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Super congruences and Euler numbers

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Science China Mathematics 2011年第12期54卷 2509-2535页

作者： sun zhi-wei Department of Mathematics Nanjing University Nanjing 210093 China

Let p 〉 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that ∑k=0^p-1（k^2k/2k）≡（-1）^（p-1）/2-p^2Ep-3（modp^3） ∑k=1^（p-1）/2（k^2k）/k≡（-1）^（p＋1）/2 8/3pEp-3（mod p^2）,∑k=0^（p-1）/2（k^2k）^2/16k≡（-1）^（p-1）/2＋p^2Ep-3（mod p^3）,where E0, E1, E2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for π2, π-2 and the constant K ：= ∑k=1^∞（k/3）/k^2 （with （-） the Jacobi symbol）, two of which are ∑k=1^∞（10k-3）8k/k2（k^2k）^2（k^3k）=π^2/2and ∑k=1^∞（15k-4）（-27）^k-1/k^3（k^2k）^2（k^3k）=K.

关键词： central binomial coefficients super congruences Euler numbers

Congruences involving generalized central trinomial coefficients

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Science China Mathematics 2014年第7期57卷 1375-1400页

作者： sun zhi-wei Department of Mathematics Nanjing University

For integers b and c the generalized central trinomial coefficient Tn(b,c)denotes the coefficient of xnin the expansion of(x2+bx+c)*** Tn=Tn(1,1)(n=0,1,2,...)are the usual central trinomial coefficients,and Tn(3,2)coincides with the Delannoy number Dn=n k=0n k n+k k in *** investigate congruences involving generalized central trinomial coefficients *** are some typical results:For each n=1,2,3,...,we have n-1k=0(2k+1)Tk(b,c)2(b2-4c)n-1-k≡0(mod n2)and in particular n2|n-1k=0(2k+1)D2k;if p is an odd prime then p-1k=0T2k≡-1p(mod p)and p-1k=0D2k≡2p(mod p),where(-)denotes the Legendre *** also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.

关键词： congruences central trinomial coefficients Motzkin numbers central Delannoy numbers

Proof of three conjectures on congruences

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Science China Mathematics 2014年第10期57卷 2091-2102页

作者： PAN Hao sun zhi-wei Department of Mathematics Nanjing University

This paper proves three conjectures on congruences involving central binomial coefficients or Lucas *** p be an odd prime and let a be a positive *** is shown that if p=1（mod 4）or a〉1then [3/4pa]∑k=0≡（2/pa）（mod p^2）where（—）denotes the Jacobi *** confirms a conjecture of the second author.A conjecture of Tauraso is also confirmed by showing that p-1∑k=1 Lk/k^2≡0（mod p） provided p〉*** the Lucas numbers Lo,L1,L2,...are defined by L_0=2,L1=1 and Ln＋1=Ln＋Ln-l（n=1,2,3,...）.The third theorem states that if p=5 then Fp^a-（p^a/5）mod p^3 can be determined in the following way： p^a-1∑k=0（-1）^k（2k k）≡（p^a/5）（1-2F p^a-（pa/5））（mod p^3）which appeared as a conjecture in a paper of sun and Tauraso in 2010.

关键词： congruences modulo prime powers Fibonacci numbers Lucas sequences

Effects of Nitrobenzenes on DNA Damage in Germ Cells of Rats

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Chemical Research in Chinese Universities 2006年第1期22卷 29-32页

作者： XU Jing-bo JING Ti-song YANG Li sun zhi-wei SHI Long Department of Environmental Science Northeast Normal University Changchun 130024 P. R. China College of Environment and Resource Jilin University Changchun 130026 P. R. China Public Health College Jilin University Changchun 130021 P. R. China

The single cell gel electrophoresis （SCGE） technique was used to detect DNA damage in germ cells of rats, which was induced by five tested nitroaromatic compounds. Mixed cultures of Sertoli and germ cells were prepared from the rats testis and their responses to rn-dinitrobenzene （ m-DNB）, 2,4-dinitrotoluene （ 2,4-DNT）, 2,6-dinitroto-luene（2,6-DNT）, 4-nitrotoluene（4-NT） and 2,4-dinitroaniline（2,4-DNAn） were studied. The results show that all the five chemicals have a reproductive toxicity. Each dose group and the control group were significantly different （ P 〈 0. 01 ）. All of them can lead to the damage to DNA in the germ cells of Kunming male rats in the definite range of concentration（m-DNB ： 0. 04-25μmol/L; 2,4-DNT, 2,6-DNT and 4-NT： 0. 032-500μmol/L; 2,4-DNAn ：0. 8-20μmol/L）. When the concentration increases, the damage rate will become higher, which shows an evident logarithm dose-effect relationship.

