The class-breadth conjecture revisited

The Mertens Conjecture Revisited

The Roller-Coaster Conjecture Revisited

A graph is well-covered if all its maximal independent sets are of the same cardinality [25]. If G is a well-covered graph, has at least two vertices, and G − v is well-covered for every vertex v, then G is a 1-well-covered graph [26]. We call G a λ-quasi-regularizable graph if λ · |S| ≤ |N (S)| for every independent set S of G. The independence polynomial I(G;x) is the generating function of i...

The Entropy Influence Conjecture Revisited

In this paper, we prove that most of the boolean functions, f : {−1, 1}n → {−1, 1} satisfy the Fourier Entropy Influence (FEI) Conjecture due to Friedgut and Kalai (Proc. AMS’96)[2]. The conjecture says that the Entropy of a boolean function is at most a constant times the Influence of the function. The conjecture has been proven for families of functions of smaller sizes. O’donnell, Wright and...

Bloch's Conjecture Revisited

Let X be a non-singular projective complex surface. We can show that Bloch's conjecture (i.e. , that if p g = 0 then the Albanese kernel vanishes) is equivalent to the following statement:

Rank Conjecture Revisited

The rank conjecture says that rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. We prove the conjecture for elliptic curves E(K) over a number field K. As a corollary, one gets a simple estimate for the rank in terms of the length of period of a continued fraction attached to the E(K). As an illustration, we consider a family of elliptic...