Volume 100, Issue 3
December 1990, pages 189-303
pp 189-202 December 1990
We establish a one-to-one correspondence between the set of all equivalence classes of affine Poisson structures (defined on the dual of a finite dimensional Lie algebra) and the set of all equivalence classes of central extensions of the Lie algebra by ℝ. We characterize all the affine Poisson structures defined on the duals of some lower dimensional Lie algebras. It is shown that under a certain condition every Poisson structure locally looks like an affine Poisson structure. As an application, we show the role played by affine Poisson structures in mechanics. Finally, we prove some involution theorems.
pp 203-220 December 1990
On the existence of a finite invariant measure
A necessary and sufficient condition is given for a Borel automorphism on a standard Borel space to admit an invariant probability measure.
pp 221-229 December 1990
Diophantinc approximation by linear forms on manifolds
M M Dodson B P Rynne J A G Vickers
The following Khintchine-type theorem is proved for manifoldsM embedded in ℝ^{k} which satisfy some mild curvature conditions. The inequality ¦q·x¦ <Ψ(¦q¦) whereΨ(r) → 0 asr → ∞ has finitely or infinitely many solutionsqεℤ^{k} for almost all (in induced measure) points x onM according as the sum Σ_{r}^{ = 1/∞}Ψ(r)r^{k−2} converges or diverges (the divergent case requires a slightly stronger curvature condition than the convergent case). Also, the Hausdorff dimension is obtained for the set (of induced measure 0) of point inM satisfying the inequality infinitely often whenψ(r) =r^{−t}. τ >k − 1.
pp 231-243 December 1990
Lebesgue-stieltjes integral inequalities in several variables with retardation
The aim of this paper is to establish some new Lebesgue-Stieltjes integral inequalities inn independent variables with retardation which generalize and unify continuous and discrete inequalities of the Gronwall-Bellman-Bihari type inn independent variables.
pp 245-253 December 1990
Integrability andL^{1}-convergence of sine series with generalized quasi-convex coefficients
In this paper we obtain a necessary and sufficient condition for the sine series with generalized quasi-convex coefficients to be a Fourier series. Also we studyL^{1}-convergence of this series under the said condition on the coefficients.
pp 255-258 December 1990
Some results involving Bessel function and Fox’s H-function of two variables
In this paper, we have presented two integrals and employed them to establish one Fourier Bessel expansion for Fox’s H-function of two variables.
pp 259-273 December 1990
Köthe spaces and topological algebra with bases
Nuclear Köthe sequence spaceλ(P) its crossdualλ(P)^{x} and their non-nuclear variants are examined as topological algebras. Modelling on them, a general theory of nuclear topological algebras with orthogonal basis is developed. As a by-product, abstract characterizations of sequence algebras ℓ^{∞} andc_{0} are obtained. In a topological algebra set-up, an abstract Grothendieck-Pietsch nuclearity criterion is developed.
pp 275-284 December 1990
Critical sobolev exponent problem in ℝ^{n}(n ≥ 4) with neumann boundary condition
In this paper we study the existence and non existence of positive solution for the critical Sobolev exponent problem − Δu =u(n + 2)/(n − 2) +λα(x)u) in Ω$$\frac{{\partial u}}{{\partial v}} = 0 on \partial B$$, where Ω is a bounded domain in ℝ^{n}(n ≥ 4).
pp 285-294 December 1990
Anderson model with decaying randomness existence of extended states
We study the Anderson model with decaying randomness inv ≥ 3 dimensions and show that there is absolutely continuous spectrum in [−2v, 2v]. The distribution of the potentials is assumed to have finite variance and the coupling constants decay at infinity at a rate α > 1.
pp 295-301 December 1990
An efficient algorithm for linear programming
A simple but efficient algorithm is presented for linear programming. The algorithm computes the projection matrix exactly once throughout the computation unlike that of Karmarkar’s algorithm where in the projection matrix is computed at each and every iteration. The algorithm is best suitable to be implemented on a parallel architecture. Complexity of the algorithm is being studied.
pp 303-303 December 1990 Erratum
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