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第20行： − 保罗·埃尔德什（Paul Erdős，1913年至1996年）是一位具有很大影响力的匈牙利数学家，他花费了一生中大量的时间与很多同事撰写论文，致力于解决困扰已久的疑难数学问题。他一生中发表的论文（至少1,525件）比历史上其他任何数学家都多。（莱昂哈德·欧拉（Leonhard Euler）发表过更多的数学论文，但单独的论文却较少：大约800篇。），而埃尔德什（Erdős）的大部分时间都生活在手提箱里，他拜访过全球500多个合作者。+ 第36行：
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第48行： − 要分配一个埃尔德什数，某人必须与另一个具有有限埃尔德什数的人共同撰写研究论文。保罗·埃尔德什的埃尔德什数为零。其他人的埃尔德什数为k+1，其中k是任何合著者中最低的埃尔德什数。美国数学学会提供免费的在线工具，可确定《数学评论》目录中列出的每个数学作者的埃尔德什数。+ 第56行：
第56行： − + 第94行：
第94行： − Fields奖牌获得者的埃尔德什中位数低至3。埃尔德什排名第二的奖牌获得者包括Atle Selberg，Kunihiko Kodaira，Klaus Roth，Alan Baker，Enrico Bombieri，David Mumford，Charles Fefferman，William Thurston，Shing-Tung Tung，Jean Bourgain，Richard Borcherds，Manjul Bhargava，Jean-Pierre Serre和陶哲轩。Fields的获得者中没有人的Erdős为1。但是，恩德雷·塞梅雷迪（Endre Szemerédi）是阿贝尔奖获得者，其埃尔德什数为1。+ 第160行：
第160行： − 截至2016年，所有费尔兹奖章获得者都有一个有限的埃尔德什数，其值在2到6之间，中位数为3。相反，所有数学家的埃尔德什数的中位数（有限的Erdős数）为5，极限值为13。下表总结了物理，化学，医学和经济学方面的诺贝尔奖获得者的埃尔德什数统计。第一列计算获奖者人数。第二列计算的是具有有限埃尔德什数的获胜者数量。第三列是具有有限埃尔德什数的获胜者的百分比。其余各列表示了这些获奖者中埃尔德什数的最小，最大，平均和中位数。+ 第340行：
第340行： − 法官理查德 · 波斯纳与阿尔文 · e · 罗斯合著了本书，本书的爱尔兰数量最多不超过4本。是一位政治家、哲学家和法律理论家，在哈佛大学法学院任教，他与 Lee Smolin 合著的《昂格尔多达4本。+ 第350行：
第350行： − 从2005年至今的德国总理安格拉·默克尔（Angela Merkel）的埃尔德什数最多为5。+ 第392行：
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第424行： − 另外，迈克尔·巴尔（Michael Barr）曾建议使用“合理的埃尔德什数”，通俗的说就是与埃尔德共同撰写过p篇论文的人应被分配埃尔德什数的1/p。根据第二种的协作多重图（尽管他也有办法处理第一种情况），即在他们所合著的每篇联合论文中，两个数学家之间都有一条边，这个边缘上都有一个1欧姆电阻器的电网。两个节点之间的总电阻表明这两个节点有多“相近”。+
无编辑摘要
The Erdős number () describes the "collaborative distance" between mathematician and another person, as measured by authorship of mathematical papers. The same principle has been applied in other fields where a particular individual has collaborated with a large and broad number of peers.
The Erdős number () describes the "collaborative distance" between mathematician and another person, as measured by authorship of mathematical papers. The same principle has been applied in other fields where a particular individual has collaborated with a large and broad number of peers.
埃尔德什数Erdős number（匈牙利语：[ˈɛrdøːʃ]）描述了数学家保罗·埃尔德什Paul Erdős与另一个人之间的“协作距离”，这是根据数学论文的著作权来衡量的。该原则应用于很多其他领域，意指特定某个人与众多同行之间的合作。
'''<font color="#ff8000"> 埃尔德什数Erdős number</font>'''（匈牙利语：[ˈɛrdøːʃ]）描述了数学家保罗·埃尔德什Paul Erdős与另一个人之间的“协作距离”，这是根据数学论文的著作权来衡量的。该原则应用于很多其他领域，意指特定某个人与众多同行之间的合作。
Paul Erdős (1913–1996) was an influential Hungarian mathematician who in the latter part of his life spent a great deal of time writing papers with a large number of colleagues, working on solutions to outstanding mathematical problems. He published more papers during his lifetime (at least 1,525) than any other mathematician in history. Erdős spent a large portion of his later life living out of a suitcase, visiting his over 500 collaborators around the world.
