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From: William Elliot on 28 Mar 2010 04:12
On Sat, 27 Mar 2010, RussellE wrote:

>>> Let M = {0, 1, 2, ..., m-1) be the domain of SumN().
>>> Let C = {0, 1, 2, ..., n(m-1) } be the Ceiling of SumN() on domain M.
>>
>> C is not the ceiling of sum_n(M), it's the range of sum_n.
>> n(m - 1) is the maximum of sum_n(M)
>
> Yes, I need a better definition for C.
> Especially since I am considering C to be variable.
>
Let c = maximum allowed value.

>> Let N be the set of non negative integers.
>> For n in N\0, define add_n:N^n -> N by
>> . . add_n(p1, p2,.. p_n) = p1 + p2 +..+ p_n
>>
>> Let N_m = { 0,1,.. m }. max add_n((N_m)^n) = nm.
>>
>> Any other profound trifles?

> How big does my workspace have to be to accommodate the mathematical
> operations I may have to perform?

Ok, there may be a point to your work for designing software.

> If my maximum workspace is |C| = 100, I can't allow more than four
> additions from domain |M| = 25. I can't allow the multiplication of two
> numbers from M where one of the numbers is greater than 4.
>
>> Are you acquainted with the notations used,
>> namely f:A -> B, f(A), A\0, max A and A^n
>> for f a function, A,B a set and n in N\0?
>
> No. Would you please explain or provide reference?
>
f:X -> Y. f is a function from X into Y.
X is the domain and Y is the codomain.
To define a function, three essentials are needed,
a definition of the domain, the codomain and the rule of correspondence.

f(A) = { f(x) | x in A }; for f:X -> Y, the range of f is f(X).

A\x = A - {x}
max A is the maximum element of A

A^n = { (a1,a2,.. a_n) | a1,a2,.. a_n in A }
(a1,a2,.. a_n) is an ordered n-tuple.
Are you familiar with them?

A problem you'd like to solve is given c and n, what is
.. . max{ m in N | max add_n((N_m)^n) <= c }

and given c, what is
.. . min{ n in N | m(n) = 0 }
where
.. . m(n) = max{ m in N | max add_n((N_m)^n) <= c }.

>>> Multiplication is a special case of SumN().
>>
>>> For x.y, all but x parameters of SumN() are 0,
>>> and all non-zero parameters equal y where n >= x.
>>
>> Clarification is needed.
>> What for example, is the subject of that incomplete sentence?
>
> I want to define multiplication as a special case of sum_n(M).
> For example 3 times 4 would be sum_6(0,0,0,4,4,4)
> where n=6 and M includes 3 and 4.
>
What advantage would that have over simple
working directly with multiplication?

Do you also want to define exponentiation
as a special case of multiplication?
From: RussellE on 28 Mar 2010 16:05
On Mar 28, 1:12 am, William Elliot <ma...(a)rdrop.remove.com> wrote:
> On Sat, 27 Mar 2010, RussellE wrote:
> >>> Let M = {0, 1, 2, ..., m-1) be the domain of SumN().
> >>> Let C = {0, 1, 2, ..., n(m-1) } be the Ceiling of SumN() on domain M.
>
> >> C is not the ceiling of sum_n(M), it's the range of sum_n.
> >> n(m - 1) is the maximum of sum_n(M)
>
> > Yes, I need a better definition for C.
> > Especially since I am considering C to be variable.
>
> Let c = maximum allowed value.
>
> >> Let N be the set of non negative integers.
> >> For n in N\0, define add_n:N^n -> N by
> >> . . add_n(p1, p2,.. p_n) = p1 + p2 +..+ p_n
>
> >> Let N_m = { 0,1,.. m }. max add_n((N_m)^n) = nm.
>
> >> Any other profound trifles?
> > How big does my workspace have to be to accommodate the mathematical
> > operations I may have to perform?
>
> Ok, there may be a point to your work for designing software.
>
> > If my maximum workspace is |C| = 100, I can't allow more than four
> > additions from domain |M| = 25. I can't allow the multiplication of two
> > numbers from M where one of the numbers is greater than 4.
>
> >> Are you acquainted with the notations used,
> >> namely f:A -> B, f(A), A\0, max A and A^n
> >> for f a function, A,B a set and n in N\0?
>
> > No. Would you please explain or provide reference?
>
> f:X -> Y.  f is a function from X into Y.
> X is the domain and Y is the codomain.
> To define a function, three essentials are needed,
> a definition of the domain, the codomain and the rule of correspondence.
>
> f(A) = { f(x) | x in A };  for f:X -> Y, the range of f is f(X).
>
> A\x = A - {x}
> max A is the maximum element of A
>
> A^n = { (a1,a2,.. a_n) | a1,a2,.. a_n in A }
> (a1,a2,.. a_n) is an ordered n-tuple.
> Are you familiar with them?

