Concerning the function equation $f(g)=f$, regular mappings and periodic mappings

Regular Mappings between Dimensions

The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by “regular mappings” (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called (s, t)-regular m...

Mappings preserving regular hexahedrons

for all x, y ∈ X . A distance r > 0 is said to be preserved (conservative) by a mapping f : X → Y if ‖ f (x)− f (y)‖ = r for all x, y ∈ X with ‖x− y‖ = r. If f is an isometry, then every distance r > 0 is conservative by f , and conversely. We can now raise a question whether each mapping that preserves certain distances is an isometry. Indeed, Aleksandrov [1] had raised a question whether a ma...

Completely Regular Mappings and Dimension

Mappings which preserve regular dodecahedrons

for all x, y ∈ X . A distance r > 0 is said to be preserved by a mapping f : X → Y if ‖ f (x)− f (y)‖ = r for all x, y ∈ X whenever ‖x− y‖ = r. If f is an isometry, then every distance r > 0 is preserved by f , and conversely. We can now raise a question whether each mapping that preserves certain distances is an isometry. Indeed, Aleksandrov [1] had raised a question whether a mapping f : X → ...

Fuzzy $e$-regular spaces and strongly $e$-irresolute mappings

The aim of this paper is to introduce fuzzy ($e$, almost) $e^{*}$-regular spaces in $check{S}$ostak's fuzzy topological spaces. Using the $r$-fuzzy $e$-closed sets, we define $r$-($r$-$theta$-, $r$-$etheta$-) $e$-cluster points and their properties. Moreover, we investigate the relations among $r$-($r$-$theta$-, $r$-$etheta$-) $e$-cluster points, $r$-fuzzy ($e$, almost) $e^{*}$-regular spaces a...