A new branch in my blog: Macronectes

When I started this blog it was intended to cover all subjects I am interested in. In the recent past there have been an increasing number of posts related to politics. Since I live in Germany they have often been concerned with topics in German politics. It was rather inconvenient writing about these themes in English. For this reason I now started a new blog Macronectes. This is intended to house posts on politics and related philosophical themes. The posts there will all be in German. The first post is related to one which I wrote in Hydrobates on grammatical gender in Germany. I will continue to write posts on all other subjects in Hydrobates as before. Anyone curious to know what the name Macronectes means can look here.

One-dimensional centre manifolds, part 2

I continue with the discussion of the previous post. There I mentioned a preliminary transformation of the coordinates. I will discuss this in more detail here. Suppose we have a two-dimensional dynamical system for unknowns and a steady state . The usual aim is to understand systems up to certain transformations. One type of transformation is of the form . A second is of the form . A third is of the form . With the assumptions which have been made we can do a translation to achieve . Assume now that the linearization of the system at the origin is diagonalizable with rank one. Then it has one eigenvalue zero and one non-zero eigenvalue. It can be diagonalized by a linear transformation of the coordinates. After these transformations the system has almost been reduced to the form in the last post, except that there is a non-zero constant in front of the first summand in the evolution equation for which might not be . It can be reduced to the latter case by rescaling the time coordinate. A similar process can be carried out when the centre manifold is of higher codimension. Then the variable becomes vector-valued and the first summand in the evolution equation for is if the form for an invertible matrix . The function defining the centre manifold is then also vector-valued. The centre manifold analysis can be done in a manner very similar to what we have seen already.

The discussion of the fold bifurcation also generalizes in a straighforward way to the case of higher codimension. There is, however, one thing about that discussion which is unsatisfactory. The bifurcation conditions are expressed in terms of the transformed coordinates. It would be more satisfactory, because more invariant, if they could be expressed in terms of the original coordinates. This leads to considering the way in which these conditions are affected by the three types of coordinate transformations previously discussed. The first type of transformation leaves the conditions involving only and its derivatives with respect to unchanged. The derivative of with respect to is rescaled by the non-zero factor and so the fact of its being non-zero is unchanged. The third type of transformation just scales all the relevant quantities by the non-zero factor and so also does not change whether these are zero or not. It remains to consider the second type of transformation. At this point a more geometrical point of view will be adopted. We change the notation for the right hand side of the equations to . These are the components of a parameter-dependent vector field which satisfies . Because of this condition the derivative defines a vector at the origin independently of the transformation of the second type. The linearization of the vector field at the origin defines a tensor. We suppose as before that it is diagonalizable and of rank one. Let be a vector spanning the kernel of , Thus using the Einstein summation convention we have . Similarly there is a one-form with . It is unique up to rescaling and we choose it to satisfy . With these notations the condition previously written in the form can be written in the invariant form . The condition previously written in the form can be written in the invariant form . Here we use the notation . It is because the vector field vanishes at the origin and is in the kernel of its first derivative that the expression involving the second derivative is invariant. These considerations may be compared with those in section 5.4 in the book of Kuznetsov.

One-dimensional centre manifolds

I have used one-dimensional centre manifolds in my research on several occasions. I now see that I always did this in quite an ad hoc way. I did not exercise due diligence in the sense that I did not take the time to get a general picture of what was going on, so as to be able to use this technique more efficiently in the future. Now I want to do so. I start with the two ODE and . This example is general enough to illustrate several important ideas. Here and are supposed to be smooth and vanish at least quadratically near the origin. The linearization of this system has the eigenvalues and zero. Its kernel is spanned by the vector with components . It follows that there exists a centre manifold of the form , where has any desired finite degree of differentiability. By definition this manifold is invariant under the flow of the system, it passes through the origin and its tangent space there is the axis. Consider the Taylor expansion . Substituting this into the evolution equation for gives . This is a leading order approximation to the flow on the centre manifold. If we can use it to read off the stability of the origin within the centre manifold. This argument uses no information about the way the centre manifold deviates from the centre subspace, i.e. how fast grows near zero. If we need to go further.

