In the paper the formulas for an evaluation of a minimum quantity of moving as a solution of a known puzzle "The Hanoi towers" are obtained. If $R$ - is a number of rings ($22$) and $Y(R,P)$ - is a minimum number of moving of rings of the tower, necessary for transposition, then \\ $$Y(R,P)=\sum_{i=0}^{k(R,P)}2^i{{P-3+i}\choose{P-3}}+2^{1+k(R,P)}(R-{{P-2+k(R,P)}\choose{P-2}}) $$ \begin{eqnarray} \mbox{where,} \quad k(R,P) = \max \left \{i|R- {{P-2+i} \choose {P-2}} \geq 0 \right \}.\nonumber \end{eqnarray} Summing up according to $Y(R,P)$ is done(formula becomes by more bulky) . The recommendations for optimal algorithm of transfer of the tower are given. 2000 Mathematics Subject Classification: 05А10. Top of the page.

The all solutions of indeterminate interpolation problems in the Stieltjes class are described by using the linear fractional transformations of the Stieltjes pairs. In this article the all solutions of the general interpolation problem in the Stieltjes class in completely indeterminate case are considered. 2000 Mathematics Subject Classification: 47A57, 42A82. Top of the page.

By the Benyamini-Lin $L_p$ theorem for $p \ne 1,2,\infty$ there exists a function $\psi_p : (0, +\infty) \to (0, +\infty)$ such that for every compact operator $T : L_p \to L_p $ holds $\| I+ T \| \ge 1 + \psi_p (\|T\| ).$ We give another proof of the Benyamini-Lin theorem, which enables us to prove that the Benyamini-Lin $L_p$-inequality ``tends" to the Daugavet equation as $p \to 1$, or more precisely that $\psi_p (t) \to t$ as $p \to 1$, where $\psi_p$ is the best possible function in the Benyamini-Lin inequality. 2000 Mathematics Subject Classification: 46B20. Top of the page.

We introduce the notion of a \sbp\ (SBP) which is a natural extension of the \dpr, give its geometric characterization, and show that a 1-unconditional sum of a sequence of Banach spaces is a \sbp\ if and only if every summand is. Then we deduce that this notion is not equivalent to the \dpr. 2000 Mathematics Subject Classification: 46B20. Top of the page.

Combined usage of iterational, structural and variational methods for solving direct and inverse non-linear heat transfer problems is proposed. Indeterminate components of approximate analytical structures of solving heat transfer problems are determined by iteration methods with solving the corresponding systems of algebraic equations. Results of solving direct and inverse non-linear heat transfer problems are presented. The results of solving an inverse heat transfer problem are compared with data for the test problem. 2000 Mathematics Subject Classification: 37M05. Top of the page.

In the paper the formulas for the reconstruction of a nonlinear control system which is a realization of the series of nonlinear power moments are given. The essential role is played by the properties of the algebra of nonlinear power moments. 2000 Mathematics Subject Classification: 93B15, 93B25. Top of the page.

We investigate functions on a tube domain that are almost periodic with respect to Besicovitch's metric. 2000 Mathematics Subject Classification: 42A75, 30B50. Top of the page.

In this note we construct a Hilbert space of analytic functions in the unit disk in which all point evaluations are not bounded. This result is a refinement of a recent construction by D.~Alpey and T.~M.~Mills who constructed a Hilbert space of functions in which there is a countable set of point evaluations that are unbounded. 2000 Mathematics Subject Classification: 42A70. Top of the page.

In this paper we receive the approximation of the solution set of linear impulsive differential inclusion with the help of the system of linear impulsive differential equations with the Hukuhara's derivative, that is equivalent to two systems of ordinary differential equations with impulses. The model examples are constructed. 2000 Mathematics Subject Classification: 34A37, 34A60. Top of the page.

Model representations for a class of operator $\left\{ A,B\right\} $ system specified in Hilbert space $H$ with certain properties of commutator \ \ $\left[ A,B\right] ,$\ $\left[ A^{\ast},B\right] $ are derived. The modes of the system of operators $\left[ A,B\right] $ for this class are integration operators by independent variables in $L^{2}\left( D_{n,m}% \right) .$This work is a continuation of investigations in the theory of constructing multidimensional triangular representations for linear operator systems [1] when the commutators $C=[A^{\ast},B]$ and $D=[A,B]$ are nilpotent, $C^{n+1}=0$, $D^{m+1}=0$ $(m,n\in \mathbb N ).$ 2000 Mathematics Subject Classification: 47A45, 47D25, 47BXX. Top of the page.

