Mathematics - 2021-09-24 (page 2 of 2)

adamaero

20:08

Maybe my question is un-answerable?

leslie townes

ada: maybe helpful to think in terms of amounts instead of percentages, because the base changes when the amount used goes up. before, in a month, person 1 used 70 units of electricity and person 2 used 30. now, in a month, person 1 will use 70 units, person 2 will use 30, and person 3 will use 30. so the new percentages are 70/(70+30+30) for person 1 and 30/(70+30+30) for persons 2 and 3.

nor that far from a 25-25-50 split because 30 is pretty close to half of 70.

adamaero

Assume my power usage is 75 kWh. Their power usage is 625 kWh. So I'm using 11% and they're using 89% total power. Third roommate comes along using 75 kWh. So now it's 75/775= 10% and 625/775= 80%

leslie townes

yeah, that kind of calculation.

Avra

Pardon me please,

Assuming that

$$
\log_2{T(n)} \approx \frac{1}{2}\log_2n+\frac 12\log_2T(\sqrt{n})
$$

making $\mathcal{T}(\cdot)=\log_2T\left(2^{(\cdot)}\right)$ and $z = \log_2 n$ we follow with the recurrence

$$
\mathcal{T}(z)=\frac z2+\frac 12\mathcal{T}\left(\frac z2\right)
$$

leslie townes

this reminds me of when my now-wife and i first moved in together. we had a number of splits that were 50/50 and some were other percentages based on our income. we had a spreadsheet where i kept a running total, to the cent, of who owed what to whom based on who paid what bills, and occasionally evened the score by writing checks to each other in weird amounts like $31.57

adamaero

20:16

lol

Avra

Could you please explain what they mean here how they get $log{T(n)} = \mathcal{T}(z)$?

adamaero

It's a bit aggravating trying to reach a resolution with this roommate. I first calculated my usage and the estimated usage of the new roommate based on the simple average of the past eight months... acted like I was doing some complicated calculation just because I used a spreadsheet

leslie townes

you gotta keep a spreadsheet. what are you gonna do, keep it on paper?

adamaero

He just wants to round the water and Internet, and do the 50/25/25 with the electric. So I would pay $10 for water, $15 for Internet, and like $35 for electric. The maximum total electric living with two of the three rooms filled was $31. It was about $21 when I lived alone.

I get that 50/25/25 are even numbers... but they are arbitrary.

leslie townes

there are other ways to even up the split. you could steal from him, or slouch on shared chores.

good pay or bum work, like the wobblies said.

adamaero

20:23

what? no way

leslie townes

it's been a while since i've had roommates. maybe things work different now.

Avra

If we have $\sum_{i=0}^{k-1} \frac{1}{k-i}$, then how we can get please $\sum_{i=1}^{k} \frac{1}{i}$?

leslie townes

a change in the index of summation. helps to introduce a new index, j = k-i

adamaero

never steal from anyone

Semiclassical

if it's not clear, try just writing it out

adamaero

20:31

(also we have no shared chores)

Semiclassical

1/k + 1/(k-1)+...+1/1 for the first

which is the same as the other one backwards

Avra

@Semiclassical. I am going crazy :/ thanks

This is the third time I am asking same question

@Semiclassical. No idea what is wrong with me

Semiclassical

@leslietownes the short way i do that is to say i'm relabeling $i\to k-i$

leslie townes

i go back and forth on whether just writing it out is a good idea with stuff like this. for simple sums it certainly helps.

Semiclassical

huzzah for dummy indices

yeah, it really depends on where you're at

ultimately you need to be able to manipulate them confidently

leslie townes

20:33

i'm fine with 'relabeling' but i'd hate to teach it to somebody that way.

Semiclassical

that's fair

you do often need some way of saying that, though

so that you can justify stuff like adding series with different index names

e.g. $\sum_{i=1}^k a_i+\sum_{j=1}^k b_j = \sum_{i=1}^k (a_i+b_i)$

Avra

@Semiclassical. Is abstract algebra on that same level though :/

Semiclassical

sometimes!

leslie townes

it's vaguely like in basic probability, where sometimes 'just write it out' causes people to start doing arithmetic with binomial coefficients and 'simplifying' and canceling things and getting numbers that shed no light on where they 'came from.' but as you add more notation to prevent people from doing that, you lose people who don't want to reason symbolically.

Semiclassical

sometimes not

Avra

20:35

@Semiclassical. Really?

