• Volume 122, Issue 3

August 2012,   pages  313-484

• 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 $\zeta(k)/\pi^k$ for odd positive integers $k&gt;1$. In this paper, we give an alternate proof of their theorem about the conditional lower bound.

• Vertex Pancyclicity and New Sufficient Conditions

For a graph $G,\delta(G)$ denotes the minimum degree of 𝐺. In 1971, Bondy proved that, if 𝐺 is a 2-connected graph of order 𝑛 and $d(x)+d(y)\geq n$ 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 $n\geq 6$ and $|N(x)\cup N(y)|+\delta(G)\geq n$ 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 $n\geq 6$ and $|N(x)\cup N(y)|+d(w)\geq n$ 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.

• Minimal Degrees of Faithful Quasi-Permutation Representations for Direct Products of 𝑝-Groups

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 $H\times K$.

• 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.

• Quotient Semigroups and Extension Semigroups

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.

• 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.

• Isoperimetric Upper Bounds for the First Eigenvalue

Let 𝑀 be a closed hypersurface in a simply connected space form $\mathbb{M}(\kappa)$ where $\kappa=0,1$ or $-1$. In this paper, we give two isoperimetric upper bounds for the first eigenvalue of the Laplacian of 𝑀.

• 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 $M\times f\mathbb{R}_\varepsilon$, where 𝑀 is a semi-Riemannian manifold, $\mathbb{R}_\varepsilon$ is the real line $\mathbb{R}$ with metric $\varepsilon dt^2(\varepsilon =\pm 1)$, and $f:M\to \mathbb{R}^+$ 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.

• Holomorphic Two-Spheres in the Complex Grassmann Manifold $G(k, n)$

In this paper, we study the non-degenerate holomorphic $S^2$ in the complex Grassmann manifold $G(k,n), 2k\leq n$, by the method of moving frame. For a non-degenerate holomorphic one, there exists globally defined positive functions $\lambda_1,\ldots,\lambda_k$ on $S^2$. We first show that the holomorphic $S^2$ in $G(k, 2k)$ is totally geodesic if these $\,\lambda_i$ 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, $\lambda_i=\frac{1}{\sqrt{k}}$, 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.

• 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.

• Uncertainty Principles for the Cherednik Transform

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.

• The Urbanik Generalized Convolutions in the Non-Commutative Probability and a Forgotten Method of Constructing Generalized Convolution

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.

• 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.

• Variational Problem with Complex Coefficient of a Nonlinear Schrödinger Equation

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.

• # Proceedings – Mathematical Sciences

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