Asian journal for mathematics education, Sep 1, 2022
Thanks to the fourth industrial revolution, 3D printing has become a fast-emerging technology tha... more Thanks to the fourth industrial revolution, 3D printing has become a fast-emerging technology that is widely applied across industries. In mathematics education, 3D printing is an innovative way to visualize mathematics concepts (e.g., geometry, calculus) that enables students to develop mathematical and design thinking, as well as digital skills and mindsets. Through digital maker education, students can apply multidisciplinary knowledge to build prototypes and create 3D objects that bring many new opportunities in mathematics formal/informal learning. However, to our knowledge, there is no existing review summarizing the existing evidence of how 3D printing has been applied in mathematics education. As such, this review aims to give a synthesis of the up-to-date literature in the burgeoning topic of using 3D printing in mathematics education. A systematic review was conducted to examine the thematic and content analysis of 30 empirical papers from 2015 to 2022. The review aims to evaluate and analyze different types of participants, methodological approaches, challenges, pedagogies, and technologies used in the selected studies. Although 3D printing has a bright prospect to revolutionize mathematics education, there are still many challenges such as hardware and software optimization, processing, formatting, printing, and maintenance issues. After all, a set of recommendations were listed to guide future researchers and educators to use 3D printing effectively in mathematics education.
Communications in Theoretical Physics, Sep 26, 2022
In this paper, we prove the existence of general Cartesian vector solutions u = b (t) + A(t) x fo... more In this paper, we prove the existence of general Cartesian vector solutions u = b (t) + A(t) x for the N-dimensional compressible Navier–Stokes equations with density-dependent viscosity, based on the matrix and curve integration theory. Two exact solutions are obtained by solving the reduced systems.
Communications in Theoretical Physics, May 1, 2015
In this paper, the Clarkson-Kruskal direct approach is employed to investigate the exact solution... more In this paper, the Clarkson-Kruskal direct approach is employed to investigate the exact solutions of the 2-dimensional rotational Euler equations for the incompressible fluid. The application of the method leads to a system of completely solvable ordinary differential equations. Several special cases are discussed and novel nonlinear exact solutions with respect to variables x and y are obtained. It is of interest to notice that the pressure p is obtained by the second kind of curvilinear integral and the coefficients of the nonlinear solutions are solitary wave type functions like tanh(kt/2) and sech (kt/2) due to the rotational parameter k ≠ 0. Such phenomenon never appear in the classical Euler equations wherein the Coriolis force arising from the gravity and Earth's rotation is ignored. Finally, illustrative numerical figures are attached to show the behaviors that the exact solutions may exhibit.
Journal of Mathematical Fluid Mechanics, Sep 9, 2019
The compressible Euler equations are the classical model in fluid dynamics. In this study, we inv... more The compressible Euler equations are the classical model in fluid dynamics. In this study, we investigate the life span of the projected 2-dimensional rotational $$C^{2}$$ non-vacuum solutions of the Euler equations. By examining the corresponding projected 2-dimensional solutions, $$\begin{aligned} (\rho (t,x_{1},x_{2}),u_{1}(t,x_{1},x_{2}),u_{2}(t,x_{1},x_{2}),0), \end{aligned}$$in $$\mathbf {R}^{3}$$, we prove that there exist the corresponding blowup results for the rotational $$C^{2}$$ solutions with a sufficiently large initial functional $$\begin{aligned} H(0)= {\displaystyle \int _{\mathbf {R}^{3}}} \vec {x}\cdot \vec {u}_{0}dV. \end{aligned}$$
In this paper, under the assumption of an initial bounded region $ \Omega(0) $, we establish the ... more In this paper, under the assumption of an initial bounded region $ \Omega(0) $, we establish the blowup phenomenon of the regular solutions and $ C^{1} $ solutions to the two-phase model in $ \mathbb{R}^{N} $. If the total energy $ E $ and the total mass $ M > 0 $ satisfy \begin{document}$ \begin{equation} \nonumber \max\limits_{\vec{x_{0}}\in\partial\Omega(0)}\sum\limits_{i = 1}^{N}u_{i}^{2}(0,\vec{x_{0}})<\frac{\min\{2,N(\Gamma-1),N(\gamma-1)\}E}{M}, \end{equation} $\end{document} where $ E = \int_{\Omega(0)}\left(\frac{1}{2}n\left\vert \vec{u}\right\vert ^{2} +\frac{1}{2}\rho\left\vert \vec{u}\right\vert ^{2}+\frac{1}{\Gamma-1}n^{\Gamma}+\frac{1}{\gamma-1}\rho^{\gamma}\right) dV $ and $ M = {\int_{\Omega(0)}} (n+\rho) dV > 0 $, then the blowup of the solutions to the two-phase model will be formed in finite time in $ \mathbb{R}^{N} $. Furthermore, under the assumptions that the radially symmetric initial data and initial density contain vacuum states, the blowup of the smooth solutions to the two-phase model will be formed in finite time in $ \mathbb{R}^{N} (N \geq2) $.
