Dirac猜想的一个反例

从约束Hamilton系统相空间中对称性分析,给出一个反例.首次用正则Noether恒等式说明Dirac猜想失效,在此反例中没有将约束线性化 关键词： 约束Hamilton系统 正则对称性 Dirac猜想     

关于丢番图方程x6±y6=3pz2

设p＞3是素数,该文运用数论方法证明了方程x6+y6=3pz2,x6±y6=6pz2均无正整数解,给出了方程x6-y6=3pz2有解的必要条件,并获得了相应的通解公式.     

The Baum–Connes Conjecture and Discrete Group Actions on Trees

We introduce a property (BC) for discrete groups, which we prove to imply the Baum–Connes conjecture with coefficients and the K-amenability of the group. Then, we show that if is a discrete group which acts on a tree X such that X/ is compact, and the stabilizers of the vertices and the stabilizers of the edges satisfy (BC), then itself satisfies (BC) Finally, we indicate a couple of applications.     

K-Theory for C*-Algebras of One-Relator Groups

We compute the K-theory groups of the reduced C*-algebra C _r ^* () of a one-relator group . We prove that every such group is K-amenable in the sense of Cuntz. For a torsion-free one-relator group =<X|r> such that r is not a product of commutators, we give a direct proof of the fact that the Baum–Connes analytical assembly map

Williamson Matrices and a Conjecture of Ito's

We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4t, 2, 4t, 2t)-difference sets in the dicyclic groups Q _8t = a, b|a ^4t = b ⁴ = 1, a ^2t = b ², b ^-1ab = a^-1 for all t of the form t = 2^a · 10 ^b · 26 ^c · m with a, b, c 0, m 1\ (mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over _m. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t, 2, 4t, 2t)-difference sets in Q _8t for every positive integer t. We also give simpler alternative constructions for relative (4t, 2, 4t, 2t)-difference sets in Q _8t for all t such that 2t - 1 or 4t - 1 is a prime power. Relative difference sets in Q _8t with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito's conjecture for all t 46.     

Tame and wild kernels of quadratic imaginary number fields

For all quadratic imaginary number fields of discriminant
we give the conjectural value of the order of Milnor's group (the tame kernel) where is the ring of integers of Assuming that the order is correct, we determine the structure of the group and of its subgroup (the wild kernel). It turns out that the odd part of the tame kernel is cyclic (with one exception, ).

     

Boundary Amenability of Groups and Positive Scalar Curvature

We show that if a finitely presented discrete group acts amenably on the boundary of a metric compactification of itself, then the Gromov–Lawson–Rosenberg conjecture holds for . Thus, if M is a compact aspherical Spin manifold with fundamental group , then M does not admit a Riemannian metric with positive scalar curvature. The class of groups having such a property includes hyperbolic groups and amenable groups and is closed under semidirect product.     

La conjecture de Novikov pour les feuilletages hyperboliques

Nous définissons la notion de bolicité pour les feuilletages, qui est une notion plus faible que l'hyperbolicité de Gromov, et nous démontrons la conjecture de Novikov pour les feuilletages boliques à base compacte dont le groupoï de d'holonomie est séparé en établissant l'injectivité de l'application de Baum–Connes. Ce résultat généralise celui de Kasparov et Skandalis obtenu dans le cas des groupes boliques.We define the notion of bolicity for foliations, which is a weaker notion than Gromov's hyperbolicity, and we prove the Novikov conjecture for foliations with compact base and whose holonomy groupoid is Hausdorff, by showing that the Baum–Connes map is injective. This result generalizes that of Kasparov and Skandalis in the case of bolic groups.     

La conjecture de Baum–Connes pour les feuilletages moyennables

We show, using the construction of Higson and Kasparov, that the Baum–Connes Conjecture holds for foliations whose holonomy groupoid is Hausdorff and amenable. More generally, for every locally compact, -compact and Hausdorff groupoid G acting continuously and isometrically on a continuous field of affine Euclidean spaces, the Baum–Connes conjecture with coefficients is an isomorphism, and G amenable in K-theory. In addition, we show that C^*(G) satisfies the Universal Coefficient Theorem.