12:04 AM
Hi DogAteMy.
There are good resources about page rank on the web.

But how could I find these resources? I'd need some crazy sort of searching algorithm…

Well, you could go to the library and look in Math Reviews. That's how I found articles back when I did my undergraduate and graduate work.

12:20 AM
Hi chat

Hey y'all!

i'm trying to work out some homological algebra and of course cartan completely understood what i'm thinking about 60 years ago

hey
I have a question for analysts
Kind of open ended/not super defined

@MikeMiller classic

i get it down to some nice looking complex but get stuck, google around for a bit and see precisely that in the original 1950 paper

12:32 AM
when reading analysis papers sometimes the sheer volume of notation can be pretty overwhelming. if it's something I need to know deeply I can sit and break down every line but this takes awhile. When I want to quickly read through a paper is there anyway to slug through some of the volume of notation quicker?
"the solution to equation (n) is (some disgusting multiline formula"

@MikeMiller "there is nothing new under the sun"

it's hard to tell which terms are important or not sometimes

@ZacharySelk i think it depends so much on the subarea that it's hard to give a meaningful answer

@EricSilva Does it though? I've found certain things are pretty universal. Inequalities give you some type of control: look for what's controlling what.

sure but that's also a very vague principle
useful but still really vague
a big principle in the stuff i like is to look for any quantity that's scale invariant and single those out

12:41 AM
yeah, it's a very vague question
What do you mean by scale invariant, can you give an example maybe?

Hey I know this has already been answered all over SE, but how long should I spend on a problem I'm struggling to solve before giving up/looking up the answer? I've got quite the busy schedule this semester and I'm going to have to be much more time-efficient than usual

2.46 minutes :)
I think it's very personal/more of a feeling

I mean, I absolutely love the challenge math brings, and if I had more time, I wouldn't move on until I'd solved each problem, but my course-load is heavy this semester and I don't want to waste time

"is spending anymore time on this problem going to be more beneficial to me than doing what I would be doing if I choose to not spend time on this problem"

so like the minimal surface equation is scale invariant because if you blow up a minimal surface it stays minimal

12:45 AM
@ZacharySelk that question is too profound for my tired brain. I need more coffee
I get the gist of what you're saying though :) thanks

@EricSilva hmm, I might have to think about that for a bit

usually i use it to mean that you have things that scale "the right way" when you blow it up or something, like if you have an estimate for something on a ball $B_{1}$ and scale to obtain a version on $B_{r}$ for general $r > 0$ i call the properly scaled form the scale-invariant estimate
sometimes scaling breaks estimates and everything is horrible and you wanna die though

sure, I've seen things like that before

there are probably loads of other vague general principles for reading analysis but im still a novice at reading papers anyway so idk

1:06 AM
@EricSilva @ZacharySelk Is there a notation to specify the domain and/or range of a function?

$f:X \to Y$, $X$ is the domain, $Y$ the codomain, and we often denote the range of $f$ by $f(X)$.

Is that only specific to certain branches of math, like set theory? Thats the only place I've seen that.
$\leftarrow$ novice here

nah dude it's all over the place

You'll find that anywhere except maybe calc @CookieToast

Why don't they use it in calc? Cause yeah I've definitely never seen that in calc

1:10 AM
it's usually not introduced yet
otherwise people would probably use it

generally in calc its understood that your functions are $f:\mathbf{R}\to \mathbf{R}$ or $f:[a,b]\to \mathbf{R}$ or $f:(a,b)\to \mathbf{R}$.

the foundational underpinnings are often elided in calculus so that the physics and engineering kids can learn how to push the right buttons on the calculator
and, as @Antonios-AlexandrosRobotis says, we typically assume that all of our functions are real

But don't most physics and engineering kids still need some upper division coursework, like analysis?

depends
not always

at my school physics kids do not need to take analysis

1:12 AM
I'll fully admit that I have less information and more assumptions

and don't forget the biology (ahem... premed...), chemistry, and CS students who also are required to take calculus
basically (in the US, at any rate) 99% of the students who are taking calculus will require no upper division mathematics ever

a lot of the calc classes at my school actually totally use $f:X \to Y$ notation but we're not normal

you're lucky if you can get them to parrot the phrase "A function passes the vertical line test! Herp derp!"

Well that explains why I've never even seen it before :) @eric what college do you go to?

@EricSilva I use use that notation in my PRECALC classes, so there!

1:14 AM
Uchicago

@Xander bonus points if they use a ruler to show what "vertical" means :P

on the other hand, UChicago is a real school

unlike my poor UC

"real"
""school""

1:14 AM
Damn Eric thats awesome!

