Combinatorial Optimization Systems Theory Prospected from
Rotational Symmetry
VOLODYMYR RIZNYK
Department of Computer Aided Control Systems
Lviv Polytechnic National University
79013, Lviv-13, Stepan Bandera Str., 12
UKRAINE
Abstract: - Combinatorial optimization systems theory prospected from rotational symmetry involves
techniques for improving the quality indices of engineering devices or systems with non-uniform structure
(e.g., controllable cyber-physical objects) concerning transformation swiftness, position accuracy, and
resolution, using designs based on extraordinary geometric properties and structural excellence of
combinatorial conformations, namely the concept of Ideal Ring Bundles. Design techniques based on the
underlying combinatorial theory provide configure one- and multidimensional systems with smaller
amounts of elements than at present, while maintaining the other substantial operating characteristics of the
systems.
Key-Words: - Cyclic group, Ideal Ring Bundle, torus reference system, GUS configuration, spatial
perfection, manifold coordinates, vector data.
Received: July 22, 2023. Revised: May 21, 2024. Accepted: June 18, 2024. Published: July 23, 2024.
1 Introduction
Combinatorial optimization theory of systems [1]
encompasses various scientific and technological
fields such as software engineering, algorithm
theory, operations research, machine learning,
computational complexity theory, applied
mathematics, and theoretical computer science. The
classical combinatorial theory involves
fundamental concepts of modern algebra and
geometry, including difference sets in a finite
group, finite projective planes, Hadamard matrices
theory, cyclic incidence matrices, and the
problematic of certain symmetrical balanced
incomplete block designs, automorphisms of
groups, and orthogonal Latin squares [2]. In paper
[3], the study of some permutations helps discover
unrelated classes. A multinomial function that
describes a system with multi-input and multi-
output systems has the coefficients for parameters
[4]. Many original models, concepts, parallel
algorithms, platforms, applications, and processing
gears relate to improving the assessment of big data
technology [5], [6], [7], artificial intelligence [8],
[9], signal processing [10], [11], and radio
engineering [12], advanced algorithms [13], [14],
and cryptography [15]. The objective of this work
is to test suitable sets of famous classes concerning
a small subset of such functions based on the
intelligence of rotational symmetry. The principle
of symmetry and asymmetry is prevalent in nature
and artificial environments, so it's crucial to
consider rotational symmetry for the development
of optimization systems theory in fundamental and
applied research. This can be achieved through
innovative methodologies based on the concept of
Ideal Ring Bundles (IRBs) [16], which involves the
idea of "perfect" multidimensional combinatorial
constructions. This paper deals with techniques for
improving the quality indices of controllable cyber-
physical systems and vector processing, such as
transformation speed, resolving ability, minimizing
machinery memory, and computing resources,
using designs based on the combinatorial
optimization systems theory. Theoretical research
into the combinatorial configuration's properties
has led to a better understanding of the role of
rotational symmetries in the theory. Modern
combinatorial theory and system design connect
with appropriate constructions such as manifolds
[17], connecting algebra through geometry [18],
and the Golden ratio [19], which involve the
relationships of rotational symmetry spatial
multidimensional configurations [16]. Symmetries
and curvature structures are embedded in general
relativity [20].
2 Optimum Combinatorial
Structures
2.1 Optimum Combinatorial Sequences
A "well-ordered" sequence of distributed elements
can be very useful in optimally solving various
technological problems, leading to high profits.
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.7
Volodymyr Riznyk
E-ISSN: 2769-2477
53
Volume 4, 2024
)3mod,2(mod
(0,2) (1,0)(1,2)
(0,0) (1,2)(1,1)
(0,1) (1,1)(1,0)
)3mod,2(mod
(1,1) )(1,2)·(1,2
(1,2) )(1,1)·(1,2
(1,0) )(1,0)·(1,2
2.1.1 Sums on a chain- ordered numbers
Let us compute all L sums of ordered-chain
numbers in the n-stage sequence of positive
integers {k1, k2, . . ., kn}, where the sums of
connected sub-sequences of the sequence
enumerate the set of integers from 1 to L. The
maximum number of distinct sums on ordered-
chain numbers is
L = n · (n+1)/2 (1)
Another type of combinatorial construction is the
use of ring structures.
