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(underline {underline {Lambda }}_{3}) and (underline {underline {Lambda }}_{2}) are block diagonal regularization matrices penalizing the squared values of the third and second, respectively, time derivatives of the radial field B r at the core surface.
(underline {underline {Lambda }}_{3}) involves integration over the full timespan of the model, while (underline {underline {Lambda }}_{2}) involves evaluating the second time derivative only at the model endpoints t=1997.0 and 2015.0.
In the data error covariance matrix (underline {underline {C}}), anisotropic errors due to attitude uncertainty (Holme and Bloxham 1996) are considered for the vector field satellite data.
where m is the model vector, the residual vector e=d obs−d mod is the difference between the vector of observations d obs and the vector of model predictions d mod, and (underline {underline {C}}) is the data error covariance matrix.
Thus, (tilde {mu }^{prime } = tilde {m}_{underline {k}^{prime }}underline {a}^{underline {k}^{prime }} + sum _{underline {i} in Lambda _{X}^{prime } smallsetminus { underline {k}^{prime } }}tilde {m}_{underline {i}}underline {a}^{underline {i}}d^{sum underline {k}^{prime } - sum underline {i}}).
Similarly, we can get (m_{underline {i}}) for (underline {i} in Lambda _{X} smallsetminus { underline {k}, underline {k}^{prime } }).
We assume that (tilde {mu } = tilde {m} underline {a}/d d^{w_{tilde {m}}} = ld tilde {m})underline {a}^{underline {k}} + sum _{underline {i} in Lambda _{X} smallsetminus { underline {k} }}tilde {m}_{underline {i}}underline {a}^{underline {i}}d^{sum underline {k} - sum underline {i}}).
To see this, let (A := left | sum _{underline {i} in Lambda _{X}^{prime }}c_{underline {i}}underline {a}^{underline {i}}d^{w_{X} - sum underline {i}} right |).
ISO solves the optimal power flow allocation problem (underline {S} underline {D}(P)), underline {p})) given fixed demands (underline {D}) and generation/supply costs (underline {p}). 4.
Let (tilde {m}^{prime } underline {x}) := sum _{underline {i} in Lambda _{tilde {m}}}tilde {m}_{underline {i}}^{prime }underline {x}^{underline {i}}), where (tilde {m}_{underline {i}}^{prime }) are variables for (underline {i} in Lambda _{tilde {m}}).
