Mathematics - 2015-02-08 (page 1 of 3)

Alex Wertheim

00:30

Hello friends.

Arkamis

00:51

I hate Folland. I hate this goddamn textbook.

JMoravitz

I found it a bit dry, but the organization was well thought out

granted, I've only made it through the first several chapters

Arkamis

I just can't put together how he's getting from one place to another.

I've spent 5 goddamn hours on one part of one problem and I feel like an idiot because the chapter is only 2.5 pages long.

And I hate how he swaps nomenclature back and forth.

Sometimes things are $E$s, sometimes they are $A$s, and that changes between definitions and problem statements.

@JMoravitz If you don't mind, what am I missing here: math.stackexchange.com/questions/1138490/…

JMoravitz

As I remember, (I don't have my copy of Folland near me at the moment), $\mu^*(E) = \inf\{\sum\limits_{k=1}^\infty \mu_0(I_k)\}$ where each $I_k\in\mathcal{A}$ and $E\subset \bigcup I_k$.

the $I$'s are the elementary sets, like intervals in the borel measure, the ones where the measure is clearly defined.

Arkamis

01:07

Right

JMoravitz

So, by definition of infimum, you can find some $A=\bigcup I_k$ such that $\sum\mu_0 (I_k) < \mu^*(E)+\epsilon$

noting also that $\mu_0(I_k)=\mu^*(I_k)$, and from subadditivity, you have

$\mu^*(A)\leq \sum \mu^*(I_k) = \sum \mu_0(I_k) < \mu^*(E)+\epsilon$

Arkamis

Wait

hang on a second

So I had all of that

but I was ignoring the definition of an infimum

bangs head against wall several times

Thanks

JMoravitz

mhmm

If I remember right, the next part of the problem should be something like If $\mu^*(E)<\infty$ then $E$ is $\mu^*$-measurable iff there exists $B\in\mathcal{A}_{\sigma\delta}$ with $E\subset B$ and $\mu^*(B\setminus E)=0$.

Arkamis

yeah

I moved onto that one.

The forward implication seems straightforward

Caratheodory's theorem guarantees the existence of a $\sigma$-algebra of $\mu^*$-measurable sets

Of which $E$ is a part

JMoravitz

Currently, I'm trying to figure out how to adequately word a proof for combinatorics. Its a really fun problem, and the solution seems intuitive enough, but I have to come up with some terminology to describe the scenario (since we are of course forbidden from searching for solutions).

So, if you know anything of graph theory, you would have heard of $\chi(G)$, the chromatic number of $G$. We are interested in $\chi_g(G)$, the "game chromatic number", which is defined in the following way:

Arkamis

01:20

I'm struggling to show that $E$ has a superset in $A_{\sigma\delta}$ But I think I'm close.

Oh man, I haven't seen that term in a while.

JMoravitz

Given a graph of currently uncolored vertices, and $r$ colors, two players, player A and player B, take turns making a legal move by coloring a currently uncolored vertex using a color that no neighbor has.

Player A wins if they succesfully color all vertices, player B wins if he can prevent that from happening (i.e. has succesfully surrounded a vertex with all available colors, leaving no legal moves for coloring that vertex).

Given optimal play by both players, $\chi_g(G)$ is the least $r$ such that player A can win.

For clarification as well, I believe that its specified that player A must go first, but I remember our professor talking about related/similar definitions where no distinction is made and you go with the max of the numbers where player A or player B goes first

Arkamis

Right

JMoravitz

Our problem is to prove that for any tree $T$, that $\chi_g(T)\leq 4$ and furthermore that there exists a tree, say $\widetilde{T}$ such that $\chi_g(\widetilde{T})=4$

From graph theory, there is a result showing that given three vertices on a tree, say $a,b,c$, there is a unique vertex where the three paths $\overline{ab}, \overline{ac}, \overline{bc}$ meet. The intuition is that if there are three colored vertices with the paths between any two of them uncolored, that player A's move will be to color that "branching point".

thus preventing any uncolored vertex from being surrounded by four colors,

@Arkamis As for $E$ having a superset in $\mathcal{A}_{\sigma\delta}$, note that $\mathcal{A}_{\sigma}\subset \mathcal{A}_{\sigma\delta}$

You can use the first part of the problem and use the $A$ that you found there

Arkamis

Ah, excellent

JMoravitz

Even more specifically, you can use an $A_\epsilon$ for decreasing values of $\epsilon$

Arkamis

01:34

Wait so we don't need Caratheodory for that then?

