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Topic: the range of values for "Pi".
Replies: 19   Last Post: Feb 4, 2014 5:02 AM

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fom

Posts: 1,969
Registered: 12/4/12
Re: the range of values for "Pi".
Posted: Feb 4, 2014 3:24 AM
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On 2/3/2014 8:37 AM, quasi wrote:
> quasi wrote:
>> Niels Diepeveen wrote:
>>> quasi wrote [edited]:
>>>>
>>>> Conjecture:
>>>>
>>>> Fix a norm on R^2.
>>>>
>>>> Let S = {v in R^2 : |v| = 1} where |v| denotes the norm
>>>> of v.
>>>>
>>>> Thus S is the standard unit circle in R^2 relative to the
>>>> given norm.
>>>>
>>>> Define the circumference C of S to be the sup of the
>>>> perimeters of all convex polygons which can be inscribed
>>>> in S (where the side lengths are computed using the given
>>>> norm).
>>>>
>>>> For the given norm, define Pi as C/2.
>>>>
>>>> Some trivial observations:
>>>>
>>>> (1) For all norms on R^2, Pi >= 2
>>>>
>>>> (2) For an appropriate choice of norm on R^2, Pi can be
>>>> made equal to any specified real number between 2 and 4
>>>> inclusive.

>>>
>>> The last one does not seem trivial to me.

>>
>> The claim was based on a miscalculation.
>>

>>> 4 is attained when S is any parallelogram,
>>
>> Yes.
>>

>>> but how do you get 2 ?
>>
>> My mistake -- I thought the max norm yielded Pi = 2,
>> but actually, it yields Pi = 4, the same as for any
>> parallelogram.
>>

>>> I can get 3 with a regular hexagon,
>>
>> Nice.
>>

>>> but then I'm out of ideas.
>>>

>>>> Conjecture: For all norms on R^2, Pi <= 4.
>>>
>>> Seems plausible, but I don't know how to prove it.

>>
>> Could be a rough one.
>>
>> So what we now know is this:
>>
>> (1) For all norms on R^2, Pi >= 2
>>
>> (2) For an appropriate choice of norm on R^2, Pi can be
>> made equal to any specified real number between 3 and 4
>> inclusive.
>>
>> Which suggests the following ...
>>
>> Conjecture: For all norms on R^2, 3 <= Pi <= 4.

>
> I can prove the following partial result ...
>
> For n even with n > 2, let Pi_n be the value of Pi when
> S is a regular n-gon centered at the origin.
>
> For n = 2 mod 4, the sequence Pi_n is strictly increasing
> and converges to ordinary Pi.
>
> For n = 0 mod 4, the sequence Pi_n is strictly decreasing
> and converges to ordinary Pi.
>
> It follows that
>
> min {Pi_n : n = 4,6,8, ...} = Pi_6 = 3
>
> max {Pi_n : n = 4,6,8, ...} = Pi_4 = 4
>
> quasi
>



I can try to give you an account of your
limits. I would hope someone has an easier
explanation. But, here goes...

First there is a sequence of definitions
that go round in circles to get back to
the codomain of your norm -- that is, |R.

In "Functional Analysis" by Kantorovich
the discussion of a norm traces back
to the relationship between absorbent
convex sets and gauge functions. This
relationship describes a family of
compatible seminorms based on convex
sets.

Convexity is defined using the vector
space relation,

z = kx + (1-k)y

The limits are related to the '1-k'
term.

A set E is convex if z is in E
whenever x and y are in E.

E is balanced if |k|x is in E
whenever x is in E and |k| <= 1.

E is absolutely convex if kx + jy is
in E whenever |k| <= 1, |j| <= 1, and
both x and y are in E.

E is absorbent if for any vector v
there is a k > 0 such that v is in
jE for j with |j| >= k. So, an
absorbent set E is a set having
an interval on every ray passing
through the origin.

Absorbent sets are associated
with a single seminorm by a
a greatest lower bound over the
multipliers for the family of
rays.

But, that is jumping a few
definitions. To continue, a
seminorm is a homogeneous gauge
function. A gauge function is
a semi-additive positive homogeneous
function from the vector space
into |R.

