The phrase "as follows given" is not correct and does not sound natural in written English.
It seems to be an attempt to introduce a list or explanation, but the phrasing is awkward and unclear.
Example: "The instructions for the project are as follows given: first, gather your materials; second, outline your plan."
Alternatives: "as follows" or "as stated below".
As noted early, the problem to be undertaken is as follows: Given a set of roads associated with capped volumes (let us call it capacity) to be tolled or subsidized, what is a pricing scheme.
In this formulation, the prediction problem can be restated as follows: given your past cell sequence up to time t, which cell will you visit at time (t+Delta t)?
The problem of texture synthesis can be stated as follows: Given a texture sample, synthesize a new texture that, when perceived by a human observer, appears to be generated by the same underlying stochastic process.
His argument went as follows: given its projected surpluses over the next decade, the federal government may not only pay off its debt, but actually find itself using surplus cash to buy private assets.
This difference might be explained as follows: Given the high surface-to-volume ratio of the nano-sized PS particles used by Besseling et al., it is likely that the kairomones become highly adsorbed.
The Common Substring Alignment Problem is defined as follows: Given a set of one or more strings S1, S2 … Sc and a target string T, Y is a common substring of all strings Si, that is, Si = BiYFi.
The even adjacency split problem is defined as follows: given an undirected graph, determine whether its vertex set can be partitioned into two dominating sets whose sizes differ by at most one.
The rule of k names can be described as follows: given a set of candidates for office, a committee chooses k members from this set by voting, and makes a list with their names.
The main result is as follows; given a target Packet Loss Rate (PLR) the Direct Garch algorithm produces parameter estimates which result in a PLR closer than other algorithms.
The Dirichlet problem for harmonic functions is as follows: Given a function defined on the boundary of a set, find an extension of the function to the interior of the set such that the function is harmonic.
We do so as follows: Given the definition of R it follows that R ∈ R ≡ ~(R ∈ R).
