Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Beckman-Quarles type theorems for mappings from R^n to C^n   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 G(f(x),f(y))=d^2. Let A(n) denote the set of all
positive numbers d such that any map f: R^n -> C^n that preserves unit distance
preserves also distance d. Let D(n) denote the set of all positive numbers d
with the property: if x,y \in R^n and |x-y|=d then there exists a finite set
S(x,y) with {x,y} \subseteq S(x,y) \subseteq R^n such that any map
f:S(x,y)->C^n that preserves unit distance preserves also the distance between
x and y. We prove: (1) A(n) \subseteq {d>0: d^2 \in Q}, (2) for n>=2 D(n) is a
dense subset of (0,\infty). Item (2) implies that each continuous mapping f
from R^n to C^n (n>=2) preserving unit distance preserves all distances.

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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
1
The categories of this paper are: cs
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The categories of this paper are: bayes-an
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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn