# Session 15: Complex Analyses

16.05.2019 | Time: 10:00-13:00 | NUUz, Tashkent | Conference Room 321, Block C

Coordinator: Prof. Azimbay Sadullaev (NUUz)

**Oscillatory integrals and Weierstrass polynomials** – Prof. Azimbay Sadullaev

Abstract

The well-known Weierstrass theorem states that if \( {f(z,w)} \) is holomorphic at a point \( {(z^0,w^0) \in \mathbb{C}^n_z \times \mathbb{C}_w} \) and \( {f(z^0,w^0)=0} \), but \( {f(z^0,w) \neq 0} \), then in some neighborhood \( {U=VW} \) of this point f is represented as

\begin{align*} {f(z,w) = [(w-w^0)^m +c\,_{m-1} (z)(w-w^0\;)\,^{m-1} + ... + c_0\, (z)] \varphi (z,w)} \end{align*} *where \( {c_k(z)} \) are holomorphic in \( {V} \) and \( {\varphi(z,w)} \) is holomorphic in \( {U} \), \( {\varphi (z,w) \neq 0} \), \( {(z,w) \in U} \).*

In recent years, the Weierstrass representation has found a number of applications in the

theory of oscillatory integrals. Using a version of Weierstrass representation the first author

obtain a solution of famous Sogge -Stein problem. He obtained also close to a sharp bound for

maximal operators associated to analytic hypersurfaces.

In the obtained results the phase function is an analytic function at a fixed critical point

without requiring a condition \( {f(z^0,w) \neq 0} \). It is natural to expect the validity of Weierstrass

theorem without requiring a condition \( {f(z^0,w) \neq 0} \), in form \begin{align*} {f(z,w) = [c\,_m(z) | (w-w^0)^m +c\,_{m-1} (z)(w-w^0\;)\,^{m-1} + ... + c_0\, (z)] \varphi (z,w)} \end{align*} Such kind of results may be useful to studying of the oscillatory integrals and in estimates

for maximal operators on a Lebesgue spaces. However, the well-known Osgood counterexample

shows that when \( {n>1} \) it is not always possible. We will show, that there is a global option, a

global multidimensional (in \( {w} \)) analogue of this theorem is proved without the condition \( {f(z^0,w) \neq 0} \).

**Cooperation Technology of Higher Education Influence of Framework Competitive Cooperation** – Dr. Kadirova Munira Rasulovna

Abstract

This article highlights the technology of family cooperation in upgrading professional skills of future professionals in higher education institutions, in which the role of the family in the education of young people today, the technology of establishing collaborative links in the formation of their professional competence, and the relationships with the family's higher education institution in improving professional competence in foreign higher education institutions analysis of the results of research and the factors that influence the quality of education imposed.

**Isometries of Ideals of Compact Operators** – Dr. Vladimir Ivanovich Chilin

Abstract

Let \( {H} \) be a complex separable Hilbert space, and let \( {K(H)} \) be the \( {C^*} \) -algebra of all compact linear operators on \( {H} \). Let \( {C_p=\{x \in K(H): Tr(|x|^p < \infty \} } \) be p-th Schatten ideal of compact operators with the norm \( {||x||_p = (Tr(|x|^p)^{1/p}, 1 \leq p < \infty} \).

In 1975, J. Arazy gave the following description of all the surjective isometries of Schatten ideals

\( {C_p} \): If \( {V: C_p → C_p, 1 ≤ p < \infty , p \neq 2} \), is a surjective isometry, then there exist unitary operators \({u_1}\) and \({u_2}\) or anti-unitary operators \({v_1}\) and \({v_2}\) on \({H}\) such that either \( {Vx = u_1 \, x \, u_2} \) or \( {Vx = v_1 \, x^* \, v_2} \) for all \({x \in C_p}\). The Schatten ideals \({C_p}\) are examples of Banach symmetric ideals \( {(C_E, \; ||.||_{CE})} \) of compact operators associated with symmetric sequence spaces \( {(E, \; ||.||_E)} \). We give description of all surjective linear isometries acting in Banach symmetric ideals \( {(C_E, \; ||.||_{CE})} \) in the case when \( {(C_E, \; ||.||_{CE})} \) is a separable or a perfect Banach symmetric ideal and \({C_E \neq C_2}\).

**Significance of Mathematical Physics in the Development of Science and Technology** – Dr. Shavkat Alimov

Abstract

The knowledge of the impact of mathematical physics on the development of science, technology and engineering is important for a new generation of scientists and engineers. This is illustrated by some examples of problems of mathematical physics, the solution of which changed our lives and the whole world.

There is the belief that the successes of computer science put mathematical physics into the slow lane, but until now the solution of the most important problems depends on the progress of the mathematical approach. The methods of modern mathematical physics do not compete with computer science techniques but help them to reach more effective results. To clarify this statement, the examples of such close cooperation are given and important role of mathematical modelling is showed.

**Local Derivations on Measurable Operators and Commutatitivity** – Prof. Shavkat Ayupov

Abstract

Given a Hilbert space \({H}\) let \({B(H)}\) be the *-algebra of all bounded linear operators on \({H}\). Consider a von *Neumann algebra* \({M}\), i.e. a *-subalgebra of \({B(H)}\), containing the identity operator, and closed in the weak operator topology. Denote by \({S(M)}\) the *-algebra of all closed unbounded linear operators on \({H}\), which are “measurable” with respect to \({M}\). A linear operator \({d}\) on \({S(L)}\) is called a *derivation *if it satisfies the Leibniz identity \( {d(xy)=d(x)y+xd(y)} \) for all \({x,y}\) from \({S(L)}\). A linear operator \({D}\) on \({L(S)}\) is called *a local derivation* if for arbitrary \({x}\) from \({S(L)}\) there exist a derivation \({d_x}\) depending on \({x}\) such that \( {D(x)=d_x(x)} \).

*The first problem* which is discussed in the talk is a description of all derivations on the algebra \({S(M)}\). We recall that this long standing problem (almost 20 years) is solved for all types of von Neumann algebras except the type \({II_1}\) case.

*The second problem* related to a problems considered by R.Kadison and is devoted to local derivations on \({S(M)}\). We show that if \({M}\) has no direct abelian summands then every local derivation is a (global) derivation. In the case of abelian von Neumann algebra \({M}\) we give necessary and sufficient conditions on the algebra \({M}\) to admit local derivations which are not derivations.

Finally, *the third problem *arises from a discussion of local derivations on algebras with professor Efim Zelmanov in California University, San Diego. Earlier we proved that in the abelian case the square \({d^2}\) of each derivation on \({S(M)}\) is a local derivation. Naturally, the question arises whether this property characterizes abelian algebras. In this talk we show that this conjecture is true for algebras \({S(M)}\), but not for arbitrary algebras. Namely, we prove that the square of every derivation on \({S(M)}\) is a local derivation if and only if the von Neumann algebra \({M}\) is commutative. We also give an example of non-commutative associative algebra on which the square of every derivation is a local derivation.

**(Thermo)dynamic systems in biology and physics** – Prof. Utkir Abdulloyevich Rozikov

Abstract

We define several biological and physical dynamical systems and give their properties depending on time and temperature. Moreover we discuss some open problems.