Interval (graph theory)

In graph theory, an interval I(h) in a directed graph is a maximal, single entry subgraph in which h is the only entry to I(h) and all closed paths in I(h) contain h. Intervals were described in 1976 by F. E. Allen and J. Cooke. Interval graphs are integral to some algorithms used in compilers, specifically data flow analyses.

The following algorithm finds all the intervals in a graph consisting of vertices N and the entry vertex n₀, and with the functions pred(n) and succ(n) which return the list of predecessors and successors of a given node n, respectively.

The algorithm effectively partitions the graph into its intervals.

Each interval can in turn be replaced with a single node, while all edges between nodes in different intervals in the original graph become edges between their corresponding nodes in the new graph. This new graph is called an interval derived graph. The process of creating derived graphs can be repeated until the resulting graph can't be reduced further. If the final graph consists of a single node, then the original graph is said to be reducible.

Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all real numbers , the set of all negative real numbers, and the empty set.

Real intervals play an important role in the theory of integration, because they are the simplest sets whose "size" or "measure" or "length" is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure.

Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations, and arithmetic roundoff.

Ash (near Salway)

Coordinates: 50°46′N 2°46′W / 50.76°N 02.76°W / 50.76; -02.76

Ash is a hamlet approximately 0.6 miles (1 km) east of the village of Salway, Dorset, England.

References

Ash (deity)

Ash was the ancient Egyptian god of oases, as well as the vineyards of the western Nile Delta and thus was viewed as a benign deity. Flinders Petrie in his 1923 expedition to the Saqqara (also spelt Sakkara) found several references to Ash in Old Kingdom wine jar seals: "I am refreshed by this Ash" was a common inscription.

In particular, he was identified by the Ancient Egyptians as the god of the Libu and Tinhu tribes, known as the "people of the oasis". Consequently Ash was known as the "lord of Libya", the western border areas occupied by the Libu and Tinhu tribes, corresponds roughly with the area of modern Libya. It is also possible that he was worshiped in Ombos, as their original chief deity.

In Egyptian mythology, as god of the oases, Ash was associated with Set, who was originally god of the desert, and was seen as protector of the Sahara. The first known reference to Ash dates to the Protodynastic Period, but by the late 2nd Dynasty, his importance had grown, and he was seen as protector of the royal estates, since the related god Set, in Lower Egypt, was regarded as the patron deity of royalty itself. Ash's importance was such that he was mentioned even until the 26th Dynasty.

Ash (near Stourpaine)

Coordinates: 50°53′N 2°12′W / 50.88°N 02.20°W / 50.88; -02.20

Ash is a hamlet approximately 0.5 miles (1 km) north of the village of Stourpaine, Dorset, England.

Ash was listed in the Domesday Book of 1086.

References

External links

Media related to Ash (near Stourpaine) at Wikimedia Commons