Significant Modification of Pairwisecontinuous
Functions with Associated Concepts
ALI A. ATOOM1, HAMZA QOQAZEH2, RAHMEH ALRABABAH1, EMAN ALMUHUR3,
NABEELA ABU-ALKISHIK4
1Department of Mathematics, Faculty of Science,
Ajloun National University,
P.O. Box 43, Ajloun, 26810,
JORDAN
2Department of Mathematics, Faculty of Arts and Science,
Amman Arab University,
P.O. Box 24 Amman, 11953,
JORDAN
3Department of Mathematics, Faculty of Science,
Ajloun National University
P.O. Box 43, Ajloun, 26810,
JORDAN
4Department of Mathematics, Faculty of Science
Jerash University
P.O. Box 2600, Jerash, 21220
JORDAN
Abstract: - The continuity is generalized by the notion of continuous functions. In this research, we present
a new weaker form for continuous functions called pairwise continuous functions. Additionally, we define
pairwise barely continuous functions, a new, weaker form of barely continuous functions. We study
the basic characteristics and impacts of pairwise continuous functions, clarifying their connection with
typical continuity and providing perspectives on the wider field of topological analysis. It explores related ideas
like the limit, which describes how sequences behave over certain conditions when the function is applied.
In addition, the concepts highlight the importance of pairwise continuous functions in theoretical and
practical conditions by discussing their relationships with other functional structures. An extensive number of
demonstrative examples will be presented, along with the new results and theorems about pairwise barely
continuous and pairwise continuous functions that generalize.
Key-Words: - Bitopological spaces, pairclosed, pairlindelöf, pair contiuous, pairalmost con-
tenuous functions, pairweakly continuous functions, pairsequentially continuous functions,
pairbarely continuous functions, pairbarely continuous functions.
Received: March 26, 2023. Revised: November 19, 2023. Accepted: December 9, 2023. Published: December 31, 2023.
1 Introduction
Pairwise-ω-continuous function analysis is seen as
an extension of a similar topic in topological spaces,
whose was a development of topological spaces
with simply one topology.
The primary goal for including ω-continuous
functions in the topology is to generalize certain
characteristics of continuous functions. In generic
topology, compactness and lindelöf are the most
basic components. Moreover, these two concepts
have a great deal of utility in implementations;
topology is employed in various fields of analysis as
well as logical mathematics, and continuous
function conceptions are utilized widely in
mathematical analysis. Since in, [1], first proposed
the idea of a bitopological space in 1963, certain
topological properties of single topology such as
compactness and paracompactness have been
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generalised to bitopological spaces, as well as
separation axioms, connectedness, function types,
and more concepts. In 1982, [2], demonstrated that
the lindelöf property is maintained by counter
images of closed mappings using lindelöf
counter images for points. He introduced a new type
of mapping, called closed mappings, that are
strictly weaker than closed mappings. Also, he
demonstrated that if the inverse image of every
point in the range is paracompact with respect to the
domain (lindelöf, respectively), then the
paracompactness (strongly paracompactness)
property is maintained by obtaining counter images
of closed mappings using regular domains.
Moreover, he defined space as a generalisation
of space and demonstrated that for any
space , the projection of is
closed if and only if is in a lindelöf space.
Additionally, he derived various product theorems
related to lindelöf for paracompact and strongly
paracompact spaces using spaces. He then
discusses a few different examples that relate to the
definitions and theorems that are provided. In 2021,
in, [3], introduces pairwise perfect functions as
well as pairwise perfect functions,
describing their features and searching for
homeomorphisms between various bitopological
spaces under their influence. Finally, he provides
the product theorem characterizations. In 2022, [4],
presented the idea of weakly continuous
functions within bitopological spaces as a
generalisation of continuous functions, and
they obtained numerous features and some of their
characteristics. Also, in the same year 2022, [5],
proposed the idea of nearly continuous
functions within bitopological spaces as a
generalization of continuous functions, and
they established several results and some of their
characteristics. While, in 2022, [6], created a new
type of function via open sets that he named
rarely continuous function and examined many
characteristics of this function. Throughout this
research, we will use paircompact, which refers to
pairwise compact and pair to denotes pairwise. If
represents a bitopological space and
; then the closure for with regard to and ,
respectively, will be indicated by the symbols
and . Let be a subset of and let
be any topological space, a point
is known as the condensation point of
if is uncountable set, for any with
. The following describes the way, [7], described
closed sets as well as open sets. If A has all
its condensation points, it is said to as closed.
open is the complement of a closed set.
