For given $k,\ell \in \mathbb Z$ we study the Diophantine system $$x+y+z=k, \quad x y z = \ell $$ for $x,y,z$ integers in a quadratic number field, which has history literature. When $\ell =1$, describe all such solutions; only $k=5,6$, do

We consider a Mean Field Games model where the dynamics of agents is given by controlled Langevin equation and cost quadratic. An appropriate change variables transforms system into two coupled kinetic Fokker–Planck equations. prove an existence result for latter system, obtaining consequently solution system.

This paper investigates a linear quadratic mean field leader-follower team problem, where the model involves one leader and large number of weakly-coupled interactive followers. The followers cooperate to optimize social cost. Specifically, for any strategy provided first by leader, would like choose minimize cost functional. Using variational analysis person-by-person optimality, we construct ...

Let F/Q be a number field. The space of positive definite binary Hermitian forms over F form an open cone in a real vector space. There is a natural decomposition of this cone into subcones. In the case of an imaginary quadratic field these subcones descend to hyperbolic space to give rise to tessellations of 3-dimensional hyperbolic space by ideal polytopes. We compute the structure of these p...