← Mathematical compendium

Published equation contexts

ϕ^=(B⊤C−1B)+B⊤C−1y\hat\phi = \left(B^\top C^{-1} B\right)^{+} B^\top C^{-1} y

Why this formula appears here

At weak field this reduces to the familiar yijy_{ij} ≈\approx (Φi\Phi_i - Φj\Phi_j)/c2c^2 for Newtonian potential Φ\Phi , exactly the quantity real fibre-linked clock comparisons already report [ 1 , 7 ] . Collect every measured edge into a vector y , one entry per link, and encode the network’s topology in an incidence matrix B : row e = (i,j) has a +1 in column i , a -1 in column j , and zero elsewhere. The model is y = Bϕ\phi + n for measurement noise n with covariance C , and the standard weighted least-squares estimate of the node potentials is ϕ^=(B⊤C−1B)+B⊤C−1y\hat\phi = \left(B^\top C^{-1} B\right)^{+} B^\top C^{-1} y. using the Moore–Penrose pseudoinverse because B has a nontrivial null space: adding any constant to every node’s ϕ\phi leaves every edge…

Read the full article-specific guide →

Read the representative guide

How to interpret it

Read it with the definitions, units, and assumptions supplied by the article.

Learn the mathematics behind it

Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

ϕ^=(B⊤C−1B)+B⊤C−1y,\hat\phi = \left(B^\top C^{-1} B\right)^{+} B^\top C^{-1} y,

Equation 27 · Evolutionary Physics

The Bend an Elevator Cannot Fake

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

At weak field this reduces to the familiar yijy_{ij} ≈\approx (Φi\Phi_i - Φj\Phi_j)/c2c^2 for Newtonian potential Φ\Phi , exactly the quantity real fibre-linked clock comparisons already report [ 1 , 7 ] . Collect every measured edge into a vector y , one entry per link, and encode the network’s topology in an incidence matrix B : row e = (i,j) has a +1 in column i , a -1 in column j , and zero elsewhere. The model is y = Bϕ\phi + n for measurement noise n with covariance C , and the standard weighted least-squares estimate of the node potentials is ϕ^=(B⊤C−1B)+B⊤C−1y\hat\phi = \left(B^\top C^{-1} B\right)^{+} B^\top C^{-1} y. using the Moore–Penrose pseudoinverse because B has a nontrivial null space: adding any constant to every node’s ϕ\phi leaves every edge…

Meanings in this article

  • ϕ^\hat\phi: the standard weighted least-squares estimate of the node potentials.
Equation guide → · Article →