Some sum-product estimates in matrix rings over finite fields

We study some sum-product problems over matrix rings. Firstly, for $A, B, C\subseteq M_n(\mathbb{F}_q)$, we have $$|A+BC|\gtrsim q^{n^2},$$ whenever $|A||B||C|\gtrsim q^{3n^2-\frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(\mathbb{F}_q)$ satisfies $|A|\geq C(n)q^{n^2-1}$ for some sufficiently large $C(n)$, then we have $$\max\{|A+A|, |AA|\}\gtrsim \min\left\{\frac{|A|^2}{q^{n^2-\frac{n+1}{4}}}, q^{n^2/3}|A|^{2/3}\right\}.$$ These improve the results due to The and Vinh (2020), and generalize the results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a recent result due to The and Vinh (2020). Our method is based on spectral graph theory and linear algebra.Comment: 18 page

On the lower bound for kissing numbers of ℓp\ell_p-spheres in high dimensions

In this paper, we give some new lower bounds for the kissing number of $\ell_p$-spheres. These results improve the previous work due to Xu (2007). Our method is based on coding theory.Comment: 15 pages, 4 figures; any comments are welcom

Covering Grassmannian Codes: Bounds and Constructions

Grassmannian $\mathcal{G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n.$ Recently, Etzion and Zhang introduced a new notion called covering Grassmannian code which can be used in network coding solutions for generalized combination networks. An $\alpha$-$(n,k,\delta)_q^c$ covering Grassmannian code $\mathcal{C}$ is a subset of $\mathcal{G}_q(n,k)$ such that every set of $\alpha$ codewords of $\mathcal{C}$ spans a subspace of dimension at least $\delta +k$ in $\mathbb{F}_q^n.$ In this paper, we derive new upper and lower bounds on the size of covering Grassmannian codes. These bounds improve and extend the parameter range of known bounds.Comment: 17 page

Abnormal Liver Function Tests Were Associated With Adverse Clinical Outcomes: An Observational Cohort Study of 2,912 Patients With COVID-19

Background and Aim: The impact of liver function test (LFTs) abnormality on adverse clinical outcomes in coronavirus disease 2019 (COVID-19) patients remains controversial. The aim of this study was to assess the impact of abnormal LFTs on clinical outcomes in a large cohort of hospitalized patients with COVID-19.Methods: We retrospectively collected data on 2,912 consecutive patients with COVID-19 who were admitted to a makeshift hospital in China between 5 February and 23 March 2020. The association between LFTs abnormalities (baseline and peak values) and clinical outcomes was measured by using Cox regression models.Results: On admission 1,414 patients (48.6%) had abnormal LFTs, with alanine aminotransferase (ALT), aspartate aminotransferase (AST), total bilirubin (TBIL), alkaline phosphatase (ALP), and gamma-glutamyltransferase (GGT) elevation in 662 (22.7%), 221 (7.6%), 52 (1.8%), 135 (4.6%), and 536 (18.5%) patients, respectively, and hypoalbuminemia in 737 (25.3%) patients. During a median 13 (IQR: 8–19) days of hospitalization, 61 patients (2.1%) died, 106 patients (3.6%) admitted to intensive care unit (ICU), and 75 patients (2.6%) required mechanical ventilation. After adjustment for confounders, baseline abnormal LFTs were independently associated with increased risks of mortality (adjusted HR 3.66, 95%CI 1.64–8.19, p = 0.002), ICU admission (adjusted HR 3.12 95%CI 1.86–5.23, p < 0.001), and mechanical ventilation (adjusted HR 3.00, 95%CI 1.63–5.52, p < 0.001), which was homogeneous across the severity of COVID-19 infection. Among the parameters of LTFs, the associations with the outcomes were more pronounced for AST and albumin abnormality. In contrast, ALT elevation was not significantly associated with those outcomes. Similar results were observed for peak values of LFTs during hospitalization.Conclusions: Abnormality of AST, albumin, TBIL, ALP, and GGT but not ALT were independently associated with adverse outcomes

Effects of inclusion level and adaptation period on nutrient digestibility and digestible energy of wheat bran in growing-finishing pigs

