Multicolor Combination Lemma

We present an extension of the Combination Lemma of Guibas et al. (1983) that expresses the complexity of one or several faces in the overlay of many arrangements (as opposed to just two arrangements in (Guibas et al. 1989)), as a function of the number of arrangements, the number of faces, and the complexities of these faces in the separate arrangements. Several applications of the new Combination Lemma are presented. We first show that the complexity of a single face in an arrangement of k simple polygons with a total of n sides is Θ(n(k)), where (·) is the inverse of Ackermann's function. We also give a new and simpler proof of the bound on the total number of edges of m faces in an arrangement of n Jordan arcs, each pair of which intersect in at most s points, where λ_s(n) is the maximum length of a Davenport–Schinzel sequence of order s with n symbols. We extend this result, showing that the total number of edges of m faces in a sparse arrangement of n Jordan arcs is , where w is the total complexity of the arrangement. Several other related results are also obtained.     

一类拟线性薛定谔方程非平凡解的存在性

本文研究一类拟线性薛定谔方程解的存在性问题.利用山路引理和集中紧性原理,得到该问题的一个非平凡解,推广和完善了已有的结果.     

Moving Intersticial Gaps

In a countable, recursively saturated model of Peano Arithmetic, an interstice is a maximal convex set which does not contain any definable elements. The interstices are partitioned into intersticial gaps in a way that generalizes the partition of the unbounded interstice into gaps. Continuing work of Bamber and Kotlarski [1], we investigate extensions of Kotlarski's Moving Gaps Lemma to the moving of intersticial gaps.     

On the structure of certain valued fields

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the n-th residue rings are isomorphic for each $n\ge 1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1\ge 1$, there is $n_2$ depending only on the ramification indices of $K_1$ and $K_2$ such that any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuation rings. Moreover, we get a functor from the category of certain principal Artinian local rings of length n to the category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. Our lifting result improves Basarab's relative completeness theorem for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields.     

Average degrees of edge-chromatic critical graphs

Let $G$ be a simple graph, and let $\Delta \left(G\right)$, $\overline{d}\left(G\right)$ and ${\chi }^{\prime }\left(G\right)$ denote the maximum degree, the average degree and the chromatic index of $G$, respectively. We called $G$ edge-$\Delta$-critical if ${\chi }^{\prime }\left(G\right)=\Delta \left(G\right)+1$ and ${\chi }^{\prime }\left(H\right)\le \Delta \left(G\right)$ for every proper subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is an edge-$\Delta$-critical graph of order $n$, then $\overline{d}\left(G\right)\ge \Delta \left(G\right)?1+\frac{3}{n}$. We prove that for any edge-$\Delta$-critical graph $G$, $\stackrel{?}{d}\left(G\right)\ge min\left\{\frac{2\sqrt{2}\Delta \left(G\right)?3?\sqrt{2}}{2\sqrt{2}+1},\frac{3\Delta \left(G\right)}{4}?2\right\},$ that is, $\overline{d}\left(G\right)\ge \left\{\begin{array}{cc}\frac{3}{4}\Delta \left(G\right)?2\hfill & \text{if}\Delta \left(G\right)\le 75;\hfill \\ \frac{2\sqrt{2}\Delta \left(G\right)?3?\sqrt{2}}{2\sqrt{2}+1}\approx 0.7388\Delta \left(G\right)?1.153\hfill & \text{if}\Delta \left(G\right)\ge 76.\hfill \end{array}\right.$This result improves the best known bound $\frac{2}{3}(\Delta \left(G\right)+2)$ obtained by Woodall in 2007 for $\Delta \left(G\right)\ge 41$.     

Blow‐up dynamics of L2−critical inhomogeneous fractional nonlinear Schrödinger equation

The purpose of this work is to investigate the blow‐up dynamics of L²?critical focusing inhomogeneous fractional nonlinear Schrödinger equation: with 0<b<1. For this, we establish a new compactness lemma related to the equation. By applying this lemma, we study the dynamical behavior for blow‐up solutions for initial data satisfying , where Q is the ground state solution of our problem.     

Fatou's identity and Lebesgue's convergence theorem

The classical Fatou lemma for bounded sequences of nonnegative integrable functions is represented as an equality. A similar result is stated for measure convergent sequences. Neither result requires a uniform integrability assumption. For the latter a converse is proven. Two extensions of Lebesgue's convergence theorem are presented.