@Semiclassical. Why a lot are talking about it so much then?

Semiclassical

i mean, it's not uncommon that you have to label a summation by some variable

and then recognize at a later time that the name of the label is irrelevant

sometimes it matters, some times it doesn't, and you have to get used to distinguishing them

Avra

@Semiclassical. Is abstract algebra mostly algebric manipulations?

leslie townes

even regular algebra isn't mostly algebraic manipulations :)

Semiclassical

nah. it's more of setting up systems with various rules

and understanding the consequences of those rules

leslie townes

that's my hot take of the day.

Semiclassical

20:37

hey, sometimes regular algebra is

understanding how to complete the square, for instance :P

Avra

@Semiclassical. So can we say that results from linear algebra and matrices are result of research in abstract algebra!

Semiclassical

by contrast, being able to prove that polynomials of degree four or less, can be solved by radicals? very much not a matter of doing the manipulation yourself

@Avra there's definitely connections, yeah

Avra

@Semiclassical. OMG. This is horrible

Semiclassical

an advanced linear algebra course, for instance, will often study matrices over fields other than complex or real numbers

(I've never had to worry about that myself, though.)

Avra

fields is even over complex or real numbers, i.e., it's after them?

Semiclassical

20:40

no

it's a matter of what kinds of numbers you allow in your matrices

Avra

like constraining the domain?

Semiclassical

Matrix field

In abstract algebra, a matrix field is a field with matrices as elements. In field theory we come across two types of fields: finite fields and infinite fields. There are several examples of matrix fields of different characteristic and cardinality. There is a finite matrix field of cardinality p for each positive prime p. One can find several finite matrix fields of characteristic p for any given prime number p. In general, corresponding to each finite field there is a matrix field. Since any two finite fields of equal cardinality are isomorphic, the elements of a finite field can be represented...

Avra

This is horrible

Semiclassical

i guess i should be talking about matrix rings as well

Matrix ring

In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication (Lam 1999). The set of all n × n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn×n). Some sets of infinite matrices form infinite matrix rings. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs. When R is a commutative ring, the matrix ring Mn(R) is an associative algebra over R, and may be called a matrix algebra. In this setting, if M is a matrix and r is in R, then...

leslie townes

make the generalizations stop

2

Avra

20:41

When you say ring theory i REMEMBER RING MOVIE

Semiclassical

so you still have matrices but now you take the matrix elements to be elements of some ring

@leslietownes lol no

i haven't done much of anything with that stuff though

leslie townes

there's the matrix film series, and there's the ring series, but nobody ever combined those two IPs into the matrix ring

Semiclassical

you do sorta see it when working with the Dirac equation in physics

specifically via the gamma matrices

geocalc33

@leslietownes im doing that shhhh

im making a movie called The matrix ring

Semiclassical

for which one way to view it is that you've got 2-by-2 matrices where the matrix elements in general don't commute

the more common way, though, is just to say that you've got 4-by-4 matrices with a certain block structure

Avra

20:44

@Semiclassical. What is your area of interest? I dropped 2 years of math after I heard about abstract algebra :/

Semiclassical

same difference

physics

mostly quantum foundations stuff now

Avra

you like quantum physics?

Semiclassical

yep

though that's a very broad range

Avra

Wow this is very nice area

Also quantum physics for physics students like abstract algebra for math students. My brother did physics mahor

Semiclassical

yeah, though a lot of it is just linear algebra

though the physicist notation takes getting used to

Avra

20:47

hahaha

geocalc33

has anyone generalized quantum field theory?

Avra

physicians are thieves b/c they keep stealing from mathematians is the battle between my brother's wife and my brother (one is math another is phycis)

Semiclassical

i think we'd need to fully understand QFT before we can imagine generalizing it

though there are certainly structures in QFT that are understood well enough to be generalized

i don't know a lot of the formalities, though

geocalc33

quantum theory is hard to understand

Semiclassical

QFT, at least the version that physicists use, is pretty well known for resisting rigorous mathematical formulation

by contrast QM is a lot more manageable

especially when you take the lazy physicist way out and reduce it to finite-dimensional linear algebra :P

Avra

20:49

math is the center of the universe. You are underestimated folks

geocalc33

I think QFT has a problem with mass gap and yang mills if I remember correctly

its one of those problems that you get 1 mil for

gauge theory or something

Semiclassical

QFT is pretty hard to wrap one's hands around

QM is sorta bounded in its mathematical weirdness

(not necessarily in philosophical weirdness, mind)

by contrast, QFT doesn't have as many safeties built in (e.g. infinitely many particles rather than finitely many)

and thus it's easier for things to go weird

geocalc33

infinitely particles in the model?

