The Σ1-provability logic of HA

In this paper we introduce a modal theory ${\mathsf{iH}}_{\sigma }$ which is sound and complete for arithmetical ${\mathrm{\Sigma }}_1$-interpretations in $\mathsf{HA}$, in other words, we will show that ${\mathsf{iH}}_{\sigma }$ is the ${\mathrm{\Sigma }}_1$-provability logic of $\mathsf{HA}$. Moreover we will show that ${\mathsf{iH}}_{\sigma }$ is decidable. As a by-product of these results, we show that $\mathsf{HA}+\square \perp$ has de Jongh property.     

Extremes of q-Ornstein–Uhlenbeck processes

Two limit theorems are established on the extremes of a family of stationary Markov processes, known as $q$-Ornstein–Uhlenbeck processes with $q\in (?1,1)$. Both results are crucially based on the weak convergence of the tangent process at the lower boundary of the domain of the process, a positive self-similar Markov process little investigated so far in the literature. The first result is the asymptotic excursion probability established by the double-sum method, with an explicit formula for the Pickands constant in this context. The second result is a Brown–Resnick-type limit theorem on the minimum process of i.i.d. copies of the $q$-Ornstein–Uhlenbeck process: with appropriate scalings in both time and magnitude, a new semi-min-stable process arises in the limit.     

Smooth distribution group and Schrödinger equation in Lp(Rn)

In College de France, Seminaire E.D.P. II, Nov. 1963–May 1964, Peetre has introduced the smooth distribution semi-group which is discussed here. This notion is redefined by introducing a functional space $\text{T}$, which measures the regularity of such a distribution. This allows us to give a spectral characterization of such groups. We show that the iterated resolvent of the infinitesimal generator of a smooth distribution group satisfies the relation

$\text{6(λ?A)}{}^{\mathrm{?q}}\text{6L(X)?cq}{}^{\mathrm{k}}\text{|λ|}{}^{\mathrm{k}}\text{|}$

, Re λ ≠ 0. The application of this notion is illustrated by the study of the Schrödinger equation in L^p($\text{R}$ⁿ).     

Classification of α-Ricci flat graphs with girth at least five

Lin, Lu and Yau classified Ricci flat graphs with girth at least 5 in Lin et al., 2014 [7] and Cushing et al., 2018 [4,5]. In Lin et al., 2014, they defined $\alpha$-Ricci curvature for the definition of Ricci curvature. We will classify $\alpha$-Ricci flat graphs with girth at least 5 for all $\alpha \in (0,1)$.     

The size of a graph is reconstructible from any n−2 cards

Let $G$ and $H$ be graphs of order $n$. The number of common cards of $G$ and $H$ is the maximum number of disjoint pairs $(v,w)$, where $v$ and $w$ are vertices of $G$ and $H$, respectively, such that $G?v?H?w$. We prove that if the number of common cards of $G$ and $H$ is at least $n?2$ then $G$ and $H$ must have the same number of edges when $n\ge 29$. This is the first improvement on the $25$-year-old result of Myrvold that if $G$ and $H$ have at least $n?1$ common cards then they have the same number of edges. It also improves on the result of Woodall and others that the numbers of edges of $G$ and $H$ differ by at most $1$ when they have $n?2$ common cards.     

A characterization for the neighbor-distinguishing total chromatic number of planar graphs with Δ=13

The neighbor-distinguishing total chromatic number ${\chi }_a^{\prime \prime }\left(G\right)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be totally colored using $k$ colors with a condition that any two adjacent vertices have different sets of colors. In this paper, we give a sufficient and necessary condition for a planar graph $G$ with maximum degree 13 to have ${\chi }_a^{\prime \prime }\left(G\right)=14$ or ${\chi }_a^{\prime \prime }\left(G\right)=15$. Precisely, we show that if $G$ is a planar graph of maximum degree 13, then $14\le {\chi }_a^{\prime \prime }\left(G\right)\le 15$; and ${\chi }_a^{\prime \prime }\left(G\right)=15$ if and only if $G$ contains two adjacent 13-vertices.     

The benefit of preemption with respect to the ℓp norm

We establish tight bounds on the benefit of preemption with respect to the ${?}_p$ norm minimization objective for identical machines and for two uniformly related machines (based on their speed ratio). This benefit of preemption is the supremum ratio between the optimal costs of non-preemptive and preemptive schedules.     

Multiplicity and concentration behaviour of positive solutions for Schrödinger-Kirchhoff type equations involving the p-Laplacian in RN

In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of Schrödinger-Kirchhoff type

$-{\in }^{p}M\left({\in }^{p-N}{\int }_{{\mathbb{R}}^N}{\left|?u\right|}^p\right){\Delta }_{p}u+V\left(x\right){\left|u\right|}^{p-2}u=f\left(u\right)$

in ${\mathbb{R}}^N$, where Δ_p is the p-Laplacian operator, 1 < p < N, M: ${\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ and V: ${\mathbb{R}}^{\text{N}}\to {\mathbb{R}}^{+}$ are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik-Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.     

