/sqt/ Stupid Questions Thread

Oh and the fixed point is the limit.
You can find it solving this [math] x=\frac{1}{2} x + \frac{1}{x} [/math] .

Use the equivalence relation that says a sequence in [math]\mathbb{R}^n[/math] is convergent iff it is Cauchy. To prove that it is Cauchy it is enough to prove that it converges. In the case of [math]x_n=\frac{1}{n}[/math], it's clear that [math]0

those things are new to me and it's supposed to be solveable with what we've already done in class seems right but don't I have to solve the recurrence to do that?

Reposting. Can anyone show how to get this solution?

Let's consider an easier example.

Only difference here is I have selected an initial condition.

Let [math]x_{1} = 1[/math] and [math]x_{n+1} = \sqrt{1+x_{n}} [/math].

First show the sequence is increasing by doing induction. Recall what it means for it to be increasing, [math]x_{n+1} \geq x_{n}[/math] for all [math]n[/math]. These problems are great for induction.

Next, the boundedness. It should be obvious that [math]x_{n} \geq x_{1}[/math] for all [math]n[/math]. The upper bound then is the challenge. Claim: [math]x_{n} \leq 2[/math] for all [math]n[/math]. This is another proof by induction.

Now you've shown it's monotone and bounded, therefore it converges, and this it's limit exists (and it's value is quite interesting).

Put x[n+1]=x[n]=x, to get
x=x/2+1/x
=> x^2=x^2/2+1
=> x^2/2-1=0
=> x^2=2
=> x=+/-sqrt(2)
If x[n] is +ve, x[n+1] must be positive. If x[n] is negative, x[n+1] must be negative.

Note that the recurrence is a specialisation of
x[n+1] = (x[n]+k/x[n])/2
for k=2. The generalised form is known as the "Babylonian method" or "Heron's method" for calculating the square root of k. If x=sqrt(k), then x=k/x=(x+k/x)/2

If [math]f : E \rightarrow F[/math] and [math]B[/math] is a subset of [math]F[/math], how does one show that [math]f^{-1}(F [/math]\[math] B) = E [/math]\[math]f^{-1}(B)[/math] ?

I guess I have to use double inclusion, but I don't really see where to go from there...

this works, thanks.

f^(-1) (F\B) = {x in E | f(x) in F\B}
= {x in E | f(x) in F} \ {x in E | f(x) in B}
= E \ f^(-1)(B)

Not really. You have to break it to two cases.
Case 1:
x0 is in the interval [math] [ \varepsilon , + \infty ) [/math] . xn converges to [math] \sqrt{2} [/math] . Contraction can be proven by the mean value theorem and the fact that the derivative is always less than 1/2.
Case 2:
x0 is in the interval [math] (- \infty, - \varepsilon] [/math] . xn converges to [math] - \sqrt{2} [/math] .

This "works" yeah, but you have to prove that it converges.
You really have to use , to actually prove it.

...

Two forms solutions are [math]e^{\pm \sqrt(\epsilon)}[/math] and then using the boundary equations you get the correct constants, so you can write it into the form cosh

>the derivative is always less than 1/2
The derivative at x=0.1 is -99.5 for example.

I meant 1/squareroot

Oh fuck it needs to be |x|>1.
Well, in [-1,1]
|f(x)-f(y)| = 1/2 |x-y| |xy-2|/|xy| bla bla bla ... bound

Prove: Suppose [math](x_{n})[/math] is monotone and has a convergent subsequence. Prove that [math](x_{n})[/math] converges.

Suppose [math]x_{n+1}\geq x_{n}[/math] . We're also given, [math]\forall \epsilon > 0 \exists N \in \mathbb{N} [/math] such that [math] \forall k \geq N , |x_{n_{k}} - L | < \epsilon [/math]

[math](x_{n})[/math] converges, so [math]\forall \epsilon > 0 \exists N \in \mathbb{N} [/math] such that [math] \forall n \geq N , |x_{n} - L | < \epsilon [/math]

How do I relate these two?

It's a basic theorem.
Fucking google it.

Proof: Think

Because the sequence is monotone you can bound every term in the sequence between two of the terms of the convergent subsequence. And then as n goes to infinity, the distance between the terms of the convergent subsequence go to 0.

