Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ...Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .展开更多
Wastewater surveillance(WWS)can leverage its wide coverage,population-based sampling,and high monitoring frequency to capture citywide pandemic trends independent of clinical surveillance.Here we conducted a nine mont...Wastewater surveillance(WWS)can leverage its wide coverage,population-based sampling,and high monitoring frequency to capture citywide pandemic trends independent of clinical surveillance.Here we conducted a nine months daily WWS for severe acute respiratory syndrome coronavirus 2(SARSCoV-2)from 12 wastewater treatment plants(WWTPs),covering approximately 80%of the population,to monitor infection dynamics in Hong Kong,China.We found that the SARS-CoV-2 virus concentration in wastewater was correlated with the daily number of reported cases and reached two pandemic peaks three days earlier during the study period.In addition,two different methods were established to estimate the prevalence/incidence rates from wastewater measurements.The estimated results from wastewater were consistent with findings from two independent citywide clinical surveillance programmes(rapid antigen test(RAT)surveillance and serology surveillance),but higher than the cases number reported by the Centre for Health Protection(CHP)of Hong Kong,China.Moreover,the effective reproductive number(R_(t))was estimated from wastewater measurements to reflect both citywide and regional transmission dynamics.Our findings demonstrate that large-scale intensive WWS from WWTPs provides cost-effective and timely public health information,especially when the clinical surveillance is inadequate and costly.This approach also provides insights into pandemic dynamics at higher spatiotemporal resolutions,facilitating the formulation of effective control policies and targeted resource allocation.展开更多
目的探讨成髓细胞瘤转录因子第2亚型(MYB proto-oncogene like 2,MYBL2)基因在子宫内膜癌组织中的表达及拷贝数变异(copy number variation,CNV)情况,评价其与临床特征的关系和对临床预后的影响。方法从癌症基因组图谱(The Cancer Genom...目的探讨成髓细胞瘤转录因子第2亚型(MYB proto-oncogene like 2,MYBL2)基因在子宫内膜癌组织中的表达及拷贝数变异(copy number variation,CNV)情况,评价其与临床特征的关系和对临床预后的影响。方法从癌症基因组图谱(The Cancer Genome Atlas,TCGA)数据库中获取临床病理特征和相应的基因组数据资料完整的子宫内膜癌患者140例,从cBio Cancer Genomics Portal(cBioPortal)数据库下载MYBL2基因CNV和基因组数据资料完整的子宫内膜癌患者137例,以各基因表达水平的中位数作为高低表达的截断值,通过χ2检验和Cox回归分析MYBL2基因高表达及CNV与子宫内膜癌临床特征的关系,Kaplan-Meier曲线分析MYBL2基因表达和CNV与患者预后之间的关系。收集2015年1月至2015年12月南昌大学第二附属医院手术切除经病理组织学确诊的60例子宫内膜癌组织及其癌旁组织。免疫组织化学法检测其MYBL2表达。结果TCGA数据库分析发现,子宫内膜癌MYBL2高表达和低表达患者在病理类型(P=0.034)和组织分级(P<0.01)方面比较,差异均具有统计学意义。cBioPortal数据库分析发现,子宫内膜癌中MYBL2存在CNV(P<0.01),主要变异形式是拷贝数增加,占26.0%。MYBL2 CNV的患者MYBL2 mRNA表达水平上调(P<0.01),MYBL2 CNV患者总生存期与无瘤生存期较短(均P<0.05)。Ⅱ型子宫内膜癌中MYBL2 CNV占比高于Ⅰ型(P<0.01)。结论子宫内膜癌组织中MYBL2呈高表达,且与不良预后相关。MYBL2基因CNV在子宫内膜癌中是一种不良预后因素,其可能成为预测子宫内膜癌临床预后的有效生物标志物。展开更多
An H（2）-move is a local move of a knot which is performed by adding a half-twisted band. It is known an H（2）-move is an unknotting operation. We define the H（2）-unknotting number of a knot K to be the minimum nu...An H（2）-move is a local move of a knot which is performed by adding a half-twisted band. It is known an H（2）-move is an unknotting operation. We define the H（2）-unknotting number of a knot K to be the minimum number of H（2）-moves needed to transform K into a trivial knot. We give several methods to estimate the H（2）-unknotting number of a knot. Then we give tables of H（2）-unknotting numbers of knots with up to 9 crossings.展开更多
AIM To investigate if the down-regulation of N-myc Downstream Regulated Gene 2(NDRG2) expression in colorectal carcinoma(CRC) is due to loss of the NDRG2 allele(s).METHODS The following were investigated in the human ...AIM To investigate if the down-regulation of N-myc Downstream Regulated Gene 2(NDRG2) expression in colorectal carcinoma(CRC) is due to loss of the NDRG2 allele(s).METHODS The following were investigated in the human colorectal cancer cell lines DLD-1, Lo Vo and SW-480: NDRG2 mRNA expression levels using quantitative reverse transcriptionpolymerase chain reaction(qRT-PCR); interaction of the MYC