A Note on non-Newtonian Isometry
OGUZ OGUR, ZEKIYE GUNES
Department of Mathematics,
Giresun University,
Giresun-Gure,
TURKEY
Abstract: In this article, we introduce non-Newtonian isometry and examine some of its basic properties. We
also give a characterization of the relationship between real isometry and non-Newtonian isometry. Finally, we
show that the ν−measure of ν−measurable sets is invariant for every generator under ν−-isometries.
Key-Words: Non-Newtonian calculus, geometric calculus, isometry, ν-isometry, measure, ν-measure, ν-inner
measure, ν-outer mesaure
Received: March 22, 2023. Revised: October 23, 2023. Accepted: November 21, 2023. Published: February 6, 2024.
1 Introduction
Non-Newtonian calculus, which is used in many
fields such as engineering, mathematics, finance, eco-
nomics, medicine and biomedicine, was developed
between 1967 and 1970 as an alternative to the classi-
cal analysis of Newton and Leibnitz, [1, 2]. The book
’Non-Newtonian Calculus’, which forms the basis of
non-Newtonian calculus, was published in 1972, [3].
The derivative and integral were investigated in the
metacalculus, [4]. Geometric calculus and their appli-
cations were investigated in [5]. Some basic topolog-
ical properties of the real non-Newtonian axis were
investigated in [6]. The non-Newtonian Lebesgue
measure for non-Newtonian open sets was defined in
[7]. Finally, the non-Newtonian measure for closed
non-Newtonian sets was defined and some related
theorems were given in [8]. For more details see,
[9],[10],[11],[12],[13],[14],[15],[16],[17],[18],[19],
[20].
Let νbe a generator, which means that νis a bijec-
tion function from Rto a subset Aof R. Let ˙p, ˙q∈A.
Then the ν−arithmetic is defined as follows;
ν−addition ˙p˙
+ ˙q=ν{ν−1( ˙p) + ν−1( ˙q)}
ν−subtraction ˙p˙
−˙q=ν{ν−1( ˙p)−ν−1( ˙q)}
ν−multiplicative ˙p˙
×˙q=ν{ν−1( ˙p)×ν−1( ˙q)}
ν−division ˙p˙
/ ˙q=ν{ν−1( ˙p)/ν−1( ˙q)}
(ν−1( ˙q)= 0)
ν−order ˙p˙
≤˙q⇔ν−1( ˙p)≤ν−1( ˙q)
The set of ν−integers is
Zν=Z(N) = . . . , ν(−1), ν(0), ν(1), . . . .
The set Rν=R(N) = {ν(a) : a∈R}is called
the set of non-Newtonian real numbers.
The absolute value of non-Newtonian number ˙a∈
A⊂Rνis denoted by |˙a|Nand define as follows;
|˙a|ν=
˙a , ˙a˙
>ν(0)
ν(0) ,˙a=ν(0)
ν(0) ˙
−˙a , ˙a˙
<ν(0)
Accordingly,
√˙a2N
N=|˙a|N=ν|ν−1( ˙a)|
is written for each ˙uin the set A⊂Rν, [21].
Definition 1. The non-Newtonian outer measure of
a non-empty ν−bounded set Kis the greatest lower
bound of the measures of all ν−bounded, ν−open
sets containing the set K. So it is defined by
m∗
NK=νinf
K⊂G{mNG},
[22].
Definition 2. The non-Newtonian interior measure of
a nonempty ν−bounded set Kis the smallest upper
bound on the measures of all ν−closed sets contained
in the set K. So it is defined by
m∗NK=νsup
F⊂K{mNF},
[22].
Theorem 1. Given a ν−bounded set K. If ∆is a
ν−open set containing the set K, then we have the
following equation;
m∗
NK˙
+m∗NCK
∆=mN∆,
[22].
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Definition 3. If the non-Newtonian inner and outer
measure of a ν−-bounded set Kare equal, the set K
is called a non-Newtonian Lebesgue-measurable set
or simply the ν−-measurable set, [22].
Theorem 2. If the set Kis the ν−measurable set in
Rν, then ν−1(K)is the measurable set in R, [22].
Theorem 3. Given a ν−bounded set E. If the set E
can be written as a combination of finite or countably
infinite sets of pairwise disjoint ν−measurable sets
Ek, then Eis ν−measurable and
mNE=νX
k
mNEk
equality is fulfilled, [23].
2 Main Results
In this section, we introduce non-Newtonian
isometry and examine some of its basic proper-
ties. We also give a characterization of the rela-
tionship between real isometry and non-Newtonian
isometry. Finally, we show that the ν−measure of
ν−measurable sets is invariant for every generator
under ν−isometries.
