Real and complex analysis, 3rd ed.

By W. Rudin. 1987

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In Pure Mathematics

Citations in the corpus

The Graph Neural Network Model (2009, methodology): “Thus, the determinant of is not null and we can apply the implicit function theorem (see [60]) to and point .”
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).”
Data-driven non-Markovian closure models (2014, background): “Actually, by the Fubini theorem [112], it suffices to assume that Ψ ∈ L(1)μ(Y ;X) for this integral to be well defined.”
Statistical behavior and consistency of classification methods based on convex risk minimization (2003, background): “, see [12], page 55), we know that f (r6 (x)) can be approximated by a continuous function a'(x) E C(U) such that Ia'(x)I < K3 and P(f (rs(x)) a'(x)) < ?/(2Ms).”
Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces (2011, methodology): “Then (by employing the Lusin theorem [53] in case (ii)) we infer that given f ∈ L: (i) the map R+ t → In(t)f ∈ L(2)loc is continuous, and (ii) limn→∞ In(t)f exists in L(2)loc for almost every t ∈ R+ and [I (·)f ] = f .”
Optimizing Expected Utility of Dividend Payments for a Brownian Risk Process and a Peculiar Nonlinear ODE (2004, background): “A precise statement in this context may be formulated like that: there does not exist a domain D in C, containing x0 as an interior point, such that the functions v(α) can all be analytically continued to D, and such that (v(α))0 α 1 is a normal family, see (Rudin, 1987, Ch.14, p.281).”
Graph properties for nonlocal minimal surfaces (2015, background): “nimal sets, see [7]. We provide full details for the convenience of the reader. First of all, we can reduce from measurable sets to Borel sets, up to sets of measure zero, see e.g. Theorem 2.20(b) in [19]. Hence we may suppose that Eis Borel. Also, we observe that, since En is relatively open in Rnn”