Cooling of the universe: Hugh Ross’s pseudo-thermodynamics revisited


On April 26, 2004 I received an email message which is reproduced here in its entirety:

Dr. Perakh, I recently picked up a copy of your Unintelligent Design and wanted to alert you to some errors. As a brief introduction, I am a graduate student in Astronomy and have followed the intelligent design movement for several years. I find debunking such creationist claims an entertaining mental exercise and overwhelmingly approve of the purpose of your book. On pages 173-178, you discuss Hugh Ross’s treatment of thermodynamics in a cosmological setting. You correctly point out that Ross’s treatment is not overwhelming rigorous–however, his claim that the universe cools because it is expanding is correct.

Ross makes the (regrettably) common slip of referring to an adiabatic expansion when he really means a reversible adiabatic expansion. Because it is assumed that the universe remained in thermal equilibrium throughout much of it’s early history, the expansion does indeed cause it to cool. A couple statements of yours that really need to be changed: on p. 177: “According to the second law, the entropy of the universe increases along with the latter’s expansion.” This is a misstatement. The second law says entropy increase OR stays constant. The entropy of the early universe, (and I think the latter universe) is dominated by the background radiation which is expanding isentropically. When measured on cosmological scales, entropy is constant.

On p. 178: “The temperature drop occurs because the initial thermal energy of the universe gradually converts…into the kinetic energy of the motion of those clumps of emerging matter moving away from the initial seed of the universe.” Again, the cooling is indeed caused by the expansion. The end of this statement hints at some muddled thinking about how the expansion proceeds–matter is not flying apart from some initial center in big bang cosmology. I have a hard time picturing how thermal energy could be converted into the proper motion of galaxies–it makes a lot more sense for that motion to come from gravitational collapses and cosmological torques. I believe Ross has very selectively gone through the scientific literature to pick out bits that support his theology, but his science is usually correct.

By saying his statements are “preposterous” and that he doesn’t understand “elementary concepts” when his explanation is much closer to correct than yours greatly undermines your credibility. I would recommend looking into a cosmology text for how the thermodynamics are treated. Peacock’s Cosmological Physics is popular, I tend to fall back on Matt Roos’ Introduction to Cosmology.

Peter Yoachim”

While I disagree with Yoachim’s remarks addressing my critique of Hugh Ross, (and I will show here, why), I appreciate his comments.

The purpose of writing up this brief article is not so much to respond to Yoachim as to clarify fro a wider audience some points raised in his message. If Yoachim seems confused by some points in my book, so may others be, and hence some crude errors in the very popular publications of Ross apparently require a little more detailed explanation. (Besides my book referred to by Yoachim, a similar critique of Ross’s misinterpretation of thermodynamics also is found in my paper titled “Intelligent Design For Dummies,” which was printed in The Skeptic magazine (Australia), vol. 23, No 4, pp 22-27, as well as at TalkReason; this article therefore should serve as a clarifying addition to all three listed sources).

Since, however, this discussion was set into motion by Yoachim’s letter, I’ll briefly address his critique per se.

First, note the ellipsis in his quotation from my book, when he writes “universe gradually converts… into the kinetic energy of the motion of those clumps of emerging matter moving away from the initial seed of the universe.” Using ellipsis is often a device aimed at changing the contents of a quotation to fit the agenda of the critic. It also is referred to as quoting out of context (although, since Yoachim’s was a personal message to me, he obviously did not intend to distort my position).

The full text of the pertinent paragraph in my book makes it clear that I suggested a hypothetical explanation of the universe’s cooling - by stating that it occurred because the thermal energy of the universe’s “seed,” starting at Planck’s time, converted into non-thermal forms of energy. As candidates for those non-thermal forms of energy I suggested several possible alternatives - the rest energy of the emerging masses, the kinetic energy of the moving masses, etc. I did not suggest any specific mechanism of the conversion of thermal energy into non-thermal forms, but based my explanation only on the law of energy conservation.

Unless we choose to discard the law of energy conservation (and I am not categorically rejecting the possibility of such an approach, especially if the discussion relates to cosmology and general relativity) we have to assume that thermal energy of the emerging universe converted into non-thermal forms thus resulting in the universe’s cooling. This is a very general statement not precluding references to any specific phenomena which could be instrumental in the universe’s cooling.

Yoachim writes that he has a hard time imagining the conversion of thermal energy into kinetic energy of masses moving away from each other. This is the argument from ignorance which has no sufficient cognitive weight. We may not know the precise mechanism of the conversion in question, but if we choose to adhere to the law of energy conservation, the mere fact of such conversion is hardly deniable.

Now turn to Yoachim’s defense of Ross, who, in Yoachim’s view, is correct when stating that universe cooled because of expansion. Let us recall exactly how Ross renders his explanation.

On page 135 of his book Creation and Time Ross writes:

As the universe expands from the creation event, it cools, like any other system obeying the laws of thermodynamics. When the heat energy of a system fills a greater volume, there is less energy per unit volume to go around.

As was mentioned in my book, that statement is preposterous. Note that Ross, in this quotation, refers to thermodynamics rather than to cosmology. Ross’s statement does not mention such points as the red shift of photons, cosmic scale factor, or any other points that one may see discussed in the literature on cosmology. His argument is fully within the framework of classical thermodynamics. And as such, it is egregiously wrong.

Indeed, “heat energy” - the term used by Ross which evidently was meant to denote “thermal energy” – is not a substance that can be diluted in a certain volume so that “the greater the volume the less energy density.” Such a view could have sounded plausible until the end of the 18th century, i.e. until Benjamin Thompson (Count Rumford) understood that “heat” is a form of motion. In the 19th century, physics clarified that thermal energy (this concept makes sense only for macroscopic systems) is in fact the kinetic energy of the particles of which the body consists. Cooling of the universe since the big bang has nothing to do with the non-existing effect imagined by Ross - the decrease of energy density when the same amount of “heat energy” spreads over an increasing volume.

Since my task was to show the errors of Ross’s discourse rather than explain why the universe cooled, I could stop discussing this point right here without going into the explanations of the universe’s behavior. Since, though, Yoachim introduced this question in his letter, I’ll briefly relate to it here in a little more detail than I did in my book, my paper in the Skeptic, and in the post on Talk Reason.

First, though, I’ll discuss Yoachim’s remark about entropy. While in my book I wrote that the entropy of the universe increases along with its expansion, Yoachim thinks that entropy remains constant (the 2nd law allows for the entropy in a closed system either to increase or to remain constant).

Let us turn again to classical thermodynamics (because Yoachim cites here the 2nd law of thermodynamics rather than some cosmological argument). Entropy, according to the 2nd law, can indeed remain constant but only either in reversible adiabatic processes or in reversible cycles. Classical thermodynamics asserts, though, that all real processes in macroscopic systems are irreversible. Classical thermodynamics further asserts that in irreversible cycles the net entropy always increases rather than remaining constant. Therefore, whatever happens in the universe, its net entropy constantly increases (remember that this argument is made without entering cosmology). This led Boltzmann to his pessimistic prediction of the imminent thermal death of the universe. Boltzmann did not know about universe’s expansion, which requires a reconsideration of his prediction.

Boltzmann was very instrumental in understanding that entropy is in fact a monotonous function of the number of accessible states. As the universe expands, more accessible states emerge. According to the modern view, at Planck’s time the universe’s entropy had the maximum possible value for the situation existing at that moment - the embryonic universe was completely disordered. As the universe expanded, two tendencies came into play: gravity worked towards creating islands of order thus causing local decreases of entropy, while expansion generated new accessible states thus enabling a continuing increase of the net entropy of the universe above the level it had at Planck’s time.

In regard to the universe’s expansion, is it legitimate to say that expansion caused cooling? It depends on the framework of our discussion. Remember that Ross stays within the framework of thermodynamics - so let us stay within thermodynamics as well when discussing Ross’s statements. Let us recall that there are many known processes where a thermodynamic system expands but temperature does not drop.

Consider conventional systems such as discussed in classical thermodynamics, say, a cylinder with a piston, filled with a gas.

The concept of a system in thermodynamics incorporates as ineliminable parts concepts of a boundary and of the surrounding. The boundary separates the system from the surrounding. The boundary may be adiabatic - that is impervious for a heat transfer to or from the surrounding. It can be diathermal, which is opposite of adiabatic - it allows a free influx to or outflow from the system, of heat. It can be anything in-between.

