James Burton Ax was an American mathematician who proved several results in algebra and number theory by using model theory. He shared, with Simon B. Kochen, the seventh Frank Nelson Cole Prize in Number Theory, awarded for a series of three joint papers on Diophantine problems. James Ax graduated from Peter Stuyvesant High School in New York City and the Brooklyn Polytechnic University, he earned his Ph. D. from the University of California, Berkeley in 1961 under the direction of Gerhard Hochschild, with a dissertation on The Intersection of Norm Groups. After one year at Stanford University, he joined the mathematics faculty at Cornell University, he spent the academic year 1965–1966 at Harvard University on a Guggenheim Fellowship. In 1969, he moved from Cornell to the mathematics department at Stony Brook University and remained on the faculty until 1977, when he retired from his academic career. In 1970 he was an Invited Speaker at the ICM in Nice with talk Transcendence and differential algebraic geometry.
In the 1970s, he worked on the fundamentals of physics, including an axiomatization of space-time and the group theoretical properties of the axioms of quantum mechanics. In the 1980s, he and James Simons founded a quantitative finance firm, Axcom Trading Advisors, acquired by Renaissance Technologies and renamed the Medallion Fund; the latter fund was named after the Cole Prize won by James Ax and the Veblen Prize won by James Simons. In the early 1990s Ax retired from his financial career and went to San Diego in California, where he studied further on the foundations of quantum mechanics and attended, at the University of California, San Diego, courses on playwriting and screenwriting; the Ax Library in the Department of Mathematics at the University of California, San Diego houses his mathematical books. Ax, James. "The intersection of norm groups". Transactions of the American Mathematical Society. 105: 462–474. Doi:10.1090/s0002-9947-1962-0141651-2. MR 0141651. Ax, James. "A field of cohomological dimension 1, not C1".
Bulletin of the American Mathematical Society. 71: 717–718. Doi:10.1090/s0002-9904-1965-11354-4. MR 0199177. Ax, James. "On the undecidability of power series fields". Proceedings of the American Mathematical Society. 16: 846. Doi:10.1090/s0002-9939-1965-0177890-2. MR 0177890. Ax, James. "Proof of some conjectures on cohomological dimension". Proceedings of the American Mathematical Society. 16: 1214–1221. Doi:10.1090/s0002-9939-1965-0188263-0. MR 0188263. "On the units of an algebraic number field". Illinois Journal of Mathematics. 9: 584–589. 1965. MR 0181630. Ax, James. "Solving diophantine problems modulo every prime". Annals of Mathematics. Series 2. 85: 161–183. Doi:10.2307/1970438. JSTOR 1970438. Ax, James. "The elementary theory of finite fields". Annals of Mathematics. Series 2. 88: 239–271. Doi:10.2307/1970573. JSTOR 1970573. Ax, James. "Zeros of polynomials over local fields—The Galois action". Journal of Algebra. 15: 417–428. Doi:10.1016/0021-869390069-4. Ax, James. "On Schanuel's conjectures". Annals of Mathematics.
Series 2. 93: 252–268. Doi:10.2307/1970774. JSTOR 1970774. Ax–Grothendieck theorem Ax–Kochen theorem Leopoldt's conjecture Schanuel's conjecture James Ax at the Mathematics Genealogy Project James B. Ax Library - at UCSD
Phitsanulok province, with a registered population as of 31 December 2018 of 866,891 is led by a governor and is divided into 9 districts. Each district is led by a districts chief. Governor, district chiefs and district clerks are appointed by the central government. There are 93 subdistricts each led by a subdistrict chief, further divided into 1,052 villages each led by a village chief. Subdistrict chiefs and village chiefs are elected by local citizens; as of 26 November 2019 there are: one Phitsanulok Provincial Administrative Organisation - PPAO and 26 municipal areas in the province. Phitsanulok has city status, Aranyik has 24 subdistrict municipalities; the non-municipal areas are administered by 76 Subdistrict Administrative Organisations - SAO. All mayors and councillors are directly elected by the local citizens. Municipalities have communities, although not directly chosen by the local citizens, which provides advice and recommendations to local administrative organisations, they promote and support community participation and enterprises at the district level and subdistrict villages.
Admin. Div. of Mueang Phitsanulok district Admin. Div. of Nakhon Thai district Admin. Div. of Chat Trakan district Admin. Div. of Bang Rakam district Admin. Div. of Bang Krathum district Admin. Div. of Phrom Phiram district Admin. Div. of Wat Bot district Admin. Div. of Wang Thong district Admin. Div. of Noen Maprang Thai district The majority of the information for this wiki-article is based on "Statistics and house statistics for the year 2018" from Registration Office Department of the Interior. Reference date is 31 December 2018