Definition 4. Let φν:Rν→Rνbe a function such
that
|φν(x)˙
−φν(y)|N=|x˙
−y|N
for every x, y ∈Rν, then the function φνis called a
non-Newtonian isometry or ν−isometry.
Example 1. Let φν:Rν→R+(N)be a ν−isometry
and let the generator νbe the function exp. Hence, we
have
φν(x)˙
−φν(y)N=νν−1(φν(x)) −ν−1(φν(y))
=exp {|ln(φν(x)) −ln(φν(y))|}
=exp
ln φν(x)
φν(y).
Also, we can write that
x˙
−yN=νν−1(x)−ν−1(y)
=exp {|ln(x)−ln(y)|}
=exp
ln x
y.
Thus, we get
exp
ln φν(x)
φν(y)=exp
ln x
y
and so
ln φν(x)
φν(y)
=
ln x
y
.
Theorem 4. Let ν:A⊆R→Rνbe the generator
function and let φν:Rν→Rνbe a non-Newtonian
function. If φνis an ν−isometry, then the function
ν−1◦φν◦νis an isometry in A⊆R.
Proof. Since the function φνis an ν−isometry, we
have
φν(x)˙
−φν(y)N=x˙
−yN.
Thus, we write
νν−1(φν(x)) −ν−1(φν(y))
=νν−1(x)−ν−1(y)
and
ν−1(φν(x)) −ν−1(φν(y))=ν−1(x)−ν−1(y).
This gives
ν−1◦φν◦νν−1(x)−ν−1◦φν◦νν−1(y)
=ν−1(x)−ν−1(y)
which completes the proof.
Theorem 5. Let φνbe a ν−isometry. Then, we have
the following properties;
a) If A⊂B, then φν(A)⊂φν(B),
b) φνS
k
Ek=S
k
φν(Ek),
c) φνT
k
Ek=T
k
φν(Ek).
d) If E0is an empty set, then φν(E0) = E0.
Proof. Since φνis a ν−isometry then ν−1◦φν◦νis
an isometry.
a) If A⊂B, then ν−1(A)⊂ν−1(B). Thus, we have
⇒ν−1◦φν◦ν(ν−1(A)) ⊂ν−1◦φν◦ν(ν−1(B))
⇒ν−1◦φν(A)⊂ν−1◦φν(B)
⇒φν(A)⊂φν(B).
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b) Since ν−1S
k
Ek=S
k
ν−1(Ek), then we get
⇒ν−1◦φν◦ν ν−1 [
k
Ek!!
=[
kν−1◦φν◦νν−1(Ek)
⇒ν−1◦φν [
k
Ek!=[
kν−1◦φν(Ek)
⇒ν−1 φν [
k
Ek!!=ν−1 [
k
φν(Ek)!
⇒φν [
k
Ek!=[
k
φν(Ek).
c) Since ν−1T
k
Ek=T
k
ν−1(Ek)and ν−1◦φν◦ν
is an isometry, we have
⇒ν−1◦φν◦ν ν−1 \
k
Ek!!
=\
kν−1◦φν◦νν−1(Ek)
⇒ν−1◦φν \
k
Ek!=\
kν−1◦φν(Ek)
⇒ν−1 φν \
k
Ek!!=ν−1 \
k
φν(Ek)!
⇒φν \
k
Ek!=\
k
φν(Ek).
d) Since ν−1◦φν◦νis an isometry, we get
⇒ν−1◦φν◦ν(E0) = E0
⇒φν(ν(E0)) = ν(E0)
⇒φν(E0) = E0.
Example 2. Consider the geometric arithmetic gen-
erated by the function ν(x) = ex. If φνis a
ν−isometry for x, c ∈R+, then either
φν(x) = x˙
+c=x.c
or
φν(x) = c˙
−x=c
x.
Proof. Let φν(1) = c. Then for every x,
•If φν(x) = x.c then
|φν(x)˙
−φν(1)|N=|x.c ˙
−c|N
=|νν−1(x.c)−ν−1(c)|N
=|exp {ln x}|N
=|exp {ln x−ln 1}|N
=|x˙
−1|N
and
•If φν(x) = c
xthen we have
|φν(x)˙
−φν(1)|N=
c
x˙
−cN
=νnν−1c
x−ν−1(c)oN
=
exp ln 1
xN
=|exp {ln 1−ln x}|N
=|exp {ln x−ln 1}|N
=|x˙
−1|N.
Let define the following function;
φν(x) = x(−1)σ(x).c [σ(x) = 0,1].
Let take xand ysuch that x= 1,y= 1 and x=y.
Thus, we have
φν(x)˙
−φν(y) = x(−1)σ(x).c ˙
−y(−1)σ(y).c
=νnν−1x(−1)σ(x).c−ν−1y(−1)σ(y).co
=exp nln x(−1)σ(x).c−ln y(−1)σ(y).co
=exp (ln x(−1)σ(x)
y(−1)σ(y))
=x(−1)σ(x)
y(−1)σ(y)
=x
yp(−1)σ(x)
Here, since p= (−1)σ(y)−σ(x),p=−1or1.
