Volume 93, Issue 2-3
December 1984, pages 67-160
pp 67-77 December 1984
On the Rogers-Ramanujan continued fraction
In the “Lost” note book, Ramanujan had stated a large number of results regarding evaluation of his continued fraction$$R(\tau ) = \frac{{exp2\pi i\tau /}}{{1 + }}\frac{{5exp(2\pi i\tau )}}{{1 + }}\frac{{exp(4\pi i\tau )}}{{1 + }}...$$ for certain values of τ. It is shown that all these results and many more have their source in the Kronecker limit formula.
pp 79-100 December 1984
pp 101-107 December 1984
Mean-value of the Riemann zeta-function on the critical line
R Balasubramanian K Ramachandra
This is an expository article. It is a collection of some important results on the meanvalue of$$\left| {\zeta (\frac{1}{2} + it)} \right|.$$
pp 109-116 December 1984
On the ratio of values of a polynomial
For given positive integersa andb, the equationa(x + 1)… (x + k) =b(y+1)… (y + k) in positive integers is considered. More general equations are also considered.
pp 117-135 December 1984
Linear and non-linear magnetoconvection in a porous medium
Linear and non-linear magnetoconvection in a sparsely packed porous medium with an imposed vertical magnetic field is studied. In the case of linear theory the conditions for direct and oscillatory modes are obtained using the normal modes. Conditions for simple and Hopf-bifurcations are also given. Using the theory of self-adjoint operator the variation of critical eigenvalue with physical parameters and boundary conditions is studied. In the case of non-linear theory the subcritical instabilities for disturbances of finite amplitude is discussed in detail using a truncated representation of the Fourier expansion. The formal eigenfunction expansion procedure in the Fourier expansion based on the eigenfunctions of the corresponding linear stability problem is justified by proving a completeness theorem for a general class of non-self-adjoint eigenvalue problems. It is found that heat transport increases with an increase in Rayleigh number, ratio of thermal diffusivity to magnetic diffusivity and porous parameter but decreases with an increase in Chandrasekhar number.
pp 137-145 December 1984
Principal bundles on the affine line
We prove that any principal bundle on the affine line over a perfect field with a reductive group as structure group comes from the base field by base change.
pp 147-160 December 1984
On estimates for integral solutions of linear inequalities
Recently, Bombieri and Vaaler obtained an interesting adelic formulation of the first and the second theorems of Minkowski in the Geometry of Numbers and derived an effective formulation of the well-known “Siegel’s lemma” on the size of integral solutions of linear equations. In a similar context involving linearinequalities, this paper is concerned with an analogue of a theorem of Khintchine on integral solutions for inequalities arising from systems of linear forms and also with an analogue of a Kronecker-type theorem with regard to euclidean frames of integral vectors. The proof of the former theorem invokes Bombieri-Vaaler’s adelic formulation of Minkowski’s theorem.
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