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# Isaac Held's Blog

## 20. The moist adiabat and tropical warming

Results from a high resolution model of horizontally homogeneous radiative-convective equilibrium, Romps 2011.  Left: equilibrium temperature profiles for 3 values of CO2 compared to an observed tropical profile.  Right: the temperature differences compared to the response to doubling CO2 in an ensemble of CMIP3 global climate models.

As a moist parcel of air ascends it cools as it expands and does work against the rest of the atmosphere.  If this were the only thing going on, the temperature of the parcel would decrease at 9.8K/km.  But once the water vapor in the parcel reaches saturation some of this vapor condenses and releases its latent heat, compensating for some of the cooling (you get about 45K of warming from latent heat release when a typical parcel rises from the tropical surface to the upper troposphere).   A warmer parcel contains more water vapor when it becomes saturated, so it condenses more vapor as it rises, and temperature decreases with height more slowly.  That is, the moist adiabatic lapse rate, $- \partial T/\partial z$, decreases with warming.

To say something about the warming of the tropical atmosphere, rather than that of a moist adiabat, we need to argue that the tropical troposphere is close to a moist adiabat and remains close as it warms.  The upper troposphere will then warm more than the lower troposphere.  This is precisely what happens in our global climate models. The consistency or inconsistency of this prediction with observations, particularly the Microwave Sounding Unit (MSU) temperatures, is a long-standing and important issue  A failure of the upper troposphere to warm as much as anticpated by this simple argument would signal a destabilization of the tropics — rising parcels would experience a larger density difference with  their environment, creating more intense vertical accelerations — affecting all tropical phenomena involving deep convection.  I like to refer to warming following the moist adiabat as the most “conservative” possible — having the least impact on tropical meteorology.

(One sometimes sees the argument that a consequence of smaller upper tropospheric warming in the tropics would be lower climate sensitivity, since a large fraction of water vapor feedback originates in this region, and the large vapor increase could not occur without the temperature increase.  But this is not the case, because of the cancellation between negative lapse rate and positive water vapor feedbacks produced by upper tropospheric warming.  In fact, the negative lapse rate feedback is generally the larger of the two, so a weaker upper level tropical warming would probably increase climate sensitivity a bit, holding everything else fixed.)  [This statement continues to be a source of confusion -- sorry -- see discussion in the comments. The water vapor feedback that I am referring to here is the part associated with the lapse rate change, if relative humidity is fixed, after subtracting off the part due to uniform warming of the troposphere (IH- Jan 26, 2013)]

The tropical atmosphere, and models of moist radiative-convective equilibrium, are dominated by concentrated saturated updrafts taking up a small fraction of the total area, with the rest of the flow experiencing very slow compensating subsidence.  The behavior of such a skewed flow field can be counterintuitive. The picture that most of us have, I think, is that within the convective updrafts themselves the temperature profile takes its moist adiabatic value; this profile is then communicated efficiently to the rest of the tropics, since the atmosphere is unable to maintain substantial horizontal temperature gradients within the tropics.  Horizontal gradients in pressure and temperature, above the boundary layer, are flattened by wave propagation rather than by mixing, a fundamentally different process than the homogenization of entropy in a dry convecting layer.

I remain somewhat confused as to how best to translate this picture into a scaling argument for how hard one has to push the tropical atmosphere to create a given departure from the moist adiabat.  Arguments of the type summarized in Emanuel, Neelin, and Bretherton, 1994 suggest that attempts to alter the free tropospheric temperature profile will modify temperatures by a rather indirect path — heating perturbations will modify the circulation in a way that then modifies the temperature and humidity of the air near the surface, which finally puts you on a different moist adiabat.  The calculations by Kuang 2010 in which a dynamic model of radiative-convective equilibrium is perturbed systematically with different heat (and moisture) sources suggests that this is the path of least resistance for a convecting atmosphere, but that there are other possibilities as well — I need to understand this paper better.

It doesn’t take much of a departure form an adiabat to be dynamically significant.  A 3K temperature difference $\delta T$ between parcel and environment averaged over a height $H = 10km$ naively produces an acceleration of $g \delta T/T \approx 0.1 ms^{-2}$ and velocities at the top of the convective layer of $\sqrt{2 gH \delta T/T} \approx 45 ms^{-1}$ — which is larger than vertical motions observed in tropical convection.  But it is not that easy to relate vertical motions quantitatively to departures from an adiabat in the tropics. There are subtleties in the definition of the moist adiabat itself associated with what happens to the condensate — temperatures are slightly different if the condensate is retained by the parcel, in which case its heat capacity must also be taken into account, or if the condensate falls out immediately, in which case we refer to the “pseudo-adiabatic” lapse rate. Real parcels are somewhere between these two extremes.  You also needs to worry about the latent heat of fusion when ice forms, the presence of supercooled water making it tricky to predict when this transition to ice occurs.  In addition, when computing the density difference between a rising parcel and its environment, and the associated vertical accelerations, you must account for the “condensate loading” — the pressures associated with the suspension of the condensate within the rising parcel.  Finally, if a parcel entrains some dry environmental air as it rises, it has less latent heat to release per unit mass, and its temperatures will fall faster with height than the temperature profile generated by an undilute parcel.

Several of these effects can be of the order of a degree or two — they are big enough to matter when trying to estimate the magnitude of the departure of the tropical atmosphere from a moist adiabat — the CAPE (Convective Available Potential Energy) of the tropics (see, for example, Xu and Emanuel 1989 and Williams and Renno 1993).  But none of them are large enough to alter the expectation that the tropical atmosphere will roughly follow a moist adiabat as it warms. One of these effects would have to change by an O(1) amount (doubling or halving its amplitude) in response to a 2K warming, say,  to have a substantial effect on the sensitivity of lapse rates to warming, but why should that happen?

This conclusion is confirmed by high resolution models of horizontally homogeneous radiative-convective equilibrium, every one of which, to my knowledge, predicts a warming profile that is more or less moist adiabatic.  The figure at the top is from Romps 2011, mentioned in the last post as well, which has 200m horizontal resolution.    The figure on the left shows the model’s equilibrated temperature profiles at three values of CO2 along with an observed tropical profile, while the panel on the right shows the changes in temperature in this model along with an average over CMIP3 global models (with grid sizes roughly 100 times larger).  The profile of temperature change is essentially identical.  There is actually a rather large fractional  increase in CAPE and increase in the magnitude of vertical motions as the climate warms in these simulations, but this increase is nowhere near large enough to compensate for the upper level maximum in warming.

(The fact  that the overall amplitude of the warming for doubled CO2 is nearly identical in the ensemble mean of GCMs and in this cloud resolving model is a coincidence — it is the vertical profile of the temperature change that I am focusing on here.  The variety of results on sensitivity with cloud resolving models of radiative-convective equilibrium is as large as that obtained with GCMs, due primarily to differing cloud feedbacks associated with differences in organization of convection.  These high resolution simulations are not necessarily more relevant to nature than GCMs, due to the idealized geometry and small domain, absence of rotation etc. The value of these dynamic radiative convective equilibrium models, even in these small idealized domains, is in testing our undersatanding of moist convection. )

The change in CO2 itself has very little to do with this moist adiabatic response; you get essentially the same temperature response if you just just prescribe and then warm the surface temperature. Here, for example, is an early attempt at a dynamic radiative-convective model, from Held et al, 1993, for a 5K surface warming with fixed CO2 (the solid lines are moist adiabats):

A dramatic change in convective organization can change the relevant moist adiabat constraining tropical temperatures.  The self-aggregation transition described in the previous post, is a good example.   The aggregated state is warmer by several degrees averaged over the troposphere than the state with more homogeneous convection, because the near surface relative humidity is higher in the region in which the convection is occurring — due, in turn, to increased winds and evaporation.  (Thanks to Caroline Muller for confirming this for me.)

As one moves upwards and convection peters out, there is presumably some potential to change local temperatures with local perturbations in ozone or aerosols, perhaps above 12 km or so.  But below that, if you are trying to change the relationship between surface and tropospheric warming,  it seems that one is better off trying to change the relevant moist adiabat, by changing low level humidities or temperatures in the convecting regions, rather than creating huge departures from a moist adiabatic profile.

In addition to the analyses of MSU temperatures (which I won’t try to summarize here) , there is other relevlant observational work that we need to focus on.  One is the study of Allen and Sherwood 2008, using thermal wind balance to relate trends in the vertical gradient of the zonal winds to trends in the north-south temperature gradient  The thermal wind equation is very accurate for zonal mean winds throughout the atmosphere, a simple consequence of the assumptions that the mean state of the atmosphere is in hydrostatic balance and that winds are in geostrophic balance.  One can use an even more accurate relation, gradient wind balance, but given the large uncertainties in atmospheric warming trends, thermal wind balance is certainly accurate enough.  I would like to see more attention on this use of wind trends, since these are totally independent of the temperature measurements, satellite or radiosonde.

Another study deserving attention  is Johnson and Xie 2010 which argues that one can look at the distribution of deep convection in the tropics and rule out a trend towards overall destabilization.  The temperature (and associated water vapor content) of near surface air has to reach a certain threshold for this air to rise close to the tropopause — currently 26-28C.  If the atmosphere follows a moist adiabat as it warms, this critical temperature will increase along with the surface warming.  If the upper troposphere does not keep up, this critical temperature would not increase as fast as the surface temperature itself, favoring more widespread convection — which, according to the paper, is not observed.  This is another paper that I have to think about more carefully.

[The views expressed on this blog are in no sense official positions of the Geophysical Fluid Dynamics Laboratory, the National Oceanic and Atmospheric Administration, or the Department of Commerce.]

### 27 Responses to “20. The moist adiabat and tropical warming”

1. Jianhua Lu says:

Isaac,

I have a question about the paragraph related to the self-aggregation transition:

—-
The aggregated state is warmer by several degrees averaged over the troposphere than the state with more homogeneous convection, because the near surface relative humidity is higher in the region in which the convection is occurring — due, in turn, to increased winds and evaporation. (Thanks to Caroline Muller for confirming this for me.)

