Reply by Jerry Avins February 19, 20052005-02-19
ben wrote:

   ...

> and those initial conditions, and those initial conditions only, were > specifically and exactly what i was asking about! :) graph one, the top > left one, that contains the initial, original temperature measurements. > so c_n(0) = 1 / sqrt(n) *does* access the initial readings after all! > > ok, thanks very much for all the replies and all your time, > > ben.
OK. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ben February 18, 20052005-02-18
In article <uMSdndviW_fdBYvfRVn-pQ@rcn.net>, Jerry Avins <jya@ieee.org>
wrote:

> ben wrote: > >>>just to be clear of what i think that means (and it's definetely > >>>contains misunderstandings somewhere, the qeustion is where/what): > >>>the above text is describing the process that allows you to go from the > >>>first (top left) graph that contains the original temperature readings, > >>>to the second (top right) graph. > >> > >>No. It's a description of the result. The process is not describes > >>anywhere ion that page. > > > > > > even though it says "The Fourier coefficients at time 0 .. are given by > > the formula c_n(0) = 1 / sqrt(n)". that's just a description of the > > results, not the process? wow. you can at least see how i might think > > that c_n(0) = 1 / sqrt(n) gives or at least should give the > > results/coefficients in graph 2, seeing as it says " *The* *Fourier* > > *coefficients* at time 0 .. *are* *given* *by* *the* *formula* c_n(0) = > > 1 / sqrt(n)" > > It does, for that particular set of initial conditions.
and those initial conditions, and those initial conditions only, were specifically and exactly what i was asking about! :) graph one, the top left one, that contains the initial, original temperature measurements. so c_n(0) = 1 / sqrt(n) *does* access the initial readings after all! ok, thanks very much for all the replies and all your time, ben.
Reply by Jerry Avins February 18, 20052005-02-18
ben wrote:
>>>just to be clear of what i think that means (and it's definetely >>>contains misunderstandings somewhere, the qeustion is where/what): >>>the above text is describing the process that allows you to go from the >>>first (top left) graph that contains the original temperature readings, >>>to the second (top right) graph. >> >>No. It's a description of the result. The process is not describes >>anywhere ion that page. > > > even though it says "The Fourier coefficients at time 0 .. are given by > the formula c_n(0) = 1 / sqrt(n)". that's just a description of the > results, not the process? wow. you can at least see how i might think > that c_n(0) = 1 / sqrt(n) gives or at least should give the > results/coefficients in graph 2, seeing as it says " *The* *Fourier* > *coefficients* at time 0 .. *are* *given* *by* *the* *formula* c_n(0) = > 1 / sqrt(n)"
It does, for that particular set of initial conditions. For a different problem, there would be different coefficients. Id a museum guide, showing one of Hals's paintings, said "Notice the interplay of light and shadow, and the folds in the curtains", that would apply to one particular image, and not to the art of painting in general. Jerry -- Engineering is the art of making what you want from things you can get. &#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
Reply by ben February 18, 20052005-02-18
> > just to be clear of what i think that means (and it's definetely > > contains misunderstandings somewhere, the qeustion is where/what): > > the above text is describing the process that allows you to go from the > > first (top left) graph that contains the original temperature readings, > > to the second (top right) graph. > > No. It's a description of the result. The process is not describes > anywhere ion that page.
even though it says "The Fourier coefficients at time 0 .. are given by the formula c_n(0) = 1 / sqrt(n)". that's just a description of the results, not the process? wow. you can at least see how i might think that c_n(0) = 1 / sqrt(n) gives or at least should give the results/coefficients in graph 2, seeing as it says " *The* *Fourier* *coefficients* at time 0 .. *are* *given* *by* *the* *formula* c_n(0) = 1 / sqrt(n)"
Reply by Jerry Avins February 18, 20052005-02-18
ben wrote:
> In article <JuSdnUHcXqU7nIvfRVn-tw@rcn.net>, Jerry Avins <jya@ieee.org> > wrote: > > >>ben wrote: > > >>>all data seems to have dissapeared at that point -- no data, oh, apart >>>from the number of waves. the number of waves is the only data that >>>makes it out of the first graph. that just can't be right. can it? >> >>You have a mistaken assumption. > > > yes that's the only thing i'm sure on :) > > i wrote the following, regarding the process that occurs between the > first (top left) graph and the second (top right) graph: > > >>"where n is the wave number" (again see >>http://www.hdbatik.co.uk/temp/waveletsbookpage.html ). so the only >>information it seems to take from the original temperature measurements >>at time 0 are the number of waves -- not the hieghts/values of those >>waves. i just must be wrong in saying that. that'd be rediculous.
