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In this article we consider the solutions of the heat equations with use of the elliptic curves over the field of rational numbers. We propose the entire functions associated with the Hasse-Weil L-function. We show the conjectures that new functions have only real zeros in the entire function plane. The obtained results are proposed as new tool to describe the complex behaviors of the heat problems as well as number theory.

In this article we consider an even entire function of order one as the solution of the classical wave equation in one-dimensional space. We suggest a conjecture that this function has only purely real zeros in the entire complex plane. This is given as a new prospective on a nice connection with number theory and wave equation.

In this article we suggest the entire functions associated with the L-functions at?tached to cusp forms. The entire function defined by the Fourier cosine transform is the solution for the diffusion equation in 1-D case. We propose three conjectures for the zeros of three entire functions of order one via theory of entire functions.

This article addresses a novel anomalous relaxation model with the new general fractional derivative of the Sonine kernel. This operator is considered in the sense of general fractional derivative proposed in the work [Yang et al., General fractional derivatives with applications in viscoelasticity, Academic Press, New York, USA, 2020]. The solution of the mathematical model is obtained with the aid of Laplace transform. The comparison among the classical and anomalous relaxation models is discussed in detail. This result is proposed as a mathematical tool to model the anomalous relaxation behavior of the complex materials.

This article addresses a new odd entire function of order one structured by the Fourier sine integral, which is the solution of the one-dimensional diffusion equation in theory of modular form.

In this article we suggest a new odd entire function of order one. We discover that it is the solution of the heat equation. We propose a new conjecture that this function has only real zeros. The obtained result gives a connection with the heat problem and number theory.

Purpose To investigate the prognostic value of NDRG1 and the relationship between NDRG1 and immunoinfiltration. Meanwhile, the expression of NDRG1 in gastric cancer was investigated.Methods A comprehensive analysis of the relationship between NDRG1 gene and prognosis of various cancers, tumor mutational burden (TMB), microsatellite instability (MSI) and immune cell infiltration was conducted by using a range of bioinformatics methods on the basis of existing public databases as a mean to investigate the potential prognostic value of NDRG1 in pan-carcinoma. At the same time, 40 gastric cancer tissues and paracancer tissue specimens were collected. The expression of NDRG1 in gastric cancer tissues and adjacent tissues were verified by experiments. R software and GraphPad Prism 8 software were used for statistical analysis and mapping.Results NGRG1 was shown to be weakly expressed in BRCA, STAD, THCA, while it was shown to be highly expressed in CHOL, HNSC, KIRC, LIHC and LUSC. The survival rate of patients with high expression of NDRG1 in BRCA, LIHC and LUAD was lower than that of patients with low expression of NDRG1. However, patients with high expression of NDRG1 in KIRC had better survival than those with low expression of NDRG1. The expression of NDPG1 was positively correlated with TMB in CESC, BRCA, ACC, UCEC, SARC, PAAD and KIRC, but negatively correlated with TMB in THCA, SKCM, PRAD, PCPG and LGG. The expression of NDRG1 was positively correlated with MSI in COAD, BRCA, UCEC, SARC, PAAD and LUSC, and negatively correlated with MSI in PRAD. In addition, the results of the study indicated that NDRG1 was low expressed in gastric cancer tissues, high expressed in paracancer tissues and mainly distributed in cytoplasm (P = 0.0002). In terms of the degree of differentiation, the expression level of NDRG1 protein in moderately differentiated and highly differentiated gastric cancer tissues was higher than that in undifferentiated and poorly differentiated groups (P = 0.038). In terms of tumor invasion, NDRG1 expression in T1-T2 group was significantly higher than that in T3-T4 group (P = 0.025). In clinicopathologic stage, the expression of NDRG1 in stage I-II was significantly higher than that in stage III-IV (P = 0.024).Conclusion NDRG1 can be used as a prognostic marker in multiple cancers. NDRG1 was low expressed in gastric cancer tissues and high expressed in paracancer tissues. High expression of NDRG1 was associated with favorable pathological features of gastric cancer, which may be involved in inhibiting the occurrence and development of gastric cancer.

Supplementary Tables 1-8 PDF file 178K, Supplementary Table 1. The expression of IGF-1R, �-catenin and POU5F1 in 200 cases of lung adenocarcinoma; Supplementary Table 2. Top 10 gene ontology classification of genes related to the biological process of LACSLCs; Supplementary Table 3. Top 10 gene ontology classification of genes related to the cellular components of LACSLCs; Supplementary Table 4. Top 10 gene ontology classification of genes involved in the molecular functions of LACSLCs; Supplementary Table 5. Expression profile of up-regulated genes identified by cDNA array and qRT-PCR in A549 LACSLCs; Supplementary Table 6. Expression profile of down-regulated genes identified by cDNA array and qRT-PCR in A549 LACSLCs; Supplemental Table 7. Antibodies used for immunostaining analysis; Supplemental Table 8. Sequences of the primers used for qRT-PCR