Where do you go @xander?

I am at UC Riverside
so, on the one hand, it is a UC, which is good
on the other hand, it is Riverside

My cousin is doing her PhD at Riverside :P

in what department?

1:15 AM
@XanderHenderson what's wrong with riverside?

Philosophy. Do you know anyone from there?

i feel like you guys are talking about a real thing not an imaginary thing...

Unless you consider math a branch of philosophy, then no, I know noone in philosophy

@EricSilva we actually had basic pointset in one of our intro calc courses LOL
but it was an anomaly.

@EricSilva we compare unfavorably to UCLA and UC Berkelely and UC Irvine and UC Davis

1:16 AM
Anyone any good at Linear Programming?

and maybe UC Santa Cruz and UC Santa Barbara

philosophy and logic sometimes overlap

BUT we can still feel superior to UC Merced!

oh when you said "on the other hand, it is riverside" i thought you meant there was a problem with the locale

we have logic classes mislabeled or pehaps correctly labelled as easy logic classes

1:17 AM
from what I've understood, philosophy and (mathematical) logic certainly do overlap

@Antonios-AlexandrosRobotis that's sooooo weird

Oh, Riverside is, to paraphrase the tangerine, a shithole county

lol

@EricSilva I wasn't in that course, but it was with one of the most notoriously difficult/strange professors at Berkeley.

but the institution is actually pretty good (though, again, we have to compete with the other UCs)

1:17 AM
I don't think that calc class did a single actual integral

that's horrible

So is it overkill to use set notation in my community college differential equations class?

@XanderHenderson shithouse*

y would u do that

because Mariusz Wodzicki works in mysterious ways

1:18 AM
@Antonios-AlexandrosRobotis I could have sworn it was shithole... :(

When defining the domain of a solution @EricSilva
I just wanna do real math :(

i dont even know what i did in calc I but there was no integrals

@XanderHenderson (just to be sure, all the republican lackeys are claiming it's shithouse now)

Math where my classmates aren't premeds :P

1:19 AM
I thought that the republican lackeys were claiming that he said nothing bad, no sir, I do not recall him using any language at all!

i mean idk what's appropriate @CookieToast, cause I'm not privy to the class

Now they're saying he said "shithouse"
LOL

christ...
MERICA! F*CK YEAH!

hmm i see

shithouse vs shithole is literally not any better

1:19 AM
@CookieToast can you take honors courses?

then perhaps i have done "real math" before

right haha, it's morally the same

indeed
right... my people are home; time to make dinner
g'night

i mean i did learn something last semester that someone who is not currently dead came up with so i am going to give it to myself.

later @XanderHenderson

1:21 AM
@XanderHenderson enjoy your sleep eating, try not to get too many crumbs in the bed

@Antonios There's "Honors Intro to Proofs", but it's one credit and only meets for six weeks. My school has less than 8000 people, and only ~400 transfer each fall, so there isn't enough interest in math to fill a class

@Antonios I'd really really like to get a PhD in mathematics

morning @MatheinBoulomenos

in that case you are going to need to learn the stuff in that honors intro to proofs course
why not take it
hi @MatheinBoulomenos

1:24 AM
Hi @Faust @Antonios (it's in the middle of the night here, btw)

yeah like 2:20, right?
you're in germany, no?

correct

Oh I am taking the proofs class, sorry @antonios. I just wish they had honors Calc or something.

is there any possible way to get to a school where you have more opportunities @CookieToast, if your school doesnt have enough math to satisfy you

i think i am going to have learn germa
german

1:25 AM
sup @Mathein

I intend to learn german this summer (or at least start on it)

hi @Eric

i would like to do my masters in germany

@Eric once I finish my lower division courses, I can take some upper-division classes at a local state school through an open-campus program, but until then I'm stuck.

@Antonios feel free to ask me for help if I'm around. Viel Erfolg!

1:26 AM
ah i see

hehe thanks

the best thing to do i guess is to try to find resources to do cool math i guess

I am self studying wherever I can in the meantime though. I'm very slowly working through TEd's Multi course

@MatheinBoulomenos you should teach us some random math in german over the summer :p

chat is a decent one although there doesnt seem to be a lot of math goin on ever...

