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Aug
26
comment Union of subspace
The union of $U$ and $W$ is a subspace of $V$ if and only if either $U\subseteq W$ or $W\subseteq U$. For a proof, suppose that neither subset relation holds. Let $x$ be in $U$ but not in $W$ and let $y$ be in $W$ but not in $U$. Show that $x+y$ is not in the union of $U$ and $W$.
Aug
26
revised Large pairwise coprime sets
added 30 characters in body
Aug
26
revised Large pairwise coprime sets
added 30 characters in body
Aug
26
answered Large pairwise coprime sets
Aug
26
revised Rearrangement and Cauchy
added 8 characters in body
Aug
26
comment Finding remainder when a function is divided by another
Hint: The remainder when $x^7$ is divided by $1+x+\cdots+x^6$ is $1$.
Aug
26
revised Rearrangement and Cauchy
added 216 characters in body
Aug
26
answered Rearrangement and Cauchy
Aug
26
revised How can $f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$ be factorized into a product of two polynomials?
added 85 characters in body
Aug
26
comment Find the maximum volume of the cylinder.
After you have the picture, draw a horizontal line at height $y$, where $y$ is fairly small, say $0.2$. This line meets the curve at two points $A$ and $B$. The cylinder has radius $y$ and length $AB$, so volume $\pi y^2(AB)$. Find $AB$ in terms of $y$, and maximize. To find $A$ and $B$ you will need to solve $x/(1+x^2)=y$ for $x$. This is the quadratic $yx^2-x+y=0$.
Aug
26
comment Where can I find this definition of “expected value”?
You are welcome. It can indeed be viewed as an expectation for a function of a uniformly distributed random variable. I do not recognize the context of your question, so cannot decide what it is about expectation you might need to know.
Aug
26
answered How can $f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$ be factorized into a product of two polynomials?
Aug
26
comment How can $f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$ be factorized into a product of two polynomials?
The polynomial is irreducible over the rationals. It is reducible over the reals, indeed over a much smaller field.
Aug
26
comment Where can I find this definition of “expected value”?
It is often called average value in calculus texts. One could think of it as ordinary expected value of the function of a random variable, where the random variable is uniformly distributed over the interval $(0,t)$.
Aug
26
comment convergence proof without finding 'N'
There are several algebra slips. It looks as if you are looking for a formal $\epsilon$-$N$ argument. Is that so?
Aug
26
comment Integration using trig substitution or substitution
Quick is $4x^2=3u^2$ or equivalently $2x=\sqrt{3}\,u$.
Aug
26
comment Integration using trig substitution or substitution
We want $2x^2=u^2$.
Aug
26
comment Integration using trig substitution or substitution
Try $\sqrt{2}\,x=u$.
Aug
26
comment How many different six digit numbers can be formed by various arrangements of the six digits: 2, 2, 2, 2, 4, 7
There are $6$ slots waiting for digits. We must choose the $4$ slots that will receive a $2$. This can be done in $\binom{6}{4}$ ways. For every such choice, there are $2$ empty slots left. We can choose which one of these will receive the $4$ in $\binom{2}{1}$ ways, for a total of $\binom{6}{4}\binom{2}{1}$.
Aug
26
revised How do I calculate probability of drawing cards in a certain order?
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