Talk Nerdy To Me ;)
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09-01-2017, 02:45 AM
RE: Talk Nerdy To Me ;)
Applies to mechanical and (analog) electronic systems, and many other: The fourier transformation of the impulse response is the transfer function. (and vice versa)

Less nerdy: If you want to find out the transfer function in the frequency domain (response over frequency) , you can either "wobble" the system in the time domain. That is: slowly sweep across the spectrum on its input, applying every frequency over time, which takes quite some time, and store the result of the output. Sum up the result of all frequency responses, and voila, you have the transfer function

You can also apply a short and hefty pulse in the time domain (the shorter and the heftier, the better), which can be made in an instant, and record the reaction over time. Then you make a (mathematical) fourier transformation, and voila you have the response in the frequency domain. This is because a short pulse contains all possible frequencies(!).

Imagine a bell, which has a certian transfer characteristic (absorb any frequency and do nothing, but when hit with your partcular *boinggg* frequency, then respond with *boinggg*. Now, if you hit that bell hard and quick (= throw all frequencies at once at it), it will respond (with its specific frequency only) and hereby reveal its transfer characteristic. You can test this with your table too, or your car. Just kick them and they will reveal everything you need to know about them.

Thats why you are kicking all defect devices as well. By applying all frequencies you hope to stimulate your TV with the one frequency needed to fix the loose contact, etc.

I guess it works with humans too, but i havent tried it yet. Probably the "impulse response" will be somewhat....hefty.

The cool thing is (and very nerdy): the fourier transformation of a square pulse in either frequency domain or time domain) results in sin(x)/x in the other domain. A single waveform in one domain results in a certian waveform in the other domain, and when you repeat that pulse in your given domain, then the result is not a continuous spectrum but a discrete spectrum, aka.: Periodisation in one domain results in discretisation in the other (and vice versa). Thats also why your gfx card for example needs anti-aliasing, because the analog-to-digital converter periodisizes the frequency spectrum by digitizing (discretizing), creating "alias" spectrums which have to be filtered out by anti-aliasing filters.

Ceterum censeo, religionem delendam esse
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12-01-2017, 12:28 AM
RE: Talk Nerdy To Me ;)
[Image: Y6h9u87G_400x400.jpeg]

"Let the waters settle and you will see the moon and stars mirrored in your own being." -Rumi
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12-01-2017, 02:27 AM
RE: Talk Nerdy To Me ;)
Lavrentiy Beria didn't wanted Czechoslovakia, Hungary and Poland as communist countries. Does not seem so shocking if one think that he was nationalist in communist guise. Hope it is nerdy enough.

The first revolt is against the supreme tyranny of theology, of the phantom of God. As long as we have a master in heaven, we will be slaves on earth.

Mikhail Bakunin.
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12-01-2017, 06:55 AM
RE: Talk Nerdy To Me ;)
XANTHIAS
Shall I crack any of those old jokes, master,
At which the audience never fail to laugh?

DIONYSUS
Aye, what you will, except "I'm getting crushed":
Fight shy of that: I'm sick of that already.

XANTHIAS
Nothing else smart?

DIONYSUS
Aye, save "my shoulder's aching."

XANTHIAS
Come now, that comical joke?

DIONYSUS
With all my heart.
Only be careful not to shift your pole,
And-

XANTHIAS
What?

DIONYSUS
And vow that you've a belly-ache.

NOTE: Member, Tomasia uses this site to slander other individuals. He then later proclaims it a joke, but not in public.
I will call him a liar and a dog here and now.
Banjo.
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12-01-2017, 08:38 AM
RE: Talk Nerdy To Me ;)
A pineapple is not a single fruit, but a group of berries that have fused together.

"Let the waters settle and you will see the moon and stars mirrored in your own being." -Rumi
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12-01-2017, 09:01 AM
RE: Talk Nerdy To Me ;)
[Image: br0cdtwieaaicks.jpg?w=700]

"Let the waters settle and you will see the moon and stars mirrored in your own being." -Rumi
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12-01-2017, 09:07 AM
RE: Talk Nerdy To Me ;)
[Image: img_6168.jpg?w=720&h=512]

"Let the waters settle and you will see the moon and stars mirrored in your own being." -Rumi
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28-01-2017, 08:58 AM (This post was last modified: 28-01-2017 09:24 AM by Kernel Sohcahtoa.)
RE: Talk Nerdy To Me ;)
*Please note that this was originally posted in the random thoughts thread and accidentally in the "would you rather thread" (my apologies for this). However, IMO, it is more logical to post it here in Jennybee's talk nerdy thread.