Assume the gas in the cylinder expands. Does it mean the gas is necessarily cooling down in the process of expansion (as Ross’s assertion implies)? Not at all. The temperature of gas in the course of expansion may increase, remain constant, or decrease. It depends on what happens at the system’s boundaries. If there is heat influx into the system through its boundaries (in this example through the cylinder’s walls) the gas’s temperature may go up despite its expansion. The expanding gas performs work against an external force - in this case the force is the weight of a load placed on the piston - at the expense of the gas’s internal energy. If the amount of work exceeds the heat influx to the gas from the surrounding, gas’s temperature will decrease. This decrease is, though, caused not by expansion as such but by the negative balance of energies: if no work is done, temperature will not go down.

Indeed, imagine a box divided by a partition into two compartments. Fill one compartment with gas at temperature T and pressure P. Keep the other compartment vacuumed (P=0). Rapidly withdraw the partition. Gas will expand into the empty compartment and this will be a very fast process. It results in the system’s entropy increasing. However, gas’s temperature remains the same - expansion into vacuum is an isothermal process.

This completely negates Ross’s ridiculous assertion that “when the heat energy of a system fills a greater volume, there is less energy per unit volume to go around.” (Btw, because of the very high speed of the expansion into vacuum, this process also is adiabatic - which negates another incorrect statement by Ross, as discussed in my book).

In the example with a piston in a cylinder, gas would cool down only if it performed work against external forces - for example against the weight of a load placed on the piston - and if the expenditure of the gas’s internal energy on doing that work is not compensated for by the heat influx from the surrounding. In the case of expansion into vacuum there are no external forces - the empty compartment does not exert any resistance to the expanding gas, so there is no decrease of the gas’s internal energy of which temperature T is a measure - and T does not decrease.

Since expansion itself is not the cause of cooling - there are many processes where a system expands but no cooling occurs - Ross’s assertion, which states that according to the laws of thermodynamics expansion causes cooling, is utterly misleading.

Go back to the universe. Extending classical thermodynamics to the universe requires caution. The concept of a thermodynamic system makes no sense without the concepts of a boundary and of the surrounding. When we talk about the universe these concepts lose their standard interpretation as used in classical thermodynamics. There supposed to be “nothing” beyond the universe - no “surrounding” and therefore no boundaries in a thermodynamic sense. That is why it is often claimed that the universe is a perfect closed system - it does not exchange either matter or energy with a surrounding because there is no surrounding. The universe expands not into vacuum, as gas into an empty compartment, but into nowhere.

If we don’t want to easily discard the law of energy conservation - which would be a rather non-parsimonious approach - we have to attribute the universe’s cooling to the conversion of thermal energy into non-thermal forms of energy. As to specific forms of non-thermal energies which are beneficiaries of the conversion, their relative contributions to cooling, and specific mechanisms of conversion - here is a space for various hypotheses and models.

In cosmology the cooling of the universe is often attributed to the cosmological red shift of photons. (There seem to be various cases of red shift. One is the red shift of light coming from remote galaxies which is usually attributed to Doppler’s effect and obeys Hubble’s law. Another is the gravitational red shift caused by the space-time’s curvature which is due to the presence of large masses. One more is the cosmological red shift caused by the expansion of the space-time per se. General relativity, however, reveals that all three cases, if explained by the properties of the metric tensor, differ only in interpretations based on various approximations and can all in fact be attributed to space-time’s expansion.)

As an illustration, this cooling effect can be juxtaposed with what is called Wien’s displacement law. It relates to the spectra of black body radiation. The dependence is as follows: T= K/λmax

where T is temperature and λmax is the wavelength corresponding to the maximum of the spectrum (i.e. the most common wavelength in the radiation). K=2.898×10^-3 Kelvin-meter is Wien’s constant.

In the course of the universe’s expansion, the wavelength of radiation increases together with the expansion of space-time (that is the red shift mentioned above). This, according to Wien’s law, means a temperature drop. It is in this sense that cosmologists say that expansion causes cooling. However, Ross’s discourse has nothing to do with the red shift mentioned. Ross attributes the universe’s cooling to its expansion, in purely thermodynamic terms (suggesting a ridiculously wrong mechanism). Since expansion of thermodynamic systems, in general, is not necessarily accompanied by cooling, assertions regarding cooling of the universe because of its expansion, in the form Ross makes them, are egregiously wrong.

