Ordering of Transformed Recorded Electroencephalography (EEG) Signals by a Novel Precede Operator

Recorded electroencephalography (EEG) signals can be represented as square matrices, which have been extensively analyzed using mathematical methods to extract invaluable information concerning brain functions in terms of observed electrical potentials; such information is critical for diagnosing brain disorders. Several studies have revealed that certain such square matrices—in particular, those related to so-called “elementary EEG signals”—exhibit properties similar to those of prime numbers in which every square EEG matrix can be regarded as a composite of these signals. A new approach to ordering square matrices is pivotal to extending the idea of square matrices as composite numbers. In this paper, several ordering concepts are investigated and a new technique for ordering matrices is introduced. Finally, some properties of this matrix order are presented, and the potential applications of this technique to analyzing EEG signals are discussed.


Introduction and Motivation
Epilepsy is a common neurological disease that affects 1% of the world's population [1]. People of all ages can be affected by this chronic brain disorder [2], which lowers sufferers' quality of life with the possibility of a seizure occurring at any time. Early diagnosis can not only improve quality of life but also prevent patients from experiencing accidents. e diagnosis of epilepsy is often not straightforward, and misdiagnosis occasionally does occur [3]. A detailed and reliable eyewitness account of the event is the most crucial piece of information for indicative assessment, but this may not be accessible [4]. In most cases, electroencephalography (EEG) is an essential diagnostic test for assessing patients with possible epilepsy. Besides providing diagnostic support [5], EEG can also assist in classifying the underlying epileptic syndrome [6].
Mathematical analysis of EEG signals offers medical professionals vital information regarding brain activity during a seizure, thus increasing the understanding of complex brain function [7,8]. In general, relevant information is extracted from EEG signals via two main methods: linear and nonlinear. Linear analysis (e.g., Fourier and wavelet transforms) has been successfully executed and has produced several good results [9][10][11][12][13][14][15]. However, since linear methods disregard the underlying nonlinear EEG dynamics, the results can only provide a limited amount of information concerning the brain's electrical activity. By contrast, it is commonly accepted that the brain is a chaotic dynamical system; therefore, EEG signals generated by the brain are considered to be chaotic in another sense [16][17][18]. Additional information can be extracted from EEG signals by progressively incorporating a nonlinear analysis that reveals features that cannot be measured via linear methods [19].
Crucial to diagnosing the disorder is to solve the neuromagnetic inverse problem to identify the location of epileptic foci [20]. erefore, fuzzy topographic topological mapping (FTTM) was introduced in [21] to determine the epileptic foci. More recently, FTTM has been extensively utilized to study the features of the recorded EEG signals of seizure patients (see [22][23][24][25][26][27]). Most notably, Yun in [28] claimed that one of the components of FTTM-namely, the magnetic contour (MC)-obeys the associative law, which is also satisfied in turn by events in time [29]. e author concluded by stating that the MC is a plane containing information. is prompted Binjadhnan in [25,30] to perform Krohn-Rhodes decomposition on a set of square matrices of EEG signals, MC n (R). e scholar found a remarkable result, namely, the EEG signals taken during an epileptic seizure (henceforth called EEG-signal square matrices for the remainder of this paper) are not chaotic, but rather exhibit ordered patterns in the form of simple algebraic structures, as expressed by eorem 1.
Theorem 1 (see [30]). Any invertible square matrix of EEGsignal readings during an epileptic seizure at time t can be written as a product of elementary EEG signals during an epileptic seizure in one and only one way. eorem 1 states that the elementary EEG signals (i.e., unipotent and diagonal EEG signals) constitute the building blocks of all EEG signals. is theorem, to a certain extent, is similar to the fundamental theorem of arithmetic, which holds that prime numbers are the multiplicative building blocks of the integers [31]. Equally significant are the results that indicate that MC n (R) has properties resembling those of prime numbers via the Jordan-Chevalley decomposition [32]. e well-ordering property of positive integers is vital in producing one of the most beautiful results in the study of prime numbers, namely, the infinitude of prime numbers. erefore, a technique of ordering matrices is required to extend the work of viewing the elementary EEG signals as prime numbers. e analogy of elementary EEG signals as prime numbers is of vital importance since the pattern of EEG signals can be investigated in terms of the pattern of prime numbers. Hence, the goal of this paper is to introduce a technique of ordering transformed EEG signals (in terms of square matrices). e remainder of this paper is organized as follows. In Section 2, a brief review of a few concepts and techniques for ordering matrices is presented, along with their viability for ordering transformed EEG signals. In Section 3, a new technique for ordering matrices, namely, the precede operator, is introduced, allowing several ordering properties to be obtained. Next, this binary relationship is shown to fulfill the partial-order properties; beyond that, it is shown to be totally ordered in Section 4. In Section 5, several results are obtained when the order is applied to symmetric matrices. en, the implementation of the precede operator to the real data of EEG signals is presented in Section 6. e interpretation of the results and their connection with the prime numbers are discussed in Section 7. Finally, we bring the paper to a close with concluding remarks concerning the need for such a partial order. roughout the following sections, every matrix is considered to be a square matrix unless otherwise stated.

