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We discuss the theory of finite orthogonal polynomials based on elementary linear algebra and its connection to the nonautonomous discrete Toda lattice with nonperiodic finite lattice boundary condition. By using the spectral transformation technique for finite orthogonal polynomials, one can give a solution to the initial value problem of the nonautonomous discrete Toda lattice. However, this construction of the solution cannot be ultradiscretized because of so-called “negative problem". In this paper, we focus on the rigged configuration technique to solve the initial value problem of the box-ball system and consider a connection between the rigged configuration and orthogonal polynomials., Mathematical structures of integrable systems, its deepening and expansion. September 9-11, 2019. edited by Takao Suzuki. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In the early Meiji period, the mathematical probability concept will be supposed to be very difficult to understand or accept at Japanese Army. In this paper, we investigate how the Japanese had understood or accepted the classical probability theory founded by Laplace, comparing several Japanese text books on probability theory with those of some European ones. Especially we focus on the words“ total probability” which is a little bit curious words from the point of view in the modern probability theory., "Study of the History of Mathematics 2020". February 1-3, 2021. edited by Naoki Osada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This paper presents three results that the author obtained in his doctoral thesis (Kümmerle, 2018a) where he investigated the topic from the viewpoints of institutionalization and the circulation of knowledge. These are: 1. While mathematical colloquia (danwakai) at the Imperial Universities in Kyōto and Tōhoku were born out of necessity, they stimulated research and helped create distinctive research traditions. In Tōkyō, where mathematicians could present at the regular meetings of the Mathematico-Physical Society, a colloquium relevant for mathematical research was created only in 1920. 2. With more than 90%, the ratio of non-high school graduates among all students at Tōhoku Imperial University specializing in mathematics was, at least in the 1910s, much higher than the literature has suggested. “Practical mathematics” was embedded in the curriculum, topics in the seminar classes went beyond pure mathematics, and many students of mathematics chose topics of applied mathematics when progressing to graduate school. The case of Iwatsuki Toranosuke, who became one of the founders of the school of “wave geometry”, can be seen as representative of this atmosphere. 3. An analysis of the chair system can explain why the focus on pure mathematics established by Fujisawa Rikitarō at Tōkyō Imperial University persisted. After the formal requirement to teach applied mathematics was passed on to the department of physics in the first years of the 20th century, Fujisawa and the other professors of mathematics, coherent with the principle of academic self-governance, did not follow other departments in establishing new chairs with clearer tasks. When they finally obtained one more chair in 1920, it was dedicated to mathematics in general. This provided the basis for continuing to ignore applications, with the exception of the lecture on probability and statistics. Moreover, based on experiences made during research on Tannaka Tadao’s paper on the duality of non-Abelian groups (Tannaka, 1938), this paper proposes three problems for further research: 1. Contextualize the economic and institutional situation of Japanese mathematicians. 2. Investigate both the reception of research results abroad and how Japanese mathematicians anticipated the reception before publication. 3. Pay close attention to Japanese-language papers and include them with translations into English in edited volumes., "Study of the History of Mathematics 2020". February 1-3, 2021. edited by Naoki Osada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed., Kümmerle, H., n.d.-a. Die Institutionalisierung der Mathematik als Wissenschaft im Japan der Meiji- und Taishō-Zeit (1868-1926), Acta Historica Leopoldina 77. Deutsche Akademie der Naturforscher Leopoldina - Nationale Akademie der Wissenschaften, Halle (Saale)., 本稿では私の博士論文の成果を報告するとともに、それに関して新たな問題提起を行う。, 博士論文は、ドイツ国民学習財団のヨハネス・ツィルケンス賞を受賞し、紹介ビデオが作成されている。単行本は、ドイツの国立科学アカデミーであるレオポルディナ・ドイツ学士院から刊行される予定である。

This paper provides a detailed study of the famous text on algebra, kitāb al-ğabr wa'l-muqābala, composed by al-Ḫwārizmī who was employed at the “House of Wisdom” (Bayt al-ḥikma) of al-Ma'mūn around the first half of the ninth century. It consists of a Japanese translation of a chapter entitled “Measurement” (misāḥa) along with a mathematical commentary., "Study of the History of Mathematics 2020". February 1-3, 2021. edited by Naoki Osada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In 1622, Mōri Shigeyoshi published the Warisansho a mathematical book with a preface. The Warisansho is an abacus-based math book that addresses practical issues following the multiplication table. The preface to this Warisansho explains the origin of numbers, incorporating the content of the Old Testament book of Genesis. There is a lot of research on this point. But there is a continuation of this preface. This is an explanation of the Yin-Yang principle that originated in Chinese Confucianism. In this paper, we will not pursue from the perspective of Christianity, but from the view of mathematics possessed by Mōri Shigeyoshi. Then, clarify the following four points. i) Why did Mōri Shigeyoshi publish a statement focusing on "division"? ii) Why did Mōri Shigeyoshi adopt the Principles of Yin and Yang as the theory that goes through the Warisansho? iii) How was Mōri Shigeyoshi influenced by mathematics in China and Western Europe? iv) What kind of influence did Mōri Shigeyoshi's view of mathematics have on Japanese academic research?, "Study of the History of Mathematics 2020". February 1-3, 2021. edited by Naoki Osada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

