Abstraction and embodiment: exploring the process of grasping a general

This paper reports from a case study which explores kindergarten children’s mathematical abstraction in a teaching–learning activity about reflection symmetry. From a dialectical perspective, abstraction is here conceived as a process, as a genuine part of human activity, where the learner establishes “a point of view from which the concrete can be seen as meaningfully related” (van Oers & Poland Mathematics Education Research Journal, 19(2), 10–22, 2007, p. 13–14). A cultural-historical semiotic perspective to embodiment is used to explore the characteristics of kindergarten children’s mathematical abstraction. In the selected segment, two 5-year-old boys explore the concept of reflection symmetry using a doll pram. In the activity, the two boys first point to concrete features of the sensory manifold, then one of the boys’ awareness gradually moves to the imagined and finally to grasping a general and establishing a new point of view. The findings illustrate the essential role of gestures, bodily actions, and rhythm, in conjunction with spoken words, in the two boys’ gradual process of grasping a general. The study advances our knowledge about the nature of mathematical abstraction and challenges the traditional view on abstraction as a sort of decontextualised higher order thinking. This study argues that abstraction is not a matter of going from the concrete to the abstract, rather it is an emergent and context-bound process, as a genuine part of children’s concrete embodied activities.

Peculiarity of Symmetric Ring Systems with Double Y-Junctions and the magnetic effects

We discuss quantum dynamics in the ring systems with double Y-junctions in which two arms have same length. The node of a Y-junction can be parametrized by U(3). Considering mathematically permitted junction conditions seriously, we formulate such systems by scattering matrices. We show that the symmetric ring systems, which consist of two nodes with the same parameters under the reflection symmetry, have remarkable aspects that there exist localized states inevitably, and resonant perfect transmission occurs when the wavenumber of an incoming wave coincides with that of the localized states, for any parameters of the nodes except for the extremal cases in which the absolute values of components of scattering matrices take 1. We also investigate the magnetic disturbance to the symmetric ring systems.

Renormalization group evolution induced leptogenesis in the minimal seesaw model with the trimaximal mixing and mu-tau reflection symmetry

In this paper, we consider the imbedding of the popular and well-motivated trimaximal mixing and μ–τ reflection symmetry (which can help us shape the forms of the neutrino mass matrix) in the minimal seesaw model (which contains much fewer parameters than the general seesaw model) with two TeV-scale right-handed neutrinos (for realizing a low-scale seesaw) of nearly degenerate masses (for realizing a resonant leptogenesis). However, either for the trimaximal mixing scenario (which is realized through the Form Dominance approach here) or for the μ–τ reflection symmetry scenario, leptogenesis cannot proceed. To address this issue, we consider the possibility that the special forms of the neutrino mass matrix for the trimaximal mixing and μ–τ reflection symmetry are slightly broken by the renormalization group evolution effect, thus allowing leptogenesis to proceed. It is found that in the normal case of the neutrino mass ordering, the baryon asymmetry thus generated can reproduce the observed value. For completeness, we have also extended our analysis to the scenario that two right-handed neutrinos are not nearly degenerate any more. Unfortunately, in this scenario the final baryon asymmetry is smaller than the observed value by several orders of magnitude.

Symmetry Detection and Analysis of Chinese Paifang Using 3D Point Clouds

The Chinese paifang is an essential constituent element for Chinese or many other oriental architectures. In this paper, a new method for detection and analysis of the reflection symmetry of the paifang based on 3D point clouds is proposed. The method invokes a new model to simultaneously fit two vertical planes of symmetry to the 3D point cloud of a paifang to support further symmetry analysis. Several simulated datasets were used to verify the proposed method. The results indicated that the proposed method was able to quantity the symmetry of a paifang in terms of the RMSE obtained from the ICP algorithm, with resistance to the presence of some random noise added to the simulated measurements. For real datasets, three old Chinese paifangs (with ages from 90 to 500 years) were scanned as point clouds to input into the proposed method. The method quantified the degree of symmetry for the three Chinese paifangs in terms of the RMSE, which ranged from 20 to 61 mm. One of the paifangs with apparent asymmetry had the highest RMSE (61 mm). Other than the quantification of the symmetry of the paifangs, the proposed method could also locate which portion of the paifang was relatively more symmetric. The proposed method can potentially be used for structural health inspection and cultural studies of the Chinese paifangs and some other similar architecture.

Radiative neutrino masses, lepton flavor mixing and muon g − 2 in a leptoquark model

We propose a leptoquark model with two scalar leptoquarks $$ {S}_1\left(\overline{3},1,\frac{1}{3}\right) $$ S 1 3 ¯ 1 1 3 and $$ {\tilde{R}}_2\left(3,2,\frac{1}{6}\right) $$ R ˜ 2 3 2 1 6 to give a combined explanation of neutrino masses, lepton flavor mixing and the anomaly of muon g − 2, satisfying the constraints from the radiative decays of charged leptons. The neutrino masses are generated via one-loop corrections resulting from a mixing between S1 and $$ {\tilde{R}}_2 $$ R ˜ 2 . With a set of specific textures for the leptoquark Yukawa coupling matrices, the neutrino mass matrix possesses an approximate μ-τ reflection symmetry with (Mν)ee = 0 only in favor of the normal neutrino mass ordering. We show that this model can successfully explain the anomaly of muon g − 2 and current experimental neutrino oscillation data under the constraints from the radiative decays of charged leptons.

Turbulence of Capillary Waves on Shallow Water

We consider the developed turbulence of capillary waves on shallow water. Analytic theory shows that an isotropic cascade spectrum is unstable with respect to small angular perturbations, in particular, to spontaneous breakdown of the reflection symmetry and generation of nonzero momentum. By computer modeling we show that indeed a random pumping, generating on average zero momentum, produces turbulence with a nonzero total momentum. A strongly anisotropic large-scale pumping produces turbulence whose degree of anisotropy decreases along a cascade. It tends to saturation in the inertial interval and then further decreases in the dissipation interval. Surprisingly, neither the direction of the total momentum nor the direction of the compensated spectrum anisotropy is locked by our square box preferred directions (side or diagonal) but fluctuate.