Mathematics - 2022-01-07 (page 1 of 2)

UnderMathUate

01:01

'Ello.

PenAndPaperMathematics

Aloe there stranger

AMDG

01:38

o/

UnderMathUate

;-;

AMDG

ded chat ecks dee

UnderMathUate

Ye, that makes sense.

AMDG

So... in terms of identities, what exactly do I have for equivalent expressions of the form $\frac{1}{\lfloor \frac{2^n}{m}\rfloor}$?

For naturals $n$ and $m$

user10478

02:38

When solving a $2$D Heat Equation, suppose I separate the solution into time and space, i.e., $f_1(T(t)) = f_2(Z(x,\ y)) = \lambda$, and then separate space into its dimensions, i.e., $f_3(X(x)) = f_4(Y(y),\ \lambda) = r$. The problem of this sort I worked seems to have two nontrivial paths, one in the case that $\lambda = r \neq 0$ and another in the case that $\lambda \neq r, \lambda \neq 0, r \neq 0$. Usually I have encountered only one nontrivial path.

After I have solved for $u$ in each of the paths, the former being a Fourier Series solution and the latter being a double Fourier Series solution, am I supposed to combine the answers into a single particular solution to the problem somehow, or are these separate particular solutions which I must choose between based on some physical measurement obtained independently of the given boundary/initial conditions?

David Choi

03:00

Hey guys I’ve been trying to prove the convergence conditions for the binomial series on my own but I’ve been having trouble with the boundary cases of the interval of convergence.

I am currently trying to show that for $x=1$, the series $\sum^{\infty}_{\nu = 0} \left( \begin{matrix} \alpha \\ \nu \end{matrix} \right)$ is absolutely convergent if $\alpha > 0$.

I’ve tried to apply the comparison test, ratio test and root test (with Stirling approximation), and all of these methods have come up empty.

I honestly think the comparison test is the way to go but I’ve been having a hard time coming up with a series to compare the power series to

The method I tried was I defined $k = floor(\alpha) + 1$ and wrote a inequality using that but it didn’t pan out

leslie townes

04:28

have you tried raabe's test? it is a refinement of the ratio test. the only general test i can think of that might work.

i think i do agree that the ratio test is inconclusive.

i'm interpreting the series as literally written, which does make sense. i don't see an $x$ for which i can put $x=1$.

David Choi

Oh sorry the original series is just the binomial series

I was studying power series and my textbook went through the process of deriving the power series directly without using Taylor’s theorem

And it was pretty easy to apply the ratio test to find it converges for all values of exponents if |x| < 1

But then I wanted to figure out what happens at the boundaries

@leslietownes I’ll definitely look into that

It seems like determining the exact convergence criterion is much more difficult than I thought… just found a semi translation of Niel Abel’s paper on his proof for the convergence of binomial series and it’s quite involved….

leslie townes

you are right in seeing that you need something more at the endpoint. i think raabe will do it.

David Choi

04:44

You know I should just stop trying to read between every line of the textbook

Taking too much time when I should just be making progress

leslie townes

which textbook?

David Choi

It’s courant’s differential and integral calculus vol 1

leslie townes

i like the idea of reading between lines of the textbook. people who write textbooks are human (except ted who is not human), and make choices. it's OK to interrogate those choices.

David Choi

I’ve been reading it every page of it as thoroughly as I can well cuz I have the time rn now

leslie townes

i've heard a ton of great sutff about courant but never read it.

David Choi

04:47

Honestly I really enjoyed it, but he is a bit terse at times

leslie townes

well you don't get a math institute named after you if you're not terse.

David Choi

Filling in the details has been difficult at times but very rewarding

@leslietownes ahahaha I guess so; he also uses the words clearly and obviously too much

I’m just like I get it Courant ur smarter than me

U don’t have to rub it in every 3 pages ahaha

leslie townes

in the mid 20th century, all white men were contractually obligated to use 'clearly' and 'obviously' no less than 1 time per 5 pages.

he did 1 in 3. that shows going above and beyond.

that's why he has his institute.

David Choi

Ahahaha who knew that was the secret

I guess if u do 1 every 2 pages u get a Fields medal :pp

leslie townes

oddly enough, one of the fields medalists i met (who has now passed away) was one of the clearest instructors i ever had.

i don't think they gave him the fields medal for that

David Choi

04:52

Huh a genius and a good teacher

Those things normally don’t normally come in pairs right

leslie townes

they generally don't, no.