Paul Erdős (1913–1996) was an influential Hungarian mathematician who in the latter part of his life spent a great deal of time writing papers with a large number of colleagues, working on solutions to outstanding mathematical problems. He published more papers during his lifetime (at least 1,525) than any other mathematician in history. Erdős spent a large portion of his later life living out of a suitcase, visiting his over 500 collaborators around the world.
保罗·埃尔德什Paul Erdős(1913年至1996年）是一位具有很大影响力的匈牙利数学家，他花费了一生中大量的时间与很多同事撰写论文，致力于解决困扰已久的疑难数学问题。他一生中发表的论文（至少1,525件）比历史上其他任何数学家都多。莱昂哈德·欧拉Leonhard Euler发表过更多的数学论文，但单独的论文却较少：大约800篇。而埃尔德什的大部分时间都生活在手提箱里，他拜访过全球500多个合作者。
Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers. The median Erdős number of Fields Medalists is 3. Only 7,097 (about 5% of mathematicians with a collaboration path) have an Erdős number of 2 or lower. As time passes, the smallest Erdős number that can still be achieved will necessarily increase, as mathematicians with low Erdős numbers die and become unavailable for collaboration. Still, historical figures can have low Erdős numbers. For example, renowned Indian mathematician Srinivasa Ramanujan has an Erdős number of only 3 (through G. H. Hardy, Erdős number 2), even though Paul Erdős was only 7 years old when Ramanujan died.
Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers. The median Erdős number of Fields Medalists is 3. Only 7,097 (about 5% of mathematicians with a collaboration path) have an Erdős number of 2 or lower. As time passes, the smallest Erdős number that can still be achieved will necessarily increase, as mathematicians with low Erdős numbers die and become unavailable for collaboration. Still, historical figures can have low Erdős numbers. For example, renowned Indian mathematician Srinivasa Ramanujan has an Erdős number of only 3 (through G. H. Hardy, Erdős number 2), even though Paul Erdős was only 7 years old when Ramanujan died.
多项研究表明，领先的数学家往往具有极低的埃尔德什数。费尔兹奖Fields Medalists的埃尔德什中位数是3。只有7,097名（拥有合作经历的数学家中约5％）的埃尔德什数为2或更低。随着时间的流逝，低埃尔德什数的数学家因死亡而无法进行协作，最小埃尔德什数（仍然存在）必然会增加。即使历史人物仍可能一直具有较低的埃尔德什数。例如，印度著名数学家Srinivasa Ramanujan的埃尔德什数仅为3（通过G. H. Hardy，Erdős数为2），尽管Ramanujan去世时保罗·埃尔德什只有7岁。
多项研究表明，领先的数学家往往具有极低的埃尔德什数。费尔兹奖Fields Medalists的埃尔德什中位数是3。只有7,097名（拥有合作经历的数学家中约5％）的埃尔德什数为2或更低。随着时间的流逝，低埃尔德什数的数学家因死亡而无法进行协作，最小埃尔德什数（仍然存在）必然会增加。即使历史人物仍可能一直具有较低的埃尔德什数。例如，印度著名数学家Srinivasa Ramanujan的埃尔德什数仅为3（通过与G. H. Hardy合作，其埃尔德什数为2），尽管Ramanujan去世时保罗·埃尔德什只有7岁。
To be assigned an Erdős number, someone must be a coauthor of a research paper with another person who has a finite Erdős number. Paul Erdős has an Erdős number of zero. Anybody else's Erdős number is where is the lowest Erdős number of any coauthor. The American Mathematical Society provides a free online tool to determine the Erdős number of every mathematical author listed in the Mathematical Reviews catalogue.
To be assigned an Erdős number, someone must be a coauthor of a research paper with another person who has a finite Erdős number. Paul Erdős has an Erdős number of zero. Anybody else's Erdős number is where is the lowest Erdős number of any coauthor. The American Mathematical Society provides a free online tool to determine the Erdős number of every mathematical author listed in the Mathematical Reviews catalogue.
