Tiny Proof @tinyproof
Proofs to famous mathematical theorems in 140 characters or less. To submit a proof, contact us: [email protected] facebook.com/pages/Tiny-Pro… Joined April 2013-
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@KenWSmith54 Nice, I didn't know this one :) It's very "Eulery", with the unjustified limit approximations, but it can be made rigorous!
@KenWSmith54 Well, this is an IVP clearly satisfying the criteria for the Picard-Lindelöf theorem, so the solution is unique :)
New proof to something we've already done: y=e^ix and y=cosx+i*sinx both solve the ODE y'=iy with y(0)=1, so they're equal. Somanycharsleft!
Happy π day! Enjoy this one-page proof by I. Niven that π is irrational: projecteuclid.org/DPubS?verb=Dis…
p = prime # slices, m = # toppings, 1 on each slice. Pizza invariant under rotations iff it's homogeneous ⇒ # pizzas = m + (m^p - m)/p ∈ Z.
Let f∈Cᵏ[a,b] s.t. (*) ∫[a,b] fg dx=0 ∀g∈Cᵏ[a,b], g(a)=g(b)=0. Plug in g=(x-a)(b-x)f. Then fg≥0, and by (*), fg=0 in (a,b), so f≡0 in [a,b].
T(v)=det(v,w2,..,wn)=v1 M1 - v2 M2 + .. where minors Mi don't depend on i is linear. So det(u+v,w2,..,wn) = det(u,w2,..,wn)+det(v,w2,..,wn).
Rewriting g(n) = ∑{d|n} f(d) as Dirichlet convolution g = f * 1, we convolve with μ to get f = f * 1 * μ = g * μ, or f(n) = ∑ g(d) μ(n/d).
Let B ⊂ ℝⁿ be the unit ball, S the unit sphere. Then Area(S) = lim[h→0](Vol((1+h)B) - Vol(B))/h = Vol(B) lim[h→0]((1+h)ⁿ - 1)/h = n Vol(B).
By definition P(A | B) P(B) = P(A ∩ B) = P(B | A) P(A), therefore after dividing both sides by P(B), we get P(A | B) = P(B | A) P(A) / P(B).
Suppose ΣP(A_n) < ∞. Then P(A_k ∪ A_(k+1) ∪ ...) → 0 as k → ∞ (tail sum). Therefore P(limsup A_n) = P(∩_k (A_k ∪ ...)) = 0. (Borel-Cantelli)
C=set of sets, P=set of all choice functions F:(subset of C)→∪C. Define p.order F<G⇔(G extends F). Zorn ⇒ ∃ maximal F:C→∪C ⇒ axiom of choice
Let f:R^n->R be positive homogeneous: f(tx)=t^m f(x) ∀x, ∀t>0. Differentiate by t: Σxi ∂f/∂xi (tx)=mt^(m-1) f(x). Let t=1: x·∇f(x) = m f(x).
Convex φ has at point x0=E[X]=∫gdμ a line ax+b≤φ(x) ∀x with equality at x0. Then ∀t ag(t)+b≤φ(g(t)); integrating dμ gives φ(E[X]) ≤ E[φ(X)].
Let n ≥ 23. The probability of n different birthdays = (1 - 1/365)*..*(1 - (n-1)/365) ≤ 1/2, so a same-birthday pair has probability > 1/2.
No further words are necessary: qph.is.quoracdn.net/main-qimg-33f2…. For this and more, check out this thread on MathOverflow: mathoverflow.net/questions/8846…
f(n)=1+1/2+..+1/n-logn ≥ ∫[1,n+1] 1/x dx - ∫[1,n] 1/x dx > 0. But f(n+1)-f(n) = 1/(n+1) - ∫[n,n+1] 1/x dx ≤ 0, so Euler's γ=lim f(n) exists.
n=minimal composite with 2 prime factorizations n=p1..pm=q1..qk. p1 divides q1..qk, and by primality ∃j:p1=qj. n/p1 contradicts minimality.
1 + 1/2 + (1/3 + 1/4) + (1/5 + .. + 1/8) + (.. + 1/16) + ... ≥ 1 + 1/2 + (1/4 + 1/4) + (1/8 + .. + 1/8) + ... = 1 + 1/2 + 1/2 + 1/2 ... = ∞.
2^(1/n) is irrational for n≥3. Proof (Schultz, 2003): Assume 2^(1/n) = p/q; then p^n = q^n + q^n, contradicting Fermat's Last Theorem.
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