I am mathematician **Bruno Alexander Dmitrievich** (Bruno/Brjuno/Bryuno) (Russian: Брюно Александр Дмитриевич) and I made substantial contributions to:

- Nonlinear Analysis and its Applications.
- Number Theory: Multidimensional generalization of the continued fraction.

More detailed:

(1) I have developed a new level of Mathematical Analysis and called it «Power Geometry». It also was applied for solution of several difficult problems in Mathematics, Mechanics, Celestial Mechanics and Hydrodynamics. The 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 (algebraic, ordinary differential and partial differential), «Power Geometry» 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 (including boundary layers and singular perturbations). Elements of plane «Power Geometry» were proposed by I.Newton for an algebraic equation (1680); and by Briot and Bouquet for an ordinary differential equation (1856). Space «Power Geometry» was proposed by me for a nonlinear autonomous system of ODEs (1962). I am is the inventor and founder of the universal «Nonlinear Analysis». It is a new level of Calculus beyond classical mathematical analysis and functional analysis. «Nonlinear Analysis» is based on «Power Geometry», which is a very powerful tool for «Asymptotic Solving» of many nonlinear problems. I also have developed the normal forms theory for both arbitrary and Hamiltonian systems of ordinary differential equations. In particulary I have found two conditions for convergency of the normalizing transformation; one of those conditions is the condition on the eigenvalues of the linear part. It is a number theoretical restriction that was named as «Bruno Condition» and it is better then «Siegel’s Condition»; in the two-dimentional case the restriction gives so colled «Brjuno Numbers». Other fields of my interests are the Painlevé equations, Mechanics, Celestial Mechanics and Hydrodynamics.

In 2016 I proposed a new method parametrization of an algebraic curve with help of «Hadamard’s polyhedron». In 2017 I Proposed a new method for computation of complicated and exotic asymptotic expansions of solutions to an ordinary differential equation.

(2) For the multidimensional generalization of the continued fraction, I proposed a modular polyhedron instead of the Klein polyhedron (that name was given by me instead of the name «Arnold polyhedron»). Preimages of vertices of the modular polyhedron give the best Diophantine approximations. The modular polyhedron can be computed by means of a standard program for computing convex hulls. It gives a solution of the problem, which majority of main mathematicians of XIX century tried to solve. In the algebraic case, using the modular polyhedron it is possible to find all fundamental units of some rings. Using them it is possible to compute all periods and to compute all solutions to Diophantine equations of some class. This approach gives also simultaneous Diophantine approximations.

# Biography

Born 26 June 1940 in Moscow, USSR (Russia).

## Education

- Mathematician:
- MS, Moscow State University, 1962;
- PhD, Institute of Applied Mathematics, 1966;
- Professor, Institute of Applied Mathematics, 1970.

## Career

- Keldysh Institute of Applied Mathematics of RAS:
- Junior Researcher, 1965;
- Senior Researcher, 1971;
- Leading Researcher, 1987;
- Head of Mathematical Department, 1995;
- Head of Sector of Singular Problems, 2008.

## Honors

- 3rd Prize at the Moscow Mathematical Olympiade, 1956;
- 1st Prize at the Moscow Mathematical Olympiade, 1957;
- 2nd Prizes for Students Papers in Moscow St University, 1960 and 1961.
- Included in list of the 500 most influential people of the XIX century.

## Memberships

- Moscow Mathematical Society;
- American Mathematical Society;
- Academy of Nonlinear Sciences.

## Research interests

- Universal Nonlinear Analysis and its applications;
- Number Theory.

## Main works

- Analytical form of differential equations (I, II). Trans. Moscow Math. Soc. 25 (1971) 131–288, 26 (1972) 199–239;
- Local Methods in Nonlinear Differential Equations. Springer-Verlag: Berlin-Heidelberg-New York-London-Paris-Tokyo, 1989. 350pp;
- The Restricted 3-Body Problem: Plane Periodic Orbits. Walter de Gruyter, Berlin-New York, 1994. 362pp;
- Power Geometry in Algebraic and Differential Equations. Elsevier Science (North-Holland), Amsterdam, 2000. 385pp.