SQT - Stupid Questions

I thought it was p^2/2m.

I'm reading Babby Rudin and he says if m^2 =2, then we could write m = p/q where p and q are not both even. How does he know that? I'm in calc 2...is this book too advanced for me?

>>8466564
Is that part of the proof that the square root of 2 is irrational?

He is probably just saying that for any number m, we can say m = p/q, where p and q are two other numbers to be determined, and you can make it so that they aren't both even.

For example, if m=5, then we could use p=15, q=3, p/q=15/3. There's never a situation where the only possible values for p and q are both even, because then you could simply divide p and q both by 2 until one isn't even, and the value m remains unchanged since you're dividing the numerator p and denominator q by the same value.

It's all very simple and pedantic, but these subtle properties are important later on in the proof.

>>8466578
Oh ok. Thanks. That last part was very helpful

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How would you go about proving this?

“Prove that if f is continuous on R and f(x + y) = f(x) + f(y) for x,y within R, then f(x) = ax for some a within R.”

>>8466785
Prove it for all rationals and then use the density of Q in R.

>>8466401
Nope. The product [math] f \lambda [/math] (called phase velocity) is, in general, not the same as [math] \displaystyle \frac{ p }{ \beta m }[/math] (group velocity) (\beta is the relativistic factor, which is almost 1 for small velocities).
But they are the same in the case of pure electromagnetic waves - both equal c.

>>8466785

This is the most basic application of density desu...

Prove that an increasing function has a countable set of non-continuous points is harder.

Is this right?
>>8466841

http://scipy-cookbook.readthedocs.io/items/SignalSmooth.html

Can anyone explain how this smooths data? or point me to a paper explaining this method.

Also what exactly is the line 45 doing:
"s=numpy.r_[x[window_len-1:0:-1],x,x[-1:-window_len:-1]]"

How does one go about doing this?
[math]\mathbf{A}[/math] is a 3x3 matrix and [math]\mathbf{b}[/math] is a 3x1 column vector. Show that all solutions of the equation
[eqn]\mathbf{Ar=d}[/eqn]
can be expressed in the form [math]\mathbf{r=p+k}[/math], where [math]\mathbf{p}[/math] is any particular solution of [math]\mathbf{Ar=d}[/math] and [math]\mathbf{k}[/math] is any solution of the related equation
[eqn]\mathbf{Ar=0}[/eqn].

>>8466981
hint: k is r-p

how can the sum of all the natural numbers be negative?
is there any case where positive plus positive can give a negative result?

>>8467012
Just think a bit about what you asked.

>>8467012
>being this stupid

Kys brainlet

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>>8467024
>>8467027
>being stupid in /sqt/
going to kill myself anyway

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tell me with reason if my working is correct?

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>>8467049
The
Question

>>8466401
E=hf is only true for electromagnetic waves. The rest is wrong, since photons do not have mass and you can not write down their momentum like that. It's p=E/c.

What's the Laplace transform of this function?

(t^2)(U(t-3))

U(t-3) would be a function expressed in terms of a unit step function, and I'm always stuck with these because I don't understand how to convert between this notation and the other one. If it wasn't expressed in the U thing notation I probably could solve this easily.

>>8467049
The limit as x -> 0 of x / (x^2 + 0^2) is not 1 / x, it's infinity. Then say 0 =/= infinity.

Then, for the conclusion, just say that since when we let y=0, the limit as x -> 0 =/= f(0), f is discontinuous at (0,0).

>>8466948
Convolution.

Each output value is a weighted average of a region of the input. If the window function (in this case, Gaussian) is some kind of "hump", this acts as a low-pass filter (i.e. higher frequencies will be attenuated more than lower frequencies).

This concept is the basis of digital signal processing. It's simplest form is referred to as a "rolling average".

> "s=numpy.r_[x[window_len-1:0:-1],x,x[-1:-window_len:-1]]
r_[] is just a short-hand for the hstack() function, i.e. it concatenates 1D arrays.

x[window_len-1:0:-1] is the first window_len elements of x but in reverse. x[-1:-window_len:-1] is the last window_len elements of x but in reverse.

The overall expression takes x and adds reflected copies of the start and end regions. This allows convolution to find the average at the ends without trying to access elements outside the array.

Copying ensures that the values of the added elements have "reasonable" magnitudes (i.e. comparable to what you'd expect to find if x was a portion of a larger array). Using a reflected copy ensures that there isn't a discontinuity at the ends (which doesn't really matter in this case, but does for other types of convolution).

Note that scipy.ndimage.convolve can do this automatically via mode='reflect'.

If P=F/A, why does a nozzle decrease Pressure? Wouldn't a smaller area equate to larger pressure?

When trying to find characteristic equation of a recurrence relation, like the following:

2^n - a(sub)n + 2a(sub)n-1 = 0

I follow that the a(sub)n's would convert as follows:

... -r + 2r^-1 = 0

However, I'm not sure what to do to 2^n when trying to find the characteristic equation.

>>8467151
t^2*U(t-3)=(q+3)^2*U(q) for q=t-3
I.e. it's (t+3)^2*U(t), time-shifted by 3.

Tables of common Laplace transforms often take the U(t) for granted, because the Laplace transform is an integral over [0,infinity], and U(t)=1 over that range.

Time-shifting by a scales the Laplace transform by e^(-a*s), i.e.
L(f(t-a)*U(t-a)) = e^(-a*s)*L(f(t)*U(t)).

L((t+3)^2*U(t))
= L((t+3)^2)
= L(t^2+6*t+9)
= 2/s^3 + 6/s^2 + 9/s
= (9*s^2 + 6*s + 2)/s^3

L(t^2*U(t-3))
= e^(-3*s)*L((t+3)^2.U(t))
= e^(-3*s)*(9*s^2 + 6*s + 2)/s^3

Alternatively, you could just evaluate the integral directly over [3,infinity], but that's more involved than using L(t^n) and time-shifting.

>>8467491
The characteristic equation is only meaningful for a linear recurrence relation.

FWIW, the solution to that is a[n]=(a[0]+n)*2^n.

>>8467478
It doesn't . A smaller nozzle with the same force behind it has a bigger force per metre exerted on it

I want to relearn every math from class six on until calculus.
Maybe add some mathematical "maturirty" preparation.
Sadly, I do not ALWAYS have Internet to just access khanacademy and I seem to be db as I can't use KA Lite as well.
Where do I start?
I am spending 32h a week just to close all my gaps and I am most of the times in places where I have to use my phone as a Hotspot.
And I am almost out of data.

>>8467723
I mean, what books do I need from libgen?
Any recommendations?
Please be honest.

>Babby Probability Problem

What is the probability that you win at least one prize given there are "n" tickets, 5 prizes are to be given, and you hold 3 tickets?

So I thought it was just one minus C(n-5, 3)/C(n,3), but the answer is supposedly one minus C(n-3, 5)/C(n,5) which is really bothering the crap out of me.

I thought the probability of no prizes is the number of distinct sets of cardinality 3 (three tickets) built from the set of tickets that contain no winning tickets (a set of cardinality n-5) all over the possible selections of "n" tickets and "k = 3 bought tickets", or just C(n-5, 3)/C(n, 3). How is this wrong?>

>forgive me I can't latex but please help

>>8467725

>rewording question

If you hold 3 tickets for which n tickets were sold and 5 prizes are to be given, what is the probability that you will win at least 1 prize?

>>8467475
Absolutely based, thank you

>>8467725
> What is the probability that you win at least one prize
It's one minus the probability that you don't win any prizes.

>>8467372
So there is no need for the working of x=0? Because y=0 is enough to prove discontinuous?

also, why is 1/x infinity?

>>8467475
Also why can you set either y=0 and x=0 if the question states that it (x,y) =/= 0

Are cohort crushes common?

How calculate sin(pi/12) and cos(pi/12) without a calculator ?

>>8468184
You just imagine sin(x) and make x = pi/12

>>8468184
sin(3x)=sin(pi/4)
Expand using angle-sum identity.
Solve cubic.
cos(x)=sqrt(1-sin(x)^2);

>>8468193
isn't just like pi/6 = 30 (remarkable value) so pi/6/2 = 30 / 2 = 15

>>8468203
it doens't help me in fact lol

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Help please

>>8468193
top kek

>>8468219
Ok, this is my solution.
F^3=F*F^2=I
let's multiply both sides by squared inverse
F^-2*F^2*F=I*F^-2
F^2*F^-2 equal to identity (or I)
thus F=I*F^-1*F^-1
lets think about F^2=I(identity)
as we know from isometries inverses, if F*G=I then G-inverse of F
F^2=F*F=I
thus F=F^-1
lets recall our previous equality:
F=I*F^-1*F^-1
and as we know F=F^-1 thus F^-1*F^-1=F^2=I
thus F=I*I=I
and this finally proves that F=I
Did I prove it right?

