• Volume 110, Issue 3

August 2000,   pages  233-345

• Poincaré polynomial of the moduli spaces of parabolic bundles

In this paper we use Weil conjectures (Deligne’s theorem) to calculate the Betti numbers of the moduli spaces of semi-stable parabolic bundles on a curve. The quasi parabolic analogue of the Siegel formula, together with the method of HarderNarasimhan filtration gives us a recursive formula for the Poincaré polynomials of the moduli. We solve the recursive formula by the method of Zagier, to give the Poincaré polynomial in a closed form. We also give explicit tables of Betti numbers in small rank, and genera.

• The algebra ofG-relations

In this paper, we study a tower {AnG: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraAnG(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraAnG(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the ‘generic case’. Finally we determine the Bratteli diagram of the tower {AnG(d): n} ≥ 1 (in the generic case).

• On the generalized Hankel-Clifford transformation of arbitrary order

Two generalized Hankel-Clifford integral transformations verifying a mixed Parseval relation are investigated on certain spaces of generalized functions for any real value of their orders (α-β).

• C2-rational cubic spline involving tension parameters

In the present paper, C1-piecewise rational cubic spline function involving tension parameters is considered which produces a monotonie interpolant to a given monotonie data set. It is observed that under certain conditions the interpolant preserves the convexity property of the data set. The existence and uniqueness of a C2-rational cubic spline interpolant are established. The error analysis of the spline interpolant is also given.

• Coin tossing and Laplace inversion

An analysis of exchangeable sequences of coin tossings leads to inversion formulae for Laplace transforms of probability measures.

• Differential equations related to the Williams-Bjerknes tumour model

We investigate an initial value problem which is closely related to the Williams-Bjerknes tumour model for a cancer which spreads through an epithelial basal layer modeled onI ⊂ Z2. The solution of this problem is a familyp = (pi(t)), where eachpi(t)could be considered as an approximation to the probability that the cell situated ati is cancerous at timet. We prove that this problem has a unique solution, it is defined on [0, +∞[, and, for some relevant situations, limt→∞Pi(t) = 1 for alli ∈ I. Moreover, we study the expected number of cancerous cells at timet.

• Suppression of instability in rotatory hydromagnetic convection

Recently discovered hydrodynamic instability [1], in a simple Bénard configuration in the parameter regime$$T_0 \alpha _2$$ under the action of a nonadverse temperature gradient, is shown to be suppressed by the simultaneous action of a uniform rotation and a uniform magnetic field both acting parallel to gravity for oscillatory perturbations whenever (12 + J/π4) &gt; 1 and the effective Rayleigh numberR(1 -T0α2) is dominated by either 27π4(1 + l/σ1/4 or 27π4/2 according as σ1 ≥1 or σ1 1 respectively. HereT0is the temperature of the lower boundary while α2 is the coefficient of specific heat at constant volume due to temperature variation and σ1,R,Q andJ respectively denote the magnetic Prandtl number, the Rayleigh number, the Chandrasekhar number and the Taylor number.

• # Proceedings – Mathematical Sciences

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June 2019