Ulugh Beg and Cahit Arf were both a what?
mathematician
Ulugh Beg
Mīrzā Muhammad Tāraghay bin Shāhrukh (Chagatai: میرزا محمد طارق بن شاہ رخ , Persian: میرزا محمد تراغای بن شاہ رخ‎ ‎ ), better known as Ulugh Beg () (March 22, 1394 in Sultaniyeh, Persia – October 27, 1449, Samarkand), was a Timurid ruler as well as an astronomer, mathematician and sultan.
 His commonly known name is not truly a personal name, but rather a moniker, which can be loosely translated as "Great Ruler" or "Patriarch Ruler" and was the Turkic equivalent of Timur's Perso-Arabic title "Amīr-e Kabīr".
 His real name was "Mīrzā Mohammad Tāraghay bin Shāhrukh".
 Ulugh Beg was also notable for his work in astronomy-related mathematics, such as trigonometry and spherical geometry.
 He built the great Ulugh Beg Observatory in Samarkand between 1424 and 1429.
 It was considered by scholars to have been one of the finest observatories in the Islamic world at the time and the largest in Central Asia.
 He built the Ulugh Beg Madrasah (1417–1420) in Samarkand and Bukhara, transforming the cities into cultural centers of learning in Central Asia.
 He was also a mathematician of the 15th century — albeit his mental aptitude was perseverance rather than any unusual endowment of intellect.
 His observatory is situated in Samarkand which is in Uzbekistan.
 He ruled Uzbekistan, Tajikistan, Turkmenistan, Kyrgyzstan, southern Kazakhstan and most of Afghanistan from 1411 to 1449.
Cahit Arf
Cahit Arf (] ; 11 October 1910 – 26 December 1997) was a Turkish mathematician.
 He is known for the Arf invariant of a quadratic form in characteristic 2 (applied in knot theory and surgery theory) in topology, the Hasse–Arf theorem in ramification theory, Arf semigroups, and Arf rings.
Zij-i Sultani
Zīj-i Sultānī (Persian: زیجِ سلطانی‎ ‎ ) is a Zij astronomical table and star catalogue that was published by Ulugh Beg in 1438-1439.
 It was the joint product of the work of a group of Muslim astronomers working under the patronage of Ulugh Beg at Samarkand's Ulugh Beg Observatory.
 These astronomers included Jamshīd al-Kāshī and Ali Qushji, among others. Battle of Tarnab (1448)
In the spring of 1448, Ala-ud-Daulah Mirza was making preparations for the celebration of his son's circumcision ceremony.
 He had announced a general relief from taxes to all citizens of Herat when he heard the news of Ulugh Beg crossing the Amu Darya with 90,000 troops.
 He hastily made his way towards the Tarnab through the Sanjab Pass.
 The two forces met at Tarnab and after a brief engagement, Ulugh Beg decisively defeated Ala-ud-Daulah Mirza who fled towards Mashad and the towards Quchan in the protection of his brother Abul-Qasim Babur Mirza.
 Ulugh Beg followed up his victory by taking Herat and then Mashad.
 Goharshad and many Tarkhans left Herat before Ulugh Beg's arrival. Ulugh Beg Observatory
The Ulugh Beg Observatory is an observatory in Samarkand, Uzbekistan.
 Built in the 1420s by the Timurid astronomer Ulugh Beg, it is considered by scholars to have been one of the finest observatories in the Islamic world.
 Some of the famous Islamic astronomers who worked at the observatory include Al-Kashi, Ali Qushji, and Ulugh Beg himself.
 The observatory was destroyed in 1449 and rediscovered in 1908. Siege of Herat (1448)
Ulugh Beg and his son Abdal-Latif Mirza marched towards Herat in the spring of 1448 in order to take Khurasan from his nephew, Ala-ud-Daulah Mirza, who had escaped to Quchan after the defeat at the Battle of Tarnab.
 They easily took the city except the Qila Ikhtiyar-al-Din and fort Neretu both of which offered stiff resistance.
 At this point only the advance troops had reached Herat, which was unable to take the citadel or the Fort Neretu.
 The Bakharz Tajik archers offered stiff resistance and beat back many assaults by the Timurids of Samarkand.
 Eventually Ulugh Beg arrived 17 days after the siege had begun; after which all resistance crumbled before him in no time.
 Abdal-Latif Mirza succeeded in capturing the citadel Qila Ikhtiyar-al-Din in which he was imprisoned after the debacle at Nishapur and now here he managed to take 4,000 Iranian toman in coins.
 They followed up their victory by taking Mashad.
