Mathematics

usukidoll

01:17

I am stuck. In a round robin tournament every team plays every other. Show that if every team wins at least one match then there are two teams with the same number of wins.

Do I just assume that if we have n teams then there's n+1 wins by the Pigeonhole Principle?

Ted Shifrin

@topologicalmagician: Here's a small hint. $d=\gcd(k,n) \iff \gcd(\frac kd,\frac nd) = 1$.

CaptainAmerica16

Is anyone still here

;-;

Ultradark

Hi friends and foes alike

CaptainAmerica16

Hi ultradark. how are u

user178403

hi. Why if I change an interval I depending on n, to n+1, and this splits I into two parts, imply that f is increasing f_n\le f_{n+1} ?

usukidoll

01:28

I'm doomed :/

2

Ultradark

Good good

Rithaniel

Evening, chat.

CaptainAmerica16

Hi Rithaniel

Rithaniel

Sup Cap, how goes it?

CaptainAmerica16

Eh, u know. Getting ready for the end of the school year and stuff. Busy.

@Ted How have u been?

Ultradark

01:30

Bulbasaur don't faint just yet use solarbeam

usukidoll

Wow why did two people star that? 🤨

Rithaniel

Same here. Have a linear programming project I'm working on. Also homework revolving around polynomial rings.

user178403

I did one star bc I feel the same

CaptainAmerica16

I wish i could come here with math questions. I've had little time for anything besides school.

Rithaniel

It's good to occasionally take a break from math.

I tend to play video games

user76284

01:34

@usukidoll Suppose there are $n$ teams. The possible win counts for any team are in $\{1,2,\dots,n-1\}$, which is of cardinality $n-1$, so by the pigeonhole principle there must be two teams with the same number of wins.

CaptainAmerica16

why does it feel so horrible tho

Rithaniel

What feels horrible? To take a break from math?

user76284

@user178403 What is your question? What is $f$ here? What is $f_n$?

CaptainAmerica16

yeah, it just feels like i'm wasting time that i could be using to improve. idk

usukidoll

@user76284 but what happens if a team loses all over the matches. Does the pigeonhole principle still apply? May be we have n teams and n-1+1 loses ?

Rithaniel

01:36

Well, you'll exhaust yourself if you never take a break, keep that in mind, and if you're exhausted, you're more liable to spin your wheels instead of improving.

user76284

@usukidoll I'm not sure I understand what you're saying.

There are $n$ teams but only $n-1$ possible win counts, so two teams must have the same win count. This makes sense, right?

CaptainAmerica16

true, i suppose.

user76284

A team cannot have $n-1+1=n$ losses because there are only $n-1$ other teams.

A team doesn't play against itself.

CaptainAmerica16

just waiting for summer, i guess

usukidoll

There were three questions but I only posted one

Rithaniel

01:39

Yeah, I'm looking forward to the summer, myself. I have a few plans in motion.

usukidoll

In a round robin tournament every team plays every other. Show that if every team wins at least one match then there are two teams with the same number of wins. Is this statement still true if some team loses all its matches? What if more than one team loses all its matches?
Is the whole thing

user178403

@user76284 To pass from $f_n$ to $f_{n+1}$, each subinterval $(k2^{-n},(k+1)2^{-n}]$ is divided in half. It follows that $f_n\le f_{n+1}$. I don't see how one could get the conclusion of $f_n$ to be increasing, why is that?

Rithaniel

I need to set up a time for me to take my GREs.

CaptainAmerica16

oh wow. How are u preparing?

user76284

It's true unconditionally, AFAIK. I don't see how the last two situations would negate the reasoning (which doesn't depend on there being an all-losing teams).

Also, the last question doesn't make sense because there can't be more than one team that loses all matches (they play each other!).

usukidoll

01:41

So last question is inconclusive

Due to how the round robin is set up

Rithaniel

Well, I plan on taking it halfway through the summer, and the month leading up to them I'll be researching and reviewing as much stuff as I possibly can.

user76284

The last question is asking "What if [contradiction]?" in which case who knows.

In classical logic, any statement can be proven from a contradiction.

Ultradark

Are you taking the general gre's or the math subject test

CaptainAmerica16

Ah, I see. I'm not sure if I want to take my SATs at the beginning of the new school year. If so, I'll have to study over the summer as well.

user76284

@user178403 What is $f$?