Differentiating the defining equation with respect to time and substituting the evolution equations into the result gives . Call this equation (*). We have Taylor expansions and . Substituting these into (*) we see that the left hand side is of order three. Thus the same must be true of the right hand side and we get and . Substituting this back into the evolution equation for gives . Thus if the stability of the origin is determined by the relative sign of and , provided these coefficients are non-zero. If one of them is zero we can do another loop of the same kind. Looking at the third order terms in the equation (*) we get . This allows the fourth order term in the evolution equation for to be determined. This procedure can be repeated as often as desired to get higher order approximations for the centre manifold and the restriction of the system to that manifold.

We can now sum up the steps involved in doing a stability analysis. First look at the coefficient of in the equation for . If it is non-zero we are done. If it is zero use the equation (*) to determine the leading term in the expansion of the centre manifold. Put this information into the equation for . If the leading term is non-zero we are done. If it is zero we can repeat the process as long as is necessary to get a case in which the leading order coefficient is non-zero. As long as this point has not been reached we cycle between using (*) and the equation for . The system I have discussed here was special. The codimension of the centre manifold was one and the system was in a form which usually could only be achieved by a preliminary linear transformation of the coordinates. The special case nevertheless exhibits the essential structure of the general case and can serve as a compass when treating examples.

These ideas can be extended to give information about bifurcations. The equations are replaced by and , where is a parameter. This can be made into a three-dimensional extended system by adding the equation . The origin is a steady state of the extended system and the centre manifold at that point is of dimension two. It is of the form . Suppose that we are in the case . Then this looks very much like the case of a generic fold bifurcation. We are just missing one condition on the parameter dependence. The dynamics on the centre manifold is given by . Of course the equation remains unchanged. We can now check the conditions for a generic fold bifurcation in the system reduced to the centre manifold. The first is . The second is . Hence is equivalent to . The third involves . We see that is equivalent to . The fourth involves . We see that is equivalent to . The first three conditions for a generic fold bifurcation of the system on the centre manifold are already satisfied and the fourth is equivalent to . In this way the bifurcation conditions can be expressed directly in terms of the coefficients of the original system. This is an illustration in a relatively simple example of a relationship discussed in much more general cases in the book of Kuznetsov.

Event with Marie-Agnes Strack-Zimmermann

I have a rather poor opinion of most current politicians. An exception is Marie-Agnes Strack-Zimmermann. I have seen her from time to time in short TV appearances and I have also read about her. All this made a positive impression on me. When I saw that she was due to talk at an event at the University of Mainz on Saturday I decided to go there and Eva, who also previously had a positive impression of the speaker, accompanied me. What we experienced at the event strengthened our previous opinion. The speaker came in with a microphone in a very modest way and just started to talk, without any introduction. Strack-Zimmermann is a member of the FDP and is their leading candidate for the coming European elections. This event was certainly part of her campaign for that election and was organized by her party. At the same time it should be emphasized that she did not say ‘vote for me’ but instead ‘go out and vote for a democratic party’, with a particular recommendation not to vote for the AfD or the party of Sahra Wagenknecht who have both explicitly said that they want Germany to leave the EU. I am not a devotee of the FDP. I find some of their policies good and others bad. I went to the event not because Strack-Zimmermann is a member of the FDP but also not in spite of that fact. My motivation was independent of the party she belongs to. We both thought that she made a milder impression than on TV. Probably the reason is that she was in a relatively friendly environment. When she is forced to defend herself against political attacks she is very capable of doing so and then she is less mild. At this event one person did shout out something about peace from the back row. This might have been due to the fact that Strack-Zimmermann is a strong and outspoken supporter of military intervention in the Ukraine by Germany and other Western countries or it might have had to do with Gaza. In any case she was easily able to handle it. In particular she repeated several times, ‘We all want peace’.

