数学と化学の学際共同研究と福井プロジェクト XVI

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1 数学と化学の学際共同研究と福井プロジェクト XVI 有本茂 1* Massou Amini 2* 福田信幸 3* Joseph E. LeBlanc 4* 村上達也 5* 成木勇夫 6* Mark Spivakovsky 7* 竹内茂 8* Keith F. Taylor 9* 山中聡 10* 横谷正明 11* Peter Zizler 12* Mathematics an Chemistry Interisciplinary Joint Research an the Fukui Project XVI Shigeru ARIMOTO, Massou AMII, obuyuki FUKUDA, Joseph E. LEBLAC Tatsuya MURAKAMI, Isao ARUKI, Mark SPIVAKOVSKY, Shigeru TAKEUCHI Keith F. TAYLOR, Satoshi YAMAAKA, Masaaki YOKOTAI an Peter ZIZLER This is the 16th part of the series of articles that recors an further evelops essentials of the Mathematics an Chemistry Interisciplinary Symposium 2013 Tsuyama, whose main themes were symmetry, perioicity, an repetition. The symposium was hel on April 5th an 6th in Tsuyama city, Okayama, Japan, in conjunction with the Fukui Project an was evote to the memory of the late Professor Kenichi Fukui (1981 obel Prize) who initiate the project. The present series also provies challenging cross-isciplinary problems which are irectly relate to the Fukui conjecture an to recent carbon nanotube research. Some of these problems are formulate using mathematical language not well known among chemists espite the importance of these notions in eluciating aitivity an high-spee asymptotic phenomena in molecules having many repeating ientical moieties. Some problems are formulate in terms of Fourier analysis connecte to the theory of analytic curves, other problems are formulate in connection with the Science-Art Multi-angle etwork (SAM etwork) Project, which seeks to brige Science an Art (visual, auial, an conceptual art) for a creative collaboration, an is an important part of the Fukui Project. Key Wors: the Fukui conjecture, Memoir of Prof. K. Fukui, Unique factorization omain (UFD), Carbon nanotube, Fourier analysis 原稿受付平成 28 年 9 月 29 日 1*, 10*, 11* 総合理工学科 3* 総合理工学科非常勤講師 2* Dept. of Math. Tarbiat Moares University, Iran 4* School of Integrate Stuies, Pennsylvania College of Technology USA 5* 富山県立大学工学部 生物工学科 6* 立命館大学理工学部 数学物理学系 数理科学科 I Introuction 7* CRS an Institute e Mathématiques e Toulouse, France 8* 岐阜大学教育学部 数学科 9* Dept. of Math. an Stat., Dalhousie University, Canaa 12* Dept. of Math., Phys., an Eng., Mount Royal University, Canaa 8. ote on the Memoir of Prof. Kenichi Fukui an His Former Supervisor an Colleague Prof. Haruo Shingu Shigeru Arimoto In recent article 1), entitle <Mathematics an Sciences> Fukui Conjecture an ew Frontier Project Interisciplinary Research, I, the author of this section gave a memoir on fragments of Prof. Kenichi Fukui s an his former colleague (an his former supervisor) Prof. Haruo Shingu s philosophy. I woul like to present some notes an raise some questions relate to them in this section

