Mathematics - 2015-12-06 (page 1 of 2)

Michael

00:00

I learned some stuff @JasperLoy

GBeau

@anon did you get the recurrence

user174558

Are you even serious about the questions you ask? You ask many, random questions at once, without studying anything seriously.

Michael

@JasperLoy Just like you told me too. I practiced all my integration and now I want to go to multi variable...

evinda

@TedShifrin We have that $f(x+1)=e^{-150 \left( x+\frac{1}{2}\right)^2}$

GBeau

@anon that one was easy if you could see how it was similar to Fibonacci recurrences, but I think you could also do it by finding a low prime that works (7 works)

anon

00:00

@GBeau I figured there was some kind of divisibility period but I ran out of time before I explored the idea. I think the answer was 181.

ooh, 7?

GBeau

@anon 181 was what I did, I found out 7 works afterward

I showed 5 divides 2015 and by a property of the recurrence a5 divides 2015 (a5 is 181)

a5 divides a2015*

@anon yeah look at the recurrence mod 7 and look for the repetition period

and note it hits 2015

user174558

So next year the questions will use 2016?

GBeau

there were 3 questions that used 2015

user174558

They must have run out of ideas.

evinda

@TedShifrin It never holds that f(x)=f(x+1).

Michael

00:03

Is there a simple book that will guide me through multi variable calculus? I tried many online videos but they are a bit too complicated. Paul's online notes was nice, but sometimes his calculations are confusing and he doesn't really explain.

evinda

@anon I am trying to see how we get that f(x-floor(x)) is a 1-periodic extension of f

GBeau

@anon I actually got that triangle sum problem (but I probably got it wrong)

anon

frac(x)=x for x in [0,1] and frac(x) is 1-periodic. have you ever seen the graph of y=frac(x)? look at it.

GBeau

I at least know how to do it, but the computation was intense and I probably made an arithmetic error

user174558

How do you pronounce Putnam?

anon

00:05

@GBeau I did that one too. I got 187/231 I think

GBeau

@anon my numerator and denominator were both in the 100s...I know my method was right, but that problem was rife with opportunities to screw up my arithmetic

anon

c could be any integer strictly between |a-b| and a+b right? (with a,b arbitrary)

Mike Miller

I know the Putnam is hard, but to get strictly between 0 and 1 points... :)

GBeau

yeah

@anon

I used the triple inequality version

a+b>c, b+c>a, c+a>b

(same thing)

anon

so I let a,b be arbitrary and inner summed from c=|a-b|+1 to a+b-1, then split it up to get rid of the absolute value and geo-sum-formulad times ten

GBeau

00:08

I'm glad I prayed to the geometric sum gods before I did that problem ;)

Michael

What are some real life applications of complex and real analysis? I might want to major in math and these fields, but my dad said he won't help me unless I can help the world with what I would learn.

anon

I also did the f+6f'+12f''+8f''' one. there was a fourth one I did in set A but I don't remember what it was

GBeau

how did you do that one

I constructed a thing like e^x*f(2x)....and you see how it works out

anon

it's $(2\frac{d}{dx}+1)^3f$, and $2\frac{d}{dx}+1=2e^{-x/2}\frac{d}{dx}e^{x/2}$

essentially the same idea

GBeau

yeah

Mike Miller

00:10

@Michael: Part of studying math is self-sufficiency, being able to help yourself on problems. Luckily, the question of "What real life applications of (blah) are there" has been well-answered on MSE and elsewhere and I'm sure an enterprising individual could find such answers.

GBeau

trying to remember the set A ones right now...

anon

oh yeah, the log of the double product, that's the fourth one I did

user174558

@Michael You need to talk to your dad about your life philosophies then. Ask him why he had you as a son. Does it help the world?

Michael

@JasperLoy He wants me to be a doctor, but I like math more...

