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  • Author or Editor: Bo He x
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Abstract  

Let A and k be positive integers. In this paper, we study the Diophantine quadruples

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\{ k,A^2 k + 2A,(A + 1)^2 k + 2(A + 1)d\} .$$ \end{document}
If d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\begin{gathered} d = (4A^4 + 8A^3 + 4A^2 )k^3 + (16A^3 + 24A^2 + 8A)k^2 + \hfill \\ + (20A^2 + 20A + 4)k + (8A + 4) \hfill \\ \end{gathered}$$ \end{document}
for A ≥ 52330 and any k. This extends our result obtained in [4].

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Abstract  

We prove that if k is a positive integer and d is a positive integer such that the product of any two distinct elements of the set {k + 1, 4k, 9k + 3, d} increased by 1 is a perfect square, then d = 144k 3 + 192k 2 + 76k + 8.

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Abstract

24p3 (also called SIP 24, uterocalin, neutrophil-gelatinase-associated lipocalin (NGAL), lipocalin 2, and siderocalin) is an acute phase protein induced under many inflammatory situations which regulates iron delivery and may play an important role in innate immunity. In this study, we examined whether 24p3 is induced in the course of Concanavalin- A (Con-A)-induced autoimmune hepatitis. Upon intravenous injection of Con-A into mice, 24p3 mRNA expression in the liver as detected by real-time quantitative RT-PCR was significantly increased by 20–28-fold. Lung and lymphoid tissues such as thymus, lymph nodes, and spleen were also found to maintain high levels of 24p3 mRNA. No induction of 24p3 mRNA was found in other tissues, including muscle, heart, brain, and kidney. Consistent with the PCR results, 24p3 protein levels detected by Western blot were increased in a time-dependent manner after Con-A injection. Our results showed that 24p3 expression is induced in Con-A-mediated acute liver failure at both transcriptional and translational levels. 24p3 might play a role in the feedback control to limit inflammation in the liver.

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Abstract

Splenic lymphocytes play an important role in host acute or chronic diseases. The abnormality of these cells in the spleens of humans might lead to some riskful diseases for human. Hence, in this study, the effects of two ginsenosides Rg1 and Rb1 on splenic lymphocytes growth were studied by microcalorimetry. Some qualitative and quantitative information, such as the metabolic power-time curves, growth rate constant k, maximum heat-output power of the exponential phase P max, total heat output Q t of splenic lymphocytes were obtained to present the effects of Rg1 and Rb1 on these cells. The values of k, P max, and Q t from the thermogenic growth curves of splenic lymphocytes were found to increase in the presence of Rg1, while the change was adverse for Rb1, illustrating that Rg1 had promotion effect and Rb1 had inhibitory effect on splenic lymphocytes growth and these promotion or inhibitory effects were enhanced with increasing the concentration of the two compounds, respectively. The microcalorimetric results were confirmed by MTT assay for determining the MTT optical density (OD) value and [3H] Thymidine incorporation assay ([3H]-TdR) for determining the count per minute (cpm) value: Rg1 could increase the MTT OD value and the cpm value of [3H]-TdR incorporation into splenic lymphocytes, and these values were increased with increasing the concentration of this compound, while Rb1 had the adverse results. The structure–activity relationships showed that the glucopyranoside and hydroxyl groups at the dammarane-type mother nucleus skeleton might play a crucial role for the opposing effects of the two ginsenosides on splenic lymphocytes. Compared with the other two assay methods, the microcalorimetric method provided more useful and reliable information for quickly and objectively evaluating the effects of drugs or compounds on the living cells, which would be a highly promising analytical tool for the characterization of the biological process and the estimation of the drugs’ efficiency.

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Abstract

In this study, the activities of four ginsenosides Rc, Re, Rd, and Rf on splenic lymphocytes growth were studied by microcalorimetry. Some qualitative and quantitative information, such as the metabolic power–time curves, growth rate constant k, maximum heat-output power of the exponential phase P max and the corresponding appearance peak time t max, total heat output Q t, and promotion rate R p of splenic lymphocytes growth affected by the four ginsenosides were calculated. In accordance with thermo-kinetic model, the corresponding quantitative relationships of k, P max, t max, Q t, R p, and c were established. Also, the median effective concentration (EC50) was obtained by quantitative analysis. Based on both the quantitative quantity–activity relationships (QQAR) and EC50, the sequence of promotion activity was Rc > Re > Rd > Rf. The analysis of structure–activity relationships showed that the number, type, and position of sugar moieties on the gonane steroid nucleus had important influences on the promotion activity of Rc, Re, Rd, and Rf on splenic lymphocytes growth. Microcalorimetry can be used as a useful tool for determining the activity and studying the quantity–activity relationship of drugs on cell.

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