Transformations of (0,1] preserving tails of Δμ-representation of numbers
In the paper, classes of continuous strictly increasing functions preserving ``tails'' of Δμ-representation of numbers are constructed. Using these functions we construct also continuous transformations of (0,1]. We prove that the set of all such transformations is infinite and forms non-c...
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| Цитувати: | Transformations of (0,1] preserving tails of Δμ-representation of numbers / T.M. Isaieva, M.V. Pratsiovytyi // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 1. — С. 102-115. — Бібліогр.: 25 назв. — англ. |
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irk-123456789-1557482019-06-18T01:25:37Z Transformations of (0,1] preserving tails of Δμ-representation of numbers Isaieva, T.M. Pratsiovytyi, M.V. In the paper, classes of continuous strictly increasing functions preserving ``tails'' of Δμ-representation of numbers are constructed. Using these functions we construct also continuous transformations of (0,1]. We prove that the set of all such transformations is infinite and forms non-commutative group together with an composition operation. 2016 Article Transformations of (0,1] preserving tails of Δμ-representation of numbers / T.M. Isaieva, M.V. Pratsiovytyi // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 1. — С. 102-115. — Бібліогр.: 25 назв. — англ. 1726-3255 2010 MSC:11H71, 26A46, 93B17. http://dspace.nbuv.gov.ua/handle/123456789/155748 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In the paper, classes of continuous strictly increasing functions preserving ``tails'' of Δμ-representation of numbers are constructed. Using these functions we construct also continuous transformations of (0,1]. We prove that the set of all such transformations is infinite and forms non-commutative group together with an composition operation. |
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Isaieva, T.M. Pratsiovytyi, M.V. |
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Isaieva, T.M. Pratsiovytyi, M.V. Transformations of (0,1] preserving tails of Δμ-representation of numbers Algebra and Discrete Mathematics |
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Isaieva, T.M. Pratsiovytyi, M.V. |
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Isaieva, T.M. |
| title |
Transformations of (0,1] preserving tails of Δμ-representation of numbers |
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Transformations of (0,1] preserving tails of Δμ-representation of numbers |
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Transformations of (0,1] preserving tails of Δμ-representation of numbers |
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Transformations of (0,1] preserving tails of Δμ-representation of numbers |
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Transformations of (0,1] preserving tails of Δμ-representation of numbers |
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transformations of (0,1] preserving tails of δμ-representation of numbers |
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Інститут прикладної математики і механіки НАН України |
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2016 |
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http://dspace.nbuv.gov.ua/handle/123456789/155748 |
| citation_txt |
Transformations of (0,1] preserving tails of Δμ-representation of numbers / T.M. Isaieva, M.V. Pratsiovytyi // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 1. — С. 102-115. — Бібліогр.: 25 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT isaievatm transformationsof01preservingtailsofdmrepresentationofnumbers AT pratsiovytyimv transformationsof01preservingtailsofdmrepresentationofnumbers |
| first_indexed |
2025-07-14T07:59:21Z |
| last_indexed |
2025-07-14T07:59:21Z |
| _version_ |
1837608428997967872 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 22 (2016). Number 1, pp. 102–115
© Journal “Algebra and Discrete Mathematics”
Transformations of (0, 1] preserving tails
of ∆µ-representation of numbers
Tetiana M. Isaieva, Mykola V. Pratsiovytyi
Communicated by A. P. Petravchuk
Abstract. Let µ ∈ (0, 1) be a given parameter, ν ≡ 1−µ. We
consider ∆µ-representation of numbers x = ∆µ
a1a2...an... belonging
to (0, 1] based on their expansion in alternating series or finite sum
in the form:
x =
∑
n
(Bn −B′
n) ≡ ∆µ
a1a2...an...,
where Bn = νa1+a3+...+a2n−1−1µa2+a4+...+a2n−2 ,
B′
n = νa1+a3+...+a2n−1−1µa2+a4+...+a2n , ai ∈N.
This representation has an infinite alphabet {1, 2, . . .}, zero redun-
dancy and N -self-similar geometry.
In the paper, classes of continuous strictly increasing functions
preserving “tails” of ∆µ-representation of numbers are constructed.
Using these functions we construct also continuous transformations
of (0, 1]. We prove that the set of all such transformations is infinite
and forms non-commutative group together with an composition
operation.
