• Volume 103, Issue 3

December 1993,   pages  209-347

• Gysin maps and cycle classes for Hodge cohomology

Iff:X→Y is a projective morphism between regular varieties over a field, we construct Gysin maps$$f_ * :H^i \left( {X,\Omega _{X/Z}^j } \right) \to H_{f(x)}^{i + d} \left( {X,\Omega _{Y/Z}^j } \right)$$ for the Hodge cohomology groups, whered-dimY-dimX. These Gysin maps have the expected properties, and in particular may be used to construct a cycle class map$$Cl_X :CH^i \left( {X,S} \right) \to H^i \left( {X,\Omega _{X/Z}^i } \right)$$ whereX is quasi-projective over a field,S is the singular locus, andCHi(X, S) is the relative Chow group of codimension-i cycles modulo rational equivalence. Simple properties of this cycle map easily imply the infinite dimensionality theorem for the Chow group of zero cycles of a normal projective varietyX overC with$$H^n \left( {X,\mathcal{O}_X } \right) \ne 0$$, wheren=dimX. One also recovers examples of Nori of affinen-dimensional varieties which support indecomposable vector bundles of rankn.

• Some remarks on subgroups determined by certain ideals in integral group rings

LetG be a group,ZG the integral group ring ofG andI(G) its augmentation ideal. Subgroups determined by certain ideals ofZG contained inI(G) are identified. For example, whenG=HK, whereH, K are normal subgroups ofG andHK⊆ζ(H), then the subgroups ofG determined byI(G)I(H)I(G), andI3(G)I(H) are obtained. The subgroups of any groupG with normal subgroupH determined by (i)I2(G)I(H)+I(G)I(H)I(G)+I(H)I2(G), whenH′⊆[H,G,G] and (ii)I(G)I(H)I(G) when degH2(G/H′, T)≤1, are computed. the subgroup ofG determined byIn(G)+I(G)I(H) whenH is a normal subgroup ofG withG/H free Abelian is also obtained

• On certain Ramanujan's mock theta functions

Ramanujan's last gift to the mathematicians was his ingeneous discovery of the mock theta functions of order three, five and seven. Recently, Andrews and Hickerson found a set of seven more functions in Ramanujan's Lost Note Book and formally labelled them as mock theta functions of order six. In this paper the complete forms of these functions have been studied and connected with the bilateral basic hypergeometric series2Ψ2. Several other interesting properties and transformations have also been studied.

• Lambert series and Ramanujan

Lambert series are of frequent occurrence in Ramanujan's work on elliptic functions, theta functions and mock theta functions. In the present article an attempt has been made to give a critical and up-to-date account of the significant role played by Lambert series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas.

• Behaviour of an associated conjugate series of a Fourier series

In this paper we obtain a generalisation of a theorem of Mohanty and Ray

• On regularity of twisted spherical means and special Hermite expansions

Regularity properties of twisted spherical means are studied in terms of certain Sobolev spaces defined using Laguerre functions. As an application we prove a localisation theorem for special Hermite expansions.

• Weakly wandering sets and compressibility in descriptive setting

A Borel automorphismT on a standard Borel space$$\left( {X,\mathbb{B}} \right)$$ is constructed such that (a) there is no probability measure invariant underT and (b) there is no Borel setW weakly wandering underT and which generates the invariant setX.

• On infinitesimal σ-fields generated by random processes

It is proved that the infinitesimal look-ahead and look-back σ-fields of a random process disagree at atmost countably many time instants.

• Absolutely continuous spectrum for sparse potentials

In this paper we show that bounded perturbation of the discrete laplacian with a potential that is sparsely supported (a notion made precise in the paper) produces absolutely continuous spectrum in the interval [−2ν, 2ν] for large dimension ν. We note that the potential need not give rise to a compact operator, let alone have decay at infinity.

• Oscillation of odd order delay differential equations

In this paper, we obtain the sufficient and necessary conditions for all solutions of the odd-order nonlinear delay differential equation.x(n)+Q(t)f(x(g(t)))=0 to be oscillatory. In particular, ifn=1, Q(t)&gt;0, f(x)=xα, where α∈(0,1) and is a ratio of odd integers andg(t)=t−ϑ for some ϑ&gt;0, then every solution of (*) oscillates if and only if ∫Q(s)ds=∞.

• # Proceedings – Mathematical Sciences

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