数学と化学の学際共同研究と福井プロジェクト XXI
有本

茂*1 Massoud Amini*2

村上達也*6 成木勇夫*7

Hao Chen*3 福田信幸*4 Joseph E. LeBlanc*5

Mark Spivakovsky *8Hong Yi Wong*11

山中 聡*12竹内茂*9

Keith F. Taylor*10

横谷正明*13 Peter Zizler*14

Mathematics and chemistryinterdisciplinary joint research and the Fukui Project XXI
Shigeru ARIMOTO, Massoud AMINI, Hao CHEN, Nobuyuki FUKUDA, Joseph E. LEBLANC
Tatsuya MURAKAMI, Isao NARUKI, Mark SPIVAKOVSKY, Shigeru TAKEUCHI
Keith F. TAYLOR, Hong Yi WONG, Satoshi YAMANAKA, Masaaki YOKOTANI and Peter ZIZLERThis is the 21st part of the series of articles that records and further develops essentials of the Mathematics andChemistry 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 wasdevoted to the memory of the late Professor Kenichi Fukui (1981 Nobel Prize) who initiated the project. The present seriesalso 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 amongchemists despite the importance of these notions in elucidating additivity and high-speed asymptotic phenomena inmolecules having many repeating identical moieties. Some problems are formulated in terms of Fourier analysis connectedto the theory of analytic curves, others 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) 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 29 年 9 月 21 日
＊1, ＊12, ＊13 総合理工学科＊4 総合理工学科非常勤講師

7. Unifying Approach in the
Repeat Space Theory (RST) II＊2 Dept. of Math. Tarbiat Modares University, Iran
＊3 Dept. of Fund. Ed., Dalian Neusoft University of Information, China＊5 School of Integrated Studies, Pennsylvania College of Technology
USA

Shigeru Arimoto, Massoud Amini, Hao Chen
Nobuyuki Fukuda, Isao Naruki, Mark Spivakovsky
Shigeru Takeuchi, Keith F. Taylor, Satoshi Yamanaka
Masaaki Yokotani and Peter Zizler＊6 富山県立大学 工学部・医薬品工学科＊7 立命館大学 理工学部・数学物理学系・数理科学科・元教授＊8 CNRS and Institute de Mathématiques de Toulouse, France＊9 岐阜大学 教育学部・数学科＊10 Dept. of Math. and Stat., Dalhousie University, Canada

I Introduction

This is a direct continuation of Section 2 of Part XIX.
In this section, we wish to unify the proofs of different
versions of the Asymptotic Linearity Theorem Extension＊11 School of Communication, Arts and Social Sciences,
Singapore Polytechnic, Singapore＊14 Dept. of Math., Phys., and Eng., Mount Royal University, Canada－17－津 山 高 専 紀 要

第５９号 （２０１７）

Conjectures (ALTECs).
C(I):

1. Introduction to the Asymptotic Linearity Theorem
Extension Conjectures (ALTECs)
To prove the Asymptotic Linearity Extension
Conjectures (ALTEC) [1-4] was a fundamental problem
in repeat space theory. In Refs. [1,4], the Asymptotic
Linearity Extension Conjectures, C(I) and CBV(I)versions, were proved for the first time by using two
types of continuous fractal functions T and K and byusing the Cantor function, by a method similar to proving
Theorem 1 via the ALT (that is, by a method similar toproving Zizler’s Theorem via the ALT).

(  : a = t0 ≤ t1 ≤ . . . ≤ tn = b).
the Banach space of all real-valued continuousfunctions on I equipped with the norm givenby||  || = sup {|  (t)|: t  I}.

(2)(In what follows, C(I) is often denoted by
C[a, b].)CBV(I) : the Banach space of all real-valued continuousfunctions of bounded variation on I equippedwith the norm given by||  || = sup {|  (t)|: t  I} + VI(  ).
AC(I):In Part XI [2], some tools for the proofs of theconjectures have been developed, but until now we could
not see clearly the above versions in a single perspective.

the Banach space of all real-valued absolutely
continuous functions on I equipped with thenorm given by||  || = sup {|  (t)|: t  I} + VI(  ).

