Chain

A chain is a series of connected links which are typically made of metal. A chain may consist of two or more links.

Two distinct chains can be connected using a quick link which resembles a carabiner with a screw close rather than a latch.

Uses for chain

Uses for chain include:

Decoration

Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is transitive, antisymmetric, and total (this relation is denoted here by infix ≤). A set paired with a total order is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain.

If X is totally ordered under ≤, then the following statements hold for all a, b and c in X:

Antisymmetry eliminates uncertain cases when both a precedes b and b precedes a. A relation having the property of "totality" means that any pair of elements in the set of the relation are comparable under the relation. This also means that the set can be diagrammed as a line of elements, giving it the name linear.Totality also implies reflexivity, i.e., a ≤ a. Therefore, a total order is also a partial order. The partial order has a weaker form of the third condition. (It requires only reflexivity, not totality.) An extension of a given partial order to a total order is called a linear extension of that partial order.

Chain (unit)

A chain (ch) is a unit of length. It measures 66 feet, or 22 yards, or 100 links, or 4 rods (20.1168 m). There are 10 chains in a furlong, and 80 chains in one statute mile. An acre is the area of 10 square chains (that is, an area of one chain by one furlong). The chain has been used for several centuries in Britain and in some other countries influenced by British practice.

By extension, chainage (running distance) is the distance along a curved or straight survey line from a fixed commencing point, as given by an odometer.

Origin

The chain was commonly used with the mile to indicate land distances and in particular in surveying land for legal and commercial purposes. Starting in the 19th centary, the chain was used as a sub division with the mile to show distances between railway stations, tunnels and bridges. In medieval times, local measures were commonly used, and many units were adopted that gave manageable units; for example the distance from London to York could be quoted in inches, but the resulting huge number would be unmemorable. The locally used units were often inconsistent from place to place.

Chain (Paul Haig album)

Chain was Edinburgh musician Paul Haig's third album and was released in May 1989 on Circa Records, a subsidiary of Virgin Records. Chain, which Haig financed himself, was recorded and completed in 1988, but it sat on the shelf after the normally accommodating Les Disques Du Crepuscule decided not to take up the option of releasing it. The album was co-produced by long-time Haig cohort, Alan Rankine, instrumentalist with celebrated Dundee band, The Associates. There was another Associates connection on the album - the track "Chained" was written by Haig's good friend, Billy Mackenzie. Haig returned the favour and gave Mackenzie the track "Reach The Top" for his album The Glamour Chase, which after many years in limbo was finally released in 2002.

One single, "Something Good", was taken from the album, but much to Circa's disappointment, neither the single nor the album sold in great numbers.

The sleeve features a shot of Audrey Hepburn, taken by the celebrated photographer, Angus McBean in 1958.

Chain (caste)

The Chain, sometimes also pronounced as Chai, are cultivating and fishing caste found in eastern Uttar Pradesh in India. They are a sub-group within the larger Kewat communinity of North India.

Origin

The Chai according to some traditions, were a community of Vaishyas, who lost caste, when they took to fishing. Other traditions make them out to be a branch of the Bind caste, and the two communities intermarry, with suggests a common origin. They are now recognized as a sub-group within the Kewat community, and they intermarry with other Kewat clans such as the Banaphar and Dhivar. The majority of the Chai are now cultivators, with a small number still employed as boatmen on the Ganges. They are Hindu, and have customs similar to other Kewat groups.

The Chai are found mainly in southern and eastern Uttar Pradesh, with concentrations in Mirzapur, Ghazipur, Varanasi, Gorakhpur, Basti and Faizabad. They speak Awadhi, although most also understand Hindi.

See also

References

Chain (algebraic topology)

In algebraic topology, a simplicial k-chain is a formal linear combination of k-simplices.

Integration on chains

Integration is defined on chains by taking the linear combination of integrals over the simplices in the chain with coefficients typically integers. The set of all k-chains forms a group and the sequence of these groups is called a chain complex.

Boundary operator on chains

The boundary of a chain is the linear combination of boundaries of the simplices in the chain. The boundary of a k-chain is a (k−1)-chain. Note that the boundary of a simplex is not a simplex, but a chain with coefficients 1 or −1 – thus chains are the closure of simplices under the boundary operator.

Example 1: The boundary of a path is the formal difference of its endpoints: it is a telescoping sum. To illustrate, if the 1-chain is a path from point to point , where , and are its constituent 1-simplices, then

Example 2: The boundary of the triangle is a formal sum of its edges with signs arranged to make the traversal of the boundary counterclockwise.