ma hema ics
A icle
A Clus e ing Pe spec i e o he Colla z Conjec u e
José A. Ten ei o Machado 1,*,† , Alexand a Galhano 1,† and Daniel Cao Labo a 2,†
Ci a ion: Machado, J.A.T.; Galhano,
A.; Cao Labo a, D. A Clus e ing
Pe spec i e o he Colla z
Conjec u e. Ma hema ics 2021,9, 314.
h ps://doi.o g/10.3390/ma h9040314
Recei ed: 24 Decembe 2020
Accep ed: 29 Janua y 2021
Published: 5 Feb ua y 2021
Publishe ’s No e: MDPI s ays neu-
al wi h ega d o ju isdic ional clai-
ms in published maps and ins i u io-
nal a ilia ions.
Copy igh : © 2021 by he au ho s. Li-
censee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and con-
di ions o he C ea i e Commons A -
ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
1Ins i u e o Enginee ing, Poly echnic o Po o, Rua D . An ónio Be na dino de Almeida,
431, 4249-015 Po o, Po ugal; [email p o ec ed]
2Depa men o S a is ics, Ma hema ical Analysis and Op imiza ion, Facul y o Ma hema ics, Ins i u e o
Ma hema ics (IMAT), Uni e sidade de San iago de Compos ela (USC), Rúa Lope Gómez de Ma zoa s/n,
15782 San iago de Compos ela, Spain; [email p o ec ed]
*Co espondence: [email p o ec ed]; Tel.: +351-228340500
† These au ho s con ibu ed equally o his wo k.
Abs ac :
This manusc ip ocuses on one o he mos amous open p oblems in ma hema ics, namely
he Colla z conjec u e. The i s pa o he pape is de o ed o desc ibe he p oblem, p o iding a
his o ical in oduc ion o i , as well as gi ing some in ui i e a gumen s o why is i ha d om he
ma hema ical poin o iew. The second pa is dedica ed o he isualiza ion o beha io s o he
Colla z i e a ion unc ion and he analysis o he esul s.
Keywo ds: mul idimensional scaling; Colla z conjec u e; clus e ing me hods
MSC: 26A18; 37P99
1. In oduc ion
The Colla z p oblem is one o he mos amous unsol ed issues in ma hema ics.
Possibly, his in e es is ela ed o he ac ha he ques ion is e y easy o s a e, bu e y
ha d o sol e. In ac , i is e en complica ed o gi e pa ial answe s. The p oblem add esses
he ollowing si ua ion.
Conside an i e a i e me hod o e he se o posi i e in ege s
N
de ined in he ollow-
ing way. I
n∈N
is e en, hen we conside he posi i e in ege
1
2n
o he nex s ep. On
he o he hand, i
n∈N
is odd, hen we conside he posi i e in ege 3
n+
1 o he nex
s ep. The Colla z conjec u e s a es ha , independen ly o he chosen ini ial alue o
n∈N
,
he numbe 1 is eached e en ually.
Rema k 1.
Obse e ha , in he case ha Colla z conjec u e does no hold, he e is a posi i e in ege
a∈Nsuch ha :
1. The o bi o a is unbounded, i.e., limn→∞Cn(a) = ∞.
2.
The o bi o
a
is pe iodic and non- i ial, i.e., he e is
N∈N
such ha
CN(a) = a
, o
a6=1, 2, 4.
Rema k 2.
Indeed, i is possible o s udy and p o ide ep esen a ions o he second possibili y in
Rema k 1. Obse e ha , essen ially, his second possibili y deals wi h he exis ence o a posi i e
in ege solu ion o ce ain linea equa ions modulo 2
N
. This idea is de eloped in Sec ion 2, and i is
used o desc ibe some in e es ing ela ions and g aphical ep esen a ions.
In [
1
], we can ind a discussion conce ning he o igin o he p oblem. Du ing he 1930s,
Lo ha Colla z ook an in e es in he i e a ions o some numbe - heo e ic unc ions. Indeed,
i is possible o ind a simila p oblem o he Colla z conjec u e in his no ebook in 1932.