How do I find a parametrization [math] X: \mathbb{R} ^2 \rightarrow \mathbb{R} ^3 [/math] of the regular surface [math] M = \{ (x,y,z) \in \mathbb{R} ^3 ; xsinz=ycosz \} [/math] ?

>>8468258
Oh Jeez, even easier.
F^3=I
this means that
F^2*F=I
and as we know F^2=I
thus I*F=I
therfore F=I
proofs for b,c are the same

>>8468258
Well,
F=F*I=F*F^2=F^3=I
I=I^2=F^4*F^4=F^8=F^7*F=I*F=F
I=I^2=F^8*F^8=F^16=F*F^15=F*(F^5)^3=F*I^3=F
I'm sure yours is fine too, but I'd rather not use inverses if I can help it.

>>8468287
thank you

>>8467638
Thanks

>>8468276
z = t
x = r cos(t)
y = r sin(t)

>>8468342
Aw, makes sense. Thank you

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What's the name of those functions?

>>8468356
http://mathworld.wolfram.com/LipschitzFunction.html

>>8468365
Thank you sir

is a de broglie wavelength just an uncertainty-type thing?
Like it's not an actual physical wave-like motion right?

>>8467846
I'm aware of that, I actually made mention to that. Can anyone actually give me a solid answer to my concern regarding the combinations?

>>8467895
>>8467895
If you let x=0 and then take the limit as y->0, as you did, you see that it's equal to 0. This does not prove discontinuity, in fact it is perfectly consistent with what a continuous function would do. It's not necessary to include that calculation.

I meant to say that the limit as x->0 of 1/x doesn't exist.

In your work, you took the limit as x->0 of x / (x^2 + 0^2), and said it's equal to 1/x. If you take the limit of something with respect to x, you can't have x in the answer. You need to continue by taking the lim x->0 of 1/x, which doesn't exist because the denominator is shrinking, and it goes to positive or negative infinity depending on which side you approach from.

>>8468429
You hold 3 tickets out of n. Initially, there are n-3 tickets which you don't hold. The probability that you don't win the first prize is (n-3)/n.

After that prize has been awarded, you hold 3 tickets out of n-1; there are n-4 tickets which you don't hold. So the probability that you don't win the second prize is (n-4)/(n-1).

The probability that you don't win any of the 5 prizes is:
(n-3)/n * (n-4)/(n-1) * (n-5)/(n-2) * (n-6)/(n-3) * (n-7)/(n-4)
= ((n-3)!/(n-8)!) / (n!/(n-5)!)
= ((n-3)!*(n-5)!) / (n!*(n-8)!)

C(n-3, 5)/C(n,5)
= ((n-3)!/(5!*(n-8)!)) / (n!/(5!*(n-5)!))
= ((n-3)!*5!*(n-5)!) / (n!*5!*(n-8)!)
= ((n-3)!*(n-5)!) / (n!*(n-8)!)

i'm doing basic thermochem and can't figure out what i'm doing wrong

i have an 80.0-gram sample of a gas
it heated from 25 °C to 225 °C.
346 J of work was done by the system and its internal energy increased by 7075 J. What is the specific heat of the gas?

here's what i did:
7075J-346J (because work was done by the system)= c(what i'm looking for) (80.0g)(225C-25C)
what i go was 0.42056J/(gC) but it says i'm wrong

>>8468436
You say if I take limit wrtx which I presume you mean x-->0 lim and x remains in the limit instead of a constant after subbing y=0 in

But. If you were to set (x,y) = (1/n,1/n) where n--> infinity then n would still exist simplifying to n/2. I have "n" in my answer and I presume you would say I can't. Why?

>>8468565
First law of thermodynamics: [math]\Delta U \,=\, W \,+\, Q[/math] then [math]Q \,=\, \Delta U \,-\, W[/math].
Always take the [math]\Delta U \,=\, W \,+\, Q[/math] convention where [math]W[/math] is the work *received* by the system, cause the other one makes no sense. It should be 7075 J + 346 J.

What does delta and epsilon mean in terms of limits?

Linear Algebra.

----------------------------------------------------------------------------
Let A be an n × n matrix such that A2 = A. Show that (AB − ABA)2 = 0 for all
n × n matrices B.
----------------------------------------------------------------------------

I can't figure this out guys. First I tried some fuckery with the fact that you can substitute B for A (since B can be ANY nxn matrix) and it works out to 0, but it only works when B = A. Then I tried shifting ABA around, but then I remembered that matrix multiplication is not communicative.

Stuck at the moment.

>>8468805
There are no tricks involved in this problem anon
(AB - ABA)^2 = (AB(1 - A))^2 = AB(1 - A)AB(1 - A)
Now notice the term (1 - A)A and draw your conclusion

>>8468832
sigh...why didn't I think to factor out the AB?

thanks anon!

In isoparametric coordinate mapping (4 nodes), let's say I have an initial configuration and a final configuration, Bo and Bt. Instead of just mapping a single configuration to a parent with coordinates (1,1) (-1,1) (-1,-1) (1,-1), what would I do to find the displacement tensor between the two? Do I map both to the parent or do I map the final config to the initial config?

>>8468554
thank you so much I appreciate it!

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I'm in a beginner proof course and I'm trying to do practice problems from my book. For some reason the autists who wrote this don't give answers for every question and only some of them

I'm stuck on this and there's no solution given. Trying to do it through induction

>>8468847
Anyone pls. I feel like this should be so simple but there are literally no examples online where it isn't just mapping a configuration to some form of the unit square.

>>8469096
just say obviously true by triangle inequality

>>8469106
Well yeah I've done the triangle inequality before but I can't leave it like that, I'm doing these to prepare for an exam

>>8469096
It's by induction, it's true for n = 2

suppose that it is true for n in N

then abs(a_1+...+a_n+1) <= abs(a_1+...+a_n)+abs(a_n+1)<= abs(a1)+abs(a2)+...+abs(a_n+1)

>>8468554
>>8469061

I actually managed to solve it another way, its one minus the number of ways that you can have five winning tickets out of a set of n-3 tickets, and the three tickets left out in the set are the ones that are bought.

Also could you recommend me a good text on probability? Thanks

>>8468676
I don't think you're taking limits correctly.

Here's a different function as an example:

The limit as x-> inf of ( x / x^2 )

= the limit as x -> inf of ( 1 / x )

'note, here I've just simplified the part inside the limit. Next, i will actually evaluate the limit.

= 0

'this is because the denominator gets arbitrarily large, shrinking the value completely.

After you evaluate the limit of something with respect to a variable, the variable is eliminated. Otherwise you're not doing it completely.

Also, with regards to the function at (0,0), the piece-wise definition of the function tells you that f(0,0) = 0.

Why is a circle the most efficient way to bound an area (uses the least perimeter while covering the most area)?

>>8469188
Try a proof by contradiction. Assume that some other shape is more efficient than a circle and, and show how that property would lead to a contradiction unless that shape is in fact a circle after all.

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how is this a proof of uncountability? isn't the basic argument being used here the same as in most common "no largest prime" proofs? by that logic isn't there an uncountable number of primes?

>>8469425
It's similar, but not the same.

In a infinite prime proof, you assume there are FINITE primes and then get a contradiction. That means you're assuming a bijection with some numbers 1...n, not with all of N like you are here.

>>8469425
The "no largest prime" proof shows that for any finite set of primes, there must be another prime outside of that set, and so the primes as a whole must be infinite.

The diagonalization argument does not assume finity, just countability. It shows that for any countable set of reals, there must be some real outside that set, and so the reals as a whole must be uncountable.

>>8469188
Every shape that is not a circle will have an arc somewhere that you can replace with a "more circular" arc that encompasses the same area in less perimeter.

What is the force pulling, say, earth outwards from sun? If the gravity of sun is pulling earth towards itself and that's why we're circling around it, then what's keeping us from getting closer?

>>8467012
It's easy to confuse yourself with this shit but it's quite simple.

All that the ramanujan summation stuff, cutoff and zeta regularization does, is look at the smoothed curve at x = 0.
What sums usually do is look at the value as x->inf.

It's just a unique value you can assign to a sum, really they have many such values.

https://en.wikipedia.org/wiki/File:Sum1234Summary.svg

>>8469465
> What is the force pulling, say, earth outwards from sun?
There isn't one.

> If the gravity of sun is pulling earth towards itself and that's why we're circling around it, then what's keeping us from getting closer?
Momentum.

If an object A is subjected to a force which is always in the direction of object B, it doesn't follow that A will get closer to B.

For an object to move in a circle (i.e. remaining at a constant distance from some point), an inward force is required. In the absence of such a force, the object would move in a straight line.

>>8469465
Even though the speed may not be changing much the velocity is constantly changing, which is a result of the force. You don't need a "counter-force" to keep things in orbit, you'd need a counter force to keep things moving at a constant velocity in a straight line.