David Choi

Alrighty thanks for the help! I’m gonna go bang my head against this problem a bit more

123

07:19

Hi All....

Thank you all, for all your support. I learnt too much from you guys and i cleared my many confusion.

Is there any simple mathematics example of ternary operation which use in daily life?

2) Why associativity is binary operation? It takes three argument and one binary operation twice?

robjohn

07:54

[Raabe's Test](https://en.wikipedia.org/wiki/Ratio_test#2%2E_Raabe%27s_test) works, or...

[Euler's Reflection Formula](https://en.wikipedia.org/wiki/Gamma_function#General) says that
$$
\begin{align}
\binom{x}{k}
&=\frac{\Gamma(x+1)}{\Gamma(k+1)\Gamma(x-k+1)}\\
&=(-1)^{k-1}\frac{\sin(\pi x)}\pi\frac{\Gamma(x+1)\Gamma(k-x)}{\Gamma(k+1)}
\end{align}
$$
Then we can use [Gautschi's Inequality](https://en.wikipedia.org/wiki/Gautschi%27s_inequality) to get the asymptotic formula
$$
(-1)^{k-1}\binom{x}{k}\sim\frac{\Gamma(x+1)\sin(\pi x)}{\pi k^{x+1}}

Alex

08:47

Hi @robjohn working about the integral of another day $\displaystyle \int_{0}^{+\infty} \cos(xt){\rm sech}^{2}(t){\rm d}t$. Using your answer (first part in the link, I didn't understand the partial fraction on $\mathbb{C}$) I got that $\displaystyle \int_{0}^{+\infty} \cos(xt){\rm sech}^{2}(t){\rm d}t=\sum_{n=1}^{+\infty}(-1)^{n}\left( \frac{-8n^{2}}{(2n)^{2}+x^{2}}\right)$

I think we can use Fourier's series for to continue. However using the test $\displaystyle \lim_{n\to +\infty}a_{n}$ for the series $\sum a_{n}$ we have the limit is undeterminated. So I don't sure if I can continue from here.

Mad Spaces

09:56

if i am using \boxed {} and in {} i have both text and equations how can i avoid presenting everything inside even the words as equations?

For example $\boxed{Here words,F \times Y}$

You see how the words are mixed cuz of $ $

i see i can just add \text

is there a way to make the height of the box fit to the text and not just extend horizontally for long texts?

robjohn

10:21

@MadSpaces $\boxed{\text{Here words},F \times Y}$. What are you trying to do?

Mad Spaces

10:38

Nvm sorry RobertJohn i have decided to do it in other way because i do not want to waste my time with this, have too much to learn and too little time, writing right now a question about the proof of inverse functions for multivariable functions, i will post it here once i finished maybe you guys could help, this proof is mean...

Vrouvrou

11:42

Hello

I have this recurrent sequence $u_{n+1}=1/(3+u_n)$ how to prove that $u_{n+1}-u_n \leq k |u_{n-1}-u_n|$?

I have |u_{n+1}-u_n|=|(1-u_n-u_n^2)/(3+u_n)|$

Vrouvrou

12:07

With $u_1=1$

And $n\in \mathbb{N}^*$

Alex

@Vrouvrou If you have a sequence $a_n=f(n)$ you can study the behavior of the mapping $x\mapsto f(x)$. In our case, you can study the mapping $f: x\mapsto \frac{1}{3+x}$. Do you know how study the monotocity of a function using derivatives?

Vio Ariton

Hey. Do you guys know if I can evaluate a transfer function at a time t on wolframalpha?

Vrouvrou

12:48

@Alex yes I know how to do

In this case f is decreasing

geocalc33

13:05

If a function is analytic for $\Re z < 1$ and not analytic for $\Re z > 1$ then is the only possible analytic continuation, to $\Re z=1$?