Is the increased evaporation that maintains a higher RH mainly in the non-convective region or in the convective region? If the former were the case, then is it possible that a higher RH in the convective region be maintained by a more efficient lower-level moisture transport from the non-convective region where the net surface radiation also favors enhanced evaporation?

• Isaac Held says:

I haven’t looked at this in detail, but thinking about it in response to your question (changing my story a bit), I am not sure that you need any change in evaporation to get this result –the moisture convergence due to the circulation generated by the localized convection is probably enough.

2. Marty Singh says:

Isaac,
Thanks for the post. Another effect that can change the ratio of upper tropospheric to surface warming on a moist adiabat is the relative humidity near the surface. I think a 3-4%/K decrease in humidity is enough to wipe out any upper tropospheric amplification. However, the required changes in winds to keep the evaporation reasonable are large, and probably fit in with your characterization of needing to change other quantities by O(1) to significantly affect the warming profile.

• Isaac Held says:

Right — that’s what’s happening in the self-aggregation transition. It is hard for me to imagine this large a reduction in tropical mean near surface humidity with warming — but one only has to get this reduction in the convecting regions.

• Isaac Held says:

Following up a bit — the SSMI measurements of vertically integrated water over the oceans and the comparison with GCMs — ie, Fig, 1 in
Soden at el 2005 — pretty much rule out any big problem in the simulations with low level humidity trends over the oceans — so the trend in the tropical ocean mean moist entropy of the sub-cloud layer must be about right as well, so there seems to be no reason to believe that one is getting on the wrong moist adiabat. It is hard to see how these measurements could be consistent with an increase in moisture that was just half of the Clausius-Clapeyron value. At a minimum, one would need to argue for a distinction between tropical mean trends and trends in the regions of deep convection.

3. Fred Moolten says:

Isaac – Among unsettled issues is the change in upper tropospheric humidity with warming – specific humdity increases, but there is conflicting information as to whether this increase is sufficient to maintain a near-constant relative humidity (RH – e.g. Minschwaner and Dessler 2004 as compared with Soden et al 2005). To what extent would a reduction in RH affect the change in the moist adiabatic lapse rate as a function of warming?

• Isaac Held says:

The Minschwawaner and Dessler study is of interannual variability. In ENSO cycles the tropics follow the moist adiabat pretty closely, as one would naively expect based on the tropical mean surface ocean warming — and models capture this temperature response very well. (This is one of the reasons why the possibility of a departure from moist adiabatic warming in the trend would be so curious, the atmospheric dynamics involved being fast enough that it should not care whether the warming is due to ENSO or slowly evolving forcing.) So I do not see this study as hinting that the modest RH variations observed could be responsible for departures from a moist adiabatic response.

4. HR says:

Here’s another independent method for looking at tropical tropospheric temperature changes.

“The Threshold Sea Surface Temperature Condition for Tropical Cyclogenesis”
http://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-10-05006.1?journalCode=clim

No temporal trends observed for the threshold cyclone temperature (the many caveats in the paper noted). So the debate rages.

I have a question. A scan of the literature seems to suggest that most of the focus is on bring observation in line with models. Can you point me to published work that does the opposite. Analysing the possible shortcomings of the models playing “what if” games to try to bring the models in line with the observations. Or are these experiments just not possible?

• Isaac Held says:

I think of the temperature threshold for tropical cyclogenesis as the same thing as the threshold for convection to the tropopause, so, based on a reading of the abstract alone, it looks like there may be a conflict between this paper and Johnson and Xie. It would be interesting to sort this out.

As for manipulating GCMs to conform to some specified data set, it depends a lot on the observations you are talking about and the “stiffness” of the model in that respect. There are a host of papers that take this approach, but mostly in the model development rather than climate change literature, in which the target is just the mean climatology. I am sure that you can just google the double ITCZ problem to see people thinking about how to fix this common model shortcoming, as an example. With regard to fitting trends, there is a tension between wanting to use observed trends to evaluate model physics and wanting to project into the future with a model that is simulating past trends optimally. This can be a source of confusion when different modeling groups take different approaches .

The tropical lapse rate’s response to warming is pretty stiff, as I was trying to emphasize in this post. I think the most plausible route to getting a different result for the lapse rate trend is through the addition of other forcing agents, other than well-mixed greenhouse gases, acting near the tropopause. If you are convinced that your model is deficient in this regard that’s what I would play with.

5. Alex Harvey says:

Dear Dr. Held,

This was a very interesting article and helped clarify some issues for me.

If you have the time I would like to ask some questions that may be helpful to me and others. If you don’t have time I would certainly understand.

1) How certain are you that a weaker than expected hotspot would lead to a slightly higher climate sensitivity? Is it not possible that this could go in various ways? That is, is there not significant uncertainty about all of the feedbacks impacted by tropical upper tropospheric warming, i.e. water vapour feedback, lapse rate feedback, and cloud feedback, e.g. Colman 2001 (Climate Dynamics, “On the vertical extent of atmospheric feedbacks”).

2) On the thermal wind equation, I wonder if you know what happened to a paper by Pielke et al. 2008, discussed at http://pielkeclimatesci.wordpress.com/2009/01/28/submitted-paper-assessment-of-temperature-trends-in-the-troposphere-deduced-from-thermal-winds-by-pielke-sr-et-al/

3) In this post you have not mentioned a recent paper by Fu, Manabe & Johanson (2011, “On the warming in the tropical upper troposphere: Models versus observations”, GRL). These authors conclude “While satellite MSU/AMSU observations generally support GCM results with tropical deep‐layer tropospheric warming faster than surface, it is
evident that the AR4 GCMs exaggerate the increase in static stability between tropical middle and upper troposphere
during the last three decades.” Do you agree with these authors? Do you regard this as a potentially controversial result?

4) There is an argument from Lindzen (2007, “Taking greenhouse warming seriously”, E&E) that has never been discussed in the literature, as far as I am aware. The argument is laid out in section 2. Lindzen himself evidently regards it as still valid today (e.g. he has cited it in the recent Lindzen/Choi 2011 paper). I assume you would regard the argument as invalid. Would you mind giving your thoughts on what he gets wrong?

With best regards,
Alex Harvey

PS. I am NOT the same Alexander Harvey who has posted here in the past!

• Isaac Held says:

Alex,
1) I was referring to the sum of water vapor and lapse rate feedbacks. If relative humidity is unchanged, the sum of these feedbacks is small but negative in response to an increase in tropical upper tropospheric temperatures. This is just a statement about radiative transfer. I am also quite confident that relative humidity will not change much ( I haven’t discussed the source of this confidence on this blog yet.) As usual, clouds are the major source of uncertainty, and I did not intend to include them in this comment. I guess the bigger point is that if the tropical lapse rate changes in such a way as to destabilize the tropical atmosphere a lot (which I am still skeptical of), this says that we are missing something important, the implications of which are hard to predict until we understand it.
2) I have no knowledge of that work
4) thanks for pointing out this Lindzen paper, which I was not aware of. Section 2 of that paper is partly devoted to an explanation for what the greenhouse effect is — this is entirely standard and exactly the explanation that I always use. The rest of this section discusses some radiosonde-based trends in lapse rate and the implications of those being correct. It might be better to return to this issue after the post that I promised above.

• Alex Harvey says:

Dear Dr. Held,

On point 2, Pielke et al. 2008 (submitted), the material may have been published in:

What Do Observational Datasets Say about Modeled Tropospheric Temperature Trends since 1979?
John R. Christy 1,*, Benjamin Herman 2, Roger Pielke, Sr. 3, Philip Klotzbach 4, Richard T. McNider 1, Justin J. Hnilo 1, Roy W. Spencer 1, Thomas Chase 3 and David Douglass 5
Remote Sensing 2010, 2, 2148-2169; doi:10.3390/rs2092148

After some discussion of the thermal wind equation, they conclude that “…these trends calculated from the TWE, as applied for AS08 and here (C10), using the current radiosonde coverage and observational limitations (consistency, accuracy, etc.) do not produce results reliable enough for studies such as ours. In particular, AS08 and C10, with TLT trends of +0.29 and +0.28 °C decade−1 are almost three times that of the mean of the directly measured systems, and are values that are, in our view, simply not consistent with the countervailing, directly-measured evidence.”

• Fred Moolten says:

Isaac – You state that the sum of water vapor and lapse rate feedbacks is net negative. Is this a different conclusion from Soden and Held 2008, or is the difference related mainly to the fact that your above statement refers only to the tropical upper tropospheric temperature?

• Isaac Held says:

Fred,

Since the lapse rate feedback wins over water vapor feedback in the global mean, and both are dominated by the tropics, I was assuming that this was true of the tropics in isolation. Typically, temperatures win because water vapor is weakened by upper level clouds — in the clear sky it would be a much closer compensation. If you haven’t seen it you might be interested in the nice discussion of water vapor feedback in Ingram, 2010.

• Joel Shore says:

Isaac,

Your statement “the lapse rate feedback wins over water vapor feedback in the global mean” confuses me. I thought that the rough numbers were that the water vapor feedback about doubles the climate sensitivity from the no-feedback value and then the lapse rate feedback takes about half of that back, leaving about a 1.5X-or-so increase in the climate sensitivity. Is this wrong…Or is your statement about the “global mean” only applied to the upper troposphere.

Could you clarify how all of these fit together?

• Isaac Held says:

Sorry for being unclear. I am thinking of the water vapor feedback as divided into two parts (let’s assume that relative humidity is fixed for simplicity). The first part is due to the increase in vapor that would occur with uniform warming of the troposphere. The second part is due to the additional vapor you get, per unit surface warming, if the lapse rate decreases (and the upper troposphere warms more that the surface). It is this second part, the water vapor feedback associated with the lapse rate change, that compensates for part, but not all, of the lapse rate feedack. So when comparing two models with more or less fixed relative humidity, I expect the one with more upper tropospheric warming to be less sensitive — ignoring clouds etc. The total water vapor feedback, including the part due to uniform warming, is larger than the lapse rate feedback, as you describe. I’ll be coming back to this in an upcoming post.

