Mathematics - 2015-06-24 (page 5 of 5)

Gato

20:00

now I know why you write $1/(1-\gamma)$

Tobias Kildetoft

@TedShifrin Neat (I hadn't thought of doing it that way)

Ted Shifrin

@Stan: No. I told you you want to reduce to the usual max/min test which tells us that $0$ is a local maximum of $F\:\Bbb R^n\to\Bbb R$ if the hessian of $\F$ at $0$ is negative-definite.

Oh @Tobias ... I think it works.

Gato

@Chris'ssistheartist for my integral over $\Bbb{R}$ of $\frac{1}{(1+x^2)(2+e^{ix}}$ it's $\frac{\pi}{2}(2+e)$

Tobias Kildetoft

@TedShifrin I agree, it looks like it will work fine

Ted Shifrin

How had you approached it? :)

Chris's sis the artist

20:01

@Gato That is another version. You asked me to calculate the one with $1$ instead of $2$.

Tobias Kildetoft

@TedShifrin Expand $(1+x)(1+y)$ in two ways

Gato

@Chris'ssistheartist yes, I said $1+exp{ix}$, sorry.

@Chris'ssistheartist I ask you for help, not to calculate for me :)

Ted Shifrin

Oh, that's easier ... @Tobias. Cute.

Gato

@TedShifrin une intégrale avec les résidus pas trop dur mais cool, pour moi ? :P

Ted Shifrin

Pas trop dur? Sais-je ce que ça veut dire? :D

Tobias Kildetoft

20:03

@TedShifrin I actually find your way simpler

Chris's sis the artist

@Gato by real analysis you mean?

Tobias Kildetoft

(though this other way also works when unity is weakened to all elements being the product of two elements)

Ted Shifrin

Yeah, but I think with that assumption you can prove that there actually is a unique $1$.

I haven't rethought all that yet.

Owatch

Something doesn't seem right

I don't feel like I get the perpendicular slope from just the reciprocal of the slope with a positive x value.

Ted Shifrin

@Gato: As-tu considéré les problèmes que je t'ai déjà envoyés il y a longtemps?

Draw the picture, @Owatch. Draw slope $2$ and draw slope $-1/2$.

You can put in right triangles and see that it works.

Tobias Kildetoft

20:06

@TedShifrin No, there need not be a $1$ (for example a simple complex Lie algebra will satisfy the conditions)

skill patrol

Did you change the sign? @Owatch

Ted Shifrin

ah, @Tobias, right. Let me look up the conditions on the problem I'm recalling.

Owatch

Only for x on the reciprocal

Michael Albanese

I'm back.

Owatch

@skillpatrol

skill patrol

20:08

@Owatch Have you never seen the formula m' = -1/m?

Gato

@TedShifrin right, j'avais pas tout fait, car des choses étaient trop avancées, je vais essayer $\cos(x)/(1+x^2)^2$, déjà c'est pair, donc j'ai 1 point sur 20 non ? :P

Ted Shifrin

Ah, @Tobias: The problem I had was this. Assume $\cdot$ is associative. Suppose that for all $a,b$ the equations $x\cdot a = b$ and $a\cdot y = b$ have solutions. Then we have a group.

Gato

@Chris'ssistheartist no, complex, in real analysis I am very bad, but I like to see some ideas anyway :)

Tobias Kildetoft

@TedShifrin Ahh, yeah

Owatch

No

skill patrol

20:10

:(

It should have been covered in high school

Ted Shifrin

Yup @skull/@skill.

But one needs only draw the right picture, as I keep trying to say. I wish I had a blackboard in here.

Owatch

Don't recal.

I see it now Ted

Michael Albanese

@skull/@skill = u/i

Owatch

I had axis wrong on excel

Ted Shifrin

@MichaelA ... $=-iu$.

Oh, use paper and pencil, @Owatch. Oy.

Michael Albanese

20:12

You have bested me yet again Ted.

Owatch

I have just blank paper

Ted Shifrin

:P

Owatch

So I can't really make good axis

Balarka Sen

sorry, @MikeMiller, I didn't think about (2) anymore. seems like too much work. I see someone has already posted an answer, but I decided adding my visualization.

Owatch

Or increments

Ted Shifrin

20:12

You should get practice drawing things roughly to scale, @Owatch. It will help you later on.

Owatch

Well, I do.