Mad Spaces

13:16

oh my god

this frkn proof is killing me ive been writing the question for 4 hours now LOL

PM 2Ring

13:50

@Vrouvrou You have a mistake. That should be $|u_{n+1}-u_n|=|(1-3u_n-u_n^2)/(3+u_n)|$. Can you do $|u_{n-1}-u_n|$ ? BTW, the sequence converges to $\frac{\sqrt{13}-3}2$

Vrouvrou

14:10

Yes

$|u_{n-1}-u_n|=|\frac{1-3u_{n-1}-u_{n-1}^2|}{3+u_{n-1}}|$

But I don't know how to find a relation

@PM2Ring

robjohn

$u_n-u_{n+1}=\frac1{3+u_{n-1}}-\frac1{3+u_n}=\frac{u_n-u_{n-1}}{(3+u_{n-1})(3+u_n)}$

@Vrouvrou this might be useful

Vrouvrou

Oh yes

But how to see that $1<(3+u_{n-1})(3+u_n)$ ?

robjohn

note also that $x\mapsto\frac1{3+x}$ maps $[0,1]$ into $[0,1]$

so $0\le u_n\le1$

@Vrouvrou isn't that at least $9$?

Vrouvrou

14:27

Why we take x from [0,1] ?

robjohn

Didn't you say that $u_1=1$?

Vrouvrou

Yes

robjohn

so $0\le u_n\le1$, no?

Vrouvrou

I don't know this

robjohn

$u_1=1$, $u_{n+1}=\frac1{3+u_n}$, $x\mapsto\frac1{3+x}$ maps $[0,1]$ into $[0,1]$

what can you deduce from that?

Vrouvrou

14:31

Yes but the function f can be defined on R-{-3}

Why it must positive?

robjohn

Who cares? we are only looking at where $u_n$ can go and $u_n$ starts at $1$

if $u_n\in[0,1]$, then $u_{n+1}\in[0,1]$, no?

Vrouvrou

Yes

robjohn

8 mins ago, by robjohn

so $0\le u_n\le1$

and now do you see that $(3+u_n)(3+u_{n-1})\ge9$?

Vrouvrou

But we can have $u_n\leq 0$

robjohn

How? if $u_1=1$

Vrouvrou

14:36

Ok ok

I understand

$(3+u_n)(3+u_{n-1})\ge9$

robjohn

25 mins ago, by robjohn

$u_n-u_{n+1}=\frac1{3+u_{n-1}}-\frac1{3+u_n}=\frac{u_n-u_{n-1}}{(3+u_{n-1})(3+u_n)}$

rb3652

Hi folks, does anyone here have any links to papers that apply Machine Learning techniques to open math problems or something similar?

Or perhaps, if anyone here has prior experience applying ML/GA (Genetic Algorithm) techniques to any fields in math, it would be great if you could shed some light or provide some resources on that subject.

rb3652

14:54

Hm, well I have another question which may be more appropriate for this forum.

If I have a bunch of integers $a$,$b$,$c$,$d$, is there any physical interpretation for $\frac{a+b+c+d}{2}$?

Perhaps something like center of mass?

Xander Henderson

15:31

The center of mass would has a $4$ in the denominator, not a $2$ (the center of mass is an average). And this interpretation would depend upon an integer $n$ representing a unit mass at the position $n$.

jay

if i have two random variables in $\mathbb{R}^d$ can i used cauchy schwartz to say $E[<X,Y>]^2\leq E[|X|^2]E[|Y|^2]$?

or am i missing something because the R.V $X,Y$ are in a higher dimension

Xander Henderson

Dimensionality should have nothing to do with it, unless you have a restrictive statement of CS. What, precisely, is the statement of the theorem you refer to as the CS inequality?

jay

or holder inequality sorry

Hölder's inequality

In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces. Theorem (Hölder's inequality). Let (S, Σ, μ) be a measure space and let p, q ∈ [1, ∞] with 1/p + 1/q = 1. Then for all measurable real- or complex-valued functions f and g on S, ‖ f g ‖ 1 ≤ ‖ f ‖ p ‖ g ‖...

you see the section for vector valued functions

Xander Henderson

Well, what you wrote originally was the CS inequality, but that can be seen as a special case of Hölder. Are you really using the definitions directly from Wikipedia, or do you have a better source?

In any event, as I said above, high dimensionality is irrelevant. These inequalities are results about normed vector spaces in general.

jay

fair enough

Mad Spaces

16:08

God almighty in heaven jesus mary and joseph i spent the wholeeeeeeeee day writing this question!!!!

0

Q: Understanding the Proof of the existence of the Inverse function of a multivariable function.

Introduction: I am having a hard time understanding this Proof. A similar scheme of the proof can be found in countable many books and lectures. However, so far the sources i looked up do not really explain every point of the proof, and the reader is left wondering about many Points. I shall writ...

