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Re: Ansatz?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg101195] Re: Ansatz?
*From*: Roland Franzius <roland.franzius at uos.de>
*Date*: Fri, 26 Jun 2009 06:52:19 -0400 (EDT)
*References*: <h1nf0q$91h$1@smc.vnet.net> <h1vmau$aj8$1@smc.vnet.net>
Vince schrieb:
> On Jun 22, 4:22 am, AES <sieg... at stanford.edu> wrote:
>> Wolfram MathWorld says:
>>
>> An ansatz is an assumed form for a mathematical statement
>> that is not based on any underlying theory or principle.
>>
>> SEE ALSO: Conjecture, Hypothesis, Principle, Proposition
>>
>> Somewhere I've picked up the idea that "ansatz" can also be used to
>> indicate the "form" or the "approach" -- more specifically, something
>> like the choice of coordinates and variables and equations -- the
>> "geometry and notation" so to speak -- in which one sets up a problem or
>> a calculation.
>>
>> The "underlying theory or principles" in my interpretation can be
>> perfectly clear, and no "Conjectures" or "Hypotheses" need be involved.
>> One is simply setting up the calculation using this ansatz, in order to
>> calculate certain consequences or numerical results (a calculation which
>> one, of course, carries out using Mathematica).
>>
>> Is my interpretation of this term off the mark? (Wikipedia's
>> explanation of the term seems to me considerably closer to my
>> understanding than to Wolfram's definition.)
>
> Also, "ansatz" might relate etymologically to "ersatz", which means
> "cheap substitute".
Ansatz in German etymologically refers to "put a tool in working
position", eg the scalpell by the doctor or the pencil of a painter.
One uses it in cocking and chemistry, where you put some ingredients in
a pot, make fire and look what is boiling out.
And finally one makes an Ansatz in order to speak a sentence or to proof
a theorem (both words mean "Satz" in German) -- but you fail because you
forgot to switch on your brain.
Always the Ansatz is meant as crucial for the outcome.
The Ansatz generally means a bundle of methods and ideas applied to
solve a given problem.
In mathematics it is used for a setting of special class of solutions
with some free parameters, if one is able to solve the complexity
reduced problem in the parameter space.
Standard examples are reductions of linear differential equations to
algebraic equations by a Fourier ansatz with a system of functions
fitting the boundary conditions or the solution of nonlinear ones
fitting the symmetries with a Bethe ansatz.
--
Roland Franzius
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