Let E/Q be an elliptic curve defined by the equation y 2 = x 3 + ax + b. For a prime p, p ∤ ∆ = −16(4a 3 + 27b 2) = 0, define N p = p + 1 − a p = |E(F p)|. As a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant c, p≤x,p∤∆ N p p ∼ c(log x) r , x → ∞. Let α p and β p be the eigenvalues of the Frobenius at p. Define ˜ c n =    α k ...

In the literature of graph recognition algorithms, two of the most extensively covered graphs are interval graphs and permutation graphs [6, 2]. One way of generalizing both interval graphs and permutation graphs is the intersection graph of a collection of trapezoids with corner points lying on two parallel lines. Such a graph is called the trapezoid graph [3, 4]; Ma and Spinrad [8] show that ...

Pins and wires offer the simplest and most effective tools for managing bone fractures and dislocations. Migration of these devices within the chest is rare, but can cause serious problems. The spontaneous migration of Kirschner wires from the right clavicle to the mediastinum resulted in penetrating injury of the esophagus and pseudo-aneurysm of the brachiocephalic artery in an 84-year-old pat...

and relational thinking and mathematical performance at the end of lower secondary school Helen C. Reed, Paul A. Kirschner and Jelle Jolles

Volar locking-plate and kirschner-wire fixation did not differ in terms of functional outcomes after dorsally displaced distal radial fracture.

BACKGROUND To evaluate internal fixation by intramedullary Kirschner wires as a surgical technique in the treatment of femoral shaft fractures in children by a prospective study. METHODS 17 femoral shaft fractures at various levels in 16 children aged 2-15 years were treated by closed intramedullary Kirschner wiring under image intensifier control between May 2000 and October 2003. No externa...

Fibroblast growth factors (FGFs) are small polypeptides that mediate embryonic induction in vertebrates. FGFs are important mediators of mesoderm induction in Xenopus and chick and of limb development in mouse and chick (Rappolee and Werb, 1994; Niswander and Martin, 1992, 1993a,b; Vogel and Tickle, 1993). There are nine members of the FGF family (Hebert et al., 1990; Basilico and Moscatelli, 1...

Tolerance graphs have been extensively studied since their introduction, due to their interesting structure and their numerous applications, as they generalize both interval and permutation graphs in a natural way. It has been conjectured by Golumbic, Monma, and Trotter in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs. Since cocompara...