数学と化学の学際共同研究と福井プロジェクト

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

^10*

横谷正明

^11*

Peter Zizler

^12*

Mathematics and Chemistry

Interdisciplinary Joint Research and the Fukui Project XIV

Shigeru ARIMOTO, Massoud AMINI, Nobuyuki FUKUDA, Joseph E. LEBLANC Tatsuya MURAKAMI, Isao NARUKI, Mark SPIVAKOVSKY, Shigeru TAKEUCHI Keith F. TAYLOR, Satoshi YAMANAKA, Masaaki YOKOTANI and Peter ZIZLER

This is the 14th part of the series of articles that records and further develops essentials of the Mathematics and Chemistry Interdisciplinary Symposium 2013 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 Nobel 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 Network (SAM Network) 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

I Introduction

1. Introduction

In April 2013, the Mathematics and Chemistry Interdisciplinary Symposium 2013 Tsuyama was held in the National Institute of Technology, Tsuyama College, Okayama prefecture, Japan. The main themes of the symposium were symmetry, periodicity, and repetition. The present series of papers records and further develops essentials of this symposium, and provides challenging cross-disciplinary problems in conjunction with the international, interdisciplinary, and inter-generational Fukui Project, which has also been called the New Frontier Project in recent years.

Here in Parts XIX to XVIII some of these problems are formulated using mathematical language not well known 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 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 7＊ CNRS 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

among chemists despite the importance of these notions in elucidating additivity and high-speed asymptotic phenomena in molecules having many repeating identical moieties. Some other problems are formulated in connection with the Science-Art Multi-angle Network (SAM Network) 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.

2. Functional Delta Existence Theorem Extension Conjecture (FDETEC)

Shigeru Arimoto

Let us first recall the Asymptotic Linearity Theorem Extension Conjectures (ALTEC C(I)-version and CBV(I)-version), [cf. refs. 1) and 2)]:

Asymptotic Linearity Theorem Extension Conjecture (ALTEC C(I)-version) The Asymptotic Linearity Theorem (ALT) cannot be extended from AC(I) to C(I), where AC(I) denotes the functional space of all real-valued absolutely continuous functions defined on the closed interval I, and C(I) denotes the functional space of all real-valued continuous functions defined on the closed interval I.

Asymptotic Linearity Theorem Extension Conjecture (ALTEC CBV(I)-version) The Asymptotic Linearity Theorem (ALT) cannot be extended from AC(I) to CBV(I), where AC(I) denotes the functional space of all real-valued absolutely continuous functions defined on the closed interval I and CBV(I) denotes the functional space of all real-valued continuous functions of bounded variation defined on the closed interval I.

Note: The Unique Factorization Domain (UFD) of the real-analytic functions and associated tools related to the resolution of singularities play an important role in the proof of the Asymptotic Linearity Theorem. In parallel with concrete physicochemical problems, there are several versions of the Asymptotic Linearity Theorem Extension Conjecture. In one of the versions, the above mentioned set of continuous functions C(I), which happened to be the underlying set of the Banach space denoted by the same symbol given below, is embedded into a larger functional space (which is not a normed space) and the high and low speed asymptoticity phenomena are investigated in conjunction with the UFD of the real-analytic functions defined on a closed interval.

Next, recall the Functional Delta Existence Theorem (FDET) [cf. ref. 3)], which is one of the most fundamental

theorems in the repeat space theory. The FDET reproduced below was indispensable for reducing the proof of the Fukui Conjecture to a problem of resolution of singularities and related methods [cf. ref. 4) and references therein].

Theorem X (FDET). Let {M_N}, {M_N}  X_r(q) be fixed repeat sequences with {M_N} – {M_N}  X_( )q . Let I be a fixed closed interval compatible with {M_N} and {M_N}.

Define the sequence of linear functionals _N: AC(I)  by

N( )

 = Tr ( M_N) – Tr ( M_N).

