Re: [bug #9259] Reduced spectral density mapping yielding bad values
Hi, I have thoroughly checked all the units of the physical constants, spectral densities, rotational correlation times, and relaxation rates and have a few important corrections about the units which are used in relax and elsewhere. Please read below for the details and a long story about SI vs. CGS units, frequency vs. angular frequency, and how it all relates to angular momentum. On 6/15/07, Sebastien Morin [EMAIL PROTECTED] wrote: Hi, Here are the different values I obtain for a residue with R1 = 1.1336 +- 0.0851 ; R2 = 12.9336 +- 0.9649 ; and NOE = 0.463921 +- 0.045 J(0) J(wN)J(wH) = == Here are the results with Leo Spyracopoulos's Mathematica notebook. 4.704231413115747e-9 2.664085520910741e-109.48428699657e-12 Here are the results with relax without multiplying frequencies in Hz by 2 pi. 4.6978912534878238e-092.6603551824374712e-10 9.478993207668287e-12 ratio 0.9986522432526923179 0.9985997677461966745 0.9993081880043085706 Here are the results with relax when multiplying frequencies in Hz by 2 pi. 3.4634030539343071e-091.9612804482358541e-10 9.478993207668287e-12 ratio 0.7362314371436068543 0.7361927508863804185 0.9993081880043085706 The very small discrepancy we get (ratio 0.999...) is due to several small differences in the definitions of constants (e.g. gn = -2.7126 in relax and -2.7108 in Leo's notebook). This is not important. However, there is a non negligible difference with the use of either frequencies in Hz or frequencies transformed to rad/s. Now, what do we do with that ? The spectral densities are in units of rad / s = rad s^-1. (these are not SI units, however) The units of radians per second, or radian Hertz, is the angular frequency (omega) rather than the frequency (nu), yet both are nevertheless SI units. For example see http://en.wikipedia.org/wiki/Angular_frequency or related sites. These SI units are the same in the CGS unit system. The reason that we use the angular frequency rather than frequency is because we are talking about angular momentum. The rates (R1, R2 and sigmaNOE) are in units of s^-1. The constant 'd' is in units of s^-2. This is not correct. The correct units are rad^2.s^-2. This can be found by doing a unit analysis on the SI dipolar constant with the mu0/4pi component (or alternatively doing the analysis in the CGS system). The reason for the radians being part of the equation is because we are using Dirac's constant (h_bar) rather than Planck's constant (h). While Planck's constant has the units of Joules per Hertz (or Joule seconds) because Dirac's constant is divided by 2pi its units are Joules per Hertz per radian. However in most cases the radian units of Dirac's constant are not stated as we are usually talking about angular momentum (the angular part means we use radians hence it is indirectly implied). The other reason is because the gyromagnetic ratio also has units of radian, and again this is usually not reported (for example see Table 1.1 of Cavanagh in which the gyromagnetic ratios are in fact in rad.s^-1.T^-1, although they are not reported as such). All of this is still in SI units - the radian components are independent of the SI or CGS systems. The relevant units in the SI system for the dipolar constant, defined as d = (mu0/4pi)^2 . (gH.gX.h_bar/r^3)^2, are mu0 - kg.m.s^-2.A^-2, h_bar - J.s.rad^-1, gx - rad.s^-1.T^-1, r - m, where tesla (T) is equal to the units kg.A^-1.s^-2. The SI units for the CSA constant, defined as c = (omegaX . csa / 3)^2, are omegaX - rad.s^-1, csa - unitless. Hence both constants have the units of rad^2.s^-2. The constant 'c' is in units of s^-2 also (or rad^2 s^-2 if we use frequencies in rad s^-1, which are not SI units). The constant c is defined by the angular frequency (omega) rather than the frequency (nu). Hence the units of this constant in the relaxation equations, in both SI and CGS units, is always rad^2.s^-2 as well. I now hesitate between 3 views. 1. Should the spectral densities be in SI units (i.e. in s, as the frequencies are in s^-1 and the rates in s^-1) ? relax currently reports the spectral densities in the SI units for angular frequency of radian Hertz. Although often reported as Hertz, the correct unit is radian Hertz. This is again because radians are implied, but this time because we are talking about rotations. The reason is as follows: The spectral density function for isotropic diffusion is J(w) = tm / (1 + (w.tm)^2). The units for w (or omega) is rad.s^-1. Because the product w.tm is unitless the units for tm are in reality s.rad^-1. Hence the units of the spectral density function J(w) is also s.rad^-1. Although not reported in text books such as Cavanagh, you can still see remnants of the radians. For example in that