Hello TTA members and anyone else. I decided to work a random proof in my math book and thought some posters here might be interested in it. Here it is:

Prove that 1^2+2^2+…+ (n-1)^2 < n^3/3 < 1^2+2^2+…+ n^2 is true for all positive integers n.

Hint: use mathematical induction.

* Please note that I've placed spoiler tags on the background info and proof for the following reasons: First, I don't want to bore those who are familiar with induction; second, this is just my attempt at the proof and I'd like to give readers the chance to do the proof for themselves if they wish.

Background on Mathematical induction

Suppose we wanted to prove that a particular statement was true for all positive integers. Manually going through each integer would take forever and there are probably better things you could do with your time. As a result, mathematical induction is a tool that allows one to prove such a statement via the following process. First, we need to plug any positive integer into the equation (usually the integer 1) in order to verify that it is true (this is the basis step). Once we have verified the basis step, then the next step is the inductive step. First, we make an inductive hypothesis by assuming that the particular statement is true for some integer, say k. Next, we must show that our inductive hypothesis implies that k+1 is also true for this same statement.

Now, induction makes more sense if we think of all of the integers as dominoes, which are set up in such a way that if you knocked any one down then all of them would go down. Hence, by assuming that the statement is true for k and verifying that it is true for k+1, this shows that k knocks down k+1, and as a result, all of the positive integers n must also get knocked down.

Proof

We will use mathematical induction to prove this statement

Basis step. Let n=1. Then (1-1)^2 < 1^3/3 < 1^2 or 0 < 1/3 < 1 which is true.

Inductive Step.

Inductive Hypothesis. Let’s assume that 1^2+2^2+…+ (k-1)^2 < k^3/3 < 1^2+2^2+…+ k^2 is true for a positive integer k. As a result, we must show that this implies 1^2+2^2+…+ (k+1-1)^2 < ( k+1)^3/3 < 1^2+2^2+…+ (k+1)^2 or 1^2+2^2+…+ k^2 < ( k+1)^3/3 < 1^2+2^2+…+ (k+1)^2. Once we demonstrate this then the proof will be complete.

Observe that 1^2+2^2+…+ (k+1-1)^2 = 1^2+2^2+…+ k^2. Now, since 1^2+2^2+…+ (k-1)^2 < k^3/3, then it must be the case that 1^2+2^2+…+ k^2 < ( k+1)^3/3. Furthermore, since k^3/3 < 1^2+2^2+…+ k^2, then it must be that (k+1)^3/3 < 1^2+2^2+…+ (k+1)^2.

Now, we can also say that 1^2+2^2+…+ (k+1-1)^2 = 1^2+2^2+…+ k^2 ≥ k^3/3 < (k+1)^3/3 ≥ 1^2+2^2+…+ k^2 < 1^2+2^2+…+ (k+1)^2.
Hence, 1^2+2^2+…+ k^2 < ( k+1)^3/3 < 1^2+2^2+…+ (k+1)^2.

Consequently, 1^2+2^2+…+ (n-1)^2 < n^3/3 < 1^2+2^2+…+ n^2 is true for all positive integers n.


"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)
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28-01-2017, 01:13 PM
RE: Talk Nerdy To Me ;)
[Image: so-true-3-16.jpg]

Don't Live each day like it's your last. Live each day like you have 541 days after that one where every choice you make will have lasting implications to you and the world around you. ~ Tim Minchin
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28-01-2017, 03:43 PM
RE: Talk Nerdy To Me ;)
(28-01-2017 01:13 PM)Commonsensei Wrote:  [Image: so-true-3-16.jpg]

I've always been more like that gentleman on the left. However, I do not possess his "decorative flair", as I'm fine with my Steve Urkel, Star Trek, or math T shirts.

"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)
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