1:27 AM
@eric yeah but you guys always answer all my questions, and I'm pretty sure you all know way more than any of my professors haha

@CookieToast that's good, im a p big advocate for learning multi really solidly (Which Ted's certainly in favor of)

thats cause no one likes me stupid questions @EricSilva

idk i know nothing about nothin

I've picked up a decent amount just by browsing through here for an hour or so a day

@CookieToast ted shifrin's online lectures are great

1:27 AM
@Faust I already talked a lot with Leaky about Sylow theorems and Galois theory (a lot of it in German) so sure, why not

@Antonios, I got his book too, and he sent me all his course materials, so I'm set :) It's just pretty slow going with everything I have going on this semester

lol I'm TAing two courses this semester... two algebra courses haha

^^ that would be awesome sauce.

i would like to TA an algebra class

I'm TAing intro algebra (I guess groups & rings stuff) and "honors algebra II" which I guess is Galois theory and more ring theory

1:31 AM
I would TA an algebra class without payment (though I'd still prefer getting paid)

Lol you all TA me all the time :P, why do you need any more than that?

i need dat $$New york is expensive LOL I TAed for multivariable analysis a couple times and diff geo and i felt like doing that made me way more solid on summoning up examples and shit i like algerbra enough that i would be fine talking about it w.o getting paid @Antonios-AlexandrosRobotis what college do you go to? NYU? 1:32 AM i'm doing a master's degree at NYU rn if you want me to teach analysis i want money though TAing is more than just talking, though. Grading isn't always fun @EricSilva teaching is super conducive to learning grading is never fun @MatheinBoulomenos luckily there is a separate grader for the course haha 1:32 AM I have super power i can read anything... i always have a bit of scotch when i grade cause i dont wanna grade completely sober @Antonios Ah I see, that's nice. Here TAs always have to grade I reckon I'll have to give quizzes and so on but those shouldn't be awful to grade I actually applied as an algebra TA, but they didn't take me because I'm still in my bachelor and the other applicants were all masters students :/ that kinda sucks you certainly seem qualified haha 1:34 AM here it depends somewhat on what the prof wants everyone said that. But intro algebra is a 3rd semester course and I was only about to finish the 4th semester when I applied, so I guess it was kinda early i did this magical thing today i ran out of hw here there's always a shortage of graders and shit shortage of shit is a +1 where u live? It depends here on the course. A lot of people wanted to TA algebra, but nobody wanted to TA functional analysis. Go figure :P 1:36 AM i love functional :'( @Faust i mean it forces them to raise the pay for graders so i guess Heidelberg is kinda weak on real/funtional analysis, especially pure analysis. The analysists we have prefer to teach applied courses. I guess that rubs off on the students, too I liked my functional analysis course tbh, but I wish we did more general things than normed vector spaces yeah lots of important TVS arent normed if i get a scholarship this semester then i may apply to mark but right now anything i made from marking id lose from bursaries money which i think is complete nonsense but w.e 1 hour later… 2:46 AM wow no one's here indeed :P 2:58 AM hi @XanderHenderson werd so... Pontryagin duals are a thing yeah I've heard about that at some point, don't know much though well, you start with a locally compact abelian topological group then stab yourself in the eye then consider the collection of "characters", which are maps from your group to the torus somewhere in there, magic happens, followed a Fourier integral of some kind 3:02 AM lol wacky. How does this relate to fractal geometry I don't really know yet I'm actually taking a class on harmonic analysis this quarter oh cool one of my friends is big into that stuff that being said, the stuff that I work on is related to the wave equation on fractal domains and solutions to the wave equation are typically best understood in terms of Fourier series and/or integrals ditto the heat equation since both deal with a Laplacian operator of some ilk hmm I need to learn more about this stuff at some point. I used to be really into analysis anal is the best uh... analysis 3:06 AM lol at berkeley the course catalog read: math 104: introduction to real anal. that was amusing real anal is okay I'm not really into complex anal but funky anal is where it's at I'm the opposite, I like complex since everything is nice and clean i like analysis, like the proof methods etc, but I find the lack of algebraic structure a bit offputting sometimes what you need, sir, is to study the glory that is \mathbb{Q}_p (and her algebraic closure) some of dem p-adics 3:11 AM one of the strangest proofs that I know uses the metric structure of the metric completion of the algebraic closure of \mathbb{Q}_p (hereon called \overline{\mathbb{Q}_p}) in order to show that \overline{\mathbb{Q}_p} is algebraically closed WHY?! WHY DOES THAT WORK?! Dark magic, voodoo, and possibly Satan are the only explanations that i can come up with tbh math might be the work of satan 3:33 AM we got anyone around here that knows how to use sage, or python, or MatLab exceptionally well? i have a system of multivariable, matrix polynomials, to grind out and 8 gb of ram @XanderHenderson @Daminark come to the world of quaternionic analysis and watch pretty things fall so... there are too many dimensions: small inductive, large inductive, Lebesgue covering, Hausdorff, Minkowski (upper and lower), box (upper and lower), Assouad, "lower", GK (wtf is that?), Krull, Lyapunov, vector basis, capacity, correlation, conformal Assouad, etc... and that is just off the top of my head @frogeyedpeas I have a friend who works in noncommutative geometry (operator algebras, planar algebras, and the like) dont forget krull dimension :P at the JMMs last week, I told him that noncommutative geometry is passe, and that he should really work in nonassociative geometry LMAOOOO @Antonios-AlexandrosRobotis reread my list... it is in there! 