JMoravitz

I don't remember having used it to solve that problem, though I may have used it implicitly,

JMoravitz

02:58

Suddenly everyone logs into chat. Good evening all.

Fargle

Evening.

Committing to a challenge

Hello @JMOra @Kaj

Kaj Hansen

Hey there

Sayan Chattopadhyay

Hi guys......morning

Axoren

"On the grassy Island of Huh, there is a magical forest. In that forest is a gate guarded by a Knight and a Knave, though they are not distinguishable from each other. Behind this gate is a fork with two paths. One path leads to immediate peril from which you cannot return. The other path leads to the island's greatest treasure from which you can easily return and have no reason not to. The paths are indistinguishable from each other other than that one is left and one is right."

Arkamis

03:11

I am still struggling on this problem :(

I've wasted all afternoon and not even made it 2/3 through the question.

JMoravitz

@Axoren I assume you are allowed to ask questions to the knight and knave? only one question? and what are the rules for how the knight and knave respond (always true / always false / some mixture of the two)

Axoren

Message was too long

Please wait

Breaking it up

JMoravitz

@Arkamis still the same one that we had started discussing earlier?

Arkamis

Yeah. I can't figure out the reverse implication.

There exists some $B \in A_{\sigma\delta}$ and $E\subset B$ such that $\mu^*(B \cap E^C) = 0$ implies that $E$ is $\mu^*$-measurable.

Axoren

"The guards together will answer no more than one question to any party that approaches the gate and will force anyone who has heard an answer to make a choice of path and take it. The knight will answer truthfully, the knave will lie. Though thousands of travelers reach this gate, none have reached the treasure and none have returned."

Arkamis

03:14

The question is posed here: math.stackexchange.com/questions/939427/…

But I have doubts re: the answer.

JMoravitz

So, if there exists $B\in\mathcal{A}_{\sigma\delta}$ with $E\subset B$ and $\mu^*(B\setminus E)=0$ then $E$ is $\mu^*$-measurable

Arkamis

Yup

JMoravitz

So, suppose such a $B$ exists. Let $B=\bigcap A_n$ for some collection $(A_n)_n$

Axoren

"A single traveler reaches the gate and asks the guards 'Are you the Knight and the Knave of the gate?' To which one of the guards says 'No'. The traveler then proceeds to make the correct choice and follows the path to the treasure. The traveler's choice was not random. How did he know?"

The riddle's done. How do you guys think?

JMoravitz

Construct a related sequence of sets, $(\widetilde{A}_n)_n$ with the property that $\widetilde{A}_n\supset \widetilde{A}_{n+1}$

You see how we can construct such a collection of $\widetilde{A}_n$ first off @Arkamis?

Folland does this a lot,

(I should mention also $B=\bigcap A_n = \bigcap \widetilde{A}_n$)

Axoren

03:20

@JMoravitz Do you want to take a crack at the riddle?

JMoravitz

@Axoren I don't personally see how there is enough information. Suppose that he did make the correct choice. Reverse time and put him in the same situation, but change which path lead to which.

If it was said that the knight always guards the path leading to the treasure, then perhaps

Arkamis

Hm

Axoren

This one requires innovative thought.

Arkamis

I don't understand your notation, @JMoravitz

Axoren

Rather than the same counter-factual tricks.

Tell me when you want the answer.

JMoravitz

03:25

@Axoren I understand that depending on your interpretation of the word "and", you can determine which of the guards is the knight and which of the guards is the knave, however there was nothing in the question that narrows down which path to take.

Axoren

Well, this is a draft of a riddle concept I thought of earlier today, so it's really rough.

Rather than regarding the answers the Knights and Knaves give you, you consider the fact that very many travelers have failed.

What happens in grassy wooded areas when a lot of people take the same path is that the grass dies and a a well-trodden path becomes visible.

The traveler chose the path with the greener fuller grass.

Because it looked like no one had taken that path compared to the other one.

Arkamis

@JMoravitz What do you mean by $(A_n)_n$?

JMoravitz

@Arkamis I mean to say that $A_n\in\mathcal{A}_{\sigma}\subset\mathcal{A}_{\sigma\delta}$ for each $n$. $(A_n)_n$ is how I refer to the entire collection at once instead of each individually. I name a collection of sets $(\widetilde{A}_n)_n$ just to avoid using more letters and since these are very closely related to the original collection.

Arkamis

ok gotcha

So I exploit the definition of $B \in A_{\sigma\delta}$.

