http://www.ccsenet.org/journal/index.php/jmr/issue/feedJournal of Mathematics Research2015-11-27T18:20:56-08:00Sophia Wangjmr@ccsenet.orgOpen Journal Systems<p><strong><em>Journal of Mathematics Research </em></strong>(ISSN: 1916-9795; E-ISSN 1916-9809) is an open-access, international, double-blind peer-reviewed journal published by the Canadian Center of Science and Education. This journal, published <strong>bimonthly</strong> (<span>February, April, June, August, October and December</span>) in <strong>both print and online versions</strong>, keeps readers up-to-date with the latest developments in all aspects of mathematics.</p><div class="Section1"><strong>The scopes of the journal </strong>include, but are not limited to, the following topics: statistics, approximation theory, numerical analysis, operations research, dynamical systems, mathematical physics, theory of computation, information theory, cryptography, graph theory, algebra, analysis, probability theory, geometry and topology, number theory, logic and foundations of mathematics. <em> </em></div><div class="Section1"><p>This journal accepts article submissions<strong> <a href="/journal/index.php/jmr/information/authors">online</a> or by <a href="mailto:jmr@ccsenet.org">e-mail</a> </strong>(jmr@ccsenet.org).</p></div><div class="Section1"><br /><br /><strong><strong><em><img src="/journal/public/site/images/jmr/jmr.jpg" alt="jmr" width="201" height="264" align="right" hspace="20" /></em></strong><strong>ABSTRACTING AND INDEXING:</strong></strong></div><div class="Section1"><strong><br /></strong></div><div class="Section1"><ul><li>BASE (Bielefeld Academic Search Engine)<strong><br /></strong></li><li><strong>EBSCOhost</strong></li><li>Google Scholar</li><li>JournalTOCs</li><li>LOCKSS</li><li><strong>MathEDUC</strong></li><li><strong><a href="http://www.ams.org/dmr/JournalList.html">Mathematical Reviews</a>® (<a href="http://www.ams.org/mathscinet">MathSciNet</a>®)</strong></li><li>MathGuide</li><li>NewJour</li><li>OCLC Worldcat</li><li>Open J-Gate</li><li><strong>ProQuest</strong></li><li>SHERPA/RoMEO</li><li>Standard Periodical Directory</li><li>Ulrich's</li><li>Universe Digital Library</li><li><strong><a href="https://zbmath.org/journals/?q=se:00006772">Zentralblatt MATH</a></strong></li></ul></div><div class="Section1"><strong><br /></strong></div><div class="Section1"> </div>http://www.ccsenet.org/journal/index.php/jmr/article/view/54132Supporting Teachers’ Learning about Mathematical Modeling2015-11-24T01:08:13-08:00June L. Gastonjgaston@bmcc.cuny.eduBarbara A. Lawrencejgaston@bmcc.cuny.edu<p>In the United States, one of the Standards for Mathematical Practice of the Common Core Curriculum (Common Core State Standards Initiative, 2010) is <em>Model with mathematics</em>. This standard requires that students be taught in a manner that will enable them to “apply the mathematics they know to solve problems arising in everyday life, society, and the workplace” (p. 7). However many prospective and practicing teachers acquire a pedagogical style that does not support this standard. To promote higher levels of student thinking associated with mathematical modeling, teachers must thus be taught not only what mathematical modeling is, but how it can be effectively incorporated in their lessons and presented to their classes. Teacher training should also include how to develop rubrics for assessment, among which are rubrics that enable students to demonstrate mathematical modeling proficiency in different ways. In this research, the topics addressed include ways professional development can help in-service teachers appreciate the importance of mathematical modeling tasks; concerns about teacher backgrounds in mathematical modeling; and the most effective ways for improving in-service teachers’ knowledge of mathematical modeling and their teaching of mathematical modeling. While the primary focus of this research is on teacher education and training in the United States, the findings from both domestic and international research are clearly significant for those who are responsible for various aspects of teacher preparation worldwide.<strong></strong></p> <p>Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.