/sci/ - Science & Math

i'm learning algebra, currently on systems of linear equations. the book i'm using did this in one of its examples. why and how did it remove the minus from the denominator?

>>15951217
Can you follow why [math]\frac {a}{-b} = \frac{1}{-1}\frac{a}{b} = -\frac{a}{b}[/math] are true?

>>15951220
i think so, i get that you could also do something like this
[math]
\frac{a}{-1*b}
=
\frac{a*1}{-1*b}
=
\frac{a}{-1}
*
\frac{1}{b}
=
-a\frac{1}{b}
[/math]

the book is also did pic related, i'm aware that the 4 outside the parenthesis and the denominator inside are canceling out out, but why does it end up as a negative before even using the distributive property? chatgpt seems to be taking another route but it makes -3x/2 into a positive for some reason

>>15951431
> but it makes -3x/2 into a positive for some reason
Because the minus sign is in front of the whole expression. It might be clearer to you if it was written as [math]\dfrac{-(2 - 3x)}{2}[/math].

>>15951431
wait after typing this i realized the book must be conveniently placing the minus sign into the denominator again before canceling the 4 and 2 out? if i don't do this the whole equation ends up as -11x = -2

>>15951217
>>15951431
I'm honestly curious who taught you math. You seem to be missing some incredibly beginner level knowledge. How anyone can not know that [math]-\frac{x}{y} = \frac{-x}{y} = \frac{x}{-y}[/math] is beyond me. This is frankly so basic it's not even math, it's notation.

Chemistry: if i have a bunch of used transdermal medication patches how would i go about extracting the left over medication from them?

what do the straight brackets generally mean in this context

>>15951616
never mind it's definitely absolute value lol sorry

>>15951528
been more than a decade since i last touched anything related to maths, and my middle school teacher probably just taught us that without any context as to why that's valid. i'm also a brainlet so fractions just mess my head up

>>15951180
>OP pic
retardanon lives on

>>15951714
seems like your negative sign confusion is just from reading the notation, other anons cleared it up. As for 'fractions messing your head up,' A*B is best intuited as a rectangle. One edge is A long, other is B long, and the area of the rectangle is A*B. Just draw out 2*3 = 6. Then 6/3 could be seen as "if a rectangle of area 6 has one edge of length 3, what is the length of the other edge?" Playing with the graphs along the algebra shows you pretty intuitively why
A/1 = A
A/0 = infinity, or 'Not a Number'
0/0 = indeterminate
0/A = 0

What is the optimal way to number an infinite list for readability? Take a driver's license number, for example. There's going to an infinite number of them. In California, a driver's license number starts with a letter and then 7 numbers. Why not two letters and 6 numbers?
I used to work in a warehouse and the boxes I would scan were labeled with purchase order numbers. A purchase order is a single purchase/transaction that a business makes(usually for a variety of goods in one order). Our purchase orders were just 1, then 2, then 3, 4 etc. Our business had been around for over a decade, so we were well past our millionth purchase order. Rather than looking at all of those digits, it seems to me that it would have been better to abbreviate our purchase order numbers into a system that uses numbers and letters, similar to what I mentioned earlier with driver's license numbers.

What is this sort of numbering called?

How do I solve this?
Rta means answer.

>>15951180
I also saved that pic, kek

>>15951180
kek I also saved that pic

About a year ago, I asked on /mg/ if there was an easy way to prove that no finite one-variable polynomial was equal to sqrt(x). The answers I got spoke about classifying clones according to signature, which is a bunch of universal algebra stuff that I don't know about since I'm an mechanical engineer.
I finally tried reading some universal algebra last week and it is really boring. Is there any easier branch of math that can answer my original problem?

>>15952111
[math]\sqrt[3]{9y^2} \times \sqrt[3]{3y^7} \\
= \sqrt[3]{9y^2 \times 3y^7} \\
= \sqrt[3]{27y^9} \\
= 3y^3[/math]

>>15952149
Thank you!!

>>15952149
Btw why did the y9 turned into y3?
I understand the 3x3x3=27
But the 9 and 3?

>>15952145
>Algebra
Use Analysis, faggot. All finite one-variable polynomial functions are differentiable. The square root function on the other hand is not differentiable in 0.

>>15952204
What do you get when you multiply [math]y^3 \times y^3 \ times y^3[/math]?

>>15951180
[math]
\text{How would society be different \\ if we communicated through our \\ sense of sight rather than our sense of \\ hearing? How would long distance \\ communicating be different? How \\ would television be different? How \\ would we even communicate? \\ Flailing our arms around? Fun question I \\ thought of today in class. Have \\ at it bros.}
[/math]

>>15952243
27? 3x3x3
Explain to me what happened to the 9 for it to turn into 3, please

>>15952254
27 is what you get when you take [math]3 \times 3 \times 3[/math].
When you multiply two exponentiations of the same number, you add their exponents together. [math]y^3 \times y^3=y^6[/math], and then [math]y^3 \times y^6=y^9[/math]. It's exactly the same idea.

>>15952254
Because of the laws for exponents: [math](x^y)^3 = x^y \cdot x^y \cdot x^y = x^{y + y + y} = x^{3y}[/math].

Now replace y by 3.

Which bones down your foot you can pull when cracking your toes?

>>15952261
>>15952262
Got it! Thank you again

>>15951972
Alphanumeric?

What about these too?

>>15952291
Try to write [math]\sqrt[n]{x}=x^{\frac1n}[/math] and use some properties of exponents. For example, [math](x^a)^b=x^{ab}[/math].

>>15952299

>>15952305
You can do it, frogposter-kun!

>>15952311
Na, fuck math, I'm quitting
Imma go study law or something

>>15952314
Cool, I want to sue my school for taking all my money and teaching me nothing.

>>15952299
[math] \sqrt[n]{x} = x^{\frac{1}{n}} [/math]
There, I tried to write that

>>15952291
[eqn]\frac{ \sqrt[3]{ \sqrt[4]{x^{19}y^{12}} } }{ \sqrt[12]{x^7} } \\
= \frac{\sqrt[12]{x^{19}y^{12}}}{\sqrt[12]{x^7}} \\
= \sqrt[12]{\frac{x^{19}y^{12}}{x^7}} \\
= \sqrt[12]{x^{12}y^{12}} \\
= xy[/eqn]

>>15952246
are you listening to this post or reading it?

>>15952329
Its so hard to do by myself but seems sooo easy when I see like this...

>>15952338
[math]
\text{I am doing your mom.}
[/math]

>>15952338
Everything is easy when someone else is doing it

>>15952368
This is true for many things in life: flying a jet plane, brain surgery, and receiving a blow job.

>>15952405
at the same time?

>>15952329
More like |xy|. Assuming x,y are real.

Is math fixed? Or can it be changed?

>>15952556
No need to take the abs value. Given the specimen answers I would assume it states elsewhere in the questions that both variables are reals and > 0.

Here's my theory on how to become rich as fuck:
Find anywhere with a fuck ton of money, figure out how they make that money.
Odds are it's some kind of super numerous transactions: coal mines sell a fuck ton of coal.
So look anywhere there are huge amounts of numbers that scale with currency and then per number get them to spend some of that number on you.
Say you know of a popular book, if you post a video about that book and have a commission link to that book you've increased your frequency for making money by interacting with the frequency of purchase.
If you did this with a book with a lower frequency that no one cares about then you get nothing.
However, if it a possible thing then there is a rate of competition which leads to portioning of your own profit attention which clearly reduces your frequency for profit.
So then there is a means to measure how exactly to make money.
Therefore, getting rich quick does indeed exist.
What do you guys think about my theory?

I love smoking cigs and doing cocaine
is it really that bad for me?
I feel great and my mood is probably better than the average coffee drinker

I'm meant to prove the Fatou's lemma. So,
[eqn]
\begin{align*}
\int_{\Omega} \left(\lim_{k \to \infty} \inf f_{k} \right)\, d\mu = \lim_{k \to \infty} \inf \int_{\Omega} f_{k}\, d\mu.
\end{align*}
[/eqn]for a sequence of measurable non negative functions [math]f_k: \Omega \to \overline{\mathbb{R}}_{+} [/math]. We already proved that [math]\lim_{k \to \infty} \inf f_{k}[/math] is measurable and we are allowed to use the Monotone Convergence Theorem. So I don't get what they want me to do here?

