We show that a finite metric space A admits an extension to a finite metric space B so that each partial isometry of A extends to an isometry of B. We also prove a more precise result on extending a single partial isometry of a finite metric space. Both these results have consequences for the structure of the isometry groups of the rational Urysohn metric space and the Urysohn metric space.

It is considered an equation for the Lyapunov exponent γ in a random metric for a scalar propagating wave field. At first order in frequency this equation is solved explicitly. The localization length L c (reciprocal of Re(γ)) is obtained as function of the metric-fluctuation-distance ∆R (function of disorder) and the frequency ω of the wave. Explicitly, low-frequencies propagate longer than hi...

results the tlr3 protein in the ach group tended toward reduction although the p value of the comparison between the ach group and hd group was not statistically significant. the tlr3 levels in the hcc, ac, and cir groups were higher than those in the hd and ach groups. tlr3 was not interrelated with all costimulation proteins in the dcs and t cells in all five groups. no group presented any in...

We start this work by demonstrating the existence of unique common fixed points for two pairs occasionally weakly biased maps type A in a id="M4"> b -metric-like space, and we end it producing illustrative examples order to support show ...

Given a bounded metric space X, the set of closed sets of X supports a metric, the Hausdorff metric. Whether X is bounded or not, there is a compact, locally compact topology on the space of closed sets. If A,B ⊂ X are closed sets, define their Hausdorff distance dH(A,B) to be the number inf { r | B is in the r − neighborhood of A andA is in the r − neighborhood of B }. We can say this more pre...

A covariant causal set (c-causet) is a causal set that is invariant under labeling. Such causets are well-behaved and have a rigid geometry that is determined by a sequence of positive integers called the shell sequence. We first consider the microscopic picture. In this picture, the vertices of a c-causet have integer labels that are unique up to a label isomorphism. This labeling enables us t...