dang, nice

@Riker Keep loose, stretch, and take a short walk.... Next time you jog six miles, before you stop, walk a quarter mile or so afterwards, and stretch, to cool down. Helps to flush the lactic acid from one's muscles.

Okay, there's the coach in me popping up. Sorry.

I did, SBA: I more or less believe it is true, because you can look at the Taylor series and see that the coefficients decrease (faster than polynomial, right?), and so it's probably easy to get a step function that does it. Then, mollify the step function to get something $C^\infty$ and if you play your cards right, you can probably get it to differ from the step function on a very small set.

Hm, okay @EricStucky

The trick is to get the relevant step function, but, since you're a fan of big numbers, you probably could get it to work with the Busy Beaver function? :P

xD

@amWhy I did :/

it was pretty steep, I think that's why

it was a mountain trail, 2k ft gain/loss total

Ouch! Take a breather tomorrow too, though moving (walking a bit) will help too. (And eat well and sleep well, too!)

So, is anyone here?

xd

hmm, maybe

lol

what's up Eric?

:)

blogging as per ususal

What are you writing about?

hehe 1sec

oks

teaching, actually

at the Joint Meetings this year, there was a talk about a first-day-of-class activity that seems really versatile

Wow, seems very interesting

So the post I just wrote is about how that activity works, and some scattered thoughts on whether I'm going to do it :P

Haha I see

That should be fun. Writing about your teaching experiences :P

I feel like I am a very boring teacher. I do my job, no worse than the next guy. But I don't think I take enough risk.

Why do you think so?

Hm.

Mostly because I don't think about it very much. And I know that it's very hard to be an interesting teacher (and effective), and so I feel like, if I'm not putting a lot of effort in, then I probably am not doing that.

I suppose in some sense I haven't been given a lot of room to take risks; just because of the boring details about how my assignments so far have run.

And maybe I'm just experiencing a kind of impatience.

I am hoping to have a more traditional class next year, where I might have some more flexibility.

Well, I cannot give you any advice bc I haven't teaching, properly speaking, but I hope you can have better opportunities to take those risks

I'm curious, what are teaching?

I've done 4 assignments

the first was business calculus; that was very hard and I tried a lot of new things, and it was quite bad.

the next was a rather more advanced course: linear algebra for scientists and engineers

Wow

and the other two were also "for scientists and engineers", the third being Calc II and most recently the linear algebra again

Calc II is a very frustrating course to teach, as a mathematician. The students basically have to do proofs but they are given no guidance about how to do proofs.

Business calculus is frustrating for different, more selfish reasons :P

Honestly, and I know you didn't ask

But Calc II is isn't about integrals?

I think it was extremely irresponsible of the department to let me teach business calculus as my first assignment

Calc II is a real mishmash

I mean just to solve integrals, no much proofs and that stuff

Oh

integral techniques, polar coordinates, improper integrals, sequences and series, and basic vectors

there's no rhyme or reason to it, it's just a whole bunch of maintenance that has to be done at some point.

Yeah now I understand how frustrating must be teaching a course where there are a lot of proofs but not being properly a course, let's say of real analysis

I mean, yes, but that's a much deeper issue than Calc II :P

that's just a foundational problem with trying to tell people how to take a limit when they don't know what a real number is.

So basically you've been teaching courses for scientists and engineers?

Yeah, I got your point

Modulo the first assignment, yeah.

Those courses are nice because you get great students, who are generally more motivated also.

But most grad students find them a bit boring, because you're mostly handing out worksheets

I see

I think that 'handing out worksheets' is something of a local maximum of effectiveness. If you want to improve a little, you have to really adjust the strategy a lot.

Anyway, what are you doing >.<

I'm reviewing some posts

ah... everyone still sane on main?

I guess :P

I like how from time on time one gets these "tests" meanwhile one is reviewing xd

>.>

Do you know user26857?

I don't.