Is it a sequence? A function? What are the domain and range?

And what is $k$?

I took my GREs last October. Good times.

Rithaniel

01:45

I'll be taking both, I believe

Ultradark

what concepts are on the math subject test gre

user178403

@user76284 $f_n$ is a sequence from X to $[0,\infty)$, $k$ is an integer

Rithaniel

General and math.

Ultradark

I hope no partial differential equations

user76284

What is $X$?

Rithaniel

01:46

Yeah, it's been a while since I did any diff eq

user76284

And what does this mean: "To pass from $f_n$ to $f_{n+1}$ , each subinterval... is divided in half."

user178403

@user76284 well I think it's an arbitrary space, most likely measurable space. It means that you change n by n+1. It's indeed (1/2) *(k2^{-n},(k+1)2^{-n}]

user76284

I'm afraid I don't understand your question.

user178403

ah ok, I don't understand the conclusion of $f_n$ to be increasing due to the half intervals

user76284

I'm taking 4 humanities classes this semester and it's killing me.

@CaptainAmerica16 I know how you feel.

Ultradark

02:02

$ \frac{\partial}{\partial s} \Psi(s,x)+\frac{\partial}{\partial t}\Psi(t,1-x)=\frac{\partial}{\partial I}\Psi(I,x)$ Any thoughts on this?

usukidoll

So if a team loses ... We can have n teams and n-1 loses so there are two teams with the same number of loses

user76284

02:17

@Ultradark No partials wrt $x$?

Ultradark

02:34

@user76284 no

Rithaniel

07:10

So, I'm currently trying to prove that given a ring $R$ and ideal $I\subseteq R$, that $R[[x]]/(I,x)\cong R/I$ which seems easy enough: Just construct a homomorphism which takes every power series to the value it takes when $x=0$, but I find myself doubting. That $x$ in the $(I,x)$. I don't believe I have a very good grasp on what that syntax indicates.

Secret

07:25

Ugh, this is so frustrating to read

The secret on why I knew when people A ignores people B is because the continuity of the message is disrupted

Here, Ultradark thinks the message flows in the way shown in the blue arrow, but in reality, it flows in the way of the red arrow

I might soon have to put him into ignore to stop seeing these disrupted flows that reminds of the ignore phenomenon

Rithaniel

What about the @ directed towards him, Secret?

Secret

Ah f888 I missed that!

Rithaniel

Yeah, you seem to have missed that he was replying to it.

(Ah, irony)

Secret

Well the Ted-Ultradark relationship is well known in this chat otherwise

Anyway, it does mean I have to improve my ability to detect my own blindness

(Don't worry, I don't put people on ignore, because I hate ignore so much to use it!)

Rithaniel

Yeah, everyone has occasional bouts of blindness. Mine tend to be related to over-analyzing things and making them more complicated than they need to be.

Secret

07:32

lol same,which is why every time people show me an extremely simply trick to solve a proof, I go O_O

Leaky Nun

@Rithaniel first show that R[[X]]/(X) = R

Rithaniel

See, the issue is that I don't know what $(x)$ is. Like, I could follow a line of reasoning at the moment which would leave me with $(x)=R[[x]]$, which I know is incorrect.

Shing

hey guys, I am a bit confused by "differentiable functions are continuous."

$(x-2)^2/(x-2)$

isn't this function differentiable at x = 2, but not continuous?

say, can't I find $lim_{x \rightarrow 2^-} f'(x)$ , and $lim_{x\rightarrow 2^+} f'(x)$

and claim the derivative "at" point x = 2 is just the limit?

Rithaniel

07:52

What matters for continuity is where the function is defined. 2 is not in the domain of your example function, so you don't need to check it, but for points arbitrarily close to 2, you can show that there is a delta interval such that all values in the domain of the function within that interval are within an epsilon interval of the function value at the point. So it is continuous.

(Though, my own knowledge with real analysis could use some brushing up. I might be getting some details wrong there)

Shing

@Rithaniel so it is just a matter of definition, we define jump points are not differentiable (probably for some nice properties)?