3:39 AM sure is lolz at some point, all ofm ath will fuse and becoem "applied set theory" according the the category theory folk, you are WRONG sir! i don't yet understand category theory as a subject like i've began to use arrows in arguments noöne does categories are pretty fun at times 3:42 AM Lmao... and diagram chasing makes some kind of sense to me. But like i've heard that there is a subject called category theory, where people jsut do pure category theory for its own sake What do you call someone who reads a paper in category theory? confused a coauthor! hahahaha that makes sense 3:42 AM BA-DUM-TISH! Category theorists are dumb... they can't tell the difference between a pecan and a tropical fruit that grows on palm trees. Even my wife can tell a nut from a coconut... lol i'm guessing co-co = id? @frogeyedpeas yarp u slept short 3:45 AM I made dinner ate dinner and put the child to sleep sleeping child I really should go to bed, but the wife is at the gym, and it is only 7:45 @XanderHenderson i was talking with my friend about fractional dmiension recently we had this ridiculous idea of trying to do topology on non continuous things continuity is defined in terms of a topology and build a notion of fractional/complex homology 3:52 AM so what do you mean by "topology on non continuous things"? we want to build surfaces, which in some abstract sense have 1.8 holes my advisor is trying to come up with a fractal homology theory... no way this is sooo cool ok so that answers half my question: "given your research interests, have u encountered that idea before? what are your thoughts" the general idea, I think, stems from the observation that if A is a set, then the complex dimensions of A with a fixed real part form an additive group these are the homology groups, which are then graded by the real numbers (in contrast to the more usual (co)homology theories, which are typically graded by the integers) but I am not a very good algebraist---fractal cohomology is way above my paygrade; I barely understand what I just wrote oh man, i wish u were on the east coast 3:57 AM I was on the east coast for a week last summer who else is working on this kind of thing we must have just missed each other :P ironically, i was in cali last summer to the best of my knowledge, Lapidus is the only one working on it so we really did 3:58 AM HAH! it is possible that some of his collaborators are also involved so there are ppl at rutgers (myself) and another undergrad who thought this might be cool *2 ppl how could we learn what u guys have done I haven't done shit :P but Michel seems to have ideas Goran Radunovic may also have ideas (though he is in Croatia now, I think), and Machial van Frankenhyusen (that might be how you spell his name; he's in Utah) might know what is going on but the idea is not very well developed and now I am really going to bed g'night take care! in The Factory Floor, 19 secs ago, by Secret One can draw some interesting philosophical connections of these with mathematics. For example, Wheel and axle work as they do because of the mathematical properties of a circle [Random] Circles are an example of a compact set (under the usual topology in \Bbb{R}^2, or when viewed as a curve, some kind of 1-manifold) Compact sets are interesting because they can "bound" infinities, meaning that you can place something infinite in them and the overall result will retain many properties of being finite. Actually, for a topologist, we distinguish between those two objects by using disk to refer to a "solid, filled in circle" and circle for the boundary, or the closed 1-manifold. Abstract circles with different values of \pi: Recall that \pi = \frac{C}{d} where C is the circumference and d is the diameter Now, suppose the paper we drew a circle on has variable number and size of pixels (paper grains?), meaning that as we change the value of \pi around, the figure will not look different to our eyes Suppose we held d fixed, then the circumference must changed. You can see that as we vary \pi, while the circumference does not look changed, given an ant that walk on it by 1 unit/s, the ant can only cover a much shorter distance compared to when it walked on the circle with smaller value of \pi Roughly speaking, this can be understood as the measure being used to calculate the arc length between the two circles are different. This makes sense because the circle has uncountably many points and thus you can always biject it with multiples of itself and still does not change the cardinality. (thus look the same in the diagram) Therefore, the difference in travel distance by the ant that is observed is because we are using a different measure If we insists we are using the same measure, then the difference can be explained by the circle with larger \pi is actually in a non-euclidean geometry Now, let us imagine the