@TedShifrin i even watched your lecture about the proof of the hypothesis but i reliazed quickly that i had pretty much the same questions i had regarding the proof of my professor, eventually i got frustrated with your students not asking the right questions that i had in mind and stopped watching lol

Xander Henderson

@MadSpaces With all due respect to the time and energy you put into it, it doesn't look like a good fit for the format. What is the actual question there?

Also, your formatting is all over the place, and a couple of the long lines run into the sidebars and are rendered unintelligible... :/

Mad Spaces

Oh i am sorry i am not really good with latex!

I try my best.. what do you mean it does not fit the format?

Xander Henderson

@MadSpaces Questions on Math SE should, ideally, be narrowly focused and clear.

Mad Spaces

Well, it is proof understanding, i tagged it as such. I wrote down the whole proof. and marked with red and green the areas of the proof which is for me not clear, and i commented separetly on them. Why is this not acceptable?

Xander Henderson

Your post is quite long, which immediately suggests that it is not very well focused. A quick skimming of the post seems to confirm that suggestion---if I understand correctly, there are a number of places where you are confused about the argument, and that you want readers here to go through everything you have written and fix up everything which is unclear. That isn't really a good fit for the SE Q&A format.

Mad Spaces

16:14

Well, i am not really forcing anyone to do that!! people can very well just leave it be or not? And i do not know how to ask these questions in another fashion, they are related to the proof, so what would you suggest i did otherwise? (or for that matter do in the future)

Xander Henderson

@MadSpaces I would suggest that you do as the commenter suggested, and ask several discrete questions, each one focussing on one issue.

Mad Spaces

But is that not exactly what i did... these questions can not be answered separetly ... because they would not make sense without the rest of the proof! . i am sorry if i am missing your point.

Xander Henderson

While you are not "forcing" anyone to do anything, I suspect that your question will not be positively received, and that you will have difficulty getting a satisfactory answer.

leslie townes

i would second xander's recommendation. i declined to answer a question last week because it was asking three or four different questions. i could have answered all of them if they were split apart. it's unsatisfying when you post a solution to something and then the OP responds by asking yet another related question that still somehow fits within the original context.

i'd love to know how you somehow broke mathjax formatting (at least in chrome on windows 11).

Xander Henderson

@leslietownes My guess is through the use of long lines with \color{} macros.

leslie townes

16:18

i would say that in general, a big part of asking a complex question is breaking it into bite-sized chunks. it's hard to comment on something that amounts to a research program.

Mad Spaces

I will try to do it better i guess next time i have such a problem.

Please do not feel forced to look into that and answer it if it is inconvienent.

Xander Henderson

@leslietownes This is a true statement in mathematics, generally. One of the fundamental principles of mathematical analysis is that of "dimensional reduction". You can't really solve a problem in 8 dimensions. Instead, you solve 8 problems in one dimension (e.g. Fubini-Tonelli).

leslie townes

you are fighting not just against the reader's inability to understand where you are (although good context should provide that). you are fighting the reader's instinct to look at several screenfuls of text and say "ahhh, nah"

rb3652

lol

Xander Henderson

Same thing here. You can't really tackle a difficult problem or theorem all at once. You have to break it down into bite-sized pieces (both for yourself, and for your readers).

leslie townes

16:20

xander in my hometown there was an old restaurant that never seemed to have any customers called Tonelli's. i always wanted to start a second sham restaurant for money laundering purposes next to it and call it Fubini's.

my advisor was incredibly funny, but almost never put anything funny in writing, and when he tried it was always a little off. one time he needed a theorem about interchangeability of iterated suprema over a product set, and called it "Supini's Theorem." that's the best (worst) example.

Derivative

my complex analysis professor forgot one of the conditions of Abel's theorem when stating it, and the textbook also misses it. Should I mention it?

leslie townes

(1) check if they're using the same 'abel's theorem' (it could be the same name is being applied to a family of related but inequivalent results), (2) sure

(1) is particularly an issue with 'old' results that might predate the modern language that is used to express them. newer treatments will express them in different and maybe inequivalent ways.

Derivative

yeah I did check. They're stating the complex variant with only the conditions of the real variant

leslie townes

if there's a hypothesis that's missing for a proof to work i would definitely bring that up.

Derivative

hmmm by email? This class has like 90 students so asking questions is a bit intimidating

leslie townes

16:27

that's how i would do it. i have done this once or twice in the past.

