• Volume 119, Issue 5

November 2009,   pages  567-712

• 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.

• Wahl's Conjecture for a Minuscule $G/P$

We show that Wahl’s conjecture holds in all characteristics for a minuscule $G/P$.

• Univalence and Starlikeness of Nonlinear Integral Transform of Certain Class of Analytic Functions

Let $\mathcal{U}(\lambda, \mu)$ denote the class of all normalized analytic functions 𝑓 in the unit disk $|z| &lt; 1$ satisfying the condition

\begin{equation*}\frac{f(z)}{z}\neq 0\quad\text{and}\quad\left|f'(z)\left(\frac{z}{f(z)}\right)^{\mu +1}-1\right| &lt; \lambda,\quad |z| &lt; 1.\end{equation*}

For $f\in\mathcal{U}(\lambda, \mu)$ with $\mu\leq 1$ and $0\neq\mu_1\leq 1$, 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)^\mu dt\right]^{-1/\mu 1},\quad z\in\Delta.\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 $\mathcal{U}(\lambda, \mu)$.

• 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.

• On the Asymptotic of an Eigenvalue Problem with $2n$ Interior Singularities

In this paper we consider the linear differential equation of the form

$$-y''(x)+q(x)y(x)=\lambda y(x),\quad -\infty &lt; a &lt; x &lt; b &lt; \infty$$

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.

• 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.

• On Linear Isometries of Banach Lattices in $\mathcal{C}_0(\Omega)$-Spaces

Consider the space $\mathcal{C}_0(\Omega)$ endowed with a Banach lattice-norm $\|\cdot p\|$ that is not assumed to be the usual spectral norm $\|\cdot p\|_\infty$ 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 $(\mathcal{C}_0(\Omega),\|\cdot p\|)$ induces a partition 𝛱 of 𝛺 into a family of finite subsets $S\subset\Omega$ along with a bijection $T:\Pi\to\Pi$ which preserves cardinality, and a family $[u(S):S\in\Pi]$ of surjective linear maps $u(S):\mathcal{C}(T(S))\to\mathcal{C}(S)$ of the finite-dimensional $C^∗$-algebras $\mathcal{C}(S)$ such that

$$(U f)|_{T(S)}=u(S)(f|_S) \quad \forall f\in\mathcal{C}_0(\Omega) \quad \forall S\in\Pi.$$

Here we endow the space 𝛱 of finite sets $S\subset\Omega$ with a topology for which the bijection 𝑇 and the map 𝑢 are continuous, thus completing the analogy with the classical result.

• Classifying Cubic Edge-Transitive Graphs of Order $8p$

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.

• 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.

• Upper Packing Dimension of a Measure and the Limit Distribution of Products of i.i.d. Stochastic Matrices

This article gives sufficient conditions for the limit distribution of products of i.i.d. $2\times 2$ 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.

• 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.

• On Kahler-Norden Manifolds

• Subject Index

• Author Index

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