Learning about multiplication by comparing algorithms: “One times one,but actually they are ten times ten”

Multiplication algorithms in primary school are still frequently introduced with little attention to meaning. We present a case study focusing on a third grade class that engaged in comparing two algorithms and discussing “why they both work”. The objectives of the didactical intervention were to foster students' development of mathematical meanings concerning multiplication algorithms, and their development of an attitude to judge and compare the value and efficiency of different algorithms. Underlying hypotheses were that it is possible to promote the simultaneous unfolding of the semiotic potential of two algorithms, considered as cultural artifacts, with respect to the objectives of the didactical intervention, and to establish a fruitful synergy between the two algorithms. As results, this study sheds light onto the new theoretical construct of “bridging sign”, illuminating students’ meaning-making processes involving more than one artifact; and it provides important insight into the actual unfolding of the hypothesized potential of the algorithms.     

Nonextensive effects on QCD chiral phase diagram and baryon-number fluctuations within Polyakov-Nambu-Jona-Lasinio model

In this paper, a version of the Polyakov-Nambu-Jona-Lasinio (PNJL) model based on nonextensive statistical mechanics is presented. This new statistics summarizes all possible factors that violate the assumptions of the Boltzmann-Gibbs (BG) statistics to a dimensionless nonextensivity parameter q. Thus, when q tends to 1, it returns to the BG case. Within the nonextensive PNJL model, we found that as q increases, the location of the critical end point (CEP) exhibits non-monotonic behavior. That is, for \begin{document}$ q<1.15 $\end{document}, CEP moves in the direction of lower temperature and larger quark chemical potential. However, for \begin{document}$ q>1.15 $\end{document}, CEP turns to move in the direction of lower temperature and lower quark chemical potential. In addition, we studied the moments of the net-baryon number distribution, that is, variance (\begin{document}$ \sigma^{2} $\end{document}), skewness (S), and kurtosis (κ). Our results are generally consistent with the latest experimental data reported, especially for \begin{document}$ \sqrt{S_{NN}}>19.6\ \mathrm{GeV} $\end{document}, when q is set to \begin{document}$ 1.07 $\end{document}.     

Assessing Earthquake Forecast Performance Based on b Value in Yunnan Province,China

Many studies have shown that b values tend to decrease prior to large earthquakes. To evaluate the forecast information in b value variations, we conduct a systematic assessment in Yunnan Province, China, where the seismicity is intense and moderate–large earthquakes occur frequently. The catalog in the past two decades is divided into four time periods (January 2000–December 2004, January 2005–December 2009, January 2010–December 2014, and January 2015–December 2019). The spatial b values are calculated for each 5-year span and then are used to forecast moderate-large earthquakes (M ≥ 5.0) in the subsequent period. As the fault systems in Yunnan Province are complex, to avoid possible biases in b value computation caused by different faulting regimes when using the grid search, the hierarchical space–time point-process models (HIST-PPM) proposed by Ogata are utilized to estimate spatial b values in this study. The forecast performance is tested by Molchan error diagram (MED) and the efficiency is quantified by probability gain (PG) and probability difference (PD). It is found that moderate–large earthquakes are more likely to occur in low b regions. The MED analysis shows that there is considerable precursory information in spatial b values and the forecast efficiency increases with magnitude in the Yunnan Province. These results suggest that the b value might be useful in middle- and long-term earthquake forecasts in the study area.     

Lattice Boltzmann simulation on thermal performance of composite phase change material based on Voronoi models

Phase change materials(PCMs) are important for sustaining energy development. For the thermal performance enhancement, the composite PCM with metal foam reconstructed by the Voronoi method is investigated in this work. The lattice Boltzmann method(LBM) is used to analyze the melting process on a pore scale. The melting interface evolution and temperature contour of the composite PCM are explored and compared with those of pure PCM. Moreover, structure parameters including the pore density, porosity and irregularity are investigated comprehensively, indicating that the additive of metal foam strengthens the melting performance of PCM obviously. Compared with pure PCM, the composite PCM has quick rates of the melting front evolution and heat transfer. The heat conduction plays a great role in the whole melting process since the convection is weakened for the composite PCM. To improve the melting efficiency, a larger pore density and smaller irregularity are recommended in general. More significantly, a suitable porosity is determined based on the requirement for the balance between the melting rate and heat storage capacity in practical engineering.     

