pp 131-137
Integral Inequalities for Self-Reciprocal Polynomials
Let $n≥ 1$ be an integer and let $\mathcal{P}_n$ be the class of polynomials 𝑃 of degree at most 𝑛 satisfying $z^nP(1/z)=P(z)$ for all $z\in C$. Moreover, let 𝑟 be an integer with $1≤ r≤ n$. Then we have for all $P\in\mathcal{P}_n$:
$$𝛼_n(r)\int^{2𝜋}_0|P(e^{it})|^2dt≤\int^{2𝜋}_0|P^{(r)}(e^{it})|^2dt≤𝛽_n(r)\int^{2𝜋}_0|P(e^{it})|^2dt$$
with the best possible factors
\begin{equation*}𝛼_n(r)=\begin{cases}\prod^{r-1}_{j=0}\left(\frac{n}{2}-j\right)^2, < \text{if 𝑛 is even},\\ \frac{1}{2}\left[\prod^{r-1}_{j=0}\left(\frac{n+1}{2}-j\right)^2+\prod^{r-1}_{j=0}\left(\frac{n-1}{2}-j\right)^2\right], < \text{if 𝑛 is odd},\end{cases}\end{equation*}
and
\begin{equation*}𝛽_n(r)=\frac{1}{2}\prod\limits^{r-1}_{j=0}(n-j)^2.\end{equation*}
This refines and extends a result due to Aziz and Zargar (1997).
pp 139-140
A Note on Two Camina's Theorems on Conjugacy Class Sizes
Let 𝐺 be a finite group. We mainly investigate how certain arithmetical conditions on conjugacy class sizes of some elements of biprimary order of 𝐺 influence the structure of 𝐺. Some known results are generalized.
pp 141-148
On 𝑠-Semipermutable Subgroups of Finite Groups and 𝑝-Nilpotency
A subgroup 𝐻 of a group is said to be 𝑠-semipermutable in 𝐺 if it is permutable with every Sylow 𝑝-subgroup of 𝐺 with $(p,|H|)=1$. Using the concept of 𝑠-semipermutable subgroups, some new characterizations of 𝑝-nilpotent groups are obtained and several results are generalized.
pp 149-161
FGT-Injective Dimensions of 𝛱-Coherent Rings and almost Excellent Extension
We study, in this article, the FGT-injective dimensions of 𝛱-coherent rings. If 𝑅 is right 𝛱-coherent, and $\mathcal{T}\mathcal{I}(\mathrm{resp.}\mathcal{T}\mathcal{F})$ stands for the class of FGT-injective (resp.FGT-flat) 𝑅-modules $(n≥ 0)$, we show that the following are equivalent:
(1) $FGT-Id_R(R)≤ n$;
(2) If $0→ M→ F^0→ F^1→\cdots$ is a right $\mathcal{T}\mathcal{F}$-resolution of left 𝑅-module 𝑀, then the sequence is exact at $F^k$ for $k≥ n-1$;
(3) For every flat right 𝑅-module 𝐹, there is an exact sequence $0→ F→ A^0→ A^1→\cdots→ A^n→ 0$ with each $A^i\in\mathcal{T}\mathcal{I}$;
(4) For every injective left 𝑅-module 𝐴, there is an exact sequence $0→ F_n→\cdots→ F_1→ F_0→ A→ 0$ with each $F_i\in\mathcal{T}\mathcal{F}$;
(5) If $\cdots→ I_1→ I_0→ M→ 0$ is a minimal left $\mathcal{T}\mathcal{I}$-resolution of a right 𝑅-module 𝑀, then the sequence is exact at $I_k$ for $k≥ n-1$.
