• Quick note - the problem with Youtube videos not embedding on the forum appears to have been fixed, thanks to ZiprHead. If you do still see problems let me know.

Lambda-CDM theory - Woo or not?

Status
Not open for further replies.
More dishonesty from you, Michael. I linked to two posts, not one, and anyone can follow those links to see for themselves the pages and pages of arguments by you where you insist that dark energy and dark matter are real and need to be accounted for when calculating the density of the sun.

My Dear D'rok. I think you need to do more reading, particularly my handle of ManInTheMirror over at Baut. My position on these issues has been entirely clear for many years. IMO you're taking a couple of posts related to a single conversation and trying to build a federal case over the issue. At no time have I been a fan of "dark energy", or "dark matter", or inflation. At times however I have to use terms that are consistent with the mainstream, even if I don't agree with them about the nature of these issues.

The only thing dishonest going on here is this stupid series of personal attacks, and conversations being taken out of context. IMO "dark energy" is composed of "current flow" and the effects of the EM field on plasma, nothing particularly exotic. You're more or less ignoring that issue only to misrepresent my overall position. I assure you I've consistently been the worlds biggest internet critics of mainstream theory, particularly their reliance upon make believe entities.

I really suggest you put things in context now with an "electric sun" model and recognize where *I* at least stand on the notion of acceleration of plasma. IMO that observation is specifically and directly related to the current flows between the sun and the heliosphere, and it's not "dark", it actually heats plasma to millions of degrees.
 
My Dear D'rok. I think you need to do more reading, particularly my handle of ManInTheMirror over at Baut. My position on these issues has been entirely clear for many years. IMO you're taking a couple of posts related to a single conversation and trying to build a federal case over the issue. At no time have I been a fan of "dark energy", or "dark matter", or inflation. At times however I have to use terms that are consistent with the mainstream, even if I don't agree with them about the nature of these issues.

The only thing dishonest going on here is this stupid series of personal attacks, and conversations being taken out of context. IMO "dark energy" is composed of "current flow" and the effects of the EM field on plasma, nothing particularly exotic. You're more or less ignoring that issue only to misrepresent my overall position. I assure you I've consistently been the worlds biggest internet critics of mainstream theory, particularly their reliance upon make believe entities.

I really suggest you put things in context now with an "electric sun" model and recognize where *I* at least stand on the notion of acceleration of plasma. IMO that observation is specifically and directly related to the current flows between the sun and the heliosphere, and it's not "dark", it actually heats plasma to millions of degrees.
Squirm all you like Mozina. The evidence is there, plain as day, in your own words. Not merely in the two posts I quoted, but in literally dozens of your posts on page after page of that thread.

Your behaviour is indistinguishable from that of a shameless intellectual fraud. Your sole value is to troll Tim and others into explaining complicated scientific concepts in layperson terms for those of us (not you) who care about learning.
 
Squirm all you like Mozina. The evidence is there, plain as day, in your own words. Not merely in the two posts I quoted, but in literally dozens of your posts on page after page of that thread.

So what? I also use the term "photosphere" when I have no faith at all that it's actually "opaque". At times one simply has to use terms that are consistent with the mainstream even if one doesn't believe they are accurate terms.

Your behaviour is indistinguishable from that of a shameless intellectual fraud. Your sole value is to troll Tim and others into explaining complicated scientific concepts in layperson terms for those of us (not you) who care about learning.

I'm sorry, but the pure denial on your part about the qualification problems with "dark" junk borders on intellectual fraud and you're it's biggest victim. There's nothing wrong with admitting the weaknesses of any and all theories and everything wrong with living in pure denial of those weaknesses.

Not a single one of you has even acknowledge the gross underestimation of normal baryonic matter! The whole religion is based on pure denial (of fact) and "faith" in the "unseen".
 
Ben probably doesn't have any invisible friends. Remember where I reminded you that when you suggest that, you're lying? Michael, you are lying.

You're the only one lying here and you're only lying to yourself. Your dark buddies do not exist except in your head. They have no physical effect on anything here on Earth or anywhere else. Whatever the physical "cause" of events in spacetime, it has nothing to do with "dark" stuff or your goofy inflation or negative pressure in vacuums. Your whole lambda-cdm religion is cobbled together with invisible friends, all of which fail to show up in a lab. Any claim to the contrary is a pure lie on your part.
 
