在${\mathbb{R}}^m$中重对数广义律的精确速率

令\{$X$, $X_n$, $n\ge 1$\}是期望为${\mathbb{E}}X=(0,\ldots,0)_{m\times 1}$和协方差阵为${\rm Cov}(X,X)=\sigma^2I_m$的独立同分布的随机向量列, 记$S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. 对任意$d>0$和$a_n=o((\log\log n)^{-d})$, 本文研究了${{\mathbb{P}}(|S_n|\ge (\varepsilon+a_n)\sigma \sqrt{n}(\log\log n)^d)$的一类加权无穷级数的重对数广义律的精确速率.     

某类$q$差分方程亚纯解的性质

对于一个有穷非零复数$q$, 若下列$q$差分方程存在一个非常数亚纯解$f$, $$f(qz)f(\frac{z}{q})=R(z,f(z))=\frac{P(z,f(z))}{Q(z,f(z))}=\frac{\sum_{j=0}^{\tilde{p}}a_j(z)f^{j}(z)}{\sum_{k=0}^{\tilde{q}}b_k(z)f^{k}(z)},\eqno(\dag)$$ 其中 $\tilde{p}$和$\tilde{q}$是非负整数, $a_j$ ($0\leq j\leq \tilde{p}$)和$b_k$ ($0\leq k\leq \tilde{q}$)是关于$z$的多项式满足$a_{\tilde{p}}\not\equiv 0$和$b_{\tilde{q}}\not\equiv 0$使得$P(z,f(z))$和$Q(z,f(z))$是关于$f(z)$互素的多项式, 且$m=\tilde{p}-\tilde{q}\geq 3$. 则在$|q|=1$时得到方程$(\dag)$不存在亚纯解, 在$m\geq 3$和$|q|\neq 1$时得到方程$(\dag)$解$f$的下级的下界估计.     

THE SUFFICIENT AND NECESSARY CONDITIONS FOR NOETHER’S SOLVABILITY OF SINGULAR INTEGRAL EQUATIONS WITH TWO CARLEMAN’S SHIFTS

In this paper we consider the problem of solvability of singular integral equtions with two Carleman's shifts \[\begin{gathered} (\mathcal{K}\varphi )(t) \equiv {a_0}(t)\varphi (t) + {a_1}(t)\varphi [\alpha (t)] + {a_2}(t)\varphi [\beta (t)] + {a_3}(t)\varphi [\gamma (t)] \hfill \ + \frac{{{b_0}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - t}}} d\tau + \frac{{{b_1}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \alpha (t)}}} d\tau + \frac{{{b_2}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \beta (t)}}d\tau } \hfill \ + \frac{{{b_s}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \gamma (t)}}} d\tau + \int_\Gamma {K(t,\tau )\varphi (\tau )d\tau = g(t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1,1)} \hfill \\ \end{gathered} \] Suppose that Г is a closed simple Lyapunoff's curve and \[\alpha (t)\], \[\beta (t)\] which satisfy Carleman's. conditions and \[\alpha [\beta (t)] = \beta [\alpha (t)]\] are two different homeomorphisms of Г onto itself, and that \[{a_k}(t),{b_k}(t)\], k = 0, 1, 2, 3 belong to the,space \[{H_\mu }(\Gamma ),g(t)\] belongs to the space \[{L_p}(\Gamma ),p > 1\]), p>l and \[K(t,\tau )\] has only weak singularity. The following main results are obtained: 1. Singular integral eqution (1.1) is solvable if and only if the Noether's conditions \[det(p(t) \pm q(t)) \ne 0\] are satisfied. 2. Index of sigular integral eqution (1.1) is calculated by the formula \[Ind{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathcal{K} = \frac{1}{{8\pi }}{\{ arg\frac{{\det (p(t) - q(t))}}{{\det (p(t) + q(t))}}\} _\Gamma }\] where p(t) and q(t) are matrices of coeffioents of so-called corresponding system of equtions. All these results have been generalized for systems of singular integral equtions with two Carleman's shifts and complex conjugate of unknown functions.     

