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Re: Re: Re: "Assuming"


On 26 Feb 2008, at 13:43, Daniel Lichtblau wrote:

> Mariano Su=E1rez-Alvarez wrote:
>> On Feb 23, 7:34 am, Daniel Lichtblau <d... at wolfram.com> wrote:
>> [...]
>
>>> [From David Cantrell:]
>>>
>>>>> [...] But I do know of a case where
>>>>> Mathematica goes even further, removing a singularity at which the
>>>>> function
>>>>> is defined as a number:
>>>
>>>>> In[17]:= FullSimplify[UnitStep[-x^2]]
>>>>> Out[17]= 0
>>>
>>>>> despite the fact that correctly
>>>
>>>>> In[18]:= UnitStep[-x^2] /. x -> 0
>>>>> Out[18]= 1
>>>
>>>>> Perhaps the simplification above is considered a bug, perhaps not.
>>>
>>>>> David
>>>
>>> A feature, really. That is, it's wrong, but FullSimplify can make
>>> mistakes on measure zero sets. We do not generally regard this
>>> phenomenon as a bug, though we reconsider on case by case basis.
>>
>>
>> How does that `measure zero' allowance work in a context
>> of something like
>>
>>  Assuming[Element[x, Integers], FullSimplify[something]]
>>
>> ?
>>
>> -- m
>
> 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


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