Derivative

alright, thanks

leslie townes

once it turned out that i was wrong (i.e. the hypothesis i thought was 'missing' was not necessary) although the instructor realized that they should have rewritten their proof to clarify the issue.

the other times i was not wrong.

is this the stuff about lim x->p [power series] with p at the edge of the disc of convergence

robjohn

=

Derivative

@leslietownes Yes. They forgot the thing about the Stolz sector

leslie townes

that might be a pedagogical choice if they only want to apply it in the real case. the whole thing with stolz angles is a lot of terminology and notions for what is ultimately a simple concept. but i'd think of this as worth checking.

i think i've commented in chat before about stolz probably being embarrassed by his name being associated with that.

Koro

16:30

Is the statement: $\frac{\partial}{\partial x}\left(\int \frac{\partial M}{\partial y}(x,y) dx\right)=\frac{\partial M}{\partial y}(x,y) $ correct?

Shing

If I have a set $lim_{n\rightarrow \inf} [1,2,3....n]$

is this the natural number set?

leslie townes

you're about to start world war III between us and europe, but ok.

more seriously, i would clarify the sense in which you are taking a limit of sets.

if you do so, it is likely that the limit will indeed be the set of positive integers. whether or not this is the set of 'natural numbers' i will leave for another day.

Mad Spaces

@Shing It seems to me that it is, without the zero, but that is a definition question.

123

Why associativity is a binary operation? It has three arguments, one operation twice.

leslie townes

associativity is a property of a binary operation, not a binary operation.

Koro

16:33

associativity is property of 'some' binary operations.

Oops, you already said that

leslie townes

apparently koro and i are sharing brains today instead of ted and i.

Shing

@MadSpaces I thought in the natural set, all the natural number is "already" there. but the "limit set", it add number one by one, it seems to me it is never going to reach the natural set.

Koro

:D

123

if associativity takes 3 argument it should be ternary not binary.

leslie townes

associativity is not an operation, but a property of an operation.

Mad Spaces

16:35

I didnt think about that, but it seems to me, you are assuming the "limit" process is somehow taking time, rather than happening instaniously. You can obviously construct a bijection between your set and the naturals., so you know they have the same cardinality. And you can assume for each number in the naturals, you will find that in your set...

leslie townes

i guess if you were doing universal algebra you could draw some distinction by which associativity is characterizable via a formula with three variables and maybe not two. people do work on that kind of thing and i don't know. but that doesn't sound like what this is.

123

@leslietownes Yes it is property... Thanks.

Is there any simple mathematical operation which ternary?

leslie townes

universal algebra is interesting. if you allow for multiple variables you can characterize the group "axioms" with a single axiom. there's lots of weird stuff like that.

123: one example i can think of is 'betweenness' in geometry, which returns true if A is between points B and C on a line.

Mad Spaces

@Shing Say lets call your set $A$ then $f(x): x\in A \rightarrow \mathbb{N}: f(x)=x$ is a bijection.

leslie townes

it's actually pretty fundamental to axiomatization of plane geometry.

hilbert realized this but maybe not euclid.

euclid certainly knew about it.

Koro

16:38

@Koro this equality has been confusing me. Why does this have to be true? It is given that the integrand is continuous; fine so the integral makes sense. But again, why the equality?

what if there were $\frac{\mathrm{d}}{\mathrm{d}x}$ outside instead of the partial derivative?

123

@leslietownes Can you pls share the link of betweenness, i found something else

leslie townes

i don't know that there is a link? do we need a link? it's a geometrical relation.

OK, i googled, and it's on page 1 of math.hkust.edu.hk/~mabfchen/Math4221/Hilbert%20Axioms.pdf but that is a random PDF.

i don't mean to criticize it. it seems to be quite good, in fact, based on a limited review of its first and second pages.

thank you, mabfchen at math.hkust.edu.hk

123

@leslietownes Thank you, I suggest you to online 24 hours everyday in this group. Because you always happy to help in this group and few other guys also. It help us

2

Shing

@MadSpaces um....thanks for answering, but I still have a feeling they are not quite exact same thing. but the time I studied native set theory was so long ago I almost forget everything.

leslie townes

123: :)

Koro

16:43

I like your message 123. :)

123

@leslietownes How do we know pi is irrational?

leslie townes

i think the first proof was via continued fractions. somebody will correct me on this if i'm wrong.

