For each real number x∈(0,1),let[a_(1)(x),a_(2)(x),…,a_n(x),…]denote its continued fraction expansion.We study the convergence exponent defined byτ(x)=inf{s≥0:∞∑n=1(a_(n)(x)a_(n+1)(x))^(-s)<∞},which reflect...For each real number x∈(0,1),let[a_(1)(x),a_(2)(x),…,a_n(x),…]denote its continued fraction expansion.We study the convergence exponent defined byτ(x)=inf{s≥0:∞∑n=1(a_(n)(x)a_(n+1)(x))^(-s)<∞},which reflects the growth rate of the product of two consecutive partial quotients.As a main result,the Hausdorff dimensions of the level sets ofτ(x)are determined.展开更多
Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pyth...Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .展开更多
随着工业化进程的加快和城市化的发展,大量污染物排入黄河流域,并被频繁检出,威胁生态系统和人类健康。为获取潜在生态环境风险污染物,该研究通过调研2000年1月1日−2022年12月31日Web of Science(WoS)和中国知网(CNKI)数据库中黄河流域...随着工业化进程的加快和城市化的发展,大量污染物排入黄河流域,并被频繁检出,威胁生态系统和人类健康。为获取潜在生态环境风险污染物,该研究通过调研2000年1月1日−2022年12月31日Web of Science(WoS)和中国知网(CNKI)数据库中黄河流域已报道的288篇污染物相关文献,使用多指标综合评分法筛选黄河流域的特征污染物,采用风险商值法获取水样和沉积物中的风险污染物。结果表明:①黄河流域共检出10类144种污染物,采用9类共13个筛选指标构建多指标综合评分法,对污染物各项指标进行评分,然后进行K-means聚类分析,按得分高低分为Ⅰ~Ⅵ级,选取得分较高的33种Ⅰ级和Ⅱ级高分值污染物作为黄河流域特征污染物,包括12种有机氯农药、10种多环芳烃、10种多氯联苯和1种邻苯二甲酸酯。②水样污染物浓度和沉积物含量前5种都是重金属、有机氯农药、邻苯二甲酸酯、多环芳烃以及药品和个人护理产品,而且二者顺序完全一致,且多数污染物的浓度之间存在显著相关性。③根据风险最大化原则,使用风险商值法(RQ)分别对水样和沉积物进行风险评估,将RQ≥0.1的污染物列为风险污染物,水样中共筛选出21种风险污染物,其中RQ≥1的高风险污染物有5种,包括硒、铅、苯并[a,h]蒽、苯并[a]蒽和邻苯二甲酸二丁酯。④沉积物中共筛选出19种风险污染物,其中有13种高风险污染物,包括8种多环芳烃(芘、蒽、荧蒽、苊、萘、芴、苯并[a]蒽、苯并[a,h]蒽)、4种重金属(汞、铅、硒、砷)和1种邻苯二甲酸酯(邻苯二甲酸二丁酯)。该研究对相关部门拟定黄河流域污染物监测方案和管控措施有重要参考意义。展开更多
基金supported by the Scientific Research Fund of Hunan Provincial Education Department(21B0070)the Natural Science Foundation of Jiangsu Province(BK20231452)+1 种基金the Fundamental Research Funds for the Central Universities(30922010809)the National Natural Science Foundation of China(11801591,11971195,12071171,12171107,12201207,12371072)。
文摘For each real number x∈(0,1),let[a_(1)(x),a_(2)(x),…,a_n(x),…]denote its continued fraction expansion.We study the convergence exponent defined byτ(x)=inf{s≥0:∞∑n=1(a_(n)(x)a_(n+1)(x))^(-s)<∞},which reflects the growth rate of the product of two consecutive partial quotients.As a main result,the Hausdorff dimensions of the level sets ofτ(x)are determined.
文摘Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .
文摘随着工业化进程的加快和城市化的发展,大量污染物排入黄河流域,并被频繁检出,威胁生态系统和人类健康。为获取潜在生态环境风险污染物,该研究通过调研2000年1月1日−2022年12月31日Web of Science(WoS)和中国知网(CNKI)数据库中黄河流域已报道的288篇污染物相关文献,使用多指标综合评分法筛选黄河流域的特征污染物,采用风险商值法获取水样和沉积物中的风险污染物。结果表明:①黄河流域共检出10类144种污染物,采用9类共13个筛选指标构建多指标综合评分法,对污染物各项指标进行评分,然后进行K-means聚类分析,按得分高低分为Ⅰ~Ⅵ级,选取得分较高的33种Ⅰ级和Ⅱ级高分值污染物作为黄河流域特征污染物,包括12种有机氯农药、10种多环芳烃、10种多氯联苯和1种邻苯二甲酸酯。②水样污染物浓度和沉积物含量前5种都是重金属、有机氯农药、邻苯二甲酸酯、多环芳烃以及药品和个人护理产品,而且二者顺序完全一致,且多数污染物的浓度之间存在显著相关性。③根据风险最大化原则,使用风险商值法(RQ)分别对水样和沉积物进行风险评估,将RQ≥0.1的污染物列为风险污染物,水样中共筛选出21种风险污染物,其中RQ≥1的高风险污染物有5种,包括硒、铅、苯并[a,h]蒽、苯并[a]蒽和邻苯二甲酸二丁酯。④沉积物中共筛选出19种风险污染物,其中有13种高风险污染物,包括8种多环芳烃(芘、蒽、荧蒽、苊、萘、芴、苯并[a]蒽、苯并[a,h]蒽)、4种重金属(汞、铅、硒、砷)和1种邻苯二甲酸酯(邻苯二甲酸二丁酯)。该研究对相关部门拟定黄河流域污染物监测方案和管控措施有重要参考意义。