JMoravitz

You can define $\widetilde{A}_1=A_1$ and $\widetilde{A}_n=A_n\setminus \widetilde{A}_{n-1}$ for $n\geq 2$

Arkamis

03:31

Right

JMoravitz

As such, $\bigcap\limits_{n=1}^\infty \widetilde{A}_n = \lim\limits_{n\to\infty}\widetilde{A}_n$

Arkamis

Alright, let me think on where to go from here.

JMoravitz

Remembering again that $B=\bigcap A_n = \bigcap \widetilde{A}_n$

Arkamis

So, we'll have some finite $\widetilde{A}_n$ such that their intersection contains $E$?

JMoravitz

@Axoren in the initial paragraph you stated "The paths are indistinguishable from each other other than that one is left and one is right."

Implying that you could not tell that one has a more welltrod path

Axoren

03:35

@JMoravitz That was a mistake then. My apologies. This is quite literally the first time I've written this down.

That line was more for "You can't see dead bodies/treasure" if you stare down the path.

JMoravitz

@Arkamis you'll use the statement that $E\subset B$ and $\mu^*(B\setminus E)=0$. Try to show then that $B$ is $\mu^*$ measurable and let subadditivity between $E$ and $B\setminus E$ take care of the rest.

Arkamis

@JMoravitz Alright, thanks.

JMoravitz

03:59

>_< As I keep typing and typing... this problem is getting longer and longer (the gamechromatic# of trees problem I mentioned earlier). The intuitive answer was quick and easy to explain I thought, but I'm three or four lemmas in at this point, having to explain each part.

Arkamis

Ugh, I don't get this at all

I simply don't see it.

JMoravitz

$\mu^*(E) + \mu^*(B\setminus E) = \mu^*(B)\Leftrightarrow \mu^*(B)-\mu^*(B\setminus E)=\mu^*(E)$. Implying that if $\mu^*(B)$ is defined and $\mu^*(B\setminus E)$ is defined, then so is $\mu^*(E)$ and $E$ is therefore $\mu^*$-measurable.

and $\mu^*(B)=\lim\limits_{n\to\infty}\mu_0(\widetilde{A}_n)$

Arkamis

Ok, that I get. I just don't get the statement "$E$ is therefore $\mu^*$-measurable."

Simply by being defined, that is.

JMoravitz

$E$ is $\mu^*$-measurable $\Leftrightarrow \mu^*(E)$ is well defined.

Arkamis

I'll have to search for that.

Ok, back to the grind.

Victor

04:57

Can someone reopen this: math.stackexchange.com/questions/1138650/…

Arkamis

@JMoravitz THanks, I think I got it. I actually got it a few different ways.

The Substitute

Under a Mobius transformation of the plane, why must an endpoint of an arc get mapped to an endpoint of a ray?

Such reasoning is used in the following: Under the map $z\mapsto (z-i)/(z+i)$ the segment $\text{Re}z = 0$ inside the unit disc gets mapped to a line containing $0,\infty$, but how do I know this ray starts at $0$? I know that under my map $i$ gets mapped to $0$ and $i$ is an endpoint of my original arc, but is there a reason for this in general?

anon

05:23

@TheSubstitute Re(z)=0 is not just a segment or an arc or a ray, it's the whole imaginary axis, which gets mapped to the whole real line excluding the point 1 under that map

oh, Re(z)=0 inside the unit circle

well, write z=ix for x between -1 and 1 and see what happens to the outputs

Axoren

05:50

@Lembik This is a really interesting problem, god damn.

Huy

06:06

@TedShifrin: Then you probably don't know about them. :(

Kaj Hansen

Gotten to the gym much lately @Committingtoachallenge ? I got some chest in today, but came too close to closing to get in my full routine :(

reversiblean

Are vectors in matrices and here mathsisfun.com/algebra/vectors.html the same?

Huy

@reversiblean: No, vectors can be thought of as matrices but not all matrices can be thought of as vectors.

Kaj Hansen

@reversiblean, sometimes it's useful to think of vectors as $n \times 1$ matrices.

But matrices, in general, are not vectors as Huy points out.

Axoren

@KajHansen Other times, it's useful to think of them as $1 \times n$ matrices.

:P

Kaj Hansen

06:11

Sure @Axoren. For the most part I actually write my vectors as columns instead of rows.

reversiblean

So how do I distinguish between the two? Both are called vectors

Axoren

@KajHansen I was mainly just being coy and making a joke.
@reversiblean When you talk about a vector, you need to have a concept of the Vector Space it belongs to.

reversiblean

"A vector has magnitude (how long it is) and direction:" .. How can this be relevant to vector in matrices? I don't understand..

anon

huh?

