Integer colorings with forbidden rainbow sums

For a set of positive integers $A \subseteq [n]$, an $r$-coloring of $A$ is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of sum-free sets, which asks for the subsets of $[n]$ with the maximum number of rainbow sum-free $r$-colorings. We show that for $r=3$, the interval $[n]$ is optimal, while for $r\geq8$, the set $[\lfloor n/2 \rfloor, n]$ is optimal. We also prove a stability theorem for $r\geq4$. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.Comment: 20 page

The minimum positive uniform Tur\'an density in uniformly dense kk-uniform hypergraphs

A $k$-graph (or $k$-uniform hypergraph) $H$ is uniformly dense if the edge distribution of $H$ is uniformly dense with respect to every large collection of $k$-vertex cliques induced by sets of $(k-2)$-tuples. Reiher, R\"odl and Schacht [Int. Math. Res. Not., 2018] proposed the study of the uniform Tur\'an density $\pi_{k-2}(F)$ for given $k$-graphs $F$ in uniformly dense $k$-graphs. Meanwhile, they [J. London Math. Soc., 2018] characterized $k$-graphs $F$ satisfying $\pi_{k-2}(F)=0$ and showed that $\pi_{k-2}(\cdot)$ ``jumps" from 0 to at least $k^{-k}$. In particular, they asked whether there exist $3$-graphs $F$ with $\pi_{1}(F)$ equal or arbitrarily close to $1/27$. Recently, Garbe, Kr\'al' and Lamaison [arXiv:2105.09883] constructed some $3$-graphs with $\pi_{1}(F)=1/27$. In this paper, for any $k$-graph $F$, we give a lower bound of $\pi_{k-2}(F)$ based on a probabilistic framework, and provide a general theorem that reduces proving an upper bound on $\pi_{k-2}(F)$ to embedding $F$ in reduced $k$-graphs of the same density using the regularity method for $k$-graphs. By using this result and Ramsey theorem for multicolored hypergraphs, we extend the results of Garbe, Kr\'al' and Lamaison to $k\ge 3$. In other words, we give a sufficient condition for $k$-graphs $F$ satisfying $\pi_{k-2}(F)=k^{-k}$. Additionally, we also construct an infinite family of $k$-graphs with $\pi_{k-2}(F)=k^{-k}$.Comment: 25 page

Existence of Local Saddle Points for a New Augmented Lagrangian Function

We give a new class of augmented Lagrangian functions for nonlinear programming problem with both equality and inequality constraints. The close relationship between local saddle points of this new augmented Lagrangian and local optimal solutions is discussed. In particular, we show that a local saddle point is a local optimal solution and the converse is also true under rather mild conditions

Hypergraphs with a quarter uniform Tur\'an density

The uniform Tur\'an density $\pi_{1}(F)$ of a $3$-uniform hypergraph $F$ is the supremum over all $d$ for which there is an $F$-free hypergraph with the property that every linearly sized subhypergraph with density at least $d$. Determining $\pi_{1}(F)$ for given hypergraphs $F$ was suggested by Erd\H{o}s and S\'os in 1980s. In particular, they raised the questions of determining $\pi_{1}(K_4^{(3)-})$ and $\pi_{1}(K_4^{(3)})$. The former question was solved recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter is still a major open problem. In addition to $K_4^{(3)-}$, there are very few hypergraphs whose uniform Tur\'an density has been determined. In this paper, we give a sufficient condition for $3$-uniform hypergraphs $F$ satisfying $\pi_{1}(F)=1/4$. In particular, currently all known $3$-uniform hypergraphs whose uniform Tur\'an density is $1/4$, such as $K_4^{(3)-}$ and the $3$-uniform hypergraphs $F^{\star}_5$ studied in [arXiv:2211.12747], satisfy this condition. Moreover, we find some intriguing $3$-uniform hypergraphs whose uniform Tur\'an density is also $1/4$.Comment: 23 page