</p>2015-11-03T02:42:54-08:00Copyright (c) http://www.ccsenet.org/journal/index.php/jmr/article/view/52041Investigation of Superluminal Motion of Free Spin-half Particles in Spacetime2015-11-24T01:08:13-08:00Emmanuel Gazoyagazoya2001@yahoo.comFrancis Odurogazoya2001@yahoo.comEdward Prempehgazoya2001@yahoo.comThe possibility of free spin-1/2 particles (also called Dirac particles)superluminal motion in spacetime, is investigated. The universal coverof the entire Lorentz group $\mathcal{L}$ consists of $SL\left(2,\mathbb{C}\right)$and the spinor map so that to obtain a relativistically invariantdescription of the state of an electron, one looks to the representationsof $SL\left(2,\mathbb{C}\right),$ that is, to the 2-valued representationof $\mathcal{L},$ known as spinors. We restrict our approach to realisticone-particle systems along with the ``positive energy'' and utilizethe free Dirac waves propagating in the $z-$direction. The Diracwave function $\psi\left(x,t\right)$ is considered as a ``classicalfield'' and the corresponding wave equation is derived from a Lagrangefunction. Using the symmetrized Dirac Lagrange density, it is observedthat variation in spin angular momentum (in the light cone) leadsto causality violation, whereas variation in orbital angular momentumdoes not. Consequently, it is shown that the expectation value of the relative translational velocitycomponent of the spinning free electron exceeds light speed.2015-11-03T02:42:54-08:00Copyright (c) http://www.ccsenet.org/journal/index.php/jmr/article/view/52435Pairs of Comparable Relations for Complete Homogeneous Symmetric Polynomial2015-11-24T01:08:13-08:00Soumendra Berasoumendra.bera@gmail.com<p>Complete homogeneous symmetric polynomial has connections with binomial coefficient, composition, elementary symmetric polynomial, exponential function, falling factorial, generating series, odd prime and Stirling numbers of the second kind by different summations. Surprisingly the relations in the context are comparable in pairs. </p>2015-11-03T02:42:54-08:00Copyright (c) 2015 http://www.ccsenet.org/journal/index.php/jmr/article/view/52745Prime Numbers: A Particle in a Box and the Complex Wave Model2015-11-24T01:08:13-08:00Raul Alberto Ribeiro Correia de Sousawavetheoryprimenumbers@outlook.com<p>Euler{'}s formula establishes the relationship between the trigonometric function and the exponential function. In doing so unifies two waves, a real and an imaginary one, that propagate through the Complex number set, establishing relation between integer numbers. A complex wave, if anchored by zero and by a defined integer number \textit{N}, only can assume certain oscillation modes. The first mode of oscillation corresponds always to a \textit{N} prime number and the other modes to its multiples.</p><p>\begin{center}<br />\(\psi (x)=x e^{i\left(\frac{n \pi }{N}x\right)}\)<br />\end{center}</p><p>Under the above described conditions, these waves and their admissible oscillation modes allows for primality testing of integer numbers, the deduction of a new formula $\pi(x)$ for counting prime numbers and the identification of patterns in the prime numbers distribution with computing time gains in the calculations. In this article, four theorems and one rule of factorizing are put forward with consequences for prime number signaling, counting and distribution. Furthermore, it is establish the relationship between this complex wave with a time independent semi-classical harmonic oscillator, in which the spectrum of the allowed energy levels are always only prime numbers. Thus, it is affirmative the reply to the question if the prime numbers distribution is related to the energy levels of a physical system.</p>2015-11-03T02:42:54-08:00Copyright (c) 2015 http://www.ccsenet.org/journal/index.php/jmr/article/view/53208Liquidity Premiums in a Levy Market2015-11-24T01:08:13-08:00Mei Xingmei.xing@kbcc.cuny.edu<p>This paper gives a theorem for the continuous time super-replication cost of European options where the stock price follows an exponential L\'{e}vy process.<br />Under a mild assumption on the legend transform of the trading cost function, the limit of the sequence of the discrete super-replication cost is proved to be greater than or equal to an optimal control problem.<br />The main tool is an approximation multinomial scheme based on a discrete grid on a finite time interval [0,1] for a pure jump L\'{e}vy model.<br />This multinomial model is constructed similar to that proposed by (Szimayer {\&} Maller, Stoch. Proce. {\&} Their Appl., 117, 1422-1447, 2007).<br />Furthermore, it is proved that the existence of a liquidity premium for the continuous-time model under a L\'{e}vy process.