Then, there exists a functional   AC(I)* = B(AC(I), ) such that

N( )

 = ( ) + o(1) as N , for all   AC(I).

The reader is asked to review ref. 5) for the notion of the repeat space and related symbols, and also for the detailed history of the Asymptotic Linearity Theorem, which proves the Fukui Conjecture. We reproduce here only the definitions relevant to the conjectures given in what follows.

Notation 1. Let I = [a, b] (a, b  , a < b) denote a closed interval.

V_I (): the total variation of a real-valued function  on I.

C(I): the Banach space of all real-valued continuous functions on I equipped with the norm given by

||||_C = sup {|(t)|: t  I }.

AC(I): the Banach space of all real-valued absolutely continuous functions on I equipped with the norm given by

||||AC = sup {|(t)|: t  I} + VI().

CBV(I): the Banach space of all real-valued continuous functions of bounded variation on I equipped with the norm given by

||||_CBV = sup {|(t)|: t  I} + V_I().

B(X, Y): the normed space of all bounded linear operators from a normed space X to a normed space Y.

B(X): = B(X, X).

Now that the two versions of the ALTEC were proved [by the author (S.A.) of this section in refs. 1) and 2)], in view of the topological analogy, it is natural to pose the following two conjectures:

Functional Delta Existence Theorem Extension Conjecture (FDETEC C(I)-version) The Functional Delta

− 36 −

Existence Theorem (FDET) cannot be extended from AC(I) to C(I), where AC(I) denotes the Banach space of all real-valued absolutely continuous functions defined on the closed interval I equipped with the

AC norm, and C(I) denotes the Banach space of all real-valued continuous functions defined on the closed interval I equipped with the

Cnorm.

Functional Delta Existence Theorem Extension Conjecture (FDETEC CBV(I)-version) The Functional Delta Existence Theorem (FDET) cannot be extended from AC(I) to CBV(I), where AC(I) denotes the Banach space of all real-valued absolutely continuous functions defined on the closed interval I equipped with the

AC norm, and CBV(I) denotes the Banach space of all real-valued continuous functions of bounded variation defined on the closed interval I equipped with the

CVB norm.

Whereas the ALTEC was made public for long, the FDETEC, in spite of its fundamental importance, was never explicitly made public before.

References

1) S. Arimoto, Proof of the Asymptotic Linearity Theorem Extension Conjecture (ALTEC),J. Math. Chem. 54 (2016) 72-84.

2)S. Arimoto, M. Amini, N. Fukuda, I. Morishima, T. Murakami, I. Naruki, K. Saito, M. Spivakovsky, S. Takeuchi, K.F. Taylor, S. Yamanaka, M.

Yokotani, and P. Zizler, "Mathematics and Chemistry Interdisciplinary Joint Research and the Fukui Project XIII", Bulletin of National Institute of Technology, Tsuyama College 57 (2015) 73-78.

3) S. Arimoto, The Functional Delta Existence Theorem and the reduction of a proof of the Fukui conjecture to that of the Special Functional Asymptotic Linearity Theorem, J. Math. Chem. 34 (2003) 287-296.

4) S. Arimoto, M. Spivakovsky, E. Yoshida, K.F. Taylor, and P.G. Mezey, Proof of the Fukui conjecture via resolution of singularities and related methods. V, J. Math. Chem. 49 (2011) 1700-1712.

5) S. Arimoto, New proof of the Fukui conjecture by the Functional Asymptotic Linearity Theorem, J. Math. Chem. 34 (2003) 259.

3. On asymptotic properties of Riemann sums for certain classes of absolutely continuous functions

Peter Zizler Abstract

Let AC[ , ]a b denote the Banach space of absolutely continuous functions defined on [ , ]a b with the norm

[ , ]

( ) var_{a b}( )

f  f a  f for f AC[ , ]a b . We establish the following

Theorem Let f AC[0,1] and let 0  1. We have

1 1

0 0

1 1

( ) ( (1) (0)) (1)

2

N

k

f k N f f f o

N  





       

   

   

 

^.

