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 Copyright (c) 2019 Winny Chepngetich Bor, Owino M. Oduor, John K. Rotich 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 Copyright (c) 2019 Halima Jibril 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 Copyright (c) 2019 Anatoliy Nikolaychuk 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 Copyright (c) 2019 Jaemin Shin 2019-08-14 2019-08-14 17 34 38 10.24297/jam.v17i0.8399 Division And Combination In Linear Algebra <p>In this paper, the relationship between matrix operation, linear equations, linear representation of vector groups and linear correlation is discussed, and the idea of division and combination in linear algebra is discussed to help learners understand the connections between various knowledge points of linear algebra from multiple angles, deep levels, and high dimensions.</p> Liang Fang Rui Chena Copyright (c) 2019 Liang Fang, Rui Chena 2019-09-06 2019-09-06 17 39 146 10.24297/jam.v17i0.8413 Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method <p>We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded&nbsp;integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration&nbsp;step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different&nbsp;nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta&nbsp;(RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we&nbsp;show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval&nbsp;where the IVP is being considered.</p> Abhinandan Chowdhury Sammie Clayton Mulatu Lemma Copyright (c) 2019 Abhinandan Chowdhury 2019-09-16 2019-09-16 17 147 154 10.24297/jam.v17i0.8408 A Parametric Approach for Solving Interval–Valued fractional Continuous Static Games <p>The aim of this paper is to show that a parametric approach can be used to solve fractional continuous static games with interval-valued in the objective function and in the constraints. In this game, cooperation among all the players is possible, and each player helps the others up to the point of disadvantage to himself, so we use the Pareto-minimal solution concept to solve this type of game. The Dinkelbach method is used to transform fractional continuous static games into non- fractional continuous static games. Moreover, an algorithm with the corresponding flowchart to explain the suggested approach is introduced. Finally, a numerical example to illustrate the algorithm’s steps is given.</p> Mervat Elshafei Copyright (c) 2019 Mervat Elshafei 2019-09-16 2019-09-16 17 155 164 10.24297/jam.v17i0.8419 Angle Trisection <p>We seek to increase the development of science, but there are several fundamental questions about what is. Without solving the question is a false reflection of the history of science and the beginning of cognition. We know that their investigation and resolution, with the exception of rooting and knowledge of morphophonemic, do not come. Research on certain natural or pure mathematical phenomena is an example of my fundamental research that will lead to the definition of general principles and scientific theories.</p> Mehryar Husyan Pour Shad Copyright (c) 2019 Mehryar Husyan Pour Shad 2019-09-16 2019-09-16 17 165 231 10.24297/jam.v17i0.8412 Fuzzy Graphs <p>In this paper, neighbourly irregular fuzzy graphs, neighbourly total irregular fuzzy graphs, highly irregular fuzzy graphs and highly total irregular fuzzy graphs are introduced. A necessary and sufficient condition under which neighbourly irregular and highly irregular fuzzy graphs are equivalent is provided. We define d2 degree of a vertex in fuzzy graphs and total d2 -degree of a vertex in fuzzy graphs and (2, k)-regular fuzzy graphs, totally (2, k)- regular fuzzy graphs are introduced. (2, k)- regular fuzzy graphs and totally (2, k)-regular fuzzy graphs are compared through various examples.</p> Huda Mutab Al Mutab Copyright (c) 2019 Huda Mutab Al Mutab 2019-10-03 2019-10-03 17 232 247 10.24297/jam.v17i0.8443 Natural Mathematics, the Fibonacci Numbers and Aesthetics in Art <p>The Mathematics of beauty and beauty in mathematics are important ingredients in learning in the liberal arts. The Fibonacci numbers play an important and useful role in this. This paper seeks to present and illustrate a grounding of visual aesthetics in natural mathematical principles, centered upon the Fibonacci numbers. The specific natural mathematical principles investigated are the Fibonacci numbers, the Fibonacci Spiral, and the Cosmic Bud.