Time

Time is a measure in which events can be ordered from the past through the present into the future, and also the measure of durations of events and the intervals between them. Time is often referred to as the fourth dimension, along with the three spatial dimensions.

Time has long been a major subject of study in religion, philosophy, and science, but defining it in a manner applicable to all fields without circularity has consistently eluded scholars. Nevertheless, diverse fields such as business, industry, sports, the sciences, and the performing arts all incorporate some notion of time into their respective measuring systems. Some simple definitions of time include "time is what clocks measure", which is a problematically vague and self-referential definition that utilizes the device used to measure the subject as the definition of the subject, and "time is what keeps everything from happening at once", which is without substantive meaning in the absence of the definition of simultaneity in the context of the limitations of human sensation, observation of events, and the perception of such events.

Interval

Interval may refer to:

Maths and physics

Music

Dramatic arts

Sport

Other

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.

Partially ordered set

In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. Such a relation is called a partial order to reflect the fact that not every pair of elements need be related: for some pairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiar total orders, in which every pair is related. A finite poset can be visualized through its Hasse diagram, which depicts the ordering relation.

A familiar real-life example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs bear no such relationship.

Formal definition