GBeau

@anon it was 2^something

anon

00:11

not after log_2ing ofc

GBeau

yeah

user174558

@Michael You should ask yourself what you want, not what he wants. Give him a piece of your mind.

user174558

It appears that Mike is ignoring me forever. So be it.

GBeau

@anon I wanted to do the arithmetic progression one...

anon

you can write the AP condition as $(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}) ( \begin{smallmatrix} -1 \\ 1 \end{smallmatrix}) = \delta( \begin{smallmatrix} 1 \\ 1 \end{smallmatrix})$ and some similar things but I ran out of time on that one

GBeau

00:14

yeah I ran out of time on that one as well

did you get the thing with 1+2+3=6 and asking if a number ending in 2015 was in the sequence?

Michael

IS there anywhere I can learn multi variable calculus from the very basics? I can't seem to find any advice from MSE, and all the online stuff I have found is too complicated.

anon

@GBeau after about 15min I thought the patter was easy and spent a long time starting a written out solution but then I realized the pattern was not as easy as I'd thought

so no

user174558

@Michael I recommend you read Kaplan and Lewis's Calculus and Linear Algebra 1 and 2. Good luck! You can download it for free if you do a google search from University of Michigan.

GBeau

@anon I felt like I should have gotten it

Michael

@JasperLoy I just read abit. Is it ok if I don't understand some of the notation?

anon

00:18

I was very disappointed when I realized I hadn't gotten it. I don't even know if the answer was yes or no.

GBeau

@anon somebody I took the exam with was saying 2015 on the dot worked, but I pointed out a flaw in his argument, but he was almost there

user174558

@Michael Notation will be defined in the book. Notation is just notation. You can call a cat a dog.

Michael

@JasperLoy Ok I will give it a read! thanks alot!

user174558

@Michael I will say this is the best calculus book in the world, after looking through over 9000 of them. The very best.

user174558

@Michael You can get very cheap copies on Amazon too.

GBeau

00:20

@anon may the graders be in your favor ;)

anon

as with you

Michael

@JasperLoy How far does it go material wise?

user174558

How do you pronounce Putnam?

anon

I pronounce it putnam, dunno about others

user174558

@Michael It covers single variable calculus, multivariable calculus, elementary linear algebra and ordinary differential equations. Very very very good.

Michael

00:21

@JasperLoy Should I read both volumes

user174558

@Michael Yes, then give your dad a lecture.

Michael

@JasperLoy Ok. I am off. Thank you so much!

user174558

@anon I emailed Rotman about when Part 2 will be out, no reply.

evinda

@anon Do you have an idea how we get to it?

anon

how we get what?

evinda

00:33

The 1-periodic extension of f, f(x-floor(x)) @anon

anon

what about it?

evinda

If $f(x)=e^{-\beta \left (x-\frac{1}{2}\right )^2}$, f(x-floor(x)) is its 1-periodic extension. But how can we justify it formally? How do we get it? @anon

anon

that's something you should do

I mean, I've already done it even

Anonymous

Achieve.

anon

36 mins ago, by anon

frac(x)=x for x in [0,1] and frac(x) is 1-periodic. have you ever seen the graph of y=frac(x)? look at it.

Ted Shifrin

00:46

@evinda Evinda, read your own post :) That's what you should be doing.

evinda

@anon @TedShifrin So do we just pick f(x-floor(x)) and show that it is 1-periodic?

misheekoh

How do you tag a user in the comments? I used to be able to do it but I can't now? it's just @_________ right?

anon

yes

evinda

@misheekoh Yes

anon

assuming your comment is under their question/answer/comment

evinda

00:48

@anon A ok...

Thanks @anon @TedShifrin

GBeau

01:05

@anon I think the median might be skewed high because I thought A1 was a lot easier than normal but one of my professors wasn't so sure (that was the hyperbola bounding the triangle problem)

user174558

02:01

Hi @ChantryCargill!

Chantry Cargill

@JasperLoy Hello there.