Axoren

@anon @reversiblean is using a very weak definition of a vector commonly presented in early physics.

So there's no background behind it belonging to a vector space.

anon

06:14

@reversiblean I'm sure there are moments where it's relevant, but why do you need an example of where it's relevant so badly?

@reversiblean Vectors are just elements of vector spaces. Matrices (whose entries are scalars from some field) are always vectors, since the set of all matrices is a vector space. Row and column vectors are vectors expressed in a certain form, and both are special cases of matrices.

Axoren

Given a vector $v$ from a vector space, calculating it's magnitude is normally just $||v|| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}$

And we can get a direction from it, too.

anon

Vectors that are elements of finite-dimensional real vector spaces have a geometric interpretation of having magnitude and direction, but to do that the vector space needs to already be equipped with a norm, or equivalently, an inner product.

Axoren

So we can always go back and forth from the direction and magnitude to a column vector.

anon

@Axoren from a coordinate vector space, to be technical

Axoren

@anon These things are still defined in a vector space, they just aren't as meaningful.

Like, the magnitude and direction of a matrix. Does that really mean anything?

For coordinate vector spaces, they are more meaningful.

anon

06:19

@Axoren Okay, the set of functions $[0,1]\to\Bbb R$ is a vector space. Let $f$ be such a function, say $x\mapsto x$. What is $f_1$? You say it is defined. In general, vectors don't have coordinates. Coordinate vectors have coordinates, and by choosing a basis we can represent all vectors as a coordinate vectors, but the coordinate vector will depend on choice of basis.

Kaj Hansen

I think context should be taken into account though @anon. @reversiblean's link was really just talking about vectors as they're used in freshman physics to represent velocity, etc. in R^2 or R^3.

anon

yes

Axoren

@anon We wouldn't be able to describe the norm in terms of coordinates, but it would still have a norm. Which is the inner product of the vector with itself.

anon

@Axoren "the" inner product? vector spaces don't come equipped with inner products, and there is more than one you can choose to equip one with in general. one could instead speak of inner product spaces.

reversiblean

yup, i have no idea about most of the things, you guys are talking about : )

Kaj Hansen

06:24

What are you studying @reversiblean ?

anon

@reversiblean it can help us visualize what a matrix does. in say R^2, pick (1,0) and (0,1). the first and second column tell you where a linear transformation sends those two vectors. if you put down the first column vector down on paper, and then the second, and then fill in the parallelogram, that tells you where and how the linear transformation maps the fundamental unit square

the magnitude+direction thing was not designed to help us understand matrices, though, so I am not sure why you're relating them together necessarily

reversiblean

IT. they taught matrices first. basics I guess. Now they are going to teach just Vectors. what i was wondering is the if they are the same thing or not

anon

I will try to summarize

Kaj Hansen

For what you are studying @reversiblean, a perhaps oversimplified way of thinking about things is tha tvectors are a direction and a magnitude, and matrices are things that can act on vectors to change their directions and magnitudes.

Axoren

@anon You're right, there can be many inner products for an vector space and not all vector spaces have an inner product.

Kaj Hansen

06:30

You should also keep in mind that "vector" means something more abstract than "a magnitude and a direction" in higher math, but ultimately that's not relevant to what you need to know right now. @reversiblean

anon

in physics, vectors represent magnitude and direction. just look at vector arithmetic - it appends one vector onto another to get a third vector, and scalar multiplication scales vectors. given a canonical choice of direction vectors like (1,0) and (0,1), one can decompose any vector into a linear combination of these basis vectors.

a linear map is a thing that acts on vectors, a function which sends vectors to vectors in a way which is linear. a matrix is an array of scalars that tells us precisely what combinations of basis vectors are obtained when one applies a linear map to basis vectors.

Kaj Hansen

I could be wrong though. I don't know the scope of the classes you're taking.

anon

in pure math, a vector space is any set of things that can be added together and multiplied by scalars, and a vector is any element of a vector space. since matrices can be added and multiplied by scalars, this also makes matrices out to be vectors, at least algebraically speaking.

Axoren

In the case of $[0,1]\to \mathbb R$, wouldn't an obvious inner product be $\langle f, g\rangle = \int_0^1 f(x)g(x)dx$?

Robert Cardona

I'm working on the following problem: $\mathcal B$ is Borel $\sigma$-algebra on $\mathbb R$, $m, n$ two measures on $(\mathbb R, \mathcal B) : m\big((a, b)\big) = n\big((a, b)\big)$ where $-\infty < a < b < \infty$. I want to show $m(A) = n(A)$ for all $A \in \mathcal B$.

anon

06:34

@Axoren yes

Axoren

@anon Are there any others? That it could be, though?

anon

you don't know if a vector space can have more than one inner product?

