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[Frumple]

I have no idea what most of those terms mean, but! If it helps any, breaking it down, it's asking this:

Let B (babble) be the basis for subspace W of the space of continuous functions (whatever the blazes that is).
Let D_x be the differential operator on W.
Find a matrix for D_x, relative to B. Plug that matrix into the 4x4 square of answers.

So that... well, I can't say it won't be hard? But it seems fairly straightforward if you can figure out how to get a matrix out of that parenthesis stuff? Presumably whatever you're using as a text/reference would tell you what those terms are and how to mathify them.

[chaotic skies]

Okay so I went off and did the rest of the nonsense that is linear algebra, and it turns out bases are important when you're doing transformations between to bases. Who'd've thought.

Anyway, I'm going to try to interpret that 'homework problem'.

Let B (babble) be the basis for subspace W of the space of continuous functions (whatever the blazes that is).

This is a set of unit 'vectors' because apparently vector spaces (the spaces where vectors live) can have entire functions as their basic components. For some unholy reason.

In this case, it means any vector (A,B,C,D) is equivalent to A+Bx+Ce^-x+Dxe^-x. Obviously.

Let D_x be the differential operator on W.

This...seems to mean that there's some matrix that you can multiply any given vector by and get the derivative of it, for the vector space defined by B. Which is...something. Certainly easier than doing it by hand.

Then we do the simple task of plugging the thing into the disaster grid.

Someone tell me if I'm horrifically wrong (I almost certainly am), but I'm working off of this understanding, even if it doesn't actually make sense.

EDIT: This was actually correct and shockingly simple. Basically, find the derivative of each vector, and re-write it in terms of the vector space, and then slot each one into its column from left to right. xe^-x gave me some trouble because I forgot the product rule for like the fiftieth time. Which was fantastic.

Edit 2: I'm now fighting with this arcane tomfuckery, which I've managed to work out from the notes multiple different ways and still get rejected by the program. Delightful.

[Ulfarr]

A bit of a necro, but I don't want to start a new thread for just one question and I think this one is a better fit than the small and/or random questions threads.

If Sum A is the result after rolling 3 six-faced dices (3d6) and Sum B is the result of 2d6. What are the chances of Sum A =< Sum B and (for future reference) how would one calculate it?

[WealthyRadish]

Ignoring the actual math for a bit, this easy online programming language is geared toward those sorts of calculations:

https://anydice.com/program/2f799

Viewing the "at least" view will show that (2d6 - 3d6) >= 0 22.15% of the time.

[Ulfarr]

Thank you !!

I didn't know about it and it'll be really useful to me. It also seems, to confirm my understanding of the subject.

Spoiler (click to show/hide)

If Ai=3d6, Bi=(2d6, where Bi>=Ai) and P(Ai), P(Bi) their respective chances, then for each Ai (ranging from 3 to 18) the chance

Pi(Bi>=Ai) = P(Ai)*P(Bi)

and the total chance is

 Pt = sum [Pi(Bi>=Ai].

[Magmacube_tr]

What is 6x3?

[McTraveller]

What is 6x3?

Strictly speaking, not enough information. For example, assuming it even means integer multiplication, the result depends on the base (your base, all belong to us).

[delphonso]

Tell me about 6x3!

[Maximum Spin]

What is 6x3?

Tell me about 6x3!

I can't answer that here. You need to ask in an 18+ restricted forum.

[MrRoboto75]

6x3!

36

[Ulfarr]

Almost in time for the yearly resurrection of this thread 

I'm having trouble tackling a system of first order partial differential equations so I'd be grateful if someone could guide me through a methodology on how to solve them. The system is in the form of:

• θG(h,t)/θh + M*X(h,t)*G(h,t) = 0
• θX(h,t)/θt + A*X(h,t)*G(h,t) = 0
• A, M are constants

For what I know the system does have an analytical solution.

[McTraveller]

Almost in time for the yearly resurrection of this thread 

I'm having trouble tackling a system of first order partial differential equations so I'd be grateful if someone could guide me through a methodology on how to solve them. The system is in the form of:

• θG(h,t)/θh + M*X(h,t)*G(h,t) = 0
• θX(h,t)/θt + A*X(h,t)*G(h,t) = 0
• A, M are constants

For what I know the system does have an analytical solution.

First step is probably going to be multiplying each of the first two equations by constants and adding them together to cancel terms.  This should give you a good starting point to then use the chain rule.

If you don't know exactly what I mean, let me know and I can explain.

Other potentially helpful context questions: did this system arise from physics or chemistry, so you can reason about what form the answer should be?  Or did this come from a math(s) class, where this is abstract and you are intended to leverage principles you learned?

[Ulfarr]

I'm not sure how the chain rule would be used here.

If I multiply the first equation with (- A/M) and then add the second would yield:

(-A/M)*[θG(h,t)/θh] + θX(h,t)/θt =0   (3)

which if I'm not wrong should integrate to

G(h,t) = (M/A)*{S[θX(h,t)/θt] dh - C1(t)}   (4)

I think I should then substitute G(h,t) in the second equation which yields

θX(h,t)/θt + M*X(h,t)*{S[θX(h,t)/θt] dh - C1(t)} = 0

But I'm not sure how I should continue from there without looping back to my initial system.

Spoiler: For context (click to show/hide)

The equations are from a mass balance between a fluid [G(h,t)] and a solid phase [X(h,t)], h and t are independent variables representing the extractor's length and the extraction's duration, respectively.  I already have a solution but I'm trying to understand how did they find it.

[McTraveller]

Since it's coming from a physical process, is there an additional constitutive equation somewhere?  Like shouldn't G+X be a constant if they are trading mass with each other?

[Ulfarr]

of course! I can't believe I forgot about that. Thanks!!