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Mahmood Iraqi Journal of Science, 2023, Vol. 64, No. 7, pp: 4352-4357
DOI: 10.24996/ijs.2023.64.7.26
__________________________________
*Email: asmahmood65@uomosul.edu.iq
4352
On GCYD-Method In e-Abacus Diagram
Ammar S. Mahmood
Department of Mathematics, College of Education for Pure Sciences, University of Mosul, Mosul, Iraq
Received: 11/3/2022 Accepted: 18/9/2022 Published: 30/7/2023
Abstract
We will provide a new method in this study that integrates two types of
applications, namely Graph Theory and Conjugate Young Diagram, the idea of
combining the graph and the Young diagram is presented by Ali And Mahmood,
which is primarily based on the idea of the e-abacus diagram, the new method is called
GCYD, it directly applies to the English letter section, which will be a two-layer
coding. It makes it difficult to detect the word or sentence.
Keywords: Partition Theory, e-Abacus Diagram, Young Diagram, Conjugate Young
Diagram, Graph Theory.
GCYDe
e
GCYD
1. Introduction
Let t be a non-negative integer number. The sequence of non-negative numbers
is called a partition of t if
& . A
Young diagram, for short, we will write (YD) [1], of a partition is the subset
of For example, if 7, 6, 5, 5, 1, 1, 1, 1), then the
YD is:
ISSN: 0067-2904
Mahmood Iraqi Journal of Science, 2023, Vol. 64, No. 7, pp: 4352-4357
4353
[
=
Diagram 1: YD of 7, 6, 5, 5, 1, 1, 1, 1)
Defining , . The set is said to be the set of - numbers
for . Let e be a positive integer number greater than or equal to 2, we can express using a
"main diagram or e- abacus diagram" see [2-3], by the following:
Table 1: e-Abacus Diagram
Runner 1
Runner 2
Runner e
0
1
e-1
e
e+1
2e-1
2e
2e+1
3e-1
where every will be represented by and the rest of the sites by (-). From the above
example then we have:
Table 2: e-Abacus Diagrams of 7, 6, 5, 5, 1, 1, 1, 1)
2-main
abacus
3-main
abacus
4-main abacus
5-main abacus
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
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-
-
-
Murphy in [4] defined Conjugate YD by
Mahmood Iraqi Journal of Science, 2023, Vol. 64, No. 7, pp: 4352-4357
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Table 3: YD and CYD of 7, 6, 5, 5, 1, 1, 1, 1)
7, 6, 5, 5, 1, 1, 1, 1)
2. GCYD-Method
In this section, we present the idea of combining the graph theory of e-abacus diagram [5]
and CYD, it is called the GCYD-Method, and it applies to the encoding of English letters and
later the word or sentence [6-7], which will be extremely difficult to decipher without prior
knowledge of the word or sentence, and this method will be a two-layer method. The merging
procedure is based on the premise that the chosen shape should be (more like a square matrix)
utilizing graph theory, which is the e-abacus diagram that is presented for each letter of the
English language and that is achieved in the initial layer of the coding process [5]. Now, we
will use the CYD to create the second layer which was demonstrated in [8] for each English
letter. To make everything clearer, we look at each step separately:
First-Layer:
This case starts with two concepts: the first one is the encoding of each English letter
according to the carefully determined model and integrated research so that each letter is
completely distinct from the others, and it is impossible to have two letters that are identical in
any way. Exactly with the second idea, which states that any graph must ultimately begin with
the first notion. A.B. Mahmood and A.S. Mahmood in [6-7] put the optimal model for each
letter based on the 5-abacus diagram, because any model that is smaller than the one chosen
will not achieve the optimal shape due to the possibility of two letters being similar in one or
the lack of clarity of the shape, and we give the coding for this in the following table:
Table 4: The Partition For Each English Letters
Letter
partition
Letter
partition
A
N
B
O
C
P
D
Q
E
R
F
S
G
T
H
U
I
V
J
W
K
X
L
Y
M
Z
Mahmood Iraqi Journal of Science, 2023, Vol. 64, No. 7, pp: 4352-4357
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Which corresponds, for example, to some letters:
Table 5: e-abacus diagram of (O, V, T, E) in YD
16
17
O
18
19
20
1
2
V
3
4
5
11
12
T
13
14
15
6
7
E
8
9
10
…
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-
-
-
-
-
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-
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-
Each letter has been converted to a graph in accordance with [5], assuming that the chosen
abacus represents "similar to a square matrix of amplitude e" and so it will satisfy all of the
hypotheses proposed in [5], for example, the abacus for the letters mentioned in Table 5 is:
1 2 3 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
Figure 1: Graph of the word (VETO) in general case
Second-Layer:
We concentrate on the work of Fatmah A. Basher in [8]. She studied the CYD on English
letters; also Mahmood and Basher in [9] used two methods to find this new partition in this case
the following table is produced:
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Mahmood Iraqi Journal of Science, 2023, Vol. 64, No. 7, pp: 4352-4357
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Table 6: Partition of any English letters (YD & CYD-methods)
YD-Method
CYD-Method
YD-Method
CYD-Method
A
N
B
O
C
P
D
Q
E
R
F
S
G
T
(9, 4,
H
U
I
V
J
W
K
X
L
Y
M
Z
Table 7: e-abacus diagram of (O, V, T, E) in CYD
O
V
T
E
…
16
17
18
19
20
1
2
3
4
5
11
12
13
14
15
6
7
8
9
10
-
-
-
-
-
-
-
-
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-
Also, if we used the GCYD for the word (VETO), then we have the following diagram:
Mahmood Iraqi Journal of Science, 2023, Vol. 64, No. 7, pp: 4352-4357
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1 2 3 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
Figure 2: The Graph of (VETO) by used GCYD-Method
Conclusion:
This application is considered safer and more difficult to encode any message. Thus, it will
achieve a kind of very important confidentiality.
Acknowledgement:
We thank the University of Mosul / College of Education for Pure Science for its great
support to complete the research.
References
[1] A. Mathas, “Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group”. University
Lecture Series, vol.15, 1999.
[2] Ammar S. Mahmood, “On the intersection of Young’s diagrams core”, Journal of Education and
Science, vol. 24, pp. 149-157, 2011.
[3] G. D. James, “Some combinatorial results involving Young diagrams”, Mathematical proceedings
of the Cambridge Philosophical Society, vol. 83, pp.1-10, 1978.
[4] G. E. Murphy, “On the representation theory of the symmetric groups and associated Hecke
algebras”, J. Algebra, vol. 152, pp. 492-513, 1992.
[5] A. M. Ali and Ammar S. Mahmood, “The graph of e-abacus diagram”, WSEAS Trans. on Math.,
vol. 19, (2224-2880), pp. 486-497, 2020.
[6] A. B. Mahmood and Ammar S. Mahmood, “Secret-word by e-abacus diagram I”, Iraqi J. of
Science, vol. 60, no. 3, pp. 638-646, 2019 .
[7] A. B. Mahmood and Ammar S. Mahmood, “Secret-text by e-abacus diagram II”, Iraqi J. of
Science, vol. 60, no. 4, pp. 840- 846, 2019.
[8] F. A Basher, “New Applications Between Young Diagram and e-Abacus Diagram”, M. Sc. Thesis,
University of Mosul, Iraq, 2021.
[9] Ammar S. Mahmood F. A. Basher, “ New method of computing the conjugate of Young diagram”,
J. of Physics Conference Series, IOP, 1879, 2021.
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