Additionally, the intersection with all closed
sets that contain will be indicated by .
Also, the space , or just refers to any
bitopological space on which, unless otherwise
specified, no separation axioms can be taken for
assumed in this resaerch. Paircontinuous
functions are a new kind of weakened continuous
function that is introduced in this study.
Additionally, we define pairbarely continuous
functions, a new, weakened type of barely
continuous functions. First, in Section 3, we
establish numerous characteristics of
paircontinuous functions. Several examples
are provided in Section 4 to show how
paircontinuous functions and certain
weakened forms of pairwise continuous functions
are connected. In the end, we show whether a
pairω-continuous image that represents a
pairlindelöf space is also a pairlindelöf space in
Section 5 by presenting a new, basic
characterization of pairlindelöf spaces.
Furthermore, as demonstrated in Section 5, a
pairwise only barely continuous image is
hereditarily. Applications of continuous functions
with different topological spaces can be found in
[8], [9], [10], [11], [12], [13], [14], [15], [16], [17],
[18], [19], [20], [21], [22], [23], [24], [25], [26],
[27], [28], [29].
2 Preliminaries
This section presents some significant concepts
along with specifics are covered that used within the
research.
Definition 2.1, [2] A function
is said to be paircontinuous if the
functions and
are continuous.
Definition 2.2, [2] A function
is said to be pairclosed if the functions
and are
closed.
Definition 2.3, [2] Let be any
bitopological space. Then, a cover of the space
is called open if .
Moreover, is called pairopen if has at least
one-nonempty member of .
Definition 2.4, [2] Let be any
bitopological space. Then, the space is called
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pairlindelöf if any pairopen cover of
has a countable subcover.
Definition 2.5, [2] Let be any
bitopological space. Then, the space is called
lindelöf if any open cover of
has a countable subcover.
Definition 2.6, [2] Let be any
bitopological space. Then, the space is called
paircountably compact if any countable pairopen
cover of has a finite subcover.
Definition 2.7, [2] Let be any
bitopological space. Then, the space is called
semicountably compact if any countably
open cover of has a finite
subcover.
Definition 2.8, [2] A function
is said to be pairlindelöf if for any
pairlindelöf closed subset of the space
, is pairlindelöf.
Definition 2.9, [10] A function
is said to be semilindelöf if for any
semilindelöf closed subset of the space
, is semilindelöf.
Definition 2.10, [10] Let be any
bitopological space, then we called represents
locally lindelöf with a respect to , if there exists
nbd of for any , such that
is pairlindelöf.
Definition 2.11, [10] Let be any
bitopological space, then the space called
pairlocally lindelöf if and only if is locally
lindelöf with respect to , where and
.
Definition 2.12, [22] A function
is said to be pairclosed if the
functions represent pairclosed sets onto
pairclosed sets.
Definition 2.13, [24] A function
is called pairweakly continuous if for
any pairopen set , there is is
pairopen.
Definition 2.14, [25] A function
is called pairalmost continuous if for
any and any open set of containing
there exists open set containing such
that .
Definition 2.15, [25] A function
is called pairweakly continuous if for
any and any open set of containing
there exists open set containing such
that .
Recall that if an arbitrary neighbourhood (in
simply nbd) for a point in a space includes an
uncountable subset of the set , then the point
is known as a condensation point of a set .
Definition 2.16, [29] Let be any
bitopological space and . The space is said to
be pairopen if for any there is a
pairopen subset of containing such that the
set is countable. Moreover, the complement
set of pairopen is called pair closed set.
The family of all pairopen (respectively
pairclosed) subsets of is
represented as pair,(respectively
pair). Additionally, the family of all
pairopen of the space containing
is represented as pair.
