In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d ( x , z ) ≤ max { d ( x , y ) , d ( y , z...

an ultrametric inequality. This leads to the notion of a generalized ultrametric. These generalizations still induce a uniform structure on the space. A...

refers to something without the Archimedean property. This includes: Ultrametric space in particular, a domain with non-Archimedean absolute value notably...

equation which contains p-adic numbers in an ultrametric space. Mathematical models that involve ultrametric pseudo-differential equations use pseudo-differential...

the ultrametric inequality means, for example, that convergence of infinite series of p-adic numbers is much simpler. Topological vector spaces over...

isomorphism Uniform property Uniformly connected space Metric topology Manhattan distance Ultrametric space P-adic numbers, p-adic analysis Open ball Bounded...

metric space such that at least one of the following holds: the metric space is a Riemannian manifold, the metric space is a locally compact ultrametric space...

|x+y|_{p}=\max(|x|_{p},|y|_{p}).} This makes the p-adic numbers a metric space, and even an ultrametric space, with the p-adic distance defined by d p ( x , y ) = | x...

, in ultrametric spaces, all of the three are simultaneously open and closed sets), and the unit sphere may even be empty in some metric spaces. If ⁠...

the metric geometry over a non-Archimedean valued field, or ultrametric space. In such a space, even more contradictions to Euclidean geometry result. For...

absolute value is called an ultrametric, non-Archimedean or p {\displaystyle p} -adic place of K {\displaystyle K} . For any ultrametric place v we have that...

comes from an ultrametric space, the optimal clustering for this quality measure follows the underlying structure of the ultrametric space. In this sense...

construct a data structure with linear space that allows the distances between pairs of points in any ultrametric space to be queried in constant time per...

maximum edge weights of minimax paths) form an ultrametric; conversely every finite ultrametric space comes from minimax distances in this way. A data...

ultrametric space, the whole space (and any of its subsets) is an isosceles set. Therefore, ultrametric spaces are sometimes called isosceles spaces. However...

e.g. in mathematical models that include ultrametric pseudo-differential equations in a non-Archimedean space. The study of pseudo-differential operators...

a ultrapseudometric or a strong pseudometric if it satisfies: Strong/Ultrametric triangle inequality: d ( x , z ) ≤ max { d ( x , y ) , d ( y , z ) }...

example of an additive distance matrix and its corresponding tree: The ultrametric distance matrix is defined as an additive matrix which models the constant...

Some Contributions to the Theory of Neural Networks. III: Embeddings of Ultrametric Spaces Doctoral advisor R. M. Wilson Website www.igb.uci.edu/~pfbaldi...

or normed space satisfying the ultrametric triangle inequality is called non-Archimedean. The concept of a non-Archimedean normed linear space was introduced...

closed sets are disjoint. Every ultra-connected space is path-connected. Ultrametric A metric is an ultrametric if it satisfies the following stronger version...

Leloup, Gérard (February 2011), "Existentially equivalent cyclic ultrametric spaces and cyclically valued groups" (PDF), Logic Journal of the IGPL, 19...

conditions for metrizability and orderability of pseudometric and ultrametric spaces. "Papić, Pavle". Croatian Encyclopedia (in Croatian). Miroslav Krleža...

"valuation" (satisfying the ultrametric inequality) is called an "exponential valuation" or "non-Archimedean absolute value" or "ultrametric absolute value"; our...

and sufficient condition for convergence due to the non-Archimedean ultrametric triangle inequality. Unlike stronger convergence tests, the term test...

research on p-adic analysis. In 1944 he introduced the concept of ultrametric spaces, to which p-adic numbers belong. In 1951, alongside Lev Kaluznin,...

max ( | x | , | y | ) {\displaystyle |x+y|\leq \max(|x|,|y|)} is an ultrametric absolute value. An absolute value induces a metric (and thus a topology)...

"Mathematical Representations of Matte Blanco's Bi-Logic based on Metric Space and Ultrametric or Hierarchical Topology: Towards Practical Application". Language...

integrals for quadratic lagrangians on p-adic and adelic spaces". P-Adic Numbers, Ultrametric Analysis, and Applications. 2 (4): 322–340. arXiv:1011.6589...

dendrograms and requires a constant-rate assumption - that is, it assumes an ultrametric tree in which the distances from the root to every branch tip are equal...