The phrase "analysis we establish the" is not correct and seems incomplete in written English.
It could be used in a context where you are discussing the process of establishing something through analysis, but it requires additional information to be clear.
Example: "In this analysis, we establish the key factors that influence market trends."
Alternatives: "through this analysis, we determine the" or "in our analysis, we identify the".
By using fixed point theory and set-valued analysis, we establish the existence theorem of equilibrium point.
By means of a delicate analysis, we establish the existence and uniqueness of a global solution to the HJB equation in two main steps.
By adopting the framework of wavelet analysis, we establish the asymptotic minimax rate of convergence under the Lp risk over Besov balls.
Here, using data on a large group of Irish manufacturing plants and econometric analysis, we establish the short- and longer-term influence of plants' adoption of quality improvement methods (QIMs) on product innovation performance.
Finally, by applying the EJA tool to our analysis, we established the iteration complexity (mathcal{O}(rlogvarepsilon^{-1})) for the NT-direction.
By this analysis we established the percentage of G1, S and G2 cells in a cell population.
In this analysis, we established the correlation between the degree of patient sickness and the sufficiency of data of their health records, for inclusion in research.
Through positional candidate analysis, we establish that the Sci phenotype is caused by a mutation in the SCN transcription factor Zfhx3.
As a consequence of our analysis, we establish three theoretical propositions that explain the link between the double fit and the system strength concept.
By a linear stability analysis, we establish some stability conditions of constant positive equilibrium for the ODE and PDE systems.
Using the exponential dichotomy techniques and a fixed point strategy with stochastic analysis theory, we establish the existence of p-mean piecewise pseudo almost periodic mild solutions for a class of impulsive nonautonomous partial stochastic evolution equations in Hilbert spaces.
