Normal approximation and fourth moment theorems for monochromatic triangles

Given a graph sequence denote by T₃(G_n) the number of monochromatic triangles in a uniformly random coloring of the vertices of G_n with colors. In this paper we prove a central limit theorem (CLT) for T₃(G_n) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T₃(G_n), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].     

Stability of Viscoelastic Wave Equation with Structural $\delta$-Evolution in $\mathbb{R}^{n}$

The aim of this paper is to study the Cauchy problem for the viscoelastic wave equation for structural $\delta$-evolution models. By using the energy method in the Fourier spaces, we obtain the decay estimates of the solution to considered problem.     

Uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings

This paper is concerned with the mixed initial–boundary value problem for semilinear wave equations with complementary frictional dampings and memory effects. We successfully establish uniform exponential and polynomial decay rates for the solutions to this initial–boundary value problem under much weak conditions concerning memory effects. More specifically, we obtain the exponential and polynomial decay rates after removing the fundamental condition that the memory-effect region includes a part of the system boundary, while the condition is a necessity in the previous literature; moreover, for the polynomial decay rates we only assume minimal conditions on the memory kernel function g, without the usual assumption of $g^{\prime }$ controlled by g.     

Rocky coast processes: with special reference to the recession of soft rock cliffs

Substantial progress in research on the recession of coastal cliffs composed of soft materials has been made in recent years and data with higher accuracy have been accumulated. This paper provides the state of the art review in the recession studies and highlights two new findings obtained from the reanalysis of existing data. The review topics are: episodic and localized nature of cliff recession; the development of cliffline; the relationship between cliff height and recession rate; mechanisms of cliff toe erosion by waves; a fundamental equation for wave-induced toe erosion; factors controlling toe erosion; and slope instabilities and mass movements. The findings are presented on (1) the temporal change in cliffline recession mode and (2) the effect of beach sediment at the cliff base on the cliff erosion.     

Precise measurement and control of the pressure-driven flows for microfluidic systems

The pressure-driven device is designed and the flow rates of the microfluidic systems can be supplied by the pressure-driven flows, which can significantly reduce the flow-rate fluctuations coming from the pump source. For pressure-driven flows, the flow rates of the fluids can be predicted by measuring the pressure drop along a polytetrafluoroethylene (PTFE) tubing. Especially, by varying the geometrical parameters of the PTFE tubing, the predicted flow rates of the fluids are compared with the experimental measurements, and the testing precision of the pressure-driven flows can be obtained. Meanwhile, the dynamic characteristics of the open-loop and closed-loop control pressure-driven device are comparatively studied. Particularly, a proportional and integral (PI) controller is integrated with the closed-loop control pressure-driven device, and the effects of the parameters of the PI controller on the dynamic characteristics of the pressure-driven devices are mainly discussed. Most importantly, by improving the dynamic characteristics of the pressure-driven devices, precise measurement and control of the pressure-driven flows can be achieved for microfluidic systems.     

Approximation by an iterative method of a low‐Mach model with temperature dependent viscosity

In this work, we prove the existence and the uniqueness of the strong solution of a low‐Mach model, for which the dynamic viscosity of the fluid is a given function of its temperature. The method is based on the convergence study of a sequence towards the solution, for which the rates are also given. The originality of the approach is to consider the system in terms of the temperature and the velocity, leading to a nonlinear temperature equation and the development of some specific tools and results.