Existence and uniqueness of global weak solutions to a Cahn-Hilliard-Stokes-Darcy system for two phase incompressible flows in karstic geometry

We study the well-posedness of a coupled Cahn-Hilliard-Stokes-Darcy system which is a diffuse-interface model for essentially immiscible two phase incompressible flows with matched density in a karstic geometry. Existence of finite energy weak solution that is global in time is established in both 2D and 3D. Weak-strong uniqueness property of the weak solutions is provided as well

The extremal genus embedding of graphs

Let Wn be a wheel graph with n spokes. How does the genus change if adding a degree-3 vertex v, which is not in V (Wn), to the graph Wn? In this paper, through the joint-tree model we obtain that the genus of Wn+v equals 0 if the three neighbors of v are in the same face boundary of P(Wn); otherwise, {\deg}(Wn + v) = 1, where P(Wn) is the unique planar embedding of Wn. In addition, via the independent set, we provide a lower bound on the maximum genus of graphs, which may be better than both the result of D. Li & Y. Liu and the result of Z. Ouyang etc: in Europ. J. Combinatorics. Furthermore, we obtain a relation between the independence number and the maximum genus of graphs, and provide an algorithm to obtain the lower bound on the number of the distinct maximum genus embedding of the complete graph Km, which, in some sense, improves the result of Y. Caro and S. Stahl respectively

Supported ITZ modification efficiencies via surface coating nanoparticles on aggregate and its influence on properties

In order to modify the porous interfacial transition zone (ITZ) microstructure of concrete more efficiently, a method of coating aggregate surfaces by using several nanoparticles was evaluated in this study. The compressive strength, chloride penetration of sound, and pre-loading samples were assessed in relation to the type of coating materials used (slag, nano-CaCO3, and nano-SiO2) and the designed coating thickness (5, 10, and 15 mu m). The ITZ microstructure was quantitatively determined via Backscattered electron (BSE) image analysis. Results showed that the overall performance of concrete is highly dependent on the coating materials and the designed coating thickness. Increasing the coating thickness of slag and nano-SiO2 could improve the chloride penetration resistance but decrease the compressive strength. Using nano-CaCO3 to coat the aggregate leads to a significant reduction in the properties of the so-prepared concrete. Though coating inert fine particles around aggregate could disturb the initial particle packing and modify the ITZ, it is not able to improve the overall concrete properties. Coating aggregate could determine the ITZ microstructure, especially within the region that is around 30 mu m away from aggregate surface

Hypergeometric SLE with κ=8\kappa=8: Convergence of UST and LERW in Topological Rectangles

We consider uniform spanning tree (UST) in topological rectangles with alternating boundary conditions. The Peano curves associated to the UST converge weakly to hypergeometric SLE$_8$, denoted by hSLE$_8$. From the convergence result, we obtain the continuity and reversibility of hSLE$_8$ as well as an interesting connection between SLE$_8$ and hSLE$_8$. The loop-erased random walk (LERW) branch in the UST converges weakly to SLE$_2(-1, -1; -1, -1)$. We also obtain the limiting joint distribution of the two end points of the LERW branch.Comment: 43 pages, 6 figures. We added Proposition 1.4 and Corollary 1.7 in V2. We also added references [Dub06] and [KW11] in V

A Monotone Discretization for the Fractional Obstacle Problem

We propose a monotone discretization method for obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over a bounded Lipschitz domain. Our approach is motivated by the success of the monotone discretization of the fractional Laplacian [SIAM J. Numer. Anal. 60(6), pp. 3052-3077, 2022]. By exploiting the problem's unique structure, we establish the uniform boundedness, existence, and uniqueness of the numerical solutions. Moreover, we employ the policy iteration method to efficiently solve discrete nonlinear problems and prove its convergence after a finite number of iterations. The improved policy iteration, adapted to the regularity result, exhibits superior performance by modifying the discretization in different regions. Several numerical examples are provided to illustrate the effectiveness of our method.Comment: 18 pages, 7 figure