THE AVERAGE NUMBER OF REAL ROOTS OF A RANDOM ALGEBRAIC EQUATION

The average number of real roots of the random algebraio equation \[{F_n}(\omega ,t) = {a_0}(\omega ) + {a_2}(\omega )t + \cdots + {a_n}(\omega ){t^{n - 1}} = 0\] has been estimated by Kao, M.[5] for the case where the \({a_i}(\omega ){\kern 1pt} {\kern 1pt} (i = 0,1, \cdots ,n - 1)\) are indenpendent Gaussian random variables with mean 0 and standard deviation 1. Let \(E{N_F}(\omega )\) be the average： aiumber of real roots of \({F_n}(\omega ,t)\) , Kao's main result is \[E{N_F}(\omega ) \le \frac{2}{\pi }{\rm{In}}n + \frac{{14}}{\pi }\] Later in (8), Stevens obtained \[\frac{2}{\pi }{\rm{In}}n - 0.6 < E{N_F}(\omega ) < \frac{2}{\pi }{\rm{In}}n + 1.4\]. The purpose of this paper is to prove the following theorem. Theorem. Let \[{F_n}(\omega ,t) = {a_0}(\omega ) + {a_2}(\omega )t + \cdots + {a_n}(\omega ){t^{n - 1}} = 0\] be a random algebraic equation where \({a_i}(\omega ){\kern 1pt} {\kern 1pt} (i = 0,1, \cdots ,n - 1)\) are indenpendent Gaussian random variables with mean 0 and standard deviation 1, Then for all \(n \ge 1\), \[\frac{2}{\pi }{\rm{In}}n \le E{N_F}(\omega ) \le \frac{2}{\pi }{\rm{In}}n + 1.2372771\].     

强奇异Calder-Zygmund积分及其交换子在 Hardy 型空间上的有界性

给出了局部 Hardy 空间 $h^{p}(\mathbb{R}^{n})$\ $\big(\frac{n}{n+1}相似文献   

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   

Zygmund 空间到 Bloch 空间上径向导数算子与积分型算子的乘积

设$H(\mathbb{B})$为单位球上全纯函数类,研究了单位球上 Zygmund 空间到 Bloch 空间上径向导数算子$\Re$与积分型算子$I_\varphi^g$乘积的有界性和紧性, 这里 $$ I_\varphi^g f(z)=\int_0^1 \Re f(\varphi(tz))g(tz)\frac{{\rm d}t}{t},\quad z\in\mathbb{B}, $$ 其中$g\in H(\mathbb{B}),\ g(0)=0$, $\varphi$ 是$\mathbb{B}$上全纯自映射.     

分数次积分交换子在非齐Morrey空间上的紧性

本文主要建立由分数次积分$I_{\gamma}$与函数$b\in\mathrm{Lip}_{\beta}(\mu)$生成的交换子$[b, I_{\gamma}]$在以满足几何双倍与上部双倍条件的非齐度量测度空间为底空间的Morrey空间上紧性的充要条件.在假设控制函数$\lambda$满足逆双倍条件下,证明了交换子$[b,I_{\gamma}]$为从Morrey空间$M^{p}_{q}(\mu)$到$M^{s}_{t}(\mu)$紧性当且仅当$b\in\mathrm{Lip}_{\beta}(\mu)$.     

连续函数之集在上半连续函数之集中的一种拓扑位置

设$(X,\rho)$是一个度量空间. 用$\dd {\rm USCC}(X)$和$\dd {\rm CC}(X)$ 分别表示从$X$ 到 $\I=[0,1]$的紧支撑的上半连续函数和紧支撑的连续函数下方图形全体. 赋予 Hausdorff 度量后, 它们是拓扑空间. 文中证明了, 如果 $X$ 是一个无限的且孤立点集稠密的紧度量空间, 则 $(\dd {\rm USCC}(X),\dd {\rm CC}(X))\approx(Q,c_0\cup (Q\setminus \Sigma))$, 即存在一个同胚 $h:~\dd {\rm USCC}(X)\to Q$, 使得 $h(\dd {\rm CC}(X))=c_0\cup (Q\setminus \Sigma)$, 这里 $Q=[-1,1]^{\omega},\,\Sigma=\{(x_n)_{n}\in Q: {\rm sup}|x_n|<1\},\, c_0=\Big\{(x_n)_{n}\in \Sigma: \lim\limits_{n\to +\infty}x_n=0\Big\}.$ 结合这个论断和另一篇文章的结果, 可以得到: 如果 $X$ 是一个无限的紧度量空间, 则 $(\uscc(X), \cc(X))\approx \left\{ \begin{array}{ll} (Q,c_0\cup (Q\setminus \Sigma)), &;\quad \text{如 果 孤 立 点 集 在} X \text{中稠密},\\ (Q, c_0), &;\quad \text{ 其他}. \end{array} \right.$ 还证明了, 对一个度量空间$X$, $(\dd {\rm USCC}(X),\dd {\rm CC}(X))\approx (\Sigma,c_0)$ 当且仅当 $X$是一个非紧的、局部紧的、非离散的可分空间.     