333pi/7 = x -pi =< x < pi

how we do ? i don't remember

>>8469187
Thanks for the clarification. I have done it correctly if you consider subbing (x,y) as (1/n,1/n) as x->inf because you end up with n/2 which is inf hence inf =/= 0 so not continous.

(I believe it also works the same if you put y->inf insread of x-> correct me if I'm wrong)

As with my first example it's wrong because 1/x as x-> 0 is 0 and the question states 0 elsewhere, hence 0=0 does not disprove continuous.

Is this correct?

what are some good books of first order logic suited suited for a grad level comp.sci course?

i need something full of formal proofs. i can learn the concepts in class but i'm bored as fuck when it comes to copy formal proofs, so my notebook lacks those parts.

I dare you to express this series in terms of n :

Un : { U0=2/3 , U(n+1)=3(Un)^2 }

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>>8466401
How come covariance of 2 vectors is a matrix? This seems to be like it returns a scalar

Let's assume that an incredibly sharp instrument clearly severs my leg at the hip. Could I beat myself to death with it before losing consciousness from blood loss/control from shock?

>>8470235
The covariance of two variables is a scalar.

The covariance matrix of a vector variable X is a matrix M where M[i,j] = Cov(X[i], X[j]), i.e. the covariance between two elements of the vector.

An element on the main diagonal M[i,i] is Cov(X[i],X[i])=Var(X[i]), i.e. the variance of a specific element of the vector.

>>8466401
Let X be a random variable with Poisson(λ) distribution, with 0 < λ < 1. Find E[X!]

What did they mean by this? Does this mean that I have to find the "Factorial Moment"?

https://en.wikipedia.org/wiki/Factorial_moment

I have not been attending classes so I don't know whether this subject has been covered or what kind of notation is the professor using.

Please help!

>>8470454
why dont you ask the professor?

>>8469976
self bump

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let's say I have a function
f(x)=(x+1)/(x-1) but when x=1 f(x)=1

is that function onto/one-to-one then?

shitty picture attached

Why do people in computational optimizations say that sometimes is difficult to calculate the Hessian/2nd derivatives?

Can't you do something like
[eqn]
f'(x) = \frac{f(x + 0.00000001) - f(x)}{0.00000001}
[eqn]
twice, and generalised to multivariate case?

>>8470692
Yes, you dumb dumb. Now prove it yourself. Set f(a)=f(b) and show a=b

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what the HELL is going on

>>8470738
What are you confused about?
I can help. All of it made sense to me.

Let X be a set with an associative binary operation * and identity e. Suppose every element of X satifsies x*x=e. Prove that * is commutative.

Holy shit please help me

>>8470742
1st it's all like
>points have no neighbors

then it's all like
>every point has an arbitrarily close neighbor

>>8470751
ab=eabe=bbabaa=b(ba)(ba)a=ba

What are the names of the 2 methods used to find particular solutions to homogeneous differential equations?

>>8470757
Ehhhhhh no. The first is like "the set has no intervals." Consider the rationals in the reals. That set has no intervals, but every point has an arbitrarily close neighbor.

>>8470809
now it makes sense, thanks

I know there are rules for the Fourier transform of f(ax) and f(x-x0), but what if you combine them to something like f(ax+b)? Can I just linearly combine the two rules?

I'm a freshman EE major and I'm taking a class on digital logic.

The circuits we end up building are pretty simple but theres a few strange things I don't really understand about them.

The biggest issue I'm having is how the hell do i know how much voltage I need in a circuit? I have a circuit with 6 LEDs, 2 chips, and some resistors. None of them use more than 5 volts. When I use exactly 5 volts of power though, nothing works. As soon as I turn it up to 5.1 volts, the circuit works as it should in multisim.

I tried googling "how much voltage" and various add ons to that search but I keep setting info about what size power supply I should use for something when really I just want some math formulas or something to tell me how much voltage a given circuit needs.

Am I being a retard by not using these constant acceleration equations? I've been getting by just fine by using displacement and standard relationships between velocity, acceleration, time, etc

Maybe I'll need them more later, but it seemed I was supposed to start using them in this chapter and Ihaven't yet.

>>8471187
What constant acceleration equations are you referring to?

>>8470724
> f(x + 0.00000001) - f(x)
This is extremely inaccurate.

First, x + 0.00000001 will be rounded, and the rounding error is a significant fraction of 0.00000001.

Second, both f(x + 0.00000001) and f(x) will be rounded. Because those two values are likely to be close, the rounding error in each value will be a significant fraction of the difference between them.

If x and y are close, and have relative error e, then the relative error in y-x is e*x/|y-x|. I.e. you're multiplying the relative error by the ratio of the value to the difference.

E.g. in the above, if f'(x) was approximately one, the difference would be ~1e-8 so you'd be multiplying the relative error by 1e8. If you did that twice (for a second derivative), you'd be multiplying the relative error by 1e16.

The relative error for double-precision floating-point is ~2.2e-16, so you'd end up with a relative error of ~2.2, i.e. the absolute error would actually be larger than the calculated value.

>>8470757
Any neighbourhood of a point in S will contain both points which are not in S (1) and points which are in S (2), no matter how small that neighbourhood.

>>8470852
> I know there are rules for the Fourier transform of f(ax) and f(x-x0), but what if you combine them to something like f(ax+b)?

f(ax+b)=g(ax) where g(x)=f(x+b).

So if you have the transform of f(x) you can find the transform of g(x)=f(x+b) by time-shifting and from that the transform of g(ax) by time-scaling.

>>8470976

There could be some losses in the circuit (e.g. a voltage drop over a resistor) that cause the voltage at the actual logic to be under 5V.

What chips are you using? Have a look at their datasheets and see what their requirements are specifically.

>>8470976
Are you using some kind of generic logic component with a threshold voltage set to 5.0V?

It's a fundamental property of digital logic that the minimum "high" output voltage is greater than the maximum "high" input voltage threshold and the maximum "low" output voltage is less than the minimum "low" input voltage threshold.

That's what allows you to ignore the voltages and treat signals as 0 or 1.

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Find a differential equation whose solution is the n parameter family.

[math] y=\sqrt{c_{1}x^{2}+c_{2}}[/math] I found [math]y'=\frac{c_{1}x}{\sqrt{c_{1}x^{2}+c_{2}}} [/math] and [math] y''=\frac{c_{1}c_{2}}{(c_{1}x^{2}+c_{2})^{\frac{3}{2}}} [/math], the answer is [math] xyy''+x(y')^{2}-yy' [/math]. It all checks out, but I have no idea how they could have arrived at that.

>>8470770
>b(ba)(ba)a=ba
what

>>8471610
[math] xyy''+x(y')^{2}-yy'=0 [/math]*

S = 1 + 11 + 111 + 1111 + ... + 111...111
S = ?

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What happens in 2050?

>>8471702
recycling

>>8471619
S = (1)+(1+10)+(1+10+100)+(1+10+100+1000)+...

S = (10^0)+(10^0+10^1)+(10^0+10^1+10^2)+..

Each term of the sum S can be written as a geometric progression

an = ((10^n)-1)/9

Now you just have to do some algebra

>>8471619
Here, I did the most naive proof for you so that your teacher will tell you that you're stupid because there are so much quicker ways to find the result but you did not bother to look for them.
Let [math]N \,\in\, \mathbf N[/math].
[eqn]\sum_{n \,=\, 1}^N \left( \sum_{k \,=\, 0}^{n \,-\, 1} 10^k \right) \,=\, \sum_{n \,=\, 1}^N \frac{10^n \,-\, 1}{10 \,-\, 1} \,=\, \frac{\sum_{n \,=\, 1}^N 10^n \,-\, \sum_{n \,=\, 1}^N 1}{9} \,=\, \frac{10 \,\frac{10^N \,-\, 1}{10 \,-\, 1} \,-\, N}{9} \,=\, \frac{10^{N \,+\, 1} \,-\, 10 \,-\, 9\,N}{81} \,=\, \frac{10^{N \,+\, 1} \,\left( 1 \,-\, \frac{1}{10^N} \,-\, \frac{9\,N}{10^{N \,+\, 1}} \right)}{81} \,\xrightarrow[N \,\rightarrow\, \infty]{}\, \infty[/eqn]
Therefore [math]S \,=\, \sum_{n \,=\, 1}^\infty \left( \sum_{k \,=\, 0}^{n \,-\, 1} 10^k \right) \,=\, \infty[/math].

A rectangular piece of aluminum is 7.60 +/- 0.01 cm long and 1.90 +/- 0.01 cm wide. Find the area of the rectangle and the uncertainty in the area.

>>8471619
9*S = 9 + 99 + 999 + 9999 + ... + 999...999
= sum[i=1..n](10^i-1)
= sum[i=1..n](10^i)-n
= 10*(10^n-1)/(10-1)-n
S = (10*(10^n-1)/9-n)/9

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Any help on v)? I feel like I should be using the linear map in iv) somehow.

