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1h
comment I need some help understanding proofs for an upside-down cycloid being the tautochrone curve. Could someone show me or point me to a simple proof?
There's a straightforward proof in the first chapter of George Simmons' Differential Equations text.
16h
comment Calculate fractional part of square root without taking square root
Related: Detecting perfect squares faster than by extracting square root
May
21
comment In topology, any relationship on boundary like $Bd(A\cap B)$ and $Bd(A) \cap Bd(B)$?
The conjecture is false even for well-behaved sets. Let $A$ be a disc and $B$ be its complement.
May
21
comment If the answer is “no” then “yes” and vice versa type of paradoxes. What are they?
I wrote about the significance of Russell's paradox here: math.stackexchange.com/questions/1086645/…
May
20
comment From Decimal to Octal
If you know how to convert to binary, do that. Then divide the binary numeral into groups of three bits, starting at the right. Each group of three bits becomes one octal digit. For example, binary 001 101 110 011 111 000 101 becomes octal 1 5 6 3 7 0 5.
May
20
comment Lambda calculus typing
Your last question doesn't make sense to me.
May
20
comment Lambda calculus typing
It could be. Depends on who you're talking to. Why is there no $b$ and $e$ for which $b=b\to e$?
May
20
comment Lambda calculus typing
Your conclusion is correct, but the reason isn't quite right. For example, suppose you found $b=p\to (q\to r)$ and also $b=(s\to t)\to u$. Does this fail? No, because you can unify these types by saying that $p=s\to t$ and $u=q\to r$. But in your example, say the type of $y$ is $b$ and the type of $yy$ is $e$. Then you need $b = b\to e$, and there is no such $b$. In this case the unification algorithm fails.
May
20
comment Special representation of a number
Experiments bear out this theory. There are 20 ways to represent 3150 in the form $n=pq+rs$, and this is the most ways for any $n$ under 10,000. But if we start looking at larger primes, there are over ten thousand such $n$ below 200,000; for example there are 20 or more ways to represent each of 170280, 170430, 170436, 170520, 170790, 170940, 170970, 171318, 171780, 171990, 172410…
May
20
comment Special representation of a number
There is a very large number of counterexamples to your uniqueness conjecture, of which the smallest is $$11\cdot13 + 19\cdot 29 = 11\cdot 43 + 13\cdot 17$$
May
20
comment graph theory-Eulerian
Something here is probably useful. math.stackexchange.com/questions/685602/…
May
20
comment Solving $e^\frac1x = x$ non-graphically?
@jack "Lambert W function" means that solving it is outside the scope of everyone's current mathematical knowledge. We don't know how to relate the solutions to anything we actually understand, so we make up a name for the solution of the equation so that we can discuss it. The name is "Lambert W function".
May
19
comment Languages in coNP
Consider for example the TAUTOLOGY problem: given a boolean formula $F$, does $F$ have a true value for every assignment of values to $F$'s variables? This problem is complete for co-NP. There is an obvious algorithm to decide it: try every possible assignment of values and check each one to see if $F$ has a true value.
May
19
comment “Rationalizing” an equation
Closely related: math.stackexchange.com/questions/1010973/… ; math.stackexchange.com/questions/359054
May
18
comment What is the nature of Conway's sequence…
It says “The terms eventually grow in length by about 30% per generation.”
May
18
comment What is the nature of Conway's sequence…
Isn't it apparent that it is increasing? Have you read the Wikipedia article about it?
May
16
comment On prime factors with $n^2+n+1$
"all divisors not greater than $x$" is confusing. Do you mean $\forall d. d\mid n^2+n+1 \implies \lnot d\gt x$, or $\lnot\forall d. k\mid n^2+n+1 \implies c\gt x$, or something else?
May
15
comment Undergraduate Research review.
@Dan math.se is not a discussion forum; it is a question-and-answer forum.
May
15
comment Showing equality of two lambda calculus expressions
Do you know how to beta-reduce a lambda-expression? For example, can you do one beta-reduction step on $(λx y z:(xz)(yz))(λu:u)$?
May
15
comment how many ways can you divide 24 people into groups of two?
7 people can't be divided into groups of 2.