We calculate Betti numbers of the framed moduli space of instantons on Ĉ/Z2, under the assumption that the corresponding torsion free sheaves E have vanishing properties(Hom(E, E(−l∞)) = Ext (E, E(−l∞)) = 0). Moreover we derive the generating function of Betti numbers and obtain closed formulas. On the other hand, we derive a universal relation between the generating function of Betti numbers o...

We propose a refinement of the Betti numbers and of the homology with coefficients in a field of a compact ANR X, in the presence of a continuous real valued function on X.The refinement of Betti numbers consists of finite configurations of points with multiplicities in the complex plane whose total cardinality are the Betti numbers and the refinement of homology consists of configurations of v...

The total Betti number of the independence complex of a graph is an intriguing graph invariant. Kalai and Meshulam have raised the question on its relation to cycles and the chromatic number of a graph, and a recent conjecture on that theme was proved by Bonamy, Charbit and Thomassé. We show an upper bound on the total Betti number in terms of the number of vertex disjoint cycles in a graph. Th...

We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well of non-positive sectional curvature, this case we prove strengthened inequality. employ these to study asymptotic behavior Betti numbers coverings Riemannian points. Finally, give vanishing result $L^{2}$-Betti c...

Topological complexity of semialgebraic sets in R has been studied by many researchers over the past fifty years. An important measure of the topological complexity are the Betti numbers. Quantitative bounds on the Betti numbers of a semialgebraic set in terms of various parameters (such as the number and the degrees of the polynomials defining it, the dimension of the set etc.) have proved use...

Sums of Betti numbers in arbitrary characteristic Nicholas M. Katz Introduction In [Mil], Milnor gave an explicit upper bound for the sum of the Betti numbers of a complex affine algebraic variety V. If V is defined in ^N, N ≥ 1, by r ≥ 1 equations Fi, i =1 to r, all of degree ≤ d, Milnor showed ‡i h i(V, $) ≤ d(2d-1)2N-1. Oleinik [Ol] and Thom [Th] gave similar results. It is standard (cf. the...

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RPn (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those of its double cover in the sphere Sn; we also give similar formulae for level sets of homogeneous quadratic maps to the plane. We discuss some applications o...

In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of t-minors of a generic pluri-circulant matrix is a transversal monomial ideal . Using a Gröbner basis for this ideal, it is sho...

We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation t...

The direct three component modified Mannich reaction via condensation of aldehydes, 2-naphthol or 2,7-naphthalendiol and piperidine to generate Betti bases has been carried out over L-proline (20 mol%) with high efficiency under solvent free conditions at 70 OC. Also, the reaction of aromatic aldehyde and 4-hydroxycoumarin in the presence of L-proline (20 mol%) in the ethanol under reflux condi...