Mean Field Game is a rather new field initially developed in applied mathematics and engineering order to deal with the dynamics of large number controlled agents or objects interaction. For class these models, there exists deep relationship between associated system equations non linear Schr\"odinger equation, which allows get insights on structure their solutions. In this work, we related asp...

The celebrated S-lemma establishes a powerful equivalent condition for the nonnegativity of a quadratic function over a single quadratic inequality. However, this lemma fails without the technical condition, known as the Slater condition. In this paper, we first show that the Slater condition is indeed necessary for the S-lemma and then establishes a regularized form of the S-lemma in the absen...

A formulation and an implementation of two-dimensional Boundary Element Method (BEM) analysis for steady state, uncoupled thermoelastic problems is presented. This approach differs from other treatments of thermal loads in BEM analysis in which the domain integrals due to the thermal gradients are to be incorporated in the analysis via particular-integrals. Thus unlike Finite Elements or Field ...

We calculate the optical properties of carbon nanotubes in an external static electric field directed along the tube axis. We predict strong Franz-Keldysh oscillations in the first and second band-to-band absorption peaks, quadratic Stark effect of the first two excitons, and the field dependence of the bound exciton ionization rate for a wide range of tube chiralities. We find that the phonon-...

A search for prime factors of the generalized Fermat numbers Fn(a, b) = a2 n + b2 n has been carried out for all pairs (a, b) with a, b ≤ 12 and GCD(a, b) = 1. The search limit k on the factors, which all have the form p = k · 2m + 1, was k = 109 for m ≤ 100 and k = 3 · 106 for 101 ≤ m ≤ 1000. Many larger primes of this form have also been tried as factors of Fn(a, b). Several thousand new fact...

To find a global optimal solution to the quadratically constrained quadratic programming problem, we explore the relationship between its Lagrangian multipliers and related linear conic programming problems. This study leads to a global optimality condition that is more general than the known positive semidefiniteness condition in the literature. Moreover, we propose a computational scheme that...