Rithaniel

Well, perhaps I should be more clear. When I say "where the function is defined" I mean "the points where the function has a logical and consistent value." We've not actually mentioned what the definition of continuity or discontinuity is.

As for why the formal definition of continuity is what it is, that can be an expansive topic.

A way to think about it is that $f(x)$ is "where you are at x" while $\lim_{a\rightarrow x}f(a)$ is "where you should be at x." In that sense, continuity is a way of talking about the "consistency" of a function.

Silent

09:33

How do we apply fundamental theorem of calculus here? That is, how do we know that antiderivative exists?

Leaky Nun

@Silent continuous => integrable

Silent

but not every integrable function has antiderivative

Ok! so every continuous function has antiderivative. Thank you leaky.

Silent

11:50

Please help me with transitivity: I thought this: $d(i,j)$ equals (minimum of all the elements among i-th row except (i,i) entry 0)+(minimum of all the elements among j-th column except (j,j) entry 0). From this observation, transitivity is obvious. Am I correct?

Also, the hypothesis 'symmetric matrix' is used only to prove $d(i,j)=d(j,i)$, right?

maths student

Basis extension theorem is true over pid? for free module L?

That is given v in L can one extend it to basis of L?

Since A = k[t] is a PID, we may “shrink” v suitably to assume that
v can be extended to an A-basis for L what is meaning of this?

@le

@LeakyNun

Mathphile

12:09

desmos.com/calculator/y7cgexvy8s

why is this so satisfying?

Silent

So beautiful!

Mathphile

Now here's a question on this

For what value of n is the maximum area enclosed between (1/n)sin(x/n), (1/n)cos(x/n), (n)sec(x/n), (n)cosec(x/n)

Secret

desmos.com/calculator/pnugzuemmk

More explosive version

Mathphile

woah

Silent

@Secret OMG!

maths student

12:20

0

Q: Extension of basis over PID

Let $k[x] = R$ be ring and $L$ be free $k[x]$ module ; let $v \in L$ be vector in $L$ then how one can extend it to $R$-basis of $L$?

Rithaniel

If you guys like mathematical spaghetti, here's some: desmos.com/calculator/hnzk8vxu08

Secret

I think the unit cell area is largest when n=1, as when n goes smaller, the sec and csc closes in and squish the area between sin and cos in between

Not sure how to prove this though

Rithaniel

You'd just need to set up an integral

Might be tedious, but so long as you're careful, not too difficult.

Secret

Rithaniel

Also, it depends on what is meant by "enclosed"

Secret

12:28

Lol what am I looking at

Rithaniel

I love using modulo when graphing functions, it leads to the most interesting chaotic sprawls.

maths student

@LeakyNun any idea ?

Rithaniel

That region isn't the densest spread of line segments I've seen, but there are a lot in there.

Mathphile

@Secret @Secret I meant the largest area enclosed by all the functions

Secret

That shaded is the area enclosed by the functions, no?

Rithaniel

12:31

What do you mean when you say "enclosed?"

Secret

Also if you mean the whole function, then there is no single largest area since this whole graph is periodic hence the area diverges to infinity

Rithaniel

It's probably best to describe your enclosure in terms of "greater than" or "less than," to avoid ambiguity.

Secret

I don;t remember if the enclosed area for sec is this

Mathphile

how do we upload pictures here?

Secret

there's a upload button next to your message box

Mathphile

12:37

all i can see is send

Rithaniel

You might need some reputation level on the main site to unlock it. Also, might not be present on mobile.

Mathphile

im on computer

i was going to explain by my picture

Rithaniel

You can always upload to imgur and link it in here.

Another form of spaghetti: desmos.com/calculator/f6yu8j2kpl

Mathphile

imgur.com/eZZoLpL

sorry it took some time

heres the image link

@Secret and @Rithaniel

Rithaniel

So, less than the absolute value of sin and cos, but greater than the absolute value of sec and csc

Secret

12:48

That area has no supremum, you can see how far it stretched as n tends to 0

though, I think I should try to calculate that, perhaps the squishing might be enough to offset the spiking to infinity

Mathphile

at n tends to 0 the area will be 0 i guess

Secret

That area also become zero somwhere around n=0.71, suggesting the derivative of the area function under n should have a maximum somewhere between 0 and 0.71

