Ультраметрлік кеңістіктер және олардың қолданылуы

Type of metric space
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to for all , , and Sometimes the associated metric is also called a non Archimedean metric or super metric.

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. 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.

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.