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18h
awarded  Enlightened
18h
awarded  Nice Answer
19h
comment Solve the following congruence: $x(x+1)(x+2) \equiv 0 \pmod{221}$
@DietrichBurde Indeed, that's when we order solutions by "come to mind" instead of "by size" or whatever the OP may possibly intend ...
19h
comment Solve the following congruence: $x(x+1)(x+2) \equiv 0 \pmod{221}$
When you are asked to find the first five solutions, where do you start? Do you want $x\ge 1$ or $x\ge 0$?
19h
answered Solve the following congruence: $x(x+1)(x+2) \equiv 0 \pmod{221}$
19h
revised Last digit of $235!^{69}$
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19h
answered Last digit of $235!^{69}$
19h
comment Proof of derivatives though first principle method
The first principle computation of the derivatives of sine and cosine typically uses addition theorems for these trigonometric functions and somewhere deep inside requires the computation of one single specific limit: $\lim_{x\to 0}\frac{\sin x}{x}=1$. One has to be careful not to fall into circular argumentation at that point! A geometrically inclined proof of that limit typically compares the areas of certain triangles and circular sectors to establish $\sin x<x<\tan x$ for $0<x<\frac\pi2$ or the like.
19h
answered Generate random variate using inverse transform technique of $ f (x) =a (1+|x-2|)$
20h
comment Generate random variate using inverse transform technique of $ f (x) =a (1+|x-2|)$
Your integral over the full domain should simply be a sum, shouldn't it? And what you need should rather be something like $\int_{-1}^xf(t)\,\mathrm dt$, I suppose
20h
comment In the definition of a limit, why do we care about all $\epsilon > 0$?
The equivalence is precisely what justifies formulations often found in proofs such as: "Let $\epsilon>0$ be given. We may assume without loss of generality that $\epsilon<\frac12$. ..."
21h
comment Planar graph with at least $6$ vertices, at least $2$ vertices have degree at most $5$.
The accepted answer to one of the similar questions you went through uses exactly that inequality.
21h
revised $a_n\geq b_n$ for $n>\bar{n}$ implies $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b_n$
added 4 characters in body; edited title
21h
answered $a_n\geq b_n$ for $n>\bar{n}$ implies $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b_n$
1d
comment Explain a physical problem by mathematics
Where does gravity play a role in (a)? Gravity of the cylinder? Or should we infer from "open to the athmosphere" in part (b) that there is a planet (Earth possibly?)nearby?
1d
comment mathematician who changed the world
That's easy: While studying von Neumann algebras, Vaughan Jones obtained results that suspiciously reminded of knot theory. It turned out that the Jones polynomial led to a break-through in knot theory (it cut a Gordian knot, so to speak), leading to a large amount of research in knot theory in the years to follow, This popularity may have been one cause that led physics to develop an interest in string theory. And without string theory the world of today would be a totally different place: no Sheldon Cooper in The Big Bang Theory.
1d
comment Prove that all three metrics induces the same topology on $X_1\times X_2$
You are absolutely right. Looking closer into that direction, you will notice however that what Zelos suggests is sufficient for this. Recall that metrics $d,d'$ are called equivalent if thee exist constants $c_1,c_2>0$ such that for all $x,y$ we have $c_1d(x,y)\le d'(x,y)\le c_2d(x,y)$. Being equivalent as metrics is the most typical (but not the only) way for two metrics to induce the same topology.
1d
answered How did the rule of addition come to be and why does it give the correct answer when compared empirically?
1d
comment Piece wise function continuity
What have you tried? What is $f(-1)$? hat is $f(-1+\epsilon)$ for small positive $\epsilon$? What happens as $\epsilon \to 0$? What is $f(1)$ and what is $f(1-\epsilon)$? What happens as $\epsilon \to 0$?
1d
revised Piece wise function continuity
added 62 characters in body