Vol 27, No 140 (2022)

The work is devoted to a survey of known results related to the study of derivations in group algebras, bimodules and other algebraic structures, as well as to various generalizations and variations of this problem. A review of results on derivations in $L_1\left(G\right)$ algebras, in von Neumann algebras, and in Banach bimodules is given. Algebraic problems are discussed, in particular, derivations in groups, $(\sigma ,\tau )$-derivations, and the Fox calculus. A review of some results related to the application to pseudodifferential operators and the construction of the symbolic calculus is also given. In conclusion, some results related to the description of derivations as characters on the groupoid of the adjoint action are described. Some applications are also described: to coding theory, the theory of ends of metric spaces, and rough geometry.

The paper considers a hyperbolic system of two first-order partial differential equations with constant coefficients, one of which is nonlinear and contains the square of one of the unknown functions. Moreover, each equation contains two unknown functions which in turn depend on two variables. Exact solutions are found for this system: a traveling wave solution and a self-similar solution. There is also defined the type of initial-boundary conditions which allow to use the constructed general solutions in order to write out a solution of the initial-boundary value problem for the system of differential equations under consideration.

We consider a regular parametric nonlinear (nonconvex) problem for constrained extremum with an operator equality constraint and a finite number of functional inequality constraints. The constraints of the problem contain additive parameters, which makes it possible to use the apparatus of the “nonlinear” perturbation method for its study. The set of admissible elements of the problem is a complete metric space, and the problem itself may not have a solution. The regularity of the problem is understood in the sense that it has a generalized Kuhn-Tucker vector. Within the framework of the ideology of the Lagrange multiplier method, a regularized nondifferential Kuhn-Tucker theorem is formulated and proved, the main purpose of which is the stable generation of generalized minimizing sequences in the problem under consideration. These minimizing sequences are constructed from subminimals (minimals) of the modified Lagrange function taken at the values of the dual variable generated by the corresponding regularization procedure for the dual problem. The construction of the modified Lagrange function is a direct consequence of the subdifferential properties of a lower semicontinuous and, generally speaking, nonconvex value function as a function of the problem parameters. The regularized Kuhn-Tucker theorem “overcomes” the instability properties of its classical counterpart, is a regularizing algorithm, and serves as a theoretical basis for creating algorithms of practical solving problems for constrained extremum.

We investigate population dynamics models given by difference equations with stochastic parameters. In the absence of harvesting, the development of the population at time points $k=1,2,\ldots$ is given by the equation $X(k+1)=f\big(X(k)\big),$ where $X(k)$ is amount of renewable resource, $f(x)$ is a real differentiable function. It is assumed that at times $k=1,2,\ldots$ a random fraction $\omega\in[0,1]$ of the population is harvested. The harvesting process stops when at the moment $k$ the share of the collected resource becomes  greater than a certain value $u(k)\in[0,1),$ in order to save a part of the population for reproduction and to increase the size of the next harvest. In this case, the share of the extracted resource is equal to $\ell(k)=\min\big\{\omega(k),u(k)\big\}, k=1,2,\ldots.$ Then the model of the exploited population has the form
$X(k+1)=f\left(\right(1-l\left(k\right) \left)X\right(k) ),k=1,2,...,$
where $x(0)$ is the initial population size.
For the stochastic population model, we study the problem of choosing a control $\overline{u}=(u(1),\ldots,u(k),\ldots)$ that limits at each time moment $k$ the share
of the extracted resource and under which the limit of the average
time profit function
$H(\overline{l},x(0\left)\right)\doteq \underset{n\to \infty }{lim}\sum _{k=1}^{n}X\left(k\right)l\left(k\right), где \overline{l}\doteq \left(l\right(1),\dots ,l(k),\dots )$
exists and can be estimated from below with probability one
by as a large number as possible.
If the equation $X(k+1)=f\big(X(k)\big)$ has a solution of the form $X(k)\equiv x^*,$
then this solution is called the equilibrium position of the equation.
For any $k=1,2,\ldots,$ we consider random variables $A(k+1,x)=f\bigl((1-\ell(k))A(k,x)\bigr),$ $B(k+1,x^*)=f\bigl((1-\ell(k))B(k,x^*)\bigr)$; here $A(1,x)=f(x),$  $B(1,x^*)=x^*.$
It is shown that when certain conditions are met, there exists a control $\overline{u}$
under which there holds the estimate of the average time profit
$\frac{1}{m}\sum _{k=1}^{m}M\left(A\right(k,x\left)l\right(k\left)\right)\le H(\overline{l},x(0\left)\right)\le \frac{1}{m}\sum _{k=1}^{m}M\left(B\right(k,x^{*}\left)l\right(k\left)\right),$
where $M$ denotes the mathematical expectation.
In addition, the conditions for the existence of control $\overline{u}$ are obtained
under which there exists, with probability one, a positive limit to the
average time profit equal to
$H(\overline{\ell},x(0)) =$
\lim\limits_{k\to\infty} MA(k,x)\ell(k) =
\lim\limits_{k\to\infty} MB(k,x^*)\ell(k).