APPENDIX
| Date | 20 January 2005 |
| DOI | https://doi.org/10.1108/S1876-0562(2005)0000005010 |
| Published date | 20 January 2005 |
| Pages | 365-395 |
APPENDIX
APPENDIX I
In this appendix we will give a self-contained proof of the characterization of scale-free
functions as power functions (basic for Lotka's law) and on the characterization of functions,
transforming products into sums, as logarithmic functions (basic for entropy). Both results are
based on the characterization of functions, transforming sums into sums, as linear functions.
Theorem A.I.I:
(i) The following assertions are equivalent for a continuous function y : K
—>
K
(a) v|/(x + y) = v|/(x) + v)/(y) (A.I.I)
for all x,y€R
(b) there is a real number c such that
i|/(x) = cx (A.I.2)
for all x e M.
(ii) The following assertions are equivalent for a continuous function \\i: K+
—>
K
(a) y(xy) = H/(x) + v|/(y) (A.I.3)
for all x,y€K+
(b) there is a real number c such that
i|;(x) = clnx (A.I.4)
for all x e K+
(c) there is a positive real number a such that
\|/(x) = logax (A.I.5)
for all x e K+.
(iii) The following assertions are equivalent for a continuous function \\i: M+
—>
M+
(a) \|/ is scale free (Definition
1.3.2.1)
366 Power laws in the information production process: Lotkaian informetrics
(b) there exist constants a E R+, cet such that
\|/(x) = axc (A.I.6)
for all x e R+.
Proof:
It is clear that all (b)s imply all (a)s and that (ii)(b) is equivalent with (ii)(c). Therefore we
only have to prove the three implications (a) => (b)
(i)(a)=»(i)(b)
Complete induction on (A.I.I) gives
vj/(nx) = nv|/(x) (A.I.7)
for all x € R and n £ N. Let now m, n E N, t E R, x = —. Hence xn = mt and
n
hence, by (A.I.7), nvj/(x) = m\|/(t) and so
m. m u\
w —t =-Vt.
n n
Putting vj;(l) = c we see that (A.I.2) is valid for all positive rational numbers. Let
x € R+ arbitrarily. Since Q+ is dense in R+, there exists a sequence (qn )neN,
qn E Q+ for all n 6 N such that lim qn = x . Since \\i is continuous we have
ll-*0O
\j/(x) = v|/(limqn)
= limv|/(qn)
n—>oo
= lim cqn
= cx,
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