Introduction
We consider representation of real numbers belonging to half-interval
(0, 1]. It depends on real parameter µ ∈ (0, 1) and has an infinite alphabet
N = {1, 2, 3, . . .}. This representation is based on the following theorem.
2010 MSC: 11H71, 26A46, 93B17.
Key words and phrases: ∆µ-representation, cylinder, tail set, function preserving
“tails” of ∆µ-representation of numbers, continuous transformation of (0, 1] preserving
“tails” of ∆µ-representation of numbers, group of transformations.
T. Isaieva, M. Pratsiovytyi 103
Theorem 1 ([19]). Let (0, 1)∋µ be a fixed real number, ν≡1−µ. For any
x ∈ (0, 1], there exists a finite tuple of positive integers (a1, a2, . . . , am)
or a sequence of positive integers (an) such that
x = νa1−1 − νa1−1µa2 + νa1+a3−1µa2 − νa1+a3−1µa2+a4 + . . . =
=
∑
n
(Bn −B′
n), (1)
where Bn = νa1+a3+...+a2n−1−1µa2+a4+...+a2n−2 , B′
n = Bn · µa2n .
We call expansion of the number x in the form of alternating series (1)
the ∆µ-expansion and its symbolic notation ∆µ
a1a2...am(∅) for finite expan-
sion of number x or ∆µ
a1a2...an... for infinite sum the ∆µ-representation.
Remark that expansion of a number in the form of alternating series (1)
first appeared in papers [23, 24] in an expression of strictly increasing
singular function ϕµ being an unique continuous solution of a system of
functional equations:
ϕµ
(
x
1 + x
)
= (1 − µ)ϕµ(x),
ϕµ(1 − x) = 1 − ϕ1−µ (x) .
This function generalizes the well-known singular Minkowski function [1–
8,10–16,25] and coincides with it for µ = 1/2. In this case the ∆µ-repre-
sentation is the ∆♯-representation studied in papers [20,21].
There exists a countable everywhere dense in [0, 1] set of numbers hav-
ing two ∆µ-representation. These numbers have a form: ∆µ
a1...[am+1](∅) =
= ∆µ
a1...am1(∅). We call these numbers ∆µ-finite. Other numbers belonging
to (0, 1] have a unique ∆µ-representation, their expansions are infinite, so
we call them ∆µ-infinite numbers. That is, ∆µ-representation has a zero
redundancy. We denote the set of all ∆µ-infinite numbers by H and the
set of ∆µ-finite numbers by S.
The ∆µ-representation of number is called the rational ∆µ-representa-
tion if µ ∈ (0, 1) is rational. In this case irrational numbers belonging to
(0, 1] have infinite non-periodic ∆µ-representation and rational numbers
have either finite or infinite periodic or infinite non-periodic ∆µ-represen-
tation [19]. So the set H contains all irrational numbers and everywhere
dense in [0, 1] subset of rational numbers.
Remark that ∆µ-representation has much in common with encod-
ing of real numbers by regular continued fraction [9, 17], namely, they
104 Transformations preserving tails
have the same topology, rules for comparing numbers etc. However, ∆µ-
representation generates other metric relations, that is, it has own original
metric theory [19].
In the paper, we construct an infinite non-commutative group of
continuous strictly increasing piecewise linear transformations of (0, 1]
preserving tails of ∆µ-representation of numbers. Analogous objects for
E-representation based on expansions of numbers in the form of positive
Engel series are discussed in paper [18]. This representation has funda-
mental distinctions from E-representation in topological as well as metric
aspects.
1. Geometry of ∆µ-representation of numbers
Geometric meaning of digits of ∆µ-representation of numbers and
essence of related positional and metric problems are disclosed by the
following important notion.
Definition 1. Let (c1, c2, . . . , cm) be a tuple of positive integers.
Cylinder of rank m with base c1c2 . . . cm is a set ∆µ
c1c2...cm
of numbers
x ∈ (0, 1] having ∆µ-representation such that ai(x) = ci, i = 1,m.
Cylinders have the following properties.
1.
⋃
a1∈N
⋃
a2∈N
. . .
