涉及例外函数与分担函数的正规族

设F是区域D内的一族亚纯函数,a(z),b(z),c(z)是区域D内三个判别的亚纯函数,其中一个可以恒为无穷,且对于任意z∈D,a(z)≠b(z),a(z)≠c(z),b(z)≠c(z),S={a(z),b(z),c(z)}.若对于任意两个函数f,g∈F,f与g在D内分担集合S,则F在D内正规.该结果推广了著名的Montel正规定则.     

涉及分担函数的正规定则

设k为正整数,M为正数;F为区域D内的亚纯函数族,且其零点重级至少为k;h为D内的亚纯函数(h(z)≠0,∞),且h(z)的极点重级至多为k.若对任意给定的函数f∈F,f与f~((k))分担0,且f~((k))(z)-h(z)=0?|f(z)|≥M,则F在D内正规.     

与例外函数和分担函数相关的亚纯函数的正规族

设F为区域D内的一族亚纯函数,对于每个f∈F,f的所有极点重数至少是2,a(z)和b(z)为两个在D内满足a(z)■b(z)的全纯函数.若对于每个f∈F,f(z)≠a(z)和f(z)≠b(z),则F在D内正规.这个结果改进了经典的Montel定则.此外,我们也讨论了亚纯函数族中每个函数与其导函数分担两个全纯函数的正规性.     

与分担全纯函数相关的正规族

设F是在区域D内的一族亚纯函数,其零点重级至少为k,k是一个正整数,a(z)(≠0)在区域D内全纯.若对于任意的f∈F,有(1)f(z)与a(z)没有公共的零点;(2)f(z)=0f(k)(z)=a(z)■0|f~((k+1))(z)-a'(x)||a(z)|,则F在D内正规.     

亚纯函数的分担值与正规族

设k(≥2)为正整数,M为一个正数,h(z)为区域D内的一个全纯函数,h≠0,F为区域D内的一族亚纯函数,其中每个函数的零点重级至少为k+1,极点重级至少为2.若任意f∈F,f~((k))(z)=h(z)|f(z)|≥M,则F在D内正规.     

涉及分担函数的正规性

设k为一个正整数,a(z)(■0,∞)为区域D的亚纯函数,F是区域D内的一族亚纯函数,其零点的重级至少为k.若对于任意f∈F,f(z)=0f~((k))(z)=a(z)?0|f~((k+1))(z)-a′(z)||a(z)|,则F在D内正规.     

关于杨乐及Schwick的一结果

设ψ■0为复平面区域D内的只有单零点的全纯函数,k为正整数,F为区域D内的亚纯函数族.如果每个f∈F满足f≠0且只有重极点;对F内任一组函数f与g,f(k)与g(k)在D内分担ψ(z),则F在D内正规.     

亚纯函数族涉及微分单项式分担移动靶的正规定则

设a(z)是一个没有零点的整函数,k≥3是个整数,F是区域D上的亚纯函数族,对每一个f∈F至少有k重零点和2重极点.若对每一对f,g∈F有ff(k)与gg(k)IM分担a(z),则F在区域D内正规.     

分担值与正规族

设F是平面区域D上的亚纯函数族,a,b是两个有穷非零复数.如果(A)f∈F,f(z)=a(=)f(k)(z)=a,f(k)(z)=b(=)f(k+1)(z)=b,且f-a的零点重数至少为k(k≥3),那么函数族F在D内正规;当k=2时,在条件a≠4b的情况下,同样有函数族F在D内正规.     

分担集合的亚纯函数正规族

对复平面C的非空有限子集S_1和S_2,记在复平面区域D内满足{z∈D:f(z)∈S_1}={z∈D:f′(z)∈S_2}的全体亚纯函数f形成的函数族为D,那么当S_1和S_2共有至少12个元素对函数族D正规.特别地,当S_1具有至少三个复数时,我们得到了准确的结果.     

分担一个亚纯函数的亚纯函数族的正规定则

Let F be a family of meromorphic functions in D,and let Ψ(≠0) be a meromorphic function in D all of whose poles are simple.Suppose that,for each f ∈F,f≠0 in D.If for each pair of functions {f,g}(?) F,f' and g' share Ψ in D,then F is normal in D.     

亚纯函数正规族与分担值

设 $k, m$ 是两个正整数, $a\ ( \ne 0)$是有穷复数. $\mathcal{F}$ 是区域 $D$ 内的一族亚纯函数, $f\in\mathcal{F}$ 的零点重数至少为 $k$, $P$ 是多项式,次数或者 ${\rm deg}\, P\geq3$ 或者 ${\rm deg}\, P=2$ 且 $P$ 只有一个不同的零点.若对于 $\mathcal{F}$ 中的任意两个函数 $f$ 和 $g$, $P(f){({f^{(k)}})^m}$ 与 $P(g){({g^{(k)}})^m}$ 在 $D$ 内 IM 分担 $a$, 则 $\mathcal{F}$ 在 $D$ 内正规.     

分担值与正规族

设F是平面区域D上的亚纯函数族,a,b是两个有穷非零复数.如果■ff∈F,f(z)=a■f~((k))(z)=a,ff~((k))(z)=b■f~((k+1))(z)=b,且f-a的零点重数至少为k(k≥3),那么函数族F在D内正规;当k=2时,在条件a≠4b的情况下,同样有函数族F在D内正规.     