Definition 2.17, [29] Let be any
bitopological space and . The space is said to
be semiopen if for any there exists
open subset of containing such that the
set is countable. Moreover, the complement
set of semi open is called semiclosed set.
The family of all semiopen (respectively
semiclosed) subsets of is
represented as semi (respectively
semi). Additionally, the family of all
semiopen of the space containing
is represented as semi.
Definition 2.18, [29] A function
is said to be semiclosed if the
functions represent semiclosed sets onto
semiclosed sets.
Theorem 2.19, [29] Every pair subset of
the pairlindelöf space is pairlindelöf.
Let function be
any pairfunction. Then the following facts are
equivalent: (a) is pair closed; (b) for any
and each pairopen set , where
, there exists pairopen set
such that and .
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(iii) Every pairlindelöf, pairopen subset of
represent , which is pairopen
and is countable set such that is set.
3 Properties of Pairwise -Continuous
Functions
Additional results regarding the topological
attributes of pairwise-ω-continuous functions are
presented in this section. Following this, we will
talk about the idea of pairwise-ω-continuous
functions, extract some of their traits, and explain
how they relate to various kinds of pairwise-ω-
continuous functions, providing examples to
demonstrate for every instance. There will also be
discussion and proof of other theories on this topic.
Definition 3.1 A function
is said to be pair continuous at the
point , if for any pair-open set
containing , there is pairopen set
containing such that . Moreover, a
function is called paircontinuous on
, if the function is paircontinuous
at any point of .
The subsequent pair of instances demonstrates
that pairwise-ω-continuous functions at a specific
point are not always pairwise almost continuous at
the identical place, and pairwise almost continuous
functions at a particular point are not always
pairwise ω-continuous at the identical location.
Example 3.2
Let the set represents the real numbers and let
is the usual topology on . Providing the function
, which is defined as
(z)=0, in the case of a z is rational, and (z)=1,
in the case of a z is irrational. Therefore at each
irrational number, ψ is pairwise-ω-continuous; at
every single real number, nevertheless, ψ is not
pairwise almost continuous.
Example 3.3
Let the set represents the real numbers and let
is the usual topology on . Consider ,
with the topology . Consider
, which is defined as
(z)=a, in the case of a z is rational, and (z)=b, in
the case of a z is irrational. nevertheless, ψ is
pairwise almost continuous at every rational
number, nevertheless, ψ is not pairwise-ω-
continuous.
Any paircontinuous function is obviously
paircontinuous. The example that follows,
nevertheless, demonstrates that this need not be
necessary.
Example 3.4 Let be any topology on where n
neighborhoods of every nonzero point are as in the
form of usual topology, whereas n neighborhoods of
will represent as , with is a neighborhoods
of in the usual topology and
. Now, let be a usual topology on . Let
the identity function be denoted by
. Then, while is clearly paircontinuous,
it cannot be paircontinuous at as a result.
Note that if represents a bitopological
space, then the family of every one of open sets
represent a topology , which is finer than .
Therefore, is pair-ω-
continuous if and only if
is paircontinuous. The next theorems
are then stated with simplicity.
Theorem 3.5 Let be a
function. Then we have the equivalent statements as
follows:
(1) the function is pair continuous,
(2) the function is
paircontinuous,
(3) for any pairopen set of ,
is pairopen set in .
(4) for any pairclosed set of ,
is pairclosed set in .
Theorem 3.6 If is
paircontinuous, and
is paircontinuous, then is
paircontinuous.
Theorem 3.7 If is
paircontinuous and , then the function
is paircontinuous.
Theorem 3.8 If is
paircontinuous and
be pair continuous, then
represents as
is paircontinuous function.
Proof Let be basics of open
subset of . Then
. Now, since
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be paircontinuous
and be
paircontinuous functions,
is
open and
is open. Thus,
is open. In the same way
for a basics open subset of
. Hence, the function
is paircontinuous.
Theorem 3.9 Let the function
is defined as
, where
. Then is pair continuous if and
only if and are two pair continuous
function.
Proof Assume that is paircontinuous.