Concepts for Ordering Matrices
Over the past few decades, mathematicians and applied scientists alike have taken a deep interest in the ordering of matrices. Several order relations for matrix algebra have been produced in connection to a series of applications relevant to different branches of mathematics and its applications. ese order relations include minus partial order [33], star partial order [34], sharp partial order [35], and matrix majorization [36]. Mitra et al. [37] wrote a comprehensive monograph in which they presented developments in the field of matrix ordering and shorted operators for finite matrices in a unified way, thus sparking research interest in this topic.
Matrix partial ordering has applications in many different areas; for instance, Liu [38] developed applications for comparing linear models. Moreover, in the field of statistics, Baksalary and Puntanen [39] presented the best linear unbiased estimator in a general Gauss-Markov model. At the same time, Dahl et al. [40] characterized a binary relation involving stochastic matrices (namely, matrix majorization), which is very useful for comparisons of statistical experiments. Additionally, in the field of finance, Fontanari et al. [41] proposed a technique called quantum majorization to compare and rank correlation matrices such that portfolio risk can be more significantly assessed. e minus partial order is the fundamental matrix partial order, of which almost all subsequent partial orders (including the star and sharp orders) constitute extensions. Such extensions have been created through the addition of restrictions to the minus partial order. e minus partial order (which was originally called the plus order) was established by Hartwig in [33] and independently by Nambooripad in [42] to generalize conventional partial orders on semigroups. Antezana et al. [43] andŠemrl [44] extended this partial order such that it could be applied in an objective way to operators on infinitedimensional spaces. Djikić et al. [45] documented a new representation of the minus order on the algebra of bounded linear operators on a Hilbert space. e natural partial order of Vagner on inverse semigroups and the star order of Drazin can be extended through minus order [37]. One key feature to note is that these partial orders are defined via the method of generalized inverses.
Another essential ordering concept for matrices is majorization, which has been applied across many fields including economics [46], statistics [47,48], and, most recently, quantum mechanics [49]. is concept was first introduced in a classical book by Hardy et al. [50]. Later, Marshall et al. [36] extensively treated both the theory and application of majorization. Torgersen in [51,52] studied the generalization of vector majorization and developed the theory of statisticalexperiment comparison. is theory is intended to answer the question "What conditions must be fulfilled in order to say that one statistical experiment provides more information than another?" A simple experiment can be found in [53], in which the conventional notion introduced by the author is closely related to that of vector majorization. However, while these generalizations evolved from statistical studies, they are not regularly discussed in the linear-algebra literature. Dahl [54] introduced and studied the generalization of (vector) majorization as it applies in the notable case of matrices with m rows. e classical concept of majorization between vectors can be generalized via matrix majorization [55].
Some of the techniques of ordering matrices found in the literature, along with their real-world applications, advantages of the techniques, and their limitations (with respect to the purpose of ordering transformed EEG signals), are summarized in Table 1.
e techniques of ordering matrices summarized in Table 1 have been deemed unfit to be used to extend the work of Binjadhnan and Ahmad [25,30] and Fuad and Ahmad [32] since there are some conditions required to be fulfilled, and some are limited to the special matrices. is offers the possibility of introducing and investigating a new partial order of square matrices as discussed in Section 3.