• Joel Shore says:

Thanks, Isaac. That makes more sense to me now! And, I just want to say that I really appreciate your blog…It had fallen off my “radar screen” for a few months and I am glad that I remembered to come back to it again.

Judging from the other comments, it looks like you are providing a great resource both for others working in the field to exchange ideas but also for some of us who are not climate scientists, but who have the necessary background to at least sort-of follow more technically-oriented discussions than are available on most blogs on the topic, to learn a lot! (In my case, I am a physicist for which reading about climate science has become a hobby.)

• Fred Moolten says:

Isaac – Thanks for the clarification, because I had been puzzled too. Incidentally, in my initial comment, I misstated the date of the relevant paper I wanted to mention. It was Soden and Held 2006, not 2008

6. Frank says:

Is there anyplace not behind a paywall that one can learn about the basics of cloud-resolving models? I was surprised to learn that this one didn’t include diurnal changes in radiation and other influences. Some convective processes are part of large organized circulation (the Hadley cell for example) that is much larger in scale than this model. Do we have any idea how far from a convective “tower” (if that is the right term) subsistence usually occurs? Rising air follows a moist adiabat once it has reached the lower condensation level, but subsiding air should follow a dry adiabat. I don’t understand how mixing under these circumstances produces a tropical atmosphere which follows a moist adiabat.

• Isaac Held says:

To access papers, I usually start by looking at author home pages and then use google scholar to find copies outside of the firewall if there is one, and if that fails I just contact the author. Possible starting points might be homepages for Chris Bretherton (UWash) , Bjorn Stevens (Max Planck, Hamburg) , Graeme Stephens (JPL), Zhiming Kuang (Harvard). I am not aware of a review paper on the subject of dynamic radiative convective equilibrium models specifically.

Many very high resolution models include diurnal cycles, but typically in more realistic settings — see, for example, here .

Radiative convective models in small domains, no matter what the resolution, do not simulate the spatial inhomogeneities and convective structures that occur in the tropics, and should not be thought of as providing quantitative simulations of tropically-averaged clouds (or relative humidity for that matter). Most research with these kinds of models are not this idealized, of course, and involve attempts to simulation specific phenomenon. I find the homogeneous radiative convective framework useful for testing ideas, despite its idealization.

The picture of the compensating subsidence in the tropics putting the atmosphere in these regions on a dry adiabat is not correct. This is part of what I referred to as the counterintuitive behavior of a flow with such dramatic skewness in its vertical motion. The upward motion is confined to such a small area, and the subsidence so slow, that one cannot ignore the radiative cooling in the subsiding air. In fact the waves that homogenize temperatures in the horizontal are so fast we think of the subsidence rate as being determined by the requirement of consistency between the radiative cooling rate and need for the parcel’s temperature, as it descends, to follow the moist adiabat determined by rising plumes. This kind of balance is formalized in what we refer to as weak-temperature gradient dynamics — see, for example, Bretherton and Sobel, 2002. In this theory, one assumes that the subsidence is spread uniformly over the entire tropics!

7. TimTheToolMan says:

Isaac writes “But once the water vapor in the parcel reaches saturation some of this vapor condenses and releases its latent heat, compensating for some of the cooling (you get about 45K of warming from latent heat release when a typical parcel rises from the tropical surface to the upper troposphere).”

Water vapour condensing is accompanied by a decrease in volume but this never seems to be mentioned in descriptions of the moist adiabat. Yours is no exception… is there a reason its ignored? And how does it figure into the explanation?

• Isaac Held says:

It’s a small effect, much smaller in the atmosphere than the expansion due to the reduction in pressure as the air rises, but the reduction in volume at a given pressure due to condensation is included in accurate computations of adiabatic lapse rates.

8. TimTheToolMan says:

Also water vapour is “lighter than air” and so doesn’t require parcels of air to be rising for the water vapour itself to be rising. If the water vapour were to be considered to be rising in its own right what implications would that have? Obviously air will come along for the ride (and have its own convection related reasons to rise too) but most explanations I’ve seen appear to treat the “parcel” as the driver of the process when that doesn’t feel right to me.

I think I need a really good reference to our current understanding of these issues. Any suggestions?

• Isaac Held says:

This question does not relate to condensation, so we can assume that there is none for simplicity. Picking a small piece of fluid — a parcel — there is then no way for the center of mass of the vapor in this parcel to move with respect to the rest of the air molecules except by molecular diffusion — in fact, you can take that relative motion to be the definition of molecular diffusion. You have to picture the water molecules as locked into the parcel along with the other molecules due to the small mean free path between collisions — ie, because molecular diffusion is small. The water molecules cannot just up and decide to do their own thing. You have to be up above 100km for the density to decrease enough so that this relative motion of different species becomes significant. An undergraduate level atmospheric thermodynamics textbook might be a good place to start. I don’t teach undergraduates, so I am not a good person to ask about introductory texts — perhaps the one by Grant Petty — I haven’t looked at how that text discusses this particular issue.

This is a very common error that unfortunately developed some traction on the web in the context of the ozone hole and CFC transport into the stratosphere — “aren’t they heavier than the rest of the air molecules, so wouldn’t they stay close to the surface?” Dispelling this notion is usually the first order of business in any class on atmospheric chemistry — which suggests that an intro textbook on atmospheric chemistry might also be good place to look for a discussion of this.

• TimTheToolMan says:

So mixing trumps diffusion. OK fair enough. Thanks for taking the time to respond Isaac.

9. Joe Kidston says:

Hi Isaac. I know very little about tropical dynamics, but have a half-thought question related to the tropical lapse rate: Is there a simple argument for why the relative humidity over land is expected to remain fixed? I could imagine that it might change, due to soil moisture, which might be affected by changing moisture convergence in the atmospheric column above, or changing runoff. I could also imagine that deep convection takes place preferentially over land, and so the relative humidity over land might be very important for setting the tropical lapse rate.

• Isaac Held says:

Joe — great question. What determines the moist entropy of the boundary layer air over land where convection occurs? In the climate change context, can the entropy over land increase more than it increases over the oceans? See Joshi et al 2008 and, in particular the conference presentation by Byrne and O’Gorman, who argue that the changes in entropy should be the same — that warming is larger over continents because the entropy increases are the same as over the oceans but relative humidities are lower. I need to think about this more myself.


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## New (pretty) SDD enclosure and backup drive.

April 20, 2021 mac osX No comments , , , ,

The SSD on my 2015 era macbook died, but once I got a new machine, I was pretty impressed how well the MacOS restore from time machine worked.  It was dog slow, and took over 24hrs, but I got back all my work, and most of the MacOS configuration too.

The new macbook has ONLY usb-C ports.  Luckily, I had the foresight to order a thunderbolt-2 to USB-C adapter with the new macbook for the monitors, so I was able to use one of my monitors as a USB hub to access my old time machine drive.  The time machine backups on my old macbook, with the drive plugged in directly, never seemed that slow, so I think that some of the slow restore was due to this indirect connection.  However, some of that must also have been due to the older magnetic drive too.

Having just gone through a harddrive failure scenerio, backup and restore speed seemed worth spending some additional resources on, so I went ahead and splurged on a new ASUS NVM SSD enclosure:

This direct connects USB-C to USB-C, and is currently equipped with a 2Tb SSD.  It is also really pretty.

I went ahead and enabled time machine encryption on this drive, and was able to do a full (0.7Tb) backup in about 4hrs, even with encryption enabled!  The big 2Tb NVM drive wasn’t cheap, but has lots of space for incremental backups of the macbook’s 1Tb drive.

It is easy to swap out the SSD, so I can use this enclosure to backup my windows machine too (provided I buy an additional drive.)

## Welcome to the Ontario police state, in the era of covid-1984

Our morbidly obese Ontario premier seems to be attempting to protect himself by instituting another phase of lockdowns and by granting additional police powers.  People can build their immune systems with exercise and fresh air, but he is probably so obese that the cardio that he would require to reduce his weight would give him a heart attack.  So, like a petulant and vindictive abused child bully, if he can’t do it, you are not allowed to either.  Gyms are now shut down for the second year of “two weeks to flatten the curve.”  Somebody who can’t exercise doesn’t care if he is inhibiting the ability of others to do so, regardless of the mental and physical health benefits of doing so, or to the livelihoods of those employed by or dependent on those gyms.

At the beginning of the covid scare, we had no idea what was going on.  We now know a great deal.  Among all the things that we now know, are the particular facts that obesity and vitamin D deficiencies are key risk factors, and that most transmission appears to be in indoor environments.  Our bodies utilize exposure to sunlight to manufacture vitamin D, so being incarcerated indoors puts us at risk.  Getting outdoors for exercise is probably the single biggest thing that we can do to keep ourselves safe, yet Ontario police have now been granted the power to stop people on the streets, demand to know their identity, their address, and their purpose of travel.  This is clearly not something that has been imposed to encourage people to get out for exercise, sunlight and fresh air.  I stopped paying attention to the so-called regulations that are being imposed$${}^1$$, so I do not know the full extent of the rules that I am supposed to be complying with.  If I were out on a bike ride by myself or with my wife, who I share a space with anyways, would I have to justify that to the police?  Do I face fines or jail for attempting to keep myself healthy and safe (not just from covid, but many other conditions that are enhanced by inactivity)?

Thankfully, it appears that there is some push back to the new dictatorial measures, and a number of police forces have stated that they will not enforce their new inquisitional power to stop people on the streets nor in their cars.  I’m not optimistic that the Toronto police will take this position, as a force that large is statistically more likely to abuse power.

