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

TRIM52 promotes proliferation, invasion, and migration of gastric cancer cells by regulating Wnt/β-catenin pathway

Purpose: This study aimed to reveal the role and mechanism of tripartite motif-containing 52 (TRIM52) in gastric cancer (GC) progression.Methods: The Cancer Genome Atlas (TCGA) database was utilized to analyze TRIM52 expression in GC samples and para-carcinoma tissue samples, and the results were confirmed by quantitative realtime polymerase chain reaction. Cell counting kit-8 and colony formation assays were used to evaluate cell viability. Wound healing assay was utilized to analyze cell migration, while Transwell assay was utilized to evaluate cell invasion. TRIM52, proliferating cell nuclear antigen, matrix metalloproteinase-2, Wnt5a, β-catenin, and c-Myc protein levels were measured by western blot.Results: TRIM52 was expressed more in GC tissue samples and cells compared to normal tissues and cells (p < 0.001). Overexpression of TRIM52 promoted growth, migration, and invasion of HGC-27 cells, and silencing inhibited growth, migration, and invasion of HGC-27 cells (p < 0.001). In addition, TRIM52 overexpression increased Wnt5a, β-catenin, and c-Myc protein expression, and silencing decreased Wnt5a, β-catenin, and c-Myc protein expression (p < 0.001 or p < 0.01), indicating that TRIM52 activates Wnt/β-catenin signaling pathway.Conclusion: These findings reveal that TRIM52 facilitates GC cell proliferation, migration and invasion, but activates Wnt/β-catenin signaling

Spillover of social responsibility associations in a brand portfolio

Extant research has established that social responsibility (SR) activity can be beneficial to companies by influencing consumers’ SR associations with the company and its product brands. However, most studies only look at the outcomes of SR initiatives implemented at the corporate level (i.e., corporate social responsibility). This research provides a new and expanded perspective by exploring how SR activity at the product brand level reverberates throughout the full brand portfolio. Based on associative network theory, it is proposed that when consumers are aware of a product brand’s SR initiatives, their social responsibility associations with this product brand will spill over to another product brand and the corporate brand through the pre-existing links between brands. This spillover effect is presumed to be stronger for companies using the monolithic rather than endorsed or stand-alone branding strategies. The spillover effect between two product brands is also expected to be influenced by their product category fit. As a result of this spillover of SR associations, consumers are expected to show a heightened purchase intent, positive word of mouth intent, and willingness to pay a higher price for other product brands offered by the company; and they are more likely to identify with the company and talk positively about it to others. Three experiments were conducted and found support for the proposed conceptual model. The findings will help managers make better decisions about which brands (product and corporate level) should be involved in SR activity.Ph.D., Marketing -- Drexel University, 201

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

Learning the hub graphical Lasso model with the structured sparsity via an efficient algorithm

Graphical models have exhibited their performance in numerous tasks ranging from biological analysis to recommender systems. However, graphical models with hub nodes are computationally difficult to fit, particularly when the dimension of the data is large. To efficiently estimate the hub graphical models, we introduce a two-phase algorithm. The proposed algorithm first generates a good initial point via a dual alternating direction method of multipliers (ADMM), and then warm starts a semismooth Newton (SSN) based augmented Lagrangian method (ALM) to compute a solution that is accurate enough for practical tasks. The sparsity structure of the generalized Jacobian ensures that the algorithm can obtain a nice solution very efficiently. Comprehensive experiments on both synthetic data and real data show that it obviously outperforms the existing state-of-the-art algorithms. In particular, in some high dimensional tasks, it can save more than 70\% of the execution time, meanwhile still achieves a high-quality estimation.Comment: 28 pages,3 figure

A theoretical framework of immune cell phenotypic classification and discovery

Immune cells are highly heterogeneous and show diverse phenotypes, but the underlying mechanism remains to be elucidated. In this study, we proposed a theoretical framework for immune cell phenotypic classification based on gene plasticity, which herein refers to expressional change or variability in response to conditions. The system contains two core points. One is that the functional subsets of immune cells can be further divided into subdivisions based on their highly plastic genes, and the other is that loss of phenotype accompanies gain of phenotype during phenotypic conversion. The first point suggests phenotypic stratification or layerability according to gene plasticity, while the second point reveals expressional compatibility and mutual exclusion during the change in gene plasticity states. Abundant transcriptome data analysis in this study from both microarray and RNA sequencing in human CD4 and CD8 single-positive T cells, B cells, natural killer cells and monocytes supports the logical rationality and generality, as well as expansibility, across immune cells. A collection of thousands of known immunophenotypes reported in the literature further supports that highly plastic genes play an important role in maintaining immune cell phenotypes and reveals that the current classification model is compatible with the traditionally defined functional subsets. The system provides a new perspective to understand the characteristics of dynamic, diversified immune cell phenotypes and intrinsic regulation in the immune system. Moreover, the current substantial results based on plasticitomics analysis of bulk and single-cell sequencing data provide a useful resource for big-data–driven experimental studies and knowledge discoveries