Investigation and Exploration of ‘Student-Centered and Teacher-Led’ Teaching Model in English Medium Instruction (EMI) Calculus Course

The internationalization of higher education in China is constantly improving with an increasing level of diversification and globalization of education. High-level international English Medium Instruction (EMI) course is crucial to the cultivation of innovative international talents. Taking the Calculus course as an example, this article first demonstrates the importance and connotation of ‘know thy enemy and know yourself’ in the construction of EMI courses. Then it elaborates on the construction methods and significance of the ‘Leaning Community’, ‘Teaching Community’, and ‘Teaching-Learning Community’ through studies of the relationship between ‘teaching’ and ‘learning’ form the student-centered aspect. Such research provides a useful reference for the teaching model reform, especially the effective construction of EMI courses in non-native English-speaking countries

Biorthogonal wavelets and tight framelets from smoothed pseudo splines

Abstract In order to get divergence free and curl free wavelets, one introduced the smoothed pseudo spline by using the convolution method. The smoothed pseudo splines can be considered as an extension of pseudo splines. In this paper, we first show that the shifts of a smoothed pseudo spline are linearly independent. The linear independence of the shifts of a pseudo spline is a necessary and sufficient condition for the construction of the biorthogonal wavelet system. Based on this result, we generalize the results of Riesz wavelets and derive biorthogonal wavelets from smoothed pseudo splines. Furthermore, by applying the unitary extension principle, we construct tight frame systems associated with smoothed pseudo splines with desired approximation order

A Variant Cubic Exponential B-Spline Scheme with Shape Control

This paper presents a variant scheme of the cubic exponential B-spline scheme, which, with two parameters, can generate curves with different shapes. This variant scheme is obtained based on the iteration from the generation of exponentials and a suitably chosen function. For such a scheme, we show its C2-convergence and analyze the effect of the parameters on the shape of the generated curves and also discuss its convexity preservation. In addition, a non-uniform version of this variant scheme is derived in order to locally control the shape of the generated curves. Numerical examples are given to illustrate the performance of the new schemes in this paper

Gibbs phenomenon for p-ary subdivision schemes

Abstract When a Fourier series is used to approximate a function with a jump discontinuity, the Gibbs phenomenon always exists. This similar phenomenon exists for wavelets expansions. Based on the Gibbs phenomenon of a Fourier series and wavelet expansions of a function with a jump discontinuity, in this paper, we consider that a Gibbs phenomenon occurs for the p-ary subdivision schemes. Similar to the method of (Appl. Math. Lett. 76:157–163, 2018), we generalize the results about the stationary binary subdivision schemes in (Appl. Math. Lett. 76:157–163, 2018) to the case of p-ary subdivision schemes. By considering the masks of subdivision schemes, we obtain a sufficient condition to determine whether there exists a Gibbs phenomenon for p-ary subdivision schemes in the limit function close to the discontinuous point. This condition consists of the positivity of the partial sums of the values of the masks. By applying this condition, we can avoid the Gibbs phenomenon for p-ary subdivision schemes near discontinuity points. Finally, some examples in classical subdivision schemes are given to illustrate the results in this paper

Stability and Hopf Bifurcation Analysis for a Phage Therapy Model with and without Time Delay

This study proposes a mathematical model that accounts for the interaction of bacteria, phages, and the innate immune response with a discrete time delay. First, for the non-delayed model we determine the local and global stability of various equilibria and the existence of Hopf bifurcation at the positive equilibrium. Second, for the delayed model we provide sufficient conditions for the local stability of the positive equilibrium by selecting the discrete time delay as a bifurcation parameter; Hopf bifurcation happens when the time delay crosses a critical threshold. Third, based on the normal form method and center manifold theory, we derive precise expressions for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to verify our theoretical analysis

Normal Vector Based Subdivision Scheme to Generate Fractal Curves

In this paper, we firstly devise a new and general p-ary subdivision scheme based on normal vectors with multi-parameters to generate fractals. Rich and colorful fractals including some known fractals and a lot of unknown ones can be generated directly and conveniently by using it uniformly. The method is easy to use and effective in generating fractals since the values of the parameters and the directions of normal vectors can be designed freely to control the shape of generated fractals. Secondly, we illustrate the technique with some design results of fractal generation and the corresponding fractal examples from the point of view of visualization, including the classical Lévy curves, Dragon curves, Sierpiński gasket, Koch curve, Koch-type curves and other fractals. Finally, some fractal properties of the limit of the presented subdivision scheme, including existence, self-similarity, non-rectifiability, and continuity but nowhere differentiability are described from the point of view of theoretical analysis. DOI: http://dx.doi.org/10.11591/telkomnika.v11i8.3025