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The Beckman-Quarles theorem for continuous mappings from C^n to C^n
Apoloniusz Tyszka
Let varphi_n:C^n times C^n->C, varphi_n((x_1,...,x_n),(y_1,...,y_n))=sum_{i=1}^n (x_i-y_i)^2. We say that f:C^n->C^n preserves distance d>=0, if for each X,Y in C^n varphi_n(X,Y)=d…
The Beckman-Quarles theorem for mappings from C^2 to C^2
Apoloniusz Tyszka
Let phi: C^2 times C^2 -> C, phi((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:C^2->C^2 preserves unit distance, if for each X,Y in C^2 phi(X,Y)=1 implies phi(f(X),f(…
The Beckman-Quarles theorem for mappings from R^2 to F^2, where F is a subfield of a commutative field extending R
Apoloniusz Tyszka
Let F be a subfield of a commutative field extending R. Let ϕ_2: F^2 \times F^2 \to F, ϕ_2((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:R^2 \to F^2 preserves distanc…
Mappings from R^n to F^n which preserve unit Euclidean distance, where F is a field of characteristic 0
Apoloniusz Tyszka
Let F be a commutative field of characteristic 0, G_n: F^n \times F^n -> F, G_n((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say that g:R^n->F^n preserves distance…
A discrete form of the Beckman-Quarles theorem for mappings from R^2 (C^2) to F^2, where F is a subfield of a commutative field extending R (C)
Apoloniusz Tyszka
Let F be a subfield of a commutative field extending R. Let phi_n:F^n \times F^n ->F, phi_n((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say that f:R^n->F^n preserv…
Beckman-Quarles type theorems for mappings from R^n to C^n
Apoloniusz Tyszka
Let G: C^n \times C^n -> C, G((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+ (x_n-y_n)^2. We say that f: R^n -> C^n preserves distance d>0 if for each x,y \in R^n G(x,y)=d^2 implies…