限制性多源点偏心距增广问题

给定一个赋权图$G=（V,E;w,c）$以及图$G$的一个支撑子图$G_{1}=（V,E_{1}）$,这里源点集合$S=\{s_{1},s_{2},\cdots,s_{k}\}\subseteq V$,权重函数$w：E\rightarrow\mathbb{R}^{+}$,费用函数$c：E\setminus E_{1}\rightarrow\mathbb{Z}^{+}$和一个正整数$B$,本文考虑两类限制性多源点偏心距增广问题,具体叙述如下：（1）限制性多源点最小偏心距增广问题是要寻找一个边子集$E_{2}\subseteq E\setminus E_{1}$,满足约束条件$c（E_{2}）$$\leq$$B$,目标是使得子图$G_{1}\cup E_{2}$上源点集$S$中顶点偏心距的最小值达到最小;（2）限制性多源点最大偏心距增广问题是要寻找一个边子集$E_{2}\subseteq E\setminus E_{1}$,满足约束条件$c（E_{2}）$$\leq$$B$,目标是使得子图$G_{1}\cup E_{2}$上源点集$S$中顶点偏心距的最大值达到最小。本文设计了两个固定参数可解的常数近似算法来分别对上述两类问题进行求解。     

带有Neumann边界条件的$k$维Minkowski平均曲率系统解的存在性

我们运用扰动方法证明了带有Minkowski平均算子非局部Neumann系统$$\begin{aligned}\begin{cases}\Big(r^{N-1}\frac{u''}{\sqrt{1-u''^{2}}}\Big)''=r^{N-1}f(r, u),\\\ r\in(0, 1),\ \ \ u''(0)=0,\ \ \ u''(1)=\int_{0}^{1}u''(s)dg(s)\\\end{cases}\end{aligned}$$解的存在性, 其中$k, N\geq1$是整数, $f=(f_{1},f_{2},\ldots,f_{k}):[0, 1]\times\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$连续且$g:[0, 1]\rightarrow\mathbb{R}^{k}$是有界变差函数.     

E1 ο Ed型距离正则图及 Q -多项式的性质

设$\Gamma$ 是一个直径$d\geq 3$的非二部距离正则图，其特征值 $\theta_{0}>\theta_{1}>\cdots>\theta_{d}.$ 设$\theta_{1'}\in\{ \theta_{1},\theta_{d}\}, $\theta_{d'}$ 是$\theta_{1'}$ 在 $\{\theta_{1},\theta_{d}\}$中的余. 又设 $\Gamma$ 是具有性质$E_{1}\circ E_{d}=|X|^{-1}(q^{d-1}_{1d}E_{d-1}+q^{d}_{1d}E_{d})$的$E_{1}\circ E_{d}$型距离正则图,$\sigma_{0},\sigma_{1},\cdots,\sigma_{d}$,$\rho_{0},\rho_{1},\cdots,\rho_{d}$和$\beta_{0},\beta_{1},\cdots,\beta_{d}$ 分别是关于$\theta_{1'}$,$\theta_{d'}$ 和 $\theta_{d-1}$的余弦序列.利用上述余弦序列,给出了 $\Gamma$关于 $\theta_{1}$ 或$\theta_{d}$是$Q$ -多项式的充要条件.     

一类具有奇异灵敏度的logistic趋化系统解的渐近行为

本文在无边界流的光滑有界区域$\Omega\subset\mathbb{R}^n~(n>2)$上研究了具有奇异灵敏度及logistic源的抛物-椭圆趋化系统$$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ 其中$\chi$, $r$, $\mu>0$, $k\geq2$. 证明了若当$r$适当大, 则当$t\rightarrow\infty$时该趋化系统全局有界解呈指数收敛于$((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$.     

增益图与其底图的正惯性指数之间的关系

设$\mathbb{T}$是模为1的复数乘法子群.图$G=(V,E)$,这里$V,E$分别表示图的点和边.增益图是将底图中的每条边赋于$\mathbb{T}$中的某个数值$\varphi(v_iv_j)$,且满足$\varphi(v_iv_j) =\overline{\varphi(v_jv_i)}$.将赋值以后的增益图表示为$(G,\varphi)$.设$i_+(G,\varphi)$和$i_+(G)$分别表示增益图与底图的正惯性指数,本文证明了如下结论: $$ - c( G ) \le {i_ + } ( {G,\varphi } ) - {i_ + }( G ) \le c( G ), $$ 这里$c(G)$表示圈空间维数,并且刻画了等号成立时候的所有极图.     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献   

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.     