When I am out of the house these days, it is usually because I am getting groceries, running errands, or walking my beast of a dog.  I can’t help wondering how I would respond if I were to get stopped by the police for any of these.  If the fear porn was truly justified, then those police officers are putting themselves in danger by approaching potential disease carriers.  How do they feel about that?  Would I be brave enough to attempt to politely ask that off the record?  Does an officer making such a stop not cringe internally against the stupidity of what they are being made to do?  I am not optimistic that would be the case.  Police depend on their jobs and paycheques, and part of their paycheque means that they have to enforce the laws, regardless of their opinion about them, and then have to justify those actions so that they can live with themselves.  There is the open question about whether these executive decrees are truly lawful, but if the police believe they are, these mandates will be enforced until challenged in the courts.  There may not be much internal Police dissent, and if there is, it is probably on the down-low, quiet, and off the record.  Is there much chance that the police who have been enforcing the “Now you do what they told ya” mandates, will end up collectively fighting back in a screaming “FUCK YOU, I WON’T DO WHAT YOU TELL ME” crescendo that pulls the power out from under the slovenly dictator who has assumed control of the province?  I doubt it, but I can hope.

Footnotes:

1. Because there are so many signs about it, I know that I can be fined or jailed for not wearing a mask when shopping indoors.

## John Cleese’s “Creativity. A short and cheerful guide.”

April 11, 2021 Incoherent ramblings No comments

This book reiterates some of the hare-brain/tortoise-brain points from Cleese’s “Professor At Large.”  However, despite the redundancy, it was worth reading just for the following remark:

“Begin with simple stuff, such as…, Who are you writing for?  You might be writing for academics, in which case you don’t have to be interesting.

I thought this was so funny, but was it intended to be funny, or just reflect reality?

## Some neighbourhood sidewalk art

April 10, 2021 Just for fun No comments

## A bitcoin ransom entrepreneur

March 25, 2021 Uncategorized No comments , ,

I call the bluff. Let’s see the video!

Hello!

Unfortunately, I have some bad news for you.
Several months ago, I got access to the device you are using to browse the internet.
Since that time, I have been monitoring your internet activity.

Being a regular visitor of adult websites, I can confirm that it is you who is responsible for this.
To keep it simple, the websites you visited provided me with access to your data.

I've uploaded a Trojan horse on the driver basis that updates its signature several times per day, to make it impossible for antivirus to detect it. Additionally, it gives me access to your camera and microphone.
Moreover, I have backed-up all the data, including photos, social media, chats and contacts.
Just recently, I came up with an awesome idea to create the video where you cum in one part of the screen, while the video was simultaneously playing on another screen. That was fun!

Rest assured that I can easily send this video to all your contacts with a few clicks, and I assume that you would like to prevent this scenario. )

With that in mind, here is my proposal:
Transfer the amount equivalent to 1350 USD to my Bitcoin wallet, and I will forget about the entire thing. I will also delete all data and videos permanently.

In my opinion, this is a somewhat modest price for my work.
You can figure out how to purchase Bitcoins using search engines like Google or Bing, seeing that it's not very difficult.

My Bitcoin wallet (BTC):
1Lg6nE6zFaMZHiPTFR9nSwuAL6hzeegToC

You have 48 hours to reply and you should also bear the following in mind:

It makes no sense to reply me - the address has been generated automatically
It makes no sense to complain either, since the letter along with my Bitcoin wallet cannot be tracked.
Everything has been orchestrated precisely.

If I ever detect that you mentioned anything about this letter to anyone - the video will be immediately shared, and your contacts will be the first to receive it. Following that, the video will be posted on the web!

P.S. The time will start once you open this letter. (This program has a built-in timer and special pixel ID - [№78853473])
Good luck and take it easy! It was just bad luck, next time please be careful.

## Introducing a fixed capacity directory (Linux).

March 23, 2021 Linux No comments , , , ,

I have a dump directory that is too easily filled up (with cores and dumps) if a programming error makes our system misbehave catastrophically.

Here’s a nice trick to create a small filesystem with fixed capacity so that the owning filesystem (in my case /var) can’t be filled up:

dd if=/dev/zero of=dump.loopback bs=1M count=10
mke2fs ./dump.loopback
mount -o loop dump.loopback ./dump
chmod 777 ./dump

## XPG laptop: how to turn off the annoying ‘lock unlock’ screen indicator.

My macbook harddrive appears to be pooched, so I’m using my personal Windows laptop for work until I can get it fixed.  There’s been an annoying feature of this laptop that I hadn’t figured out, but after trying to use it all day, it was well past time.  In particular, if I hit Caps-Lock, I get the following screen indicator:

close to the top left corner of the screen, which often obscures what I am trying to type!  This indicator is extremely stupid.  I know when I hit caps lock: it’s when I hit caps lock, and don’t need something to tell me that I’ve done it.  If I did not know what state it was in, I can look the keyboard caps lock LED.

I found a couple Q&A’s about similar issues:

I tried the Settings configurations, and disabled the toggle stuff, which did not help.  This suggested that there was vendor (XPG) supplied software that was controlling this annoyance.

To track this down, I ran the sysinternals Process Explorer, and searched for xpg:

Sure enough, after brute force killing all these xpg processes, the annoying Lock-Unlock indicator goes away. After a restart, I found that there’s an xpg application running in the background, and sure enough there’s an option to be annoying:

It turns out that there’s also a pop up indicator that occurs if you press Num-Lock. I also won’t miss the XPG application notifications for Num-Lock either — there is also a keyboard LED for that!

## Hardcover physics class notes.

Amazon’s kindle direct publishing invited me to their hardcover trial program, and I’ve now made hardcover versions available of most of my interesting physics notes compilations:

Instead of making hardover versions of my classical mechanics, antenna theory, and electromagnetic theory notes, I have unpublished the paperback versions. These are low quality notes, and I don’t want more people to waste money on them (some have.) The free PDFs of all those notes are still available.

My geometric algebra book is also available in both paperback and hardcover (black and white). I’ve unpublished the color version, as it has a much higher print cost, and I thought it was too confusing to have all the permutations of black-and-white/color and paperback/hardcover.

## Debugging a C coding error from an XPLINK assembly listing.

There are at least two$${}^1$$ z/OS C calling conventions, the traditional “LE” OSLINK calling convention, and the newer$${}^2$$ XPLINK convention.  In the LE calling convention, parameters aren’t passed in registers, but in an array pointed to by R1.  Here’s an example of an OSLINK call to strtof():

*  float strtof(const char *nptr, char **endptr);
LA       r0,ep(,r13,408)
LA       r2,buf(,r13,280)
LA       r4,#wtemp_1(,r13,416)
L        r15,=V(STRTOF)(,r3,4)
LA       r1,#MX_TEMP3(,r13,224)
ST       r4,#MX_TEMP3(,r13,224)
ST       r2,#MX_TEMP3(,r13,228)
ST       r0,#MX_TEMP3(,r13,232)
BASR     r14,r15
LD       f0,#wtemp_1(,r13,416)

R1 is pointed to r13 + 224 (a location on the stack). If the original call was:

float f = strtof( mystring, &err );

The compiler has internally translated it into something of the form:

STRTOF( &f, mystring, &err );

where all of {&f, mystring, &err} are stuffed into the memory starting at the 224(R13) location. Afterwards the value has to be loaded from memory into a floating point register (F0) so that it can be used.  Compare this to a Linux strtof() call:

* char * e = 0;
* float x = strtof( "1.0", &e );
400b74:       mov    $0x400ef8,%edi ; first param is address of "1.0" 400b79: movq$0x0,0x8(%rsp)       ; e = 0;
400b82:       lea    0x8(%rsp),%rsi       ; second param is &e
400b87:       callq  400810 <strtof@plt>  ; call the function, returning a value in %xmm0

Here the input parameters are RDI, RSI, and the output is XMM0. Nice and simple. Since XPLINK was designed for C code, we expect it to be more sensible. Let’s see what an XPLINK call looks like. Here’s a call to fmodf:

*      float r = fmodf( 10.0f, 3.0f );
LD       f0,+CONSTANT_AREA(,r9,184)
LD       f2,+CONSTANT_AREA(,r9,192)
L        r6,=A(__fmodf)(,r7,76)
L        r5,=A(__fmodf)(,r7,72)
BASR     r7,r6
NOP      9
LDR      f2,f0
STE      f2,r(,r4,2144)
*
*      printf( "fmodf: %g\n", (double)r );

There are some curious details that would have to be explored to understand the code above (why f0, f2, and not f0,f1?), however, the short story is that all the input and output values in (floating point) registers.

The mystery that led me to looking at this was a malfunctioning call to strtof:

*      float x = strtof( "1.0q", &e );
LA       r2,e(,r4,2144)
L        r6,=A(strtof)(,r7,4)
L        r5,=A(strtof)(,r7,0)
LA       r1,+CONSTANT_AREA(,r9,20)
BASR     r7,r6
NOP      17
LR       r0,r3
CEFR     f2,r0
STE      f2,x(,r4,2148)
*
*      printf( "strtof: v: %g\n", x );

The CEFR instruction converts an integer to a (hfp32) floating point representation, so we appear to have strtof returning it’s value in R3, which is an integer register. That then gets copied into R0, and finally into F2 (and after that into a stack spill location before the printf call.) I scratched my head about this code for quite a while, trying to figure out if the compiler had some mysterious way to make this work that I wasn’t figuring out. Eventually, I clued in. I’m so used to using a C++ compiler that I forgot about the old style implicit int return for an unprototyped function. But I had included <stdlib.h> in this code, so strtof should have been prototyped? However, the Language Runtime reference specifies that on z/OS you need an additional define to have strtof visible:

#define _ISOC99_SOURCE
#include <stdlib.h>

Without the additional define, the call to strtof() is as if it was prototyped as:

int strtof( const char *, char ** );

My expectation is that with such a correction, the call to strtof() should return it’s value in f0, just like fmodf() does. The result should also not be garbage!

Footnotes:

1.  There is also a “metal” compiler and probably a different calling convention to go with that.  I don’t know how metal differs from XPLINK.
2. Newer in the lifetime of the mainframe means circa 2001, which is bleeding edge given the speed that mainframe development moves.