<br />This paper concentrates on the L\'{e}vy processes with infinitely many jumps in any finite time interval.<br />The approach overcomes some difficulties that can be encountered when the L\'{e}vy process has infinite activity.</p>2015-11-03T02:42:54-08:00Copyright (c) 2015 http://www.ccsenet.org/journal/index.php/jmr/article/view/54405Residual Symmetry for Linear Algebraic Equations and the Lorentz-Group Algorithm2015-11-24T01:08:13-08:00Chein-Shan Liuliucs@ntu.edu.tw<p>In the iterative solution of $n$ linear algebraic equations ${\bf B}{\bf x}={\bf b}$<br />by using the steepest descent method, i.e.,<br />${\bf x}_{k+1}={\bf x}_k-\alpha_k {\bf B}^{\mbox{\scriptsize T}}{\bf r}_k$,<br />it is known that the steplength<br />$\alpha_k:={\bf r}_k^{\mbox{\scriptsize T}} {\bf A}{\bf r}_k/\|{\bf A}{\bf r}_k\|^2$ causes<br />a slow convergence, where ${\bf r}={\bf B}{\bf x}-{\bf b}$ is the residual vector and ${\bf A}={\bf B}{\bf B}^{\mbox{\scriptsize T}}$.<br />In this paper we study the residual symmetry of the residual dynamics<br />for a scaled residual vector ${\bf y}\in {\mathbb S}^{n-1}_{\|{\bf r}_0\|}$, which<br />as expressed in the augmented space is a nonlinear Lorentzian dynamical system, and is endowed with a cone structure in the Minkowski<br />space with the Lorentz group $SO_o(n,1)$ being the internal symmetry group. Consequently,<br />we can modify the steplength to $\alpha_k={\bf y}_k^{\mbox{\scriptsize T}} {\bf A}{\bf y}_k/\|{\bf A}{\bf y}_k\|^2$<br />with ${\bf y}_k$ being computed by a Lorentz group algorithm (LGA)<br />based on $SO_o(n,1)$,<br />which can significantly improve the convergence speed and enhance the stability.<br />Several linear inverse problems are used to assess the numerical performance of the LGA.</p>2015-11-03T02:42:54-08:00Copyright (c) 2015 Journal of Mathematics Researchhttp://www.ccsenet.org/journal/index.php/jmr/article/view/54586Kink Multi-soliton Solutions for the (2+1)-Dimensional Bogoyavlenskii-Schiff Equation2015-11-24T01:08:13-08:00Xichao Dengxuzhenhui19@163.comZhenhui Xuxuzhenhui19@163.comHanlin Chenxuzhenhui19@163.comIn this work, (2+1)-dimensional Bogoyavlenskii-Schiff equation, which cann't be converted to a complete form of Hirota's bilinear operator, is considered. By using modification of extended homoclinic test approach new kink breather soliton and kink multi-solitons solutions are obtained and exhibited, respectively.2015-11-03T00:00:00-08:00Copyright (c) 2015 Journal of Mathematics Researchhttp://www.ccsenet.org/journal/index.php/jmr/article/view/52887Irregularity Strength of Corona Product of a Graph with Star Graph2015-11-24T01:08:13-08:00Santhosh Kumar K Rsanthoshkumargwc@gmail.comIf positive weights are assigned to the edges of a graph $G$, then degree of a vertex is the sum of the weights of edges that are incident to the vertex. A graph with weighted edges is said to be irregular if the degrees of the vertices are distinct. The irregularity strength of a graph is the smallest $s$ such that the edges can be weighted with $\{1,2,3, \cdots, s\}$ and be irregular. This notion was defined by Chartrand et al. (G Chartrand, M. S. Jacobson, J. Lehel, O. R. Ollerman, S. Ruiz \& Saba. 1988). In this paper, we discuss the irregularity strength of corona product of a graph with star graph $K_{1,n}$. We obtain a sufficient condition on the minimum degree of a graph $H$ which determines the irregularity strength of a graph $H$ having $p$ vertices with the star graph $K_{1,n}$.2015-11-03T02:42:54-08:00Copyright (c) 2015 http://www.ccsenet.org/journal/index.php/jmr/article/view/54583Performance of Robust Linear Classifier with Multivariate Binary Variables2015-11-24T01:08:13-08:00I. Egboegboike@gmail.comM. Egboegboike@gmail.comS. I. Onyeaguegboike@gmail.com<p>This paper focuses on the robust classification procedures in two group discriminant analysis with multivariate binary variables. A normal distribution based data set is generated using the R-software statistical analysis system 2.15.3 using Barlett’s approximation to chi-square, the data set was found to be homogenous and was subjected to five linear classifiers namely: maximum likelihood discriminant function, fisher’s linear discriminant function, likelihood ratio function, full multinomial function and nearest neighbour function rule. To judge the performance of these procedures, the apparent error rates for each procedure are obtained for different sample sizes. The results obtained ranked the procedures as follows: fisher’s linear discriminant function, maximum likelihood, full multinomial, likelihood function and nearest neigbour function.