That is you want to maximise the following integral:

$$2\int_a^b n \csc (\frac{x}{n}) -\frac{1}{n}\sin (\frac{x}{n}) dx$$

wrt n

where $a,b$ are solutions to $n \csc (\frac{x}{n}) = \frac{1}{n}\sin (\frac{x}{n})$ and $n \csc (\frac{x}{n}) = n \sec (\frac{x}{n})$ respectively

In the range of $n$ you are interested in, there is no risk of division by zero. Thus we have:

$n^2 = \sin^2 (\frac{x}{n})$

and $\tan (\frac{x}{n}) = 1$

Thus we have:

$b = n(\frac{1}{4} + m)\pi, m \in \Bbb{Z}$

and

Mathphile

13:03

I am having trouble interpreting this as chatjax aint working for me

Secret

$a = \pm n(\text{arcsin} (n) + m \pi), m \in \Bbb{Z}$

Taking $m=0$ since we only need one value of the intersections, we have

$a = n \text{arcsin} (n)$

$b = \frac{n\pi}{4}$

Thus the integral becomes:

$$2\int_{n \text{arcsin}(n)}^{\frac{n\pi}{4}} n \csc (\frac{x}{n}) -\frac{1}{n}\sin (\frac{x}{n}) dx$$

I will let you to compute the rest

pics for chatjax fails

user131753

@Rithaniel The book is freely available to download here.

Mathphile

thank you @secret

RaphX

Hi guys what would be the correct room to ask permutations and combination problems?

Leaky Nun

your teacher's office room /s

Rithaniel

13:13

Ah, thank you, @user170039
I probably will still be getting a physical copy. I love to have those, but it's great to be able to look through it ahead of actually getting one.

Current problem I'm working on: "Show that $\mathfrak{M}\subseteq R[[x]]$ is a maximal ideal if and only if $\mathfrak{M}=(M,x)$ for some maximal ideal $M\subset R$ of $R$."

I've gotten started multiple times, but each time the proof grows to unwieldy sizes, and so I start over.

Secret

♦ $Mathematics$ Basic Mathematics

This room is meant for all basic mathematical discussion, incl...

user131753

In any case I would suggest you to take a look at this book also @Rithaniel. But as you can easily see from the contents, this book is rather huge.

user131753

@Secret Why not this room?

user131753

Is this room not "for both general discussion & math questions alike"?

user131753

:49921297

user131753

13:19

$Mathematics$ Mathematics

Associated with Math.SE; for both general discussion & math qu...

6

3

Rithaniel

Phew, 600 pages is quite a few.

user131753

@Rithaniel Yep. But you will get some very interesting insights in the book.

Secret

Well for starters, combinitorics problems that is asked here tend to be problem solving like (based also on Leaky's sarcarstic response). Secondly, we don't have combinitorics people in the room right now, so unless the problem is very interesting, people are going to miss the problem

If that is the wrong room, user21820 and co. will automatically move the message back here

Rithaniel

I will keep an eye on it, then. Need to prioritize which I go with first.
(Also, a lot of questions get asked here and never get answered)

user131753

@Secret Leaky's sarcasm?? Interesting!!

Secret

13:22

9 mins ago, by Leaky Nun

your teacher's office room /s

user131753

$\ddot\smile$

Secret

that's what /s stood for

Rithaniel

(So, if nothing else, it's good to ask in multiple locations to "cast a wide net")

RaphX

Ok, I asked in the Basic Mathematics room

Mathphile

does anyone believe that we will reach a point where no development in math and science will be possible as reaching that level of education where further development is possible will take longer than the life span of an individual?

guys?

Rithaniel

13:29

Nope, if we ever get to the point that there's too much information for a mind to hold before progress can be made, then the next goal will naturally evolve toward consolidation of information.

Also, we're already at the point where it happens that a result which was published in a paper decades previously is "re-discovered" independent of the original author and published a second time.

Mathphile

well im pretty sure that if there is a graph of scientific discoveries vs time from the 1900 to present, it would be exponentially decreasing

is there a graph like that tho

Rithaniel

I'd actually argue against that. Lots of papers are published every year.

Not every one is a huge, world-redefining accomplishment, but things are still moving.