Mm. He/she is a very well-known user in the tags of abstract-algebra, commutative-algebra and ring-theory

He/she knows a lot of commutative algebra

hm

But, on the other hand, he/she is not very polite sometimes.

haha so I'm reviewing my notes from the Joint Meetings and the next post I'm supposed to write is "Pitching Coxeter Groups to a Curious Undergraduate". It strikes me more as a talk about how I'm supposed to write a sequence on Coxeter groups, rather than a talk I should actually write up itself >.<

Idk what to say xd

jeje

mostly I am lazy and don't want to have to figure out how to write about Coxeter groups, tbqh

haha I see

xd

His suggested program is (1) cube group and symmetry groups of a simplex, (2) classification theorem (3) group presentations and (4) infinite coxeter groups

The laziness is really in writing (1). Which has the unfortunate property of being the first step...

Wow, that seems complicated :/ ... at least for me

Yeah, overcome the first step is very important

The thing is, if you feel good about group presentations, it's very easy to just dive straight into Coxeter stuff. But he argued that most students aren't, so you have to get to it somehow

Mmm

I haven't studied group presentations so I can't say a word about it

Well, you sound like the perfect audience for the book, then :P

Hahaha :P

Well, I gotta go

Have to sleep :(

See you Eric ;)

arrrighty

good to talk

to you*

>.<

Hello

Good morning

Good morning

More like good afternoon though for most of us

afternoon

::waves in the background::

o/

Good evening

/yawn

It's 2:37PM and we still haven't had lunch...

@DeependraDhakal Hello and welcome to my realm!

@Mithrandir I'll eat it for you if you don't

oh

ho ho ho

Merry Christmas?

@SimplyBeautifulArt eherm. That's not what I said.

:-)

:}

That's like an evil smile...

oh

@Riker PCG has lots of locked questions lol

@SimplyBeautifulArt The age of men is over. The time of the @JoE has come.

lol

How is everyone in the Realm doing today?

eh

normal

It's weird, when I read Realm I can't help but think about this:

Just saying... :P

:P

Hey @Mithrandir

@SimplyBeautifulArt o/

Hey

@EricStucky "everyone still sane on main?" I didn't realize they ever were sane on main, let alone "still sane." ;-)

now, now

be nice :P

:}

Could you please teach me something that I don't know yet? I'm curious

not again

I'm not expecting poor jokes, sorry

Poor jokes are essential for living a happy life...

SBM, you have a topic in mind?

@SBM Logic lesson: Let the domain of x, y be the set of all people. Let $L(x, y)$ mean "x loves y". How do the following statements differ?: $\forall x, \exists y (L(x, y))$ vs. $\exists y, \forall x(L(x, y))$

I was thinking Calculus or Linear Algebra

First statement

for all x there exists a y such that x loves y

I guess the second one just makes there exists a y common to all x

Maybe

Correct, so far (also can be written : "Everyone loves someone".)

@SBM Indeed! There is someone (in particular) who is loved by all.

Oh

this is physically uncomfortable d27t3nufpewl0w.cloudfront.net/lichess/…

Does $\sup \mathbb{N}^2$ mean the same as $(\sup \mathbb{N})^2$ ?

To the best of my understanding, neither of those mean anything

so... yes?

@SBM No need to be a snob.

Ph

Oh

sorry

:)

$\sup (\mathbb N^2)$ would be $\sup\{ (x, y) \mid x, y\in \mathbb N\}$ so unless you have an ordering relation which compares any two ordered pairs (a, b), (c, d) and defining when one is greater than the other, vice versa, is undefined. $\sup \mathbb N$ does not exist, so nor does its square.

oh

only if N were finite

sorry, I misread that as "if only N were finite" >.<

I get it now, thanks

@SBM If we have a finite subset $A$ of natural numbers, say $A = \{1, 2, 3, \ldots, 96, 97\}$, then $\sup(A) = 97,$ and so $(\sup A)^2 = 97^2$.

oh

Does 2 plus 2 really equal 5?

Sorry, let me rephrase in Math.SEspeak, does $2 + 2$ really $= 5$?

now who's the snob?

@EricStucky Still you man. I've got nothing against poor jokes. I find them as amusing as rich jokes. :P

Just to be clear: I am joking. :)

I know :P

But yeah sure, it's true mod 1.

Wait, what is $2 % 1$?

2

or 1

or 0

Exactly.

but not, say, 2.4

so it's not a completely useless concept.

