Mathematics - 2022-09-13

Curious Mind

00:24

Hello.

Isn't standard linear form of -kx=mx" x''+kx/m=0?

Leading coefficient should be 1.

why is it x"m+kx=0?

Ted Shifrin

Because mass times acceleration? What’s the big deal?

robjohn

Mathematically, they are the same unless $m=0$

Ted Shifrin

Massless Newtonian physics?

robjohn

Mathematically

@Unknownx how are you defining the order of convergence?

never mind, I read the post which has an image of the definition

robjohn

00:53

@Unknownx: the post received a downvote and a close vote. I imagine it is because the definition is given in an image. If possible, it would be best to convert the content of the image to MathJax. A link to an image is fine for verification, but MathJax is more easily readable and can be searched.

Curious Mind

01:19

@TedShifrin I don't understand what you mean.

Ted Shifrin

01:42

This is Newton’s second law. This is how it naturally shows up. Who cares about standard form?

Akiva Weinberger

02:11

Do you know a name for multisets with finite but possibly negative multiplicities?

1

Q: What is the name of this multiset-like object?

My understanding is that a multiset (roughly, a set where we care about multiplicity) can be modeled as a function $V\to\Bbb{Card}$ with set-sized support, where $V$ is the class of all sets, $\Bbb{Card}$ is the class of all cardinalities, and set-sized support means the class of sets on which th...

It feels like such a natural concept that it should have a name already but I don't think I've ever seen one

leslie townes

i haven't heard of one.

ZaWarudo

02:28

If $f:\mathbb{R} \to \mathbb{R}$ is two times differentiable and such that $f(x)=1$ for any $x \ge 1$ and $f'(x)=-1$ for any $x \le -1$, can I conclude that there exists $c \in (-1,1)$ such that $f''(c)=0$ using the mean value theorem or I must use Rolle's theorem?

Ted Shifrin

03:08

Negative multiplicities? Yikes.

Rolle is a special cass of MVT.

Do you have a typo in the question, @ZaWarudo?

It’s false as stated.

Oh, I misread. It is correct. Rolle is fine, after a little work.

Koro

There is no product rule for derivatives in several variables calculus?

I ask because I never saw one.

Ted Shifrin

03:29

If it makes sense, there is. It’s in my book.

Not Tfue

A disk 2 inches in diameter is thrown at random on a tiled floor, where each tile
is a square with sides 4 inches in length. Let C be the event that the disk will land
entirely on one tile. In order to assign a value to P(C), consider the center of the disk.
In what region must the center lie to ensure that the disk lies entirely on one tile?
If you draw a picture, it should be clear that the center must lie within a square
having sides of length 2 and with its center coincident with the center of a tile.

I don't think it is necessary that center concides

And I don't understand why P(C)=4/16. It doesn't make sense.

Ted Shifrin

What you say is clear seems wrong to me.

Thorgott

the product rule is $\partial^{\alpha}(fg)=\sum_{\beta+\gamma=\alpha}{\alpha\choose\beta,\gamma}\partial^{\beta}f\cdot\partial^{\gamma}g$, where the greek letters are multi-indices

or, if you mean the total differential, there is $d(fg)=df\cdot g+f\cdot dg$

ZaWarudo

@TedShifrin thank you.

Ted Shifrin

No, it’s right. I was thinking about the corners wrong.

Ted Shifrin

03:49

@NotTfue By symmetry it has to. Just put the center at least one unit from all the edges.

Not Tfue

@TedShifrin Oh yes my bad. The book is right I see a square now inside the square.

Ted Shifrin

04:09

Cool. Pictures help.

OneRaynyDay

Hey, I've been lurking around a while but was curious about something: ever since undergrad I've been interested in going to grad school for math that's more on the abstract side. I've been in the industry for more than 3 years now and I'm wondering what I should do if I want to plan to go back to grad school in the next few years :) does anyone have advice?

Ted Shifrin

What pure math courses did you take in undergrad and how did you do?

