Ganikhodzhaev, Nasir Nabievich

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Total publications: 14
Scientific articles: 14

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Doctor of physico-mathematical sciences (1991)
Speciality: 01.01.03 (Mathematical physics)
Keywords: dynamical systems; trajectory theory of dinamical systems; quadratic stochastic operators and processes; topological entropy of quadratic operators; random quadratic operators; the random model of heredityin random environment; the Gibbs states; lattice models of statistical mechanics.


It was proved that any two ergodic countably-countinous sequences of measurable partitions are lacunarily isomorphic, whence any two ergodic countably-continuous partitions are isomorphic. The translation-invariant extemal Gibbs states for Ising and Potts models on the Cayley tree were described and continuum extremal Gibbs states for these models were constructed. The conditions when disordered phase in the ferromagnetic Potts model on the Bethe lattice be extreme were determined. A exact solution of Ising model with competing ternary and binary interactions on Cayley tree was obtained. The quadratic stochastic processes both commutative and noncommutative were determined, the several ergodic theorems for these processes were proved and correlation between its and Markov processes was established. The constructions of quadratic stochastic operators by means of Gibbs states and Haar measures were proposed and ergodicity of quadratic operators in the second case was proved. The models of heredity described by random quadratic stochastic operators in the random environment were studied. The topological entropy of some class of quadratic stochastic operators were colculeted. The following conjecture was formulated: any Volterra quadratic stochastic operator has zero topological entropy.


Graduated from Faculty of Mathematics and Mechanics of Tashkent State University (MSU) in 1971 (department functional analysis). Ph.D. thesis was defended in 1975. D.Sci. thesis was defended in 1992. A list of my works contains more than 80 titles.

Main publications:
  • Ganikhodjaev N. N. On stochastic processes generated by quadratic operators // Jour. of Theor. Prob., 1990, 3(1), 51–70.
  • Ganikhodjaev N. N., Rozikov U. A. On disordered phase in the ferromagnetic Potts model on the Bethe lattice // Osaka journal Math. 2000. Vol. 37(2), 373–383.
List of publications on Google Scholar
List of publications on ZentralBlatt

Publications in Math-Net.Ru
1. N. N. Ganikhodzhaev, U. A. Rozikov, “On quadratic stochastic operators generated by Gibbs distributions”, Regul. Chaotic Dyn., 11:4 (2006),  467–473  mathnet  mathscinet  zmath
2. N. N. Ganikhodzhaev, C. H. Pah, “Phase diagrams of multicomponent lattice models”, TMF, 149:2 (2006),  244–251  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 149:2 (2006), 1512–1518  isi  scopus
3. N. N. Ganikhodzhaev, F. M. Mukhamedov, “Some Properties of a Class of Diagonalizable States of von Neumann Algebras”, Mat. Zametki, 76:3 (2004),  350–361  mathnet  mathscinet  zmath; Math. Notes, 76:3 (2004), 329–338  isi  scopus
4. N. N. Ganikhodzhaev, D. V. Zanin, “On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex”, Uspekhi Mat. Nauk, 59:3(357) (2004),  161–162  mathnet  mathscinet  zmath; Russian Math. Surveys, 59:3 (2004), 571–572  isi  scopus
5. N. N. Ganikhodzhaev, U. A. Rozikov, “Group representation of the Cayley forest and some of its applications”, Izv. RAN. Ser. Mat., 67:1 (2003),  21–32  mathnet  mathscinet  zmath; Izv. Math., 67:1 (2003), 17–27  isi  scopus
6. N. N. Ganikhodzhaev, F. M. Mukhamedov, U. A. Rozikov, “$\mathbb {Z}$Existence of a Phase Transition for the Potts $p$-adic Model on the Set $\mathbb {Z}$”, TMF, 130:3 (2002),  500–507  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 130:3 (2002), 425–431  isi
7. N. N. Ganikhodzhaev, “Exact Solution of the Ising Model on the Cayley Tree with Competing Ternary and Binary Interactions”, TMF, 130:3 (2002),  493–499  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 130:3 (2002), 419–424  isi
8. N. N. Ganikhodzhaev, F. M. Mukhamedov, “Ergodic properties of discrete quadratic stochastic processes defined on von Neumann algebras”, Izv. RAN. Ser. Mat., 64:5 (2000),  3–20  mathnet  mathscinet  zmath; Izv. Math., 64:5 (2000), 873–890  isi  scopus
9. N. N. Ganikhodzhaev, U. A. Rozikov, “On unordered phases of certain models on a Cayley tree”, Mat. Sb., 190:2 (1999),  31–42  mathnet  mathscinet  zmath; Sb. Math., 190:2 (1999), 193–203  isi  scopus
10. N. N. Ganikhodzhaev, F. M. Mukhamedov, “Ergodic properties of quantum quadratic stochastic processes defined on von Neumann algebras”, Uspekhi Mat. Nauk, 53:6(324) (1998),  243–244  mathnet  mathscinet  zmath; Russian Math. Surveys, 53:6 (1998), 1350–1351  isi  scopus
11. N. N. Ganikhodzhaev, U. A. Rozikov, “Discription of periodic extreme Gibbs measures of some lattice models on the Cayley tree”, TMF, 111:1 (1997),  109–117  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 111:1 (1997), 480–486  isi
12. N. N. Ganikhodzhaev, “Pure phases of the ferromagnetic Potts model with three states on a second-order Bethe lattice”, TMF, 85:2 (1990),  163–175  mathnet  mathscinet; Theoret. and Math. Phys., 85:2 (1990), 1125–1134  isi
13. P. M. Blekher, N. N. Ganikhodzhaev, “On pure phases of the Ising model on the Bethe lattices”, Teor. Veroyatnost. i Primenen., 35:2 (1990),  220–230  mathnet  mathscinet  zmath; Theory Probab. Appl., 35:2 (1990), 216–227  isi
14. V. G. Vinokurov, N. N. Ganikhodzhaev, “Conditional functions in the trajectory theory of dynamical systems”, Izv. Akad. Nauk SSSR Ser. Mat., 42:5 (1978),  928–964  mathnet  mathscinet  zmath; Math. USSR-Izv., 13:2 (1979), 221–252  isi

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