In this section, we unify the two proofs of the
Asymptotic Linearity Theorem Extension Conjecture(ALTEC) C(I) and CBV(I) versions and give solutions to
the Challenging Problems V and VI in Ref. [3].

(3)

(4)

Before recalling the following practical version of the
Asymptotic Linearity Theorem [5], the reader is referredto the Appendix to Part XX for the definition of the repeat
space Xr(q).

We retain the notation of Refs. [2-4], and recommend
the reader to review these references before proceeding
further. In what follows, we recall only essential notationand definitions from Refs. [2-4].
Throughout this section, let + anddenote,respectively, the sets of all positive integers and real
numbers.
Recall that a real sequence EN is said to have an
asymptotic line if there exist ,   such that EN – (N+ )  0 as N   , that is, if there exist ,   suchthat EN = N +  + o(1), as N   , where o(1) denotes
the Landau notation. Note that for a real sequence EN tohave an asymptotic line, it is necessary for the sequence

Theorem 1 (Practical ALT, Xr(q)-version). Let {MN} 
Xr(q) be a fixed repeat sequence, let I be a fixed closed
interval compatible with {MN}. Then, for any  AC(I), there exist (  ), (  )  R such thatTr  (MN) = (  )N + (  ) + o(1)(5)
as N   .
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.

EN/N to have a limit   lim  EN / N   . If this limit
N 

exists, the sequence EN has an asymptotic line if and only
if   lim  EN   N  exists in .
N 

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 boundedvariation defined on the closed interval I.

In the present article, a real sequence EN is said to be
on a line if there exist ,   such that EN – (N + ) =0 for all N  +.
Notation Let I = [a, b] (a, b  , a < b) denote a closed
interval.
VI (  ):the total variation of a real-valued function

 on I, that is,
VI (  ) = supn|  (ti ) –  (ti - 1)|

2. Unification of the proofs of the ALTECs
In recent publications [1] and [4], one of the authors

(1)

i 1

－18－

数学と化学の学際共同研究と福井プロジェクトXXI 有本・Amini・Chen・福田・LeBlanc・村上・成木・Spivakovsky・竹内・Taylor・Wong・山中・横谷・Zizler(S.A.) has proved the above two conjectures by using thefollowing two Lemmas A0, A1, which were established,respectively, in Refs. [1,4]. (In Ref. [1], Lemmas A0 and
A1 were called Lemmas 2.1 and A1, respectively.)

Assume (i). Since f(N) has an asymptotic line, there
exist ,0 0  such thatf(N) = 0N + 0 + o(1),(10)
as N   . Hence, we havef(mN) = 0mN + 0 + o(1),(11)
as N   . By (10) and (11), we see that

Lemma A0. If there exists a   C[0, 1] such that thereal sequence EN(  ) :=  k 1  (k/N) does not havef (mN )  f ( N )1  0 (m  1)  o   ,
N
N

N

an asymptotic line, then the Asymptotic Linearity

as N   .
Since the right-hand side of (12) is A by (6), letting N  , we infer that

Theorem Extension Conjecture (ALTEC) C(I)-version istrue.
Lemma A1. If there exists a   CBV[0, 1] such that0 

the real sequence EN(  ) :=  k 1  (k/N) does notNhave an asymptotic line, then the Asymptotic Linearity

is true.
The purpose of this section is to recall the prototypal
approach given in [1] and develop new unifying tools forgetting a deeper insight into the both versions of ALTEC.
For this purpose, we prove the following two key lemmas
B and C.

