By Karl Menninger
Desk of Contents
Calculator's crafty: The paintings of speedy Reckoning
I: trying out a Calculation
II: units for the 4 Operations, or Arithmetical Tricks
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Additional resources for Calculator's cunning: the art of quick reckoning
Php 42 Chapter 2. 5 Complementarity in infinite dimensions CPs in infinite dimensions arise in many situations, such as in connection with partial differential equations. The framework for CPs starts with a Banach space X and its dual space X . Then for K a closed convex cone we have the dual cone given in terms of the duality pairing between X and X . We need the function F : X → X (rather than F : X → X) to be continuous. Actually requiring F : X → X is an advantage, since for second order elliptic partial differential equations we can take X = H 1( ) for to be a bounded open set in Rd , and X = H −1( ).
127. php 26 Chapter 2. Static Problems Without pointedness, this result fails in finite dimensions. Consider, for example, the half-space K 0 := x = [x, y]T ∈ R2 | x ≥ 0 . The only convex cone that strictly contains a half-space is the whole space, so K η = R2 for any η > 0. Without strong pointedness, this result fails in infinite dimensions. For example, we could take K 0 := x ∈ 2 | x 1 ≥ x j /j for j = 2, 3, . . Now 0 ∈ co K 0 ∩ S 2 since x j = e1 /j + e j / 1 + 1/j 2 ∈ K 0 ∩ S 2 , which converges weakly to zero in 2 .
Suppose otherwise. Then there is a sequence x k → x 0 as k → ∞ in and yk ∈ (x k ) with yk ∈ Hξ ,α for some −ξ ∈ int dom σ K and α > σ K (−ξ ). Since −ξ ∈ int dom σ K and σk is a convex lower semicontinuous function, there is a closed neighborhood of −ξ + θ B X on which σ K is continuous. By choosing θ > 0 sufficiently small, we can ensure that for −ξ ∈ −ξ + θ B X we have σ K (−ξ ) − σ K (−ξ ) ≤ 1 (α − σ K (−ξ )) , 2 and so σ K (−ξ ) ≤ 12 (α + σ K (−ξ )) < α for all such ξ . Let α := 12 (α + σ K (−ξ )) < α.
Calculator's cunning: the art of quick reckoning by Karl Menninger