No. Each "wave" is a term in the infinite series that describes the temperature distribution along the bar at a particular time. The number of "waves" used is determined by how accurately one wants to approximate the true distribution.
> you wrote: > > >>Not ridiculous, just (for the moment) incomprehensible. > > > also you said the fact that the coefficient graph (the top right, > second in the process graph) happend to follow the shape of the > original data was a coincidence. > > so, against my common sense, i was taking it that the only information > that was being used from the original data was the number of waves.
See above.
> so there was a misunderstanding somewhere. doesn't matter at all. > > so, as originally suspected (it'd be ridiculous if this wasn't the > case) the initial temperature values (the heights of the waves) are > used and carried forwards from the first graph, to the second graph.
There are no "waves" in the first graph.
> ------------------------------------------------ > > > the text says: > > > [......] Next one goes into Fourier space, calculating its Fourier > transform f^, which tells us the coefficient for each wave number > making up the function f at time 0. [...] The Fourier coefficients at > time 0, c_n(0) (where n is the wave number , 1, 2, 3...), are given by > the formula c_n(0) = 1 / sqrt(n) [...... see > http://www.hdbatik.co.uk/temp/waveletsbookpage.html for orginal source] > > > just to be clear of what i think that means (and it's definetely > contains misunderstandings somewhere, the qeustion is where/what): > the above text is describing the process that allows you to go from the > first (top left) graph that contains the original temperature readings, > to the second (top right) graph.
No. It's a description of the result. The process is not describes anywhere ion that page.
> the formula c_n(0) = 1 / sqrt(n) has the description "The Fourier > coefficients at time 0, c_n(0) [...] are given by the formula c_n(0) = > 1 / sqrt(n)" << so that formula works out the values, the coefficients > that are displayed and make up the second (top right) graph. > apparently.
Yes. None of the math that produces that formula is described. There are a few pages of it.
> what would great is a direct, simple answer to, or clarification of > (and it's bound to contain a misunderstanding): > > c_n(0) = 1 / sqrt(n) doesn't seem to access the wave values/heights of > the original temperature readings depicted in the top left graph.
Right. They are the coefficients of an infinite series that describes the shape of the curve. I've been trying to tell you that for a few days now. I seem to be getting through.
> is > that correct or wrong? in particular 1 / sqrt(n) doesn't seem to access > wave values. and the c_n(0) part is i think what gets assigned, > receives the answer(s) so that doesn't access/read the initial readings > displayed in the top left graph. > > > > > to further, in a detailed way, illustrate what i think c_n(0) = 1 / > sqrt(n) does so there can't be any assumed things: > > 1 / sqrt(1) > 1 / sqrt(2) > 1 / sqrt(3) > .... > 1 / sqrt(12) (stopping at 12 because there's 12 waves)
> > eg the second one is 1 / 1.4142.. (1.4142.. being the square root of 2) > which equals 0.7071... > > and all that gets assigned to c_n(0)
You need to learn the lingo, too. Collectively, the c_n(0) are all the coefficients {c_1, c_2, c_3 ... c_1000, c_1001, ...} at time=0.
> and the main thing for me is, i don't think at any point in carrying > out c_n(0) = 1 / sqrt(n) the initial temperature readings are accessed. > that formula doesn't seem to access the initial temperature readings > data (apart from the count/number of waves, 12, in this case) <<< > correct or not? (it can't be right but that's what the basic > methematical symbols are telling me)
The formula is derived from the initial conditions by pages of math not shown.
