pp 395-402
On Zero Sum Subsequences of Restricted Size
Let 𝐺 be a finite abelian group with $\exp(G)=e$. Let $s(G)$ be the minimal integer 𝑡 with the property that any sequence of 𝑡 elements in 𝐺 contains an 𝑒-term subsequence with sum zero. Let 𝑛, 𝑚 and 𝑟 be positive integers and 𝑚 ≥ 3. Furthermore, $𝜂(C^r_m)=a_r(m-1)+1$, for some constant $a_r$ depending on 𝑟 and 𝑛 is a fixed positive integer such that
$$n≥\frac{m^r(c(r)m-a_r(m-1)+m-3)(m-1)-(m+1)+(m+1)(a_r+1)}{m(m+1)(a_r+1)}$$
and $s(C^r_n)=(a_r+1)(n-1)+1$. In the above lower bound on $n,c(r)$ is the Alon-Dubiner constant. Then $s(C^r_{nm})=(a_r+1)(nm-1)+1$.
pp 403-407
The Rational Maps $F_𝜆(z) = z^m + 𝜆/z^d$ have no Herman Rings
It is proved that the rational maps in the family $\{z→ z^m+𝜆/z^d:𝜆\in\mathbb{C}\backslash\{0\}\}$ for integers $m,dgeq 2$ have no Herman rings.
pp 409-415
𝐾-Theory for Certain Extension Algebras of Purely Infinite Simple 𝐶*-Algebras
We consider extension algebras of unital purely infinite simple 𝐶*-algebras by purely infinite simple stable 𝐶*-algebras. 𝐾-theory of such extension algebras is described.
pp 417-428
Geometric Structures on Loop and Path Spaces
Vicente Muñoz Francisco Presas
The loop space associated to a Riemannian manifold admits a quasi-symplectic structure (that is, a closed 2-form which is non-degenerate up to a finite-dimensional kernel). We show how to construct a compatible almost-complex structure. Finally conditions to have contact structures on loop spaces are studied.
pp 429-440
𝐿_{𝑝}-Mixed Intersection Bodies and Star Duality
Zhao Chang-jian Wing-Sum Cheung
The paper extends the two notions of the dual mixed volumes and 𝐿_{𝑝}-intersection body to 𝑞-dual mixed volumes and 𝐿_{𝑝}-mixed intersection body, respectively. Inequalities for the star dual of 𝐿_{𝑝}-mixed intersection bodies are established.
pp 441-456
Mixed Norm Estimate for Radon Transform on Weighted $L^p$ Spaces
We will discuss about the mapping property of Radon transform on $L^p$ spaces with power weight. It will be shown that the Pitt’s inequality together with the weighted version of Hardy–Littlewood–Sobolev lemma imply weighted inequality for the Radon transform.
pp 457-464
Rigidity of Minimal Submanifolds with Flat Normal Bundle
Let $M^n(n≥ 3)$ be an 𝑛-dimensional complete immersed $\frac{n-2}{n}$-superstable minimal submanifold in an $(n+p)$-dimensional Euclidean space $\mathbb{R}^{n+p}$ with flat normal bundle. We prove that if the second fundamental form of 𝑀 satisfies some decay conditions, then 𝑀 is an affine plane or a catenoid in some Euclidean subspace.
pp 465-479
Lifshitz Tails for the Interband Light Absorption Coefficient
In this paper we consider the interband light absorption coefficient (ILAC) for various models. We show that at the lower and upper edges of the spectrum the Lifshitz tails behaviour of the density of states implies similar behaviour for the ILAC at appropriate energies. The Lifshitz tails property is also exhibited at some points corresponding to the internal band edges of the spectrum.
pp 481-493
Ting-Bin Cao Hong-Yan Xu Chuan-Xi Zhu
In this paper, we investigate the complex oscillation of differential polynomials generated by meromorphic solutions of differential equations
$$f^{(k)}+A(z)f=0,\quad k≥ 2,$$
where the coefficient 𝐴 is meromorphic in the unit disc $\mathbb{D}=\{z:|z| < 1\}$.
pp 495-506
Stabilization for the Vibrations Modeled by the `Standard Linear Model' of Viscoelasticity
We study the stabilization of vibrations of a flexible structure modeled by the `standard linear model’ of viscoelasticity in a bounded domain in $\mathbb{R}^n$ with a smooth boundary. We prove that amplitude of the vibrations remains bounded in the sense of a suitable norm in a space $\mathbb{X}$, defined explicitly in (22) subject to a restriction on the uncertain disturbing forces on $\mathbb{X}$. We also estimate the total energy of the system over time interval [0,𝑇] for any 𝑇>0, with a tolerance level of the disturbances. Finally, when the input disturbances are insignificant, uniform exponential stabilization is obtained and an explicit form for the energy decay rate is derived. These results are achieved by a direct method under undamped mixed boundary conditions.
pp 507-513
Mean Value Estimates of the Error Terms of Lehmer Problem
Let 𝑝 be an odd prime and 𝑎 be an integer coprime with 𝑝. Denote by $N(a,p)$ the number of pairs of integers 𝑏, 𝑐 with $bc≡ a(\mathrm{mod} p),1≤ b, c < p$ and with 𝑏, 𝑐 having different parity. The main purpose of this paper is to study the mean square value problem of $N(a,p)-\frac{1}{2}(p-1)$ over interval (𝑁, 𝑁+𝑀] with 𝑀,𝑁 positive integers by using the analytic methods, and finally by obtaining a sharp asymptotic formula.
Current Issue
Volume 127 | Issue 4
September 2017
© 2017 Indian Academy of Sciences, Bengaluru.