‎By the Mordell‎- ‎Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎This paper studies the rank of the family Epq:y2=x3-pqx of elliptic curves‎, ‎where p and q are distinct primes‎. ‎We give infinite families of elliptic curves of the form y2=x3-pqx with rank two‎, ‎three and four‎, ‎assuming a conjecture of Schinzel ...

A classical R-matrix structure is described for the Lax representation of the integrable n-particle chains of Calogero-Olshanetski-Perelomov. This R-matrix is dynamical, non antisymmetric and non-invertible. It immediately triggers the integrability of the Type I, II and III potentials, and the algebraic structures associated with the Type V potential.

A new spectral parameter independent R-matrix (that depends on all of the dynamical variables) is proposed for the elliptic Calogero-Moser models. Necessary and sufficient conditions for this R-matrix to exist reduce to an equality between determinants of matrices involving elliptic functions. The needed identity appears new and is still unproven in full generality: we present it as a conjecture.

with constants C ;p1;p2 depending only on ; p1; p2 and p := p1p2 p1+p2 hold. The rst result of this type is proved in [4], and the purpose of the current paper is to extend the range of exponents p1 and p2 for which (2) is known. In particular the case p1 = 2, p2 =1 is solved to the a rmative. This was originally considered to be the most natural case and is known as Calderon's conjecture [3]. ...

This paper proves the reconstruction conjecture for graphs which are isomorphic to the cube of a tree. The proof uses the reconstructibility of trees from their peripheral vertex deleted subgraphs. The main result follows from (i) characterization of the cube of a tree (ii) recognizability of the cube of a tree (iii) uniqueness of tree as a cube root of a graph G, except when G is a complete gr...

Given a finite field k of characteristic p ≥ 5, we show that the Tate conjecture holds for K3 surfaces over k if and only if there are finitely many K3 surfaces defined over each finite extension of k.

Denote by Fq a field of q elements, F̄q an algebraic closure of Fq, φ ∈ Gal(F̄q/Fq) the Frobenius substitution x 7→ xq and F = φ−1 the “geometric Frobenius”. Denote by X a scheme (separated of finite type) over Fq, and denote by X̄ the scheme over F̄q obtained by extension of scalars. For all closed points x of X, let deg(x) = [k(x) : Fq] be the degree over Fq of the residue extension. The zeta fun...

Given a partition λ of n, a k-minor of λ is a partition of n − k whose Young diagram fits inside that of λ. We find an explicit function g(n) such that any partition of n can be reconstructed from its set of k-minors if and only if k ≤ g(n). In particular, partitions of n ≥ k2 + 2k are uniquely determined by their sets of k-minors. This result completely solves the partition reconstruction prob...

We prove that a large family of graphs which are decomposable with respect to the modular decomposition can be reconstructed from their collection of vertex-deleted subgraphs.

The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n−d. That is, any two vertices of the polytope can be connected by a path of at most n− d edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope wit...