Four letters from Confucian Ogyū Sorai (1666-1728) to calendrical calculator Nakane Genkei (1662-1733) have been preserved in the Kansai University Library. In each of these letters, the date and month are only written, but not the year. Although it is not clear when the two met, it is likely that the letters were written between the 10th and 12th year of Kyōhō period (1725-1727). The main content of the letter is related to the request for a preface to Genkei's work, the Kowa-tsūreki (the Japanese Emperor's Almanac) and the Chinese literature, the Jun shi that Genkei planned to publish. The letter also mentions that the two had a disagreement over the evaluation of the Gakuritsu zensho (the Complete Works of Chinese classical Music and Calendrer) written by Zhū zǎi yù (1536 - 1611), which was introduced to Japan from China in July of 10th year of Kyōhō period (1725). In this paper, we will first introduce the contents of the letter, converted into modern printing, as well as the process of writing the preface for the two books and the conflicts between the two sides regarding the value of 𝜋 discovered by Zhū zǎi yù., "Study of the History of Mathematics 2020". February 1-3, 2021. edited by Naoki Osada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

本稿では, 古典的なイデアル類群のp=2に対する岩澤理論について紹介する.§2では岩澤加群の有限部分加群について, §3では岩澤主予想について, §4ではイデアル類群のFittingイデアルについてpが奇素数の時と比較しながら知られている結果を述べる., In this paper, we introduce classical Iwasawa theory for class groups with p = 2. In §2, we discuss about non-trivial finite submodules of Iwasawa modules. In §3, we discuss about the Iwasawa main conjecture. In §4, we discuss about Fitting ideals of class groups., Algebraic Number Theory and Related Topics 2018. November 26-30, 2018. edited by Takao Yamazaki and Shuji Yamamoto. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This is an announcement of the original paper [4](Kodai Math. J. 41 (2018), No.1, 167{ 200). In this paper, we give an overview about a map from the additive higher Chow group of 1-cycles to the module of absolute Kähler differentials, called a regulator map., Algebraic Number Theory and Related Topics 2017. December 4-8, 2017. edited by Hiroshi Tsunogai, Takao Yamazaki and Yasushi Mizusawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This is a report on our work on the anticyclotomic Iwasawa main conjecture for modular forms. In this paper, we explain the main result on a half part of the conjecture., Algebraic Number Theory and Related Topics 2017. December 4-8, 2017. edited by Hiroshi Tsunogai, Takao Yamazaki and Yasushi Mizusawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In this paper, we explain the results of the author's article "Fitting invariants in equivariant Iwasawa theory." We develop an algebraic theory to compute "Fitting invariants" (a generalization of Fitting ideals), and apply it to equivariant Iwasawa theory. This generalizes a result of Greither and Kurihara., Algebraic Number Theory and Related Topics 2017. December 4-8, 2017. edited by Hiroshi Tsunogai, Takao Yamazaki and Yasushi Mizusawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In the present article, the author gives an outline of the paper "The semi-absolute an- abelian geometry of geometrically pro-p arithmetic fundamental groups of associated low- dimensional configuration spaces". The detail of the arguments appearing in the present paper may be found in [Hgsh2]., Algebraic Number Theory and Related Topics 2017. December 4-8, 2017. edited by Hiroshi Tsunogai, Takao Yamazaki and Yasushi Mizusawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This article is a research announcement of my upcoming joint paper with Minoru Hirose on a certain class of Q-linear relations among the multiple zeta values (MZVs), which we call confluence relations. As is well known, MZVs are iterated integrals of meromorphic one-forms dt/t and dt/t-1 on a projective line. Here we consider more general iterated integrals of three different one-forms dt/t, dt/t-1 and dt/t-z and define the confluence relations as limits as z → 1 of Q-linear relations among these iterated integrals. At first, we define standard relations among the iterated integrals which naturally arise by regarding them as functions of z and thus using their differential structure with respect to z, and then we consider their limits as z → 1. The confluence relations seem to give a very rich family of Q-linear relations among MZVs and we even propose a conjecture that they exhaust all the Q-linear relations among MZVs. As a good reason for our conjecture, we prove that the confluence relations imply the extended double shuffle relations (also the duality realtion). A small table up to weight 4 of the confluence relations is given at the end., Algebraic Number Theory and Related Topics 2017. December 4-8, 2017. edited by Hiroshi Tsunogai, Takao Yamazaki and Yasushi Mizusawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

Rikitaro Fujisawa who was the second Japanese professor of mathematics of the Imperial University insists in the preface of his book that he wrote his book for repaying an obligation to the country. In this paper, we show that really he made the best effort to apply the life insurance to the society by using of talent of the self from the world level documents concerning life insurance. Especially we show that he refers De Morgan's book([3]) and the article the Life Insurance written by Kenzo Wadagaki([26]) who is appreciated in the preface of Fujisawa's book., Study of the History of Mathematics 2019. September 2-4, 2019. edited by Naoki Osada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

The Chūsan-shiki is a small monograph of bout Napier's bones which was written by Arisawa Munesada (1690-1752) in 1725. Author Arisawa was a samurai who served as a tactician under the Kaga domain (in present day Ishikawa prefecture). He wrote many military treatises in his life, also he has been studied mathematics and astronomical calendar in his young age. In this monograph he classified calculation method into three types of Wasan, Karasan and Kōmōsan. According to his emphasize Wasan is calculation method using Soroban (Chinese abacus), Karasan is Chinese classical calculation method using counting rods for solving equation and Kōmōsan is calculation method depend on Napier's bones. And Arisawa Munesada stressed that calculation using Napier's bones is the easiest way to get the answer quickly and without making error even if one does not kwon multiplication. In this paper we discuss that, first, what kind of mathematical ideas as a samurai or a tactician did he has, second why did he classify calculation method into three types, finally what mathematical book did he learn Napier's bones from., Study of the History of Mathematics 2019. September 2-4, 2019. edited by Naoki Osada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