Re: [bug #9259] Reduced spectral density mapping yielding bad values
Hi, I'll do the patch right away. Sorry for the attachment, I'll put a link next time... Cheers Séb Edward d'Auvergne wrote: Hi, In this IUPAC report, on page 11, the radian unit is described as The units radian (rad) and steradian (sr), for plane angle and solid angle respectively, are described as 'SI supplementary units' [3]. Since they are of dimension 1 (i.e. dimensionless), they may be included if appropriate, or they may be omitted if clarity is not lost thereby, in expressions for derived SI units. This is the part meaning that radians are implied if you are doing anything angular. I don't know what they mean by clarity because by omitting them it complicates things. Maybe you have to be a physicist before you can see this clarity. Séb, would you be able to create a single patch that contains your bug fixes, the changes to the system tests for the reduced spectral density mapping, and with the multiplication by 2pi added back (the first patch removed it), that would be very much appreciated. I can then apply a single patch with a single commit message saying that bug #9259 (http://gna.org/bugs/?9259) has been fixed (by you of course). Thanks, Edward P. S. As a side note, could you provide a link rather than attach a file to a post to a mailing list. Thanks. Because this mailing list is archived in many different internet repositories and because the message is sent out to all those subscribed to this list, the system is not designed to handle large attachments. On 6/19/07, Sebastien Morin [EMAIL PROTECTED] wrote: Hi, I agree quite well with what you say. However, I would have thought that, even if radians are often implied and not discussed, they should be present in the IUPAC reports (see attached file). See, for example, that the Planck constant divided by 2 pi has units of J s, and that the magnetogyric ratio has also units devoid of radians (s^-1 T^-1), and even the Larmor angular frequency has units of s^-1 (without radians). Maybe even the IUPAC treats the radians as implied and doesn't bother with them... I just checked with data from the Lefevre paper (1996) and I can approximately reproduce their data (approximately, since I don't know the exact values and precisions they used for the different constants) when I multiply the frequencies by 2 pi... However, the discrepancy between spectral densities calculated with frequencies multiplied or not by 2 pi is small... so this verification is not really that precise... Here are the values : Res R1R2 NOE J(0) J(wN) J(wH) My values... J(0) J(wN) J(wH) 4 1.8 4.92 0.162 1.23 0.32 0.027-1.50 0.40 0.024 x 2 pi 1.24 0.33 0.024 - Best 142.08 11.16 0.757 3.15 0.42 0.009-3.86 0.51 0.008 x 2 pi 3.18 0.42 0.008 - Best It seems that Ed is right and that radians are part of the units and that the frequencies in Hz should be multiplied by 2 pi (as in the Lefevre paper). Cheers Séb Edward d'Auvergne wrote: Hi, I have thoroughly checked all the units of the physical constants, spectral densities, rotational correlation times, and relaxation rates and have a few important corrections about the units which are used in relax and elsewhere. Please read below for the details and a long story about SI vs. CGS units, frequency vs. angular frequency, and how it all relates to angular momentum. On 6/15/07, Sebastien Morin [EMAIL PROTECTED] wrote: Hi, Here are the different values I obtain for a residue with R1 = 1.1336 +- 0.0851 ; R2 = 12.9336 +- 0.9649 ; and NOE = 0.463921 +- 0.045 J(0) J(wN)J(wH) = = = Here are the results with Leo Spyracopoulos's Mathematica notebook. 4.704231413115747e-9 2.664085520910741e-10 9.48428699657e-12 Here are the results with relax without multiplying frequencies in Hz by 2 pi. 4.6978912534878238e-092.6603551824374712e-10 9.478993207668287e-12 ratio 0.9986522432526923179 0.9985997677461966745 0.9993081880043085706 Here are the results with relax when multiplying frequencies in Hz by 2 pi. 3.4634030539343071e-091.9612804482358541e-10 9.478993207668287e-12 ratio 0.7362314371436068543 0.7361927508863804185 0.9993081880043085706 The very small discrepancy we get (ratio 0.999...) is due to several small differences in the definitions of constants (e.g. gn = -2.7126 in relax and -2.7108 in Leo's notebook). This is not important. However, there is a non negligible difference with the use of either frequencies in Hz or frequencies transformed to rad/s. Now, what do we do with that
Re: [bug #9259] Reduced spectral density mapping yielding bad values
