Two-dimensional space

In physics and mathematics, two-dimensional space or bi-dimensional space is a geometric model of the planar projection of the physical universe. The two dimensions are commonly called length and width. Both directions lie in the same plane.

A sequence of n real numbers can be understood as a location in n-dimensional space. When n = 2, the set of all such locations is called two-dimensional space or bi-dimensional space, and usually is thought of as a Euclidean space.

History

Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics.

Later, the plane was described in a so-called Cartesian coordinate system, a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

Dito

Eduardo José Gomes Camassele Mendes (born 18 January 1962 in Barcelos), known as Dito, is a retired Portuguese footballer who played as a central defender, and a manager.

Club career

During 16 professional seasons Dito appeared in 358 Primeira Liga matches, starting his career with S.C. Braga, for which he was already an important first-team member at the age of 18, and signing with S.L. Benfica in 1986 after six years in Minho.

During his two season-spell with Benfica, Dito played in 27 league games in his first year as the Lisbon club won the double, then partnered Carlos Mozer in his second (no silverware won). He then moved to rivals and title holders FC Porto for one season, with Benfica regaining its domestic supremacy at the expense of precisely the northerners.

From 1989 to 1994, always in the top division, Dito represented Vitória de Setúbal, S.C. Espinho and Gil Vicente FC. After an unassuming stint with A.D. Ovarense in the second level he retired from football, aged 34.

Dito's biggest achievement as a coach was managing S.C. Salgueiros over the course of three top flight campaigns, being sacked after the tenth round of 1999–2000 as the Paranhos team eventually retained its status. In 2009 he returned to his first team Braga, being appointed at its junior sides.

Dimension (data warehouse)

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.

A dimensional data element is similar to a categorical variable in statistics.

Typically dimensions in a data warehouse are organized internally into one or more hierarchies. "Date" is a common dimension, with several possible hierarchies:

Dimension (song)

"Dimension" is a song by Australian hard rock band Wolfmother, featured on their 2005 debut studio album Wolfmother. Written by band members Andrew Stockdale, Chris Ross and Myles Heskett, it was released as the second single from the album in Europe (and the third single overall) on 17 April 2006, charting at number 49 on the UK Singles Chart.

Music video

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 gonzo metal not unlike Black Sabbath playing Motown".

Track listing

All songs written and composed by Andrew Stockdale, Chris Ross, Myles Heskett. 

References

Krull dimension

In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.

The Krull dimension has been introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal I in a polynomial ring R is the Krull dimension of R/I.

A field k has Krull dimension 0; more generally, k[x₁, ..., x_n] 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 .