> if c_n(0) = 1 / sqrt(n) doesn't access the initial readings can you see > how (from the book) "The Fourier coefficients at time 0, c_n(0) [...] > are given by the formula c_n(0) = 1 / sqrt(n)" seems illogical?
To learn how Fourier analysis works, go to http://www.dspguide.com/ and download chapter 8. The page you're trying to understand doesn't explain how it works, just what it does. You've convinced me that it's not intelligible to someone who doesn't already know it. Jerry -- Engineering is the art of making what you want from things you can get. &#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
Reply by ben February 18, 20052005-02-18
In article <JuSdnUHcXqU7nIvfRVn-tw@rcn.net>, Jerry Avins <jya@ieee.org>
wrote:

> ben wrote:
> > all data seems to have dissapeared at that point -- no data, oh, apart > > from the number of waves. the number of waves is the only data that > > makes it out of the first graph. that just can't be right. can it? > > You have a mistaken assumption.
yes that's the only thing i'm sure on :) i wrote the following, regarding the process that occurs between the first (top left) graph and the second (top right) graph:
> "where n is the wave number" (again see > http://www.hdbatik.co.uk/temp/waveletsbookpage.html ). so the only > information it seems to take from the original temperature measurements > at time 0 are the number of waves -- not the hieghts/values of those > waves. i just must be wrong in saying that. that'd be rediculous.
you wrote:
> Not ridiculous, just (for the moment) incomprehensible.
also you said the fact that the coefficient graph (the top right, second in the process graph) happend to follow the shape of the original data was a coincidence. so, against my common sense, i was taking it that the only information that was being used from the original data was the number of waves. so there was a misunderstanding somewhere. doesn't matter at all. so, as originally suspected (it'd be ridiculous if this wasn't the case) the initial temperature values (the heights of the waves) are used and carried forwards from the first graph, to the second graph. ------------------------------------------------ the text says: [......] Next one goes into Fourier space, calculating its Fourier transform f^, which tells us the coefficient for each wave number making up the function f at time 0. [...] The Fourier coefficients at time 0, c_n(0) (where n is the wave number , 1, 2, 3...), are given by the formula c_n(0) = 1 / sqrt(n) [...... see http://www.hdbatik.co.uk/temp/waveletsbookpage.html for orginal source] just to be clear of what i think that means (and it's definetely contains misunderstandings somewhere, the qeustion is where/what): the above text is describing the process that allows you to go from the first (top left) graph that contains the original temperature readings, to the second (top right) graph. the formula c_n(0) = 1 / sqrt(n) has the description "The Fourier coefficients at time 0, c_n(0) [...] are given by the formula c_n(0) = 1 / sqrt(n)" << so that formula works out the values, the coefficients that are displayed and make up the second (top right) graph. apparently. what would great is a direct, simple answer to, or clarification of (and it's bound to contain a misunderstanding): c_n(0) = 1 / sqrt(n) doesn't seem to access the wave values/heights of the original temperature readings depicted in the top left graph. is that correct or wrong? in particular 1 / sqrt(n) doesn't seem to access wave values. and the c_n(0) part is i think what gets assigned, receives the answer(s) so that doesn't access/read the initial readings displayed in the top left graph. to further, in a detailed way, illustrate what i think c_n(0) = 1 / sqrt(n) does so there can't be any assumed things: 1 / sqrt(1) 1 / sqrt(2) 1 / sqrt(3) .... 1 / sqrt(12) (stopping at 12 because there's 12 waves) eg the second one is 1 / 1.4142.. (1.4142.. being the square root of 2) which equals 0.7071... and all that gets assigned to c_n(0) and the main thing for me is, i don't think at any point in carrying out c_n(0) = 1 / sqrt(n) the initial temperature readings are accessed. that formula doesn't seem to access the initial temperature readings data (apart from the count/number of waves, 12, in this case) <<< correct or not? (it can't be right but that's what the basic methematical symbols are telling me) if c_n(0) = 1 / sqrt(n) doesn't access the initial readings can you see how (from the book) "The Fourier coefficients at time 0, c_n(0) [...] are given by the formula c_n(0) = 1 / sqrt(n)" seems illogical?