From another standpoint, pointing to the red shift and the concomitant spectrum’s shift toward lower temperature does not contradict the more general statement which refers to the law of energy conservation and the transformation of thermal energy into non-thermal forms. Therefore, since Ross appealed only to thermodynamics but not to cosmology, and his thermodynamic explanation (of energy diluting in increasing volumes) is absurd, my critique of Ross (which was entirely about his use of thermodynamics) is valid.

Perhaps we might note that the law of energy conservation is not clearly understood within the framework of the general theory of relativity and cosmology. While cosmologists usually have no problem with applying laws of physics, as they are known under the earthly conditions, locally, they face uncertainty when these laws are applied to the universe as a whole (which is connected to the curvature of space-time). The law of conservation of energy is no exception. However, this peculiarity of the cosmological interpretation of the energy conservation law is not negating the explanation of the cooling effect which attributes it to the conversion of thermal energy into non-thermal form. Indeed, this explanation does not require extending the conservation law to the universe as a whole. Acts of conversion occur locally and that is where the law of conservation of energy works in cosmology as it does anywhere else.

Also, the uncertainty related to the extension of the law of energy conservation to the universe as a whole does not mean that the law in question is invalid for the universe as a whole - it only means that there is a possibility that the law in question may need modification if applied to the universe as a whole. Except for some special discussions limited to the narrow points of cosmology, the law of conservation of energy is invariably construed as one of the most fundamental postulates of physics (although the concept of energy itself is one of the least understood concepts in physics).

My main point is, however, that I did not delve into the cosmological problem of the universe’s cooling which would require mobilizing concepts of modern cosmology and general relativity, but only addressed Ross’s abjectly erroneous, purely thermodynamic explanation of the universe’s cooling.

After I shared with Yoachim the above discourse, he sent me the following second message dated May 10, 2004 (reproduced here in full):

That makes your argument much cleaner–now your beef is clearly that Ross went and used classical thermodynamics on the universe without discussing redshift and scale factors and all that good stuff which really needs to be there for the discussion to be even close to correct. You’re welcome to post it if you like, I don’t mind. Keep up the good fight, cheers, Peter.

I have nothing to add to this message, and hopefully the above argument will further clarify my critique of Ross’ errors for those readers who may still need such a clarification.


While I haven’t read Ross’ book, so I don’t know the full context of the “Page 135” remark, I still think it is reasonable to say that “the expansion of the Universe causes its cooling”. As a co-moving volume of the universe expands, the radiation in it has to do work against the pressure of the surrounding radiation; therefore, it cools. If the conservation of energy bothers you, remember that in general relativity the conservation of energy is a local law. In general, there isn’t a global conserved quantity called “energy”. (It might be possible to construct one, using a term analagous to Newtonian gravitational potential energy; I don’t immediately see how to do this for radiation.)

Also, it is true that in general relativity the expansion of the radiation-dominated universe is generally supposed to be isentropic; entropy is only generated when things get out of equilibrium. This gives another way to see that the temperature decrease is a consequence of expansion: an isentropic expansion of a gas must result in a decrease of temperature. I agree that gas expansions don’t *have* to be isentropic, but the radiation in our Universe is believed to have expanded in this way since early times.

Finally, the “easy” way to understand the temperature fall - the stretching out of electromagnetic waves as the Universe expands - also makes explicit the connection between expansion and cooling.

What’s the fuss about? Yoachim’s initial critcisms seem fair to me. (As I’ve said, I’ve no idea what Ross wrote, so I’m not defending him.)

Nigel, what is the radiation whose pressure must supposedly be overcome by tne expanding radiation? There is nothing beyond the universe, so there is no radiation beyond it which can exert pressure. Perhaps you simply did not care to express yourself clearly, but as it is phrased in your comment, it does not seem to make sense. The rest of your comment, IMHO, does not add anything beyond what I have myself stated in my post - in regard to energy conservation etc. Regarding entropy, I stand by my intepretation which is in agreement with thermodynamics; to assert otherwise, you first must show that thermodynamics is wrong. As to Yoachim’s comments, my disagreement with him relates mainly to the question of Ross’s errors - otherwise we are largely in agreement as can be seen from his second message. Since you do not relate to my critique of Ross, what essentially did you want to assert besides the unclear reference to the radiation pressure and reiterating some of the statements I myself made?

A small bug report, in that your link to your paper at talkreason in the above article gives a 404.