Strack-Zimmermann is chair of the defence committee in the German Parliament. Correspondingly her appearances in the media are often related to military themes. She was in the news recently because of her support of providing the Ukraine with the Taurus cruise missile, thus opposing the policy of Chancellor Scholz. She voted in favour of an initiative of the opposition party CDU that Taurus should be provided to the Ukraine. She was the only member of the government to do so. In her presentation yesterday she discussed many political themes and in particular how they all relate to each other. She is qualified to talk about these things because she has been more than once in the Ukraine during the present war, because she has been in other hotspots such as Mali and Niger, because she has spoken personally with one of the Israeli women taken hostage by Hamas and meanwhile released etc. For me it was refreshing to hear a politician talking in a way which struck me as honest, well-informed, experienced, rational and courageous. One thing which surprised me was what she said about the population of Europe compared to that of the world. She gave the figure 5% and I found that very low. In the internet I found the figure 10% which would have surprised me almost as much. Perhaps the explanation for the discrepancy in the figures is that I was not paying enough attention and she mentioned the population of the EU and in that case 5% could be correct. She talked about many political themes, including the Ukraine, China and Taiwan, the US and NATO, the Red Sea and the Houthis and of course Gaza. At the end of her presentation she took questions. An interesting one came from a young woman who identified herself as being in the army. She asked why the German army was not recruiting people from other European countries. Strack-Zimmermann pointed out the following problem. Soldiers in Germany are paid significantly better than soldiers in many other European countries. Thus the danger exists that if Germany tried to recruit in this way this might seriously weaken the armies of allied countries by draining the human resources. She indicated that discussions were underway to find an alternative.

This speech was not recorded but another presentation by Strack-Zimmermann can be found here:

There is quite a lot of overlap in the topics but it was more defiant in tone that what we heard live, as befits an election speech made to politicians.

Nobel lecture of Harvey Alter

In 2020 the Nobel prize for medicine was awarded to Harvey Alter, Michael Houghton and Charles Rice for their role in the discovery of the hepatitis C virus. I now watched the videos of the corresponding Nobel lectures. For my taste the lecture of Alter was by far the most interesting of the three. I think that he was also the one who played the most fundamental role in this discovery. At the beginning of his lecture he emphasizes the point that the most important discoveries in science often come as a complete surprise and not as a result of planned research programmes. Alter was 85 when he got the prize and so he had to wait a long time for it. The papers documenting his fundamental contributions were published in 1989. A central part of this work was the collection and preservation of blood samples from patients undergoing open heart surgery. Why was this group chosen? One of the most important modes of infection with hepatitis B and C used to be blood transfusions. This continued to be the case until tests were available to screen donors for these diseases. This kind of surgery involves extensive blood transfusions and so the chances of infection were relatively high in these patients. Also these patients suffered from relatively few other diseases which could have been confounding factors. These blood samples were an invaluable resource in the search for the virus. They were the basis of painstaking analysis over many years.

One important feature of hepatitis C is that it becomes chronic in 70 per cent of cases. This looks like a failure of the immune system to handle this disease. What are the reasons for this failure? One concerns quasispecies. The hepatitis C virus has an RNA genome and the copying of RNA is very error-prone. This leads to a huge variety in the genomes of virions in a single patient. This in turn results in rapid mutations of the virus. If an antibody has developed to combat the virus then selective pressure will quickly cause a new form to become dominant which is not vulnerable to that antibody. It seems to me that if this type of effect is to be captured using mathematical model it will require a stochastic model. Deterministic models of the type I have studied in the past are probably not helpful for that. In the lecture it is also mentioned that the number of T cells (CD4+ and CD8+) declines very much in chronically infected hepatitis C patients. No explanation is offerred as to why that is the case. Deterministic mathematical models might be able to contribute some understanding in that case.

The lecture contains the following interesting story. There was a time at which liver cancer was much more common in Japan than in the West. The reason for this was that that cancer was in many cases a late stage effect of hepatitis C. During wars in the early part of the 20th century many Japanese soldiers injected drugs with shared needles and this was what spread the disease. It was observed that there were many cases of jaundice (the most striking symptom of hepatitis) on the battlefield. Decades later many of these men developed serious liver disease, including cancer. Japanese doctors predicted that a similar phenomenon would be seen in the West when the effects of recreational drug use became manifest. They were right.