I is also known, and i has been con i med by many o he ma hema icians, ha Colla z
discussed se e al p oblems o his kind in he In e na ional Cong ess o Ma hema icians in
1950 (Camb idge, MA, USA). Ne e heless, i is no clea whe he he 3
n+
1 p oblem was
Ma hema ics 2021,9, 314. h ps://doi.o g/10.3390/ma h9040314 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2021,9, 314 2 o 14
men ioned in hese discussions o no . In any case, wha is clea is ha he p oblem sp ead
apidly du ing ha decade. Acco ding o Richa d Guy and Shizuo Kaku ani, he p oblem
was s udied in Camb idge and Yale o some ime be ween he la ely 1950s and he ea ly
1960s, bu wi h no ema kable esul s. In he las 50 yea s, he ma hema ical communi y
has ied di e en app oaches o he Colla z conjec u e, bu none o hem is belie ed o
p o ide a de ini i e pa h ha would allow o sol e he p oblem. Possibly, i should be
adequa e o highligh con ibu ions in wo di ec ions:
•
On he one hand, he e a e heo e ical a gumen s ha allow p o ing s a emen s
which a e simila o he conjec u e, bu a bi weake . In his sense, we can ind he
con ibu ions in [2,3].
•
On he o he hand, he e a e nume ical expe imen s ha show ha he conjec u e
holds o numbe s which a e smalle han a ce ain h eshold
N∈N
, o ha he
Colla z unc ions does no ha e non- i ial cycles wi h leng h less o equal o
m∈N
.
The alues o
m
and
N
ha e been con inuously imp o ed, and, nowadays, we can
ensu e ha he conjec u e holds o
N<
5.78
×
10
18
(see [
1
]) o ha he leng h o a
non- i ial cycle is, a leas , 1.7
×
10
7
(see [
4
]). These compu a ional a gumen s ha e
been eeding con inuously he opinion ha he Colla z conjec u e is ue, and ha
he e is no non- i ial cycle.
Besides, some au ho s ha e de o ed e o s o ew i e he conjec u e in o he e ms
(see, e.g., [
5
] o an app oach in e ms o algeb aic and boolean ac als). Fu he mo e,
some wo k has been de eloped conce ning he ep esen a ion and s udy o he Colla z
conjec u e in e ms o g aphs [6–12].
The isualiza ion o he Hails one sequences is o en pe o med by means o di ec ed
g aphs. Howe e , while hese ep esen a ions p oduce simple o ead plo s, he ques ion
a ises on how he ‘ ules’ adop ed o he g aphical ep esen a ion pu some addi ional
condi ions on he inal plo . Ha ing his idea in mind, we p opose he adop ion o wo
clus e ing compu a ional echniques, namely he hie a chical clus e ing (HC) and mul idi-
mensional scaling (MDS) me hods, o compu a ional clus e ing and isualiza ion [
13
–
21
].
The i s p oduces g aphical po ai s o da a known as dend og ams and ees in a wo-
dimensional space, while he second consis s o poin loci. In p ac ical e ms, he MDS
se o poin s a e plo ed in ei he wo- o h ee-dimensional cha s. These compu a ional
schemes ha e been adop ed success ully in a numbe o scien i ic a eas and allow un eiling
pa e ns embedded in he da ase [
22
–
26
]. Fo he case o MDS, he possibili y o ha ing
h ee dimensions allows an addi ional deg ee o eedom ha is o u mos impo ance
when handling complex phenomena.
This pape is o ganized as ollows. Sec ion 2discusses he Colla z conjec u e and he
di icul ies posed by his appa en ly simple p oblem. Sec ion 3in oduces he MDS ech-
nique and analyzes he esul s o he Hails one sequences. Finally, Sec ion 4summa izes
he main conclusions.
2. Why Is Colla z P oblem Di icul ?
The e a e se e al easons ha could be conside ed enough o claim ha he Colla z
conjec u e is a eally di icul ma hema ical p oblem. P obably, one o he mos impo an
ones is ha i has been a popula p oblem o a leas i y yea s and i has no been
sol ed ye . Ne e heless, he e a e heu is ic a gumen s ha can gi e a hin on why i is
complica ed o p o ide an answe , o why he mos easonable pa hs o acing he p oblem
end up in no hing.
Obse e ha , due o Rema k 1, o p o e he Colla z conjec u e, i would be enough
o ensu e ha he e a e nei he unbounded o bi s no non- i ial cycles o he Colla z
i e a ion map.
2.1. Numbe Theo y A gumen s
We gi e a hin on why would i be e y ha d o p o e he Colla z conjec u e wi h
s anda d numbe heo y a gumen s. I we imagine ha he Colla z p oblem would admi
Ma hema ics 2021,9, 314 3 o 14
non- i ial pe iodic o bi s, hen we would be able o ind
a∈N
and
N∈N
such ha
CN(a) = a. Fi s , we show how hese equa ions look o small alues o N∈N.