 Ulugh Beg was unable to pursue his nephews and instead decided to return to Herat leaving his son Abdal-Latif Mirza in charge at Mashad. Revolt of Abdal-Latif Mirza
It is not clear what led Abdal-Latif Mirza to revolt against his father Ulugh Beg in 1449 C.E.
 Many theories abound; one being that he was raised by Goharshad in Herat and not by his father Ulugh Beg who was governor of Samarkand during the reign of Shahrukh Mirza, therefore Abdal-Latif Mirza was not attached to his father.
 Ulugh Beg on the other hand preferred his second son Abdal-Aziz Mirza to be his heir, whom he had raised himself and groomed to take over.
 Another reason was that when the Baysonqor brothers invaded Khurasan, Ulugh Beg left his son Abdal-Latif Mirza, to deal with a difficult situation making it look as if it were his failures that led to the loss of Khurasan; this Abdal-Latif Mirza resented deeply.
 And finally, during a revolt in Balkh in Spring of 1449 by a certain Miranshah, which was crushed by Abdal-Latif Mirza, was discovered a letter from Ulugh Beg to this Miranshah ordering him to instigate the revolt.
 This story is not corroborated by any other source than Mirkhond but is given as a last excuse for all out war with his father.
 The real reason of this revolt may never be fully understood but Abdal-Latif Mirza marched in haste towards Samarkand after abolishing the state taxes on trade through Balkh and taking over the river boats on Amu Darya. Occupation of Balkh (1447)
With the death of Shahrukh Mirza in 1447 began the long drawn out Second Timurid Succession Crisis.
 His only surviving heir was his son Ulugh Beg who was at that time viceroy of Central Asia at Samarkand.
 Goharshad and Abdal-Latif Mirza were with Shahrukh Mirza when he died on his way back to Khurasan from Iran.
 Abdal-Latif Mirza became the commander of his grandfather's army and in conjunction with his father Ulugh Beg began operations against his cousins.
 As soon as Ulugh Beg heard of his father's death, he mobilized his forces and reached Amu Darya in order to take Balkh from his nephews.
 Balkh belonged to Ulugh Beg's brother Mirza Muhammad Juki who died in 1444.
 Balkh was divided among his sons Mirza Muhammad Qasim and Mirza Abu Bakr.
 However, Mirza Abu Bakr took his older brothers' possessions when Shahrukh Mirza died.
 Ulugh Beg summoned Abu Bakr to his court and promised him his daughter in marriage.
 But while there he had him convicted of plotting against him and imprisoned at Kok Serai in Samarkand where he was later executed.
Ulugh Beg then marched on Balkh and took that province unopposed. Battle of Farhadgerd
While Abul-Qasim Babur Mirza was away from Herat crushing the revolt of Amir Hendugha in Asterabad, Ala-ud-Daulah Mirza, his older brother, managed to escape from prison in Herat and went straight to his youngest brother Sultan Muhammad Mirza's province of Fars seeking his protection.
 Sultan Muhammad Mirza and Ala-ud-Daulah Mirza then marched with a large army and invaded Khurasan in 1449.
 This was the same time as the revolt of Abdal-Latif Mirza in Balkh against his father Ulugh Beg at Samarkand.
 While the father and son were busy facing off at the Amu Darya in the north, the Baysonqor brothers were about to engage in battle in Khurasan.
 Abul-Qasim Babur Mirza marched to face his brothers in battle and the two armies met at Farhadgerd.
 Sultan Muhammad Mirza and Ala-ud-Daulah Mirza defeated Abul-Qasim Babur Mirza who fled to the castle of Omad.
 Sultan Muhammad Mirza entered Herat and freed Ibrahim Mirza son of Ala-ud-Daulah Mirza.
 Abdal-Latif Mirza, who before the battle had sent an envoy to Abul-Qasim Babur Mirza professing peace during his revolt against his father Ulugh Beg, now congratulated Sultan Muhammad Mirza in taking Herat.
 But Sultan Muhammad Mirza was saddened by Ulugh Beg's defeat at Dimishq and his murder en route to Makkah by his son; however, he decided not to pursue a war in Transoxiana and instead wanted to concentrate on his holdings in Iraq-i-Ajam, Fars and now Khurasan as well. Arf invariant of a knot
In the mathematical field of knot theory, the Arf invariant of a knot, named after Cahit Arf, is a knot invariant obtained from a quadratic form associated to a Seifert surface.
 If "F" is a Seifert surface of a knot, then the homology group H("F", Z/2Z) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an imbedded circle representing an element of the homology group.
 The Arf invariant of this quadratic form is the Arf invariant of the knot.