Semiclassical

yep

though

a better way to put it

is that there's no limit on how many particles can exist

and thus you can't say "i've never got more than N particles"

for instance

suppose you have an electron moving along. it comes in and comes out on the same line

simplest description of that is that the electron just went on that straight trajectory

but another option is that it emitted a photon which was subsequently reabsorbed by that same electron

if you keep going with the options you basically have no limit

most of those will not matter, insofar as they're very rare and change the final answer very little

geocalc33

is this path integral formulation?

Semiclassical

21:00

but they're still part of the theory

yeah

combined with Feynman diagrams and virtual particles

(one probably shouldn't think of those virtual particles as "real" but the math you use sorta forces you to pretend they exist while computing)

Avra

@Semiclassical. Do you like measure theory?

@Semiclassical. You don't know much about MT?

leslie townes

it's comforting to me that things are apparently the same as they were the last time i thought about this

Semiclassical

i don't know as much of it as I should

leslie townes

20 years ago: "yeah, basically none of this QFT stuff has a mathematical foundation, although we have these notions and formalisms that do work"

Semiclassical

for me measure theory is relevant insofar as it's the foundation of mathematical probability theory

leslie townes

21:02

now seems to be the same

Avra

@Semiclassical. Wow!

geocalc33

lol

Semiclassical

i'd have a hard time disagreeing

Avra

@Semiclassical. OMG

Semiclassical

@Avra see for instance en.wikipedia.org/wiki/Probability_axioms

Avra

21:03

@Semiclassical. Probability theory is founded based on measure theory!!

Semiclassical

yeah

Avra

OMG

They never thought this is math departments though

Semiclassical

a measure space is a set + a sigma-algebra on that set + a measure on those two

geocalc33

yeah they have

Avra

They just go over real, complex, PDE, ODE, AL,

geocalc33

21:05

oh

Avra

that is mostly it in pure math sectionb

Semiclassical

you do see Kolmogorov foundations if you do grad probability theory

Avra

This is how it goes in my prev department :(

Ryan Unger

sigma algebra hah

Semiclassical

but if you're just learning about probability and statistics then measure theory will never make an appearance

Avra

21:06

We always felt that we don't understand

geocalc33

they lied to me

Semiclassical

anyways, a probability space is a kind of measure space where the set is the sample space, the sigma-algebra is the event space, and the probability of an event is the measure

Avra

@Semiclassical. Yeah! Never saw it taught but I realized only by skimming wiki how important it its!!

@Semiclassical. Why they do this in math departments?

Semiclassical

there's a bit more structure than a generic measure space, hence the axioms

Avra

Why they don't include that in the syllabus

Semiclassical

21:08

probably because they want people to be able to use probability concepts without having seen measure theory

that said

once you get to grad school math, the two are very much presented together

Avra

OMG

I see, so they prefer to keep it at grad level

Semiclassical

yeah

knowing measure theory is a high threshold

whereas knowing how to use Bayes' theorem is a pretty low one :P

Avra

@Semiclassical. Nothing in quantum physics has to do with measure theory :-|

Semiclassical

well

quantum physics does deal in probabilities

quantum physics can be roughly viewed as a generalization of classical probability

i can't say i'm much of an expert on this though

Avra

@Semiclassical. I will enroll in math department and start all over :+

I always get excited about math, but when I start I run away :3

Semiclassical

21:12

for instance, i'd really like to understand this bit from nCatLab: ncatlab.org/nlab/show/state+on+a+star-algebra#Examples

Avra

I saw Halbert space, I closed the web page immediately

Semiclassical

lol. hilbert space is QM bread and butter

that said

finite-dimensional Hilbert space is just linear algebra

and most of what you do in quantum computing applications is finite-dimensional

Avra

the big bang for me is all atoms around abstract algebra

Semiclassical

just vector spaces with inner products

Avra

@Semiclassical. Abstract algebra is the reason for big bang theory

SmokenSieEinBitteChebaHitBits

22:51

Ted's not here. Party at Ted's house :)

Rithaniel

23:30

Will Ted be cooking?

SmokenSieEinBitteChebaHitBits

He was in here just a second ago. lol

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