As an example, one can consider the operation of adding integers Z. The doubling function φ(x) = 2x maps elements of Z to elements of the even integers 2Z. Since φ(a+b) = 2(a+b) = 2a+2b = φ(a)+φ(b), adding in Z is structurally identical as adding in 2Z (which makes this an example of isomorphism).
Automorphism
in abstract algebra an automorphism is an isomorphism from a group, or set of elements onto itself.
Group center
In abstract algebra the center of a group G denoted by Z(G) is the set of elements which commute with every element of G, meaning that for any element x of a group G, and an element y of Z(G), xy = yx. The center is equal to the intersection of the centralizers of the group elements.
The notation Z(G) comes from the fact that in German the word for center is "zentrum".
Caroline of Ansbach
Caroline of Ansbach was the Queen Consort of George II of Great Britain. During her time as queen she had a lot of power.
Caroline of Ansbach was born in Ansbach in Germany, the daughter of Johann Friedrich, Margrave of Brandenburg-Ansbach, and his second wife, Princess Eleanor Erdmuthe Louise of Saxe-Eisenach. Orphaned at an early age, Caroline grew up an intelligent, cultured and attractive woman, and was much sought-after as a bride.
She turned down the King of Spain because it would cause her to renounce the Protestant faith. Shortly after she met the son of the elector of Hanover They married in 1705. Over the next 30 years they had nine children.
George I of Great Britain died 11 June 1727. George Augustus ascended the throne as George II of Great Britain. Caroline held George on a string. When the Prime Minister Sir Robert Walpole met with the King he and Caroline had signs to communicate with each other. George never noticed.Both the King and Queen truly disliked their eldest son Frederick, Prince of Wales. Caroline once called him the Greatest Ass the world has ever known. They preferred Prince William, Duke of Cumberland over Fredrick.
Caroline died 20 November 1737. George was at her side. Caroline asked him to remarry on her deathbed, to which he replied "No, I shall only have mistresses" or in French, "Non, j'aurai seulement des maîtresses!" in tears.
George had Caroline buried in Westminster Abbey. He bought two coffins with removable sides, so after he died they would lie together again.
Caroline's nine pregnancies (from 1707-1724) resulted in eight live births. One of them, Prince George William (13 November 1717-17 February 1718), died in infancy. Seven of them lived to adulthood.
Caroline alongside Prince Albert, and Mary of Modena is regarded one of the most powerful British consorts in History.
Coset
If G is a group and H a subgroup of G, x is an element in G, and h is an element in H the cosets of H in G are defined as follows:
When xH = Hx we say that H is normal in G.
Lagrange's theorem (group theory)
Lagrange's theorem in group theory states if G is a finite group and H is a subgroup of G, then |H| (how many elements are in H, called the order of H) divides |G|. Moreover, the number of distinct left (right) cosets of H in G is |G|/|H|.
Applications
Label (philately)
Label or coupon or tab in philately is a part of sheet of stamps separated from them with perforation (or narrow white margin in imperforate stamps). It cannot be used for postage because it does not have face value and any indication of a postal administration that issued such stamps with labels. The notion of label should not be messed up with the term "gutter" or with a margin of a stamp sheet.
Sometimes, label is also a stamp-like adhesive of no postal value, often used for promotional purposes.
Liz Cohen
Liz Cohen (born 1973) is a performance artist. Her best known project is called "Bodywork". She changed an old East German small car (a Trabant), so that it can transform into a larger American car (a Chevrolet). She hired a personal trainer and dieted so she could appear in a bikini to be the model for the car.
She currently teaches at Cranbrook Academy of Art and lives in Detroit, Michigan.
Rainbow flag (LGBT)
The Rainbow flag has been the symbol of the LGBT (lesbian, gay, bisexual, and transgender) community since its creation in 1978. It is also called the 'Pride Flag.' The flag was created by Gilbert Baker of San Francisco, California in 1978.
The flag was first used at a LGBT pride parade in San Francisco in 1978, at that time it had eight colors. The next year it was changed to six colors, so it could be easier to make.
The flag was first made by Gilbert Baker, to be used instead of the pink triangle symbol used before. The pink triangle was used by Nazis to show who was homosexual. Homosexual people were sent to concentration camps in Nazi Germany, where they made homosexual men wear a pink triangle. This was like the yellow Star of David that Jews had to wear to show who was Jewish. LGBT people were not ashamed of the pink triangle, but they wanted a more positive, happier symbol.
In the LGBT version of the rainbow flag, each color has a different meaning:
It used to have a pink stripe, which meant "sexuality", but was removed because the pink cloth was unable to be mass produced.
Program specification
In the computer science, a program specification is the definition of what a computer program is expected to do. It can be informal, in which case it can be considered as a blueprint or user manual from a developer point of view, or formal, in which case it has a definite meaning defined in mathematical or programmatic terms. In practice, most successful specifications are written to understand and fine-tune applications that were already well-developed, although safety-critical software systems are often carefully specified prior to application development. Specifications are most important for external interfaces that must remain stable.
In systems engineering a specification is a document that clearly and accurately describes the essential technical requirements for items, materials, or services including the procedures by which it can be determined that the requirements have been met. Specifications help avoid duplication and inconsistencies, allow for accurate estimates of necessary work and resources, act as a negotiation and reference document for engineering changes, provide documentation of configuration, and allow for consistent communication among those responsible for the eight primary functions of Systems Engineering. They provide a precise idea of the problem to be solved so that they can efficiently design the system and estimate the cost of design alternatives. They provide guidance to testers for verification (qualification) of each technical requirement.
Debugging
In computer science, debugging is the process of finding (and correcting) errors in a computer program. When programmers think that the program may have an error, they debug the program. They look for an error, and when they find it, they try to correct it so the program will work correctly. This improves the quality of the product.
Hubble's law
Hubble's law or Hubble—Lemaître's law is the name for the observation that:
In effect, the space-time volume of the observable universe is expanding and Hubble's law is the direct physical observation of this. It is the basis for believing in the expansion of the universe and is evidence often cited in support of the Big Bang model.
Although widely attributed to Edwin Hubble, the law was first derived from the General Relativity equations by Georges Lemaître in a 1927 article. There he proposed that the Universe is expanding, and suggested a value for the rate of expansion, now called the Hubble constant. Two years later Edwin Hubble confirmed the existence of that law and determined a more accurate value for the constant that now bears his name. The recession velocity of the objects was inferred from their redshifts, many measured earlier by Vesto Slipher in 1917 and related to velocity by him.