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.     

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.     

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.     

Automorphisms of Hilbert schemes of points on a generic projective K3 surface

We study automorphisms of the Hilbert scheme of n points on a generic projective K3 surface S, for any . We show that is either trivial or generated by a non‐symplectic involution and we determine numerical and divisorial conditions which allow us to distinguish between the two cases. As an application of these results we prove that, for any 348-24cd170585f7/mana201800557-math-0003.png">, there exist infinitely many admissible degrees for the polarization of the K3 surface S such that admits a non‐natural involution. This provides a generalization of the results of [7] for .     

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

We study the well‐posedness of the fractional differential equations with infinite delay on Lebesgue–Bochner spaces and Besov spaces , where A and B are closed linear operators on a Banach space X satisfying ,  344b-45cb-bd77-31694df35b10/mana201800104-math-0005.png"> and . Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of in the above vector‐valued function spaces on by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied.     

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.     

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 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 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).     

A new method for exact solutions of variant types of time‐fractional Korteweg‐de Vries equations in shallow water waves

The current article devoted on the new method for finding the exact solutions of some time‐fractional Korteweg–de Vries (KdV) type equations appearing in shallow water waves. We employ the new method here for time‐fractional equations viz. time‐fractional KdV‐Burgers and KdV‐mKdV equations for finding the exact solutions. We use here the fractional complex transform accompanied by properties of local fractional calculus for reduction of fractional partial differential equations to ordinary differential equations. The obtained results are demonstrated by graphs for the new solutions. Copyright © 2016 John Wiley & Sons, Ltd.