On the annihilators and attached primes of top local cohomology modules

Let ${\mathfrak{a}}$ be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that ${{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))= {\rm Ann}_R(M/T_R(\mathfrak{a}, M))}$ , where ${T_R(\mathfrak{a}, M)}$ is the largest submodule of M such that ${{\rm cd}(\mathfrak{a}, T_R(\mathfrak{a}, M)) < {\rm cd}(\mathfrak{a}, M)}$ . Several applications of this result are given. Among other things, it is shown that there exists an ideal ${\mathfrak{b}}$ of R such that ${{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))={\rm Ann}_R(M/H_{\mathfrak{b}}^{0}(M))}$ . Using this, we show that if ${ H_{\mathfrak{a}}^{{\rm dim} R}(R)=0}$ , then ${{{\rm Att}_R} H^{{\rm dim} R-1}_{\mathfrak a}(R)= \{\mathfrak{p} \in {\rm Spec} R | \,{\rm cd}(\mathfrak{a}, R/\mathfrak{p}) = {\rm dim} R-1\}.}$ These generalize the main results of Bahmanpour et al. (see [2, Theorem 2.6]), Hellus (see [7, Theorem 2.3]), and Lynch (see [10, Theorem 2.4]).     

有限域上幂剩余正规元的存在性

GF(q)是q个元的有限域,q是素数的方幂,n是正整数,GF(q~n)为GF(q)的n次扩张.用指数和估计的方法给出了3种情形下幂剩余正规元存在的充分条件,即(1)GF(q~n)中存在元ξ为GF(q)上的幂剩余正规元;(2)GF(q~n)中存在元ξ与ξ~(-1)同时为GF(q)上幂剩余正规元;(3)对GF(q~n)~*中任意给定的非零元a和b,GF(q~n)中存在元ξ与ξ~(-1)同时为GF(q)上d次幂剩余正规元,且满足Tr(ξ)=a,Tr(ξ~(-1))=b.     

Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem

Let K be a totally real number field, π an irreducible cuspidal representation of ${{\rm GL}_{2}(K){\backslash}{\rm GL}_{2}(\mathbb{A}K)}$ with unitary central character, and χ a Hecke character of conductor ${\mathfrak{q}}$ . Then ${L(1/2, \pi\oplus\chi) \ll (\mathcal{N}\mathfrak{q})^{\frac{1}{2}-\frac{1}{8}(1-2\theta)+\epsilon}}$ , where 0 ≤ θ ≤ 1/2 is any exponent towards the Ramanujan–Petersson conjecture (θ =  1/9 is admissible). The proof is based on a spectral decomposition of shifted convolution sums and a generalized Kuznetsov formula.     

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”.     

On Deformations of Lie Algebroids

For any Lie algebroid A, its 1-jet bundle ${\mathfrak{J} A}$ is a Lie algebroid naturally and there is a representation ${\pi:\mathfrak{J} A\longrightarrow\mathfrak{D} A}$ . Denote by ${{\rm d}_{\mathfrak{J}}}$ the corresponding coboundary operator. In this paper, we realize the deformation cohomology of a Lie algebroid A introduced by M. Crainic and I. Moerdijk as the cohomology of a subcomplex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A,A)_{{\mathfrak{D}} A}),{\rm d}_{\mathfrak{J}})}$ of the cochain complex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A, A)),{\rm d}_\mathfrak{J})}$ .     

