Volume 111, Issue 3
August 2001, pages 249-370
pp 249-262 August 2001
On totally reducible binary forms: I
Letv(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 formf that is a product ofl > 2 distinct linear factors with integral coefficients. We prove that$$v(n) = O(n^{2/\ell - \eta _\ell + \in } )$$, where$$\eta \ell = \left\{ \begin{gathered} 1/\ell ^2 , if \ell = 3, \hfill \\ (\ell - 2)/\ell ^2 (\ell - 1), if \ell > 3 \hfill \\ \end{gathered} \right.$$, thus demonstrating in particular that it is exceptional for a number represented byf to have essentially more than one representation.
pp 263-269 August 2001
Stability of Picard bundle over moduli space of stable vector bundles of rank two over a curve
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.
pp 271-291 August 2001
PrincipalG-bundles on nodal curves
LetG be a connected semisimple affine algebraic group defined over C. We study the relation between stable, semistable G-bundles on a nodal curveY and representations of the fundamental group ofY. This study is done by extending the notion of (generalized) parabolic vector bundles to principal G-bundles on the desingularizationC ofY and using the correspondence between them and principal G-bundles onY. We give an isomorphism of the stack of generalized parabolic bundles onC 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 onY is isomorphic to ⊕_{m} Z,m being the number of components ofY.
pp 293-318 August 2001
Uncertainty principles on two step nilpotent Lie groups
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.
pp 319-336 August 2001
On property (β) in Banach lattices, Calderón-Lozanowskii and Orlicz-Lorentz spaces
The geometry of Calderón-Lozanowskii 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-Lozanowskii function spaces is studied. Finally, it is shown that in Orlicz-Lorentz function spaces property (β) and uniform convexity coincide.
pp 337-350 August 2001
In this paper, sufficient conditions have been obtained under which every solution of$$\left[ {y(t) \pm y(t - \tau )]'} \right. \pm Q(t)G(y(t - \sigma )) = f(t), t \geqslant 0$$, oscillates or tends to zero or to ±∞ → ∞. Usually these conditions are stronger than$$\int\limits_0^\infty {Q(t)dt = \infty } $$. 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$$[y(t) - y(t - \tau )]' + Q(t)G(y(t - \sigma )) = f(t)$$ is considered.
pp 351-363 August 2001
Monotone iterative technique for impulsive delay differential equations
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.
pp 365-370 August 2001
On initial conditions for a boundary stabilized hybrid Euler-Bernoulli beam
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 displacementy^{0}(x) and initial velocityy^{1} (x). By investigating the evolution of the motion by Laplace transform, it is proved (in dimensionless units of length and time) that$$\smallint _0^1 y_{xt}^2 dx \leqslant \smallint _0^1 y_{xx}^2 dx,t > t_0 $$, wheret_{0} may be sufficiently large, provided that {y^{0},y^{1}} 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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