pp 495-503
New Inequalities for the Hurwitz Zeta Function
We establish various new inequalities for the Hurwitz zeta function. Our results generalize some known results for the polygamma functions to the Hurwitz zeta function.
pp 505-516
An Optimal Version of an Inequality Involving the Third Symmetric Means
Wen Jiajin Yuan Jun Yuan Shufeng
Let $(GA)^{[k]}_n(a), A_n(a), G_n(a)$ be the third symmetric mean of 𝑘 degree, the arithmetic and geometric means of $a_1,\ldots,a_n(a_i>0,i=1,\ldots,n)$, respectively. By means of descending dimension method, we prove that the maximum of 𝑝 is $\frac{k-1}{n-1}$ and the minimum of 𝑞 is $\frac{n}{n-1}\left(\frac{k-1}{k}\right)^{\frac{k}{n}}$ so that the inequalities
$$(G_n(a))^{1-p}(A_n(a))^p≤ (GA)^{[k]}_n(a)≤ (1-q)G_n(a)+q A_n(a) (2≤ k≤ n-1)$$
hold.
pp 517-524
On Ideals and Quotients of $A\mathcal{T}$-Algebras
Some results on $A\mathcal{T}$-algebras are given. We study the problem when ideals, quotients and hereditary subalgebras of $A\mathcal{T}$-algebras are $A\mathcal{T}$-algebras or $A\mathcal{T}$-algebras, and give a necessary and sufficient condition of a hereditary subalgebra of an $A\mathcal{T}$-algebra being an $A\mathcal{T}$-algebra.
pp 525-536
Local Duality for 2-Dimensional Local Ring
We prove a local duality for some schemes associated to a 2-dimensional complete local ring whose residue field is an 𝑛-dimensional local field in the sense of Kato–Parshin. Our results generalize the Saito works in the case 𝑛=0 and are applied to study the Bloch–Ogus complex for such rings in various cases.
pp 537-546
Some Augmentation Quotients of Integral Group Rings
Deepak Kumar Gumber Ram Karan Indu Pal
Let 𝐺 be a group and 𝐻 be a subgroup of 𝐺. A complete description of $𝛥(G)𝛥^n(H)/𝛥^{n+1}(H)$ is given, and as a consequence the structures of 𝛥(𝐺)/𝛥(𝐻) and $𝛥^2(G)/𝛥^2(H)$ are determined. Also, the structure of $𝛥^n(G)/𝛥^n(H)$ for all 𝑛 ≥ 1 is determined when 𝐺 is a free group.
pp 547-555
On 𝑛-Weak Amenability of Rees Semigroup Algebras
Let 𝑆 be a Rees matrix semigroup. We show that $l^1(S)$ is $(2k+1)$-weakly amenable for $k\in\mathbb{Z}^+$.
pp 557-567
On 𝑃-Coherent Endomorphism Rings
A ring is called right 𝑃-coherent if every principal right ideal is finitely presented. Let $M_R$ be a right 𝑅-module. We study the 𝑃-coherence of the endomorphism ring 𝑆 of $M_R$. It is shown that 𝑆 is a right 𝑃-coherent ring if and only if every endomorphism of $M_R$ has a pseudokernel in add $M_R; S$ is a left 𝑃-coherent ring if and only if every endomorphism of $M_R$ has a pseudocokernel in add $M_R$. Some applications are given.
pp 569-572
Hypersurfaces in Simply Connected Space Forms
Let 𝑀 be a hypersurface in a simply connected space form $\mathbb{M}(𝜅)$. We prove some rigidity results for 𝑀 in terms of lower bounds on the Ricci curvature of the hypersurface 𝑀.
pp 573-581
Harmonic Riemannian Maps on Locally Conformal Kaehler Manifolds
We study harmonic Riemannian maps on locally conformal Kaehler manifolds ($lcK$ manifolds). We show that if a Riemannian holomorphic map between $lcK$ manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the $lcK$ manifold is Kaehler. Then we find similar results for Riemannian maps between $lcK$ manifolds and Sasakian manifolds. Finally, we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.
pp 583-612
Planar Algebra of the Subgroup-Subfactor
We give an identification between the planar algebra of the subgroup-subfactor $R \rtimes H \subset R \rtimes G$ and the 𝐺-invariant planar subalgebra of the planar algebra of the bipartite graph $\star_n$, where $n=[G:H]$. The crucial step in this identification is an exhibition of a model for the basic construction tower, and thereafter of the standard invariant of $R \rtimes H \subset R \rtimes G$ in terms of operator matrices.
We also obtain an identification between the planar algebra of the fixed algebra subfactor $R^G \subset R^H$ and the 𝐺-invariant planar subalgebra of the planar algebra of the `flip’ of $\star_n$.
pp 613-625
Some Properties of Unbounded Operators with Closed Range
S H Kulkarni M T Nair G Ramesh
Let $H_1, H_2$ be Hilbert spaces and 𝑇 be a closed linear operator defined on a dense subspace 𝐷(𝑇) in 𝐻_{1} and taking values in 𝐻_{2}. In this article we prove the following results:
(i) Range of 𝑇 is closed if and only if 0 is not an accumulation point of the spectrum $𝜎(T^* T)$ of $T^* T$,
In addition, if $H_1=H_2$ and 𝑇 is self-adjoint, then
(ii) $\inf \{\| Tx\|:x\in D(T)\cap N(T)^\perp \| x\|=1\}=\inf\{| 𝜆|:0≠𝜆\in𝜎(T)\}$,
(iii) Every isolated spectral value of 𝑇 is an eigenvalue of 𝑇,
(iv) Range of 𝑇 is closed if and only if 0 is not an accumulation point of the spectrum $𝜎(T)$ of 𝑇,
(v) 𝜎(𝑇) bounded implies 𝑇 is bounded.
We prove all the above results without using the spectral theorem. Also, we give examples to illustrate all the above results.
pp 627-635
On Existence and Stability of Solutions for Higher Order Semilinear Dirichlet Problems
We provide existence and stability results for semilinear Dirichlet problems with nonlinearity satisfying general growth conditions. We consider the case when both the coefficients of the differential operator and the nonlinear term depend on the numerical parameter. We show applications for the fourth order semilinear Dirichlet problem.
pp 637-643 Subject Index
pp 645-647 Author Index
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Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.