pp 411-421
On the 2𝑚-th Power Mean of Dirichlet 𝐿-Functions with the Weight of Trigonometric Sums
Rong Ma Junhuai Zhang Yulong Zhang
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_{𝜒≠ 𝜒_0}\left| \sum\limits^{p-1}_{a=1}𝜒(a)e\left( \frac{f(a)}{p}\right)\right|^2 |L(1,𝜒)|^{2m},\end{equation*}
where $e(y)=e^{2𝜋 iy}$. 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.
pp 423-429
Good Points for Diophantine Approximation
Daniel Berend Artūras Dubickas
Given a sequence $(x_n)^∞_{n=1}$ of real numbers in the interval [0,1) and a sequence $(𝛿_n)^∞_{n=1}$ 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 $|y-x_n| < 𝛿_n$ holds for infinitely many positive integers 𝑛. We show that the set of `well approximable’ points by a sequence $(x_n)^∞_{n=1}$, which is dense in [0,1], is `quite large’ no matter how fast the sequence $(𝛿_n)^∞_{n=1}$ converges to zero. On the other hand, for any sequence of positive numbers $(𝛿_n)^∞_{n=1}$ tending to zero, there is a well distributed sequence $(x_n)^∞_{n=1}$ in the interval [0,1] such that the set of `well approximable’ points 𝑦 is `quite small’.
pp 431-452
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.
pp 453-458
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 ≥ 0.
pp 459-468
The Poincaré Series of a Local Gorenstein Ring of Multiplicity up to 10 is Rational
Gianfranco Casnati Roberto Notari
Let 𝑅 be a local, Gorenstein ring with algebraically closed residue field 𝑘 of characteristic 0 and let $P_R(z):=\sum^∞_{p=0}\dim_k(\mathrm{Tor}^R_p(k, k))z^p$ 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.
pp 469-485
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(𝜏) admits a semistable point for the action of the torus 𝑇 with respect to a non-trivial line bundle on 𝐺/𝐵 .
pp 487-499
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 $r=0,\ldots,n-1,R\in \mathbb{R}^{(n+2)×(n+2)}$. If the 𝑟-th mean curvature $H_r$ is constant, we prove that the only hypersurfaces satisfying that condition are 𝑟-minimal $(H_{r+1}≡ 0)$ or isoparametric. In particular, we locally classify such hypersurfaces which are not 𝑟-minimal.
pp 501-512
Relations between Bilinear Multipliers on $\mathbb{R}^n, \mathbb{T}^n$ and $\mathbb{Z}^n$
Debashish Bose Shobha Madan Parasar Mohanty Saurabh Shrivastava
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.
pp 513-519
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.
pp 521-529
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.
pp 531-539
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 $\int fh_id_𝜇=𝜆_i$ 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.
pp 541-557
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.
pp 559-566
Real Moments of the Restrictive Factor
Andrew Ledoan Alexandru Zaharescu
Let 𝜆 be a real number such that 0 < 𝜆 < 1. We establish asymptotic formulas for the weighted real moments $\sum_{n≤ x}R^𝜆(n)(1-n/x)$, where $R(n)=\prod^k_{v=1}p^{𝛼 v-1}_v$ is the Atanassov strong restrictive factor function and $n=\prod^k_{v=1}p^{𝛼 v}_v$ is the prime factorization of 𝑛.
Current Issue
Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.