Robert Cardona

I have a specific version of the Monotone Class Theorem at hand: $\mathcal A_0$ is an algebra, $\mathcal A$ is the smallest $\sigma$-algebra containing $\mathcal A_0$, and $\mathcal M$ is the smallest monotone class containing $\mathcal A_0$, then $\mathcal M = \mathcal A$.

Axoren

@anon I can't think of any other one that's drastically different.

Obviously if there's some sort of scalar in there somewhere, it's suddenly a new inner product.

Robert Cardona

I have been unable to apply this version of the monotone class theorem to show what I want to show.

anon

06:37

@Axoren I assume by not drastically different you mean $\int_0^1 f(x)g(x)w(x)dx$ are not drastically different? or what about $\int_0^1f(x)g(1-x)dx$? or $\int_0^1 f(x)g(a\pm x)w(x)dx$ where we've extended $f$ and $g$ to $\Bbb R$ periodic mod $1$?

Axoren

@anon I hadn't even thought about products like those.

anon

and I'd bet there are inner products that can't be written in those forms too

Axoren

@anon Now that I think about it, you're right. We can consider the functions $f = \{(x_1,, y_1)\mapto z_1, (x_2, y_2) \mapto z_2, ...\}$

And then we can just find a function like that that acts as an inner product.

anon

also, we are technically only defining these inner product for integrable functions. if we restrict further to sufficiently differentiable functions, we can introduce derivatives as well, e.g. $\int f^{(n)}\cdot g^{(m)}$ (but then we'd need to worry about non-degeneracy of the bilinear form), or involve other linear operators on functions

Axoren

I ran out of time to find the proper latex notation for the map arrow

I was just missing an s

$\mapsto$

reversiblean

06:44

Here's our curriculum

Linear Algebra: Definition of a matrix, Column and row vectors, scalar product, Matrix addition and multiplication, Determinant of a matrix, Row operations for solving linear equations, Rank, Linear dependence, Inverse matrix, Eigenvalues and Eigenvectors

This is what they gonna teach this semester.

Vectors: Scalars Vs. vectors, Directed line segments, Direction cosines, Adding, subtracting and scalar multiplication, Magnitude of a vector, Components: i,j, and k. Scalar product and projection, Concept of work done, Cross product and applications, Scalar and vector triple products, evaluation as a determinant and applications

Axoren

@reversiblean They didn't start with anything on Linear Transformations?

reversiblean

Nope. those are for the first and second semester

Axoren

So this is a multisemester course? You're in the third semester of it?

reversiblean

we're in first year second semester. its an IT course

Kaj Hansen

By IT, do you mean Information Technology? @reversiblean

reversiblean

06:48

yup yup : )

Axoren

Do you know what a system of equations is already?

reversiblean

yeah

Kaj Hansen

I'd imagine linear algebra alone would be extremely dry. I had it with Ted, who taught it alongside vector calculus. It blended quite well.

Maybe that's just me though.

Huy

@KajHansen: I did Linear Algebra alone, three times. It was awesome.

Axoren

Linear Algebra alone is dry when it's slow.

Kaj Hansen

06:51

I'd be interested in taking a linear algebra only course now that I have some broader perspective though. I know I could use the brush-up.

Axoren

But when your instructor manages to teach fast enough for you to learn how to do linear regression by hand, it's fun.

We almost managed to cover compression when I took it.

Huy

@KajHansen: It's a standard compulsory course for maths and physics freshmen at my uni. I rather enjoyed it and it helped me a lot more on the path to understanding how to come up with proofs by myself, way more than analysis did.

Kaj Hansen

Yeah, it's compulsory here too @Huy, but the linear algebra / vector calc blend satisfied it for me. At the same time, I never took a "how to do proofs" course, so it also had that effect for me.

Huy

@reversiblean: Seems like a bit little material for two semesters, in my opinion. Over here, the IT linear algebra course is only a one-semester course and covers all of that, plus some more matrix decompositions (QR, SVD) and an introduction to MATLAB.

@KajHansen: There are "how to do proofs" courses? :D

Axoren

@Huy At UMass, it's a course called "Proofs".

Kaj Hansen

06:55

As I understand, they are quite common @Huy here in the US. Even compulsory.

