Maximal Temporal Period of a Periodic Solution Generated by a One-Dimensional Cellular Automaton

One-dimensional cellular automata evolutions with both temporal and spatial periodicity are studied. The main objective is to investigate the longest temporal periods among all two-neighbor rules, with a fixed spatial period σ and number of states n. When σ = 2, 3, 4 or 6, and the rules are restricted to be additive, the longest period can be expressed as the exponent of the multiplicative group of an appropriate ring. Non-additive rules are also constructed with temporal period on the same order as the trivial upper bound n σ . Experimental results, open problems and possible extensions of the results are also discussed.

Correspondences between parameters in a reaction-diffusion model and connexin functions during zebrafish stripe formation

Different diffusivities among interacting substances actualize the potential instability of a system. When these elicited instabilities manifested as forms of spatial periodicity, they are called Turing patterns. Simulations using general reaction-diffusion (RD) models have demonstrated that pigment patterns on the body trunk of growing fish follow a Turing pattern. Laser ablation experiments performed on zebrafish revealed apparent interactions among pigment cells, which allowed for a three-components RD model to be derived. However, the underlying molecular mechanisms responsible for Turing pattern formation in this system had been remained unknown. A zebrafish mutant with a spotted pattern was found to have a defect in Connexin41.8 (Cx41.8) which, together with Cx39.4, exists in pigment cells and controls pattern formations. Here, molecular-level evidence derived from connexin analyses was linked to the interactions among pigment cells described in previous RD modeling. Channels on pigment cells were generalized as “gates,” and the effects of respective gates were deduced. The model used partial differential equations (PDEs) to enable numerical and mathematical analyses of characteristics observed in the experiments. Furthermore, the improved PDE model included nonlinear reaction terms, enabled the consideration of the behavior of components.

Correspondences between parameters in a reaction-diffusion model and connexin functions during zebrafish stripe formation

Different diffusivities among interacting substances actualize the potential instability of a system. When these elicited instabilities manifested as forms of spatial periodicity, they are called Turing patterns. Simulations using general reaction-diffusion (RD) models have demonstrated that pigment patterns on the body trunk of growing fish follow a Turing pattern. Laser ablation experiments performed on zebrafish revealed apparent interactions among pigment cells, which allowed for a three-components RD model to be derived. However, the underlying molecular mechanisms responsible for Turing pattern formation in this system had been remained unknown. A zebrafish mutant with a spotted pattern was found to have a defect in Connexin41.8 (Cx41.8) which, together with Cx39.4, exists in pigment cells and controls pattern formations. Here, molecular-level evidence derived from connexin analyses was linked to the interactions among pigment cells described in previous RD modeling. Channels on pigment cells were generalized as “gates,” and the effects of respective gates were deduced. The model used partial differential equations (PDEs) to enable numerical and mathematical analyses of characteristics observed in the experiments. Furthermore, the improved PDE model included nonlinear reaction terms, enabled the consideration of the behavior of components.

2D Dynamic Directional Amplification (DDA) in Phononic Metamaterials

Phononic structures with unit cells exhibiting Bragg scattering and local resonance present unique wave propagation properties at wavelengths well below the regime corresponding to bandgap generation based on spatial periodicity. However, both mechanisms show certain constraints in designing systems with wide bandgaps in the low-frequency range. To face the main practical challenges encountered in such cases, including heavy oscillating masses, a simple dynamic directional amplification (DDA) mechanism is proposed as the base of the phononic lattice. This amplifier is designed to present the same mass and use the same damping element as a reference two-dimensional (2D) phononic metamaterial. Thus, no increase in the structure mass or the viscous damping is needed. The proposed DDA can be realized by imposing kinematic constraints to the structure’s degrees of freedom (DoF), improving inertia and damping on the desired direction of motion. Analysis of the 2D lattice via Bloch’s theory is performed, and the corresponding dispersion relations are derived. The numerical results of an indicative case study show significant improvements and advantages over a conventional phononic structure, such as broader bandgaps and increased damping ratio. Finally, a conceptual design indicates the usage of the concept in potential applications, such as mechanical filters, sound and vibration isolators, and acoustic waveguides.