2.2 Ring Numerical Structures
Let's consider a sequence of positive integers {k1,
k2, ..., kn} arranged in a specific order such that kn is
followed by k1, forming a chain structure. As we
continue to add integers to this sequence, it
eventually turns into a numerical ring structure with
n stages. In contrast to an ordered chain, a ring
numerical structure allows for a sum of connected
sub-sequences to have any length from 1 to n–1 as
its starting point, with the final sum including all n
integers. Therefore, the maximum number of
distinct sums S that can be obtained from the ring
numerical structure is
S = n (n –1) +1 (2)
Comparing the equations (1) and (2), easy to see
that the number of sums S for connected terms in
the ring topology is nearly double the number of
sums L in the daisy-
chain
topology,
for the same sequence
of n terms.
2.2.1 Ideal Ring Bundles
Ideal Ring Bundles are cyclic sequences of positive
integers that form perfect partitions of a finite
interval [1, S] of integers. The sums of consecutive
sub-sequences of an Ideal Ring Bundle (IRB)
enumerate the set of integers [1, S] exactly once.
Here is an example of an IRB with n=5 and S=5(5–
1) +1=21, namely {1,5,2,10,3}. To see this, we
observe:
1=1 6=1+5 11=3+1+5+2 16=2+10+3+1
2=2 7=5+2 12=2+10 17=5+2+10
3=3 8=1+5+2 13=10+3 18=1+5+2+10
4=3+1 9=3+1+5 14=10+3+1 19=10+3+1+5
5=5 10=10 15=2+10+3 20=5+2+10+3
21=1+5+2+10+3
We understand that each ring sum from 1 to S =21
occurs exactly once.
2.2.2 Relative of Ideal Ring Bundles to
Rotational Symmetry
For a better understanding of the role of geometric
structures in the combinatorial optimization
systems theory, we regard Ideal Ring Bundles with
informative parameters S and n as cyclic numerical
relationships followed by equation (2) based on the
idea of “generative” rotational symmetry of order
S. Employing 21-fold rotational symmetry, the IRB
can be configured using complementary
asymmetries relations of geometric structure.
2.2.3 Two-Dimensional Ideal Ring Bundles
Let’s consider a cyclic sequence of n -stages,
denoted as {K1, K2, …, Ki, …, Kn}, K1= (k11, k12),
K2 =(k21, k22), ..., Ki=(ki1, ki2), …., Kn=(kn1, kn2). This
sequence consists of 2-stage (t=2) sub-sequences.
We require that all two-dimensional modular vector
sums (mod m1, mod m2) must form a two-
dimensional coordinate grid of sizes m1×m2 over a
toroidal surface, where m1·m2 = S–1. This
configuration is known as the two-dimensional
Ideal Ring Bundle (2-D IRB). Here are four
variants of 2-D IRBs with parameters S =7, n = 3,
m1= n –1 = 2, and m2 = n = 3:
(a) {(1,0),(1,1),(1,2)}; (b) {(0,1),(0,2),(1,0)};
(c) {(0,1),(0,2),(1,2)}; (d) {(0,1),(0,2),(1,1 )}
The group {(1,0),(1,1),(1,2)} in two-dimensional
IRB allows for addition and multiplication
operations modulo m1=2, m2=3.
