一类非线性分数阶薛定谔-泊松系统非零解的存在性

本文研究了分数阶薛定谔-泊松系统$$\left\{\begin{array}{l}(-\Delta)^su+u+\phi u=\lambda f(u)\ \text {in} \ \mathbb {R}^3, \\ (-\Delta)^{\alpha}\phi =u^2\ \text {in} \ \mathbb {R}^3\emph{},\end{array}\right. $$ 非零解的存在性, 其中$s\in (\frac{3}{4},1), \alpha\in(0,1),\lambda$ 是正参数, $(-\Delta)^s,(-\Delta)^{\alpha}$是分数阶拉普拉斯算子. 在一定的假设条件下, 利用扰动法和Morse迭代法, 得到了系统至少一个非平凡解.     

Two solutions for a class of singular Kirchhoff‐type problems with Hardy–Sobolev critical exponent II

In this article, we devote ourselves to investigate the following singular Kirchhoff‐type equation: $$\left\{\begin{array}{l}?\left(a+b{\int }_{\mathrm{\Omega }}|?u{|}^{2}dx\right)\mathrm{\Delta }u=\frac{u^{5?2s}}{|x{|}^s}+\frac{\lambda }{|x{|}^{\beta }{u}^{\gamma }},x\in \mathrm{\Omega },\\ u>0,x\in \mathrm{\Omega },\\ u=0,x\in ?\mathrm{\Omega },\end{array}\right.$$ where $\mathrm{\Omega }?{?}^3$ is a bounded domain with smooth boundary ?Ω,0∈Ω,a≥0,b,λ>0,0<γ,s<1, and $0\le \beta <\frac{5+\gamma }{2}.$ By using the variational and perturbation methods, we obtain the existence of two positive solutions, which generalizes and improves the recent results in the literature.     

Improving the accuracy of discontinuous Galerkin schemes at boundary layers

The resolution of boundary layers typically requires fine grids in the wall normal direction, which leads to anisotropic elements being refined towards the wall. Best practice guidelines for the mesh generation of stretched boundary layer grids exist for the finite volume or finite difference discretizations. A similar resolution of boundary layers with DG schemes can be achieved with a coarser grid because of the subgrid resolution of the DG scheme. High order schemes incorporate the possibility of high order element mappings, resulting in different resolution properties inside the element. In this paper, we show that the use of an internal element mapping in combination with a stretched grid can be used to reduce the error of the boundary layer approximation by an order of magnitude in comparison with the classical linear internal element mapping. The boundary layer is modeled by a one‐dimensional singular perturbation problem. In addition, we discuss the construction of the element mappings by interpolation and investigate the limits of the stretching function such that the resulting element Jacobian remains positive. A parameter study shows the influence of the element mapping for different polynomial degrees on the solution. Copyright © 2014 John Wiley & Sons, Ltd.     

The Hopf Algebra Structure of Connes and Kreimer in Epstein–Glaser Renormalization

We show how the Hopf algebra structure of renormalization discovered by Kreimer can be found in the Epstein–Glaser framework without using an analogue of the forest formula of Zimmermann.     

Pionic and electromagnetic decays ofb-hadrons

The chiral perturbation theory has been used to calculate pionic and radiative decays ofb-hadrons. These particles are being studied and their decays have been predicted.     

Singular Index Pairs

Using Conley's idea of slow exit points, we construct index pairs for singularity perturbed families of lows. Some applications are also presented.     

Positivity and Convergence in Fermionic Quantum Field Theory

We derive norm bounds that imply the convergence of perturbation theory in fermionic quantum field theory if the propagator is summable and has a finite Gram constant. These bounds are sufficient for an application in renormalization group studies. Our proof is conceptually simple and technically elementary; it clarifies how the applicability of Gram bounds with uniform constants is related to positivity properties of matrices associated to the procedure of taking connected parts of Gaussian convolutions. This positivity is preserved in the decouplings that also preserve stability in the case of two-body interactions.     

Calculation of the optical characteristics of atoms with a filled shell on the basis of the Hartree-Fock method

For He, Be, Ne, and Mg atoms and their isoelectronic series the dipole dynamic polarizability, the oscillator energies, and the strengths of lower transitions are calculated with the use of the nonstationary perturbation theory in the Hartree-Fock-Roothaan method. The problem of choosing the dimension and composition of the basis of atomic orbitals of the Slater type is studied, within the framework of which the Hartree-Fock limit of the quantities calculated is reached. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 67, No. 1, pp. 5–10, January–February, 2000.