关键词： Nitroaromatic compound Sertoli-germ cell coculture SCGE DNA damage

CO liberated from CORM-2 modulates the inflammatory response in the liver of thermally injured mice

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World Journal of Gastroenterology 2008年第4期14卷 547-553页

作者： Bing-wei sun Yan sun zhi-wei sun Xi Chen Department of Bums and Plastic Surgery Affiliated Hospital Jiangsu UniversityZhenjiang 212001 Jiangsu Province China

AIM： To explore the effects of CO-releasing molecules [tricarbonyldichlororuthenium （Ⅱ） dimer, CORM-2]- liberated CO on attenuation of inflammatory responses in liver of an experimental animal model of thermal injury and to investigate the associated potential mechanisms. METHODS： Thirty-six mice were assigned to three groups in three respective experiments. In each experiment, mice in sham group （n = 4） received sham thermal injury, whereas mice in burn group （n = 4） received a 15% of total body surface area （TBSA） fullthickness thermal injury, and mice in burn ＋ CORM-2 group （n = 4） received the same thermal injury with immediate administration of CORM-2 （8 mg/kg, iv）. Hepatic tissue sections were stained with hematoxylin and eosin and examined under a light microscope. Levels of aminotransferases （ALT and AST） and nitric oxide （NO） were measured by biochemical methods. Tumor necrosis factor-α （TNF-α） and interleukin （IL-1β） activity, and the protein expression of iNOS and HO-1 in serum and tissue homogenates were assessed. In in vitro experiments, Kupffer cells were stimulated with LPS （10 μg/mL） for 4 h in the presence or absence of CORM-2 （10-100 μmol/L）. Subsequently, the expression levels of TNF-α and NO production were assessed. RESULTS： Pro-inflammatory mediators （TNF-α, IL- 1β, NO） in serum and liver homogenates of thermally injured mice were significantly reduced by CORM-2 administration. This was accompanied by a decrease in the expression of iNOS while an increase in the expression of HO-1 in the liver tissue. In parallel, the concentrations of TNF-α and NO in supernatants of LPS-stimulated Kupffer cells co-incubated with CORM-2 （10-100 μmol/L） were also markedly *** examination demonstrated that CORM-2 could attenuate the leukocytes infiltration to the liver tissue. CONCLUSION： CORM-released CO modulates liver inflammation and significantly protects liver injury in burn mice by inhibiting the expression of iNOS and NO production, dow

关键词： Carbon monoxide Thermal injury Liver Inflammation

Contact Finite Determinacy of Relative Map Germs

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Communications in Mathematical Research 2010年第1期26卷 1-6页

作者： CHEN LIANG sun wei-zhi PEI DONG-HE Sehool of Mathematics and Statistics Northeast Normal University Changchun 130024 School of Science Changchun University of Science and Technology Changchun 130022

The strong contact finite determinacy of relative map germs is studied by means of classical singularity theory. We first give the definition of a strong relative contact equivalence （or KS,T equivalence） and then prove two theorems which can be used to distinguish the contact finite determinacy of relative map germs, that is, f is finite determined relative to KS,T if and only if there exists a positive integer k, such that M^5（n）ε（S;n）p ∩→ TKs,T（f）.

关键词： KS,T equivalent the tangent space of an orbit relative deformation finite determined relative to KS,T

Therapeutic Effects of Different Animal Bile Powders on Lipid Metabolism Disorders and Their Composition Analysis

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Chinese Journal of Integrative Medicine 2022年第10期28卷 918-923页

作者： CHEN Da-xin CHU Jian-feng LIN Shan ZHANG Ling CHEN Hong-wei sun zhi-wei XU Jian-feng CAI Qiao-yan WANG Li-li PENG Jun Academy of Integrative Medicine Cardiovascular Research Institute Fuzhou(350122)China Fujian Key Laboratory of Integrative Medicine on Geriatric Fujian University of Traditional Chinese MedicineFuzhou(350122)China Chen Ke-ji Academic Thought Heritage Studio Fuzhou(350122)China Fujian Guizhentang Pharmaceutical Co. Ltd.HuianFujian Province(362142)China

Objective: To compare the therapeutic effect of different animal bile powders on lipid metabolism disorders induced by high-fat diet in rats, and analyze the bioactive components of each animal bile powder. Methods: Sixty Sprague-Dawley rats were randomly divided into 6 groups(n=10): normal diet control group, high-fat diet model group, high-fat diet groups orally treated with bear, pig, cow and chicken bile powders, respectively. Serum biochemical markers from the abdominal aorta in each group were analyzed. Changes in the body weight and liver weight were recorded. Pathohistological changes in the livers were examined. High performance liquid chromatography coupled with quadrupole time-of-flight tandem mass spectrometry was used to determine the composition of bioactive components in each animal bile powder. Results: Treatment with different types of animal bile powders had different inhibitory effects on high-fat diet-induced increase of body weight and/or liver weight in rats, most notably in bear and pig bile powders(Pweight of 496.2738 Da, whose function has not yet been reported, was identified only in bear bile powder. Conclusions: Different animal bile powders had varying therapeutic effect against lipid metabolism disorders induced by high-fat diet, and bear bile powder demonstrated th