Asian journal for mathematics education, Sep 1, 2022
Thanks to the fourth industrial revolution, 3D printing has become a fast-emerging technology tha... more Thanks to the fourth industrial revolution, 3D printing has become a fast-emerging technology that is widely applied across industries. In mathematics education, 3D printing is an innovative way to visualize mathematics concepts (e.g., geometry, calculus) that enables students to develop mathematical and design thinking, as well as digital skills and mindsets. Through digital maker education, students can apply multidisciplinary knowledge to build prototypes and create 3D objects that bring many new opportunities in mathematics formal/informal learning. However, to our knowledge, there is no existing review summarizing the existing evidence of how 3D printing has been applied in mathematics education. As such, this review aims to give a synthesis of the up-to-date literature in the burgeoning topic of using 3D printing in mathematics education. A systematic review was conducted to examine the thematic and content analysis of 30 empirical papers from 2015 to 2022. The review aims to evaluate and analyze different types of participants, methodological approaches, challenges, pedagogies, and technologies used in the selected studies. Although 3D printing has a bright prospect to revolutionize mathematics education, there are still many challenges such as hardware and software optimization, processing, formatting, printing, and maintenance issues. After all, a set of recommendations were listed to guide future researchers and educators to use 3D printing effectively in mathematics education.
Communications in Theoretical Physics, Sep 26, 2022
In this paper, we prove the existence of general Cartesian vector solutions u = b (t) + A(t) x fo... more In this paper, we prove the existence of general Cartesian vector solutions u = b (t) + A(t) x for the N-dimensional compressible Navier–Stokes equations with density-dependent viscosity, based on the matrix and curve integration theory. Two exact solutions are obtained by solving the reduced systems.
Communications in Theoretical Physics, May 1, 2015
In this paper, the Clarkson-Kruskal direct approach is employed to investigate the exact solution... more In this paper, the Clarkson-Kruskal direct approach is employed to investigate the exact solutions of the 2-dimensional rotational Euler equations for the incompressible fluid. The application of the method leads to a system of completely solvable ordinary differential equations. Several special cases are discussed and novel nonlinear exact solutions with respect to variables x and y are obtained. It is of interest to notice that the pressure p is obtained by the second kind of curvilinear integral and the coefficients of the nonlinear solutions are solitary wave type functions like tanh(kt/2) and sech (kt/2) due to the rotational parameter k ≠ 0. Such phenomenon never appear in the classical Euler equations wherein the Coriolis force arising from the gravity and Earth's rotation is ignored. Finally, illustrative numerical figures are attached to show the behaviors that the exact solutions may exhibit.
Journal of Mathematical Fluid Mechanics, Sep 9, 2019
The compressible Euler equations are the classical model in fluid dynamics. In this study, we inv... more The compressible Euler equations are the classical model in fluid dynamics. In this study, we investigate the life span of the projected 2-dimensional rotational $$C^{2}$$ non-vacuum solutions of the Euler equations. By examining the corresponding projected 2-dimensional solutions, $$\begin{aligned} (\rho (t,x_{1},x_{2}),u_{1}(t,x_{1},x_{2}),u_{2}(t,x_{1},x_{2}),0), \end{aligned}$$in $$\mathbf {R}^{3}$$, we prove that there exist the corresponding blowup results for the rotational $$C^{2}$$ solutions with a sufficiently large initial functional $$\begin{aligned} H(0)= {\displaystyle \int _{\mathbf {R}^{3}}} \vec {x}\cdot \vec {u}_{0}dV. \end{aligned}$$
In this paper, under the assumption of an initial bounded region $ \Omega(0) $, we establish the ... more In this paper, under the assumption of an initial bounded region $ \Omega(0) $, we establish the blowup phenomenon of the regular solutions and $ C^{1} $ solutions to the two-phase model in $ \mathbb{R}^{N} $. If the total energy $ E $ and the total mass $ M > 0 $ satisfy \begin{document}$ \begin{equation} \nonumber \max\limits_{\vec{x_{0}}\in\partial\Omega(0)}\sum\limits_{i = 1}^{N}u_{i}^{2}(0,\vec{x_{0}})<\frac{\min\{2,N(\Gamma-1),N(\gamma-1)\}E}{M}, \end{equation} $\end{document} where $ E = \int_{\Omega(0)}\left(\frac{1}{2}n\left\vert \vec{u}\right\vert ^{2} +\frac{1}{2}\rho\left\vert \vec{u}\right\vert ^{2}+\frac{1}{\Gamma-1}n^{\Gamma}+\frac{1}{\gamma-1}\rho^{\gamma}\right) dV $ and $ M = {\int_{\Omega(0)}} (n+\rho) dV > 0 $, then the blowup of the solutions to the two-phase model will be formed in finite time in $ \mathbb{R}^{N} $. Furthermore, under the assumptions that the radially symmetric initial data and initial density contain vacuum states, the blowup of the smooth solutions to the two-phase model will be formed in finite time in $ \mathbb{R}^{N} (N \geq2) $.
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