not a prerequisite but useful. in my humble opinion, introductory real analysis is almost never necessary for anything that isn't more real analysis. but discrete stuff is so much harder than continuous stuff, it certainly helps.
i generally think of the discrete world as more complicated than the continuous world, so it seems logical to use real analysis as a stepping stone. others may disagree.
well I need to eat something before dinner. Copper thanks for the help. I have something to build upon. My conceptualization of these sets was nowhere what I was thinking. Going to have to play with more concrete sets to get some better intuition.
most informal definitions i have seen of the limit are very close to formal definitions, just some of the quantifiers have been taken out and the user is not expected to operationalize the definition.
i guess maybe it's a word game. a lot of what people would regard as informal "definitions" of the limit are not definitions at all. they're descriptions, or something slightly less than definitions.
once i answered a question and the guy said there was a little extra if i solved another problem (computing the volume of a wine hopper). so i went the extra mile and sent the answer. so he said next time you are area drop by and i will give you a bottle of wine.
unfortunately the winery is north of vallejo, like near vancouver or something
@copper.hat it happens and very probable. That is why I have decided so long ago to do things for my own pleasure, writing codes and solving problems for others helping them. I do not know why they offer things when they can not do it! may be they think if they do not say so, they will not receive volunteer offers...
you need something resembling completeness. how people axiomatize it can vary. but it's definitely a big part of it. you can't get anything resembling real analysis without it.
@zacts in maths, you can get off at any floor. if you want to descend into ZFC hell you can, but most folks are willing to get off at floor "i know enough about real numbers already".
you need a hammer like archimedes property and a few other tools, but that's about it for most hiking.
I guess my curiosity with real analysis is that you have a framework that you are tinkering into concepts like the limit. I'm not quite as curious into the formality aspect of it, as much as I'm curious into tinkering a framework into those ideas.
i had a funny conversation with one of my friends about this. how basically every narrative or dramatic device prior to 2000 completely breaks down if one character is given a cell phone.
@copper.hat why? that was almost the thing I have always liked about physics and maths! I am in engineering field and I can feel how realistic the quote is!
andrew, i'd like to give you some kind of smoke and mirrors thing in exchange for money. there's a ledger. it's permanent, there's crypto in it. everyone's doing it. please pay me for it.
andrew, i'll re-describe it. it's indestructible. there's something called a blockchain. nobody can change its entries. it involves you giving money to me. the money will not be refunded. i can also assign you personally signed leslie townes certificates but if i were you i'd trust the blockchain.
there really is something such as culture. there's so much of that in my grand-dad.
the old irish tales growing up had lots of powerful kings/queens etc, but the real power was help by the smart assed poets. i never really understood until i read behan.
its happening now, boith sides of the political divide are busy printing money
@copper.hat Makes me think of that lovely Yogi beaar quote that stephen pinker always repeats: "Making predictions is difficult, especially about the future"
@leslietownes Well you will just have to take this "IOU 1 happy meal" that I am waiting to redeem from an angry collegue
that's true, i am actually a member of the state bar of california and its federal courts, and also a member of the bar of the federal court for the district of colorado.
@zacts Depends upon what you want to do with knots. If you just want to do the combinatorial aspect of knots, you do not need much prereq. But again personally combinatorial stuff seems magic hogwash to me, so knowing some homology, fundamental groups and what homology theories are can help you go beyond the combinatorial picture
category people can get into it more than i can but a lot of knot invariants appear when you have functors from the space in which the knot exists to something simpler.
hey leslie I got another one for you, is it appropriate to use $\equiv$ in this setting ($\Psi$ is the piecewise function I asked about earlier) $$ \int\limits_{-\infty}^{+\infty}{\Psi^* \Psi}dx \equiv \int\limits_{-\infty}^{+\infty}{\Psi^2}dx \equiv \int\limits_{0}^{a}{A^2(x/a)^2}dx + \int\limits_{a}^{b}{\big(\frac{A(b-x)}{(b-a)}\big)^2}dx = 1.$$?
if the three-bar stuff is equals by definition, this seems OK. assuming of course that psi is real valued, which it seemed to be, but generally wouldn't.
it's quite late. i think we inhabit the same time zone. but if he were to offer pedagogical advice on this issue, some time tomorrow or later, i'd say, follow it.
i'm so rarely up at this hour, my daughter usually gets me up early in the morning, so the idea of me being up at 12:30am is like, me being back in my 20s again.