You're the only one lying here and you're only lying to yourself. Your dark buddies do not exist except in your head. They have no physical effect on anything here on Earth or anywhere else. Whatever the physical "cause" of events in spacetime, it has nothing to do with "dark" stuff or your goofy inflation or negative pressure in vacuums. Your whole lambda-cdm religion is cobbled together with invisible friends, all of which fail to show up in a lab. Any claim to the contrary is a pure lie on your part.


Oh, another tantrum.

:dl:

Your qualifications to communicate in a sane, rational, honest, or intelligent manner on the subject of physics has been challenged. You've made thousands of posts and not once in all those words have you been able to demonstrate that you do indeed possess any such qualifications.
 
Not a single one of you has even acknowledge the gross underestimation of normal baryonic matter!
What are you talking about?

I asked you this earlier (more than once?), but you didn't respond.

For example:
DeiRenDopa said:
Michael Mozina said:
How long did you intend to ignore that information about the amount of "dust" in the universe in terms of your mass estimation techniques?
Huh? Perhaps you could remind us all of what you are referring to?

Dust is a very minor component of the mass-energy of the universe; even if the estimates were out by a factor 10, they'd likely be no more than a small fraction of the uncertainty in the estimate of the total baryonic component.

How long did you intend to ignore that information about the gross *UNDERESTIMATION* of smaller stars in a given galaxy compared to the larger ones we can observe?
Huh? Perhaps you could remind us all of what you are referring to?

Stars - indeed whole galaxies - comprise only a modest fraction of the estimated total baryonic mass of the universe (most of it is in the WHIM, and the plasma which pervades clusters of galaxies), so even doubling their estimated total mass would barely register, in terms of the estimated uncertainty of the baryonic component.

How long did you simply intend to ignore that information DRD? Forever? What need do I have for "exotic matter" when I know for a fact you *GROSSLY* underestimate the normal mass of a galaxy and I know for a fact you've done nothing about it for years?
(bold added)

How do you "know" this, MM?

I have "underestimated the normal mass of a galaxy"? What are you talking about?

Care to answer now?

Again, it's impossible for me (and I'm sure others who've posted here too) to "acknowledge" something which only you seem to understand (and which, despite many requests for clarification and explanation, you continue to not try to help us understand).
 
I'm sorry, but the pure denial on your part about the qualification problems with "dark" junk borders on intellectual fraud and you're it's biggest victim. There's nothing wrong with admitting the weaknesses of any and all theories and everything wrong with living in pure denial of those weaknesses.
What "qualification problems"?

If you don't explain what you mean, how do you expect anyone to have a meaningful dialogue with you?

OK, I admit that any and all theories have weaknesses; happy now? No, I didn't think so; what you're looking for is an acknowledgement that "any and all theories" have the specific "weaknesses" that you think you've identified. Trouble is, no one can understand what these are (other than you, of course). Why is that?
 
Curious.

There are days when this thread has added several pages to its considerable length.

And others where nothing is posted at all.

There seems to be just one JREF member trying to make a case that "Lambda-CDM theory" is scientific woo - that's MM - I wonder why that is? Didn't there used to be several who not only felt it was, but who were also not shy about saying so? And, IIRC, there were not all proponents of "EU/PC theory" (whatever that might be).
 
So what? I also use the term "photosphere" when I have no faith at all that it's actually "opaque".

1) It's opaque by definition. Which means that you're wrong by definition, which is as wrong as you can possibly be.
2) It's observably opaque.

At times one simply has to use terms that are consistent with the mainstream even if one doesn't believe they are accurate terms.

The problem is not that you don't believe, the problem is that you don't even understand.
 
Dark Matter and Ultra Faint Dwarf Galaxies II

Not a single one of you has even acknowledge the gross underestimation of normal baryonic matter!
Does this count as a deliberate lie? Of course it is not true. We have already seen this from DRD...

What are you talking about? I asked you this earlier (more than once?), but you didn't respond.

For example: ... Care to answer now?
Follow the link to DRD's post to see the omitted material.

But I also responded directly to Mozina on this very issue.

http://arxiv.org/abs/1002.0504

FYI, they found some more of that "missing mass" in our own galaxy, and oh ya, it's "baryonic material" we just never identified until now, not some new and exotic type of mass. Get used to that scenario. It's happens a lot and it's going to keep happening.
To which I responded ...
Your ignorance is showing again.

