pp 455-467
Hyperbolicity in Median Graphs
If 𝑋 is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1 x_2],[x_2 x_3]$ and $[x_3 x_1]$ in 𝑋. The space 𝑋 is 𝛿-hyperbolic (in the Gromov sense) if any side of 𝑇 is contained in a 𝛿-neighborhood of the union of the two other sides, for every geodesic triangle 𝑇 in 𝑋. If 𝑋 is hyperbolic, we denote by 𝛿(𝑋) the sharp hyperbolicity constant of 𝑋, i.e.,$𝛿(X)=\inf\{𝛿≥ 0: X \quad\text{is}\quad 𝛿-\text{hyperbolic}\}$. In this paper we study the hyperbolicity of median graphs and we also obtain some results about general hyperbolic graphs. In particular, we prove that a median graph is hyperbolic if and only if its bigons are thin.
pp 469-478
On the Number of Generators of a Projective Module
Sumit Kumar Upadhyay Shiv Datt Kumar Raja Sridharan
In this article we give a bound on the number of generators of a finitely generated projective module of constant rank over a commutative Noetherian ring in terms of the rank of the module and the dimension of the ring. Under certain conditions we provide an improvement to the Forster–Swan bound in case of finitely generated projective modules of rank 𝑛 over an affine algebra over a finite field or an algebraically closed field.
pp 479-490
Let 𝐸 be a vector bundle and 𝐿 be a line bundle over a smooth projective variety 𝑋 . In this article, we give a condition for the vanishing of Dolbeault cohomology groups of the form $H^{p,q}(X, S^𝛼 E \otimes {\wedge}^𝛽 E \otimes L)$ when $S^{𝛼+𝛽} E \otimes L$ is ample. This condition is shown to be invariant under the interchange of 𝑝 and 𝑞. The optimality of this condition is discussed for some parameter values.
pp 491-498
Let 𝐺 be a finite group, and $n(G)$ be the set of the number of subgroups of possible order of 𝐺. We investigate the structure of 𝐺 satisfying that $n(G) = \{1, m\}$ for any positive integer 𝑚 > 1. At first, we prove that the nilpotent length of 𝐺 is less than 2. Secondly, we investigate nilpotent groups with $m = p + 1$ or $p^2 + p + 1$ (𝑝 is a prime), and we get the classification of such kinds of groups. At last, we investigate non-nilpotent groups with $m = p + 1$ and get the classification of the groups under consideration.
pp 499-513
We consider a class of nonlinear viscous Cahn–Hilliard equations with gradient dependent potentials and sources. By a Galerkin approximation scheme combined with the potential well method, we prove the global existence of weak solutions.
pp 515-524
Sign Changing Solutions of the 𝑝(𝑥)-Laplacian Equation
This paper deals with the variational and Nehari manifold method for the 𝑝(𝑥)-Laplacian equations in a bounded domain or in the whole space. We prove existence of sign changing solutions under certain conditions.
pp 525-537
Strichartz Estimates for the Schrödinger Propagator for the Laguerre Operator
We obtain homogeneous Strichartz estimate for the Schrödinger propagator $e^{-itL_𝛼}$ for the Laguerre operator $L_𝛼$ on $\mathbb{R}^n_+$. We follow regularization technique as introduced in J. Funct. Anal. 224(2) (2005) 371–385. We also establish inhomogeneous Strichartz estimates for different admissible pairs.
pp 539-546
Completely Continuous and Weakly Completely Continuous Abstract Segal Algebras
Let $\mathcal{A}$ be a Banach algebra. It is obtained a necessary and sufficient condition for the complete continuity and also weak complete continuity of symmetric abstract Segal algebras with respect to $\mathcal{A}$, under the condition of the existence of an approximate identity for $\mathcal{B}$, bounded in $\mathcal{A}$. In addition, a necessary condition for the weak complete continuity of $\mathcal{A}$ is given. Moreover, the applications of these results about some group algebras on locally compact groups are obtained.
pp 547-575
Arup Chattopadhyay Kalyan B Sinha
In J. Funct. Anal. 257 (2009) 1092–1132, Dykema and Skripka showed the existence of higher order spectral shift functions when the unperturbed self-adjoint operator is bounded and the perturbation is Hilbert–Schmidt. In this article, we give a different proof for the existence of spectral shift function for the third order when the unperturbed operator is self-adjoint (bounded or unbounded, but bounded below).
pp 577-586
Determination of Star Bodies from 𝑝-Centroid Bodies
In this paper, we prove that an origin-symmetric star body is uniquely determined by its 𝑝-centroid body. Furthermore, using spherical harmonics, we establish a result for non-symmetric star bodies. As an application, we show that there is a unique member of $𝛤_p\langle K \rangle$ characterized by having larger volume than any other member, for all real 𝑝 ≥ 1 that are not even natural numbers, where $𝛤_p\langle K \rangle$ denotes the 𝑝-centroid equivalence class of the star body 𝐾.
pp 587-593 Subject Index
pp 595-597 Author Index
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Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.