Reply by Jerry Avins February 18, 20052005-02-18
ben wrote:
> when the book says n ("where n is the wave number, 1, 2, 3...") doesn't > it really mean the value/height of the line at the n'th wave? > > >>>> c_n(0) = 1 / sqrt(n) > > > so: > > c_n(0) = 1 / sqrt( waveheight(n) ) > > > then they'd be a direct relation between the height of a wave and a > coefficient, rather than there being absolutely no relation > what-so-ever between the height of a wave and a coefficient.
Start over. In imagining what the author means, you seem to have constructed a completely distorted view. The nth "wave" at time t is c_n*sin(nx) where c_n is given by the formula in the web page. Jerry -- Engineering is the art of making what you want from things you can get. &#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
Reply by Jerry Avins February 18, 20052005-02-18
ben wrote:
> In article <idCdnUTzJoViwIjfRVn-hQ@rcn.net>, Jerry Avins <jya@ieee.org> > wrote: > > >>ben wrote: > > >>>something new i've found confusing about that page (which is something >>>i could probably keep up for eternity): the formula that's for >>>calculating the various coeffecients at the time 0 doesn't even seem to >>>use the heights (or values) of the waves! the formula is >>> >>> c_n(0) = 1 / sqrt(n) >> >>That's a derived result for this particular case. Implicit in it is that >>the temperature at the hot end is normalized to 1. (The wavy curve has a >>value of zero there, proving that it's artificial.) >> >> >>>"where n is the wave number" (again see >>>http://www.hdbatik.co.uk/temp/waveletsbookpage.html ). so the only >>>information it seems to take from the original temperature measurements >>>at time 0 are the number of waves -- not the hieghts/values of those >>>waves. i just must be wrong in saying that. that'd be rediculous. >> >>Not ridiculous, just (for the moment) incomprehensible. Anyhow, I think >>you better nail down Fourier analysis before you tackle wavelets. > > > the direction and flow of the (i've thought all along; fourier, but > even that's in question now) process goes top left graph, top right > graph, bottom right graph, bottom left graph. the data in the line in > the first graph doesn't seem to be used at all (not called upon, not > read/accessed/looked at), and yet there it is in the final graph, so > the data must be used somehow. it obviously isn't a concidence that the > bendy line in the last graph is the same as the bendy line in the first > graph. where does the final graph's bendy line data come from? how is > it ending up there in the last graph (as the formula that starts the > process doesn't seem to use that data at all)?
Top left: the temperature distribution along the bar at t=0 as calculated from a series with enough of the higher terms omitted to introduce significant waviness. The author should have retained more terms or used the actual distribution. I now understand why she was seduced into making that error, but let that pass. Top right: The relative values of the coefficients of the retained terms that produce the curve of top left when the series of sines is summed. Bottom left: a superposition of curves showing temperature along the bar at various times. All but t=0 are actual, rather than computed. Bottom right: a superposition of the relative values of the coefficients that that produce the several curves at bottom left.
> putting the same thing above in another way: the text says about the > bottom right graph "the information on space seesm to have > dissapeeared" -- yes, but so has the values of the waves it seems. so
The waves are in space. What do you mean? If you mean what the graph calls "wave number", their magnitudes are the coefficients. See below.
> all data seems to have dissapeared at that point -- no data, oh, apart > from the number of waves. the number of waves is the only data that > makes it out of the first graph. that just can't be right. can it?
You have a mistaken assumption.
