Subject: Re: EXTRAORDINARY PI
From: bstan@datasync.com (BLStansbury)
Date: Wed, 01 Jan 1997 09:03:10 GMT
On Wed, 01 Jan 1997 01:11:46 GMT, juanvp@impsat1.com.ar (JuanVP)
wrote:
>>>Sure. How about 6*arcsin(1/2)?
>>Approximation to pi.
>No, it's the real McCoy. :)
I am sorry, you are correct. I meant to say it is an infinite
number--like pi.
>>>We just don't restrict ourselves to writing answers entirely in
>>>decimal, get it? A circle of radius 2 has area 4pi, exactly.
>>But a square can't have an area of exactly 4pi unless its a circle.
>Is this a flame bait or just a lapsus?
Probably just flame bait.
>>>Again, just because we can't write it in base-10 doesn't mean we
>>>can't use its exact value.
>>You can't write its exact value in any base.
>How about in base Pi? It would be 10, wouldn't it?
Two things: if you are going to use pi as the base for a number
system, then you are going to either use the exact value of pi as it
is determined by the relation of the radius and area of a circle OR
you are not.
In the first case, there is not an exact value for pi yet.
In the second case, you can use whatever value of pi you wish to use,
but you could not "circle the square" to get the same area exactly.
BLS
Subject: Fourier transform of non-linearities...
From: tph1001@cus.cam.ac.uk (T.P Harte)
Date: 1 Jan 1997 20:16:21 GMT
Hi!
Not being a mathematician, I have what may seem to many to be quite
an obvious question.
At one stage of an algorithm on which I am working I have a finite
real sequence, f(t), represented by its DFT, let's call it F(k).
[I shall refer to ``the sequence f(n)'', although in reality this
refers to the nth number in the sequence {x}.]
For the sake of computational economy I wish to keep F(k) in the
Fourier domain until the very last step.
One of the steps, significntly before the end of the algorithm, requires
me to apply the following function to each point of the __real__ domain
sequence f(t):
1
g(\sigma) = ---------------
1 + e^{-\sigma}
where \sigma = f(t). Thus y = g(\sigma) = g(f(t)).
g(\sigma) is clearly non-linear.
The problem lies in the fact that would rather some form of g(\sigma)
were applied to F(k), i.e., the sequence f(t) in the __Fourier__ domain.
Is this possible?
[===>>QUESTION<<===]
Can I apply, say, the Fourier transform of g(\sigma) to each
point of F(k), i.e.,
if DFT[g(\sigma)] = G(k), then is IDFT[G(F(k))] = g(f(t)),
where DFT is the discrete Fourier transform and IDFT
is the inverse discrete Fourier transform? (*)
[===>>QUESTION<<===]
Does this violate the axioms of linearity on which the Fourier transform
rests? It is fine to say that a_{1}f_{1}(t) + a_{2}f_{2}(t) produces output
a_{1}F_{1}(t) + a_{2}F_{2}(t), where a_{1} and a_{2} are contants; however,
it is completely different to assert that the same holds where a_{1}
and a_{2} are replaced by non-linear functions which operate on f_{1}(t)
and f_{2}(t).
The only way that I can think of validating or invalidating (*) is by
finding the Fourier transform of g(\sigma) and then applying it to
a sequence in the Fourier domain, whose real domain sequence I know, and then
taking the inverse Fourier transform. However, my maths isn't up to scratch
and I'm having trouble with the integration involved, which I'll now detail.
First, I need to get the Fourier transform (FT) of g(\sigma). For convenience,
I switch the variable \sigma to t. Thus,
FT[g(t)] = \int_{-infty}^{+infty} \frac{1}{1 + e^{-t}} \exp(i2\pi ft) dt
FT[g(t)] = \int_{-infty}^{+infty} \frac{1}{1 + e^{-t}} \cos(2\pi ft) dt +
\int_{-infty}^{+infty} \frac{1}{1 + e^{-t}} i\sin(2\pi ft) dt)
and the second (sin) part disappears due to the integral of the sin being 0.