Semi-additivity is defined as

p( x + y ) <= p(x) + p(y)

Positive homogeneity is defined as

p( k * x ) = k * p(x)

while homogeneity is defined as

p( k * x ) = |k| * p(x)

To every convex absorbent set E there
corresponds the Minkowski functional,

p_E(x) = inf{ k : k > 0 and x in kE }

so that

{ x : p_E(x) < 1 } subset E subset { x : p_E(x) <= 1 }

p_E(x) is a seminorm if E is absolutely
convex.


So, the usual definition of a norm,

|| x || = 0 if and only if x = 0

|| k * x || = |k| * || x ||

|| x + y || <= || x || + || y ||

induces a topology on the vector space
in terms of the family of seminorms
associated with Minkowski functionals.

Now, "attaching" such a function is
discussed in "Vectors and Tensors" by
Bowen and Wang.

As had been pointed out to me in the
past, there is a distinction between
a mere vector space and an inner product
space. If this construction is done
with a vector space, it yields an affine
space. If it is done with an inner
product space, it yields a Euclidean
point space.

The construction above requires a mapping
into |R. So, there are actually two
constructions occurring here. First, |R
must be understood as an inner product
space "attaching" to itself. Then, the
domain for the gauge functions associated
with a norm must be thought of as the
set to which the inner product space |R
is being "attached".

Here is their definition:

<begin quote>

Consider an inner product space V and a
set E. The set E is a Euclidean point
space if there exists a function

f: E x E :=> V

such that:

1) f(x,y) = (f,z) + f(z,y) for all x,y,z in E

2) For every x in E and v in V there exists
a unique element y in E such that

f(x,y) = v

<end quote>


They then write:

"A Euclidean point space is not a vector
space but a vector space with inner product
is made a Euclidean point space by defining

f(v_1, v_2) = v_1 - v_2

for all v in V. For an arbitrary point
space the function f is called the point
difference, and it is customary to use the
suggestive notation,

f(x,y) = x - y"


They then go on to define a distance
function in terms of the inner product
of their definition,

d(x,y) = || x - y || = [( x - y )( x - y )]^1/2


Going back to Kantorivich and the notions
from functional analysis, attaching a
norm to a topological vector space induces
the norm topology. The norm topology is
a locally convex space by virtue of its
system of neighborhoods about the origin.
And, a normed space is metrizable. Hence,
the last step from Bowen and Wang is
precisely the step that attaches a metric
to a normed space.


Now, consider the formula with which
this began,

z = kx + (1-k)y

The definition of a Minkowski functional
depended upon an absorbent set. This is
a notion of betweenness. So, in the
equation above x, y, and z must be
distinct. Therefore k cannot be 0
or 1.

Consider the form,

y = ( z - kx ) / ( 1 - k )

The primary concern here is what occurs
in |R since this is the codomain of a
norm. And, since x and y are supposed
to be arbitrary points in a set with
respect to the construction of a Euclidean
point domain, take x and y to be
different from 0. Then, we may
write,

|y| = | z - kx | / ( 1 - k )

so that

|y| > | z - kx |

|y| > | z - kx | + ( k * | z - kx | )

|y| > | z - kx | + ( k * | z - kx | ) + ( k^2 * | z - kx | )

and so on

by virtue of

Sum[n=0..oo] x^n = 1/(1-x)


The purpose for all of this has been to
convince you that the limits from your
analysis of the question arise from an
infintary process. If I have made a mistake
and wasted your time, then you have no
need to continue. My experience, however,
is that tracing these definitions through
different subjects often leads to this
kind of tedious analysis to identify the
point at which "foundational" concerns
must be brought to bear.

With that, consider the initial set of
intervals from my recent post,

0 < 1/32 < 2/32 < 3/32 < 4/32 < 5/32 < 1/6 < 1/4 < 1/3

You will see both 1/3 and 1/4 in the sequence
of intervals. Their multiplicative inverses
are your limits.

Now, for me 1/3 had been significant in
relation to Cantor sets. But, when you
recently informed me about Wilson's theorem,
I have been learning about primes as
quickly as possible.

