scholarly journals Decomposing the Krohn-Rhodes Form of Electroencephalography (EEG) Signals Using Jordan-Chevalley Decomposition Technique

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 10
Author(s):  
Amirul Aizad Ahmad Fuad ◽  
Tahir Ahmad
Keyword(s):  
Decomposition Method ◽  
Building Blocks ◽  
Prime Numbers ◽  
Decomposition Technique ◽  
Eeg Signals ◽  
Nilpotent Matrices

This paper explores how electroencephalography (EEG) signals in the Krohn-Rhodes form can be decomposed further using the Jordan-Chevalley decomposition technique. First, the recorded EEG signals of a seizure were transformed into a set of matrices. Each of these matrices was decomposed into its elementary components using the Krohn-Rhodes decomposition method. The components were then further decomposed into semisimple and nilpotent matrices using the Jordan-Chevalley decomposition. These matrices—which are the extended building blocks of elementary EEG signals—provide evidence that the EEG signals recorded during a seizure contain patterns similar to that of prime numbers.

QUANTUM-STATISTICAL SIMULATIONS FOR QUANTUM CIRCUITS

2002 ◽  
Vol 13 (07) ◽  
pp. 931-945 ◽  
Author(s):  
KURT FISCHER ◽  
HANS-GEORG MATUTTIS ◽  
NOBUYASU ITO ◽  
MASAMICHI ISHIKAWA
Keyword(s):  
Prime Numbers ◽  
Quantum Circuits ◽  
Decomposition Technique ◽  
Shor's Algorithm ◽  
Quantum Statistical ◽  
Classical Computer

Using a Hubbard–Stratonovich like decomposition technique, we implemented simulations for the quantum circuits of Simon's algorithm for the detection of the periodicity of a function and Shor's algorithm for the factoring of prime numbers on a classical computer. Our approach has the advantage that the dimension of the problem does not grow exponentially with the number of qubits.

Connections: Prime decomposition using tools

2010 ◽  
Vol 17 (4) ◽  
pp. 256-259
Author(s):  
Terri Kurz ◽  
Jorge Garcia
Keyword(s):  
Building Blocks ◽  
Prime Numbers ◽  
Prime Decomposition ◽  
Square Roots ◽  
Important Concept

Factoring numbers through multiplication using primes is an important concept for students to understand. Those in grades 3–5 should be able to find equivalent representations of the same number by de composing and composing numbers (NCTM 2000). One aspect of this essential skill is prime decomposition, which relates to such mathematical topics as simplifying, divisibility, square roots, and fractions. Prime numbers are crucial building blocks for understanding numbers and their multiplicative relationships (Zazkis and Liljedahl 2004).

Modal Decomposition Solution of Lamb Wave Scattering at Step-Like Discontinuity

2011 ◽  
Vol 368-373 ◽  
pp. 2398-2401 ◽  
Author(s):  
Fei Long Feng ◽  
Yan Liu
Keyword(s):  
Lamb Wave ◽  
Wave Scattering ◽  
Decomposition Method ◽  
Symmetric Mode ◽  
Decomposition Technique ◽  
Modal Decomposition ◽  
Model Decomposition ◽  
Infinite Model ◽  
Different Thickness ◽  
Flux Conservation

Modal decomposition method is utilized to investigate Lamb wave scattering at step-like discontinuity. The structure is regard as a special delamination structure where one or two delamination layers are missing and only one delamination layer extended to infinite. Model decomposition technique is then combined with the boundary condition equations to get the scattering solution of the whole discontinuity region. The efficacy of the method is illustrated by calculation examples where the scattering of the second symmetric mode at step-discontinuity with different thickness are evaluated and Energy flux conservation is confirmed.

