JOURNAL OF ADVANCES IN MATHEMATICS Interested in submitting to this journal? We recommend that you review the About the Journal page for the journal's section policies, as well as the Author Guidelines. Authors need to register with the journal prior to submitting or, if already registered, can simply log in and begin the five-step process. KHALSA PUBLICATIONS en-US JOURNAL OF ADVANCES IN MATHEMATICS 2347-1921 <p>Authors retain the copyright of their manuscripts, and all Open Access articles are distributed under the terms of the&nbsp;<a href="">Creative Commons Attribution License</a>, which permits unrestricted use, distribution, and reproduction in any medium, provided that the original work is properly cited.</p> <p>&nbsp;</p> A Lie Symmetry Solutions of Sawada-Kotera Equation <p>In this article, the Lie Symmetry Analysis is applied in finding the symmetry solutions of the fifth order Sawada-Kotera equation. The technique is among the most powerful approaches currently used to achieveprecise solutions of the partial differential equations that are nonlinear. We systematically show the procedure to obtain the solution which is achieved by developing infinitesimal transformation, prolongations, infinitesimal generatorsand invariant transformations hence symmetry solutions of the fifth order Sawada-Kotera equation.</p> <p><strong><em>Key Words</em></strong>- Lie symmetry analysis. Sawada-Kotera equation. Symmetry groups. Prolongations. Invariant solutions. Power series solutions. Symmetry solutions.</p> Winny Chepngetich Bor Owino M. Oduor John K. Rotich ##submission.copyrightStatement## 2019-07-30 2019-07-30 17 1 11 10.24297/jam.v17i0.8364 Breaking Boundaries: Discovering the Impossible Counterproof of Beal’s Conjecture <p>This paper will attempt to logically differentiate between two types of fractions and discuss the idea of Zero as a neutral integer. This logic can then be followed to create a counterexample and a proof for Beal’s conjecture.</p> Halima Jibril Mohamed Adela Zyfi Ghedlawit Futzum ##submission.copyrightStatement## 2019-07-23 2019-07-23 17 12 18 10.24297/jam.v17i0.8279 Convergence of the Collatz Sequence <p>For any natural number was created the supplement sequence, that is convergent together with the original Collatz sequence. The numerical parameter - index was defined, that is the same for both sequences. This new method provides the following results:</p> <ol> <li>All natural numbers were distributed into six different classes;</li> <li>The properties of index were found for the different classes;</li> <li>For any natural number was constructed the bounded sequence of increasing numbers,</li> </ol> <p>&nbsp;&nbsp;&nbsp; that is convergent together with the regular Collatz sequence.</p> Anatoliy Nikolaychuk ##submission.copyrightStatement## 2019-07-23 2019-07-23 17 19 33 10.24297/jam.v17i0.8336 Karp's Theorem in Inverse Obstacle Scattering Problems <p>In this work, we provide a proof of the so-called Karp's theorem in a different approach. We&nbsp;use the unique continuation principle together with the monotonicity of eigenvalues for the&nbsp;negative Laplace operator. This method is new and would be applicable to other types of&nbsp;inverse scattering problems.</p> Jaemin Shin ##submission.copyrightStatement## 2019-08-14 2019-08-14 17 34 38 10.24297/jam.v17i0.8399