Balarka Sen

if it doesn't get any vote (i.e., if it's not useful), I'll delete it later on.

Owatch

I was.

Ted Shifrin

ah, it's Balarka's bedtime :P

skill patrol

Get some graph paper pal :-)

Owatch

20:13

Things just looked ever so slightly wrong. Which is what had me paranoid.

Chris's sis the artist

Use $$f(a)=\int_{-\infty}^{\infty} \frac{e^{-a x}}{(1+x^2)(2+e^{bx})} \ dx$$ with the fact that $\displaystyle f''(a)+f(a)= \frac{1}{a-b}+\frac{2}{a}$. Maybe it helps.

Owatch

I can go buy some.

Ted Shifrin

Ever so slightly is ok ... @Owatch.

Pictures aren't a proof, anyhow. They're for you to understand.

skill patrol

It helps A LOT

Michael Albanese

@Owatch: You can print some.

Ted Shifrin

20:14

Oh, that's cool, @MichaelA. I made my own in Illustrator some years ago ...

Owatch

Oh, sweet. I'll print some then.

Ted Shifrin

Thanks for the mini-challenge, @Tobias :)

Tobias Kildetoft

@TedShifrin Your welcome

Ted Shifrin

Don't want my brain atrophying completely ... just yet.

skill patrol

Have fun making sketches @Owatch :D

To scale

Ted Shifrin

20:17

By the way, when I do that in the third day of my multivariable or linear algebra class, @skill/@skull, none of the students remembers why the slope is the negative reciprocal. So I always show them the picture with the right triangles and this motivates the algebraic definition of dot product.

skill patrol

Great approach Professor @TedShifrin

People remember pictures.

Stan Shunpike

@TedShifrin okie dokie. Danke :)

Owatch

I have some difficulty

The distance between object A and B is described in terms of a (dx,dy)

It is (100,30).

Object A is stationary, but Object B must move at velocity V

I require object B to move perpendicular to the surface of A.

So, the (direction), it must move is the reciprocal of the distance (dx,dy) with -dx.

But integrating V into the direction is where it becomes a bit more difficult..

Gato

@TedShifrin ici $i$ un pole double, to compute the residue it's a big long but I get $-i/4e$,so the integral is $\frac{\pi}{2e}$

Owatch

V is basically the hypotenuse of the direction described by dx, dy.

As I think of it. So now I must find a way to put it in terms of dx, and dy.

Chris's sis the artist

20:38

@Gato by residue theorem, we have that $$\int_{-\infty}^{\infty} \frac{1}{(1+x^2)(2+e^{i x})} \ dx=\frac{e \pi }{1+2 e}$$

Gato

@Chris'ssistheartist this is what I have no ?

Chris's sis the artist

@Gato Isn't this the integral you needed to calculate?

@Gato Mathematica says that my calculations are perfect.

Gato

@Chris'ssistheartist Yes, I succeed too :), multumesc.

Chris's sis the artist

@Gato I can generalize it as I want. Welcome. :-)

Gato

Now, I need to sleep ^^, good night @Chris'ssistheartist

Chris's sis the artist

20:43

@Gato Good night.

Agawa001

bonne nuit @Gato

Owatch

If I got the distance to be (100,30). And then I said that it must go in direction (-30,100) to be perpendicular, than how can I determine a vector who's hypotenuse is 90 from that?

Ah ha, I got it.

Let us say distance (dx, dy) is (100,30). I know I need to go in direction (-30,100) to be perpendicular. This is good. But if I need to go in that direction at 90 units/s. I should have a vector for that too.

So the hypotenuse of the triangle created when you draw (100,30) is the magnitude?

And then you divide your wanted magnitude (90) by that, and multiple both dx, and dy to obtain the appropriate change in x and y to get that.

Chris's sis the artist

21:11

$$\int_{-\infty }^{\infty } \frac{1}{\left(x^2+1\right) \left(1+e^{e^{i x}}\right)} \, dx=\frac{\pi }{1+e^{1/e}}$$

Just easy stuff.

BBL

Danny

j

$\infty$

e

"In any system in which customers arrive and depart one at a time" / can someone help me to understand this statement? does it mean that(1) there is first an arrival and then a departure, or(2) is it that it arrives one at the time and departs one at a time?

Ted Shifrin

You're doing Poisson random variables?