OneRaynyDay

Unfortunately, I have a cs degree and I'm not sure which professor would take someone who came from industry with no graduate math course experience besides complex analysis. I've thought about doing a masters but haven't found any programs for "non-applied math", i.e. I don't want to take applied math courses as a means to an end to find a career, but rather just to understand things from a more general perspective

Unsure what defines as "pure", but I went to UCLA, took terrence tao's (or rather, his postdoc's) real analysis honors course, his graduate complex analysis, honors linear algebra, probability theory & stochastic processes (but not from a measure theoretic perspective, so I wonder if this can be considered pure), computability theory

Ted Shifrin

You need to learn basic analysis, algebra, topology quite well.

OneRaynyDay

I've gotten some basic topology & analysis from my undergrad experience, but not so much algebra

Ted Shifrin

04:14

Have you done solid proof-based linear algebra?

OneRaynyDay

I've been overextending a bit in the analysis side, currently learning measure theory (as a means to an end to understand stochastic calculus to e.g. price options at work). Do you have suggestions on what's the most effective way to get started on algebra?

yes, the honors series is always focused on proofs: math.ucla.edu/~totaro/115ah.1.15f (I didn't take specifically this professor's course, but the syllabus should be uniform)

Ted Shifrin

You need to read and do lots, lots of proof problems, as well as play with many concrete examples.

OneRaynyDay

I understand. What do you recommend as a starter text that's heavy on proof exercises?

Ted Shifrin

Get Artin’s Algebra if you want an interesting treatment.

2

There are plenty of easier books, but that won’t help.

OneRaynyDay

Nice, I just ordered it on amazon. When it comes to "interesting", what do you mean? The author has a sense of humor? Their approach is aggressive in how general it attempts to be at the start? Writing style(terseness, etc) or a mix of all of the above?

Ted Shifrin

04:21

Linear algebra integrated, interesting things in math as a whole. Farthest thing from just symbol pushing.

leslie townes

you want lang for maximal aggression in how general it attempts to be at the start. and middle, and end. (you do not want lang)

OneRaynyDay

Great! Thanks for the recommendation :) I'm really excited to read it when I have the chance

Ted Shifrin

Have you done something like Rudin?

OneRaynyDay

Yeah Rudin was a required text

It was a bit dry but it got the job done

Ted Shifrin

Very dry, but for grad school ….

OneRaynyDay

04:24

compared to typical grad school texts it's not dry?

In real analysis honors I also read Metric Spaces, Cambridge University Press, by E. T. Copson which I found to be much more enjoyable

currently reading another cambridge press book by David Williams (probability theory & martingales) and really liking the presentation style and humor

Not Tfue

I want to speed run undergrad probability in a week...

leslie townes

roll a lot of dice.

Ted Shifrin

Korner’s Fourier Analysis is another gorgeous book.

2

leslie townes

ted may have been implying that in grad school, 'very dry' is about as good as it gets. many textbooks act like there's a law against motivation, and when you're reading papers, you're at the very dry mercy of people who are generally not as good writers as rudin.

i love korner's fourier analysis book.

OneRaynyDay

oh no :(

Curious Mind

04:28

that is how a mathematician learn probability but not students

OneRaynyDay

Why is this the case? Why can't authors just add a bit of personality into their texts to make things more entertaining for the reader? Maximizing utility can't be the only objective here

leslie townes

i think that some authors don't want to stand in the way of their work reaching the largest possible audience, and worry that if they throw in anything too specific (whether to personality, or to one application) they will lose audience.

at the grad level there may also be a concern of, if i present this result too much as a machine for working in one context, it may prevent someone with a completely different perspective from realizing that it is applicable in some other context i haven't even thought of.

OneRaynyDay

hmm I see

leslie townes

also i think many mathematicians are at least somewhat afraid of looking foolish in public, and if they ventilate too much of how they think about stuff, they worry that they will look foolish. "i always think of this as just a trick, i learned it in 1985 but it is still just a magic trick" would be helpful to know, but someone's gonna respond with some paper on how it isn't a trick and is the fundamental theorem of something-or-other.

better just to march through definition, theorem, proof. then nobody's cards are on the table.

2

OneRaynyDay

I'm only curious about pure math as a means of peeling back the onion on the applied layers. If the grad text authors approach it by separating the two (fear of presenting it as a machine for one applied context) it sounds demotivating

leslie townes

04:39

there is also an element of efficiency. grad school would be impossible if you had to learn the origin and historical development of every idea that you might end up using. to truly stand on the shoulders of giants, it sometimes helps just to jump up on those shoulders.