First, the paper in question:
Big fish, small fish: Two New Ultra-Faint Satellites of the Milky Way
Belokurov, et al., accepted for publication in Astrophysical Journal Letters
Abstract: We report the discovery of two new Milky Way satellites in the neighboring constellations of Pisces and Pegasus identified in data from the Sloan Digital Sky Survey. Pisces II, an ultra-faint dwarf galaxy lies at the distance of ~180 kpc, some 15 degrees away from the recently detected Pisces I. Segue 3, an ultra-faint star cluster lies at the distance of 16 kpc. We use deep follow-up imaging obtained with the 4-m Mayall telescope at Kitt Peak National Observatory to derive their structural parameters. Pisces II has a half-light radius of ~60 pc, while Segue 3 is twenty times smaller at only 3pc.

Now, our Milky Way has a baryonic mass no more than 1011 solar masses and a total virial mass of about 1012 solar masses, so the baryons constitute only about 10% of the total mass of the Galaxy, assuming of course that we are using the right law of gravity (Xue, et al., 2008). So if the "dark matter" is made up of all these ultra faint systems, they have to cover 90% of the mass. But of course, the reason they are ultra faint is that they are ultra small, meaning ultra low in mass. The Belokurov, et al., paper cited by Mozina compares Picsces II to other dwarf galaxies, Leo IV & Leo V. Those two show dynamical masses of 1.5x106 solar masses & 3x105 solar masses respectively (Moretti, et al., 2009, Walker, et al., 2009). But these masses could be to low because they are derived from stellar samples that may reveal only the central mass (the motions of stars can only reliably determine the dynamical mass at lesser, but not greater distance from the center of the system). It has been proposed that the combination of Leo IV + Leo V could be as high as 8x109 solar masses (de Jong, et al., 2009), but this does require an abnormally high mass to light ratio. Segue 3, on the other hand, carries only a "few tens of stars", as the Belokurov, et al., paper tells us, and so is insignificant.

It should come as no surprise to anyone that smaller, less massive systems are dimmer, and larger, more massive systems as brighter. So we face the dilemma of 10% of the stellar mass (i.e., the baryonic mass of the Galaxy) being hugely bright, while 90% of the stellar mass is in such ultra faint systems that they are barely visible at all. I take it the unbiased reader can understand that this is a hard thing to believe.

Meanwhile, I noted above that "Your ignorance is showing again". In this case, specific ignorance of the "missing satellites problem":
Where Are the Missing Galactic Satellites?
Klypin, et al., The Astrophysical Journal, Volume 522, Issue 1, pp. 82-92; September 1999
Abstract: According to the hierarchical clustering scenario, galaxies are assembled by merging and accretion of numerous satellites of different sizes and masses. This ongoing process is not 100% efficient in destroying all of the accreted satellites, as evidenced by the satellites of our Galaxy and of M31. Using published data, we have compiled the circular velocity (Vcirc) distribution function (VDF) of galaxy satellites in the Local Group. We find that within the volumes of radius of 570 kpc (400 h-1 kpc assuming the Hubble constant h=0.7) centered on the Milky Way and Andromeda, the average VDF is roughly approximated as n(>Vcirc)~55+/-11(Vcirc/10 km s-1)-1.4+/-0.4 h3 Mpc-3 for Vcirc in the range ~10-70 km s-1. The observed VDF is compared with results of high-resolution cosmological simulations. We find that the VDF in models is very different from the observed one: n(>Vcirc)~1200(Vcirc10 km s-1)-2.75 h3 Mpc-3. Cosmological models thus predict that a halo the size of our Galaxy should have about 50 dark matter satellites with circular velocity greater than 20 km s-1 and mass greater than 3x108 Msolar within a 570 kpc radius. This number is significantly higher than the approximately dozen satellites actually observed around our Galaxy. The difference is even larger if we consider the abundance of satellites in simulated galaxy groups similar to the Local Group. The models predict ~300 satellites inside a 1.5 Mpc radius, while only ~40 satellites are observed in the Local Group. The observed and predicted VDFs cross at ~50 km s-1, indicating that the predicted abundance of satellites with Vcirc~50 km s-1 is in reasonably good agreement with observations. We conclude, therefore, that unless a large fraction of the Local Group satellites has been missed in observations, there is a dramatic discrepancy between observations and hierarchical models, regardless of the model parameters. We discuss several possible explanations for this discrepancy including identification of some satellites with the high-velocity clouds observed in the Local Group and the existence of dark satellites that failed to accrete gas and form stars either because of the expulsion of gas in the supernovae-driven winds or because of gas heating by the intergalactic ionizing background.

