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Applications of the Combinatorial Configurations for Optimization of Technological Systems V

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Applications of the Combinatorial Configurations for Optimization of Technological Systems V ECONTECHMOD. AN INTERNATIONAL QUARTERLY JOURNAL – 2016. Vol. 5. No. 2. 27–32 Applications of the combinatorial configurations for optimization of technological systems V. Riznyk Lviv Polytechnic National University, e-mail: [email protected] Received March 5.2016: accepted May 20.2016 Abstract. This paper involves techniques for improving mathematical methods of optimization of systems that the quality indices of engineering devices or systems with non- exist in the structural analysis, the theory of combinatorial uniform structure (e.g. arrays of sonar antenna arrays) with configurations, analysis of finite groups and fields, respect to performance reliability, transmission speed, resolving algebraic number theory and coding. Two aspects of the ability, and error protection, using novel designs based on matter the issue are examined useful in applications of combinatorial configurations such as classic cyclic difference sets and novel vector combinatorial configurations. These symmetrical and non-symmetrical models: optimization design techniques makes it possible to configure systems with of technology, and hypothetic unified “universal fewer elements than at present, while maintaining or improving informative field of harmony” [1]. on the other operating characteristics of the system. Several factors are responsible for distinguish of the objects depending THE ANALYSIS OF RECENT RESEARCHES an implicit function of symmetry and non-symmetry interaction AND PUBLICATIONS subject to the real space dimensionality. Considering the significance of circular symmetric field, while an asymmetric In [2] developed Verilog Analog Mixed-Signal subfields of the field, further a better understanding of the role simulation (Verilog-AMS) model of the comp-drive sensing of geometric structure in the behaviour of system objects is element of integrated capacitive micro- accelerometer. This developed. This study, therefore, aims to use the appropriate model allows simulate the reaction of the sensing elements algebraic results and techniques for improving such quality effected by the applied force of acceleration, changes of its indices as combinatorial varieties, precision, and resolving comb-drive capacities, output voltages and currents for ability, using remarkable properties of circular symmetry and determining its constructive parameters and for analysis of non-symmetry mutual penetration as an interconnection cyclic mechanical module the integrated device, but precision are relationships, and interconvertible dimensionality models of of very important indices for these models. Proposed in [3] optimal distributed systems. Paper contains some examples for method of adaptive data transmission in telecommunication the optimization relating to the optimal placement of structural elements in spatially or temporally distributed technological access networks with a combine modulation types ensures systems, to which these techniques can be applied, including the lowest possible bit error rate during data transmission at applications to coded design of signals for communications and some ratio of signal power to noise power. General problem radar, positioning of elements in an antenna array, and of systems optimization relates to finding the best placement development vector data coding design. of its structural elements and events. Research into Key words: Ideal Ring Bundle, circular sequence, underlying mathematical area involves the appropriate circular symmetry, model, optimal proportion, vector data algebraic structures and modern combinatorial analysis [4], coding, self-correcting, resolving ability. finite projective geometry [5], difference sets and finite groups in extensions of the Galois fields [4]. We’re seeing a INTRODUCTION remarkable progress in developing of innovative techniques The latest advances in the modern theory of systems in systems optimization and design combinatorial of certifying the existence of a direct link of circular Sequencing Theory, namely the concept of Ideal Ring symmetry and non-symmetry relationships with Bundles (IRB)s [6–16]. The concept of the IRBs can be used delivering the totality of philosophical, methodological, for finding optimal solutions for wide classes of specifically-scientific and applied problems, which technological problems. A new vision of the concept with allowed her to gain the status of theoretical foundation of point of view of the role of geometric circular symmetry and system engineering in modern science. Particularly non-symmetry mutual relation laws allows better understand relevant emerging study of the physical laws of nature, the idea of “perfect” combinatorial constructions, and to what in their treatises paid attention even the ancient apply this concept for multidimensional systems philosophers. Studies include the use of modern optimization [17–20]. Lviv Polytechnic National