由$q$-积分算子定义的某些亚纯函数类的Fekete-Szeg\"{o}问题

本文首先引入满足如下条件$$-\frac{qzD_{q}f(z)}{f(z)}\prec \varphi (z)$$和$$\frac{-(1-\frac{\alpha }{q})qzD_{q}f(z)+\alpha qzD_{q}[zD_{q}f(z)]}{(1-\frac{\alpha}{q})f(z)-\alpha zD_{q}f(z)}\prec \varphi (z)~(\alpha \in\mathbb{C}\backslash (0,1],\ 0相似文献   

一列连续函数的遍历优化

设$T:X\rightarrow X$是紧度量空间$X$上的连续映射, $\mathcal{F}=\{f_n\}_{n\geq 1}$是$X$上的一族连续函数. 如果 $\mathcal{F}$是渐近次可加的, 那么$\sup\limits_{x\in \mathrm{Reg}(\mathcal{F},T)}\lim\limits_{n\rightarrow\infty}\frac 1 n f_n (x)=\sup\limits_{x\in X} \limsup\limits_{n\rightarrow\infty}\frac 1 n f_n (x) =\lim\limits_{n\rightarrow\infty}\frac 1 n \max\limits_{x\in X}f_n (x)=\sup\{\mathcal{F}^*(\mu):\mu\in\mathcal{M}_T\}$, 其中$\mathcal{M}_T$表示$T$-\!\!不变的Borel概率测度空间, $\mathrm{Reg}(\mathcal{F},T)$ 表示函数族$\mathcal{F}$的正规点集, $\mathcal{F}^*(\mu)=\lim\limits_{n\rightarrow\infty}\frac 1 n \int f_n \mathrm{d}\mu$. 这把Jenkinson, Schreiber 和 Sturman 等人的一些结果推广到渐近次可加势函数, 并且给出了次可加势函数从属原理成立的充分条件, 最后给出了 一些相关的应用.     

图加一条边后的带宽和

令$BS(G,f)=\sum\limits_{uv\in E(G)}|f(u)-f(v)|$, 其中$f$为$V(G)\rightarrow\{1,2,\cdots,|V(G)|\}$的双射, 并称$BS(G)=\min\limits_{f}BS(G,f)$为图$G$的带宽和. 讨论顶点数为$n$的简单图$G$加上一条边$e\in\overline{E(G)}$后, 带宽和$BS(G+e)$与$BS(G)$的关系, 得其关系式$BS(G)+1\leq BS(G+e)\leq BS(G)+n-1$. 并证明此不等式中等号可取到, 即存在图$G_{1}$和$G_{2}$使得$BS(G_{1}+e)=BS(G_{1})+1$, $BS(G_{2}+e)=BS(G_{2})+n-1$.     

Projective equivalence for the roots of unity

Journal of Algebraic Combinatorics - Let $$\mu _{\infty }\subseteq \mathbb {C}$$ be the collection of roots of unity and $$\mathcal {C}_{n}:=\{(s_{1},\ldots ,s_{n})\in \mu _{\infty }^{n}:s_{i}\ne...     

Extremal Functions for Adams Inequalities in Dimension Four

Let $\Omega\subset \mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(\Omega)$ be the usual Sobolev space. For any positive integer $\ell$, $\lambda_{\ell}(\Omega)$ is the $\ell$-th eigenvalue of the bi-Laplacian operator. Define $E_{\ell}=E_{\lambda_1(\Omega)}\oplus E_{\lambda_2(\Omega)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Omega)}$, where $E_{\lambda_i(\Omega)}$ is eigenfunction space associated with $\lambda_i(\Omega)$. $E^{\bot}_{\ell}$ denotes the orthogonal complement of $E_\ell$ in $W_0^{2,2}(\Omega)$. For $0\leq\alpha<\lambda_{\ell+1}(\Omega)$, we define a norm by $\|u\|_{2,\alpha}^{2}=\|\Delta u\|^2_2-\alpha \|u\|^2_2$ for $u\in E^\bot_{\ell}$. In this paper, using the blow-up analysis, we prove the following Adams inequalities$$\sup_{u\in E_{\ell}^{\bot},\,\| u\|_{2,\alpha}\leq 1}\int_{\Omega}e^{32\pi^2u^2}{\rm d}x<+\infty;$$moreover, the above supremum can be attained by a function $u_0\in E_{\ell}^{\bot}\cap C^4(\overline{\Omega})$ with $\|u_0\|_{2,\alpha}=1$. This result extends that of Yang (J. Differential Equations, 2015), and complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv: 1701.08249, 2017).     