</p> Anthony G Shannon Seamus A. Power Copyright (c) 2019 Anthony G Shannon, Seamus A. Power 2019-10-28 2019-10-28 17 248 254 10.24297/jam.v17i0.8479 Decrypting the Central Mystery of Quantum Mathematics: <p class="p1">This article proposes a solution to the double slit experiment of Quantum Mechanics. We attack the problem from a previously untried angle. Unsolved math problems must be attacked from unexpected angles because every conventional approach has already been tried and failed. Richard Feynman warned that the quantum world is such a strange place that humans can’t understand it. There is empirical evidence of particles following zero energy waves backwards, although that is counterintuitive. Schr˝odinger waves carry zero energy: they carry probability amplitudes instead. In our proposed model zero energy Schr˝odinger waves emanating from every point on the target screen pass backwards through the two slits, interfere at the particle gun, and a particle randomly chooses which wave to follow backwards. Once that decision is made the particle follows its wave with a probability of one, through only one slit (it doesn’t matter which slit) and inevitably strikes that point from which its wave emanates. This produces the same math and same pattern on the target screen. We propose three Axioms of the Theory of Elementary Waves (TEW) as a better platform for mathematics in this experiment than the Axioms of QM. This constitutes a paradigm shift.</p> Jeffrey Boyd Copyright (c) 2019 Jeffrey Boyd 2019-11-09 2019-11-09 17 255 282 10.24297/jam.v17i0.8475 Decrypting the Central Mystery of Quantum Mathematics: <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>The Theory of Elementary Waves (TEW) is based on three new Axioms that lead to a different understanding of quantum mathematics. There is a massive amount of research data that supports TEW. This article will take six well established experiments from mainstream scientific journals and re-interpret their axioms from a TEW point of view. Although it is usually asserted that QM explains all existing quantum experiments, that is only true if you can convince yourself that the quantum world is weird. If you adopt TEW axioms, suddenly the quantum world transforms itself into looking ordinary, like everyday Nature. If, for example, time only goes forwards, never backwards; if there is no such thing as a quantum eraser; if nothing is transmitted faster than the speed of light, then TEW axioms allow you to make sense of a quantum world which QM can only explain if you allow for weirdness throughout Nature. TEW consists of axioms that allow us to understand the quantum world in a way that makes sense from the viewpoint of our everyday experience.</p> </div> </div> </div> Jeffrey Boyd Copyright (c) 2019 Jeffrey Boyd 2019-11-09 2019-11-09 17 283 314 10.24297/jam.v17i0.8489 Decrypting the Central Mystery of Quantum Mathematics: <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>The fact that loophole-free Bell test experiments have proved Einstein’s local realism wrong, does not prove that the quantum mechanical (QM) model is correct, because the Theory of Elementary Waves (TEW) Axioms can also explain the Bell test experiments. Bi-Rays are a pair of coaxial elementary rays traveling at the speed of light in countervailing directions. In a Bell test experiment a Bi-Ray stretches from Alice’s equipment, through the fiberoptic cable, across the 2-photon source, through more fiberoptic cable, to Bob’s equipment. A pair of entangled photons is born into that Bi-Ray. Each photon follows the same Bi-Ray in opposite directions. This model produces the same Bell test results found by QM. According to QM this would be classified as a “non-local” model, so it is no surprise that it can explain the Bell test results. But it is a different model than QM. TEW supports a more realistic view of Nature, based on better Axioms. Although QM can explain quantum experiments, it requires that you believe the quantum world is weird. TEW Axioms explain the quantum world in a way that is more intuitively similar to the world of everyday experience.</p> </div> </div> </div> Jeffrey Boyd Copyright (c) 2019 Jeffrey Boyd 2019-11-09 2019-11-09 17 315 331 10.24297/jam.v17i0.8490 Decrypting the Central Mystery of Quantum Mathematics: <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>We live in a world, half of which consists of invisible elementary waves, of which we know very little. They are not electromagnetic waves: they travel in the opposite direction and convey no energy. What is the medium in which they travel? Franco Selleri (1936-2013) of University of Bari, Italy, devoted his career to answering that question. He developed his own theory of relativity. Zero energy quantum waves travel in Lorentz aether at rest. His relativity differs from Einstein’s Theory of Special Relativity (TSR) in terms of Absolute Simultaneity. If two events are simultaneous for one observer, they are simultaneous for all observers. Although this contradicts TSR, international treaties have adopted Absolute Simultaneity as the basis for coordinating all atomic clocks to the nanosecond. Atomic clocks control all other clocks. Absolute simultaneity is essential for commerce and computer networks.. Selleri’s relativity can be divided into two parts: time and aether. Time can be understood without ever speaking of the speed of light. When it comes to aether, a subject rarely mentioned today, it appears to be Isaac Newton’s absolute time and space, modified to fit the Lorentz transformations and the non-Euclidean curved space of Einstein’s General Relativity.