• Volume 107, Issue 2

May 1997,   pages  101-222

• Degenerations of the moduli spaces of vector bundles on curves I

LetY be a smooth projective curve degenerating to a reducible curveX with two components meeting transversally at one point. We show that the moduli space of vector bundles of rank two and odd determinant on Ydegenerates to a moduli space onX which has nice properties, in particular, it has normal crossings. We also show that a nice degeneration exists when we fix the determinant. We give some conjectures concerning the degeneration of moduli space of vector bundles onY with fixed determinant and arbitrary rank.

• Generic hypersurface singularities

The problem considered here can be viewed as the analogue in higher dimensions of the one variable polynomial interpolation of Lagrange and Newton. Let x1,...,xr be closed points in general position in projective spacePn, then the linear subspaceV ofH0 (⨑n,O(d)) (the space of homogeneous polynomials of degreed on ⨑n) formed by those polynomials which are singular at eachxi, is given by r(n + 1) linear equations in the coefficients, expressing the fact that the polynomial vanishes with its first derivatives at x1,...,xr. As such, the “expected” value for the dimension ofV is max(0,h0(O(d))−r(n+1)). We prove thatV has the “expected” dimension for d≥5 (theorem A). This theorem was first proven in [A] using a very complicated induction with many initial cases. Here we give a greatly simplified proof using techniques developed by the authors while treating the corresponding problem in lower degrees.

• Bredon cohomology of cyclic geometric realization ofG-cyclic sets

We define equivariant cyclic and Hochschild cohomology modules of a cyclic objectX in the category ofG-sets and relate them with the Bredon cohomologies of the cyclic geometric realization ¦X¦cy.

• A note on equivariant Euler characteristic

We give a new equivariant cohomological characterization of the equivariant Euler characteristic of aG-simplicial set as defined by Brown. This implies in particular that the equivariant Euler characteristic is aG-homotopy invariant.

• Degree of approximation of functions in the Hölder metric by Borel’s means

After establishing the Fourier character of the series the authors have studied the degree of approximation of functions associated with the same series in the Hölder metric using Borel’s mean.

• On a general theorem concerning some absolute summability methods

In this note a general theorem covering several absolute summability methods e.g, ¦N,pn¦k¦R,pn¦k, is proved.

• Inequalities for the derivative of a polynomial

Let P(z) be a polynomial of degreen which does not vanish in ¦z¦ &lt;k, wherek &gt; 0. Fork ≤ 1, it is known that$$\mathop {\max }\limits_{|z| = 1} |P'(z)| \leqslant \frac{n}{{1 + k^n }}\mathop {\max }\limits_{|z| = 1} |P(z)|$$, provided ¦P’(z)¦ and ¦Q’(z)¦ become maximum at the same point on ¦z¦ = 1, where$$Q(z) = z^n \overline {P(1/\bar z)}$$. In this paper we obtain certain refinements of this result. We also present a refinement of a generalization of the theorem of Tuŕan.

• On self-reciprocal polynomials

In this paper we establish a sharp result concerning integral mean estimates for self-reciprocal polynomials.

• Random commutation

We investigate the commutation between a continuous linear random operator and a continuous linear deterministic operator on a Banach space. From this we obtain probabilistic versions of theorems by Fuglede and Putnam, both of them dealing with the commutation between continuous linear operators with continuous normal operators on a Hilbert space.

• Surface instability in a finite thickness fluid saturated porous layer

The Rayleigh-Taylor (RT) instability at the interface between fluid and fluid saturated sparsely packed porous medium has been investigated making use of boundary layer approximation and Saffmann [8] boundary condition. An analytical solution for dispersion relation is obtained and is numerically evaluated for different values of the parameters. It is shown that RT instability can be controlled by a suitable choice of the thickness of porous layer, ratio of viscosities and the slip parameter.

• Erratum to Quasi-parabolic Siegel formula

The main result of the above paper is mistaken, because of a defective lemma. Here we replace the defective lemma, and derive the corrected quasi-parabolic analogue of the Siegel formula.

• # Proceedings – Mathematical Sciences

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• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019