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...

TImis note is a continuation of [Nl], Wbere we have discusged tbe unknotting number of knots With rspect tía knot diagrams. Wc wilI show that for every minimum-crossing knot-diagram among ah unknotting-number-one two-bridge knot there exist crossings whose exchangeyields tIme trivial knot, ib tbe tbird Tait conjecture is true.

We argue that the large n limit of the n-particle SU(1, 1|2) superconformal Calogero model provides a microscopic description of the extreme Reissner-Nordström black hole in the near-horizon limit.

We show that the Calogero and Calogero-Sutherland models possess an N -body generalization of shape invariance. We obtain the operator representation that gives rise to this result, and discuss the implications of this result, including the possibility of solving these models using algebraic methods based on this shape invariance. Our representation gives us a natural way to construct supersymm...