>sit in on signal processing course to see how much of a joke

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>be member of the CS master race
>take course "programming for mathematicians" just for lulz
>the people struggle with simple concepts like for loop

Brainlets, when will they learn?

>>8404468
>From a purely mathematical viewpoint, the Dirac delta is not strictly a function, because any extended-real function that is equal to zero everywhere but a single point must have total integral zero.[5][6] The delta function only makes sense as a mathematical object when it appears inside an integral. From this perspective the Dirac delta can usually be manipulated as though it were a function. The formal rules obeyed by this "function" are part of the operational calculus, a standard tool kit of physics and engineering. The operational calculus, and in particular the delta function, was viewed with suspicion by mathematicians of the early 20th century, until a satisfactory rigorous theory was introduced by Laurent Schwartz in the 1950s.[7] Formally, the delta function must be defined as the distribution that corresponds to a probability measure supported at the origin. In many applications, the Dirac delta is regarded as a kind of limit (a weak limit) of a sequence of functions having a tall spike at the origin. The approximating functions of the sequence are thus "approximate" or "nascent" delta functions.

is that so hard? really? because the delta dirac function is far more useful than your shit-tier autism.

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>>8404468
>tfw to intelligent for Delta functions

>>8404468
But it is a function if you aren't a brainlet.

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>professor called gravity a force

>>8404479
>thinks he's smart
>can't come up with a better use of his time

>>8404666
But it fucking is

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>>8404701

>>8404701
Nope, gravity is acceleration along a geodesic caused by curved spacetime

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>a system of linear equations either has 0, 1, or infinitely many solutions
>the order of integration in iterated integrals does not matter
>every vector space has a basis
>not every group is surjunctive
>the assembly map [math] H_*^G(E_{FIN}(G),K_{l^1}^{top})\rightarrow H_*^G(\{\cdot\},K_{l^1}^{top})=K_*(l^1(G)) [/math] is an isomorphism

>>8404764
haha totally

>>8404480
>From a purely mathematical viewpoint, the Dirac delta is not strictly a function, because any extended-real function that is equal to zero everywhere but a single point must have total integral zero
This doesn't imply in any way that it isn't a function, what the fuck?

>>8404820
Dirac delta has a non-zero integral and is zero at all points except 0 so yes it does imply it's not a function

>>8404764
don't forget
>[math] \partial_x\partial_y=\partial_y\partial_x [/math]

>>8404468
It is a function. It maps a function f in a suitable class to f(0).

>>8404820
Yes, it does. Functions to the extended reals admit "infinity" or "-infinity" as outputs (this lets us talk about integrating something like 1/sqrt(x) from 0 to 1 in the Lebesgue sense).

The issue is that even if dirac delta outputs "infinity" at a point, the Lebesgue integral does not care what goes on at a single point. If a set has measure zero (e.g. {0}), you can remove it from the domain of your integral, and it necessarily won't change the value of your integral. Thus, since the delta "function" is zero everywhere except a single point, there is no way it can have integral 1, since it is (from the point of view of Lebesgue integration) equivalent to an identically-zero function (which would have integral 0).

In short, if the dirac delta were a function to the extended reals, its integral should be 0, and not 1.

>>8404833
>>8404848
Why the fuck the integral of the function even matters?

Oh yeah, let's define a function on the integers, oh wait, we can't because we can't fucking integrate there.

>>8404847
No, the "dirac delta" the kernel of the functional you're describing. The "dirac delta" itself is not outputting f(0).

>>8404850
Do you know anything about math?

>>8404850
The dirac-delta is defined by its integral. This definition does not work for functions, hence it can't actually be a function.

>>8404850
Because the fact that it integrates to zero is one of the defining features of the "dirac delta." Of course integration matters; essentially, what I'm saying is that the fact that it integrates to 1 is incompatible with the definition of extended-real functions, as no extended real-value function can be zero except on a set of measure zero, and integrate to anything but zero.

As for your second statement, learn some basic measure theory, and you'll see that you can in fact "integrate" over any set. With an appropriate measure, "integrating" over the integers is just summation.

>people in this thread acting like the Dirac delta "function" is some dark-magic mathematical object
>it's really a perfectly well-defined concept in the context of Hilbert spaces and the Riesz representation theorem.

>>8404873
No one is saying it's not well defined, we are simply stating that it is not a function

>>8404857
>The dirac-delta is defined by its integral.
I've always seen it defined as function by parts, and I see no problem with that.

>>8404858
Aren't you just saying that it isn't a measurable function?
How does that correlate with it not being a function?

What if I want to define a function on a non-measurable set?

>>8404881
How do you define the dirac-delta without its integral?

>>8404886
>0 for x =\= 0
>inf at x = 0

>>8404881
Okay, assume the dirac delta is in fact an extended real-valued function. What's it's value at 0? Well, it doesn't matter, since it's defined to be zero everywhere else, and the set R \ {0} is measurable. Such a "function" would in fact be measurable, and in fact integrable. But, the integral (in the usual Lebesgue notion of integration) would necessarily be zero. This contradicts the "integral = 1" property of dirac delta, so we cannot have this object be an extended real valued function at all.

