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Extra resources for Combinatorial Mathematics IX, Brisbane, Australia: Proceedings, 1981

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For example, the projection onto the x-axis is itself a slim Cantor-like set of dimension 1/2. A diagonal line (with slope 1/2) gives an exceptional case: The projection is a solid interval. If A is any rectifiable set, then H 1 (A ∩ E) = 0. 1. A purely unrectifiable one-dimensional set E. E is invisible from almost all directions. 1 Give an example of a Lipschitz function f : [0, 1] → R such that f is not differentiable at any rational point. 3 to deduce that a Lipschitz function is approximately differentiable almost everywhere.

To check that S2 is indeed a normal current, compute ∂S2 from the definition: ∂S2 (f(x, y)) = S2 (d f ) = S2 = e1 , A = A 1 = ∂f ∗ ∂f ∗ e + e ∂x 1 ∂y 2 ∂f ∗ ∂f ∗ e + e dH ∂x 1 ∂y 2 2 ∂f dx dy ∂x [f(1, y) − f(0, y)] dy 0 1 = 1 f(1, y) dy − 0 f(0, y) dy. 0 Therefore, ∂S2 = H 1 {(1, y) : 0 ≤ y ≤ 1} − H 1 {(0, y) : 0 ≤ y ≤ 1}, and M(∂S2 ) = 2 < ∞. If B = {(x, 0) : 0 ≤ x ≤ 1}, T = H denotes translation by (x, y), then 1 S2 = τ(0, y) T dy. 0 1 B ∧ e1 , and τ(x, y) Normal and Rectifiable Currents 49 Thus, S2 is an integral of integral currents.

1. The current T is not flat; its prescribed vectorfield is not tangent to the underlying set. The suggestiveness of the term flat is a happy accident. H. Whitney, also a student of music, coined the term for the smaller of his flat and sharp norms, originally and . designated Outline of Proof I. Smoothing. A smooth normal current in Rn is one of the form L n ∧ ξ, with ξ a smooth m-vectorfield of compact support. Any normal current T can be approximated in the flat norm by a smooth normal current Tε = L n ∧ ξ as follows.

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Combinatorial Mathematics IX, Brisbane, Australia: Proceedings, 1981 by Billington E.J. (ed.), Oates-Williams S.(ed.), Street A.P.

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