In this instance, 1/3 also relates to the
Markov-Lagrange spectrum. This spectrum is
normally thought to converge at 3. However,
there is a particularly relevant analysis in
the paper,

http://arxiv.org/pdf/0906.0611v1.pdf

Roy formulates an analysis of the Markov
numbers in such a way that he replaces the
usual triples with 2 x 2 matrices. Working
in the interval [0,1/3] he demonstrates that
the Lagrange spectrum converges simultaneously
with two conjugate values. What is important
is that there is a simultaneous convergence
of both linear and quadratic forms in this
convergence.

The simultaneous convergence is important
because area is non-linear. This is the
source of incommensurability. So, the fact
that infinitary processes arise in your
question follows from the problem of definition
for irrational numbers.

Now, the value of 1/4 enters the sequence above
from my personal considerations. The purpose
of the Dedekind-Cantor program had been to
provide an exact identity criterion for real
numbers. I went back and read Bolzano's proof
of the intermediate value theorem. It is
prior to modern methods, and, it depends upon
a fixed-point argument. So, the intervals
above are based on a system of fixed-points
overlapping one another.

First, the Cantor intervals [0,1/3^i] are
divided to have midpoints,


[ 0, ..., 1/( 2 * 3^i ), ..., 1/3^i ]


Next, the rightmost subinterval is divided
to have a "between point" which is not
necessarily related by a direct mathematical
equation. So, the largest dyadic power
that lies in the interval is used,


[ 1/( 2 * 3^i ), ..., 1/2^j, ..., 3^i ]


The leftmost subinterval is divided into
6 subintervals. My reasoning for this
arose from reading Sierpinski's monograph
on "Pythagorean Triangles". In the last
chapter he discusses certain relationships
associated with using Pythagorean triples
to form parallelepipeds. It is significant
that the numbers 5 and 2 (in all powers for
2) occur as algebraic limits associated
with those constructions. So, I used
that idea to partition the subinterval.

My intuitions on the matter seem to have
been correct. I found this while preparing
this reply:

http://en.wikipedia.org/wiki/Five-point_stencil

Although that construction appears to use
an "initial" partition, it is actually a
"final" configuration. If you look at what
is actually being done, a neighborhood system
is being generated around the least prime,
namely 2.

Consider the "residual" partition in the
first 3 iterations,


interval_1 :=> { ..., 33 }
interval_2 :=> { 31, ..., 17 } [5 primes] [length: 14] [gap: 2]
interval_3 :=> { 13, 11 } [2 primes] [length: 2] [gap: 4]
interval_4 :=> {} [0 primes] [length: null] [gap: null]
interval_5 :=> { 7 } [1 prime] [length: 0] [gap: 4]
interval_6 :=> {} [0 primes] [length: null] [gap: null]
interval_7 :=> { 5 } [1 prime] [length: 0] [gap: 2]
interval_8 :=> {} [0 primes] [length: null] [gap: null]

residual :=> { 2 } [1 prime] [length: 0] [gap: 3]
omitted :=> { 3 } [1 prime] [length: 0] [gap: 0]



interval_1 :=> { ..., 131 }
interval_2 :=> { 127, ..., 67 } [13 primes] [length: 60] [gap: 4]
interval_3 :=> { 61, ..., 43 } [5 primes] [length: 18] [gap: 6]
interval_4 :=> { 41, 37 } [2 primes] [length: 4] [gap: 2]
interval_5 :=> { 29 } [1 prime] [length: 0] [gap: 8]
interval_6 :=> { 23, 19 } [2 primes] [length: 4] [gap: 6]
interval_7 :=> { 17 } [1 prime] [length: 0] [gap: 2]
interval_8 :=> { 13, 11 } [2 primes] [length: 2] [gap: 4]

residual :=> { 7, 5, 3, 2 } [4 primes] [length: 5] [gap: 4]
omitted :=> { 31 } [1 prime] [length: 0] [gap: 0]



interval_1 :=> { ..., 521 }
interval_2 :=> { 509, ..., 257 } [49 primes] [length: 252] [gap: 2]
interval_3 :=> { 251, ..., 173 } [15 primes] [length: 78] [gap: 6]
interval_4 :=> { 167, ..., 131 } [8 primes] [length: 36] [gap: 6]
interval_5 :=> { 127, ..., 103 } [5 primes] [length: 24] [gap: 4]
interval_6 :=> { 101, ..., 59 } [10 primes] [length: 42] [gap: 2]
interval_7 :=> { 53, ..., 37 } [5 primes] [length: 16] [gap: 6]
interval_8 :=> { 31, 29 } [2 primes] [length: 2] [gap: 6]

residual :=> { 23, ..., 2 } [9 primes] [length: 21] [gap: 6]
omitted :=> {} [0 primes] [length: null] [gap: null]