The equa ion C(n) = n eads
n
2=n, i n≡0(mod 2),
3·n+1=n, i n≡1(mod 2).
(1)
The equa ion C2(n) = n eads
n
4=n, i n≡0(mod 4),
3·n
2+1=n, i n≡2(mod 4),
3·n+1
2=n, i n≡1, 3 (mod 4).
(2)
The equa ion C3(n) = n eads
n
8=n, i n≡0(mod 8),
3·n
4+1=n, i n≡4(mod 8),
3·n
2+1
2=n, i n≡2, 6 (mod 8),
3·n+1
4=n, i n≡1, 5 (mod 8),
3·3·n+1
2+1=n, i n≡3, 7 (mod 8).
(3)
We obse e ha , o small alues o
m∈N
, he s udy o he equa ion
Cm(n) = n
can be di ided in o he s udy o
Fm+1
linea equa ions wi h cons ain s modulo 2
m
, whe e
Fk
is he
k
h Fibonacci numbe . This can be gene alized o a bi a ily la ge alues o
m
,
as ollows.
Fi s , obse e ha any linea equa ion would be
m
j=1 kj(n) = n, (4)
whe e
n∈N
,
kj∈ {
0, 1
}
,
0(n) = n/
2,
1(n) =
3
n+
1 and
deno es he composi ion
ope a o . Thus, in p inciple, we would ha e as many exp essions o
Cm(n)
as di e en
choices o
~
k:= (k1
,
. . .
,
km)∈ {
0, 1
}m
, bu we ecall ha no all choices o
~
k
a e possible.
Indeed, i is no possible o apply
1
wo imes in a ow wi hou any
0
be ween hem,
since, gi en any odd numbe
n
, he esul ing numbe
1(n) =
3
·n+
1 will always be
e en. Hence, in gene al, he numbe o equa ions
Cm(n) = n
will be s ic ly ewe han he
ob ious es ima e 2m.
Why is
Fm+1
he numbe o possible linea equa ions o
Cm(n) = n
? As men ioned
abo e, o small alues o
m
, we ha e al eady checked his p ope y. To p o ide a igo ous
p oo o la ge alues o
m
, we only need o use ma hema ical induc ion. Suppose ha
we know ha o
Cm−2(n) = n
we ha e
Fm−1
equa ions. I is always possible o apply
0
o each o he
Fm−1
le -hand sides, ge ing
Fm−1
new equa ions o
Cm−1(n) = n
.
Besides, we ge some addi ional equa ions by applying
1
o some o he le -hand sides
(no all o hem). Due o he induc ion hypo heses, his
1
can be applied o
Fm−Fm−1
le -hand sides. Hence, o
Fm−1
equa ions ha appea in
Cm−1(n) = n
, we can apply
ei he
0
o
1
, and, o
Fm−Fm−1
equa ions, we can only apply
0
. In conclusion, we ha e
Fm+1=2·Fm−1+Fm−Fm−1equa ions o Cm(n) = n.
Why does each equa ion appea ing in
Cm(n) = n
ha e some cons ain modulo 2
m
?
Again, we can use a ma hema ical induc i e a gumen in o de o cla i y his poin . I we
Ma hema ics 2021,9, 314 4 o 14
assume ha all equa ions appea ing in
Cm−1(n) = n
ha e ce ain cons ain s modulo 2
m−1
,
and since we can de e mine
C(n)
modulo 2
m−1
, p o ided we know
n
modulo 2
m
, hen i is
clea ha all equa ions in Cm(n) = Cm−1(C(n)) = nha e some cons ain s modulo 2m.
Acco ding o he wo p e ious pa ag aphs, seeking non- i ial cycles is equi alen
o looking o non- i ial solu ions o one o hese
Fm+1
linea equa ions wi h cons ain s
modulo 2
m
. I we o ge abou he modula condi ion, each o hese equa ions has a
unique solu ion. Besides, i would be possible o compu e hese solu ions induc i ely, a e
de eloping a ecu ence ha compu es he new solu ions in e ms o he p e ious ones.
A e doing his, he idea would be o use some a gumen in ol ing in ege a i hme ic
(cong uences,
p
-adic alua ions, e c.) in o de o show ha he equa ion does no admi
solu ions apa om 1, 2 and 4. This a emp would ail, since he Colla z i e a ion map
is known o ha e mo e cycles han
(
4, 2, 1
)
when de ined on in ege numbe s, allowing
nega i e alues. Thus, he e is no clea obs uc ion in e ms o elemen a y numbe heo y
o ha ing solu ions o Cm(n) = ndi e en om n∈ {1, 2, 4}.