scenario when \pi \to \infty, what will happen: and now suddenly, the ant gone stuck and can go nowhere Therefore, one immediate consequence of being in a universe with a different value of \pi is that you need to use more/less material to build some circle with a desired circumference and this can affect the length of rope you need to use for your pulley system, or the number of charges you can place in a storage ring as well many other things... (NB The word "immediate" is used because we all knew that when the fundamental constants were varied just a little bit (without varying other constants to compensate for it), there will even be intelligent life developed to e.g. discuss about circles) In other words, the physical significance of \pi (I think) is it controls the geometry of circles and periodic phenomenon in the universe 4:45 AM (To be done later) Think of a geometry such that if the ant walked at some velocity < n, then it got stuck, otherwise it walked some distance x > n 2 hours later… 7:01 AM @AkivaWeinberger I kinda like this Most Unwanted Song. It's good fun :P It's clearly very tongue and cheek @Secret x mod 1 controls all perioic functionality B-) @AkivaWeinberger Check out "Trout Mask Replica" by Captain Beefheart @TheGreatDuck how does that help? I want the object to basically act like it is infinite until a threshold speed is reached @Secret im correcting your statement that sin does. Btw you ever have a moment where a professor gives you a word range for a paper and you just have the urge to multiply it by a factor of 10. @TheGreatDuck I don't understand, I said nothing about sines in my paragraph above? @TheGreatDuck Rare, I have the opposite problem when writing essays: I always don't have enough words 7:09 AM @Secret well every semester whether as a project or by leisure I write some kind of math paper that approaches 20,000 words. eithout fail unless im just bored of it so when a professor says "go look up the definition of the real numbers and write 100-300 words about it" i feel tempted to change my latex document so that I 'accidentally' read it as 1000-3000 XD @Secret Im going to prove first thing tomorrow morning that the set of all dedekind cuts is a complete ordered field. it will be easy to do In dynamical systems theory, the baker's map is a chaotic map from the unit square into itself. It is named after a kneading operation that bakers apply to dough: the dough is cut in half, and the two halves are stacked on one another, and compressed. The baker's map can be understood as the bilateral shift operator of a bi-infinite two-state lattice model. The baker's map is topologically conjugate to the horseshoe map. In physics, a chain of coupled baker's maps can be used to model deterministic diffusion. As with many deterministic dynamical systems, the baker's map is studied by its action... hmm... Ah ok, I see... x \mod 1 contains orbits of different periods and periodicity indeed does not necessary have anything to do with \pi x mod 1 the remainder when x is divided by 1 its the sawtooth wave the main periodic function for periodic functions of a rational period So I guess \pi only controls circles and periods that involves circles yeah @Secret you should look at my recent question 0 I have been working on some math proofs and general theory stuff in a paper linked below and I have hit a mental block on something. Piecewise Constant Functions in Differential and Functional Equations On page 22 there is a definition given for the set of alternate derivatives associated with ... 7:26 AM lol, I am still on page 10 Your latest version changed a bit of the main text thus I cannot afford to miss out things which is why I am reading page by page 7:48 AM hmm... Given the set of all differential operators \mathcal{D} and the set of all functions \Bbb{R}^{\Bbb{R}}, a differential fork S is a set containing d \in \mathcal{D} and f \in \Bbb{R}^{\Bbb{R}} such that$$S = \{d \in \mathcal{D}, f \in \Bbb{R}^{\Bbb{R}}: df =0\}
and $\forall n \in \Bbb{N} \cup \{\infty\} (\forall h \in \mathcal{C}^{n}(\Bbb{R}))$ such that $h(f,g) \in S$
In other words, the most important property of differential forks is it is an equivalence class of functions together with some differential operator such that $df = 0$
This is a generalisation of differential algebra such that the axiom $dc = 0$ is replaced by a weaker axiom where $dC = 0$ where $C$ forms an equivalence class with the constant elements
I need to read typhon's paper in more detail later, this can be interesting to ponder about the generalisation
The notion of implied differential operator can potentially be generalised to a notion of differential analysis which allow us to define a notion of generalised differential forms and generalised set of tangent spaces for non manifolds and other non smooth geometries and spaces...
however, we will wonder about this later cause I don't even have a background on differential analysis yet
...actually on deeper thoughts, implied differential operators is just a differential algebra where the constant elements are not just constant functions
Still, the notion of differential forks can be useful to select different differential algebras in function space that can be used to solve certain ODEs involving discontinuous function
> A differential fork is any set of single variate functions S that is a superset
of all constant functions such that for any two elements f1 and f2 and any continuous
function g(x, y) we have that g(f1(x), f2(x)) ∈ S.
ok screw all of that above, I misread stuff