Isn't it trivial or am I missing something?

>>15952706
Fatou's lemma and the MCT are very closely related. If you've proven one the other is an easy corollary.

What's it called when we split a fraction like this

[math]
\frac{5+4}{5}
[/math]

to this?

[math]
\frac{5}{5}*\frac{4}{5}
[/math]

And is there all there is to it? we can split any fraction like so?

>>15952776
Sorry meant to type + not *

>>15952776
>>15952780

Formally it's called the "distributive property," but you can just say "split the fraction" sounds fine. Yes you can split any fraction like that

I was curious what the Riemann one is referring to, wasn't able to find anything on my own. Anyone more versed in math history that knows?

>>15952805
https://en.wikipedia.org/wiki/Dirichlet%27s_principle

>>15952854
Much appreciated Anon!

>>15952405
Hence the use of the word "everything"...
I guess you still haven't figured out how to make reading easy for you yet and it's still just easy when someone else is doing it

>>15951180
oh boy I remember that mentally challenged anon, good times

I recently created a model for cancer but I swear to god, I was looking at studies earlier on nature cancer and now they aren't there. How is that even possible? They basically never take down studies.

can someone go to nature cancer and list what studies have been published in 2024, I feel like I'm going crazy ;3

>>15951972
For human-readability, words. Tens of thousands of words, 2 word identifiers gets you hundreds of millions. For practical applications, just recycle identifiers. Who cares if the same one is used on an order from 5 years ago.

suicide suicide suicide

>>15953345
U good anon?

The justification for (2.12) in Apostol is nonsensical to me.
How would multiplying by the reciprocal of [math]\frac{\sin\theta}{\theta}[/math], which is shown by the fundamental inequality to be greater than [math]\cos \theta[/math], result in a greater value?

>>15953692
Start from the fact [math]\cos{\theta} \le 1[/math], then if you integrate this expression you get the inequality [math]\sin{\theta} \le \theta \implies 1 \le \dfrac{\theta}{\sin{\theta}}[/math].
So if you combine them you get: [math]\cos{\theta} \le 1 \le \dfrac{\theta}{\sin{\theta}}[/math].

How do I learn to read DC circuits? Any good resources? I know all the math and theory but I don't know how to read the circuits to solve what is required by the problem.

if a=b±c and b and c are integrable, then does that automatically mean that a is integrable? Can someone link me the rule if it is?

>>15954117
It's true because of the triangle inequality.
[eqn] \|a\|_{L^1} \leq \|b\|_{L^1} + \|c\|_{L^1} [/eqn]

>>15954117
Yes indeed. Look at Thm. 7.1.8 part 2 and its proof:

https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch7-riemann-integration.html

>>15951591
Fentanyl patches? Just use isopropyl alcohol.

>>15952246
the concept of "body language" would be bound to talking and listening

if i pee on a poop stain in a toilet will the poop germs ride the pee stream up to my penis?

I see that both are indeed equal by plugging something for u, but I don`t know what mathematical rule was used to take that top u out of the fraction and turn the divide by u/sqrt u into just a + sqrt u/1.

>>15955294
[math]\frac{u-1} {\sqrt{u}} = \frac{u} {\sqrt{u}} - \frac{1} {\sqrt{u}} = \frac{ \sqrt{u} \sqrt{u} }{\sqrt{u}} - \frac{1} {\sqrt{u}} = \sqrt{u} - \frac{1}{\sqrt{u} }[/math]

If playing rock-paper-scissors against a computer who only uses random picks (as random as computers can get), can probability be used to ultimately win more against the computer?
For example if the computer picks rock, would i be correct in assuming that rock is less likely to be picked in the next round, allowing me to adjust my pick accordingly? Or is trying to predict future events based on the previous sequence useless here?

>>15955348
Depends on how it's determining its inputs, but since it'd most likely be pseudorandom, the answer is sort of a vacuous "yes". You could technically figure out the most likely outcome... but it would be extremely impractical and you're better off just writing it off as the gambler's fallacy than trying to put in the effort, unless you're dealing with a very, very rudimentary RNG

>>15955348
> is trying to predict future events based on the previous sequence useless here?
Yes. That is what random means.

>>15953635
I am now

>>15955348
You could program a computer to be exploitable like that but it wouldn't happen by default, it doesn't happen with dice.

Why does this 3D presentation of the smith chart involve two torodial shapes meeting at the point of infinity or whatever the fuck it's actually called?

I suppose each toroid represents electricity and magnetism respectively, but them meeting at this precise point seems to heavily suggest there actually is a nonphysical ether (with infinite potential contained within it) driving the force of electromagnetism, and can hardly be called a coincidence

>mdpi.com/2073-8994/10/10/458

In a congruence-permutable variety V is the Malcev term unique, in the sense that for any two Malcev terms m and n, V satisfies the identity m(x,y,z)=n(x,y,z)?

>>15951180
why is the earth's magnetic field weaker than a common fridge magnet ? I read in my physics book that size of an object affects the magnitude of the magnetic field but if thats the case shouldnt earth's field be stronger? Does it have to do with the amount of density of moving electrons in the fridge magnet comparted to the earth?

>>15956592
magnetic field strength weakens drastically as the separation of the two objects in question increases, just like gravity.
Now consider how close your fridge magnet is to your iron filings or whatever, in contrast to their distance to Earth's core

>>15956594
so if im understanding you correctly, if our moon had a magnetic field which interacted with earth's. the earths field would be stronger

>>15956592
Because the mechanism producing each of those magnetic fields is completely different. One is from liquid metal convection currents inside the Earth, the other is from macro-scale spin alignment.

How do I do this?
the hint doesn't help because I'm not sure what n and n+1 (or n-1 and n) would mean in terms of proving associativity. I guess it would be continually placing more and more parentheses but how would that be represented by induction

>>15956753
Use strong induction.
Any product of the a_i is of the form
(...)*(...)
if you look at outermost multiplication. In each of those (...) brackets there are less than n+1 terms so you can use the induction hypothesis to sort them so you get something like
(a_1 * (...)) * (...) = a_1 * ((...)*(...))
Now you use the induction hypothesis again on everything but a_1 and you're done.

>A line passes through the points (3,3) and (1,10). Find the equation of the line and write it in slope-intercept form.

[math]
m=\frac{10-3}{1-3}=-\frac{7}{2}
[/math]

[math]
y=-\frac{7}{2}x+b
[/math]

[math]
3=-\frac{7}{2}3+b
[/math]

[math]
b=\frac{27}{2}
[/math]

[math]
y=-\frac{7}{2}x+\frac{27}{2}
[/math]

What in the fuck am I doing wrong here? The book's answer is

[math]
y=-4x+14
[/math]

>>15953136
>Who cares if the same one is used on an order from 5 years ago.
this is a shitty xkcd comic waiting to happen

>>15956594
>magnetic field strength weakens drastically as the separation of the two objects in question increases, just like gravity.
well unless I'm mistaken gravity weakens with the second power of distance whereas magnetism weakens with the EIGHTH power of distance. so not quite the same, lol.
that's quite handy because the "base strength" of magnetism is about 100 million billion trillion times stronger than gravity anyway (hence why an ordinary fridge magnet can lift iron filings off the floor against the entire strength of Earth's gravity)

>>15957303
Nothing, you are correct and the textbook is false (or you are looking at the wrong answer lol)

>>15957353
You are mistaken. Very badly so.

It might help your understanding to recall you have no capability to meaningfully vary the distance associated with gravity without spaceflight. Or you might be very drunk.

>>15957311
it's common practice to recycle employee numbers, and I'm not sure what you mean.

>>15957353
>>15957393

Both gravitational and electrostatic force follow the inverse-square law with respect to distance between 2 points.

[math]Fgravity=G⋅m1⋅m2 / r^2[/math]
[math]Felectrostatic=k⋅∣q1⋅q2∣ / r^2[/math]

If I put a white hot nickel ball in space will it cool much more slowly since the only way for transferring its energy is through light/radiation?

I assume the radiation from the sun would actually keep it warm longer as well but I'm not sure

>>15957505
yes, things in an atmosphere cool faster because of convection. it's why blowing on hot things works

Not sure where to start but I need to relearn math.
Whatever a computer scientist would need. I have no idea what logarithms are or the big O.

all i literally know is middle school tier math, maybe grade 9.

is Khan academy decent for this?