Objective This experiment was to evaluate the effect of different inclusion levels and adaptation periods on digestible energy (DE) and the apparent total tract digestibility (ATTD) of chemical constituents in diets supplemented with wheat bran. Methods Thirty-six crossbred barrows with an initial body weight of 85.0±2.1 kg were allotted to 6 diets in a completely randomized block design with 6 pigs per diet. Diets included a corn-soybean basal diet and 5 additional diets which were formulated by replacing corn and soybean meal in control diet with 15%, 25%, 35%, 45%, or 55% wheat bran. The experiment lasted for 34 d, and feces were collected from d 8 to 13, 15 to 20, 22 to 27, and 29 to 34 respectively. Results The results showed no interaction effects between inclusion level and adaptation period on the concentration of DE and the ATTD of gross energy (GE) and crude protein (CP) in wheat bran. The DE value and ATTD of GE in wheat bran decreased (p<0.05) significantly as the inclusion level of wheat bran increased, but no difference in the ATTD of CP was observed. The ATTD of CP in wheat bran increased (p<0.10) significantly as the adaptation period for pigs was prolonged. In addition, the concentration of DE and the ATTD of GE in wheat bran decreased linearly (p<0.05) when pigs were fed either an increased level of wheat bran or given an increased adaptation period to the diets. Conclusion Wheat bran showed a negative effect on the concentration of DE and ATTD of GE and CP as the inclusion level increased. A longer adaptation period can gradually increase the DE value and ATTD of GE and CP in wheat bran, and at least 14 to 21 d of adaptation might be recommended for growing-finishing pigs fed the high-fiber diets with wheat bran

HDAC2- and EZH2-Mediated Histone Modifications Induce PDK1 Expression through miR-148a Downregulation in Breast Cancer Progression and Adriamycin Resistance

Background: Breast cancer has one of highest morbidity and mortality rates for women. Abnormalities regarding epigenetics modification and pyruvate dehydrogenase kinase 1 (PDK1)-induced unusual metabolism contribute to breast cancer progression and chemotherapy resistance. However, the role and mechanism of epigenetic change in regulating PDK1 in breast cancer remains to be elucidated. Methods: Gene set enrichment analysis (GSEA) and Pearson&rsquo;s correlation analysis were performed to analyze the relationship between histone deacetylase 2 (HDAC2), enhancer of zeste homologue 2 (EZH2), and PDK1 in database and human breast cancer tissues. Dual luciferase reporters were used to test the regulation between PDK1 and miR-148a. HDAC2 and EZH2 were found to regulate miR-148a expression through Western blotting assays, qRT-PCR and co-immunoprecipitation assays. The effects of PDK1 and miR-148a in breast cancer were investigated by immunofluorescence (IF) assay, Transwell assay and flow cytometry assay. The roles of miR-148a/PDK1 in tumor growth were investigated in vivo. Results: We found that PDK1 expression was upregulated by epigenetic alterations mediated by HDAC2 and EZH2. At the post-transcriptional level, PDK1 was a new direct target of miR-148a and was upregulated in breast cancer cells due to miR-148a suppression. PDK1 overexpression partly reversed the biological function of miR-148a&mdash;including miR-148a&rsquo;s ability to increase cell sensitivity to Adriamycin (ADR) treatment&mdash;inhibiting cell glycolysis, invasion and epithelial&ndash;mesenchymal transition (EMT), and inducing apoptosis and repressing tumor growth. Furthermore, we identified a novel mechanism: DNMT1 directly bound to EZH2 and recruited EZH2 and HDAC2 complexes to the promoter region of miR-148a, leading to miR-148a downregulation. In breast cancer tissues, HDAC2 and EZH2 protein expression levels also were inversely correlated with levels of miR-148a expression. Conclusion: Our study found a new regulatory mechanism in which EZH2 and HDAC2 mediate PDK1 upregulation by silencing miR-148a expression to regulate cancer development and Adriamycin resistance. These new findings suggest that the HDAC2/EZH2/miR-148a/PDK1 axis is a novel mechanism for regulating cancer development and is a potentially promising target for therapeutic options in the future