Danny

queuing theory

by the way ted it was a long time ago

how is tennis going

Ted Shifrin

just saying that the people don't clump up ... either in arrival or in departure

what was a long time ago? :D

Danny

long time ago i talked to you

Ted Shifrin

21:21

I remember your name ... but no more

Danny

you probabably don't remember, anyhow back to the question

Owatch

Hmm

Danny

"In any system in which customers arrive and depart one at a time" then the rate at which arrivals find n = the rate at which departure leave n

Owatch

Stuff doesn't behave entirely right in my simulator :(

velocity is too high, things spin off into the distance.

Ted Shifrin

that sentence didn't quite make sense, but yeah, Danny

you're probably not scaling right, @Owatch

Danny

21:23

the proposition is

"In any system in which customers arrive and depart one at a time"

implies

the rate at which arrivals find n = the rate at which departure leave n

Ted Shifrin

"arrivals find n" ?

Owatch

Look. All I want to do is have an object move around another object. I have used the well known V = sqrt ( G * M / r) equation to get that speed I need. Yet it slowly spirals outwards.

Ted Shifrin

whoa whoa ... where is that well known?

Danny

@TedShifrin they find "n" in the system

they can see n persons in the system

Owatch

Google "orbit velocity equation", and that pretty much shows up all the time.

Ted Shifrin

21:25

how is the number of people you see a rate, Danny?

Owatch

It was also taught to me in high school physics.

Ted Shifrin

That's for escape velocity, @Owatch. You'd better know what you're googling.

Hippalectryon

He's right though

It's the speed required to orbit at a radius $r$

Owatch

I don't think so.

Ted

Danny

@TedShifrin let me find a URL link

to the book i would appriciate if you could explain

Ted Shifrin

21:26

So we're just doing conservation of energy here.

I don't want to be in the middle of both of these.

Hippalectryon

@Owatch What are you doing exactly ?

Ted Shifrin

thanks @Hippa for taking over

Owatch

PUT TED BACK

It's fine.

jokes

Danny

books.google.se/…

Hippalectryon

xD

Danny

21:27

@TedShifrin books.google.se/…

Owatch

I am building a simulator that is supposed to simulate the way objects with mass move in a gravity field.

People sometime call it an n-body simulator I think?

Hippalectryon

I've done that a while ago :D

Ted Shifrin

oh, a person arriving will see $n$ people there ... and a person departing will leave $n$ people behind.

Owatch

I am using Velocity Verlet Integration equations to determine the positions of objects each time interval.

Hippalectryon

ok

And what's the issue ?

Owatch

21:29

Because they apparently converse energy, which is what I want.

Ted Shifrin

You're only doing two bodies with one fixed, right, @Owatch?

Danny

@TedShifrin but does the Proposition mean that : there is first an arrival and then a departure

Semiclassical

for reference

Orbital speed

The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body. they can be used to refer to either the mean orbital speed, i.e. the average speed as it completes an orbit, or the speed at a particular point in its orbit. The orbital speed at any position in the orbit can be computed from the distance to the central body at that position, and the specific orbital energy, which is independent of position: the kinetic energy is the total energy...

Ted Shifrin

no, nothing sequential like that, Danny

Owatch

@TedShifrin One is supposed to be fixed, but it is slightly moved by the other depending on the mass of the satellite.

The problem is I placed the satellite at close range, and instead of maintaining a fast but steady orbit, it spiraled outwards.

Ted Shifrin

21:30

@Danny: All they're saying is that two people never arrive together and two people never leave together. But there might be several arrivals and then perhaps a departure, etc.

Hippalectryon

What initial speed did it get ? What magnitude, what unit vector ? @Owatch

Danny

then i don't know how they come to that conclusion

Owatch

I get the initial speed using V = sqrt(GM/r). Where r is the hypotenuse of the distance vector (dy,dx)

Danny

they state in the proof: "in any interval of length T the number of transitions from n + n+1 must equal to within 1 the number from n +1 to n"

why must that be so?

@TedShifrin

Owatch

The distance vector is (dy,dx). So to go perpendicular to the surface. It must go (-dx,dy)

Ted Shifrin

21:32

because we're talking about the same $n$, @Danny.

Semiclassical

what's the initial direction of the velocity?

Hippalectryon

@Owatch R=r ?

Owatch

Then, I get the magnitude of (-dx,dy) which is sqrt(dx^2 + dy^2)

@Hippalectryon My bad, it is just r. Fixed it

Hippalectryon

@Owatch Ok, and what's M ?