2.2. P obabilis ic A gumen s
O he possible way o ace he p oblem would be he ollowing one. Suppose ha
we a e gi en a numbe in he Colla z sequence ha has been ob ained a e applying he
map
1
: Wha is he expec ed con ac ion ac o o such a numbe un il we apply
1
again?
Obse e ha , in he case ha such a ac o would be smalle han
1
3
, one could y o
de elop p obabilis ic a gumen s ensu ing ha any numbe e en ually sh inks in size and
a i es o he i ial cycle. In he case ha such a ac o would be la ge han
1
3
, one could
y o de elop p obabilis ic a gumen s showing ha unbounded sequences do exis .
A e applying
1
, we ha e a numbe
n
which is known o be e en. Thus, a leas we
will apply
0(n) = n/
2 one ime. Indeed, he numbe
n
will be e en, bu no a mul iple o
4, wi h p obabili y 1
/
2. Analogously, i will be a mul iple o 4, bu no a mul iple o 8, wi h
p obabili y 1
/
4, e c. Obse e ha , i
n
is a mul iple o 2
s
, bu no 2
s+1
, hen we will apply
0
exac ly
s
imes. Thus, he expec ed alue o he i e a ion o
n
which is p e ious o he
s ep o applying 1is: ∞
∑
j=1
1
2j·n
2j=n·
∞
∑
j=1
1
4j=n
3.
Hence, in p obabilis ic e ms, he Colla z i e a ion map is expec ed o nei he sh ink
no expand a numbe in he long- e m.
3. Clus e ing Analysis and Visualiza ion
3.1. Hie a chical Clus e ing
The HC is a compu a ional echnique ha assesses a g oup o
N
objec s in a
n
-dim
space
A
and po ays hem in a g aphical ep esen a ion highligh ing hei main simila i ies
unde he ligh o some me ic [16,27].
The me hod s a s by ga he ing he da ase
A
ha cha ac e izes he phenomenon in
some sense. Usually, we ob ain a numbe
N
o objec s ha ing a high dimensional na u e
which makes i s analysis di icul . The nex s ep is o de ine some me ic o compa ison o
all objec s be ween hemsel es. I is possible o adop measu es o simila i y o , al e na i ely,
o ‘dis ance’. We adop dis ances
d
ha obey he axioms o : (i) iden i y o indisce nibles
d(x,y)=
0
⇔x=y
; (ii) symme y and sub-addi i i y
d(x,y)=d(y,x)
; and (iii) iangle
inequali y
d(x,y)≤d(x,z)+d(z,y)
, whe e
x
,
y
,
z∈ A
. Based on his me ic, an
N×N
ma ix
D=dij
,
i
,
j=
1,
. . .
,
N
, o objec - o-objec dis ances is cons uc ed. The ma ix
D
is symme ic and has main diagonal wi h ze os when adop ing dis ances. The HC uses
he inpu in o ma ion in ma ix
D
and p oduces a g aphical ep esen a ion consis ing in a
dend og am o a hie a chical ee.
The HC equi es using ei he he agglome a i e o di isi e clus e ing i e a i e com-
pu a ional scheme. In he i s , each objec s a s in i s own clus e and he algo i hm
me ges he mos simila i ems un il ha ing jus one clus e . In he second, all objec s s a
in a common clus e and he algo i hm sepa a es hem un il each has i s own clus e . In
Ma hema ics 2021,9, 314 5 o 14
bo h schemes, a linkage c i e ion, based on he dis ances be ween pai s, is equi ed o
calcula ing he dissimila i y be ween clus e s. The maximum, minimum and a e age
linkages a e possible c i e ia [
28
]. The clus e ing quali y can be assessed by means o
he cophene ic co ela ion [
29
]. When he cophene ic co ela ion is close o 1 ( o 0), we
ha e a good (weak) clus e ep esen a ion o he o iginal da a. In Ma lab, he cophe-
ne ic co ela ion is compu ed by means o he command
cophene
. None heless, we
adop he agglome a i e clus e ing and he a e age-linkage [
30
,
31
], wi h he p og am
Phylip h p://e olu ion.gene ics.washing on.edu/phylip.h ml, o p ocessing he ma ix
o dis ances D.
3.2. Mul idimensional Scaling
The MDS is a compu a ional echnique ha ies o ep oduce and isualize in a space
o dimension
n
objec s desc ibed in a space o dimensional
m>n
, whe e he objec s a e
ep esen ed by poin s.
