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Bar-invariant bases of the quantum cluster algebra of type A 2 ( 2 )

Xueqing Chen, Ming Ding, Jie Sheng (2011)

Czechoslovak Mathematical Journal

We construct bar-invariant [ q ± 1 / 2 ] -bases of the quantum cluster algebra of the valued quiver A 2 ( 2 ) , one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.

Betti numbers of some circulant graphs

Mohsen Abdi Makvand, Amir Mousivand (2019)

Czechoslovak Mathematical Journal

Let o ( n ) be the greatest odd integer less than or equal to n . In this paper we provide explicit formulae to compute -graded Betti numbers of the circulant graphs C 2 n ( 1 , 2 , 3 , 5 , ... , o ( n ) ) . We do this by showing that this graph is the product (or join) of the cycle C n by itself, and computing Betti numbers of C n * C n . We also discuss whether such a graph (more generally, G * H ) is well-covered, Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum, or S 2 .

Bifurcations in symplectic space

G. Ishikawa, S. Janeczko (2008)

Banach Center Publications

In this paper we take new steps in the theory of symplectic and isotropic bifurcations, by solving the classification problem under a natural equivalence in several typical cases. Moreover we define the notion of coisotropic varieties and formulate also the coisotropic bifurcation problem. We consider several symplectic invariants of isotropic and coisotropic varieties, providing illustrative examples in the simplest non-trivial cases.

Biliaisons élémentaires en codimension 2

Mireille Martin-Deschamps (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

Un théorème de Strano montre que si une courbe gauche localement Cohen-Macaulay n’est pas minimale dans sa classe de biliaison, elle admet une biliaison élémentaire strictement décroissante. R. Hartshorne a récemment donné une nouvelle preuve de ce résultat en le plaçant dans un contexte plus général. Dans cet article on apporte une précision, en utilisant les techniques introduites par Hartshorne : on montre que si un sous-schéma de codimension 2 localement Cohen-Macaulay de N n’est pas minimal...

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