Hi, That is awesome work tracking down this problem. Thank you! I'll apply your patch and then hopefully make a new relax 1.2 release with your fixes very soon. I do have a important question first though. My question relates to the multiplication of the frequency by 2pi to convert from Hz to rad/s units. The symbol for the frequency in Hz is nu whereas the frequency in rad/s is omega. In all the relaxation equations composed of spectral density components, the frequencies are in rad/s and are represented by the omega symbol. This includes the CSA constant defined in SI units as c = (omegaX.CSA)^2/3, where CSA is the chemical shift anisotropy and omegaX = gammaX.B0. To get nuX which is the frequency of the X nucleus in Hz, omegaX measured in rad/s should be divided by 2pi. So my question is, do you get the same results as the Mathematica notebooks of Leo Spyracopoulos if you retain the multiplication of the frequency by 2pi? Thanks, Edward P.S. The problem with the list of frequencies is probably the major issue. I'm not sure why I attempted to fill out the entire list of frequencies as the reduced spectral density mapping code only uses the value in self.data.frq_list[0, 1], the frequency of the heteronucleus, to calculate the CSA constant. The higher frequencies are never used in the calculation. Anyway, your patch fixes this problem. On 6/13/07, Sebastien Morin [EMAIL PROTECTED] wrote: Hi ! I've checked the equations used for reduced spectral density mapping in relax. They're all right... The assumption about the factor of (mu0 / (4pi))^2 is ok since the old equations were written in Gaussian units (cgs) and now we use SI units. However, 2 things seem to be wrong. 1. The frequencies need not to be scaled by a factor of 2 pi since the unit of frequency in the SI is Hz. Thus, line 52 of 'maths_fns/jw_mapping.py' must be removed. 2. The frequency used for calculating the CSA seems not to be the heteronuclear frequency. In fact, there is an error in lines 57 to 60 from 'maths_fns/jw_mapping.py' since the same item in the list is assigned different values one after the other. Changing those lines from : self.data.frq_list[0, 1] = frqX self.data.frq_list[0, 1] = frq - frqX self.data.frq_list[0, 1] = frq self.data.frq_list[0, 1] = frq + frqX to : self.data.frq_list[0, 1] = frqX self.data.frq_list[0, 2] = frq - frqX self.data.frq_list[0, 3] = frq self.data.frq_list[0, 4] = frq + frqX should work. The important thing is that item 1 stays the heteronuclear frequency so it matches with line 1020 of 'maths_fns/ri_comps.py' : data.csa_const_fixed[j] = data.frq_sqrd_list[j, 1] / 3.0 where the constant 'c' is calculated using the squared heteronuclear frequency. With those two modifications, I now get the same values as when calculating manually or using Leo Spyracopoulos's Mathematica notebooks (http://www.bionmr.ualberta.ca/~lspy/index_7.html). Bye ! Sébastien :) Edward d'Auvergne wrote: Hi, For the reduced spectral density mapping in relax, I have used equations 10 to 12 from: Markus M. A., Dayie K. T., Matsudaira P., and Wagner G. Local mobility within villin 14T probed via heteronuclear relaxation measurements and a reduced spectral density mapping. Biochemistry. 1996, 35(6):1722-32. The equations themselves are derived from: Lefevre J. F., Dayie K. T., Peng J. W., and Wagner G. Internal mobility in the partially folded DNA binding and dimerization domains of GAL4: NMR analysis of the N-H spectral density functions. Biochemistry. 1996, 35(8):2674-86. One problem may be that I made the assumption that the dipolar constant of equation 7 of the first reference was missing the factor of (mu0 / (4pi))^2! I based this assumption on the SI units formulation of the R1, R2, and NOE equations and how the CSA constant is defined. I think this is a fairly safe assumption though if you look at equations 1, 2, and 8 of that paper. Could the problem be the definition of the equations used? I've looked at the code in relax and it seems to replicate these equations correctly. Are the equations of Markus et al., (1996) correct? Is my assumption about the dipolar constant correct? If you manually calculate the reduced spectral density values using these alternative equations, does relax produce the same values? I'm sorry that I can't exactly pinpoint the problem, but something is seriously amiss. Regards, Edward On 6/1/07, anonymous [EMAIL PROTECTED] wrote: URL: http://gna.org/bugs/?9259 Summary: Reduced spectral density mapping yielding bad values Project: relax Submitted by: None Submitted on: Friday 06/01/2007 at 17:15 CEST Category: relax's source code Severity: 4 -
Re: [bug #9259] Reduced spectral density mapping yielding bad values
