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An interval graph is an undirected graph G formed from a family of intervals , =,,, … by creating one vertex v i for each interval S i, and connecting two vertices v i and v j by an edge whenever the corresponding two sets have a nonempty intersection.
The Heaviside step function is an often-used step function.. A constant function is a trivial example of a step function. Then there is only one interval, =. The sign function sgn(x), which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.
The smallest constant is sometimes called the (best) Lipschitz constant [4] of f or the dilation or dilatation [5]: p. 9, Definition 1.4.1 [6] [7] of f. If K = 1 the function is called a short map , and if 0 ≤ K < 1 and f maps a metric space to itself, the function is called a contraction .
The graph of the Cantor function on the unit interval. In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero ...
An independent set in an interval graph is just a set of non-overlapping intervals. The problem of finding maximum independent sets in interval graphs has been studied, for example, in the context of job scheduling : given a set of jobs that has to be executed on a computer, find a maximum set of jobs that can be executed without interfering ...
A constant function is both monotone and antitone; conversely, if f is both monotone and antitone, and if the domain of f is a lattice, then f must be constant. Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are found in these places.
3. A proper interval graph or proper circular arc graph is an intersection graph of a collection of intervals or circular arcs (respectively) such that no interval or arc contains another interval or arc. Proper interval graphs are also called unit interval graphs (because they can always be represented by unit intervals) or indifference graphs.
A function with period P will repeat on intervals of length P, and these intervals are sometimes also referred to as periods of the function. Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry , i.e. a function f is periodic with period P if the graph of f is invariant under translation ...