Fixed Points Results in Algebras of Split Quaternion and Octonion
Abstract
:1. Introduction
2. Main Results
2.1. Some Fixed Points Results of Quadratic Functions in Split Quaternions over the Prime Field
2.2. Some Algebraic Consequences about
2.3. Some Fixed Points Results of Quadratic Functions in Split Octonions over the Prime Field
3. Some Algebraic Consequences about
4. Examples
5. Conclusions and Further Directions
6. Data Availability Statement
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Computer Codes
Appendix A.1. Program for Finding Solutions of the Quadratic Equation in
#include<iostream> #include<conio.h> using namespace std; main() { int a1, a2, a3, p, n, a, b, c, count; count=0; cout<<"Enter value for p: "; cin>>p; cout<<"Enter value for a: "; cin>>a; cout<<"Enter value for b: "; cin>>b; cout<<"Enter value for c: "; cin>>c; n=((p*p-b*b)/(4*a*a))+(c/a); while(n<0) n=n+p; for(int i=0; i<p; i++) { a3=i; for(int j=0; j<p; j++) { a2=j; for(int k=0; k<p; k++) { a1=k; int sum=(a1*a1)-(a2*a2)-(a3*a3); while(sum<0) sum=sum+p; if(sum%p==n) { count++; cout<<a1<<" "<<a2<<" "<<a3<<endl; } } } } cout<<"\nCount: "<<count; getch(); }
Appendix A.2. Program for Finding Roots of the Quadratic Equation in
#include <iostream> #include <fstream> using namespace std; int main(){ int a1,a2,a3,a4,a5,a6,a7; int sum=0; int p; int n=0; int count=2; int totalCount=0; cout<<"Enter value of p: "; cin>>p; n = ((p*p -1)/4)%p; for(int i=0;i<p;i++) { a1 = i; for(int j=0; j<p; j++) { a2 = j; for(int k=0; k<p; k++) { a3 = k; for(int l=0; l<p;l++) { a4 = l; for(int m=0; m<p; m++) { a5 = m; for(int q=0; q<p; q++) { a6 = q; for(int r=0; r<p;r++) { a7 = r; totalCount++; cout<<a1<<" "<<a2<<" "<<a3<<" "<<a4<<" "<<a5 <<" "<<a6<<" "<<a7<<endl; //dataFile << a1 << endl; sum = a1*a1+a2*a2+a3*a3-a4*a4-a5*a5-a6*a6-a7*a7; if(sum%p == n) count++; } } } } } } } cout<<"Total Count is: "<<totalCount<<endl; cout<<"Count is: "<<count<<endl; system("pause"); return 0; }
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Munir, M.; Naseem, A.; Rasool, A.; Saleem, M.S.; Kang, S.M. Fixed Points Results in Algebras of Split Quaternion and Octonion. Symmetry 2018, 10, 405. https://doi.org/10.3390/sym10090405
Munir M, Naseem A, Rasool A, Saleem MS, Kang SM. Fixed Points Results in Algebras of Split Quaternion and Octonion. Symmetry. 2018; 10(9):405. https://doi.org/10.3390/sym10090405
Chicago/Turabian StyleMunir, Mobeen, Asim Naseem, Akhtar Rasool, Muhammad Shoaib Saleem, and Shin Min Kang. 2018. "Fixed Points Results in Algebras of Split Quaternion and Octonion" Symmetry 10, no. 9: 405. https://doi.org/10.3390/sym10090405
APA StyleMunir, M., Naseem, A., Rasool, A., Saleem, M. S., & Kang, S. M. (2018). Fixed Points Results in Algebras of Split Quaternion and Octonion. Symmetry, 10(9), 405. https://doi.org/10.3390/sym10090405