/sci/ - Science & Math

>>2492372
> too hard
Oh wow your cock is so huge.

> NSA basically reduces to exotic logic and we have seen enough times what has come of that in the past i.e. little.
What comes out of it is the notion that your notation isn't abuse. Perhaps you consider that little.

> exotic logic
Yeah, and transcendentals are only specially-concocted numbers, and negative numbers don't have square roots.

You can take your epsilon delta and shove it.

>>2492302
I didn't claim non-standard analysis invented the notation. It does, however, elevate the "useful abuse of notation" to something that is no longer abuse. Since the "abuse" is so useful, it is a wonder people so stubbornly cling to standard analysis.

If the OP had learned non-standard analysis, then he wouldn't be confused at all.

But, please cry more about why your favorite branch is better even though it confuses people and you have to remember what notation is abuse and what isn't. I find it very enlightening.

>>2492049
> doesn't know about non-standard analysis

>>2491860
This is why there's non-standard analysis: so that your intutions aren't considered abuses.

>>2491931
The ones are also horrible. But every sensible person on earth slashes his or her sevens.

Rectangles.

>>2491407
No it isn't.

...

>>2491378
protip: hallucinations are real ghosts, the undead do not exist

inb4 traps (mostly because I haven't posted any)

what the

I leave /sci/ alone for one day and look what happens

scheme

>>2477787
> there

I can. But I won't. Do your own homework.

> why do math majors focus so much on analysis/algebra when there is geometry, logic, discrete mathematics, etc
Watch this thread sink like a brick, just like it did earlier today:
>>2474779

>>2477228
sorry, I wrote this little routine to print a polynomial with latex and it defaults to x, clearly the variable should be the limit of the summation.

For much better details, please see
"The Finite Difference Calculus and Applications to the Interpolation of Sequences" by Kunin, a short 10-page piece that uses finite difference ideas to develop the closed-form formula for the nth Fibonacci number. (Google it, pdf)

As the discrete integral is just a kind of summation, we can use this interpolation by integrating one more time to find an explicit formula for our summation.

So, for instance, suppose we wanted to find the sum of n cubes. We calculate out our preliminary values and whip out our difference table
1 9 36 100 225 441 784
8 27 64 125 216 343
19 37 61 92 127
18 24 30 36
6 6 6 ...

Suddenly we see that this derivative is a constant. So, let's integrate, taking the constant of integration, as before, as the leftmost value.
<span class="math">6 \rightarrow 6x+18[/spoiler]
<span class="math">6x+18 \rightarrow 3x^{\downarrow 2} + 18x + 19[/spoiler]
etc. When you have reached the top column, you integrate just one more time to serve as the sum, with a constant of integration of zero, and you have
<div class="math">\sum_{i=1}^n i^3 = \frac{1}{4}x^{4} + \frac{1}{2}x^{3} + \frac{1}{4}x^{2}</div>

This can be easily put into code to come up with the closed form sum of the OP, which would be way too tedious by hand.

So, suppose we have <span class="math">f(0) = 4, f(1) = 7, f(2)=12[/spoiler] and we wish to interpolate this, and we forgot the stupid lagrangian interpolation shit. Well, we know we want to fit this with a second order, so, calculate the discrete derivative until we're left with just a constant, then reintegrate.
4 7 12
goes to
3 5 (notice it is f(x+1)-f(x) as mentioned)
goes to
2.

Now we reintegrate
<span class="math">2 \rightarrow 2x^{\downarrow 1} + 3[/spoiler]
<span class="math">2x^{\downarrow 1} + 3 \rightarrow x^{\downarrow 2} + 3x^{\downarrow 1} + 4[/spoiler]
<span class="math">x^{\downarrow 2} + 3x^{\downarrow 1} + 4 = x(x-1) + 3x + 4 = x^2 + 2x + 4[/spoiler] which, if you check, indeed interpolates the three points given. (2 pts determine a line, 3 a parabola, etc)

A little drunk now, forgive me for stupidity.

Finite derivatives:
<span class="math">\delta_x f(x) = f(x+1) - f(x)[/spoiler]
Falling powers: (I don't know how to do underlines so...)
<span class="math">x^{\downarrow m} = x(x-1)\cdots(x-m+1)[/spoiler] eg. <span class="math">4^{\downarrow 3} = 4(4-1)(4-2)[/spoiler].

Using algebra and the derivative definition you can then show that, similar to the power rule of analysis,
<span class="math">\delta_x x^{\downarrow n} = nx^{\downarrow n-1}[/spoiler].

So, suppose you have a series of values, <span class="math">f(0), f(1), \ldots[/spoiler]. Then you can use these to calculate the discrete derivative until you are left with just a constant, and then reintegrate.

(more coming)

>>2475321
I hope you failed calculus.

> says there's no objective morals
> gives criteria for ruling a society anyway

>>2471623
> implying a mass calculation wouldn't be primitive recursive and therefore trivial to perform tail call optimization on

>>2475261
ever hear of sage