Mary Star

Hello!! Could someone of you take a look at my question:

3

Q: Proof that the solutions are algebraic functions

I am looking at the following: $$$$ $$$$ I haven't really understood the proof... Why do we consider the differential equation $y'=P(x)y$ ? Why does the sentence: "If $(3)_{\mathfrak{p}}$ has a solution in $\overline{K}_{\mathfrak{p}}(x)$, then $(3)_{\mathfrak{p}}$ h...

and explain to me how we conclude that $\beta_i \in \mathbb{Q}$ ?

Michael

02:33

what does 3 (mod 10) mean?

robjohn

@Michael that means any of the numbers $\{\dots,-17,-7,3,13,23,33,\dots\}$

@Michael It means that when you divide by $10$ you get a remainder of $3$

or the difference from $3$ is a multiple of $10$

Michael

@robjohn oh Ok. Thank you

What is the formal definition of a limit?

@robjohn What is the formal definition of a limit? Not the "the value of a point as it approaches the line" definition

robjohn

$$\lim\limits_{x\to a}f(x)=b\iff \forall\varepsilon\gt0, \exists\delta\gt0: \left|x-a\right|\le\delta \implies\left|f(x)-b\right|\le\varepsilon$$

Michael

@robjohn Can you explain what the symbols mean? Quick before jasper kills me

robjohn

$\forall$ means "for all" and $\exists$ means "there exists"

$:$ means "such that" and $\implies$ means "implies"

$\iff$ means "is equivalent to"

GBeau

02:47

your explanations for some of those are ironic when you haven't enabled chatjax yet

robjohn

@GBeau indeed. I am assuming that they have ChatJax running.

Michael

@robjohn what about the weird symbols?

ε and δ

robjohn

@Michael which weird symbols?

Michael

@robjohn δ and ε

robjohn

@Michael $\varepsilon$ and $\delta$ are epsilon and delta, Greek letters

@Michael just variables

Michael

02:50

@robjohn why not use l or m or n? Are these variables used in other areas of math too?

robjohn

delta and epsilon are classically used for epsilon-delta proofs of convergence.

Michael

@robjohn oh convergence. Ok.

robjohn

@Michael Greek letters are often used in math. There are some that are used in certain places very commonly, such as in the definition I gave above.

(ε, δ)-definition of limit

In calculus, the (ε, δ)-definition of limit ("epsilon-delta definition of limit") is a formalization of the notion of limit. It was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy never gave an () definition of limit in his Cours d'Analyse, but occasionally used arguments in proofs. The definitive modern statement was ultimately provided by Karl Weierstrass. == HistoryEdit == Isaac Newton was aware, in the context of the derivative concept, that the limit of the ratio of evanescent quantities was not itself a ratio, as when he wrote: Those ultimate ratios ... are not actually ratios...

user174558

Rob John is very kind to answer the random questions of Michael which I shan't answer from now, not that mine are any good.

user174558

I cannot understand how someone can jump from one idea to the next to the next to the next...

user174558

02:53

I regret answering some of your questions in the past now.

Michael

@JasperLoy I suffer from that issue in everyday life too.

user174558

@Michael Are you suffering from a psychological disorder?

Michael

@JasperLoy Severe curiosity disorder.

user174558

@Michael Rubbish.

user174558

I think I should ignore you from now, just like many people have ignored me already.

Michael

02:56

@JasperLoy Do as your heart seeks good sir.

Oh ok I get Euler's identity now

user174558

Nope, you do not get it yet, but you won't realise it yet.

robjohn

How does $e^{i x}$ produce rotation around the imaginary unit circle?