So we see that one of the problems that has faced the Lambda-CDM concordance cosmology has long been that we see far fewer dwarf satellite galaxies than we expected to see.

Fast forward to ...
The Kinematics of the Ultra-faint Milky Way Satellites: Solving the Missing Satellite Problem
Simon & Geha, The Astrophysical Journal, Volume 670, Issue 1, pp. 313-331; November, 2007
Abstract: We present Keck DEIMOS spectroscopy of stars in eight of the newly discovered ultra-faint dwarf galaxies around the Milky Way. We measure the velocity dispersions of Canes Venatici I, Canes Venatici II, Coma Berenices, Hercules, Leo IV, Leo T, Ursa Major I, and Ursa Major II from the velocities of 18-214 stars in each galaxy and find dispersions ranging from 3.3 to 7.6 km s-1. The six galaxies with absolute magnitudes MV<-4 are highly dark matter dominated, with mass-to-light ratios approaching 1000 Msolar/Lsolar,v. For the fainter galaxies we find tentative evidence for tidal disruption. The measured velocity dispersions of the ultra-faint dwarfs are correlated with their luminosities, indicating that a minimum mass for luminous galactic systems may not yet have been reached. We also measure the metallicities of the observed stars and find that the new dwarfs have mean metallicities of [Fe/H]=-2.0 to -2.3 these galaxies represent some of the most metal-poor stellar systems known. The six brightest of the ultra-faint dwarfs extend the luminosity-metallicity relationship followed by more luminous dwarfs by a factor of ~30 in luminosity. We detect metallicity spreads of up to 0.5 dex in several objects, suggesting multiple star formation epochs. UMa II and Com, despite their exceptionally low luminosities, have higher metallicities that suggest they may once have been much more massive. Having established the masses of the ultra-faint dwarfs, we re-examine the missing satellite problem. After correcting for the sky coverage of the Sloan Digital Sky Survey, we find that the ultra-faint dwarfs substantially alleviate the discrepancy between the predicted and observed numbers of satellites around the Milky Way, but there are still a factor of ~4 too few dwarf galaxies over a significant range of masses. We show that if galaxy formation in low-mass dark matter halos is strongly suppressed after reionization, the simulated circular velocity function of CDM subhalos can be brought into approximate agreement with the observed circular velocity function of Milky Way satellite galaxies.

Note from the abstract above: Having established the masses of the ultra-faint dwarfs, we re-examine the missing satellite problem. After correcting for the sky coverage of the Sloan Digital Sky Survey, we find that the ultra-faint dwarfs substantially alleviate the discrepancy between the predicted and observed numbers of satellites around the Milky Way, but there are still a factor of ~4 too few dwarf galaxies over a significant range of masses. Far from being a problem for dark matter, as Mozina says, the discovery of ultra faint dwarf galaxies like this is in fact a big deal in confirming that dark matter cosmology is consistent with observations, by bringing the number of observed dwarf satellite galaxies up to the number expected from theory. As noted in the Simon & Geha paper, there remained a significant deficit of observed versus expected galaxies. But as we are able to see deeper & dimmer, we continue to discover more ultra faint dwarf galaxies, closing the gap between observation and theory.
Get used to that scenario. It's happens a lot and it's going to keep happening.
That's right Mozina, get used to it. The more of these ultra faint dwarf systems we find, the better dark matter cosmology looks and the worse you look.

This exchange took place early last February, on another thread: Michael Mozina's thread on Dark Matter, Inflation and Cosmology. Mozina constantly ignores other people's arguments, and then goes on to arrogantly & dishonestly claim that "not one of you has ...".

Let me make it clear for the record, if it is not already clear, that Mozina's claims about the mainstream underestimate of baryonic mass are not at all true, not even close. It's all in his dreams, but not in the truth. At most the underestimate is on the order of a few percent, and it is constantly acknowledged in the mainstream literature, where it is in fact an active topic of research (e.g., Geha, et al., 2006; Sinha & Holley-Bocklemann, 2010). But the dynamical mass estimates, the ones that include dark matter, show that the visible baryonic matter is out-massed by a factor of 10 by the dark matter. There is no way Mozina can even begin to justify any claim that the baryonic mass estimate for dwarf systems can be off by an entire order of magnitude.

