pp 137-143
A Note on 𝑝-Groups of Order $≤ p^4$
Houshang Behravesh Hamid Mousavi
In [1], we defined $c(G),q(G)$ and $p(G)$. In this paper we will show that if 𝐺 is a 𝑝-group, where 𝑝 is an odd prime and $|G|≤ p^4$, then $c(G)=q(G)=p(G)$. However, the question of whether or not there is a 𝑝-group 𝐺 with strict inequality $c(G)=q(G) < p(G)$ is still open.
pp 145-148
Clean Elements in Abelian Rings
Let 𝑅 be a ring with identity. An element in 𝑅 is said to be clean if it is the sum of a unit and an idempotent. 𝑅 is said to be clean if all of its elements are clean. If every idempotent in 𝑅 is central, then 𝑅 is said to be abelian. In this paper we obtain some conditions equivalent to being clean in an abelian ring.
pp 149-158
On a Class of Semicommutative Modules
Nazim Agayev Tahire Özen Abdullah Harmanci
Let 𝑅 be a ring with identity, 𝑀 a right 𝑅-module and $S=\mathrm{End}_R(M)$. In this note, we introduce 𝑆-semicommutative, 𝑆-Baer, $S-q.$-Baer and $S-p.q.$-Baer modules. We study the relations between these classes of modules. Also we prove if 𝑀 is an 𝑆-semicommutative module, then 𝑀 is an $S-p.q.$-Baer module if and only if $M[x]$ is an $S[x]-p.q.$-Baer module, 𝑀 is an 𝑆-Baer module if and only if $M[x]$ is an $S[x]$-Baer module, 𝑀 is an $S-q.$-Baer module if and only if $M[x]$ is an $S[x]-q.$-Baer module.
pp 159-164
A Generalization of the Finiteness Problem in Local Cohomology Modules
Let $\mathfrak{a}$ be an ideal of a commutative Noetherian ring 𝑅 with non-zero identity and let 𝑁 be a weakly Laskerian 𝑅-module and 𝑀 be a finitely generated 𝑅-module. Let 𝑡 be a non-negative integer. It is shown that if $H^i_{\mathfrak{a}}(N)$ is a weakly Laskerian 𝑅-module for all $i < t$, then $\mathrm{Hom}_R(R/\mathfrak{a},H^t_{\mathfrak{a}}(M, N))$ is weakly Laskerian 𝑅-module. Also, we prove that $\mathrm{Ext}^i_R(R/\mathfrak{a},H^t_{\mathfrak{a}}(N))$ is weakly Laskerian 𝑅-module for all $i=0,1$. In particular, if $\mathrm{Supp}_R(H^i_{\mathfrak{a}}(N))$ is a finite set for all $i < t$, then $\mathrm{Ext}^i_R(R/\mathfrak{a},H^t_{\mathfrak{a}}(N))$ is weakly Laskerian 𝑅-module for all $i=0,1$.
pp 165-177
We begin the study of arbitrary split Lie triple systems by focussing on those with a coherent 0-root space. We show that any 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 $j≠ k$. Under certain conditions, it is shown that 𝑇 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. The key tool in this job is the notion of connection of roots in the framework of split Lie triple systems.
pp 179-186
Sign (di)Lemma for Dimension Shifting
There is a surprising occurrence of some minus signs in the isomorphisms produced in the well-known technique of dimension shifting in calculating derived functors in homological algebra. We explicitly determine these signs. Getting these signs right is important in order to avoid basic contradictions. We illustrate the result – which we call as the sign lemma for dimension shifting – by some de Rham cohomology and Chern class considerations for compact Riemann surfaces.
pp 187-195
A Note on a Paraholomorphic Cheeger-Gromoll Metric
The aim of this note is to study a paraholomorphic Cheeger–Gromoll metric on the tangent bundle of Riemannian manifolds.
pp 197-201
An Elementary Approach to Gap Theorems
Using elementary comparison geometry, we prove: Let (𝑀, 𝑔) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature 𝐾 satisfies $-1-s(r)≤ K≤ -1$, where 𝑟 denotes distance to a fixed point in 𝑀. If $\lim_{r→∞} e^{2r}s(r)=0$, then (𝑀, 𝑔) has to be isometric to $\mathbb{H}^n$.