Ted Shifrin

So in a particular time interval, we might have several arrivals and several departures. But if we're keeping $n$ people in the room, there must be a departure for every arrival or vice versa.

Owatch

21:33

Mass of parent object. (Object to be orbited).

= M

Hippalectryon

It should be the sum of the masses of the two objects

Owatch

Should it be..

?

Hippalectryon

Check en.wikipedia.org/wiki/Orbital_speed#Mean_orbital_speed

Owatch

Oh.

You're right

Danny

@TedShifrin let me think about it, Iam a bit slow

Owatch

21:34

Well. Then that should be an easy fix.

Ted Shifrin

That's the same author who wrote the probability book I used, @Danny. Sometimes he is confusing.

Hippalectryon

It probably won't be enough, but it should be better.

Owatch

Spiral still occurs

But I guess it will be slightly improved.

Danny

@TedShifrin ok right now, I am doing a course called "Stochastic processes II" ...

Hippalectryon

Ok now, doesn't it lose its circular orbit rather slowly ? @Owatch

Owatch

21:36

It does not.

Hippalectryon

Doesn't it follow a quasi-circular motion at the beginning ?

Owatch

Well. It might, but I don't have any line tracing it's path.

Hippalectryon

Do it and see. Btw what are you coding in ?

Owatch

I can only observe the distance between it and the parent object increase each pass.

I'm writing this in Objective -C.

Adding line tracing is pretty problematic. I might do so eventually..

It sounds simple to add. But it's been something I've tried putting in months ago without any success. I will try again for testing purposes I guess.

Hippalectryon

Could you graph the distance to the center in function of t/T where T is the expected period ?

Owatch

21:39

Graph period over expected period?

Hippalectryon

Over, say, 20T

Owatch

Hm. It may be possible yes.

I can log positions.

But they mostly go to the console.

Plus, the position updates many times per second.

Hippalectryon

I'll write my own code, give me 5 mins.

Owatch

I'll be back in 20`

Danny

@TedShifrin i still don't get it : he says that that is enough that customers arrive and depart one at a time

implies that departure and arrival rates are equal

Ted Shifrin

21:45

But he's talking about rates with the same $n$ ... If more people arrive, $n$ will go up a lot and then to get back to the same $n$, the same number of people will have to depart.

Nickolas

Hey

Anyone could verify this? math.stackexchange.com/questions/1337839/…

Clarinetist

Afternoon, @Ted. I have this strange liquid mixture of mayo, tarragon, lemon juice, salt, pepper, and olive oil. Any ideas what I could do with this? [Probably shouldn't have rushed, the olive oil wasn't needed :P ]

and add chicken broth to that (since it was used to cook chicken breast) xp

Ted Shifrin

22:00

LOL, hi @Clarinet

reduce it a bit, add a bit more acid, maybe, and use it as a salad dressing or marinade for vegetables

Hippalectryon

@Owatch I get a perfect orbit

Clarinetist

@TedShifrin Thanks! Heh, I was trying to make chicken salad for my first time but there ended up being too much liquid remaining

Ted Shifrin

well, the poaching liquid is unrelated to the chicken salad, although I wouldn't object to reducing it (before the mayo) and then whisking some mayo into it and having that be the dressing.

Hippalectryon

Ugh cooking O_o

Ted Shifrin

@Nickolas: I don't understand your write-up at all, but I haven't figured out the right way to do the problem. However, I think this is a standard topic in standard probability books.

We don't say ugh to you when you show up, @Hippa.

Well, maybe I do.

Hippalectryon

22:06

@TedShifrin I'm just saying that because I'm awful at cooking q_q

Ted Shifrin

You're smart. If you want to learn, you will.

Hippalectryon

If I were interested, maybe :-)

Ted Shifrin

If you weren't interested in math, you wouldn't work so hard at it, either.

Hippalectryon

True

Owatch

Huh..

Not sure how you're calculating it.

Hippalectryon

22:09

pastebin.com/RaSx9e8Q

I took the earth-moon system for the values

(it's not circular because the scales aren't the same)

Owatch

Hmm. I do mine differently.

Hippalectryon

What did you do differently exactly ?

Owatch

It shouldn't have any effect on it.

Wait

Hippalectryon

What timestep did you take ?