关键词： bile powders lipid metabolism active ingredient liver

Peroxisome proliferator-activated receptor gamma inhibits hepatic fibrosis in rats

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Hepatobiliary & Pancreatic Diseases International 2011年第1期10卷 64-71页

作者： ZhengWang,Jia-Peng Xu,Yong-Chao Zheng,wei Chen,Yong-wei sun,zhi-YongWu and Meng Luo Department of General Surgery,Renji Hospital,Shanghai Jiaotong University School of Medicine,Shanghai 200127,China Department of General Surgery Renji Hospital Shanghai Jiaotong University School of Medicine Shanghai 200127 China

BACKGROUND:Hepatic fibrosis is a necessary step in the development of hepatic *** this study we used lentiviral vector-mediated transfection technology to evaluate the effect of peroxisome proliferator-activated receptor gamma(PPAR-γ) on rat hepatic fibrosis. METHODS:Hepatic fibrosis in rats was induced by CCl4 for 2 weeks(early fibrosis)and 8 weeks(sustained fibrosis).The rats were randomly divided into four groups:normal control, fibrosis,blank vector,and PPAR-γ.They were infected with the recombinant lentiviral expression vector carrying the rat PPAR-γgene by portal vein *** liver of the rats was examined histologically and hydroxyproline was *** vitro primary hepatic stellate cells(HSCs)were infected with the recombinant lentiviral expression vector carrying the rat PPAR-γ*** status of HSC proliferation was measured by the MTT *** protein levels of PPAR-γ,α-smooth muscle actin(α-SMA)and type I collagen expression were evaluated by the Western blotting method. RESULTS:In vitro studies revealed that expression of PPAR-γ inhibited expression ofα-SMA and type I collagen in activated HSCs(P<0.01)as well as HSC proliferation(P<0.01).In vivo experiments indicated that in the early hepatic fibrosis group,the hydroxyproline content and the level of collagen I protein in the liver in the PPAR-γtransfected group were not significantly different compared to the hepatic fibrosis group and the blank vector group;whereas the expressions of PPAR-γ andα-SMA were different compared to the hepatic fibrosis group(P<0.01).In the sustained hepatic fibrosis group,there were significant differences in the hydroxyproline content and the expression of PPAR-γ,α-SMA,and type I collagen between each ***:PPAR-γcan inhibit HSC proliferation and hepatic fibrosis,and suppressα-SMA and type I collagen expression.

关键词： peroxisome proliferator-activated receptor gamma hepatic fibrosis hepatic stellate cells lentiviral vector

Cytotoxicity and DNA Damage Effect of TGA-capped CdTe Quantum Dots

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Chemical Research in Chinese Universities 2012年第2期28卷 276-281页

作者： LI Yan-bo ZHANG Hai-xia GUO Cai-xia HU Gui-qin DU Hai-ying JIN Ming-hua HUANG Pei-li sun zhi-wei YANG Wen-sheng School of Public Health and Family Medicine Capital Medical UniversityBeijing 100069P.R.China Public Health Institute Beijing Chaoyang District Centre for Disease Control and PreventionBeijing 100021P R.China School of Public Health Jilin UniversityChangchun 130021P.R.China Key Laboratory of Surface and Interface Chemistry of Jilin Province College of ChemistryJilin UniversityChangchun 130012P.R.China

The cytotoxicity and DNA damage caused by thioglycolic acid（TGA）-capped cadmium telluride（CdTe） quantum dots（QDs） to hepatocyte line HL-7702 were *** viability was measured by 3-（4,5-dimethylthiazol-2-yl）-2,5-diphenyltetrazolium bromide（MTT） assay;DNA damage was detected by single cell gel electrophoresis（SCGE）;the change of cell cycle progression was examined by propidium iodide（PI）-flow cytometry（FCM）;apoptosis was measured by acridine orange/ethidium bromide（AO/EB） assay and Annexin V-FITC/PI-FCM（FITC：fluorescein isothiocyanate）.The results show that the cytotoxicity induced by CdTe QDs was increased in a dose-dependent and time-dependent manner;after exposure to QDs for 24 h,as the exposure dose increased,the rate of DNA damage was significantly increased（P0.05）,and the degree of DNA damage was *** the dose of CdTe QDs increased,the percentage of G 0 /G 1 phase cells was significantly decreased（P0.001）,while the percenttages of S and G 2 /M phases cells were significantly increased（P0.001）.In AO/EB assay,apoptotic cells could be observed under a fluorescence microscope,and apoptotic rate was increased as exposure dose *** Annexin V-FITC/PI-FCM assay,the apoptotic rates of CdTe QDs treated groups were significantly increased compared with that of control group（P0.05）.Our studies indicate that CdTe QDs could influence cell viability,and induce DNA damage,the S and G2 /M phases arrest and apoptosis of HL-7702.

关键词： Apoptosis CdTe quantum dot(QD) Cell cycle Cell viability DNA damage