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Topic: Math question: (Read 92 times) previous topic - next topic

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Math question:
In an overdetermined system in the cases where there are no solutions and where there are (as presented in the wiki link), are we looking at error, chaos (or some math equivalent), or faulty definitions? Or what?

I am starting from ground zero so I am probably asking the question wrong. I encountered the term in a meeting this morning and did not get a good response when I asked what it meant in context. The people were all mathy types and seemed ok with the analogy but I really don't get it. The analogy was to the known factors that contribute to the direction our institution is taking. Since there was an empirical or at least nominally empirical dimension to the statement, I assume this means the people in the meeting who understood the analogy got some utility from the understanding. Any thoughts?
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • el jefe
  • asleep till 2020 or 2024
Re: Math question:
Reply #1
in linear algebra (whose tentacles run through diff eqs and most applied higher math), overdetermined means you have given more information than is necessary for the problem to have a unique answer.

it often (though not always) means the problem is inconsistent, and therefore isn't solvable.

to give a concrete (and prototypical) example....  in linear algebra, if you have n unknown variables, you need n equations to solve for them.  (these equations must further be "linearly independent"....).  if you have n+1 or more equations, the problem is overdetermined.  if you have n-1 or fewer, it is underdetermined and cannot have a unique answer.  (though might be guaranteed to have an infinite number of solutions?)

to see it geometrically....  suppose you have 3 equations in 3 unknowns.  each equation defines a plane in 3-dimensional space.  solving the system of equations is equivalent to finding the point where all three planes intersect.  here are the possible ways that might get weird and/or not work:

- if you have more than 3 equations, that means more than 3 planes.  you don't need a 4th plane to define a point.  that's overdetermined.  and indeed there's a good chance the 4th plane doesn't run through the common point of the other three planes, in which case the system has no solution.  still, if the 4th plane runs through the common point of the other three, then the system is consistent - overdetermined, but consistent.

- if you have only 2 equations, that means only 2 planes, and the intersection of 2 planes cannot be a single point.  that's underdetermined.  it can be either a line (infinite number of intersection points / solutions) or nothing (no solution).

- even with 3 planes, there's no guarantee of intersecting at a single point.  they could all be parallel.  or they could define an infinite triangular prism - each plane intersecting each other at a line, but all three never meeting anywhere.  that's an inconsistent system - no solution.
  • Last Edit: June 21, 2018, 02:55:59 PM by el jefe

  • Brother Daniel
  • Global Moderator
  • predisposed to antagonism
Re: Math question:
Reply #2
The analogy was to the known factors that contribute to the direction our institution is taking.
So these "known factors" are like equations, in the analogy?

Sounds like the sort of analogy I might use if I were describing a set of constraints with the property that they are impossible to obey all at the same time.

Re: Math question:
Reply #3
The analogy was to the known factors that contribute to the direction our institution is taking.
So these "known factors" are like equations, in the analogy?

Sounds like the sort of analogy I might use if I were describing a set of constraints with the property that they are impossible to obey all at the same time.
That's what I was thinking but then I got a little bit in that analogy and decided that inconsistency would be due to the equations using different axioms or something else that makes the solution  inconsistent. As in, why would there be multiple true equations ( in this case having some empirical corollary) that together produce inconsistent results? Are they different logics? Is it a recursive problem?
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • uncool
Re: Math question:
Reply #4
Multiple true equations should not be able to produce inconsistent results. Can you provide an example of what you are talking about?

  • Brother Daniel
  • Global Moderator
  • predisposed to antagonism
Re: Math question:
Reply #5
As in, why would there be multiple true equations ( in this case having some empirical corollary) that together produce inconsistent results?
Measurement error?  When approximate constraints are treated as exact, they can be inconsistent.

  • el jefe
  • asleep till 2020 or 2024
Re: Math question:
Reply #6
testy, did my explanation make it clearer or even more opaque?  be blunt

Re: Math question:
Reply #7
it was good. The guy who used the term sounded like he was intending a technical use of the term which didn't make sense but now I think he meant it in some closer to technical but still metaphorical way.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor