BACKGROUND Hepatitis B cirrhosis(HBC)is a chronic disease characterized by irreversible diffuse liver damage and aggravated by intestinal microbial imbalance and metabolic dysfunction.Although the relationship between...BACKGROUND Hepatitis B cirrhosis(HBC)is a chronic disease characterized by irreversible diffuse liver damage and aggravated by intestinal microbial imbalance and metabolic dysfunction.Although the relationship between certain single probiotics and HBC has been explored,the impact of the complex ready-to-eat Lactobacillus paracasei N1115(LP N1115)supplement on patients with HBC has not been determined.AIM To compare the changes in the microbiota,inflammatory factor levels,and liver function before and after probiotic treatment in HBC patients.METHODS This study included 160 HBC patients diagnosed at the General Hospital of Ningxia Medical University between October 2018 and December 2020.Patients were randomly divided into an intervention group that received LP N1115 supplementation and routine treatment and a control group that received routine treatment only.Fecal samples were collected at the onset and conclusion of the 12-wk intervention period.The structure of the intestinal microbiota and the levels of serological indicators,such as liver function and inflammatory factors,were assessed.RESULTS Following LP N1115 intervention,the intestinal microbial diversity significantly increased in the intervention group(P<0.05),and the structure of the intestinal microbiota was characterized by an increase in the proportions of probiotic microbes and a reduction in harmful bacteria.Additionally,the intervention group demonstrated notable improvements in liver function indices and significantly lower levels of inflammatory factors(P<0.05).CONCLUSION LP N1115 is a promising treatment for ameliorating intestinal microbial imbalance in HBC patients by modulating the structure of the intestinal microbiota,improving liver function,and reducing inflammatory factor levels.展开更多
The Owen’s T function is presented in four new ways, one of them as a series similar to the Euler’s arctangent series divided by 2π, which is its majorant series. All possibilities enable numerically stable ...The Owen’s T function is presented in four new ways, one of them as a series similar to the Euler’s arctangent series divided by 2π, which is its majorant series. All possibilities enable numerically stable and fast convergent computation of the bivariate normal integral with simple recursion. When tested computation on a random sample of one million parameter triplets with uniformly distributed components and using double precision arithmetic, the maximum absolute error was 3.45 × 10<sup>-</sup><sup>16</sup>. In additional testing, focusing on cases with correlation coefficients close to one in absolute value, when the computation may be very sensitive to small rounding errors, the accuracy was retained. In rare potentially critical cases, a simple adjustment to the computation procedure was performed—one potentially critical computation was replaced with two equivalent non-critical ones. All new series are suitable for vector and high-precision computation, assuming they are supplemented with appropriate efficient and accurate computation of the arctangent and standard normal cumulative distribution functions. They are implemented by the R package Phi2rho, available on CRAN. Its functions allow vector arguments and are ready to work with the Rmpfr package, which enables the use of arbitrary precision instead of double precision numbers. A special test with up to 1024-bit precision computation is also presented.展开更多
Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynom...Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.展开更多
For any given positive integer n≥1,the Euler functionφ(n) is defined to be the number of positive integers not exceeding n which are relatively prime to n.ω(n) is defined to be the number of different prime divisor...For any given positive integer n≥1,the Euler functionφ(n) is defined to be the number of positive integers not exceeding n which are relatively prime to n.ω(n) is defined to be the number of different prime divisors of n.Some kind of equations involving Euler's function is studied in the paper.展开更多
Let φ(n) denote the Euler-totient function, we study the distribution of solutions of φ(n) ≤ x in arithmetic progressions, where n ≡ l(mod q) and an asymptotic formula was obtained by Perron formula.
基金Supported by The Health System Research Project of Ningxia Hui Autonomous Region of China,No.2022-NWKY-061.
文摘BACKGROUND Hepatitis B cirrhosis(HBC)is a chronic disease characterized by irreversible diffuse liver damage and aggravated by intestinal microbial imbalance and metabolic dysfunction.Although the relationship between certain single probiotics and HBC has been explored,the impact of the complex ready-to-eat Lactobacillus paracasei N1115(LP N1115)supplement on patients with HBC has not been determined.AIM To compare the changes in the microbiota,inflammatory factor levels,and liver function before and after probiotic treatment in HBC patients.METHODS This study included 160 HBC patients diagnosed at the General Hospital of Ningxia Medical University between October 2018 and December 2020.Patients were randomly divided into an intervention group that received LP N1115 supplementation and routine treatment and a control group that received routine treatment only.Fecal samples were collected at the onset and conclusion of the 12-wk intervention period.The structure of the intestinal microbiota and the levels of serological indicators,such as liver function and inflammatory factors,were assessed.RESULTS Following LP N1115 intervention,the intestinal microbial diversity significantly increased in the intervention group(P<0.05),and the structure of the intestinal microbiota was characterized by an increase in the proportions of probiotic microbes and a reduction in harmful bacteria.Additionally,the intervention group demonstrated notable improvements in liver function indices and significantly lower levels of inflammatory factors(P<0.05).CONCLUSION LP N1115 is a promising treatment for ameliorating intestinal microbial imbalance in HBC patients by modulating the structure of the intestinal microbiota,improving liver function,and reducing inflammatory factor levels.
文摘The Owen’s T function is presented in four new ways, one of them as a series similar to the Euler’s arctangent series divided by 2π, which is its majorant series. All possibilities enable numerically stable and fast convergent computation of the bivariate normal integral with simple recursion. When tested computation on a random sample of one million parameter triplets with uniformly distributed components and using double precision arithmetic, the maximum absolute error was 3.45 × 10<sup>-</sup><sup>16</sup>. In additional testing, focusing on cases with correlation coefficients close to one in absolute value, when the computation may be very sensitive to small rounding errors, the accuracy was retained. In rare potentially critical cases, a simple adjustment to the computation procedure was performed—one potentially critical computation was replaced with two equivalent non-critical ones. All new series are suitable for vector and high-precision computation, assuming they are supplemented with appropriate efficient and accurate computation of the arctangent and standard normal cumulative distribution functions. They are implemented by the R package Phi2rho, available on CRAN. Its functions allow vector arguments and are ready to work with the Rmpfr package, which enables the use of arbitrary precision instead of double precision numbers. A special test with up to 1024-bit precision computation is also presented.
文摘Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.
基金Foundation item： Supported by the National Natural Science Foundation of China（10671056）
文摘For any given positive integer n≥1,the Euler functionφ(n) is defined to be the number of positive integers not exceeding n which are relatively prime to n.ω(n) is defined to be the number of different prime divisors of n.Some kind of equations involving Euler's function is studied in the paper.
基金Supported by the National Natural Science Foundation of China（11271249） Supported by the Scientific and Technological Research Program of Chongqing Municipal Education Commission（1601213） Supported by the Scientific Research Program of Yangtze Normal University（2012XJYBO31）
文摘Let φ(n) denote the Euler-totient function, we study the distribution of solutions of φ(n) ≤ x in arithmetic progressions, where n ≡ l(mod q) and an asymptotic formula was obtained by Perron formula.