>>15957573
Professor Leonard is way better than Khan academy for learning undergrad math. Just find a good algebra 1 or 2 book and start from there.


https://openstax.org/details/books/college-algebra-2e

https://youtube.com/playlist?list=PLDesaqWTN6ESsmwELdrzhcGiRhk5DjwLP&feature=shared

>>15957505
You are correct. Which is why the whole Hollywood trope of someone instantly freezing when thrown into space is a myth. It would take a day or more for the body to turn to ice. It's the lack of air that would kill you, while in the mean time you would suffer depressurisation symptoms which would be rather unpleasant (but you wouldn't explode, another movie myth).

>>15956257
bump

How regularly do you use slope intercept, point slope and standard form equations in more advanced math?

>>15958032
Don't remember the last time "standard" form came up
Point-slope is mostly in the context of calculating tangent lines
Slope-intercept is still by far the most common, at least in my experience, probably because it's the one that plays the nicest with calculus as a whole

Are stochastic processes and functionals analogous?
Aka, are what stochastic processes are to random variables analogous to what functionals are to functions?

>>15958032
To be honest, they don't directly come up, as in you won't be asked to find them or be quizzed on them.

You're taught them and made to do exercises with them to develop your intuition and skill with algebra and linear functions; to give you practice with manipulating algebraic expressions and to show that the forms really are equivalent to each other

[eqn]\zeta(3)[/eqn]How do you evaluate it without sounding mad?

>>15958301
It's defined as an infinite sum. If you want an approximation, truncate the sum. Pretty simple

1+ 1/8 + 1/27 ~ 1.2
Good to two sig figs already

anyone know where i could find this?

>>15958349
http://library.lol/scimag/10.2307/27843007

>>15958365
thanks, i think thats the right book, my uni's library has it.

>>15958294
I wouldn't say so. It depends on your definition of functional, but to me that means a function from some space [math]X[/math] into, say, [math]\mathbb R[/math]. A regular function would be a map [math]f:X \to Y[/math].
Random variables are more like functionals in that often [math]X:\Omega \to \mathbb R[/math].
Stochastic processes are just multivariable mappings: [math]W: I \times \Omega \to \mathbb R[/math] could be a stochastic process, or, say, [math]W:\Omega \to \{w: I \to \mathbb R\}[/math].

>>15958365
<3

>>15958466
forgot the fuggin picture omg

What steps do I have to follow to get that result?

>>15958714
try raising both sides to the power of 4 and then raising that result to the power of 3

>>15958714
Anyone? Bump

>>15958883
Dont know how to. Do it for me please

>>15958895
no

>>15958960
lol dumb ass
""Try raising both sides to the power" yeah really fucking helpful, kys

>>15958714
>Simplify the LHS:
[eqn]\sqrt[4]{3 x^2} \sqrt[3]{9x^8} = 3^{\frac{1}{4}} x^{\frac{2}{4}} 3^{\frac{2}{3}} x^{\frac{8}{3}} = 3^{\frac{1}{4} + \frac{2}{3}} x^{\frac{1}{2} + \frac{8}{3}} = 3^{\frac{11}{12}} x^{\frac{19}{6}} [/eqn]
>Simplify the RHS:
[eqn]x^3 \sqrt[12]{3^{11}x^2} = x^3 3^{\frac{11}{12}} x^{\frac{2}{12}} = 3^{\frac{11}{12}} x^{3 + \frac{1}{6}} = 3^{\frac{11}{12}} x^{\frac{19}{6}} [/eqn]
They both simplify to the same expression so they are equal.

>>15958983
I know they are equal.
X = Y
The point of the exercise is to transform X into Y. Not to solve X and Y.
You are solving x and y, I need to turn x into y

>>15958996
have you tried, say... undoing the steps once you show them to be equal?

>>15958998
Thats the point. I dont know the steps, I have to apply the steps myself but I dont know how to.

>>15959000
do exactly what >>15958983 said and go through the second one in reverse

>>15959005
I DONT KNOW HOW TO.
Am trying to learn what properties to apply to get the first expression to look like the second one.
Dont tell me to do it because I dont know how, I'm here to learn

>>15959014
for nonnegative a and b, [math]\sqrt[a]{b} = b^{ \frac{1}{a}}[/math]
everything else follows from normal rules of exponentiation

>>15959021
Solve it for me please, I'm stuck
Transform the first expression (4√3x2.3√9x8) into (X312√311X2)
>>15958714

>>15959022
I'll go through, step-by-step, on how to take the second term in the first part to the form presented in >>15958983 as an example.
[math]\sqrt[3]{9x^8} = (9x^8) ^ { \frac { 1 } { 3 } } = ( 9 ) ^ { \frac { 1 } { 3 } } ( x^8 ) ^ { \frac { 1 } { 3 } } = ( 3 ^ 2 ) ^ { \frac { 1 } { 3 } } x ^ { ( 8 ) ( \frac { 1 } { 3 } ) } = ( 3 ^ { (2) ( \frac { 1 } { 3 } ) } ) x ^ { \frac { 8 } { 3 } } = (3 ^ { \frac { 2 } { 3 } } ) ( x ^ { \frac { 8 } { 3 } } )[/math]

And then, as another example, going from that form to the x-term in the presented problem. You should be able to follow by example and apply similar steps to the parts I'm leaving untouched.
[math]x^{\frac{19}{6}} = x ^ { \frac {18}{6} + \frac{1}{6} } = x ^ { \frac { 18 } { 6 } } x ^ { \frac { 1 } { 6 } } = x ^ { 3 } x ^ { ( \frac { 1 } { 6 } ) ( 1 ) } = x ^ { 3 } x ^ { ( \frac { 1 } { 6 } ) ( \frac { 2 } { 2 } ) } = x ^ { 3 } x ^ { \frac { 2 } { 12 } } = x ^ { 3 } \sqrt[12]{x^2}[/math]

the switch from 1/6 to 2/12 is only to get a common root with the 3^11 term, before you ask. If it were on its own there'd be no reason to do it.

>>15959040
You are not solving the exercise

>>15958714
Anyone? How to turn the first expression into the second one?

What is the most useless math topic?

>>15957615
Wait wouldn't rapid depressurization cause your lungs to implode?

>>15959063
i fear for the next generation
picrel starts with four basic properties that allow you transform the top expression into the bottom expression, i tried to use a single property per step

may god bless your soul anon

>>15959123
googology i.e. really big numbers
certain large infinities can be useful sometimes but for finite numbers? literally a dick measuring contest on who can set theory their way into the "largest" "non-trivial" (you can't just +1 or factorial another number) hall of fame
contributes 0 value to society at large and solves nothing in the space of pure problems

>>15959132
Thank you! May God bless yours too

>>15959132
Where did the X18/6 came from?

why are there so many atheists on a science board when science and math have already proven God's existence?

>>15951180
How would you solve the midwit crisis?

>>15959174
By impregnating your mother. Hopefully she won't birth another midwit

>>15959164
>Philosophy
>"Nothing exists"
Cogito, ergo sum you fucking idiot.

>>15959126
If you tried to hold your breath the air inside would expand and rupture your lungs. However if you let the air out, because of the pressure differential, unlike in an atmosphere oxygen would diffuse *out* of your lungs. You would suffer hypoxia and rapidly become unconscious.

>>15959380
>If you tried to hold your breath the air inside would expand and rupture your lungs. However if you let the air out, because of the pressure differential, unlike in an atmosphere oxygen would diffuse *out* of your lungs. You would suffer hypoxia and rapidly become unconscious.


There are worse ways to go. Thanks for the response

So finding X and Y in systems of linear equations is basically finding the points where they intersect?

Let [math]T: \mathbb{R} \to \mathbb{R}[/math] be the translation [math]Tx = x+1[/math], which leaves the Lebesgue measure [math]\lambda[/math] invariant. Suppose there exists a measurable automorphism [math]\pi: \mathbb{R} \to \mathbb{R}[/math] such that [math]\pi \circ T = T \circ \pi[/math] and [math]\lambda \circ \pi^{-1} = c \lambda[/math] for some positive real number [math]c[/math].

I need to show that [math]c = 1[/math]. The hint is to use the decomposition [math]\mathbb{R} = \bigsqcup_{n\in \mathbb{Z}} T^n [0,1)[/math]. I have no clue, though...

Where did that X come from? What propertie?

