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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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The finite field analog of the classical Kakeya problem asks the smallest possible size for a set of points in the Desarguesian affine plane which contains a line in every direction. This problem has been definitively solved by Blokhuis and Mazzocca (2008), who show that in , a prime power, the smallest possible size of such a set is . In this paper we examine a new construction of an infinite family of sets in containing a line in every direction, where with a prime power and an integer with . These sets have size , which is small in the sense that the lower bound is also plus smaller order terms. In addition, we discuss the minimality of our sets, showing that they contain no proper subset containing a line in every direction. 相似文献
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Very recently, Thomassé et al. (2017) have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time . In this article we improve this running time to . As a byproduct, we also improve the previous Turing-kernel for this problem from to . Furthermore, for the subclass of bull-free graphs without holes of length at most for , we speed up the running time to . As grows, this running time is asymptotically tight in terms of , since we prove that for each integer , Weighted Independent Set cannot be solved in time in the class of -free graphs unless the ETH fails. 相似文献
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We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to where denotes the standard weak Lebesgue space. 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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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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Recently, Mubayi and Wang showed that for and , the number of -vertex -graphs that do not contain any loose cycle of length is at most . We improve this bound to . 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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In this paper integration method is used to derive the enumeration formulas of the standard Young tableau (SYT) of truncated shape. An integration representation of the number of SYT of truncated shape is given, which implies a positive proof of the conjecture of Adin et al. on the number of SYT of truncated shape . Furthermore, the enumeration formulas of the numbers of SYTs of truncated shapes and have been obtained. 相似文献