Axoren

But by the time a Computer Scientist can take it, it's already too late to take it. Lol

Huy

Okay, that seems really odd to me. Do they not show you how to do proofs often enough in any other course or at least the corresponding exercise classes?

Axoren

Curse the pumping lemma to all hell.

Kaj Hansen

@Huy, it sounds like the setup for courses are quite different in your country.

infinitesimal

Algebra should be taught as an extension of arithmetic.

Axoren

06:56

@Huy Knowing that something follows from another thing doesn't mean you can formally present the reason why you know it's true.

Huy

@KajHansen: Everytime I hear about courses in the US, I find out they are even more different than I had previously thought.

Kaj Hansen

Without the proofs course, I taught myself all sorts of proof techniques on my own. This includes the usual contradiction, contrapositive, etc. as well as various common tricks.

Huy

@Axoren: No, it doesn't, but that's what we have exercise classes for.

Axoren

@Huy Exercise classes?

infinitesimal

Tutorials.

Axoren

06:58

@infinitesimal I understand that, but there were classes dedicated to tutorials on proofs?

As part of the course or supplementary?

Huy

@Axoren: For almost every course, we have lectures and exercises. Say in the first semester we have 4 hours of lectures about linear algebra. Each week, we get a problem set and then we discuss them briefly in so called exercise classes, 2 hours per week. Then, we hand in our solutions next week and they will be corrected and returned a week after.

Axoren

@Huy You mean like a discussion session. I've only had that for early physics in which proofs consisted of almost single-layer non-formal logic.

Huy

@Axoren: By discuss them briefly I mean that the TA would typically solve a similar exercise (especially for freshmen) or just give a few pointers. Sometimes, we'd also just repeat what we've seen in the lecture and go through an important proof another time, just so that everything becomes clear. We also discuss previous problem sets to see where people typcally struggled and why.

Kaj Hansen

@Axoren, what country are you in?

Axoren

For other courses in computer science and math, the curriculum doesn't have much time to allot discussion sessions. @KajHansen USA, Massachusetts.

Kaj Hansen

07:00

Interesting. I've actually never had a recitation hour as seems common elsewhere, even in the states.

Huy

@Axoren: I can imagine for CS students at my uni the math courses will be a little less proof-based, but still are, since I have tutored a lot of CS students and they were required to prove a lot of things in their exercises.

Axoren

@KajHansen It's rare for anything other than early sciences in UMass.

@Huy We were always required, but never formally prepared for them.

For example, when I was presented an exercise on the pumping lemma, I understood mechanically how the lemma worked.

Huy

@Axoren: And the formal preperation is exactly what we did in exercise classes. It is clear to the profs over here that it doesn't suffice for them to just present the proofs to students in the lectures. This is why TAs have to explain over and over and then there are tons of exercises to practice.

Kaj Hansen

What is the pumping lemma @Axoren ?

Huy

I always imagined the tuition fees in the US are somewhat justified by much closer guidance by profs and TAs, is that wrong?

Axoren

07:04

It's a lemma in Computation Theory that describes a property of any context free grammar.

That a part of the grammar can be "pumped" within a string of that language to produce another string in the same language.

Kaj Hansen

Huh. Interesting

Axoren

The only way I survived that was "Hope the problem requires a contradiction, then try pumping 0 or 2 times."

If the proof needed to be more complicated, at the time I wouldn't have been able to do anything with it.

Even now, I need the lemma in front of me to work with it.

Pumping lemma for context-free languages

In computer science, in particular in formal language theory, the pumping lemma for context-free languages, also known as the Bar-Hillel lemma, is a lemma that gives a property shared by all context-free languages. It generalizes the pumping lemma for regular languages. == Formal statement == If a language L is context-free, then there exists some integer p ≥ 1 (called a "pumping length") such that every string s in L that is longer or equal than p symbols (i.e. with |s| ≥ p) can be written as s = uvwxy with substrings u, v, w, x and y, such that 1. |vwx| ≤ p, 2. |vx| ≥ 1, and 3. uvnwxny is in...

I can safely say that I sucked at proving things until I reached Grad School, which is terrifying.

Because I made it into Grad School like that.

Huy

@Axoren: If you don't intend to go into theoretical CS, I don't think that's very uncommon, do you?

Kaj Hansen

Are you doing a lot of CS theory? Or more of the applied side? At my university, the CS department seems very, very applied. I don't know for certain though.

Axoren

@Huy @KajHansen Now I'm getting into a lot more theory and a lot more Math.

Our department is very CS theory heavy, I'd say.

UMass Boston is very research focused.

