We study the supremal p-negative type of finite metric spaces. An explicit expression for the supremal p-negative type ℘ (X, d) of a finite metric space (X, d) is given in terms its associated distance matrix, from which the supremal p-negative type of the space may be calculated. The method is then used to give a straightforward calculation of the supremal p-negative type of the complete bipartite graphs Kn,m endowed with the usual path metric. A gap in the spectrum of possible supremal p-negative type values of path metric graphs is also proven.
The Magnus expansion is a frequently used tool to get approximate analytic solutions of time-dependent linear ordinary differential equations and in particular the Schrödinger equation in quantum mechanics. However, the complexity of the expansion restricts its use in practice only to the first terms. Here we introduce new and more accurate analytic approximations based on the Magnus expansion involving only univariate integrals which also shares with the exact solution its main qualitative and geometric properties.
We introduce a class of metric spaces called p-additive combinations and show that for such spaces we may deduce information about their p-negative type behaviour by focusing on a relatively small collection of almost disjoint metric subspaces, which we call the components. In particular we deduce a formula for the p-negative type gap of the space in terms of the p-negative type gaps of the components, independent of how the components are arranged in the ambient space. This generalizes earlier work on metric trees by Doust and Weston [DW08b,DW08a]. The results hold for semi-metric spaces as well, as the triangle inequality is not used.2010 Mathematics Subject Classification. 46B85, 46T99,05C12.
Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitative versions of these inequalities involving the so-called p-negative type gap. In particular, we focus our attention on the class of finite ultrametric spaces which are important in areas such as phylogenetics and data mining.Let (X, d) be a given finite ultrametric space with minimum non-zero distance α. Then the p-negative type gap Γ X (p) of (X, d) is positive for all p ≥ 0. In this paper we compute the value of the limit Γ X (∞) := lim p→∞
Metric spaces of generalized roundness zero have interesting non-embedding properties. For instance, we note that no metric space of generalized roundness zero is isometric to any metric subspace of any L p -space for which 0 < p ≤ 2. Lennard, Tonge and Weston gave an indirect proof that ℓ (3) ∞ has generalized roundness zero by appealing to non-trivial isometric embedding theorems of Bretagnolle, Dacunha-Castelle and Krivine, and Misiewicz. In this paper we give a direct proof that ℓ (3) ∞ has generalized roundness zero. This provides insight into the combinatorial geometry of ℓ (3) ∞ that causes the generalized roundness inequalities to fail. We complete the paper by noting a characterization of real quasi-normed spaces of generalized roundness zero.
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