2 津山高専紀要第 58 号 (2016) The article 1) in Japanese appeare in the Japanese journal calle Sugaku (Mathematics) publishe by the Mathematical Society of Japan. I ha been requeste by a former Eitor-in-chief of the journal to write an article that contains an introuctory review of the Fukui conjecture. The article 1) contains, among other topics, (i) Review of the origin of the Fukui Conjecture, (ii) Review of the funamental structure of the repeat space, which is the motherboar of the conjecture. (iii) Description of the philosophy of Prof. K. Fukui an Prof. Haruo Shingu. scientists, artists, musicians, philosophers, an those who are intereste in crossing conventional bounaries in a variety of Sciences an Arts. To those who are associate with this extene form of network, I woul like to pose the following Questions: Questions. (1) What is the origin of the notion of complementarity viewe from the point of cultural anthropology? (2) To which extent i the philosophy of complementarity [avocate by iels Bohr, Werner Heisenberg, Wolfgang Pauli, an others] in the Copenhagen School give philosophical an methoological influence to quantum chemists (like Prof. Fukui) an experimental chemists (like Prof. Shingu), after the avent of Quantum Mechanics? (3) Is there a parallelism between (a) Bohr, Heisenberg, an Pauli s ialectic an complementary approach to science, (b) ishia s ialectic philosophy, (c) Complementary aspects foun in Taoism an other Practice an Thoughts in the East? Fig. 1. Professor Haruo Shingu ( ) courtesy of the late Mrs. Emiko Shingu Here, I woul like to focus on (iii), especially on Prof. Fukui s philosophical trens of ialectic an complementary approaches to science, which I witnesse many times as I was his stuent in Kyoto. As inicate in article 1), Prof. Fukui s an his former supervisor an research colleague Prof. Shingu s philosophical trens may have been influence by the Japanese philosopher Kitaro ishia ( ), who exerte strong philosophical influence not only on scholars but also on many other people from a variety of walks of life in Japan. As mentione in 1) an 2), the Fukui Project is currently calle the ew Frontier Project, after being recognize that the Fukui Conjecture was on the extension of Fukui s pattern oriente methoology in Frontier Orbital Theory to aitivity-oriente material science in chemistry. In the SAM et Program of the ew Frontier Project, we have extene the communication network of the project beyon the science an mathematics community to inclue social ote: As was mentione in 1) an 2), Prof. Haruo Shingu ( ) was an eminent experimental (organic) chemist, who playe an important role in the formation of Fukui s Frontier Electron Theory. Prof. Shingu was also the name-giver of Frontier Electron (ote: In the earliest perio, this is how Fukui s theory was so calle instea of Frontier Orbital Theory ). Prof. Shingu also playe a crucial role in the formation of the Fukui Conjecture. In fact, this conjecture was strongly base on Shingu s an his ingenious research collaborator Takehiko Fujimoto s empirical formula for the zero-point energies of hyrocarbons [cf. 1) an 3)-5)], an references therein]. As mentione in 1) an 2), the zero-point energy is a manifestation of Heisenberg s uncertainty principle an Shingu often mentione this principle of quantum mechanics uring his lectures on experimental chemistry, which accompanie a consierable amount of stimulating philosophical talks. While Prof. Fukui was an unergrauate stuent at Kyoto University, Prof. Shingu, then Associate Professor, supervise young Fukui to open his eyes to the fascinating worl of organic chemistry. Prof. Shingu s critical remarks on the then prevailing classical electron theories later gave Prof. Fukui a strong impact that ultimately lea Prof. Fukui to form his celebrate Frontier Electron Theory, for which he was aware obel Prize in [Prof. Shingu was a coauthor of the first paper 6) of the Frontier Electron Theory.]