Nakane Genkei presented a way to arrive at an 12 equal temperament by opening one octave into the 12th root in his Ritsugen-hakki [律原発揮]. His achievement was that his calculation-methods were groundbreaking at the world history level. It can be explained because he had clearly more advanced computational power than other music researchers in his era and had the ability to handle higher-order and exponential calculations freely. In this paper, I consider based on the comparison of Nakane Genkei 's Ritsugen-hakki and Cai Yuan-ding's Lülüxinshu [律呂新書]. And, I clarify the following four points by investigating historical documents. i)Why did Genkei, a famous mathematician , participate in the study of system of tuning? And what was his purpose? ii)What was the underlying mathematical thought of his pursuit? iii)What are the characteristics of the mathematics used in his pursuit? iv)What was his influence on academic research in Japan?, Study of the History of Mathematics 2019. September 2-4, 2019. edited by Naoki Osada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This is an announcement of the original paper [8]. We first review some problems and known results in Iwasawa theory for elliptic curves at supersingular primes. Then we discuss the non-existence of non-trivial finite ∧-submodules of the plus and the minus Selmer groups and prove a result which is a generalization of B.D. Kim's result in 2013. This article is written in Japanese., Algebraic Number Theory and Related Topics 2016. November 28 - December 2, 2016. edited by Yasuo Ohno, Hiroshi Tsunogai and Toshiro Hiranouchi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

There are 4 kinds of Jinkōki published from 1641(*the 18th year of Kan'ei) to 1643(*the 20th year of Kan'ei). Three kinds of Jinkōki were published in 1641. One of them was the so-called Idaibon. The second was published by YASUDA Jūbei and the third was published by TENNŌJIYA Ichirōbyōe. NISHIMURA Matazaemon published Jinkōki in 1643. YASUDA, TENNŌJIYA and NISHIMURA were commercial publishers. In addition to that, Idaibon was a mathematics book which had only problems without answers. The term "Idai" has been used since Edo era. And in early Edo era the word "Konomi" was used. I think it is possible that Idaibon Jinkōki was written by YOSHIDA Mitsuyoshi himself and published by SUMIKURA. Jinkōki published by YASUDA is a similar book of Jinkōki published in 1627(*the 4th year of Kan'ei ) and 1631(*the 8th year of Kan'ei). I think that Jinkōki by TENNŌJIYA is so-called "atozuri" later edition. It was reprinted original Idaibon Jinkōki after about 100 years. NISHIMURA Matazaemon published Jinkōki in 1643 based on Idaibon Jinkōki. There were misprint and miscalculate. But it became reference of after ages and similar books were reprinted. When I review Jinkōki in term of publisher, I am able to analyze differently. In this paper, I investigated Jinkōki in term publisher. Until now, TENNŌJIYA Jinkōki has not received any attention or investigation. I found that the TNNŌJIYA Jinkōki had been the same as the original Idaibon Jinkōki. However, it turned out that the Jinkōki by TENNŌJIYA had been republished about 100 years after original Idaibon Jinkōki., Study of the History of Mathematics 2019. September 2-4, 2019. edited by Naoki Osada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

The open problems of Kazuyuki Sawaguchi's Kokon Sampōki are famous for being solved in Hatsubi Sampō by Takakazu Seki using the method of elimination of unknowns. In this paper, we consider how Sawaguchi solved the open problems proposed by Masaoki Satō in the Sampō Kongenki , and how he made the open problems his book, Kokon Sampōki. We will reveal what kind of solutions Sawaguchi had expected. We will also describe Sawaguchi's contribution to the development of early-modern Japanese mathematics (wasan) that were beyond Chinese traditional mathematics of that time., Study of the History of Mathematics 2019. September 2-4, 2019. edited by Naoki Osada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In this report, I summarize results in the paper (Kodama, Matsutani, Previato, Ann. Inst. Fourier 63 (2013) 655-688) to pose a problem to give an explicit relation between periodic and quasi-periodic solutions of Toda lattice. For a hyperelliptic curve Xg of genus g, we have a quasi-periodic solution of Toda lattice in terms of the hyperelliptic σ function and its addition theorem. Using the division polynomial of Xg, we find 2N-division points in its Jacobi variety and then have N-periodic solution of Toda-lattice. It is well-known that the N-periodic solution is associated with a hyperellptic curve ˆXg, N-1 of genus N-1 rather than g. However it is not clear how Xg and ˆXg, N-1 are connected geometrically, though the problem is very simple and natural. In this report, after I give a review of the recent development of σ function theory of higher genus and show the summary of our previous work, I give some comments on the problem., "Mathematical structures of integrable systems and their applications". September 5-7, 2018. edited by Shinsuke Iwao. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

