Makoto Tamura

“The Nine Chapters on the Mathematical Art” was the oldest book of mathematics in China before the unearthing of “Suan-shu shu.” The aim of our research is to provide a complete translation and annotation of it including annotations of Liu Hui（劉徽）and Li Chunfeng（李淳風）from the viewpoint of our previous work on “Suan-shu shu.” This is the thirtieth article based on our research and results in which we studied the problems 9 to 16 of Chapter 9, Gougu（句股）

九章算術』は『算数書』出土以前は数学書としては中国最古のものであった。我々は、我々の『算数書』研究を起点に、『九章算術』の劉徽注、李淳風注を含めた訳注を完成させることを目的としている。 本論文では、方程章の算題［一八］に対する訳注を与える。

九章算術』は『算数書』出土以前は数学書としては中国最古のものであった。我々は、我々の『算数書』研究を起点に、『九章算術』の劉徽注、李淳風注を含めた訳注を完成させることを目的としている。 本論文では、句股章の算題［一］～［八］に対する訳注を与える。

九章算術』は『算数書』出土以前は数学書としては中国最古のものであった。我々は、我々の『算数書』研究を起点に、『九章算術』の劉徽注、李淳風注を含めた訳注を完成させることを目的としている。 本論文では、方程章の算題［四］～［九］に対する訳注を与える。

九章算術』は『算数書』出土以前は数学書としては中国最古のものであった。我々は、我々の『算数書』研究を起点に、『九章算術』の劉徽注、李淳風注を含めた訳注を完成させることを目的としている。 本論文では、方程章の算題［一〇］～［一七］に対する訳注を与える。

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu Hui(劉徽)and Li Chunfeng(李淳風)from the viewpoint of our previous work on "Suan-shu shu." This is the twenty-fourth article based on our research and results in which we studied the problems 16 to 20 of Chapter 7, Yingbuzu(盈不足).

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu Hui(劉徽)and Li Chunfeng(李淳風)from the viewpoint of our previous work on "Suan-shu shu." This is the twenty-second article based on our research and results in which we studied the problems 11 to 15 of Chapter 7, Yingbuzu(盈不足).

We discuss a problem posed by Gersten: Is every automatic group which does not contain Z x Z subgroup, hyperbolic? To study this question, we define the notion of "n-track of length n", which is a structure like Z x Z, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, we show that if a group acts freely, cellularly, properly discontinuously and cocompactly on a CAT(0) cube complex and its quotient is "weakly special", then the above question is answered affirmatively.

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu Hui(劉徽) and Li Chunfeng(李淳風) from the viewpoint of our previous work on "Suan-shu shu." This is the thirteenth article based on our research and results in which we studied the problems 1 to 7 of Chapter 5, Shang gong(商功).

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu Hui(劉徽)and Li Chunfeng(李淳風)from the viewpoint of our previous work on "Suan-shu shu." This is the twelfth article based on our research and results in which we studied the problems 23 and 24 of Chapter 4, Shaoguang(少広).

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu Hui(劉徽)and Li Chunfeng(李淳風)from the viewpoint of our previous work on "Suan-shu shu." This is the eleventh article based on our research and results in which we studied the problems 19 to 22 of Chapter 4, Shaoguang(少広).

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu Hui(劉徽) and Li Chunfeng(李淳風) from the viewpoint of our previous work on "Suan-shu shu." This is the tenth article based on our research and results in which we studied the problems 12 to 18 of Chapter 4, Shaoguang(少広).

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu Hui(劉徽) and Li Chunfeng(李淳風) from the viewpoint of our previous work on "Suan-shu shu."This is the ninth article based on our research and results, in which we studied problems 1 to 11 of Chapter 4, Shaoguang(少広).

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu Hui(劉徽) and Li Chunfeng(李淳風) from the viewpoint of our previous work on "Suan-shu shu." This is the seventh article based on our research and results, in which we studied problems 10 to 20 of Chapter 3, Shuaifen(衰分).

The Research Group on Ancient Chinese Mathematics went to China in December 2009 to investigate the Qin bamboo strips "Shu" housed at the Yuelu Academy and the Han bamboo strips "Suan-shu" in the Hubei Provincial Institute of Cultural Relics and Archaeology. In this paper, we report the results of our investigation of the "Shu" and the "Suan-shu", and introduce and criticize an article on the "Shu."

