• Volume 99, Issue 3

December 1989,   pages  191-286

• 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.

• 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.

• Graded subrings of ℂ[X, Y]

We prove the following resultTheorem.Let R be an affine normal 2-dimensional subring of ℂ[X, Y] generated by homogeneous polynomials, say, F1,…,Fn. If the g.c.d. (F1,…,Fn) has at most two distinct linear factors (possibly occurring with multiplicities), then R is isomorphic to a ring of invariants ℂ[v, Y]Wfor 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].

• 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[X1…Xn]=B[Y1…Yn] whereXi,YiF are variables overA andB respectively, then there exists an injective ring homomorphism ϕ:AB such thatB is finitely generated over ϕ(A).

• 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 groupM24 and the automorphism group of the Leech lattice. This leads to the Monster and speculations on its geometric origin and current physics.

• Transcendence conjectures about periods of modular forms and rational structures on spaces of modular forms

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)/π2k-1 (ζ(s)=Riemann zeta function;k∈ℕ,k≥2) is irrational or even transcendental.

• On Shintani correspondence

For an even natural numberm we will construct a Shintani lifting fromS2k(m,ψ2) toSk+1/20(2m),ψ0), 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$$c(m_1 )\overline {c(m_2 )}$$ (m2 squarefree) of two Fourier coefficients of a new form inSk+1/2new0(2m)) in terms of certain cycle integrals of the corresponding new form inS2knew(m). Ifm1=m2 we will get Waldspurger's result expressing the squarec(m1)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 inSk+1/2old0(2m)) andSk+1/2+,old0(2m)) and of a Hecke eigenform inSk+1/2+,old0(4q)) (q⦻3(4) is a prime) are also given.

• 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:av→0 and Σv=1v22(av/v)|&lt;∞.

• 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.

• Steady-state thermoelastic stresses in an infinite transversely isotropic medium containing an external circular crack

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.

• Authors' note on the paper “Rheology of polarizable non-piezoelectromagnetic material in relativity”

• Proceedings of the Indian Academy of Sciences Mathematical Sciences - Notes on the preparation of papers

• # Proceedings – Mathematical Sciences

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Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019