3 数学と化学の学際共同研究と福井プロジェクト XVI 有本 Amini 福田 LeBlanc 村上 成木 Spivakovsky 9. Fukui Project, iagara Project, an the Science-Art Multi-angle etwork (SAM et) an Challenging Problem DDS Shigeru Arimoto, Joseph E. Leblanc, MasaakiYokotani The metho of sonification (auiolization) of real number sequences associate with repeat sequences (i.e., matrix sequences in the repeat spaces) has playe a heuristic an peagogical role in the Fukui Project. The Rosetta software system, which combines visual an auial heuristic tools was initially evelope by the first author (S.A.) in 1990s in the University of Saskatchewan, Canaa [cf. ref. 1)]. This unifying metho has been revitalize through the iagara Project which was initiate by S. Arimoto an J. LeBlanc in The iea of this project was born after the authors visite Corning Glass Museum, ew York State, on the way to iagara Falls, USA. The reaer is referre to refs. 2),3) for the backgroun an the evelopment of the iagara Project. This project was later incorporate into the Fukui Project an is now vigorously eveloping into what is calle Science-Art Multi-angle etwork (SAM et). The reaer is invite to visit the web site of SAM et [cf. link 4) an references in the website]: Along with the new evelopment of the SAM et, we are planning to construct a web site calle iagara SAM et, or iagara for short, which will provie both art-linke auio an visual ata relate to the Repeat Space Theory (RST). [ote: Here iagara oes not symbolize Falls but it symbolizes an interesting Bounary between two regions.] Let f enote the -imensional Magic Mountain [0, ] efine in refs. 5) an 6). The function f is a real-value continuous function efine on the -imensional cube: I = {(x 1, x 2,, x ): 0 x 1, x 2,, x 1}. See figs. 1 for schematic representation of how one can make Magic Mountain 0. See figs. 2 an 3 for the graphics 2 of the 2-imensional Magic Mountain 0 an, i.e., f 0 2 an f. (We remark that the Magic Mountain, or MagicMt( ), was nickname Tsuyama-castle function, since it was iscovere in the castle town of Tsuyama, Japan.) Challenging Problem DDS. Let g : I 1 I be a continuous function, an consier the composite function f g : I 1 I 1. Develop a computer program that prouces -imensional (fractal) soun of music by using the real number sequence where h = k 1 ( h) : h h(x)x k 1 0 f g. We remark that by the efinition of T(u) = 0,,..., f u u u, K(u) = f u, u,..., u for all 0 u 1. Thus, if g is given by then we have g ( u) u, u,..., u, T = f0 K = f f, T, an K, we have g, g. Thus, the above challenging problem is a -imensional generalization of the Sonification of the Sequences (T), (K) iscusse in section (3) by the authors of the present section. References 1) S. Arimoto, <Mathematics an Sciences> Fukui Conjecture an ew Frontier Project Interisciplinary Research (in Japanese), Sugaku (Mathematics, The Mathematical Society of Japan), 68-3 (2016) ) S. Arimoto, M. Spivakovsky, E. Yoshia, K.F. Taylor, an P.G. Mezey, Proof of the Fukui conjecture via resolution of singularities an relate methos. V, J. Math. Chem. 49 (2011) ) S. Arimoto, M. Amini, M. Spivakovsky, J. LeBlanc, K.F. Taylor, T. Yamabe, Repeat space theory applie to carbon nanotubes an Matrix Art, Bulletin of Tsuyama ational College of Technology, 54 (2012) ) S. Arimoto, Science-Art Multi-angle etwork, LIK: 5) S. Arimoto, Multiimensional Magic Mountains an Matrix Art for the Generalize Repeat Space Theory, J. Math. Chem. 50 (2012) ) S. Arimoto, Tsuyama-castle Function an Matrix Art, Bulletin of Tsuyama ational College of Technology, 53E (2011) 1-5. ow we can state

4 津山高専紀要第 58 号 (2016) How to make Magic Mountain Magic Mountain = Pyrami(n) n0 In other wors, Magic Mountain is the infinite sum of the pyrami functions calle Pyrami(n) which are given as follows: Pyrami function: Pyrami(n) n = 0, 1, 2, 3 The 4th an 10th Approximations of Magic Mountain The 4th Approximation of Magic Mountain = Pyrami(0) + Pyrami(1) + Pyrami(2) + Pyrami(3) Figs. 1. How to make Magic Mountain

5 数学と化学の学際共同研究と福井プロジェクト XVI 有本 Amini 福田 LeBlanc 村上 成木 Spivakovsky Fig. 2. Matrix Art of the graph of Magic Mountain 0 an that of the contour map of the graph both with 200 times horizontally rescale an 100 times vertically rescale Fig. 3. Matrix Art of the graph of Magic Mountain, anaglyph picture of part of the graph, an Matrix Art of the contour map of the graph