Seki Takakazu (?-1708) has long been known as one of the founders of mathematics in Pre- modern Japan. His personal profile, documentary legacy, and view of mathematics have been studied by numerous scholars in numerous papers. Here, I confrm that his primary interest was in the theory of algebraic equations, a topic on which he left several manuscripts. He was the first person to grasp equations systematically and abstractly as mathematical objects per se. His theory was based on the side writing" system that he developed to solve the problems set in Sawaguchi Kazuyuki's Mathematical Method Old and New (Kokon Sanpouki, 1671). It has previously been pointed out that this side writing system allowed problems to be resolved into simultaneous equations of differing degrees (simple, quadratic, etc.) which led immediately to elimination theory as a new subject of study. Seki's theory on algebraic equations has two aspects. One is still of mathematical interest today while the other belongs more to the historical context. I discuss the former quickly and the latter in rather more detail, referring to a problem treated in Seki's Method for Restoring Defective Problems (Byoudai Meichi)., Study of the History of Mathematics 2019. September 2-4, 2019. edited by Naoki Osada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In the two decades from 1930 to 1950, Japanese national curricula for schools, from elementary to post-secondary, were revised several times to cope with the rapidly changing situations both at home and abroad. Especially, the national curriculum for mathematics were revised drastically in the 1940s to cope with the situation. However, teaching of mathematics (and also other subjects) at each school according to the curriculum was impossible in the 1940s due to the war. Mathematics actually taught at school at that time was different in each school and year. This paper is a record of mathematics actually taught at school at that time., "The study of the history of mathematics 2018". September 18-21, 2018. edited by Shigeru Jochi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In the present paper, the author introduces results concerning the indecomposability of various profinte groups related to hyperbolic curves. For instance, we discuss the indecomposability of the étale fundamental group of the configuration space of a hyperbolic curve over an algebraically closed field of characteristic zero, and the pro‐l Grothendieck - Teichmüller group. The details of the results and their proofs are given in [M]., 本稿では, 双曲的曲線に関連した様々な副有限群の非分解性についての結果を紹介する. 例えば, 標数 0 の代数閉体上の双曲的曲線に付随した配置空間のエタール基本群や副l グロタンディーク・タイヒミューラー群の非分解性について議論する. 結果や証明の詳細は[M]で与えられている., "Algebraic Number Theory and Related Topics 2015". November 30 - December 4, 2015. edited by Hiroki Takahashi, Yasuo Ohno and Takahiro Tsushima. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This article includes a brief explanation of the ordinariness of dormant GL_{n}‐opers and a research announcement for results in the papers titled "Abelian étale coverings of generic curves and ordinariness of dormant opers" (cf. [28]) and "The symplectic nature of the space of dormant indigenous bundles on algebraic curves" (cf. [26]) . In this article, we first recall the notion of a (dormant) GL_{n} ‐oper on a proper smooth curve and give some examples of dormant GL_{n} ‐opers arising from certain objects in characteristic p geometry (including Tango structures). Then, we give the definition of the ordinariness of dormant GL_{n}‐opers and propose two related assertions; the first assertion concerns the relationship between the ordinariness of dormant GL_{n}‐opers and Galois coverings of the underlying curves; the second assertion concerns the symplectic structure on the moduli stack classifying ordinary dormant GL_{2} ‐opers., "Algebraic Number Theory and Related Topics 2015". November 30 - December 4, 2015. edited by Hiroki Takahashi, Yasuo Ohno and Takahiro Tsushima. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This is an expository article of the author's paper [17], where under some assumptions we prove the rank-part of the Mazur‐Tate refined conjecture of BSD type. More precisely, we show that the rank of the Mordell‐Weil group of an elliptic curve over mathbb{Q} is less than or equal to the order of zeros of Mazur - Tate elements, which refine the p‐adic L function in a sense. In this article, we explain the proof of the main result., "Algebraic Number Theory and Related Topics 2015". November 30 - December 4, 2015. edited by Hiroki Takahashi, Yasuo Ohno and Takahiro Tsushima. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In this paper, we give a brief description of an explicit construction of generators of the degree-zero part of the Chow group of zero-cycles on some diagonal cubic surfaces over 3-adic fields. An important point is to check that such cycles are linearly independent by using the Hilbert symbol., "Algebraic Number Theory and Related Topics 2015". November 30 - December 4, 2015. edited by Hiroki Takahashi, Yasuo Ohno and Takahiro Tsushima. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This paper is a summary of [6] in which we formulate the refinement of Gross's Leading Term Conjecture. Here, Gross's Leading Term Conjecture is a certain generalization of Gross - Stark conjecture and Gross conjecture. To formulate our conjecture, we introduce the notion of Shintani data, which axiomatizes algebraic aspects of Shintani zeta functions. In this paper, we also discuss the idea behind the conjecture., "Algebraic Number Theory and Related Topics 2015". November 30 - December 4, 2015. edited by Hiroki Takahashi, Yasuo Ohno and Takahiro Tsushima. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

The Jinko-ki played the role important for development of mathematics in Japan., and many researches have also been done. However, since spirit of innovation is not observed in the contents of the Jinko-ki, the evaluation from a mathematics viewpoint is low. Rather, the Jinko-ki is evaluated highly in respect of cultural history, history of education, an economic history, etc. In research from this viewpoint, they make portions other than a solution applicable to consideration. Moreover, in the research from a mathematics viewpoint, the portion of a solution is not historical reading which is carrying out the interpretation which uses contemporary This paper aims at historical reading of the solution portion in the Jinko-ki. And I considered it as the object about the commercial problem in the Jinko-ki. This paper surveys and compares with the treatment of the Jinko-ki the treatment of the commercial problem in the China mathematics and Western mathematics which had big influence on the Jinko-ki formation. The commercial problem in the Jinko-ki specialized in the practical commercial problem, and the result became clear that it was a high mathematics document of convenience.., "The study of the history of mathematics 2017". September 19-22, 2017. edited by Shigeru Jochi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This paper is a brief survey of a history of the teaching and learning of Western mathematics and of the spread of mathematicical knowledge in Japan from 1870s to 1930s, making some suggestions for further studies in a history of mathematics and for the development of mathematical education in Japan., "The study of the history of mathematics 2017". September 19-22, 2017. edited by Shigeru Jochi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