</p> </div> </div> </div> Jeffrey Boyd Copyright (c) 2019 Jeffrey Boyd 2019-11-09 2019-11-09 17 332 351 10.24297/jam.v17i0.8491 A Parabolic Transform and Averaging Methods for General Partial Differential Equations <p>Averaging method of the fractional general partial differential equations and a special case of these equations are studied, without any restrictions on the characteristic forms of the partial differential operators. We use the parabolic transform, existence and stability results can be obtained.</p> Mahmoud Mohammed Mostafa El-Borai Hamed Kamal Awad Awad Randa Hamdy. M. Ali Ali Copyright (c) 2019 Mahmoud Mohammed Mostafa El-Borai, Hamed Kamal Awad Awad, Randa Hamdy. M. Ali Ali 2019-11-09 2019-11-09 17 352 361 10.24297/jam.v17i0.8481 Some Structural Resuits on Prime Graphs <p>Given a graph G = (V,E), a subset M of V is a module [17] (or an interval [10] or an autonomous [11] or a clan [8] or a homogeneous set [7] ) of G provided that x ∼ M for each vertex x outside M.&nbsp; So V,φ and {x}, where x ∈ V , are modules of G, called trivial modules. The graph G is indecomposable [16] if all the modules of G are trivial. Otherwise we say that G is decomposable<strong> . </strong>The prime graph G is an indecomposable graph with at least four vertices. Let G and H be two graphs. Let If G has no induced subgraph isomorphic to H, then we say that G is H-free. In this paper, we will prove the next theorem</p> Ibtesam Ali Alrowily Copyright (c) 2019 Fairouz Tchier 2019-11-28 2019-11-28 17 362 369 10.24297/jam.v17i0.8519 On Three Dimensional Pseudosymmetric Alpha-Kenmotsu Manifolds <p>The main purpose of this paper is to study pseudosymmetric conditions on alpha-Kenmotsu manifolds with dimension . In particular, we obtain some results satisfying some certain curvature conditions on such manifolds depending on.</p> Hakan Öztürk Sunil Kumar Yadav Copyright (c) 2019 Hakan Öztürk, Sunil Kumar Yadav 2019-11-28 2019-11-28 17 370 377 10.24297/jam.v17i0.8528 The Trisection of an Arbitrary Angle: A Condensed Classical Geometric Solution <p>This paper presents a short version of an elegant geometric solution of angle trisection that was published by this author on 2018-04-30 in Volume:&nbsp; 14 Issue:&nbsp; 02 of the Journal of Advances in Mathematics.</p> <p>The style of writing for the above paper was based on how teaching geometry was taught in high schools from 1940 to 1942.&nbsp; Proofs of a problem consisted of a statement that was followed by a valid reason why the statement was made.&nbsp; If the proof was many lines in length, the teacher wanted the students to show each step.&nbsp; The students were not allowed to skip a step or steps to reach the final line of the proof.</p> <p>This short version was generated when a copy of the above paper was reviewed by a retired school teacher, who suggested the proof of the trisection of an arbitrary angle could be shortened.</p> <p>The exposed methods of proof have not changed from the Euclidean postulates of classical geometry.</p> Arthur Rediske Copyright (c) 2019 Arthur Rediske 2019-12-10 2019-12-10 17 378 389 10.24297/jam.v17i0.8487 Almost Paracontact 3-Submersions <p>In this paper, we discuss some geometric properties of Riemannian submersions whose total space is an almost paracontact manifold with 3-structure. The study is focused on the transference of structures, the geometry of the fibres and sectional curvature tensor.</p> T.Tshikuna Matamba Copyright (c) 2019 TSHIKUNA MATAMBA 2019-12-10 2019-12-10 17 390 400 10.24297/jam.v17i0.8507 A Classical Geometric Relationship That Reveals The Golden Link in Nature <p>This paper introduces the perfect complementary relationship between the 3-4-5 Pythagorean triangle and the 1:2:&nbsp;right-angled triangle. The classical geometric intimacy between these two right triangles not only provides for the ultimate geometric substantiation of Golden Ratio, but it also reveals the fundamental Pi: Phi correlation (π: φ), with an extreme level of precision, and which is firmly based upon the classical geometric principles.</p> Chetansing K. Rajput Copyright (c) 2019 Chetansing Rajput 2019-12-11 2019-12-11 17 401 413 10.24297/jam.v17i0.8498