Representations of the Fermion-Virasoro algebras

Let $\delta=0$ or $\frac{1}{2}$. In this paper, we introduce the Fermion algebra $F(\delta)$ and the Fermion-Virasoro algebra $\mathcal S(\delta)$. They are infinite-dimensional Lie superalgebras. All simple smooth $F(\delta)$-modules, all simple weight $F(\delta)$-modules, all simple smooth $\mathcal S(\delta)$-modules of nonzero level, and all simple Harish-Chandra $\mathcal S(\delta)$-modules are determined. Surprisingly, we have found four different $\mathcal S(\delta)$-module structures on free $\mathbb C[L_0]$-modules of rank $2$

Tensor modules over Witt superalgebras

In this paper, we study the tensor module $P\otimes M$ over the Witt superalgebra $W_{m,n}^+$ (resp. $W_{m,n}$), where $P$ is a simple module over the Weyl superalgebra $K_{m,n}^+$ (resp. $K_{m,n}$) and $M$ is simple weight module over the general linear Lie superalgebra $\mathfrak{gl}(m,n)$. We obtain the necessary and sufficient conditions for $P\otimes M$ to be simple, and determine all simple subquotient of $P\otimes M$ when it is not simple. All the work leads to completion of some classification problems on the weight representation theory of $W_{m,n}^+$ and $W_{m,n}$.Comment: The paper has been accepted for publication in SCIENCE CHINA Mathematic

Influence of Intergenerational Parenting on Gross Motor Skills Among Children Aged 3-6 Years Old

The stage of 3-6 years old is a critical period for the development of children\u27s gross motor skills, influencing their growth and development. Intergenerational parenting is a common way of family education in China. Studies have shown that the rough intergenerational parenting concept and limited energy of the elderly reduce the quality of life of the children under care (Qimeng Jiang, Nan Zhou, 2020). However, no research focused on the influence of intergenerational parenting on gross motor skills. This study aimed to explore the influence of different intergenerational parenting style on gross motor skills among children aged 3-6 years old. The participants were 62 children (25 boys and 37 girls) aged 3-6 years old from Liaoning Province. All of the participants were under intergenerational parenting. Gross motor skills were assessed using the Test of Gross Motor Development-Third Edition (TGMD-3), which includes locomotor skills and ball skills components. The intergenerational parenting status was divided into parent-dominated intergenerational parenting and grandparent-dominated intergenerational parenting according to a questionnaire (Lu Ye, 2020). The parenting style included authoritative, authoritarian and tolerant styles. The scores of three styles were determined by the grandparents-reported Chinese version of Parental Authority Questionnaire. Descriptive statistics, independent t-test and Pearson correlation were employed and the significant levels were set at 0.05. The results showed that participants had lower mean scores in both locomotor skills (Mboy = 12.58±4.42, Mgirl =13.51 ±3.03) and ball skills (Mboy = 8.46±2.9, Mgirl = 8.04±3.34) compared to the Chinese norm. There was no significant difference between parent-dominated and grandparent-dominated intergenerational parenting (M Parent-dominated =22.90±4.15, M grandparent-dominated = 20.48±4.47; t = 1.269, p = 0.209). Correlation analysis indicated a small association between the score of locomotor skills and authoritative style (r = 0.269, p \u3c 0.05). No significant relationship was found between other parenting styles and the scores of TGMD-3. It is concluded that intergenerational parenting may negatively influence children’s gross motor development. Parent-dominated and grandparent-dominated intergenerational parenting may not have differences in children’s motor development. The authoritative parenting style of intergenerational education has certain impacts on children\u27s gross motor skills, especially on children\u27s locomotor skills

Threshold Characteristics of Slow-Light Photonic Crystal Lasers

The threshold properties of photonic crystal quantum dot lasers operating in the slow-light regime are investigated experimentally and theoretically. Measurements show that, in contrast to conventional lasers, the threshold gain attains a minimum value for a specific cavity length. The experimental results are explained by an analytical theory for the laser threshold that takes into account the effects of slow-light and random disorder due to unavoidable fabrication imperfections. Longer lasers are found to operate deeper into the slow-light region, leading to a trade-off between slow-light induced reduction of the mirror loss and slow-light enhancement of disorder-induced losses.Comment: 5 pages, 7 figure