Volume 100, Issue 2
August 1990, pages 95-188
pp 95-102 August 1990
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.
pp 103-106 August 1990
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.
pp 107-132 August 1990
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.
pp 133-136 August 1990
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.
pp 137-145 August 1990
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.
pp 147-156 August 1990
Riesz means for the sublaplacian on the Heisenberg group
The uniform boundedness of the Riesz means for the sublaplacian on the Heisenberg groupH^{n} is considered. It is proved thatS_{R}^{α} are uniformly bounded onL^{p}(H^{n}) for 1≤p≤2 provided α>α(p)=(2n+1)[(1/p)−(1/2)].
pp 157-177 August 1990
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 II_{1} factors (M, N) satisfyingN⊂M, [M:N]<∞ andM∮N′=C.
pp 179-184 August 1990
Existence of solution for nonlinear Volterra integral equations
We prove an existence theorem for a class of nonlinear Volterra integral equations.
pp 185-188 August 1990
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.
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