Ajima Naonobu (1732-1798) studied the quantitative relationship on Steiner Chain in Wasan. He pursued the figures on circles which contact with one after another. We showed that Ajima's figures correspond to crest of family, trademark of shops, and picture of kimono. We raised the hypothesis that Ajima researched the establishment conditions in the design of crest and picture of kimono At first in this paper, we will show that the roots of such figures are three patterns from foreign countries. From the arrival of them, we will show that Japanese patterns were international at that time and Wasan had been under the influence of foreign culture of patterns beyond borders and language., "The study of the history of mathematics 2016". August 29～September 1, 2016. edited by Shigeru Jochi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

Since it is the writing which accomplishes the core of the Seki style mathematics developed in Japan, Taisei Sankei by Kowa Seki, Kataakira Takebe, and Katahiro Takebe is investigated with much concern. However, about the relation between evaluation of the Front of Volume which described the algorithm by abacus calculation systematically, and evaluation of the Middle of Volume which investigated the algorithm which solves high order simultaneous equations, and the End of Volume, It is not solved enough. In this paper, I summarized the actual condition of the Taisei Sankei from comparison with the Unified Lineage of Mathematical Methods which is the classic document which described the algorithm by abacus calculation systematically. And I considered the directivity of algebraic investigation from comparison with the Tongwen Suanzhi which introduced Western arithmetic to China., "The study of the history of mathematics 2016". August 29～September 1, 2016. edited by Shigeru Jochi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In Wasan which is old mathematics in Japan, Steiner chain was studied by Ajima Naonobu (1732-1798) earlier than Jakob Steiner (1796-1863). Geometry was the major areas of Wasan. Now in practice, it takes a lot of time to solve beautiful geometry problems in the time of the method, but it is difficult to explain them using sophisticated way. If we can show the relevance among the problems clearly in a simple way, we can show more charm of Wasan. This paper will be such an experimental approach. The second purpose of this paper is related to the fact that one of the invariant holds on Steiner chain problems in Japan. This fact is known among people who are interested to solve Wasan geometry problems. Probably for the first time, it was submitted collectively in Tatsuhiko Kobayashi's paper. Although it has passed more than 30 years from that pointing out, the elucidation of it has not been made. We already made another simple proof to Ikeda's theorem and Kenmochi's theorem, using the inversion method. Now, the idea of proof and the other one have been found to be effective to clarify that. This paper will be an effort to answer the problem., "The study of the history of mathematics 2016". August 29～September 1, 2016. edited by Shigeru Jochi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This paper is a brief survey of a history of teaching and learning of geometry in secondary education in Japan in the Showa era (1926-1989). In the Showa era, drastic changes of the school curriculum were made several times to cope with the situation at that time, and school mathematics, especially teaching and learning of geomety was greatly affected by these changes., "The study of the history of mathematics 2016". August 29～September 1, 2016. edited by Shigeru Jochi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This paper is a brief survey of a history of teaching and learning of geometry in secondary education in Japan from the 1870s to the 1920s. Geometry was regarded as the science which treats of the properties of the space and as a subject suitable for mental discipline, and Eu-clidean geometry was taught in the normal way. Dairoku Kikuchi made great efforts to draw up the syllabus of geometry and to improve the teaching of geometry in the last two decades of the 19th century and the first decade of the 20th century., "The study of the history of mathematics 2016". August 29～September 1, 2016. edited by Shigeru Jochi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

As the domination of Nazis in Germany from 1933 to 1945 had a tremendous impact on all aspects of German society, so mathematical community was no exception in this regard. In the same way as the period of Nazis was related to the previous and latter Era, mathematical community in Germany also had a close relationship with the previous and latter times and has continued to our times. The most obvious example of this continuity would be Mathematical Research Institute of Oberwolfach (Mathematisches Forschungsinstitut Oberwolfach) in Southern Black Forest, which is, so far, regarded as the most important institute in the world. Mathematical Research Institute was established in September 1944, survived post-war Allies' occupation and was highly acclaimed through the West German Era until today. The aim of this paper is to discuss what led to the establishment of this institute in difficult times of the end of the war and how it has been maintained until today., "The study of the history of mathematics 2016". August 29～September 1, 2016. edited by Shigeru Jochi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In this paper, we give an introduction to the positive characteristic multizeta values. We state several results of their relations and linear/algebraic independence and an interpretation as periods of t-motives. We also sketch how we prove algebraic independence results., "Various aspects of multiple zeta values". July 23～26, 2013. edited by Kentaro Ihara. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This article includes a partial explanation of the diversity of (dormant) opers (and their moduli) as well as a research announcement for some results of the paper titled "A theory of dormant opers on pointed stable curves--a proof of Joshi's conjecture--" (cf. [18]). First, we recall the notion of a projective structure (as well as an indigenous bundle) defined on a Riemann surface, which is equivalent to the notion of an sl2-oper. Next, we give the definition of a dormant g-oper (for each semisimple Lie algebra g), which is defined on an algebraic curve in positive characteristic. Finally, we propose some results, concerning an explicit computation of the number of dormant opers on a sufficiently general curve., "Algebraic Number Theory and Related Topics 2014". December 1～5, 2014. edited by Takeshi Tsuji, Hiroki Takahashi and Yuichiro Hoshi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