Therefore, two-dimensional IRB {(1,0),(1,1),(1,2)}
generates a coordinate grid 2×3 over a toroidal
surface with a common reference point (0,0):
(1,0)
(1,1)
(1,2)
(0,0)
(0,1)
(0,2)
The next we see result of multiplying IRB
{(1,0),(1,1),(1,2)} by vector (1,2):
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.7
Volodymyr Riznyk
E-ISSN: 2769-2477
54
Volume 4, 2024
Here we see transformation IRB {(1,0), (1,1), (1,2)}
into myself. Taking the same conversion for
variants (b), (c), and (d), we finally obtain the next
result: (a) × (1,2) (a); (b) × (1,2) (b); (c) ×
(1,2) (d); (d) × (1,2) (c).
Hence, the set of four 2-D IRBs {(a), (b), (c), (d)}
form both two isomorphic (a, b), and two non-
isomorphic (c, d) modifications of the 2-D IRB. We
call this the cyclic two-dimensional IRB group.
Note, that each of these variants makes it possible
to obtain m1 ∙ m2 = 2 ∙ 3 = 6 varied 2-D
IRBs.
Table 1 demonstrates optimized two-dimensional
binary code, based on the 2-D IRB {(1,0), (1,1),
(1,2)} with informative parameters S=7, n=3.
Table 1
Optimized two-dimensional binary code, based on
the 2-D IRB {(1,0), (1,1), (1,2)} with informative
parameters S=7, n=3
№
Vector
Digit weights of the 2-D code
(1,0)
(1,1)
(1,2)
1
(0,0)
0
1
1
2
(0,1)
1
1
0
3
(0,2)
1
0
1
4
(1,0)
1
0
0
5
(1,1)
0
1
0
6
(1,2)
0
0
1
Table 1 defines a two-dimensional binary code
system as a torus surface coordinate grid (n –1) × n
= 2 × 3 with two (t=2) circle axes m1 =2, and m2= 3.
Here is an example of code system design, using the
combinatorial optimization systems theory
prospected from rotational symmetry of order seven
(S =7). The example belongs design of an optimized
data system processing two categories and three
attributes concurrently (Table 2).
Table 2
Optimized data system processing two categories
and three attributes concurrently.
№
Category
Digit weights of the 2-D code
1
2
(1,0)
(1,1)
(1,2)
1
0
0
0
1
1
2
0
1
1
1
0
3
0
2
1
0
1
4
1
0
1
0
0
5
1
1
0
1
0
6
1
2
0
0
1
Table 2 contains 6 binary 2-D (t = 2) 3- 3-digit (n =
3) combinations (n2 –n = 6) for coding two-
dimensional (t = 2) data sets both with two (m1 = 2)
category of the first, and three (m2 = 3) – the second
attribute concurrently. More practical examples for
combinatorial optimization of vector data
processing are proposed in [16].
2.2.4 Multidimensional Ideal Ring Bundles
Multidimensional ideal ring bundles form a t-
manifold coordinate system immersed in (t+1)-
dimensional no real space without self-intersection
of coordinate axes. A t-dimensional coordinate
system (t > 2) with t axes is named the manifold
coordinate system m1×m2 ×…×mt. The principal
property of coordinate grid m1 m2 … mt over a t-
manifold surface is n-stage sequence {K1, K2, …,
Ki, …, Kn}, K1= (k11, k12,…, k1t), K2 =(k21, k22,…,
k2t), ..., Ki=(ki1, ki2, …, kit), …., Kn=(kn1, kn2,…, knt)
of t-stage sub-sequences of the sequence, where we
require a set modulo sums taking t- modulo (m1, m2
,.…, mt ) enumerates all coordinates of the t-
manifold surface. This is perfect t-manifold
coordinate system m1 m2 … mt with information
parameters S, n, mi (i = 1, 2, …, t). It is a t-
dimensional image surface involving spatially
disjointed reference t-axes. A planar projection of
t-dimensional manifold coordinate axes m1, m2, …,
mt for grid m1×m2 ×…×mt with common point
illustrates Fig. 1.
Fig.1: A planar projection of t-dimensional manifold
coordinate axes m1, m2, …, mt for grid m1×m2
×…×mt with common point.