>>15959653
Because [math]\pi[/math] is a bijection and [math]\pi,\pi^{-1}[/math] commute with [math]T[/math], you can write [math][0,1)=\pi^{-1}(\mathbb R)\cap[0,1)=\bigcup\limits_{n \in \mathbb Z}\pi^{-1}([-n,-n+1))\cap[0,1)=\bigcup\limits_{n \in \mathbb Z}T^{-n}\left(\pi^{-1}([0,1))\cap[n,n+1)\right)[/math] as a disjoint union.
Then [math]1=\lambda([0,1))=\sum\limits_{n \in \mathbb Z}\lambda(T^{-n}\left(\pi^{-1}([0,1))\cap[n,n+1)\right))=\sum\limits_{n \in \mathbb Z}\lambda(\pi^{-1}([0,1))\cap[n,n+1))=\lambda(\pi^{-1}([0,1)))=c\lambda([0,1))=c[/math], as desired.
>>15959774
Can you show that [math]x^{18/6}\cdot x^{2/12} = x^{19/6}[/math]?

>>15959786
>Can you show that
Yeah, thank you, now I see where it came from. Now can you tell me the why? Not let it the way it was, why split it in two?

>>15959899
*shrug* it's what the question wanted. For practical purposes it makes no difference.

>>15959912
Thank you

>>15951180
I loved that post that described it as him trying to sneak up on the theorem

Book wants me to decide for which [math]\alpha > 0[/math] the function [eqn]f_\alpha(x)= \left(\frac{\sin x}{x}\right)^\alpha[/eqn]can be integrated on [math]\mathbb{R}_+[/math] using first the Riemann and then the Lebesgue integral.

Problem is, the function isn't defined for [math]\alpha \in \mathbb{R}\backslash\mathbb{N}[/math]. So I don't get, what I'm supposed to do here. Do you have any ideas, what they want me to do?

>>15960005
*[math]\alpha \in \mathbb{R}\backslash\mathbb{N}[/math]

>>15959581
yeah

>>15960005
Why wouldn't the function be defined?

>>15960053
Because the sin is negative on some values along [math]\mathbb{R}_+[/math]. If you chose [math]\alpha = 1/2[/math], then you'd end up with a complex number in the denominator. I don't think that can be the author's intention, because it's a book about real analysis.

>>15960005
They probably want you to take the absolute value inside the exponent.

>>15960069
Ah right. Probably `integrable' means that [math]\int_{\mathbb R_+} |f_\alpha|<\infty[/math].
There is one interesting thing going on here, for [math]\alpha=1[/math]. The integral above doesn't exist but [math]\lim\limits_{b\to\infty}\int_{(0,b)}f_\alpha[/math] does.

Now roots, how do I solve this?

>>15951528
You sound like a faggot. Do you just go into "stupid questions" threads and call peoples' questions stupid?

>>15960150
Anyone? I have a test in two months and need to learn this to start studying the rest, please help me

>>15960150
>sqrt(10) = sqrt(5)sqrt(2)
>sqrt(20) = 2sqrt(5)
Factor it out and see if you can get rid of some terms

>>15960204
Sqrt? Niggers just do it for me so I can see how its done, I know its easy but I'm 100% new to this math stuff and its hard af

>>15960150
>>15960216
[math](\sqrt{10} - \sqrt{5})(\sqrt{5} + \sqrt{2}) - \sqrt{20} + \sqrt{10} + 2 \\
= (\sqrt{10}\sqrt{5} + \sqrt{10}\sqrt{2} - \sqrt{5}\sqrt{5} - \sqrt{5}\sqrt{2}) - \sqrt{20} + \sqrt{10} + 2 \\
= (\sqrt{50} + \sqrt{20} - 5 - \sqrt{10}) - \sqrt{20} + \sqrt{10} + 2 \\
= \sqrt{50} - 3 \\
= 5\sqrt{2} - 3[/math]

>>15960216

>>15960255
Thank you very much
Could explain to me as if you were a teacher, what happened there? What did you do, step by step, please

>>15960150
Just expand it out and then do addition and subtraction.
Is the problem multiplying the stuff in the brackets?

>>15960266
Yeah, kinda. I need somebody to teach me step by step how to solve it because I dont what happened in the first step of >>15960255 with those new + and -

>>15960262
You can do it quicker but I tried to make it as clear as possible with each step. What don't you get??

Multiply out the brackets, combine the square roots, cancel terms, simplify.

>>15960272
It is the distributive law: [math](a+b)(c+d) = ac + ad + bc + bd[/math]

But note that in your question [math]b = -\sqrt{5}[/math].

>>15960273
Shit I just got it. √10.√2=√10√2

√5.√10 = √5√10 right? So when I got a root multiplying another root I just put them together?

>>15960285
>>15960255
One last question, I'm starting to understand.
How did all that turned into √50-3
And how did the √50 turned into 5√2
Explain please

>>15960286
Yes. The [math]\cdot[/math] to denote multiply is optional.

>>15960297
50 = 25 x 2. What is the sqrt of 25?

> How did all that turned into √50-3
You do know how to add right? What is sqrt(20) - sqrt(20)? What is -5 + 2?

>>15960286
Yep.
The rule is √a * √b is √a√b, but then you can also go the other way and decompose, say √10 into √2√5.
So √10√2 is also √5√2√2. And then you could multiply out the √2√2 in √5√2√2 to make √5 * 2.

>>15960300
Thanks. So when I got something like √50 I can write it as √25.2 and then take the 25 out as 5 because 5x5=25. But can I take the 25 out without touching it?
Something like 25√2?

>>15960308
> But can I take the 25 out without touching it?
These are all identical: [math]\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25}\sqrt{2} = 5\sqrt{2}[math]

>>15960308
> But can I take the 25 out without touching it?
These are all identical: [math]\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25}\sqrt{2} = 5\sqrt{2}[/math]

>>15960308
Note [math] \sqrt{ab} = \sqrt{a} \sqrt{b} [/math] for any nonnegative reals [math]a,b[/math]

To prove this just square both sides

I'm a HS dropout relearning math. Are there any books worth getting or am I good just sticking with Khan Academy?

If alien blood is green, does that mean it doesn't have a lot of iron?

>>15960664
Not necessarily. It would just mean that it isn't based upon hemoglobin.

>>15960664
arent iron compounds usually green? like iron chloride?

>>15960768
Chlorocruorin, the reason for green blood in some marine animals, is a hemeprotein (iron based).

>>15960503
professor Leonhard on youtube is better than khan academy. look through his playlists and try to see where you should start. then just start doing problems until it sinks in. No shortcuts

>>15960768
Iron(II) compounds are often green, yes.
And iron(III) compounds are often red (think rust).
But one has to keep in mind that iron is a transition metal, and transition metal chemistry is... odd. Those statements are true when the ion is relatively free, but the colour of transition metal compounds, like many of their other properties, is very heavily influenced by what ligands are present, if any. (The most obvious example of comes with the hydrates of various transition metal salts, which can often be wildly different in colour from their dehydrated forms.)

>>15960822
Fuck, I shouldn't answer questions when half-asleep. I fell victim to the folly myself.
Iron(II) compounds are usually green... when hydrated. When anhydrous, they tend to be more of the usual white/off-white/colourless

I'm graphic inequalities of linear systems, this one came up

[math]
3x-2y≥-6
[/math]

When I turn it into point-slope I get

[math]
y≥\frac{3}{2}x+3
[/math]

So I graphed it like the one on the left, but the problem's answer shows right graph. Where did I mess up? I can't find my mistake

>>15960978
>point-slope
y intercept slope*

>>15960978
You subtracted 3x from both sides and divided by -2, I assume?
When you multiply/divide by a negative number, your inequalities switch orientation. Think of it this way: -1<0, but (-1)(-1)=1 > 0(-1)

>>15960978
When you divide an inequality by a negative number, the inequality must be inverted.

For example:
3 > 2
By mutiplying by -1, we get -3 and -2, and so we must write:
-3 < -2

>>15960815
I'll check him out. Does he provide practice problems or is there somewhere else I can get them?

>>15961046
he goes through an example or two then he has a set of problems that get increasingly harder in most videos which he walks you through (though not too hard). So with him its best to see the examples then pause the videos and do the problems yourself the unpause to check your answers.

https://www.youtube.com/@ProfessorLeonard/playlists

as for books. open stax has a decent algebra book for free

https://openstax.org/details/books/college-algebra-2e

for linear equations, can i only get the point where they intersect if i solve for x and y in their standard form? if i get their intersection point what's the need for finding slopes or another point? shouldn't their y-intersect and the point when both meet be enough?

