Mollweide's formula in teaching trigonometry

Trigonometry is one of the topics in mathematics that the students in both high school and pre-undergraduate levels need to learn. Generally, the topic covers trigonometric functions, trigonometric equations, trigonometric identities and solving oblique triangles using the Laws of Sines and Cosines. However, when solving the oblique triangles, Mollweide's formula is most likely to be omitted from the discussion. Mollweide's formula--which exhibits a cyclical nature--is particularly useful in checking one's result after solving an oblique triangle since all six components of the triangle are involved. It is interesting to note that proving Mollweide's formula can be performed without words. Furthermore, the Law of Tangents can be derived straightforwardly from this equation. In this article, we revisit Mollweide's formula and provide classroom examples where this equation comes into alive. In addition, we suggest that this seemingly less-known equation is to be included in the mathematics syllabus on the topic of Trigonometry.Comment: 5 pages, 1 figur

Flipped classroom in Introductory Linear Algebra by utilizing Computer Algebra System {\sl SageMath} and a free electronic book

This article describes the authors' teaching experience in flipping the class of a basic undergraduate mathematics course Introductory Linear Algebra. We utilize a full-featured free electronic textbook, online lecture notes, an intranet learning management system (LMS) {\sl icampus}, the video-sharing website {\sl YouTube} and a Computer Algebra System (CAS) {\sl SageMath} in our flipped classroom approach. Prior coming to the class, the students are assigned to complete a portion of reading assignments from the free e-book and to watch recorded video lectures from {\sl YouTube} that cover the relevant materials. Announcements are made through the LMS {\sl icampus}, as well as disseminating teaching materials, assigning tasks, and fostering online communication outside the class hours. We dedicated in-class activities with problem-solving sessions, question and answer, discussion and providing instant feedback to the students. Hence, both out-of-class and in-class sessions are filled with active learning and interactive engagement. From the students' feedback and opinion, we discovered that some of them preferred the traditional classroom rather than the flipped classroom pedagogy. A possible reason is either they are used to the traditional lecture style where receiving information occurs inside the classroom or the flipped classroom approach that we implement does not satisfy students' learning styles. Nevertheless, several students enjoyed problem-solving activities where they have an opportunity to communicate with the instructors whenever they encounter some difficulty.Comment: 12 pages, 4 figure

First- and second-order wave generation theory

The first-order and the second-order wave generation theory is studied in this paper. The theory is based on the fully nonlinear water wave equations. The nonlinear boundary value problem (BVP) is solved using a series expansion method. Using this method, the problem becomes a set of linear, signalling problems according to the expansion order. The first-order theory leads to a homogeneous BVP. It is a system with the first-order steering of the wavemaker motion as input and the surface wave field with propagating and evanescent modes as output. The second-order theory leads to a nonhomogeneous BVP. It is a system where the second-order steering of the wavemaker motion is prescribed in such a way that the second-order part of the surface elevation far from the wavemaker contains only the bound wave component and the free wave component vanishes. The second-order surface wave elevation consists of a superposition of bichromatic frequencies.Comment: 12 pages, 2 figure

Calculus teaching and learning in South Korea

This article discusses an experience of teaching Calculus classes for the freshmen students enrolled at Sungkyunkwan University, one of the private universities in South Korea. The teaching and learning approach is a balance combination between the teacher-oriented traditional style of lecturing and other activities that encourage students for active learning and classroom participation. Based on the initial observation during several semesters, some anecdotal evidences show that students' learning is improved after implementing this student-oriented active learning approach, albeit a longer period of time is definitely needed to transform general students' attitude from passive learners to active ones.Comment: 10 pages, 1 tabl

Newton's method's basins of attraction revisited

In this paper, we revisit the chaotic number of iterations needed by Newton's method to converge to a root. Here, we consider a simple modified Newton method depending on a parameter. It is demonstrated using polynomiography that even in the simple algorithm the presence and the position of the convergent regions, i.e. regions where the method converges nicely to a root, can be complicatedly a function of the parameter.Comment: 9 pages, 3 figure

Maximum temporal amplitude and designs of experiments for generation of extreme waves