By the last equality, we get
x
yp(−1)σ(x)N
=|x˙
−y|N
and
exp (
ln x
yp(−1)σ(x))=exp
(−1)σ(x).ln x
yp
=exp
ln x
yp.
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Also, we have
exp
ln x
yp=exp
ln x
y
ln x
yp
=
ln x
y
and so we get
ln x
yp=ln x
y
or
ln x
yp=−ln x
y.
But the second equality is impossible since
⇒ln x−ln yp=−ln x+ln y
⇒2ln x=ln yp+ln y
⇒ln x2=ln yp+1.
which gives if p=−1,x=yand if p= 1,
x=ywhich is a contradiction. Thus; we get
⇒(−1)σ(y)−σ(x)= 1
⇒σ(y)−σ(x) = 0
⇒σ(x) = σ(y)
Therefore, the function σ(x)be as follows;
σ(x) = σ(σ= 0,1), foreveryx = 1
Finally, since
φν(x) = x(−1)σ.c
and φν(1) = cwe get x= 1.
Theorem 6. Let x∈Rνand let φνbe a ν−isometry.
Then, there are some c∈Rνsuch that
φν(x) = x˙
+c
or
φν(x) = c˙
−x .
Proof. Since φνis a ν−isometry, then the func-
tion ν−1◦φν◦νis a isometry. Thus, we
have ν−1◦φν◦ν(ν−1(x)) = ν−1(x) + dor
ν−1◦φν◦ν(ν−1(x)) = −ν−1(x) + dfor some
d∈R. Let d=ν−1(c). Then, we get
⇒ν−1◦φν◦ν(ν−1(x)) = ν−1(x) + ν−1(c)
⇒ν−1◦φν(x) = ν−1(x) + ν−1(c)
⇒νν−1◦φν(x)=νν−1(x) + ν−1(c)
⇒φν(x) = x˙
+c
or
⇒ν−1◦φν◦ν(ν−1(x)) = −ν−1(x) + ν−1(c)
⇒ν−1◦φν(x) = ν−1(c)−ν−1(x)
⇒νν−1◦φν(x)=νν−1(c)−ν−1(x)
⇒φν(x) = c˙
−x
which completes the proof.
Theorem 7. If the function φνis an ν−isometry, then
its inverse is an ν−isometry.
Proof. Since φνis a ν−isometry, then ν−1◦φν◦ν
is an isometry. Since the inverse of isometry is also
isometry; ν−1◦φν◦ν−1=ν−1◦φ−1
ν◦νis an
isometry. Thus, we get φ−1
νis a ν−isometry.
Theorem 8. Under a ν−isometry the following is
true;
a) Every ν−open interval maps to an ν−-open inter-
val of the same measure, and the endpoints of the im-
age interval are images of the endpoints of the origi-
nal interval.
b) The image of a ν−-bounded set is a ν−-bounded
set.
Proof.
a) Let ∆=(a, b)Nbe a ν−open inter-
val. Let φν(x) = x˙
+c. Then, we have
φν(∆) = (a˙
+c, b ˙
+c)N. Thus, we get
mNφν(∆) = mN(a˙
+c, b ˙
+c)N
= (b˙
+c)˙
−(a˙
+c)
=ν{ν−1(b˙
+c)−ν−1(a˙
+c)}
=ν{ν−1(ν(ν−1(b) + ν−1(c)))
−ν−1(ν(ν−1(b) + ν−1(c)))}
=ν{ν−1(b) + ν−1(c)−ν−1(a)−ν−1(c)}
=ν{ν−1(b)−ν−1(a)}
=b˙
−a
=mn∆.
Let φν(x) = c˙
−x. Then φν(∆) = (c˙
−b, c ˙
−a)N.
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thus, we get
mNφν(∆) = mN(c˙
−b, c ˙
−a)N
= (c˙
−a)˙
−(c˙
−b)
=ν{ν−1(c˙
−a)−ν−1(c˙
−b)}
=ν{ν−1(ν(ν−1(c)−ν−1(a)))
−ν−1(ν(ν−1(c)−ν−1(b)))}
=ν{ν−1(c)−ν−1(a)−ν−1(c) + ν−1(b)}
=ν{ν−1(b)−ν−1(a)}
=b˙
−a
=mn∆.
In both cases, we get
mNφν(∆) = b˙
−a=mn∆.
b) Let Ebe a ν−bounded set and let ∆be a ν−open
interval contaning the set E. Then, we have
φν(E)⊂φν(∆)
and so the set φν(E)is a ν−bounded. Indeed, Since
Eis ν−bounded, we have |x|N˙
<k for every x∈E.