Here S is an order of spatial symmetry, n- number of
t-stage sub-sequences of the n-sequence, and
mt
m1
m2
mi
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.7
Volodymyr Riznyk
E-ISSN: 2769-2477
55
Volume 4, 2024
number of basic attribute-categories subsets forming
a complete set of t-dimensional vector data arrays.
Hence, the t-dimensional IRB forms a manifold t-
dimensional coordinate system. A t-dimensional
perfect manifold coordinate system can be designed
for configuring t-dimensional optimized control
systems or CAD. Therefore, all information about
the t-dimensional vector data array of sizes m1 m2
… mt is embedded into the coordinate system.
3 Glory to Ukraine Stars Ensembles
Of very exciting property has been discovered in
“Glory to Ukraine Star” (GUS) ensembles as a new
type of spatial combinatorial configuration [16].
Graphic representation one of paired seven-pointed
(n=7) GUS-configurations {(4,2), (0,2), (1,2), (0,4),
(2,2), (3,2), (5,2), (4,2)} (black ring line) and
{(4,2), (1,2), (2,2), (5,2), (3,2), (0,4), (0,2), (4,2)}
(color broken line) are shown in Fig.2.
Fig.2: Graphic representation of paired seven-
pointed (n=7) GUS-configurations {(4,2), (0,2),
(1,2), (0,4), (2,2), (3,2), (5,2)} (black ring line) and
{(4,2), (1,2), (2,2), (5,2), (3,2), (0,4), (0,2)} (color
broken line).
The GUS-configuration {(4,2), (0,2), (1,2), (0,4),
(2,2), (3,2), (5,2)} (black ring line) generates all n
(n –1)= 42 two-dimensional ring sums, taking
modulo (mod 6, mod 7) as follows:
1. (0,0) ≡
((5,2)+(4,2)+(0,2)+(1,2)+(0,4)+(2,2));
2. (0,1) ≡ ((3,2) +(5,2)+(4,2)+(0,2));
3. (0,2) ≡ (0,2);
4. (0,3) ≡ ((1,2)+(0,4)+(2,2) +(3,2));
5. (0,4) ≡ (0,4);
6. (0,5) ≡ ((0,2)+ (1,2)+ (0,4)+ (2,2)+
(3,2));
7. (0,6) ≡ ((3,2)+(5,2)+(4,2));
8. (1,0) ≡
((3,2)+(5,2)+(4,2)+(0,2)+(1,2)+(0,4));
9. (1,1) ≡ ((0,2)+ (1,2)+ (0,4));
10. (1,2) ≡ (1,2);
11. (1,3) ≡ ((3,2)+(5,2)+(4,2)
+(0,2)+(1,2));
12. (1,4) ≡ (0,2)+(1,2));
13. (1,5) ≡ ((4,2)+ (0,2)+ (1,2)+(0,4)+
(2,2));
14. (1,6) ≡ ((1,2)+(0,4));
15. (2,0) ≡
((0,4)+(2,2)+(3,2)+(5,2)+(4,2)+(0,2));
16. (2,1) ≡ ((2,2)+(3,2)+(5,2)+(4,2));
17. (2,2) ≡ (2,2);
18. (2,3) ≡
((2,2)+(3,2)+(5,2)+(4,2)+(0,2));
19. (2,4) ≡ ((3,2)+(5,2));
20. (2,5) ≡
