Volume 110, Issue 3
August 2000, pages 233-345
pp 233-261
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 Harder-Narasimhan 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.
pp 263-292
Vijay Kodiyalam R Srinivasan V S Sunder
In this paper, we study a tower $\{A^G_n(d):n≥ 1\}$ of finite-dimensional algebras; here, 𝐺 represents an arbitrary finite group, 𝑑 denotes a complex parameter, and the algebra $A^G_n(d)$ has a basis indexed by `𝐺-stable equivalence relations' on a set where 𝐺 acts freely and has 2𝑛 orbits.
We show that the algebra $A^G_n(d)$ is semi-simple for all but a finite set of values of 𝑑, 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 $\{A^G_n(d): n≥ 1\}$ (in the generic case).
pp 293-304
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).
pp 305-314
𝐶^{2}-rational Cubic Spline Involving Tension Parameters
In the present paper, 𝐶^{1}-piecewise rational cubic spline function involving tension parameters is considered which produces a monotonic interpolant to a given monotonic 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 𝐶^{2}-rational cubic spline interpolant are established. The error analysis of the spline interpolant is also given.
pp 315-322
Coin Tossing and Laplace Inversion
An analysis of exchangeable sequences of coin tossings leads to inversion formulae for Laplace transforms of probability measures.
pp 323-334
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 on 𝐼 ⊂ 𝑍^{2}. The solution of this problem is a family 𝑝=(𝑝_{𝑖}(𝑡)), where each 𝑝_{𝑖}(𝑡) could be considered as an approximation to the probability that the cell situated at 𝑖 is cancerous at time 𝑡. We prove that this problem has a unique solution, it is defined on [0, + ∞], and, for some relevant situations, lim_{𝑡 → ∞} 𝑝_{𝑖}(𝑡)=1 for all 𝑖 ∈ 𝐼. Moreover, we study the expected number of cancerous cells at time 𝑡.
pp 335-345
Suppression of Instability in Rotatory Hydromagnetic Convection
Recently discovered hydrodynamic instability [1], in a simple Bénard configuration in the parameter regime 𝑇_{0}𝛼_{2} > 1 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 $(\mathscr{Q}𝜎_1/𝜋^2+\mathscr{J}/𝜋^4) > 1$ and the effective Rayleigh number $\mathcal{R}$(1-𝑇_{0}𝛼_{2}) is dominated by either 27𝜋^{4}(1 + 1/𝜎_{1})/4 or 27𝜋^{4}/2 according as 𝜎_{1} ≥ 1 or 𝜎_{1} ≤ 1 respectively. Here 𝑇_{0} is the temperature of the lower boundary while 𝛼_{2} is the coefficient of specific heat at constant volume due to temperature variation and 𝜎_{1}, $\mathcal{R}, \mathscr{Q}$ and $\mathscr{T}$ respectively denote the magnetic Prandtl number, the Rayleigh number, the Chandrasekhar number and the Taylor number.
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