User:Tossh eng/mizutani

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

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.

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,

${\begin{cases}y=a_{1}x+b_{1}\cdots (1)\\y=a_{2}x+b_{2}.\cdots (2)\end{cases}}$

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

$y=mx+n\cdots (3)$

where $m$ and $n$ are defined with known constants.

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

$a_{2}x+b_{2}=m(a_{1}x+b_{1})+n$

$(a_{2}-ma_{1})x=mb_{1}+n-b_{2}\cdots$ (4)

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

${\begin{cases}a_{2}-ma_{1}=0\\mb_{1}+n-b_{2}=0\end{cases}}$

which results in

${\begin{cases}m={\dfrac {a_{2}}{a_{1}}}\cdots (5)\\n=b_{2}-{\dfrac {a_{2}b_{1}}{a_{1}}}.\cdots (6)\end{cases}}$

$m$ and $n$ are expressed with known constants.

　Derivation of Mizutani's formula　

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

${\begin{cases}y={\dfrac {X_{1}-X_{0}}{p}}x+X_{0}\cdots (7)\\y={\dfrac {Y_{1}-Y_{0}}{p}}x+Y_{0},\cdots (8)\end{cases}}$

respectively, where only literary works A and C are considered, the points corresponding to $X_{0}$ and $Y_{0}$ are put on the y-axis, and $p$ 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 $(X_{0},Y_{0})$ and $(X_{1},Y_{1})$ becomes

$y={\frac {p}{X_{1}-X_{0}}}{\frac {Y_{1}-Y_{0}}{p}}x+\left\{Y_{0}-{\frac {Y_{1}-Y_{0}}{X_{1}-X_{0}}}X_{0}\right\}$

which reduces to

${\frac {y-Y_{0}}{Y_{1}-Y_{0}}}={\frac {x-X_{0}}{X_{1}-X_{0}}}.$

This is just the formular Mizutani defined.