Proof that π is irrational

In the 1760s, Johann Heinrich Lambert proved that the number π (pi) is irrational. Thus, it cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven, and Nicolas Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich. In 1882, Ferdinand von Lindemann proved that π is not just irrational, but transcendental as well. == Lambert... ==

123

@Koro I mentioned there are also few other peoples which helped me a lot in past 2 ,3 days before that i was not happy with this group.

leslie townes

i think the proof i saw in school was niven's proof, also on that page.

123

@leslietownes I have seen this article before. But problem is that where pi comes from , after that we need to prove this is irrational. Where pi comes from?

leslie townes

16:46

you do need a definition of pi to do that. there are a lot of arbitrary but equivalent choices to be made there.

123

Pi

The number π (; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159. It is defined in Euclidean geometry as the ratio of a circle's circumference to its diameter, and also has various equivalent definitions. The number appears in many formulas in all areas of mathematics and physics. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in 1706. It is also referred to as Archimedes' constant.Being an irrational number, π cannot be expressed as a common fraction, although...

leslie townes

the proofs on that page use characterizations of pi as roots of various trigonometric functions.

which can be defined via integrals or series if you don't like to take them as primitives.

123

I have read above article , but don't understand where pi comes from.

Mad Spaces

@shing maybe smarter people here can provide a better insight. I am glad i could offer some kind of insight or something to consider.

123

This is once again confusion. If sin and cos are the ratios of right triangle . why sin cos are transcendental ?

@leslietownes Above pi article did not mentioned where the story of pi started and why it is important for us and we stick with this number. What is the core problem of calculating pi from where the story of pi started. After that we used different techniques. But what is core first problem of pi where this number comes in first place.

UnderMathUate

16:53

pi = e = 3.

leslie townes

i think many people first encounter pi as half the circumference of the unit circle.

if you express that in terms of integrals you get one potential definition of pi.

123

I understand it has many applications and we defined many methods for proof of pi is irrational. But all these proofs are secondary. What is the first main core problem where pi started , which is entirely related to the pi.

leslie townes

under: boooo

UnderMathUate

I kid, I kid.

123

@leslietownes I have read this. The problem of approximating square with circle... Does this first problem from which we can not measure it with 100% accuracy?

leslie townes

16:59

you can define numbers in all sorts of ways. some of them don't easily lend themselves to calculation. ease of calculation is a separate issue.

consider the examples i was using the other day, 2^35434875348

we know exactly what that is, and in many instances we can decide whether other numbers are equal to that number or not, without calculation.

we don't necessarily have that recipe as a homework problem that we have to figure out as a precursor to reasoning with the number.

123

@leslietownes Yes i remember. I was confused because pi and e have special place in math. There should be first problem where the story pi started , then we made different methods after that.

In our text book we have question. How do we know pi is irrational? It is not fair answer to say , because it is non-terminating & non-recurring decimal number. Because all irrational numbers have this property. Also in my opinion we can answer we have proof pi is irrational.

leslie townes

it's a difficult problem. pi in some form or another, and the concept of irrationality, go back at least to euclid and ancient greece. it was proved irrational in 1760.

Ted Shifrin

@UnderMathUate A shorn kid.

leslie townes

even though it's a known result, i don't think most people with phds in math who haven't seen a proof could come up with one on their own.

it's just hard. deciding rationality is hard.

UnderMathUate

@TedShifrin Hm? 🤔

123

17:06

I think better answer should be if it is true. We can not calculate circumference of circle with 100% accuracy in calculation, after that we proved with other methods that circle circumference is irrational. Is it correct?

Ted Shifrin

For $e$ we luck out, leslie.

leslie townes

yeah, e is a joke.

Ted Shifrin

@Under A kid is a baby sheep?

UnderMathUate

Ooh, I was thinking of a different kid.

leslie townes

123: i don't think it's helpful to think of 'calculating' anything with '100% accuracy.' see the example of 2^23597349574 which i can conjure up just by typing keys on a keyboard. perfectly good number, perfectly good subject of reasoning. nobody has to figure out its decimal expansion.

you keed, you keed. like triumph.

PenAndPaperMathematics

17:08

In a topos $\mathcal{E}$, a subobject classifier $\Omega$ is an injective object.

Proof. enjoysmath.blogspot.com/2022/01/…

Ted Shifrin

Most mathematicians don’t think of numbers as decimals, but the rest of the world understands little but those.

PenAndPaperMathematics

@Koro

123

@TedShifrin What mathematician think for numbers?

Ted Shifrin

@Koro I suppose, but bad abuse of notation. I don’t write $\frac d{dx} \int f(x)dx$.