The call of the north and the voyage of the Vega

I grew up in the Orkney Islands, a place which is further north than most people live. As a schoolboy I had a map of the world on my bedroom wall and I was fascinated by faraway places and travel. A natural consequence of my place of birth is that when I heard about people I knew travelling they were almost always travelling towards the south. For this reason the north seemed to me to be the direction which was most exotic. At one time I started reading books about arctic exploration. One of the first of these, and probably the best, was the book of Fridtjof Nansen about his voyage with his ship Fram. It was a perfect book to capture my imagination about the far north. On the basis of the fact that wreckage from a ship which sank in the Bering Strait was found in Greenland Nansen was convinced that there was a flow of ice in this direction. He decided to let a ship get frozen into the ice near Siberia in the hope that this current would carry it near the North Pole. The Fram was a ship specially built so that when it was squeezed by the surrounding ice it would be lifted to the surface of the ice instead of being crushed and sunk. His plan worked and the Fram was eventually released by the ice near Spitzbergen. He himself left the ship at what he judged to be the most northerly point of its trajectory in an attempt to be the first to reach the North Pole. He did not reach the pole and turned around to reach Franz Josef Land. There he met another expedition which was able to bring him back to civilisation.

Recently a book came into my hands about another arctic explorer, Adolf Erik Nordenskiöld. It is called ‘Nordostpassage’ [northeast passage] and is by Friedrich-Franz von Nordenskjöld, a descendant of the explorer. As the title indicates, the most famous achievement of Nordenskiöld was that he led the first expedition through the northeast passage, i.e. this was the first time that someone had travelled by ship from Europe to the Bering Strait along the north coast of Siberia. This was something which was a worldwide sensation at that time. For instance at the end of his journey he was invited to visit the emperor of Japan. The success of an expedition of this kind depends a lot on luck but I think this particular expedition depended essentially on the personal qualities of its leader. At the same time I had the feeling that he was often stubborn in an unreasonable way. At least he was apparently a lot more competent than Scott on his attempt to reach the South Pole. (Of course once I had started to read about arctic explorers I also had to read about antarctic ones. Cf. my post on my trip to Ushuaia.) The Vega was the ship which successfully achieved the northeast passage. In fact almost the whole voyage was completed in one summer. Unfortunately, when already close to the Bering Strait the ship got caught in the ice and had to spend the winter a short way from its goal. It was necessary to wait until late July before further progress was possible. On the expedition nobody died and no ships sunk, which is not to be taken for granted for an expedition of this kind.

The starting point for the voyage (and for other arctic expeditions of Nordenskiöld) was Tromsø. It was interesting to read that during one visit to Tromsø Nordenskiöld saw the Admiral Tegethoff, an Austrian ship on an unknown mission. I read an account of that expedition a long time ago but I think it was more of a literary work than a documentary one. I do not remember the title. What that expedition actually did was that it discovered Franz Josef Land. I visited Tromsø myself on my first trip to the far north in 1986. In that year I attended my first international conference in Stockholm as a PhD student and the temptation was great to use the opportunity to travel to the north afterwards. I took an overnight train to Kiruna and then travelled further to Abisko. I had a tent with me and camped there, being almost eaten by mosquitoes during the night. After that I switched to youth hostels despite my limited finances. My best memories of Abisko are numerous Bluethroats (I do not think I have seen another one since) and my first Long-Tailed Skuas. I travelled on to the end of the train line in Narvik. From there I took a bus to Tromsø. At midnight I boarded the Hurtigrute and crossed to Svolvaer in the Lofotens. In 1997 I passed Tromsø again on a cruise after visiting Iceland and Spitzbergen but did not spend any time there. I previously wrote something about that cruise here. That cruise also brought me to the most northerly point I have reached up to now in my life which is the Magdalenenfjorden at the north-west corner of Spitzbergen, about 79.5 degrees north. There was a picnic there for the passengers from the ship with sausages and mulled wine. It was not exactly a sublime experience but I was excited to have set foot on Spitzbergen. Another high point of that trip was Jan Mayen. That island is notorious for being covered with fog and I did not expect to see much of it. When we arrived about midnight the fog rose and we had excellent views. The conditions were so good that the ship circled for an hour to let us enjoy it. Most of Jan Mayen is a huge volcano rising straight out of the sea, the Beerenberg which is more than 2000 metres high. It is spectacular sight. The cruise was also due to pass close to Bear Island but I did not realise that. Nordenskiöld was one of the first to make scientific observations on Bear Island. Now Bear Island is probably much less spectacular than Jan Mayen, also usually covered in fog and I would probably have had to get up some time in the middle of the night to see it. Despite that, if I had known I had a chance of that type I would have taken it. The far north exerts an irresistible attraction on me. I have been in Iceland again and spent time in Vardø in the extreme north east of Norway, where I saw a White-Billed Diver. Maybe I will return to the north this summer.

