I am not sure if it is even possible to sum that alternate series in this fashion:
$(((1-1)+1)-1)+1-\cdots$
One thing I am interested under this framework is to have some notation of measuring how diverging a divergent series is by checking if for each different value obtained due to nonassociativity, whether there is a correlation between some values and the pattern of the nesting of the brackets
Another thing I am interested in is whether there exists series that have countably infinite many diverging values corresponding to countably many ways of summing it
What I am interested in is then to query how divergent a given series is, and whether it makes sense to treat the different ways of grouping terms together as some notation of path dependence in some space
hmm... actually now that I think about it, it appears that grouping terms in different ways for the above alternating series can give as any interger number as its value is controlled by how many 1s and -1s are left behind and where that infinite tail of 0 starts
e.g. $((1-1)+1)+((1-1)+1)+((1-1)+1)+(-1+1)+(-1+1)+\cdots = 3$
I see, so there is indeed some kind of pattern in the divergence. Hmm, I am not sure if this property is significant enough to have a name, probably people will explain it in terms of subsequence like what you said
In mathematics, 1 − 2 + 3 − 4 + ··· is the infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as
∑
n
=
1
m
n
(
−
1
)
n
−
1
.
{\displaystyle \sum _{n=1}^{m}n(-1)^{n-1}.}
The infinite series...
hmm...
(Obviously not realated to Cesàro sum). We cannot say that some given divergent series is equal to the average value of all its possible divergent values weighted by the number of ways it reached said divergent value by picking subsequences, because there are continuumly many thus the average will not work out.
I am not sure if it is even meaningful to sort of construct a space with points indexed by the ways subsequences were picked since at least in a natural sense, there seemed to be no obvious way on how to establish a partial ordering in such space to meaningfully talk about paths
My original idea is trying to came up some notion of topology generated by the different ways brackets are placed and arranged and see if that will give some ideas on how the different diverging values of divergent series depends on topology (if any), however it seemed to be a bit complicated as it now seems
Grothendieck's relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on parameters, as the basic field of study, rather than a single such object. It is named after Alexander Grothendieck, who made extensive use of it in treating foundational aspects of algebraic geometry. Outside that field, it has been influential particularly on category theory and categorical logic.
In the usual formulation, the language of category theory is applied, to describe the point of view...
Lol, then I think like Grothendieck given how I tend to explosively overgeneralise things
@Secret I tend to think that algebra tries to generalise more and more while analysis tends to specify more and more, going down to every epsilon and delta.
So, I want to prove that $$\sup_{n \in \Bbb N}{\sin (n)} = 1$$
I was thinking of proving that some set related to $\pi$ is dense in $\Bbb R$ that will then imply there is some $n \in \Bbb N$ s.t. $\sin(n)$ is as close to $1$ as desired. ($\left( \frac m n \right) \times \pi\quad \forall m,n \in \...
Like, for the first one, pick those natural numbers $n$ such that $n \in [\pi/2-1,\pi/2 +1]$
(modulo $2\pi$ of course)
And for the second one, the same with $-\pi/2$ everywhere
For the first sequence, you'll have $\sin(\pi/2 -1) \leq \sin(n) \leq 1$
and for the second $-1 \leq \sin(n) \leq \sin(-\pi/2 -1)$
Well, technically they're all strict inequalities but that doesn't matter. You then just have to note that $\sin(\pi/2 - 1) > 0$ and $\sin(-\pi/2 - 1) < 0$
So definitely those two subsequences can never converge to the same limit
@WillHunting reminds me of a Terry Tao blog post, where he discusses ultra filters / non-standard analysis as a way to "automate" some of the minutiae of epsilon-delta arguments
(Though if memory serves there's a price, in that the results one gets in this way usually aren't as strong/precise as possible)
Even for Gaussian curvature, that's defined as the ratio of determinant of the Ist and the IInd fundamental form, it's not obvious that's a local isometry invariant (i.e. is an "intrinsic property").
i.e. the number of roots can be written as $\frac{1}{2\pi i}\int_\gamma \frac{P'(x)}{P(x)}\,dx$ where $\gamma$ winds around the first quadrant of the complex plane.
Explanation: should learn the subject before (you can start) asking the questions incorrectly vs should learn the subject, before asking the questions incorrectly
well, all positive even cases (a,b) should have no roots since the $x^a$ and $x^b$ wil be simple concave up curves thus +1 will make the whole thing leave the x axis
The odd cases and case involving negatives is where it gets difficult
If I take fourth powers of all those polynomials (in the sense of $(e^{i\theta})^{1/4}=e^{i\theta/4}$) then those 2^2015 roots all all land in the first quadrant.
Topological spaces are another example. A homomorphism of topological spaces S -> T maps open sets in T to open sets in S (although homomorphisms are usually not considered to explicitly contain such a mapping).
@WillHunting Well, I'm trying to describe what I mean by giving several examples.
(tbh one could probably use the info I've said throughout the years here to figure out who I am off-line, and from there it's easy to get my address. but eh)
Although I do know the futility of emails. I have had so many contacts that simply vanish after a while. They don't bother to respond for several months.
Probably what one should do is write the time in air for a given bounce as a function of the max rise of that bounce, which is itself proportional to the max potential energy (and thus max kinetic energy) in each bounce.
Since the time for an object of mass $m$ to descend from a height $h$ undergoing gravitational acceleration satisfies $h=\frac{1}{2}gT^2$, the total time for one bounce would be $2T=\sqrt{2h/g}$.
But the potential energy at max bounce is $U_{max}=mgh$, and $U_{max}=K_{max}$. So $2T= \sqrt{2h/g}=\sqrt{2/mg}\sqrt{mgh}=\sqrt{2K_{max}/mg}$. Hence the time in air for each bounce varies as the square root of the kinetic energy after rebound.
If no energy is lost on each rebound, then each bounce would have the same initial position and the same initial velocity. Hence each bounce would be identical.
This is not affected by gravity, since that's a conservative force.
So if your bounces are not identical, then you are not properly simulating a purely elastic rebound.
I've got another category theory question, and this time it's very broad.
How can categories of algebras be built out of Set?
One example of a "category of algebras" is the category of pointed sets.
This category is just the coslice category 1/Set.
Another example is the category of sets equipped with functions into themselves.
This seems to be a little more complicated to build. Start with the comma category id_Set/id_Set. Take the full subcategory of this where the left and right sets are the same.