>>15959653
>>15959786

Thank you anon! Very neat.

Is AdS/CFT correspondence just the stokes theorem?

>>15961435
A correspondence is a 1:1 reversible mapping, that isn't true for Stoke's theorem which just gives you an alternative method to calculate a particular quantity.

>>15952628
yet you are insecure to need to ask on a burmese snail farming forum whether its bad for you

favorite study techniques?
i cant seem to do more than the bare minimum which has always worked, but i really need to figure out how to study longterm since i have a difficult/comprehensive test in 2months that requires me to sit down and grind hours of material. wouldnt be in this state if i couldve just studied before now.

>>15961517
It sounds just like it though
>For example, a single particle in the gravitational theory might correspond to some collection of particles in the boundary theory. In addition, the predictions in the two theories are quantitatively identical so that if two particles have a 40 percent chance of colliding in the gravitational theory, then the corresponding collections in the boundary theory would also have a 40 percent chance of colliding.[24]
>Notice that the boundary of anti-de Sitter space has fewer dimensions than anti-de Sitter space itself. For instance, in the three-dimensional example illustrated above, the boundary is a two-dimensional surface.

>>15961517
It sounds just like it though
>For example, a single particle in the gravitational theory might correspond to some collection of particles in the boundary theory. In addition, the predictions in the two theories are quantitatively identical so that if two particles have a 40 percent chance of colliding in the gravitational theory, then the corresponding collections in the boundary theory would also have a 40 percent chance of colliding.[24]
>Notice that the boundary of anti-de Sitter space has fewer dimensions than anti-de Sitter space itself. For instance, in the three-dimensional example illustrated above, the boundary is a two-dimensional surface.
And they give the correspondence of the stoke's theorem as
>Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains. It says that the pairing of differential forms and chains, via integration, gives a homomorphism from de Rham cohomology to singular cohomology groups
>De Rham's theorem, proved by Georges de Rham in 1931, states that for a smooth manifold M, this map is in fact an isomorphism.

I want to show that[eqn]
\begin{align*}
\lim_{\epsilon \to 0} \int_{\mathbb{R}_{+}} e^{-\epsilon x} \sin\left(x^{\alpha}\right)\, dx = \int_{\mathbb{R}_{+}} \sin\left(x^{\alpha}\right)\, dx,
\end{align*}[/eqn]This is what I know:
>the values of the involved improper Riemann integrals exist
>the integrals on the LHS are L-integrable, but the integral on the RHS is not
>the functions in the integral on LHS converge pointwise but not uniformly to the function on the RHS; therefore, this one theorem for integrals is not applicable
>Dominated convergence theorem is not applicable, because the supremum of the sequence of the absolute values of the functions is the absolute value of the function in the integral on the RHS, which is not L-integrable

Can you give me a hint on how I could solve this?

My physics professor got crazy and he started teaching us kinematics but with the axis flipped or tilted on an angle
are there any resources for that subject? I cannot find shit

Why we don't use ambient heat as energy source?

>>15961636
Split it into an integral from 0 to 1 and an integral from 1 to infinity.
In the integral over [0,1] you just pull the limit in without issues and in the other integral write
[eqn]e^{-\epsilon x} \sin\left(x^{\alpha}\right) = \left( \frac{e^{- \epsilon x}}{\alpha x^{\alpha - 1}} \right) \left(\alpha x^{\alpha - 1} \sin(x^\alpha) \right)[/eqn]
and then integrate by parts. Again you get an integral on which you can apply Lebesgue's theorem to pull the limit inside.

>>15961688
oh great, thanks. I'll try that

>>15961678
You need a temperature differential to do any kind of work to produce energy. If everything is at the ambient temperature, which is kind of its definition, that isn't possible.

>>15961677
Are you talking about Galilean Relaitivity?

Is there a countable set [math]A[/math] of positive reals such that the set of all convergent partial sums [math]\{ \sum_{b \in B} b \mid \emptyset \ne B \subset A, \sum_{b \in B} b < \infty\}[/math] equals precisely the set [math]\{2^k \mid k \in \mathbb{Z}\}[/math]?

Is a linear equation's slope of [math]-\frac{2}{3}[/math] always -2 on Y and +3 on X? or can the negative sign be on the 3? If so how do I know?

>>15962440
If the negative is on the 3, then the 2 must be positive.
If you graph out a line like this, you'll notice it's exactly the same line as when you take 3 to be positive and 2 to be negative. The only difference is the direction you draw it.
So, technically speaking, yes, you could have the 3 be negative. But it would change absolutely nothing.

>>15956753
This is pure retardation, when the book asked me back in the day to prove the definition it just gave me on the line up above i just went bananas for two minutes straight. Like wtf.

I’m taking intro to pdes 2yrs after taking ode’s. What should I review?

Why is it when atoms lose electrons, their charge becomes positive? If particles are being subtracted, shouldn't that make it negative?

I know it's because "electrons are positive" but why are they positive? What reason is there besides it was already decided and people are too lazy to change. I know it's difficult to change now because it'd confuse people, but to be honest it's no more confusing than changing from fahrenheit to celsius or imperial to metric. I feel next time someone makes fun of someone from the US for using fahrenheit or imperial units they should mention electricity would make more sense if negative and positive were reversed, yet we never do it.

Consider the integral of (x+2)^2+4
(x+2)^2 = x^2+2(2)x+2^2 = x^2+4x+4
Hence (x+1)^2+4 = x^2+4x+8
Therefore the integral is x^3/3+2x^2+8x+C
But surely the substitution u=x+1 hence du=dx gives
(x+2)^2+4 = u^2+4 whose integral is u^3/3+4u+C = (x+2)^3/3+4(x+2) = x^3/3+2x^2+8x+32/3+C.
Did I calculate wrong? Both methods seem legitimate here.

>>15962900
You didn't calculate anything wrong but note that the constant 'C' in each equation will in general not the same, they are for two different integrals. You could just merge that 32/3 into the second C.

>>15963372
Of course! Honestly, I can't believe I missed that. I'm afraid the point of the +C must have passed completely from my mind and I was just mindlessly integrating symbols robotically. Thank you, I'll have to remind myself to ignore constant terms in my studies going forward. Thank you so much.

>>15961697
But you can generate temperature difference artificially with less energy than flows trough difference.

>>15963445
Also remember that when calculating definite integrals any constants never matter since they will just vanish when calculating the difference of the limits.

>>15963461
Right, don't think I ever considered that but yes I'll have to remember it. Thanks again.

I'm kind of embarassed of my first few articles I published as an undergrad. They are unrelated to what I do now. Can I just not mention them for applications and such?

>>15963624
Of course. Most places won't even check the references you list let alone going to the effort to find any you omitted.

6,336.1 sq miles
I´m kinda stumped right now, Normally should get this, but am kinda stumped right now. Is this sixty threetousand and threehundredsixtyone miles or another number? Honestly dont even know what the commas there mean. Usually an kilometre kind of guy.

>>15963722
six thousand three hundred thirty-six and one-tenth
I assume you're a Eurofriend? , is used to delineate every power of 1000 like how a number of Europeans use .

>>15963722
https://docs.oracle.com/cd/E19455-01/806-0169/overview-9/index.html

>>15951180
How come no one use AI or any program to farm winning sweepstakes numbers?

As for preparation on college maths, do you memorize first the greek alphabet?

Help pls, I need to get the result in the red circle.
What step am I missing or what am I doing wrong? Dont know what to do next

>>15964218
Is this right? lol

>>15964218
anon, serious question, but are you genuinely retarded? what is 11 - 9?