Using integration by parts for the cos integral:
\int u dv = uv - \int v du
let u = \frac{1}{1 + e^{-t}} and let
v = \int_{-\infty}^{+\infty} \cos(2\pi ft) dt
= 1/{2\pi f} \sin(2\pi ft) \]_{-\pi/2}^{+\pi/2}
= 1/{2\pi f} \sin(2\pi f +\pi/2) - \sin(2\pi f -\pi/2) = C
[the integral only needs to be evaluated between these limits:
it integrates to zero elsewhere] v is, therefore, a constant, C.
C = 1/{2\pi f} \sin(\pi^{2} f) - \sin(-\pi^{2} f)
= \sin(-\pi^{2} f)/(\pi f)
d/dt(u) = d/dt \frac{1}{1 + e^{-t}} =
= (1 + e^{-t}) x 0 - 1 x d/dt(1 + e^(-t)
---------------------------------------
(1 + e^{-t})^{2}
= -(-1) x e^{-t} = e^{-t}
-------------- ------------
(1 + e^{-t})^{2} 1 + e^{-t})^{2}
Thus:
FT[g(t)] = \int_{-\pi/2}^{+\pi/2} \frac{1}{1 + e^{-t}} \cos(2\pi ft) dt =
{\frac{1}{1 + e^{-t}} \times C} -
{\int_{-\pi/2}^{+\pi/2 C \times {e^{-t} \over {1 + e^{-t})^{2}}}}
= C \times ({\frac{1}{1 + e^{-t}} -
(\pi/2 - (-\pi/2) \times {e^{-t} \over {1 + e^{-t})^{2}}})
= C \times (1 + e^{-t} - \pi e^{-t}
-----------------------
(1 + e^{-t})^{2}
where C = \sin(-\pi^{2} f)/(\pi f)
which looks something(!) like a sinc function, i.e., sin(x)/x.
Now I have found it impossible to get the inverse Fourier transform of this to
validate what I questioned above.
Any ideas? Can anyone get the inverse FT of the above expression for G(if)
(assuming that it is the correct FT of g(t)!)
Thomas.
Subject: chi square assignment
From: Ulrike Hassold
Date: Wed, 1 Jan 1997 22:48:03 +0000
Hello everybody
I’ve got a big problem, I have to do an assignment for university, it's
got to be handed in on the 6th of January, and don’t understand a thing.
The first assignment (Standard Deviation) was easy as the lecturer
really went into it but he just rushed through the chi square-test and
now I find myself not being able to do the assignment even though I sat
down with about 6 Statistics books... I’m not lazy or anything, I did
do well in the last assignment and got 97% and I wouldn’t ask for help
if it wasn’t necessary so please don’t flame me. So if anybody could
help me, I’d be really, really grateful.
Ulrike
************************************************************************
The assignment is as follows:
1. What is meant by a "test of significance"?
2. What is meant by "degrees of freedom"?
3. What is meant by a "null hypothesis"?
4. What is meant by "goodness of fit"?
5. The management of a firm wants to know how their employees feel
about working conditions, particularly whether there are differences in
sentiment between various departments. A study based on random samples
of the employees of four departments yielded the results shown in this
table:
Working Department Department Department Department Total
Conditions A B C D
Very Good 65 112 85 80 342
Average 27 67 60 44 198
Poor 8 21 15 16 60
Total 100 200 160 140 600
a) What would be the null hypothesis in this example?
b) Using chi square-test, would you reject or accept the null
hypothesis (at 0.05 level).
6. Assuming that the expected normal curve frequencies given below were
calculated using the mean and standard curvation of the observed
frequencies, test for goodness of fit at a level of significance of
0.05:
ObservedFrequencies Expected NormalCurve Frequencies
29 25
160 156
314 312
202 215
42 40
3 2
--
Ulrike Hassold
Subject: Re: EXTRAORDINARY PI
From: Richard Mentock
Date: Wed, 01 Jan 1997 21:46:38 -0500
Ronald Bruck wrote:
>
> [Snipped bunch of stuff. What *were* you talking about?]
>
> Historically, the representation of real numbers as decimals, or to any
> base, is a very late addition to the question. By the way, how DOES one
> write a real number to base pi? Using WHAT as digits?
Maybe we use base 10 digits, like they (we) do with sexigesimal (base 60).
