The Thai language is the national and official language of Thailand. It is the first language of the Thai people, Thailand's main ethnic group. Thai is closely related to the Lao language. In fact, some dialects of Thai and Lao are mutually intelligible, meaning that speakers of one language can talk with and understand speakers of the other language without knowing the other's language.
The Thai alphabet has 44 consonants:
The alphabet has Thai numbers:
New England Patriots
The New England Patriots is an American football team in the National Football League or the NFL. They are in the American Football Conference or the AFC. Their starting quarterback is Cam Newton. Cody Kessler plays the second string quarterback. The Patriots play in Gillette Stadium on Foxborough, midway between Boston, Massachusetts and Providence, Rhode Island.
In the 2007 season, the Patriots have won all 16 games and became the first team with 16 wins and no losses. They would win 2 more playoff games and enter the Super Bowl at 18-0. However, they did not complete a historic 19-0 season, because they lost to the Giants. The Giants scored 17 points when David Tyree made a helmet catch and the Patriots scored 14 points.
The New England Patriots won the Super Bowl in 2001, 2003, 2004, 2014, 2016, and 2018. They play in Foxboro, Massachusetts. They lost to the New York Giants in Super Bowl XLVI on February 5, 2012 in a rematch of Super Bowl XLII which their perfect season got spoiled. This time, the Giants scored 21 points and the Patriots scored 17 points.
The Patriots also won Super Bowl XLIX against the Seattle Seahawks when Russell Wilson threw an interception on the final play. The Patriots scored 28 points and the Seahawks scored 24 points.
The Patriots were originally called the "Boston Patriots", and played in Fenway Park. Their name comes from the city of Boston's importance in the American Revolution.
Digenea
Digenea are a type of flatworms. They are parasites. They have two suckers, usually. One of them is ventral ("on the belly"), and the other is oral (where the mouth is).
Aspidogastrea
The Aspidogastrea (gr. "aspid" - shield, "gaster" - stomach/pouch) is a small group of flukes. There are about 80 species.
The group is a subclass of the Trematoda, and sister group to the Digenea. Species are between approximately one mm to several cm long. They are parasites of freshwater and marine molluscs and vertebrates (cartilaginous and bony fishes and turtles). Maturation may occur in the mollusc or vertebrate host.
None of the species has any economic importance, but the group is of very great interest to biologists because it has several characters which appear to be basal.
Theropod
Theropods ('beast foot') are a group of bipedal saurischian dinosaurs.
Although they were primarily carnivorous, a number of theropod groups evolved herbivory, omnivory and insectivory.
Today, they are represented by the 9,300 living species of birds, which evolved in the Upper Jurassic from small feathered coelurosaurian dinosaurs.
Among the features linking theropods to birds are bipedalism, the three-toed foot, a furcula (wishbone), air-filled bones, feathers and brooding of the eggs.
Theropods first appear in the earliest part of the Upper Triassic about 230 million years ago. They were the sole large terrestrial carnivores from the Lower Jurassic until the close of the Cretaceous, about 65 million years ago.
The earliest and most primitive of the theropod dinosaurs were:
Iwasawa theory
In number theory, Iwasawa theory is a Galois module theory of ideal class groups, started by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur thought about generalizations of Iwasawa theory to Abelian Varieties. Later, in the early 90s, Ralph Greenberg has suggested an Iwasawa theory for motives.
The first thing Iwasawa noticed was that there are towers of fields in algebraic number theory, having Galois group isomorphic with the additive group of p-adic integers. That group, usually written Γ in the theory and with multiplicative notation, can be found as a subgroup of Galois groups of infinite field extensions (which are by their nature pro-finite groups). The group formula_1 is the inverse limit of the additive groups formula_2, where "p" is the fixed prime number and formula_3. We can express this by Pontryagin duality in another way: Γ is dual to the discrete group of all formula_4-power roots of unity in the complex numbers.
Let formula_5 be a primitive formula_4-th root of unity and look at the following tower of number fields:
where formula_8 is the field generated by a primitive formula_9-th root of unity. This tower of fields has a union formula_10. Then the Galois group of formula_11 over formula_12 is isomorphic with formula_1; because the Galois group of formula_14 over formula_15 is formula_16. In order to get an interesting Galois module here, Iwasawa took the ideal class group of formula_8, and let formula_18 be its formula_4-torsion part. There are norm mappings formula_20 when formula_21, and so an inverse system. Letting formula_22 be the inverse limit, we can say that formula_23 acts on formula_24, and it is good to have a description of this action.
The motivation here was undoubtedly that the formula_4-torsion in the ideal class group of formula_15 had already been identified by Kummer as the main obstacle to the direct proof of Fermat's last theorem. What Iwasawa did that was new, was to go 'off to infinity' in a new direction. In fact, formula_22 is a module over the group ring formula_28. This is a well-behaved ring (regular and two-dimensional), meaning that it is quite possible to classify modules over it, in a way that is not too coarse.
From this beginning, in the 1950s, a good-sized theory has been built up. A basic connection was noticed between the module theory, and the p-adic L-functions that were defined in the 1960s by Kubota and Leopoldt. The latter begin from the Bernoulli numbers, and use interpolation to define p-adic analogues of the Dirichlet L-functions. It became clear that the theory might be able to move ahead finally from Kummer's century-old results on regular primes.
The main conjecture of Iwasawa theory was formulated as an assertion that two ways of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was eventually proved by Barry Mazur and Andrew Wiles for Q, and for all totally real number fields by Andrew Wiles. These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (so-called Herbrand-Ribet theorem).
More recently, also modeled upon Ribet's method, Chris Skinner and Eric Urban have announced a proof of a "main conjecture" for GL(2). An easier proof of the Mazur-Wiles theorem can be found by using Euler systems as developed by Kolyvagin (see Washington's book). Other generalizations of the main conjecture proved using the Euler system method have been found by Karl Rubin, amongst others.
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two operations: addition (+) and multiplication (•). These two operations must follow special rules to work together in a ring.
Mathematicians use the word "ring" this way because a mathematician named David Hilbert used the German word "Zahlring" to describe something he was writing about. The integers, the rational numbers, the real numbers and the complex numbers are all famous examples of rings. There are other, more unusual examples of rings, however they all obey the special rules below.
The operations are used to combine two elements to form a third element. There are a few rules the set and the operations must obey to qualify as a ring. These are called ring axioms:
Some rings have additional properties from those mentioned above, these rings get special names: