Wikipedia:Reference desk/Archives/Mathematics/2011 July 15

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July 15

Where do we find linear equations?

The article about them says " rise quite naturally when modeling many phenomena,". What are these "many phenomena"?

One example is Hooke's law, another is the relationship between time and velocity under a constant force. Basically just open a science textbook at random and start looking. I tried a physics textbook and found the relationship between area and the force of pressure in a body of fluid.--RDBury (talk) 13:06, 15 July 2011 (UTC)

A huge number of situations where you have oscillations about an equilibrium point are well described by linear equations when the amplitude of oscillation is small. These include waves, vibrations, the movement of a pendulum or spring, etc. Looie496 (talk) 17:46, 15 July 2011 (UTC)

Another couple of examples: Ohm's law gives a linear relationship between voltage and current, assuming a constant resistance, and the hydrostatic pressure of an incompressible fluid, such as water pressure in the ocean, is a linear function of depth. —Bkell (talk) 20:22, 16 July 2011 (UTC)

Typically everything behaves linearly for small deviations when away from special points. That's practically the first assumption any scientist makes in any discipline. Singularity theory is the study of the next more complicated situation in general. Dmcq (talk) 20:37, 16 July 2011 (UTC)

Linear equations give first order approximations to any system. That is why they are so ubiquitous. The truth is that no physical theory is totally correct, and that they are all approximations. Every known physical theory breaks down outside of some set of parameters. Look at Newton's Laws. These can be viewed as (very, very good) approximations to what actually happens; but when things get really, really small, Newton's Laws fall apart. The truth is that linear equations turn up in every system: as a linear approximation to that system. — Fly by Night (talk) 23:03, 16 July 2011 (UTC)

That's a little bit of an overstatement. Lots of important systems (particularly in biology) show limit-cycle oscillations, which are not usefully modeled by linear equations. Looie496 (talk) 23:39, 16 July 2011 (UTC)

I didn't comment on the validity of the approximations. — Fly by Night (talk) 02:56, 17 July 2011 (UTC)

:-) Looie496 (talk) 17:00, 17 July 2011 (UTC)

What are these number tringles called?

When looking at Floyd's triangle on Wikipedia, i mentally replaced each counting number n with the nth odd number and found the kth row added to k^3. (This is easy to prove)

• 1
• 3 5
• 7 9 11
• 13 15 17 19
• 21 23 25 27 29
• ....

But related is

• 1
• 5 9 13
• 17 21 25 33 37
• 41 45 49 53 57 61 65

....

and

• 1
• 7 13 19 25
• 31 37 43 49 55 61 67
• ....

and generally

• 1
• 2m+1 4m+1 6m+1 ... 2(m)(m+1)+1
• ..... ... 2(m)(3m+2)+1
• ...

All of these triangles have nice cube sums on the rows. These aren't hard to prove either. I'm certain this is wellknown but I haven't found a name for them. In any case, they seem at least as interesting as floyd's triangle, so i'd like to get them on Wikipedia.Thanks, Rich Peterson199.33.32.40 (talk) 18:41, 15 July 2011 (UTC)

See [1] for the first traingle. We've got most of that material in Squared triangular number but with different emphasis. I searched the OEIS for the others but didn't turn up anything. The OEIS turns up most sequences that have been published.--RDBury (talk) 20:26, 15 July 2011 (UTC)