1. G(n) is a graph of n vertices and G(n ; e) is a graph of n vertices and e edges. Is it true that if every induced subgraph of a G(10n) of 5n vertices has more than 2n 2 edges then our G(10n) contains a triangle? It is easy to show that if true this result is best possible . To see this let A i , i =1, 2, . . . , 5, be sets of 2n vertices, put A, = A 6 and join every vertex of A, to every ver...

We consider the totally real cyclic quintic fields Kn = Q(θn), generated by a root θn of the polynomial fn(x) = x 5 + nx − (2n + 6n + 10n + 10)x + (n + 5n + 11n + 15n + 5)x + (n + 4n + 10n + 10)x + 1. Assuming that m = n4 + 5n3 + 15n2 + 25n + 25 is square free, we compute explicitly an integral basis and a set of fundamental units of Kn and prove that Kn has a power integral basis only for n = ...

For each integer n ≥ 0, there is a closed, unknotted, polygonal curve Kn in R3 having less than 10n + 9 edges, with the property that any Piecewise-Linear triangulated disk spanning the curve contains at least 2n−1 triangles.

The nth Ramanujan prime is the smallest positive integer Rn such that if x ≥ Rn, then the interval ( 1 2x, x ] contains at least n primes. We sharpen Laishram’s theorem that Rn < p3n by proving that the maximum of Rn/p3n is R5/p15 = 41/47. We give statistics on the length of the longest run of Ramanujan primes among all primes p < 10n, for n ≤ 9. We prove that if an upper twin prime is Ramanuja...

An ion microprobe ha~ been. used to observe Mg/"Mg in excess of normal values in anorthite crystals from Allende refractory mclus10ns. In one crystal, Mg excesses up to 40% above normal, correlated with AI/Mg, were. obser_ved. These ~xc~sses are many times larger than those observed by other techniques. T_he _correla~10n with Al(Mg mdicated tha_t the excess '"Mg in the Allende anorthite is unif...

Linear time solv-able optimization problems on certain structured graph families, extended abstract. On the xed parameter complexity of graph enumeration problems deenable in monadic second order logic. in preparartion, 1998. 35 the rst case there is no other vertex z of L such that w and z occur in the same equivalence class of R red , or else z and w are adjacent to both u and v, a contradict...

Consider a geometric graph given by n points in the plane (in general position) andm line segments, each segment joining a pair of the given points. We show that: if m ≥ 3n + 1 then there are 3 pairwise disjoint segments; if m ≥ 10n + 1 then there are 4 disjoint segments; and if m ≥ ckn(log n) k−4 then there are k disjoint segments.

ABSTRACT :Let MISimp,IP2n+1(k) be the moduli space of stable symplectic instanton bundles on IP with second Chern class c2 = k (it is a closed subscheme of the moduli space MIIP2n+1(k)) We prove that the dimension of its Zariski tangent space at a special (symplectic) instanton bundle is 2k(5n − 1) + 4n − 10n + 3 , k ≥ 2. It follows that special symplectic instanton bundles are smooth points fo...