@Michael ^^ then plug in $x=\pi$

user174558

@Michael First you would need the analytic definition of e and the analytic definition of pi and then complex analysis to truly understand it. Understanding can be of many levels.

user174558

For example, to truly understand 1+1=2 we will need some axiomatic set theory and mathematical logic.

user174558

03:12

@Michael Good night, and good luck!

user174558

@robjohn I will see you in my dreams.

robjohn

@JasperLoy mathematical dreams, of course...

user174558

@robjohn Who knows what kind of dreams.

GBeau

why has nobody posted the putnam problems on the internet yet!

I need to waste my time working the ones I didn't get during exam time

dREaM

04:00

Finally done with my Cat homework

now I can study for my Calc exam

skull petrol

04:26

**Countdown has begun

Europe - The Final Countdown (Official Video)

►

Ben Lim

@JasperLoy hey.

Michael

"@Michael First you would need the analytic definition of e and the analytic definition of pi and then complex analysis to truly understand it. " What does he mean the analytic efinition of e?

skull petrol

Have you heard of Wikipedia?

E (mathematical constant)

The number e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828, and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series The constant can be defined in many ways. For example, e can be defined as the unique positive number a such that the graph of the function y = ax has unit slope at x = 0. The function f(x) = ex is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

Michael

04:46

I was on the website

I couldn't find the analytic part

user174558

07:25

The stupid thunder woke me up, so loud.

user174558

@BenLim Hello, good luck.

SAWblade

1

Q: String of 0's and 1's in combinatorics

Here is a question in one of my combinatorics homework. We want to know the number of string of $n$ ones and zeros in which no zeros are next to one another and out of those $n$ digits we have $m$ zeros. We need to find the formula that gives us this number of strings but I have no idea how to d...

combinatorics

I think I solved this. :0

Sab

07:43

hello

This is a programming question codechef.com/DEC15/problems/PLANEDIV Q :find out the size of the largest perfect subset of this set. a set of lines to be perfect if there does not exist a point that belongs to two or more distinct lines of the set. In the example there where a)3x+4y+5 =0, b)30x+40y = 0, c)30x+40y+50 = 0,

3*x + 4*y + 5 = 0 and 30*x + 40*y + 0 = 0 is considered as perfect set why no 30*x + 40*y + 50 = 0 included in that list?

Ben Lim

09:20

@JasperLoy hello.

@JasperLoy what you up to?

Frank Science

10:11

@TedShifrin Thanks for this useful piece of info. I heard that when Siu taught undergrad real analysis, Nash-Moser iteration was on the syllabus. What I'm interested in is the (both for undergrads and for grads) programs in the top universities (like these of Ivy Leagues, UCB, UCLA, Chicago, etc), and that how much I am weaker than students from these programs, and how they study. U.S. is a big country (just like China) and criteria are diverted, of course. General standard means nothing.

user174558

@BenLim Nothing much, still trying to get better, still not working.

user174558

@FrankScience The undergrad programs in the US are nothing compared to those in the UK.

Frank Science

@TedShifrin And I should admit that what I suggested is quite biased to a general undergrad student. I suggested it because I thought that if I had been taught with these books when I was learning first year analysis, it would have been a great profit.

@JasperLoy What do you mean by that?

user174558

@FrankScience The avergage UK undergrad learns much more math than the avergae US undergrad.

Frank Science

@JasperLoy What about these top students? Comparing Ivy Leagues with Oxford, Cambridge, etc?

user174558

10:23

@FrankScience Same thing.

Frank Science

@JasperLoy OK. In France, students don't learn a lot in the first two years of university (more precisely, in prépa).

user174558

Of course, I am only a banana. Every time I comment on something in this chat, people will make tons of noise.

user174558

@FrankScience Don't look at those lists of university rankings. They are mostly rubbish.

Huy

Those university rankings aren't based on how good education is, but how "good" research is.

or let's say the latter is weighted a lot more than the former.

user174558

There is just no sensible way to rank these things. Therefore, all ranking is crap.

Huy

10:36

no, people just need to realize how these rankings are made and therefore what they could mean. but many people interpret them wrongly.

