Mathematical analysis : a modern approach to advanced calculus / Tom M. Apostol

By T. Apostol. 1957

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Optimal Taxation without State‐Contingent Debt (2002, background): “and (21) revenue and The government wants to conﬁne x t”
Sparse inverse covariance matrix estimation using quadratic approximation (2011, background): “. According to Lemma 3, the level set U deﬁned in (25) contains all the iterates, and it is in turn contained in the compact set S ≡{X |mI X MI}. According to the Weierstrass extreme value theorem [Apostol, 1974], any continuous function in a compact set attains its minimum. Furthermore, ∇2g(X) = X−”
Midpoint-Type Rules from an Inequalities Point of View (2019, methodology): “It is well known [30] that if p : [a, b] → R is continuous on [a, b] and v : [a, b] → R is of bounded variation on [a, b] , then ∣∣∣∣ ∫ b a p (x) dv (x) ∣∣∣∣ ≤ sup x∈[a,b] |p (x)| b ∨”
Random Fields and Geometry (2007, background): “A proof can be found, for example, in [13].”
The semi normed space defined by a double gai sequence of modulus function (2010, unknown): “The double series ∞ m,n=1 x mn is said to be convergent if and only if the double sequence (S mn )is convergent, where S mn = m,n i,j=1 x ij (m, n = 1, 2, 3, ...) (see[1]).”