数学と化学の学際共同研究と福井プロジェクト XVIII

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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 XVIII 有本茂 1* Massoud 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 1* 横谷正明 11* Peter Zizler 12* Mathematics and Chemistry Interdisciplinary Joint Research and the Fukui Project XVIII Shigeru ARIMOTO, Massoud AMII, obuyuki FUKUDA, Joseph E. LEBLAC Tatsuya MURAKAMI, Isao ARUKI, Mark SPIVAKOVSKY, Shigeru TAKEUCHI Keith F. TAYLOR, Satoshi YAMAAKA, Masaaki YOKOTAI and Peter ZIZLER This is the 18th part of the series of articles that records and further develops essentials of the Mathematics and Chemistry Interdisciplinary Symposium 213 Tsuyama, whose main themes were symmetry, periodicity, and repetition. The symposium was held on April 5th and 6th in Tsuyama city, Okayama, Japan, in conjunction with the Fukui Project and was devoted to the memory of the late Professor Kenichi Fukui (1981 obel Prize) who initiated the project. The present series also provides challenging cross-disciplinary problems which are directly related to the Fukui conjecture and to recent carbon nanotube research. Some of these problems are formulated using mathematical language not well known among chemists despite the importance of these notions in elucidating additivity and high-speed asymptotic phenomena in molecules having many repeating identical moieties. Some problems are formulated in terms of Fourier analysis connected to the theory of analytic curves, other problems are formulated in connection with the Science-Art Multi-angle etwork (SAM etwork) Project, which seeks to bridge Science and Art (visual, audial, and conceptual art) for a creative collaboration, and is an important part of the Fukui Project. Key Words: the Fukui conjecture, Memoir of Prof. K. Fukui, Unique factorization domain (UFD), Carbon nanotube, Fourier analysis 原稿受付平成 28 年 9 月 29 日 1*, 1*, 11* 総合理工学科 3* 総合理工学科非常勤講師 2* Dept. of Math. Tarbiat Modares University, Iran 4* School of Integrated Studies, 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 Introduction 7* CRS and Institute de Mathématiques de Toulouse, France 8* 岐阜大学教育学部 数学科 9* Dept. of Math. and Stat., Dalhousie University, Canada 12* Dept. of Math., Phys., and Eng., Mount Royal University, Canada 12. A unifying approach to the spectral-symmetry and additivity problems in carbon nanotubes and related molecular networks via the repeat space theory II Shigeru Arimoto, Keith F. Taylor, Massoud Amini obuyuki Fukuda, Mark Spivakovsky, Satoshi Yamanaka Masaaki Yokotani and Peter Zizler 1. Formulation of the Problem Theorem A (functional alpha existence theorem, X Hr (q, 1)-C(I) version). Let { M } be a fixed element of