As , where the projection function
, so by theorem
(3.4) we have is paircontinuous. In the
same way, as we can demonstrate that is
paircontinuous.
Assume that the functions and are two
paircontinuous. Let be subset of
such that , where the
set is a open of . Thus
. But the function is
paircontinuous, so the set
is
pairopen. Therefore, the function
is
paircontinuous.
Theorem 3.10 Let be
any function and . If there exist a
pairopen set of such as , and
the function
is paircontinuous at .
Proof Assume that the subset is open of
containing . Since the function
is paircontinuous at , then there is a
open set of containing such that
. Since the set is pairopen of
and , so since If
is pair continuous and , then
the function
is paircontinuous , we have
containing . In the same way of the proof,
for the subset is open of containing
. Therefore, the function is
paircontinuous at .
Corollary 3.11 Let the function
and be any pairopen
cover of , such as for any .
If the function
is paircontinuous for any
, then is paircontinuous.
It can be observed in the following example that two
pairopen sets do not necessarily have to be
pairopen products.
Example 3.12 Let ,
where the set represents the irrational numbers of
and is subset of . Then is
pairopen set in and is pairopen in , so it
is pairopen set. Moreover, is not
pairopen set.
Question: Let the functions
and be
two paircontinuous. Is the function
paircontinuous.
In fact, the answer will be negative. The example
below demonstrates this.
Example 3.13 Let the set represents the real
numbers and let is the usual topology on .
Assume that such that ,
and the function
represents as is irrational, is
rational. Now, let the identity function is
. Thus, it clearly that
the functions and are two
paircontinuous. Now, suppose that the
function
represents as
. Let the set
is open in . So,
,
but by previous example (3.11), we have
is
not pairopen. Therefore, the function is
not paircontinuous.
Definition 3.14 A space is called
paircompact, if any pairopen cover of
the space has a finite subcover.
Definition 3.15 A space is called
pairlindelöf, if any pairopen cover of
the space has a countable subcover.
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Definition 3.16 A space is called
pairclosed compact, if any pair closed
cover of the space has a finite subcover.
Definition 3.17 A space is called a
countably pairclosed compact, if any
countable pairclosed cover of the space has
a finite subcover.
Theorem 3.18 Let the surjection function
is pair continuous.
Then the next statements are hold:
(1) If is pairlindelöf space, then
is pairlindelöf,
(2) If is countably paircompact
space, then is paircountably compact.
Proof (1) Let be any
pairopen cover of , since is
surjection function and paircontinuous, then
for any , there exists , for any
is countable subcover such that
, where the
set is open, and
is open. Now, let the set
is a open containing ,
and the set
is a
open containing , where
,
. Let the cover
is pairopen of .
Since is pairlindelöf space, then
. Therefore,
.
That is the space is pairlindelöf.
(2) Similar to the prove of part (1).
Corollary 3.19 Let the surjection function
is pair continuous.
Then the next statements are hold:
(1) If is pairlindelöf space, then
is pairlindelöf,
(2) If is countably pair compact
space, then is paircountably compact.
Theorem 3.20 Let the surjection function
is pair continuous.
Then the next statements are hold:
(1) If is pairclosed compact space,
then is paircompact,
(2) If is pairclosed lindelöf space,
then is pairlindelöf,
(3) If is pairclosed compact space,
then is paircountably compact.
Proof We can prove this theorem in the same way
as the previous theorem (3.18).
Corollary 3.21 Let the surjection function
is pair continuous.
Then the next statements are hold:
(1) If is pairclosed compact space,
then is paircompact,
(2) If is pair closed lindelöf space,
then is pairlindelöf,
(3) If is pairclosed compact space,
then is paircountably compact.
Definition 3.22 A space is called
pair space, if for any two distinct points
and of , there exists two sets and
is pairopen containing and , respectively,
where and .
Definition 3.23 A space is called
pair space, if for any two distinct points
and of , there exists two sets and
is pairopen, where and .
Theorem 3.24 Let the injection function
is pair continuous
and is pair space. Then the space
is pair.
Proof Assume that is pair. For any
two distinct points and of , there
exists open set and open set such that
, , , .