Just thought that you might want to fix it.…



Thanks for pointing to the glitch. We still have some problems with the kwickcode. Hopefully it’ll be fixed soon. In the meantime, the link is . Mark


Yep, I get the same 404 message. Until corrected, The article referred to is here I believe.


Re: my previous comment.

Just think of a volume of radiation: any volume, 1 m^3, anywhere in space. The pressure against which it must do work is the pressure of the radiation surrounding it. Since (in space described by the standard Robertson-Walker metric) there is no boundary to the Universe, whether finite or infinite, this argument applies everywhere.

I agree that this seems paradoxical, but I promise you it is very standard stuff. The Einstein field equations, plus assumptions of homogeneity and isotropy, plus a standard coordinate choice, give this equation:

d/dt ( rho a^3 ) + p d/dt ( a^3 ) = 0,

where a can be interpreted as the distance between two co-moving points in the expanding Universe, rho is the energy density and p the pressure of whatever is filling space. Thinking of a^3 as a volume, this equation is saying

“The rate of change of the energy contained in any volume of the Universe” is “minus the rate at which the matter or radiation in that volume is doing work on its surroundings”.

I agree that if we take the volume being considered to the whole Universe then this interpretation becomes less clear, but that’s our problem, not Einstein’s!

Regarding thermodymanics: I certainly don’t think it is wrong! It is simply the case that unless the matter in the expanding Universe gets out of thermal equilibrium the expansion will take place at constant entropy. Of course entropy is being generated here today, and it will be continue to be generated until all the hydrogen now present has fused to iron-56 (or whatever might be the equilibrium state of matter). But in the early Universe the expansion really and truly is believed to be isentropic - thermodynamically reversible. Given this, an increase in the volume of space necessarily requires a decrease in temperature, to keep the total number of states accessible to the Universe constant.

It is my sincere belief that what I am putting forward here is completely standard cosmology dating from the 1930s or so, but still the simple basis for today’s work. I’m not trying to be controversial or to support any creationist ideas (I recognise that you haven’t implied this, but I thought I’d state it explicitly.)

I appreciate Nigel’s elaboration of his earlier comment. It seems that now he agrees that the universe’s entropy is increasing - which was my contention (in fact, it is a rather common view). Even if the radiation’s expansion can be assumed to be isoentropical (I am not discussing here the validity of that assumption), the universe is full of irreversible (i.e. non-equilibrium) processes wherein the net entropy is constantly generated. Since expansion is accompanied by the increase in the number of accessible states, there always is enough space for the overall entropy increase despite the local creation of order by gravity. I’d like to stress once again that my thesis was about Ross’s quasi-thermodynamic interpretation of the cooling (attributing it to the alleged dilution of what he called heat energy in the ever increasing volume). This explanation is patently absurd. Neither in my book nor in the article in Australian Skeptic, nor in the post on Talk Reason did I delve into the cosmological matters and into the real explanation of the universe’s cooling. I touched on these matters (and only very briefly) in my post here only in response to Yoachim’s remarks. In my post above I briefly addressed the cosmological explanation of the universe’s cooling in accordance with the common understanding and not offering any idiosyncratic concepts or iconoclastic views. I mentioned there the uncertainty related to the energy conservation within the framework of general relativity and cosmology (and Nigel for some reasons chose to point to it again as if I expressed some contrary view) etc. I mentioned briefly the conventional reference to the red shift of photons as related to the universe’s cooling, etc. Hence I can only repeat the words of Nigel in his first message: What is this fuss about? Remember this is Panda’s Thumb site where most readers are expected to be interested in evolution-related matters, and thus it is hardly an appropriate forum for the discussion of intricasies of general relativity, with formulas from general relativity, and more so since I touched on them only in passing and very briefly. Hopefully this discussion will terminate at this point. Best wishes, Mark

My cat’s breath smells like cat food.

There is no uncertainty regarding the energy conservation in General relativity. In most cases it simply does not conserve. The law of energy conservation is not a fundamental principle, it follows from Nether’s theorem - if the time is homogeneous, energy is conserved. Generally, in GR this is not the case, the space-time curvature violates time homogeneity. This results in gravitational waves, for which it is not possible to consistently define energy.

Also, what do you mean by the statement that energy is not understood in physics? This is quite new for me and I am PhD. student in physics.

