Some results of the local theory of smooth functions

We present results of the investigation of the local behavior of smooth functions in neighborhoods of their regular and critical points and prove theorems on the mean values of the functions considered similar to the Lagrange finite-increments theorem. We also study the symmetry of the derivative of an analytic function in the neighborhood of its multiple zero, prove new statements of the Weierstrass preparation theorem related to the critical point of a smooth function with finite smoothness, determine a nongradient vector field of a function in the neighborhood of its critical point, and consider one critical case of stability of an equilibrium position of a nonlinear system. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 231–267, February, 2007.     

Dynamic systems inJ m× En

Expanded text of paper read by the author to a meeting of the Kiev Mathematical Society, February 26, 1991.     

Existence and uniqueness of bounded (on the whole axis) solutions of systems of nonlinear differential-difference equations of neutral type

The existence and uniqueness of the bounded (on the whole axis) solution of a certain class of systems of nonlinear differential-difference equations of neutral type are proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 390–394, March, 1991.     

Phase Diagram and Properties of Glasses of the System Sb<Subscript>2</Subscript>S<Subscript>3</Subscript>-AgI

The phase diagram of the system Sb₂S₃-AgI was constructed from data of the differential thermal and X-ray phase analyses. The physicochemical properties of glassy and crystalline alloys were studied.     

1,2,4-Triazine and triazole derivatives from α-keto acids and thiosemicarbazides

Thiosemicarbazides of 2-hydroxyphenylglyoxylic acid, which, depending on the conditions, undergo cyclization to interconvertible 5-(2-hydroxyphenyl)-6-oxo-1,2,4-triazine-3-thiones or (2-hydroxy-benzoyl)-²-1,2,4-triazoline-5-thiones, are formed by the action of thiosemicarbazide and its homologs on coumarindione. The thiosemicarbazones of 2-hydroxyphenylglyoxlyic and other 2-substituted phenylglyoxylic acids undergo cyclization to 6-(2-hydroxyphenyl)-5-oxo-1,2,4-triazine-3-thiones.Deceased.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 1, pp, 116–121, January, 1977.     

Synthesis and Heterocyclization of 3-Arylaminothiocrotonanilides

Reactions of acetylthioacetanilide with arylamines in acetic acid in the presence of sodium acetate give 3-arylaminothiocrotonanilides in good yields. When treated with -bromoacetophenone in acetone, these products are converted to substituted 4-hydroxy-²-thiazolinium bromides, one of which was dehydrated to obtain the corresponding thiazolium bromide. The structure of the heterocyclization products was confirmed by single crystal X-ray diffraction and NMR study of 2-acetonylidene-3,4-diphenyl-2,3-dihydrothiazole formed by dehydration of the corresponding ²-thiazolinium salt with simultaneous hydrolysis.     

Reducibility of nonlinear almost periodic systems of difference equations given on a torus

We establish sufficient conditions for a nonlinear system of difference equations x(t + 1) =x(t) + + P(x(t),t)+ to be reducible to the system y(t + 1) =y(t) + . Here, P(x, t) is a function 2-periodic in x_i(i = 1, ...,n) and almost periodic int with a frequency basis .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 404–412, April, 1994.This work was supported by Ukrainian State Committee on Science and Technology.     

Synthesis of an inorganic-organic polymer blend from orthoboric acid and caprolactam

The possibility of modifying boron polyoxide with an oligomeric amide upon thermal dehydration of orthoboric acid and oligomerization of caprolactam in a common melt is shown. The products obtained after thermal pretreatment of the initial blend containing 30 wt % caprolactam are investigated. It is shown that the main processes at T < 200°C are dehydration of orthoboric acid and hydrolysis of caprolactam with the formation of ε-aminocaproic acid. At temperatures of 225–260°C, the predominant process is the formation of boron polyoxide and a caprolactam-based oligomeric product. The data of ¹¹B NMR spectroscopy show that the chemical transformations of caprolactam occur against the background of the N:B donor-acceptor interaction. The two-dimensional [¹¹B-¹H] heteronuclear correlation spectrum indicates that the systems obtained upon thermal treatment are solid solutions.     

Thermal explosion in a batch reactor charged with a liquid–solid heterogeneous system

The characteristics of a thermal explosion in an ideal mixing batch reactor charged with a liquid–solid heterogeneous system are studied. The reactor initially contains both phases. The solid reagent dissolves and reacts in the liquid phase. A strong dependence of the critical value of the Semenov parameter on the dimensionless time of complete dissolution of the solid reagent is established. It is shown that, at short times of complete dissolution, the critical value of the Semenov parameter is practically independent on this time, and the thermal explosion occurs as in a homogeneous system, according to Semenov theory. The heterogeneous properties of the reaction system manifest themselves only at long times of complete dissolution.