Nodal standing waves for a gauged nonlinear Schrödinger equation in R2

This paper investigates the existence and asymptotic behavior of nodal solutions to the following gauged nonlinear Schrödinger equation

$\{\begin{array}{c}?\mathrm{\Delta }u+\omega u+(\frac{h^2(\left|x\right|)}{{\left|x\right|}^2}+\underset{\left|x\right|}{\overset{+\infty }{\int }}\frac{h(s)}{s}{u}^2(s)ds)u=\lambda {\left|u\right|}^{p?2}u,x\in {\mathbb{R}}^2,\hfill \\ u(x)=u(|x|)\in H^1({\mathbb{R}}^2),\hfill \end{array}$

where $\omega ,\lambda >0$, $p>6$ and

$h(s)=\frac{1}{2}\underset{0}{\overset{s}{\int }}r{u}^2(r)dr$

is the so-called Chern–Simons term. We prove that for any positive integer k, the problem has a sign-changing solution ${u^{\lambda }}_k$ which changes sign exactly k times. Moreover, the energy of $u_k^{\lambda }$ is strictly increasing in k, and for any sequence $\{{\lambda }_n\}\to +\infty (n\to \infty )$, there exists a subsequence $\{{\lambda }_{n_s}\}$, such that ${({\lambda }_{n_s})}^{\frac{1}{p?2}}{u}_k^{{\lambda }_{n_s}}$ converges in $H^1({\mathbb{R}}^2)$ to $w_k$ as $s\to \infty$, where $w_k$ also changes sign exactly k times and solves the following equation

$?\mathrm{\Delta }u+\omega u=|u{|}^{p?2}u,u\in H^1({\mathbb{R}}^2).$

     

Polynomial hulls and H￥H^\infty control for a hypoconvex constraint

We say that a subset of is hypoconvex if its complement is the union of complex hyperplanes. Let be the closed unit disk in , . We prove two conjectures of Helton and Marshall. Let be a smooth function on whose sublevel sets have compact hypoconvex fibers over . Then, with some restrictions on , if Y is the set where is less than or equal to 1, the polynomial convex hull of Y is the union of graphs of analytic vector valued functions with boundary in Y. Furthermore, we show that the infimum is attained by a unique bounded analytic f which in fact is also smooth on . We also prove that if varies smoothly with respect to a parameter, so does the unique f just found. Received: 18 December 1998 / Published online: 28 June 2000     

Curved fronts in the Belousov–Zhabotinskii reaction–diffusion systems in R2

In this paper we consider a diffusion system with the Belousov–Zhabotinskii (BZ for short) chemical reaction. Following Brazhnik and Tyson [4] and Pérez-Muñuzuri et al. [45], who predicted V-shaped fronts theoretically and discovered V-shaped fronts by experiments respectively, we give a rigorous mathematical proof of their results. We establish the existence of V-shaped traveling fronts in ${\mathbb{R}}^2$ by constructing a proper supersolution and a subsolution. Furthermore, we establish the stability of the V-shaped front in ${\mathbb{R}}^2$.     

稳态热传导的二阶一致无网格法

将具二阶一致性的3点积分方法(quadratically consistent 3-point integration method,QC3)拓展到稳态热传导问题的无网格法分析中.数值结果表明,与标准三角形积分方法以及已存在的仅满足线性一致性的1点积分方法相比,所建议的QC3方法不仅能精确地通过二阶分片试验,而且在精度、收敛性以及计算效率等方面都表现出显著优势.     

新三维非线性系统的动力学分析

通过代数方法,构造出来一个具有复杂混沌吸引子的非线性混沌自治三维系统．从理论和数值两方面对吸引子进行了分析和仿真,得到了系统在平衡点处不稳定的参数范围．通过分岔图和Ｌｙａｐｕｎｏｖ指数谱进一步揭示了系统丰富的动力学行为．     

Global existence of solutions for a nonlinearly perturbed Keller–Segel system in \mathbbR2{\mathbb{R}}^2

We show the global existence of small solution to the perturbed Keller–Segel system of simplified version. Our system has a perturbed nonlinear term of worse sign, therefore the existence and uniqueness of solution is not really obvious. The local existence theorem is obtained by a variational observation for the elliptic part.     

向量优化问题有效点集的稳定性

在不需要紧性假设下,利用拟C-凸函数及回收锥的性质,建立了向量优化问题有效点集的稳定性,获得了一列目标函数和可行集均扰动情形下的向量优化问题与对应的向量优化问题有效点集的Painlevé-Kuratowski内收敛性结果。所得结果推广和改进了相关文献(Attouch H,RiahiH.Stability results for Ekeland’s-variational principle and cone extremal solution;Huang X X.Stabilityin vector-valued and set-valued optimization)中的相应结果,并给出例子说明了所得结果的正确性。     

传递辛矩阵群收敛于辛Lie群

通过作用量变分原理,给出了Hamilton正则方程离散积分的传递辛矩阵表示,利用Hamil-ton正则方程给出了其对应的Lie代数。说明了当时间区段长度趋近于0时,离散系统积分的传递辛矩阵群收敛于连续时间Hamilton系统微分方程分析积分得到的辛Lie群。