Топологічна еквівалентність кусково-лінійних функцій

Анотація

TOPOLOGICAL EQUIVALENCE PIECEWISE-LINEAR FUNCTIONS

T. Krivorot

An important direction is the problem of studying the conditions of the topological equivalence of piecewise linear functions. The main research is solving the problem of the existence of a polynomial topologically equivalent to a piecewise linear function. Many scientists have been involved in the classification and research of the conditions of the topological equivalence of functions. We consider a general case, but we restrict ourselves to more narrow classes of continuous mappings-polynomials.

The purpose of the study is to investigate topological equivalence for piecewise linear functions.

If we consider an arbitrary continuous function  with a finite number of local extrema on a segment [a, b], then by virtue of the homeomorphism of a numerical straight line , the function  can be replaced by a function       . If the point y_i is a local maximum (minimum) for the function f, then it will be a local maximum (minimum) for the function  and vice versa.

Let a complete alternating sequence be given . Then the sequence  can be fitted to a piecewise linear curve .

If the interval [a, b] has a continuous function f with a skinned number of extreme points. Let's put it in accordance with its complete alternating sequence . Then the functions f and  will be topologically equivalent.

We can assume that all extreme points  of functions f and  coincide, and the values of functions f and  in extreme points also coincide. If this is not the case, then with the help of homeomorphism  that preserves the orientation, we can achieve this for a function . The narrowing of the function f at intervals  is a strictly monotonic function, which is topologically equivalent to the narrowing of the function  to this interval. Consequently, function f is topologically equivalent to a function .

Two continuous functions  and  on the segment [a, b] with finite number of local extrema will be topologically equivalent if and only if their complete alternating sequences  and .

For an arbitrary complete alternating sequence , we can construct a continuous function  jn [a, b] with a finite number of local extrema such that its complete alternating sequence  is .

Let a continuous function f with a finite number of extreme points be given on     [a, b]. Let's put it in line with its periodic alternating sequence . Then the functions f and  will be topologically equivalent.

Suppose that  is a continuous function with a finite number of local extrema, then f is topologically equivalent to a piecewise linear function.

For each piecewise-linear function  having n-1 local extrema, there exists a polynomial  of degree n, which is topologically equivalent to the function l.

Consequently, the invariant of the function that answers the questions posed is the alternating sequences by which the sequence of the values of the function given on the segment is constructed in the extreme points. The polynomial  is topologically equivalent to the piecewise-linear function. Polynomial functions of different degrees can be topologically equivalent.

Invariants of continuous functions with finite number of extreme points given on a segment are considered. The piecewise-linear function, called PL, is realized by the implementation of an alternating sequence, and it is proved that the function considered in this paper is topologically equivalent to PL – the realization of its complete alternating sequence. It is investigated that for each piecewise-linear function having n–1 local extrema, there exists a polynomial of degree n, which is topologically equivalent to this function.