Recyclable MoO3 nanobelts for photocatalytic degradation of Rhodamine B by near infrared irradiation

Molybdenum trioxide (MoO₃) represented an excellent photocatalytic performance with many applications, including degradation of organic contaminants and splitting of water. This paper presented a new route to synthesize MoO₃ nanobelts with high aspect ratios and crystallinity by a hydrothermal technique. This work showed that the as-synthesized nanobelts exhibited strong photocatalytic activity to degrade an organic dye of Rhodamine B (RhB) in aqueous solution under the exposure of the light source in the near infrared wavelength range, significantly improving the photocatalytic activity of the nanobelts. The results also showed that for a small concentration of RhB at 7.5 mg/L a complete photodegradation (for a given MoO₃ nanobelts quantity of 0.1 g) can be reached after exposing for 60 min. For all concentrations of the RhB solution, the photodegradation exhibited an exponential dependence on the exposure time followed by a sudden shutdown, but no complete photodegradation can be reached. Also, the residual quantity of RhB in solution after the photocatalytic reaction was determined by the initial RhB concentration. The photocatalytic degradation can be interpreted by the pseudo–first-order equation for the absorption of liquid/solid based on solid capacity; thus, photocatalytic degradation can be attributed to the interaction between the photoexcited electrons in the substrate and the antibonding orbital of the RhB in solution. The sudden shutdown was due to the inability of the photoexcited electrons in the substrate hopping to the antibonding orbital of RhB in the presence of the RhB intermediate products from the degraded RhB. In addition, this work showed that the photocatalytic reaction can be recovered after a thermal treatment of postreacted MoO₃ nanobelts, enhancing the utilization efficiency of the catalysis.     

Transition probabilities of Lévy‐type processes: Parametrix construction

We present an existence result for Lévy‐type processes which requires only weak regularity assumptions on the symbol with respect to the space variable x. Applications range from existence and uniqueness results for Lévy‐driven SDEs with Hölder continuous coefficients to existence results for stable‐like processes and Lévy‐type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of Lévy and Lévy‐type processes. The paper includes an extensive list of Lévy(‐type) processes satisfying the assumptions of our results.     

Sharp oscillation and nonoscillation tests for delay dynamic equations

In this paper, we give new sufficient conditions for both oscillation and nonoscillation of the delay dynamic equation where and satisfy τ(t) ≤ σ(t) for all large t and . As an important corollary, we obtain the time scale invariant integral condition for nonoscillation: for all large t. Also, with some examples, we show that newly presented results are sharp.     

PDα‐type distributed learning control for nonlinear fractional‐order multiagent systems

This paper considers the consensus tacking problem for nonlinear fractional‐order multiagent systems by presenting a PD^α‐type iterative learning control update law with initial learning mechanisms. The asymptotical convergence of the proposed distributed learning algorithm is strictly proved by using the properties of fractional calculus. A sufficient condition is derived to guarantee the whole multiagent system achieving an asymptotic output consensus. An illustrative example is given to verify the theoretical results.     

Improving the approximation of the probability density function of random nonautonomous logistic‐type differential equations

In this paper, we address the problem of approximating the probability density function of the following random logistic differential equation: P^′(t,ω)=A(t,ω)(1?P(t,ω))P(t,ω), t∈[t₀,T], P(t₀,ω)=P₀(ω), where ω is any outcome in the sample space Ω. In the recent contribution [Cortés, JC, et al. Commun Nonlinear Sci Numer Simulat 2019; 72: 121–138], the authors imposed conditions on the diffusion coefficient A(t) and on the initial condition P₀ to approximate the density function f₁(p,t) of P(t): A(t) is expressed as a Karhunen–Loève expansion with absolutely continuous random coefficients that have certain growth and are independent of the absolutely continuous random variable P₀, and the density of P₀, , is Lipschitz on (0,1). In this article, we tackle the problem in a different manner, by using probability tools that allow the hypotheses to be less restrictive. We only suppose that A(t) is expanded on L²([t₀,T]×Ω), so that we include other expansions such as random power series. We only require absolute continuity for P₀, so that A(t) may be discrete or singular, due to a modified version of the random variable transformation technique. For , only almost everywhere continuity and boundedness on (0,1) are needed. We construct an approximating sequence of density functions in terms of expectations that tends to f₁(p,t) pointwise. Numerical examples illustrate our theoretical results.     

A problem of determining a special spatial part of 3D memory kernel in an integro‐differential hyperbolic equation

The inverse problem of determining 2D spatial part of integral member kernel in integro‐differential wave equation is considered. It is supposed that the unknown function is a trigonometric polynomial with respect to the spatial variable y with coefficients continuous with respect to the variable x. Herein, the direct problem is represented by the initial‐boundary value problem for the half‐space x>0 with the zero initial Cauchy data and Neumann boundary condition as Dirac delta function concentrated on the boundary of the domain . Local existence and uniqueness theorem for the solution to the inverse problem is obtained.