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Let R ⊂ T be an extension of integral domains and ∗ be a semistar operation stable of finite type on R. We define a semistar operation ∗1 on T in the following way: for each nonzero T-submodule E of the quotient field K1 of T, let E∗1 = ∪ {E :K1 JT | J ∈ F∗}, where K1 denotes the quotient field of T and F∗ the localizing system associated to ∗. In this paper we investigate the basic properties of ∗1. Moreover, we show that the map φ which associates to a semistar operation ∗ stable and of finite type on R, the semistar operation ∗1 is continuous. Furthermore, we give sufficient conditions for φ to be a homeomorphism.
This paper is devoted to the study of the effects of indefinite weights on the following nonlinear Neumann problems
(P±){−Δu=λa(x)u±|u|p−1u in Ω⊂ℝN∂u∂ν= 0 on ∂Ω
The function a=a(x) is assumed to be continuous and sign-changing. Then the linear part has two sequences of eigenvalues. Our results establish a relation between the position of the parameter λ and the number of nontrivial classical solutions of these problems. The proof combines spectral analysis tools, variational methods and the Clark multiplicity theorem.
In [7], A.Bahri introduced two topological invariants μ and τ to study the prescribed scalar curvature problem on standard spheres of high dimensions. In this paper we first extend μ and τ to the problem on general riemannian manifolds. Second we analyze, as suggested in [7], the relation between these two quantities and we prove under topological conditions that μ = τ.
Let d ≥ 1 be an integer and Kd be a contravariant functor from the category of subgroups of (ℤ/2ℤ)d to the category of graded and finite 𝔽2-algebras. In this paper, we generalize the conjecture of G. Carlsson [C3], concerning free actions of (ℤ/2ℤ)d on finite CW-complexes, by suggesting, that if Kd is a Gysin-(ℤ/2ℤ)d-functor (that is to say, the functor Kd satisfies some properties, see 2.2), then we have:
(Cd):∑i≥0dimF2(Kd(0))i≥2d
We prove this conjecture for 1 ≤ d ≤ 3 and we show that, in certain cases, we get an independent proof of the following
results (for d = 3 see [C4]):
If the group (ℤ/2ℤ)d, 1 ≤ d ≤ 3, acts freely and cellularly on a finite CW-complex X, then ∑i≥0dimF2Hi(X;F2)≥2d
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