Modeling of ocean wave propagation across the crevasse-ridden ice shelf: focus on the comparison of two models

<p>The propagation of high-frequency elastic-flexural waves through an ice shelf was modeled by a full 3-D elastic models. These models based on the momentum equations that were written as the differential equations (model#1) and as the integro-differential equations (model#2). The integro-differential form implies the vertical integration of the momentum equations from the ice surface to the current coordinate z like, for instance, in the Blatter-Pattyn ice flow model. The sea water flow under the ice shelf is described by the wave equation. The numerical solutions were obtained by a finite-difference method. Numerical experiments were undertaken for a crevasse-ridden ice shelf with different spatial periodicities of the crevasses. In this research the modeled positions of the band gaps in the dispersion spectra dependently on the spatial periodicities of the crevasses is investigated from the point of view of agreement of these positions with the Bragg’s law. The investigation of the dispersion spectra shows that different models reveal different sensitivities of the dispersion spectra (in relation to the appearance of the band gaps in the spectra) dependently on the spatial periodicity of the crevasses and on the crevasses depth.</p>

Focusing magnetohydrodynamic waves as a trigger of star formation

Propagation of fast-mode magnetohydrodynamic (MHD) waves in interstellar space is simulated, and a focusing MHD wave model is proposed for triggered star formation (SF). Waves from an SF region are trapped by nearby molecular clouds and converge onto their focal points, causing implosive compression. Even an isolated cloud suffers from long-distance invasion of waves from remote sources. Echoing SF occurs inside a cloud as well as between clouds. Repetitive refocusing in a filamentary cloud suggests spatial periodicity in SF sites along the filament. The model is applied to the SF regions M 16 and M 17, where MHD waves produced by M 16 are shown to converge onto the focal point of the nearby giant molecular cloud and trigger the SF in M 17.

Inline LIPSS Monitoring Method Employing Light Diffraction

Laser-induced ripples that are also known as laser-induced periodic surface structures (LIPSS) have gained a considerable attention by researchers and industry due to their surface functionalization applications. However, texturing large areas or batch manufacture of parts that incorporate LIPSS surfaces require the development of tools for monitoring the LIPSS generation and potentially for controlling their main geometrical characteristics, i.e., spatial periodicity, orientation, and amplitude. In this context, the focus of the research reported in this paper is on developing process monitoring and inspection methods for identifying shifts and changes in these characteristics. One of the well-known and widely used by industry method for characterizing and inspecting surfaces is light scattering, and this research investigates the capabilities of this method for inline monitoring of LIPSS optical response. A simple setup was designed and implemented for measuring the diffraction angle and intensity of the reflected light from LIPSS surfaces. The capabilities of this concept for determining relative shifts in the optical response on surfaces processed with known disturbances, such as incident angle deviations and focus offset, were investigated. Sensitivity of the method proved to be sufficient to detect shifts/deviations from LIPSS reference and thus potentially to monitor their generation in line with a simple sensor, e.g., the LIPSS treatment of larger tool surfaces or serial manufacture of holograms.

Improving the Performance of 2-To-4 Optical Decoders Based on Photonic Crystal Structures

In this study, a novel, two-dimensional photonic crystal-based structure for the 2-to-4 optical decoder is presented. The structure consists of 23 rows and 14 columns of chalcogenide rods that are arranged in a square lattice with a spatial periodicity of 530 nm. The bias and the optical signals are guided toward the main waveguide through the three waveguides. Two unequal powers are applied to the input ports to approach the different intensities proportional to four working states into the main waveguide. Four cavities including the nonlinear rods are in response to drop the optical waves toward the output ports. To calculate the band diagram and the spatial distribution of the electric and magnetic fields, the plane wave expansion and the finite difference time domain methods have been used. The delay time of the designed structure is obtained around 220 fs, which is less than one for the previous structures. Furthermore, the gap between the margins for logic 0 and 1 is equal to 83%, which is higher than one for other works. Besides, the area of the structure is reduced to 90 µm2 in comparison to all reported structures. Based on the mentioned results, it seems that an improvement of the performance for 2-to-4 optical decoders has been obtained in this research.

Rethinking calcium profiles around single channels: the exponential and periodic calcium nanodomains

Many fundamental calcium-dependent physiological processes are triggered by high local calcium levels that are established around the sites of calcium entry into the cell (channels). They are dubbed as calcium nanodomains but their exact profiles are still elusive. The concept of calcium nanodomains stems from a linear model of calcium diffusion and is only valid when calcium increases are smaller than the concentration of cytoplasmic buffers. Recent data indicates that much higher calcium levels cause buffer saturation. Therefore, I sought explicit solutions of a nonlinear reaction-diffusion model and found a dichotomous solution. For small fluxes, the steady state calcium profile is quasi-exponential, and when calcium exceeds buffer concentration a spatial periodicity appears. Analytical results are supported by Monte-Carlo simulations. I also imaged 1D- and radial calcium distributions around single α-synuclein channels in cell-free conditions. Measured Ca profiles are consistent with theoretical predictions. I propose that the periodic calcium patterns may well arise under certain conditions and their specific functional role has to be established.