Cancer stem-like cells (CSLC) are crucial in tumor initiation and progression; however, the underlying mechanism for the self-renewal of cancer cells remains undefined. In the study, immunohistochemical analysis of specimens freshly excised from patients with lung adenocarcinoma showed that high expression of insulin-like growth factor I receptor (IGF-IR) in lung adenocarcinoma cells was positively correlated with the expressions of cancer stem cell markers CD133 and aldehyde dehydrogenase 1 family member A1 (ALDH1A1). IGF-IR activation enhanced POU class 5 homeobox 1 (POU5F1) expression on human lung adenocarcinoma stem-like cells (LACSLC) through PI3K/AKT/GSK3β/β-catenin cascade. POU5F1 could form a novel complex with β-catenin and SOX2 to bind Nanog promoter for transcription to maintain self-renewal of LACSLCs, which was dependent on the functional IGF-IR. Genetic and pharmacologic inhibition of IGF-IR abrogated LACSLC capabilities for self-renewal and tumorigenicity in vitro. In an in vivo xenograft tumor model, knockdown of either IGF-IR or POU5F1 impeded tumorigenic potentials of LACSLCs. By analyzing pathologic specimens excised from 200 patients with lung adenocarcinoma, we found that colocalization of highly expressed IGF-IR with β-catenin and POU5F1 predicted poor prognosis. Taken together, we show that IGF-IR—mediated POU5F1 expression to form a complex with β-catenin and SOX2 is crucial for the self-renewal and oncogenic potentials of LACSLCs, and the integrative clinical detection of the expressions of IGF-IR, β-catenin, and POU5F1 is indicatory for predicting prognosis in the patients of lung adenocarcinoma. Cancer Res; 73(10); 3181–9. ©2013 AACR.

Supplementary Table 9 PDF file 167K, Supplementary Table 9. Clinicopathological features and IHC staining index of ALDH1A1, IGF-1R, �-catenin and POU5F1 in 200 patients with LAC

Supplementary Figures 1-8 PDF file 1850K, Supplementary Figure 1. LACSLCs derived from A549 and H358 cell lines; Supplementary Figure 2. Sphere LACSLCs derived from A549 cell line expressed high levels of CSLCs transcription factors including SOX2 and POU5F1 with low expression of differentiated markers CK8/CK18;Supplementary Figure 3. Immunofluorescence staining of CD133 and ALDH1A1;Supplementary Figure 4. Correlation of IGF-1R and ALDH1A1 with poor DSS in LAC patients;Supplementary Figure 5. Insulin signaling/IGF pathway is highly correlated with expression genes in LACSLCs;Supplementary Figure 6. Disruption of IGF-1R signaling pathway inhibits the self-renewal and the tumorigenic potential of human LACSLCs;Supplementary Figure 7. MEK inhibitor PD98059 has no effect on POU5F1; Expression of IGF-1, �-catenin and POU5F1 in fresh LAC specimens and A549 LACSLCs

Hepatocellular carcinoma (HCC) is one of the most common cancers worldwide. Its invasiveness and ability to metastasize contributes to an extremely high patient mortality. However, the molecular mechanisms that underlie the characteristics of HCC progression are not well understood. The kinesin superfamily 21 B (KIF21B) has been shown to be involved in hepatocellular carcinoma (HCC). Nevertheless, its role in HCC has not yet been thoroughly examined. Tumor Genome Atlas (TCGA) was used to analyze the difference of KIF21B expression between HCC and normal tissues and its relationship with clinicopathological features. For further validation, immunohistochemistry (IHC) was used to detect the expression of KIF21B in 116 cases of liver cancer. In vitro studies, we used loss-of-function assays to evaluate the effects of KIF21B and its direct target NF-κB(nuclear factor-κB) p65 subunit on cell growth, proliferation, apoptosis, migration, invasion, and epithelial-mesenchymal transition (EMT).Then the effect of KIF21B inhibition on tumor growth was studied in nude mice in vivo. We demonstrated that elevated KIF21B expression is associated with adverse clinicopathological features, such as tumor size, in HCC patients. In vitro experiments showed that silencing KIF21B could inhibit the proliferation, migration, invasion and EMT of HCC cells, and induce apoptosis, while overexpression of p65 could reverse these effects. In addition, silencing of KIF21B inhibited the growth of HCC in xenografts in nude mice. In this study, we identified and validated that KIF21B plays a carcinogenic role in HCC through NF-κB/p65 signaling pathway, providing a new insight into HCC pathogenesis and therapeutic possibilities.

In this paper, we investigate some new inequalities based on the Riesz-type fractional integral operator for synchronous and bounded functions.

A modified aluminum phosphate sealing agent was prepared by using aluminum dihydrogen phosphate and silica sol as raw materials, and was used for sealing treatment of iron-based amorphous coating. The phase of sealing agent was analyzed by XRD. SEM and TG-DSC were used to characterize the surface morphology of the coating before and after sealing and the heat resistance of the sealing agent. The corrosion resistance of the sealing coating was evaluated by electrochemical measurements. The results show that the modified aluminum phosphate sealing agent has good heat resistance, and fine and close sealing layer was obtained, thus the corrosion resistance is significantly improved.

In this article, we address a new model for the scaling law heat conduction problem by using the scaling law vector calculus associated with the Korcak scaling law. The scaling law heat conduction equations are discussed in detail. The scaling law vector calculus formulas are proposed as an efficiently mathematical tool to describe the Korcak scaling -law phenomena in heat transport system.