2

and that often means not elaborating on at least some of the motivation.

i guess it depends on how abstract you wanna go. certainly in applied areas there are folks who are more than happy, and not shy, about stressing one application, or a set of related applications.

bioinformatics comes to mind.

OneRaynyDay

When you say it's impossible, is it under some certain time frame?

Are you expected to have been presented with Every Big Idea TM at the end of the first two/three years

leslie townes

well no, but, the basic grad curriculum (to the extent one exists) is a pretty huge toolkit all by itself. and the history and motivations of that would be its own subject matter almost.

2

OneRaynyDay

Ah ok understood.

leslie townes

UCLA would have given you a solid background in a lot of it. as far as 'selling yourself' to a grad program, who knows. note that in math it is fairly common to have to sell yourself to a department/program before specializing to a particular person. although i guess if you had the say-so of someone with a big grant in a department reading your application that would help.

or if you had external funding, that would also help. i think that is pretty rare though.

OneRaynyDay

I see. I'm not even sure which grad programs I would be interested in. I'm not opposed to going back to UCLA since some of the professors still remember me from back then

I've saved up quite a bit slaving for quant finance, so I was hoping I could alleviate some of that funding pressure with that

leslie townes

04:47

i found it fairly freeing that the math application process didn't require a relationship (or proposed relationship) with a specific person at the school. i know it isn't like this in other areas (and not even in math, in other countries)

OneRaynyDay

i.e. I don't really care if the professors pay me while I stay there

leslie townes

i was thinking more that a department would see/use the existence of external funding as a kind of proxy for vetting your potential for success in the program. not so much that the money would matter.

what's wrong with slaving for quant finance? you're moving in the wrong direction, money wise.

OneRaynyDay

Nothing wrong, I just don't care about money

(as much)

leslie townes

i get that.

OneRaynyDay

Everything at work, works great due to the simplicity of the ideas and the execution. I'm a little bored of all the approximations we have to do to do things practically

I want to dig a little deeper to understand why things really work and not just "we see this pattern", "oh ya just assume brownian motion and use ito's lemma", or "shrink that estimator"

it's like listening to 3 songs in your spotify playlist on repeat for the rest of your life

leslie townes

04:51

but i love shrink that estimator. i have all their records.

OneRaynyDay

oh then you'd do great in manhattan!

anyways, thanks for the context, I really appreciate it :) see you guys around

leslie townes

cheers

Alexander Gruber

05:09

Horrific to see how many people like Rudin as a writer

I know this is an unpopular opinion but good god i hated that blue rag he wrote

leslie townes

no, that's the popular opinion that everyone thinks is an unpopular opinion.

Alexander Gruber

The minority is vocal

leslie townes

i didn't mean to promote rudin to the status of good writer above, i just meant to say that the average quality of published papers is lower than that

2

Alexander Gruber

That is true and also says a lot

leslie townes

at least with rudin there's the possibility of a cult member pointing to all the clues rudin left you for how the proof explains the theorem. it's sufficiently structured that this kind of argument is possible.

and the default for publication is, not unstructured mess, but less than that

Alexander Gruber

05:16

Ive often thought there should be another proof writing class you have to take later in grad school that emphasizes exposition

leslie townes

the 'teaching statement' of the academic job search could be replaced with a representative sample of expository work

copper.hat

06:02

I think part of the goal of writing proofs is just that little tweak that is obvious in retrospective but generates angst and insecurity on the first 20 readings.

2

Curious Mind

07:02

I am curious if it is normal that most math students have hard time socializing.

2

user1057909

That doesn't sound normal or true to me

I'm preparing an hour-long workshop for a group of gifted primary school-children... do you think a session on infinities (comparing cardinalities of $\Z$, $\Q$, and $\R$, looking at bijections, hilbert's hotel, etc...) would be suitable?

The past few of these workshops have covered have included modular arithmetic, invariants/colouring, proofs and they've been received very well.

Jakobian

10:30

I suppose, clever reading is a thing. You can't just read, you need to read, consider possibilities of what people mean. Understand them right. It's hard to do

2

maybe textbooks that don't just grab you by the hand with all the arguments, are a good step in that direction. It's a negative for your time, but it might benefit you in the future

Well it's not like anyone has a choice

Not Tfue

11:51

Probability is hard to swallow stuff.

