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Group Theory
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
William Cocke, Meng-Che ‘Turbo’ Ho
The origin of modern group theory comes from E. Galois's study of the symmetries of solutions to equations (Stillwell, 2002). Galois used group theory as a tool to address field extensions of the rational numbers. The now-named Fundamental Theorem of Galois Theory gives a correspondence between subfields of certain field extensions and subgroups of a group known as Galois group of the extension. The question of which groups occur as the Galois groups of rational field extensions is known as the Inverse Galois Problem, and is a major open problem in modern number theory. While it is unknown if all groups occurs in the original context introduced by Galois, all groups do occur as the group of symmetries of certain graphs. In Section 14.2, we give some formal examples of groups.
New explicit exact traveling wave solutions of Camassa–Holm equation
Published in Applicable Analysis, 2021
In general, we can only expect explicit solutions for the cases where a are positive rational numbers. Let be the lowest form with positive integers m and n. We notice that implies that the resulting equation is a polynomial equation of degree m + n. However by the Galois theory, we know that there is no general formula for the roots of a polynomial of degree greater than or equal to 5. Therefore we can only expect explicit solutions for the cases where with . It is easy to see that m + n = 3 implies or a = 2 and m + n = 4 implies or a = 3. All in all, the work on the explicit single peak traveling wave solutions for the CH equation have essentially been done. However there are no much explicit solutions provided for each type of traveling wave solution classified by Lenells in [11]. We encourage the readers to find all possible other types of explicit traveling wave solutions. Moreover, we have some similar cases for DP equation that remain open and should be solvable similarly. We leave it to the readers.
A multi-dock, unit-load warehouse design
Published in IISE Transactions, 2019
Axiom 2.Expressing the expected dual-command distance as a function of the warehouse’s width and taking the first derivative with respect to the warehouse’s width, a cubic equation is obtained. For reasonable parameter values (the necessary condition for each scenario is provided in the proof of Corollary 4), the discriminant of the cubic equation is greater than zero. Therefore, the cubic equation has three distinct real roots, but there exist no rational roots, as the cubic equation is irreducible polynomial (from Galois Theory). Solving an irreducible cubic equation requires taking the roots of complex quantities. Therefore, reducing the cubic equation to a depressed form, setting the depressed cubic equation equal to zero, and solving for the warehouse’s width, the viable root can be obtained using Viète's trigonometric solution (Nickalls, 2006). The viable root is the first root, as results with the second and third roots are infeasible (the value of the expected distance is negative for the second root and the width of the warehouse is zero for the third root). Taking the second derivative of the expected dual-command distance with respect to the warehouse’s width and finding the second derivative is greater than zero for all reasonable values of the warehouse’s width establishes that the expected dual-command distance is a convex function of the warehouse’s width and the viable root is the optimal width of the warehouse.
An Analytic Benchmark for Neutron Boltzmann Transport with Downscattering—Part I: Flux and Eigenvalue Solutions
Published in Nuclear Science and Engineering, 2021
Vladimir Sobes, Pablo Ducru, Abdulla Alhajri, Barry Ganapol, Benoit Forget
Once we know how many poles to look for, Galois theory states that if , then they are a priori not solvable by radicals (see the Abel-Ruffini theorem). Therefore, a numerical algorithm is needed to find all the poles in the complex plane. One such algorithm tested in this work was the vector-fitting algorithm as explored by Peng et al.21 However, for greater accuracy and certainty of convergence, numerical root finding of the denominator polynomial was performed in this work in the spirit of Ducru et al.22 The MATLAB symbolic toolbox was leveraged to analytically calculate the polynomials for the numerator and the denominator of the factor. All of the cross-section definitions and the resulting polynomials are functions of . The exact rational fraction equations used for the cross sections will be presented in Sec. VI. After obtaining the analytic expressions for the polynomial in of the denominator of the factor, an exact function for the Durand-Kerner method23,24 was used. The Durand-Kerner method is an iterative method for the simultaneous determination of all the zeros of a polynomial . The following fixed-point iteration is used to sequentially improve the estimate of the root until convergence: