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

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8:08 PM

Maybe my question is un-answerable?

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.

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%

yeah, that kind of calculation.

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

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

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

8:16 PM

lol

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

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

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

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.

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.

8:23 PM

what? no way

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

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

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

never steal from anyone

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

8:31 PM

(also we have no shared chores)

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

which is the same as the other one backwards

@Semiclassical. I am going crazy :/ thanks

This is the third time I am asking same question

@Semiclassical. No idea what is wrong with me

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

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.

huzzah for dummy indices

yeah, it really depends on where you're at

ultimately you need to be able to manipulate them confidently

8:33 PM

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

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

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

sometimes!

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.

sometimes not

8:35 PM

@Semiclassical. Really?

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

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

@Semiclassical. Is abstract algebra mostly algebric manipulations?

even regular algebra isn't mostly algebraic manipulations :)

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

and understanding the consequences of those rules

that's my hot take of the day.

8:37 PM

hey, sometimes regular algebra is

understanding how to complete the square, for instance :P

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

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

@Semiclassical. OMG. This is horrible

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.)

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

8:40 PM

no

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

like constraining the domain?

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...

This is horrible

i guess i should be talking about matrix rings as well

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...

make the generalizations stop

2

8:41 PM

When you say ring theory i REMEMBER RING MOVIE

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

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

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

specifically via the gamma matrices

@leslietownes im doing that shhhh

im making a movie called The matrix ring

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

8:44 PM

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

same difference

physics

mostly quantum foundations stuff now

you like quantum physics?

yep

though that's a very broad range

Wow this is very nice area

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

yeah, though a lot of it is just linear algebra

though the physicist notation takes getting used to

8:47 PM

hahaha

has anyone generalized quantum field theory?

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)

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

quantum theory is hard to understand

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

8:49 PM

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

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

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

infinitely particles in the model?

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

is this path integral formulation?

9:00 PM

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)

@Semiclassical. Do you like measure theory?

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

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

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

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

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

9:02 PM

now seems to be the same

@Semiclassical. Wow!

lol

i'd have a hard time disagreeing

@Semiclassical. OMG

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

9:03 PM

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

yeah

OMG

They never thought this is math departments though

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

yeah they have

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

9:05 PM

oh

that is mostly it in pure math sectionb

you do see Kolmogorov foundations if you do grad probability theory

This is how it goes in my prev department :(

sigma algebra hah

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

9:06 PM

We always felt that we don't understand

they lied to me

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

@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?

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

Why they don't include that in the syllabus

9:08 PM

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

OMG

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

yeah

knowing measure theory is a high threshold

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

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

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

@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

9:12 PM

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

I saw Halbert space, I closed the web page immediately

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

the big bang for me is all atoms around abstract algebra

just vector spaces with inner products

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

2 hours later…

SmokenSieEinBitteChebaHitBits

10:51 PM

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

11:30 PM

Will Ted be cooking?

SmokenSieEinBitteChebaHitBits

He was in here just a second ago. lol

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