Positivity for systems of reaction-diffusion equations

Here I consider a system of reaction-diffusion equations of the form , . The functions are defined on where is a bounded domain in with smooth boundary. Let the be denoted collectively by . I assume that the diffusion coefficients are non-negative. If some of them are zero then the system is degenerate. In particular there is an ODE special case where all are zero. If this system really describes chemical reactions and the are concentrations then it is natural to assume that for all . It should then follow that in the presence of suitable boundary conditions for all . I assume that is a classical solution and that it extends to the boundary with enough smoothness that the boundary conditions are defined pointwise. It is necessary to implement the idea that the system is defined by chemical reactions. This can be done by requiring that whenever and it follows that . (This means that if a chemical species is not present it cannot be consumed in any reaction.) It turns out that this condition is enough to ensure positivity.

I will now explain a proof of positivity. The central ideas are taken from a paper of Maya Mincheva and David Siegel (J. Math. Chem. 42, 1135). Thanks to Maya for helpful comments on this subject. The argument is already interesting for ODE since important conceptual elements can already be seen. I will first discuss that case. Consider a solution of the equation on the interval where and are continuous functions with and suppose that . I claim that for all . Let . Since it follows by continuity that . Assume that . By continuity . If were greater than zero then by continuity it would also be positive for slightly larger than , contradicting the definition of . Thus . The evolution equation then implies that . This implies that for slightly less than , which also contradicts the definition of . Hence in reality and this completes the proof of the desired result.

Suppose now that we weaken the assumptions to and . We would like to conclude that for all . To do this we define a new quantity for positive constants and . Then . Hence and . Now and so . It follows that if is large enough satisfies the conditions satisfied by in the previous argument and it can be concluded that for all . Letting tend to zero shows that for all , the desired result.

This is different, and perhaps a bit more complicated than, the proof I know for this type of result. That proof involves considering the derivative of on . It also involves approximating non-negative data by positive data. A difference is that the proof just given does not use the continuous dependence of solutions of an ODE on initial data and in that sense it is more elementary. In Theorem 3 of the paper of Mincheva and Siegel the former proof is extended to a case involving a system of PDE.

Now I come to that PDE proof. The system of PDE concerned is the one introduced above. Actually the paper requires the be positive but that stronger condition is not necessary. This equation is solved with an initial datum and a boundary condition . Here is a diagonal matrix with non-negative entries and the function is non-negative. The derivative is that in the direction of the outward unit normal to the boundary. We assume that is a classical solution, i.e. all derivatives of appearing in the equation exist and are continuous. Moreover has a continuous extension to and a extension to . We now replace the differential equation by the differential inequality . We assume that the initial data are non-negative, . The assumption that is non-negative, together with the boundary condition, gives rise to the inequality . The aim is to show that all solutions of the resulting system of inequalities are non-negative. We assume the condition for a system of chemical reactions already mentioned.

The proof is a generalization of that already given in the ODE case. The first step is to treat the case where each of the inequalities is replaced by the corresponding strict inequality. In contrast to the proof in the paper we assume that that is strictly positive on . We define as in the ODE case so that is the longest interval where the solution is non-negative. We suppose that is finite and obtain a contradiction. Note first that, as in the ODE case, by continuity. Now . If were strictly positive on then by continuity would be strictly positive for slightly greater than , contradicting the definition of . Hence there is an index and a point with and for all . Suppose first that . Then and . This contradicts the strict inequality related to the evolution equation for and so cannot happen. Suppose next that is on the boundary of . Then and . This contradicts the strict inequality related to the boundary condition for . Thus in fact and is strictly positive for all time.

Now we do a perturbation argument by considering . Here is the vector all of whose components are and is a positive function. It is obvious that . We now choose which ensures its positivity and require that the outward normal derivative of on the boundary is equal to one. (Here I will take for granted that a function of this kind exists. A source for this statement is cited in the paper. For me the positivity statement is already very interesting in the case that is a ball and there the existence of the function is obvious.) Then the fact that satisfies the non-strict boundary inequality implies that satisfies the strict boundary inequality. It remains to derive an evolution equation for . A straightfoward calculation gives where is a Lipschitz constant for on the image of the compact region being considered. Choosing large enough ensures that the right hand side and hence the left hand side of this inequality is strictly positive. We conclude that and, letting , that .

If we want to prove an inequality for solutions of a PDE it is common to proceed as follows. We deform the problem continuously as a function of a small parameter so as to get a simpler problem. When that has been solved we let tend to zero to get a solution of the original problem. Often it is the equations which are deformed. Then we need a theorem on existence and continuous dependence to get a continuous deformation of the solution. The above proof is different. We perturb the solution in a way whose continuity is obvious and then derive a family of equations of which that is a family of solutions. This is easy and comfortable. The hard thing is to guess a good deformation of the soluion.

 

An evening with Dieter Nuhr

Yesterday Eva and I went to a performance by the comedian Dieter Nuhr in Mainz. It was in a building called Halle 45. According to the ticket it was possible to enter the building 30 minutes before the event. We arrived 40 minutes before the event and most of the people were already inside. The capacity of the building is 2000 people but the chairs are moveable and so this can be varied depending on the event. I do not think that there were so many people there yesterday but it was full and there were many hundreds. Due to the circumstances we ended up sitting quite far back, a long way from the stage. This was a bit disappointing since it meant that there was little sense of intimacy with the performer. We had better experiences with performances we have attended by Lisa Eckhart and Josef Hader (twice) in the past in other (smaller) venues.

In recent years, especially since the pandemic, our society has become more and more polarized in its opinions. It would be an oversimplifcation to think that this is only a polarization along one axis, although there are remarkable correlations between opinions on issues which at first sight are quite unrelated. In fact there are fault lines running in different directions and two people who are on opposite sides in one case may be on the same side in another case. In comedy in Germany (and Austria) there is a clear polarization in the political views expressed. This might be formulated in terms of the words ‘left’ and ‘right’ but this is too simple. Often people with extreme right wing views and those with extreme left wing views seem to agree on many subjects. It seems that the topology of the political spectrum is rather that of a circle than that of the real line. Another word which might be used to characterize one direction on this axis is ‘woke’. That direction is the one often propagated by established political parties and state media in Germany. It is fortunately the case that this bias is not complete. There are two state-run TV companies in Germany, ARD and ZDF and while the principal representative of the ‘woke’ direction among the comedians, Jan Böhmermann, has a show with ZDF Dieter Nuhr, the principal representative of the other direction, has a show on ARD. In his show he performs himself part of the time and he also invites other comedians to take the stage. Recently it was often the case that the last person to appear, and in some sense the highlight, was Lisa Eckhart who I have written about in a previous post.

In the description of the event on the web page of Halle 45 it says ‘Ein Abend mit Dieter Nuhr ist Spaß und Therapie zugleich’ [An evening with Dieter Nuhr is fun and therapy at the same time]. Interestingly, during his performance Nuhr said that he did not consider his performance as therapy, or that he did not like it to be considered in that way. Was this statement sincere or was it intended as a joke? In any case it is worth taking a moment to think about in which sense it might be considered therapy. There are some people who at least some of the time (and I consider myself one of these people) do not feel at home in the public consensus presented by politics and media, which could also be associated with the word ‘woke’. They feel themselves confined by certain barriers in an unpleasant way. The ’therapeutic’ aspect of Dieter Nuhr is that in his texts he breaks through these barriers. (The same is true of Eckhart and Hader in their own ways.) I appreciate Dieter Nuhr because I like his humour. At the same time it is not the only reason I appreciate him. I think he has an important role to play in German politics by contributing to a certain balance and helping to prevent public opinion from becoming too extreme in certain directions. For example, he stepped forward to publicly support Lisa Eckhart when she was attacked in the way I described in a previous post. He also exerts a significant influence by his regular appearances on TV.

There is one point I want to mention before ending this post. In the last post I wrote about Karl Lauterbach. In his programme yesterday Nuhr made jokes about Lauterbach in a rather insulting way. (He has also behaved similarly in other places.) For instance he refers to him as the Sensenmann (‘grim reaper’), referring to his pessimism and his appearance. Now I tend to believe that a comedian should have a lot of freedom in making jokes about public figures, even if they might be unpleasant for the people concerned. Correspondingly, Nuhr’s jokes about Lauterbach do not cross the boundary of what is acceptable for me. At the same time I wonder if Nuhr has a particular grievance against Lauterbach and if so what it is. The jokes about Lauterbach were among the few yesterday I did not laugh at.

