End
End or Ending may refer to:
End
END
Ending
See also
End (graph theory)
In the mathematics of infinite graphs, an end of a graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as equivalence classes of infinite paths, as havens describing strategies for pursuit-evasion games on the graph, or (in the case of locally finite graphs) as topological ends of topological spaces associated with the graph.
Ends of graphs may be used (via Cayley graphs) to define ends of finitely generated groups. Finitely generated infinite groups have one, two, or infinitely many ends, and the Stallings theorem about ends of groups provides a decomposition for groups with more than one end.
Definition and characterization
Ends of graphs were defined by Rudolf Halin (1964) in terms of equivalence classes of infinite paths. A ray in an infinite graph is a semi-infinite simple path; that is, it is an infinite sequence of vertices v0, v1, v2, ... in which each vertex appears at most once in the sequence and each two consecutive vertices in the sequence are the two endpoints of an edge in the graph. According to Halin's definition, two rays r0 and r1 are equivalent if there is another ray r2 (not necessarily different from either of the first two rays) that contains infinitely many of the vertices in each of r0 and r1. This is an equivalence relation: each ray is equivalent to itself, the definition is symmetric with regard to the ordering of the two rays, and it can be shown to be transitive. Therefore, it partitions the set of all rays into equivalence classes, and Halin defined an end as one of these equivalence classes.
End (film)
End is a 1984 fiction film by Mahmoud Shoolizadeh; is a story about the children who live near the railways and their lives are full of ups and downs. This film displays the story of a small child who lives in south of Tehran during times of social problems. The film analyses an unjust and unfair society.
Director's View
Inflation, unemployment, divorce and social poverty are the major problems in modern Iran. If the officials, social and cultural experts and planners do not find solutions to these problems, the future society would face crisis due to the consequences.
Technical specifications and Film crew
Empire (1962 TV series)
Empire is an hour-long Western television series set on a 1960s 500,000-acre (2,000 km2) ranch in New Mexico, starring Richard Egan, Terry Moore, and Ryan O'Neal. It ran on NBC from September 25, 1962, to May 14, 1963.
In the second abbreviated season, from September 24 to December 31, 1963, it was renamed Redigo after Egan's title character, Jim Redigo, the general manager of the fictitious Garrett ranch in Empire, and reduced to a half-hour.
Cast
Egan starred in the series at the age of forty-one, having previously been in the hit film A Summer Place, with the catchy theme song. Redigo was a rare ranch manager, having a Master of Business Administration degree. The ranch was located somewhere in the American Southwest, but the exact location was never pinpointed. The Garretts did have an empire. Besides ranching they were involved in oil, agriculture, and mining. The series has unusually- titled episodes.
Empire also featured 22-year-old Ryan O'Neal, some two years before he gained greater recognition as Rodney Harrington in ABC's Peyton Place. O'Neal, who began acting in 1959, played the son, Tal Garrett. Terry Moore portrayed O'Neal's 33-year-old sister, Connie, who had a romantic interest in Redigo. Their mother and ranch matriarch, Lucia, was played by 53-year-old Anne Seymour (1909–1988).
Empire (EP)
Empire is the first label release by American metalcore band The Word Alive. It was released on July 21, 2009 on Fearless Records. The album was produced by Andrew Wade and charted at number 15 on Billboard's Top Heatseekers. After this EP, founding drummer Tony Aguilera would be kicked from the band, it is however, the first release with lead vocalist Tyler "Telle" Smith.
Re-recordings
The song "Casanova Rodeo" was originally a part of the unreleased The Word Alive EP. After the release of Empire, the track "Battle Royale" became one of The Word Alive's most known and notable songs and was re-recorded for the band's debut full-length album, Deceiver.
Track listing
Personnel
References
Empire (Super8 & Tab album)
Empire is the debut studio album by Finnish Trance duo Super8 & Tab, released on September 13, 2010.
Track listing
Singles
Black Is the New Yellow (Feat. Anton Sonin)
Empire (Feat. Jan Burton)
Empire (The Remixes - Feat. Jan Burton)
Irufushi
Mercy (Feat. Jan Burton)
My Enemy (Feat. Julie Thompson)
Image (category theory)
Given a category C and a morphism
in C, the image of f is a monomorphism 
satisfying the following universal property:
such that 
.
and a monomorphism 
such that 
, there exists a unique morphism 
such that 
.Remarks:
The image of f is often denoted by im f or Im(f).
One can show that a morphism f is epic if and only if f = im f.
Examples
In the category of sets the image of a morphism is the inclusion from the ordinary image 
to 
. In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets.
In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism 
can be expressed as follows:
