A 'math notation' question

Hi,
  One of the guys here recently provided a link 
to an "FFT" paper written by some researchers at 
Apple Computer.  In that paper the authors had 
two different equations of the form:

    Z = max[x(n) - y(n)]
         n

where x(n) and y(n) are two sequences.  Notice the 
lowercase "n" under the letters "max".  

I've seen that kind notation once or twice over the 
years, but I haven't been able to dig up any clear 
definition for that notation that helped me 
understand the two equations in the Apple paper.

Can any of you tell me how to interpret the meaning 
of that lowercase "n" under the letters "max"?

As Elvis would say, "Thang ya ver mush",
[-Rick-]

>Hi,
>  One of the guys here recently provided a link 
>to an "FFT" paper written by some researchers at 
>Apple Computer.  In that paper the authors had 
>two different equations of the form:
>
>    Z = max[x(n) - y(n)]
>         n
>
>where x(n) and y(n) are two sequences.  Notice the 
>lowercase "n" under the letters "max".  
>
>I've seen that kind notation once or twice over the 
>years, but I haven't been able to dig up any clear 
>definition for that notation that helped me 
>understand the two equations in the Apple paper.
>
>Can any of you tell me how to interpret the meaning 
>of that lowercase "n" under the letters "max"?
>
>As Elvis would say, "Thang ya ver mush",
>[-Rick-]

Hi Rick,

If there is an equation of the form
   a = max_n f(n)
this should mean a is the maximum value of f(n) over n.  

So in the equation you wrote above, Z should be the maximum difference
[x(n) -y(n)] over all possible n.

Note that there is also a different notation with "arg" preceding "max". 

   b = arg max_n f(n)
In this case b is the value of n that maximizes f(n).

For example, take f(n) = 1 - n^2, for all real n.  Then, for the above
definitions of a and b, one has a=1, and b=0.

Hope this helps.

Emre

Rick Lyons wrote:

>     Z = max[x(n) - y(n)]
>          n

> Can any of you tell me how to interpret the meaning 
> of that lowercase "n" under the letters "max"?

It means that Z becomes the maximal function value along the n 
variable. That's of course more useful when there's more than one 
variable. Did you think it might make Z become the maximizing index? 
If so, that would be written argmax_n f(n).


Martin

-- 
Quidquid latine scriptum est, altum videtur.

Rick Lyons wrote:

> In that paper the authors had 
> two different equations of the form:

>     Z = max[x(n) - y(n)]
>          n

> where x(n) and y(n) are two sequences.  Notice the 
> lowercase "n" under the letters "max".  

It looks like an extension to the usual sigma notation
for sum with an index variable.  I suppose there should
be a greek letter, except that both min and max start
with m.

-- glen

On Nov 2, 11:25&#4294967295;am, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
> Hi,
> &#4294967295; One of the guys here recently provided a link
> to an "FFT" paper written by some researchers at
> Apple Computer. &#4294967295;In that paper the authors had
> two different equations of the form:
>
> &#4294967295; &#4294967295; Z = max[x(n) - y(n)]
> &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295;n
>
> where x(n) and y(n) are two sequences. &#4294967295;Notice the
> lowercase "n" under the letters "max". &#4294967295;
>
> I've seen that kind notation once or twice over the
> years, but I haven't been able to dig up any clear
> definition for that notation that helped me
> understand the two equations in the Apple paper.
>
> Can any of you tell me how to interpret the meaning
> of that lowercase "n" under the letters "max"?
>
> As Elvis would say, "Thang ya ver mush",
> [-Rick-]

What's the difference between max and sup?

On Nov 1, 10:40&#4294967295;pm, HardySpicer <gyansor...@gmail.com> wrote:

> What's the difference between max and sup?

My experience: "max" is usually used in a totally ordered domain, such
as the integers or reals. "sup" (for supremum) is usually used over a
partial order (or more properly a join semilattice, I think) . When
the partial order is also a total order, it's the same as max.  Of
course, this is probably subject to as many exceptions as any other
rule or definition used in the literature :-)

 - Kenn

>What's the difference between max and sup?
>

Supremum (sup) is the least upper bound on a set, wherever that makes
sense. Supremum itself may or may not be in the set.  Maximum has to be
*in* the set.

For example, 1 is both the maximum and the supremum of all real numbers in
the interval [0,1]; whereas it is only the supremum for the open interval
[0,1). 

Emre

On Sat, 01 Nov 2008 15:25:15 -0700, Rick Lyons
<R.Lyons@_BOGUS_ieee.org> wrote:

>
>Hi,
>  One of the guys here recently provided a link 
>to an "FFT" paper written by some researchers at 
>Apple Computer.  In that paper the authors had 
>two different equations of the form:

   (snipped by Lyons)

Hi Guys,
  Thanks for your thoughts.

[-Rick-]

"emre" <eguven@ece.neu.edu> writes:

>>What's the difference between max and sup?
>>
>
> Supremum (sup) is the least upper bound on a set, wherever that makes
> sense. Supremum itself may or may not be in the set.  Maximum has to be
> *in* the set.
>
> For example, 1 is both the maximum and the supremum of all real numbers in
> the interval [0,1]; whereas it is only the supremum for the open interval
> [0,1). 

That's my understanding, too, emre. The previous poster may also want to
google "epsilon neighborhood" and "deleted neighborhood."
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