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu Hui(劉徽) and Li Chunfeng(李淳風) from the viewpoint of our previous work on "Suan-shu shu." This is the seventh article based on our research and results, in which we studied problems 1 to 9 of Chapter 3, Shuaifen(衰分).

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu-Hui(劉徽) and Li Chunfeng(李淳風) from the viewpoint of our previous work on "Suan-shu shu." This is the sixth article based on our research and results, in which we studied problems 32 to 46 of Chapter 2, Sumi(粟米).

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu-Hui(劉徽) and Li Chunfeng(李淳風) from the viewpoint of our previous work on Suan-shu shu." This is the fifth article based on our research and results, in which we studied problems 1 to 31 of Chapter 2, Sumi(粟米).

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in Chinabefore the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu-Hui(劉徽) and Li Chunfeng(李淳風) from the viewpoint of our previous work on "Suan-shu shu." This is the fourth article based on our research and results, in which we studied problems 33 to 38 of Chapter 1, Fangtian(方田).

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of "Suan-shu shu." The aim of our research is to provide a complete translation and annotation of it including annotations of Liu-Hui (劉徽) and Li Chunfeng (李淳風) from the viewpoint of our previous work on "Suan-shu shu." This is the third article based on our research and results in which we studied the problems 31 and 32 of Chapter 1, Fangtian (方田).

"The Nine Chapters on the Mathematical Art" was the oldest book of mathematics in China before the unearthing of the "Suan-shu shu." The aim of our research is to provide a complete translation and annotation including annotations of Liu-Hui(劉徽) and Li Chunfeng(李淳風) from the viewpoint of our previous work on the "Suan-shu shu." This is the second article based on our research and results in which we studied problems 10 to 30 in Chapter 1, Fangtian(方田).

The unearthed book "Suanshu-shu (算数書)" is the oldest book of mathematics in China. In order to report this book's proper collation, translation, and annotation, the very first procedure was to decipher the letters from photographs with the following investigation of the results with the mathematical and historical viewpoints. This is the seventh released article based on our research and results in which we studied 9 sections named "Shuitian (税田)", "Wuquan (誤券)", "Zuwuquan (租誤券)", "Qucheng (取程)", "Quxicheng (取〓程)", "Xing (行)", "Qiguang (啓広)", "Qizong (啓縦)" and "Haozu (耗租)".

The unearthed book "Suanshu-shu (算数書)" is the oldest book of mathematics in China. In order to report this book's proper collation, translation, and annotation, the very first procedure was to decipher the letters from photographs with the following investigation of the results with the mathematical and historical viewpoints. This is the sixth released article based on our research and results in which we studied 9 sections named "Zengjianfen (増減分)", "Fendangbanzhe (分当半者)", "Fenbanzhe (分半者)", Chuanma(伝馬)", "Yushi (羽矢)", "Zengfu (繪幅)", "Qiqian (漆銭)", "Yinqi (飲漆)", and "Chengzhu (程竹)".

The unearthed book "Shanshu-shu (算数書)" is the oldest book of mathematics in China. In order to report this book's proper collation, translation, and annotation, the very first procedure was to decipher the letters from photographs with the following investigation of the results with the mathematical and historical viewpoints. This is the fifth released article based on our research and results in which we studied 10 sections named "Silian (糸練)", "Chongsu (春粟)", "Tongmao (銅耗)", "Futan (負炭)", "Lutang (〓〓)", "Yushi (羽矢)", "Fumi (負米)", "Fenqian (分銭)", "Michuqian (米出銭)", and "Fangtian (方田)".

Though a number of schoolars made an effort to explain in the section "yinqi (飲漆)" of the underthed book "Suanshu-shu (算数書)", no proper result has yet been obtained. In this article, we attempt to clarify its meaning by examing the phenomenon that lacquer can be mixed homogeneously with the utmost water volume of up to three times.

中国関係論説資料に採録

Feng Luo and Richard Stong introduced the average edge order mu(0)(K) of a triangulation K and showed in particular that for closed 3-manifolds mu(0)(K) being less than 4.5 implies that K is on S-3. In this paper, we establish similar results for 3-manifolds with non-empty boundary; in particular it is shown that mu(0)(K) being less than 4 implies that K is on the 3-ball.