Hi, I made a patch for the test-suite so the spectral density test passes (patch_2007-06-15)... This patch should go with the one for solving the bug that I uploaded yesterday. Cheers Sébastien P.S. I have a question about the test-suite. Should the test-suite files be modified when a patch is sent as an answer to a bug report ? The patch from yesterday made the test-suite fail, I believe it is thus fine to also make a patch for the test-suite. Do you prefer making 2 different patches or a single one with everything in it (as the output from svn diff) ? Edward d'Auvergne wrote: Hi, That is awesome work tracking down this problem. Thank you! I'll apply your patch and then hopefully make a new relax 1.2 release with your fixes very soon. I do have a important question first though. My question relates to the multiplication of the frequency by 2pi to convert from Hz to rad/s units. The symbol for the frequency in Hz is nu whereas the frequency in rad/s is omega. In all the relaxation equations composed of spectral density components, the frequencies are in rad/s and are represented by the omega symbol. This includes the CSA constant defined in SI units as c = (omegaX.CSA)^2/3, where CSA is the chemical shift anisotropy and omegaX = gammaX.B0. To get nuX which is the frequency of the X nucleus in Hz, omegaX measured in rad/s should be divided by 2pi. So my question is, do you get the same results as the Mathematica notebooks of Leo Spyracopoulos if you retain the multiplication of the frequency by 2pi? Thanks, Edward P.S. The problem with the list of frequencies is probably the major issue. I'm not sure why I attempted to fill out the entire list of frequencies as the reduced spectral density mapping code only uses the value in self.data.frq_list[0, 1], the frequency of the heteronucleus, to calculate the CSA constant. The higher frequencies are never used in the calculation. Anyway, your patch fixes this problem. On 6/13/07, Sebastien Morin [EMAIL PROTECTED] wrote: Hi ! I've checked the equations used for reduced spectral density mapping in relax. They're all right... The assumption about the factor of (mu0 / (4pi))^2 is ok since the old equations were written in Gaussian units (cgs) and now we use SI units. However, 2 things seem to be wrong. 1. The frequencies need not to be scaled by a factor of 2 pi since the unit of frequency in the SI is Hz. Thus, line 52 of 'maths_fns/jw_mapping.py' must be removed. 2. The frequency used for calculating the CSA seems not to be the heteronuclear frequency. In fact, there is an error in lines 57 to 60 from 'maths_fns/jw_mapping.py' since the same item in the list is assigned different values one after the other. Changing those lines from : self.data.frq_list[0, 1] = frqX self.data.frq_list[0, 1] = frq - frqX self.data.frq_list[0, 1] = frq self.data.frq_list[0, 1] = frq + frqX to : self.data.frq_list[0, 1] = frqX self.data.frq_list[0, 2] = frq - frqX self.data.frq_list[0, 3] = frq self.data.frq_list[0, 4] = frq + frqX should work. The important thing is that item 1 stays the heteronuclear frequency so it matches with line 1020 of 'maths_fns/ri_comps.py' : data.csa_const_fixed[j] = data.frq_sqrd_list[j, 1] / 3.0 where the constant 'c' is calculated using the squared heteronuclear frequency. With those two modifications, I now get the same values as when calculating manually or using Leo Spyracopoulos's Mathematica notebooks (http://www.bionmr.ualberta.ca/~lspy/index_7.html). Bye ! Sébastien :) Edward d'Auvergne wrote: Hi, For the reduced spectral density mapping in relax, I have used equations 10 to 12 from: Markus M. A., Dayie K. T., Matsudaira P., and Wagner G. Local mobility within villin 14T probed via heteronuclear relaxation measurements and a reduced spectral density mapping. Biochemistry. 1996, 35(6):1722-32. The equations themselves are derived from: Lefevre J. F., Dayie K. T., Peng J. W., and Wagner G. Internal mobility in the partially folded DNA binding and dimerization domains of GAL4: NMR analysis of the N-H spectral density functions. Biochemistry. 1996, 35(8):2674-86. One problem may be that I made the assumption that the dipolar constant of equation 7 of the first reference was missing the factor of (mu0 / (4pi))^2! I based this assumption on the SI units formulation of the R1, R2, and NOE equations and how the CSA constant is defined. I think this is a fairly safe assumption though if you look at equations 1, 2, and 8 of that paper. Could the problem be the definition of the equations used? I've looked at the code in relax and it seems to replicate these equations correctly. Are the equations of Markus et al., (1996) correct? Is my assumption about the dipolar constant correct? If you manually
