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Re: Re: Re: "Assuming"


Andrzej Kozlowski wrote:
> 
> On 26 Feb 2008, at 13:43, Daniel Lichtblau wrote:
> 
>> [...]
>>
>> I've seen cases where the FullSimplify[something] result differs from
>> something on a finite set of integers. This motivated me several  months
>> ago to alter assumptions of integrality, to reality (realness?
>> realhood?), in processing of Integrate.
>>
>> Daniel Lichtblau
>> Wolfram Research
>>
> 
> Formally speaking, reasonable measures (e.g. Radon measures) are  either 
> diffuse or  Dirac measures (or linear combinations of these).  However, 
> for the former,  the entire set of integers has measure zero,  and for 
> the latter certain finite sets will have a non-zero measure.  So none of 
> these seems to fit the intended meaning of "measure zero".
> 
> More seriously; I think the intended meaning is that in  
> Simplify[thing1] -> thing2 thing1 and thing2 should both be functions  
> of some variable that is defined on an uncountable set, then they may  
> be are allowed to differ for a finite number of values. But, if the  
> functions are defined only on countable sets (e.g. the set of all  
> integers, as is the case with many number theoretic functions) then  the 
> failure of thing2 to be equal to thing1 on a finite set could be  very 
> serious. I think in such situations  the "set of measure zero"  should 
> really be the empty set, or perhaps in really exceptional cases  a set 
> that contains no more than a single point.
> 
> Andrzej Kozlowski

Here is an example of the behavior in question. I do not pass judgement 
on whether it should be regarded as a bug or a feature. I simply wanted 
to give a concrete example where the behavior arises and is difficult to 
supress.

In[2]:= i1 = Integrate[Sin[m*x]*Sin[n*x], {x,0,2*Pi}, 
Assumptions->Element[{m,n},Reals]];

Check what happens when we assign n->1 and then take limit as m->1.

In[3]:= l1 = Limit[i1 /. n->1, m->1]
Out[3]= Pi

That was fine. Now see what happens if we assign n->1 and simplify under 
assumption that m is an arbitrary integer.

In[4]:= l2 = Simplify[i1 /. n->1, Element[m,Integers]]
Out[4]= 0

Daniel Lichtblau
Wolfram Research




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