>>15964239
yes

>>15962115
I don't think so. Here is a (somewhat ugly) proof by contradiction:
write [math]\Sigma(B)[/math] for [math]\sum\limits_{b \in B}b[/math].
By assumption there are two subsets of [math]A[/math], say [math]B[/math] and [math]C[/math], so that [math]\Sigma(B)=1[/math] and [math]\Sigma(C)=4[/math].
We cannot have that [math]D:=B\cap C=\emptyset[/math], as then [math]\Sigma(B\cup C)=\Sigma(B)+\Sigma(C)=5[/math], which isn't a power of 2.
Also, if [math]D=B[/math] then [math]\Sigma(C \setminus D)=\Sigma(C)-\Sigma(D)=3[/math], which cannot be either.
We cannot have that [math]D=C[/math] since then [math]C \subseteq B[/math] so that [math]\Sigma(C)\leq\Sigma(B)[/math], another contradiction.
This all implies that [math]0<\Sigma(D)<1[/math] since [math]D \subsetneq B[/math].
Set now [math]E=C\setminus D[/math] and notice that [math]\Sigma(E)<\Sigma(E)+\Sigma(D)=\Sigma(C)=4<\Sigma(E)+1[/math] so that [math]\Sigma(E) \in(3,4)[/math].
Of course there is no power of 2 in [math](3,4)[/math].
>>15962761
Probably a little bit of functional analysis.

How do I rationalize this one?

>>15964677
[eqn]\frac{63}{\sqrt{27} - \sqrt{108}} \\
= \frac{63}{\sqrt{27} - \sqrt{108}} \cdot \frac{\sqrt{27} + \sqrt{108}}{\sqrt{27} + \sqrt{108}}\\
= \frac{63(\sqrt{27} + \sqrt{108})}{27 - 108} \\
= -\frac{63(3\sqrt{3} + 6\sqrt{3})}{81} \\
= -\frac{63\cdot9\sqrt{3}}{81} \\
= -7\sqrt{3}[/eqn]

>>15964705
Thank you. Can you teach me how to do this part? Or whats the step called so I can google it.

>>15964726

[math]\sqrt{27} - \sqrt{108} = \sqrt{9 \times 3} - \sqrt{36 \times 3} = \sqrt{9}\cdot\sqrt{3} - \sqrt{36}\cdot\sqrt{3} = 3\sqrt{3} - 6\sqrt{3}[/math]

>>15964726
don't think there's an actual name for it
you just factor out some square that divides the number and gets you to a convenient form
in this case, [math]27=3 \times 3 \times 3[/math], so [math]9=3^2[/math] is a factor of 27. that lets us write [math]\sqrt{27}= \sqrt{ 9 \times 3 } = \sqrt{9}\sqrt{3} = 3\sqrt{3}[/math]
likewise, [math]108 = 3 \times 3 \times 3 \times 2 \times 2 = 3 \times 9 \times 4 = 3 \times 36 = 3 \times 6^2[/math], so [math]\sqrt{108} = \sqrt{36}\sqrt{3} = 6\sqrt{3}[/math]

>>15964737
>>15964741
So the root doesnt just change form. Its separated and partially resolved, no?

>>15964766
Correct. You are just simplifying the expression. In this particular question you know the answer contains [math]\sqrt{3}[/math] so that is a big clue about what to factor.

What about this one

>>15951180
if you took a fridge and were to cool the coolant with the ice produced in the very same fridge, what would be its efficiency?

sorry for not being technical enough, I honestly have no idea about the specifics of heat pumps (or even the theory behind them)... I only know that what produces ice is the coolant being vaporized after passing through a compressor to liquify it and then cooling it in the radiator(?) and dumping heat to the air. why am I asking this, then? because this could be useful to save energy in a desalination system.

>>15964978
>do my homework
ok...
x/5 - 2/3 + x = 0 <=>
x/5 + x = 2/3 // add 2/3 to both sides of the equation
x/5 + 5*x/x = (x + 5*x)/5 = 6x/5
6x = 5*2/3 // multiply both sides by 5
6x = (5*2)/(3*6) // multiply both sides by 5
x = (5*2)/(3*3*2) = (2/2)*(5/[3*3]) = 5/9
you can guess the operations I applied in the last 2 steps

>>15965010
>Homework
I'm trying to get into college and I finished highschool like ten years ago, so its extra hard. Plus I got no money for a substitute teacher. I understand most exercises but those I dont I come to ask here

>>15965010
>add 2/3 to both sides of the equation
But you didnt? Or at least im not seeing it.
Can you write it like this dude writes it?
>>15960255

>>15965042
my man, I finished HS some 20 years ago
I wish you the best, anon. I really do. I hope you learn this stuff so you become a better person and are able to understand how this shitty society works.

my recommendation is to look into the properties of addition, substraction, multiplication, division, sums of fractions and so on.

btw, remember: to check for results, you can simply replace x with the number that you get as result. if the equation matches, then your results are correct.

>>15965071
sorry anon, I'm too lazy for that shit and don't really know much of latex notation
but, see:
left side:
-2/3 + 2/3 = 0
right side:
0 + 2/3 = 2/3

>>15964458
Wonderful (not an ugly argument at all)! Thanks anon.

>>15965042
[eqn]\frac{8x^2 - 14 + 8x}{4x-3} = 2x \\
\implies 8x^2 - 14 + 8x = 2x(4x-3) \\
\implies 8x^2 - 14 + 8x = 8x^2 - 6x \\
\implies 14x = 14 \\
\implies x = 1[/eqn]

3x + 1 = 1 has solution x = 0 but if i rewrite it as x(3 + 1/x) = 1 i have to assume x =/= 0 but now i have no solutions.
the 2 equations are still the same though, where's my mistake? apart from not learning latex

>>15965846
> I have to assume x =/= 0
Why do you think that? It is an incorrect assumption.

>>15965863
Probably the part where it means division by 0.
>>15965846
The incredibly unsatisfying answer is that they don't actually describe the same equation. It's the same at almost every point, yes... but as you have acknowledged, one is defined at x=0 and one is not. This is an example of what is informally known as a hole, and more formally as a removable discontinuity. (To be pedantic, it is only a "hole" when applied to the graph that the function describes and not to the function itself, but I digress.)
The nice thing is, as the formal name implies, we can... well, not quite ignore these, but acknowledge them and then move on with our lives anyway. Because we know what values the function takes at points arbitrarily close to 0 (say, x=0.01, x=0.00001, x=0.00000001, etc.), and because these values more-or-less agree for positive and negative x alike (compare, say, x=0.000000001 with x=-0.000000001), we know the value that the function "should take" at 0, if it were defined there, and so we can easily "remove the discontinuity" by considering a function that is identical besides ACTUALLY being defined there (your 3x+1=1).

This process of redefining a function to one that agrees at all points of definition, but is defined at more points, is called analytic continuation, though it is in general considerably more complicated than this

>>15965846
You have to assume x=/=0 to do the rewriting, so the equations are only the same if x=/=0.
>>15965908
It's not analytic continuation. Analytic continuation only results in analytic functions, removing removable singularities can give you any continuous function.

>>15951180
Why do junkies in exUSSR bend backwards, but junkies in the US fold forward?

>>15951180
Is it ok to cite wikipedia in a paper or is it full retard? Won't be published, it's for uni and simply shares information on a subject (think phagocytes).

if you drink nothing but clear liquids, do you shit water?

>>15961678
we do, Gibbs' free energy

>>15961543
>skim notes
>jack off
>ace test
>remember everything for decades to come

>>15966028
>>15965908
so if it's not the same equation why am i allowed to do this kind of manipulations when calculating limits?

>>15966316
Precisely because they're limits.
They're identical everywhere except x=0, including any arbitrary neighbourhood of that point, so their limit as x tends to 0 is also identical, even though their actual value at that point is not.

>>15966208
Junkies in exUSSR take GHB while junkies in america take fentanyl.

>>15966643
break down the fractions: [math]\frac{2x - 1 } { 3 } = \frac{ 2x } { 3 } - \frac { 1 } { 3 }[/math], and likewise, [math] \frac { x - 3 } { 2 } = \frac { x } { 2 } - \frac{ 3 } { 2 } [/math]
then you can rearrange the terms so all of your x-values are on one side and everything else is on the other
[math]\frac{ 2x } { 3 } - \frac { 1 } { 3 } + 1 = \frac { x } { 2 } - \frac{ 3 } { 2 } - x[/math]
[math]\frac{ 2x } { 3 } + x - \frac { x } { 2 } = - \frac{ 3 } { 2 } - 1 + \frac { 1 } { 3 }[/math]
rewrite so everything on a given side has a common denominator
[math]\frac{ 4x } { 6 } + \frac { 6x } { 6 } - \frac { 3x } { 6 } = - \frac{ 9 } { 6 } - \frac { 6 } { 6 } + \frac { 2 } { 6 }[/math]
[math]\frac{ 7x } { 6 } = - \frac{ 13 } { 6 }[/math]
[math]7x = -13[/math]
[math]x= \frac{-13}{7}[/math]

>>15966669

>>15966669
But were did the 6 go?

