Mathematical modelling of the development of mechanical instability in fracturing systems

The dynamics of processes accompanying a loss of stability in a mechanical system are investigated. The mechanical system is in the form of an elastic rod, stretched by an axial load, with one of its lateral surfaces “glued” to a rigid wall. The “glue” is a low-strength elastic material which is subject to brittle fracture at a certain value of the load acting on it. In a fractured segment, the rod surface slides over the wall surface under the action of a dry friction force which is less than the breaking stress. The high sensitivity of the process of the development of instability to small perturbations which initiate the development of instability is established. The system considered is the simplest model of the zone of contact between lithospheric plates which generate earthquakes.     

On behavior of Fourier coefficients by Walsh system

The paper proves that for any ε > 0 there exists ameasurable set E ? [0, 1] with measure |E| > 1 ? ε such that for each f ∈ L¹[0, 1] there is a function \(\tilde f \in {L^1}\left[ {0,1} \right]\) coinciding with f on E whose Fourier-Walsh series converges to \(\tilde f\) in L¹[0, 1]-norm, and the sequence \(\left\{ {\left| {{c_k}\left( {\tilde f} \right)} \right|} \right\}_{n = 0}^\infty \) is monotonically decreasing, where \(\left\{ {{c_k}\left( {\tilde f} \right)} \right\}\) is the sequence of Fourier-Walsh coefficients of \(\left\{ {\left| {{c_k}\left( {\tilde f} \right)} \right|} \right\}_{n = 0}^\infty \).     

Nonlinear approximation of functions from the class L r with respect to the Vilenkin system

In this paper we prove that for any function f from the class L ^r on [0, 1) one can find a function g from the same class (which differs from f on a set of arbitrarily small measure) whose greedy algorithm with respect to the Vilenkin system converges to f.     

Necessary conditions and a test for the stability of a system of two linear ordinary differential equations of the first order

We use the Riccati equation method for the derivation of necessary conditions and a test for the stability of a system of two linear first-order ordinary differential equations. We consider an example in which our results are compared with the results obtained by the Lyapunov and Bogdanov methods by estimating the norms of solutions via Lozinskii logarithmic norms, and by the freezing method.     

Almost Everywhere Convergence of Greedy Algorithm with Respect to Vilenkin System

In this paper, we prove that for any ε ∈ (0, 1) there exists ameasurable set E ∈ [0, 1) with measure |E| > 1 ? ε such that for any function f ∈ L¹[0, 1), it is possible to construct a function \(\tilde f \in {L^1}[0,1]\) coinciding with f on E and satisfying \(\int_0^1 {|\tilde f(x) - f(x)|dx < \varepsilon } \), such that both the Fourier series and the greedy algorithm of \(\tilde f\) with respect to a bounded Vilenkin system are almost everywhere convergent on [0, 1).     

Quasiunconditional Basis Property of the Faber–Schauder System

Ukrainian Mathematical Journal - We prove that, for any 0 &lt; ?? &lt; 1, there exists a measurable set E?? ? [0, 1], mes (E??) &gt; 1...     

Formation of the crystal structure of intermetallic compounds in mechanically activated Ni-Al and Ti-Al systems during self-propagating high-temperature synthesis

The dynamics of structural and phase transitions during heterogeneous reaction in mechanically activated mixtures has been investigated by X-ray diffraction, scanning electron microscopy, and dynamic synchrotron radiation diffraction. It is shown that the mechanisms of formation of the structure of reaction products are significantly different, depending on the temperature mode of the synthesis. Upon slow heating, the process is multistage and includes several intermediate crystalline phases. Upon fast heating, the transformation in the combustion wave occurs much faster, with only one intermediate phase observed.     

Specific features of the distribution of protons at small altitudes in near-equatorial latitudes according to data of satellite experiments. I. Energy spectrum

The spectra of protons with energies from 10 keV to 1 MeV as measured in experiments conducted at satellites and the MIR orbital station are presented. These data were obtained in the area of the geomagnetic equator at L < 1.15 and at altitudes up to 1000 km. The proton spectrum was determined during periods of geomagnetic perturbations and quite periods. These spectra were approximated using a kappa-function. The proton spectra in the near-equatorial area and in the ring current are compared. The proton’s life time in the near-equatorial area is estimated, taking into account charge exchange and Coulomb scattering. It is confirmed that the ring current is a main source of protons in the near-equatorial area at low altitudes.