要分配一个埃尔德什数，某人必须与另一个具有有限埃尔德什数的人共同撰写研究论文。保罗·埃尔德什的埃尔德什数为零。其他人的埃尔德什数为''k+1''，其中''k''是任何合著者中最低的埃尔德什数。美国数学学会提供免费的在线工具，可确定《数学评论》目录中列出的每个数学作者的埃尔德什数。
Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 511 direct collaborators;), those who have collaborated with people who have an Erdős number of 2 (but not with Erdős or anyone with an Erdős number of 1) have an Erdős number of 3, and so forth. A person with no such coauthorship chain connecting to Erdős has an Erdős number of infinity (or an undefined one). Since the death of Paul Erdős, the lowest Erdős number that a new researcher can obtain is 2.
Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 511 direct collaborators;), those who have collaborated with people who have an Erdős number of 2 (but not with Erdős or anyone with an Erdős number of 1) have an Erdős number of 3, and so forth. A person with no such coauthorship chain connecting to Erdős has an Erdős number of infinity (or an undefined one). Since the death of Paul Erdős, the lowest Erdős number that a new researcher can obtain is 2.
埃尔德什一生撰写了约1500篇数学文章，其中大部分是合作的。他有511个直接合作者；这些是埃尔德什数为1的人。与这些人合作（但未与埃尔德什本人合作）的人所拥有的埃尔德什数为2（截至2020年8月7日为12,600人），而与埃尔德什数为2的人合作的人（但与埃尔德什或埃尔德什数为1的任何人无合作关系），其埃尔德什数为3，依此类推。没有此类共同作者链接能指向埃尔德什的人，其埃尔德什数为无穷大（或未定义）。自保罗·埃尔德什逝世以来，新研究员可获得的最低Erdős数为2。
埃尔德什一生撰写了约1500篇数学文章，其中大部分是合作的。他有511个直接合作者；这些是埃尔德什数为1的人。与这些人合作（但未与埃尔德什本人合作）的人所拥有的埃尔德什数为2（截至2020年8月7日为12,600人），而与埃尔德什数为2的人合作的人（但与埃尔德什或埃尔德什数为1的任何人无合作关系），其埃尔德什数为3，依此类推。没有此类共同作者链接能指向埃尔德什的人，其埃尔德什数为无穷大（或未定义）。自保罗·埃尔德什逝世以来，新研究员可获得的最低埃尔德什数为2。
The median Erdős number among Fields medalists is as low as 3.Fields medalists with Erdős number 2 include [[Atle Selberg]], [[Kunihiko Kodaira]], [[Klaus Roth]], [[Alan Baker (mathematician)|Alan Baker]], [[Enrico Bombieri]], [[David Mumford]], [[Charles Fefferman]], [[William Thurston]], [[Shing-Tung Yau]], [[Jean Bourgain]], [[Richard Borcherds]], [[Manjul Bhargava]], [[Jean-Pierre Serre]] and [[Terence Tao]]. There are no Fields medalists with Erdős number 1; however, Endre Szemerédi is an Abel Prize Laureate with Erdős number 1.
The median Erdős number among Fields medalists is as low as 3.Fields medalists with Erdős number 2 include [[Atle Selberg]], [[Kunihiko Kodaira]], [[Klaus Roth]], [[Alan Baker (mathematician)|Alan Baker]], [[Enrico Bombieri]], [[David Mumford]], [[Charles Fefferman]], [[William Thurston]], [[Shing-Tung Yau]], [[Jean Bourgain]], [[Richard Borcherds]], [[Manjul Bhargava]], [[Jean-Pierre Serre]] and [[Terence Tao]]. There are no Fields medalists with Erdős number 1; however, Endre Szemerédi is an Abel Prize Laureate with Erdős number 1.
