Well. To prove any real consequence of this equivalence you'd need to show that every power set can be made into a Noetherian module over some commutative ring.
However, since every module is a group, this means that every set can be given a group structure. This last one is equivalent to the full axiom of choice.
Not all is lost, though. It is possible that you can get from this equivalence that for sets which can be given a group structure you can define some choice function.
I don't know if I would say that the association is with rows or columns, but definitely, dC/ds measures the change along each column and dC/dt measures the change along each row.
Perhaps that is what they are trying to ask (kind of poorly in my opinion)
I dislike how essays/articles/books/etc. are titled "How X" (where X = some mysterious phenomena, or human activity the public is ill-informed about, etc), but the major thesis is actually establishing X exists in the first place (and implications of it, and so forth) - rather actually explaining or describing the mechanisms responsible for X. Very important semantic distinction.
Oh, I see. Sivaram has posted a more detailed solution. Of course, it goes without saying that you can accept any answer that you feel helped you most, @Victor.
How does one prove that a regular $n$-gon with perimeter $1$, approaches (becomes) a circle as $n$ goes to (or if $n$ is) infinity?
It is not enough to prove that all points become equal distance to origin, since this also holds for the limiting object of the graph of largest area drawn on the s...
@robjohn If it's any consolation, I upvoted yours (long time back), but not Mike's. That answer is swimming in votes already, so he won't miss mine I imagine... =)
@robjohn What does pointwise convergence mean for a sequence of polygons? As QED wrote, Hausdorff metric makes sense, but we should be a bit careful in defining the sequence of polygons.
Quick question that came up in my reading: how does one prove that if U is simply connected in the complex plane and if there exists a continuous bijective function mapping U to V, then V is simply connected. Obviously since they are homemorphic it should be preserved, but how does one prove this?
This problem illustrates the fact that two functions can be very close: $|f(x)-g(x)|<\epsilon$
for all $x\in [0,1]$, but their derivatives can still be far apart, $|f'(x)-g'(x)|>c$ for some
constant $c>0$.
In our case, let $x=a(t),y=b(t),0\le t\le 1$ and $x=c(t),y=d(t), 0\le t\...
let gamma_n converge as a point set to { gamma(t) in R^2 | t in R }
Suppose for all epsilon > 0, there exists n such that gamma_n is within epsilon of gamma (as a point set). Then for what proportion of the curve is it possible that gamma'_n - gamma' > c?
@robjohn You commented in your answer to the question "How to prove this particular limit as x->0 (sin x ) / x = 1" "understood, to be proven later". In any definition, there will be things that are left as understood. & a little bit of hand-waving can make believable... These are excellent examples of a recurring theme in mathematics found from the earliest elementary school days all the way up through Calculus and beyond...
In my university I'm the only student in set theory. So there are not enough people to open interesting courses. Furthermore, I already took all the courses this university can offer.
I am still missing two courses for my degree. So I was forced to take courses in alg. top. and commutative alg.
@AsafKaragila What do think of the comment: In any definition, there will be things that are left as understood. & a little bit of hand-waving can make believable..
So assuming it's the other way around, where dot{x} stands for the canonical p-name and check{s} stands for any old p-name, then dot{x}[G] = x for any G?
@AsafKaragila Would you agree with the comment: In any BASIC definition, there will be things that are left as understood. & a little bit of hand-waving can make believable..
@Skullpatrol There are several points of view about this. Euclid for example thought that mathematics could be divided into arithmetic and geometry and that was it.
@Skullpatrol Then there was the point of view that mathematics is about taking a set of axioms and then deduce theorems based on those axioms regardless of whether the axioms and the deduced things could actually be true in reality.
If you will correct your question, I will correct my answer. If not, do note there that there was a typo in the source because otherwise people can argue about my answer :-)
At the very basic level there must be certain undefinable elements that a little bit of hand-waving can make believable. this is the essence of doing mathematics
He studied under Frege and Russell, however he found mathematics irrelevant for philosophy since it is composed only of statements of "assume ... then ..." which can be vacuously true, and without much content.
I am agnostic about these things. I focus my energy on mathematics, rather on its philosophical interpretations and implications. I did that during my undergrad and that led me nowhere interesting really. You always get stuck at the dogmatic stage that one has to believe.
"the dogmatic stage that one has to believe" is exactly what I'm talking about in At the very basic level there must be certain undefinable elements that a little bit of hand-waving can make believable.
@Skullpatrol How dare you tell me what I will and won't be able to do? You don't even know me, let alone know what and how I know things. Who do you think you are, telling me these sort of things??
@Skullpatrol This is exactly why mathematics is not a part of life. It has its own accord, disjoint from this of life. Until you understand this, you can never really do mathematics.
Irrespectively, my point is that it's not based on some arbitrary definition somewhere; while certain aspects of mathematics may be unprovable, they are unquestionably true.
This problem illustrates the fact that two functions can be very close: $|f(x)-g(x)|<\epsilon$
for all $x\in [0,1]$, but their derivatives can still be far apart, $|f'(x)-g'(x)|>c$ for some
constant $c>0$.
In our case, let $x=a(t),y=b(t),0\le t\le 1$ and $x=c(t),y=d(t), 0\le t\...