## Hardcover edition of Geometric Algebra for Electrical Engineers.

I was invited to Kindle Direct Publishing‘s hardcover beta program, and have made my geometric algebra book available in black and white hardcover.

As always, the PDF, leanpub edition, and latex sources are also available.

I thought that it was too confusing to have color and black-and-white editions of the book (color has a significantly higher printing cost), so I have unpublished the color editions of the book (softcover, and hardcover). There is one copy of the color edition left, and once that is sold, it will show as out of print.

format: plain


text:
# 11.10: Drawing and Naming Cycloalkanes

## Learning Objectives

• To name cycloalkanes given their formulas and write formulas for these compounds given their names.

The cycloalkanes—cyclic hydrocarbons with only single bonds—are named by adding the prefix cyclo- to the name of the open-chain compound having the same number of carbon atoms as there are in the ring. Thus the name for the cyclic compound C4H8 is cyclobutane. The carbon atoms in cyclic compounds can be represented by line-angle formulas that result in regular geometric figures. Keep in mind, however, that each corner of the geometric figure represents a carbon atom plus as many hydrogen atoms as needed to give each carbon atom four bonds.

Some cyclic compounds have substituent groups attached. Example $$\PageIndex{1}$$ interprets the name of a cycloalkane with a single substituent group.

## Example $$\PageIndex{1}$$

Draw the structure for each compound.

1. cyclopentane
2. methylcyclobutane

### Solution

1. The name cyclopentane indicates a cyclic (cyclo) alkane with five (pent-) carbon atoms. It can be represented as a pentagon.
• The name methylcyclobutane indicates a cyclic alkane with four (but-) carbon atoms in the cyclic part. It can be represented as a square with a CH3 group attached.
• ## Exercise $$\PageIndex{1}$$

Draw the structure for each compound.

1. cycloheptane
2. ethylcyclohexane

11.10: Drawing and Naming Cycloalkanes is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.


text:
# 6.2 Properties of power series

 Page 1 / 10
• Combine power series by addition or subtraction.
• Create a new power series by multiplication by a power of the variable or a constant, or by substitution.
• Multiply two power series together.
• Differentiate and integrate power series term-by-term.

In the preceding section on power series and functions we showed how to represent certain functions using power series. In this section we discuss how power series can be combined, differentiated, or integrated to create new power series. This capability is particularly useful for a couple of reasons. First, it allows us to find power series representations for certain elementary functions, by writing those functions in terms of functions with known power series. For example, given the power series representation for $f\left(x\right)=\frac{1}{1-x},$ we can find a power series representation for ${f}^{\prime }\left(x\right)=\frac{1}{{\left(1-x\right)}^{2}}.$ Second, being able to create power series allows us to define new functions that cannot be written in terms of elementary functions. This capability is particularly useful for solving differential equations for which there is no solution in terms of elementary functions.

## Combining power series

If we have two power series with the same interval of convergence, we can add or subtract the two series to create a new power series, also with the same interval of convergence. Similarly, we can multiply a power series by a power of x or evaluate a power series at ${x}^{m}$ for a positive integer m to create a new power series. Being able to do this allows us to find power series representations for certain functions by using power series representations of other functions. For example, since we know the power series representation for $f\left(x\right)=\frac{1}{1-x},$ we can find power series representations for related functions, such as

$y=\frac{3x}{1-{x}^{2}}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=\frac{1}{\left(x-1\right)\left(x-3\right)}.$

In [link] we state results regarding addition or subtraction of power series, composition of a power series, and multiplication of a power series by a power of the variable. For simplicity, we state the theorem for power series centered at $x=0.$ Similar results hold for power series centered at $x=a.$

## Combining power series

Suppose that the two power series $\sum _{n=0}^{\infty }{c}_{n}{x}^{n}$ and $\sum _{n=0}^{\infty }{d}_{n}{x}^{n}$ converge to the functions f and g , respectively, on a common interval I .

1. The power series $\sum _{n=0}^{\infty }\left({c}_{n}{x}^{n}±{d}_{n}{x}^{n}\right)$ converges to $f±g$ on I .
2. For any integer $m\ge 0$ and any real number b , the power series $\sum _{n=0}^{\infty }b{x}^{m}{c}_{n}{x}^{n}$ converges to $b{x}^{m}f\left(x\right)$ on I .
3. For any integer $m\ge 0$ and any real number b , the series $\sum _{n=0}^{\infty }{c}_{n}{\left(b{x}^{m}\right)}^{n}$ converges to $f\left(b{x}^{m}\right)$ for all x such that $b{x}^{m}$ is in I .

## Proof

We prove i. in the case of the series $\sum _{n=0}^{\infty }\left({c}_{n}{x}^{n}+{d}_{n}{x}^{n}\right).$ Suppose that $\sum _{n=0}^{\infty }{c}_{n}{x}^{n}$ and $\sum _{n=0}^{\infty }{d}_{n}{x}^{n}$ converge to the functions f and g , respectively, on the interval I . Let x be a point in I and let ${S}_{N}\left(x\right)$ and ${T}_{N}\left(x\right)$ denote the N th partial sums of the series $\sum _{n=0}^{\infty }{c}_{n}{x}^{n}$ and $\sum _{n=0}^{\infty }{d}_{n}{x}^{n},$ respectively. Then the sequence $\left\{{S}_{N}\left(x\right)\right\}$ converges to $f\left(x\right)$ and the sequence $\left\{{T}_{N}\left(x\right)\right\}$ converges to $g\left(x\right).$ Furthermore, the N th partial sum of $\sum _{n=0}^{\infty }\left({c}_{n}{x}^{n}+{d}_{n}{x}^{n}\right)$ is

$\begin{array}{cc}\hfill \sum _{n=0}^{N}\left({c}_{n}{x}^{n}+{d}_{n}{x}^{n}\right)& =\sum _{n=0}^{N}{c}_{n}{x}^{n}+\sum _{n=0}^{N}{d}_{n}{x}^{n}\hfill \\ & ={S}_{N}\left(x\right)+{T}_{N}\left(x\right).\hfill \end{array}$

Because

$\begin{array}{cc}\hfill \underset{N\to \infty }{\text{lim}}\left({S}_{N}\left(x\right)+{T}_{N}\left(x\right)\right)& =\underset{N\to \infty }{\text{lim}}{S}_{N}\left(x\right)+\underset{N\to \infty }{\text{lim}}{T}_{N}\left(x\right)\hfill \\ & =f\left(x\right)+g\left(x\right),\hfill \end{array}$

we conclude that the series $\sum _{n=0}^{\infty }\left({c}_{n}{x}^{n}+{d}_{n}{x}^{n}\right)$ converges to $f\left(x\right)+g\left(x\right).$

#### Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
Abdul Reply
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul

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# Computation of extreme rays of rational polyhedral cones - Hemmecke's project and lift algorithm

I am working on an implementation of Raymond Hemmecke's algorithm for finding generating sets of cones: http://arxiv.org/abs/math/0203105

Unfortunately I am struggling to make the algorithm work on the simple examples. Consider a cone given by a bunch of vertices in $\mathbb{R}^2$:

$$\begin{bmatrix} 4&2\\ 2&1\\ 2&2\\ 0&1\\ \end{bmatrix}$$

There are several things blocking me at the moment:

1. In his paper Raymond Hemmecke assumes that any cone has generators given in Hermite Normal Form. Since HNF are a result of a unimodal transformation of a matrix, I should be able to transform any set of generators into HNF invertibly. But does that also hold for the minimal generating sets obtained from the HNF? I.e. if I use a HNF of a cone to obtain the minimal generating set, will I always be able to transform the result back to the minimal generating set of the original cone? Here's how the generators of cones and lattices are described by Hemmecke in $\mathbb{R}^n$: $$\begin{matrix} p_1 &= &(p_{1,1},&p_{1,2},&...,&...,&p_{1,r},&...s,&p_{1,n})\\ p_2 &= &(0,&p_{2,2},&...,&...,&p_{2,r},&...,&p_{2,n}) \\ p_3 &= &(0,&0,&p_{2,2},&...,&p_{2,r},&...,&p_{2,n})\\ &\vdots&(\vdots,&\vdots,&...,&\ddots,&\vdots,&...,&\vdots)\\ p_r &= &(0,&0,&...,&0, &p_{r,r}, &..., &p_{r,n})\\ \end{matrix}$$

2. For the computation of the extreme rays of a cone, I end up with misshapen cones/lattices Hemmecke defined a cone $\Gamma$ the following way

Let $$L = span(p_i),\quad p_i \in \mathbb{Z}^n.$$ Then: $$\Gamma = L \cap \mathbb{R}^n_+$$

But my cone above does not include all of the points in the positive orthant! So when I span a lattice with the generators $p_i$ and intersect it with the positive orthant, I still end up with lattice points that are not in my cone.

Does anyone have any experience with the ray algorithm? I have looked at the Macaulay implementation, but it uses Fourier-Moutzkin, I would like to use the method described by Hemmecke, but unfortunately I think I am missing something. I have tried running 4ti2 on these examples, but I get no results at all - the program returns an empty generating set.

 The goal is to use the algorithm to compute the Hilbert basis given the generators of a cone, like in section 5.5 of the paper. I.e. given the picture above, get me the Hilbert basis of the red area. With the homogeneity restriction I struggle to define my generating sets in a sensible way - I am given the vectors in the kernel, but I don't have the linear map $A$. If my input consists of 4 2-dimensional vectors like above, I can't possibly come up with a map from $RR^2$ to $RR^2$ that the red cone as its kernel. My supervisor said I am going to need to embed it in a higher dimensional space, but I don't really see how to do it sensibly.

One possible set of rays or $\mathbb{R}$-generators of the cone above is (reading as rows) \begin{bmatrix} 2&1\\ 0&1\\ \end{bmatrix} This doesn't satisfy $Ax= 0\quad x \geq 0$. On another hand this cone is uniquely defined by the y axis and the ray bisecting the cone into two equal pieces (or a halfspace defined by the normal vector that bisects the cone), is that how I bring it to a homogenous system?