Frank Science

11:29

@JasperLoy I'm not reading lists of rankings. However, I want to know how best students in the world study and what they study, so students from universities with good rankings are samples of best students right? I mean, even if it's not accurate, I cannot find anything more accurate?

Paradox 101

11:45

@Huy how do we show the equivalence of matrices?

*metrics

I'm mostly just an idiot

12:10

@Paradox101 Show that there are two positive numbers $C,D$ where $C d_1(x,y)\leq d_2(x,y) \leq D d_1(x,y)$

Two norms are equivalent if (there are $C,D\gt 0$ s.t.) $C\|x\|_1 \leq \|x\|_2 \leq D \|x\|_1$

evinda

Hello!!!

I'm mostly just an idiot

Hi!

evinda

I want to find the solution of the following problem: $ \left\{\begin{matrix}
u_t+2tx^2u_x=0, & x \in [0,2], & t \in [0,1] \\
u(0,x)=u_0(x)=e^{-\beta \left(x-\frac{b-a}{2} \right )^2} &, x \in [0,2] & \\
u(t,0)=0 &, t \in [0,1] &
\end{matrix}\right.$.

I found that the solution is this: $u(t,x)=e^{-\beta \left(\frac{x}{1+x t^2}-1 \right)^2}$.

I'm mostly just an idiot

Looks like some sort of boundary problem, so count me out :D.

evinda

Ok @I'mmostlyjustanidiot

But this doesn't satisfy u(t,0)=0 .
The initial condition value neither does.

So is there no solution?

user174558

12:21

@FrankScience Yes, I agree with you.

I'm mostly just an idiot

Hello @JasperLoy, you are even lighter in my absence!

user174558

@I'mmostlyjustanidiot Hello. Do you have a name?

Paradox 101

@I'mmostlyjustanidiot so if we need to prove whether the manhattan and standard metrics are equivalent we have to sandwich one between a multiple of the other?

I'm mostly just an idiot

@JasperLoy I have three :).

user174558

@I'mmostlyjustanidiot Can you tell me your first name?

I'm mostly just an idiot

12:24

@Paradox101 Yep, both positive values.

@JasperLoy It starts with T, if you guess it :D.

user174558

@I'mmostlyjustanidiot Ted?

I'm mostly just an idiot

I'll give you 5 guesses in one message.

evinda

Hi @robjohn
Do you maybe have an idea?

I'm mostly just an idiot

@JasperLoy It's not Ted, you have 4 more guesses

user174558

@I'mmostlyjustanidiot I don't like to guess. No more guessing.

I'm mostly just an idiot

12:27

Orl Korrect

Krijn

Tim, Tom, Toinne?

user174558

Toidi.

I'm mostly just an idiot

Nope, one guess left :D

Are those last two real guesses?

user174558

Timothy.

I'm mostly just an idiot

Nope(I would have given that to Tim)

Krijn

12:35

Toinne is a common name over here and in France as well

I'm mostly just an idiot

No more guesses, you'll get it by exhaustion :D.

Krijn

Does it start with Ta?

I'm mostly just an idiot

No, but no more information :P.

Huy

telephone?

I'm mostly just an idiot

@Huy :P

Huy

12:38

that's not a no

user174558

Thomas?

Frank Science

I can't recall who said no more guessing a moment ago.

user174558

Me.

Paradox 101

If a set $|X|< \infty$ then will all the metrics on $X$ be convergent to zero?

Huy

what should "a metric is convergent to zero" mean?

Paradox 101

12:48

@Huy i mean $d(x_n,x) \rightarrow 0$

Huy

what is $x_n$, what is $x$?

user174558

LOL

I'm mostly just an idiot

@Paradox101 Do you mean if a sequence converges under one metric, does it converge under any equivalent metric?

Paradox 101

@Huy a sequence

Huy

$x$ is a sequence?

Paradox 101

12:51

@I'mmostlyjustanidiot yes

I'm mostly just an idiot

Look at the definition of convergence under a metric, and look at the definition of equivalent metric

Paradox 101

@Huy no it's supposed to be $x_0$ instead of $x$

Huy

and what is $x_0$?

I'm mostly just an idiot

@Paradox101 He's nitpicking because you're being unrigorous

Huy

it's because he just started with it and if you don't nitpick he won't learn how to express himself in a way that (most) people will understand what he means

I'm mostly just an idiot

12:54

@Huy I know, it's a good idea, I probably shouldn't have tried to translate :D

Paradox 101

@Huy it's the point that the sequence converges to

I'm mostly just an idiot

@Paradox101 I.e. you can write what that means with 3 'symbols'(where $x_0$ is a symbol I mean)

Paradox 101

@I'mmostlyjustanidiot yes I'm still not really good at this

Huy

@Paradox101: Well if a sequence converges to a point it's kind of redundant saying that the metric goes to zero.

I'm mostly just an idiot

@Paradox101 ($\{x_n\}_{n=1}^\infty$) $x_n\to x_0$ I mean

Paradox 101

12:58

@Huy yes I suppose it is. I need to show whether all metrics on $X$ are equivalent if $|X|< \infty$. Also I'm a girl :p

Huy

@Paradox101: do you mean $\dim X < \infty$ and $X$ is a vector space? hi girl, sorry. :P

oh, nevermind, I was thinking norms.

Paradox 101

@Huy I'm not sure as the question doesn't state anything except what I wrote above

Huy

yea, I mixed something up

I'm mostly just an idiot

I know that all norms on a finite dimensional normed space are equivalent. I think all metrics on a finite set are the discrete metric, but not super sure

Paradox 101

And I'm not sure what to do with the information on X

Huy

13:01

@I'mmostlyjustanidiot are you sure this is sufficient (or necessary) for equivalence of metrics?

I'm mostly just an idiot

Actually my discrete metric thing is false since we could take $kd(x,y)$(where $d$ there is the discrete metric)

Huy

@Paradox101: did you somewhere define what it means for two discrete metrics to be equivalent in your lecture?

Frank Science

@Huy It's a theorem that finite dimensional TVS is isomorphic to $K^n$ for some $n$.

Huy

TVS?

topological vector space?

Frank Science

Yes

I'm mostly just an idiot

13:02

If its a normed space, its an infinite set

Huy

I know

I'm mostly just an idiot

I don't know anything about topological vector spaces I guess, or not by that name.

Huy

@I'mmostlyjustanidiot: but they're just defining a metric on a set, not necessarily a vector space.

Paradox 101

@Huy not for discrete metrics, no. Just the standard equivalence

Huy

@Paradox101: can you quote your specific definition?

it's usually best to work with what you were given in your specific course rather than use different definitions on the internet

I recall two metrics being called equivalent if they induce the same topology

but I don't know if that's what you're doing

Paradox 101

13:04

@Huy the question states: If $|X|< \infty$, show that all metric on $X$ are equivalent.

Huy

@Paradox101: do you have a definition of "two metrics are equivalent" in your lecture notes somewhere?

Frank Science

Since he mentioned all norms on a finite dimensional v.s. are equivalent, I just stated a more general statement.

Paradox 101

@Huy yes it simply states that two metrics $d_1$ and $d_2$ are equivalent s.t $d_1(x_n,x_0)\rightarrow 0$ iff $d_2(x_n,x_0)\rightarrow 0$

Huy

@Paradox101: I think you could try to see (and then prove) why any metric on a finite set will be a multiple of the discrete metric.

if I'm not mistaken, that should be true

Paradox 101

@Huy but here when they say all metric don't they mean any metric? So shouldn't it also include other metrics which aren't a discrete one?

Frank Science

13:13

What's a discrete metric?