Mathphile

well like you said many of those papers have results that are just rediscovered

Rithaniel

That's true, but that's doesn't mean that there aren't ones which are legitimately new discoveries. It just means that things are coming out so fast that not all of it can be accounted for as easily as we would want.

Mathphile

well what i mean like just like there is a limit to how small we can make transistors after which development of cpus will be incredibly tough, a time will come when development possible by humans will reach a threshold after which ai will have to take place

well that sounded kinda stupid :P

Rithaniel

13:46

Well, that kind of assumes that there is a maximum degree of complexity an abstract concept can represent.

Like, today, I need to learn complex analysis and algebraic topology in different classes. If tomorrow someone discovers a construct which encapsulates both topics, then I only need to attend one class.

Mathphile

that is true

Rithaniel

I always think of Gödel's incompleteness theorems when talks like this come up.

Mathphile

how should i learn higher level math if i currently just finished high school

i am going to pursue my undergrad in cs

Rithaniel

Depends on where your interests lie.

Mathphile

but i am really interested in math too

do you think i should double major?

or can i attain the same level of knowledge by learning myself?

Rithaniel

13:54

Like, for me, I played a lot of video games back when I was starting out college, and ones like TIS-100 and Kerbal Space Program geared me up for math.

It's always better to have a guide, so don't just study by yourself. Study with someone who will engage with you and help you understand concepts

That's where college excels.

Mathphile

but im not sure if ill be able to manage a double major

Rithaniel

As for a double major, I can't say yes or no, particularly because I don't know you well enough to know if that would be advisable.

You can always go for a minor, or go for a major in cs with a specialization in mathematics.

Mathphile

yeah

i think a minor will be better

Rithaniel

Though, you should talk to the people at your college about this stuff, as they're going to be the ones best equipped to guide you on that.

Mathphile

well college starts in august so I cant ask anyone right now

Rithaniel

14:00

So you're already accepted and signed up for the fall?

Mathphile

yup

Rithaniel

Excellent.

Mathphile

do you teach somewhere?

Rithaniel

Nope, I'm a senior gearing up for graduate school.

Mathphile

ahh

what are you planning to specialize in?

Rithaniel

14:04

Currently abstract algebra, with additional focuses likely in geometry or topology.

Mathphile

nice

Rithaniel

But, I'm not going to say that's absolutely going to be the case. I'm still learning stuff.

Silent

14:20

Please help me with transitivity: I thought this: $d(i,j)$ equals (minimum of all the elements among i-th row except (i,i) entry 0)+(minimum of all the elements among j-th column except (j,j) entry 0). From this observation, transitivity is obvious. Am I correct?
Also, the hypothesis 'symmetric matrix' is used only to prove $d(i,j)=d(j,i)$, right?

nbro

15:03

Any idea of why the Laplacian is a "local operator"?

I've seen this statement in the context of graphs (and thus Laplacian matrix of the graphs)

anakhro

15:28

@nbro can you give an example of that occurring somewhere?

Googling it, I can't find anything.

nbro

15:45

@anakhro Yes, section 2 (specifically, under the subsection "Chebyshev Spectral CNN (ChebNet).") of this paper: Geometric deep learning on graphs and manifolds using mixture model CNNs.

Silent

16:10

@LeakyNun, please confirm if i am correct above!

Junglemath

16:36

Can someone explain the last sentence in the answer to this question? math.stackexchange.com/questions/1788345/…

Namely, why does the subgroup in $G$ that corresponds to the order $p$ subgroup in $G/H_k$ have order $p^{k + 1}$?

Dair

18:11

hmm... I'm kind of debating on getting "An Illustrated Theory Of Numbers"... Looks like a good book, and I do find it hard to visualize number theory in a lot of cases, but 60$...

Mathphile

hey

im pretty sure that tan(x) has some relation with the tetration of x

desmos.com/calculator/2fuedid6v7

Aaron Hall

what's up?

Mathphile

nothing much rn

chat not much active

Aaron Hall

can I say $y = f(x) = 3 + 0x = 3$???

can 3 be an $f(x)$?

Mathphile

well i guess so

yes

Aaron Hall

18:25

ok, I need some trivial examples...

that's why I ask...

Mathphile

what do you mean examples?

Aaron Hall

I'm teaching, and it's for a class.