The MDS algo i hm ies o ep oduce he o iginal dis ances by calcula ing a
N×N
ma ix
˜
∆=˜
dij
so ha he eplica ed dis ances
˜
d
minimize some quad a ic index, called
s ess
S
. Consequen ly, he p oblem is con e ed o a nume ical op imiza ion o some
index such as
S=h∑i,j=1,...,Ndij −˜
dij2i1/2
and we ob ain a se o
N
objec s in a space
o dimension
n
ha app oxima e he o iginal ones. Usually, use s adop
n=
2 o
n=
3
since hey allow a di ec isualiza ion. We adop
n=
3 because he plo s allow be e
app oxima ions han he simpe case o
n=
2, bu his equi es some o a ion, shi and
ampli ica ion o ob aining he bes pe spec i e o he isualiza ion.
The ob ained loci o objec s is called a ‘map’ and he quali y can be assessed by means
o he so-called Sheppa d and s ess plo s. The i s one d aws he o iginal e sus he
eplica ed dis ances. A low/high sca e means a good/poo ma ch be ween he dis ances.
Mo eo e , a collec ion o poin s nea a 45 deg ee s aigh (cu ed) line means a linea
(non-linea ) ela ionship. None heless, in bo h cases, he key poin is o ha e a low sca e .
The second ool o assessing he MDS quali y consis s o he plo o
S
e sus
n
. Usually,
we ob ain a mono onic dec easing cu e wi h a signi ican educ ion o
S
a e he ini ial
alues. The inal s ep o he p ocess equi es he use o analyze he MDS map since he
axes ha e no physical meaning and he e is no a p io i assignmen o some good/bad o
high/low in e p e a ion o he coo dina e alues o he poin s.
The in e p e a ion o he map mus ha e in mind he clus e s ha may eme ge and he
pa e ns o med by he poin s. This in e p e a ion is no based on an asce ic pe spec i e, bu
a he in he sense ha hey e lec some ela ionship embedded in he o iginal da ase . The
use can es se e al dis ances because each one may ha e i s owns me i s and d awbacks
in cap u ing he cha ac e is ics o he phenomena unde s udy. In o he wo ds, hese loci
a e usually di e en since each one ollows a dis inc me ic. Consequen ly, we can ha e
mo e han one dis ance p oducing a ‘good’ MDS map. On one hand, his means ha we
may ha e o es a numbe o dis ances o ob ain an eclec ic o e iew, while, on he o he
hand, we may use mo e han one map o isualize and in e p e he esul s.
We calcula e he MDS echnique using he Ma lab classical mul idimensional scaling
command cmdscale.
3.3. The Adop ed Compu a ional Algo i hm
He ea e , we apply he HC and MDS echniques o un a el he e olu ion o he
Hails one sequences. Fo cap u ing he dynamics o he Hails one sequences, we eco d
he successi e numbe s un il eaching he inal alue o 1. To ha e ec o s o iden ical
leng h all emaining alues a e conside ed 0. Finally, he ec o s a e o de ed in he
in e se sequence. This means ha , o example, numbe 6 is ep esen ed as he ec o
x= (
1, 2, 4, 8, 16, 5, 10, 3, 6, 0,
. . .
, 0
)
. We conside a es -bed o six dis ances, namely he
A cCosine, Manha an, Euclidean, Canbe a, Cla k and Lo en zian, gi en by [32,33]:
Ma hema ics 2021,9, 314 6 o 14
dAC =a ccos
m
∑
k=1
xi(k)xj(k)
q∑m
k=1xi(k)2∑m
k=1xj(k)2, (5a)
dMa =
m
∑
k=1xi(k)−xj(k), (5b)
dEu =sm
∑
k=1xi(k)−xj(k)2, (5c)
dCa =
m
∑
k=1xi(k)−xj(k)
|xi(k)|+xj(k)
, (5d)
dCl =
u
u
m
∑
k=1"xi(k)−xj(k)
|xi(k)|+xj(k)#2
, (5e)
dLo =
m
∑
k=1
ln1+xi(k)−xj(k), (5 )
whe e
xj(k)
and
xj(k)
a e he
k
h componen s o he
i
,
j=
1,
. . .
,
N
objec s. Mo eo e ,
he undamen al idea unde lying he Hamming dis ance, usual in in o ma ion heo y, is
adop ed [
34
]. The e o e, when compa ing wo componen s, he esul is 0/1 i hey a e
iden ical/dis inc .
