I first moved to Germany in 1987 and I have spent most of the time since then here. The total time I have spent elsewhere since my first arrival in Germany does not add up to more than two years. There is every reason to expect I will spend the rest of my life here. I am married to a German, I have a job here I like and a house. I could have applied for German citizenship a long time ago but I never bothered. Being an EU citizen living in Germany I had almost almost all privileges of a native. The only exception is that I could not vote except in local elections but since I am not a very political person that was not a big issue for me. It was also the case that for a long time I might have moved to another country. For instance I applied for a job in Vienna a few years ago and I might well have taken it if it had been offered to me. Now the chances of my moving are very small and so there is no strong argument left against becoming a German citizen.
What is more important is that there are now arguments in favour of doing so. With the EU showing signs of a possible disintegration the chance that I could lose the privileges I have here as an EU citizen is not so small that it should be neglected. The referendum in which the Scots voted on the possibility of leaving the UK was the concrete motivation for my decision to start the application process. Scotland stayed in the UK but then the Brexit confirmed that I had made the right decision. At the moment there is no problem with keeping British citizenship when obtaining German citizenship and I am doing so. This may change sometime, meaning that I will have to give up my British citizenship to keep the German one, but I see this as of minor importance.
As prerequisites for my application I had to do a number of things. Of course it was necessary to submit a number of documents but I have the feeling that the amount of effort was less than when obtaining the documents needed to get married here. I had to take an examination concerning my knowledge of the German language, spoken and written. It was far below my actual level of German and so from that point of view it was a triviality. It was just a case of investing a bit of time and money. I also had to do a kind of general knowledge test on Germany and on the state where I live. This was also easy in the sense that the questions were not only rather simple for anyone who has lived in the country for some time but they are also taken from a list which can be seen in advance. Again it just meant an investment of time and money. At least I did learn a few facts about Germany which I did not know before. In my case these things were just formalities but I think it does make sense that they exist. It is important to ensure that other applicants with a background quite different from mine have at least a minimal knowledge of the language and the country before they are accepted.
After all these things had been completed and I had submitted everything it took about a year before I heard that the application had been successful. This time is typical here in Mainz – I do not know how it is elsewhere in Germany – and it results from the huge backlog of files. People are queueing up to become German citizens, attracted by the prospect of a strong economy and a stable political system. Yesterday I was invited to an event where the citizenship of the latest group of candidates was bestowed in a ceremony presided over by the mayor. There were about 60 new citizens there from a wide variety of countries. The most frequent nationality by a small margin was Turkish, followed by people from other middle eastern countries such as Iraq and Iran. There were also other people from the EU with the most frequent nationality in that case being British. My general feeling was one of being slightly uneasy that I was engaged in a futile game of changing horses. It is sad that the most civilised countries in the world are so much affected by divisive tendencies instead of uniting to meet the threats confronting them from outside.
and a steady state 
, i.e. a solution of 
. It is well-known that this steady state is asymptotically stable if all eigenvalues 
of the linearization 
have negative real parts. This property of the eigenvalues is of course a property of the roots of the characteristic equation 
. It is always the case here that 
but I prefer to deal with a general polynomial with real coefficients 
and a criterion for the situation where all its roots have negative real parts. It is tempting to number the coefficients in the opposite direction, so that, for instance, 
becomes 
but I will stick to this convention. Note that it is permissible to replace 
by 
in any criterion of this type since if we multiply the polynomial by 
we get a polynomial in 
where the order of the coefficients has been reversed. Moreover, if the real part of 
to the integers so that these coefficients are zero for 
and 
.
the Hurwitz matrix is an 
th diagonal element is 
, with 
. Starting from a diagonal element and proceeding to the left along a row the index increases by one in each step. Similarly, proceeding to the right along a row the index decreases by one. In the ranges where the index is negative or greater than 
. The Hurwitz criterion says that the real parts of all roots of the polynomial are negative if and only if 
and 
for all 
. Note that a necessary condition for all roots to have negative real parts is that all 
and so the last condition can be replaced by 
. Note that the form of the 
we get the conditions 
and 
. For 
we get the conditions 
and 
. Note that the third condition is invariant under the replacement of 
. When 
, 
we get the conditions 
and 
.
is the region in matrix space where the Routh-Hurwitz criteria are satisfied, what happens on the boundary of 
. Let us look at the situation where the boundary is approached while 
must become zero. In terms of eigenvalues what happens is that a number of complex conjugate pairs reach the imaginary axis away from zero. The generic case is where it is just one pair. An interesting question is whether and how this kind of event can be detected using the 
while the 
. In a paper of Liu (J. Math. Anal. Appl. 182, 250) it is shown that the condition for a Hopf bifurcation that the derivative of the real part of the eigenvalues with respect to a parameter is non-zero is equivalent to the condition that the derivative of 
with respect to the parameter is non-zero. In a paper with Juliette Hell (Math. Biosci. 282, 162), not knowing the paper of Liu, we proved a result of this kind in the case 
of phosphorylation sites on the T cell receptor complex which are included in the model. The results just mentioned on steady states were obtained for 
.
of the maximally phosphorylated state of the T cell receptor. The response function describes how 
of the ligand and the constant 
describing the rate at which the ligand unbinds from the T cell receptor. A widespread idea (the lifetime dogma) is that the quantity 
, the dissociation time, determines how strongly an antigen signals to a T cell. It might naively be thought that the response should be an increasing function of 
and for still larger values it once again becomes a line of slope one, shifted with respect to the original one. Finally the curve levels out as it must do, since the function is bounded. The proofs do not make heavy use of general theorems and are in general based on doing certain estimates by hand.
happens’. Of course this is just a first step. In the future we would like to understand better to what extent biologically motivated restrictions on the parameters lead to restrictions on the dynamical behaviour.
Hydrobates 