(13)

and let   lim N  g ( N )  . Then f(N)  (N +)  0 as N   . Thus, we have (i).
Now under the assumption (i), suppose that for someN0  +
g(N0)  0.
(15)
Then we would have by (9),
0  g(N0) = g(mN0) = g(m2N0) = g(m3N0) = …, (16)which contradicts (14). Thus, we haveg(N) = 0(17)
for all N  +, that is,f(N) = N + 0(18)
for all N  +. We therefore infer that the sequence is ona line. We conclude that (i) => (iv). Clearly, (iv) => (ii)=> (i). This completes the proof.
//

Lemma B. Let f: Z+  R be a real sequence. Supposethat there exist a positive integer m  2 and a real
number A such that the equalityf(mN) = f(N) + AN
(6)+holds for all N  Z . Let  := A/(m  1) and let g(N) :=f(N)   N. Theng(mN) = g(N)(7)+. Moreover, the following conditions areequivalent.
(i)
(ii)
(iii)
(iv)

A
= .
m 1

Note that (10), (13) and the definition of g(N) imply
thatg(N) = f(N)  N = 0 + o(1)(14)
as N   . Thus, the sequence g(N) must converge to 0.
We conclude that (i) => (iii).
The converse (iii) => (i) is easy to prove: Assume (iii)

Theorem Extension Conjecture (ALTEC) CBV(I)-version

for all N (12)

The sequence f(N) has an asymptotic line.
The sequence f(N) is on a line.
The sequence g(N) converges.
The sequence g(N) is a constant sequence, thatis, g (1)  g (2)  ... .

The following Lemma B’ is an improvement of Lemma
B. This improvement is due to H. Chen:
Lemma B’. Let f: Z+  R be a real sequence. Supposethat there exist a positive integer m  2 and a real
number  such that the equalityf(mN) = f(N) +  (m  1)N(19)

Proof. First, observe that the following equalities hold for
all N  +.
g(mN)  g(N) = f(mN)   mN  f(N) +  N= AN   (m  1)N= AN  AN = 0.
(8)
Thus,
g(mN) = g(N)(9)
for all N  +.

holds for all N  Z+. Let g(N) := f(N)   N. We haveg(mN) = g(N)(20)
for all N  +. Moreover, the following conditions areequivalent.
(i) The sequence f(N) has an asymptotic line.
(ii) The sequence f(N) is on a line.

－19－津 山 高 専 紀 要

第５９号 （２０１７）

In what follows, we provide solutions to the following
challenging problems stated in [3]. See Figs 1 and 2 on
the next page for the graphs of EN(T + K) and EN(T K) numerically calculated by MATLAB.

(iii) The sequence g(N) converges.
(iv) The sequence g(N) is a constant sequence, i.e.,g (1)  g (2)  ... .
Proof. The proof proceeds in the same way as the proof of
Lemma B.
//

Challenging problem V. Does EN(T + K) have an
asymptotic line?

Note that in Lemma B’,  plays a role of ‘mthdifference coefficient’ sincef(mN) = f(N +(m  1)N)= f(N) +  (increment from N)= f(N) +  (m  1)N.

Challenging problem VI. Does EN(T  K) have an
asymptotic line?
Solution: From [3] references therein, recall the relations
E2N(T) = EN(T) + N,
(22)
E4N(K) = EN(K) + N,
(23)+
valid for all N  .
By (22), we have
E4N(T) = E2N(T) + 2N = EN(T) + 3N.
(24)
By the linearity of E4N and EN, we easily see that
E4N(T + K) = EN(T + K) + 4N,(25)
E4N(T  K) = EN(T  K) + 2N,(26)+
for all N  .
Now, it is easily checked that neither of EN(T + K) andEN(T  K) is on a line. Therefore, neither of EN(T + K)
and EN(T  K) has an asymptotic line.

Although Lemma B’ is better to geometrically understandthe meaning of the assertion of the above two versions B
and B’, we keep old version B to highlight therelationship between both versions and our earlier
arguments in Refs. [1-3].
Lemma C. Fix either X = C[0, 1] or X = CBV[0, 1]. For
any   X, let EN(  ) denote the real sequence definedby EN(  ) :=



N

k1

 (k/N). Suppose that there exists a

positive integer m  2, real number A, and   Xsuch that
EmN(  ) = EN(  ) + AN
for all N (21). Let  N ( ) := EN(  )  (A/(m  1))N.

