let $e$ be an elliptic curve over $bbb{q}$ with the given weierstrass equation $ y^2=x^3+ax+b$. if $d$ is a squarefree integer, then let $e^{(d)}$ denote the $d$-quadratic twist of $e$ that is given by $e^{(d)}: y^2=x^3+ad^2x+bd^3$. let $e^{(d)}(bbb{q})$ be the group of $bbb{q}$-rational points of $e^{(d)}$. it is conjectured by j. silverman that there are infinitely many primes $p$ for which $...

The classification of the nilpotent Jacobians with some structure has been an object study because its relationship Jacobian conjecture. In this paper we classify polynomial maps in dimension n form H=(u(x,y),u2(x,y,x3),…,un−1(x,y,xn),h(x,y)) JH nilpotent. addition prove that X+H are invertible, which shows for kind conjecture is verified.

A famous conjecture (usually called Ryser’s conjecture) that appeared in the PhD thesis of his student, J. R. Henderson, states that for an r-uniform r-partite hypergraph H, the inequality τ(H) 6 (r − 1)·ν(H) always holds. This conjecture is widely open, except in the case of r = 2, when it is equivalent to Kőnig’s theorem, and in the case of r = 3, which was proved by Aharoni in 2001. Here we ...

In this paper, we give a new and direct proof for the recently proved conjecture raised in Soltani and Roozegar (2012). The conjecture can be proved in a few lines via the integral representation of the Gauss-hypergeometric function unlike the long proof in Roozegar and Soltani (2013).

The Erdős-Hajnal conjecture states that for every graph H, there exists a constant δ(H) > 0, such that if a graph G has no induced subgraph isomorphic to H, then G contains a clique or a stable set of size at least |V (G)|. This conjecture is still open. We consider a variant of the conjecture, where instead of excluding H as an induced subgraph, both H and H are excluded. We prove this modifie...

The Hanna Neumann conjecture states that if F is a free group, then for all finitely generated subgroups H,K 6 F , rank(H ∩ K) − 1 6 [rank(H)− 1] [rank(K)− 1] In this paper, we show that if one of the subgroups, say H , has a generating set consisting of only positive words, then H is not part of any counterexample to the conjecture. We further show that if H ≤ F2 is part of a counterexample to...