</p>2015-11-03T02:42:54-08:00Copyright (c) 2015 Journal of Mathematics Researchhttp://www.ccsenet.org/journal/index.php/jmr/article/view/54584Accelerated Bidirectional Modifications of the Steepest Descent Method for Ill-posed Linear Algebraic Systems with Dynamics-theoretical and Optimization Interpretation2015-11-24T01:08:13-08:00Li-Wei Liuliweiliu@ntu.edu.twHong-Ki Hongliweiliu@ntu.edu.twChein-Shan Liuliweiliu@ntu.edu.tw<p>It is well known that the classical numerical algorithm of the steepest descent method (SDM) is effective for well-posed linear systems, but performs poorly for ill-posed ones. In this paper we propose accelerated and/or bidirectional modifications of SDM, namely the accelerated steepest descent method (ASDM), the bidirectional method (2DM), and the accelerated bidirectional method (A2DM). The starting point is a manifold defined in terms of a minimum functional and a fictitious time variable; nevertheless, in the end the fictitious time variable disappears so that we arrive at purely iterative algorithms. The proposed algorithms are justified by dynamics-theoretical and optimization interpretation. The accelerator plays a prominent role of switching from the situation of slow convergence to a new situation that the functional tends to decrease stepwise in an intermittent and ceaseless manner. Three examples of solving ill-posed systems are examined and comparisons are made with exact solutions and with the existing algorithms of the SDM, the Barzilai-Borwein method, and the random SDM, revealing that the new algorithms of ASDM and A2DM have better computational efficiency and accuracy even for the highly ill-posed systems.</p>2015-11-03T02:42:54-08:00Copyright (c) 2015 Journal of Mathematics Researchhttp://www.ccsenet.org/journal/index.php/jmr/article/view/54653Quasi-arithmetic Means Inequalities Criteria for Differentiable Functions2015-11-24T01:08:13-08:00Bovzidar Ivankovicivankovb@fpz.hrQuasi-arithmetic means are defined for continuous, strictly monotone functions. In the case that functions are twice differentiable, we obtained criteria for inequalities between finite number of quasi-arithmetic means in additional and multiplicative case. Applications for H\"older and Minkowski type inequalities are given.2015-11-06T00:00:00-08:00Copyright (c) 2015 Journal of Mathematics Researchhttp://www.ccsenet.org/journal/index.php/jmr/article/view/53279An Embedding into a Substructure of the r.e. Turing Degrees2015-11-24T01:08:13-08:00Patrizio Cintiolipatrizio.cintioli@unicam.itLet $(I_m,\le)$ be the partial ordering of the $m$-introimmune r.e. Turing degrees.We wonder if such structure is an upper semi-lattice. We give a partial answer, by embedding some Boolean algebras in $(I_m,\le)$.2015-11-18T22:12:01-08:00Copyright (c) 2015 http://www.ccsenet.org/journal/index.php/jmr/article/view/53422The Application of Distributional Henstock-Kurzweil Integral on Third-order Three-Point Nonlinear Boundary-Value Problems2015-11-24T01:08:13-08:00Dakai Zhouzhoudakai24@163.comGuoju Yezhoudakai24@163.comWei Liuzhoudakai24@163.comBing Liangzhoudakai24@163.comWe consider a type of form such as $u'''=-f$ with three-point<br />nonlinear boundary-value problems (NBVPs). We verified the existence<br />of solutions of the (NBVPs) when $f$ is distributional<br />Henstock-Kurzweil integral but not Henstock-Kurzweil integral.We use<br />the distribution derivative and fixed point theorem to deal with the<br />problem. The results obtained generalize the known results. For<br />this reason, it is conducive to the further study of NBVPS.2015-11-24T01:07:10-08:00Copyright (c) 2015 http://www.ccsenet.org/journal/index.php/jmr/article/view/55139On The Minimal Number of Elements of Markov Partitions for Pseudo-Anosov Homeomorpisms2015-11-24T01:13:55-08:00Alexey Zhirovalexei_zhirov@mail.ruIn the paper an estimation of the minimal number of elements of Markov partition for generalized pseudo-Anosov homeomorphism of closed non necessary orientable surface is given. It is formulated in terms of characteristic of invariant foliation of generalized pseudo-Anosov homeomorphism.2015-11-24T00:00:00-08:00Copyright (c) 2015 Journal of Mathematics Researchhttp://www.ccsenet.org/journal/index.php/jmr/article/view/55140A Strong Class of Lifted Valid Inequalities for the Shortest Path Problem