Simulation of conditioned diffusions

In this paper, we propose some algorithms for the simulation of the distribution of certain diffusions conditioned on terminal point. We prove that the conditional distribution is absolutely continuous with respect to the distribution of another diffusion which is easy for simulation, and the formula for the density is given explicitly

The Lagrangian multiplier method of finding upper and lower limits to critical stresses of clamped plates

The theory of Lagrangian multipliers is applied to the problem of finding both upper and lower limits to the true compressive buckling stress of a clamped rectangular plate. The upper and lower limits thus bracket the true stress, which cannot be exactly found by the differential-equation approach. The procedure for obtaining the upper limit, which is believed to be new, presents certain advantages over the classical Rayleigh-Ritz method of finding upper limits. The theory of the lower-limit procedure has been given by Trefftz, but, in the present application, the method differs from that of Trefftz in a way that makes it inherently more quickly convergent. It is expected that in other buckling problems and in some vibration problems the Lagrangian multiplier method of finding upper and lower limits may be advantageously applied to the calculation of buckling stresses and natural frequencies

The Lagrangian Multiplier Method of Finding Upper and Lower Limits to Critical Stresses of Clamped Plates

The theory of Lagrangian multipliers is applied to the problem of finding both upper and lower limits to the true compressive buckling stress of a clamped rectangular plate. The upper and lower limits thus bracket the truss, which cannot be exactly found by the differential-equation approach. The procedure for obtaining the upper limit, which is believed to be new, presents certain advantages over the classical Raleigh-Rite method of finding upper limits. The theory of the lower-limit procedure has been given by Trefftz but, in the present application, the method differs from that of Trefftz in a way that makes it inherently more quickly convergent. It is expected that in other buckling problems and in some vibration problems problems the Lagrangian multiplier method finding upper and lower limits may be advantageously applied to the calculation of buckling stresses and natural frequencies

Rainbow triangles in three-colored graphs

Erdős and Sós proposed the problem of determining the maximum number F(n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F(n) = F(a) + F(b) + F(c) + F(d) + abc + abd + acd + bcd, where a + b + c + d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4k for all k ≥ 0. These results imply that lim F(n)/((n)(3)) = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments