Mathematics - 2021-08-03 (page 1 of 2)

AMDG

00:01

Anyways, I'm going to get back to working on this quotient algorithm and finding the greatest common integer among the set of integers that make up the interval defined by the quotient of the msb and lsb for all bits.

dc3rd

Had a quick question about framing a question in the right way to get the solution. The question asks me to find the angle that a stream makes with the horizontal at point $(\mathbf{a}, c)$ bold if intentional. if it flows in the $e_{2}$ direction at the point. Pertinent information to this is $z = f(\mathbf{x})$, $\mathbf{x} \in U \subset \mathbb{R}^{2}$. Also $f)\mathbf{a}) = c$ and $Df(\mathbf{a}) = (3, -4)$.

Should also add that the solution is $-\arctan(4)$.

My problem is I don't see how to "visualize" the stream against the horizontal properly.

this is perfect for a mathematica picture...........

shintuku

are quantifiers basic, or is there a way to prove, e.g., $\exists x P(x) \iff \neg \forall x \neg P(x)$? it makes sense in natural language, but I'm wondering if there is some sort of logical calculus for this

Ted Shifrin

01:05

@dc3rd keyword directional derivative

shintuku

@shintuku there's a tree-system in formal logic

oh cool thanks shintuku

dc3rd

hmm, so the directional derivative in this case is the vector $(3,-4)$.............so you saying directional derivative just now triggered the memory of you saying in one of your lectures of having to specify it as a unit vector....

leslie townes

i need a job where we do not record time in tenths of an hour. this is driving me crazy.

o.9

can someone explain this ansa to mi ?

1

A: Chance of getting a date

Use the fact that expectation is linear. A single coin flip has an expected value of $1/100$th of a date. To get an expected value of 1 date, flip the coin 100 times.

Ted Shifrin

@dc3rd huh?

dc3rd

01:13

never mind the second part about unit vector...I thought about what I had said...

Ted Shifrin

No, the first…

Expectation is linear. $E(nX) = n E(X)$.

Avra

@leslietownes. Which is better zoom or in class?

o.9

that allows us to conclude for $100$ there is $1$ expected date

Avra

@TedShifrin. Expectation is linear transformation?

o.9

Does that mean $100$ is the expected number of attempts we need to get our first date.

leslie townes

01:17

avra, in class. zoom is OK for presentation but not for conversation.

Ted Shifrin

Go read, $E(X+Y) = E(X)+E(Y)$ whether or not independent random variables.

leslie townes

which is what class ought to be.

o.9

I agree that the expected value is $1$ when we have $100$ attempts.

But that isn't really what is being asked

Ted Shifrin

It isn’t?

o.9

No we want the expected number of attempts till the first success

Ted Shifrin

01:19

No.

leslie townes

the linearity of expectation might go on my list of top 10 profound thoughts.

Avra

@o.9. What about this answer please?

Use the fact that expectation is linear. A single coin flip has an expected value of 1/100th of a date. To get an expected value of 1 date, flip the coin 100 times.

It's from the same page above

It seems logical

Since each date is independent, so we should multiply 1/100 100 times

In case date are dependent (don't think this is true), so we sum 1/100 100 times

dc3rd

Oh... I see what you meant Ted $D(f(a)) v = (3, -4) (0,1)^t$ you know what I mean in terms of the directional derivative...pardon the formatting

Avra

@dc3rd. This is directional deritivative in the direction of $v$

dc3rd

yes. with $v = e_2$

Avra

01:23

(0,1) is the unit vector and (3, -4) is the vector along it

@leslietownes. They might continue the closure, which means mooore zooms

@leslietownes. COVID cases are striking again

leslie townes

yeah, it's not good. my office had a plan to return to the office after labor day. they are reconsidering it.

o.9

I thought you got some days to rest after labour

to take care of the child

Avra

@leslietownes. It's serious! The virus has polarized itself to a new form

leslie townes

it doesn't help that i work in orange county, which is a hotbed of people who disbelieve in the virus. even if we're all vaxxed, the people delivering stuff to the office or visiting might not be.

Avra

@leslietownes. If they reopened it, it might be catastrophic

leslie townes

01:27

o.9 i got four weeks after my child was born. if i were a girl it would have been six.

our relatively small business has already lost two people to the coronavirus and it has been hard to replace them.

o.9

would you also get six if it was your wife delivering the child and you were a girl?

leslie townes

yes, i think so. there are probably legal issues with the agreement.

o.9

pog

leslie townes

almost anything i have ever gotten at work was due to the gay rights movement, and even if it wasn't the right thing to do, i would support it for that reason alone.

i think the smart thing to do if you run a large business is to make everything gender-neutral.

it's just easier and you don't get into the weeds of people's lives.

once you say 6 weeks for this and 8 weeks for that, you are asking for trouble.

i'm an attorney but i'm not offering legal advice.

i'm just commenting on what's out there.

o.9

why not?

I do not agree with the answer I linked previously :/

leslie townes

01:33

my now-wife and i were 'domestic partners' in iowa for a few years. thank you gay rights movement although we are hetero. it let her get on my health insurance. big thing in the USA. huge thing. hard to explain to people in civilized countries.

o.9

Glad you were able to do that :)

leslie townes

insurance is a weird creature in the USA. it is state-specific so what you get in iowa might not be what you can get in maine.

o.9

that answer also only works when $p$ is the reciprocal of a positive integer

Avra

haha

In one country, woman take 3 months

Now they shrink it to 4 weeks

o.9

in which country?

Avra

01:38

@leslietownes. It seems it's standardized !

leslie townes

companies do uniformize their policies to the extent they are able.

Avra

Russian

o.9

they don't standardize pay tho :'(

they try to scam you and you have to negotiate it

leslie townes

uniformization is generally a good idea. it's the concept of insurance.

o.9

fuckin' pain

but I guess they save a lot of money that way

Avra

01:40

Yep!

Here in the US is $$

o.9

what does that mean?

Avra

So people would like to come back to work ASAP ASAP SSSS

o.9

that cash is king?

Avra

It's very expensive

o.9

oh

Avra

01:41

So you can not take unpaid vacation for 3 months

o.9

I remember when I went to CA when I was 12 with my parents we had to bargain for a ton of stuff

for the house

cellphone plans, matresses, cars, car insurance, laptops

leslie townes

it's called freedom

shintuku

comrades, it is time to say we've had enough

Avra

@o.9. Yes. Here it's FREE as @leslietownes with BUTS

leslie townes

freedom of choice is what you got, freedom from choice is what you want

i'm going to spend the next 3 hours approximating devo riffs

o.9

01:44

You can also buy stuff from regular folk in Mexico

well, I guess salesman aren't regular folk

Salesmen are professionals

I don't really know what I'm saying

Avra

You have been practicing math for life and they have been bargining for life. Do $E(n)$

o.9

I guess I just don't like bargaining with professionals

shintuku

if only distribution and retail was the mandate of centralized public agency

o.9

no that would suck

but you can get people who have actual jobs to also do the sales

instead of having people that just talk shit for a living

leslie townes

my wife just discouraged my daughter from pulling something out of the refrigerator by saying "no, that's a wellness shot" which to me seems peak west coast

shintuku

01:48

fine comrade, if you enjoy being sold things by people who are trained to make you want things you aren't sure you need, suit yourself

leslie townes

i'm an american, i buy what i'm sold

Avra

true

o.9

I guess many contractors are not salesman but will also overcharge you if they think they can get away with it.

knowing the price of many things can be very helpfull I guess.

Avra

@o.9. 100%

NO MERCY

Unfortunately.

You have to be circumspect

shintuku

woah these logical trees are so damn useful

why haven't I seen these before

Avra

01:56

@shintuku. Which one ?

shintuku

Avra

MM

this is expressions tree?

What course?

shintuku

learning first-order logic

Avra

*expression tree?

WOW!

I did not expect that?

What course?

shintuku

i think people see this in an introduction to logic class

Avra

01:59

Oh! I thought introductory to graph theory

or compiler theory

shintuku

the above tree shows the following argument is valid:
$P \iff Q$
$Q \iff R$
$\therefore P \iff R$

which, well, is obvious but it's nice to have a sort of calculus for these things

Avra

@leslietownes. Have you seen these in any math class before?

Is $(\log(n)) ^ {\frac{n}{3}}$ same as $\log (n^{\frac{n}{3}})$? I guess there is a typo here?

leslie townes

02:26

i've seen those trees, not in any class i was in but in general, yes.

copper.hat

@TedShifrin Sorry to hear that. Hope it goes away quickly.

robjohn

02:50

@Avra Those are definitely not the same

o.9

I don't understand how that expression can be $\mathcal O(n)$

robjohn

03:23

what expression?

o.9

Either of them

2

Q: Growth rate of $(\log{n})^{3n}$ and $(n)^{\frac{n}{3}}$

To find the growth rate of $(\log{n})^{3n}$ and $(n)^{\frac{n}{3}}$, I took the log of both: $(\log{n})^{3n}$: $$\log (\log{n})^{3n} = 3n \log{\log{n}} \in O(n \log n)$$ but the answer I have says it's $O(n)$. $(n)^{\frac{n}{3}}$: $$\log (n)^{\frac{n}{3}} =\frac{n}{3} \log{n} \in O(n \log n) ...

This must be trolling right? math.stackexchange.com/questions/4215481/…

Wolgwang

> The diagonals of a parallelograms are given by the vectors $\vec A=3\vec {i} + \vec {j} + 2\vec {k}$ and $\vec{B}=\vec {i} - 3\vec {j} + 4\vec {k}$. Find the area of the parallelogram.

$$| \vec A|=\sqrt{14}\\| \vec B|=\sqrt{26}$$

Area=$\frac12 d_1 \times d_2=\frac12\times \sqrt{14}\times \sqrt{26}=\sqrt {91}$

Is this wrong?

robjohn

$(a-b)\times(a+b)=2a\times b$,

the area is $\frac12$ the cross product of the diagonals

Wolgwang

That gives $5\sqrt 3$

robjohn

yep

Wolgwang

03:37

But what is wrong in the above approach? :-(

copper.hat

you need more $\sin$

robjohn

I don't see what you are doing above. explain

@copper.hat more sin... so hedonistic!

copper.hat

that's me to a tee :-)

always looking for the indulgent angle

Wolgwang

I calculated the magnitude of diagonals (which gives the length) and used the formula :Area of parallelogram=1/2 product of diagonals.

robjohn

not the product, the cross product

copper.hat

03:39

hence the need for $\sin $ :-)

robjohn

as copper says.

in a rhombus, you would take the straight product.

or a kite

Kite (geometry)

In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other instead of being adjacent. Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape and which are in turn named for a bird. Kites are also known as deltoids, but the word "deltoid" may also refer to a deltoid curve, an unrelated geometric object. A kite, as defined above, may be either convex or concave, but...

Note that the diagonals are perpendicular

so the sin is $1$

full sin

Wolgwang

Oh! So stupid of me. Thanks :-)

fia presheaf

I can not figure out how small to make the inner radius of an annulus to make a holed cross-cap from a Mobius strip.

Can someone help me with the cross-cap please?

Im cutting the along the radius and putting the ends on like an intersection, but in every picture i find of the cross-cap there is just 1 loop for a boundary at the posterior side or behind the intersection. My paper has a hole there in a figure 8 and at the front of the intersection....

fia presheaf

04:09

I dont think i was self intersecting the inner radius. That would mean that I've just got to lengthen some parts of the figure 8 if im right.... Im still looking for how that part is 1 loop though!!

littleO

I've never used chat before so let me know if I'm doing this wrong. I am wondering if anyone agrees or disagrees with the following idea: Whenever you have a real symmetric matrix $A$, you should always think of $A$ as representing a quadratic form ($x \mapsto x^T A x$ ) rather than as representing a linear transformation ($x \mapsto Ax$).

fia presheaf

What I read is 'chatting' and 'asking questions' have different guidelines -_-

leslie townes

it is helpful to think of the two things together.

particularly if A is unbounded.

in many applications the linear operator is of central importance and the quadratic form is secondary.

littleO

For example, the Hessian $Hf(x)$ is a good example of a symmetric matrix, and it appears as a quadratic form in the Taylor expansion of $f$. It seems that it is more illuminating to think of $Hf(x)$ as representing this quadratic form rather than as representing a linear transformation.

Is there a good example where a symmetric matrix $A$ arises in a context where it's more illuminating to think of $A$ as representing a linear transformation?

leslie townes

i agree there but would not extend this vibe to all linear operators.

littleO

04:16

not even to all symmetric linear operators?

leslie townes

yeah, not even there.

but i'm an idiot, so i should point that out.

littleO

haha. No, it's helpful to hear what you think about this

leslie townes

d/dx is unbounded on basically every domain. i don't see what you gain by emphasizing any quadratic form.

littleO

d/dx isn't self-adjoint, though

leslie townes

you can tweak parameters so that it is, but i see your point

o.9

04:20

more illuminating to think of $A$ as a linear transformation instead of a symmetric matrix?

leslie townes

any step away from the matrix world is welcome.

o.9

You can think of $A$ as a flow network

Is that a good thing?

littleO

I'm not familiar with the flow network viewpoint. Would $A_{ij}$ tell you like the rate of flow from node $i$ to node $j$ in a graph?

o.9

yes I think an example of that viewpoint is like when you have a doubly stochastic matrix in a transition matrix for a finite markov chain or something like that.

fia presheaf

Whoa

o.9

04:28

but I think you wanted examples where it's better to just view something as a linear transformation right?

littleO

yeah, I'd like an example where a symmetric matrix $A$ arises and it's more illuminating to think of $A$ as representing a linear transformation than as representing a quadratic form

fia presheaf

the Timestamps are not even showing up now.. .

littleO

My hypothesis is that it is always more illuminating to think of a symmetric matrix $A$ as representing a quadratic form. I could be wrong of course.

o.9

But why look at it simply as a linear transform when you can see it as at least a diagonal matrix with orthonormal basis.

fia presheaf

Is pro stack exchange like this I might check it out agian

o.9

04:31

Would spectral graph theory be an example of what you are looking for?

adjacency matrices are symmetric

and they work with the spectra, maybe that counts as looking at it as a symmetric matrix?

littleO

That could be a good example, but I don't know much about spectral graph theory

That is worth looking into though. I should probably read more about that.

Do you have any favorite books that introduce spectral graph theory? I know Strang at least touches on it in some of his books.

fia presheaf

Is non spectral sequences in Strangs book

o.9

Sorry I don't I just know the stuff that my advisor needed me to learn

I know David Godsil has a book and he also answers stuff on this site, I think I used it once to check something and it worked for me.

SAJW

I have found a trick.

littleO

Oh good to know, thanks

SAJW

04:35

In some university tests you are allowed to bring either a book or a readable piece of paper with you.

Print your own book, gg

but probably there are only some books allowed, so...

o.9

are you allowed to bring a non-readable piece of paper?

SAJW

well, of course, but you can't bring a magnifying glass

wonder if you can bring glasses without a seeing-weakness

o.9

When I went to the ACM-ICPC finals there was a page limit for the written stuff you could take in.

But we took less pages cuz we were kind of nooby :'/

and didn't really know what else to put in there

SAJW

print out the internet as a book

o.9

but there was a page limit

fia presheaf

04:41

@SAJW how can you print out the internet as a book

SAJW

but not a font-limit

o.9

It was 25 pages max

SAJW

i'm only joking ;)

o.9

This document may contain up to 25 pages of reference materials, single-sided, letter or A4 size, with pages numbered in the upper right-hand corner and your university name and team name printed in the upper left-hand corner. Text and illustrations must be readable by a person with correctable eyesight without magnification from a distance of 1/2 meter.

It may include hand-written comments and corrections on the fronts of pages only. The document must be in some type of notebook or folder with the name of your institution on the front.

apparently they took that shit kinda serious :/

fia presheaf

you already made a joke rofl

o.9

04:45

what is a person with correctable eyesight?

fia presheaf

there mongrels then

SAJW

I'm not sure, if I can read this chat from 1/2 meter

o.9

why did you delete your account fia? now everything renders weird :'(

SAJW

sure text, but not $e^{e^x}$

fia presheaf

Im probably banned for beginning topology here

I just care about learning about the cross cap so i bet koob face tracked my information to here

o.9

04:49

It says you have an account on math.se with 20 rep but when I click it it says it doesn't exist.

fia presheaf

thats odd

I blame koob face

SAJW

cu later

o.9

that's a virus?

fia presheaf

because i dont have another reason > _<

o.9

I just learned about koobface 30 seconds ago

cu later ali gater

fia presheaf

04:51

What is a virus - yes koob face is one

o.9

I don't know, malware

fia presheaf

Okay pce.

Absolutely

It is a worm-virus

o.9

what makes it a worm?

fia presheaf

its the way of the cross caps

o.9

Sorry I'm sleepy I'm going to sleep now

Shashaank

07:29

@robjohn Ok I see what you mean. (1+i, 1+i) is not in that vector space, so closure property won’t be satisfied. But what I cannot understand is how can (i,-i) be a basis vector in R^2. “i” itself isn’t a real number so the vector (i,-i) is itself not in R^2 space just like (1+i, 1+i) isn’t in R^2……What I can’t get is how did you get (i, -i) to be a basis vector because to me it isn’t in R^2 just like (1+i, 1+i) isn’t……Could you elaborate this a bit, please

Omni man

09:34

Hello I was banned for like 7 years. Nice to come back.

I need to go back bye guys. I will be banned for like 3 more m****s years.

robjohn

@Shashaank The vector space has dimension $2$ because it has $2$ basis vectors, but it is not a subspace of $\mathbb{R}^2$. It is a vector space over $\mathbb{R}$.

robjohn

09:58

This is just the same as Hermitian matrices being a vector space over $\mathbb{R}$. $2\times2$ Hermitian matrices have conjugate elements in the $(1,2)$ and $(2,1)$ positions; that is, if you really mean Hermitian.