Fighting homeopathy

Most health systems in the world have financial problems and Germany is no exception to this. As a scientifically educated person it seems to me outrageous that the public health system here in Germany should spend money on homeopathic treatments. There is hardly anything in the area of ‘alternative medicine’ which is based on ideas which so blatantly contradict scientific reasoning. I was thus very glad to see that the health minister Karl Lauterbach has announced that he wants to ban the public health system from paying for homeopathic treatments. It is not the first time that a politician has tried this and suggestions of this type are usually greeted by a storm of opposition. I am not very optimistic that the idea of stopping public funds being used to finance homeopathy will actually find a sufficiently strong political consensus so as to result in legislation. If the idea actually succeeded I would be very happy. Of course the loudest opposition comes from those who earn money with homeopathy. One argument used is that the money involved is a small part of the total costs of the public health system. Of course it is the duty of those responsible to make sure that government money spent on health is well spent. The standards are usually very high. Often people cannot get treatments paid which are probably valuable but where the level of evidence required has not been reached. Thus I see the issue of homeopathy is one of principle. It is an insult to someone who is seriously ill and has to accept that the health insurance cannot pay for a treatment which would probably be effective (but is not yet provably so) while it pays for quackery. Germany has made a lot of important contributions to medicine but it should not be forgotten that although the disease homeopathy (as I see it) is widespread in the world it had its origin in Germany.

Karl Lauterbach is quite open about the fact that it is not the amount of money which is the central point. He is a scientist and qualified as a medical doctor and acts according to the ethical principles of these disciplines. He became publicly known through his role in the COVID-19 pandemic. (His academic speciality is epidemiology and public health.) During the pandemic he tended towards recommending strict measures and in this way he made himself unpopular with many people. At the same time his role was appreciated by many others (including myself) and this led to his appointment as health minister. He is a member of the SPD, having switched a long time ago from the CDU. Some politicians of other parties have accused him of using the issue of homeopathy to direct people’s attention away from other parts of his health policy which are seen as unsuccessful. I feel sure that this is not true. It is typical of Lauterbach, during the pandemic and otherwise, that he publicly says what he believes to be the truth, even when that results in unpopularity and attacks on him in the media. Although there are many aspects of Lauterbach’s politics I do not agree with I tend to identify with him as a person. I see him as a representative of honesty and rationalism in the public domain. The fact that many of the attacks on him involve personal antipathy and making fun of his appearance only tend to strengthen my feeling of solidarity with him.

It might be said that politicians have more important things which they should concentrate on than homeopathy but I am not prepared to accept that. I think that in the long term the struggle between science and superstition is a matter of key importance for the future of our civilization.

Postpositive adjectives

As a scientist whose native language is English it often happens that I end up correcting English texts by collaborators, other colleagues or students. There is one type of correction which I often had to make where I was not able to identify what exactly was going on. Thus I had difficulties communicating this in any other way than by lists of the examples where the problem occurred. This resulted in my wasting a lot of time. I was able to identify that certain phrases were wrong and give the corresponding corrections but I did not understand, and thus could not explain, why they were wrong. Now I have got some more insight into this problem. In particular I have found a name for the construction that some people were failing to use. It is ‘postpositive adjective’ or ‘postnominal adjective’. Anyone with some knowledge of English knows that an adjective very often comes immediately before the noun it describes, for instance ‘a black cat’. This works the same way in German, ‘eine schwarze Katze’ but in a different way in French, ‘un chat noir’. The last phrase is an example of a postnominal adjective, the adjective comes immediately after the noun. The problem is that postnominal adjectives can also occur in English. An example is ‘the method used’. This is the same in French, ‘la méthode utilisée’ but different in German, ‘die verwendete Methode’. In English it occurs when the adjective is a past participle. On the other hand that does not seem to be universal, as shown by the example ‘an escaped prisoner’. Maybe the point is that while the method is the object of the verb concerned (this is also an example of a postpositive adjective which I used without thinking) the prisoner is the subject of the verb concerned. With this start I will continue to observe.