• Volume 119, Issue 4

September 2009,   pages  411-566

• On the $2m$-th Power Mean of Dirichlet 𝐿-Functions with the Weight of Trigonometric Sums

Let 𝑝 be a prime, 𝜒 denote the Dirichlet character modulo $p,f(x)=a_0+a_1 x+\cdots+a_kx^k$ is a 𝑘-degree polynomial with integral coefficients such that $(p, a_0,a_1,\ldots,a_k)=1$, for any integer 𝑚, we study the asymptotic property of

\begin{equation*}\sum\limits_{\chi\neq \chi_0}\left| \sum\limits^{p-1}_{a=1}\chi(a)e\left( \frac{f(a)}{p}\right)\right|^2 |L(1,\chi)|^{2m},\end{equation*}

where $e(y)=e^{2\pi iy}$. The main purpose is to use the analytic method to study the $2m$-th power mean of Dirichlet 𝐿-functions with the weight of the general trigonometric sums and give an interesting asymptotic formula. This result is an extension of the previous results.

• Good Points for Diophantine Approximation

Given a sequence $(x_n)^\infty_{n=1}$ of real numbers in the interval [0,1) and a sequence $(\delta_n)^\infty_{n=1}$ of positive numbers tending to zero, we consider the size of the set of numbers in [0,1] which can be well approximated’ by terms of the first sequence, namely, those $y\in[0,1]$ for which the inequality $|y-x_n| &lt; \delta_n$ holds for infinitely many positive integers 𝑛. We show that the set of well approximable’ points by a sequence $(x_n)^\infty_{n=1}$, which is dense in [0,1], is quite large’ no matter how fast the sequence $(\delta_n)^\infty_{n=1}$ converges to zero. On the other hand, for any sequence of positive numbers $(\delta_n)^\infty_{n=1}$ tending to zero, there is a well distributed sequence $(x_n)^\infty_{n=1}$ in the interval [0,1] such that the set of well approximable’ points 𝑦 is `quite small’.

• Cohomology with Coefficients for Operadic Coalgebras

Corepresentations of a coalgebra over a quadratic operad are defined, and various characterizations of them are given. Cohomology of such an operadic coalgebra with coefficients in a corepresentation is then studied.

• On Artinian Generalized Local Cohomology Modules

Let 𝑅 be a commutative Noetherian ring with non-zero identity and $\mathfrak{a}$ be a maximal ideal of 𝑅. An 𝑅-module 𝑀 is called minimax if there is a finitely generated submodule 𝑁 of 𝑀 such that $M/N$ is Artinian. Over a Gorenstein local ring 𝑅 of finite Krull dimension, we proved that the Socle of $H^n_a(R)$ is a minimax 𝑅-module for each $n\geq 0$.

• The Poincaré Series of a Local Gorenstein Ring of Multiplicity up to 10 is Rational

Let 𝑅 be a local, Gorenstein ring with algebraically closed residue field 𝑘 of characteristic 0 and let $P_R(z):=\sum^\infty_{p=0}\dim_k(\mathrm{Tor}^R_p(k, k))z^p$ be its Poincaré series. We compute $P_R$ when 𝑅 belongs to a particular class defined in the Introduction, proving its rationality. As a by-product we prove the rationality of $P_R$ for all local, Gorenstein rings of multiplicity at most 10.

• Torus Quotients of Homogeneous Spaces - Minimal Dimensional Schubert Varieties Admitting Semi-Stable Points

In this paper, for any simple, simply connected algebraic group 𝐺 of type $B,C$ or 𝐷 and for any maximal parabolic subgroup 𝑃 of 𝐺, we describe all minimal dimensional Schubert varieties in $G/P$ admitting semistable points for the action of a maximal torus 𝑇 with respect to an ample line bundle on $G/P$. We also describe, for any semi-simple simply connected algebraic group 𝐺 and for any Borel subgroup 𝐵 of 𝐺, all Coxeter elements 𝜏 for which the Schubert variety $X(\tau)$ admits a semistable point for the action of the torus 𝑇 with respect to a non-trivial line bundle on $G/B$.

• Hypersurfaces Satisfying $L_rx = Rx$ in Sphere $\mathbb{S}^{n+1}$ or Hyperbolic Space $\mathbb{H}^{n+1}$

In this paper, using the method of moving frames, we consider hypersurfaces in Euclidean sphere $\mathbb{S}^{n+1}$ or hyperbolic space $\mathbb{H}^{n+1}$ whose position vector 𝑥 satisfies $L_r x=Rx$, where $L_r$ is the linearized operator of the $(r+1)$-th mean curvature of the hypersurfaces for a fixed $r=0,\ldots,n-1,R\in \mathbb{R}^{(n+2)\times(n+2)}$. If the 𝑟-th mean curvature $H_r$ is constant, we prove that the only hypersurfaces satisfying that condition are 𝑟-minimal $(H_{r+1}\equiv 0)$ or isoparametric. In particular, we locally classify such hypersurfaces which are not 𝑟-minimal.

• Relations between Bilinear Multipliers on $\mathbb{R}^n, \mathbb{T}^n$ and $\mathbb{Z}^n$

In this paper we prove the bilinear analogue of de Leeuw’s result for periodic bilinear multipliers and some Jodeit type extension results for bilinear multipliers.

• Hypercyclicity of the Adjoint of Weighted Composition Operators

In the present paper we investigate the hypercyclicity of the adjoint of weighted composition operator in special function spaces.

• Weighted Composition Operators between Different Bergman Spaces of Bounded Symmetric Domains

In this paper, we consider the boundedness and compactness of the weighted composition operators between different Bergman spaces of bounded symmetric domains in terms of the Carleson measure. As an application, we study the multipliers between different Bergman spaces.

• Entropy Maximization

It is shown that (i) every probability density is the unique maximizer of relative entropy in an appropriate class and (ii) in the class of all pdf 𝑓 that satisfy $\int fh_id_\mu=\lambda_i$ for $i=1,2,\ldots,\ldots k$ the maximizer of entropy is an $f_0$ that is proportional to $\exp(\sum c_i h_i)$ for some choice of $c_i$. An extension of this to a continuum of constraints and many examples are presented.

• Logarithm Laws and Shrinking Target Properties

We survey some of the recent developments in the study of logarithm laws and shrinking target properties for various families of dynamical systems. We discuss connections to geometry, diophantine approximation and probability theory.

• Real Moments of the Restrictive Factor

Let 𝜆 be a real number such that $0 &lt; \lambda &lt; 1$. We establish asymptotic formulas for the weighted real moments $\sum_{n\leq x}R^\lambda(n)(1-n/x)$, where $R(n)=\prod^k_{v=1}p^{\alpha v-1}_v$ is the Atanassov strong restrictive factor function and $n=\prod^k_{v=1}p^{\alpha v}_v$ is the prime factorization of 𝑛.