Owatch

My satellite affects the parent object too.

Hippalectryon

22:11

It's negligible if the difference in mass is great enough

Owatch

10,000 to 1.

Hippalectryon

Do you still have your problem when the parent object is much much heavier ? If so, the problem lies elsewhere

Yeah, it's as if the parent was not affected with that ratio.

Owatch

Well. Let me look more at it later. Since I can't right now. I bookmarked the pastern file! so I'll take a look at what I am doing it a bit.

I have to eat, then I have a swimming thing to go to.

Hippalectryon

Ok :-) I'll probably be here at the same time (and earlier) tomorrow

Owatch

Thanks!

Ted Shifrin

22:14

Night, guys.

Clarinetist

Night @Ted

Hippalectryon

Night. beware the knight of the night muhahahahahaha owait that's totally lame :/

Ted Shifrin

I'm not quite gone, @Clarinet, but soon. Going to dinner at friends'.

Clarinetist

Ah k. Now how to make this acidic... maybe a bit of sugar

Idk

:P

Ted Shifrin

No, no, not sugar. More lemon or a bit of vinegar.

Clarinetist

22:15

Oh k

Ted Shifrin

Don't you know any chemistry? :D

You might also use a bit of Dijon mustard in the dressing. It'll help it emulsify.

Clarinetist

It's been too long :P

Ted Shifrin

Just don't tell me sodium hydroxide is an acid.

wb @Stan

Tobias Kildetoft

@TedShifrin But also please don't put that into your food. It will not taste good

Clarinetist

LOL

Hippalectryon

22:17

Well, in some cases, it can be :P

Ted Shifrin

oh, @Hippa is the chemistry expert.

Stan Shunpike

@TedShifrin hola!

Clarinetist

Looks beautiful. :) Thanks @Ted. That was my first reduction on purpose :P

Ted Shifrin

LOL ...

It's more about taste than looks, though.

Stan Shunpike

@TedShifrin so I was talking to my econ professor, and he was groaning about how laborious writing a textbook is. How long did yours take you on average?

Ted Shifrin

22:19

Oh hell, many years.

Clarinetist

Indeed

Stan Shunpike

Yes, that's what he said.

Clarinetist

Oh, I was responding to Ted about taste vs. looks

Ted Shifrin

The first draft of the multivariable I did in about a year, but I pilfered much of the linear algebra material from an earlier book.

All the responses make sense ... not to worry.

But I kept adding exercises and revising the text for five years or so before I went to the publisher.

Stan Shunpike

IIRC, you have written 4. Which took the longest?

Ted Shifrin

22:21

Dunno.

I had a coauthor for the linear algebra book.

The diff geo notes have been around for about 10 years and I still revise them in little ways.

Danny

@TedShifrin where can i buy your book :)

Iam in sweden ,tough

Ted Shifrin

They're all too expensive, @Danny, except for the free diff geo notes.

Clarinetist

Mmm, tastes very good

Ted Shifrin

Yippee @Clarinet

Hippalectryon

@TedShifrin In France they write them in some months (they're forced to do so) :( no wonder why people think they are terrible

Ted Shifrin

22:22

LOL, @Hippa.

Hippalectryon

Well, since we have a new education reform with each new president ...

Tobias Kildetoft

@Danny Where in Sweden?

Ted Shifrin

I've usually taught out of the text four or five times and revised continuously before publishing, @Hippa.

Clarinetist

@Hippalectryon Happens in actuarial too.

Stan Shunpike

@Hippalectryon write a textbook in a few month?

Danny

22:23

@TobiasKildetoft Stockholm

Tobias Kildetoft

@Danny Neat. I am in Uppsala (well, at UU, I don't actually live in Uppsala)

Hippalectryon

@TedShifrin Since I'm in prep school, we haven't used a single book in any subject (except French). I find that as good.

Danny

@TobiasKildetoft really, i live in Upplands Väsby, so we are not so far from eachother

Hippalectryon

@StanShunpike Yep. With the reforms, they have to write the books very quickly so that they can be printed and ready for the upcoming year

Tobias Kildetoft

@Danny I live in Bålsta, so yeah

Danny

22:25

@TobiasKildetoft what do u study

Tobias Kildetoft

@Danny I am a postdoc

Danny

oh really

Stan Shunpike

@Hippalectryon that's interesting. Are they digital?

