On modeling bodies with delaminating coatings taking into account the fields of prestresses

The paper presents a model of steady-state oscillations of an inhomogeneous body with a prestressed exfoliating coating based on a general linearized statement of the problem of the motion of a prestressed-strained elastic body. On its basis, the statement of the problem of oscillations of an inhomogeneous strip consisting of a substrate and a prestressed coating is formulated, between which there is a delamination in a certain region. Steady oscillations are caused by a load applied to the upper boundary of the coating. To calculate the oscillations of the two-dimensional structure under consideration, the Fourier transform in the longitudinal coordinate was used and the original problem was reduced to solving a number of auxiliary boundary value problems with respect to transformants of the desired functions. From the conditions that the stress functions vanish (the cover is modeled as a mathematical section) of the substrate and the coating, the operator relations are constructed in the area of delamination to calculate the opening functions. The kernels of these operator relations are singular and are integrals over an infinite interval. A study was made of the behavior of their integrands at infinity, on the basis of which special approaches were used to calculate the kernels. As a result of solving the obtained hypersingular equations with difference kernels, for which the collocation method is used, the originals of the disclosure functions are constructed. Using a similar approach for inverting the Fourier transformations, we constructed relations to calculate the originals of the displacement functions at the upper boundary of the coverage. Based on the computational experiments, an analysis is made of the influence of the initial geometric and mechanical parameters of the substrate and coating on the values of the disclosure functions in the delamination region and the displacement functions at the upper boundary of the layer. The influence of the prestress level on the amplitude-frequency characteristics (AFC) was also investigated. It was found that the most significant effect on the frequency response is in the vicinity of the frequencies of the thick resonances. Based on the information on the displacement fields, it is possible to construct schemes for identifying delamination characteristics.

ORGANIZATION OF THE MATHEMATICAL CYCLE DISCIPLINES PROPAEDEUTIC STUDY OF NATURAL SCIENTIFIC AND ENGINEERING DIRECTIONS

Currently, the main trends in the reform of higher education naturally lead to a significant reduction of the part of classroom studies in the educational process, which is extremely negative for the study of the mathematical cycle disciplines due to their abstraction and often cumbersome presentation of the material. Thus, there is a need to increase the effectiveness of contact work and to strengthen the role of students’ independent work. One of the factors contributing to an increase in the effectiveness of contact work is independent propaedeutic consideration of the material under study. The approach to organizing extracurricular independent work in the propaedeutic study of the mathematical cycle disciplines by students of natural sciences and engineering should include the following steps: recalling the facts previously studied, presenting the natural science (engineering) applications in their historical retrospective and presenting the main ideas of this section. The latter is supposed to be performed the most clearly, with the maximum use of interdiscipline relations. To increase the effectiveness of propaedeutic independent work, it is advisable to use appropriate methodological support. As the latter (depending on the section under study), you can use interactive study guide integrated with applications written in various high-level languages, presentations, educational films and study guide in written form. Thus, the preliminary acquaintance with the new mathematical section should be during extracurricular independent work with the widespread use of interactive teaching technologies.

Temperature Variations Analysis for Condensed Matter Micro- and Nanoparticles Combustion Burning in Gaseous Oxidizing Media by DTM and BPES

Combustion process for iron particles burning in the gaseous oxidizing medium is investigated using the Boubaker polynomial expansion scheme (BPES) and the differential transformation method (DTM). Effects of thermal radiation from the external surface of burning particle and alterations of density of iron particle with temperature are considered. The solutions obtained using BPES technique and DTM are compared with those of the fourth-order Runge-Kutta numerical method. Results reveal that BPES is more accurate and reliable method than DTM. Also the effects of some physical parameters that appeared in mathematical section on temperature variations of particles as a function of time are studied.