>>8404743
>>8404666
>>8404701
then can't gravity be considered the force that curves spacetime?

>>8404887
And that, in the theory of Lebesgue, would have integral zero.

>>8404887
Whoever is teaching you this shit is doing it wrong. That's only half of the definition. The dirac-delta is useless if you don't define its integral, which inherently prevents it from being a valid function.

>>8404899
Not him, but I remember encountering dirac delta first in an ODEs class in undergrad, in relation to the Laplace transform.

I remember the professor made a huge deal about it. He put a wizard hat on, and claimed the "dark magician" Dirac came up with a "function" like this whose integral is 1.

He even put it up in the "piecewise" sense, saying that delta(0) = infinity (but he really blackened in the infinity on the board, and made a big deal that it's a "special kind" of infinity that makes the integral 1).

It was a memorable lecture, and he emphasized how it's not actually a function (despite the intuitive direction he was giving us with the piecewise wizard demonstration). idk it worked well enough for an entry-level ODEs class.

>>8404895
What's the problem with that?

>>8404893
>>8404899
I don't like the definition with the integral. I like things nice and square and no integrals in my functions.

But I get the problem now, some twat decided to define a function with an integral that has value 1, and surprise, it turns out it isn't a function, yet called it that anyways.
Fucking physicists.

>>8404928
>I don't like the definition with the integral
Well, that's the definition, so get used to it.

>>8404720
>>8404743
>4 fundamental forces
>strong, weak, electromagnetic, gravity
>gravity
>force

>>8404932
Oh no, everything is alright with the world again. Since I only viewed it as a function by parts, I couldn't fathom why it wasn't a function.

But if some retard defined it as function by parts with an extra condition then it all makes sense.
The retarded function with the condition isn't a function at all, and the definition by parts still is, as it should be, a function.

Im pretty sure he meant *a* linear equation, not a system.

>>8404936
>I'm right and literally everybody else in math is wrong because I'm right XD

>>8404944
Actually
>I'm right and literally everybody else in physics is wrong

I'm agreeing with everyone here now, I'm at peace.

>>8404468
The computer you are using to post this was designed partially thanks to that Delta Dirac Function, you autist fag

>>8404935
t. undergrad

or more likely

t. dropout

>>8404468
Lmao! You're joking right? Please, tell me you're just yankin' our chains op.

>>8404966
That was an unnecessarily hurtful thing to say

>>8404928
>What's the problem with that?
Because, part of the dirac delta's definition (really, part of its main point) is that it has integral one. Your piecewise function is a perfectly good function into the extended reals, but it isn't the Dirac delta.

>>8404852
It's not a proper kernel though. It's a linear functional, mapping a test function f to f(0).

>>8405004
It depends on the space you're considering. Dirac delta is not in L^2, but you can consider other spaces of "distributions," which it is, and in which case it could be considered a "kernel."

>>8404764
>Euclid
>Fucking
>Implying
>Lmao

All dirac delta shit is so much simpler if you just think of it as a measure. All the properties are simply proved and fully rigorous.

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>>8404480
>the delta dirac function is far more useful than your shit-tier autism
← this

>>how the fuck is delta function not a function we can define any function we fucking want.
>>how the fuck is this not a group it is just a generic term

>>8404935
4 fundamental forces, weak force, strong force, Electromagnetic force, gravitational force.

Gravity an be thought of as a field where the mass of an object deforms it. Kinda...

[eqn] \delta(f) = f(0) [/eqn]

>>8404764
When can a system of linear equations have two solutions?

>>8404468
you will never be a good engineer if that's a problem for you

>>8405615
OP sounds like a butt blasted math major

>dirac
>delta
>function

>>8405588
Vector spaces over a fucking finite field you trippel noob

>>8404835
Why are you derivating twice while not in C2?

>>8404479
Sorry, this is a Science and Mathematics board, you must be lost.

>>8404480
>is that so hard? really? because the delta dirac function is far more useful than your shit-tier autism.
This.

>>8404764
A linear system may behave in any one of three possible ways:
1.The system has infinitely many solutions.
2.The system has a single unique solution.
3.The system has no solution.
https://en.wikipedia.org/wiki/System_of_linear_equations#Solution_set

>>8404939
Nope, he was right.

>>8405399
That's because it is a measure

>>8406471
You missed the point of finite fields in general.

Just because you are working "mod 5" doesn't mean that 0, 5, etc. are all solutions. They are literally the same element.

>>8406809
Are you retarded? It means that there's finitely many linear dependent vectors to a given vector, which means finite number of solutions in any case

>>8406732
Retard

>>8406809
I'm afraid it's you who missed the point.

Consider the linear equation x-y=0 over the finite field with two elements. This equation has two solutions: [0,0] and [1,1].