Marian, thanks for the comment. Don’t you think that you have yourself answered your question by providing an example of a situation where (as you wrote) “it is not possible to consistently define energy?” If you can’t even define energy consistently (and I agree with that) does it not mean there is uncertainty? This can be said though not only about the situation you refer to - there is no real definition of energy in physics in general (unless you acquiesce in such a meaningless “definition” as “energy is the ability to do work.” Feynman, in his highly acclaimed Lectures on Physis abstained from giving any definition of energy, saying instead (I am quoting from memory) that energy is “something” that remains constant while everything else changes. This substitute for a definition also illustrates that the law of energy conservation is indeed a generally adopted first postulate (in a logical rather than a chronological context). Even within the framework of classical mechanics the energy conservation law is far from being a simple and straightforward concept. Consider a simple conservative mechanical system. The conservation law asserts that in such a system the sum of kinetic and potential energy is constant. However, this sum has no definite value. Kinetic energy depends on the frame of reference, potential energy depends on the choice of the zero energy location. Then what is that constant value which is conserved? It is undefined. Today it is common to read in discussions of the history of the universe etc that the net energy of the universe is zero (see, for example the popular book by Hawking)- and this implies the conservation law regardless of any theorems. Otherwise, since I wrote that within GTR the conservation law is believed to work locally but there is uncertainty in its application to the university as a whole because of the space-time curvature, it seems to me that, apart from the above points, you just say again the same thing I wrote. Going deeper into this discussion will be IMHO inappropiate for this forum. Cheers, Mark

Mark, just a last comment. Trust me, the energy conservation law is not a first principle. Whether the energy can be defined and if it is conserved follows from the definition of the system (typically Lagrangian or Hamiltonian). This is very similar to momentum, angular momentum, etc. If there is a symmetry of the system, as time homogeneity, a conserved quantity emerges. This can be considered as the definition of the quantity, in our case energy. There are systems, where the energy is not conserved or, as in GTR, it cannot be globally defined. However, it is perfectly ok and there is no uncertainty about it.

There is much more uncertainty about applying thermodynamics on the universe and I would be very cautious myself to make any statements without taking a deeper look on the subject. However, what Peter Yoachim says looks solid, at the end he is an astronomer, so I would take his advice more seriously. He is definitely right with the universe cooling issue, it’s certainly cooling because of the expansion and not because thermal energy would be somehow converted. The curvature of the space-time changes, so you cannot apply the energy conservation law and this change is what is responsible for the lenghtening of the photon waves.

Hi, Marian: Thanks for your comments and for your evident desire to educate me. Regarding your appeal to trust you, I have no reason not to trust you but may still disagree with you. You refer to theoretical concepts. This is fine, but sometimes theoreticians tend to forget that physics is after all an experimental science. The law of energy conservation was introduced first as a postulate based on observed and experimentally obtained data and it is still viewed as the most fundamental and most seminal postulate in physics (which is not just my personal opinion but rather a commonly accepted notion). All those abstract theoretical interpretations you refer to are of a much later origin. There is still no real definition of energy, its behavior is still not fully understood, and all those beautiful theoretical constructions you mention, while highly satisfactory for theoreticians, often only create an illusion of clarifying the matter and do not really shed light on the fundamental understanding of the matter. Theoreticians are sometimes prone to overevaluate their mental constructs, viewing formulas as the reality itself (although I like and admire theoretical physics). Sometimes simpler notions are more revealing than sophisticated exercises of the high priests of science. Regarding Yoachim’s remarks, I think we have clarified the points of disagreement with him, and the fact that he is a graduate student in astronomy and you are a PhD student (apparently in theoretical physics?) while impressive, is as irrelvant as the fact that I have taught physics on various levels (including various parts of theoretical physics) for more than half a century and published nearly 300 papers - appeals to authority have little weight. Despite all our degrees and records of achievements, we all are prone to err. Finally, your point about the caution necessary when applying thermodynamic to the universe sounds odd given the fact that this is precisely what I wrote - so why do you want to warn me against doing something which I myself warned against?

I’d like to repeat that this discussion is IMHO improper for PT, so if you still feel a desire to educate me, please send you thought to my personal email address. Cheers, Mark

Found this link while searching Google, thanks

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This page contains a single entry by Mark Perakh published on May 14, 2004 1:02 PM.

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