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17h
comment Image of open set is not open?
By "one-to-one" did you mean "a bijection"? Because "one-to-one" doesn't imply "onto" ("one-from-one"), but it seems like that's what you intended.
2d
accepted Continuous trapdoor functions?
2d
awarded  Yearling
Dec
13
comment Continuous trapdoor functions?
+1 nice arguments
Dec
13
comment Continuous trapdoor functions?
The first reason seems questionable to me (I didn't see natural numbers anywhere on the Wikipedia page... citation needed?) and the second one is pretty handwavy. I only asked for continuous, not analytic or smooth or even differentiable. A function like the Weierstrass looks rather unpredictable to me despite its continuity (but I'm not sure how I should make this statement rigorous).
Dec
13
asked Continuous trapdoor functions?
Dec
10
accepted Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
Dec
10
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
Oh wow you multiplied it out and simplified it massively haha. Yeah I agree, I think choosing $w$ is harder, especially for more complicated examples. Thanks!
Dec
9
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
Also, how did you differentiate $\frac{y'}{\sqrt{(1+ (y')^2)}}$ to get $\frac{y''}{{(1+ (y')^2)}^{3/2}}$? Aren't you missing some terms there?
Dec
9
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
Could you explain how you used Cauchy-Schwarz to prove what you mentioned?
Dec
9
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
I like the answer, I can't help but wonder if there's a simpler way to do it than requiring integration by parts though!
Dec
9
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
Oh, it's the difference between $w(x_1) = 0$ and $w'(x_1) = 0$, I see now!
Dec
9
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
Oh so you're basically saying it's a Dirac delta centered at $x_0$?
Dec
9
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
Oh, interesting! Which assumption does it break...?
Dec
9
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
Hm okay that's fine, I'm not too worried about that. What I'm worried about more is, doesn't the fundamental lemma imply $C = 0$?
Dec
9
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
I'm confused what $n$ is doing in your definition, it doesn't seem to be present in the definition anywhere.
Dec
9
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
Actually, just kidding, I just realized that's the fundamental lemma of the calculus of variations. However, am I missing something or does that lemma imply $C = 0 \implies y'(x) = 0 \implies y(x) = C_0$?
Dec
9
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
How did you get the last implication (that $g(x)$ is constant)? It's not obvious to me.
Dec
9
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
Ok thanks just checking.
Dec
9
comment Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation
When you say "if $u$ satisfies $\int u\,dx = 0$" do you mean "if $u$ satisfies $\int_{x_1}^{x_2} u\,dx = 0$"?