The A cCosine dis ance is no sensi i e o ampli ude and jus p o ides a measu e o
he angle be ween wo ec o s. The Manha an and Euclidean dis ances a e special cases o
he Minkowski dis ance
dMi =h∑m
k=1xi(k)−xj(k)
pi1/p
o
p=
1 and
p=
2, espec i ely.
The Canbe a and Cla k dis ances a e he wo p e ious ones when we subs i u e he
‘absolu e’ di e ence
xi(k)−xj(k)
by he ‘ ela i e’ di e ence
|xi(k)−xj(k)|
|xi(k)|+|xj(k)|
. The e o e, he
Canbe a and Cla k dis ances p o ide a be e iew o alues close o ze o, while he
Manha an and Euclidean dis ances o en ‘sa u a e’ in he p esence o la ge and small
alues. Simila o hese ones, he Lo en zian dis ance adjus s he compa ison o small and
la ge alues by means o he log(·) unc ion.
3.4. MDS Analysis o he Hails one Sequences
We s a by a limi ed se o numbe s, which a e ep esen ed by poin s and iden i ied
by a label co esponding o he numbe .
Figu es 1and 2show he dend og am and he hie a chical ee o he i s 100 numbe s
using he A cCosine–Hamming dis ance dAC, espec i ely.
Figu e 3shows he MDS h ee-dimensional cha o he i s
N=
100 numbe s using
he A cCosine–Hamming dis ance, whe e e en and odd numbe s a e ep esen ed by blue
and ed ma ks, espec i ely. We obse e: (i) he eme gence o a clea h ee-dimensional
s uc u e; (ii) ha e en and odd numbe s a e no de ining he ‘b anches’ in he plo ; and
(iii) well known sequences such as 1
↔
2
↔
4
↔
8
↔
16
. . .
. In a p ac ical pe spec i e,
he poin labels educe signi ican ly he eadabili y and, he e o e, a e no conside ed in
he ollow-up o MDS plo s ackling a la ge da ase . The Sheppa d and s ess plo s a e
no ep esen ed he e o he sake o pa simony and because hey a e o mino ele ance.
None heless, he clus e ing quali y o he achie ed plo was con i med.
Ma hema ics 2021,9, 314 7 o 14
24
816 21 42 84
64
32 75
85 153060
4692
23
70 61 81
35 93
53 27
54
41
82
31
62
47
94
71
55
83
73
97
63
95
91
3612 24 48 96
5
10
20
40 80 13
26
52 69
17 7918 36 72
28 56 37 74 43
86 57
65
4998 87
99
14
22 19 25 3366
100
50
76
38
58
29 51
77 39
78
59
89
67 79
44 88
11
34 45 90
68
1
e en
odd
Figu e 1. The dend og am o he i s 100 numbe s using he A cCosine–Hamming dis ance dAC.
2
4
8
16
21
42
84
64
32
75
85
15
30
60 46
92 2370
61
81
35
93
53
27
54
41
82
31
62
47
94 71
55 83 73 97 63
95
91
3612
24
48
96
5
10
20
40 80
13
26
52
69
17
7
918 36 72
28
56 37 74 43
86
57
65
49
98
87
99
14
22
19
25
33
66
100
50
76
38
58
29
51
77
39
78
59
89
67
79
44
88
11
34
45
90
68
1
e en
odd
Figu e 2.
The hie a chical ee o he i s 100 numbe s using he A cCosine–Hamming dis ance
dAC
.
Ma hema ics 2021,9, 314 8 o 14
Figu e 3.
The MDS h ee-dimensional cha o he i s
N=
100 numbe s using he A cCosine–
Hamming dis ance dAC.
We now es he six dis ances (5) o
N=
10
4
numbe s. The esul ing h ee-dimensional
MDS maps a e depic ed in Figu es 4–9, o he dis ances
dAC
,
dMa
,
dEu
,
dCa
,
dCl
and
dLo
,
espec i ely.
Figu e 4.
The MDS h ee-dimensional cha o he i s
N=
10
4
numbe s using he A cCosine–
Hamming dis ance dAC.
Ma hema ics 2021,9, 314 9 o 14
Figu e 5.
The MDS h ee-dimensional cha o he i s
N=
10
4
numbe s using he Manha an–
Hamming dis ance dMa.
Figu e 6.
The MDS h ee-dimensional cha o he i s
N=
10
4
numbe s using he Euclidean–
Hamming dis ance dEu.