Then, for every y∈φν(E), if φν(x) = x˙
+c, then
|y|N=|x˙
+c|N˙
<|x|N˙
+|c|N< k ˙
+|c|N
and if φν(x) = c˙
−x, then
|y|N=|c˙
−x|N=|x˙
−c|N˙
<|x|N˙
+|c|N< k ˙
+|c|N.
which gives that the set φν(E)is ν−bounded.
Theorem 9. Under a ν−isometry the following prop-
erties are true;
a) The image of a ν−-closed set is a ν−-closed set.
b) The image of a ν−-open set is a ν−-open set.
Proof.
a) Let φν(F)be the image of ν−closed set F. Let y0
be a ν−limit point of the set φν(F)and let ynbe a
sequence such that
νlim yn=y0yn∈φν(fν).
Also, let define
x0=φ−1
ν(y0), xn=φ−1
ν(yn)
and so (xn)⊂F.
Since φνis a ν−isometry, φ−1
νis a ν−isometry. Thus,
we have
⇒ |xn˙
−x0|N=|φ−1
ν(yn)˙
−φ−1
ν(y0)|N
⇒ |xn˙
−x0|N=|yn˙
−y0|N
and so
xn
ν
→x0.
Since Fis a ν−closed, x0∈Fand thus
y0=φν(x0)∈φν(F).
which completes the proof.
b) Let Gbe a ν−open set and let define
F=Gc.
Then, Fis a ν−closed set and
G∪F=Rν, G ∩F=∅.
Thus, we get
φν(G∪fν) = φν(Rν), φν(G∩fν) = φν(∅)
φν(G)∪φν(fν) = Rν, φν(G)∩φν(fν) = ∅.
which shows that φν(G)is complemet of φν(F)
ν−closed. This completes the proof.
Theorem 10. The ν−measure of a ν−bounded open
set is invariant under all ν−isometries.
Proof. Let Gbe a ν−bounded open set. Then, φν(G)
is a ν−bounded open set. Let δk= (ak, bk)N(k=
1,2,···)and let define
G=[
k
δk.
Thus, we have
φν(G) = φν [
k
δk!=[
k
φν(δk).
Therefore, we get
mNφν(G) =NX
k
mN(φν(δk))
=NX
k
mNφν((ak, bk)N)
=NX
k
bk˙
−ak
=NX
k
mNδk
=mNG.
This gives that
mNφν(G) =NX
k
mNφν(δk)
=NX
k
mNδk
=mNG.
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Theorem 11. ν−isometries do not change the
ν−outer and ν−inner measures of a ν−bounded set.
Proof.
a) Let Ebe a ν−bounded set. For every ϵ˙
>˙
0, there is
aν−bounded open set Gsuch that
G⊂E, mNG˙
<m∗
NE˙
+ϵ.
Then, φν(G)is ν−bounded open set containing the
set φν(E). Thus, we get
m∗
Nφν(E)˙
≤mNφν(G) = mNG˙
<m∗
NE˙
+ϵ
which gives
m∗
Nφν(E)˙
≤m∗
NE.
This shows that the ν−outer measure of a ν−-
bounded set does not increase under a ν−isometry.
Otherwise, the ν−inverse isometry is non-decreasing
since it leads to an increase in the ν−outer measure.
Therefore we get
m∗
Nφν(E) = m∗
NE.
b) Let ∆be a ν−open interval contaning E. Then,
φν(∆) is a ν−open interval contaning φν(E). Let
A=CE
∆.
Since
E∪A= ∆, E ∩A=∅
we have
φν(E)∪φν(A) = φν(∆), φν(E)∩φν(A) = ∅.
Thus, we write
m∗
Nφν(A)˙
+m∗Nφν(E) = mNφν(∆)
and so
m∗
NA˙
+m∗Nφν(E) = mN∆
This shows that
m∗Nφν(E) = mN∆˙
−m∗
NCE
∆.
Finally, we get
⇒m∗Nφν(E) = m∗
NCE
∆˙
+m∗NE˙
−m∗
NCE
∆
⇒m∗Nφν(E) = m∗NE.
3 Conclusion
In this article, we have introduced ν−isometry and
gave some of its properties using examples. First, we
showed that the necessary and sufficient condition for
the function φνto be a ν−isometry is that the func-
tion ν−1◦φν◦νis a real isometry. Using this the-
orem, we showed that the inverse of a ν−isometry is
aν−isometry. Finally, we show that the ν−measure
of ν−measurable sets is invariant for every generator
under ν−isometries.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.10
Oguz Ogur, Zekiye Gunes
E-ISSN: 2224-2880
86
Volume 23, 2024