Introduction. Let BAC[0,1] denote the vector space of absolutely continuous functions on [0,1] . We equip

B with the following three norms. For f B we let,

1 1

0 0

BV( ); ;

T W

f  f _ f f 



f 



f

BV (0) BV( )

f  f  f

where BV( )f var_[0,1]( )f denotes the variation of f on [0,1] and f _sup_x[0,1] f x( ).

All three norms are equivalent. For a reader's convenience we outline the proof. First we showTBV. Clearly

BV T

f  f . On the other hand, ( ) for some [0,1]

( ) (0) (0)

BV( ) (0) ,

f f x x

f x f f

f f

  

  

 

thus we obtain

2 BV

f T  f . Next we show BVW. Note that

1

0 f BV( )f



^and

1 1

0 f  0 f(0) 0^x f f(0)BV( )f

  

Thus

2 BV

f W  f . On the other hand, ( ) (0) 0^x

f x  f 



f ^thus f(0) f x( )



0^xf which implies

1 1 1 1

0 0 0 0 0

(0) (0) ^x BV( )

f 



f 



f 

 

f



f  f thus

BV 2

f  f W.

The space B is a Banach space. Let

 

fn be a Cauchy sequence in B with respect to one (and hence to all three) of the norms. Then it is Cauchy in all three norms. We extract

n m 0

f f _ and ¹

0 f_nf_m 0



^and

(0) (0) 0

n m

f f  .

We conclude that there exists f AC[0,1] such that

0 0

( ) (0) ^x (0) ^x ( )

n n n

f x f 



f f 



ff x for each x[0,1].

Note that B is in fact a Sobolev space W^1,1[0,1] and a standard result states that polynomials are dense in W^1,1[0,1], see 3) for example. The following result was proven in 1)

1 1 0

(1) (0) 2 (1)

N

k

k f f

f N f o

 N

     

 

 

 

^,

for f AC[0,1].

In the rest of the paper B will denote the Banach space of absolutely continuous functions on [0,1] equipped with the the BV norm.

If we allow shifts in the Riemann sums we obtain the following.

Theorem Let fB and let 0  1, then we have

1 1

0 0

1 1

( ) ( (1) (0)) (1)

2

N

k

f k N f f f o

N  





       

   

   

 

^.

Proof. Denote

1

0

( ) 1( )

N

k

f N f k

N

 ^ 



 





  ^.

Define a sequence

 

^{FN }_N1of linear functionals on B by

1

( ) ( ) 0

FN^ f f^ N N



f ^,

where f B. Clearly each F_N^ is linear functional on B. Let f B with f 1. Note by the Mean Value Theorem with _k lying in the appropriate intervals, we have

1 0

1 1

0 1

1

0

( ) ( )

1( ) ( )

1( ) ( )

BV( ).

N

N N

k

k k

N

k k

F f f N N f

f k f

N

f k f

N f

 

 

 

 

 





 

 

   

 

 

   

 





 



Thus each F_N^ is a bounded, hence a continuous linear functional on B. In fact we have F_N^ 1 for all N. Consider now f x^p on [0,1] and note that

(1) (0) 1

f f  . Direct calculations show that

lim ( ) 1 ( (1) (0))

N 2

N F^ f  f f



 

    . This can be shown using the Faulhaber's formula, 2),

1

1 0

1 1 1 ( 1)

p N

p j p j

j

k j

k p B N

j p

 

 

  

    

 

with B₁^₂¹ . We expand the following expression in powers of N:

1 1 1

1

0 0 0

( )

N N N

p p p p

k k k

k  k p k  N

  



  

    

  

^.

Observe the coefficient of N^p is equal to 1

2. Define a bounded linear functional F on B as

( ) 1 ( (1) (0))

F^ f ^2^ f  f where f B. Clearly

1 F^   2 . We extend by linearity and claim

lim _N( ) ( )

N F^ p F^ p

 

for all pP, where P denotes the set of all polynomials.