So (all numbers base pi):
1+1=2 still
and
1+2=3
but
2+2=10.220122...
On the other hand, Pi = 1.0, exactly.
--
D.
mentock@mindspring.com
http://www.mindspring.com/~mentock/index.htm
Subject: Re: chi square assignment
From: James Tuttle
Date: Wed, 01 Jan 1997 17:08:53 +0000
Ulrike Hassold wrote:
>
> Hello everybody
>
> I’ve got a big problem, I have to do an assignment for university,
> it's got to be handed in on the 6th of January, and don’t understand a
> thing. The first assignment (Standard Deviation) was easy as the
> lecturer really went into it but he just rushed through the chi
> square-test and now I find myself not being able to do the assignment
> even though I sat down with about 6 Statistics books... I’m not lazy
> or anything, I did do well in the last assignment and got 97% and I
> wouldn’t ask for help if it wasn’t necessary so please don’t flame me.
> So if anybody could help me, I’d be really, really grateful.
>
> Ulrike
>
> **********************************************************************
There's nothing I can say that's not explained better in one or more of
your six statistics books, but I'll give it a try. Also, I'm no expert
here, so maybe someone will correct me. In any event, what I say here
is better than having no clue at all.
>
> The assignment is as follows:
>
> 1. What is meant by a "test of significance"?
The idea is that if you repeat a "pass/fail" experiment many times and
you pass x% and fail (100-x)%, then you pass with an x% significance
level. Example -- throw 100 darts at a large target while blindfolded,
count the number of hits, and repeat that experiment 100 times. You can
figure x from this.
> 2. What is meant by "degrees of freedom"?
Practically, it's the number of categories you have less the number of
parameters in your model, less 1. A linear least squares model,
y = ax+b, has two parameters -- one for a and one for b. The number of
categories depends on how you group your results. See below.
> 3. What is meant by a "null hypothesis"?
You assume the truth of the hypothesis and then see if it's disproven.
> 4. What is meant by "goodness of fit"?
How well does the model fit the data.
> 5. The management of a firm wants to know how their employees feel
> about working conditions, particularly whether there are differences
> in sentiment between various departments. A study based on random
> samples of the employees of four departments yielded the results shown
> in this table:
>
> Working Department Department Department Department Total
> Conditions A B C D
>
> Very Good 65 112 85 80 342
> Average 27 67 60 44 198
> Poor 8 21 15 16 60
> Total 100 200 160 140 600
>
> a) What would be the null hypothesis in this example?
That there are no significant differences among the departments.
> b) Using chi square-test, would you reject or accept the null
> hypothesis (at 0.05 level).
I don't know ... you figure it out. Assume no differences. Take the
total (600) and spread it proportionally among the departments and
conditions. Dept. A gets (100/600) (342 198 60) = (57 33 10). Then
calculate the chi-sq values from the formula (act - exp)^2/exp where act
is actual and exp is expected. For Dept. A the numbers are (1.123 1.091
0.400) where (65 - 57)^2/57 = 1.123. Repeat that for all 12 of your
values and add up the 12 results to get your chi-sq total. Degrees of
freedom for a 4x3 table is 3x2=6. (Four categories across and three
categories down, less one parameter in each direction because of
averaging or weighting).
Then take your chi-sq result and look it up in a table. You want the
.05 level, so use the table value for 6 d.f. and .025 because you have a
two-tailed test. If your result is less than the table value, you do
not reject your hypothesis that there is no significant difference.
> 6. Assuming that the expected normal curve frequencies given below
> were calculated using the mean and standard curvation of the observed
> frequencies, test for goodness of fit at a level of significance of
> 0.05:
>
> ObservedFrequencies Expected NormalCurve Frequencies
> 29 25
> 160 156
> 314 312
> 202 215
> 42 40
> 3 2
Do the same as above with 2 degrees of freedom (5 - 3). The first
chi-sq calculation is (29 - 25)^2/25 = 0.64. Ignore the last value
because chi-sq is not reliable when the expected value is less than 5.
That also explains why you use 5 - 3 = 2 d.f. instead of 6 - 3 = 3. Use
of the mean and standard deviation takes away two degrees of freedom.
James
