On Applications of Quasi-Monotone Sequences and Quasi Power Increasing Sequences

In this article, we prove a general theorem dealing with an application of quasi-f-power increasing sequences and δ-quasi monotone sequences. This theorem also includes some known and new results.     

Unconditionally Superclose Analysis of a New Mixed Finite Element Method for Nonlinear Parabolic Equation

This paper develops a framework to deal with the unconditional superclose analysis of nonlinear parabolic equation. Taking the finite element pair $Q_{11}/Q_{01} × Q_{10}$ as an example, a new mixed finite element method (FEM) is established and the $τ$ -independent superclose results of the original variable $u$ in $H^1$-norm and the flux variable $\mathop{q} \limits ^{\rightarrow}= −a(u)∇u$ in $L^2$-norm are deduced ($τ$ is the temporal partition parameter). A key to our analysis is an error splitting technique, with which the time-discrete and the spatial-discrete systems are constructed, respectively. For the first system, the boundedness of the temporal errors is obtained. For the second system, the spatial superclose results are presented unconditionally, while the previous literature always only obtain the convergent estimates or require certain time step conditions. Finally, some numerical results are provided to confirm the theoretical analysis, and show the efficiency of the proposed method.     

Malliavin and Dirichlet structures for independent random variables

On any denumerable product of probability spaces, we construct a Malliavin gradient and then a divergence and a number operator. This yields a Dirichlet structure which can be shown to approach the usual structures for Poisson and Brownian processes. We obtain versions of almost all the classical functional inequalities in discrete settings which show that the Efron–Stein inequality can be interpreted as a Poincaré inequality or that the Hoeffding decomposition of $U$-statistics can be interpreted as an avatar of the Clark representation formula. Thanks to our framework, we obtain a bound for the distance between the distribution of any functional of independent variables and the Gaussian and Gamma distributions.     

Sharp Reilly-type inequalities for a class of elliptic operators on submanifolds

Let M be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form ${\mathbb{R}}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on M defined by $L_{T}f=-\text{div}(T\mathrm{\nabla }f)$, where T is a general symmetric, positive definite and divergence-free $(1,1)$-tensor on M. The upper bound is given in terms of an integration involving tr T and $|H_T{|}^2$, where tr T is the trace of the tensor T and $H_T={\sum }_{i=1}^{n}A(T{e}_i,e_i)$ is a normal vector field associated with T and the second fundamental form A of M. Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous “Reilly inequality”. The operator $L_T$ can be regarded as a natural generalization of the well-known operator $L_r$ which is the linearized operator of the first variation of the $(r+1)$-th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of Grosjean [17] and Li–Wang [20] in codimension one to arbitrary codimension.     

In situ green synthesis of Cu‐Ni bimetallic nanoparticles supported on reduced graphene oxide as an effective and recyclable catalyst for the synthesis of N‐benzyl‐N‐aryl‐5‐amino‐1H‐tetrazoles

In the present work, for the first time we have designed a novel approach for the synthesis of N‐benzyl‐N‐aryl‐5‐amino‐1H‐tetrazoles using reduced graphene oxide (rGO) decorated with Cu‐Ni bimetallic nanoparticles (NPs). In situ synthesis of Cu/Ni/rGO nanocomposite was performed by a cost efficient, surfactant‐free and environmentally benign method using Crataegus azarolus var. aronia L. leaf extract as a stabilizing and reducing agent. Phytochemicals present in the extract can be used to reduce Cu²⁺ and Ni²⁺ ions and GO to Cu NPs, Ni NPs and rGO, respectively. Analyses by means of FT‐IR, UV–Vis, EDS, TEM, FESEM, XRD and elemental mapping confirmed the Cu/Ni/rGO formation and also FT‐IR, NMR, and mass spectroscopy as well as elemental analysis were used to characterize the tetrazoles. The Cu/Ni/rGO nanocomposite showed the superior catalytic activity for the synthesis of N‐benzyl‐N‐aryl‐5‐amino‐1H‐tetrazoles within a short reaction time and high yields. Furthermore, this protocol eliminates the need to handle HN₃.     

Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Let be the class of all sense‐preserving homeomorphic self‐mappings of . The aim of this paper is twofold. First, we obtain Heinz‐type inequality for (K,K′)‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δω) = g in unit disk with associated boundary value conditions and . Second, we establish biLipschitz continuity for (K,K′)‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when and are small enough.