I have a 2D linear system [math] \dot{\mathbf{x}}=A\mathbf{x} [/math] where [math] A=[3,4;-\frac{9}{4},-3] [/math]. I have computed that the eigenvalues of [math] A [/math] are [math] \lambda_{1,2}=0 [/math] and the eigenvector of [math] A [/math] is [math] \mathbf{u}_1=[-4,3]^T [/math]. I have also deducted that all points on the line [math] y=-\frac{3}{4}x [/math] are fixed points. If there is only one fixed point where [math] A [/math] has one eigenvector you would call such a fixed point a [math] \textit{degenerate node} [/math]. In this case I have a line of fixed points, as I said, instead of only one fixed point. Would I call such a line a [math] \textit{degenerate line} [/math]?

The set of functions R -> R

df/dx + 2 f = 0

prove it is a vector space. Im having trouble.

Sin(x) doesnt jive. What am I doing wrong?

Is the memory palace method really the way to go? I mean is it worth whatever amount of time you put into it?

I'm in studying math and physics, and exams require to be fast so I don't have much time to remember everything.

Lim x->a = L

What the fug does L mean?

>>8466401
THREAD MOVED HERE
>>8471873

>>8471869
The set itself is [math]\left\{ f:\, x \,\longmapsto\, \lambda\,\mathrm e^{-2\,x} \mid \lambda \,\in\, \mathbf R \right\}[/math] which is trivially a vector space.

Alternatively, you suppose [math]f[/math] and [math]g[/math] are solutions and you calculate [math]\left(f \,+\, \lambda\,g \right)' \,+\, 2 \left(f \,+\, \lambda\, g \right)[/math].

>>8471901
it means "limit"

are all homogeneous diff eqns solvable by assuming y=e^(kx) and then either analytically or numerically finding the roots of the polynomial that results from the substitution?

>>8471901
Limit as x approaches a of what?

>>8472190
Not if the coefficients aren't constants.

Otherwise yes, as long as it's linear, by using the kernel lemma

>>8471817
by rank nullity theorem you have dim (ker T_U) + dim (Im T_U) = dim (W)

so since ker (T_U) = (U^perp) you have
dim (U^perp)+dim (Im T_U) = dim (W).

since Im (T_U) is inside (U*) which has the same dimension as U, dim (Im (T_U)) <= dim (U^*) = dim U

therefore dim U + dim U^perp >= dim W

>>8471612
b(ba)(ba)a
=b(ba)^2a
=bea
=ba

does that help?

File: sci.png (8KB, 553x290px)

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No idea how to do this, I'm sure I'm being totally fucking retarded as well.

Say you're trying to minimise [math]f(x)[/math]

I know about FONC, SONC, SOSC, etc. but what about checking at each iteration: [math]x^{i+1} - x^{i}[/math] and [math]f(x^{i+1}) - f(x^i)[/math]?, and see if they're smaller than a super small threshold?

I know they're not great, but what are these optimality conditions? Do they have a name? Were they used/ are they used? Where can I read up on them

>>8472316
Btw [math]x^{i+1}[/math] is found via gradient descent

>>8472315
when the variable resistor is low, you have 1 volt. that means that R2 has a voltage drop of 1V. when it is up, yiu have 9V. therefore the middle resistance has an 8 volt drop. then you know the current. with the current you can get the values of the resistors

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Pls halp

>>8471299
>There's like 6 of them for 1D motion, I was referring to all of them.

File: 1471803690368.jpg (992KB, 2048x1365px)

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Do you guys surround yourself with intelligent people for good discussion? How do you do it?

The people I live with are all horribly retarded, and I can feel my articulation skills getting softer and softer. How do I stop this regression?

>>8472361
Thanks anon.

>>8472381
my guess: if m isn't a multiple of 12, then for n the smallest natural such that n*m=0 mod 12, throw the dice n times and add up all the shots modulo 12
dunno if this gives an even distribution

>>8472388
this wasn't supposed to be green texted lol

>>8472392
Meet new people. Try volunteering or picking up new hobbies and network with people.

>>8472381
Is 13 much larger than 12?

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>>8466858
i made some notes for you anon. I won't solve your problem for you, but these should help with your understanding of the question. If you are still having trouble message me and I'll walk through your problem for you.

>>8472381
Is this supposed to be exact? Why would M have to be much larger if you need it exact?

If m is the roll of the M-sided die, you can take t=mod(m-1,12)+1
and get approximately each 1,...,12 approximately 1/12 of the time.

If M is a multiple of 12, this will work exactly.

>>8472381
If you roll a number larger than 12 reroll
eventually you must terminate on a number 1-12 and all numbers are equally likely.

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>>8472580
Cool notes, anon. It helps. Also, I would appreciate if you have notes on proving a limit exists for f(x) and f(x,y) with all the delta epsilon stuff?

My TA and professor insist I use the method in my pic related. Where (x,y)=(1/n,1/n). Although, it would obviously be of benefit knowing a variety of techniques. When 1/n may not work.

Also if: lim x->a f(x)=L where L means limit?
Then I presume if you replace L with l0 it would also mean the same thing. I am just trying to clarify what these letters mean in my textbook.

How do I evaluate [eqn] \int \frac{\mathrm{e}^{ i \mathbf{c \cdot r}}}{|\mathbf{r}|^2} \mathrm{d}^3\mathbf{r} [/eqn] where [math]\mathbf{c}[/math] is an arbitrary constant vector, over all space in spherical polar coordinates?

>>8472707
What is d^3r? I never seen that notation.

>>8472723
It means a volume element independent of coordinate system I think, pretty much just equivalent to dV.

>>8472665
The word "constant" in the problem appears to preclude this approach.

>>8472707
assume c is pointing in the z direction and has modulo k (constant) plug the Jacobian in and you have all set for integration with respect to θ

>>8472723
Standard notation in physics courses

>>8472747
Thanks anon, thank works. I think my homework is screwy because it doesn't make any reference to the direction of c, but I assume it's impossible or very difficult to solve analytically in a general direction.

>>8472361
R1 = 3750, R2 = 1250?

>>8472707

do you mean

[math]\int\int\int \frac{\exp\big( i (c_1 x+c_2 y+ c_3 z) \big)}{x^2 +y^2 + z^2} dx dy dz [/math]

Do you need to use spherical coordinates? I might assume c = (0,0,k), i.e. k=|c| and use cylindrical.

>>8472758
Spherical is suggested but I don't think it matters. I'll try that, thanks anon.

>>8472767
i found it to be the integral of sen(x)/x from zero to infinity, Plus some constants.

>>8472841
pi then?

http://math.stackexchange.com/questions/891812/integration-of-sinc-function

can a heat pump between equal temperatures work?

>>8472665
>>8472642
All the answers so far are mathematically wrong. Returning rand() % N does not uniformly give a number in the range [0, N) unless N divides the length of the interval into which rand() returns (i.e. is a power of 2). Furthermore, one has no idea whether the moduli of rand() are independent: it's possible that they go 0, 1, 2, ..., which is uniform but not very random. The only assumption it seems reasonable to make is that rand() puts out a Poisson distribution: any two nonoverlapping subintervals of the same size are equally likely and independent. For a finite set of values, this implies a uniform distribution and also ensures that the values of rand() are nicely scattered.

This means that the only correct way of changing the range of rand() is to divide it into boxes; for example, if RAND_MAX == 11 and you want a range of 1..6, you should assign {0,1} to 1, {2,3} to 2, and so on. These are disjoint, equally-sized intervals and thus are uniformly and independently distributed.

The suggestion to use floating-point division is mathematically plausible but suffers from rounding issues in principle. Perhaps double is high-enough precision to make it work; perhaps not. I don't know and I don't want to have to figure it out; in any case, the answer is system-dependent.

The correct way is to use integer arithmetic.

Source: stackoverflow

>>8472920
2*π^2/|c| is my result

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This is kind of stupid, but I was trying to do a problem in a form similar to pic related.

All the things online I've found are similar to my pic, in that the step in the middle literally consists of "check the numbers randomly from the pool of 16(or however many) until you find one that works".

Is there any better method to do this or do you literally worst-case have to just try all 16 of the different possibilities?

File: SLP.png (353KB, 1992x924px)

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Somebody please want to help walk me through this? I'll answer some others in the meantime.

>>8472979
your other option is the cubic equation so yeah, that's the best way to do it.

>>8472954
>unless N divides the length of the interval

This is what was said.

> moduli of rand()
nobody mentioned rand() at all

>>8473024
walk-through is pretty much here

https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory

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Pls help I am lost

Why must an object have a constant velocity if it is in equilibrium? I get that it should, but why is that necessary?

>>8473156
Newton's second law
If it's not in equilibrium (I'm assuming you're talking forces) then...

>>8473156
Equilibrium means no net force on the object. Since F=ma there's no acceleration.