It generates the system of neighborhoods,


{ { 2 }, { 7, 5, 3, 2 }. { 23, ..., 2 } }


Obviously, if the generation of partitions
is taken to infinity, this is a particular
system of neighborhoods converging to 2.

So, in fact, the '1/4' in the system of
intervals,


0 < 1/32 < 2/32 < 3/32 < 4/32 < 5/32 < 1/6 < 1/4 < 1/3


is a terminal value just as '1/3' is a
terminal value in Roy's paper.

Now, my other post simply dealt with generating
primes. But, as always, I am working at the
sense of a foundation which drives my researches.
The system of intervals,


0 < 1/32 < 2/32 < 3/32 < 4/32 < 5/32 < 1/6 < 1/4 < 1/3


Should actually be viewed as half-open intervals,


( 0, 1/32 ], ( 1/32, 2/32 ], ( 2/32, 3/32 ], ( 3/32, 4/32 ], ...

... ( 4/32, 5/32 ], ( 5/32, 1/6 ], ( 1/6, 1/4 ], ( 1/4, 1/3 ]


And, after the first partition, they are interlaced
with oppositely directed half-open intervals. So,
the second partition,


0 < 1/128 < 2/128 < 3/128 < 4/128 < 5/128 < 1/18 < 1/16 < 1/9


is overlayed with the intervals,


[ 1/49, 6/49 ), [ 5/49, 24/49 )

where the denominator is the square of the largest
prime from the corresponding residual. In this
case, one has 7^2 as the denominators because the
residual is

{ 7, 5, 3, 2 }


The sequence < 24/49, 264/529, 3120/6241, ... >
approaches 1/2 while the the sequence of Cantor
intervals < [ 0, 1/3 ], [ 0, 1/9 ], [ 0, 1/27 ], ... >
is driving the sequence of midpoints to 0. That
is,

< 24/49, 264/529, 3120/6241, ... > -> 1/2

< 1/6, 1/18, 1/54, ... > -> 0

Once again, the notion of "between points" is
implmented. So, the second partition would
actually look like,


0 < 1/128 < 2/128 < 3/128 < 4/128 < 1/49 < 5/128 ...

... < 1/18 < 1/16 < 5/49 < 1/9 < 6/49 < 24/49


Relative to the first partition, 1/4 is omitted
since it demarcates an interval.

And, in each partition the endpoints for the
squared residuals are supposed to be as near
as possible to specific points. So, one has


5/49 < 1/9 < 6/49

4/128 < 1/49 < 5/128


with the first interval being as small as possible
around the value 1/3^i and the second interval
having the squared residual value as near to the
larger endpoint as possible.

There is also this subinterval,

5/49 < 6/49 < 24/49

In the limit, then, one obtains

( 0, 1/2 )

But, in fact, the values 1/3 and 1/4 arise from
the "infinite descent" intended to form the neighborhood
system around 2.

Now, I could talk about the relationships to various
Diophantine equations and ideas from Sierpinski's
monograph on Pythagorean triangles that motivated this.
But, I have written enough.

I know it is a lot. I should add that for every
odd prime p, 4p is representable as the sum of
four odd integral squares. That is where your modulo 4
arithmetic arises. With that in mind, you should
look at Marion's theorem and Morgan's theorem,

http://mathworld.wolfram.com/MarionsTheorem.html

Also, with a handful of removable discontinuities,
one gets the postive reals as the codomains of


r(x) = x , x in ( 0, 1/2 )

s(x) = 1/2 + ( 1/2 - x ), x in ( 0, 1/2 )

t(x) = 1/( 1/2 + ( 1/2 - x ) ), x in ( 0, 1/2 )

u(x) = 1/x , x in ( 0, 1/2 )


But, there are other ways of trying to extend it.
And, I have to learn the algebra to put this all
together properly.








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