To put it more clearly, I want to compute the Hilbert basis of the red cone, but I do not know the linear transformation that has the red cone as its kernel. All I have is a (not necessarily minimal) generating set of points of the cone.

• I think the software Normaliz home.uni-osnabrueck.de/wbruns/normaliz might have an implementation of Hemmecke's algorithm. – Tony Huynh Mar 17 '15 at 16:30
• There is a similar algorithm implemented in Normaliz. As far as I know, the only implementation of my algorithm is by Matthias Walter in 4ti2 (www.4ti2.de), which still gives the state-of-the-art for the computation of Graver bases. When it comes to Hilbert bases, you should probably first try Normaliz. For extreme ray computations you may wish to try 4ti2's function 'rays', a very efficient implementation (by Peter Malkin) of my algorithm (which is in fact equivalent to the double description method). – Raymond Hemmecke Mar 26 '15 at 21:52

Tomasz, I think you are mixing something here. You wish to compute the extreme rays of a cone given by generators? (This would mean you wish to throw away redundant generators, right?)

However, the algorithm that you wish to implement does something completely different! It takes a system $Ax=0$, $x\ge0$ (defining a pointed rational cone) and computes the cone generators or the Hilbert basis, in case you are interested in the integer situation.

In the case of $Ax=0$, $x\ge0$, it takes generators of $Ax=0$ and brings those into HNF. Then all works out nicely.

Hope that helps.

Raymond

PS: If you wished to compute the Hilbert basis of a cone given by cone generators, I recommend using Normaliz.

• Many thanks for the reply, we want to use your algorithm to see if there is any way to program it in parallel (split the cones symmetrically and compute the intersection) or run it on a GPU. I have updated the question to make it clearer what I am stuck with – swistak Apr 15 '15 at 14:44
• @TomaszNguyen, let me briefly restate your problem: Given: rays generating a cone Find: Hilbertbasis of cone Solution: Use normaliz. It does exactly what you want: It runs in parallel. (It does not use my algorithmic approach, as this is a different problem.) My project-and-lift algorithm solves the following problem: Given: matrix A Find: Hilbertbasis of cone ker(A)\cap\Z^n_+ Solution: use normaliz or 4ti2 – Raymond Hemmecke Apr 22 '15 at 15:14
• So in the section 5.5 when you say I can compute the Hilbert basis of a cone (now I know it is Ax = 0) what do you mean by generators? Are these rays? (that's what I initially thought) or halfspace normals (would seem to make more sense, but still gives me an inequality) I might sound really stubborn, but my project brief was to use your algorithm to produce HB of any cone and after getting stuck trying to produce a sensible A out of a (not necessarily minimal) generating set, I would like to have absolute clarity about what you mean there. Many thanks – swistak Apr 25 '15 at 11:15


text:
/ hep-ex FERMILAB-PUB-16-038-E

Observation of a new $B_s^0 \pi^\pm$ state

Published in: Phys.Rev.Lett.
Pages: 8

Abstract: We report the observation of a narrow structure, $X(5568)$, in the decay sequence $X(5568) \rightarrow B_s^0 \pi^{\pm}$, $B_s^0 \rightarrow J/\psi \phi$, $J/\psi\rightarrow \mu^+ \mu^-$, $\phi \rightarrow K^+K^-$. This is the first observation of a hadronic state with valence quarks of four different flavors. The mass and natural width of the new state are measured to be $m = 5567.8 \pm 2.9 {\rm \thinspace (stat)} ^{+0.9}_{-1.9} {\rm \thinspace (syst)}$ MeV/$c^2$ and $\Gamma = 21.9 \pm 6.4 {\rm \thinspace (stat)} ^{+5.0}_{-2.5} {\rm \thinspace (syst)}$ MeV/$c^2$, and the significance including look-elsewhere effect and systematic uncertainties is 5.1$\sigma$. The observation is based on$10.4~\rm{fb^{-1}}$ of $p \overline p$ collision data at $\sqrt{s}$ = 1.96 TeV collected by the D0 experiment at the Fermilab Tevatron collider.

Note: 8 pages, 4 figures, submitted to Phys. Rev. Lett

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This is my quick look of recent “Brest” amendment proposal to Tezos blockchain, which was injected and proposed by OCamlPro.

Data

Proposer
“TzScan Baker” tz1LWub69XbTxdatJnBkm7caDQoybSgW4T3s ran by OCamlPro
The social announcement
https://www.reddit.com/r/tezos/comments/by5xy8/proposal_for_amendment_brest_a/ It calls the proposal “Brest A”.
The code with the same announcement
https://gitlab.com/tzscan/brest-amendment

As of 2019-06-17T04:35:34Z, this is the only one proposed in this Tezos voting period.

At a glance

As stated by the above announcements, I see 3 modifications between 004_Athens (or just Athens) and 005_Brest (or just Brest):

• A security fix to make some DoS attack to the storage layer of Tezos nodes more difficult.
• A fix of the proposal rewards trackability issue.
• The reward is ꜩ8000 to TzScan Baker, which is equivalent to 1 roll (i.e. 1 voting power) in Athens.

I did not find any other undocumented changes. Let’s see their details.

Security fix

Contract index database

Blockchain implementations often have Git style ledger databases. In Tezos it is called “context”. Tezos’ context is a huge file, where its storage subsystem builds its own directory structure using Irmin and LMDB.

The context carries account and contract related data stored in key-value mappings where keys are accounts and contracts. This KVS is encoded using the storage subsystem’s internal file-directory structure.

There is a vulnerability around this KVS in the storage subsystem, and up to 003_PsddFKi3 (the previous protocol of Athens) it was possible to perform DoS attacks to the Tezos nodes. It can slow down the network, but attackers cannot steal tokens just with this vulnerability as far as I understand.

I am not going to disclose the full details for now. The full detailed report is unlocked when the issue will be really fixed. Meantime, you can see the BLAKE2 hash of my draft:

$b2sum draft.md 245c031ebf72745c1d6550aa15623d29565f6ec7cece5221162bcca607e485c238bd834782f3ed8a25dc1180dca1c31927064e039618ebdd0b1e651aefd4cecd draft.md  Contract reindexing of Athens A mitigation for this DoS vulnerability was introduced by Athens, but quietly. Let’s see the explanation of “Context reindexation” in Nomadic Lab’s announcement of Athens: It is a preliminary to future improvements of the account system, such as making key handling more flexible for bakers. The announcement never mentioned it as a security mitigation because Tezos’ protocol amendment is a slow process takes 3 months. Nomadic labs carefully chose words not to send a signal to hackers who otherwise would have attacked the previous protocol running with the issue at that time. I guess it also aimed to prepare the possible emergency situation: if someone would have found the vulnerability and attacked the network, Tezos could have asked its community an emergency security hard fork, by showing the fix available in the Athens’ proposal. Luckily such events did not happen. This mitigation did not solve the vulnerability entirely but just made the attack very difficult adding more calculation for it. The real definitive solution is to fix the storage subsystem itself. Then, why Athens did not fix the storage subsystem once and for all? Since storage is not in the protocol, but in the shell, and shell fix and upgrade are not targets of protocol amendment. Famous "el Pulpo" of Tezos. The green brain of the octopus is the protocol, which requires amendment voting to upgrade (except emergency security hard forks). The rest of the octopus is called shell, where the storage subsystem lies. The purpose of this reindexation in Athens was to protect Tezos until the storage subsystem is properly fixed. I believe now people are working hard on the fix of the storage. What Brest is going to do, and what it has done already Now let’s see the explanation of the security fix of Brest: A security issue. The rehashing performed during Athens protocol change was not enough to prevent some kinds of attacks. This amendment performs a new rehashing that makes these attacks ineffective. The path length of addresses is increased from 7 to 9, making the attack 65536 times more difficult. The modification is trivial: it changes a parameter Athens has introduced from 7 to 9, to make the attack 65536 times more difficult. Questions remain to me. As of today (2019-06-17), we have not observed any attack attempt using this vulnerability. The problem itself remains in the storage subsystem even after Brest. Nothing is explained why 7 are not enough but 9 are. (Since this is a security issue, the reason might be explained privately to Tezos Foundation.) Unfortunately Brest has announced the existence of the vulnerability to public. It could have been done in a more silent manner as Athens did; the proposer should have known what they are doing because they have the perfect understanding of the vulnerability and how Athens has introduced the mitigation with careful wording. Protocol upgrade reward in a block What happened between 003_PsddFKi3 and Athens Athens has charged ꜩ100 to the proposer’s account tz1iSQEcaGpUn6EW5uAy3XhPiNg7BHMnRSXi for the upgrade reward. As you can see in block explorers, this reward was minted from nothing. There is no transaction or anything about it: neither the last block of 003_PsddFKi3 nor the first block of Athens tells you about the reward. But you can check the reward is indeed added to the account after the last block of protocol 003_PsddFKi3: At block level 458751, just before the last block by 003_PsddFKi3: $ ./tezos-client -b BMXd3qGXLKjJjdUjqCFG4HA78od8dL7JvC284gunZ55An6eWR1N-1 get balance for tz1iSQEcaGpUn6EW5uAy3XhPiNg7BHMnRSXi
0 ꜩ

After the next block at level 458752, when Athens starts working:

\$ ./tezos-client -b BMXd3qGXLKjJjdUjqCFG4HA78od8dL7JvC284gunZ55An6eWR1N get balance for tz1iSQEcaGpUn6EW5uAy3XhPiNg7BHMnRSXi
100 ꜩ

Brest claims this current behaviour makes block explorers difficult to track the protocol update rewards and they need ad-hoc fixes for each protocol update. It is true, but it only happens once three months at most, and the explorers themselves often have to be upgraded at protocol changes in any way.

Brest’s approach

Brest makes this reward payment recorded in a block explicitly.

For this, it cannot invoice the reward to its proposer just after the last block of the previous protocol, as Athens did. This is since the new protocol cannot write the balance update to the last block of the previous protocol.

