卷 223, 编号 1 (2017)

In the present paper, we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, ω -boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) topological Brandt \( {\lambda}_i^0 \) -extensions of semitopological monoids with zero. In particular, we show that if \( \left\{\left({B}_{\uplambda_i}^0\left({S}_i\right),\kern0.5em {\uptau}_{B\left({S}_i\right)}^0\right):i\in \mathrm{\mathcal{I}}\right\} \) is a family of Hausdorff pseudocompact topological Brandt \( {\uplambda}_i^0 \) -extensions of pseudocompact semitopological monoids with zero such that the Tychonoff product \( \prod \left\{{S}_i:i\in \mathrm{\mathcal{I}}\right\} \) is a pseudocompact space, then the direct product \( \prod \left\{\left({B}_{\uplambda_i}^0\left({S}_i\right),\kern0.5em {\uptau}_{B\left({S}_i\right)}^0\right):i\in \mathrm{\mathcal{I}}\right\} \) endowed with the Tychonoff topology is a Hausdorff pseudocompact semitopological semigroup.

For matrix linear equations AX + BY = C and AX + YB = C over quadratic rings \( \mathbb{Z}\left[\sqrt{k}\right] \), we establish necessary and sufficient conditions for the existence of integer solutions, i.e., solutions X and Y over the ring of integers \( \mathbb{Z} \). We also present the criteria of uniqueness of the integer solutions of these equations and the method for their construction.

On the basis of the three-dimensional elasticity theory, we study a problem of free axisymmetric vibrations of inhomogeneous hollow cylinders of finite length made of functionally graded materials under different boundary conditions imposed on their end faces. The elastic properties of the material continuously vary in the radial direction. We propose a numerical-analytic approach for the solution of this problem. The original problem of the elasticity theory containing partial differential equations is reduced to a boundary-value problem for the systems of ordinary differential equations of high order in the radial coordinate by using spline approximations and collocation methods. The obtained one-dimensional problem is solved by using a stable numerical method of discrete orthogonalization together with the method of step-by-step search. We also present the results of determination of the frequencies and modes of vibrations of the cylinder made of a functionally graded material obtained as a composition of stainless steel and nickel for different types of boundary conditions imposed at the end faces and different values of temperature.