Recall that PB with respect to the given norm. Let f B , not necessarily a polynomial, and choose a polynomial p sufficiently close to f . Let

( ) ( ) ( ) ( )( )

N N N

G^ f F^ f F^ f  F^F^ f . We need to show G_N^( )f 0 as N . To this end consider

( ) ( ) ( ) ( )

( ) ( )

( )

N N N N

N N

N N

G f G f G p G p

G f p G p

G f p G p

   

 

 

  

  

  

and recall that G_N^( )p 0 as N . Moreover, the sequence

 

GN^ N ₁



 is uniformly bounded by 1 2 1

  . This shows that

( ) ( )

FN^ f F^ f as N 

for all fB. The result follows. //

The interval [0,1] can be replaced by [ , ]a b and the following can be stated.

Corollary. Let B_{a b,} AC[ , ]a b denote the Banach space of absolutely continuous functions defined on [ , ]a b with the norm f  f a( )var_{[ , ]a b}( )f for f B. Let f B_{a b,} and let 0  1, then we have

1

0

( )

1 ( ( ) ( )) (1)

2

N

k

b a

f a b a k

N

N f f b f a o

b a









    

 

 

 

      





Proof. Direct calculation by change of variables

( )

x  a b a x yields the result. //

References

− 38 −

1) S. Arimoto, The Second Generation Fukui Project and a New Application of the Asymptotic Linearity Theorem, Bulletin of Tsuyama National College of Technology, 52 (2010) 49-56.

2) Donald E. Knuth (1993). Johann Faulhaber and sums of powers. Math- ematics of Computation 61 (203): 277294.

3) Adams, Robert A.; Fournier, John (2003) [1975]. Sobolev Spaces. Pure and Applied Mathematics. 140 (2nd ed.). Boston, MA: Academic Press.

4. Recent Advances in Chirality Enrichment of Single-Walled Carbon Nanotubes

Tatsuya Murakami

This is a direct continuation of the article entitled

“Near-Infrared Spectrophotometers for Carbon Nanotubes Characterization” [1] by the present author. In this previous article, I discussed how semiconducting and metallic single-walled carbon nanotubes (s-SWNTs and m-SWNTs, respectively) were identified by near-infrared (NIR) spectroscopies. Briefly, s-SWNTs have intense NIR absorption derived from S₁₁ and S₂₂ van Hove transitions and fluoresce upon exitation at S₂₂. Each component of s-SWNTs is defined by the roll-up vector (n,m) on a graphene sheet, in which n – m is not divisible by 3, thereby showing different absorption maxima for the transition. Kataura et al. reported that E11 is inversely propotional to the diameter of SWNTs, which is closely correlated with the vector (n,m) [2]. Then, a more comprehensive study on more than 30 different components of s-SWNTs was reported [3].

Latest chirality enrichment procedures from s-SWNTs are the following three: density-gradient ultracentrifugation (DGU) [4], DNA-controlled aqueous two-phase extraction (ATP) [5,6], and gel chromatography [7,8]. In the DGU method, HiPco SWNTs dispersed with sodium cholate were centrifuged at 268,000 × g for 18 h in a density-gradient medium of 15.0 – 30.0% (w/v) iodixanol containing 0.7%

(w/v) sodium cholate. Ten different (n,m) species were obtained at moderate purity as follows: (6,4) at 47%, (7,3) at 66%, (6,5) at 83%, (9,1) at 36%, (8,3) at 55%, (9,2) at 47%, (7,5) at 46%, (8,4) at 34%, (10,2) at 40%, and (7,6) at 48%. It should also be noted that this method allows separation of the enantiomers of some species. The ATP method was developed based on the authors’ previous finding of short DNA sequences that wrapped around a particular species.

These DNA-wrapped SWNTs were mixed with a variety of the two immiscible aqueous phases rich in polyethylene glycols and other polymers, respectively, and then a particular species moved to either of the two phases, according to the solvation energy difference between them. This method enabled separation of 15 different (n,m) species, (5,5), (7,4), (8,4), (7,5), (6,5), (9,1), (6,4), (6,6), (5,4), (7,6), (8,3), (10,0), (9,2), (10,2), and (7,3), at high purity (data not shown).