⋃
am∈N
∆µ
a1a2...am
=(0, 1]; 2. ∆µ
c1c2...cm
=
∞⋃
i=1
∆µ
c1c2...cmi;
3. Cylinder ∆µ
c1c2...cm
is a closed interval, moreover,
if m is odd, then ∆µ
c1c2...c2k−1
= [a− δ, a], where
δ = νc1+c3+...+c2k−1−1 · µc2+c4+...+c2k−2+1;
a = νc1−1 − νc1−1µc2 + . . .+ νc1+c3+...+c2k−1−1µc2+c4+...+c2k−2 ,
if m is even, then ∆µ
c1c2...c2k
= [a, a+ δ], where
δ = νc1+c3+...+c2k−1 · µc2+c4+...+c2k .
a = νc1−1 − νc1−1µc2 + . . .+
+νc1+c3+...+c2k−1−1µc2+c4+...+c2k−2 −νc1+c3+...+c2k−1−1µc2+c4+...+c2k ,
4. The length of cylinder of rank m is calculated by the formulae:
|∆µ
c1...cm
|=
{
νc1+c3+...+c2k−1−1 · µc2+c4+...+c2k−2+1 if m=2k−1,
νc1+c3+...+c2k−1 · µc2+c4+...+c2k if m=2k.
T. Isaieva, M. Pratsiovytyi 105
5. If ∆µ
c1c2...cm
is a fixed cylinder, then the following equality (basic
metric relation) holds:
|∆µ
c1c2...cmi|
|∆µ
c1c2...cm |
=
{
νµi−1 if m = 2k − 1,
µνi−1 if m = 2k.
6. min∆µ
c1...c2k−1i =max∆µ
c1...c2k−1(i+1); max∆µ
c1...c2k
=min∆µ
c1...c2k(i+1);
7. Cylinders of the same rank do not intersect or coincide. Moreover,
∆µ
c1c2...cm
= ∆µ
c′
1c′
2...c′
m
⇐⇒ ci = c′
i i = 1,m;
8. For any sequence (cm), cm ∈ N, intersection
∞⋂
m=1
∆µ
c1c2...cm
= x ≡ ∆µ
c1c2...cm...
is a point belonging to half-interval (0, 1].
In paper [19], it is proved that geometry of ∆µ-representation of
numbers is N -self-similar and foundations of metric theory are laid. In
paper [22], functions with fractal properties defined in terms of ∆µ-repre-
sentation are considered. Geometry plays an essential role in studies of
such functions.
2. Tail sets and functions preserving tails
of ∆µ-representation of numbers
Let Zµ
H be the set of all ∆µ-representations of numbers belonging to
set H. We introduce binary relation “has the same tail” (symbolically: ∼)
on the set Zµ
H .
Two ∆µ-representations ∆µ
a1a2...an... and ∆µ
b1b2...bn... are said to have
the same tail (or they are ∼-related) if there exist positive integers k and
m such that ak+j = bm+j for any j ∈ N.
It is evident that binary relation ∼ is an equivalence relation (i.e., it is
reflexive, symmetric and transitive) and provides a partition of the set Zµ
H
into equivalence classes. Any equivalence class is said to be a tail set. Any
tail set is uniquely determined by its arbitrary element (representative).
We say that two numbers x and y belonging to setH have the same tail
of ∆µ-representation (or they are ∼-related) if their ∆µ-representations
are ∼-related. We denote this symbolically as x ∼ y.
Theorem 2. Any tail set is countable and dense in (0, 1]; quotient set
F ≡ (0, 1]/ ∼ is a continuum set.
106 Transformations preserving tails
Proof. Suppose K is an arbitrary equivalence class, x0 = ∆µ
c1c2...cn... is its
representative. Then it is evident that, for any m ∈ Z0, there exists set
Km =
{
x : x = ∆µ
a1...akcm+1cm+2..., ai ∈ N, k = 0, 1, 2, . . .
}
of numbers
x such that for some k ∈ Z0
ak+j(x) = cm+j for any j ∈ N and K =
⋃
m∈Z0
Km .
The set K is countable because it is a countable union of countable sets.
Now we prove that K is a dense in (0, 1] set. Since number x belongs
or does not belong to the set K irrespective of any finite amount of first
digits of its ∆µ-representation, we have that any cylinder of arbitrary
rank m contains point belonging to K. Thus K is an everywhere dense
in half-interval (0, 1] set.