费尔兹Fields奖获得者的埃尔德什中位数低至3。埃尔德什排名第二的奖牌获得者包括Atle Selberg，Kunihiko Kodaira，Klaus Roth，Alan Baker，Enrico Bombieri，David Mumford，Charles Fefferman，William Thurston，Shing-Tung Tung，Jean Bourgain，Richard Borcherds，Manjul Bhargava，Jean-Pierre Serre和陶哲轩。Fields的获得者中没有人的Erdős为1。但是，恩德雷·塞梅雷迪（Endre Szemerédi）是阿贝尔奖获得者，其埃尔德什数为1。
, all Fields Medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3. In contrast, the median Erdős number across all mathematicians (with a finite Erdős number) is 5, with an extreme value of 13. The table below summarizes the Erdős number statistics for Nobel prize laureates in Physics, Chemistry, Medicine and Economics. The first column counts the number of laureates. The second column counts the number of winners with a finite Erdős number. The third column is the percentage of winners with a finite Erdős number. The remaining columns report the minimum, maximum, average and median Erdős numbers among those laureates.
, all Fields Medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3. In contrast, the median Erdős number across all mathematicians (with a finite Erdős number) is 5, with an extreme value of 13. The table below summarizes the Erdős number statistics for Nobel prize laureates in Physics, Chemistry, Medicine and Economics. The first column counts the number of laureates. The second column counts the number of winners with a finite Erdős number. The third column is the percentage of winners with a finite Erdős number. The remaining columns report the minimum, maximum, average and median Erdős numbers among those laureates.
截至2016年，所有费尔兹奖章获得者都有一个有限的埃尔德什数，其值在2到6之间，中位数为3。相反，所有数学家的埃尔德什数的中位数（有限的埃尔德什数）为5，极限值为13。下表总结了物理，化学，医学和经济学方面的诺贝尔奖获得者的埃尔德什数统计。第一列计算获奖者人数。第二列计算的是具有有限埃尔德什数的获胜者数量。第三列是具有有限埃尔德什数的获胜者的百分比。其余各列表示了这些获奖者中埃尔德什数的最小，最大，平均和中位数。
Judge Richard Posner, having coauthored with Alvin E. Roth, has an Erdős number of at most 4. Roberto Mangabeira Unger, a politician, philosopher and legal theorist who teaches at Harvard Law School, has an Erdős number of at most 4, having coauthored with Lee Smolin.
Judge Richard Posner, having coauthored with Alvin E. Roth, has an Erdős number of at most 4. Roberto Mangabeira Unger, a politician, philosopher and legal theorist who teaches at Harvard Law School, has an Erdős number of at most 4, having coauthored with Lee Smolin.
与Alvin E. Roth合作的法官Richard Posner的埃尔德什数最多为4。政治家，哲学家和法律理论家Roberto Mangabeira Unger与Lee Smolin曾经合作过，其埃尔德什数最多为4。
Angela Merkel, Chancellor of Germany from 2005 to the present, has an Erdős number of at most 5.
Angela Merkel, Chancellor of Germany from 2005 to the present, has an Erdős number of at most 5.
从2005年至今的德国总理安格拉·默克尔Angela Merkel的埃尔德什数最多为5。
Erdős numbers have been a part of the folklore of mathematicians throughout the world for many years. Among all working mathematicians at the turn of the millennium who have a finite Erdős number, the numbers range up to 15, the median is 5, and the mean is 4.65; For example, political scientist Steven Brams has an Erdős number of 2. In biomedical research, it is common for statisticians to be among the authors of publications, and many statisticians can be linked to Erdős via John Tukey, who has an Erdős number of 2. Similarly, the prominent geneticist Eric Lander and the mathematician Daniel Kleitman have collaborated on papers, and since Kleitman has an Erdős number of 1, a large fraction of the genetics and genomics community can be linked via Lander and his numerous collaborators. Similarly, collaboration with [[Gustavus Simmons]] opened the door for Erdős numbers within the cryptographic research community, and many linguists have finite Erdős numbers, many due to chains of collaboration with such notable scholars as Noam Chomsky (Erdős number 4), William Labov (3), Mark Liberman (3), Geoffrey Pullum (3), or Ivan Sag (4). There are also connections with arts fields.
