A fractal tiling (f-tiling) is a kind of rarely explored tiling by similar polygonal tiles which possesses self-similarity and the boundary of which is a fractal. Based on a tiling by similar isosceles right triangles, Dutch graphic artist M. C. Escher created an ingenious print Square Limit in which fish are uniformly reduced in size as they approach the boundaries of the tiling. In this article, we present four families of f-tilings and propose an easy-to-implement method to achieve similar Escher-like drawings. By systematically investigating the local star-shaped structure of f-tilings, we first enumerate four families of f-tilings admitted by kite-shaped or dart-shaped prototiles. Then, we establish a fast binning algorithm for visualising f-tilings. To facilitate the creation of Escher-like drawings on the reported f-tilings, we next introduce one-to-one mappings between the square, and kite and dart, respectively. This treatment allows a pre-designed square template to be deformed into all prototiles considered in the article. Finally, we specify some technical implementations and present a gallery of the resulting Escher-like drawings. The method established in this article is thus able to generate a great variety of exotic Escher-like drawings.

Self-Similar Fractal Drawings Inspired by M. C. Escher’s Print Square Limit

In this paper, we show some results about the existence and the uniqueness of the positive solution for a $p$-Laplacian fractional differential equations with fractional derivative boundary condition. Our results are based on Krasnosel'skii's fixed point theorem, the nonlinear alternative of Leray-Schauder type and contraction mapping principle. Three examples are given to illustrate the applicability of our main results.

Positive solutions of $p$-Laplacian fractional differential equations with fractional derivative boundary condition

&lt;p style='text-indent:20px;'&gt;In this study, we develop a diffusive HIV-1 infection model with intracellular invasion, production and latent infection distributed delays, nonlinear incidence rate and nonlinear CTL immune response. The well-posedness, local and global stability for the model proposed are carefully investigated in spite of its strong nonlinearity and high dimension. It is revealed that its threshold dynamics are fully determined by the viral infection reproduction number &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathfrak{R}_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and the reproduction number of CTL immune response &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \mathfrak{R}_1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. We also observe that the viral load at steady state (SS) fails to decrease even if &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \mathfrak{R}_1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; increases through unit to lead to a stability switch from immune-inactivated infected SS to immune-activated infected SS. Finally, some simulations are performed to verify the analytical conclusions and we explore the significant impact of delays and CTL immune response on the spatiotemporal dynamics of HIV-1 infection.&lt;/p&gt; 

Spatiotemporal dynamics for a diffusive HIV-1 infection model with distributed delays and CTL immune response

In this study, we develop a diffusive HIV-1 infection model with intracellular invasion, production and latent infection distributed delays, nonlinear incidence rate and nonlinear CTL immune response. The well-posedness, local and global stability for the model proposed are carefully investigated in spite of its strong nonlinearity and high dimension. It is revealed that its threshold dynamics are fully determined by the viral infection reproduction number \begin{document}$ \mathfrak{R}_0 $\end{document} and the reproduction number of CTL immune response \begin{document}$ \mathfrak{R}_1 $\end{document} . We also observe that the viral load at steady state (SS) fails to decrease even if \begin{document}$ \mathfrak{R}_1 $\end{document} increases through unit to lead to a stability switch from immune-inactivated infected SS to immune-activated infected SS. Finally, some simulations are performed to verify the analytical conclusions and we explore the significant impact of delays and CTL immune response on the spatiotemporal dynamics of HIV-1 infection.

In this paper, using both hand-drawn and computer-drawn graphics, we establish a method to generate advanced Escher-like spiral tessellations. We first give a way to achieve simple spiral tilings of cyclic symmetry. Then, we introduce several conformal mappings to generate three derived spiral tilings. To obtain Escher-like tessellations on the generated tilings, given pre-designed wallpaper motifs, we specify the tessellations’ implementation details. Finally, we exhibit a rich gallery of the generated Escher-like tessellations. According to the proposed method, one can produce a great variety of exotic Escher-like tessellations that have both good aesthetic value and commercial potential.

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Generation of advanced Escher-like spiral tessellations

In this paper, we consider the following mixed fractional resonant boundary value problem with p(t)-Laplacian operator $$\begin{aligned} \left\{ \begin{array}{lll} ^{C} D^{\beta }_{0^{+}}\varphi _{p(t)}(D^{\alpha }_{0^{+}}u(t))=f(t, u(t), D^{\alpha }_{0^{+}}u(t)),~~t\in [0, T],\\ t^{1-\alpha }u(t)\mid _{t=0}=0,~~D^{\alpha }_{0^{+}}u(0)=D^{\alpha }_{0^{+}}u(T), \end{array}\right. \end{aligned}$$
 where $$^{C} D^{\beta }_{0^{+}}$$
 is Caputo fractional derivative, $$D^{\alpha }_{0^{+}}$$
 is Riemann–Liouville fractional derivative, $$\varphi _{p(t)}$$
 is p(t)-Laplacian operator, $$p(t)>1$$
 , $$p(t)\in C^{1}[0, T]$$
 with $$p(0)=p(T)$$
 . Under the appropriate conditions of the nonlinear term, the existence of solutions for the above mixed fractional resonant boundary value problem is obtained by using the continuation theorem of coincidence degree theory, which enrich the existing literatures. In addition, an example is included to demonstrate the main result.

The existence of solutions for mixed fractional resonant boundary value problem with p(t)-Laplacian operator

Automatic generation of hyperbolic drawings

The distributed delay was firstly proposed by Volterra in the 1930s since it is more realistic than discrete delay and has been introduced in many dynamical systems. In this paper, we establish a diffusive viral infection model with general incidence function and distributed delays subject to the homogeneous Neumann boundary conditions. Firstly, we prove the existence, uniqueness, positivity and boundedness of solutions of the model. Then, by using the linearization method and constructing appropriate Lyapunov functionals, we show that the global dynamics of the model is determined by the reproductive numbers for viral infection 
$$\mathcal {R}_{0}$$

, which implies that the global stability of the model precludes the existence of complex dynamical behaviors such as Hopf bifurcation and patter formation. Furthermore, an example is presented and numerical simulations are also carried out to illustrate the main results.

Global dynamics of a diffusive viral infection model with general incidence function and distributed delays

A fractal tiling or f-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal By substitution rule of tilings, this short paper presents a very simple strategy to

Xinchang Wang

Papers

The existence of solutions for mixed fractional resonant boundary value problem with p(t)-Laplacian operator

Automatic generation of hyperbolic drawings

Global dynamics of a diffusive viral infection model with general incidence function and distributed delays

Fractal Tilings from Substitution Tilings