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1h
comment Prove that $(5x^3+9x^2-27x+3)$ is a maximal ideal in $Q[x]$
Rational Root Theorem.
5h
comment Integrating a square's perimeter to get its area
I will be going to sleep soon. If you have further questions, do ask, but there will be no response until tomorrow.
5h
comment Integrating a square's perimeter to get its area
Maybe it is best if you draw special cases and compute, like the squares of side $40$ and $48$ that I mentioned earlier. Let me use another example. We have a square picture of perimeter $120$, so side $30$. We put a border around it so that the square given by picture plus border has perimeter $132$. What is the width of the border? The sides of picture plus border have length $132/4=33$. Let the top border has width $w$. The bottom border also has width $w$. And $2w+30=33$. So $w=3/2$. Note that $132-120$, which is $dP$ in our case, divided by $8$, is equal to $3/2$, the border width.
5h
comment On splitting a number as the sum of two squares.
The posts so far mentioned as possible duplicates do not deal with the algorithmic part of the question.
5h
reviewed Leave Open On splitting a number as the sum of two squares.
5h
comment On splitting a number as the sum of two squares.
For a number $n$ which is a product of two large primes of the form $4k+1$, if we can find two really different representations, then we can efficiently factor $n$. Since the factorization problem appears to be computationally difficult, the general two squares problem may also be.
5h
comment On splitting a number as the sum of two squares.
There are good algorithms for representing primes. Names to look for are Hermite, Serret, and (in the computer age) Brillhart.
6h
reviewed Leave Open Generated subring and finiteness
6h
reviewed Leave Open Using disjunction to prove that $A \setminus (A \setminus B) = A \cap B$
6h
comment Integrating a square's perimeter to get its area
Yes, that's it. I am thinking of the scaled up version of the square as the original square together with a border of uniform width. The original square is a picture, and the border is the (thin) frame around it. The bigger square is picture plus frame.
6h
comment Can a real value function, defined for every real number, have finite (or numerable) points of continuity?
And a mild modification gives a function whose points of discontinuity are the natural numbers.
6h
comment A singleton set $\{g\}$ can be regarded as a unary relation in $G$. Why?
It was explained in the quoted passage. Let $G$ have $n$ elements. We can say just using $=$ and logical symbols that there are exactly $n$ objects. Call the sentence that does this $\varphi$. Then $\varphi$ is true in any elementary extension of $G$, so an elementary extension of $G$ has $n$ elements, and contains $G$, so it must be $G$.
6h
revised Integrating a square's perimeter to get its area
added 530 characters in body
7h
comment Confidence Interval - Cigarette HW Question
@SuperRhinocerus: Yes, $2.86$ is about right for the $t$. But as is now clear to you, for known variance we should be using the normal.
7h
comment Confidence Interval - Cigarette HW Question
It would be better to use the normal. And for the $t$, the number $2.539$ is not right.
7h
comment Confidence Interval - Cigarette HW Question
In the first problem, from the wording it sounds as though the (population) standard deviation is known, On the assumption that the distribution is normal, one can use the normal distribution instead of the $t$-distribution, giving a narrower confidence interval. But the normality assumption is not necessarily reasonable, By the way, I doubt the number used, since for the normal one has $2.57$.
7h
revised Given a finite collection of disjoint subsets of $I$ must every ultrafilter on $I$ contain exactly one?
added 463 characters in body
7h
comment Given a finite collection of disjoint subsets of $I$ must every ultrafilter on $I$ contain exactly one?
You are welcome.
8h
comment Integrating a square's perimeter to get its area
Draw the original square and the slightly bigger square. I assume you agree that the side increases by $dP/4$. So the width of the border is $dP/8$. Imagine a square garden of perimeter $40$. Expand the garden on all sides to make the perimeter $48$. So now the sides are $12$ each. That means that we have added a border of width $1$. The North border plus the South border have combined width $1+1$, which is exactly the increase in the side.
8h
comment Given a finite collection of disjoint subsets of $I$ must every ultrafilter on $I$ contain exactly one?
Is it the part $O$ "outside" $X_1\cup X_2$ that bothers you? If $X_1$ is not in $D$, then its complement $X_2\cup O$ is in $D$, and by assumption $X_1\cup X_2$ is in $D$, so the intersection of $X_2\cup O$ and $X_1\cup X_2$ is in $D$, that is, $X_2$ is in $D$.