Huy

@Paradox101: but it will turn out that all metrics on a finite set are a multiple of the discrete one.

and that's the thing you should try to prove

@FrankScience: what's the most obvious way to define a metric on a finite set?

Frank Science

Wiki told me that it's $d(x,y)=1$ if $x\neq y$.

Apparently there is a metric on a finite set which isn't a multiply of a discrete one.

Huy

really?

Frank Science

Consider just three points on $\mathbb R$.

Its topology is discrete, but it's not a discrete metric.

Huy

oh

yeah, you can choose different numbers for distinct points

Frank Science

13:18

Any $T_1$ topology on a finite set should be discrete, however.

evinda

Hi @anon
Are you familiar with initial and boundary value problems?

Paradox 101

@Huy from this question $X$ is a finite normed vector space right?

Huy

@Paradox101: I think in your question $X$ is just a finite set.

Paradox 101

@Huy ok but why is it absolute?

Huy

absolute?

$|X|$ just denotes the cardinality of the set, and $|X| < \infty$ means $X$ contains finitely many elements.

Paradox 101

13:21

Not absolute exactly i mean it's $|X|$ instead of just $X$

Ohhhh ok

Frank Science

@evinda What you asked is just a question of first-order PDE. It could be solved systematically by, say, characteristic lines.

evinda

https://i.sstatic.net/Qp9tY.jpg
I found that the solution is the following:
https://i.sstatic.net/IrjTX.jpg
But this doesn't satisfy u(t,0)=0, but neither u_0(x) does.
So doesn't the problem have a solution?
Or do we use u(t,0)=0 only for the approximation of the solution if we want to apply the upwind-method? @FrankScience

robjohn

13:40

@evinda You get that $u$ is constant on the characteristic curve $t^2+\frac1x=\frac1{x_0}$, right?

@FrankScience indeed.

$u(t,x)=u\left(0,\frac1{t^2+\frac1x}\right)=u_0\left(\frac{x}{xt^2+1}\right)$

@evinda However, you have a problem since your initial conditions don't match at $(0,0)$

$u(t,x)=u\left(\sqrt{t^2+\frac1x},0\right)=0$

So the initial conditions are incompatible.

evinda

13:56

What initial conditions could we pick so that they are compatible? @robjohn

robjohn

@evinda get rid of the initial condition $u(t,0)=0$

evinda

I want to apply the upwind-method at this problem. Isn't then an initial condition necessary? @robjohn

robjohn

@evinda If you know the initial condition for $u(0,x)$, that and the differential equation force what $u(t,0)$ has to be.

Oops... sorry, for $(t,0)$, $t^2+\frac1x=\infty$...

evinda

@robjohn Isn't this: https://i.sstatic.net/IrjTX.jpg the right exact solution?
So does it hold that $u(t,0)=e^{-\beta}$ ?

r9m

@robjohn do you have a simple proof (not using Poisson Distribution) for this in mind? :-)

robjohn

14:05

@evinda if $b-a=1$, that is the solution.

@r9m let me look

evinda

@robjohn Yes we have a=0, b=1.

Semiclassical

14:34

Is this the same question you had before? I was curious as to how you resolved the matter @evinda

evinda

@Semiclassical No, it is an other problem. At the previous one, I took the 1-periodic extension of the initial condition, so that the solution is also 1-periodic.

Semiclassical

Ah, okay

Semiclassical

14:48

@evinda Though, what did the solution actually look like in that case? If memory serves, doing that on your original solution would give a function which is 1-periodic but not continuous at $x=0,1$

r9m

The Hell was that?!! There's a bounty from Cleo in this question for an exemplary answer! =P

evinda

@Semiclassical $e^{-100 \left((x-2t)-\lfloor x-2t \rfloor-\frac{b-a}{2} \right)}$

@Semiclassical So can we now pick this solution?

