pp 567-570
Sums of Powers of Fibonacci Polynomials
Using the explicit (Binet) formula for the Fibonacci polynomials, a summation formula for powers of Fibonacci polynomials is derived straightforwardly, which generalizes a recent result for squares that appeared in Proc. Ind. Acad. Sci. (Math. Sci.) 118 (2008) 27--41.
pp 571-592
Wahl's Conjecture for a Minuscule 𝐺/𝑃
We show that Wahl’s conjecture holds in all characteristics for a minuscule 𝐺/𝑃.
pp 593-610
Univalence and Starlikeness of Nonlinear Integral Transform of Certain Class of Analytic Functions
M Obradović S Ponnusamy P Vasundhra
Let $\mathcal{U}(𝜆, 𝜇)$ denote the class of all normalized analytic functions 𝑓 in the unit disk $|z| < 1$ satisfying the condition
\begin{equation*}\frac{f(z)}{z}≠ 0\quad\text{and}\quad\left|f'(z)\left(\frac{z}{f(z)}\right)^{𝜇 +1}-1\right| < 𝜆,\quad |z| < 1.\end{equation*}
For $f\in\mathcal{U}(𝜆, 𝜇)$ with 𝜇 ≤ 1 and $0≠𝜇_1≤ 1$, and for a positive real-valued integrable function 𝜑 defined on [0,1] satisfying the normalized condition $\int^1_0\varphi(t)dt=1$, we consider the transform $G_\varphi f(z)$ defined by
\begin{equation*}G_\varphi f(z)=z\left[\int^1_0\varphi(t)\left(\frac{zt}{f(tz)}\right)^𝜇 dt\right]^{-1/𝜇 1},\quad z\in𝛥.\end{equation*}
In this paper, we find conditions on the range of parameters 𝜆 and 𝜇 so that the transform $G_\varphi f$ is univalent or star-like. In addition, for a given univalent function of certain form, we provide a method of obtaining functions in the class $\mathcal{U}(𝜆, 𝜇)$.
pp 611-618
Riemannian Foliations on Quaternion 𝐶𝑅-Submanifolds of an almost Quaternion Kähler Product Manifold
The purpose of this paper is to study the canonical foliations of a quaternion 𝐶𝑅-submanifold of an almost quaternion Kähler product manifold.
pp 619-627
On the Asymptotic of an Eigenvalue Problem with 2𝑛 Interior Singularities
In this paper we consider the linear differential equation of the form
$$-y''(x)+q(x)y(x)=𝜆 y(x),\quad -∞ < a < x < b < ∞$$
where 𝑦 satisfies Dirichlet boundary conditions and 𝑞 is a real-valued function which has even number of singularities at $c_1,\ldots,c_{2n}\in(a, b)$. We will study the asymptotic eigenvalue near the singularity points.
pp 629-634
A Distinguished Real Banach Algebra
We present a new and elementary approach to characterize the maximal ideals and their associated multiplicative linear functionals for a classical real Banach algebra of analytic functions.
pp 635-645
On Linear Isometries of Banach Lattices in $\mathcal{C}_0(𝛺)$-Spaces
Consider the space $\mathcal{C}_0(𝛺)$ endowed with a Banach lattice-norm $\|\cdot p\|$ that is not assumed to be the usual spectral norm $\|\cdot p\|_∞$ of the supremum over 𝛺. A recent extension of the classical Banach-Stone theorem establishes that each surjective linear isometry 𝑈 of the Banach lattice $(\mathcal{C}_0(𝛺),\|\cdot p\|)$ induces a partition 𝛱 of 𝛺 into a family of finite subsets $S\subset𝛺$ along with a bijection $T:𝛱→𝛱$ which preserves cardinality, and a family $[u(S):S\in𝛱]$ of surjective linear maps $u(S):\mathcal{C}(T(S))→\mathcal{C}(S)$ of the finite-dimensional 𝐶*-algebras $\mathcal{C}(S)$ such that
$$(U f)|_{T(S)}=u(S)(f|_S) \quad \forall f\in\mathcal{C}_0(𝛺) \quad \forall S\in𝛱.$$
Here we endow the space 𝛱 of finite sets $S\subset𝛺$ with a topology for which the bijection 𝑇 and the map 𝑢 are continuous, thus completing the analogy with the classical result.
pp 647-653
Classifying Cubic Edge-Transitive Graphs of Order 8𝑝
Mehdi Alaeiyan M K Hosseinipoor
A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let 𝑝 be a prime. It was shown by Folkman (J. Combin. Theory 3(1967) 215--232) that a regular edge-transitive graph of order 2𝑝 or 2𝑝^{2} is necessarily vertex-transitive. In this paper, an extension of his result in the case of cubic graphs is given. It is proved that, every cubic edge-transitive graph of order 8𝑝 is symmetric, and then all such graphs are classified.
pp 655-667
Measure Free Martingales and Martingale Measures
Let $T\subset\mathbb{R}$ be a countable set, not necessarily discrete. Let $f_t,t\in T$, be a family of real-valued functions defined on a set 𝛺. We discuss conditions which imply that there is a probability measure on 𝛺 under which the family $f_t,t\in T$, is a martingale.
pp 669-677
Arunava Mukherjea Ricardo Restrepo
This article gives sufficient conditions for the limit distribution of products of i.i.d. 2 × 2 stochastic matrices to be continuous singular, when the support of the distribution of the individual random matrices is countably infinite. It extends a previous result for which the support of the random matrices is finite. The result is based on adapting existing proofs in the context of attractors and iterated function systems to the case of infinite iterated function systems.
pp 679-698
A Further Note on the Force Discrepancy for Wing Theory in Euler Flow
Uniform steady potential flow past a wing aligned at a small angle to the flow direction is considered. The standard approach is to model this by a vortex sheet, approximated by a finite distribution of horseshoe vortices. In the limit as the span of the horseshoe vortices tends to zero, an integral distribution of infinitesimal horseshoe vortices over the vortex sheet is obtained. The contribution to the force on the wing due to the presence of one of the infinitesimal horseshoe vortices in the distribution is focused upon. Most of the algebra in the force calculation is evaluated using Maple software and is given in the appendices. As in the two previous papers by the authors on wing theory in Euler flow [E Chadwick, A slender-wing theory in potential flow, Proc. R. Soc. A461 (2005) 415–432, and E Chadwick and A Hatam, The physical interpretation of the lift discrepancy in Lanchester–Prandtl lifting wing theory for Euler flow, leading to the proposal of an alternative model in Oseen flow, Proc. R. Soc. A463 (2007) 2257–2275], it is shown that the normal force is half that expected. In this further note, in addition it is demonstrated that the axial force is infinite. The implications and reasons for these results are discussed.
pp 699-699 Erratum
pp 701-708 Subject Index
pp 709-712 Author Index
Current Issue
Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.