Therefore, the balance update of the reward must be recorded in the first block of the new protocol. For this purpose, Brest introduces an integer state Storage.Pending_protocol_init and set it to 1 (or true) at the end of Athens. This integer is used as a flag to trigger the reward payment and its balance update injection to the first block of Brest. The flag is reset to 0 (or false) after the reward payment of course.

I am not sure whether this implementation with a new flag state is the best here. Is it hard to give the reward simply when the new protocol handles its first block?


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## Absence of caution: The European vaccine suspension fiasco

[Repeated from commoninfirmities blog.]

Multiple European countries have now suspended use of the Oxford/AstraZeneca vaccine, because of scattered reports of rare clotting disorders following vaccination. In all the talk of “precautionary” approaches the urgency of the situation seems to be suddenly ignored. Every vaccine triggers serious side effects in some small number of individuals, occasionally fatal, and we recognise that in special systems for compensating the victims. It seems worth considering, when looking at the possibility of several-in-a-million complications, how many lives may be lost because of delayed vaccinations.

I start with the case fatality rate (CFR) from this metaanalysis, and multiply them by the current overall weekly case rate, which is 1.78 cases/thousand population in the EU (according to data from the ECDC). This ignores the differences between countries, and differences between age groups in infection rate, and certainly underestimates the infection rate for obvious reasons of selective testing.

Let’s assume now that all of the blood clotting problems that have occurred in the EEA — 30 in total, according to this report — among the 5 million receiving the AZ vaccine were actually caused by the vaccine, and suppose (incorrectly) that all of those people had died.* That would produce a vaccine fatality rate (VFR) of 6 per million. We can double that to account for possible additional unreported cases, or other kinds of complications that have not yet been recognised. We can then calculate how many days of delay would cause as many extra deaths as the vaccine itself might cause.

The result is fairly clear: even the most extreme concerns raised about the AZ vaccine could not justify even a one-week delay in vaccination, at least among the population 55 years old and over. (I am also ignoring here the compounding effect of onward transmission prevented by vaccination, which makes the delay even more costly.) As is so often the case, “abundance of caution” turns out to be fundamentally reckless.

* I’m using only European data here, to account for the contention that there may be a specific problem with European production of the vaccine. The UK has used much more of the AZ vaccine, with even fewer problems.

## What’s the matter with Santa Clara?

I can’t get out of my head the instantly infamous paper by Bendavid et al. at Stanford Medical School from a few weeks back, claiming to estimate the prevailing exposure to SARS-Cov-2 in Santa Clara County, California, by antibody tests on a random sample of the population. The study has garnered a lot of attention because of its optimistic outcome: Something between 1.5 and 4 percent of the population were estimated to have been already infected, far more than the official number of cases, which, if true, would imply an infection fatality rate below 0.2%.

The paper has attracted a great deal of criticism, and has already been revised. Much of the critique was targeted at the problematic sampling procedure — which I think is not such a big deal — and the weird ethical lapses. But the gaping hole in the paper is in its treatment of test specificity.

The problem, in a nutshell, is this: 1.5% of the tests they carried out (50 out of 3,330) yielded positive results. In order to estimate the true rate of seropositivity in the population, we need to know approximately how many of those positive test results were correct. They estimate the specificity from their laboratory tests of samples known to be negative (because they were produced before the origin of the virus, or, at least, its entrance into the US), of which 368 out of 371 were indeed negative. (Strangely, the first version of the paper reported 369 out of 371, which is a pretty substantial difference. This figure has not been consistently updated.) This suggests the test specificity is 99.2%, which would make about half the positive results false. But it’s even worse than that, because 3 false results out of 371 might be a randomly low number for a much higher failure rate. The paper recognises this, appealing to the Clopper-Pearson method for computing confidence intervals for binomial probabilities, which yields at a 95% confidence intervals of about (97.6%, 99.9%). If the specificity is anything below 98.5% and they measured 1.5% positives, then the maximum likelihood estimate for the true positive rate is 0. This raises two questions:

1. They claim to have taken the specificity into account, saying “We use a bootstrap procedure to estimate confidence bounds for the unweighted and weighted prevalence, while accounting for sampling error and propagating the uncertainty in the sensitivity and specificity.” They produced a 95% CI of (0.7%, 1.8%). What did they think they were doing?
2. How should this be analysed?

The bootstrap is a brilliant tool, but since it resamples from the data, or from the fitted model, it tends to miss out on uncertainty from what could have been observed but wasn’t. It’s not clear, though, whether the bootstrap has anything to do with the problem here. The updated version includes an appendix with a calculation from Bayes’ Theorem that the probability of a sampled person being infected is $\frac{q+s-1}{r+s-1}$, where s is the specificity, r is the sensitivity (the probability that a seropositive subject receives a positive test) and q is the fraction of the population with a positive test. They claim to have generated a bootstrap estimate sampling their s from the “empirical distribution” of specificity, but they don’t say what they do with the negative estimates for the infection incidence that must inevitably arise. Instead, they give (in the appendix to the updated version of the paper) a long and (to my mind) incoherent argument against critics who say that the confidence interval should include 0 — since the false positive rate has a reasonable chance of being larger than the fraction of positive test results — mainly because we know from other evidence that the infection incidence can’t actually be 0. True enough, but the point is not that the rate is actually 0%, but that these data give us no reason to believe in any particular lower bound above 0.

How should the data be analysed? The problem is, Bayes’ Theorem doesn’t play well with frequentist confidence intervals, so it’s easy to end up in a logical muddle if we try to calculate a confidence interval for the true probability of seropositive, given that we’ve observed 1.5% positive. What does it even mean? It would be quite reasonable to say, values of false-positive rate above 1.5% are perfectly plausible, in which case true positive rates down to 0% are plausible, hence there is no justification for claiming evidence for any positive lower bound on the true incidence of seropositivity, and then call it a day.

The “bootstrap” approach that they seem to have made up without much justification is a kind of incoherent mixed-Bayesian-frequentist approach: Inverting the normal approximation to the sampling distribution of the parameters (I think that’s what they mean by the “empirical distribution”) as though it were a posterior, that they then sample from to generate .

Instead, it would be most natural to analyse this in a purely Bayesian framework. You could start off with a uniform prior on the true seropositive rate π and the specificity s. Then the posterior will be proportional to $\bigl( 1 - s(1-\pi)\bigr)^{50}\bigl(s(1 - \pi) \bigr)^{3280} s^{368}(1-s)^3$. This doesn’t provide a nice closed-form solution, but it can be calculated numerically with various software tools, such as the Wolfram Alpha double integral calculator. The posterior distribution of π has mean 0.0067, and a symmetric 95% credible interval (0.049%, 1.41%) for the seropositive incidence. This could be repeated for each of their demographic groups, which I’m loath to take the time to do, but it would presumably inflate both ends of the confidence interval by a similar factor, producing something like (0.06%,3.8%).

The updated version of the paper includes a large number of additional tests on pre-Covid samples, bringing the total to 3324, with just 16 false positives, so again about a 99.5% specificity. If we believe this, we get a much more favourable credible interval — though still not as favourable as reported in the paper — of (0.55%,1.52%). (They report a 95% confidence interval for the unweighted incidence (0.7%, 1.8%).)

Should we believe their new results? I’m not sure. One argument against: They report on 13 different sources of negative samples, but these produce wildly variable specificities. The most extreme is a sample of 150 pre-Covid serum samples that yielded 4 false positives, and on the other side 1102 samples that yielded no false positives — both have substantially less than 1% probability if specificity is 99.5%, but they pull in different directions.

For the aficionados: Actually, there’s also a sensitivity that needs to be considered. In tests of 157 known seropositive samples, only 130 produced positive results. This isn’t as problematic as the specificity — which has the potential for wiping out all of the positive tests — but it will increase our estimates of the true positive rate, while widening the uncertainty. Writing r for the sensitivity (the probability of a positive test given that the individual is seropositive) the posterior likelihood now becomes

$\bigl( 1-s+ \pi(r+s-1)\bigr)^{50}\bigl(s - \pi(r+s-1) \bigr)^{3280} s^{368}(1-s)^3 r^{130}(1-r)^{27}$

with the original sensitivity data, or

$\bigl( 1-s+ \pi(r+s-1)\bigr)^{50}\bigl(s - \pi(r+s-1) \bigr)^{3280} s^{3308}(1-s)^{16} r^{130}(1-r)^{27}$

with the updated sensitivity data.

The 95% credible intervals for π (without demographic weighting) are then (.06%,1.7%) with the original sensitivity data, and (0.67%, 1.9%) with the updated sensitivity data. This last is fairly close to the reported results in the revised paper. Unsurprising, since with 3000 instead of 300 samples, the uncertainty in the sensitivity is no longer a major issue.

## Fluctuations in reports of daily Covid deaths

As we are all watching the daily reports of Covid-19 cases and deaths, one conspicuous fact is that there are enormous fluctuations from day to day. It’s clear that we shouldn’t be too focused on the counts from an individual day to understand the trend. But why do they fluctuate so much?

Here are the numbers for the UK since the start of April (data from the European Centre for Disease Prevention and Control):

We have 381 deaths one day, then 743 deaths the next day, then 389, then 684. A natural explanation is that it has to do with delays in reporting weekends and/or holidays. But the 381 and 389 were on Wednesday and Friday, and the next peak came on a Saturday and Sunday. I don’t have an explanation, but I wanted to investigate whether this is a real issue. Are the day-to-day fluctuations more than you would expect purely by chance?

It’s not immediately obvious how many we should expect, since there is clearly a trend. I decided to estimate it in two different ways:

1. Choose a range s of days — most naturally s=3 — and estimate the predicted cumulative number of events C[x] on day x as the geometric average of the cumulative number of events to day x+s, and the cumulative number to day x-s. This also gives us a natural estimate of the growth rate r over the period, r=(C[x+s]/C[x-s])^(1/2s) -1, and we calculate the Predicted number of events on day x as rC[x]/(1+r). We call this the geometric average method.
2. As above, but we estimate r by the maximum likelihood estimator, assuming that the number events on day x is Poisson distributed with parameter rC[x-1], where C[x-1] is the observed cumulative number on day x-1. We call this the cumulative MLE method.
3. An approach that addresses more directly the fluctuations in daily events computes the maximum likelihood estimators for a two-parameter model with parameters λ and r, where the number of events on day x+k is Poisson(λr^k), taking as observations the numbers on days x-s,x-s+1,…,x-1,x+1,…,x+s. We call this the single-day MLE method.