Haven't even finished chapter 1 today lol

And makes me tired

Not Tfue

13:05

I am curious how can you derive a equation of collection of events given all the pairwise independence equation

For ex: P(A_1 intersect A_2 intersect A_3)=P(A_1)P(A_2)P(A_3)

given P(A_1 intersect A_2)=P(A_1)P(A_2)

P(A_2 intersect A_3)=...

and ...

3. Independence

►

At 33:23 I didn't understand both the intutive definition and proof of mathematical definition

sorry not proof I mean I couldn't derive it

UnderMathUate

15:42

How come in probability, we count every single outcome even when it seems like two may be equivalent? For example, if you are tasked with randomly selecting 2 out of 3 electronic items, with one being defective, what's the probability that you select the defective one?

With a problem like this, why aren't the outcomes D1 N1 and N1 D1 equivalent? With D1 being the defective item and N1 being the "first" non-defective item.

When I do problems like this, it makes it seem as if the order matters when in reality, I don't see why it would.

leslie townes

here it seems the notion of "equivalent" will glue together different selections of 2 items from 3 items.

but the selection process itself is still potentially sensitive to distinctions that you don't know about while you do the selecting.

very loose analogy but it might help to consider tossing a coin 10 times and counting the number of heads. it should be clear that there's 10 ways you can get exactly one head, but more ways of getting exactly two heads. it's not like "two heads either happens or it doesn't." you do need to think about the flip sequences that will get you there.

even if you can ultimately replace the space in which all head-tail sequences are equally probable with a different space in which all observed numbers of heads have potentially different probabilities.

in these examples if you focus too much on the observable thing "defective or not", "heads or tails" you miss that you're choosing from a bag where you don't know what the observable thing is. thinking of two sequences of draws or coin flips as 'the same' is gluing together distinct events (in my toy example, of equal probability).

UnderMathUate

Hm...I see how that makes sense, then. I've never heard it explained that way before.

leslie townes

properly counting the number of events often requires enforcing distinctions between things that, when you observe whatever you're observing, vanish. "one head" doesn't care which head it is. but it matters that the event "one head" arises only in ten ways.

same with "two heads."

seems like a lot of probability problems implicitly involve a choice between a vast event space (e.g. all sequences, all draws from a bag) with simple probabilities (e.g. all things are equally probable or close to it), or a small event space (e.g. all categories of observable outcomes) with complex probabilities (where you have to do math or counting to figure out how likely something is).

UnderMathUate

So what about when something like "order doesn't matter" is stated? Is it then ok to ignore these distinctions or are the situations the same?

leslie townes

15:58

'order doesn't matter' for me often suggests that you are evaluating some function for which order doesn't matter, on a conceivable set of possible outcomes that might be more easily modeled in a way where order does matter.

e.g. X = {all sequences of H and T}, and f(x) = number of heads. "order doesn't matter" for the thing i care about, but if i model the problem on X for a fair coin, all sequences are equally likely.

UnderMathUate

Could you expand a little more on what you mean by that?

I guess X is the sample space, so you're saying every possible ordering has to be in there, which is why the ordering matters.

By if you're counting the number of heads, you can conflate the two events HT and TH?

Koro

@TedShifrin yeah, I proved it using Chain Rule.

Xander Henderson

I tend to think of things a little differently: the sample space consists of all possible outcomes. If we are flipping a coin 10 times, and are interested in only the counts of heads and tails, then there are actually two different sample spaces we could use to model what's happening: there is a sample space which contains all $2^{10}$ possible sequences of tosses, and there is a sample space which contains only the numbers $0$ through $10$ (i.e. the number of possible heads).

The advantage of the second sample space is that it is smaller. The example of the first is that all of the outcomes are equiprobable.

Like, how do you assign probabilities to the second sample space?

Koro

in the case of $f,g: U (\subset R^n)\to R^n$ being differentiable on non empty open set U. Then, the total derivative of $fg(t)=f(t).g(t)$ is $D(fg)_x$ defined as $D(fg)_xv=f(x).Dg_x (v)+g(x).Df_x(v), \, \forall v\in R^n$.