We give a brief survey on number theoretic applications of the theory of perfectoid spaces recently obtained by Peter Scholze and his collaborators. We explain basic strategies and some ideas of the proofs of the weight-monodromy conjecture for complete intersections, comparison theorems for torsion coefficients for rigid analytic varieties, and the construction of Galois representations associated with regular algebraic cuspidal automorphic representations of GLn over totally real or CM fields as well as torsion cohomology classes of locally symmetric varieties. In this paper, no new results are given, and most of the arguments explained are only briefly sketched., "Algebraic Number Theory and Related Topics 2014". December 1～5, 2014. edited by Takeshi Tsuji, Hiroki Takahashi and Yuichiro Hoshi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

The Eichler-Deuring mass formula says that the weighted number of isomorphism classes of supersingular elliptic curves over an algebraically closed field of characteristic p is expressed as a simple polynomial in p. In 2009, C.-F. Yu and J.-D. Yu generalized this formula for supersingular abelian surfaces. In this paper, we show a mass formula for supersingular abelian three-folds of a-number equal to two. This is a short version of [11]., "Algebraic Number Theory and Related Topics 2014". December 1～5, 2014. edited by Takeshi Tsuji, Hiroki Takahashi and Yuichiro Hoshi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This paper is a Japanese expository version of my joint paper with S.Wood [TW]. In occasion to write a Japanese version paper of [TW], I explained the fundamental concept of conformal field theories and description of vertex operator algebras by using language of conformal field theories. I hope this will help researchers who are not familiar with conformal field theories. Now I explain the contents of this paper. Capter 2. Energy momentum tensors and field operators. In the first part of this paper, I explained relationship between the representation theory of Virasoro algebra and the energy momentum tensor. In the second part, I explained the concepts of field operators, the mutually locality of two field operators, and operator product expantions. Finally I explained the concepts of propagation of locality of operators. This is important to formulate vertex operator algebras and their modules. Chapter3. Vertex operator algebras and their representations. In this chapter, by using the concept of energy momentum tensor, mutually locality of two field operators and the propagation of the locality, I introduce the concept of vertex operator algebra (VOA). Then I introduce the universal enveloping algebra U(V) of a vertex operator algebra V. Then I define the zero mode algebra A0(V) as a subquotient of U(V). An finiteness condition, called Zhu's C2 cofiniteness condition, and Theorem 3-2 due to I. Frenkel and Zhu state that when Zhu's C2 condition is satisfied, the dimension of A0(V) is finite, and the set of irreducible V-modules and the set of irreducible A0-modules are in one to one correspondance. So the number of irreducible V-modules is finite. Chapter 4. Bosonic vertex operator algebras and lattice vertex operator algebras In this chapter I introduce the most important VOAs, the Bosonic VOA and the lattice VOA. Bosonic VOA has parameters k+, k- ∈ C* with k+k- = 1, and will be denoted by IIk+, k-. The central charge of energy momentum tensor of IIk+, k- is ck+, k- = 13 - 6(k+ + k-). So the Bosonic VOA IIk+, k- contains the Virasoro vertex operator algebra Virck+, k-. The abelian category of IIk+, k--mod consisting of IIk+, k--modules is semi-simple, and the simple objects of IIk+, k--modules are Fock modules Fβ ∋ uβ, β ∈ C. So we get Virck+, k--module Fβ. If k+ ∈ C＼Q, ⇔ k- ∈ C＼Q, then Fβ is not simple as Virck+, k--module if and only if β ∈ Z1/2α+ ⊕ Z1/2α-, where α+ = √k+/2, α- = √k-/2, α+・α- = -2. And for β ∈ Z1/2α+ ⊕ Z1/2α-, the structure of Virck+, k--module is very simple, which can be analyzed by Virasoro screening operators S+(z), S-(z). When k+ ∈ Q>0, ⇔ k- ∈ Q>0, the structure of Fβ, β ∈ Z1/2α+ + Z1/2α-, is very complicated. In this case the rank of the free abelian group Z1/2α+ + Z1/2α- is one. For k+ = p-/p+, k- = p+/p-; p+, p- ≥ 2 and p+ and p- are relatively prime integers, we can define lattice vertex operator algebra Vp+, p- which has sub-VOA IIk+, k- ⊂ Vp+, p-. When k+ = p-/p+, k- = p+/p-, it can be proved that the abelian category Vp+, p--mod consisting of Vp+, p--modules is semi-simple and has 2p+p- simple modules. Chapter 5. Free field realization of Virasoro vertex operator algebra. For k+, k- ∈ C*, k+k- = 1, the Virasoro vertex operator algebra Virck+, k- is obtained as sub-VOA of bosonic VOA IIk+, k- by using the screening operators S+(z), S-(z). When k+ = p-/p+, k- = p+/p-, p+, p- ≥ 2, p+, p- mutually prime, then the central charge is ck+, k- = 13 - 6(p+-p-)2/p+p-, which we write cp+;p- in this case. In this case the Virasoro vertex operator algebra Virck+, k- is not irreducible as VOA, and it has quotient VOA which is called minimal Virasoro algebra and we denote it MinVirck+, k-. The abelian category MinVirck+, k--mod consisting of MinVirck+, k--modules is semi-simple