>>15966696
I mean before you got 7x=-13 you had two 6 as denominators, what happened to them?

>>15966714
multiply both sides by 6

>>15966716
Please explain

Ok, I need to understand if I'm retarded or not.
For commutative rings with 1, Z_n rings have no subrings, right? Since 1 always generates the whole Z_n ring and any subring would have to include it.

>>15966723
>>15966716
Got it, thank you

>>15966729
I will pedantically note that any ring is a subring of itself.
But besides that, your understanding is precisely correct

why did no burrowing dinosaurs survive the extinction event

>>15966791
To begin with they were not common. From that period there is evidence of only a single type of dinosaur that burrowed (Oryctodromeus). Just because they could burrow didn't make them any more likely to live since the after effects of the event (lack of food & water, environmental changes, etc) would have lasted for years, maybe decades.

>>15966258
>it's for uni
If it's a uni worth anything then fuck no, why not just use the references the wikipedia article itself uses?

>>15966739
This hurts my brain. Even kids still at kindergarten know better than this.

>>15964979
anyone? pls

If you put a large inductive load at the earth's poles, would the magnetic field get stronger?
Would this accelerate the spinning of the core, thus heating of the mantle?

>>15964979
>>15967278
You don't seem to know how refrigerators work. It is the phase change from liquid to gas of the coolant passing through the expansion value that extracts heat. If you drop the temperature of the coolant too far this process becomes less efficient or will not happen at all.

Can someone point me to resources to learn how to do simulations such as
https://www.youtube.com/watch?v=TZfh8hpJIxo
https://www.youtube.com/watch?v=PNtKXWNKGN8
https://www.youtube.com/watch?v=tVNoetVLuQg
https://www.youtube.com/watch?v=sEPh6bAQVP0
My background (BS) is in math. I have done some programming in various languages.

>>15967834
I guess I was not clear enough?

>It is the phase change from liquid to gas of the coolant passing through the expansion value that extracts heat
...I know that. as I said here >>15964979:
>I only know that what produces ice is the coolant being vaporized

what I'm trying to say is this: you take the ice from the freezer and move it to the radiator, to transfer heat from the compressed, hot liquid to that ice (to melt it).
if you do that, would the efficiency of the system improve? if so, by how much? I guess it seems pointless, but it might not be in some contexts where you want liquid water and not ice, but you need to generate ice to get water.

>>15951180
I've never seem the summation and integral switched before. I'm confused as to why it became what it did after the summation and integral are interchanged in pic related.

>>15968533
Forgot pic related

more or less a year ago i bought vitamin pills, it has calcium, magnesium, vitamin C, iron, zinc, B2 B6 copper and several other things

is it true that they go bad, or lose their nutrition properties? it has an expiry date and i bought them on bulk so there's no way i can eat them quick enough

>>15968536
You can insert an indicator function to make the step clearer. [eqn]1_{[1,k)} (x) := \begin{cases} 1 & x \in [1,k) \\ 0 & \text{else} \end{cases} [/eqn]
The swap is then just justified by Fubini's theorem. Remember that a summation is just integration with respect to the counting measure.


[eqn]\sum_{k=1}^n \int_1^k \frac{dx}{x} = \sum_{k=1}^n \int_1^n \frac{1_{[1,k)} (x)}{x} dx = \int_1^n \frac{\sum_{k=1}^n 1_{[1,k)} (x)}{x} dx = \int_1^n \frac{\sum_{k=\lfloor x \rfloor + 1}^n 1}{x} dx [/eqn]

>>15968708
Thanks alot for the response. The only thing I'm still not sure about is how the last step become the floor of x + 1.

> fubinis theorem
Pic related

>have to take out more loans
How do you guys deal with this kind of shit? I feel so fucking demoralized being a loan pig.
And if I don't get a job I'm going to fucking kill myself.

>>15968708
how did you write latex on 4chan?

>>15969364
[math] tags do wonders

>>15969349
why do you *have* to take more loans? if you need loans to survive, you are doing something very wrong...

My textbook says that molecules in monoatomic ideal gases have only 3 degrees of freedom. Is that an approximation, bacause of the size of the molecule? I imagine it in such a way, that since they're small, they have a small moment of inertia and thus can't store too much rotational kinetic energy. But how does this relate to the equipartition theorem? Whats the reason the energy isn't equally distributed into those rotational degrees anyway?

>>15969847
If you are lucky enough to have no idea about such things, thank your parents from the bottom of your heart next time you meet them

>>15969857
Monoatomic means a single atom, not a molecule. They can't have vibrational or rotational kinetic energy. So the only degrees of freedom are the three x,y,z axes.

>>15969862
1) Wrong board for this kind of question
2) Get a job, any job.

>>15969867
Why can't they? They're not a mathematical infinitely small point in the space

>>15969869
1) I'm not asking, I'm answering
2) You wouldn't get most loans without a job anyway

>>15969872
They are also not tiny balls. They are quantum objects and don't have those properties.

>>15969914
Yea, that's what we know now, but in the kinetic ideal gas model they kinda are, aren't they?

>>15969936
No. In an ideal gas they are treated as point-like. It worked but at the time they had no idea *why* it worked.

>>15969944
Ah, I see, that makes sense now. Thanks.

What is the most Maths heavy chemistry field, and what kind of Maths does it use?

>>15969862
>thank your parents from the bottom of your heart next time you meet them
yeah, I thank my narcissistic mom for at least teaching me:
- work ethics
- not to get loans from jewish banks
and for having at least SOME common sense.
I was born poor though, and that was HER fault (and she assumed her responsibility multiple times. still didn't do shit about it though).

anyway, as the other anon said, this isn't the board for these topics. go ask >>>/biz/ . don't treat them seriously, though, they are a bunch of clowns.

>>15969349
there are plenty of good reasons for taking out a loan. education is one of them.

>>15970096
paying for education is unethical. paying with fucking LOANS is even more unethical.
only amerimutts would accept this reality without questioning it

>>15970128
>paying for education is unethical.
why?
>paying with fucking LOANS is even more unethical.
why?

>>15970134
>why is paying to transfer knowledge unethical in societies that are supposed to look for self-development?
>why is giving loans to kids unethical?
I dunno anon, it's a mystery

>>15970291
do you think its unethical to charge for food and water? clothes? shelter? i sure hope youre job doesnt involve exchanging goods and/or services for money, that would be unethical.
>>why is giving loans to kids unethical?
the whole idea behind student loans (and specifically the nationalization of them) was to give children the opportunity for self-development. you literally cannot be too poor to go to college. and despite how much some people bitch about it, the 1965 Higher Education Act has been wildly successful. americans are more educated than ever, and extremely few people do not see a positive ROI on student loans. even if you get a degree in underwater basket weaving, youre more likely than not going to out-earn americans with no post-secondary degree. i fail to see how any of it is unethical.

>>15969944
> In an ideal gas they are treated as point-like
Not necessarily. The number of internal degrees of freedom just shows up in the heat capacity, not the ideal gas equation of state PV=nRT.

>>15970322

>do you think its unethical to charge for food and water? clothes? shelter?
actually... yes. I think these things should be provided by governments when needed. and indeed they do, as subsidies. but I think govts should actually have enterprises producing and giving these things for free to people in need instead of giving them cash.

>i sure hope youre job doesnt involve exchanging goods and/or services
yeah, it does, but I learned most of my stuff by myself.
I learned some basic stuff in uni, for which the government itself provided me with loans that I'm NOT PAYING :^)

> for money, that would be unethical.
... for some reason you mentioned charging for "food and water? clothes? shelter?". you obviously know the difference between these things and every other goods and/or services.

>self-development
I was talking about the self-development of the whole society, not just individuals. even in america you still have states, and a national state (aka federal government) and those are supposed to be for the people. yet getting people indebted, particularly when uni careers are so expensive and some are provided by for-profit universities, doesn't really look like a great idea, for multiple reasons, one of which is the obvious fact that these people might not be able to pay them in time and therefore debts accumulates and they practically become slaves to that debt. also you might end up wasting a lot of resources on useless "professionals" if the resources aren't spent in a planned manner. also, if too many people become indebted then you create instabilities in the system. and so on...