To prove that quotient set F ≡ (0, 1]/ ∼ is continuum set, we assume
the converse. Suppose that F is a countable set. Then half-interval (0, 1]
is a countable set as a countable union of countable sets (equivalence
classes of quotient set F ). This contradiction proves the theorem.
Remark that it is easy to introduce a distance function (metric) in
the quotient set F .
Definition 2. Suppose function f is defined on the setH and takes values
from this set. We say that function f preserves tails of ∆µ-representations
of numbers if for any x ∈ (0, 1] there exist positive integers k = k(x) and
m = m(x) such that
ak+n(x) = am+n (f(x)) for all n ∈ N.
It is clear that functions preserving tails of ∆µ-representations of
numbers form an infinite set. However, only continuous functions are
interested for us. Identity transformation y = e(x) is a simplest example
of such function.
By X we denote the set of all functions satisfying Definition 2. In the
sequel, we consider some representatives of this class.
3. Function σ1(x)
We consider function defined on the set H by equality
y = σ1(x) = σ1
(
∆µ
a1(x)a2(x)a3(x)a4(x)...an(x)...
)
= ∆µ
[a1+a2+a3]a4...an...
.
T. Isaieva, M. Pratsiovytyi 107
This function is well-defined due to uniqueness of ∆µ-representation
of numbers belonging to the set H. It is evident that it preserves tails of
∆µ-representation of numbers.
Lemma 1. Analytic expression for function y = σ1(x) is given by formula
σ1(x) =
(
ν
µ
)a2(x)
· x+ νa1(x)+a2(x)−1
(
1 −
1
µa2(x)
)
, (2)
this function is linear on every cylinder of rank 2 and has the following
properties:
1) it is continuous strictly increasing function;
2) sup
x∈∆µ
ij
σ1(x) = νi+j, inf
x∈∆µ
ij
σ1(x) = 0;
3)
∫
∆µ
ij
σ1(x)dx =
1
2
ν2i+jµj; 4)
1∫
0
σ1(x)dx =
1
2
·
ν3
1 + ν3
.
Proof. 1. Indeed, if x = ∆µ
a1a2a3a4a5...an..., then
x = νa1−1 − νa1−1µa2 + νa1+a3−1µa2 − νa1+a3−1µa2+a4 + . . . =
= νa1−1 − νa1−1µa2 +
µa2
νa2
· σ1(x).
Whence it follows that
σ1(x) =
(
ν
µ
)a2(x)
· x+ νa1(x)+a2(x)−1
(
1 −
1
µa2(x)
)
.
It is evident that function σ1(x) is linear. Therefore it is continuous strictly
increasing on the set H ∩ ∆µ
a1a2
. Extending by continuity in ∆µ-finite
points we obtain continuous function on the whole cylinder ∆µ
a1a2
.
2. Boundary values of function σ1(x) on cylinder ∆µ
ij can be calculated
by formulae:
sup
x∈∆µ
ij
σ1(x) = lim
k→∞
σ1
(
∆µ
ij1(k)
)
= ∆µ
[i+j+1](∅) = νi+j .
inf
x∈∆µ
ij
σ1(x) = lim
k→∞
σ1
(
∆µ
ij(k)
)
= lim
k→∞
∆µ
[i+j+k](k) = 0.
3. Calculate integral on cylinder ∆µ
ij :
∫
∆µ
ij
σ1(x)dx =
∆µ
i[j+1](∅)∫
∆µ
ij(∅)
σ1(x)dx =
νi−1(1−µj+1)
∫
νi−1(1−µj)
σ1(x)dx =
1
2
ν2i+jµj .
108 Transformations preserving tails
4. Calculate integral on the unit interval:
∫ 1
0
σ1(x)dx =
1
2
∞∑
i=1
ν2i
∞∑
j=1
νjµj =
1
2
·
ν2
1 − ν2
·
νµ
1 − νµ
=
1
2
·
ν3
1 + ν3
.
4. Function ds(x)
Let s be a fixed positive integer. We consider function depending on
parameter s, well-defined on half-interval (0, 1] by equality
y = ds(x) = ds
(
∆µ
a1(x)a2(x)a3(x)...
)
= ∆µ
[s+a1]a2a3...
.
Since s is an arbitrary positive integer, we have a countable class of
functions y = ds(x).
Theorem 3. Function ds is analytically expressed by formula:
ds(x) = νs · x
and has the following properties:
1) it is linear strictly increasing, 2) inf
x∈(0,1]
ds(x)=0, sup
x∈(0,1]
ds(x)=νs.