+

References

Then, the following conditions are equivalent.

[1] S. Arimoto, Proof of the Asymptotic Linearity Theorem Extension(i) The sequence EN(  ) has an asymptotic line.
(ii) The sequence EN(  ) is on a line.
(iii) The sequence  N ( ) converges.
(iv) The sequence  N ( ) is a constant sequence,i.e., 1 ( )   2 ( )  ... .

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 XI", Bulletin of National Institute ofTechnology, Tsuyama College 57 (2015) 59-66.

Moreover, if one of the conditions (i), (ii), (iii), (iv) is not
fulfilled, then the Asymptotic Linearity Theorem
Extension Conjecture (ALTEC X version) is true.

[3] 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 XII", Bulletin of National Institute

Proof. Suppose that there exists a   X, a positive
integer m  2, and a real number A such that the real
sequence EN(  ) satisfies (21) for all N  +. Fix such a  X, an m  2, and real number A, and set EN =EN(  ) and  N =  N ( ) . We know from Lemmas A0
and A1 that if (i) is not fulfilled, then the Asymptotic
Linearity Theorem Extension Conjecture (ALTEC X
version) is true. Thus, for the proof of the lemma, wehave only to show the equivalence of the conditions (i),(ii), (iii), and (iv). Set f(N) = EN(  ) and g(N) =  N ( ) ,and apply Lemma B, then the conclusion follows.
//of Technology, Tsuyama College 57 (2015) 67-72.
[4] 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 Instituteof Technology, Tsuyama College 57 (2015) 73-78.
[5] S. Arimoto and K.F. Taylor, Practical Version of the Asymptotic
Linearity Theorem with Applications to the Additivity Problems of
Thermodynamic Quantities, J. Math. Chem. 13 (1993) 265-275.

－20－

数学と化学の学際共同研究と福井プロジェクトXXI 有本・Amini・Chen・福田・LeBlanc・村上・成木・Spivakovsky・竹内・Taylor・Wong・山中・横谷・Zizler

Fig. 1. Sequence  N (T  K ) calculated by using MATLAB

Fig. 2. Sequence  N (T  K ) calculated by using MATLAB－21－

津 山 高 専 紀 要

8.

第５９号 （２０１７）

Mindful Frontier International Project(MFI Project)

Shigeru Arimoto, Nobuyuki Fukuda, Hong Yi Wong
Satoshi Yamanaka and Masaaki Yokotani
1.

Origin of the Mindful Frontier International
Project

A new project called Mindful Frontier International
Project (MFI Project) started within the framework of the
New Frontier Project, along with the Science-Art Multi-angle
Network (SAM Network). This MFI project originates fromearlier teachings of Prof. K. Fukui, his teacher Prof. Haruo
Shingu, and Prof. Fukui’s English acquaintance, Prof. G.G.
Hall (Applied Mathematics). [Remarks. Prof. Hall is apioneer in Quantum Chemistry, noted for his discovery of the
Roothaan-Hall equations. Prof. Shingu was also acollaborator in the initial development of Fukui’s FrontierElectron Theory. It is also well-known that Prof. Shingu gave
the name of the ‘Frontier Electron Theory’ to Fukui’s
pioneering theory. Prof. G.G. Hall, Nottingham University,
England, was invited to Kyoto University, in 1983, as thefirst full-time non-Japanese professor of national university in
post-war Japan, through the efforts of Prof. Fukui and his
Japanese colleagues.]

2.

MFI Project for Innovative Thinking and Global
Cross-disciplinary Education

The MFI Project originally aimed at enhancing the lively
and creative activities of students and researchers using theassets of teaching mainly from the above-mentioned threepioneers in science. The first author of this section (S.A.)
who was a student of all the three professors in Kyoto
University, recalled their lectures, seminars, publications, andpersonal conversations which were especially related toinnovative thinking and global cross-disciplinary education.
The first author began to store and share the assets from theseteachers with the members of the New Frontier Project andalso with people outside the circle of this project.