Precede Operator
As mentioned in Section 2, the set of square matrices of EEG signals during a seizure, MC n (R), has properties similar to those of prime numbers. erefore, EEG-signal square matrices can be assumed to be analogs of natural numbers. It can be said that one matrix is "greater" than another matrix, just as any natural number can be either greater than or less than another natural number since R is a complete ordered field and N ⊂ R. With this in mind, the precede operator, denoted by ≻, is introduced as defined by Definition 1.

Definition 1.
Let C and C ′ be n × n matrices and C ≠ C ′ . Matrix C is said to precede C ′ , written as C≻C ′ , whenever the first c ij > c ij ′ exists for some i, j. e comparison must be made in the sequence of rows, i.e., R 1 , R 2 , . . . , R n , until the first c ij > c ij ′ is discovered and denoted as ≻(C, C ′ ) � c ij . Otherwise, if c ij ′ > c ij , then C ′ ≻ C. When C � C ′ , i.e., all the corresponding entries for each matrix are the same, then ≻(C, C ′ ) � c 11 .
In other words, let us consider C, C (1) In other words, rearrange and reidentify the entries of  [36,40] (ii) Comparing the information of classical or quantum physical states [56] (iii) Network flow theory [55] (iv) Measuring income inequalities [57][58][59] (v) Measuring experimental designs and survey sampling [60] (i) More than one attribute of a system, such as income inequality, can be compared (ii) Comparison between matrices that have different dimensions Requires the existence of a doubly stochastic matrix Quantum majorization (i) Comparing and ranking correlation matrices to assess portfolio risk in a unified framework [41] (ii) Comparing quantum processes in which a complete set of entropic conditions for state transformations in resources theories of asymmetry and quantum thermodynamics is derived [49] ( Finally, ≻(C, C ′ ) � ω * . Definition 1 is introduced as a map between a matrix and a real number in R, since R is a complete ordered set.

Example 1. Consider two matrices A and B such that
As can be clearly seen, the first a ij > b ij is found. In this case, a 22 > b 22 or 0.2785 > 0.0318. en, A≻B.
Hence, the mapping of ≻: e execution of this definition can be summarized by Algorithm 1.
In short, ≻: e composition of mappings ≻ is best illustrated by

Ordered Matrices
In this section, several results are obtained to show that any square matrices together with the precede operator are ordered matrices.
Proof. Let A≻B. en, there exists a first a ij and a first b ij for some i, j ∈ N such that a ij ≥ b ij . Suppose that B⊁A is false; then, B≻A is true. erefore, there exists a first b ij and a first a ij for some i, j ∈ N such that b ij ≥ a ij . is is impossible, since A ≠ B according to Definition 1, and it also contradicts with the assumption that says there exists a first a ij and a first b ij for some i, j ∈ N such that a ij ≥ b ij , as noted earlier.

