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.     

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.     

Carbon Paste Composite with Co3O4 as a New Electrochemical Sensor for the Detection of Allura Red by Reduction

The detection of Allura red (AR) by electrochemical reduction using a different electrode from the conventional mercury electrode is presented. A carbon paste with cobalt (II, III) oxide composite electrode (CoO_x/CPE) is reported for the first time for the detection of AR. Moreover, others dyes such as tartrazine (TZ), sunset yellow (SY), amaranth (AM), Ponceaut 4‐R (P‐4R), and Sudan (SD) as well as pharmaceutical agents such as paracetamol (PMC) that are present in samples that contained AR did not show a reduced signal between 0.0 and ?0.3 V, which is the potential range where AR reduction was observed. The surface electroactivity was studied using cyclic voltammetry (CV), electrochemical impedance spectroscopy (EIS) and scanning electron microscopy (SEM). The presence of CoO_X increased the cathodic peak current for AR by more than 50 % and 65 % via CV and square wave voltammetry (SWV), respectively, compared with an unmodified carbon paste electrode. Under the optimal parameters, (pH=3.0, accumulation time (t_ACC)=60 s and accumulation potential (E_ACC)=0.50 V), the detection limit for AR was 0.05 μmol L^?1. The new sensor was sensitive and stable for the detection of AR. Moreover, it was easily manufactured and very convenient for food samples such as soft and isotonic drinks as well as chili sauce.     

Robustly topological mixing of Kan's map

In 1994, I. Kan constructed a smooth map on the annulus admitting two physical measures, whose basins are intermingled. In this paper, we prove that Kan's map is $C^2$ robustly topologically mixing.     

Singular solutions of elliptic equations on a perturbed cone

In this work we obtain positive singular solutions of

$\{\begin{array}{ccc}?\mathrm{\Delta }u(y)\hfill & \hfill =\hfill & u{(y)}^p\text{ in }y\in {\mathrm{\Omega }}_t,\hfill \\ u\hfill & \hfill =\hfill & 0\text{ on }y\in ?{\mathrm{\Omega }}_t,\hfill \end{array}$

where ${\mathrm{\Omega }}_t$ is a sufficiently small $C^{2,\alpha }$ perturbation of the cone $\mathrm{\Omega }:=\{x\in {\mathbb{R}}^N:x=r\theta ,r>0,\theta \in S\}$ where $S?S^{N?1}$ has a smooth nonempty boundary and where $p>1$ satisfies suitable conditions. By singular solution we mean the solution is singular at the ‘vertex of the perturbed cone’. We also consider some other perturbations of the equation on the unperturbed cone Ω and here we use a different class of function spaces.     

Frame properties of systems arising via iterated actions of operators

Motivated by recent progress in dynamical sampling we prove that every frame which is norm-bounded below can be represented as a finite union of sequences ${\{{(T_j)}^n{\phi }_j\}}_{n=0}^{\infty },j=1,\dots ,J$ for some bounded operators $T_j$ and elements ${\phi }_j$ in the underlying Hilbert space. The result is optimal, in the sense that it turns out to be problematic to replace the collection of generators ${\phi }_1,\dots ,{\phi }_J$ by a singleton: indeed, for linearly independent frames we prove that we can represent the frame in terms of just one system ${\{T^n\phi \}}_{n=0}^{\infty }$, but unfortunately this representation often forces the operator T to be unbounded. Several examples illustrate the connection of the results to typical frames like Gabor frames and wavelet frames, as well as generic constructions in arbitrary separable Hilbert spaces.