Namangan regional department of the V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, was founded in accordance with item 17 of the Decree № PQ-4708 on 7^th May 2020 of the President of the Republic of Uzbekistan. In accordance with Appendix 2 to this Decree, the department was formed with 9 staff units and located at the Namangan State University. At present, the department is headed by DSc M.Rakhmatullayev. There are principal researcher F.Arzikulov, leading researchers R.Khakimov, A.Urinov, V.Khodzhibaev, B.Samatov, E.Karimov, senior researchers Yu.Apakov, I.Irgashev, I.Karimjanov and junior researchers A.Akbaev, I.Juraev, D.Egamov are engaged in scientific activities in the department.

In the present researchers of the branch are involved in the following scientific directions:

-         Gibbs measures on a Cayley tree;

-         Enveloping  *-algebras of special AJW-algebras and derivations of AJW-algebras;

-         Local and 2-local derivations, local and 2-local automorphisms of low-dimension Jordan algebras;

-         Initial and boundary problems for the degenerate hyperbolic type equations of second kind;

-         Problems with the integral conditions for the parabolic-hyperbolic equations;

-         Fractional order integral-differential operators;

-         Boundary problems for stochastic processes with independent increments: asymptotic expansions and two-sided estimates for the distributions and characteristics of some boundary functionals.

-         Boundary problems for an equation with degeneration in one variable including fractional derivative of higher order;

-         The Tricomi problem for mixed parabolic-hyperbolic equation;

-         Differential games with Langenhop, Gronwall and non-stationary integral constraints on controls;

-         Classification of automorphisms, local and 2-local automorphisms of filiform Leibniz algebras, which are naturally graded and with a nulfiliform nil radical;

-         Dynamical systems of quadratic operators defined on finite-dimensional simplex of idempotent measures;

- The existence of weakly periodic Gibbs measures for the Ising model on a Cayley tree of order two.

- We find the explicit form of the unique TIGM and determine the regions where this measure is or is not extremal for fertile HC models with three states in the cases of hinge and wand graphs on the Cayley tree of order three. In addition, for the wand graph, we find the conditions under which the extremal measure is not unique on the third-order Cayley tree.

- A complete description of two-periodic Gibbs measures on the Cayley tree of orders two and three for HC model with two states is obtained and using the reconstruction method, the extremality of these measures in the area of their existence is proved.

- In the theory of differential games, the concept of Langenhop type constraint on controls is introduced. A relationship of the classes of control with Geometric, Langenhop and Integral constraint has been studied and for each case, attainability domains of objects have been defined. In the case of Langenhop constraint, a parallel pursuit strategy (П-strategy) is constructed for the pursuit problem of differential game and sufficient conditions of pursuit have been determined. In the evasion problem, the least lower boundary of the distance between players has been found.

- A unique solution of the boundary problem for higher-order fractional differential equations with the Riemann-Liuoville derivative ina rectangular domain has been found.

- Conditions on solvability of the Dirichlet type problem for higher-order mixed type equation with variable coefficients have been found.

-Studied the probability of the first exit from a strip through its upper boundary  for the stationary stochastic processes with independent increments. Found the two-sided inequalities for this probability under various conditions on the process.

 - it is proved that every 2-local derivation on a finite dimensional semisimple Jordan algebra over an algebraically closed field of characteristic different from 2 is a derivation;

- An infinite Peirce decomposition of the algebra of compact operators on an infinite dimensional separable Gilbert space is constructed, using the norm and an analog of the criterion of the fundamentality of a sequence. It is proved that this infinite Peirce decomposition is  a  C*-algebra.

- it is proved that every local and 2-local derivation on conservative algebras of 2-dimensional algebras are derivations. Also, we prove that every local and 2-local automorphism on conservative algebras of 2-dimensional algebras are automorphisms;

- Unique solvability of the Tricomi problem for mixed parabolic-hyperbolic type equation in a space has been proved.

-For the third-order equation with repeated characteristics using the construction of Green function the uniqueness of solution of the second boundary problem is shown.

- The uniqueness of solution of the first boundary problem is proved for the elliptic equation with three singular coefficient in a parallelepiped. The existence of solution of the problem  is proved imposed by the energy integral.

- In proving the existence of solution of the problem, the spectral method of Fourier relying on separation of variables has been applied. A solution of the set problem is constructed by the doubled series of Fourier-Bessel. The asymptotic methods are used to prove smooth convergence. Obtained estimation allows to prove convergence of the series and that of its derivatives up to the second order and also to prove the existence theorem in the class of regular solutions of that equation.