@Daminark Let $H$ be a codim 1 subspace of your ambient space. Find a function f_H: M \to R which singular points p of f_H correspond to points with T_pM=H.
I think I've figured sometihng out in math, but I'm not sure if anybody else has seen this pattern before or thought of this before. How can I find more information on what I'm working on?
Hopefully, it's an ethical or moral discussion, rather than an essay about the practical aspects.
@SimplyBeautifulArt In regards to triangular matrices, you're talking about square matrices which have entirely zeros either below or above the diagonal, right?
@Anheuser Now that he says he's in middle school, it's more clear now that he's in that age-group where they ask for that argue for/against style of work.
Those were the worst.
Because you generally need to pick a side on a topic you don't feel strongly about at all.
@Axoren I honestly cannot relate, my middle school days are slightly hazy and the memories I do have paint a bleak picture of the educational system I was in at the time.
I had to write a thesis about Corn in my Freshman year of college. I can say that regardless of what Middle School is like, if you end up taking the wrong seminars, you're in for a snorefest.
@BalarkaSen Because it could end this line of questioning very quickly :P
But generally, being able to debate allows you to resolve subjective biases by defending them against another person and losing, or strengthen beliefs and conjectures by being able to successfully defend them.
@Axoren I just think if someone feels the need to resolve certain subjective biases or strengthen beliefs about certain things, they will automatically develop a line of reasoning in their defense. Making someone "practice" this by debating about topics he or she feels ambivalent about sounds off the point.
@BalarkaSen I see, debate skills vs. debating uninteresting points are two different matters. I agree with you that forcing someone to debate something they are opposed to is definitely off point.
Agreed. I actually believe that a logic system should be taught before algebra, because it seems kind of astounding that children are expected to just get it.
There definitely should be a course in logic, perhaps as a subset of the mathematics curriculum, for every year in high school in increasing order of complexity/abstractness. And the rest of the syllabus should draw frequent references to logic in that specific grade.
You know, I understood it as well. But absolutely hated it. I would not study, I wouldn't do any homework. I failed grade 11 and 12 and graduated with an average of 65%.
Logic is more beneficial because it allows people to find flaws in reasoning regardless of context. You'd solve so many deep and philosophical Facebook posts if you just educated the common man in basic logic.
Maybe even then, Shaq wouldn't believe the world was flat.
Even I find myself using logical falacies, sometimes. Because when I was younger, they taught me how to find the roots of a 2nd-degree polynomial instead of logic.
On the other hand it's just an utopia. Even if there was logic, the teachers - most of them incompetent at logic themselves - would teach those chapters mechanically and make it seem like the logical ideas are a rule of thumb which can only be applied to the math test, not the physical world.
@BalarkaSen One of the biggest problems with lower education is that the teachers are not specialized in the subjects they teach. With only a valence understanding of each topic, they can only teach from their curriculuum as written, not as intended.
In fact, my grade 6 teacher was such a bitch (excuse the language, there's no other way to describe her) I didn't self-study at all because I just couldn't stand math
We need to dissolve the fact that teaching is a certificate-level position and incentivize qualified professionals to teach in elementary and secondary education positions.
Not just for money; because they can get a good environment (for their research, say). In most high schools qualified professionals don't get the right environment to teach what they want to teach and soon gets frustrated and stops interacting
I've been in that practical-less hole a few times in my comp-sci studies, where a colleague of mine and I worked together to formalize a protocol for transmitting data across space-time in the time-axis.
We spend months formalizing sending a problem into the future, solving it later, and retrieving the answer in the past, so that you didn't have to wait.
@MeowMix Find some meaning in the stuff you do. Or try to find it. I don't know how to live through the days doing what I do without hoping for some meaning to it all... being fully aware that it's a dubious idea.
Java since Highschool. Python, but not fluently. I'm a tool of the job kind of guy, so I'll pick up a new language in a day if it has language features that make solving the problem I have easier to do.
I'm trying to perform a sanity check. If a group G has subgroups A, B, C so that B is normal in C, what is it that can make <A, B> not normal in <A, C>?