>>8473166
>>8473165
I get that, but why does the world work like that?

>>8473181
well your question is why does Newton's 2nd Law exist.
Which is the same thing as asking why momentum is conserved.

Why is momentum conserved? For the lulz? idk

>>8473181
Are you asking why [math]F = ma[/math] is a thing? It's because we've made lots of tests over hundreds of years which indicate that Newton's second law is true. (Pedants will insist on the relativistic version, but whatever -- the idea is the same).

I'm afraid you may have confused science with philosophy.

>>8473156
>>8473181

This is actually a pretty deep fact. It results from the geometry and symmetry properties of flat spacetime and the coordinate-invariance of the laws of physics.

One way to look at this is to note flat spacetime has Poincare symmetry, which includes certain transformations that relate "moving" frames to each other (the so-called Lorentz transformations, or the Galilean transformations for low speeds). Consequently this constrains the paths of particles (which are geodesics) to be straight lines in spacetime.

Of course, as the others pointed out, you can also just solve the equations of motion, which will give a straight line.

>>8473197
engineer detected
if you're math/phys you should be disgusted with yourself
>Newton's 2nd law is as deep as it goes, that's all, and it's only empirically derived
sad.jpg

>>8473195
>Why is momentum conserved?
Translational symmetry of the laws of physics

https://en.wikipedia.org/wiki/Noether%27s_theorem

Next question pls

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I'm not sure how to show that this is not possible for all cases. It clearly doesn't work when I try to do it on a board.

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>>8473252
My professor is referencing this equation. I'm not sure how to apply it to this problem though.

>>8473252
>>8473257
What class is this?

>>8473215
how's freshman mechanics, kiddo? :^)

>>8473261
It's a problem solving class. We look at a variety of problems and different techniques to solve them. I

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Is this basically just saying that random variables are morphisms in the category of measurable spaces?

Also when it says that [math]\mathcal{B}(E)[/math] is the sigma algebra generated from the collection of open sets, does it mean generated by the batty process described here?
https://en.wikipedia.org/wiki/Borel_set#Generating_the_Borel_algebra

What is the intuition behind said process? I am having trouble grasping it.

>>8473329
> random variables are morphisms in the category of measurable spaces

well a random variable is in particular such a morphism, but not all such morphisms are random variables, because the domain of a rv has to be the probability space (so in particular they can't be composed)

>>8473347
That's another thing. The definition sets up the domain as being specifically a p.space but then doesn't make any further use of that fact.

Could you instead say that random variables are [math]X \circ F_0[/math] where X in Mor(Measble) and F is the forgetful functor between probability spaces and measurable spaces?

>>8472407
> dunno if this gives an even distribution
Adding up dice will give you something approaching a Gaussian distribution (i.e. a bell curve).

>>8472381
For a roughly-even distribution, you'd use e.g.
1+((x1-1) + M*((x2-1) + M*((x3-1) + M*(x4-1)))*12/(M^4)
where the xi are the results of the individual rolls.

But that's only going to be perfectly uniform if M is a multiple of 6. If the number of rolls is large, it will be close to uniform.

If you need perfect uniformity, you'd choose a number of rolls n so that M^n is large, choose the largest K such that 12*K<M^n and if the result is >=K, start over.

While there's no limit to the number of attempts required, the expected number of attempts will be finite, as the infinite sum of n*p(n) converges.

>>8473252
If a square has exactly two contaminated neighbours, the square becoming contaminated doesn't change the length of the perimeter. Upon becoming contaminated, the two edges between the cell and its contaminated neighbours are removed from the perimeter, while the other two edges are added.

If a square has 3 contaminated neighbours, then becoming contaminated will reduce the length of the perimeter by 2 (removing 3 edges, adding 1). And for four neighbours, it's reduced by 4 (removing 4, adding none).

So the length of the perimeter never increases.

9 1x1 contaminated cells have a perimeter of at most 9*4=36. The entire 10x10 board has a perimeter of 10*4=40, so for the entire board to be contaminated, the perimeter would have to grow.

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Any ideas on how to go about B?

>>8473419
An example of the most contamination possible is to contaminate the diagonal of a 9x9 "corner" (i.e. ignoring two edges).

You start with a "line" of width 1. After the neighbours have been contaminated, you have a line of width 3, then 5, then 7, then ending up with a 9x9 square bounding the original contaminated cells.

>>8473378
well there's nothing stopping you from studying the category of measurable spaces and maps, but the definition of a rv is what it is for a reason. The idea of a random variable is supposed to formalize the idea of "the outcome of an experiment". For example, X1=the outcome of a dice roll, and X2=the outcome of a second dice roll. Then what would we mean by [math] X_1\circ X_2[/math]? The setup doesn't exactly lend itself to the language of category theory.

>>8471187
The constant-acceleration equations are just the result of solving the differential equations v=ds//dt, a=dv/dt (which are effectively the definitions of velocity and acceleration) and re-arranging (i.e. choosing which of the variables to solve the resulting system for).

You can always just solve each specific case from first principles, rather than solving the general case and substituting values for the independent variables.

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how do i do this

Not really sci related, but what am I doing wrong here?

function b = atest(fc)
fc=str2func('fc');
b=2
end

function dudt = g(t,u)
a=3;
b=9;
c=15;
d=15;
dudt = [ a*u(1)-b*u(1)*u(2); c*u(1)*u(2)-d*u(2) ];
end


function yp=f(~,yn)
l=0.8;
yp=l.*yn;
end


atest(f) works, but atest(g) gives
>Not enough input arguments.
>Error in lotka (line 6)
>dudt = [ a*u(1)-b*u(1)*u(2); c*u(1)*u(2)-d*u(2) ];

>>8473499
nvm I found the solution

>>8473447
1. Let N be the plane normal vector [1,1,-2].
2. Choose any two vectors U and V such that N,U,V are linearly independent. U=[1,0,0] and V=[0,1,0] will work.
3. Let X=N×U and Y=N×V (where × is the cross product operator). These are guaranteed to be perpendicular to N (not necessarily to each other, but that doesn't matter).
4. Let M be the matrix whose rows are N,X,Y.
5. Let R be the matrix [[-1,0,0],[0,1,0],[0,0,1]], i.e. the reflection in the first dimension.
6. (M^-1).R.M is the reflection in the plane.

FWIW, the result is (1/3)*[[2,-1,2],[-1,2,2],[2,2,-1]].

Note that (like any reflection matrix), this is orthonormal, its own inverse and, and its own transpose (i.e. it's a square root of the identity matrix).

I have a test 10 hours from now that I forgot about until 3 hours ago. It's entirely rote memorization. Should I study for 10 hours or get some sleep and study for 4? I don't think 4 hours is quite enough for me to memorize everything but I don't know if that would be worse than the brain fog that comes from exhaustion or not.

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Uh. Am I on the right track?

how come I'm quite good at linear algebra, but suck at analysis?

t.brainlet

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why exactly is it ok to use this -1/12 nonsense here?

>>8473637
The divergent series, senpai.

https://en.wikipedia.org/wiki/File:Sum1234Summary.svg

>>8466401
A vertical cylinder with a frictionless piston, which is burdened by a weight. The piston diameter is 3.5 cm. The effect of gravity on the weight and the piston causes along with the atmospheric pressure, a pressure in the gas under the piston, which is 1.35 * 10 ^ 5 Pa. The volume is 0.25 liter and the temperature is 22 * C.

What is the mass of the weight and the stamp?

I really have no idea what to do here..

>>8473699
Piston*, not stamp.

>>8473699
Simple version: Atmospheric pressure is ~101 kPa. The compression results in an extra 34 kPa. Multiply by the area of the piston to get the force (i.e. weight). Multiply by g to get the mass.

More complex version: Boyle's law.

Hi sci, i have essentially lost all physics/math knowledge at this point, but just wondering
is pressure proportional(?) to volume?
Lets say i have a cylinder with volume of 1cubic meter, filled with a gas at a pressure of 1 bar, and connect it to another cylinder with nothing in it(lets say this is in a vacuum), the total accessible volume for the gas now becomes 2 cubic meters, does the pressure halve?

TLDR: What is the relation between pressure and volume?

>>8473737
Thanks for your answer. I have read Boyle's law a few times now, but I can't see how it's relevant here, since none of the variables used are mass.

>>8473783
Boyle's law relates to pressure, but I don't think that it's relevant here.

It might become relevant if the question has more parts.

>>8473793
I'm asking about question B in the assignment. It starts by asking about the amount of mol gas in the cylinder, where I have used p=((n*R*T)/(V))=solve(1.35*10^(5)*_Pa=((n*_Rc*295.15*_°K)/(0.25*_l)),n) ▸ n=0.013753*_mol
It then asks for the mass of the weight and the piston.

>>8473774
> is pressure proportional(?) to volume?
Inversely proportional.