gene-regulatory protein with the NDRG2 promoter using chromatin immunoprecipitation; and NDRG2 promoter methylation using bisulfite sequencing.Furthermore, we performed qPCR to analyse the copy numbers of NDRG2 and MYC genes in the above three cell lines, 8 normal colorectal tissue samples and 40 CRC tissue samples.RESULTS As expected, NDRG2 mRNA levels were low in the three colorectal cancer cell lines, compared to normal colon.Endogenous MYC protein interacted with the NDRG2 core promoter in all three cell lines.In addition, the NDRG2 promoter was heavily methylated in these cell lines, suggesting an epigenetic regulatory mechanism.Unaltered gene copy numbers of NDRG2 were observed in the three cell lines.In the colorectal tissues, one normal and three CRC samples showed partial or complete loss of one NDRG2 allele.In contrast, the MYC gene was amplified in one cell line and in more than 40% of the CRC cases.CONCLUSION Our study suggests that the reduction in NDRG2 expression observed in CRC is due to transcriptional repression by MYC and promoter methylation, and is not due to allelic loss.展开更多
A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set ...A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set of G. In this paper we calculate the k-domination number (for k = 2) of the product of two paths Pm × Pn for m = 1, 2, 3, 4, 5 and arbitrary n. These results were shown an error in the paper [1].展开更多
In this paper, we obtain the strong law of large numbers for a 2-dimensional array of pairwise negatively dependent random variables which are not required to be identically distributed. We found the sufficient condit...In this paper, we obtain the strong law of large numbers for a 2-dimensional array of pairwise negatively dependent random variables which are not required to be identically distributed. We found the sufficient conditions of strong law of large numbers for the difference of random variables which independent and identically distributed conditions are regarded. In this study, we consider the limit as which is stronger than the limit as m× n→?∞ when m, n →?∞?are natural numbers.展开更多
文摘Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .
基金financially supported by the Health and Medical Research Fund(COVID1903015)the Food and Health Bureau,the Government of the Hong Kong Special Administrative Region(SAR),China+1 种基金supported by the AIR@InnoHK(KL,GML,and JTW)Health@InnoHK(MP and LLMP)administered by the Innovation and Technology Commission of the Government of the Hong Kong SAR.
文摘Wastewater surveillance(WWS)can leverage its wide coverage,population-based sampling,and high monitoring frequency to capture citywide pandemic trends independent of clinical surveillance.Here we conducted a nine months daily WWS for severe acute respiratory syndrome coronavirus 2(SARSCoV-2)from 12 wastewater treatment plants(WWTPs),covering approximately 80%of the population,to monitor infection dynamics in Hong Kong,China.We found that the SARS-CoV-2 virus concentration in wastewater was correlated with the daily number of reported cases and reached two pandemic peaks three days earlier during the study period.In addition,two different methods were established to estimate the prevalence/incidence rates from wastewater measurements.The estimated results from wastewater were consistent with findings from two independent citywide clinical surveillance programmes(rapid antigen test(RAT)surveillance and serology surveillance),but higher than the cases number reported by the Centre for Health Protection(CHP)of Hong Kong,China.Moreover,the effective reproductive number(R_(t))was estimated from wastewater measurements to reflect both citywide and regional transmission dynamics.Our findings demonstrate that large-scale intensive WWS from WWTPs provides cost-effective and timely public health information,especially when the clinical surveillance is inadequate and costly.This approach also provides insights into pandemic dynamics at higher spatiotemporal resolutions,facilitating the formulation of effective control policies and targeted resource allocation.