UnderMathUate

Oooh, with the smaller sample space things would be much more complicated. Because then you would have to consider things like what's the probability you get a heads on the 1st toss, 2nd toss, etc. Whereas with the first one, it's just # wanted outcomes/2^10 since they're equiprobable

Xander Henderson

16:15

@UnderMathUate No, the point of the smaller space is that you don't consider the order at all. But, again, you have no way of understanding the probabilities in that smaller space (at least, not without first dealing with the bigger space).

If the problem is modeled using the second space, an outcome is a number from $0$ to $10$. Where did this number come from? Who knows. It just appeared. What is the probability of any particular outcome? Uh... hard to say.

UnderMathUate

Oh, the smaller space is literally just numbers.

Xander Henderson

A similar problem would be to compare the space $\{HH, HT, TH, TT\}$ and $\{0,1,2\}$. In the first space, all outcomes are equiprobable. For the second space, I assign the probabilities $P(0) = P(2) = 1/4$ and $P(1) = 1/2$.

It turns out that either space can be used to answer the question "If a fair coin is tossed twice, what is the probability of getting two heads?"

In the first space, each outcome is equiprobable, there are four total outcomes, so each outcome occurs with probability $1/4$. There is only one outcome with two heads, so the probability of getting two heads is $1/4$.

In the second space, $P(2) = 1/4$. Where did that come from? Well... uh... by analyzing the first space.

UnderMathUate

Lol, ok I see

leslie townes

i love it, i was on a phone call and come back to dr. x saying (what i would have said less clearly).

Xander Henderson

And now I am teaching.

Bye.

leslie townes

16:26

that's why they pay him to do it

UnderMathUate

Bye. Thanks for the help @XanderHenderson

You too, @leslietownes

leslie townes

i added parentheticals above so it doesn't look like an insult

Ted Shifrin

English syntax distinguishes between the two, and what you had is unambiguous. The problem is that most people abuse English syntax.

leslie townes

yeah! whoo! i love abusing english syntax

Ted Shifrin

You love being abusive.

leslie townes

16:37

my daughter yelled at the cat this morning. the cat often vocalizes back when she knows you're talking to her, so i heard 'ahhh how could you do this' meow 'you ruined my puzzle' meow 'stay off my puzzle' meow it's a miracle i'm not more out of it than i am

Ted Shifrin

I think Olivia is getting ready to take action.

Koro

16:51

$ch_{AB}(x)=ch_{BA}(x)$ proof for the case A,B have entries from C: If A is invertible, then we are done. So suppose that A is not invertible. Then, by Schur's lemma, A =$PUP^{-1}$, where U is an upper triangular matrix. U has atleast one zero on its diagonal. Let S= set of absolute values of all non zero diagonal entries of U. Choose $m\in \mathbb N$ such that 1/m < inf S. Define $U_m:= U+I/m$ and $A_m:= PU_m P^{-1}$.

Let's consider M_n (C) to be equipped with topology induced by operator norm.

Then, $|| A_m- A||= ||I/m||\le 1/m$, whence $\lim_{m\to \infty} A_m= A$

Since all $A_m$'s are invertible (because all entries on diagonal of U are non zero), it follows that $ch_{A_m B}(x)= ch_{BA_m}(x)$. Taking $\lim m\to \infty$ on both sides, noting that $ch_{PQ}(x):=det (xI-PQ)$, and continuity of the determinant, the desired result follows.

Note: It doesn't matter if S is empty because $\inf \emptyset= +\infty$

Is my proof correct? Thanks.

$ch_{A}(x)$ is notation for characteristic polynomial of matrix A.

Xander Henderson

17:48

I have to go teach another class, but, because it hurts my eyes: @Koro use \|, not ||. You could also use \Vert, I suppose. In either case, $\Vert$ or $\|$ is better than $||$. :D

Akiva Weinberger

Easy win

1

A: $f:[0,1)\to \mathbb{R}$ invertible and surjective (onto)?

Here is one example. Note that the graph has infinitely many pieces; in fact, this is necessary for this problem.

Ted Shifrin

The key realization is that it must fail to be everywhere increasing.

UnderMathUate

18:05

We learned about rings in Abstract Algebra yesterday. We didn't get super far in the examples and stuff, but my professor presented one where the elements where matrices whose entries were commutative rings with identities.

It was at the end of class, so I didn't really get a chance to think about it until right now, but they made the statement that this new ring would now be a ring with identity, but not commutative. I don't know if I understand why, yet.