一类二阶线性微分方程解的增长性

本文研究一类二阶齐次线性微分方程f"+A_1(z)e~(P(z))f'+A_0(z)e~(Q(z))f=0,解的增长性,其中P(z)=az~n,Q(z)=bz~n,ab≠0,a=cb(c1),A_j(z)(j=0,1)是非零多项式,证明了该方程的每个非零解满足σ(f)=∞并且σ_2(f)=n.     

Schur–Weyl Theory for C*‐algebras

To each irreducible infinite dimensional representation $(\pi ,\mathcal {H})$ of a C*‐algebra $\mathcal {A}$, we associate a collection of irreducible norm‐continuous unitary representations $\pi _{\lambda }^\mathcal {A}$ of its unitary group ${\rm U}(\mathcal {A})$, whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group ${\rm U}_\infty (\mathcal {H}) = {\rm U}(\mathcal {H}) \cap (\mathbf {1} + K(\mathcal {H}))$ are. These are precisely the representations arising in the decomposition of the tensor products $\mathcal {H}^{\otimes n} \otimes (\mathcal {H}^*)^{\otimes m}$ under ${\rm U}(\mathcal {A})$. We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which ${\rm U}(\mathcal {A})$ acts transitively and that the corresponding norm‐closed momentum sets $I_{\pi _\lambda ^\mathcal {A}}^{\bf n} \subseteq {\mathfrak u}(\mathcal {A})^{\prime }$ distinguish inequivalent representations of this type.     

混合径角$\lambda$中心有界平均振荡空间的一个特征

在这篇文章中,我们通过Hardy算子交换子$\mathrm{H}_b$与它的对偶算子交换子$\mathrm{H}^*_b$, 其中$b\in {\mathrm{CMOL}^{p_2, \lambda}_{\rm rad}L^{p_1}_{\rm ang}(\mathbb R^n)}$,建立了混合径角$\lambda$中心有界平均振荡空间的一个特征.     

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Let denote the linear space over spanned by . Define the (real) inner product , where V satisfies: (i) V is real analytic on ; (ii) ; and (iii) . Orthogonalisation of the (ordered) base with respect to yields the even degree and odd degree orthonormal Laurent polynomials , and . Define the even degree and odd degree monic orthogonal Laurent polynomials: and . Asymptotics in the double-scaling limit such that of (in the entire complex plane), , and (in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on , and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].     

The q-Variation of Functions and Spectral Integration from Dominated Ergodic Estimates

For $1\leq q < \infty$, let $\mathfrak{M}_{q}\left( \mathbb{T}\right)$, (respectively, $\mathfrak{M}_{q}\left( \mathbb{R}\right) $) denote the Banach algebra consisting of the bounded complex-valued functions having uniformly bounded $q$-variation on the dyadic arcs of the unit circle, (respectively, on the dyadic intervals of the real line). Suppose that $(\Omega,\mu)$ is a $\sigma$-finite measure space, $1< p < \infty$, and $T:L^{p}(\mu)\rightarrow L^{p}(\mu)$ is a bounded, invertible, separation-preserving linear operator such that the two-sided ergodic means of the linear modulus of $T$ are uniformly bounded in norm. We show that there is a real number $q_{_{0}} > 1$ such that whenever $1\leq q < q_{_{0}}$, $T $ has a norm-continuous functional calculus associated with $\mathfrak{M}_{q}\left(\mathbb{T}\right) $. Our approach is rooted in a dominated ergodic theorem of Mart\{\i}n--Reyes and de la Torre which assigns $T$ a canonical family of bilateral $A_{p}$ weight sequences. We first establish the relevant multiplier properties of $\mathfrak{M}_{q}\left( \mathbb{R}\right) $ classes in weighted settings, transfer the outcome to $\mathfrak{M}_{q}\left(\mathbb{T}\right) $, and then apply the consequent $\mathfrak{M}_{q}\left(\mathbb{T}\right) $ multiplier theorem for weighted settings to the spectral decomposition of $T$. The desired $\mathfrak{M}_{q}\left(\mathbb{T}\right)$-functional calculus for $T$ then results from an extension criterion for spectral integration obtained in the general Banach space setting. The multiplier result for $\mathfrak{M}_{q}\left( \mathbb{R}\right) $ shown at the outset of this process expands the scope of the weighted Marcinkiewicz multiplier theorem from $q=1$ to appropriate values of $q > 1$