Reports read at the annual meeting of the mathematics section of the academy of sciences of the USSR

Abstract

In the last decade, when mathematicians encountered the actual necessity of solving complicated applied problems, most often "nonrigorous," or the so-called heuristic, methods of solution were used.Very often these methods were used for the solution of multiextremal problems of optimization and problems of classification and prediction.The application of nonrigorous methods was stipulated with difficulties of the construction of precise mathematical models for the description of the processes under investigation (especially in the application of mathematics in weakly formalized domains of natural science), heterogeneity of information, describing real objects or processes. Therefore, a stage is produced perfectly naturally when for the investigation of a problem or a class of problems a nonrigorous, but substantially reasonable, method of solution and an algorithm based on it are suggested on the basis of plausible reasonings, and then the justification is carried out by performing a direct experiment with the problems. Because of high result-orientedness, the method and algorithm were put to use without constructing justifications with the universally recognized standard mathematical rigor. The application of these "plausible" procedures led to great practical success in the solution of many applied problems and was, consequently, completely regular. The heuristic methods were specially often applied for the solution of problems of classification and discrete prediction. These problems got the apt, in our view, name of "pattern recognition," coming from the first works on the automatic reading of texts and automatic perception of information from voice.Indeed, a wide class of problems approach this title, when according to some, usually very heterogeneous, information it is required to establish, whether the given objects possess a fixed finite collection of properties-the problems of classification, or according to the information about a finite set of sufficiently same-type processes it should be clarified, in which domain out of a finite number of domains (classes) will these processes be found over the period of time At. The problems of geological prospecting, prediction of properties of chemical compounds, the means of development of cities and regions, course of construction of large objects, etc. reduce to problems of this form. A mathematical description of the class of problems indicated above can be carried out in the following manner.

The surface chemistry of amorphous silica. Zhuravlev model

“…Point A indicates the maximum of kinetic curves 1 -8; points A i , the maxima of curves 9 -13 and 15 -17 (i =I, II, III,…VIII); curves 8 and 14 are boundary curves (see Table 1 and text). 303,306,311,329,332]. In our complex thermographic investigations of several different silica samples, we have detected endothermic peak at 150°C on a DTA curve [329], and this effect, as it noted before, is related to the removal of physically (molecularly) adsorbed water.…”

“…The versions of the method of deuterium exchange (DE method) developed by the author [304][305][306]311,312,321,322,328] have been used for determining the concentration of the hydroxyl groups on the surface of dispersed oxide adsorbents. The advantage of this method is that, under certain conditions, deuterium exchange is limited to the surface and does not involve structurally bound water inside silica.…”

“…One of the variants of this DE method [311,322] made it possible to determine the silanol number and simultaneously to study the kinetics of water adsorption and isotopic exchange between D 2 O and the surface OH groups. The minimal measured amount of water on the SiO 2 surface formed from 2 · 10 − 4 to 2 · 10 − 6 moles H 2 O with a relative error of 9 1-5% (the different versions).…”

“…The samples differed strongly in the following characteristics: specific surface area, types of pore, distribution of pores according to their size, and the density of packing of the particles. The study of such structures proved to be an important stage in our work [301][302][303][305][306][307][308][309][310][311][317][318][319][320][323][324][325][326][327]. To determine the structure of different samples, besides the adsorption of Kr and N 2 , we measured the total isotherms of adsorption and desorption -including capillary condensation, of water vapor, methanol and benzene, we used the method of apparent density, the mercury-porosimetric method, and the electron-microscopic and the kinetic methods.…”

“…In those cases where the biporous samples of SiO 2 contained very narrow pores (ultramicropores, dB 1 nm) in addition to wide pores (mesopores) [307,311,[318][319][320]331], the samples were considered only to be wide-pore ones, and the very narrow pores were excluded from consideration in determining h OH (see above).…”