But unless you can provide a partial order on ordered pairs of such integers (finite or infinite) under an ordering relation on the natural numbers that defines when $(a, b)\leq(c, d),\;\; a, b, c, d \in \mathbb N,$ and that is reflexive, antisymmetric and transitive, $$((a, b) \leq (a, b),$$ $$((a, b) \leq (c, d)) \rightarrow (c, d)\leq (a, b),$$ $$((a, b) \leq (c, d), \land (c, d) \leq (e, f)) \rightarrow (a, b)\leq (e, f)$$ there can be no sup $\mathbb N\times N = \mathbb N^2$.

@EricStucky I count two mods

>.<

that was good

@Mithrandir Dat buuuuuurn.

:D

Apply @amWhy to burnt area.

amWhy doesn't count as half a mod?

Let's see. @amWhy are you half a mod?

@EricStucky Now that's just mean. Calling someone half a mod.

@EricStucky Actually, last word is that I'm $\frac {499}{1000}$ of a mod

that is a suspiciously large number of significant figures...

@Mithrandir Well, it doesn't contradict of your claim! ;-)

oh

@SimplyBeautifulArt yea, most are old pop-cons

@EricStucky Just promoted, this moment to $\dfrac{49999}{100000}$ of a mod :P

jeje

Now I count 3 mods

a right natural combinatorialist, you are ;)

@SBM See the article in Wikipedia, on "partially ordered set". Also, the second of the three properties I listed above should be "antisymmetric": If $(a, b)\leq (b, a),$ and $(b, a)\leq (a, b),$ then $(a, b) = (b, a)$.

yes that seems obvious

I'm confused. Why are the mods a partially ordered set?

I thought we were all minions of the SE higher-ups.

@EricStucky I count p mods, but I'm a left natural combinatorialist.

Hence a fully ordered set?

@amWhy Wow why bring politics into the equation?!

um

ok

@JoErNanO I was simply playing on Erics comment: "a right natural combinatorialist" ... no political angle intended. E.g., If we favor right cosets over left cosets, or vice versa, this isn't a political bias, but a group theory bias.

?

group theory?

@amWhy Isn't politics a group theory?

@JoErNanO Hahahaha! Now we're getting into Sociology, as well as politics!!

@amWhy Leave sociopaths out of this.

You have ever more bad jokes than me :P

*even

@Mithrandir Do you have something against *odd (numbers, or people)?

@amWhy Yeah @Mithrandir is clearly a oddophobe.

What is group theory?

He's an evensupremacist.

@SBM First, take a look through this Wikipedia Article

o.O

why would you pick that one instead of en.wikipedia.org/wiki/Group_(mathematics)

Why not en.wikipedia.org/wiki/Flock_%28birds%29

^ this guy gets it

What, the last one seems so irrelevant?

(it is)

Well, maybe @Mith champions $2$ (quintessential factor of every even number) because of 2's inferiority complex, since it is the only even prime), because prime numbers are dominated by odds with only one exception, by the odds!!

@SBM A flock is a group.

@amWhy How odd.

So (+,-,×,÷) are binary operations defined for the group $\mathbb{R}$

ehh careful

We've got quite the crowd going here!

$\Bbb R$ is only a group under $+$

oh

the $-$ operation comes for free because inverses exist

And, $\times$ isn't a group operation because $0$ doesn't have an inverse.

ok

now I see

But in any case, the point is that a group by definition only has one operation (it gets a second one from the inverses, but that one is not associative)

oh

For instance, you can tell that $(a-b)-c$ is different from $a-(b-c)$.

So in some sense, addition is "more natural"

oh

yes

@SBM Best to start with reviewing material on binary relations which are well defined. And also, only an associative binary relation on a set which is closed under that binary relation, can be group operations.

well defined in the sense that ÷is undefined when divisor is zero?

The first one is everyone loves one person

Key ideas in defining a group is the identity element (for example, the real numbers under addition have 0 as it's identity), because every real number, added to $0$, remains unchanged. We also need that each element in the group has an inverse (in this case, with $0$ as the identity of the reals under addition, that means inverse of some element $x \in \mathbb R$ is $-x$, because $x+(-x) = 0$.

