<P> The system of bureaucracy created by Kublai Khan reflected various cultures in the empire, including that of the Hans, Khitans, Jurchens, Mongols, and Tibetan Buddhists . While the official terminology of the institutions may indicate the government structure was almost purely that of native Chinese dynasties, the Yuan bureaucracy actually consisted of a mix of elements from different cultures . The Chinese - style elements of the bureaucracy mainly came from the native Tang, Song, as well as Khitan Liao and Jurchen Jin dynasties . Chinese advisers such as Liu Bingzhong and Yao Shu gave strong influence to Kublai's early court, and the central government administration was established within the first decade of Kublai's reign . This government adopted the traditional Chinese tripartite division of authority among civil, military, and censorial offices, including the Central Secretariat (Zhongshu Sheng) to manage civil affairs, the Privy Council (Chinese: 樞 密 院) to manage military affairs, and the Censorate to conduct internal surveillance and inspection . The actual functions of both central and local government institutions, however, showed a major overlap between the civil and military jurisdictions, due to the Mongol traditional reliance on military institutions and offices as the core of governance . Nevertheless, such a civilian bureaucracy, with the Central Secretariat as the top institution that was (directly or indirectly) responsible for most other governmental agencies (such as the traditional Chinese - style Six Ministries), was created in China . At various times another central government institution called the Department of State Affairs (Shangshu Sheng) that mainly dealt with finance was established (such as during the reign of Külüg Khan or Emperor Wuzong), but was usually abandoned shortly afterwards . </P> <P> While the existence of these central government departments and the Six Ministries (which had been introduced since the Sui and Tang dynasties) gave a Sinicized image in the Yuan administration, the actual functions of these ministries also reflected how Mongolian priorities and policies reshaped and redirected those institutions . For example, the authority of the Yuan legal system, the Ministry of Justice, did not extend to legal cases involving Mongols and Semuren, who had separate courts of justice . Cases involving members of more than one ethnic group were decided by a mixed board consisting of Chinese and Mongols . Another example was the insignificance of the Ministry of War compared with native Chinese dynasties, as the real military authority in Yuan times resided in the Privy Council . </P> <P> Advances in polynomial algebra were made by mathematicians during the Yuan era . The mathematician Zhu Shijie (1249--1314) solved simultaneous equations with up to four unknowns using a rectangular array of coefficients, equivalent to modern matrices . Zhu used a method of elimination to reduce the simultaneous equations to a single equation with only one unknown . His method is described in the Jade Mirror of the Four Unknowns, written in 1303 . The opening pages contain a diagram of Pascal's triangle . The summation of a finite arithmetic series is also covered in the book . </P> <P> Guo Shoujing applied mathematics to the construction of calendars . He was one of the first mathematicians in China to work on spherical trigonometry . Gou derived a cubic interpolation formula for his astronomical calculations . His calendar, the Shoushi Li (授時 暦) or Calendar for Fixing the Seasons, was disseminated in 1281 as the official calendar of the Yuan dynasty . The calendar may have been influenced solely by the work of Song dynasty astronomer Shen Kuo or possibly by the work of Arab astronomers . There are no explicit signs of Muslim influences in the Shoushi calendar, but Mongol rulers were known to be interested in Muslim calendars . Mathematical knowledge from the Middle East was introduced to China under the Mongols, and Muslim astronomers brought Arabic numerals to China in the 13th century . </P>

Which group held the lowest rank in yuan china