We don't have a single textbook that covers everything, so I'm working with the dept head on the lecture material, and I don't want to say something that's considered wrong...

Mathphile

what grade do you teach?

Aaron Hall

I'm an adjunct, the class is computational math and statistics, and it's for an MS in Data Analytics and Visualization...

we're doing the linear algebra material, and we need lots of small examples they can work by hand in class, and that we can use Python to check our work...

Mathphile

this is pretty elementary you know

according to your defination, for any number you plug into x, f(x)=3

so f(x) will output 3 for any input yo give it

*you

hey simplifire

Aaron Hall

18:32

exactly so, it's going to be my first example of f(x). We start small and trivial and quickly complicate it. I'm going to use this to introduce systems of linear equations.

TheSimpliFire

@Mathphile It is mainly due to the fact that both very quickly tend to infinity after $x=1$; it is possible to construct functions that have these similar properties.

Aaron Hall

someone ping me if they think I'm wrong to say f(x) = a constant with something that justifies their objection, please! :)

TheSimpliFire

In this graph, ${}^7x$ only looks like it has an asymptote at $\pi/2$, but of course it is defined over $x\in \Bbb R^+$.

Mathphile

could we find an n for which $^nx=tan(x)$

MrAP

Points which represent the complex number $z$ for which $\frac{z-1/4}{z}=1$ all lie on :(a) Parabola (b) Circle (c) Ellipse (d) A straight line.

TheSimpliFire

18:37

@Mathphile That isn't possible, tan can't be written as such

MrAP

I think it's a straight line. What do you all think?

TheSimpliFire

You sure it's $(z-1/4)/z=1$ @MrAP? Multiplying through by $z$ leads to the absurd equation $z-1/4=z$

or do you mean (z-1)/4

Mathphile

how would we find an imaginary number x where $x=e^x$

TheSimpliFire

Usually we denote it by z = x + iy

Use Euler's identity

and equate

MrAP

@TheSimpliFire, I don't know. This is what was printed. May be it should be what you have suggested.

TheSimpliFire

18:42

*formula, not identity

Mathphile

hmm...

i got what you mean

TheSimpliFire

$\exp(x+iy)=\exp(x)\exp(iy)=e^x(\cos y+i\sin y)$

Secret

18 mins ago, by Aaron Hall

can I say $y = f(x) = 3 + 0x = 3$???

I will be really happy if $x=3$ can be expressed in the form $y=mx+c$

TheSimpliFire

This means that you need to solve $x=e^x\cos y$ and $y=e^x\sin y$, and at first glance you can get the equation $\frac yx=\tan y$

@Secret that's correct, the line $y=3$ is parallel to the $x$-axis and thus has zero gradient. The intercept is 3.

$x=3$ gives you an undefined gradient

Secret

No that isn't my question, I just quote Aaron's to aid my question

ok

TheSimpliFire

18:45

$x=3$ is parallel to the $y$-axis

Secret

yeah I knew, still it will be nice to understand how it arises from taking the limit of $y=mx+c$

so somehow $m$ has to blow up and yielding $c$ without consuming the $x$

Mathphile

im bringing up this question again

1

Q: Evaluating: $\lim_{n \to 0} \prod_{\substack{i=nk \\k \in \Bbb Z_{\geq 0}}}^{2-n} \left( 2-i \right) $

How would we evaluate: $$\lim_{n \to 0} \prod_{\substack{i=nk \\k \in \Bbb Z_{\geq 0}}}^{2-n} \left( 2-i \right) $$ Is it possible to evaluate this manually? Or do we have to make a program to get an approximation? EDIT: Since people are getting confused in the comments, here is an example: ...

limits arithmetic infinite-product

Secret

Currently, I am thinking about getting some initial function and then take the limit of $m$ to approach $x=3$

Mathphile

is it possible to evaluate this?

it looks to me like the factorial function with decrements of n tending to 0 instead of 1

*instead of decrements of 1

Secret

I am smelling Gamma functions, but I am terrible at calculating product sequences

TheSimpliFire

18:53

@Secret If $x=a$, $y=mx+c\implies\frac ym=x+\frac cm$ and as it by definition goes through $(a,0)$, we get $\frac ym=x-a=0$.

