Characterization of intrinsic ultracontractivity for one dimensional Schrödinger operators

We consider a Schrödinger operator $L=?d^2/d{x}^2+V(x)$ on $\mathbb{R}$, where V is a real-valued measurable function, and give an explicit and simple characterization of intrinsic ultracontractivity (IU) of the Schrödinger semigroup generated by L for a wide class of potentials. By making use of it, we also give new examples of potentials for which the semigroups satisfy (IU) or non-(IU).     

Existence of homoclinic solutions for nonlinear second-order coupled systems

This work presents sufficient conditions for the existence of homoclinic solutions for second order coupled discontinuous systems of differential equations on the real line without the usual growth condition in the literature.The arguments apply the fixed point theory, Green's functions technique, $L^1$-Carathéodory functions, lower and upper solutions and Schauder's fixed point theorem.     

Boundedness of semilinear Duffing equations at resonance in a critical situation

In this paper, we propose a sufficient and necessary condition for the boundedness of all the solutions for the equation $\ddot{x}+n^{2}x+g(x)=p(t)$ with the critical situation that $|{\int }_0^{2\pi }p(t)e^{?int}dt|=2|g(+\infty )?g(?\infty )|$ on g and p, where $n\in {\mathbb{N}}^{+}$, $p(t)$ is periodic and $g(x)$ is bounded.     

Perturbative manifolds and the Noether generators of nth-order Poisson equations

A manifold that contains small perturbations will induce a perturbed partial differential equation. The partial differential equation that we select is the Poisson equation – in order to explore the interplay between the geometry of the manifold and the perturbations. Specifically, we show how the problem of symmetry determination, for higher-order perturbations, can be elegantly expressed via geometric conditions.     

Coleman automorphisms of finite groups and their minimal normal subgroups

In this paper, we show that all Coleman automorphisms of a finite group with self-central minimal non-trivial characteristic subgroup are inner; therefore the normalizer property holds for these groups. Using our methods we show that the holomorph and wreath product of finite simple groups, among others, have no non-inner Coleman automorphisms. As a further application of our theorems, we provide partial answers to questions raised by M. Hertweck and W. Kimmerle. Furthermore, we characterize the Coleman automorphisms of extensions of a finite nilpotent group by a cyclic p-group. Finally, we note that class-preserving Coleman automorphisms of p-power order of some nilpotent-by-nilpotent groups are inner, extending a result by J. Hai and J. Ge, where p is a prime number.     

General non local boundary value problem for second order elliptic equation

In this paper we study a class of second order coefficient operators differential equation with general (possibly non local) boundary conditions. We obtain new results extending those given in a previous paper 1 . Existence, uniqueness and optimal regularity of the strict solution are proved in UMD spaces, using the well‐known Dore–Venni theorem.     

Synthesis and Applications of SAPO-34 Molecular Sieves

Silicoaluminophosphate zeolite (SAPO-34) has been attracting increasing attention due to its excellent form selection and controllability in the chemical industry, as well as being one of the best industrial catalysts for methanol-to-olefin (MTO) reaction conversion. However, as a microporous molecular sieve, SAPO-34 easily generates carbon deposition and rapidly becomes inactivated. Therefore, it is necessary to reduce the crystal size of the zeolite or to introduce secondary macropores into the zeolite crystal to form a hierarchical structure in order to improve the catalytic effect. In this review, the synthesis methods of conventional SAPO-34 molecular sieves, hierarchical SAPO-34 molecular sieves and nanosized SAPO-34 molecular sieves are introduced, and the properties of the synthesized SAPO-34 molecular sieves are described, including the phase, morphology, pore structure, acid source, and catalytic performance, in particular with respect to the synthesis of hierarchical SAPO-34 molecular sieves. We hope that the review can provide guidance to the preparation of the SAPO-34 catalysts, and stimulate the future development of high-performance hierarchical SAPO-34 catalysts to meet the growing demands of the material and chemical industries.     

Synthesis and stability evaluation of hierarchical silicoaluminophosphates with different structural frameworks in the methanol to olefins process

Silicoaluminophosphates (SAPOs) with different pore structures were synthesized through the implementation of polyethylene glycol (PEG) as a mesopores impregnation agent. Using PEGs with different molecular weights (MWs) and concentrations in the synthesis precursor, several samples were synthesized and characterized. Applying a PEG capping agent to the precursors led to the formation of tuned mesopores within the microporous matrix of the SAPO. The effects of the PEG molecular weight and PEG/Al molar ratio were investigated to maximize the efficiency of the catalyst in the methanol-to-olefin (MTO) process. Using PEG with a MW of 6000 resulted in the formation of both Zeolite Rho and chabazite structural frameworks (i.e., DNL-6 and SAPO-34). Pure SAPO-34 samples were successfully prepared using PEG with a MW of 4000. Our results showed that the PEG concentrations affect the porosity and acidity of the synthesized materials. Furthermore, the SAPO-34 sample synthesized with PEG (MW of 4000) and a PEG/Al molar ratio of 0.0125 showed a superior catalytic stability in the MTO reaction owing to the tuned bi-modal porosity and tailored acidity pattern. Finally, through reactivation experiments, it was found that the catalyst is stable even after several regeneration cycles.     

One-pot two step synthesis of unsymmetrically substituted indenes from 3,4-diarylbutadiene sulfones

A new one-pot two step synthesis of unsymmetrically substituted indenes from available 3,4-diarylbutadiene sulfones involves SO₂ thermal extrusion followed by acid- catalyzed cyclization of the diene formed, the cyclization proceeding selectively at the more electron-rich aryl rings. The procedure is efficient for substrates bearing donor, acceptor, as well as bulky substituents.     

Synthesis of organosilyl compounds-containing 1,2,4,5-tetraaryl imidazoles sonocatalyzed by M/SAPO-34 (M = Fe,Co, Mn,and Cu) nanostructures

The one-step synthesis of silylated 1,2,4,5-tetraaryl imidazoles by use of a series of M/SAPO-34 (M: Fe, Co, Mn, and Cu) nanocatalysts and subsequent silylation reactions is described. Cu/SAPO-34 catalyst has the highest activity in improving the efficiency of the heterogeneous cyclo-condensation of an aldehyde, benzil, ammonium acetate and a primary aromatic amine in water under ultrasonic irradiation. Some of imidazole derivatives are studied with a view to the synthesis of a series of new, multi-substituted imidazoles containing organosilyl groups including carbosilanes (Si–C) and silyl ethers (Si–O).