So you can proceed in either of two ways: Use this to prove the general sequence is Cauchy or even better: conjecture that the limit of the sequence is the limit of convergent subsequence and then do your epsilon magic.

Bolzano-Weierstrass theorem.

By the way, which book are you working out of?

Please help...

Proof: Think.

If the charge is distributed along the sphere, a Gaussian sphere with the same center but smaller radius will enclose exactly 0 charge. Apply Gauss law and then think.

*If the charge is distributed along the surface

Why do they make the distinction of considering a uniform and a non-uniform charge distribution as separate cases then?

To test if you are stupid

The answer on my exam key is 2, so that is incorrect.

Pic related is the problem. I set my bounds as stated in my stack exchange post here: math.stackexchange.com/questions/2506874/triple-integral-bounded-by-a-cylinder-and-a-plane

Can anyone tell me why my bounds are wrong?

The book says the answer to this problem is e^7 - 1
I get where the e^7 comes from but not the 1

It is a telescopic series.
Write out a partial sum and you'll see why. Everything except the first term e^7 and the last term -e^(n+1) cancel out.

>last term -e^(n+1)
-e^(7/(n+1))

Someone can explain to my why the phase spectrum of the sinc function is [math]\pm \pi[/math] when [math]\sinc(x) < 0[/math]?
Sinc is real so it shouldn't be all 0?

Can you expand conditional probability expressions like that ?

P(A) = P(c2) + P(c3) - P(c2) x P(c3)

P(c2|A) = [P(A|c2) x P(c2)]
P(A)

Can I then expand - P(A|c2) by replacing where c2 occurs in A with 1?

1+ P(c3) - P(c3)x1

Is there anything wrong with this ?

Simple question, why is [math]\phi_2=180°-\phi_0[/math] in the second quadrant of a unit circle? I understand if ex. you have 180°, and the first angle fills one portion, while the other fills the rest of the 180°. But doesn't make sense if both have the same starting point.

I'm new to general relativity and I am confused by notation

what's the difference between

[math] \Gamma^a_{\ bc} \quad \quad \Gamma_{ab}^{\ \ \ c} [/math]
?

Am I doing this wrong, feel like I am doing this wrong.

They want me to show if I know how to get a trig identity. I am trash at them so I came here. The stuff on top is the original identity.

How does one derive the recurrence formula [math]T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)[/math] using the generating function [math]g(x, t) = \frac{1 - tx}{1 - 2tx + t^2} = \sum\limits_{n=0}^\infty T_n(x)t^n[/math] without using cosines in any way? The basic idea is clear: derivative wrt t and then playing with the coefficients, but I can't do this in practice for some reason, or maybe I'm just too blind to see how to do it.

It's literally multiplying both sides by the denominator. Then you see they are equal.
After that just work backwards: adding the stuff and then dividing it.

Anyway, you're doing it right.

?

Thx

Did you try anything?

By what method can I evaluate this limit?

derivative

there was an attempt

1/n^(1/3)=1/ε^3
n>N ==> n>1/ε^3 ==> 1/n 1/n^(1/3)

If two expressions are equal, is it a guarantee you can manipulate either one into the other form algebraically?

An example would be (n-1)*(n-3)*(n-5)...*(n-(n-1)) == n! / ((2/n)! * 2^(n/2)).

Is there a rigorous definition for equality between algebraic statements? Is the left hand side of that equality I posted even an algebraic statement if I have to include "..." to show what I mean?

Also: f(n) = (n/n) has a discontinuity at 0 but n/n = 1 and g(n) = 1 does not have a continuity.

If 1 = n/n then doesn't f(n) = g(n) but if they're equal why does f(0) not equal g(0)? Or does it?

Is this just a case of attributing more to the idea of equality than is there?

I would ask one of my profs this but I feel like this is something I should understand from elementary school....

Using l'hopital's rule?

no
definition of derivative, the limit

"In the card game of poker, each player is dealt 5 cards. Assume the game is played with all 52
cards. In how many ways can a player have a ‘four of a kind’ (that is, 4 of the 5 cards must be
from the same ‘denomination’ (e.g. the four 8s) - the fifth card can be any other card)?"

The answer is 624 but I don't know how this was reached.
Can someone do like a step-by-step?

no

There are 13 denominations in a 52 card deck so you can get 13 different sets of 4 cards of a kind.

