• Volume 100, Issue 2

August 1990,   pages  95-188

• Singular pencils of quadrics and compactified Jacobians of curves

LetY be an irreducible nodal hyperelliptic curve of arithmetic genusg such that its nodes are also ramification points (char ≠2). To the curveY, we associate a family of quadratic forms which is dual to a singular pencil of quadrics in$$\mathbb{P}^{2g + 1}$$ with Segre symbol [2...21...1], where the number of 2's is equal to the number of nodes. We show that the compactified Jacobian ofY is isomorphic to the spaceR of (g−1) dimensional linear subspaces of$$\mathbb{P}^{2g + 1}$$ which are contained in the intersectionQ of quadrics of the pencil. We also prove that (under this isomorphism) the generalized Jacobian ofY is isomorphic to the open subset ofR consisting of the (g−1) dimensional subspaces not passing through any singular point ofQ.

• Decomposition of the de Rham complex

We give a reformulation of Deligne and Illusie's characteristicp proof of the degeneration of the “Hodge-to-de Rham” spectral sequence, which replaces patching in the derived category by an explicit quasi-isomorphism.

• On the ratio of two blocks of consecutive integers

Under certain assumptions, it is shown that eq. (2) has only finitely many solutions in integersx≥0,y≥0,k≥2,l≥0. In particular, it is proved that (2) witha=b=1, l=k implies thatx=7,y=0,k=3.

• Extension of certain types of generating relations

This paper gives a new expansion formula which essentially involves a double sum. As a consequence of our main result, eq. (6), various types of generating relations are seen to emerge and relevance with some known results is pointed out briefly. The usefulness of our main result is also indicated by considering its application to a probabilistic method for a lattice path enumeration problem.

• Some uncertainty inequalities

We prove an uncertainty inequality for the Fourier transform on the Heisenberg group analogous to the classical uncertainty inequality for the Euclidean Fourier transform. Inequalities of similar form are obtained for the Hermite and Laguerre expansions.

• Riesz means for the sublaplacian on the Heisenberg group

The uniform boundedness of the Riesz means for the sublaplacian on the Heisenberg groupHn is considered. It is proved thatSRα are uniformly bounded onLp(Hn) for 1≤p≤2 provided α&gt;α(p)=(2n+1)[(1/p)−(1/2)].

• Pairs of II1 factors

A class of objects—that are best described as being actions ofgroup-like objects of von Neumann algebras—is axiomatised and it is shown that there exists a bijective correspondence between isomorphism classes of suchcovariant systems and isomorphism classes of pairs of II1 factors (M, N) satisfyingN⊂M, [M:N]&lt;∞ andMN′=C.

• Existence of solution for nonlinear Volterra integral equations

We prove an existence theorem for a class of nonlinear Volterra integral equations.

• Effect of a viscous fluid flow past a spherical gas bubble on the growth of its radius

A nearly spherical gas bubble expands adiabatically in a viscous incompressible fluid flowing past it. The Rayleigh-Plesset formula for the growth of the bubble radius is modified due to the flow of the viscous fluid.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 4
September 2019