University Institutional Repository http://ena.lp.edu.ua 28 V. RIZNYK OBJECTIVES In general the order S of the circular symmetry may be chosen arbitrarily. The objectives of the proposed The objectives of the underlying concept are as research and applications of circular symmetric and non- research into the underlying mathematical principles symmetric relationships are improving such quality of relating to the optimal placement of structural elements in information technology as precision, resolving ability, and spatially or temporally distributed systems, including a better understanding of the role of geometric structure in appropriate algebraic constructions based on cyclic the behaviour of natural objects. Underlying models help groups. Development of the scientific basis for to understand the evolutionary aspects of the role of technologically optimum systems theory, and the geometric structure in the behaviour of natural and man- generalization of these methods and results to the made objects. In this reasoning, we precede from a visible improvement and optimization of technological systems. geometric circular symmetry. A plane circular S-fold symmetry, is known can be THE MAIN RESULTS OF THE RESEARCH depicted graphically as a set of S lines diverged from a IRBs are cyclic sequences of positive integers which central point O uniformly (to be equal in spacing angles). form perfect partitions of a finite interval [1, N] of Regarding n1 and n2 of S lines as being complementary integers. The sums of connected sub-sequences of an IRB sets (S= n1 + n2), we require all angular distances between enumerate the set of integers [1, N-1] exactly R-times [8]. n1 lines enumerate the set of spacing angles [αmin, Example: The IRB {1, 2, 6, 4} containing four elements 360º– α min] exactly R1 times, while between n2 – allows an enumeration of all numbers from 1 to exactly R2 times, we call this a Perfect Circular Relation 12= 1+2+6+4 exactly once (R=1). The chain ordered (PCR). Our reasoning proceeds from the fact, that approach to the study of sequences and events is known to minimal and maximal angular distances relation initiated be of widespread applicability, and has been extremely by S-fold circular symmetry to be of prime importance for effective when applied to the problem of finding the finding the PCR origin (Fig. 2). optimum ordered arrangement of structural elements in a sequence. Let us regard n-fold symmetric sequence as to an ability to reproduce the maximum number of o combinatorial varieties in the sequence using two-part αmin=360 /S αmax = N·αmin distribution. Clearly, the maximum number N of such O n variants in n-stage sequence is taken two connected sub- sequences of the sequence: Nnn =-1. (1) The maximum number of variants N in ring ordered Fig. 2. The minimal- maximal relationship of spacing angles of (closed loop) sequence divided of two connected a plane S –fold natural PCR origin subsequence is a number of ordered combinations of n elements taken 2 at a time as below: Clearly the set of all N=n(n–1) angular distances [α , N α ] of S-fold PCR plane quantized field allows N=-nn(1) . (2) min · min an enumeration of all integers [1, S–1] exactly R-times Comparing the equations (1) and (2), we see that the (Fig. 2): number N of ordered combinations for binary consecutive N=-(SR1) . (3) sub-sequences of close-loop topology is n as many the From equations (2) and (3) follows the integer number N of combination in the non-closed topology, for n relation, the PCR for 7 ≤ S ≤ 19, n1,2 ³ R1,2 +2 are the same sequence of n elements. tabulated (Table 1): To extract meaningful information from the nn(-1) underlying comparison let us apply to circular S-fold S=+1. (4) R symmetry as a quantized planar field of two comple- mentary completions of the symmetric field. Table 1. PCR for 7 ≤ S ≤ 19, n1,2 ³ R1,2 +2 Example: The 3-fold (S=3) circular symmetry № S n1 R1 n2 R2 combined with two complementary completions (Fig. 1): 1 7 3 1 4 2 2 11 5 2 6 3 n = n1 + n 2. 3 13 9 6 4 1 4 15 7 3 8 4 5 19 9 4 10 5 Here are some examples of the simplest PCR. The elementary is 3-fold (S=3) circular symmetry that splits n= 3 into 1-fold (n1=1, R1=1), and 2-fold (n2=2, R2=1) n 1=1 n2= 2 complementary asymmetries. The first of them enumerates the set {1} by quantization level α min=360º, Fig. 1. The 3-fold circular symmetry (left) combined with two while the second {1, 2} by α min=120º exactly once. complementary completions The 7-fold rotational symmetry (S=7) splits into 3-fold Lviv Polytechnic National University Institutional Repository http://ena.lp.edu.ua APPLICATIONS OF THE COMBINATORIAL CONFIGURATIONS FOR OPTIMIZATION OF… 29 (n1=3) asymmetry, which allows an enumeration the set sets with the smallest possible number of intersections of all angular intervals [360º/7, 6×360º/7] of n1=3, exactly twice. Here we can see that underlying R1=1, and n2 =4, R2 =2 by quantization level segmentation provide an ability to reproduce the α min=360º/7, etc.[13]. maximum number of harmonious combinatorial varieties From Table 1 we can see that PCR is two in the system using sequential parting technology.
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