This paper aims to describe a deterministic generation of extreme waves in a typical towing tank. Such a generation involves an input signal to be provided at the wavemaker in such a way that at a certain position in the wave tank, say at a position of a tested object, a large amplitude wave emerges. For the purpose, we consider a model called a spatial nonlinear Schr\"odinger equation describing the spatial propagation of a slowly varying envelope of a signal. Such a model has an exact solution known as (spatial) Soliton on a Finite Background (SFB) that is a nonlinear extension of Benjamin-Feir instability. This spatial-SFB is characterized by wave focusing leading to almost time-periodic extreme waves that appear in between phase singularities. Although phase singularities and wave focusing have been subject to a number of studies, this spatial-SFB written in the field variables has many interesting properties among which are the existence of many critical values related to the modulation length of the monochromatic signal in the far fields. These properties will be used in choosing parameters for designing experiments on extreme wave generation. In doing so, a quantity called maximum temporal amplitude (MTA) is used. This quantity measures at each location the maximum over time of the wave elevation. For a given modulation length of SFB and desired maximum amplitude at a position in a towing tank, the MTA readily shows the maximum signal that is required at the wavemaker and the amplitude amplification factor of the requested signal. Some examples of such a generation in realistic laboratory variables will be displayed.Comment: 6 pages, 5 figures, one table, Proceedings of the 11th Asian Congress of Fluid Mechanics, 22-25 May 2006, Kuala Lumpur, Malaysia, pages 978-98

Exploiting bifurcations in waveguide arrays for light detectors

An array of a finite number waveguides, driven laterally by injecting light at the outer waveguides, is considered. The array is modeled by a discrete nonlinear Schr\"{o}dinger equation. It has been shown [Phys. Rev. Lett. 94, 243902 (2005)] that, when the injected light is in the proximity of a bifurcation point, such a system can be sensitive to small disturbances, making it possible to act as a light detector. Here, the optimum intensity of the injected light is discussed, and an analytical approximation is presented.Comment: 3 pages, 2 figure

On varying coefficients of spatial inhomogeneous nonlinear Schr\"odinger equation

A nonlinear evolution equation for wave packet surface gravity waves with variation in topography is revisited in this article. The equation is modeled by a spatial inhomogeneous nonlinear Schr\"odinger (NLS) equation with varying coefficients, derived by Djordjevi\'c and Redekopp (1978) and the nonlinear coefficient is later corrected by Dingemans and Otta (2001). We show analytically and qualitatively that the nonlinear coefficient and the corresponding averaging value, stated but not derived, by Benilov, Flanagan and Howlin (2005) and Benilov and Howlin (2006) are inaccurate. For a particular choice of topography and wave characteristics, the NLS equation alternates between focusing and defocusing case and hence, it does not admit the formation of a classical soliton, neither bright nor dark one.Comment: 8 pages, 4 figures, Appendix, presented in the 8th International Conference on Applied Physics and Mathematics (ICAPM), Phuket, Thailand, 27-29 January 201

On the transition period of implementing new mathematics curriculum for Foundation Engineering students

An overview on several mathematics modules in the transition period of introducing a new curriculum for the Foundation programme in Engineering at the University of Nottingham Malaysia Campus is discussed in this paper. In order to progress to Undergraduate programmes in Engineering, previously the students must complete three mathematics modules of 40 credit points in total, for which one of them was a year-long module with 20 credit points. Currently under the new curriculum, the students are required to complete five mathematics modules with 10 credit points each. The new curriculum gives positive impacts for both the lecturers and the students in terms of material organization, fully utilizing textbooks and a new arrangement for tutorial sessions. The new curriculum also provides the students with stronger mathematical background in critical thinking and problem solving skills to equip them to embark the Undergraduate programmes in Engineering.Comment: 17 page

Oligomers with complex couplings as PT\cal{PT}-symmetric systems

We consider an array of double oligomers in an optical waveguide device. A mathematical model for the system is the coupled discrete nonlinear Schr\"odinger (NLS) equations, where the gain-and-loss parameter contributes to the complex-valued linear coupling. The array caters to an optical simulation of the parity-time ($\cal{PT}$)-symmetry property between the coupled arms. The system admits fundamental bright discrete soliton solutions. We investigate their existence and spectral stability using perturbation theory analysis. These analytical findings are verified further numerically using the Newton-Raphson method and a standard eigenvalue-problem solver. Our study focuses on two natural discrete modes of the solitons: single- and double-excited-sites, also known as onsite and intersite modes, respectively. Each of these modes acquires three distinct configurations between the dimer arms, i.e., symmetric, asymmetric, and antisymmetric. Although both intersite and onsite discrete solitons are generally unstable, the latter can be stable, depending on the combined values of the propagation constant, horizontal linear coupling coefficient, and gain-loss parameter.Comment: 11 pages, 10 figures, 43 reference