in Digraphs with Negative Cost Cycles2015-11-24T01:19:11-08:00Mamane Souleye Ibrahimibrah_dz@yahoo.frIn this paper, we address a strong class of lifted valid inequalities for the shortest path problem in digraphs with possibly negative cost cycles. We call these lifted inequalities the $incident \ lifted \ valid \ inequalities$ ($ILI$) as they are based on the incident arcs of a given vertex. The $ILI$ inequalities are close in spirit of the so-called \textit{simple lifted valid inequalities} ($SLI$) and $cocycle \ lifted \ valid \ inequalities$ ($CLI$) introduced in Ibrahim et al. (2015). However, as we will see the $ILI$ inequalities are stronger than the first ones in term of linear relaxation strengthening. Indeed, contrary to $SLI$ and $CLI$ inequalities, consider the same instances, in a cutting plane algorithm, the computational results prove that the $ILI$ inequalities provide the optimal integer solution for all the considered instances within no more than three iterations except one case for which after the first strengthening iteration, there exists no generated inequality.2015-11-24T00:00:00-08:00Copyright (c) 2015 Journal of Mathematics Researchhttp://www.ccsenet.org/journal/index.php/jmr/article/view/55155The Mathematics and Applications behind Image Warping and Morphing2015-11-24T18:15:37-08:00Tanvir Princetprince@hostos.cuny.eduMaria Maliktprince@hostos.cuny.eduIldefonso Salvatprince@hostos.cuny.eduAriel Mazortprince@hostos.cuny.eduSakhr Aldaylamtprince@hostos.cuny.edu<p class="Normal1">This research is conducted in the summer of 2015 and is possible by the support of various agency, in particular, by the grant of Prof. Angulo Nieves and the New York City Research Initiative.</p><p class="Normal1">The purpose of this research is to reveal the mathematics and applications of the computer animation techniques of warping and morphing. A warp is a twist or distortion in the form of an object in an image while a morph is the smooth and gradual transformation of an object in one image into the object in another image. Linear algebra makes these computer animation techniques possible; the first phase of this research delves into how those mathematical processes translate into image warps and morphs. Image morphs and morphs were identified as affine transformations of original images. The second part of this study requires the analysis and application of image warping and morphing techniques in an array of fields. The team utilized the computer software, Abrosoft Fantamorph and Morpheus in order to create a series of warps and morphs. This shows an example of the use of technology in undergraduate research. The final phase of this research was to identify what uses NASA can have for these computer animation techniques and what further research can be done to expand our knowledge of warps and morphs. By identifying the mechanics of warps and morph, we can discover how they can assist scientists and organization, such as NASA, to create depictions of objects, ideas, places, and events. Ultimately, studying morphing and warping techniques allows us to find better ways to represent visual data - whether it is images of the ozone hole or maps of the ever-changing weather in a region. The limitations that were found during the study can be used to conduct further research about warps and morphs - such as distorting images using quadratics or varying the rate at which each part of a transformation happens.</p>2015-11-25T00:00:00-08:00Copyright (c) 2015 Journal of Mathematics Researchhttp://www.ccsenet.org/journal/index.php/jmr/article/view/55230Solution of HIV and Malaria Coinfection Model Using Msgdtm2015-11-27T18:20:56-08:00Bonyah Ebenezerebbonya@yahoo.comKwasi Awuah-Werekohebbonya@yahoo.comJoseph Acquahebbonya@yahoo.com<p>In this paper, we investigate an epidemic model of HIV and Malaria co-infection using fractional order Calculus (FOC). The multistep generalized differential transform method (MSGDTM) is employed to obtain an accurate approximate solution to the epidemic model of HIV and Malaria co-infection disease in fractional order. A unique positive solution for HIV and Malaria co-infection is presented in fractional order form. For the integer case derivatives, the approximate solution of MSGDTM and the Runge–Kutta–order four scheme are compared. Numerical results are produced for the justification for this method.</p>2015-11-27T00:00:00-08:00Copyright (c) 2015 Journal of Mathematics Research