Influence of Helicobacter pylori culture supernatant on the ecological balance of a dual-species oral biofilm

Dental caries is a chronic progressive disease occurring in the tooth hard tissue due to multiple factors, in which bacteria are the initial cause. Both Streptococcus mutans and Streptococcus sanguinis are main members of oral biofilm. Helicobacter pylori may also be detected in dental plaque, playing an important role in the development of dental caries. Objective The aim of this study was to investigate the effect of H. pylori culture supernatant on S. mutans and S. sanguinis dual-species biofilm and to evaluate its potential ability on affecting dental health. Material and methods The effect of H. pylori supernatant on single-species and dual-species biofilm was measured by colony forming units counting and fluorescence in situ hybridization (FISH) assay, respectively. The effect of H. pylori supernatant on S. mutans and S. sanguinis extracellular polysaccharides (EPS) production was measured by both confocal laser scanning microscopy observation and anthrone-sulfuric acid method. The effect of H. pylori supernatant on S. mutans gene expression was measured by quantitative real-time PCR (qRT-PCR) assays. Results H. pylori supernatant could inhibit both S. mutans and S. sanguinis biofilm formation and EPS production. S. sanguinis inhibition rate was significantly higher than that of S. mutans. Finally, S. mutans bacteriocin and acidogenicity related genes expression were affected by H. pylori culture supernatant. Conclusion Our results showed that H. pylori could destroy the balance between S. mutans and S. sanguinis in oral biofilm, creating an advantageous environment for S. mutans, which became the dominant bacteria, promoting the formation and development of dental caries

Teachers’ experiences of developing the teacher-student relationships during COVID-19 in China

The purpose of this study was to express middle school teachers’ experiences of developing teacher-student relationships (TSRs) in online teaching during COVID-19 in China. The study used a qualitative phenomenological design to offer an in-depth description of the experience’s teachers perceive as strategies to develop TSRs. Participants (N=6) were from several middle schools in Nanning, GuangXi, China. Information was collected with semi-structured interviews and analyzed using a five-step phenomenological method in mental research. Data analysis revealed five general themes about teachers’ experiences. The study found that teachers’ proactively initiating dialogues, responding to student needs, emotion-based information exchange, treating students with respect, and expressing motivation and expectations for students were the strategies to develop TSRs. The results of this study can be used in middle school to create positive interventions to enhance educators’ understanding of strategies associated with TSRs

Cloning of a gene encoding glycosyltransferase from Pueraria lobata (Wild.) Ohwi and its expression in Pichia pastoris

The key enzyme of puerarin biosynthesis in Pueraria lobata (Willd.) Ohwi was unclear but may involve glycosylation. To investigate the regulation of puerarin biosynthesis, a putative UDP-dependent glycosyltransferase (UGT) gene, PlUGT1 was isolated from P. lobata root, which contained abundant puerarin. PlUGT1 encoded 480 deduced amino acid residues with a conserved UDP-glucose-binding domain, which has 61 to 84% similarity to homologues from other plant species. SDS polyacrylamide gel electrophoresis and western blotting results showed that, fusion protein migrated as a single protein band with a molecular weight of 55 kDa. A yeast expression vector pPICZA-PlUGT1 was constructed and was transformed into Pichia pastoris strain GS115. Several recombinants containing multi-copy expression cassettes were obtained on the zeocin-YPD plate and confirmed by southern dot blotting. The yield of PlUGT1 attained 0.05 g/l when recombinant cells were cultured at pH 5.5, 30°C and induced with 0.5% methanol for 72 h. The expression of PlUGT1 protein correlates positively with the copy numbers of PlUGT1 in transformed yeast cells. These results suggest that, the PlUGT1 protein can be expressed efficiently in the P. pastoris expression system and may supply a new economic and convenient way for the production of PlUGT1 protein.Keywords: Pueraria lobata (Willd.) Ohwi, glycosyltransferase, cloning, expression, Pichia pastori