Huy

15:04

@Semiclassical: If we have a smooth homomorphism $G \to H$ of some Lie groups and take the derivative at the identity, with the exponential map we get a commuting diagram. In general, is the derivative a Lie algebra homomorphism? I know it's true for the adjoint.

nevermind, yes it is.

Mary Star

16:05

When we have the following:

$L_1(y)=0 \land L_2(y) \neq 0$

Is this equivalent with $L_1(y)+L_2(y) \neq 0$ ?

Or doesn't this stand?

Huy

no

only implies

Mary Star

So it is $L_1(y)=0 \land L_2(y) \neq 0 \Rightarrow L_1(y)+L_2(y) \neq 0$, right? @Huy

Huy

yes

Mary Star

16:21

Ok... Thanks!! :-) @Huy

@Huy But it stands that $L_1(y)=0 \land L_2(y) \neq 0 \Leftrightarrow L_1(y)=0 \land dL_1(y)+L_2(y) \neq 0$, doesn't it?

Mike Miller

16:36

morning

Pedro Tamaroff

noon

Huy

yes @MaryStar, without the $d$

Mary Star

16:55

So it stands only when it is $L_1$, and not with a mulitple of $L_1$ ? @Huy

Huy

@MaryStar: If $L_1= 0$ then any multiple of it is as well.

I rather thought your $dL_1(y)$ was a typo considering you never wrote that $d$ before.

Mary Star

Ok!!

Moses

Hi @DanielFischer

Alec Teal

Other than looking nightmarish, is there anything wrong with:
$$\forall(x_n)_{n=1}^\infty\subseteq X\left[\left(\exists x\in X\ \forall\epsilon>0[\vert B_\epsilon(x)\cap(x_n)_{n=1}^\infty\vert=\aleph_0]\right)\implies\left(\exists(k_n)_{n=1}^\infty\subseteq\mathbb{N}\left[(\forall n\in\mathbb{N}[k_n<k_{n+1}])\implies\left(\exists x'\in X\left[\lim_{n\rightarrow\infty}(x_{k_n})=x'\right]\right)\right]\right)\right]$$

Ted Shifrin

17:31

mr @Pedro: You been at a conference? ... goodnight, @MikeM

user685252

18:16

Huy

22 hours ago, by Huy

user685252

22 hours ago, by L33ter

I spit on your theorem

haha

:-)

Mateon1

Hello. Is it true that the composition of a monotonic function with another monotonic function is also monotonic?
While doing my homework (this isn't my homework) I thought that might be true and simplify my calculations. Intuitively it seems wrong to me, not sure why, but I can't find any counterexample.

I don't think this question is worthy of asking on the main site, so I ask here in chat.

Alec Teal

Yes @Mateon1

Think of monotonicity as $\forall x,y[x\le y\implies f(x)\le f(y)$

Mateon1

@AlecTeal Thanks. I do wonder why it seems to be wrong to me. Probably confusing composition with something else.

Alec Teal

18:27

If $g$ is also monotonic (in the same way) we see that $u\le v\implies g(u)\le g(v)$ use $u:=f(x)$ $v:=f(y)$ to complete the theorem.

Mateon1

Ah, thanks! I don't know why it seemed wrong to me when it's that obvious.

Alec Teal

@Mateon1 think of a monotonic function. As you go forward it only goes up. (this is very "visualising"-y not formal-y) If you feed this into another function .... good

Mateon1

I felt that the composition of a rising and a 'falling' function could not work for some reason. Probably confused with multiplication.

Sorry for the typos, my wireless keyboard is failing and it's not the batteries.

Owatch

Hello..

Alec Teal

rising into a falling should be a falling and a falling into a rising should be falling too (because at most it goes backwards)

falling into a falling goes down backwards - which is up.

@Mateon1 isn't that a bit like multiplication, -1x1=1x-1=-1 and -1x-1=1x1=1?

Mateon1

18:35

@AlecTeal Yeah, probably why I thought there's some counterexample.

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