Analysis on Manifolds

By M. Marias. 2005

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Optimality of Matched-Pair Designs in Randomized Controlled Trials (2022, background): “2 (implicit function theorem) of Munkres (1991), there exists an open set U containing u = (x1, . . . , xp−1), an open ball B(z) containing z and an open set O in R containing xp, and a function k : U×B(z) → R of class C((2)) such that h(u, k(u, z)) = z for all u ∈ U , z ∈ B(z) and x ∈ O. Moreover, ”
Random Coefficients on Endogenous Variables in Simultaneous Equations Models (2014, methodology): “This mapping from the parameters {πki/πii : k 6= i} to {γk : k 6= i} is one-to-one and differentiable and hence the joint distribution of {γk : k 6= i} can be written in terms of the joint distribution of {πki/πii : k 6= i} via the change of variables formula (e.g., Munkres (1991) theorem 17.2).”
Understanding the (un)interpretability of natural image distributions using generative models (2019, methodology): “If G is differentiable and bijective, then the change of variables formula [16] provides a method for determining the exact density of the warped distribution.”
ON IDENTIFIABILITY OF NONLINEAR ODE MODELS AND APPLICATIONS IN VIRAL DYNAMICS (2011, background): “For a general introduction to the implicit function theorem, the reader is referred to [73].”