In mathematics, an ultrametric space is a metric space in which the triangle inequality. Sometimes the associated metric is also called a non-Archimedean metric or super-metric.

Formal definition

An ultrametric on a set is a real-valued function

denotes the real numbers:

1. (non-degeneracy)

2. (symmetry)

3. (strong triangle inequality or ultrametric inequality).

An ultrametric space, which is called the space's associated distance function (also called a metric).

is called an ultrapseudometric. An ultrapseudometric space.

is an Abelian group is generated by a length function), the last property can be made stronger using the Krull sharpening to:

1. .

. Without loss of generality.

Properties

holds. That is, every triple of points in the space forms an isosceles triangle, so the whole space is an isosceles set.

Defining the (open) ball, we have the following properties:

• .

• is non-empty.

• All balls of strictly positive radius are both open and closed sets in the induced topology) are also open.

• forms a partition.

Proving these statements is an instructive exercise. All directly derive from the ultrametric triangle inequality. Note that, by the second statement, a ball may have several center points that have non-zero distance. The intuition behind such seemingly strange effects is that, due to the strong triangle inequality, distances in ultrametrics do not add up.

Examples

• The discrete metric is an ultrametric.

• The p-adic numbers form a complete ultrametric space.

• Consider the set of words is the first place at which the words differ. The resulting metric is an ultrametric.

• The set of words is an ultrametric space with respect to the p are p-close if any substring of p.

• is a sequence of real numbers induces an ultrametric on the space of all complex sequences for which it is finite. (Note that this is not a seminorm since it lacks homogeneity (Zero to the power of zero).

• is an edge-weighted undirected graph is the weight of the minimax path, form an ultrametric space, and all finite ultrametric spaces may be represented in this way.

Applications

• A contraction mapping may then be thought of as a way of approximating the final result of a computation (which can be guaranteed to exist by the Banach fixed-point theorem). Similar ideas can be found in domain theory. p-adic analysis makes heavy use of the ultrametric nature of the p-adic metric.

• In condensed matter physics, the self-averaging overlap between spins in the SK Model of spin glasses exhibits an ultrametric structure, with the solution given by the full replica symmetry breaking procedure first outlined by Giorgio Parisi and coworkers. Ultrametricity also appears in the theory of aperiodic solids.

• In taxonomy and phylogenetic tree construction, ultrametric distances are also utilized by the UPGMA and WPGMA methods. These algorithms require a constant-rate assumption and produce trees in which the distances from the root to every branch tip are equal. When DNA, RNA and protein data are analyzed, the ultrametricity assumption is called the molecular clock.

• Models of intermittency in three dimensional turbulence of fluids make use of so-called cascades, and in discrete models of dyadic cascades, which have an ultrametric structure.

• In geography and landscape ecology, ultrametric distances have been applied to measure landscape complexity and to assess the extent to which one landscape function is more important than another.

References

Bibliography

Further reading

• .

External links