> Lets say i have a cylinder with volume of 1cubic meter, filled with a gas at a pressure of 1 bar, and connect it to another cylinder with nothing in it(lets say this is in a vacuum), the total accessible volume for the gas now becomes 2 cubic meters, does the pressure halve?
If all other factors (e.g. temperature) remain constant, then yes.

>>8473796
The pressure difference between the enclosed section and the atmosphere exerts an upward force equal to the pressure difference multiplied by the area of the piston. The weight of the piston exerts a downward force. These two must be equal.

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I want to simulate (relatively) realistic bullet physics in a game. I'm doing it from scratch, so no Havok or whatever. Could someone suggest what and where I should do my research to learn these things?

>>8473808
Sorry to ask this, but can you possibly line up the equation, because I honestly can't think my way through this..

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Can anybody help me solve a problem with my code? Just started using Python last month so I'm really struggling...

>>8473844
Atmospheric pressure is roughly 101 kPa. If the pressure inside the cylinder is 135 kPa, that's a differential of 34 kPa.

The area of the piston is pi*0.035^2/4 = 9.62e-4 m^2. The force on the piston is 34e3 * 9.62e-4 = 32.71 N. Thus the mass of the piston is 32.71/9.81 = 3.33 kg.

I have a 100 unit circle and I am 50 units on the x axis. How do I get the intensity or height of the circle at x=50 ?

>>8473886
sp.integrate is not a function, it's a module, and therefore not callable.

https://docs.scipy.org/doc/scipy-0.18.1/reference/integrate.html

Here is a list of the functions included in the scipy.integrate module. These are the actual functions that you can call.

>>8474025
Tried quad and got another error.
>error: Supplied function does not return a valid float.

wat do
thanks for the help btw

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>>8474030
>>8474025
Scrap that, made a change to my limits and now I have a new error...

how do i solve a function at a point in SAGE
like if i have
D = x^2 + y^3
What do i input for say, x = 1 and y = 2 to give me the answer for D?

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I want to read about electronics and circuts. Where do I begin? In my country I did not have physicis in highschool but I have done math up untill like calc 2. So sitting thru the physics classroom feels retarded. And tips?

>>8472979
If you use a programmable calculator, you could make a simple program that automatically calculates all the candidate values and tests them and returns the ones that work.

>>8473606

cos(1/x2) is bounded and sqrt(x) --> 0 as x -> 0, so the limit of the product is also 0.

Could you use the resulting enthalpy of a determinate chemical reaction to boil water?

This shit question came in a recent Thermo exam as a bonus and I dont know the answer. Shit prof wouldnt give it to us, either.

The enthalpy in mention is -241.25 kj/mol

File: _20161114_233328.jpg (124KB, 1944x393px)

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I'm having a brainfart rn.
Can anyone solve this?

>>8474157
The answer is A.

The velocity is Aω*cos(ωt).
The period T is 2π/ω.
At t=0, the velocity is Aω*cos(0)=Aω.
At t=T/4=π/2ω, the velocity is Aω*cos(ω(π/2ω)) = Aω*cos(π/2) = 0.
The velocity changes from Aω to 0 over a period of π/2ω, the average acceleration is -Aω/(π/2ω) = -2(ω^2)A/π.

>>8474180
Thx
Idk what's happening to me.

>>8474152
Any Thermofags can help?

>>8474192
Yes you can I think. What is your reasoning against it?

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How would you go about solving this/is it solvable?

>>8474283
actually ignore the last 3 lines, that's.. bad
the subscripts correspond to the same state(sub 1 - velocity, accel, temp at point 1)

>>8474238
My main argument is that the enthalpy is molar enthalpy. Henceforth it may not be enough.

Why do think you could boil water withvit?

Learned about integrals in Calc I today. Professor said the + C is any real number.

Is it possible to have an integral on the complex plane with an imaginary/ complex constant of integration?

>>8474152
Any exothermic reaction could be used to boil water, but the enthalpy you give corresponds to nothing out of context. It could be the enthalpy needed to boil one mole of water.

>>8474384
You can define an integral in any [math]\mathbb R[/math]-Banach space (i.e. normed vector space with a non-fucked up notion of convergence and continuity) with constants of integration. If the dimension of the vector space is finite, then the integral is defined by simply integrating each coordinate.

>>8466401
On ProofWiki they define the limit of a function as:

Let M1=(A1,d1) and M2=(A2,d2) be metric spaces.

Let c be a limit point of M1.

Let f:A1→A2 be a mapping from A1 to A2 defined everywhere on A1 except possibly at c.

Let L∈M2.

f(x) is said to tend to the limit L as x tends to c and is written:

f(x)→L as x→c or lim_{x→c} f(x)=L

if and only if the following equivalent conditions hold:

∀ ϵ ∈ R_{>0} ∃ δ ∈ R_{>0} : 0 < d1(x,c)< δ ⟹ d2(f(x),L) < ϵ

This makes sense except they say c is a limit point of the metric space M1, and L is in M2. Wouldn't it be correct to say c is a limit point of A1, and L is in A2? Sorry for the pleb-tier question, my Intro to Analysis class doesn't even go over metric spaces, and I'm trying to get a better understanding of what is really going on.

>>8474470
It's understood to mean the same thing.

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How did you foster your love for mathematics/science in your youth?

Is x/0 an intermediate? Where x =/= 0. Or can only 0/0 be an intermediate?

Hey I'm sure this is a really stupid question but I often commit "simple" mistakes such as not paying attention to signs (+, *, -) and I find myself having to re-do problems because of that.
I think doing the exercise all over again doesn't make me improve in paying attention to that kind of details, how could I improve my operator/sign/general awareness?

Thanks for reading.

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>>8474619
How do I find the orthogonal complement of this set? I think I must rref the matrix but what then? What values of the matrix are the orthogonal complement. (My teacher is pretty bad and I'm worse)

>>8474665
I think you mean indeterminate.

And if you're talking about limits for constant nonzero [math]x[/math], [math]\frac{x}{t}[/math] will tend to positive or negative infinity as [math]t \rightarrow 0[/math] depending on the sign of [math]x[/math] and whether you're approaching from the left or right.

>>8474551
thank you

>>8474665
x/0 is undefined for all real x.

I believe you're looking for indeterminate forms for things like l'hospital's rule.
In that case you're looking for the limit of some ratio of functions
[eqn] \lim_{x\to a} \frac{f(x)}{g(x)} [/eqn]
If [math] \displaystyle \lim_{x\to a} g(x) = \lim_{x to a} f(x) = 0 [/math], then the limit of the ratio is said to be indeterminant.
However if [math] \displaystyle \lim_{x\to a} g(x) = 0 \neq \lim_{x to a} f(x) [/math], then the limit of the ratio is not indeterminant, it is divergent.

>>8474734
if you take the 2nd one and subtract the 1st one you get the 3rd one. So just drop the third one. Take the cross product of the first 2 to get a vector that spans the orthogonal complement.

>>8474619
I actually wasn't very good at math as a child. Wasn't until a young adult, actually. I blame it on several factors: parents weren't really invested in education; TV / youth culture made learning or being smart "uncool"; friends who had a similar upbringing; and the way mathematics is taught. You are taught to memorize rules from getting to point A to point B, which was really boring for a child who spent most of his time in front of a television or video games. For most of my life I simply thought I was just bad at math.

I am now 28 and wrapping up a bachelor's in math and minor in statistics. I suspect many people who think they can't do math had a similar upbringing. When it comes down to it I can really only put the blame on myself. However I think if more teacher's tried to motivate their students, and perhaps set up camps to get the one's struggling caught up, our nation's performance in math wouldn't be so bad. However after taking a few classes with the "Math Ed." students, i feel the prospects are pretty bleak. They are seriously almost as bad as pre-med majors in terms of academic integrity.

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>>8474759
If they were all linear independent what would I do? (Thanks for you answer, I'm trying to understand thing but the guy teaches only tricks and speaks german. Most books I come across are horrible)
What about pic related? How'd I go to find it then? (You don't need to do it just say to me)

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We give in order to free our selves from the stigma of poverty

>>8474738
So as lim_x->0 2x/x^2 = + infinity

Hence, limit does not exist?

Could you explain why (if above is correct) limit does exist

>>8474445
Im late but thanks for the answer anon.

>>8474777
dude weed bro

>>8474283
bls resbond

Is there a general way to use the Mean Value Theorem for proving inequalities?

>>8474619
I didnt
I sucked at math as a kid and once I started college i started loving math. I have a strong feeling that elementary schooling takes the wrong approach at math which is why i think lots of kids (even those who do well in their classes) have some form of math anxiety or inability to see the bigger picture as a result of the schooling
now im a physics major with a math minor

>>8474937
I guess I should elaborate: I know that if we want to show f(x) <= g(x) for all x in [a,b], if f and g satisfy the conditions of the mean value theorem, we construct a function h(x) = g(x) - f(x), and then show that this function is greater than 0 for all x in [a,b]. How do we approach an inequality with three functions f(x) <= g(x) <= h(x)?