Kaj Hansen

07:13

That's like our math department. Just not our CS department.

Axoren

Our Math department doesn't have a grad program. They seem to repulse math majors out of the field into management degrees.

And I don't always hear good things about the current head of the department from the undergrads.

The courses he teaches seem to either be taught poorly, or attract really unprepared students.

Kaj Hansen

That's unfortunate

Axoren

So people come into the degree hearing horror stories which may or may not be true, morale gets hit, and no one really progresses on to want a graduate degree in Math.

Kaj Hansen

There are very passionate people at my school, and at the same time there's also a lot of people who are majoring in math who don't seem to like math and are content taking the bare minimum of courses. It's bewildering.

Axoren

Honestly, when you look at it objectively, a lot of people come into college wanting to do mathematics because they think "Math in Highschool was AWESOME"

Huy

07:17

@KajHansen: Why would anyone major in math if they don't like it?

Axoren

And then end up getting more than they bargained for.

@Huy People don't know what higher level math is actually like until they're already in too deep.

Kaj Hansen

I was actually the opposite. I hated math in high school -- good at it, but didn't enjoy the classes. I didn't even know I'd be a math major in college.

Huy

@Axoren: Well, I didn't either, but we have a lot of people dropping out, more than 60% after the first year.

(by exams or because they don't like it)

Axoren

@Huy Unfortunately, the first years here tend to be in classes like Calculus

Which puts them 2 semesters behind the Hefty math that forces people to make Change-of-Major decisions

Huy

Oh, I remember. And that's basically high school maths, or maybe a bit more difficult but without (or with little) proofs, right?

Axoren

07:20

^Yeah, I had Calculus in High School.

Very few proofs, more procedural stuff

Huy

That's what I found weird about having this sort of calculus in university. We just go straight to linear algebra and analysis very proof-heavily.

Axoren

I had the idea in my head from freshman year of high school that I wanted to be a computer scientist because I wanted to make video games.

BOY was I wrong

Huy

Because that way everyone knows what to expect to in the future (at least more than with calc courses) and some will learn to love it and some hate it and drop out.

Axoren

@Huy Honestly, I agree that Linear should be taught first for a math major

Kaj Hansen

scontent-a-iad.xx.fbcdn.net/hphotos-prn2/v/t1.0-9/…

3

Huy

07:21

@Axoren: Not gonna lie, at some point in my life in high school, that was also my dream.

Axoren

BUT the fact that so many people change out of the major, they make people take the Calc line because it's a pre-req for so many other STEM degrees

If you're going to change out of Math, at least you can take some credits with you

@KajHansen Lol

I've been programming for 8+ years now, I think at least 9 if I had to count.

I've not made a single video game.

Other than a shitty assignment back in Visual Basic class in High School that didn't count.

Kaj Hansen

Ha :)

Vrouvrou

@TedShifrin for example $\mathbb{N}^*\setminus\{1,2,...\}=\emptyset$ this is when $n=1$ when $n=2$ $\mathbb{N}^*\setminus\{2,...\}=\{1\}$ i want to genereliz this

Kaj Hansen

I made Pong on my TI-84 back in high school.

Huy

@Axoren: I've been "programming" for at least 12 years. I "made" games using RPG maker, back when I was 9. Did you ever play around with that? I thought that's what programming was and I loved it. :D

Axoren

07:29

@Huy I never counted that. I made a version of Pac-Man where he shot the balls he picked up at ghosts and shat fire as he ran.

Huy

@Axoren: Hence the "".

Axoren

@KajHansen TI-83 was so powerful... I wrote a BASIC program on it that would calculate digits of pi and print them to the screen in the order it found them.

Huy

I played so much Super Mario and Falldown on my TI-89.

Axoren

I left it running over the summer in my backpack and when I went to use it for the AP Exam for Calculus, I saw that the batteries had exploded.

Kaj Hansen

I played sooo much PuzzPack on my TI-84

Axoren

07:31

My highschool didn't have a single calculator to loan me for the AP exam, so I think I got a 2 because I couldn't answer questions fast enough by hand.

Kaj Hansen

All that wasted time....
I could be doing some insane math right now if we actually did stuff in HS math.

Axoren

I feel like we should teach children formal logic, but for fun things.

That way, they'll be better suited for all types of mathematics in the future.

And school playground bullying becomes funnier.

Huy

@Axoren: I don't think they or their parents would appreciate it.

Axoren

"I've written a 5 page proof as to why you're a loser, Jimmy."

"Oh yeah? Well the reason you're a loser is trivial!"