文摘目的探讨成髓细胞瘤转录因子第2亚型(MYB proto-oncogene like 2,MYBL2)基因在子宫内膜癌组织中的表达及拷贝数变异(copy number variation,CNV)情况,评价其与临床特征的关系和对临床预后的影响。方法从癌症基因组图谱(The Cancer Genome Atlas,TCGA)数据库中获取临床病理特征和相应的基因组数据资料完整的子宫内膜癌患者140例,从cBio Cancer Genomics Portal(cBioPortal)数据库下载MYBL2基因CNV和基因组数据资料完整的子宫内膜癌患者137例,以各基因表达水平的中位数作为高低表达的截断值,通过χ2检验和Cox回归分析MYBL2基因高表达及CNV与子宫内膜癌临床特征的关系,Kaplan-Meier曲线分析MYBL2基因表达和CNV与患者预后之间的关系。收集2015年1月至2015年12月南昌大学第二附属医院手术切除经病理组织学确诊的60例子宫内膜癌组织及其癌旁组织。免疫组织化学法检测其MYBL2表达。结果TCGA数据库分析发现,子宫内膜癌MYBL2高表达和低表达患者在病理类型(P=0.034)和组织分级(P<0.01)方面比较,差异均具有统计学意义。cBioPortal数据库分析发现,子宫内膜癌中MYBL2存在CNV(P<0.01),主要变异形式是拷贝数增加,占26.0%。MYBL2 CNV的患者MYBL2 mRNA表达水平上调(P<0.01),MYBL2 CNV患者总生存期与无瘤生存期较短(均P<0.05)。Ⅱ型子宫内膜癌中MYBL2 CNV占比高于Ⅰ型(P<0.01)。结论子宫内膜癌组织中MYBL2呈高表达,且与不良预后相关。MYBL2基因CNV在子宫内膜癌中是一种不良预后因素,其可能成为预测子宫内膜癌临床预后的有效生物标志物。
基金supported by Grant-in-Aid (20540079) for Scientific Research (C),Japan Society for the Promotion of Science
文摘An H（2）-move is a local move of a knot which is performed by adding a half-twisted band. It is known an H（2）-move is an unknotting operation. We define the H（2）-unknotting number of a knot K to be the minimum number of H（2）-moves needed to transform K into a trivial knot. We give several methods to estimate the H（2）-unknotting number of a knot. Then we give tables of H（2）-unknotting numbers of knots with up to 9 crossings.
文摘AIM To investigate if the down-regulation of N-myc Downstream Regulated Gene 2(NDRG2) expression in colorectal carcinoma(CRC) is due to loss of the NDRG2 allele(s).METHODS The following were investigated in the human colorectal cancer cell lines DLD-1, Lo Vo and SW-480: NDRG2 mRNA expression levels using quantitative reverse transcriptionpolymerase chain reaction(qRT-PCR); interaction of the MYC gene-regulatory protein with the NDRG2 promoter using chromatin immunoprecipitation; and NDRG2 promoter methylation using bisulfite sequencing.Furthermore, we performed qPCR to analyse the copy numbers of NDRG2 and MYC genes in the above three cell lines, 8 normal colorectal tissue samples and 40 CRC tissue samples.RESULTS As expected, NDRG2 mRNA levels were low in the three colorectal cancer cell lines, compared to normal colon.Endogenous MYC protein interacted with the NDRG2 core promoter in all three cell lines.In addition, the NDRG2 promoter was heavily methylated in these cell lines, suggesting an epigenetic regulatory mechanism.Unaltered gene copy numbers of NDRG2 were observed in the three cell lines.In the colorectal tissues, one normal and three CRC samples showed partial or complete loss of one NDRG2 allele.In contrast, the MYC gene was amplified in one cell line and in more than 40% of the CRC cases.CONCLUSION Our study suggests that the reduction in NDRG2 expression observed in CRC is due to transcriptional repression by MYC and promoter methylation, and is not due to allelic loss.
文摘A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set of G. In this paper we calculate the k-domination number (for k = 2) of the product of two paths Pm × Pn for m = 1, 2, 3, 4, 5 and arbitrary n. These results were shown an error in the paper [1].
文摘In this paper, we obtain the strong law of large numbers for a 2-dimensional array of pairwise negatively dependent random variables which are not required to be identically distributed. We found the sufficient conditions of strong law of large numbers for the difference of random variables which independent and identically distributed conditions are regarded. In this study, we consider the limit as which is stronger than the limit as m× n→?∞ when m, n →?∞?are natural numbers.
