• Volume 111, Issue 3

      August 2001,   pages  249-370

    • On Totally Reducible Binary Forms: I

      C Hooley

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      Let 𝜐(𝑛) be the number of positive numbers up to a large limit 𝑛 that are expressible in essentially more than one way by a binary form 𝑓 that is a product of 𝑙 > 2 distinct linear factors with integral coefficients. We prove that

      $$𝜐(n) = O\left(n^{2/l-𝜂_l+\epsilon}\right),$$

      where

      \begin{equation*}𝜂_l=\begin{cases}1/l^2, \quad\text{if}\quad l=3,\\ (l-2)/l^2(l-1), \quad\text{if}\quad l>3,\end{cases}\end{equation*}

      thus demonstrating in particular that it is exceptional for a number represented by 𝑓 to have essentially more than one representation.

    • Stability of Picard Bundle Over Moduli Space of Stable Vector Bundles of Rank Two Over a Curve

      Indranil Biswas Tomás L Gómez

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      Answering a question of [BV] it is proved that the Picard bundle on the moduli space of stable vector bundles of rank two, on a Riemann surface of genus at least three, with fixed determinant of odd degree is stable.

    • Principal 𝐺-bundles on Nodal Curves

      Usha N Bhosle

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      Let 𝐺 be a connected semisimple affine algebraic group defined over 𝐶. We study the relation between stable, semistable 𝐺-bundles on a nodal curve 𝑌 and representations of the fundamental group of 𝑌. This study is done by extending the notion of (generalized) parabolic vector bundles to principal 𝐺-bundles on the desingularization 𝐶 of 𝑌 and using the correspondence between them and principal 𝐺-bundles on 𝑌. We give an isomorphism of the stack of generalized parabolic bundles on 𝐶 with a quotient stack associated to loop groups. We show that if 𝐺 is simple and simply connected then the Picard group of the stack of principal 𝐺-bundles on 𝑌 is isomorphic to ⊕𝑚 𝑍, 𝑚 being the number of components of 𝑌.

    • Uncertainty Principles on Two Step Nilpotent Lie Groups

      S K Ray

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      We extend an uncertainty principle due to Cowling and Price to two step nilpotent Lie groups, which generalizes a classical theorem of Hardy. We also prove an analogue of Heisenberg inequality on two step nilpotent Lie groups.

    • On Property (𝛽) in Banach Lattices, Calderón–Lozanowskiĭ and Orlicz–Lorentz Spaces

      Paweł Kolwicz

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      The geometry of Calderón–Lozanowskiĭ spaces, which are strongly connected with the interpolation theory, was essentially developing during the last few years (see [4, 9, 10, 12, 13, 17]). On the other hand many authors investigated property (𝛽) in Banach spaces (see [7, 19, 20, 21, 25, 26]). The first aim of this paper is to study property (𝛽) in Banach function lattices. Namely a criterion for property (𝛽) in Banach function lattice is presented. In particular we get that in Banach function lattice property (𝛽) implies uniform monotonicity. Moreover, property (𝛽) in generalized Calderón–Lozanowskiĭ function spaces is studied. Finally, it is shown that in Orlicz–Lorentz function spaces property (𝛽) and uniform convexity coincide.

    • On Oscillation and Asymptotic Behaviour of Solutions of Forced First Order Neutral Differential Equations

      N Parhi R N Rath

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      In this paper, sufficient conditions have been obtained under which every solution of

      $[y(t)± y(t-𝜏)]'±\mathcal{Q}(t)G(y(t-𝜎)) = f(t),\quad t≥ 0$,

      oscillates or tends to zero or to ± ∞ as 𝑡 → ∞. Usually these conditions are stronger than

      \begin{equation*}\int\limits_0^∞\mathcal{Q}(t)dt=∞.\tag{*}\end{equation*}

      An example is given to show that the condition $(*)$ is not enough to arrive at the above conclusion. Existence of a positive (or negative) solution of

      $[y(t)-y(t-𝜏)]'+\mathcal{Q}(t)G(y(t-𝜎))=f(t)$

      is considered.

    • Monotone Iterative Technique for Impulsive Delay Differential Equations

      Baoqiang Yan Xilin Fu

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      In this paper, by proving a new comparison result, we present a result on the existence of extremal solutions for nonlinear impulsive delay differential equations.

    • On Initial Conditions for a Boundary Stabilized Hybrid Euler–Bernoulli Beam

      Sujit K Bose

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      We consider here small flexural vibrations of an Euler–Bernoulli beam with a lumped mass at one end subject to viscous damping force while the other end is free and the system is set to motion with initial displacement 𝑦0(𝑥) and initial velocity 𝑦1(𝑥). By investigating the evolution of the motion by Laplace transform, it is proved (in dimensionless units of length and time) that

      $$\int_0^1 y_{xt}^2 dx ≤ \int_0^1 y_{xx}^2 dx, \quad t>t_0,$$

      where 𝑡0 may be sufficiently large, provided that {𝑦0, 𝑦1} satisfy very general restrictions stated in the concluding theorem. This supplies the restrictions for uniform exponential energy decay for stabilization of the beam considered in a recent paper.

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