Results of comparative research of separation of a pressure wave that is propagated in a viscoelastic tube filled with a liquid, into two components – forward and backward waves are given. The problem concerns investigation of pressure waves in arteries. The parameters of the forward and backward waves carry information on state (normal or pathological) of the heart and the downstream arterial bed respectively. In the framework of the quasi one-dimensional and axisymmetric flow models the comparative study of the parameters of the waves is carried out. It is shown, that the separation method gives characteristic dependences of resistive and capacitor properties of the vascular bed on frequency of the wave. At that time different models give curves which slightly differ from one another. Thus, biomechanical interpretation of the dependences with the aim of receiving diagnostically important information can be carried out on the basis of the simple 1d-models. 2000 Mathematics Subject Classification: 76D33, 76Z05, 74F10, 92C35. Top of the page.

The processes of magnetic particles aggregation in a layer of a magnetic fluids which is in equilibrium in the presence of steady homogeneous mag-netic field are studied. It is shown that the equations determining evolution of a magnetic state of the fluid have the solutions such as hot and cold auto-solitons. These solutions describe generation of a sub-layer more or less ag-gregated continuum in the fluid layer between nonmagnetic plates. 2000 Mathematics Subject Classification: 76W05. Top of the page.

The homogeneous Lame equation in an elastic semi-space with a circular cylindrical cavity is solved by means of the generalized Fourier method in case of the displacements are given on boundary surfaces. The distribution of normal and tangential stresses is studied for the concrete parameter values and the boundary functions. 2000 Mathematics Subject Classification: 35J25. Top of the page.

The auxiliary differential equations of the first and second orders are deduced from boundary conditions for the fields and Debye potentials. It is shown that these equations are the degenerated equations for associated Legendre functions. Solutions of these equations are found and analyzed. As the result we obtain the simplified non-connected boundary conditions for Debye potentials. It is shown that this approach permits, in some sense, to decrease of the computational dimension of the diffraction problem. 2000 Mathematics Subject Classification: 65N12, 35A25,78A45. Top of the page.

Algebraic structures generated by the presentation of affine system as the series of nonlinear power moments are considered. Free-generated gradings of the algebra of nonlinear power moments correspond to asymptotically different controls. The method of construction canonical form for nonlinear control system with asymptotically different controls is proposed. 2000 Mathematics Subject Classification: 93B10, 93B25. Top of the page.

In this paper some complete minimal surfaces in $3$-dimensional Heisenberg group $Nil$ are considered. Stability of analogues of planes and exponential images of planes in the Lie algebra and instability of $SO(2)-$invariant surfaces which are analogues of catenoid are proved. 2000 Mathematics Subject Classification: 53C40,53C42. Top of the page.

We study the existence problem for totally geodesic contact structures on 3-dimensional compact geometries. It was shown, that compact geometries of non-positive Ricci curvature don't admit totally geodesic contact structures. In the case of Lie groups $S^3$, $\widetilde{SL_2(\mathbb{R})}$ and $Nil$, horizontal distributions were shown to be totally geodesic contact structures. It was also shown, that there are no totally geodesic contact structures on $S^2 \times \mathbb{R}$. 2000 Mathematics Subject Classification: 53С15. Top of the page.

We propose the solution of the synthesis problem for a linear autonomous control system with the multidimensional control which is based on the controllability function method. The examples are given. 2000 Mathematics Subject Classification: 93B05, 93B50, 93B52, 93C05. Top of the page.

The article examines the ring of vectors defined with circular matrixes. It find sufficient conditions of existence of solution of Burger's nonlinear evolutionary equation in that ring. 2000 Mathematics Subject Classification: 35Q53. Top of the page.

The functional models for the Lie algebra of the linear limited operators $A_1,$ $ A_2$ have been built in the case $ A_1x_1 + A_2x_2 \; $ is not a dissipative operator for every $ x_1, x_2 \in R, $ and when $\dim G = 2,$ where G equals next $G = {\rm span} (A_k - A_k^*)h,$ $k = 1,2;$ $h\in H.$2000 Mathematics Subject Classification: 47A45. Top of the page.

Estimates for a sum of series $\sum\limits_{a\neq\infty}\widehat{\beta}(a,f)$ are obtained in term of the Valiron defect of a derivative at zero for a meromorphic functions in the disk of finite lower order. 2000 Mathematics Subject Classification: 30D30, 30D35. Top of the page.