@123 Typically, closed-form expressions or properties. $\sqrt 2$, $\root5\of\pi$.

UnderMathUate

Anyone got any problems? Kind of bored.

Ted Shifrin

17:17

I gave you plenty :P

123

@leslietownes I don't know how to explain my question , i think answer may be hidden in their after that we create other methods for calculating pi but first problem is the pi number arise in first place for calculation of circumference of circle or area of circle.

Ted Shifrin

@Under If you're tired of baby number theory, name a topic.

123

@TedShifrin But not every irrational number can be written in this form.

Ted Shifrin

Give me an irrational number, @123.

UnderMathUate

@TedShifrin I did all except the last one. Well, I'm not counting the n, -n one because I sucked at that one.

Ted Shifrin

17:18

LOL

123

@TedShifrin It means you said, it is possible to write every irrational number in above form...

Ted Shifrin

Well, I sent you that whole sheet of practice problems for your final. But, as I said, give us a topic. Oh, here's a cool one (in a minute).

@123 I certainly did NOT say that.

UnderMathUate

Oh yeah, there's still some on there I could do. I'll work on that and the one you're about to send.

123

@TedShifrin What exactly you mean sir?

leslie townes

under: isn't it time to think about operators on hilbert space?

Ted Shifrin

17:21

This one was given to me by a former student, @Under, and it stumped me for a little while. But it's cool. If $f$ is a polynomial of even degree $n$ and $f\ge 0$, then prove that $f+f'+f''+\dots+f^{(n)}\ge 0$.

UnderMathUate

and i oop

123

Okay last , Why sin and cos are transcendental , but sin = per/ hyp , it is ratio (rational) not irrational

UnderMathUate

Alright, I'll take a crack at it, lol.

Ted Shifrin

Numbers are more typically given by equations that they satisfy or properties that they have. For example, I might define a number $a$ by saying $a$ is the smallest positive number so that $\int_0^a \sin x\,dx = 0$.

123

Transcendental function

In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and root extraction.Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions. == Definition == Formally, an analytic function f(z) of one real or complex variable z is...

Ted Shifrin

17:24

@123 Ratio of what? It's not a ratio of integers!!!!!!!

Again, as with our discussion the other day, you are taking words totally out of context.

leslie townes

transcendental means something different for functions than it does for numbers.

123

@TedShifrin No no.... not out of context. You gave me thinking point. It means sin and cos are the ratio of irrational.

Ted Shifrin

Yes, out of context. Rational functions are different from rational numbers. Transcendental functions are different from transcendental numbers.

But, yes, in general $\sin(\theta)$ is a ratio (if you want to think of it that way) of two real numbers. That's all you can say.

123

It means all three or at least two sides of triangle are always irrational?

Ted Shifrin

That has nothing to do with the function being a transcendental function.

What means? There are plenty of right triangles with integer sides.

123

17:29

@leslietownes If this is the case. sin and cos become rational number, not irrational/transcendental

leslie townes

12345

Ted Shifrin

Do you understand that sin and cos are actually FUNCTIONS?

123

@TedShifrin They are functions. It's domain -infinity to +infinity and range [-1,1]

Ted Shifrin

OK, so as a function that hits every number between $-1$ and $1$, it takes on all rational values in that interval and all irrational values in that interval. What are we talking about, then?

123

@TedShifrin Aaah Okay Thank you i got your point. One confusion arose if this is the case , f(x) = x , it should also irrational function.

Koro

17:37

hey @PenAndPaperMathematics :)

123

It means all polynomials are irrational functions.

leslie townes

ted, you're good at this stuff. you should consider a career in teaching.

123

All continuous functions are irrational in all or some interval.

Ted Shifrin

Failing at teaching, you mean, @leslie?

123

Pls correct me.

Ted Shifrin

17:38

Again, @123, you're using words out of context. A rational function is by definition a ratio of two polynomials. Every polynomial is a rational function. This has NOTHING to do with its values.

Koro

@TedShifrin it was written like this. It is met in the proof that shows that the ODE $M+Ny'=0$ is an exact ODE iff $M_y=N_x$.

leslie townes

ted, this is the kind of skepticism of the whole project that leads people to law school.

123

Yes i know rational function are ratio of two polynomials. But why you define sin cos irrational from its values of range?

Ted Shifrin

@Koro The right way to understand that is from multivariable calculus. We're saying the vector field $(M,N)$ is a gradient (locally) if and only if ... or, better, the $1$-form $M\,dx+N\,dy$ is exact (locally) if and only if it is closed. .... I don't expect differential equations textbooks to be careful with their notation.

robjohn

@Ted frustration

Ted Shifrin

17:41

@robjohn I think that word is universal.

robjohn

yep. The only way to win is not to play.

Ted Shifrin

@123 We keep going around and around the same thing.

@robjohn Next you'll tell me I didn't win the election.

robjohn

There was an election?

Ted Shifrin

A little while ago, yes.

123

@TedShifrin You said here , that's why i confused. I am not very intelligent as you guys. That's why i asked many times silly questions.

Ted Shifrin

17:46

But you have to sit and think about what we are telling you. You keep repeating the same thing as if I haven't explained it two or three times already. And then you complain that we are not helpful. shrug

robjohn

@TedShifrin Oh, that election. You played, so...

leslie townes

not to be noodge, but, WHAT ABOUT THE EXPLANATIONS

rob: i'm not convinced that there was an election.

123

No no you are all helpful. And really thanks to you, robjohn , leslie , ADMG they helped me a lot .

PenAndPaperMathematics

@Shaun chat :D ?

robjohn

@leslietownes see National Treasure recently?

leslie townes

17:49

you know i've never dipped into that franchise.

123

Let me read all the comments once again... Thank you.

Ted Shifrin

Anyhow, @123, one final thing. It is "obvious" that $\sin$ cannot be a polynomial (because no polynomial can have infinitely many roots), so it cannot be a rational function. It is not quite so obvious that it must be a transcendental function.

robjohn

@leslietownes Ah, "not to be a noodge" is a line from that movie

Koro

But I still didn't understand the equality. What if there were $\frac{\mathrm {d}}{\mathrm {d}x}$ outside instead of the partial derivative? :(

leslie townes

rob: oh, is it? i have internalized it from my environment.

123

17:50

@TedShifrin Thank you... That's my answer

Ted Shifrin

@Koro It can't be $d/dx$ because you're differentiating a function of both $x$ and $y$.

leslie townes

i'm thinking of the curb your enthusiasm scene where larry realizes his lawyer isn't jewish. god, that's funny.

Ted Shifrin

I only saw a few of those, back when I had HBO.

123

If any function can not be write as a finite polynomial , so the function is irrational. This is the criteria of being a function irrational. Not from its range. Thank you......

leslie townes

a very regional joke is that my place of employment is jews pretending to be goys and an arguable rival of our place of employment is goys pretending to be jews.

Koro

17:52

can't be written as a finite polynomial? What does that mean?

leslie townes

i haven't seen the new episodes of curb. my dad said it was OK.

Ted Shifrin

We don't talk about irrational functions. I guess we can say NOT a rational function. $\sqrt x$ is not a rational function, but it is an algebraic function. It is NOT a transcendental function. $\sin$ is a transcendental function, because it cannot be algebraic.

Koro

Ohh, Ted: I thought that what if $d/dx$ is the "total derivative" (like we define in multivariable calc.).

Thanks a lot. I think my confusion arose due to the notations only.

Ted Shifrin

"We" do not define total derivative. That is a much maligned notion.

Koro

we do define total derivative in multivariable calc. What do you mean?

:(

Alex

17:55

9 hours ago, by Alex

I think we can use Fourier's series for to continue. However using the test $\displaystyle \lim_{n\to +\infty}a_{n}$ for the series $\sum a_{n}$ we have the limit is undeterminated. So I don't sure if I can continue from here.

Ted Shifrin

I never never never use that terminology.

Alex

I don't know as to continue :-(

Koro

Ahh, OK Ted :).

Ted Shifrin

@leslie So what is the easy argument that $\sin$ is transcendental?

123

@TedShifrin Sorry . Yes you are right algebraic if any function can written as finite polynomial and transcendental function . Rational function are ratio of two polynomials.

leslie townes

17:57

ted: i don't know of one. ask some AG folks.

Ted Shifrin

@Koro Applied types use it in a totally different way (like with $t$ as an independent variable and with $x,y,\dots$ dependent on $t$).

Ha ha ha @leslie

leslie townes

it's not irrelevant to my former field of study. you can do a lot of stuff for operators that are roots of an analytic function that maybe don't work in general. but it was not my focus.

Transcript for

$Mathematics$ Mathematics