Erdős numbers have been a part of the folklore of mathematicians throughout the world for many years. Among all working mathematicians at the turn of the millennium who have a finite Erdős number, the numbers range up to 15, the median is 5, and the mean is 4.65; For example, political scientist Steven Brams has an Erdős number of 2. In biomedical research, it is common for statisticians to be among the authors of publications, and many statisticians can be linked to Erdős via John Tukey, who has an Erdős number of 2. Similarly, the prominent geneticist Eric Lander and the mathematician Daniel Kleitman have collaborated on papers, and since Kleitman has an Erdős number of 1, a large fraction of the genetics and genomics community can be linked via Lander and his numerous collaborators. Similarly, collaboration with [[Gustavus Simmons]] opened the door for Erdős numbers within the cryptographic research community, and many linguists have finite Erdős numbers, many due to chains of collaboration with such notable scholars as Noam Chomsky (Erdős number 4), William Labov (3), Mark Liberman (3), Geoffrey Pullum (3), or Ivan Sag (4). There are also connections with arts fields.
多年以来，埃尔德什数在数学家之间一直盛行。在千年之交的所有在职数学家中，都伴随着一个有限埃尔德什数，数字范围最大为15，中位数为5，平均值为4.65。几乎每个具有有限埃尔德什数的人其数字都小于8。由于当今科学领域跨学科合作的频率很高，因此许多其他科学领域的大量非数学家也具有有限的埃尔德什数。例如，政治学家Steven Brams的埃尔德什数为2。在生物医学研究中，统计学家通常是出版物的作者，许多统计学家可以通过John Tukey（其埃尔德什数为2）与埃尔德什链接。同样，著名的遗传学家Eric Lander和数学家Daniel Kleitman在论文上进行了合作，由于Kleitman的埃尔德什数为1，因此可以通过Lander及其众多合作者将遗传学和基因组学领域的大部分联系起来。另外，与Gustavus Simmons的合作为密码研究界内的埃尔德什数打开了大门，许多语言学家拥有有限的埃尔德什数，这许多是由于与Noam Chomsky（埃尔德什数为4），William Labov（3）等著名学者的合作产生，Mark Liberman（3），Geoffrey Pullum（3）或Ivan Sag（4）。同时与艺术领域也有联系。
多年以来，埃尔德什数在数学家之间一直盛行。在千年之交的所有在职数学家中，都伴随着一个有限埃尔德什数，数字范围最大为15，中位数为5，平均值为4.65。几乎每个具有有限埃尔德什数的人其数字都小于8。由于当今科学领域跨学科合作的频率很高，因此许多其他科学领域的大量非数学家也具有有限的埃尔德什数。例如，政治学家Steven Brams的埃尔德什数为2。在生物医学研究中，统计学家通常是出版物的作者，许多统计学家可以通过John Tukey（其埃尔德什数为2）与埃尔德什链接。同样，著名的遗传学家Eric Lander和数学家Daniel Kleitman在论文上进行了合作，由于Kleitman的埃尔德什数为1，因此可以通过Lander及其众多合作者将遗传学和基因组学领域的大部分联系起来。另外，与Gustavus Simmons的合作为密码研究界内的埃尔德什数打开了大门，许多语言学家拥有有限的埃尔德什数，这许多是由于与Noam Chomsky（埃尔德什数为4），William Labov（埃尔德什数为3）等著名学者的合作产生，类似有Mark Liberman（3），Geoffrey Pullum（3）或Ivan Sag（4）。同时与艺术领域也有联系。
Also, Michael Barr suggests "rational Erdős numbers", generalizing the idea that a person who has written p joint papers with Erdős should be assigned Erdős number 1/p. From the collaboration multigraph of the second kind (although he also has a way to deal with the case of the first kind)—with one edge between two mathematicians for each joint paper they have produced—form an electrical network with a one-ohm resistor on each edge. The total resistance between two nodes tells how "close" these two nodes are.
Also, Michael Barr suggests "rational Erdős numbers", generalizing the idea that a person who has written p joint papers with Erdős should be assigned Erdős number 1/p. From the collaboration multigraph of the second kind (although he also has a way to deal with the case of the first kind)—with one edge between two mathematicians for each joint paper they have produced—form an electrical network with a one-ohm resistor on each edge. The total resistance between two nodes tells how "close" these two nodes are.
另外，迈克尔·巴尔Michael Barr曾建议使用“合理的埃尔德什数”，通俗的说就是与埃尔德共同撰写过p篇论文的人应被分配埃尔德什数的1/p。根据第二种的协作多重图（尽管他也有办法处理第一种情况），即在他们所合著的每篇联合论文中，两个数学家之间都有一条边，这个边缘上都有一个1欧姆电阻器的电网。两个节点之间的总电阻表明这两个节点有多“相近”。