Kaj Hansen

haha

It's intuitively obvious.

Axoren

07:33

It's an axiom in my theory!

Kaj Hansen

:D

John Jack

09:26

Hi @DanielFischer, I have a very simple notation question, if you have a sequence of functions $\phi_{n}: [a,b] \rightarrow \mathbb{R}$. To show the derivative of $\phi_{n}$ at $z \in [a,b]$ for any $z \in [a,b]$, for any $n$, are the following all equivalent notational representations $\phi_{n}'(z)$, $\frac{\partial \phi_{n}(z)}{\partial z}$ and $\frac{d \phi_{n}}{d z}$? Thanks.

Hanno

hi everyone

John Jack

The last one is $\frac{d \phi_{n}(z)}{d z}$.

HI @Han how's it going?

Hanno

@JohnJack: Thanks, I'm fine :) What about you?

Fred Kline

Q: What do mathematicians do? A: They sit in front of computers, reading books and papers.

Sayan Chattopadhyay

Solve amazing problems @FredKline

John Jack

09:38

@Han Not bad also, what do you think of my notational question above, do you think they are all equivalent?

Fred Kline

@Axoren You should have brought your slide-rule! That's what I used in highschool.

Sayan Chattopadhyay

@JulianRachman did u get the proof for the question I asked u

Forever Mozart

10:14

hi

hello

bonjour

Sayan Chattopadhyay

Hi

Forever Mozart

hi

are you good with matrices?

@SayanChattopadhyay

Sayan Chattopadhyay

@ForeverMozart nope

Well tellme the question

I will try

Forever Mozart

10:34

it is here

2

Q: Integer eigenvalues

Suppose $A$ is an $n×n$ symmetric matrix with all entries $0$ or $1$, and with diagonal $0$. Are all of the eigenvalues of $A$ integers? It works for all the cases I have tried so far.

A Beautiful Mind

Hi

Sayan Chattopadhyay

Hi jasper

Sorry@ForeverMozart I know what is aatrix but not eigenvalues

Sayan Chattopadhyay

10:59

Jasper how r u

Lucio D

@ABeautifulMind How's it going?

Hi @Sayan

Danny

11:16

@robjohn are

you alive

where is pedro is he still hanging around this place

$\gamma$

Sayan Chattopadhyay

Yes he is alive@Danny

Danny

@SayanChattopadhyay hey ok thank god i taught he was dead :)

Sayan Chattopadhyay

Why

Danny

i need some help with statistics

Sayan Chattopadhyay

Oh......

U use $\gamma$ in statistics

Danny

11:27

hehe Iam trying to

fix latex

i can't manage to do it for this chatroom

lets see if it works now $\Gamma$

Lucio D

HI @Danny

Danny

hello @LucioD

yes it works :)

Lucio D

@Danny Do you know what the gradient of a single variable function will be?

Danny

i dont, the derivative?

usukidoll

11:44

anyone here

Sayan Chattopadhyay

@LucioD can I find derivatives for functions with complex numbers

Hi @usukidoll

usukidoll

how do I graph dx/dt = x (a^2 - x^2) if a is the varying parameter

do I set a = to something or/>!

Danny

$\alpha$

usukidoll

alpha?!

Danny

hehe sorry usukidoll i was just trying my late

x

Sayan Chattopadhyay

11:47

@usukidoll u watch anime

usukidoll

ya

Danny

@usukidoll why you could plot it in 3$d$ ?

Sayan Chattopadhyay

Death note

usukidoll

huh?!

Consider the model equation\\
$\frac{dx}{dt}=x(a^2-x^2)$\\
where $a$ is a real parameter. Find the equilibria, their stability, and draw a bifurcation diagram where $a$ is the varying parameter. \\

ignore equilibria and stability... I got those... I just have to graph the equation

Sayan Chattopadhyay

U have seen death note

usukidoll

11:49

-_- is my latex broken? I don't think so kicks

bifurcation diagram = need phase lines -_-

Sayan Chattopadhyay

I don't know how to use latex on android cn someone help me

usukidoll

if a is a varying parameter... that would be a = 0,1,2,3, etc etc

?!

wait a sec
$xa^2-x^3$

Sayan Chattopadhyay

Anyone knows how to use latex on android

usukidoll

oh boy those graphs may be expanding!

Danny

@usukidoll what is really your issue

usukidoll

11:51

points up drawing a bifurcation diagram

a = 0
$-x^3$
a=1
$x-x^3$
a=2
$4x-x^3$?!

a=3
9x-x^3

Chris's sis