• Volume 100, Issue 3

December 1990,   pages  189-303

• Affine Poisson structures

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.

• 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.

• Diophantinc approximation by linear forms on manifolds

The following Khintchine-type theorem is proved for manifoldsM embedded in ℝk which satisfy some mild curvature conditions. The inequality ¦q·x¦ &lt;Ψq¦) whereΨ(r) → 0 asr → ∞ has finitely or infinitely many solutionsqεℤk for almost all (in induced measure) points x onM according as the sum Σr = 1/∞Ψ(r)rk−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) =rt. τ &gt;k − 1.

• 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.

• Integrability andL1-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 studyL1-convergence of this series under the said condition on the coefficients.

• 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.

• 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 andc0 are obtained. In a topological algebra set-up, an abstract Grothendieck-Pietsch nuclearity criterion is developed.

• Critical sobolev exponent problem in ℝn(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 ℝn(n ≥ 4).

• 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 &gt; 1.

• 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.

• Verschiebung and Frobenius operators

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