Kurosu Konosuke

Kurosu Kōnosuke ( Japanese 黒 須 康 之 介 ; * February 1, 1893 in Ageo , Saitama Prefecture ; † February 18, 1970 ) was a Japanese mathematician.

Life

Kōnosuke graduated from Tōhoku University in 1917 and taught at the Marine Engineering School ( 海軍 機関 学校 , Kaigun kikan gakkō) from 1920 , and from 1925 to 1949 at the First High School in Tokyo (Dai-ichi kōtō gakkō), as well from 1939 to 1967 at the University of Natural Sciences Tokyo and from 1949 to 1970 at the Rikkyō University in Tokyo.

From 1943 to 1948 he was a director of the Japan Society of Mathematical Education (adviser from 1948, honorary member from 1952).

Kurosu dealt with analysis (for example with the Laplace transformation ) and number theory ( theory of continued fractions ). He wrote 11 mathematical articles between 1913 and 1925. The sentence contained in his article Notes on some points in the theory of continued fractions (1924) was also found independently by Blagovest Sendov in 1959 .

Kurosu-Sendov's theorem

Definitions

A continued fraction is called semi-regular if the partial counters , which are partial denominators , apply to all : ${\displaystyle b_{0}+{\frac {a_{1}|}{|b_{1}}}+{\frac {a_{2}|}{|b_{2}}}+{\frac {a_{3}|}{|b_{3}}}+\cdots }$ ${\displaystyle a_{n}=\pm 1}$ ${\displaystyle b_{1},b_{2},\ldots \in \mathbb {N} }$ ${\displaystyle n\geq 1}$

${\displaystyle b_{n}+a_{n+1}\geq 1.}$

If the continued fraction is not finite, it is usually also required that for infinitely many n even holds. ${\displaystyle b_{n}+a_{n+1}\geq 2}$

A semi-regular continued fraction with for all is called singular , ${\displaystyle b_{n}\geq 2,b_{n}+a_{n}\geq 2}$ ${\displaystyle n\geq 1}$

and a semi-regular chain break with for everyone is called a chain break after the next whole . ${\displaystyle b_{n}\geq 2,b_{n}+a_{n+1}\geq 2}$ ${\displaystyle n\geq 1}$

sentence

Kurosu-Sendov's theorem is an analogous statement to Vahlen's theorem for regular continued fractions and says: is

${\displaystyle \alpha =b_{0}+{\frac {a_{1}|}{|b_{1}}}+{\frac {a_{2}|}{|b_{2}}}+{\frac {a_{3}|}{|b_{3}}}+\cdots }$

a singular continued fraction expansion or a continued fraction expansion according to the nearest whole of the real number , at least one of every two successive approximate fractions satisfies the inequality ${\displaystyle \alpha }$

${\displaystyle \left|\alpha -{\frac {p_{n}}{q_{n}}}\right|<{\frac {1}{(1+{\sqrt {5}}/2)\,q_{n}^{2}}}.}$

Because of this, this is a stronger statement than in the regular case. ${\displaystyle {\frac {1}{1+{\sqrt {5}}/2}}=0{,}4721\ldots <{\frac {1}{2}}}$

Fonts

Items:

• On the convergence-abscissa of a certain definite integral , Tōhoku Math. J. 16, 291-298, 1919
• Note on the theory of approximation of irrational numbers by rational numbers , Tōhoku Math. J. 21, 247-260, 1922
• Notes on some points in the theory of continued fractions , Japanese J. Math. 1, 17-21, 1924, Corrigendum Volume 2, 1926, p. 64

Books:

• Saidai saishō ( 最大 最小 , maximum and minimum )
• Insūbunkai ( 因数分解 , factorization )
• Shin bisekibun enshū ( 新 微 積分 演習 , exercises on the new differential and integral calculus )

Web links

• Journal of Japan Society of Mathematical Education, 1970: To the Memory of Our Late Honorary Member Mr. Kōnosuke Kurosu
• Entry at kotobank

Remarks

1. ↑ See also Cor Kraaikamp: A new class of continued fraction expansion (PDF; 4.1 MB), Acta Arithmetica 57 , 1991, p. 32 and Iosifescu / Kraaikamp: Metrical Theory of Continued Fractions , Springer, 2002, p. 287 .

Japanese name: As is common in Japan, the family name comes before the first name in this article . Thus Kurosu is the family name, Kōnosuke the first name.