It's all bluff, bluster and arrogant swaggering. It would not be quite so bad if a little basic honesty were included.
 
Dark Matter and Science II.5

For instance, as I have pointed out before, there is a dispute amongst scientists over the correct interpretation of the data from the XENON100 WIMP dark matter detection experiment. (e.g., ).
Of course, there was supposed to be something besides "(e.g., )" in the quote above. Sorry about that. See my earlier post XENON100 and Dark Matter: The Real Story for references & links to the dispute over the correct interpretation of the XENON100 experimental results.
 
Dark Matter and Science V

I am carrying the conversation from the religious scientists thread in an attempt to restore that thread to its OP and move this topic back to a relevant thread where it belongs.

Today's new and improved brand of "dark matter" is evidently based on SUSY theory (a non mainstream particle physics theory), a wing and a prayer. :)
Stupid.

To begin with, the modern idea of dark matter is not based on SUSY, it is based on the observation that normal matter is nowhere seen to be sufficient to account for the observed effect of gravity. It is the very empirical act of observation which leads to the conclusion that nothing already known can account for what is observed. It's all based on observation.

Now, once we have eliminated everything that we know about, it's time to consider the things we guess about. Here it must be pointed out that "SUSY" is an acronym for supersymmetry, which contrary to Mozina's claim, is very much a mainstream theory of particle physics (230 (as of now) papers with "SUSY" in the title 2009-2010; 1221 papers with "supersymmetry" in the title, 2009-2010; that's a lot of coverage for something that is not mainstream). Supersymmetry was introduced into particle physics in 1971 by Pierre Ramond and the team of John Schwarz & André Neveu, solving the fermion problem in string theory (see my TLA award winning post String Theory: An Historical Narrative, which qualifies me to be a "Muse"). The idea has become a force in mainstream physics, leading to several proposed modifications to the standard model of particle physics, and supergravity, which resulted from applying supersymmetry to general relativity. The attentive reader must remember that ignorance is Mozina's calling card, and this is just one more in a long list of examples of Mozina not knowing what he is talking about.

So, what does SUSY have to do with dark matter? Easy. We already know that no form of matter that we know about will solve the missing mass or dark matter problem. Therefore the obvious choice is a form of matter that we don't know about, but have reason to believe exists. What might that be? SUSY predicts that the particles we know about should have high mass counterparts called supersymmetric partners. None of these have been observed yet, but there are active research programs looking for them. Now, Mozina wants you to think that the fact they have not been observed is proof that they don't exist, can't exist, and we should not consider such things in any explanation of dark matter. I think That's a stupid idea, extremely contrary to the very basis of science, which is supposed to be directed by reason, not by prejudice. If what you know fails, why not appeal to what you don't know? How in the world is that supposed to be unscientific? Or contrary to empiricism?

I guess you guys just don't like to admit that the term "I don't know" is a valid scientific answer.
Quite the opposite. Everybody working on the dark matter & dark energy problems know very well that they don't know what it is. All anyone has to do is read a paper or two and it's right there. In fact, "we don't know" is why supersymmetric particles are the leading candidate for dark matter. We don't know what dark matter is, myself included, and that's a fact acknowledged by everyone you choose to insult by claiming they say otherwise. What you are too ignorant to realize is that while we don't know what dark matter is, we do know what it is not. It is not normal matter, a fact settled by observation, and that's why your railing against it comes across as hypocritical; you seem to think that observations are important, and yet you always reject the observations you don't like. Why is that? Is it because those observations challenge your religious doctrine of faith?

I have already addressed Mozina's multiple incompetencies in dealing with dark matter in many posts:

Dark Matter and Science IV
Dark Matter and Science III
Dark Matter and Ultra Faint Dwarf Galaxies II
Dark Matter and Science II
XENON100 and Dark Matter: The Real Story IV
XENON100 and Dark Matter: The Real Story III
XENON100 and Dark Matter: The Real Story II
XENON100 and Dark Matter: The Real Story I
Dark Matter, Dark Energy, Inflation and Real Science
Dark Matter: Direct Detection?
Dark Matter and Ultra Faint Galaxies

Mozina is completely wrong about dark matter, completely wrong about dark energy, completely wrong about magnetic reconnection, completely wrong about any iron surface on the sun, completely wrong about what constitutes empirical science, just plain completely wrong about everything. And I suspect that the vast majority of all you "lurkers" out there have figured that out for yourself. In any case, I stand by what I have posted, and I note that Mozina has never been able to muster an objective argument against any of it.
 