((0,4)+(2,2)+(3,2)+(5,2)+(4,2));
21. (2,6) ≡ ((0,4)+(2,2));
22. (3,0) ≡ ((1,2)+(0,4)+(2,2)
+(3,2)+(5,2)+(4,2));
23. (3,1) ≡ ((1,2)+(0,4)+ (2,2));
24. (3,2) ≡ (3,2);
25. (3,3) ≡ ((0,2)+ (1,2)+ (0,4)+ (2,2));
26. (3,4) ≡ ((5,2)+(4,2));
27. (3,5) ≡ ((2,2)+(3,2)+(5,2)+(4,2)+
(0,2)+ (1,2));
28. (3,6) ≡ ((5,2)+(4,2)+(0,2));
29. (4,0) ≡ ((4,2)+(0,2)+ (1,2)+ (0,4)+
(2,2)+ (3,2));
30. (4,1) ≡ ((5,2)+(4,2) +(0,2)+(1,2));
31. (4,2) ≡ (4,2);
32. (4,3) ≡ ((0,4)+(2,2)+(3,2)+(5,2));
33. (4,4) ≡ ((4,2)+(0,2));
(0,4)
(0,2)
(2,2)
(3,2)
(5,2)
(1,2)
(4,2)
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DOI: 10.37394/232028.2024.4.7
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E-ISSN: 2769-2477
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Volume 4, 2024
34. (4,5) ≡
((5,2)+(4,2)+(0,2)+(1,2)+(0,4));
35. (4,6) ≡ ((2,2)+(3,2)+(5,2));
36. (5,0) ≡ ((0,2)+ (1,2)+ (0,4)+
(2,2)+(3,2)+(5,2));
37. (5,1) ≡ ((0,4)+ (2,2)+(3,2));
38. (5,2) ≡ (5,2);
39. (5,3) ≡ ((4,2)+(0,2)+(1,2)+(0,4));
40. (5,4) ≡ ((2,2)+(3,2));
41. (5,5) ≡ ((1,2)+ (0,4)+
(2,2)+(3,2)+(5,2));
42. (5,6) ≡ ((4,2)+(0,2)+(1,2));
The calculation procedures form m1 × m2 = 6×7
grid, embracing a two-dimensional (t=2) toroid
surface as being a coordinate system, where each
point node from (0,0) to (5,6) occurs exactly once
(R=1).
The second of the paired GUS-configurations
{(4,2), (1,2), (2,2), (5,2), (3,2), (0,4), (0,2)} (color
broken line) forms the same set of sums, taking 2D
modulo (mod 6, mod 7):
1. (0,0) ≡
((0,4)+(0,2)+(4,2)+(1,2)+(2,2)+(5,2));
2. (0,1) ≡ ((4,2)+(1,2)+(2,2)+(5,2));
3. (0,2) ≡ (0,2);
4. (0,3) ≡ ((0,2)+(4,2)+(1,2)+(2,2)+(5,2));
5. (0,4) ≡ (0,4);
6. (0,5) ≡ ((5,2)+(3,2)+(0,4)+(0,2)+(4,2));
7. (0,6) ≡ ((0,4)+(0,2));
8. (1,0) ≡
((5,2)+(3,2)+(0,4)+(0,2)+(4,2)+(1,2);
9. (1,1) ≡ ((0,2)+(4,2)+(1,2)+(2,2));
10. (1,2) ≡ (1,2);
11. (1,3) ≡ ((3,2)+(0,4)+(0,2)+(4,2));
12. (1,4) ≡ (2,2)+(5,2));
13. (1,5) ≡
((0,4)+(0,2)+(4,2)+(1,2)+(2,2));
14. (1,6) ≡((4,2)+(1,2)+(2,2));
15. (2,0) ≡ ((2,2)+
(5,2)+(3,2)+(0,4)+(0,2)+(4,2));
16. (2,1) ≡ ((5,2)+(3,2+(0,4));
17. (2,2) ≡ (2,2);
18. (2,3) ≡ ((5,2)+(3,2)+(0,4)+(0,2));
19. (2,4) ≡ ((5,2)+(3,2));
20. (2,5) ≡ (3,2)+(0,4)+(0,2)+(4,2)+(1,2);
…………………………………………………
42. (5,6) ≡ ((0,2)+(4,2)+(1,2)).