>>8474772
if you have k linear independent vectors in an n dimensional inner-product space, then the orthogonal complement will be of dimension n-k. So if you have 3 linearly independent vectors in R^3, the orthogonal complement has dimension zero... it just the zero vector space.

In your latest pic, you have 3 independent vectors in R^5, so the complement will be of dimension 2. [1,0,0,0,0] is obviously orthogonal to all of 3, so you need to find only one more.

>>8475097
Looks like [0,1,-1,0,0] will work.

>>8475097
>>8475105
Thank you anon.

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pls help a brainlet

I can do everything correct to get the left side but I have no idea how to simplify it to get the right side.

>>8475161
you just find a common denominator for the term in parentheses then absorb the 1/12 into it and you get (8x -1)/112
and of course you don't touch the other terms

>>8475185
i feel like such a retard that i get hung up on stupid thing like that


it wouldn't take me so long to do math but my basic algebra whatever is so bad and shaky i always have problems with things that are assumed to be known

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I want to integrate the region W bounded by y=0, x+y=3, x^2+3z^2=9.
Book tells me the limits of integration are z \in [-sqrt(3-x^(2)/3), sqrt(3-x^(2)/3))], y \in [0, 3-x], and x \in [0, 3].
I have no problem with this except for the x limits. Why the fuck is x not in [-3,3]?
pic related

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I'm not sure if this is stupid or hard but can't figure it out

>>8475235
you diagonalize it
M=RDR^-1
As you take successive powers you get RD(R^-1)RD(R^-1), which is just R(D^2)(R^-1),
so to find the nth power of M you just have to transform it to D, find the nth power of D, which is easy af because it's diagonal, then apply the inverse transform and get back M^n

idk, it's easy as shit, my explanation probably sucks, but your book or the internet will have tons of info on diagonalizing a matrix

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>>8475225
If everything you've said is correct then it seems like the book is wrong.

>>8475262
Thanks a lot.

I'm sort of confused on some details of the equivalence relation congruence mod n.

I know that the actual relation says that a ≡ b (mod n) iff n | (a-b). This relation partitions the integers into equivalence classes. This makes sense to me.

Now when someone says "x mod n equals y" (so maybe 11 (mod 5) = 1) are they really just saying x ≡ y (mod n)? Or if they ask, x mod n equals what? are they simply asking for the equivalence class representative of x mod n?

>>8475322
yep you're right. the equivalence classes of the relation are conveniently indexed by the numbers 0,1, ..., n-1, but these numbers are not themselves the equivalence classes. So instead of saying x mod n=y it's more correct to say x mod n =[y], where [y] is the equivalence class of the integer y under the equivalence relation, but this is apparently too pedantic for most people.

>>8475262

I'm sure this is what they are after, but seriously I wonder if it wouldn't be easier to compute it as

M^64 M^32 M^4 = M^100

>>8475264
Jesus Christ no. The boundary of integration is a circle, not a 3x3 rectangle you pleb idiot.

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I've been sitting on this all night and i'm starting to feel really stupid for not getting it right.

Can anyone tell me what i am missing?
There must be a basic step i just did not think of

I started forming the equation 4-6 times and always seemed to do it wrong, pls help.

sorry for the bad lighting on pic

Is double and triple integration the same things as taking one integral 2 and 3 times, respectively? If not, then what's different about it and why do a lot of people here say its difficult (or is it just a meme)?

>>8475387

How many individual matrix multiplications do you need to do that?
I count 9
There are cases where that method is fastest. I'd say that M^100 is about on the line of whether or not it's worth it.

>>8475443
the right inequality is true since

0
<= (x-y)^2
= x^2-2xy+y^2

implies 4xy<=x^2+2xy+y^2=(x+y)^2

implies 4xy/(x+y)^2 <= 1

implies (2xy/(x+y))^2<=xy

>>8475443
left inequality is true since

x<= y
implies x+y<= 2y
implies (x+y)^2 <= (2y)^2
implies 1<= (2y/(x+y))^2
implies x^2<=(2xy/(x+y))^2

>>8475452
>Is double and triple integration the same things as taking one integral 2 and 3 times

In practice yeah, but it's usually defined slightly differently. Just like you can define a single integral by dividing up the range into a lot of small intervals and taking a limit of Riemann sums, you can define a double integral by dividing up the domain into a lot of small squares, and taking a limit of Riemann sums, where you replace the length of the interval with the area of the square. It's not entirely obvious that this should give the same answer as integrating one variable at a time, but fortunately it does.

> why do a lot of people here say its difficult (or is it just a meme)?

it's just a meme

>>8475488
Oh damn you did the whole thing
I cannot thank you enough, kind anon
Thy true science has been noted

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can someone explain how they unified the two series?

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also, everything after coefficient makes no sense to me. The 5th term of the Maclauren series should be the 4th derivative of f(x)/4! right? But if x is zero shouldnt equal zero? on a side note, graded homework is retarded.

>>8475545
when you have sum x^{n+2}/n! + sum 4x^{n+1}/n! and you want to turn it into one sum you find the coefficient of x^{n+1} to be 1/(n-1)!+4/(n!), where the first term comes from the first sum and the second term from the second sum

>>8475549
the coefficient of x^4 is when n=1, so (-1)/2!

since this is a maclaurin series this coefficient is equal to f^(4)(0)/4!

How to prove that every isometry has an inverse?

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In a banked turn with the angle relative to the horizontal, why is the horizontal normal force mgtan(x) instead of mg/tan(x)?

>>8475341
Most appreciated anon

>>8475322
If someone says "x mod y", they mean the remainder after dividing x by y.

This is a primitive operation in most programming languages; C (and languages derived from it) use "%" as the modulo (remainder) operator.

So most programmers tend to view modular arithmetic in terms of an arithmetic operator rather than as the ring of integers modulo N.

>>8475452
> Is double and triple integration the same things as taking one integral 2 and 3 times, respectively?
Yes.

> why do a lot of people here say its difficult (or is it just a meme)?
Multiple integrals create an opportunity for additional problems which can't occur when you only have one integral.

E.g. integration often occurs over a region which is defined implicitly rather than with explicit limits. Integration over a unit disc might have the limits expressed as "x^2+y^2<=1".

In that case, it's fairly simple to determine that the limits are [-1,1] for x and [-sqrt(1-x^2),sqrt(1-x^2)] for y. For other cases, it may be more involved, or even impossible without re-formulating the integral (e.g. converting from Cartesian to polar coordinates).

Double integrals are often over a closed surface in 3 dimensions, e.g. integrating f(x,y,z) over the unit sphere, so you have to find some way to parameterise the surface, re-formulate the function in terms of the parameters, then integrate (and the choice of parameterisation will affect how easy the re-formulated function is to integrate).

>>8475618
> In a banked turn with the angle relative to the horizontal, why is the horizontal normal force mgtan(x) instead of mg/tan(x)?
The angle is relative to level flight, i.e. an angle of zero means that the wings are horizontal, not that the normal force is horizontal.

>>8474720
Blease respond to a brainlet.

More ODE shit

Find a 1 parameter family of solutions.

[math] yx^{2}dy-y^{3}dx=2x^{2}dy [/math]

So far I have divided out by [math] x^{2}y^{3} [/math]

which gets me

[math] \frac{dy}{y^{2}}-\frac{dx}{x^{2}}=\frac{2}{y^{3}}dy \implies (\frac{1}{y^{2}}-\frac{2}{y^{3}})dy-\frac{dx}{x^{2}}=0 \implies \frac{y-2}{y^{3}}dy-\frac{dx}{x^{2}}=0[/math] I integrate and get [math] \frac{1-y}{y^{2}}+\frac{1}{x} = c [/math] Now I don't know what to do. The book answer is [math] (cx+1)y^{2}=(y-1)x [/math]. It is a separable equation. There must be some simple fuck up I am doing.

>>8475800
what? just use basic algebra to get the same answer? set c= -c'

Ah I see it now, multiply out the fractions. I think I'll leave it there, as long as I do the differential equation correctly I don't care. Good to know my answer is equivalent.

>>8475608
>>8475608
shoe it is injective, every injective function is invertible, if you restrict the codomain

>>8474720
>>8475767
One thing that helped me quite a bit with this is to deliberately try to be neater. Most people who make a lot more minor errors than usual (myself included) tend to have sloppy, fast handwriting.

Work in relatively straight lines across and down your paper (don't put each separate chunk of a problem in it's own little vaguely bounded bubble floating off somewhere on the side), make sure you don't blaze through problems at 1000mph when it's not necessary (there's nothing wrong with taking an extra 15 minutes on an exam if it means your grade goes from 85 to 100), write a little bit bigger and more spaced so you can see your own writing more clearly (letting your lines bleed into each other because they're too close isn't a good idea), those sort of things help.