The same proof also yields that if 𝐾 satisfies $-s(r)≤ K≤ 0$ where $\lim_{r→∞}r^2 s(r)=0$, then (𝑀, 𝑔)) is isometric to $\mathbb{R}^n$, a result due to Greene and $Wu$.
Our second result is a local one: Let (𝑀, 𝑔) be any Riemannian manifold. For a $\in\mathbb{R}$, if $K≤ a$ on a geodesic ball $B_p(R)$ in 𝑀 and $K=a$ on $𝜕 B_p(R)$, then $K=a$ on $B_p(R)$.
pp 203-220
Transferring Strong Boundedness among Laguerre Orthogonal Systems
I Abu-Falahah R A Macías C Segovia J L Torrea
Given the family of Laguerre polynomials, it is known that several orthonormal systems of Laguerre functions can be considered. In this paper we prove that an exhaustive knowledge of the boundedness in weighted $L^p$ of the heat and Poisson semigroups, Riesz transforms and 𝑔-functions associated to a particular Laguerre orthonormal system of functions, implies a complete knowledge of the boundedness of the corresponding operators on the other Laguerre orthonormal system of functions. As a byproduct, new weighted $L^p$ boundedness are obtained. The method also allows us to get new weighted estimates for operators related with Laguerre polynomials.
pp 221-230
2-Summing Operators on $C([0, 1], l_p)$ with Values in $l_1$
Let 𝛺 be a compact Hausdorff space, 𝑋 a Banach space, $C(𝛺,X)$ the Banach space of continuous 𝑋-valued functions on 𝛺 under the uniform norm, $U:C(𝛺,X)→ Y$ a bounded linear operator and $U^\#,U_\#$ two natural operators associated to 𝑈. For each $1≤ s <∞$, let the conditions $(𝛼)U\in 𝛱_s(C(𝛺, X), Y);(𝛽)U^\#\in 𝛱_s(C(𝛺), 𝛱_s(X, Y));(𝛾)U_\#\in 𝛱_s(X, 𝛱_s(C(𝛺), Y))$. A general result, [10,13], asserts that (𝛼) implies (𝛽) and (𝛾). In this paper, in case 𝑠=2, we give necessary and sufficient conditions that natural operators on $C([0,1],l_p)$ with values in $l_1$ satisfies (𝛼), (𝛽) and (𝛾), which show that the above implication is the best possible result.
pp 231-249
Domain Decomposition Methods for Hyperbolic Problems
In this paper a method is developed for solving hyperbolic initial boundary value problems in one space dimension using domain decomposition, which can be extended to problems in several space dimensions. We minimize a functional which is the sum of squares of the 𝐿^{2} norms of the residuals and a term which is the sum of the squares of the 𝐿^{2} norms of the jumps in the function across interdomain boundaries. To make the problem well posed the interdomain boundaries are made to move back and forth at alternate time steps with sufficiently high speed. We construct parallel preconditioners and obtain error estimates for the method.
The Schwarz waveform relaxation method is often employed to solve hyperbolic problems using domain decomposition but this technique faces difficulties if the system becomes characteristic at the inter-element boundaries. By making the inter-element boundaries move faster than the fastest wave speed associated with the hyperbolic system we are able to overcome this problem.
pp 251-265
A New Fenchel Dual Problem in Vector Optimization
Radu Ioan Boţ Anca Dumitru Gert Wanka
We introduce a new Fenchel dual for vector optimization problems inspired by the form of the Fenchel dual attached to the scalarized primal multiobjective problem. For the vector primal-dual pair we prove weak and strong duality. Furthermore, we recall two other Fenchel-type dual problems introduced in the past in the literature, in the vector case, and make a comparison among all three duals. Moreover, we show that their sets of maximal elements are equal.
Current Issue
Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.