We observe either of the GUS-configurations
{(4,2), (0,2), (1,2), (0,4), (2,2), (3,2), (5,2)} (black
ring line) and {(4,2), (1,2), (2,2), (5,2), (3,2), (0,4),
(0,2)} (color broken line) forms m1 × m2 = 6×7 grid,
embracing toroid surface as 2-D coordinate system.
Here's another example of paired seven-pointed
(n=7) GUS configurations.{(1,1), (1,3), (1,5), (1,0),
(1,2), (1,4), (1,6)} (ring cycle), and {(1,1), (1,5),
(1,2), (1,6), (1,3), (1,0), (1,4)} (star cycle) presents
in Fig.3.
Fig.3: Paired seven-pointed (n=7) GUS-
configurations, namely the {(1,0), (1,2), (1,4), (1,6),
(1,1), (1,3), (1,5)} (ring cycle), and {(1,0), (1,4),
(1,1), (1,5), (1,2), (1,6), (1,3)} (star cycle).
GUS-configuration {(1,1), (1,3), (1,5), (1,0), (1,2),
(1,4), (1,6)} (ring cycle) forms the set of 2-D
vector sums (clockwise), taking 2-D modulo (6, 7):
1. (0,0) ≡
((1,2)+(1,4)+(1,6)+(1,1)+(1,3)+(1,5));
2. (0,1) ≡ ((1,1)+(1,3)+ (1,5)+(1,0)+
(1,2)+(1,4));
3. (0,2) ≡
((1,0)+(1,2)+(1,4)+(1,6)+(1,1)+(1,3));
4. (0,3) ≡ ((1,6)+ (1,1)+
(1,3)+(1,5)+(1,0)+ (1,2));
(1,3)
(1,5)
(1,0)
(1,2)
(1,4)
(1,6)
(1,1)
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.7
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E-ISSN: 2769-2477
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Volume 4, 2024
5. (0,4) ≡ ((1,5)+(1,0)+ (1,2)+(1,4)+
(1,6)+(1,1));
…………………………………………………
…………………………………………………
41. (5,5) ≡ ((1,4)+(1,6)+ (1,1)+
(1,3)+(1,5);
42. (5,6) ≡ ((1,0)+ (1,2)+(1,4)+
(1,6)+(1,1)).
The second of the paired GUS configuration {(1,1),
(1,5), (1,2), (1,6), (1,3), (1,0), (1,4)} (star cycle)
forms the set of 2-D vector sums, taking modulo
(mod 6, mod 7):
1. (0,0) ≡ ((1,4)+(1,1)+(1,5)+
(1,2)+(1,6)+(1,3));
2. (0,1) ≡ ((1,3)+ (1,0)+(1,4)+(1,1)+
(1,5)+(1,2));
3. (0,2) ≡ ((1,2)+(1,6)+(1,3)
+(1,0)+(1,4)+(1,1));
4. (0,3) ≡ ((1,1)+(1,5)+
(1,2)+(1,6)+(1,3)+(1,0));
5. (0,4) ≡ ((1,0)+(1,4)+ (1,1)+(1,5)+
(1,2)+(1,6));
…………………………………………………
…………………………………………………
41. (5,5) ≡ ((1,0)+(1,4)+ (1,1)+
(1,5)+(1,2);
42. (5,6) ≡ ((1,3)+ (1,0)+(1,4)+
(1,1)+(1,5)).
Each of the paired seven-pointed (n=7) GUS-
configurations, {(1,1), (1,3), (1,5), (1,0), (1,2), (1,4),
(1,6)} (ring cycle), and {(1,1), (1,5), (1,2), (1,6),
(1,3), (1,0), (1,4)} (star cycle) forms m1 × m2 = 6×7
grid, embracing two-dimensional (t=2) toroid
surface as coordinate system exactly once (R=1).