Yes. Thanks for the explaination.

> A problem you'd like to solve is given c and n, what is
> . . max{ m in N | max add_n((N_m)^n) <= c }
>
> and given c, what is
> . . min{ n in N | m(n) = 0 }
> where
> . . m(n) = max{ m in N | max add_n((N_m)^n) <= c }.
>
> >>> Multiplication is a special case of SumN().
>
> >>> For x.y, all but x parameters of SumN() are 0,
> >>> and all non-zero parameters equal y where n >= x.
>
> >> Clarification is needed.
> >> What for example, is the subject of that incomplete sentence?
>
> > I want to define multiplication as a special case of sum_n(M).
> > For example 3 times 4 would be sum_6(0,0,0,4,4,4)
> > where n=6 and M includes 3 and 4.
>
> What advantage would that have over simple
> working directly with multiplication?

I think what I am most interested in is N.
How many iterations (parameters) do I need
for these mathematical operations?
Converting everything to addition allows
me to compare different operations directly
in terms of number of iterations required.

Another advantage is I don't have to define
another domain for multiplication.

> Do you also want to define exponentiation
> as a special case of multiplication?

Yes. For example, consider the infinite set
2^^1, 2^^2, 2^^3, ...

Set theory assumes we can iterate as many
times as we want. How many times do we
have to iterate addition to compute the series above?


Russell
- 2 many 2 count
From: RussellE on 28 Mar 2010 16:24
On Mar 28, 1:12 am, William Elliot <ma...(a)rdrop.remove.com> wrote:
> On Sat, 27 Mar 2010, RussellE wrote:

> What advantage would that have over simple
> working directly with multiplication?
>
> Do you also want to define exponentiation
> as a special case of multiplication?

Consider 3^3. = Sum_9(3,3,3,3,3,3,3,3,3)
Defining 3^3 with this system requires
n >= 9 and m >= 3 and c >= 27.
From: RussellE on 28 Mar 2010 16:52
On Mar 28, 1:12 am, William Elliot <ma...(a)rdrop.remove.com> wrote:
> On Sat, 27 Mar 2010, RussellE wrote:

> What advantage would that have over simple
> working directly with multiplication?

There may be no advantage.
It might be simpler to define a new domain
for multiplication and assume the co-domains
are the same for addition and multiplication.


Russell
- Zeno was right. Motion is impossible.
From: William Elliot on 29 Mar 2010 03:54
On Sun, 28 Mar 2010, RussellE wrote:

> On Mar 28, 1:12�am, William Elliot <ma...(a)rdrop.remove.com> wrote:
>> On Sat, 27 Mar 2010, RussellE wrote:
>
>> What advantage would that have over simple
>> working directly with multiplication?
>
> There may be no advantage.
> It might be simpler to define a new domain
> for multiplication and assume the co-domains
> are the same for addition and multiplication.
>
Simplest is just to set a maximum integer c.
Then ask what's the maximum number of times,
with integers <= b can an operation O be done
without exceeding c; max(O,b) or max(O,b,c).
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