We summarize the recently-developed framework of linearly-solvable stochastic optimal control. Using an exponential transformation, the (Hamilton-Jacobi) Bellman equation for such problems can bemade linear, giving rise to efficient numericalmethods. Extensions to game theory are also possible and lead to linear Isaacs equations. The key restriction that makes a stochastic optimal control probl...

A classical theorem on character degrees states that if a finite group has fewer than four degrees, then the is solvable. We prove corresponding result values by showing eight in its table, This confirms conjecture of T. Sakurai. also classify non-solvable groups with exactly values.

We review some old and new methods of reduction of the number of degrees of freedom from ∼ N 2 to ∼ N in the multi-matrix integrals.

A 2-dimensional framework is a straight line realisation of a graph in the Euclidean plane. It is radically solvable if the set of vertex coordinates is contained in a radical extension of the field of rationals extended by the squared edge lengths. We show that the radical solvability of a generic framework depends only on its underlying graph and characterise which planar graphs give rise to ...

We discuss distortion of solvable linear groups over a locally compact field and provide necessary and sufficient conditions for a subgroup to be undistorted when the field is of characteristic zero.

In this paper we give a geometric realization of free solvable groups, and study its Poisson-Furstenberg boundaries, we also discuss the construction of normal forms in the solvable groups.

A finite word is called Petri net solvable if it is isomorphic to the reachability graph of some unlabelled Petri net. In this paper, the class of two-letter Petri net solvable words is studied.