We study the expected behavior of Betti numbers arrangements zeros random (distributed according to Kostlan distribution) polynomials in R P n $\mathbb {R}\mathrm{P}^n$ . Using a spectral sequence, we prove an asymptotically exact estimate on number connected components complement s $s$ such hypersurfaces also investigate same problem case where are defined by quadratic polynomials. In this cas...

Let φ be a w-continuous homomorphism from a dual Banach algebra to C. The notion of φ-Connes amenability is studied and some characterizations is given. A type of diagonal for dual Banach algebras is dened. It is proved that the existence of such a diagonal is equivalent to φ-Connes amenability. It is also shown that φ-Connes amenability is equivalent to so-called φ-splitting of a certain short...

1. Exact sequences A sequence of R-module maps A φ → B π → C is exact if the kernel of π and the image of φ are the exact same submodule of B. A longer sequence is exact by definition if every subsequence of two consecutive maps is exact. a). Show that B π → C → 0 is exact if and only if π is surjective. The kernel of C → 0 is all of C, so it is exact if the image of π is all of C—that is, if π...

To a short exact sequence of triangulated categories with duality, we associate a long exact sequence of Witt groups. For this, we introduce higher Witt groups in a very algebraic and explicit way. Since those Witt groups are 4-periodic, this long exact sequence reduces to a cyclic 12-term one. Of course, in addition to higher Witt groups, we need to construct connecting homomorphisms, hereafte...

This paper presents two new results on the enumeration of all extreme equilibria of the sequence form of a two person extensive game. The sequence form of an extensive game is expressed, for the first time to our knowledge, as a parametric linear 0− 1 program. Considering Ext(P ) as the set of all of the sequence form extreme Nash equilibria and Ext(Q) as the set of all the parametric linear 0 ...