>>>the shape of the top right graph does reflect the shape of the top left >>>graph, >> >>Coincidence, to the extent that any coincidence is possible in math. > > > i see. > > > > so the coefficient has nothing to do with the wave's height? that seems > somewhat, to say the least, contradictory to something that was > established a few messages ago, something i thought i'd understood. > (i'm drawing that statement from the fact(??) that the formula does not > use the wave's heights, only the number of waves <<<<< is that > correct?)
The relative values of the coefficients determine the curve's shape. A single scaling factor sets the size.
> it would appear right now, that the calculation that gives the > coefficients don't use the heights of waves, only the number of waves > in the data. is that correct?
No. There are in fact an infinite number of terms in the series, but they decrease in size as they go up. Terms too small to matter can be discarded. The curve for t=0 shows the reason not to discard too many. The first term is a constant; a coefficient on its own. The second term is the product of a coefficient and sin(x), where x relates to position along the bar. The third term has its own coefficient and sin(2*x) The nth term is c_n*sin(x*n). Sines are wavy. The graphs refer n to as the wave number. If this is still mysterious, I suggest that you tackle it from the other end. Instead of trying to understand someone else's not very adroit presentation of results, actually do a computation yourself or follow a worked-out step-by-step example. Jerry -- Engineering is the art of making what you want from things you can get. &#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
Reply by ben February 18, 20052005-02-18
when the book says n ("where n is the wave number, 1, 2, 3...") doesn't
it really mean the value/height of the line at the n'th wave?

> > > c_n(0) = 1 / sqrt(n)
so: c_n(0) = 1 / sqrt( waveheight(n) ) then they'd be a direct relation between the height of a wave and a coefficient, rather than there being absolutely no relation what-so-ever between the height of a wave and a coefficient.
Reply by ben February 18, 20052005-02-18
In article <idCdnUTzJoViwIjfRVn-hQ@rcn.net>, Jerry Avins <jya@ieee.org>
wrote:

> ben wrote:
> > something new i've found confusing about that page (which is something > > i could probably keep up for eternity): the formula that's for > > calculating the various coeffecients at the time 0 doesn't even seem to > > use the heights (or values) of the waves! the formula is > > > > c_n(0) = 1 / sqrt(n) > > That's a derived result for this particular case. Implicit in it is that > the temperature at the hot end is normalized to 1. (The wavy curve has a > value of zero there, proving that it's artificial.) > > > "where n is the wave number" (again see > > http://www.hdbatik.co.uk/temp/waveletsbookpage.html ). so the only > > information it seems to take from the original temperature measurements > > at time 0 are the number of waves -- not the hieghts/values of those > > waves. i just must be wrong in saying that. that'd be rediculous. > > Not ridiculous, just (for the moment) incomprehensible. Anyhow, I think > you better nail down Fourier analysis before you tackle wavelets.
the direction and flow of the (i've thought all along; fourier, but even that's in question now) process goes top left graph, top right graph, bottom right graph, bottom left graph. the data in the line in the first graph doesn't seem to be used at all (not called upon, not read/accessed/looked at), and yet there it is in the final graph, so the data must be used somehow. it obviously isn't a concidence that the bendy line in the last graph is the same as the bendy line in the first graph. where does the final graph's bendy line data come from? how is it ending up there in the last graph (as the formula that starts the process doesn't seem to use that data at all)? putting the same thing above in another way: the text says about the bottom right graph "the information on space seesm to have dissapeeared" -- yes, but so has the values of the waves it seems. so all data seems to have dissapeared at that point -- no data, oh, apart from the number of waves. the number of waves is the only data that makes it out of the first graph. that just can't be right. can it?
> > > the shape of the top right graph does reflect the shape of the top left > > graph, > > Coincidence, to the extent that any coincidence is possible in math.
i see. so the coefficient has nothing to do with the wave's height? that seems somewhat, to say the least, contradictory to something that was established a few messages ago, something i thought i'd understood. (i'm drawing that statement from the fact(??) that the formula does not use the wave's heights, only the number of waves <<<<< is that correct?) it would appear right now, that the calculation that gives the coefficients don't use the heights of waves, only the number of waves in the data. is that correct? thanks, ben