Ted Shifrin

Entries were elements of a commutative ring ….

Just think about the matrices you know from linear algebra.

UnderMathUate

Oh, duh. Matrix multiplication.

Koro

@XanderHenderson 👍 noted. Thanks.

UnderMathUate

@TedShifrin Did I misunderstand? Are you saying the entries are elements of the ring, rather than rings themselves? I'll admit my notes are a little sloppily written...

Ted Shifrin

Yes, your notes are sloppy !

UnderMathUate

18:11

Ha, lol

Koro

Suppose that f is defined on an open subset U of R^n and suppose that f is real valued.

PNDas

Product of two non-zero linear maps is non-linear right?

Colombeau algebra

In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this. Such a multiplication of distributions has long been believed to be impossible because of L. Schwartz' impossibility result, which basically states that there cannot be a differential algebra containing the space of distributions and preserving the product of continuous functions. However, if one only wants to preserve the product...

Koro

nvm, I got it.

PNDas

Can anyone please explain the scwartz impossibility result?

Here he wanted to define product of two distributions

copper.hat

@Koro You could base a proof on the following: Start with $\begin{bmatrix} I & P \\ -Q & I \end{bmatrix}$. Premultiply by $\begin{bmatrix} I & 0 \\ Q & I \end{bmatrix}$. Postmultiply by $\begin{bmatrix} I & 0 \\ Q & I \end{bmatrix}$. Compute determinants to conclude that $\det (I+PQ) = \det (I + QP)$.

PNDas

18:20

s.t TS(\phi)=T(\phi)S(\phi). But then TS may not be a distribution

Ted Shifrin

@copper.hat I was too lazy to type this earlier, and I thought Koro was appreciating the density/continuity proofs :)

PNDas

Ok, I got it.

Ted Shifrin

@PNDas For example, the product of two delta distributions isn’t a distribution, is it?

PNDas

That's what I was thinking.

I think what they want is to define product even if the product is not a distribution

and which satisfies some properties

and Scwartz shows that this is not possible

Colombeau shows that product is possible if we weaken some condition

Jakobian

19:02

Yes, that's precisely what is happening here

Akiva Weinberger

19:31

$\tanh\left(\tan\left(x\right)\right)\operatorname{sign}\left(\cos\left(x\right)\right)$

Fun graph

Copy/paste it into Desmos

user 726941

19:46

Thanks for sharing 🙂

anak

20:46

that mixed feeling you get when you spend all that time to write a good SE question for something you are struggling with, but then in doing so, you figure it out

copper.hat

@TedShifrin I would have been too lazy as well, but found it effective as a means of procrastinating from essential work.

user 726941

that mixed feeling is called learning by clarifying your thoughts

copper.hat

AE Housman wrote that "Perfect understanding will sometimes almost extinguish pleasure."

user 726941

> The art of asking questions is more valuable than solving problems.

user 726941

21:31

You have to know the problem before you can solve it.

Xander Henderson

23:26

Whelp... I have now had my Omicron booster. And a shiny new flu shot!

Xander Henderson

23:43

Also, it hailed very hard in the middle of my class today. I feel like I had to yell to be heard. :/

But the precip is nice.

user51462

Is there a symbol for "is a"?

Ted Shifrin

I'm getting the booster Thursday, @Xander. I was supposed to meet two friends for dinner tonight, but one canceled because the booster yesterday knocked him on his a**.

@user51462 This doesn't sound mathematical. Context?

Xander Henderson

@TedShifrin Yeah, about halfway through my shots, I realized that I am going to feel like sh*t on Thursday.

And I have to teach until 9pm on Thursday.

Not my best moment.

But maybe I'll get lucky?

user51462

I came across the sentence "P ∧ (a contradiction) is a contradiction." I think I could use "implies" here but I was just wondering if there was a symbol for "is a", since I've been using ⟺ to say "is equivalent to".

Xander Henderson

23:59

In other news: I am frying tonight. I've got cod, I've got potatoes.

Ted Shifrin

Well, maybe it'll be tomorrow you feel like sh*t, instead. I had no problems with my previous boosters, but who knows ...

Xander Henderson

It is nigh certain that I will end up with more batter than I need.

What else should I fry?

Transcript for

$Mathematics$ Mathematics