Moreover, equation σ1(x) = ds(x) does not have solutions if a2 > s,
and has a countable set of solutions:
E =
{
x : x = ∆µ
a1(a2[s−a2]), where a1 ∈ N, a2 ∈ {1, 2, . . . , s− 1}
}
if a2 < s.
Proof. By definition of function ds, we have
ds(x) = ∆µ
[s+a1]a2a3...
= νs+a1−1 − νs+a1−1µa2 + . . . = νs · x,
Thus ds(x) = νs·x. It is evident that function ds is linear strictly increasing
on half-interval (0, 1]. Moreover,
inf
x∈(0,1]
ds(x) = lim
x→0+0
ds(x) = lim
k→∞
ds
(
∆µ
(k)
)
= lim
k→∞
∆µ
[s+k](k) = 0;
sup
x∈(0,1]
ds(x) = lim
x→1−0
ds(x) = lim
k→∞
ds
(
∆µ
1(k)
)
= ∆µ
[s+1](∅) = νs.
We can write equation σ1(x) = ds(x) in the form
∆µ
[a1(x)+a2(x)+a3(x)]a4(x)... = ∆µ
[s+a1(x)]a2(x)a3(x)a4(x)... .
T. Isaieva, M. Pratsiovytyi 109
From uniqueness of ∆µ-representation of numbers belonging to set H
it follows that following equalities hold simultaneously:
a1(x) + a2(x) + a3(x) = s+ a1(x), a4(x) = a2(x),
a5(x) = a3(x) = s− a2(x), . . . a2k(x) = a2(x),
a2k+1(x) = s− a2(x), k ∈ N.
It is evident that this system is inconsistent if a2 > s. However, for a2 < s,
equation has a countable set of solutions x = ∆µ
a1(a2[s−a2]), where a1, a2
are independent positive integer parameters.
5. Left shift operator on digits
of ∆µ-representation of number
Let Zµ
H be the set of all ∆µ-representations of numbers belonging to
set H. We consider shift operator ω2 on digits defined by equality
ω2
(
∆µ
a1a2a3a4...an...
)
= ∆µ
a3a4...an....
This operator generates function y = ω2(x) = ∆µ
a3(x)a4(x)...an(x)... on the
set H. It is evident that operator ω2 is surjective but not injective.
Any point ∆µ
(ij) =
νi−1(1 − µj)
1 − νiµj
, where (i, j) is any pair of positive
integers, is an invariant point of the mapping ω2.
Lemma 2. Function y = ω2(x) is analytically expressed by formula
ω2(x) =
x
νa1(x)µa2(x)
−
1 − µa2(x)
νµa2(x)
(3)
and is continuous monotonically increasing on any cylinder of rank 2.
Proof. Let x ∈ ∆µ
ij . Then x = ∆µ
ija3a4... and
x = νi−1 − νi−1µj + νi+a3−1µj − νi+a3−1µj+a4 + . . . =
= νi−1 − νi−1µj + νiµj · ω2(x).
Whence, ω2(x) =
x
νiµj
−
1 − µj
νµj
.
Since function ω2 is linear, we have that this function is continuous
strictly increasing on the set H ∩ ∆µ
a1a2
. Extending by continuity in the
points of the set S we obtain continuous function on the whole cylinder
∆µ
a1a2
.
110 Transformations preserving tails
Lemma 3. Equation ds(x) = ω2(x) has a countable set of solutions
having the form x = ∆µ
a1(a2[s+a1]), where a1, a2 are arbitrary positive
integers.
Proof. We can write equation ds(x) = ω2(x) in the form
∆µ
[s+a1(x)]a2(x)a3(x)a4(x)... = ∆µ
a3(x)a4(x)... .
From uniqueness of ∆µ-representation of numbers belonging to set H
it follows that the following equalities hold simultaneously:
s+ a1(x) = a3(x), a2(x) = a4(x), a3(x) = a5(x) = s+ a1(x),
a4(x) = a6(x) = a2(x), . . . , a2k+1(x) = s+ a1(x),
a2k(x) = a2(x), k ∈ N.
Then solutions of equation are numbers having the form x=∆µ
a1(a2[s+a1]),
where a1, a2 ∈ N.