Last edited:
Dark Energy and Empirical Science II

Dark energy is a figment of your imagination. It doesn't "cause" anything to happen, not ever. That is part of your religious dogma, it is not something that you can empirically demonstrate. Your "dark energy" deity is empirically impotent.
Mozina said that after reading this ...
Because "dark energy" is by definition the "anything" which is responsible for the accelerated expansion of the universe. If it is a problem with GR, then "dark energy" is that problem with GR. If it is electromagnetism, then "dark energy" is that electromagnetism. If it is a repulsive term in Einstein's equations, then "dark energy" is that repulsive term. How can you be so dense as to fail to understand such a simple aspect of ordinary English?
Definitions are hardly a figment of the imagination, so one can only wonder at the tortured process that passes for "thinking" in the mind of Mozina.

I have addressed Mozina's incompetencies regarding dark energy in many posts ...

Dark Energy Defined
Empirical Evidence for Dark Energy II
Empirical Evidence for Dark Energy
Accelerating Universe & Gravity
Cosmological Constant III
Dark Energy and Empirical Science
Dark Matter, Dark Energy, Inflation and Real Science
Dark Energy is not Classical Electromagnetism VI
Dark Energy is not Classical Electromagnetism V
Dark Energy is a Cosmological Constant IV
Dark Energy is a Cosmological Constant III
Dark Energy is a Cosmological Constant II
Inflationary Cosmology is Real Science
Dark Energy is a Cosmological Constant
Dark Energy is not Classical Electromagnetism III
Dark Energy is not Classical Electromagnetism II
Dark Energy is not Classical Electromagnetism
Inflationary Cosmology and Science

Dark energy, like dark matter, is an unknown in cosmology. We know it is there because we can see how it effects the universe in the form of an accelerated expansion, but we don't know what it is. However, just as is the case for dark matter, even if we don't know what it is, we know what it is not. Physics recognizes the existence of 4 and only 4 fundamental forces: gravity, electromagnetism, the weak force and the strong force (with the caveat that general relativity does not recognize gravity as a genuine force, treating it as geometry). So which of these 4 is dark energy? It can't be either the weak or strong force, they are extremely short range forces that show up only over nuclear distances, roughly 10-15 meters. It can't be gravity as we know it because gravity is attractive not repulsive. And it can't be classical electromagnetism, as illustrated in my posts above (electromagnetic fields are not at all "dark" in any sense). At once we see that all of the known forces are ruled out by observation.

Does Mozina know what causes the accelerated expansion? No, he does not. Do I know what causes the accelerated expansion? No, I do not. And I have seen no published paper or preprint, or reference to one, which provides a definitive explication of the cause of the accelerated expansion. However, I do know that the bulk of the published evidence shows that the observed accelerated expansion is consistent with a cosmological constant term in Einstein's equations (see my posts listed above). Does Mozina know what a cosmological constant really is? No, he does not. Do I know what a cosmological constant really is? No, I do not. However, I do know that the cosmological constant is, curiously, both cosmological and constant.

Mozina thinks that dark energy is a figment of the imagination, but at the same time agrees that there is an accelerated expansion, and that the accelerated expansion must have a cause. But the cause is, by definition, "dark energy", so Mozina contradicts himself, claiming that the cause which he knows to exist must be a figment of imagination.

The real truth, of course, comes in a few parts: (1) dark energy is entirely empirical in concept, (2) dark energy is most consistent with a cosmological constant, (3) dark energy is not any of the 4 known forces. If Mozina thinks that dark energy is in fact one of the 4 known forces, then let him say which one, and explain why anybody should believe him. Otherwise, it's just one more in a long line of failures for Mozina and his pseudoscience.
 
... And I suspect that the vast majority of all you "lurkers" out there have figured that out for yourself. ...


Tim, I can only speak for myself but in my case you are absolutely correct.

And I would like to add that I have greatly enjoyed your posts. Your writing skill is exceptional and you have a wonderful ability to write about extremely technical topics in language that even we "lurkers" can comprehend. (Well, I will admit that I do not understand most of the higher mathematics but I have no trouble understanding the broader points you are making).

Your posts along with those of the many other knowledgeable posters here are greatly appreciated by this lurker.
 
Status
Not open for further replies.

Back
Top Bottom