@EricStucky That's one reason "minus" is a problematic group operator, unless treated as the negation of a number. Since minus is not an associative binary operator.

Getting a bit impatient here. I want to get to predicate deduction already. xD

... why did you ping me on this?

He just wants your attention

:p

Well, division is a problem on many sets, for example, the set of integers. There does not exist closure of division on integers. For example let $a=2, b=3$. The problem becomes that $2\div 3 = \frac 23 \notin \mathbb Z$.

@shredalert Are you referring to my example statement $\forall x, \exists y(L(x, y))$, and its translation? If so, you are being naughty! Although it is true that $\forall x(L(x, \text{amWhy}))$

hehe

everyone loves one person is still correct ;)

I didn't say it was the same person :p

@shredalert :P

The ambiguities of English! xD

Ima read my novel, take a nap, go to the gym after, and then do more logic until bedtime.

That's my gameplan for the rest of the day

Thanks for sharing, @shredalert! Sounds like a good game-plan!

A man has a problem.

He decides to solve it with a bash script.

He now has two problems.

@JoErNanO Well, Men always have multiple problems!

So your man with two problems doesn't surprise me!

@amWhy I wish I had just two problems.

Maybe I've lost you; What about a person that has a problem so s/he decides to solve it with a bash script. As a result, s/he has two problems (in addition to the problems s/he invariably has). :P

@JoErNanO This would all make more sense if you were to phonetically sound out loud my user name. a....m....Why

@amWhy Did you just assume the numbers of problem s/he might have?

Perhaps I did so assume.

On a different note: youtube.com/playlist?list=PLfimnwaZdumgdg-BPofqK8h6cdkxIxe2H

So. 11 is a prime number. Dammit!

Hi, @Mithrandir ! (Mith-reindeer?)

Awww, I'm being silly.

Wonder where @SimplyBeautifulArt is, hey?

@SimplyBeautifulArt Another ping.

:D I'm here

@SimplyBeautifulArt Yet another ping :P

@amWhy @amWhy @amWhy

Playing Adventure Quest Worlds right now

That makes $3 \times $@amWhy.

@SimplyBeautifulArt Cool... What are your plans for summer, besides the comparative politics course?

Hm, math? I suppose that's not a plan though...

As long as you're doing it, and exploring it, all the more power to you!

:-)

...and as long as your having fun doing it, exploring it, (creating it?), that's a perfect summer!

Yup

@SimplyBeautifulArt Do you have any ideas for what you'd like delve into? (in addition to big numbers and computing)?

logic + set theory + real analysis

Fun, fun, fun, each and every one!!

:-)

@Mithrandir my last comment was simply in fun... Thought you were around these parts! How's the mod election going?

:O

@SimplyBeautifulArt oops, did I do something bad?

nah, I just forgot that election was going on

@amWhy sorry, busy for a bit in real life

I'll be available soon

@Mithrandir :D

@amWhy votes aren't public, so I don't know :P

@Mithrandir wrong reply

@SimplyBeautifulArt whoops

@SimplyBeautifulArt two days left to vote..

Sh...

@Mithrandir Yay, Mythrandeer!!

@amWhy yep, exactly :P

@Mithrandir My mom played along with me and my brother and sister, (way back when) by "illustrating" our last name (as it might appear in a cave, using stick figures and such): the sketch of what was clearly meant to represent a young woman; (sounds like) the sketch of a farm animal (I'll leave this as a puzzle), and a stick figure wearing "skis" on a slope.

At the time I thought it was so cool.

Girl - lamb - ski? No idea...

Ending three letters, correct "ski".

I'll let the rest sit...for now. As an MSE/Puzzling collaboration.

If you post it as a Puzzling rebus...

Final hint: Well known mathematician (historically) with $n \leftrightarrow s$

(I don't speak MathJax...: P)

Hmmm...okay: exchange "n" with "s".

@Mithrandir The use of the address to such a young (unmarried) woman (sketched by my mom) is rather outdated in contemporary America.

Miss?

@amWhy ^

@Mithrandir Quick, delete that, you are right! (Don't want everyong to know)!!!

@amWhy heh

You win, you clever one!

I enjoyed the challenge :)