Main results of L.D. Pustyl'nikov

in mathematics and physics

1) The solution of Lagrange problem on finding of explicit analytic expressions of solutions of ordinary differential equation describing of vibrations of discrete string, and the investigation of its stochastic analogue ([40], [101])

2) The solution of Fermi-Ulam problem on the behaviour of trajectories in the Fermi-Ulam model ([16], [27], [47], [53])

3) The proof of Fermi conjecture justifying the Fermi acceleration mechanism in the relativistic Fermi-Ulam model and in the accelerating model with one wall in a constant force field ([30], [33], [53], [6])

4) Rigorous justification of the second law of thermodynamics from mechanics on the base of the proof of the law of entropy increase for nonequilibrium gas ([35], [53])

5) Solutions of Weyl problems on finding of strong estimates of trigonometric sums and of the remainder term in the distribution law for the fractional parts of the values of a polynomial ([39], [46], [50], [72], [106])

6) The proof of the central limit theorem for the fractional parts of sums of the values of polynomials ([85])

7) The construction of an operator acting in a Hilbert space such that the Riemann problem on zeros of zeta-function is equivalent to the problem of the existence of its eigenvector with eigenvalue $\lambda = -1$ (Hilbert problem) ([36])

8) The construction of a dynamical system such that the Riemann hypothesis is equivalent to the problem of the existence of its trajectory of order two having a special type ([120])

9) The representation of Riemann zeta-function in the critical strip by means of infinite product of second order matrices of the explicit form ([120], [36])

10) The finding of necessary condition for the validity of the Riemann hypothesis by means of signs of even derivatives of the function $\xi(s)$

11) The proof of the positiveness of even derivatives of the function $\xi(s)$ at the point $s=\frac{1}{2}$ and the finding of the asymptotic formula for these derivatives ([73], [81], [84])

12) The refutation of the Riemann hypothesis for an arbitrarily sharp approximation of the zeta-function satisfying the same functional equation ([107])

13) The application of ergodic theory to obtaining new results in classical problem on distribution of quadratic residues and non-residues ([45], [64], [104], [106])

14) The introduction of a new notion of a generalized continued fraction and the construction of a new theory of generalized continued fraction having applications in some problems of analysis, number theory and in theory of quantum chaos ([50], [64], [104], [106], [96], [109], [114])

15) The proof of quantum chaos conjecture for a class of quantum systems including a popular ''kicked rotator'' model as a very special case ([75], [96], [109], [114])

16) The solution of Frenkel-Kontorova problem and the construction of infinite-dimensional KAM theory for Frenkel-Kontorova equation ([25], [65])

17) The solution of Fermi-Pasta-Ulam problem and the construction of infinite-dimensional KAM theory for Fermi-Pasta-Ulam equation ([54], [65])

18) The construction of stable phase transition in Frenkel-Kontorova model ([29], [51])

19) The definition of a generalized billiard and the construction of theories of newtonian and relativistic billiards having applications in statistical physics ([33], [35], [53], [74], [110], [112], [124])

20) Proofs of conjectures associated with the standard map ([22], [82], [105])

21) The proof of unbounded growth of an action variable in popular physical systems with $\delta$-interaction ([9], [15])

22) The construction of a massive set of stable oscillatory motions in accelerator model of charge particles ([20], [28], [117])

23) The solution of nonautonomous Birkhoff problem on the stability of a fixed point for a nonautonomous dynamical system in a general elliptic case ([3], [6], [15], [53])

24) The solution of nonautonomous functional-theoretical problem on a reducibility of a sequence of conformal maps in the linear form ([2], [5], [53], [70])

25) The solution of a capture problem and the construction of final motions in n-body problem for $n > 3$ ([34])

26) The estimate of a measure of one-sided oscillatory motions in the model of A.N. Kolmogorov and in its generalization in n-body problem ([71])

27) The discovery of infinite-dimensional chaos and bifurcations in lattice systems ([41], [102])

28) The construction of an probabilistic invariant measure for hyperbolic equation describing the vibration of a string in the framework of Cauchy problem ([83])

29) Proofs of structural algebraic theorems in theories of Toeplitz and Hankel matrices and applications to fast computations in some problems of linear algebra connected with Toeplitz and Hankel matrices and to fast prediction of random processes ([10], [11], [18], [31])

30) Proofs of structural algebraic theorems in theories of Toeplitz and Hankel integral operators ([18])