Ultrametric space explained
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to
d(x,z)\leqmax\left\{d(x,y),d(y,z)\right\}
for all
,
, and
. 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
(where denote the
real numbers), such that for all :
- ;
- (symmetry);
- ;
- if then ;
- (strong triangle inequality or ultrametric inequality).
An ultrametric space is a pair consisting of a set together with an ultrametric on, which is called the space's associated distance function (also called a metric).
If satisfies all of the conditions except possibly condition 4 then is called an ultrapseudometric on . An ultrapseudometric space is a pair consisting of a set and an ultrapseudometric on .
(so that
), the last property can be made stronger using the
Krull sharpening to:
\|x+y\|\lemax\left\{\|x\|,\|y\|\right\}
with equality if
.
We want to prove that if
\|x+y\|\lemax\left\{\|x\|,\|y\|\right\}
, then the equality occurs if
.
Without loss of generality, let us assume that
This implies that
. But we can also compute
\|x\|=\|(x+y)-y\|\lemax\left\{\|x+y\|,\|y\|\right\}
. Now, the value of
max\left\{\|x+y\|,\|y\|\right\}
cannot be
, for if that is the case, we have
contrary to the initial assumption. Thus,
max\left\{\|x+y\|,\|y\|\right\}=\|x+y\|
, and
. Using the initial inequality, we have
and therefore
.
Properties
From the above definition, one can conclude several typical properties of ultrametrics. For example, for all
, at least one of the three equalities
or
or
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
centred at
as
B(x;r):=\{y\inM\midd(x,y)<r\}
, we have the following properties:
- Every point inside a ball is its center, i.e. if
then
.
- Intersecting balls are contained in each other, i.e. if
is
non-empty then either
or
.
- All balls of strictly positive radius are both open and closed sets in the induced topology. That is, open balls are also closed, and closed balls (replace
with
) are also open.
- The set of all open balls with radius
and center in a closed ball of radius
forms a
partition of the latter, and the mutual distance of two distinct open balls is (greater or) equal to
.
Proving these statements is an instructive exercise.[1] 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 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.
- The set of words 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.[2]
- If r = (rn) is a sequence of real numbers decreasing to zero, then |x|r := lim supn→∞ |xn|rn 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 rn are allowed to be zero, one should use here the rather unusual convention that 00 = 0.)
- If G is an edge-weighted undirected graph, 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.[3]
Applications
Further reading
Notes and References
- Web site: Stack Exchange . Ultrametric Triangle Inequality .
- .
- .
- Mezard, M; Parisi, G; and Virasoro, M: SPIN GLASS THEORY AND BEYOND, World Scientific, 1986.
- Rammal . R. . Toulouse . G. . Virasoro . M. . Ultrametricity for physicists . Reviews of Modern Physics . 1986 . 58 . 3 . 765–788 . 10.1103/RevModPhys.58.765 . 20 June 2011. 1986RvMP...58..765R .
- Legendre, P. and Legendre, L. 1998. Numerical Ecology. Second English Edition. Developments in Environmental Modelling 20. Elsevier, Amsterdam.
- Benzi. R. . Biferale . L. . Trovatore . E. . Ultrametric Structure of Multiscale Energy Correlations in Turbulent Models . Physical Review Letters . 1997 . 79 . 9 . 1670–1674 . 10.1103/PhysRevLett.79.1670 . chao-dyn/9705018 . 1997PhRvL..79.1670B . 53120932 .
- Papadimitriou. Fivos. 2013. Mathematical modelling of land use and landscape complexity with ultrametric topology. Journal of Land Use Science. en. 8. 2. 234–254. 10.1080/1747423x.2011.637136. 121927387. 1747-423X. free.