Топологічна еквівалентність функцій двох змінних

Т. Г. Криворот

Анотація


TOPOLOGICAL EQUIVALENCE OF FUNCTIONS TWO VARIABLES

T. G. Krivorot

 

Many processes that arise in the natural sciences are described by smooth and continuous functions. When creating a classification of continuous functions, many difficulties arise, since the behavior of the function in the neighborhood of the critical point can be complex and not analogous to the smooth case. The classification and investigation of the conditions for the topological equivalence of functions of two variables on manifolds is an important direction of research in topology.

The questions of topological equivalence of continuous functions of two variables are considered in the article. Their topological classification is completely carried out in the case of smooth functions on manifolds. It is proved that in a neighborhood of an isolated critical point the function is topologically equivalent to Rezn, and in the neighborhood of a local minimum (maximum) it is equivalent to x2+y2. One of the classes of continuous functions that have only a singularity of the saddle type is illustrated, and also the case of functions with an isolated local minimum (maximum) is considered. The case of continuous functions with a point of an isolated local minimum (maximum) in the interior of a domain is characterized, examples are constructed that determine the features that can arise and become an obstacle in the topological classification of functions of two variables.

Among smooth and continuous functions, harmonic functions occupy an important place, which are the main apparatus for solving problems in physics and mechanics. Thus, for example, the gravitational potential in a region that does not contain masses attracts the potential of a constant electric field in the region, does not contain electric charges, the velocity potential of the irrotational motion of the liquid, the body temperature under the condition of stable heat distribution, the strain of the membrane stretched to an arbitrary contour - all these processes are described by harmonious functions. Therefore, it is very important to investigate the topological behavior of such functions.

The problems of classification and investigation of the conditions for topological equivalence of functions are devoted to the works of A. Bolsinov, S. Maksimenko, A. Oshemkov, V. Sharko. In the solution of the problem of the outlined direction, A. Andriyuk's work became the first, where a criterion for topological equivalence of functions from the class F(D2) was found, which coincides with the class of pseudo harmonic functions that take at most one critical value. To each such function, the author associates its combinatorial invariant. Topological classification of smooth functions was carried out in the works of V. Arnold, namely, the mappings R1R1 are classified. However, the classification problem for the multidimensional case remains unresolved.

The aim of the research is to investigate the existence of a criterion for the topological equivalence of functions of two variables with an isolated local extremum.

Consequently, with topological equivalence of functions, the property of a point to be a local minimum (maximum) or an isolated local minimum (maximum) is invariant. If z = f (x, y) is a continuous function defined in a neighborhood of zero, (0,0) is its isolated local minimum and f (0, 0) = 0. The function z = f (x, y) is topologically equivalent to the cone over Jordan closed curve γ, which bounds a         star-shaped domain if and only if for any value c such that 0 < c < a0, f 1(с) is homeomorphic to a closed Jordan curve; the set of sets {f 1(с)} forms a regular family of curves in a neighborhood (0,0).

Thus, the questions of topological equivalence of continuous functions of two variables are considered in the article. In particular, conditions are found under which a continuous function defined in a neighborhood of zero and which is its local minimum (maximum) will be topologically equivalent to a cone. It is clear that in the case when the graph of a continuous function z = f (x, y) is a cone over a curve that bounds a starlike region, then z = f (x, y) is topologically equivalent to a smooth function z = x2+y2.


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Bolsinov, A., Oshemkov, S., Sharko, V. (1996). On classification of flows on manifolds, 2 (2), 51–60.

Andriiuk, O. P. (2006). Funktsiyi na odnovymirnykh mnohovydakh [Functions on one-dimensional manifolds]. K., – 19s.

Maksimenko, S. Y. (1999). Klasifikatsiya m-funkciyi na poverhnostyah [Classification of m-functions on surfaces] Ukrainian mathematical journal, 8, 1129–1135.

Oshemkov, A. A., Sharko, V. V. (1998). O klassifikatsiyi potokov Morsa-Smejla na dvumernyh mnogoobraziyah [On the classification of Morse-Smale flows on two-dimensional manifolds]. Mathematical collection, 7, 93–140.

Sharko, V. V. (2003). Gladkaya i topologicheskaya ehkvivalentnost' funkcij na poverhnostyah [Smooth and topological equivalence of functions on surfaces]. Ukrainian mathematical journal, 5, 687–700.


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