1 citations · 5 across the 10 of their papers we have counts for
5 papers · 1 filter
A discrete form of the theorem that each field endomorphism of R (Q_p) is the identity
Apoloniusz Tyszka
Let K be a field and F denote the prime field in K. Let \tilde{K} denote the set of all r \in K for which there exists a finite set A(r) with {r} \subseteq A(r) \subseteq K such th…
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…