Lemma 3. If A≻B and B≻C, then A≻C.
Proof. Let A≻B. en, there exists a first a ij and a first b ij for some i, j ∈ N such that a ij ≥ b ij . If B≻C, then there exists a first b ij ′ and a first c ij ′ for some i, j ∈ N such that b ij ′ ≥ c ij ′ . Suppose that A≻C is false; therefore, C≻A. In other words, there exists a first c ij ″ and a first a ij ″ for some i, j ∈ N, such that c ij ″ ≥ a ij ″ . ere are three cases to consider: ′ is no longer the first to be found, such that b ij ′ ≥ c ij ′ and B ≠ C, according to Definition In other words, B≻A, which is a contradiction ( → ←).
All three cases lead to contradictions; thus, if A≻B and B ≻ C, then A ≻ C.
Consequently, the binary relation is a partial order. □ eorem 5 is best illustrated by Example 2.
Similarly, by Definition 1, A + C ≻ B + D is obtained.
en, by Definition 1, A≻B. Next, which, by Definition 1, implies that A − B≻B − A.
Theorem 6. C≻ − C for every positive symmetric matrix C (i.e., a positive symmetric matrix precedes its negation).
Proof. Suppose that A and B are positive symmetric matrices and that A ≻ B. Notice that, A + A T is a positive symmetric matrix since Similarly, B + B T is also a positive symmetric matrix. Now, Notice that if F and G are positive symmetric matrices, then en, by Definition 1, we obtain C≻ − C.
Clearly, by Definition 1, C ≻ A. Next, Journal of Mathematics 7

Implementation
As an example of the implementation of the precede operator on real data, two readings of EEG signals' square matrices are presented. e EEG signals of epileptic seizure patients could be recorded and composed into a set of square matrices (see Figure 2). Firstly, the EEG data is gathered from the hospital by the EEG technologists and the threedimensional data are transformed into two-dimensional data. e transformation of the EEG data into a lower-dimensional MC is executed via a novel technique called flattening the EEG, where the information can be preserved and conveniently analyzed [70]. e coordinate system of EEG signals, depicted in Figure 3(a), is defined as , y, z), e p |x, y, z, e p ∈ R and Moreover, a function S t : C EEG ⟶ MC (see Figure 3(b)) is defined as S t (x, y, z), e p � rx + iry r + z , e p � rx r + z , ry r + z e p (x,y,z) .
(18) e mapping S t is an injective mapping of a conformal structure since both C EEG and MC were designed and proven as two manifolds [70]. Hence, the mapping S t can keep up the data in a specific angle and orientation of the surface throughout the recorded EEG signals. e technique of flat EEG is executed on three groups of EEG signals recorded from three different epileptic patients [70]. e author digitized the EEG signals during epileptic seizures at 256 samples per second using the Nicolet One EEG software. Next, each APD at every second was stored in a file that contained the position of an electrode on a magnetic-contour (MC) plane. Subsequently, the stored data were used to compose a set of square matrices.
Differences in surface potential could be recorded using an array of electrodes appended to the scalp; the computed voltages between pairs of electrodes are then clarified, amplified, and recorded. e most widely used system of electrode placement is the International Ten-Twenty System; this is a recommended standard method for characterizing the locations of electrodes at particular time intervals along with the head for recording scalp EEG [71]. e Ten-Twenty system depends upon the connection between the position of an electrode and the underlying area of the cerebral cortex (the "Ten" and "Twenty" refer to 10% and 20% interelectrode distances, respectively) [72]. e electrode position of this system is shown systematically in Figure 4. Figure 4(a) illustrates the case where almost all of the electrodes are positioned 40% or below from vertex C z . On the contrary, Figure 4(b) shows the electrode position from the top view of the head by modeling the head as a sphere. We assume that the hemisphere is 80% from the top of the head [70]. In other words, from the front to the back is from F pz to O z and from the left to the right is from T 3 to T 4 . In general, every APD at each second is stored in a file containing the position of electrodes on the MC plane, as tabulated in Table 2.
en, the readings in Table 2 are rewritten in terms of a 5 × 5 matrix, as shown below.
Let x 1 ≤ x 2 ≤ · · · ≤ x 21 , i, j � 1, 2, 3, 4, 5 { } and a function β ij be defined as e mapping of β ij can be rewritten as the following matrix: Specifically, x 5 , y 5 x 6 , y 6 x 7 , y 7 x 8 , y 8 x 9 , y 9 x 10 , y 10 x 11 , y 11 x 12 , y 12 x 13 , y 13 x 14 y 14 x 15 , y 15 x 16 , y 16 x 17 , y 17 x 18 , y 18 x 19 , y 19 x 20 , y 20 e corresponding square matrix is generated by substituting the analogous average potential difference of every element into the above matrix. In particular, every single second of the APD is stored in a square matrix that contains the positions of electrodes on the MC plane. erefore, the MC plane becomes a set of n × n matrices (EEG signals), which is written as follows: (22) where β ij (z) t is a potential-difference reading for EEG signals from a particular ij sensor at time t. For instance, the recorded EEG signals data during seizures at times t � 2 and t � 3 are tabulated in Tables 3 and 4. e data in Tables 3 and 4 are then reordered in ascending order of X values and tabulated in Tables 5 and 6, respectively, through the MATLAB programming developed by Binjadhnan [30]. Typically, the program exhibits twenty-one sensor readings and four readings of zero (i.e., no recorded data from the "ghost" sensors at the specified time t); thus, a 5 × 5 matrix is obtained from each program.   Figure 4: e international Ten-Twenty system seen from (a) left and (b) above the head [73].   x 9 y 9 z 9 C 4

Journal of Mathematics
x 10 y 10 z 10 P 3 x 6 y 6 z 6 P 4 x 7 en, by Definition 1, the first a ij that is greater than b ij (a ij > b ij ) is a 12 > b 12 (i.e., 234.7169922 > 112.3018); hence, A(3)≻A (2). Consequently, the precede operator can be used in the set of EEG signals' square matrices, allowing one matrix to be said to precede another. e set of EEG signals' square matrices, along with the precede operator, enables one to see the set of matrices as analogous to the set of real numbers, as related to each other with the "greater than," > operator. is analogy is best demonstrated in Figure 5.

Discussion
pt e key advantage of using the precede operator over the techniques discussed in Section 2 is that it does not require to fulfill the necessary conditions (such as the existence of doubly stochastic matrix and generalized inverses), rather, the precede operator "inherits" the totally-ordered property of real numbers.
us, a similar result provides a piece of evidence that the elementary EEG signals during seizure contain similar attributes to the distribution of prime numbers among positive integers. e similarity of elementary EEG signals with prime numbers is summarized in Table 7. e resemblance of elementary EEG signals with the prime numbers in terms of its ordering properties corroborates the assertions made by Binjadhnan [30], Barja [27], and Ahmad Fuad and Ahmad [74] that the EEG signals during seizures contain a similar pattern to that of the prime number distribution among positive integers. e premise of viewing the EEG signals as prime numbers enables one to study the dynamics of EEG signals during seizures in terms of the pattern of prime numbers. More importantly, the deduced pattern of seizures is critical to devise a methodology that is capable of predicting a seizure, which in turn would significantly improve the patients' quality of life [75]. Conversely, it is instructive to explore the property of elementary EEG signals that could possibly exist in the distribution of prime numbers.

Conclusions
is study introduces a new technique for ordering square matrices, called precede operator, and this binary relation is R, > MC n (R), > Figure 5: e analogy between the set of EEG-signal square matrices and the precede operator and the set of real numbers and the greater-than operator.  (1) Twin-prime conjecture [31] (2) Well-ordered proven to exhibit partial ordering. In addition, several results obtained by using this new ordering technique are presented. Furthermore, the newly introduced matrix partial order is applied to the square matrices of EEG signals, giving opportunities to further develop EEG-signal square matrices' structure by studying those features that resemble some of the rich properties of prime numbers. In particular, it will be of interest to extend the totally ordered property of transformed EEG signals to the well-ordered as well. We are working on this problem and hoping to present our findings in a future paper. Furthermore, as this paper is limited to the precede operator implemented only on the transformed EEG signals, it would be intriguing to investigate the feasibility of the proposed technique to other structures, such as, among others, the measurement of income inequality, statistical experiment, and thermodynamics, as previously summarized in Table 1.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.