Kataura’s group has been developing a series of chirality enrichment procedures by gel chromatography. At first, it was found that s- and m-SWNTS dispersed with sodium dodecyl

sulfate (SDS) showed different affinity to agarose gel, which could be attributed to the different degree of SDS binding between s- and m-SWNTs. This technique was further evolved to achieve chirality enrichment of s-SWNTs by using temperature-controlled gel (dextran beads) filtration chromatography. This rationale is that the binding of s-SWNTs to the gel is dependent on temperature and the degree of SDS coating and the latter is dependent on the chirality. Thirteen different (n,m) species were separated at high purity as follows: (6,4) at 66%, (6,5) at 91%, (7,5) at 58%, (8,3) at 52%, (8,4) at 71%, (7,6) at 73%, and (8,6) at 65%. A modified method recently allowed enrichment of (5,4)-SWNTs of 94% purity, which were the smallest diameter SWNTs obtained in sufficient purity to date [9], as well as improvement of the purity of (6,4)-SWNT (91%). At present, this modified gel-based method is likely advantageous to obtain smaller diameter chirality species in high purity.

The chirality enrichment of s-SWNTs is of pivotal importance in order to reliably assess their detailed characteristics. In biomedical fields, small diameter s-SWNTs, including (6,5)-, (6,4)-, and (9,4)-SWNTs, have been attracting much attention as in vivo NIR imaging agents [10–

12]. Another significant point to be emphasized is that chirality-enriched s-SWNTs would greatly reduce therapeutic doses, leading to alleviation of their toxicity concerns.

References

[1] Arimoto, S., Amini, M., Fukuda, N., Morishima, I., Murakami, T., Naruki, I., Saito, K., Spivakovsky, M., Takeuchi, S., Taylor, K. F., Yamanaka, S., Tokotani, M. and Zizler, P., Mathematics and Chemistry, 2015, 1–2.

[2] Kataura, H., Kumazawa, Y., Maniwa, Y., Umezu, I., Suzuki, S., Ohtsuka, Y.

and Achiba, Y., Syn. Met. 1999, 103, 2555–2558.

[3] Bachilo, S. M., Strano, M. S., Kittrell, C., Hauge, R. H., Smalley, R. E. and Weisman, R. B., Science 2002, 298, 2361–2366.

[4] Ghosh, S., Bachilo, S. M. and Weisman, R. B., Nat. Nanotechnol. 2010, 5, 443–450.

[5] Ao, G., Khripin, C. Y., Zheng, M., J. Am. Chem. Soc. 2014, 136, 10383–

10392.

[6] Tu, X., Manohar, S., Jagota, A., Zheng, M., Nature 2009, 460, 250–253.

[7] Liu, H., Nishide, D., Tanaka, T. and Kataura, H., Nat. Commun. 2011, 2, 309.

[8] Liu, H., Tanaka, T., Urabe, Y. and Kataura, H., Nano Lett. 2013, 13, 1996–

2003.

[9] Wei, X., Tanaka, T., Akizuki, N., Miyauchi, Y., Matsuda, K., Ohfuchi, M.

and Kataura, H., J. Phys. Chem. C 2016, 120, 10705–10710.

[10] Antaris, A. L., Robinson, J. T., Yaghi, O. K., Hong, G., Diao, S., Luong, R.

and Dai, H., ACS Nano 2013, 7, 3644–3652.

[11] Antaris, A. L., Yaghi, O. K., Hong, G., Diao, S., Zhang, B., Yang, J., Chew, L. and Dai, H., Small 2015, 11, 6325–6330.

[12] Yomogida, Y., Tanaka, T., Zhang, M., Yudasaka, M., Wei, X. and Kataura, H., Nat. Commun. 2016, 7, 12056.