The underlying examples of paired seven-pointed
(n=7) GUS-configurations evident that either of the
combinatorial configurations {(4,2), (0,2), (1,2),
(0,4), (2,2), (3,2), (5,2)}, {(4,2), (1,2), (2,2), (5,2),
(3,2), (0,4), (0,2)}, {(1,1), (1,3), (1,5), (1,0), (1,2),
(1,4), (1,6)}, {(1,1), (1,5), (1,2), (1,6), (1,3), (1,0),
(1,4)} forms complete coordinate system m1 × m2 =
6×7 over toroid surface.
A graphic representation of a set of paired seven-
pointed (n=7) GUS ensembles is illustrated (Fig.4).
Fig. 4: Graphic representation of a set of paired
seven-pointed (n=7) GUS ensembles.
The cardinal number P of paired 2-D GUS
configurations depending on the n-pointing cyclic
structures with m1 × m2 grid sizes for n = 2,3,…7 is
given in Table 3.
Table 3
The cardinal number P of paired 2-D GUS
configurations depending on the n-pointing cyclic
structures with m1 × m2 grid sizes for n = 2,3,…7
Table 3 shows an increasing number of paired 2-D
GUS configurations, arranged by n-pointing cyclic
structures.
4 Conclusion
The theory of combinatorial optimization systems,
explored through the lens of rotational symmetry,
n
Grid sizes m1 × m2
P
2
1×2
1
3
2×3
4
4
3×4
24
5
4×5, 3×7
272
6
5×6, 3×10
256
7
6×7, 3×14
360
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.7
Volodymyr Riznyk
E-ISSN: 2769-2477
58
Volume 4, 2024
offers a novel way to conceptualize engineering
devices and technical systems in an optimized
manner. This approach allows for the optimization
to be integrated into the model itself, thereby
enabling the configuration of systems with fewer
elements than currently used, while still maintaining
or even improving other characteristics of the
system. The theoretical connection between cyclic
groups and IRBs presents significant opportunities
for the advancement of systems theory in
configuring innovative devices and process
engineering. This is due to the exceptional
mathematical properties and structural perfection of
IRBs. The use of optimized perfect manifold
coordinate systems in information technologies
offers new conceptual techniques to improve the
quality of technology and management systems.
This includes improving the transmission and
compression of vector data and ensuring the
reliability of vector data coding and processing
using a minimized basis of manifold coordinates.
The essence of the technology is processing vector
information in the database of manifold coordinate
systems, where the basis is a set of coordinates
smaller than the total number of coordinates of this
coordinate system, which generates it by adding the
latter. The theoretical connection between cyclic
groups and IRBs presents significant opportunities
for the advancement of systems theory in
configuring innovative devices and process
engineering. This is due to the exceptional
mathematical properties and structural perfection of
IRBs. The exceptional mathematical properties and
structural perfection of IRBs create significant
opportunities for the advancement of systems theory
in configuring innovative devices and process
engineering through their theoretical connection
with cyclic groups. Multidimensional systems
engineering can be improved by researching
combinatorial optimization systems theory, with a
focus on rotational symmetry. This improvement
can lead to better quality indices such as
information capacity, reliability, transmission speed,
positioning precision, and the ability to reproduce
the maximum number of combinatorial varieties in
the system with a limited number of elements and
bonds. The GUS combinatorial configurations have
remarkable properties and structural perfection,
which can be utilized for direct applications in
information and computational technologies,
telecommunications, radio and electronic
engineering, radio physics, and other engineering
areas, as well as in education. By using these design
techniques, you can configure optimum two- and
multidimensional vector data processing, using
innovative methods based on the underlying
combinatorial models, which offers ample scope for
progress in systems engineering, cybernetics,
computational and applied mathematics, and
industrial informatics.
Declaration of Generative AI and AI-assisted
technologies in the writing process
During the preparation of this work the author used
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author reviewed and edited the content as needed
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publication.
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International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2024.4.7
Volodymyr Riznyk
E-ISSN: 2769-2477
60
Volume 4, 2024
