• Volume 95, Issue 1

September 1986,   pages  1-77

• Axisymmetric melting of a long cylinder due to an infinite flux

By employing a new embedding technique, a short-time analytical solution for the axisymmetric melting of a long cylinder due to an infinite flux is presented in this paper. The sufficient condition for starting the instantaneous melting of the cylinder has been derived. The melt is removed as soon as it is formed. The method of solution is simple and straightforward and consists of assuming fictitious initial temperature for some fictitious extension of the actual region.

• Instability of a planar liquid layer in an alternating longitudinal magnetic field with non-zero mean

The conducting liquid interface is found to undulate in an alternating magnetic field. It was shown earlier that ifM =B02/μηω, B0, ω, μ andη being the amplitude (complex) of the alternating longitudinal magnetic field imposed at the interface, the angular frequency of the field, the magnetic permeability and the viscosity respectively, and ifMc was the critical value ofM then the planar layer was stable or unstable according asM &lt; Mc orM &gt; Mc. In this paper we have determined the stability criterion when in addition to the alternating longitudinal field there acts a uniform field in the same direction. After comparing our results with those obtained earlier, in the absence of the uniform field, we find that the additional uniform field has a significant destabilizing effect.

• Convective instability of a rotating layer of ferromagnetic fluid

The instability of a hot horizontal layer of ferromagnetic fluid rotating about a vertical axis has been investigated when the Prandtl numberP &lt; 1. Earlier it was shown that forP &gt; 1 the overstability cannot occur. In this paper the convective and overstable marginal states have been investigated separately forP &lt; 1 and it is found that though convective marginal state is possible for alla, the non-dimensional wave number, and N the Taylor number, the overstability is possible only ifN &gt; (1 +P)π4/(1 −P) and in case the condition is satisfied, overstability is possible for all those values ofa which satisfya2 &lt; [N(1 −P)π2/(1 +P)]1/3 − π2. IfRc(con) andRc(o.s) are the critical values of the convective and the overstable marginal states respectively, then it is also found thatRc(con) &lt;Rc(o.s) providedN is not sufficiently large.

• Effect of aspect ratio on the meridional circulation of a rotating homogeneous fluid

The effect of aspect ratio on the meridional circulation of a homogeneous fluid is analyzed. Aspect ratio is allowed to range between zero and unity. Relationships between possible horizontal and vertical length scales are obtained by length scale analysis as well as by solving an idealized problem. It is found that whenE1/2 ≪ Z ≪ E1/2/δ, whereE is the Ekman number, the stream lines are closely packed near the sidewall within a thickness ofO(E1/2). The effect of stratification is unimportant within this depth range. In the depth rangeE1/2/δ ≪ Z ≪ 1/ the vertical boundary layer in which the streamlines are packed is ofO(EZδ)1/3. WhenZ ≫ 1/Eδ it is shown that the circulation decays algebraically with depth as 1/Z.

• Analytic and harmonic maps into a topological space

The relationship between the harmonicity and analyticity of a continuous map from the open unit disc to the underlying space of a real algebra is investigated.

• On subnormal operators

LetS be a pure subnormal operator such thatC*(S), theC*-algebra generated byS, is generated by a unilateral shiftU of multiplicity 1. We obtain conditions under which 5 is unitarily equivalent toα + βU, α andβ being scalars orS hasC*-spectral inclusion property. It is also proved that if in addition,S hasC*-spectral inclusion property, then so does its dualT andC*(T) is generated by a unilateral shift of multiplicity 1. Finally, a characterization of quasinormal operators among pure subnormal operators is obtained.

• Computation ofg(1,5; 10)

It is established that the exact covering numberg(1,5; 10) is 102. It is further shown that this configuration is unique. It can be obtained from the unique Steiner systemS(5, 6, 12).

• Some generating functions of Jacobi polynomials

In this paper, Weisner’s group-theoretic method of obtaining generating functions is utilized in the study of Jacobi polynomialsP&gt;n(a,ß)(x) by giving suitable interpretations to the index (n) and the parameter (β) to find out the elements for constructing a six-dimensional Lie algebra.

• Cohomology of the moduli of parabolic vector bundles

The purpose of this paper is to compute the Betti numbers of the moduli space ofparabolic vector bundles on a curve (see Seshadri [7], [8] and Mehta & Seshadri [4]), in the case where every semi-stable parabolic bundle is necessarily stable. We do this by generalizing the method of Atiyah and Bott [1] in the case of moduli of ordinary vector bundles. Recall that (see Seshadri [7]) the underlying topological space of the moduli of parabolic vector bundles is the space of equivalence classes of certain unitary representations of a discrete subgroup Γ which is a lattice in PSL (2,R). (The lattice Γ need not necessarily be co-compact).

While the structure of the proof is essentially the same as that of Atiyah and Bott, there are some difficulties of a technical nature in the parabolic case. For instance the Harder-Narasimhan stratification has to be further refined in order to get the connected strata. These connected strata turn out to have different codimensions even when they are part of the same Harder-Narasimhan strata.

If in addition to ‘stable = semistable’ the rank and degree are coprime, then the moduli space turns out to be torsion-free in its cohomology.

The arrangement of the paper is as follows. In § 1 we prove the necessary basic results about algebraic families of parabolic bundles. These are generalizations of the corresponding results proved by Shatz [9]. Following this, in § 2 we generalize the analytical part of the argument of Atiyah and Bott (§ 14 of [1]). Finally in § 3 we show how to obtain an inductive formula for the Betti numbers of the moduli space. We illustrate our method by computing explicitly the Betti numbers in the special case of rank = 2, and one parabolic point.

• # Proceedings – Mathematical Sciences

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