2 津山高専紀要第 58 号 (216) XHr(q, 1), let I be a fixed closed interval compatible with { M }. Then, there exists a functional C(I)* = B(C(I), ) such that Tr ( M ) as, for all C(I). = o1 (3.1) (This theorem was proved in Part II of ref. [1].) Theorem B. Let q, q + and let q = q + q. Let { M } = { Q } be a fixed element of X i (q, 1)((q, q)) - P XH(q, 1), let I = [-a, a] with a > be a fixed closed interval compatible with { M }. Then, for any C o (I), we have Tr Q. (3.2) Moreover, the spectrum of Q (which is contained in the real line) is symmetric around the origin; moreover, the algebraic multiplicities of the pairing eigenvalues coincide. Theorem C. Let q be a positive integer, let t, s 1, s 2,, s t be positive integers such that q = s 1 + s s t. Let s = (s 1, s 2,, s t ). Let { M } = { P Q } be a fixed element of X i (q, 1)(s) - XH(q, 1), let I = [-a, a] with a > be a fixed closed interval compatible with { M }. Then, for any C o (I), we have Tr Q. (3.3) Moreover, the spectrum of Q (which is contained in the real line) is symmetric around the origin; moreover, the algebraic multiplicities of the pairing eigenvalues coincide. Problem I. To prove theorems B and C via theorem A. 2. Solution of the Problem In this section, we provide a solution to problem I, by using theorem A. Solution: Theorem B is an immediate consequence of theorem C. Thus, we shall establish theorem C, by using theorem A. First recall from [3]: Proposition 2. Let a > and I = [-a, a]. Let C o (I) denote the subset of C(I) of all odd functions and let P o (I) denote the subset of P(I) of all odd polynomial functions. Let C e (I) denote the subset of C(I) of all even functions and let P e (I) denote the subset of P(I) of all even polynomial functions. Then, we have: (i) Po () I = C o (I), (4.1) Proof. A proof is given in [3]. ow use theorem A in section 2, and let C(I)* = B(C(I), ) be such that Tr ( M ) = o1 (4.3) as, for all C(I). Then, by using the argument of section 3 in [3] and proposition 2 above, we can easily get the conclusion via (i) and (ii): (i) (P o (I)) = {}, (4.4) (ii) (C o (I)) = {}. (4.5) [ote that (ii) directly follows from proposition 2 above and from the continuity of.] Thus, the proof of theorem C is reduced to the proof of (i). Let P o (I). We have only to show that Tr (4.6) M for all. But, by the linearity of Tr, it suffices to show that k Tr M (4.7) for each positive odd integer k. ote that theorem 2 easily implies that (X i (q, 1)(s) - ) k X i (q, 1)(s) -, (4.8) for each positive odd integer k. It then follows that (4.7) is true, therefore we see that (i) is true. // ote: We present in the Appendix some Matrix Art using Albrecht Dürer s etching Melancholia. Challenging Problem D given there can be easily answered by using the techniques given in Part XVII and the present part of this series. References [1] S. Arimoto, Repeat space theory applied to carbon nanotubes and related molecular networks. I, II J. Math. Chem. 41 (27) ; J. Math. Chem. 43 (28) [2] S. Arimoto, M. Spivakovsky, M. Amini, E. Yoshida, M. Yokotani, T. Yamabe, Repeat space theory applied to carbon nanotubes and related molecular networks. III, J. Math. Chem. 5 (212) [3] S. Arimoto and K.F Taylor, Aspects of Form and General Topology: Alpha Space Asymptotic Linearity Theorem and the Spectral Symmetry of Alternants, J. Math. Chem. 13 (1993) [4] S. Arimoto, ew proof of the Fukui conjecture by the Functional Asymptotic Linearity Theorem, J. Math. Chem. 34 (23) [5] S. Arimoto, K. Fukui, P. Zizler, K.F. Taylor, and P.G. Mezey, Structural Analysis of Certain Linear Operators Representing Chemical etwork Systems via the Existence and Uniqueness Theorems of Spectral Resolution. V, Int. J. Quantum Chem. 74 (1999) (ii) Pe () I = C e (I). (4.2)

3 数学と化学の学際共同研究と福井プロジェクト XVIII 有本 Amini 福田 LeBlanc 村上 成木 Spivakovsk Appendix: A = Fig. 1. Matrix Art Picture of Albrecht Dürer s etching Melancholia, created by using MATLAB. This picture is a visualization of a matrix with entries of integers ranging from 11 to 228. A = = + Fig. 2. Let A denote this matrix, then A = (1/2)(A + A T ) + (1/2)(A A T ) ; this addition of matrices is shown by pictures above

4 津山高専紀要第 58 号 (216) n Fig. 3. Pictures of M and M ( n 2,3,4,5), and the picture of spectrum of M, which is symmetric with respect to the y axis

5 数学と化学の学際共同研究と福井プロジェクト XVIII 有本 Amini 福田 LeBlanc 村上 成木 Spivakovsky Concluding Remarks Shigeru Arimoto On behalf of my co-authors and myself, I would like to express sincere gratitude to all who helped in enhancing the Fukui Project. Special thanks are due to Professor T. Yamabe and Professor J. LeBlanc, who have been collaborators in the program related to (1) the Fukui conjecture, carbon nanotube, and (2) visual-audial Matrix Art. As a complementary aspect to the visual Matrix Art, we have begun to target heuristic and innovative sonification (audiolization) of various scientific data and their Fourier analysis with the help of computer scientists and musicians. By the 1 th anniversary of the birth of Professor Kenichi Fukui in 218, heuristic visualization and sonification programs in the Fukui Project currently separated as modular units shall be united. The Science-Art Multi-angle etwork (SAM etwork) Project, which is an important part of the Fukui Project, will be playing a central role for this unification. The reader is referred to the link: and sub-links therein especially for referential data of the SAM etwork, which seeks to bridge natural science, art, literature, and philosophy for a creative collaboration. Acknowledgements The first author (S.A) was supported by the Tsuyama College President Fund. The second author (M.A.) was partially supported by Iran ational Science Foundation (ISF Grant o ). On behalf of the authors of this series of papers, the first author (S.A.) would like to express sincere gratitude to Dr. T. oritsugu, President of Tsuyama College, ational Institute of Technology, his predecessor Dr. H. Inaba, and members of the Fukui Project Association for their help in holding our Symposium 213 Tsuyama and promoting our interdisciplinary and international collaborations. The Symposium was also supported by the Tsuyama City and the Tsuyama City Educational Board, which we would also like to thank cordially