Hermitian matrices have complex elements, yet they form a vector space over $\mathbb{R}$ of dimension $6$.

Samyak Marathe

10:40

Greetings

Shashaank

11:09

@robjohn Ok, I see what you are saying. So is it totally correct to say that this vector space given by you, having a basis vector (i,-i) and (1,1) i

*is a perfectly good vector space over reals

robjohn

cool

Shashaank

*of reals. The only thing is that this vector space is different from R^2….is that correct

I was thinking that when we say a vector space over reals, we mean that not only the coefficients, the scalars that multiplies with the vector has to be real but even the number that we write inside “(“ and “)” (like (1,0) or (0,1) ) have to be also real numbers. That is why I was thinking that how can (i,-i) can be a basis vector in the space over reals. But I see you are saying that it could be a vector space but it’s just that it’s not the same as R^2 ..

@robjohn I find it strange. I always thought that if we say, a vector space over reals we also mean that the elements of the tuple or the numbers which appear between “(“ and “)” have also got to be real number

robjohn

11:33

The complex numbers form a vector space over the reals.

Related question: Complex numbers as vector spaces

Peter John

13:38

Can we view the ordinary derivative (one variable case) a special case of total derivative?

Avra

14:27

@robjohn. Similarily to my question I posted, growth rate of $n^{a}$ is $a\log{n}$

Peter John

14:46

2

A: Why can't I take the derivative of $x^x$ as $x(x^{x-1})$?

Taking the basis as variable, the derivative is $x\cdot x^{x-1}$, taking the exponent as such the derivative is $x^x\cdot\ln(x)$. Adding both gives the derivative of $x^x$, namely $$\left(x^x\right)'=x^x+x^x\cdot\ln(x).$$ Astonishing at the first glance, isn't it?

Is this a valid argument?

leslie townes

you're right that you can do it via the multivariable chain rule.

d/dt F(t,t) = F_1(t,t) + F_2(t,t) where F_1 and F_2 are partial derivatives with respect to the first and second variable.

F_1 meaning hold y constant and F_2 meaning hold x constant. it feels like magic.

Peter John

@leslietownes You mean $F(a,b) = a^b$?

Xander Henderson

@PeterJohn Yes, that is how $F$ would be defined.

leslie townes

yeah.

Peter John

15:04

@leslietownes Why do we say 'total' derivative not just derivative? Is there any difference except the dimension?

leslie townes

terminology varies a lot. i think 'total' is often used if partial derivatives are also in the picture. when context is clear i just say 'derivative.'

thankfully i don't teach anymore, so the context is always clear :)

Peter John

@leslietownes So it's just a terminology. I can just say total derivative for one variable function right?

leslie townes

i wouldn't stop you from saying that. :)

in one variable there's basically only one thing any type of derivative could mean, so one tends not to use qualifiers.

but you can put them there.

shintuku

if you said total derivative for one variable function, i'd think you made a typo because I only see it in a multivariable context

but from context i'd understand

leslie townes

if you were teaching a single variable calculus class people would wonder why you were saying 'total' and what non-total derivatives you weren't talking about.

but in the comfort and privacy of the internet i am fine with it.

robjohn

15:29

we need total concentration

@PeterJohn I have added an answer that is a bit more explicit, but follows what that author is trying to say.

leslie townes

that's a good answer.

no offense to other respondents but a lot of junk in that chain of answers

robjohn

16:12

@leslietownes ~~would I have written it otherwise?~~ Thanks!

leslie townes

haha

all of your stuff is great, as you well know.

which brings me to the key point, which is: where's grandad?

copper.hat

hiding in the pool

123

Hi All..

If marks in Eng = 12/25 , Maths = 40/50 , Science = 15/25 then Marks obtained/Total maks = 67/100 = 100%

copper.hat

going back in a few days

123

If equivalent fractions are taken for all subjects to out of 100 then, Eng = 48/100 , Maths = 80/100 , Science = 60/100 , Marks obtained/ Total Marks = 188/300 = 62.66% .

Why??? They are unequal

Hello @copper.hat

copper.hat

16:16

hi 123 :-)

robjohn

@123 the weights are different in the first computation

123

How we take weight factor in calculation. I also thought about weight is the problem.

robjohn

16:51

In the first computation, the math is weighted twice as much as the English and Science.

in the second computation, the math is weighted equally with the other subjects

since your math score was the highest, the first computation gives 67% and while the second gives 62.66%

@123 it's as easy as 1-2- oh, nevermind.

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$Mathematics$ Mathematics