$c$-半层空间与几乎完全正则空间一点注记

In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces(CSS) by semi-continuous functions. We mainly show that:(1)Let X be a space. Then the following statements are equivalent:(i) X is almost completely regular.(ii) Every two disjoint subsets of X, one of which is compact and the other is regular closed, are completely separated.(iii) If g, h : X → I, g is compact-like, h is normal lower semicontinuous, and g ≤ h, then there exists a continuous function f : X → I such that g ≤ f ≤ h;and(2) Let X be a space. Then the following statements are equivalent:(a) X is CSS;(b) There is an operator U assigning to a decreasing sequence of compact sets(Fj)j∈N,a decreasing sequence of open sets(U(n,(Fj)))n∈N such that(b1) Fn■U(n,(Fj)) for each n ∈ N;(b2)∩n∈NU(n,(Fj)) =∩n∈NFn;(b3) Given two decreasing sequences of compact sets(Fj)j∈N and(Ej)j∈N such that Fn■Enfor each n ∈ N, then U(n,(Fj))■U(n,(Ej)) for each n ∈ N;(c) There is an operator Φ : LCL(X, I) → USC(X, I) such that, for any h ∈ LCL(X, I),0 Φ(h) h, and 0 Φ(h)(x) h(x) whenever h(x) 0.     

On reflection of stationary sets in

Let be an inaccessible cardinal, and let and is regular and . It is consistent that the set is stationary and that every stationary subset of reflects at almost every .

     

Minimizers for the embedding of Besov spaces

Using the profile decomposition, we will show the relatively compactness of the minimizing sequence to the critical embeddings between Besov spaces, which implies the existence of minimizer of the critical embeddings of Besov spaces $\dot{B}^{s_1}_{p_1,q_1}\hookrightarrow \dot{B}^{s_2}_{p_2,q_2}$ in $d$ dimensions with $s_1-d/p_1=s_2-d/p_2$, $s_1>s_2$ and $1 \leq q_1相似文献   

纯无限单的C*-代数通过某些C*-代数扩张的非稳定K-理论

设0→B■E■A→0是有单位元C~*-代数E的一个扩张,其中A是有单位元纯无限单的C~*-代数,B是E的闭理想.当B是E的本性理想并且同时是单的、可分的而且具有实秩零及性质(PC)时,证明了K_0(E)={[p]| p是E\B中的投影};当B是稳定C~*-代数时,证明了对任意紧的Hausdorff空间X,有■(C(X,E))/■_0(C(X,E))≌K_1(C(X,E)).     

带参数的一阶泛函差分方程的定号周期解

In this paper,the authors obtain the existence of one-signed periodic solutions of the first-order functional difference equation ?u(n) = a(n)u(n)-λb(n)f(u(n-τ(n))),n ∈ Z by using global bifurcation techniques,where a,b:Z → [0,∞) are T-periodic functions with ∑T n=1 a(n) 0,∑T n=1 b(n) 0;τ:Z → Z is T-periodic function,λ 0 is a parameter;f ∈ C(R,R) and there exist two constants s_2 0 s_1 such that f(s_2) = f(0) = f(s_1) = 0,f(s) 0 for s ∈(0,s_1) ∪(s_1,∞),and f(s) 0 for s ∈(-∞,s_2) ∪(s_2,0).     

整数集的加权表示的渐进基

Let k1, k2 be nonzero integers with(k1, k2) = 1 and k1k2≠-1. Let Rk1,k2(A, n)be the number of solutions of n = k1a1 + k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A  Z such that Rk1,k2(A, n) = 1 for all n ∈ Z. Let f : Z-→ N0∪ {∞} be a function such that f-1(0) is finite. In this paper, we generalize Xiong's result and prove that there exist uncountably many sets A  Z such that Rk1,k2(A, n) = f(n) for all n ∈ Z.     

Toeplitz算子代数的归纳极限

设(G_1,E_1),(G_2,E_2)为两个拟格序群,记■~(E_1),■~(E_2)为相应的Toeplitz算子代数．设■：G_1→G_2为一个保单位的群同态,使得■(E_1)■E_2．本文给出了上述两个Toeplitz算子代数间的自然同态映照成为C~*-代数的单同态的充要条件,刻画了Toeplitz算子代数的归纳极限,证明了任何自由群上的Toeplitz算子代数是顺从的．