scholarly journals SOME TYPICAL PROPERTIES OF SYMMETRICALLY CONTINUOUS FUNCTIONS, SYMMETRIC FUNCTIONS AND CONTINUOUS FUNCTIONS

1995 ◽  
Vol 21 (2) ◽  
pp. 708
Author(s):  
Shi
1980 ◽  
Vol 6 (1) ◽  
pp. 67 ◽  
Author(s):  
Kostyrko ◽  
Neubrunn ◽  
Smital ◽  
Šalát

2016 ◽  
Vol 09 (03) ◽  
pp. 1650069
Author(s):  
Tammatada Khemaratchatakumthorn ◽  
Prapanpong Pongsriiam

We give the definition of uniform symmetric continuity for functions defined on a nonempty subset of the real line. Then we investigate the properties of uniformly symmetrically continuous functions and compare them with those of symmetrically continuous functions and uniformly continuous functions. We obtain some characterizations of uniformly symmetrically continuous functions. Several examples are also given.


1967 ◽  
Vol 1 (4) ◽  
pp. 257-260
Author(s):  
S. P. Ponomarev

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Paweł Pasteczka

AbstractWe extend some approach to the family of symmetric means (i.e. symmetric functions $$\mathscr {M} :\bigcup _{n=1}^\infty I^n \rightarrow I$$ M : ⋃ n = 1 ∞ I n → I with $$\min \le \mathscr {M}\le \max $$ min ≤ M ≤ max ; I is an interval). Namely, it is known that every symmetric mean can be written in a form $$\mathscr {M}(v_1,\dots ,v_n):=F(f(v_1)+\cdots +f(v_n))$$ M ( v 1 , ⋯ , v n ) : = F ( f ( v 1 ) + ⋯ + f ( v n ) ) , where $$f :I \rightarrow G$$ f : I → G and $$F :G \rightarrow I$$ F : G → I (G is a commutative semigroup). For $$G=\mathbb {R}^k$$ G = R k or $$G=\mathbb {R}^k \times \mathbb {Z}$$ G = R k × Z ($$k \in \mathbb {N}$$ k ∈ N ) and continuous functions f and F we obtain two series of families (depending on k). It can be treated as a measure of complexity in a family of means (this idea is inspired by theory of regular languages and algorithmics). As a result we characterize the celebrated families of quasi-arithmetic means ($$G=\mathbb {R}\times \mathbb {Z}$$ G = R × Z ) and Bajraktarević means ($$G=\mathbb {R}^2$$ G = R 2 under some additional assumptions). Moreover, we establish certain estimations of complexity for several other classical families.


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