Re: [bug #9259] Reduced spectral density mapping yielding bad values
Hi ! I've checked the equations used for reduced spectral density mapping in relax. They're all right... The assumption about the factor of (mu0 / (4pi))^2 is ok since the old equations were written in Gaussian units (cgs) and now we use SI units. However, 2 things seem to be wrong. 1. The frequencies need not to be scaled by a factor of 2 pi since the unit of frequency in the SI is Hz. Thus, line 52 of 'maths_fns/jw_mapping.py' must be removed. 2. The frequency used for calculating the CSA seems not to be the heteronuclear frequency. In fact, there is an error in lines 57 to 60 from 'maths_fns/jw_mapping.py' since the same item in the list is assigned different values one after the other. Changing those lines from : self.data.frq_list[0, 1] = frqX self.data.frq_list[0, 1] = frq - frqX self.data.frq_list[0, 1] = frq self.data.frq_list[0, 1] = frq + frqX to : self.data.frq_list[0, 1] = frqX self.data.frq_list[0, 2] = frq - frqX self.data.frq_list[0, 3] = frq self.data.frq_list[0, 4] = frq + frqX should work. The important thing is that item 1 stays the heteronuclear frequency so it matches with line 1020 of 'maths_fns/ri_comps.py' : data.csa_const_fixed[j] = data.frq_sqrd_list[j, 1] / 3.0 where the constant 'c' is calculated using the squared heteronuclear frequency. With those two modifications, I now get the same values as when calculating manually or using Leo Spyracopoulos's Mathematica notebooks (http://www.bionmr.ualberta.ca/~lspy/index_7.html). Bye ! Sébastien :) Edward d'Auvergne wrote: Hi, For the reduced spectral density mapping in relax, I have used equations 10 to 12 from: Markus M. A., Dayie K. T., Matsudaira P., and Wagner G. Local mobility within villin 14T probed via heteronuclear relaxation measurements and a reduced spectral density mapping. Biochemistry. 1996, 35(6):1722-32. The equations themselves are derived from: Lefevre J. F., Dayie K. T., Peng J. W., and Wagner G. Internal mobility in the partially folded DNA binding and dimerization domains of GAL4: NMR analysis of the N-H spectral density functions. Biochemistry. 1996, 35(8):2674-86. One problem may be that I made the assumption that the dipolar constant of equation 7 of the first reference was missing the factor of (mu0 / (4pi))^2! I based this assumption on the SI units formulation of the R1, R2, and NOE equations and how the CSA constant is defined. I think this is a fairly safe assumption though if you look at equations 1, 2, and 8 of that paper. Could the problem be the definition of the equations used? I've looked at the code in relax and it seems to replicate these equations correctly. Are the equations of Markus et al., (1996) correct? Is my assumption about the dipolar constant correct? If you manually calculate the reduced spectral density values using these alternative equations, does relax produce the same values? I'm sorry that I can't exactly pinpoint the problem, but something is seriously amiss. Regards, Edward On 6/1/07, anonymous [EMAIL PROTECTED] wrote: URL: http://gna.org/bugs/?9259 Summary: Reduced spectral density mapping yielding bad values Project: relax Submitted by: None Submitted on: Friday 06/01/2007 at 17:15 CEST Category: relax's source code Severity: 4 - Important Priority: 5 - Normal Status: None Privacy: Public Assigned to: None Originator Name: Sébastien Morin Originator Email: [EMAIL PROTECTED] Open/Closed: Open Discussion Lock: Any Release: Repository: 1.2 line Operating System: GNU/Linux ___ Details: Hi I performed spectral density mapping on data recorded at three magnetic fields (500, 600, 800). The values I get are erroneous (when compared with Leo Spyracopoulos' Mathematica notebook which were manually verified using equations from the method 1 of Farrow et al., 1995, JBNMR, 6 : 153) and scaled depending on the magnetic field as shown in the table below (for which values calculated using either Leo's notebook or relax are divided by the value calculated manually). FieldMethodJ(0) J(wN) J(wH) ==== ====== 500 Farrow1 (ref) 1 (ref) 1 (ref) Leo 1 1 1 relax 0.04758 0.04757 0.999 600 Farrow1 (ref) 1 (ref) 1 (ref) Leo 1 1 1 relax 0.03361 0.03361 0.999 800 Farrow1 (ref) 1 (ref) 1 Leo 1 1 1 relax 0.01932 0.01932
Re: [bug #9259] Reduced spectral density mapping yielding bad values
Hi,
For the reduced spectral density mapping in relax, I have used
equations 10 to 12 from:
Markus M. A., Dayie K. T., Matsudaira P., and Wagner G. Local
mobility within villin 14T probed via heteronuclear relaxation
measurements and a reduced spectral density mapping. Biochemistry.
