• Volume 91, Issue 1

      January 1982,   pages  1-71

    • Ramanujan and dirichlet series with euler products

      S S Rangachari

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      Ramanujan, in his unpublished manuscripts, had written down without proof, explicit linear combinations of cusp forms whose Mellin transforms possess Euler products in the sense of Hecke. All these results are proved here and their connection with the work of Hecke, Rankin and Serre is pointed out.

    • Oberbeck convection through vertical porous stratum

      N Rudraiah M Venkatachalappa M S Malashetty

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      Natural convection through a vertical porous stratum is investigated both analytically and numerically. Analytical solutions are obtained using a perturbation method valid for small values of buoyancy parameterN and the numerical solutions are obtained using Runge-Kutta-Gill method. It is shown that analytical solutions are valid forN < 1 and several features of the effect of large values ofN are reported. The combined effects of increase in the values of temperature difference between the plates and the permeability parameter on velocity, temperature, mass flow rate and the rate of heat transfer are reported. It is shown that higher temperature difference is required to achieve the mass flow rate in a porous medium equivalent to that of viscous flow.

    • Long waves in inviscid compressible atmosphere II

      P L Sachdev V S Seshadri

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      Solitary waves have been found in an adiabatic compressible atmosphere which, in ambient state, has winds and temperature gradient, generalizing our earlier results for the isothermal atmosphere. Explicit results are obtained for the special case of linear temperature and linear wind distributions in the undisturbed conditions. An important result of the study is that the number of possible critical speeds of the flow depends crucially on whether the maximum Richardson number (which is variable in the present example) is greater or less than 1/4.

    • On minimizing the duration of transportation

      Satya Prakash

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      The problem of minimizing the duration of transportation has been studied. The problem has been reduced to a goal programming-type problem which readily lends itself to solution by the standard transportation method. This approach to the solution of the problem is very much different from all other existing ones.

    • Weyl’s theorem and thin spectra

      Shanti Prasanna

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      Using the notion of thin sets we prove a theorem of Weyl type for the Wolf essential spectrum ofTβ (H). *Further we show that Weyl’s theorem holds for a restriction convexoid operator and consequently modify some results of Berberian. Finally we show that Weyl’s theorem holds for a paranormal operator and that a polynomially compact paranormal operator is a compact perturbation of a diagnoal normal operator. A structure theorem for polynomially compact paranormal operators is also given.

    • On the normality of the rings of Schubert varieties

      C Huneke V Lakshmibai

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      We prove that the cone over a Schubert variety inG/P (P being a maximal parabolic subgroup of classical type) is normal by exhibiting a 2-regular sequence inR(w) (the homogeneous coordinate ring of the Schubert varietyX(w) inG/P under the canonical protective embeddingG/P ⊂→ (p (H° G/P,L)),L being the ample generator of (PicG/P), which vanishes on the singular locus ofX(w). We also prove the surjectivity ofH° (G/Q, L) H° (X(w), L), whereQ is a classical parabolic subgroup (not necessarily maximal) ofG andL is an ample line bundle onG/Q.

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