pp 313-317
On the Dimension of Chowla–Milnor Space
In a recent work, Gun, Murty and Rath defined the Chowla–Milnor space and proved a non-trivial lower bound for these spaces. They also obtained a conditional improvement of this lower bound and noted that an unconditional improvement of their lower bound will lead to irrationality of $𝜁(k)/𝜋^k$ for odd positive integers 𝑘>1. In this paper, we give an alternate proof of their theorem about the conditional lower bound.
pp 319-328
Vertex Pancyclicity and New Sufficient Conditions
For a graph 𝐺,𝛿(𝐺) denotes the minimum degree of 𝐺. In 1971, Bondy proved that, if 𝐺 is a 2-connected graph of order 𝑛 and $d(x)+d(y)≥ 𝑛$ for each pair of non-adjacent vertices 𝑥,𝑦 in 𝐺, then 𝐺 is pancyclic or $G=K_{n/2,n/2}$. In 2001, $Xu$ proved that, if 𝐺 is a 2-connected graph of order $n≥ 6$ and $|N(x)\cup N(y)|+𝛿(G)≥ n$ for each pair of non-adjacent vertices 𝑥,𝑦 in 𝐺, then 𝐺 is pancyclic or $G=K_{n/2,n/2}$. In this paper, we introduce a new sufficient condition of generalizing degree sum and neighborhood union and prove that, if 𝐺 is a 2-connected graph of order $n≥ 6$ and $|N(x)\cup N(y)|+d(w)≥ n$ for any three vertices 𝑥,𝑦,𝑤 of $d(x,y)=2$ and 𝑤𝑥 or 𝑤𝑦 $\notin E(G)$ in 𝐺, then 𝐺 is 4-vertex pancyclic or 𝐺 belongs to two classes of well-structured exceptional graphs. This result also generalizes the above results.
pp 329-334
Minimal Degrees of Faithful Quasi-Permutation Representations for Direct Products of 𝑝-Groups
Ghodrat Ghaffarzadeh Mohammad Hassan Abbaspour
In [2], the algorithms of $c(G), q(G)$ and $p(G)$, the minimal degrees of faithful quasi-permutation and permutation representations of a finite group 𝐺 are given. The main purpose of this paper is to consider the relationship between these minimal degrees of non-trivial 𝑝-groups 𝐻 and 𝐾 with the group 𝐻 × 𝐾.
pp 335-337
Finite Groups with Three Conjugacy Class Sizes of some Elements
Let 𝐺 be a finite group. We prove as follows: Let 𝐺 be a 𝑝-solvable group for a fixed prime 𝑝. If the conjugacy class sizes of all elements of primary and biprimary orders of 𝐺 are $\{1,p^a,n\}$ with 𝑎 and 𝑛 two positive integers and (𝑝,𝑛)=1, then 𝐺 is 𝑝-nilpotent or 𝐺 has abelian Sylow 𝑝-subgroups.
pp 339-350
Quotient Semigroups and Extension Semigroups
Rong Xing Changguo Wei Shudong Liu
We discuss properties of quotient semigroup of abelian semigroup from the viewpoint of 𝐶*-algebra and apply them to a survey of extension semigroups. Certain interrelations among some equivalence relations of extensions are also considered.
pp 351-373
Isometric Coactions of Compact Quantum Groups on Compact Quantum Metric Spaces
We propose a notion of isometric coaction of a compact quantum group on a compact quantum metric space in the framework of Rieffel, where the metric structure is given by a Lipnorm. Within this setting we study the problem of the existence of a quantum isometry group.
pp 375-384
Isoperimetric Upper Bounds for the First Eigenvalue
Let 𝑀 be a closed hypersurface in a simply connected space form $\mathbb{M}(𝜅)$ where 𝜅=0,1 or -1. In this paper, we give two isoperimetric upper bounds for the first eigenvalue of the Laplacian of 𝑀.
pp 385-397
On the Mean Curvature of Semi-Riemannian Graphs in Semi-Riemannian Warped Products
We investigate the mean curvature of semi-Riemannian graphs in the semi-Riemannian warped product $M× f\mathbb{R}_𝜀$, where 𝑀 is a semi-Riemannian manifold, $\mathbb{R}_𝜀$ is the real line $\mathbb{R}$ with metric $𝜀 dt^2(𝜀 =± 1)$, and $f:M→ \mathbb{R}^+$ is the warping function. We obtain an integral formula for mean curvature and some results dealing with estimates of mean curvature, among these results is a Heinz–Chern type inequality.
pp 399-409
Holomorphic Two-Spheres in the Complex Grassmann Manifold 𝐺(𝑘, 𝑛)
Xiaoxiang Jiao Xu Zhong Xiaowei Xu
In this paper, we study the non-degenerate holomorphic $S^2$ in the complex Grassmann manifold 𝐺(𝑘, 𝑛), 2𝑘 ≤ 𝑛, by the method of moving frame. For a non-degenerate holomorphic one, there exists globally defined positive functions $𝜆_1,\ldots,𝜆_k$ on $S^2$. We first show that the holomorphic $S^2$ in $G(k, 2k)$ is totally geodesic if these $\,𝜆_i$ are all equal. Conversely, for any totally geodesic immersion 𝑓 from $S^2$ into $G(k, n)$, we prove that $f(S^2)\subset G(k, 2k)$ up to $U(n)$-transformation, $𝜆_i=\frac{1}{\sqrt{k}}$, the Gaussian curvature $K=\frac{4}{k}$ and 𝑓 is given by $(z_0,z_1)\mapsto(z_0 I_k,z_1 I_k,0)$, in terms of homogeneous coordinate.
pp 411-427
Finsler Metrics with Constant (or Scalar) Flag Curvature
By finding Killing vector fields of general Bryant’s metric we give a lot of new Finsler metrics of constant (or scalar) flag curvature and determine their scalar curvature.
pp 429-436
Uncertainty Principles for the Cherednik Transform
R Daher S L Hamad T Kawazoe N Shimeno
We shall investigate two uncertainty principles for the Cherednik transform on the Euclidean space $\mathfrak{a}$; Miyachi’s theorem and Beurling’s theorem. We give an analogue of Miyachi’s theorem for the Cherednik transform and under the assumption that $\mathfrak{a}$ has a hypergroup structure, an analogue of Beurling’s theorem for the Cherednik transform.
pp 437-458
Barbara Jasiulis-Gołdyn Anna Kula
The paper deals with the notions of weak stability and weak generalized convolution with respect to a generalized convolution, introduced by Kucharczak and Urbanik. We study properties of such objects and give examples of weakly stable measures with respect to the Kendall convolution. Moreover, we show that in the context of non-commutative probability, two operations: the 𝑞-convolution and the (𝑞,1)-convolution satisfy the Urbanik’s conditions for a generalized convolution, interpreted on the set of moment sequences. The weak stability reveals the relation between two operations.
pp 459-467
Fair Partitions of Polygons: An Elementary Introduction
We introduce the question: Given a positive integer 𝑁, can any 2D convex polygonal region be partitioned into 𝑁 convex pieces such that all pieces have the same area and the same perimeter? The answer to this question is easily `yes’ for 𝑁=2. We give an elementary proof that the answer is `yes’ for 𝑁=4 and generalize it to higher powers of 2.
pp 469-484
Variational Problem with Complex Coefficient of a Nonlinear Schrödinger Equation
Nigar Yildirim Aksoy Bunyamin Yildiz Hakan Yetiskin
An optimal control problem governed by a nonlinear Schrödinger equation with complex coefficient is investigated. The paper studies existence, uniqueness and optimality conditions for the control problem.
Current Issue
Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.