Volume 114, Issue 1
February 2004, pages 1-97
pp 1-6 February 2004
We will study the solution of a congruence,x ≡g1/2)ωg(2n) mod 2n, depending on the integersg andn, where ωg(2n) denotes the order ofg modulo 2n. Moreover, we introduce an application of the above result to the study of an estimation of exponential sums.
pp 7-14 February 2004
LetX be a projective scheme over a noetherian base schemeS, and letF be a coherent sheaf onX. For any coherent sheaf ε onX, consider the set-valued contravariant functor Hom(ε,F)S-schemes, defined by Hom(ε,F) (T)= Hom(εT,FT) where εT andFT are the pull-backs of ε andF toXT =X xS T. A basic result of Grothendieck ([EGA], III 7.7.8, 7.7.9) says that ifF is flat over S then Komε,F) is representable for all ε.
We prove the converse of the above, in fact, we show that ifL is a relatively ample line bundle onX over S such that the functor Hom(L-n,F) is representable for infinitely many positive integersn, thenF is flat overS. As a corollary, takingX =S, it follows that ifF is a coherent sheaf on S then the functorT ↦H°(T, Ft) on the category ofS-schemes is representable if and only ifF is locally free onS. This answers a question posed by Angelo Vistoli.
The techniques we use involve the proof of flattening stratification, together with the methods used in proving the author’s earlier result (see [N1]) that the automorphism group functor of a coherent sheaf onS is representable if and only if the sheaf is locally free.
pp 15-31 February 2004
LetX (Δ) be the real toric variety associated to a smooth fan Δ. The main purpose of this article is: (i) to determine the fundamental group and the universal cover ofX (Δ), (ii) to give necessary and sufficient conditions on Δ under which π1(X(Δ)) is abelian, (iii) to give necessary and sufficient conditions on Δ under whichX(Δ) is aspherical, and when Δ is complete, (iv) to give necessary and sufficient conditions forCΔ to be aK (π, 1) space whereCΔ is the complement of a real subspace arrangement associated to Δ.
pp 33-38 February 2004
This note proves that, forF = ℝ, ℂ or ℍ, the bordism classes of all non-bounding Grassmannian manifoldsGk(Fn+k), withk <n and having real dimensiond, constitute a linearly independent set in the unoriented bordism group Nd regarded as a ℤ2-vector space.
pp 39-54 February 2004
SupposeH is a hyperbolic subgroup of a hyperbolic groupG. Assume there existsn > 0 such that the intersection ofn essentially distinct conjugates ofH is always finite. Further assumeG splits overH with hyperbolic vertex and edge groups and the two inclusions ofH are quasi-isometric embeddings. ThenH is quasiconvex inG. This answers a question of Swarup and provides a partial converse to the main theorem of .
pp 55-63 February 2004
In this paper, we first characterize reflexive one-sided A-submodulesU of a unital operator algebraA inB(H) completely. Furthermore we investigate the invariant subspace lattice LatR and the reflexive hull RefR, whereR is the submodule generated by rank-one operators inU; in particular, ifL is a subspace lattice, we obtain when the rank-one algebraR of AlgL is big enough to determined AlgL in the following senses: AlgL = Alg LatR and AlgL = RefR.
pp 65-78 February 2004
In this paper we consider the formally symmetric differential expressionM [.] of any order (odd or even) ≥ 2. We characterise the dimension of the quotient spaceD(Tmax)/D(Tmin) associated withM[.] in terms of the behaviour of the determinants det [[frgs](∞)] where 1 ≤n ≤ (order of the expression +1); here [fg](∞) = lim [fg](x), where [fg](x) is the sesquilinear form in f andg associated withM. These results generalise the well-known theorem thatM is in the limit-point case at ∞ if and only if [fg](∞) = 0 for everyf, g ε the maximal domain Δ associated withM.
pp 79-96 February 2004
Assuming certain forms of the stream function inverse solutions of an incompressible viscoelastic fluid for a porous medium channel in the presence of Hall currents are obtained. Expressions for streamlines, velocity components and pressure fields are described in each case and are compared with the known viscous and second-grade cases.
pp 97-97 February 2004 Erratum
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