### Deaths

Here are the results for the UK. The discrepancy between observed and predicted has been standardised to standard deviation 1, so we should not expect to see results outside of the range (-2.5,+2.5) by chance.

There’s no obvious pattern. The geometric-average increase-rate is always larger, hence the discrepancies are smaller. Not surprisingly, the growth rates estimated from single-day counts are more variable, but the discrepancies are, surprisingly, less extreme. With any method the discrepancies are extreme in both directions, and don’t seem to have an obvious pattern with respect to the weekends. We see extreme positive deviations on weekends, and extreme negative deviations on weekdays. (I have shaded Sunday and Monday as weekends, supposing that most of the deaths in hospitals on Saturday and Sunday would appear in national totals the following day. According to some estimates, the most common delay is 2 days, suggesting that Monday and Tuesday counts should be expected to be low.)

Here is Germany:

These show, perhaps, a more consistent weekly pattern, with negative deviations on Sundays and Mondays.

The US:

Italy shows an odd alternating pattern:

### Cases

The dynamics of new cases are expected to be different. Here is the equivalent plot for new cases in the UK

The fluctuations seem to be increasing, with huge excess number of cases reported on Easter, for some reason — 8719 new cases, where the largest reported on any other day was 5903.

The US also shows this intensification of fluctuations

Germany, Italy, and Canada don’t seem to show this pattern. The plots for Germany and Canada each is dominated by a single very extreme day, possibly corresponding to a shift in counting criteria.

## Putting Covid-19 mortality into context

[Cross-posted from Common Infirmities blog.]

The age-specific estimates of fatality rates for Covid-19 produced by Riou et al. in Bern have gotten a lot of attention:

These numbers looked somewhat familiar to me, having just lectured a course on life tables and survival analysis. Recent one-year mortality rates in the UK are in the table below:

Depending on how you look at it, the Covid-19 mortality is shifted by a decade, or about double the usual one-year mortality probability for an average UK resident (corresponding to the fact that mortality rates double about every 9 years). If you accept the estimates that around half of the population in most of the world will eventually be infected, and if these mortality rates remain unchanged, this means that effectively everyone will get a double dose of mortality risk this year. Somewhat lower (as may be seen in the plots below) for the younger folk, whereas the over-50s get more like a triple dose.


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# Factorization of polynomials

factoring polynomials

Since C.F. Gauss it is known that an arbitrary polynomial over a field or over the integers can be factored into irreducible factors, essentially uniquely (cf. also Factorial ring). For an efficient version of Gauss' theorem, one asks to find these factors algorithmically, and to devise such algorithms with low cost.

Based on a precocious uncomputability result in [a13], one can construct sufficiently bizarre (but still "computable" ) fields over which even square-freeness of polynomials is undecidable in the sense of A.M. Turing (cf. also Undecidability; Turing machine). But for the fields of practical interest, there are algorithms that perform extremely well, both in theory and practice. Of course, factorization of integers and thus also in $\mathbf Z [ x ]$ remains difficult; much of cryptography (cf. Cryptology) is based on the belief that it will remain so. The base case concerns factoring univariate polynomials over a finite field $\mathbf{F} _ { q }$ with $q$ elements, where $q$ is a prime power. A first step is to make the input polynomial $f \in \mathbf{F} _ { q } [ x ]$, of degree $n$, square-free. This is easy to do by computing $\operatorname { gcd } ( f , \partial f / \partial x )$ and possibly extracting $p$th roots, where $p = \operatorname { char } \mathbf{F} _ { q }$. The main tool of all algorithms is the Frobenius automorphism $\sigma : R \rightarrow R$ on the $\mathbf{F} _ { q }$-algebra $R = \mathbf{F} _ { q } [ x ] / ( f )$. The pioneering algorithms are due to E.R. Berlekamp [a1], [a2], who represents $\sigma$ by its matrix on the basis $1 , x , x ^ { 2 } , \ldots , x ^ { n - 1 } ( \operatorname { mod } f )$ of $R$. A second approach, due to D.G. Cantor and H. Zassenhaus [a3], is to compute $\sigma$ by repeated squaring. A third method (J. von zur Gathen and V. Shoup [a8]) uses the so-called polynomial representation of $\sigma$ as its basic tool. The last two algorithms are based on Gauss' theorem that $x ^ { q ^ { d } } - x$ is the product of all monic irreducible polynomials in $\mathbf{F} _ { q } [ x ]$ whose degree divides $d$. Thus, $f _ { 1 } = \operatorname { gcd } ( x ^ {q } - x , f )$ is the product of all linear factors of $f$; next, $f _ { 2 } = \operatorname { gcd } ( x ^ { q ^ { 2 } } - x , f / f _ { 1 } )$ consists of all quadratic factors, and so on. This yields the distinct-degree factorization $( f _ { 1 } , f _ { 2 } , \ldots )$ of $f$.

The equal-degree factorization step splits any of the resulting factors $f_i$. This is only necessary if $\operatorname { deg } f _ { i } > i$. Since all irreducible factors of $f_i$ have degree $i$, the algebra $R _ { i } = \mathbf{F} _ { q } [ x ] / ( f _ { i } )$ is a direct product of (at least two) copies of $\mathbf{F} _ { q ^ { i } }$. A random element $a$ of $R_i$ is likely to have Legendre symbol $+ 1$ in some and $- 1$ in other copies; then $\operatorname { gcd } ( a ^ { ( q ^ { i } - 1 ) / 2 } - 1 , f _ { i } )$ is a non-trivial factor of $f_i$. To describe the cost of these methods, one uses fast arithmetic, so that polynomials over $\mathbf{F} _ { q }$ of degree up to $n$ can be multiplied with operations in $\mathbf{F} _ { q }$, or $O ^ { \sim } ( n )$ for short, where the so-called "soft O" hides factors that are logarithmic in $n$. Furthermore, $\omega$ is an exponent for matrix multiplication, with the current (in 2000) world record $\omega < 2.376$, from [a4].

All algorithms first compute $x ^ { q }$ modulo $f$, with $O ^ { \sim } ( n \operatorname { log } q )$ operations in $\mathbf{F} _ { q }$. One can show that in an appropriate model, $\Omega ( \operatorname { log } q )$ operations are necessary, even for $n = 2$. The further cost is as follows:'

 ${\bf F} _ { q } [ x ] / ( f )$ as Cost in $O ^ { \sim }$ Berlekamp $\mathbf{F} _ { q }$-vector space $n ^ { \omega }$ Cantor–Zassenhaus multiplicative semi-group $n ^ { 2 } \operatorname { log } q$ von zur Gathen–Shoup $\mathbf{F} _ { q }$-algebra $n ^ { 2 }$

For small fields, even better algorithms exist.

The next problem is factorization of some $f \in \mathbf Q [ x ]$. The central tool is Hensel lifting, which lifts a factorization of $f$ modulo an appropriate prime number $p$ to one modulo a large power $p ^ { k }$ of $p$. Irreducible factors of $f$ will usually factor modulo $p$, according to the Chebotarev density theorem. One can then try various factor combinations of the irreducible factors modulo $p ^ { k }$ to recover a true factor in $\mathbf{Q} [ x ]$. This works quite well in practice, at least for polynomials of moderate degree, but uses exponential time on some inputs (for example, on the Swinnerton-Dyer polynomials). In a celebrated paper, A.K. Lenstra, H.W. Lenstra, Jr. and L. Lovász [a12] introduced basis reduction of integer lattices (cf. LLL basis reduction method), and applied this to obtain a polynomial-time algorithm. Their reduction method has since found many applications, notably in cryptanalysis (cf. also Cryptology). A method in [a10] promises an even faster factorizing method.

The next tasks are bivariate polynomials. It can be solved in a similar fashion, with Hensel lifting, say, modulo one variable, and an appropriate version of basis reduction, which is easy in this case. Algebraic extensions of the ground field are handled similarly.

Multivariate polynomials pose a new type of problem: how to represent them? The dense representation, where each term up to the degree is written out, is often too long. One would like to work with the sparse representation, using only the non-zero coefficients. The methods discussed above can be adapted and work reasonably well on many examples, but no guarantees of polynomial time are given. Two new ingredients are required. The first are efficient versions (due to E. Kaltofen and von zur Gathen) of Hilbert's irreducibility theorem (cf. also Hilbert theorem). These versions say that if one reduces many to two variables with a certain type of random linear substitution, then each irreducible factor is very likely to remain irreducible. The second ingredient is an even more concise representation, namely by a black box which returns the polynomial's value at any requested point. A highlight of this theory is the random polynomial-time factorization method in [a11].

Each major computer algebra system has some variant of these methods implemented. Special-purpose software can factor huge polynomials, for example of degree more than one million over $\mathbf{F} _ { 2 }$. Several textbooks describe the details of some of these methods, e.g. [a9], [a5], [a6], [a14].

Factorization of polynomials modulo a composite number presents some surprises, such as the possibility of exponentially many irreducible factors, which can nevertheless be determined in polynomial time, in an appropriate data structure; see [a7].

For a historical perspective, note that the basic idea of equal-degree factorization was known to A.M. Legendre, while Gauss had found, around 1798, the distinct-degree factorization algorithm and Hensel lifting. They were to form part of the eighth chapter of his "Disquisitiones Arithmeticae" , but only seven got published, due to lack of funding.

#### References

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How to Cite This Entry:
Factorization of polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Factorization_of_polynomials&oldid=50659
This article was adapted from an original article by Joachim von zur Gathen (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article