MrAP

@TheSimpliFire, I am getting $z=-1/3$ from what you have suggested.

TheSimpliFire

@MrAP OK, that is only one point, so the question is probably incorrect

Mathphile

the lim n tends to 0 part really confuses me

Secret

@TheSimpliFire how do you get $\frac{c}{m}=-a$?

TheSimpliFire

@Secret (a, 0) gives 0 = a + c/m

Secret

18:58

ah right

so that means in the limiting process, c and m vary together to maintain the constant $a$

TheSimpliFire

@Mathphile If I take $0.01$ then it is $\prod_{t=0}^{199}\left(2-0.01t\right)$ which is something of magnitude $10^{-26}$

Mathphile

so i think that it is safe to assume that when n temds to 0 then to product also tends to 0?

TheSimpliFire

@Mathphile That's my guess

@Secret sort of, put in an extremely unrigorous manner, $y=\infty x-\infty$ that kind of thing

Anyone want to have a go at proving the lower bound in this question?

Mathphile

desmos.com/calculator/ho4yzaoung

why are these equations so similar?

Lozansky

Look at the power series

Mathphile

19:13

well i cant isolate y so i cant generate the power series

TheSimpliFire

They seem to differ by 1/2 for large x

@Mathphile another recreational one: desmos.com/calculator/uoipolfyro

Mathphile

nice

user178403

19:28

hi .To say 'the function converges pointwise' is the same as to say 'the function converges everywhere' ?

Mathphile

19:41

desmos.com/calculator/elbuer3yfl

i found these functions pretty amazing^

Junglemath

Can someone explain the last sentence in the answer to this question? math.stackexchange.com/…
Namely, why does the subgroup in $G$ that corresponds to the order $p$ subgroup in $G/H_k$ have order $p^{k + 1}$?

Ted Shifrin

20:11

@user178403 This makes no sense. It should be a sequence of functions converges pointwise. If $f_n\to f$ pointwise, this means that for every $x$ in the domain, $\lim_{n\to\infty}f_n(x)=f(x)$.

Dair

hi chat

Ultradark

hi dair

Dair

how goes it?

Ultradark

not bad, thinking about buying an intro textbook today with my tax return refund

Dair

intro to what?

Ultradark

20:23

I'm thinking intro to abstract algebra

Dair

which book?

Ultradark

Introduction to abstract algebra by W. Keith Nicholson

Dair

out of curiosity, why that book in particular?

user178403

@TedShifrin indeed. My point was about the terms only ''converges pointwise'' equivalent to ''converges everywhere'' . Seems like they are indeed the same using different words

Ultradark

20:44

@Dair I looked at a few different books and this one is in my price range and looks like it covers the topics i want. Also it has a lot of exercises

with some answers

MrAP

21:13

If for a complex number $z$, $|z-3|=1$, what is the maximum value of $|z|$? I got 4. What do you think?

Daniel ML

21:39

Hi i want to conclude that the inverse of a triangulable operator is triangulable

Probing first that if $\betha$ is also a fan basis for $T^-1$

With the hypothesis that T:V--->V a linear operstor over V invertible and we suppose that $\betha$ Is a fan basis for T

How can i prove that?

I try doing the proof by induction

To prove that $T^-1(V_i) \subseteq V_i$

Shaun

Does anyone else get depressed about academic job prospects? I'm doing an average PhD at an average university. I doubt I'll get an academic career :/

What are the odds of getting a postdoc, honestly?

nbro

What does this notation $D_j(x) f$ mean? This should be an operator, x is a node of a graph and f is a function defined on the vertices of the graph

How should I read that notation?

Shaun

I specialise in group theory. That's a competitive area of research. Most positions ask for geometric group theory, too, instead of combinatorial group theory, my area.

nbro

21:54

I think that I should read it as follows: the operator D is applied to vector f of x

I guess that each x, node of the graph, has its own vector f, representing the output of the signal/function defined on the vertices of the graph

Shaun

$\Psi$ Nevermind, I'll have a chat with my supervisor. He knows what's what. Thank you nonetheless.

♦ Basic Mathematics

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Mathematics

♦ $Mathematics$ Basic Mathematics

$Mathematics$ Mathematics

$Mathematics$ Mathematics