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Let q be a positive integer. Recently, Niu and Liu proved that, if , then the product is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and , the product is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer such that, for any positive integer , the product is not a powerful number. 相似文献
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Pierre Deligne 《Comptes Rendus Mathematique》2018,356(7):717-719
Let G be a reductive group over a field k of characteristic . For and , let be deduced from G by the extension of scalars . If k is perfect, this keeps making sense for . We show that, if k is perfect, there exists such that the algebraic groups G and over k are isomorphic. The isomorphism class of , as a reductive group over k, then depends only on n modulo m. For k not necessarily perfect, we show that such a periodicity remains true for n large enough. 相似文献
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《Journal of Mathematical Analysis and Applications》2014,419(2):783-795
We study restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension d is even, then it is conjectured that the Stein–Tomas restriction result can be improved to the estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in dimensions. 相似文献
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Quốc Anh Ngô 《Comptes Rendus Mathematique》2017,355(5):526-532
In this note, we mainly study the relation between the sign of and in with and for . Given the differential inequality , first we provide several sufficient conditions so that holds. Then we provide conditions such that for all , which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to and with in . 相似文献
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Anton Alekseev Nariya Kawazumi Yusuke Kuno Florian Naef 《Comptes Rendus Mathematique》2017,355(2):123-127
We define a family of Kashiwara–Vergne problems associated with compact connected oriented 2-manifolds of genus g with boundary components. The problem is the classical Kashiwara–Vergne problem from Lie theory. We show the existence of solutions to for arbitrary g and n. The key point is the solution to based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman–Turaev Lie bialgebra . In more detail, we show that every solution to induces a Lie bialgebra isomorphism between and its associated graded . For , a similar result was obtained by G. Massuyeau using the Kontsevich integral. For , , our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction. 相似文献
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We extend the notion of a partial cohomology group to the case of non-unital A and find interpretations of and in the theory of extensions of semilattices of abelian groups by groups. 相似文献