Extremal numbers for odd cycles

We describe the C_{2k+1}-free graphs on n vertices with maximum number of edges. The extremal graphs are unique except for n = 3k-1, 3k, 4k-2, or 4k-1. The value of ex(n,C_{2k+1}) can be read out from the works of Bondy, Woodall, and Bollobas, but here we give a new streamlined proof. The complete determination of the extremal graphs is also new. We obtain that the bound for n_0(C_{2k+1}) is 4k in the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turan graph.Comment: 6 page

Model for Triplet State Engineering in Organic Light Emitting Diodes

Engineering the position of the lowest triplet state (T1) relative to the first excited singlet state (S1) is of great importance in improving the efficiencies of organic light emitting diodes and organic photovoltaic cells. We have carried out model exact calculations of substituted polyene chains to understand the factors that affect the energy gap between S1 and T1. The factors studied are backbone dimerisation, different donor-acceptor substitutions and twisted geometry. The largest system studied is an eighteen carbon polyene which spans a Hilbert space of about 991 million. We show that for reverse intersystem crossing (RISC) process, the best system involves substituting all carbon sites on one half of the polyene with donors and the other half with acceptors.Comment: 7 Pages; 10 Figure

Universality and m_X cut effects in B -> Xs l+ l-

The most precise comparison between theory and experiment for the B -> Xs l+ l- rate is in the low q^2 region, but the hadronic uncertainties associated with an experimentally required cut on m_X potentially spoil the search for new physics in these decays. We show that a 10-30% reduction of d\Gamma(B -> Xs l+ l-) / dq^2 due to the m_X cut can be accurately computed using the B -> X_s gamma shape function. The effect is universal for all short distance contributions in the limit m_X^2 << m_B^2, and this universality is spoiled neither by realistic values of the m_X cut nor by alpha_s corrections. Both the differential decay rate and forward-backward asymmetry with an m_X cut are computed.Comment: 5 pages, journal versio

Extracting short distance information from b-->s[script-l]+[script-l]- effectively

We point out that in inclusive B-->Xs[script-l]+[script-l]- decay an angular decomposition provides a third (q2 dependent) observable sensitive to a different combination of Wilson coefficients than the rate and the forward-backward asymmetry. Since a precise measurement of q2 dependence requires large data sets, it is important to consider the data integrated over regions of q2. We develop a strategy to extract all measurable Wilson coefficients in B-->Xs[script-l]+[script-l]- from a few simple integrated rates in the low q2 region. A similar decomposition in B-->K*[script-l]+[script-l]-, together with the B-->K*gamma rate, also provides a determination of the Wilson coefficients, without reliance on form factor models and without having to measure the zero of the forward-backward asymmetry