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For the system of equations of the integral theory of a fire in a room with open embrasures, we obtain comparison theorems that permit one to obtain relationships between solutions corresponding to various ignition conditions and hence compare the values of dangerous factors arising in fires under such conditions. These results are used for the practical problem of the replacement of a complicated system of embrasures by a single rectangular embrasure without diminishing the level of the fire danger. [ABSTRACT FROM AUTHOR]

The article discusses the instability theorem for a class of eighth-order differential equations. The article starts with an introduction and statement of results. An assertion is presented as the main result of the study. The author use the Lyapunov function in the proof of the theorem. The main properties of the functions imply that the zero solution of system is unstable.

Illustrates error estimates in the Galerkin numerical method for an abstract second-order evolution equation with nonsmooth right-hand side. Consideration of the Cauchy problem for the second-order abstract evloution equation; Original conditions of the Galerkin method; Presentation of integral theorems.

Illustrates the steepest descent numerical method with an adaptive alternating-triangular preconditioner. Consideration of an operator equation in a definite-dimensional Hilbert space; Optimization of the stationary modified alternating-triangular method with the use of a priori information; Presentation of integral theorems.

Discusses a two-sided functional-discrete method for second-order differential equations with general boundary conditions. Illustration of the Green function; Presentation of integral theorems; Examples of boundary conditions.

Illustrates a two-sided approximation of eigenvalues of some elliptic operators. Estimation of eigenvalue error; Process of obtaining upper and lower eigenvalue bounds; Presentation of integral theorems.

This article presents a solution for a hyperbolic equation with a transmission condition for limits of fractional derivatives. Theorems used for the solution are provided in the article along with their derivations.

Discusses Dirac operators generated on a finite interval by differential equations. Illustration of differential expressions; Construction of minimal and maximal operators; Analysis of the Sturm-Liouville operator with singular coefficients; Presentation of integral theorems.