1996, 35(6):1722-32.
The equations themselves are derived from:
Lefevre J. F., Dayie K. T., Peng J. W., and Wagner G. Internal
mobility in the partially folded DNA binding and dimerization domains
of GAL4: NMR analysis of the N-H spectral density functions.
Biochemistry. 1996, 35(8):2674-86.
One problem may be that I made the assumption that the dipolar
constant of equation 7 of the first reference was missing the factor
of (mu0 / (4pi))^2! I based this assumption on the SI units
formulation of the R1, R2, and NOE equations and how the CSA constant
is defined. I think this is a fairly safe assumption though if you
look at equations 1, 2, and 8 of that paper.
Could the problem be the definition of the equations used? I've
looked at the code in relax and it seems to replicate these equations
correctly. Are the equations of Markus et al., (1996) correct? Is my
assumption about the dipolar constant correct? If you manually
calculate the reduced spectral density values using these alternative
equations, does relax produce the same values? I'm sorry that I can't
exactly pinpoint the problem, but something is seriously amiss.
Regards,
Edward
On 6/1/07, anonymous [EMAIL PROTECTED] wrote:
URL:
http://gna.org/bugs/?9259
Summary: Reduced spectral density mapping yielding bad
values
Project: relax
Submitted by: None
Submitted on: Friday 06/01/2007 at 17:15 CEST
Category: relax's source code
Severity: 4 - Important
Priority: 5 - Normal
Status: None
Privacy: Public
Assigned to: None
Originator Name: Sébastien Morin
Originator Email: [EMAIL PROTECTED]
Open/Closed: Open
Discussion Lock: Any
Release: Repository: 1.2 line
Operating System: GNU/Linux
___
Details:
Hi
I performed spectral density mapping on data recorded at three magnetic
fields (500, 600, 800).
The values I get are erroneous (when compared with Leo Spyracopoulos'
Mathematica notebook which were manually verified using equations from the
method 1 of Farrow et al., 1995, JBNMR, 6 : 153) and scaled depending on the
magnetic field as shown in the table below (for which values calculated using
either Leo's notebook or relax are divided by the value calculated manually).
FieldMethodJ(0) J(wN) J(wH)
==== ======
500 Farrow1 (ref) 1 (ref) 1 (ref)
Leo 1 1 1
relax 0.04758 0.04757 0.999
600 Farrow1 (ref) 1 (ref) 1 (ref)
Leo 1 1 1
relax 0.03361 0.03361 0.999
800 Farrow1 (ref) 1 (ref) 1
Leo 1 1 1
relax 0.01932 0.01932 0.999
Then, if you take the different values for J(0) and J(wN) and compare from
field to field, you get this :
J(0) J(wN) J(wH)
500/600 -1.415 1.415 1
500/800 -2.462 2.462 1
Those ratios are similar to what you get when comparing fields quadratically
:
(600/500)^2 = (1.2)^2 = 1.44 ~ 1.415
(800/500)^2 = (1.6)^2 = 2.56 ~ 2.462
So there seems to be a problem somewhere in the calculations of J(0) and
J(wN) and, to a lesser extent, J(wH)...
I first thought the problem was related with bug #9238... In fact, before
this bug was solved, the problem was worst by a factor of ~2... Still, the
skewing of Jw mapping results is quite important. Maybe is this something
with the units or constants values...
Thanks for helping me !
Sébastien :)
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