as an abelian category, and the number of simple objects is 1/2(p+ -1)(p- - 1). Using the minimal Virasoro VOA MinVirck+, k-, three Russian physicists Belavin, Polyakov and Zamolodchikov developed the conformal field theory, and get numbers of remarkable results about two dimensional critical phenomena [BPZ]. Since Vircp+, p- is a sub-VOA of k+, k-, for k+ = p-/p+, k- = p+/p-, and in this case the bosonic IIVOA k+, k- is a sub-VOA of the lattice VOA Vp+, p-. Using the screening operators S+(z), S-(z), we can define the sub-VOA Mp+, p- of Vp+, p- and Mp+, p- has Vircp+, p- as sub-VOA. The main purpose of our paper [TW] is analysis of the vertex operator algebra Mp+, p-, which we call the extended Virasoro algebra of type sl2 at positive rational levels. The results of our paper [TW] are the following. (1) The vertex operator algebra Mp+, p- is generated by T(z), W1, a(z), a = ±, 0 and the conformal dimension of W1, a(z) is Δ1 = (2p+ - 1)(2p- - 1). (2) The VOA Mp+, p- satisfies Zhu's C2 finiteness condition and we can determined the structure of finite dimensional algebra A0(Mp+, p-) explicitly. (3) We can determine all the simple modules of Mp+, p-, the number of them is 1 2 (p+ -1)(p- -1)+2p+p-, and we can determine the structure of these simple Mp+, p- -modules. (4) The VOA Mp+, p- has sub-VOA Vircp+, p- and has the quotient VOA Mp+, p-- module. (5) The simple MinVircp+, p-- modules can be considered as simple Mp+, p-- modules, and the number of simple modules is 1/2 (p+ - 1)(p- - 1). The method of the proof of these results are very complicated, and mainly use the lifting of base field C to the discrete valuation ring O = C[[ϵ]]. Chapter 6. On the lifting to discrete valuation ring. At first we lift to the valuation field K = C((ϵ)). And then by using the properties of Jack polynomial defined on K, we prove the C[[ϵ]] integrality of the lifting. Chapter 7. The representation theory of Vcp+, p- and of Wcp+, p-. We analyze the structure of Fβ, β ∈ Z1/2α+ + Z1/2α-, as Vcp+, p- - module carefully. The results are stated as Felder complex of Virasoro Fock modules Fβ, and we construct Vircp+, p- -modules X±[r, s] ⊂ K[r, s], 1 ≤ r ≤ p+, 1 ≤ s ≤ p-. Chapter 8. The extended W algebra Mp+, p- and their representations. First we construct operators E and F which we call Frobenius operators. The construction of E and F and analysis of their properties in [TW] are very complicated, and we need some simple method to analyze this part. In 2016 we succeeded this simplification. We are now prepareing a paper and it will appear in 2017. So in this resume we state the properties of E and F, and skip the proof of it. Using the properties of E and F, we prove that X±[r;s] ⊂ K±[r;s] are indeed Mp+, p- -modules and X±[r;s] are irreducible as Mp+, p- -modules. Then we prove the C2-cofiniteness of Mp+, p-, and we can determine the structure of the zero mode algebra A0(Mp+, p-) explicitly. Then using the structure of A0(Mp+, p-), we can prove that the set of irreducible modules is {L[r;s] = L[p+-r, p-s] 1 ≤ r ≤ p+ -1, 1 ≤ s ≤ p- -1} and {X±[r, s] 1 ≤ r ≤ p+, 1 ≤ s ≤ p-}, and L[r;s] = L[p+-r, p-s] are the lifting of the simple MinVircp+, p- -modules to simple Mp+, p- modules by quotient VOA map Mp+, p- → MinVircp+, p- → 0. In Chapter 8 we simply state the results and their proof are omited. These are the statements of our Japanese version of [TW]., "String theory, integrable systems and representation theory". July 30～August 2, 2013. edited by Koji Hasegawa and Yasuhiko Yamada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This paper presents a numerical solution to multi-objective shape optimization problem in steady-state viscous flow fields. In this study, a multi-objective shape optimization problem using normalized objective functional is formulated for the flow velocity distribution prescribed problem and the total dissipated energy minimization problem in the viscous flow fields. In addition, another multi-objective shape optimization problem is formulated for the flow velocity distribution prescribed problem, while the total dissipated energy is constrained to less than a desired value, in the viscous flow fields. Shape gradients of these multi-objective shape optimization problems are derived theoretically using the Lagrange multiplier method, adjoint variable method, and the formulae of the material derivative. Reshaping is carried out by the traction method proposed as an approach to solving shape optimization problems. The validity of proposed method is confirmed by results of 2mathrm{D} numerical analysis., "Topology optimization theory and applications toward wide fields of natural sciences". May 7～9, 2014. edited by Takashi Nakazawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In a previous work, the author introduced non-linear q-difference systems which include q-Garnier systems. These systems were given as a compatibility condition of linear q-difference equations whose coefficients were products of N × N matrices. In this paper, we consider a special case with N = 3 and derive a 3×3 matrix Lax form of the q-Painlevé equation of type $E^{(1)}_6$ and two kinds of characterizations of a 3rd-order scalar equation related to it. They seems to be new.

