Ultrametric space

In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to $d(x,z)\leq \max \left\{d(x,y),d(y,z)\right\}$ for all $x$ , $y$ , and $z$ . Sometimes the associated metric is also called a non-Archimedean metric or super-metric.

Every point inside a ball is its center, i.e. if $d(x,y)<r$ then $B(x;r)=B(y;r)$ .

Intersecting balls are contained in each other, i.e. if $B(x;r)\cap B(y;s)$ is then either $B(x;r)\subseteq B(y;s)$ or $B(y;s)\subseteq B(x;r)$ .

non-empty

All balls of strictly positive radius are both and closed sets in the induced topology. That is, open balls are also closed, and closed balls (replace $<$ with $\leq$ ) are also open.

open

The set of all open balls with radius $r$ and center in a closed ball of radius $r>0$ forms a of the latter, and the mutual distance of two distinct open balls is (greater or) equal to $r$ .

partition

From the above definition, one can conclude several typical properties of ultrametrics. For example, for all $x,y,z\in M$ , at least one of the three equalities $d(x,y)=d(y,z)$ or $d(x,z)=d(y,z)$ or $d(x,y)=d(z,x)$ 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 of radius $r>0$ centred at $x\in M$ as $B(x;r):=\{y\in M\mid d(x,y)<r\}$ , we have the following properties:

Proving these statements is an instructive exercise.^[2] 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.

The is an ultrametric.

discrete metric

The form a complete ultrametric space.

p-adic numbers

Consider the of arbitrary length (finite or infinite), Σ^*, over some alphabet Σ. Define the distance between two different words to be 2⁻ⁿ, where n is the first place at which the words differ. The resulting metric is an ultrametric.

set of words

The with glued ends of the length n over some alphabet Σ is an ultrametric space with respect to the p-close distance. Two words x and y are p-close if any substring of p consecutive letters (p < n) appears the same number of times (which could also be zero) both in x and y.^[3]

set of words

If r = (r_n) is a sequence of decreasing to zero, then |x|_r := lim sup_n→∞ |x_n|^r_n 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 — If the r_n are allowed to be zero, one should use here the rather unusual convention that 0⁰ = 0.)

real numbers

If G is an edge-weighted , all edge weights are positive, and d(u,v) is the weight of the minimax path between u and v (that is, the largest weight of an edge, on a path chosen to minimize this largest weight), then the vertices of the graph, with distance measured by d, form an ultrametric space, and all finite ultrametric spaces may be represented in this way.^[4]

undirected graph

A 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.

contraction mapping

In , 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.^[5] Ultrametricity also appears in the theory of aperiodic solids.^[6]

condensed matter physics

In and phylogenetic tree construction, ultrametric distances are also utilized by the UPGMA and WPGMA methods.^[7] 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.

taxonomy

Models of in three dimensional turbulence of fluids make use of so-called cascades, and in discrete models of dyadic cascades, which have an ultrametric structure.^[8]

intermittency

In 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.^[9]

geography

Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. 978-1584888666. OCLC 144216834.

ISBN

; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

Schaefer, Helmut H.

(1977), Set Theory and Metric Spaces, AMS Chelsea Publishing, ISBN 978-0-8218-2694-2.

Kaplansky, I.

Media related to Non-Archimedean geometry at Wikimedia Commons