Ivan Matveevich Vinogradov, 14 September 1891 - 20 March 1983

Ivan Matveevich Vinogradov was born on 14 September (New Style) 1891. His father Matveǐ Avraam’evich was the priest of the village church ( pogost ) of Milolyub in the Velikie Luki district of Pskov province in western Russia. His mother was a teacher. From an early age he showed an aptitude for drawing and, instead of an ecclesiastical school (as would have been normal for a son of the clergy), his parents sent him in 1903 to the modern school ( real´noe uchilishche : i.e. one with a scientific as opposed to a classical orientation) in Velikie Luki, whither his father had moved with his family on his translation to the Church of the Holy Shroud ( Pokrovskaya Tserkov ’) there. In 1910 on completing school, Vinogradov entered the mathematical section of the Physico-mathematical Faculty of the University at the Imperial capital, St Petersburg. Among the staff were A. A. Markov, whose lectures on probability he is said to have known by heart, and Ya. V. Uspenskiǐ ( = J. V. Uspensky, later of Stanford University, U.S.A.), both with interests in number theory and probability theory. There had been a long tradition in these subjects (Chebyshëv in both; Korkin, Zolotarëv and Voronoǐ in number theory). Vinogradov was attracted to number theory and showed such ability that on completing the course in 1914 he was retained at the University for training as an academic. He successfully completed the extensive Master’s examination and in 1915, on the initiative of V. A. Steklov, was awarded a bursary.

An Epigraphical Note-book of Sir Arthur Evans

Among the papers of the late Sir Arthur Evans was found a note-book, measuring 5¾ X 3⅝ X ⅝ in., bound in red leather and with a metal clasp, comprising 122 pages, inclusive of the inner sides of the cover. Sixteen of these are blank, eight contain notes on Greek coinage and numerals, a list of the Attic tribes, bibliographical references and some mathematical problems, and the remainder bear copies, in Evans' characteristically microscopic handwriting, of a large number of Greek inscriptions, with brief notes added in many cases. All are written in pencil save the mathematical section, which is in ink and appears to be in a different hand. At the request of Mr. E. Thurlow Leeds, Keeper of the Ashmolean Museum, I have examined the copies of inscriptions and find that, with very few exceptions, the originals are in the British Museum. These exceptions are IG i2. 929, 11. 1-5 (in Paris), IG iii. 1418, IG ii.2 3765 and CIG 3333 (all three in the Ashmolean Museum). Of the rest 105 appear in the Collection of Ancient Greek Inscriptions in the British Museum; 91 of them are Attic, three (Nos. 159, 160, 162) Boeotian, three (Nos. 373, 375, 376) Tenian, five (Nos. 1003, 1012, 1022-4) Anatolian, one (No. 1398) Italian, and two (Nos. 1107,1123 a) of uncertain provenance.

IV. On the logocyclic curve, and the geometrical origin of logarithms

In a paper read before the Mathematical Section of the British Association during its meeting at Cheltenham in 1856, and which was printed among the reports for that year, I developed at some length the geometrical origin of logarithms, and showed that a trigonometry exists as well for the parabola as for the circle, and that every formula in the latter may be translated into another which shall indicate some property of parabolic arcs analogous to that from which it has been derived. I showed, moreover, that the hole theory of logarithms was founded on a basis as purely geometrical as the trigonometry of the circle, and I pointed out how the imaginary expressions in the latter—such as DeMoivre’s theorem—have real counterparts in the trigonometry of the parabola. As the principle of duality is of the widest application in geometrical investigation, and as every property of circumscribed space has its dual, it would be strange if the dual of circular trigonometry had no existence. In the paper.to which I have referred, I showed how, by the help of certain arbitrary lines drawn about the parabola, numbers and their logarithms might be exhibited. But as there was something conventional in this representation, I was not quite satisfied with the construction. I suspected that the geometrical theory of logarithms was just as little conventional as the trigonometry of the circle. With this view I have again lately considered the whole subject, and by the help of a new curve, which I have called the Logocyclic Curve , from the similarity of many of its properties to those of the circle, and from its use in representing numbers and their logarithms , I have succeeded in exhibiting the whole theory of logarithms in a geometrical form as complete as it is comprehensive, and simple as it is beautiful.

III. On the application of parabolic trigonometry to the investigation of the properties of the common catenary

Some time ago, on the publication of a paper read by me last summer at Cheltenham before the Mathematical Section of the British Association on Parabolic Trigonometry and the Geometrical origin of Logarithms, Sir John Herschel called my attention to the analogy which exists between the equation of the common catenary referred to rectangular coordinates, and one of the principal formulæ of parabolic trigonometry. Since that time I have partially investigated the subject, and find, on a very cursory examination, that the most curious analogies exist between the properties of the parabola and those of the catenary,—that in general for every property of the former a corresponding one may be discovered for the latter. In this paper I cannot do more than give a mere outline of these investigations, but I hope at some future time, when less occupied with other avocations than at present, I may be permitted to resume the subject. I will only add, that the properties of this curve appear to be as inexhaustible as those of the circle or any other conic section.