Oh god, go to sleep man, I feel like it was false recollections that made that lack of sleep OK
I used to get by on barely any sleep in my 20's but I was a caffien addict, and I had no attention during the day. I am surprised I didn't have more traffic accidents in retrospect
i wake up, check my phone, and if nothing needs immediate response, right back to bed. it's genius. i don't know why we didn't do this before the pandemic.
when she's home she's the one who wakes me up. "get up, dada!" i tell her to get up. it doesn't work. she says "i'm up, you get up." it's a who's on first kind of situation.
is a square with adjacent sides identified isomorphic to the klein bottle? they have the same fundamental group so I can't tell them apart using that, this is the space im talking about imgur.com/a/N9TiwM9
ah okay I think they are homotopy equivalent, you can make a diagonal cut along the diagonal of my diagram, and then extend the corners of the square perpendicular to the diagonal into lines (and remember to reidentify them at the end), and you should get a union of two mobius strips glued along their boundaries
sorry what I meant to say for that is, I can draw another graph where I extend those opposite points to opposite lines that have been identified, and then this new edge I've added to the graph (the identified opposite edges) is a contractible subcomplex of my new graph, and so my new graph is homotopy equivalent to the old one via the map collapsing those opposite edges
oh wait a sec, yeah you're right that isn't contractible, its just $S^1$
whoops
so I don't have a homotopy equivalence, still interested if this is homotopy equivalent to $K$ though
rotate the square 45 degrees so it looks like a kite, and deformation retract the 'bottom' two edges of the square onto the horizontal line through the center of the kites (via a orthorgonal projection), so that one gets the following figure shown on the right of this image: (taken from hathcher page 51) imgur.com/a/G8n2yZo
one should get exactly the triangle shown in the right, which is a klein bottle
i dont see whats wrong with this one
@AndrewMicallef nah, I was just hasty
ah crap, the orientations are slightly different
its still wrong :(
oh actually, its still a klein bottle isn't it? In the imgur link I posted, it should make no difference to what the identifications are if I make the $c$ identifications clockwise rather than anticlockwise..
If we interpret an $n \times n$ matrix $A$ as a linear transformation in n-dimensional space, is there any easy way to visualize/interpret $A^T$? So far all the standard properties of matrices, determinants seem to pop right out of the geometric interpretation but I got stuck at arriving at $(AB)^T = B^TA^T$
Hey guys I have a quick question: When calculating the mean of grouped data what happens if the last class includes both term(say 12-14 includes both 12 and 14) but other classes don't would the formula of mean, median and mode change somehow?
@robjohn I meant that generally classes like 8-10 mean [8,10)(includes 8 but not 10) but in my case I have all classes like [0-2); [2-4); [4-6) but last class is [12-14]
I don't know if it'll affect the formula for median and mode like if I have to divide data as [12-14); [14-16) and for that I'll need to know how many '14's are there....
does anyone know if there's a link between the existence of complex (i.e. not real or pseudo-real) representations of a Lie group and its fifth homotopy group being non-trivial?
In my commutative algebra class, professor taught primary decomp. on monday and integral dependence on wednesday. HW problem only consists of integral dependence
@user2236 hmm...I don't really think the answer will have much to do with homotopy theory
at any rate, my question arises from a physics context (anomalies in gauge theory), so there might not actually be a direct mathematical relation between the two - but I'm just checking to see if I missed anything obvious
What should I say to refer to the following diagrams? Do they have a special name in mathematics?
If there are a cartesian product {(a,1),(b,2),(c,1)} and I want my students to draw this relation with the above diagram rather than plotting the dots on the Cartesian coordinate system, what should I say in the question sheet?
the ones on the left are called functions. i hate to be literal about this.
i don't use the term 'many-to-one,' i just say 'not one-to-one,' because who knows if it's 'many' (e.g. there, it's just one value that has more than one thing mapped to it, and it's just two things being mapped to that value).
oh you mean the graphic form of representation? i don't think it has a name.
i can't think of a way of doing it other than just giving examples or even a diagram of the sets to fill in with the arrows.
you can also use the diagrams to illustrate function composition, which i like. you just put another set to the left or right, and more arrows. for obvious reasons it's fairly rare to see these written out for sets having more than, like, six elements.
really good way to illustrate permutations. you compose a string of those by just thinking of all the intermediate arrows as string and pulling them tight.
you can even make meaningful sense of some of the graphical properties, at least some of the time. like the numbers of times arrows cross.