Tobias Kildetoft

I do representation theory and categorification

Stan Shunpike

That would make it easy to add / edit

Hippalectryon

22:25

@StanShunpike nope

We're pretty late in terms of new technolgies in our education system

Stan Shunpike

Wow, that's very inefficient. :/

Ted Shifrin

I wonder if they've heard of LaTeX, @Hippa.

Hippalectryon

@TedShifrin Hopefully, yes

Ted Shifrin

Well, can't you tell from what you get printed?

Hippalectryon

I don't have any book with me atm. I don't have my old books anymore O_o

Stan Shunpike

22:29

@TedShifrin okay can you settle a debate we had in my econ study sessions. Is it L-AH-TEK , LAY-TEK or L-AH-TEX?

Ted Shifrin

About 1/2 my students just used the free .pdf file, rather than paying for a $12 printed version.

I say the second, @Stan.

Danny

@TedShifrin what about the last statement about $a_n = d_n = 0$ if the overall arrival rate will exceed the overall departure rate

Stan Shunpike

Okay me too

Hippalectryon

@TedShifrin Our school gave us the books with a financial guarantee (which is why I don't have them anymore)

Clarinetist

In Actuarial, if you know LaTeX, you are an asset to any actuarial ed company.

Ted Shifrin

22:30

@Danny: Sorry, I have to leave in a few seconds. Maybe tomorrow ... :)

Danny

books.google.se/… @TedShifrin

ok..

Ted Shifrin

Bye, all.

Clarinetist

Bye @Ted

Danny

bye

Hippalectryon

Bye.

Ted Shifrin

22:31

Bon appétit, @Clarinet.

Clarinetist

Heh thanks again @Ted

Owatch

22:47

What do you do when members of your external testing group ask where you put certain App features that don't in fact exist..

Clarinetist

You tell them they don't exist?

Owatch

I have to sneak it in next update.

Clarinetist

Oh boy

Tell them you'll have a status update to them in a little bit and work overtime to get it done

Owatch

Someone mentioned it to me like it was an "oh, by the way" comment.

My sim allows planets and small moons to be played with in a sandbox.

He asked where the stars were

And where he had to go to load and save the "worlds"

:S

Clarinetist

Yeah, just tell them you'll get back to them ASAP when you find out

and work overtime to get it done

That's what I would do, at least

Owatch

22:51

I will put it in.

I already worked overtime this week to refine it.

But it shouldn't be too bad to put that in too.

Clarinetist

Well, if it's done, might as well put them in

Owatch

Cool. Will do then

Alec Teal

@AlexClark you there?

Nickolas

@TedShifrin sorry but english is not my first language so some math terms may be poorly translated. What I did is assumed my random variable is following the geometric distribution (since what matters is the number of trials for first success, in this case number of flips to get heads when previously we got heads) then calculated the probability (I hope correctly) and then $E(X) = \sum_{i=1}^{5}E(X_i)$ where $E(X) = \frac{1}{p}$

Clarinetist

23:08

@Nickolas Was that the 5-consecutive-heads question?

Chris's sis the artist

23:37

@robjohn I discovered a nice integral

robjohn

@Chris'ssistheartist looks interesting.

Chris's sis the artist

@robjohn ooooo, that is nice! When you say interesting, it really is! :-)

@robjohn Some hours ago someone asked a question about a nasty integral, and I said to myself to spend some time in the area of complex analysis. So, I calculated more integrals like this one.

@robjohn In a way I'm a bit anxious about not affecting my real analysis thinking, it's very important for my book. :-)

robjohn

@Chris'ssistheartist do you use contour integration to solve?

Chris's sis the artist

@robjohn It's a more sophisticated approach.

@robjohn Yes, I used contour integration. I do it for some hours, but I hope to cease soon, I'll use the proper complex tools at the proper moment.

But it was fun to play with some tough integrals while using complex tools.

robjohn

23:53

@Chris'ssistheartist: this was an interesting question.

Chris's sis the artist

@robjohn Yeah, I see. Nice the series you got.

robjohn

@Chris'ssistheartist I am surprised that it has not gotten more answers. I read it before my son and I went out for lunch, and it was still unanswered when we got back.

Chris's sis the artist

@robjohn There were some comments there, I suppose there were some people that tried to answer it.

@robjohn It's nice to see in what kind of other situations the series I like can appear.

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$Mathematics$ Mathematics