Since is paircontinuous and injection
function, then there are subsets of
is open and of
is open, such that
, , , .
Thus, the space is pair.
Corollary 3.25 Let the injection function
is pair continuous
and is pair space. Then the space
is pair .
Theorem 3.26 Let the injection function
is pair continuous
and is pair space. Then the space
is pair.
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Proof Let is pair space, and the
injection function is
paircontinuous, and in ,
then , are two disjoint subsets of
. Since the space is pair,
there exists two sets open of , and
open , such that , ,
. Since is pair continuous and
injection function, then the sets be
open in and containing ,
be open in and
containing , such that
. Thus, the space
is pair .
Corollary 3.27 Let the injection function
is paircontinuous
and is pair space. Then the space
is pair .
4 Relations Pairwise Continuous
Function with Forms Continuity
Observe that the paircontinuous functions at
any given point may not always be pairalmost
continuous functions at the same point, as
demonstrated by the next two examples, which also
demonstrate that pairalmost continuous functions
at any given point do not always have to be
paircontinuous at the same point.
Example 4.1 Let the function
defined as , if is rational
number and if is irrational number.
Then the function is pairalmost continuous at
any rational number. Nevertheless, the function is
not paircontinuous at each real number.
Example 4.2 Let the real number set be denoted by
, and represents the usual topology on by .
Assume that with .
Now, let is a function
such that for irrational and
for rational . Then, for every rational number, is
a paircontinuous function. At any real
number, the function cannot be pairalmost
continuous.
Observe that the pair continuous functions
at any given point may not always be pairweakly
continuous functions at the exact same point, as
demonstrated by the next two examples, which also
demonstrate that pairweakly continuous functions
at any given point do not always have to be
paircontinuous at the exact same point.
Example 4.3 At every irrational number, the
paircontinuous function , as defined in
example [4.1], exists. Nevertheless, the function
cannot be pairweakly continuous.
Example 4.4 Let the real number set be denoted by
, and represents the co-countable topology on
by . Assume that with
. Now, let
is a function such that for
rational and for irrational . Then, for
any irrational number, is a pair
continuous function. At any real number, the
function cannot be pair continuous.
Definition 4.5 A function
is called pairsequentially continuous if
for any pairsequence such that
we have , for .
Note that the function that is defined in example
[4.5] is not pair continuous; more so, it is
pairweakly and pairsequentially continuous.
Conversely, the next example demonstrates that a
paircontinuous function does not necessarily
imply a pairsequentially continuous function.
Example 4.6 Let the function
is identity. Then, the function is
paircontinuous (pair continuous). But the
function cannot be pairsequentially continuous.
. Thus is a
finite subcover of for . Hence is
compact.
5 Pairwise Barely Continuous
Functions and Pairwise
Lindelöf Space
Theorem 5.1 The following are hold for every
bitopological space :
(1) the space is pair-lindelöf,
(2) any pair open cover of contains a
countable pairsubcover.
Proof Assume that is
pairopen cover of , since
is pairlindelöf, so we have countable
subsets , of the family , then contains
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pairopen cover
,
where is open,
is
open, for any , is countable;
, and is countable. Now, the space is
pairlindelöf gives that contains a countable
pairsubcover
. Therefore, the set
covers by
but the set
is countable, thus
is covered with countably members of . Therefore,
we get the result.
is obviously.
Theorem 5.2 Let the function
is paircontinuous. If is
pairlindelöf, then is so.
Proof Let
be any
pairopen cover of , where the set
is open and
is
open. Since the function is
paircontinuous, then
is pair open cover of the
space , which the set
is open, and
is
open. Now, since is pairlindelöf,
by theorem (5.1) we have contains a countable
pairsubcover such that
. Therefore,
is countably pairsubcover
of . Thus, the result.
Corollary 5.3 Let the function
is paircontinuous. If is
pairlindelöf, then is so.
Corollary 5.4 Let the function
is paircontinuous. If is
pairlindelöf, then is so.
Theorem 5.5 The following are hold for every
bitopological space :
(1) the space is pair-lindelöf,
(2) any family pairclosed sets of
which contains a countable intersection property
that has nonempty intersection.