3.

A new element began to be added to the MFI Project aroundthe time when the first author visited Singapore Polytechnic
in 2015 together with students from Tsuyama College forinternational global education. Singapore Polytechnic (SP)provides multi-cultural international education for students
from non-English speaking countries. Singapore itself is amulti-racial and culturally diverse society. SP is aware of
the challenges that come with learning a foreign language,and it seeks to help students (both from Singapore andabroad) to speak and express themselves in English
effectively regardless of their mother tongue. SP positively
accepts and celebrates accents of English beyond those of a
British or North American origin. To the non-native speakerof English, accents such as Japanese, Filipino or
Singaporean ought to be seen as a distinctive feature, not a
weakness, given the diverse world we live in. We witness
Asian students with accents having a psychologicalinhibition that hinders their natural flow of speaking.
Combined with other public speaking techniques, thispsychological recognition and positive acceptance ofdiversity can instill greater confidence in the student andenable more effective use of English as a tool for global
communication.
Through contacts with the teachers and instructors in
Singapore Polytechnic and through actual participation in
their English lectures for foreign (Japanese) students, thefirst author recalled Prof. Fukui’s and Prof. Hall’s seminars
in Kyoto in 1980s, in which all the participants had to speakin English.
Through cooperative discussions, the authors of thissection noticed the possibility of adding the pedagogicalmethods of Singaporean teachers to those of Prof. Fukui,Prof. Hall, and Prof. Shingu, for international global
education and also for improving public speeches andlectures in English.
Here in this section, we pose the following questions, andwe will give our answers to these questions in a future
publication elsewhere.

MFI Project for International Global Educationand for Improving Public Speeches and Lectures inEnglish－22－(1) How can one give international global education
effectively to students?
(2) How can one give lectures in English effectively to
students whose mother tong is not English? [Aspecific question belonging to category (2): For
mathematics or science lectures in English, the firstauthor developed English & mother tong paralleltext method and the ‘variable bilingual method’. Can

数学と化学の学際共同研究と福井プロジェクトXXI 有本・Amini・Chen・福田・LeBlanc・村上・成木・Spivakovsky・竹内・Taylor・Wong・山中・横谷・Zizler

these methods be combined with Singaporeanteachers’ more visual and schematic methods?]

N

EN =  (  /2)(i(AN))1/2,

(1)

i 1

(3) How can one train students for presentations inEnglish?
In a ‘Challenge Seminar’ in Tsuyasma College, the firstauthor taught techniques of brainstorming, analysis,organization, and structuring of data for presentation.
Besides these standard techniques he also taught, especiallyfor ‘rigid students’ with mental inhibitions, the methods
involving (i) Yoga-Zen-type exercises, (ii) breathing
exercises, (iii) meditative exercises, which were all
non-religious and neutral and thus suitable for any student
from all over the world. In our future publication, we will
also discuss these methods, which Prof. Shingu practiced in
his later years, and which potentially enhance innovativeand organizing power of those who practice the above
exercises regularly.

9.

Interdisciplinary Challenging Problems Related
to Fourier Analysis

Shigeru Arimoto, Nobuyuki Fukuda, Joseph E. LeBlanc
Satoshi Yamanaka and Masaaki Yokotani
1.

Quantum Linear Chain ChN (m, k)

Let ChN (m, k) denote the quantum mechanical network
system of linear chain with free ends consisting of N
particles each of mass m and separation 1 that can vibrate
harmonically under a restoring force due to thefirst-neighbor interaction k > 0. The network ChN (m, k) iscompletely characterized by the N  N positive-semidefinite
real symmetric matrix AN which is referred to as the
mass-weighted Hessian matrix of ChN (m, k):

 1

1


AN  (k / m)










0


1 2 


  


 2 10
1 2 1
1 1 .
1
2 1[Note. This matrix sequence with m = k = 1 is nothing but
the matrix sequence we used for proving the ALTECs andZizler’s Theorem via the ALT.] The zero-point vibrationalenergy of the ChN(m, k) with fixed m, k > 0, is expressed interms of the N eigenvalues of AN:

where i(AN) denotes the i-th eigenvalue of AN, or,equivalently, in terms of the trace of the function of thematrix:EN = Tr(  /2)(AN)1/2.

