Let X be a smooth projective variety over C of dimension d. Let ∆ ⊂ X × X be the diagonal. There is a cohomology class cl(∆) ∈ H(X ×X). In this paper we use Betti cohomology with rational coefficients. There is the Künneth decomposition H(X ×X) ≃ ⊕2d i=0 H (X)⊗H (X). We write cl(∆) = ∑2d i=0 π hom i according to this decomposition. Here π hom i ∈ H (X)⊗ H (X). If the Hodge conjecture is true, t...

Answering a question of Kalai and Meshulam, we prove that graphs without induced cycles of length 3k have bounded chromatic number. This implies the very first case of a much broader question asserting that every graph with large chromatic number induces a graph H such that the sum of the Betti numbers of the independence complex of H is also large.

The analysis of existing methods of construction of prime numbers is carried out in the presented paper. New concepts related to pseudoprime numbers are introduced. Theorems are formulated and proved on the basis of which such numbers are constructed. Using pseudoprime numbers can significantly reduce the number of candidates viewed as primes. The proposed corollaries of the theorems allow to s...

We initiate a classification of polynomials f : C → C of degree d having the top Betti number of the general fibre close to the maximum. We find a range in which the polynomial must have isolated singularities and another range where it may have at most one line singularity of general Morse transversal type. Our method uses deformations into particular pencils with non-isolated singularities. 2...

For any moduli space of stable representations of quivers, certain smooth varieties, compactifying projective space fibrations over the moduli space, are constructed. The boundary of this compactification is analyzed. Explicit formulas for the Betti numbers of the smooth models are derived. In the case of moduli of simple representations, explicit cell decompositions of the smooth models are co...

We prove an extension of the following result of Lubotzky and Madid on the rational cohomology of a nilpotent group G: If b1 < ∞ and G⊗Q = 0,Q,Q then b2 > b1/4. Here the bi are the rational Betti numbers of G and G ⊗ Q denotes the Malcev-completion of G. In the extension, the bound is improved when the relations of G are known to all have at least a certain commutator length. As an application ...

For every fixed l > 0, we describe a singly exponential algorithm for computing the first l Betti number of a given semi-algebraic set. More precisely, we describe an algorithm that given a semi-algebraic set S ⊂ Rk a semi-algebraic set defined by a Boolean formula with atoms of the form P > 0, P < 0, P = 0 for P ∈ P ⊂ R[X1, . . . ,Xk], computes b0(S), . . . , bl(S). The complexity of the algor...

We describe an algorithm for constructing Carmichael numbers N with a large number of prime factors p1, p2, . . . , pk. This algorithm starts with a given number Λ = lcm(p1 − 1, p2 − 1, . . . , pk − 1), representing the value of the Carmichael function λ(N). We found Carmichael numbers with up to 1101518 factors.

We introduce structural condition numbers and discuss their signiicance for the proper scaling of nonsymmetric and symmetric matrices.