>>15970383
Sure, but that depends on if the gas is monatomic, diatomic, or polyatomic. So rather than an 'idealised' gas used in the original model derivation the value is for a 'real' gas.

>>15970039
Bump

>>15970467
>loans that I'm NOT PAYING :^)
that tanks your credit, ya know
>one of which is the obvious fact that these people might not be able to pay them in time and therefore debts accumulates and they practically become slaves to that debt.
student loans are pretty forgiving. you just said you arent paying them, do you feel like a slave to them? your credit might tank and your wont be able to get a mortgage, but if you cant make student loan payments you probably werent making a house note either.
>also you might end up wasting a lot of resources on useless "professionals" if the resources aren't spent in a planned manner.
i would hardly describe that as "unethical" behavior, but in the US we give student loans to quite literally anyone and we have a pretty good white-collar sector. compare that to, say, china, where something like >90% degrees are STEM and yet they arent surpassing us technologically.
>also, if too many people become indebted then you create instabilities in the system.
if everyone is dirt poor, maybe, but if most people are paying off their debts in a responsible and sustainable manner then i dont see what the issue is. you wouldnt say everyone having a mortgage is a bad thing, would you? (inb4, the 2008 recession was caused by people *not* paying their mortgage.)

>>15969847
I'm jewmerican...
>>15970096
lol no wtf imagine being so buckbroken you thinking usury is good.
>>15970128
I question it and I'm treated like a mad man.

>>15971053
you need to think about this seriously. you are going to fuck up your life if you dont nut up like a big boy and take out another loan. even if you think you wouldve made more money long-term if you never took out a loan (nb: you likely wouldnt have made more money even in the short-term), your circumstances now are that you have debts, and finishing school is by far the best method of paying them (as well as, you know, paying for other things).

>>15970990
>that tanks your credit, ya know
I'm not muttmerican. here only the state and the university can charge you anything at all (but they can't force you), so it doesn't get reported in commercial scoring system.
I owe like $15k US dollars. I'm not paying until I have a decent life.

>you just said you arent paying them, do you feel like a slave to them?
while they can't force me to pay for now, politicians could one day decide to enforce it by law and I could get fucked.
plus it's still debt. I don't like having debt. I feel ashamed of that. also the info is public record (you can find it on google),
I also know that signing for those loans was a mistake. I was a poor, ignorant and enthusiastic 17 years old student at the time. why did they even offer such loans to me? why didn't they tell me to wait until I was 18? why didn't they actually inform me what the consequences could be? I know, I should I informed myself... but they were as irresponsible as me.

>(inb4, the 2008 recession was caused by people *not* paying their mortgage.)
this is misleading. banks were giving loans to people that couldn't pay and they KNEW it.
then the US government went and saved those banks. the US government could have subsidized people or given them cheap loans but instead decided to help the banks. corruption disguised as "economic reasons"

>>15971103
>no payment enforcement mechanism
>doesnt show up on credit score
can i ask what country this is? is there literally no incentive to pay?
>why did they even offer such loans to me?
its a numbers game. even if a bank is being extremely cautious with who they loan money to, some people will always default. but the goal isnt for nobody to go bankrupt, the goal is just a net positive return on *all* their investments. the same is true for your government handing out educational loans. the goal is for *most* people to eventually contribute more in taxes than the cost of their education. its a good plan that historically works very well, even if it didnt pan out for you personally. or maybe youre doing fine and youre just being melodramatic, iunno. from the sound of things, it seems like youre not actually in real debt.

If we know what causes all cancers, why is it so hard to develop a treatment or early detection to stop any future cancer deaths?

>>15971185
>why is it so hard to develop a treatment
Cancer is, when you boil it down to the basics, a condition where your cells mutate to multiply at an uncontrolled rate.
They're still your cells, though. Anything that kills them (i.e. treats the cancer) is perfectly capable of killing the rest of you.

>>15971195
So early detection could be more dangerous because allowing growth makes surgical removal easier?

>>15971185
even instantaneous detection would not stop all cancers.

Development of treatment to cure cancer deterministically is like having a treatment that could administer genetic therapy to every single strand of DNA in the entire adult body within the span of the expected patient survival time. That's a supertechnology we haven't developed yet.

>>15971216
That was kind of my assumption, but I was wondering if there was a simpler and closer generalized approach that could leverage our understanding

>>15971223
Probably, but I don't know enough about the mechanisms of particular cancers to find a more elegant solution for all cancer in general than an extremely thorough genetic therapy.

Then again neither do any doctors in the entire world though so...

>>15971228
Ample elegance underbrush opulence.

>>15971185
>If we know what causes all cancers,
I'm not an expert (not even close), but no, we don't. also, AFAIU most cancers occur very rarely.

>why is it so hard to develop a treatment or early detection
causes and illness are two completely different things, though? and all cancers are different.

>>15971115
>can i ask what country this is?
chile

>is there literally no incentive to pay?
actually, they changed the student loans system while I was in uni, and now students owe their loans to banks. the state acts as a guarantor. which means the debtors do indeed get reported when they don't pay their loans. I saved myself from this fate for only a couple of years...
though apparently (I just googled it) they enacted a new law making these credit report companies "forget" student loan debts.

Me and some friends want to make a gift for a girl who collects minerals
I was thinking an UV lanttern for fluorescent minerals

Do the cheap ones work, or what? Any rec?

>>15969847
You have no idea how many people are one bad dentist visit away from getting into a debt spiral. Seen that exact thing happen. And it's not like you can skip or wait through that. Having a cusion like that in those kind of random situations changes everything. So yeah, thank your parents.

>>15970134
>paying with fucking LOANS is even more unethical.
>why?
Because the first and foremost goal of education is fighting and prevention of poverty. Everything else should be considered an extra.
Having to get into a debt in order to get an education is a mechanism that impoverishes the population, sabotaging the idea of education itself

>>ignore shitpost replies

>>15971773
the whole point of student loans is to fight poverty. they enable people to go to college who otherwise could not have afforded to. the data shows they work. you make more money in the long-term, the bank makes interest, and you pay more in extra taxes than it cost the government to back your loan.
>>15971777
i sure hope you arent calling my post a shitpost.

>>15971758
>You have no idea how many people are one bad dentist visit away from getting into a debt spiral. Seen that exact thing happen
that's fucking sad. I wonder if that happens in europe too...

>>15971859
not all students get rich though. hell, I'd guess a high % of people simply can't pay them, probably. https://www.forbes.com/advisor/student-loans/average-student-loan-debt-statistics/ says 5.1 million people already have defaulted on loans. of course that does not mean all of them are unable to pay... but still

>>15964979
anyone?

>>15971946
>https://www.forbes.com/advisor/student-loans/average-student-loan-debt-statistics/
we might want some pre (or post) pandemic statistics before we start making any judgments. according to this only 1 out of 80 people are actually paying back their student loans, which seems kind of low to me.

>>15971992
well, yeah, the data on that article doesn't seem to be up to date, and they seem to use data from different periods (wtf?)

>>15960005
The function is continuous when [math]\alpha \in \mathbb{N}[/math] and the function is continuous when [math]\alpha \not\in \mathbb{N}[/math]

>>15972121
Not continuous when [math]\alpha \not\in \mathbb{N}[/math], I meant.

Is there integration by parts for scalar fields? I.e. if [math]u, v \in C^1(\Omega)[/math] where [math]\Omega \subseteq \mathbb{R^n}[/math] is some domain what can we say about [eqn]\int_{\Omega}\nabla uv[/eqn] or alternatively about [eqn]\int_{\Omega} \partial_j uv[/eqn] for [math]1 \leq j \leq n[/math]

New thread when?

>>15972325
https://www.youtube.com/watch?v=RPgE991VDwM

>>15972224
Yes, this is used all the time in physics. Since you can break up the multiple integral into several one-dimensional integrals, one of which is the jth coordinate, this follows directly from the one-dimensional case.

I can look at a compound and determine what it's IR spectrum would be, but not vice versa. Countless practice problems have not fixed this issue, even in scenarios where it is combined with NMR and mass spectra. I can identify the functional groups, sure enough, but once I get past a certain point and the question becomes "at least some of the remaining carbons have to be equivalent—how?" I flounder. Is there a way to train specifically that skill?

>>15972871