(2)

We know that the number sequence EN has an asymptotic
line, namely:Tr(  /2)(AN)1/2 =  N +  + o(1),(3)

as N  , where = (  /2)(k/m)1/2(4/), =  (  /2)(k/m)1/2,
and o denotes the Landau notation.
2.

Classical Linear Chain CChN (m, k)

Let x(t) = (x1(t), x2(t), …, xN(t))T. The above quantum
mechanical network system of linear chain ChN (m, k) has aclassical mechanical analogue, which we denote byCChN(m, k). This system can be studied theoretically bysolving the following differential equation with given initialconditions.
d2x(t)/dt2 =  ANx(t).
3.

Relation between ChN (m, k) and CChN (m, k), andChallenging Problems

It is well known that the above quantum and classicalsystem involve the same eigenvalue analysis of the matrix
AN. We are especially interested in the case where N is large
enough. In such a case, one can study Ch(m, k) and its
classical analogue CChN(m, k) in connection with the Fukui
Conjecture and the Asymptotic Linearity Theorem
Extension Conjectures (ALTECs), (cf. [1-4]). Moreover, as
is well known, if N is large enough, the above oscillatory
system CChN(m, k) can be regarded as a continuum
oscillator with a good approximation. The reader is referred
to LeBlanc’s Physics Laboratory Manual [5] forexperiments involving transversal and longitudinal waves ina continuum oscillatory system and for experiments
involving other interesting wave phenomena. Bearing in
mind the above-mentioned quantum and classicalvibrational systems, and the limiting correspondence
between discrete and continuum systems, we pose thefollowing interdisciplinary－23－

津 山 高 専 紀 要

第５９号 （２０１７）[10] G.G. Hall and S. Arimoto, Eigenvalue Distributions and Asymptotic

Challenging Problems CQF:

Lines of the Energy in Alternant Hydrocarbons, Int. J. Quantum Chem. 45(1993) 303-328.

(i)

(ii)

To study the above classical oscillator system
CChN(m, k) for a large enough N, referring to [5-7]and to the technique of Fourier analysis given inarticles [8-10] for quantum chemistry problems.
To study the above quantum oscillator system
ChN(m, k) using the method of difference
equations (not differential equation), referring to[5-10] and recalling the Fukui Conjecture.

It is hoped that the above challenging problems help enrich
the interdisciplinary investigation in physics andmathematical chemistry related to Fourier analysis. We
remark that Fourier analysis is one of our common
platforms for our interdisciplinary investigations in the New
Frontier (Fukui) Project.

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 XI", Bulletin of National Institute ofTechnology, Tsuyama College 57 (2015) 59-66.
[3] 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 XII", Bulletin of National Institute ofTechnology, Tsuyama College 57 (2015) 67-72.
[4] 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 ofTechnology, Tsuyama College 57 (2015) 73-78.
[5] J.E. LeBlanc, PHYSICS Laboratory Manual, Physics with TechnologicalApplications (Kendall Hunt, Dubuque, 2016).
[6] J.E. LeBlanc, Private notes and communications.
[7] L.A. Pipes, L.R. Harvill, Applied Mathematics for Engineers andPhysicists (McGraw-Hill, Singapore, 1971).
[8] G.G. Hall, Private notes and communications.
[9] S. Arimoto and G.G. Hall, Integral Representation of a Fundamental
Functional for the Study of the Zero-Point Vibrational Energy ofHydrocarbons and the Total Pi-Electron Energy of Alternant Hydrocarbons,Int. J. Quantum Chem. 46 (1992) 612-635.

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