A dimension is a structure that categorizes facts and measures in order to enable users to answer business questions. Commonly used dimensions are people, products, place and time.
In a data warehouse, dimensions provide structured labeling information to otherwise unordered numeric measures. The dimension is a data set composed of individual, non-overlapping data elements. The primary functions of dimensions are threefold: to provide filtering, grouping and labelling.
These functions are often described as "slice and dice". Slicing refers to filtering data. Dicing refers to grouping data. A common data warehouse example involves sales as the measure, with customer and product as dimensions. In each sale a customer buys a product. The data can be sliced by removing all customers except for a group under study, and then diced by grouping by product.
Typically dimensions in a data warehouse are organized internally into one or more hierarchies. "Date" is a common dimension, with several possible hierarchies:
Directed by The Malloys, the music video for "Dimension" was first aired in the week of 13 February 2006. Prior to this, the video was featured on the 2006 extended play (EP) Dimensions.
Critical reception
In a review of Wolfmother for Blender, writer Jonah Weiner identified "Dimension" as an example of the band "at [their] hardest", describing it as an "acid anthem".NME reviewer James Jam described the song as "a throb of gonzometal not unlike Black Sabbath playing Motown".
A field k has Krull dimension 0; more generally, k[x1, ..., xn] has Krull dimension n. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent.
Explanation
We say that a chain of prime ideals of the form
has length n. That is, the length is the number of strict inclusions, not the number of primes; these differ by 1. We define the Krull dimension of to be the supremum of the lengths of all chains of prime ideals in .
Blizzard Entertainment's bestselling real-time strategy game series StarCraft revolves around interstellar affairs in a distant sector of the galaxy, with three species and multiple factions all vying for supremacy in the sector. The playable species of StarCraft include the Terrans, humans exiled from Earth who excel at adapting to any situation; the Zerg, a race of insectoids obsessed with assimilating other races in pursuit of genetic perfection; and the Protoss, a humanoid species with advanced technology and psionic abilities, attempting to preserve their civilization and strict philosophical way of living from the Zerg. Each of these races has a single campaign in each StarCraft real-time strategy game. In addition to these three, various non-playable races have also been part of the lore of the StarCraft series; the most notable of these is the Xel'Naga, a race which features prominently in the fictional histories of the Protoss and Zerg races.
The original game has sold over 10 million copies internationally, and remains one of the most popular games in the world. One of the main factors responsible for StarCraft's positive reception is the attention paid to the three unique playable races, for each of which Blizzard developed completely different characteristics, graphics, backstories and styles of gameplay, while keeping them balanced in performance against each other. Previous to this, most real-time strategy games consisted of factions and races with the same basic play styles and units with only superficial differences. The use of unique sides in StarCraft has been credited with popularizing the concept within the real-time strategy genre. Contemporary reviews of the game have mostly praised the attention to the gameplay balance between the species, as well as the fictional stories built up around them.
Valkyrie III, officially named Valkyrie, was the unsuccessful British challenger of the ninth America's Cup race in 1895 against American defender Defender.
Design
Valkyrie III, a keel cutter, was designed by George Lennox Watson and built at the D&W Henderson on the River Clyde in 1894-1895 for a syndicate including Lord Londsale, Lord Wolverton, Captain Henry McCalmont and headed by Lord Dunraven of the Royal Yacht Squadron.
Valkyrie III had a steel frame, a hull planked with American elm and teak, and a pine deck.
Career
Valkyrie III was launched on May 27, 1895. After a few mixed test races (for which she was later criticized for lack of previous competition) she sailed to New York to prepare for the ninth America's Cup.
Valkyrie III was beaten by Defender in the first race of the 1895 America's Cup. Valkyrie III fouled the leeward Defender during the prestart to the second race, breaking her starboard shrouds, but the latter did not protest and the race took place nevertheless, with Valkyrie III finishing ahead of Defender on corrected time. In turn the America's Cup committee ruled to disqualify the Valkyrie III because of the foul and dismissed Lord Dunraven counter-proposal to re-race. Valkyrie III was withdrawn immediately after the start of the third race and Defender ran over the course unchallenged and successfully defended the America's Cup. Lord Dunraven claimed the Americans had cheated, creating a public controversy that jeopardized the future challenges for the America's Cup race until Sir Thomas Lipton issued his own Shamrock challenge in 1898. In 1899 Valkyrie III' hull was refaired and repainted to serve as a trial horse for Shamrock, but eventually she did not serve that purpose. She was broken up in 1901.
has length n. That is, the length is the number of strict inclusions, not the number of primes; these differ by 1. We define the Krull dimension of 
to be the supremum of the lengths of all chains of prime ideals in 
