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

Rainbow

A rainbow is a meteorological phenomenon that is caused by reflection, refraction and dispersion of light in water droplets resulting in a spectrum of light appearing in the sky. It takes the form of a multicoloured arc. Rainbows caused by sunlight always appear in the section of sky directly opposite the sun.

Rainbows can be full circles; however, the average observer sees only an arc formed by illuminated droplets above the ground, and centred on a line from the sun to the observer's eye.

In a primary rainbow, the arc shows red on the outer part and violet on the inner side. This rainbow is caused by light being refracted when entering a droplet of water, then reflected inside on the back of the droplet and refracted again when leaving it.

In a double rainbow, a second arc is seen outside the primary arc, and has the order of its colours reversed, with red on the inner side of the arc.

Overview

A rainbow is not located at a specific distance from the observer, but comes from an optical illusion caused by any water droplets viewed from a certain angle relative to a light source. Thus, a rainbow is not an object and cannot be physically approached. Indeed, it is impossible for an observer to see a rainbow from water droplets at any angle other than the customary one of 42 degrees from the direction opposite the light source. Even if an observer sees another observer who seems "under" or "at the end of" a rainbow, the second observer will see a different rainbow—farther off—at the same angle as seen by the first observer.

Lullaby and the Ceaseless Roar

Lullaby and the Ceaseless Roar (rendered as Lullaby and … The Ceaseless Roar on the album cover) is the tenth solo album by English rock singer Robert Plant. It was released on 8 September 2014 on Nonesuch/Warner Bros. Records. It was also Plant's first studio album with his backing band the Sensational Space Shifters, although the name of the band is not credited on the front cover.

Critical reception

Lullaby and the Ceaseless Roar was received favourably by the majority of music critics. At Metacritic, which assigns a normalised rating out of 100 to reviews from a selection of critics, the album received an average score of 81, based on 23 reviews. The album was named one of the 50 best albums of 2014 by NPR Music.

Track listing

Sources:

Personnel

Rainbow (2005 film)

Rainbow (simplified Chinese: 我心飞翔; traditional Chinese: 我心飛翔; pinyin: Wǒ Xīn Fēi Xiáng; literally: "my heart is flying") is a 2005 Chinese film written and directed by Gao Xiaosong, starring Chen Daoming.

Cast

External links