Background of Mizutani's revision of Ohno's lexical law

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Original Ohno's lexical law had some ambiguity in setting a vertical line for the set of points of the rate for a literature. And it had, in some cases, an extraordinarily highly error-sensitive part in the practical plotting procedure. Thus a more general description of the law was required.[1] [2] [3]

Mizutani's revision is based on the following mathematical ground by which two defaults of the original Ohno's law could be accomplished:

 
Fig. 2. Two lines and a vertical line.

Consider two lines,

 

When a vertical line crosses with these two lines (1) and (2) at the points of the y-coordinate   and  , respectively, the quantity   and  , plotted as a point   on a seperate plane, determines another line

 

where   and   are defined with known constants.

Proof. Substitute   of (1) and (2) for   and   of (3), respectively, we obtain

 
  (4)

The condition of the identical equation with respect to x for (4) is

 

which results in

 

  and   are expressed with known constants.

 Derivation of Mizutani's formula 

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In the setting of Mizutani's revision, lines for the noun and a different word class are expressed as

 

respectively, where only literary works A and C are considered, the points corresponding to   and   are put on the y-axis, and   is designatd to be the distance along the x-axis between A and C. Then from (7) and (8), a line connecting the two points   and   becomes

 

which reduces to

 

This is just the formular Mizutani defined.

Literatures Cited and Footnotes

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  1. ^ Shizuo Mizutani (1965) On Ohno's lexical law. Keiryo-Kokugo-gaku (Mathematical Linguistics of Japanese) 35: 1-12. (in Japanese)
  2. ^ Shizuo Mizutani (1982) Mathematical Linguistics (Lectures on modern mathematics D-3) Baifukan Publisher, 204pp. (in Japanese)
  3. ^ Shizuo Mizutani (1989) Ohno's lexical law: its data adjustment by linear regression. In "Quantitative Linguistics Vol. 39, Japanese Quantitative Linguistics" (ed. Shizuo Mizutani) pp. 1-13, Bochum: Studienverlag Dr. N. Brockmeyer.