6. Right shift operator on digits
of ∆µ-representation of number
Let i, j be fixed positive integers. We consider operator depending on
parameters i, j, well-defined on half-interval (0, 1] by equality
δij(x) = δij
(
∆µ
a1(x)a2(x)...
)
= ∆µ
ija1a2....
This operator defines a countable set of functions y = δij(x), i ∈ N, j ∈ N.
Lemma 4. Function y = δij(x) is analytically expressed by formula
y = δij(x) = νiµj · x+ νi−1
(
1 − µj
)
and is linear strictly increasing on half-interval (0, 1], moreover,
inf
x∈(0,1]
δij(x) = ∆µ
ij(∅) = νi−1
(
1 − µj
)
,
sup
x∈(0,1]
δij(x) = ∆µ
ij1(∅) = νi−1
(
1 − µj+1
)
.
Proof. In fact, by definition of function δij , we have:
y=δij(∆µ
a1a2...)=∆µ
ija1a2... =νi−1−νi−1µj+νi+a1−1µj−νi+a1−1µj+a2+. . .=
T. Isaieva, M. Pratsiovytyi 111
= νi−1−νi−1µj +νiµj
(
νa1−1 − νa1−1µa2 + . . .
)
︸ ︷︷ ︸
x
= νi−1−νi−1µj +νiµj ·x.
Therefore, y = δij(x) = νiµj · x+ νi−1
(
1 − µj
)
.
From linearity of function δij it follows that it is a continuous strictly
increasing function on (0, 1] for any pair of positive integers (i, j). More-
over,
inf
x∈(0,1]
δij(x) = lim
x→0+0
δij(x) = lim
k→∞
δij
(
∆µ
(k)
)
= lim
k→∞
∆µ
ij(k) =
= ∆µ
ij(∅) = νi−1
(
1 − µj
)
;
sup
x∈(0,1]
δij(x) = lim
x→1−0
δij(x) = lim
k→∞
δij
(
∆µ
1(k)
)
=
= ∆µ
ij1(∅) = νi−1
(
1 − µj+1
)
.
For functions ω2 and δij , the following equalities are obvious:
ω2 (δij) = x, δa1(x)a2(x) (ω2(x)) = x.
Theorem 4. For function δij, the following propositions are true.
1. Equation σ1(x) = δij(x) does not have any solution if a1 + a2 > i and
has a countable set of solutions
E=
{
x : x = ∆µ
(a1a2[i−a1−a2]j), a1 ∈ N, a2 ∈ N, a1+a2 ∈{1, 2, . . . , i−1}
}
if a1 + a2 < i.
2. Equation ds(x) = δij(x) does not have any solution if s > i and
has a countable set of solutions
E =
{
x : x = ∆µ
([i−s]j), s ∈ N, s ∈ {1, 2, . . . , i− 1}
}
if s < i.
3. Equation ω2(x) = δij(x) has infinitely many solutions having a
general form
x = ∆µ
(a1a2ij), where (a1, a2) is an arbitrary pair of positive integers.
Proof. 1. We can write equation σ1(x) = δij(x) in the form
∆µ
[a1(x)+a2(x)+a3(x)]a4(x)a5(x)... = ∆µ
ija1(x)a2(x)a3(x)a4(x)... .
112 Transformations preserving tails
From uniqueness of ∆µ-representation of numbers belonging to H it
follows that following equalities holds simultaneously:
a1(x) + a2(x) + a3(x) = i, a4(x) = j, a5(x) = a1(x), a6(x) = a2(x),
a7(x)=a3 = i− (a1 + a2), a8(x)=a4 =j, . . . , a4k−1(x)= i− (a1 + a2),
a4k(x) = j, a4k+1(x) = a1, a4k+2(x) = a2, k ∈ N.
Then this system does not have any solution if a1 + a2 > i and have a
countable set of solutions E =
{
x : x = ∆µ
(a1a2[i−a1−a2]j)
}
, where a1, a2
are independent positive integer parameters, if a1 + a2 < i.
Similarly, we can prove statements 2 and 3 of the theorem.
7. Transformations preserving tails
of ∆µ-representation of numbers
Recall that transformation of non-empty set E is any bijective (i.e.,
both injective and surjective) mapping of this set onto itself.
