Wikipedia:Reference desk/Archives/Mathematics/2012 November 5

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

November 5

Teach integration

Hi. I realize that mathematical integration contains dozens of human variants of the How-wise, including the Quadrature methods, Archimedes' quadruplets and likewise. For example, I may need to integrate a piecewise function, and require to explanify the various modulus on f(x) to derive the antiderivative in two, three, n and n-1.5 dimensions. My brain works best in acute visual calculus, and so to predict a subspace or manifold I must first visualize, and then derive, otherwise my dopamine level approaches 0 as t approches 1. Since I will need to use integrative methods for processing the universe within the next three weeks, I would great-infinitely appreciate some explanatory guidance unto this matter. As it stands, I have absolutely no background in integration, and only limited edusophical background in calculus. My understanding of complex erudite logic is far superior to my thought-attention span in numerical and differential mathematics. Some history of integration across the lands of the English, the Arabs and the Chinese would also serve interest. You may recall that I asked a question in Science many years ago on the flow of water into a geo-sedimentary basin of lower height, yet this is likely a common application of integral calculus. I find the quadrature of the Parabola highly interesting, and the proposition of self-same continuity across x as the scale rotates about the x-y plane is also. Please understand the meaning of this post, and give any resources based on my understanding or lack thereof of the subject that which I would highly very like to learn about regarding. Thank you! Verily. ~AH1 ^(discuss!) 19:54, 5 November 2012 (UTC)

The phrasing of this question is so clever that I can't figure out what the question is. I get that it has something to do with integration, but what exactly? Looie496 (talk) 22:29, 5 November 2012 (UTC)

Yes, unfortunately he has integrated so many different fields into his Q that we can't differentiate between them. In the hopes that my answer won't seem derivative, let me offer this calculus: "If you want to be able to visualize integration, consider that 'the area under a curve is the integral of the equation of that curve' ". So, for example, what's the area between the X-axis and curve y=x², between x = 1 and x = 2 ? The integral is C + x³/3. So, ignoring the constant C (which is the area below the X-axis, in this case), and evaluating between x = 1 and x = 2, we get 2³/3 - 1³/3 = 8/3 - 1/3 = 7/3. Plot the curve out (on graph paper, if you must), and see if that isn't correct. StuRat (talk) 03:01, 6 November 2012 (UTC)

You may recall that I asked a question in Science many years ago on the flow ..... Er, in a word, no. -- Jack of Oz ^[Talk] 03:03, 6 November 2012 (UTC)

I thought you guys had infinite memory. At that time I wanted to know the equation for calculating how quickly water fills a depression of lower height, yet now I know there are many complex non-linear effects such as Darcy's laws that make an approximation difficult - I had assumed that the forward surface speed at the entry point was equal to the speed of sound in air. Now, I want to know whether this might be a possible application of integration, and how much integrative knowledge I'd need to know in order to derive an equation for it. ~AH1 ^(discuss!) 18:47, 6 November 2012 (UTC)

The Ramanujan master theorem. Suppose f(x) is analytic in a neighborhood of zero and $f(0)\neq 0$. Then we have:

$\int _{0}^{\infty }x^{s-1}f(x)dx=\Gamma (s)\phi (s)$

where $\phi (s)$ is defined as follows. For integer values it is given in terms of the series expansion coefficients of f(x):

$f(x)=\sum _{n=0}^{\infty }(-1)^{n}{\frac {\phi (n)}{n!}}x^{n}$

This is then analytically continued to the complex plane. Count Iblis (talk) 19:20, 7 November 2012 (UTC)

localization

in the article about localization it reads:

The ring Z/nZ where n is composite is not an integral domain. When n is a prime power it is a finite local ring, and its elements are either units or nilpotent. This implies it can be localized only to a zero ring. But when n can be factorised as ab with a and b coprime and greater than 1, then Z/nZ is by the Chinese remainder theorem isomorphic to Z/aZ × Z/bZ. If we take S to consist only of (1,0) and 1 = (1,1), then the corresponding localization is Z/aZ.

I don't agree with the statement that for n = prime power, the localization must be a zero ring. For example:

Z/pZ where p is prime is a field. So any element a!=1 in Z/pZ* generates Z/pZ*.

Take for example: Z/5Z and a = 2, then S:=<2> = {2,4,8=3,6=1} = Z/5Z* but then S^-1 Z/5Z ~= Z/5Z (because the field of fraction of a field is the same field)

--helohe (talk) 21:36, 5 November 2012 (UTC)

I agree; it should somehow be rephrased. As you said, if you localize a field with respect to all nonzero elements, you end up with the same field but not a zero ring. I suppose it should say a zero ring or the same field. -- Taku (talk) 12:35, 9 November 2012 (UTC)

I assume non-trivial localization was intended.--80.109.106.49 (talk) 22:23, 9 November 2012 (UTC)