You're still going to do it every once in a while though (everyone does, there's no avoiding it).

How we can see than 2sin^2(x) + sin(x) - 1 is equal to (2sin(x)-1 ) (sin(x) +1) ?

There is a remarkable identity that i don't see ?

>>8475873
It has nothing to do with any identities. You're just factoring a quadratic.

2a^2+a-1 = (2a-1)(a+1)

but in this case a = sin(x)

>>8475877
ok, thanks !

i saw a method where when we have ax^2 + bx + c, we can find the couple of factors who give a and those who give c and it exists (n,m): nm = a; (o,p): op = c (nx+o)(mx+p) = ax^2 + bx + c

There are no other way than test for find the good couples when we have all the couples in front of us and the equation, right ?

I know there is an easy method with the disciminant for factoring a quadratic but i ask that by curiosity.

Does anyone know a formula for this mathematical problem:
I want to invest in stocks. I put in 10 000€ in it every year and then get 4% dividend. That is 10 400 after one year. the second year i put another 10 000 in stocks. That means i have 20 400*1,04(+4%). Is there an easier way to calculate this, or do you have to do it manually?

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>>8475898
This is called a recurrence relation or sometimes a delay equation. They're the discrete time analog of differential equations and have a similarly rich theory, though it is rarely taught.

>>8475890
Quadratic equation. (-b±sqrt(b^2-4ac))/2a

If b^2-4ac is a perfect square, then it can be factored with rational coefficients.

In this case, b^2-4ac=9=3^2, so it has a rational factorisation.

More generally, if b^2-4ac=q^2 for some q, then b^2-q^2=4ac => (b+q)(b-q)=4ac. Note that (b+q)+(b-q)=2b. So if you can factor 4ac into two numbers x,y whose average is b, then sqrt(b^2-4ac)=(x-y)/2 and the roots are -x/2a and -y/2a.

Here, b=1, 4ac=-8, factor into -2 and 4 (average=1=b), so the roots are 1/2 and -1.

>>8475898
x=10000
r=1.04
x*r + x*r^2 + x*r^3 + ...
Sum of a geometric series.
= a*(r^n-1)/(r-1)
where a is the first term (x*r=10400), r is the ratio (1.04), n is the number of terms (i.e. the number of full years).

>>8473424
Did you already solve part A? It seems like you should be able to use very similar techniques.

E ⊆ F ⊆ X. V ∩ E = (X\F)\(X\E) what is V?

Halppls

>>8476000
The set containing all female presidents of the US

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>>8476003
damn thats no good. ok pic related is what im trying to prove, ive done the first bit and this is what i have for the second:

F is closed in E
F^c is open in E
F^c = E\F is in τ_E

Also E is closed in X
E^c = X\E is in τ_E

Since E\F is in τ_E there must exist some V in τ_X s.t. V ∩ E = E\F

Why must V=X\F??

>>8476018
>>8476000
just noticed how retarded i am:

F ⊆ E ⊆ X not
E ⊆ F ⊆ X

Also F^c is supposed to be the compliment of F and t_E is the topology of E

Is it easier to pull an object up a ramp at 10° 40° or 70°?

>>8475719
Wait so is it an operation on the set of integers that forms a ring? What other operation makes up the ring structure?

>>8473606
Looks like you misunderstood the piecewise function information and the constant a. The definition of f(x) is telling you that for all x<=0, f(x) is 2^x - a. You incorrectly concluded that 1 - a <= 0. It sounds like you thought f(x) had to be <= 0, but that's not necessarily true. That is only a restriction on x, not f(x). You can't make any conclusions about what the value of a is. However, if, for example, you knew that the function was continuous everywhere, you could determine that a must be equal to 1, because then the lim of the first part of the function would be 1 - a = 1 - 1 = 0 which is equal to the lim of the other part of the function, which is the only way it could be continuous.

On the second part of the limit, you also made a note that x=/= 0, but the function definition already tells you that x > 0 for that portion of f(x).

>>8475719
>>8476153
Never mind, I think I see my goof.

The relation "equivalence mod n" isn't literally an operation; it is an equivalence relation and thus partitions the integers into equivalence classes (whose elements are those integers such that when you their remainder upon division by n is the equivalence class representative from 1 through (n-1)). We can then define operations addition and multiplication (sort of) on these equivalence classes, and this forms a ring? Or is it literally an operation that maps two integers, one you mod and the other you use as the mod (the n) to the remainder? So mod:ZxZ->Z?

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|x|fx dx < inf

Why is there a modulus of x and why less the infinity?

What did they mean by this?

When doing epsilon delta limit proofs for a function at a point, we typically start the proofs by saying let epsilon > 0, and then proceed working with the inequality |f(x) - L| < epsilon. However in the limit definition this inequality is in the consequent of the implication. Typically we aren't allowed to assume the consequent as true, so why can we here?

>>8476267
Using the definition here:
>>8474470

>>8475871
Thank you anon I'll try to actively put more thought into what I write.

>>8476267
>Typically we aren't allowed to assume the consequent as true, so why can we here?
you cant

can you give an example of such a proof you think uses this?

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>>8476350
Perhaps I'm mistaken, but it seems to me every limit proof is like this.

>>8466401
When factoring exponents, why do you choose two exponents that add up to the exponent which you're attempting to factor instead of two numbers that multiply together to equal the exponent?

>>8476267
By definition, L is a limit of f iff:
∀ε>0.∃δ.∀x>δ.|f(x)-L|<ε

The first quantifier is "for all positive epsilon ...". So epsilon is what you're given, delta is what you need to find (or at least show that a suitable delta must exist, even if actually finding it isn't straightforward).

>>8476376
x^(a+b)=(x^a)*(x^b)

If you're factoring x^k, you want a+b=k so (x^a)*(x^b)=x^k. If you're factoring k, you want a*b=k.

>>8476487
holy shizz. I need to get on this level. I understood what you just said, but I don't think I can answer a question like I asked on this level.

>>8476193
Assuming [math]f[/math] is positive, which you usually have for pdfs, the requirement
[math]
\left| \int_{-\infty}^\infty x f(x) dx \right| \leq
\int_{-\infty}^\infty |x| |f(x)| dx =
\int_{-\infty}^\infty |x| f(x) dx,
[/math]
so if the latter is finite, the modulus of the integral we started with is also finite, and if the modulus of your integral is finite, so is the integral itself.
It is a slightly stronger requirement than just assuming that the integral itself is finite, but it also gives your more mathematical freedom by ignoring a few pathological cases.
I've never done any stochastics, but I believe the integrals there are mostly done in the Lebesgue-sense, and not in the Riemann-sense, and for improper integrals like this one, you need this absolute convergence (i.e. integrating the modulus of the integrand gives a finite result, like the condition in your screenshot) for the Lebesgue integration to make sense.
The underlying reason is that Lebesgue integration theory is way better behaved, making possible things like the "Dominated Convergence Theorem" and similar stuff.

TL;DR: If you only required the integral itself to be finite, and not this specific modulus version, you'd run into problems at some point with Lebesgue integration theory. Accept and move on!

>>8473898
Is 100 units the radius or diameter? You can simply represent the point that you're at with the right angle vertex of a right triangle inscribed inside the circle. The hypotenuse is equal to the radius, and the leg length depends on x. Now just use the Pythagorean theorem to calculate the other leg of the triangle, which will be the height of the circle.

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Guys confused asf by the wording here. What does continuity mean WITH RESPECT TO d_a?

I get that f:(X,dx)->(Y,dy) is continuous if:

for all e>0 there exists a b>0 s.t for all x' in X

dx(x',x)<b implies dy(f(x'),f(x))<e

But what do we mean when it says wrt d_a??

does it mean ||.||_a: (V,da)->(R,da) or what??

>>8476640
The definition of continuity depends on the metrics you use in the domain and image, like you correctly wrote down. So now you're supposed to prove that the norm induced by [math]d_a[/math] is continuous, no matter if you use [math]d_a[/math] or [math]d_b[/math] as the metric in [math]V[/math].
In particular, you don't need to adjust the metric in the image [math]\mathbb{R}[/math], because using [math]d_a[/math] in there doesn't even make sense in general.

>>8476546
So when would I use the modulus version over the "normal" one, as in, which type of question?

Or do I just ignore this statement and pretend I never saw the modulus of x

>>8476734
If you ever need to prove any statements with mathematical rigor, you'll need to remember that condition. If you're just jerking around with well-behaved functions and integrals, you can usually forget about the condition. That'd be my best guess.

For an acid, what's the difference between its acidity and its reactiveness?

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How to solve? And would I use limit definition of differentiation

>>8475235
>elegently

>>8475800
(y-1)x^2 dy = y^3 dx
1/x^2 dx = (y-1)/y^3 dy

Hope this helps