Further, we characterize such homological dimension in terms of $\mathcal{T}\mathcal{I}-syzygy$ and $\mathcal{T}\mathcal{F}-cosyzygy$ of modules. Finally, we consider almost excellent extensions of rings. These extend the corresponding results in [10] as well.
pp 163-168
Deficiently Extremal Cohen-Macaulay Algebras
The aim of this paper is to study homological properties of deficiently extremal Cohen–Macaulay algebras. Eagon–Reiner showed that the Stanley–Reisner ring of a simplicial complex has a linear resolution if and only if the Alexander dual of the simplicial complex is Cohen–Macaulay. An extension of a special case of Eagon–Reiner theorem is obtained for deficiently extremal Cohen–Macaulay Stanley–Reisner rings.
pp 169-183
Segal-Bargmann Transform and Paley-Wiener Theorems on $M(2)$
We study the Segal–Bargmann transform on $M(2)$. The range of this transform is characterized as a weighted Bergman space. In a similar fashion Poisson integrals are investigated. Using a Gutzmer’s type formula we characterize the range as a class of functions extending holomorphically to an appropriate domain in the complexification of $M(2)$. We also prove a Paley–Wiener theorem for the inverse Fourier transform.
pp 185-198
On Split Lie Triple Systems II
Antonio J Calderón Martín M Forero Piulestán
In [4] it is studied that the structure of split Lie triple systems with a coherent 0-root space, that is, satisfying $[T_0,T_0,T]=0$ and $[T_0,T_𝛼,T_0]≠ 0$ for any nonzero root 𝛼 and where $T_0$ denotes the 0-root space and $T_𝛼$ the 𝛼-root space, by showing that any of such triple systems 𝑇 with a symmetric root system is of the form $T=\mathcal{U}+\sum_j I_j$ with $\mathcal{U}$ a subspace of the 0-root space $T_0$ and any $I_j$ a well described ideal of 𝑇, satisfying $[I_j,T,I_k]=0$ if $j≠ k$. It is also shown in [4] that under certain conditions, a split Lie triple system with a coherent 0-root space is the direct sum of the family of its minimal ideals, each one being a simple split Lie triple system, and the simplicity of 𝑇 is characterized. In the present paper we extend these results to arbitrary split Lie triple systems with no restrictions on their 0-root spaces.
pp 199-207
$A\mathcal{T}$-Algebras and Extensions of $AT$-Algebras
Lin and Su classified $A\mathcal{T}$-algebras of real rank zero. This class includes all $A\mathbb{T}$-algebras of real rank zero as well as many 𝐶*-algebras which are not stably finite. An $A\mathcal{T}$-algebra often becomes an extension of an $A\mathbb{T}$-algebra by an 𝐴𝐹-algebra. In this paper, we show that there is an essential extension of an $A\mathbb{T}$-algebra by an 𝐴𝐹-algebra which is not an $A\mathcal{T}$-algebra. We describe a characterization of an extension 𝐸 of an $A\mathbb{T}$-algebra by an 𝐴𝐹-algebra if 𝐸 is an $A\mathcal{T}$-algebra.
pp 209-215
Equivalence Relations of 𝐴𝐹-Algebra Extensions
In this paper, we consider equivalence relations of 𝐶*-algebra extensions and describe the relationship between the isomorphism equivalence and the unitary equivalence. We also show that a certain group homomorphism is the obstruction for these equivalence relations to be the same.
pp 217-241
Splittings of Free Groups, Normal Forms and Partitions of Ends
Siddhartha Gadgil Suhas Pandit
Splittings of a free group correspond to embedded spheres in the 3-manifold $M=\sharp_k S^2× S^1$. These can be represented in a normal form due to Hatcher. In this paper, we determine the normal form in terms of crossings of partitions of ends corresponding to normal spheres, using a graph of trees representation for normal forms. In particular, we give a constructive proof of a criterion determining when a conjugacy class in $𝜋_2(M)$ can be represented by an embedded sphere.
pp 243-248
Discreteness Criteria in 𝑃𝑈(1, 𝑛; 𝐶)
In this paper, we study the discreteness for non-elementary subgroups in 𝑃𝑈(1, 𝑛; 𝐶), and several discreteness criteria are obtained.
pp 249-258
On Qualitative Analysis of Delay Systems and $x^𝛥 = f (t, x, x^𝜎)$ on Time Scales
Here we solve two problems presented in paper [9] (C C Tisdell and A Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal. 68 (2008) 3504–3524). We study existence and uniqueness of solutions for delay systems and first-order dynamic equations of the form $x^𝛥=f(t,x,x^𝜎)$ on time scales by using the Banach’s fixed-point theorem. Some examples are presented to illustrate the efficiency of the proposed results.
Current Issue
Volume 127 | Issue 4
September 2017
© 2017 Indian Academy of Sciences, Bengaluru.