Abelian groupIn mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.
HomomorphismIn algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the Ancient Greek language: ὁμός () meaning "same" and μορφή () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).
Isomorphism theoremsIn mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.
Ken GoodmanKenneth Goodman (December 23, 1927 - March 12, 2020) was Professor Emeritus, Language Reading and Culture, at the University of Arizona. He is best known for developing the theory underlying the literacy philosophy of whole language. Goodman began teaching at Wayne State University in 1962. His research focused on reading in public schools. While at Wayne State University, Goodman developed miscue analysis, a process of assessing students' reading comprehension based on samples of oral reading.
Tallahassee, FloridaTallahassee (ˌtæləˈhæsi ) is the capital city of the U.S. state of Florida. It is the county seat and only incorporated municipality in Leon County. Tallahassee became the capital of Florida, then the Florida Territory, in 1824. In 2022, the population was 201,731, making it the eighth-most populous city in the state of Florida. The population of the Tallahassee metropolitan area was 385,145 . Tallahassee is the largest city in the Florida Big Bend and Florida Panhandle region, and the main center for trade and agriculture in the Florida Big Bend and Southwest Georgia regions.
MonomorphismIn the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of , a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow f : X → Y such that for all objects Z and all morphisms g1, g2: Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below.
Structure theorem for finitely generated modules over a principal ideal domainIn mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.
Order isomorphismIn the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that either of the orders can be obtained from the other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are order embeddings and Galois connections.
Quotient ringIn ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring R and a two-sided ideal I in R, a new ring, the quotient ring R / I, is constructed, whose elements are the cosets of I in R subject to special + and ⋅ operations.
Upsilon OctantisUpsilon Octantis (Upsilon Oct), Latinized from υ Octantis, is a solitary star in the southern circumpolar constellation Octans. It is faintly visible to the naked eye with an apparent magnitude of 5.75 and is located 343 light years away from the Solar System. However, it is receding with a heliocentric radial velocity of 19km/s. Upsilon Oct has a stellar classification of K0 III, indicating that it is a red giant. At present it has 2.12 times the mass of the Sun but has expanded to 10.3 times its girth.
Gamma2 OctantisDISPLAYTITLE:Gamma2 Octantis γ2 Octantis, Latinized to Gamma2 Octantis (Gamma2 Oct), is a solitary star in the southern circumpolar constellation Octans. It has an apparent magnitude of 5.72, allowing it to be faintly seen with the naked eye. Parallax measurements place the object at a distance of 320 light years and is currently receding with a heliocentric radial velocity of 27km/s. Gamma2 Oct has a stellar classification of K0 III, indicating that it is a red giant. At present it has 115% the mass of the Sun but has expanded to 10.
World Championship of Online PokerThe World Championship of Online Poker (WCOOP) is an online poker tournament series sponsored by PokerStars. It is played on the PokerStars website in September. Established in 2002, WCOOP is PokerStars' attempt to establish the online equivalent of the World Series of Poker. The WCOOP tournament series is the largest of its kind on the Internet. The fifteen WCOOP events in 2005 generated $12,783,900 in prize money, making it not only the biggest ever online poker event, but the third biggest poker series (live or online) in all of 2005.