Feature Extraction of EEG Signals Based on Local Mean Decomposition and Fuzzy Entropy

2020 ◽  
Vol 34 (12) ◽  
pp. 2058017
Author(s):  
Yanping Li ◽  
Qi Wang ◽  
Tao Wang ◽  
Jian Pei ◽  
Shuo Zhang
Keyword(s):  
Feature Extraction ◽  
Empirical Mode Decomposition ◽  
Extraction Method ◽  
Decomposition Method ◽  
Fuzzy Entropy ◽  
Eeg Signals ◽  
Feature Extraction Method ◽  
Mode Decomposition ◽  
Empirical Mode Decomposition Method ◽  
Local Mean

An improved feature extraction method is proposed aiming at the recognition of motor imagined electroencephalogram (EEG) signals. Using local mean decomposition, the algorithm decomposes the original signal into a series of product function (PF) components, and meaningless PF components are removed from EEG signals in the range of mu rhythm and beta rhythm. According to the principle of feature time selection, 4[Formula: see text]s to 6[Formula: see text]s motor imagery EEG signals are selected as classification data, and the sum of fuzzy entropies of second-and third-order PF components of [Formula: see text], [Formula: see text] lead signals is calculated, respectively. Mean value of fuzzy entropy [Formula: see text] is used as input element to construct EEG feature vector, and support vector machine (SVM) is used to classify and predict EEG signals for recognition. The test results show that this feature extraction method has higher classification accuracy than the empirical mode decomposition method and the total empirical mode decomposition method.

Noise removal and classification of EEG signals using the Fourier decomposition method

2020 ◽  
Author(s):  
Virender Kumar Mehla ◽  
Ashish Kumar ◽  
Amit Singhal ◽  
Pushpendra Singh
Keyword(s):  
Decomposition Method ◽  
Noise Removal ◽  
Eeg Signals ◽  
Fourier Decomposition

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

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Amirul Aizad Ahmad Fuad ◽  
Tahir Ahmad
Keyword(s):  
Prime Numbers ◽  
New Technique ◽  
Brain Disorders ◽  
Eeg Signals ◽  
Mathematical Methods ◽  
New Approach ◽  
Brain Functions ◽  
Electrical Potentials ◽  
Potential Applications ◽  
A New Technique

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.

Mathematical Roots: Get Primed to the Basic Building Blocks of Numbers

2007 ◽  
Vol 13 (2) ◽  
pp. 122-127
Author(s):  
Christiana Robbins ◽  
Thomasenia Lott Adams
Keyword(s):  
Natural Number ◽  
Building Blocks ◽  
Number System ◽  
Prime Numbers ◽  
Natural Numbers

In his book Elements, Euclid established that certain numbers are the building blocks of our natural number system. He revealed that these natural numbers could be “decomposed” into their smallest units as products of specific numbers. Numbers that can only be factored by themselves and 1 are called “prime numbers” and comprise part of the basic building blocks of numbers.

scholarly journals Exact solutions of the Laplace fractional boundary value problems via natural decomposition method

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1178-1187
Author(s):  
Hajira ◽  
Hassan Khan ◽  
Yu-Ming Chu ◽  
Rasool Shah ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Boundary Value Problems ◽  
Decomposition Method ◽  
Graphical Representation ◽  
Simple Procedure ◽  
Boundary Value ◽  
Decomposition Technique ◽  
Approximate Analytical Solutions ◽  
Natural Decomposition ◽  
Fractional Boundary Value Problems

Abstract In this article, exact solutions of some Laplace-type fractional boundary value problems (FBVPs) are investigated via natural decomposition method. The fractional derivatives are described within Caputo operator. The natural decomposition technique is applied for the first time to boundary value problems (BVPs) and found to be an excellent tool to solve the suggested problems. The graphical representation of the exact and derived results is presented to show the reliability of the suggested technique. The present study is mainly concerned with the approximate analytical solutions of some FBVPs. Moreover, the solution graphs have shown that the actual and approximate solutions are very closed to each other. The comparison of the proposed and variational iteration methods is done for integer-order problems. The comparison, support strong relationship between the results of the suggested techniques. The overall analysis and the results obtained have confirmed the effectiveness and the simple procedure of natural decomposition technique for obtaining the solution of BVPs.

1. What is number theory?

2020 ◽  
pp. 1-13
Author(s):  
Robin Wilson
Keyword(s):  
Number Theory ◽  
Historical Context ◽  
Building Blocks ◽  
Prime Numbers ◽  
Sum Of Squares ◽  
Whole Numbers ◽  
Perfect Numbers

‘What is number theory?’ puts number theory in its historical context, from the Pythagoreans to the present, explaining integers (whole numbers), prime numbers (the building blocks of number theory) squares and cubes, and perfect numbers (numbers whose factors add up to the number itself). How long can gaps between prime numbers be? Is there a formula for producing perfect numbers? Which primes can be expressed as a sum of squares? Other questions arise when we start adding primes.

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