限制零点个数的亚纯函数的正规定则

设k,n(≥k+1)是两个正整数,a(≠0),b是两个有穷复数,F为区域D内的一族亚纯函数.如果对于任意的f∈F,f的零点重级大于等于k+1,并且在D内满足f+a[L(f)]~n-b至多有n-k-1个判别的零点,那么F在D内正规·这里L(f)=f~((k))(z)+a_1f~((k-1))(z)+…+a_(k-1)f'(z)+a_kf(z),其中a_1(z),a_2(z),…,a_k(z)是区域D上的全纯函数.     

亚纯函数与其微分多项式的分担函数与正规族

设${\cal F}$为开平面内的区域$D$上的亚纯函数族, ${\cal F}$中任何函数$f(z)\in{\cal F}$, $f$的零点竽数至少为$k+1$.对于$D$内不等于零的解析函数$a(z)$.若$f(z)$与其微分多项式$D(f)$ IM分担$a(z)$,本文不仅得到${\cal F}$在$D$上正规, 而且得到相应于正规函数的结果.     

参数函数$\epsilon$取边际值时的$GCF_\epsilon$集

Let ∈ :N → R be a parameter function satisfying the condition ∈(k) + k + 1 > 0and let T∈ :(0,1] →(0,1] be a transformation defined by T∈(x) =-1 +(k + 1)x1 + k-k∈x for x ∈(1k + 1,1k].Under the algorithm T∈,every x ∈(0,1] is attached an expansion,called generalized continued fraction(GCF∈) expansion with parameters by Schweiger.Define the sequence {kn(x)}n≥1of the partial quotients of x by k1(x) = ∈1/x∈ and kn(x) = k1(Tn-1∈(x)) for every n ≥ 2.Under the restriction-k-1 < ∈(k) <-k,define the set of non-recurring GCF∈expansions as F∈= {x ∈(0,1] :kn+1(x) > kn(x) for infinitely many n}.It has been proved by Schweiger that F∈has Lebesgue measure 0.In the present paper,we strengthen this result by showing that{dim H F∈≥12,when ∈(k) =-k-1 + ρ for a constant 0 < ρ < 1;1s+2≤ dimHF∈≤1s,when ∈(k) =-k-1 +1ksfor any s ≥ 1where dim H denotes the Hausdorff dimension.     

Ranks of backward shift invariant subspaces of the Hardy space over the bidisk

Let $\{\varphi _n(z)\}_{n\ge 0}$ be a sequence of inner functions satisfying that $\zeta _n(z):=\varphi _n(z)/\varphi _{n+1}(z)\in H^\infty (z)$ for every $n\ge 0$ and $\{\varphi _n(z)\}_{n\ge 0}$ has no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace $\mathcal{M }$ of $H^2(\mathbb{D }^2)$ . The ranks of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }_z$ and $\mathcal{F }^*_z$ respectively are determined, where $\mathcal{F }_z$ is the fringe operator on $\mathcal{M }\ominus w\mathcal{M }$ . Let $\mathcal{N }= H^2(\mathbb{D }^2)\ominus \mathcal{M }$ . It is also proved that the rank of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }^*_z$ equals to the rank of $\mathcal{N }$ for $T^*_z$ and $T^*_w$ .     

Moments of L-Functions Attached to the Twist of Modular Form by Dirichlet Characters

Let f(z) be a holomorphic cusp form of weight κ with respect to the full modular group SL2(Z). Let L(s, f) be the automorphic L-function associated with f(z) and χ be a Dirichlet character modulo q. In this paper, the authors prove that unconditionally for k =1/n with n ∈ N,and the result also holds for any real number 0 k 1 under the GRH for L(s, f ■χ).The authors also prove that under the GRH for L(s, f ■χ),for any real number k 0 and any large prime q.     

Compact and Loeb Hausdorff spaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ZF}$\end{document} and the axiom of choice for families of finite sets

Given a set X, $\mathsf {AC}^{\mathrm{fin}(X)}$ denotes the statement: “$[X]^{<\omega }\backslash \lbrace \varnothing \rbrace$ has a choice set” and $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$ denotes the family of all closed subsets of the topological space $\mathbf {2}^{X}$ whose definition depends on a finite subset of X. We study the interrelations between the statements $\mathsf {AC}^{\mathrm{fin}(X)},$ $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega })},$ $\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$ $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ and “$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set”. We show:

• (i) $\mathsf {AC}^{\mathrm{fin}(X)}$ iff $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega } )}$ iff $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set iff $\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$.
• (ii) $\mathsf {AC}_{\mathrm{fin}}$ ($\mathsf {AC}$ restricted to families of finite sets) iff for every set X, $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set.
• (iii) $\mathsf {AC}_{\mathrm{fin}}$ does not imply “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set($\mathcal {K}(\mathbf {X})$ is the family of all closed subsets of the space $\mathbf {X}$)
• (iv) $\mathcal {K}(\mathbf {2}^{X})\backslash \lbrace \varnothing \rbrace$ implies $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ but $\mathsf {AC}^{\mathrm{fin}(X)}$ does not imply $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$.

We also show that “For every setX, “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every setX, $\mathcal {K}\big (\mathbf {[0,1]}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every product$\mathbf {X}$of finite discrete spaces,$\mathcal {K}(\mathbf {X})\backslash \lbrace \varnothing \rbrace$ has a choice set”.