For each of the 13 possibilities, you will have 52 - 4 = 48 cards remaining. So then there are 48 possible 5th cards for each 4 of a kind.

If there are 48 "versions" of each and 13 possible 4 of a kind hands, then you have 13 * 48 = 624 possible 5 card hands with 4 of a kind.

If you didn't follow that then you need some basic combinatorics knowledge: en.wikipedia.org/wiki/Rule_of_product

Judging by the answer, the order that the cards were dealt doesn't matter.
How many tetrads are there? 13
Pick one of them. 13 possible ways.
You are left with 52-4=48 cards.
Pick one of them. 48 ways.
13*48=624

There are two types of Christoffel symbols, one with only lower indexes (first kind) and one with one upper index (second kind). So long as you don't mix them, whichever way you write it is fine. I've seen it written both ways and also with the upper index in the middle.

so [math] \Gamma^a_{\ bc} = \Gamma_{bc}^{\ \ \ a} [/math] then?

why is every SQT just a bunch of math proofs questions that can be googled?

ask questions here that cant be googled and get ppls opinions

ill start: what causes cell apoptosis?

Sometimes having a dialogue with someone over a topic provides a deeper understanding of the concept, or allows understanding to take place at all.

>ill start: what causes cell apoptosis?
What do you mean by "causes" it? The mediating mechanism or the initiating one? Maybe neither, since both of these questions can be easily googled or found in the Wikipedia page on apoptosis and you said your question can't be.

13 dollar bet you go play again after big win

Can someone explain math/formulas? What are we actually doing here? Are we trying to find relations between some elements of a system and connect them through a formula?

I wouldn't know how to get a formula on my own, even though it may make sense.

Yes, but don't quote me on that.

An addendum: just look at the definition of the Christoffel symbol, it really doesn't care where you write the upper index. The only thing it cares about is the order of the lower indexes.

why not l hopital?

Equality is essentially defined as a relation where, if two things are equal, then you can substitute them for each other arbitrarily and everywhere:
[math](x=y)\Longrightarrow \left(P(x)\Leftrightarrow P(y)\right)[/math], no matter what proposition P represents.
In your case however, [math]f(n) = \frac{n}{n} = 1[/math] only when [math]n\not = 0[/math], and so [math]f(n)=g(n)[/math] doesn't hold when n is zero.

The generating function of a standard tetahedra die is: x^1 +x^2 +x^3 + x^4 and a standard cubic die is x^1 +x^2 +x^3 + x^4 +x^5+x^6.

So the sum would be: 2x^1+2x^2+2x^3+2x^4+x^5+x^6

How do I come up with the other dice that would generate this?

Am I even on the right track?

It can only be used in 0/0 etc type cases.

Should I study ODE with Boyce book or Tenenbaum book?

>Update
Well I figured out that the probability distribution of the sum of the two standard dice would be:
x^2+2x^3+3x^4+4x^5+4x^6+4x^7+3x^8+2x^9+x^10

Is there any way to find possibilities other than guessing and checking. Sorry if I am dumb.

Simmons if you're not a brainlet.

Your bonds are wrong. It tells you z is bound by the plane z = 0. That means your first set of bonds should be 0, sqrt(4-y^2). That gives you the right answer.

Write neater. Always. it doesn't matter if this is your rough work. That's illegible.

but I am a brainlet

I evaluated it using l'hopital's rule.
Given that F(gamma) = F0(cos(gamma*t) - cos(omega*t))
and G(gamma) = omega^2 - gamma^2
limit as gamma approaches omega for F(gam)/G(gam) evaluates to 0/0.

Entirely depends on what your talking about. 2+2 = 4 is simply simplifying the equation on the LHS into the RHS. This is often the case for equations including calculus.

An integral is solved from its integral form into an equation to make it understandable.

When we make a formula given elements of a system we are finding a relation between the elements and then expression that relation as a mathematical formula.

If for every 3 apples I have I can trade those apples for 4 pears a formula would be:

3a = 4p

Figured it out. There are four possibilities including the standard dice.

