By Samuel Arbesman
New insights from the technological know-how of science
Facts swap forever. Smoking has long past from physician advised to lethal. We used to imagine the Earth was once the guts of the universe and that the brontosaurus was once a true dinosaur. briefly, what we all know concerning the global is continually changing.
Samuel Arbesman exhibits us how wisdom in so much fields evolves systematically and predictably, and the way this evolution unfolds in a desirable manner which may have a robust influence on our lives.
He takes us via a wide selection of fields, together with those who switch fast, over the process many years, or over the span of centuries.
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Extra info for The Half-Life of Facts: Why Everything We Know Has an Expiration Date
N Proof: The implication (i) =⇒ (ii) is trivial. To see that (ii) implies (iii), first observe that if ψ is absolutely monotone on (a, b) and h ∈ (0, b − a), then ∆h ψ is absolutely monotone on (a, b − h). Indeed, because D ◦ ∆h ψ = ∆h ◦ Dψ on (a, b − h), we have that t+h h Dm ◦ ∆h ψ (t) = Dm+1 ψ(s) ds ≥ 0, t ∈ (a, b − h), t for any m ∈ N. Returning to the function ϕ, we now know that ∆m h ϕ is absolutely monotone on (0, 1 − mh) for all m ∈ N and h > 0 with mh < 1. In particular, m [∆m h ϕ](0) = lim [∆h ϕ](t) ≥ 0 t and so ∆m h ϕ (0) ≥ 0 when h = 0 1 n if mh < 1, and 0 ≤ m < n.
Indeed, because D ◦ ∆h ψ = ∆h ◦ Dψ on (a, b − h), we have that t+h h Dm ◦ ∆h ψ (t) = Dm+1 ψ(s) ds ≥ 0, t ∈ (a, b − h), t for any m ∈ N. Returning to the function ϕ, we now know that ∆m h ϕ is absolutely monotone on (0, 1 − mh) for all m ∈ N and h > 0 with mh < 1. In particular, m [∆m h ϕ](0) = lim [∆h ϕ](t) ≥ 0 t and so ∆m h ϕ (0) ≥ 0 when h = 0 1 n if mh < 1, and 0 ≤ m < n. Moreover, since [∆n1 ϕ](0) = lim1 [∆nh ϕ](0), n h n we also know that ∆nh ϕ (0) ≥ 0 when h = n1 , and this completes the proof that (ii) implies (iii).
5) |y|2 1 , g(y) ≡ √ exp − 2 2π y ∈ R, and recall that a random variable X is standard normal if P X∈Γ = g(y) dy, Γ ∈ BR . Γ In spite of their somewhat insultingly bland moniker, standard normal random variables are the building blocks for the most honored family in all of probability theory. Indeed, given m ∈ R and σ ∈ [0, ∞), the random variable Y is said to be normal (or Gaussian) with mean value m and variance σ 2 (often this is abbreviated by saying that X is an N m, σ 2 -random variable) if and only if the distribution of Y is γm,σ2 , where γm,σ2 is the distribution of the variable σX + m when X is standard normal.
The Half-Life of Facts: Why Everything We Know Has an Expiration Date by Samuel Arbesman