Alexander Dmitrievich Bruno

Who is Alexander Dmitrievich Bruno?

Mathematician Bruno, Alexander Dmitrievich made substantial contribution to the normal forms theory. Bruno developed a new level of Mathematical Analysis and called it "Power Geometry". He also applied it for solution of several difficult problems in Mathematics, Mechanics, Celestial Mechanics, and Hydrodynamics. Traditional differential calculus is effective for linear and quasilinear problems. It is less effective for essentially nonlinear problems. A linear problem is the first approximation to a quasilinear problem. Usually a linear problem is solved by methods of functional analysis, then the solution to the quasilinear problem is found as a perturbation of the solution to the linear problem. For an essentially nonlinear problem, we need to isolate its first approximations, to find their solutions, and to construct perturbations of these solutions. This is what Power Geometry is aimed at. For equations and systems of equations, PG allows to compute asymptotic forms of solutions as well as asymptotic and local expansions of solutions at infinity and at any singularity of the equations. Elements of plane PG were proposed by I.