Definition 5.6 The function
is called pairbarely continuous if for
any nonempty pairclosed , the
restriction
contains there exists one point of
paircontinuity.
Definition 5.7 The function
is called pairbarely continuous if
for any nonempty pairclosed , the
restriction
contains there exists one point of
paircontinuity.
Theorem 5.8 Let the function
is pairbarely continuous. If the
space is a hereditarily lindelöf, then
is pairlindelöf.
Proof Assume that
is pairopen cover of , which the sets
is open,
is open
and
is
pairopen cover of the space , which
is open,
is open set. Now, let
, the set is open,
is open, where is
pairopen subset of that covered by a
countably numbers of members of . Also, let
. Since the space is
hereditarly lindelöf, we get . Thus, the
elements of is maximal of . Assume that
, then the nonempty subset of that
is pairclosed. Thus, the function
is
paircontinuous at some . Now, let
such as , then there a set is
pairopen containing as .
Additionally, there is a set is pairopen
containing as
is countable. Therefore,
is covered with countably numbers of members of
. Thus, that is contradiction with is a
maximal element of , . So, is covered by
countably subcolletion of . Hence, contains
countable subcover, and so the space is
pairlindelöf.
Corollary 5.9 Let the function
is pair-barely continuous, then if the
space is a hereditarily lindelöf, then
is pairlindelöf.
Remark 5.10 Notice that although the function in
example (4.1) is pairbarely continuous, it
cannot be pairbarely continuous.
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6 Conclusions
As we have seen in this research, the functions of
continuous are a generalization of continuous
functions. They have a way to hold onto sequence
limits and are specified on topological spaces. This
implies that the image of a series under the function
will likewise converge to the image of the point if it
contains a sequence of points in the function's
domain that converge to a point. It's a means of
applying the idea of continuity to more complex
circumstances such as weakened these functions, so
based on that we get and explored in this research
their master features, such that the concepts of
pairwise continuous as well as pairwise barely
continuous functions. We have looked at these ideas'
key characteristics and revealed how they relate to
other circumstances. We identified their principal
characteristics as a whole and defined the conditions
that must be met in order to attain comparable
connections between them. We talked about their
main traits and demonstrated how they work
together. The report also highlighted the
characteristics of these functions and offered
numerous instances of them. These functions will
serve as a springboard for research into all of these
functions' possible futures. Future studies might
look into investigating other variations of these roles
such as in fuzzy, soft and group, [30], [31], [32].
Acknowledgements:
We deeply appreciate each person who contributed
to this study. Their teamwork, instructions, and
encouragement have been essential to finishing this
study.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.105
Ali A. Atoom, Hamza Qoqazeh,
Rahmeh Alrababah, Eman Almuhur,
Nabeela Abu-Alkishik
E-ISSN: 2224-2880
969
Volume 22, 2023
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Transactions on mathematics,Vol. 19, 2020,
No.10, pp. 37394/23206
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Ali Atoom: presented the main idea of the
research, provided the basic definition of the
research, and enriched it with several theories and
illustrative examples.
- Hamza Qoqazeh: wrote the introduction of the
research and added some theories and scientific
facts to the research.
- Rahmeh Alrababah: wrote the introduction of the
research and added some theories and scientific
facts to the research.
- Eman Almuhur: conducted scientific and
linguistic scrutiny of the research, as well as
reviewed and verified the academic references.
- Nabeela Abu-Alkishik: conducted scientific and
linguistic scrutiny of the research, as well as
reviewed and verified the academic references.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research was funded by Ajloun National
University.
Conflict of Interest
The authors acknowledge the work's authorship and
declare that no personal gain is intended from its
publication. The primary aim of this study is to
make a scientific contribution to the area of general
topology. Furthermore, every researcher certifies
that this work was not published, submitted to, or
accepted for publication in any journal and that it
was entirely new research.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.105
Ali A. Atoom, Hamza Qoqazeh,
Rahmeh Alrababah, Eman Almuhur,
Nabeela Abu-Alkishik
E-ISSN: 2224-2880
970
Volume 22, 2023