It is clear that continuous transformations of [0, 1] are strictly mono-
tonic (increasing or decreasing) functions such that f(0)=0 and f(1)=1
or f(0) = 1 and f(1) = 0.
If f is a transformation of [0, 1], then ϕ(x) = 1 − f(x) is also trans-
formation of this set. Therefore, to study continuous transformations of
[0, 1], we can consider only strictly increasing functions, i.e., continuous
probability distribution functions.
Simple examples of continuous strictly increasing transformations pre-
serving tails of ∆µ-representation of numbers are the following functions:
ϕτ (x) =
di(x) if 0 < x 6 x1 ≡ ∆µ
a1(a2[i+a1]),
ω2(x) if x1 < x 6 x2 ≡ ∆µ
(a1a2),
e(x) if x2 < x 6 1,
where τ = (i, a1, a2) is an arbitrary triplet of positive integers;
ψ(x) =
d1(x) if 0 < x 6 x1 ≡ ∆µ
1(12),
ω2(x) if x1 < x 6 x2 ≡ ∆µ
(1112),
δ12(x) if x2 < x 6 x3 ≡ ∆µ
(12),
e(x) if x3 < x 6 1;
T. Isaieva, M. Pratsiovytyi 113
γ(x) =
d3(x) if 0 < x 6 x1 ≡ ∆µ
1(12),
σ1(x) if x1 < x 6 x2 ≡ ∆µ
(1111),
δ31(x) if x2 < x 6 x3 ≡ ∆µ
(1231),
ω2(x) if x3 < x 6 x4 ≡ ∆µ
(1212),
e(x) if x4 < x 6 1.
Theorem 5. The set G of all continuous strictly increasing transfor-
mations of half-interval (0, 1] preserving tails of ∆µ-representation of
numbers together with an operation ◦ (function composition) form an
infinite non-commutative group.
Proof. The set of continuous transformations of (0, 1] is a subset of all
transformations of (0, 1] forming a group. Thus we use a subgroup test.
It is evident that set G is closed under the composition operation. For
continuous strictly increasing function, inverse function is continuous and
strictly increasing too. If transformation f preserves “tails” of ∆µ-rep-
resentations, then inverse transformation preserves them too. Therefore,
for transformation f ∈ G, inverse transformation belongs to G too.
Since set of transformations ϕτ , τ ∈ N × N × N, is countable, we see
that set G is infinite.
To prove that group (G, ◦) is non-commutative, we provide an example
of two transformations f1 and f2 such that they are not commute, i.e.,
f2 ◦ f1 6= f1 ◦ f2. Consider two transformations ϕτ1(x) and ϕτ2(x), where
τ1 = (1, 2, 3), τ2 = (1, 1, 2), i.e.,
ϕτ1(x) =
d1(x) if 0 < x 6 x1 ≡ ∆µ
2(33),
ω2(x) if x1 < x 6 x2 ≡ ∆µ
(23),
e(x) if x2 < x 6 1;
ϕτ2(x) =
d1(x) if 0 < x 6 x3 ≡ ∆µ
1(22),
ω2(x) if x3 < x 6 x4 ≡ ∆µ
(12),
e(x) if x4 < x 6 1.
Then, for x0 = ∆µ
12(3), tacking into account inequalities x0 > x2 = ∆µ
(23)
but ϕτ1(x0)<x3 =∆µ
1(22) and x0<x3 =∆µ
1(22) but ϕτ2(x0)<x1 =∆µ
2(33),
we obtain
ϕτ2
(
ϕτ1
(
∆µ
12(3)
))
= ϕτ2
(
∆µ
12(3)
)
= ∆µ
22(3);
ϕτ1
(
ϕτ2
(
∆µ
12(3)
))
= ϕτ1
(
∆µ
22(3)
)
= ∆µ
32(3) 6= ∆µ
22(3).
Therefore ϕτ2 ◦ϕτ1 6= ϕτ1 ◦ϕτ2 and (G, ◦) is a non-commutative group.
114 Transformations preserving tails
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Contact information
T. Isaieva,
M. Pratsiovytyi
Institute of Physics and Mathematics,
National Pedagogical Mykhailo Drahomanov
University,
9 Pyrohova St., Kyiv, 01601, Ukraine
E-Mail(s): isaeva_tn@ukr.net,
prats4444@gmail.com
Received by the editors: 10.04.2016
and in final form 10.08.2016.
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