Real and Complex Analysis

By R. Cooke. 2011

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Citations in the corpus

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Normalizing Flows for Probabilistic Modeling and Inference (2019, methodology): “Under these conditions, the density of x is well-defined and can be obtained by a change of variables (Rudin, 2006; Bogachev, 2007): px(x) = pu(u) |det JT (u)|−1 where u = T−1(x).”
Breaking the Curse of Dimensionality with Convex Neural Networks (2014, background): “In order to define our space of functions from X → R, we need real-valued Radon measures, which are continuous linear forms on the space of continuous functions from V to R, equipped with the uniform norm (Rudin, 1987; Evans and Gariepy, 1991).”
Least Squares Temporal Difference Methods: An Analysis under General Conditions (2012, neutral): “Deﬁne functions ¯ L k : Then, by a theorem on integration [28, Theorem 1.38, pp. (cid:14) (cid:15) 28–29], we can deﬁne an (cid”
Kernel methods in system identification, machine learning and function estimation: A survey (2014, background): “In particular, L ∞ denotes the space of the essentially bounded functions with respect to the Lebesgue measure (Rudin, 1987).”
Some properties of a class of elliptic partial differential operators (2014, unknown): “ins a univalent disk DR0 with R0 ≥ 2ρ0 3 β M(2 +α) − Mρ0(2−ρ0) π(1− ρ0)2 . The proof of this lemma is complete. Let us recall the following result which is referred to as Jensen’s inequality (cf. [28]). 14 Sh. Chen and M. Vuorinen LemmaD.Let (Ω,A,µ) be a measure space such that µ(Ω) = 1.If gis a r”
New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications (2013, background): “6 ) is called the Radon–Nikodym derivative of μ with respect to the Lebesgue measure on ℝ 17 , p.”
Existence of solutions to fractional order ordinary and delay differential equations and applications. (2011, methodology): “Now we verify that the conditions of the Hausdroff maximum principle ([35]) hold and hence that S contains a maximal element, say x̄.”
Equilibria of homogeneous functionals in the fair-competition regime (2016, methodology): “Using a variant of Vitali’s Lemma [44], we see that the sequence (p∗jl) actually converges strongly to ρ̃ in L 1 RN on all finite balls in RN .”