In this survey paper, we explain the proof of the p-adic local Langlands correspondence for mathrm{G}mathrm{L}_{2}(mathbb{Q}_{p}) following Paskunas and Colmez-Dospinescu-Paskunas which are based on the groundbreaking works by Colmez and Berger-Breuil., "Algebraic Number Theory and Related Topics 2013". December 9～13, 2013. edited by Tadashi Ochiai, Takeshi Tsuji and Iwao Kimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

本稿の目的は, 有限体上の(特異な)多様体のゼータ関数の特殊値を, 高次チャウ群を用いて記述する筆者の結果([15])を概説することである. 正確には, 高次チャウ群から重みホモロジー群とよばれる不変量への写像の核と余核の位数の商の交代積として, 特殊値を記述する公式を与える. 証明の主なアイディアは, 重みスペクトル系列を用いて問題をスムーズかつ固有な場合に帰着することである., The aim of this paper is to survey the author s result (to appear in [15]), which describes special values of zeta functions of (singular) varieties over finite fields by using higher Chow groups. To be precise, we show that the special value at s = 0 is equal to the alternating product of the ratio of the cardinalities of kernels and cokernels of maps from higher Chow group to another invariant which is called weight homology. The principal idea of proof is to reduce the problem to the case that the variety is smooth and proper by using weight spectral sequences., "Algebraic Number Theory and Related Topics 2013". December 9～13, 2013. edited by Tadashi Ochiai, Takeshi Tsuji and Iwao Kimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This is a research announcement of the paper [O] by the author. We explain the construction and an outline of the proof of the log global Cartier transform of higher level, which is a log and higher level version of the global Cartier transform of Ogus-Vologodsky., "Algebraic Number Theory and Related Topics 2013". December 9～13, 2013. edited by Tadashi Ochiai, Takeshi Tsuji and Iwao Kimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In this survey paper, we explain a local-global compatibility for p-adic Langlands correspondences in the completed cohomology of modular curves after Emerton. We also explain its application to the Fontaine-Mazur conjecture and a compatibility between the local Langlands correspondence and the p-adic local Langlands correspondence., "Algebraic Number Theory and Related Topics 2013". December 9～13, 2013. edited by Tadashi Ochiai, Takeshi Tsuji and Iwao Kimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

本稿の目的は, 平行な重さ2の実2次体上のHilbertカスプ形式とHilbert Eisenstein級数の間の合同式からそれらに伴うL関数の特殊値の間の合同式を導いた結果[Hi]を報告することである. この結果は, Greenberg氏とVatsal氏によるp進Galois表現が剰余して可約となる素数pにおけるmathbb{Q}上の楕円曲線の岩澤主予想が成り立つ多くの新しい例をもたらしたVatsal氏の結果の一般化である., The purpose of this paper is to announce the result of [Hi] in which we obtain congruences between special values of L-functions from a congruence between a Hilbert cusp form and a Hilbert Eisenstein series of parallel weight 2 over a real quadratic field. This is a generalization of a result of Vatsal. By using the result of Vatsal, Greenberg and Vatsal give many examples of elliptic curves over mathbb{Q} satisfying the Iwasawa main conjecture when the associated p-adic Galois representations are residually reducible., "Algebraic Number Theory and Related Topics 2013". December 9～13, 2013. edited by Tadashi Ochiai, Takeshi Tsuji and Iwao Kimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

本稿では, 多重ゼータ値とモジュラー形式の間の関係について議論する. 最近, Francis Brownにより, 3以上の奇数をインデックスにもつ多重ゼータ値(純奇多重ゼータ値)の空間の次元に関する新しい予想が提唱された. 彼の予想は, 純奇多重ゼータ値の線形関係式がモジュラー形式と密接に関わっていることを示唆している. 以下では, 純奇多重ゼータ値の線形関係式と偶周期多項式を対応させる事により, 彼の予想の深さ4の場合を部分的に解決した我々の仕事[17]を紹介する. また, 関連する多重Eisenstein級数の話題についても触れる., In this paper, we will be interested in connections between the theory of elliptic modular forms and multiple zeta values (MZVs). Recently, Francis Brown has proposed a new conjecture on the dimension of the space spanned by MZVs at the sequences indexed by odd integers greater than 1 (called totally odd MZV). His conjecture suggests that the modular forms and the linear relations among totally odd MZVs are deeply related with each other. We will give a partial answer to his conjecture in the case of depth 4 by showing that there is an injective linear map from the space of even period polynomials to the space of linear relations among totally odd MZVs. We shall also announce a result related with multiple Eisenstein series., "Algebraic Number Theory and Related Topics 2013". December 9～13, 2013. edited by Tadashi Ochiai, Takeshi Tsuji and Iwao Kimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This is an announcement of another paper by the author (published in Math. Z.). We prove that whether a K3 surface has potential good reduction can be determined from the Galois representation defined from the l-adic or p-adic étale cohomology groups of the K3 surface. This is an analogue of the Néron-Ogg-Shafarevich criterion for abelian varieties. We also have an application to the period map of K3 surfaces in mixed characteristics., "Algebraic Number Theory and Related Topics 2013". December 9～13, 2013. edited by Tadashi Ochiai, Takeshi Tsuji and Iwao Kimura. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In this paper, we give an overview of a method of explicit 2-descent for hyperelliptic Jacobian varieties ([1]), and relate it to the set of complete intersections of two quadrics. This is an announcement of the results contained in the paper [4]., "Algebraic Number Theory and Related Topics 2012". December 3～7, 2012. edited by Atsushi Shiho, Tadashi Ochiai and Noriyuki Otsubo. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

The Taisei Sankei is an encyclopaedic work of Japanese mathematics compiled by three mathematicians, Seki Takakazu and his disciples, the Tatebe brothers. The compilation project began in 1683 on the initiative of the younger Takebe, Katahiro, and was completed by the elder Takebe, Kata'akira, in 1710. An interim edition in twelve volumes, entitled Sanpō Taisei, was drafted by Katahiro around 1695, but no manuscript of this edition survives. However, it is important to study the Sanpō Taisei because it is considered to be the culmination of Seki Takakazu's mathematical works. By studying the mathematical manuscripts left behind by the three mathematicians, this paper deduces which volumes of the Taisei Sankei succeed the Sanpō Taisei and how the Sanpō Taisei was compiled. We also identify the author(s) of the four manuscripts Kyūketsu Henkeisōkai, Kyūseki, Kaihō Sanshiki and Kaihō whose contents are common to some parts of the Taisei Sankei.