So then the statement "n/n = 1" is incomplete, or at least the "for n not equal to 0" is implicit. It seems like there is an analogy to the congruence relation in modular arithmetic, like a ≡ b (mod n), b ≡ c (mod n) then a ≡ c (mod n). Similarly f(n) = n/n = 1 = g(n), when we say f(n) = g(n) by the transitive property, we can't do so without maintaining the implicit restriction that they are equal WHEN n != 0, which arose when we set n/n = 1.

I've seen congruence modulo written with the modulus over the triple bar, so there is some sense in which the restrictions under which an equality holds are "attached" to the equality?

I think this question is hard to answer both well and concisely.

If you think a bunch about the rules of chess, you can come up with some theorems about how chess works. Maybe you can develop some new rules which follow logically from the "basic" rules which are set out in the definition of the game. Furthermore, maybe you can extend those idea outside of chess to tell you things about systems which are related to chess. Mathematics is about doing that, in general, for all kinds of different sets of rules.

You may find this channel interesting? youtube.com/watch?v=R7p-nPg8t_g

Any help on this would be appreciated. Obviously a rectangle (2n, 3m) or (3n, 2m) would work. What about other possibilities?

Both n and m have to be at least 2 and their product must be divisible by 6.

Key observation: A rectangle can be tiled by [math] 2 \times 3 [/math] if and only if that rectangle can be constructed by stitching together [math] 2\times 3 [/math] tiles. So lets instead investigate all the ways to stich together these tiles.

The easiest way would be by putting a [math] 2 \times 3 [/math] tile on the plane and calling that the main corner. And then stitching together some tiles next to it horizontally and some other tiles next to it vertically. You can then proof that the rest of the rectangle can be filled in properly.

If we let there be [math] x [/math] horizontal rectangles and [math] y [/math] vertical rectagles, the rectangle we created will have side lengths [math] m = 2x, n=3y [/math] so the set of pairs we are looking for is simply

[eqn] \{ (2x,3y), x,y \in \mathbb{Z} \} [/eqn]

You can very easily tile a [math]5 \times 6[/math] rectangle.

I found a story here some months ago about a hypothetical Russian rocket that used some insanely dangerous fuel/oxidizer combo because the CIA leaked that the US had found a way to make the combination work with use of a top secret stabilizer.
Anyone have a link to the story?

The probability density function

fx(x) = 1/((1+x)^2) for x>0, and 0 elsewhere.

How do I show that 1/x has the same distribution as x?

Brainlet incoming, shit deflectors activated.

What do you do when you're on the verge of failing (D or F) towards the end of the semester? I have three classes that are giving me a huge death wish. For one class I may get an instructor withdrawal because of circumstances, the other two have just been really difficult for me to balance with my stupid life and on top of that getting intuition has been difficult to me.

If anyone has any good resources that take things a bit slow for an engineering statistics and first course in linear algebra, please send. YouTube can be helpful sometimes, but there seems to be so many different approaches to these courses that not every video is directly relevant. LA has had a really bad exam average, but the hw is super long and I will probably have 2/3 of the total grade for that. Going to office hours for both classes tomorrow to be a little crybaby.

Anybody here fail out of uni before? If I actually pull my shit together I could be fine, but long term gf just left me so last week I did almost nothing (made my situation a lot worse).

[eqn] P(X \leq t) = \int_0^t \frac{1}{(1+x)^2}dx \\
= \int_{\frac{1}{t}}^\infty \frac{1}{(1+x)^2}dx \\
= P\left( X \geq \frac{1}{t} \right) \\
= P\left( \frac{1}{X} \leq t \right) [/eqn]

I see. Thank you...these are a little confusing for me.

How about..

If X has distribution R(0,1) and Y has distribution (-1/a)(lnX) with a>0,
how can I show that Y has the same distribution as the exponential distribution of a?

Stop asking other to do your homework.
Fucking do it yourself.
It's done the same way the other guy showed you.

I don't know how to.

You know the definition of a cummulative distribution function is.
Compute the cummulative distribution function of Y.
P(Y

= int from 0 to y of (-1/a)(lnX)dx, no?

then what?

>then what?
Then solve the integral.

But you can also do this
y ranges from 0 to infinity.
P(Ye^(-ay)) = 1-P(X

Anyone can recommend me a book on power electronic? About step down/up converter and half/full bridge inverter and fly-back SMPS converter?