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:

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 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

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

$$ \begin{cases} a_2 - m a_1 = 0 \\ m b_1 + n - b_2 =0 \end{cases} $$

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

which reduces to This is just the formular Mizutani defined.