evidence that the billiard in a square is ergodic for some large enough values of the magnetic field. A numerical study reveals that the scattering on two circles is essentially chaotic.     

Krull’s theory for the double gamma function

The Barnes’ G-function G(x) = 1/Γ₂, satisfies the functional equation logG(x + 1) − logG(x) = logΓ(x). We complement W. Krull’s work in Bemerkungen zur Differenzengleichung g(x + 1) − g(x) = φ(x), Math. Nachrichten 1 (1948), 365-376 with additional results that yield a different characterization of the function G, new expansions and sharp bounds for G on x > 0 in terms of Gamma and Digamma functions, a new expansion for the Gamma function and summation formulae with Polygamma functions.     

On a Construction of Fast Direct Solvers

Fast direct solvers for the Poisson equation with homogeneous Dirichlet and Neumann boundary conditions on special triangles and tetrahedra are constructed. The domain given is extended by symmetrization or skew symmetrization onto a rectangle or a rectangular parallelepiped and a fast direct solver is used there. All extendable domains are found. Eigenproblems are also considered.     

Observation of insulating–insulating monoclinic structural transition in macro‐sized VO2 single crystals [Phys. Status Solidi RRL 5, No. 3, R107–R109 (2011)]

In our article, we reported the observation of monoclinic M2 to M1 structural phase transition in VO₂ single crystal near the temperature of ～49 °C. However, the re‐examination of Laue patterns reveals that previously defined monoclinic M1 and M2 phases can be interpreted as monoclinic M2 and triclinic T phases instead. Careful experimental geometry calibration and further refinement of the lattice parameter ratios and angles show that monoclinic M2 and triclinic T phases fit better with the experimental data. On the other hand, our previous misidentification of the insulating phases does NOT affect the conclusions of our article. (© 2015 WILEY‐VCH Verlag GmbH &Co. KGaA, Weinheim)     

Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangle

A boundary value problem for the Laplace equation with Dirichlet and Neumann boundary conditions on an equilateral triangle is transformed to a problem of the same type on a rectangle. This enables us to use, e.g., the cyclic reduction method for computing the numerical solution of the problem. By the same transformation, explicit formulae for all eigenvalues and all eigenfunctions of the corresponding operator are obtained.     

Ornamental Plant Efficiency for Heavy Metals Phytoextraction from Contaminated Soils Amended with Organic Materials

Accumulation of heavy metals (HMs) by ornamental plants (OPs) from contaminated agriculture soils is a unique technique that can efficiently reduce the metal load in the food chain. Amaranthus tricolor L. has attractive characteristics acquiring a higher growth rate and large biomass when grown at heavy metal contaminated soils. Site-specific detailed information is not available on the use of A. tricolor plant in metal phytoremediation from the polluted sites. The study aimed to enhance the uptake of HMs (Pb, Zn, and Cu) via amending poultry litter extract (PLE), vinasse sugarcane (VSC), and humic acid (HA) as natural mobilized organic materials compared to ethylene diamine tetraacetic acid (EDTA), as a common mobilized chemical agent by A. tricolor plant. The studied soils collected from Helwan, El-Gabal El-Asfar (Cairo Governorate), Arab El-Madabeg (Assiut Governorate), Egypt, and study have been conducted under pot condition. Our results revealed all organic materials in all studied soils, except EDTA in EL-Gabal El-Asfar soil, significantly increased the dry weight of the A. tricolor plant compared to the control treatment. The uptake of Pb and Zn significantly (p > 0.05) increased due to applying all organic materials to the studied soils. HA application caused the highest uptake as shown in Pb concentration by more than 5 times in Helwan soil and EDTA by 65% in El-Gabal El-Asfar soil while VSC increased it by 110% in El-Madabeg soil. Also, an increase in Zn concentration due to EDTA application was 58, 42, and 56% for Helwan, El-Gabal El-Asfar, and El-Madabeg soil, respectively. In all studied soils, the application of organic materials increased the remediation factor (RF) than the control. El-Madabeg soil treated with vinasse sugarcane gave the highest RF values; 6.40, 3.26, and 4.02% for Pb, Zn, and Cu, respectively, than the control. Thus, we identified A. tricolor as a successful ornamental candidate that, along with organic mobilization amendments, most efficiently develop soil health, reduce metal toxicity, and recommend remediation of heavy metal-contaminated soils. Additionally, long-term application of organic mobilization amendments and continued growth of A. tricolor under field conditions could be recommended for future directions to confirm the results.     

DNA fragment separations by on‐line combination of capillary isotachophoresis–capillary zone electrophoresis with UV detection

A high‐speed DNA fragment separation system based on an on‐line combination of capillary ITP with CZE (CITP‐CZE) and using UV detection at 260 nm was developed. A novel CITP‐CZE buffer system of pH 6.1 was designed for the separation of ten DNA fragments with sizes ranging from 100 to 1000 bp. An effect of underivatized α‐, β‐ and γ‐cyclodextrins on the resolution of DNA fragments in the CZE step of the CITP‐CZE combination was systematically investigated. Methylhydroxyethylcellulose present in the BGE was used to eliminate the EOF. DNA ladder fragments were separated within 10 min with LODs in the range of 1–5 ng/μL (S/N = 3). The RSDs of the migration time and peak area of individual DNA fragments were in the range of 1–3 and 3–9%, respectively. The developed CITP‐CZE system was further applied to the analysis of digest plasmid DNA samples.     

Alphabet‐Inspired Design of (Hetero)Aromatic Push–Pull Chromophores

Push–pull molecules represent a unique and fascinating class of organic π‐conjugated materials. Herein, we provide a summary of their recent extraordinary design inspired by letters of the alphabet, especially focusing on H‐, L‐, T‐, V‐, X‐, and Y‐shaped molecules. Representative structures from each class were presented and their fundamental properties and prospective applications were discussed. In particular, emphasis is given to molecules recently prepared in our laboratory with T‐, X‐, and Y‐shaped arrangements based on indan‐1,3‐dione, benzene, pyridine, pyrazine, imidazole, and triphenylamine. These push–pull molecules turned out to be very efficient charge‐transfer chromophores with tunable properties suitable for second‐order nonlinear optics, two‐photon absorption, reversible pH‐induced and photochromic switching, photocatalysis, and intercalation.

     

Understanding the role of surfactants on the preparation of ZnS nanocrystals

We have synthesized surface modified ZnS nanoparticles of size 2-3 nm using non-ionic surfactant-stabilized reverse emulsions. The non-ionic surfactants in the Span series, i.e. sorbitan monolaurate (Span 20) and sorbitan monooleate (Span 80) of hydrophilic-lipophilic balance (HLB) values of 8.6 and 4.3, respectively, have been used for the stabilization of emulsions. The role of these surfactants in controlling the size and properties of the ZnS nanoparticles has been discussed. The triethylamine (TEA) has been proved to be the effective surface modifying (capping) agent for the preparation of free-standing ZnS nanoparticles. The Span 20 with the higher HLB value of 8.6 has been found to be highly suitable in synthesizing TEA-capped ZnS nanoparticles of smaller size and higher photophysical characteristics compared to that of the Span 80 of lower HLB value of 4.3. A mechanism for the formation of TEA-capped ZnS nanoparticles from the surfactant-stabilized reverse emulsions has been proposed.