Volume 99, Issue 3
December 1989, pages 191-286
pp 191-196 December 1989
Cycles on the generic abelian threefold
H Clemens and C Schoen gave examples of three-folds where the group of codimension two cycles modulo algebraic equivalence has infinite rank. This paper provides yet another example of the same phenomenon.
pp 197-208 December 1989
Mumford's example and a general construction
We construct a line bundle on a complex projective manifold (a general ruled variety over a curve) which is not ample, but whose restriction to every proper subvariety is ample. This example is of interest in connection with ampleness questions of vector bundles on varieties of dimension greater than one. The method of construction shows that a stable bundle of positive degree on a curve is ample. The example can be used to show that there is no restriction theorem for Bogomolov stability.
pp 209-215 December 1989
We prove the following resultTheorem.Let R be an affine normal 2-dimensional subring of ℂ[X, Y] generated by homogeneous polynomials, say, F_{1},…,F_{n}. If the g.c.d. (F_{1},…,F_{n}) has at most two distinct linear factors (possibly occurring with multiplicities), then R is isomorphic to a ring of invariants ℂ[v, Y]^{W}for some finite group W.
This generalizes a result of D Anderson on rings generated by monomials. As a corollary of the theorem, we prove some special cases of a conjecture of CTC wall. We also prove some results on general graded affine subrings of ℂ[X, Y].
pp 217-220 December 1989
A note on the coefficient rings of polynomial rings
LetA andB be two reduced commutative rings with finitely many minimal prime ideals. If the polynomial algebrasA[X_{1}…X_{n}]=B[Y_{1}…Y_{n}] whereX_{i},Y_{iF} are variables overA andB respectively, then there exists an injective ring homomorphism ϕ:A→B such thatB is finitely generated over ϕ(A).
pp 221-229 December 1989
Ramifications of Ramanujan's work on η-products
We follow, the evolution of ideas arising from Ramanujan's 1916 paper ‘On certain arithmetical functions’ by examining multiplicative η-products and quotients and their relation with the characters of the Mathieu groupM_{24} and the automorphism group of the Leech lattice. This leads to the Monster and speculations on its geometric origin and current physics.
pp 231-233 December 1989
The conjecture is made that the rational structures on spaces of modular forms coming from the rationality of Fourier coefficients and the rationality of periods are not compatible. A consequence would be that ζ(2k-1)/π^{2k-1} (ζ(s)=Riemann zeta function;k∈ℕ,k≥2) is irrational or even transcendental.
pp 235-247 December 1989
M Manickam B Ramakrishnan T C Vasudevan
For an even natural numberm we will construct a Shintani lifting fromS_{2k}(m,ψ^{2}) toS_{k+1/2}(Γ_{0}(2m),ψ_{0}), ψ a Dirichlet character modulo2m,$$\psi _0 = \left( {\frac{{\psi ( - 1)}}{ \cdot }} \right)\psi $$, which is adjoint to the Shimura lifting. Whenm is squarefree we will combine this result with the multiplicity 1 theorem proved in [3] to give a formula for the product$$c(m_1 )\overline {c(m_2 )} $$ (m_{2} squarefree) of two Fourier coefficients of a new form inS_{k+1/2}^{new}(Γ_{0}(2m)) in terms of certain cycle integrals of the corresponding new form inS_{2k}^{new}(_{m}). Ifm_{1}=m_{2} we will get Waldspurger's result expressing the squarec(m_{1})^{2} in terms of anL-series off at the centre of the critical strip. Similar product formulas for the Fourier coefficients of a Hecke-Pizer eigenform inS_{k+1/2}^{old}(Γ_{0}(2m)) andS_{k+1/2}^{+,old}(Γ_{0}(2m)) and of a Hecke eigenform inS_{k+1/2}^{+,old}(Γ_{0}(4q)) (q⦻3(4) is a prime) are also given.
pp 249-253 December 1989
On the integral modulus of continuity of Fourier series
We obtain an estimate for the integral modulus of continuity of orderk of Fourier series with coefficients satisfying:a_{v}→0 and Σ_{v=1}^{∞}v^{2}|Δ^{2}(a_{v}/v)|<∞.
pp 255-261 December 1989
On orbit equivalence of borel automorphisms
LetE andF be two Borel sets of the countable productZ of the two point space {0,1}. Assume thatE andF are invariant sets for the odometer transformationR and thatE andF are of measure zero with respect to the unique finiteR-invariant measure onZ. We show thatE andF areR-orbit equivalent in a strict sense.
pp 263-272 December 1989
Geetarashmi Basu Bikash Ranjan Das
The temperature and the normal components of stress and displacement around an external circular crack in an infinite transversely isotropic body have been calculated in the present paper. The stress intensity factor has been found and a comparison of the results with those for the isotropic case has been presented graphically.
pp 273-273 December 1989 Note
Authors' note on the paper “Rheology of polarizable non-piezoelectromagnetic material in relativity”
pp 283-286 December 1989
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