Note on unicyclic graphs with given number of pendent vertices and minimal energy

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let G(n,p) denote the set of unicyclic graphs with n vertices and p pendent vertices. In [H. Hua, M. Wang, Unicyclic graphs with given number of pendent vertices and minimal energy, Linear Algebra Appl. 426 (2007) 478-489], Hua and Wang discussed the graphs that have minimal energy in G(n,p), and determined the minimal-energy graphs among almost all different cases of n and p. In their work, certain case of the values was left as an open problem in which the minimal-energy species have to be chosen in two candidate graphs, but cannot be determined by comparing of the corresponding coefficients of their characteristic polynomials. This paper aims at solving the problem completely, by using the well-known Coulson integral formula.     

Approximate solution of the system of nonlinear integral equation by Newton-Kantorovich method

The Newton-Kantorovich method is developed for solving the system of nonlinear integral equations. The existence and uniqueness of the solution are proved, and the rate of convergence of the approximate solution is established. Finally, numerical examples are provided to show the validity and the efficiency of the method presented.     

Well-posedness of a stochastic phase-field problem with multiplicative noises

In this paper, we prove the existence and uniqueness of the solution of a stochastic phase-field problem with multiplicative noises. Phase-field models are typically used to describe melting and solidification processes. We consider here the case of multiplicative noises induced by a Q-Brownian motion.     

Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

We study semi-linear elliptic PDEs with polynomial non-linearity in bounded domains and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives, we extend previous results in the literature by showing that our probabilistic representation provides a solution to the PDE without assuming its existence. In the general case, we derive a new representation of the solution by using marked branching diffusion processes and automatic differentiation formulas to account for the non-linear gradient term. We consider several examples and estimate their solution by using the Monte Carlo method.     

A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.     

A recurrence formula with respect to the Cameron–Storvick type theorem of the 𝒯-transform

ABSTRACT

In this paper, we define a transform which has the kernel in its definition and a concept of derivative for functionals on Wiener space. We then establish some results and formulas for the transforms of functionals on Wiener space. We also establish the Cameron–Storvick type theorem for the transform. Finally, we obtain the recurrence formula for the transforms to evaluate formulas involving the multi-dimensional derivative.     

On the Carey–Helton–Howe–Pincus trace formula

In this article, we give a new proof of the Carey–Helton–Howe–Pincus trace formula using Kato's theory of “relatively-smooth” operators and Krein's trace formula.     

泰勒公式及泰勒级数之妙用

泰勒公式及泰勒级数是非常重要的数学工具,除了读者熟知的应用方面外,在其他问题的解决中也有妙用.举例介绍了应用泰勒公式及泰勒级数解决判断级数的敛散性、证明与积分有关的不等式等问题.     

计算数学期望和概率的条件化方法

借助于条件数学期望和随机事件A的示性函数IA,通过对随机变量的适当"条件化"处理,应用全期望公式和推广的全概率公式,讨论了计算数学期望和概率的条件化方法.