So you'd need to take into consideration item acquisition sequencing. Each chest depends on certain items to reach them (in the case of OoT, glitches can resolve this), ensure that there exists some reachable sequence of chests before that chest which contain those items, and there's no deadlocks.
the reason I thought this worked for a moment is because B normal in C gives a natural projection map C -> C/B whose kernel is B, and it seems I can use this to define a map <A, C> -> C/B by taking all elements of A to 1, and elements of C to the same as the natural projection. and that map's kernel looks like it's <A, B>
but it seems like the most likely breaking point in an otherwise possible proof of something I have a counterexample for in front of me
It's to do with subnormality. If H and K are both subnormal in G, then J = <H, K> is not always subnormal. But
since K is subnormal then I have a subnormal sequence K < K_1 < K_2 <...< K_n = G, and if K = G then obviously J = G is subnormal in G. So I can induct and say <H, K_1> is subnormal and <H, K> is normal in <H, K_1>, so <H, K> is subnormal in G. actually this is even worse since I don't appear to have used H's subnormality at all
the counteexample to that is a kinda longwinded construction of a group where you can get two subnormal groups with series of length just 3 or so but their join isn't subnormal
I think there's something wrong with my conclusion about the kernel but I'm not sure what
Well, my losing my temper and speaking intemperately earlier to the French person who insists on asking everyone for help was probably not the right way. But I honestly don't think the graph of $|x|+|y|\le C$ is that difficult for someone doing relatively advanced mathematics :(
Demonark, are you rewriting some of Shakespeare these days?
@BalarkaSen I was thinking learning <insert math topic I don't know about yet here> while listening to show tunes and watching avant-garde festival films.
Well I have to finish up a proposal that works in a nice medley of finite fields, and then when I come here it's always some crazy groups that leave me feeling like I wouldn't question it if a physical fucking apple was one of the elements
I mean you'd think Z is all the numbers, but then you're like oh! in-betweens! boom Q no problem.... but wait, in-between those in-betweens... ok fine.... R... but wait! NO. NO WAIT JUST R. R.
That just has me considering: I know it's possible to extend $X^J$ to arbitrary sets $J$, but is it possible, or for that matter, even sensible, to extend Cartesian powers to, say, any real number?
Last night dream: Saw a strange algebraic system which seemed to have some kind of topological properties:
Given $ax=x$ Multiply on the right by $x$ $(ax)x=xx$ Then $x=x^2$
The above is the example in the dream, which is quite straightforward as a and x form a pseudoinverse. However, there's an implication that the dream use this example to illustrate, and it is the following:
Consider the string of elements: $axxxxcxxb$ This algebraic system has an unusual axiom that $a(any 5 elements)c=e$ where $e$ is the identity
Therefore if you have $axxxc$, it does not simplify under this rule. However, if you have $axxxxxc$, then it becomes $exxxxx=xxxxx=x^5$
In a sense, it is a generalisation of pseudoinverse, a "position dependent" pseudoinverse
Of course, we can go even further: Let a string of elements $s$. There's an axiom $a(\text{some string from a set S})b=c(\text{some string from a set S})$
Now if $xxxxx=y$, then one can see something interesting because $ayb=cy$ but $axc$ stays as $axc$
meaning that for the special case where elements combined to form the identity, you now hae selective pseudoinverse, and more generally, you have algebra that is dependent on the position in the string
More investigation is needed to determine what are the axiomatic systems required to preserve associativity
Now, because of the position dependent nature of operations, this means one can easily encode a topology in this algebraic system because the string only simplify if the elements are in the correct relative positions
@TedShifrin "tourner" would be much much more colloquial. Except in the phrases "virer de bord", "virer à droite/gauche", people will understand "virer" as "remove" or "throw away"
Hi, I am a discrete math noob and I have a following question: Given numbers from 1,2,3,...,10 , if we are allowed to pick 3 of them, in how many cases does the sum of those numbers a) equals nine b) is less then one ???
yes, it does not matter....I recognize it has to do with partition of numbers where there will be 3 disjoint sets with those sums, under the equivalence relation