PKP simulation of size effect on interaction field distribution in highly ordered ferromagnetic nanowire arrays

Perpendicular structured nanowire arrays interaction field distributions (IFDs), as revealed from first-order reversal curves (FORC) diagrams, are related to the presence of the demagnetizing field in the system. Despite the similarity between the geometric properties of bit patterned media and mentioned nanowire arrays, FORC diagrams of these two types of systems are not similar essentially due to the different number of magnetic entities influencing the switch of an individual element. We show that one Preisach–Krasnosel'skii–Pokrovskii (PKP) symmetrical hysteron can be representative of an ideal infinite nanowire array when the field is applied along the wires. Starting from this observation, we present a very simple model based on PKP symmetrical hysterons that can be applied to real finite ferromagnetic nanowire arrays, and is able to describe a wide class of experimentally observed FORC distributions, revealing features due to size effects. We also present IFDs modeled for different geometric characteristics such as array size, interwire distance, and nanowire dimensions, and an identification procedure for the proposed model.     

Influence of hydrogen on structural and magnetic properties of the hexagonal Laves phase HoMn2

The HoMn₂ compound crystallizes in the cubic C15 or hexagonal C14 Laves phases depending on preparation. The effect of hydrogen absorption on structural and magnetic properties of HoMn₂H_x hydrides for the C14 phase has been investigated by XRD and AC/DC magnetometry in the temperature ranges of 75-380 K and 4-390 K, respectively. In addition to general features revealed by RMn₂H_x compounds (R=rare earth or Yttrium), unusual behavior of these hydrides was found. In particular, a transformation from the hexagonal to the monoclinic structure was detected, the same as that observed for cubic HoMn₂H_x compounds. The structural transformations are correlated to the magnetic behavior. The presented results are compared mainly with the properties of the cubic HoMn₂H_x hydrides as well as with those of other RMn₂H_x hydrides. Tentative magnetic and structural phase diagrams are proposed.     

Spin waves in the block checkerboard antiferromagnetic phase

Motivated by the discovery of a new family of 122 iron-based superconductors, we present the theoretical results on the ground state phase diagram, spin wave, and dynamic structure factor obtained from the extended J₁-J₂ Heisenberg model. In the reasonable physical parameter region of K₂Fe₄Se₅, we find that the block checkerboard antiferromagnetic order phase is stable. There are two acoustic spin wave branches and six optical spin wave branches in the block checkerboard antiferromagnetic phase, which have analytic expressions at the high-symmetry points. To further compare the experimental data on neutron scattering, we investigate the saddlepoint structure of the magnetic excitation spectrum and the inelastic neutron scattering pattern based on linear spin wave theory.     

Control and synchronization of a hyperchaotic system based on passive control

In this paper, a new hyperchaotic system is proposed, and the basic properties of this system are analyzed by means of the equilibrium point, a Poincar map, the bifurcation diagram, and the Lyapunov exponents. Based on the passivity theory, the controllers are designed to achieve the new hyperchaotic system globally, asymptotically stabilized at the equilibrium point, and also realize the synchronization between the two hyperchaotic systems under different initial values respectively. Finally, the numerical simulation results show that the proposed control and synchronization schemes are effective.     

Shape analysis of cubic trigonometric Bézier curves with a shape parameter

For the cubic trigonometric polynomial curves with a shape parameter (TB curves, for short), the effects of the shape parameter on the TB curve are made clear, the shape features of the TB curve are analyzed. The necessary and sufficient conditions are derived for these curves having single or double inflection points, a loop or a cusp, or be locally or globally convex. The results are summarized in a shape diagram of TB curves, which is useful when using TB curves for curve and surface modeling. Furthermore the influences of shape parameter on the shape diagram and the ability for adjusting the shape of the curve are shown by graph examples, respectively.