I just saw an article by the chemistry professor Anna Krylov from October 2025 where she explains why she refused to review an article for a journal belonging to the Nature publishing group and more generally communicated to them that she will not cooperate with them in the future, unless they change their policies. The reason is that there are grounds to believe that they are sacrificing truth to political ideology. In this way they are undermining science and truth. She gives three examples to support this view. The first is that reviewers are to be selected ‘with diversity in mind’. In other words, at this point the criterion of competence is being replaced by one of identity politics. The second is so-called ‘citation justice’. This means that the criterion for citing a paper is not its relevance but biological and social characteristics of the authors. It suggests replacing criteria in science like ‘is it true’ or ‘is it interesting’ by ‘who said it’. The third is that results of research work should be suppressed if they are potentially harmful to certain groups, whatever harmful means.
I am very conscious that the influence of truth and science in society is declining but I was shocked to see that this process has affected science itself to such an extent. I suppose that I did not notice this directly since my field, mathematics, is presumably one of the least affected. The problems pointed out by Krylov do not apply equally to all journals. Despite this, since my most recent article appeared in a journal belonging to the Nature publishing group, I feel I have to pay more attention to this type of issue in the future. When I started publishing scientific papers there was no reason to doubt the integrity of any well-known publisher. The choice of a journal to publish in was based on the quality of the scientific content alone. More recently it has become necessary to more careful due to harmful economic influences and the rise of predatory publishing. Now we have an additional factor, where the quality of the scientific literature is under attack not only due to monetary influences but also due to political ideology.
. When I asked the lecturer who was teaching us a course on thermodynamics he was not able to give me an explanation which I found satisfactory. As a schoolboy physics was the subject which interested me most. After my second year at university I had to decide between doing a degree in physics, a degree in mathematics or a joint degree in both. My decision for the second alternative was strongly influenced by that thermodynamics conundrum. Another experience which contributed to my decision was that at one time I happened to have two courses on the same topic, Fourier series, in the same term, one in physics and one in mathematics. The second was quite transparent to me, the first obscure. The fact that I had
or open subsets thereof. The considerations are local, i.e. whenever necessary the functions will be restricted to arbitrarily small neighbourhoods of the origin. A relatively simple case is 
. Suppose that 
for a smooth function 
. Then the derivatives 
vanish for all 
. The question whether the converse holds is the simplest example of the type of questions to be considered here. In fact it does hold but the proof is not quite trivial. It is easy to construct an inductive proof once the case 
is known. That case can be treated using the fundamental theorem of calculus and a change of variables. This leads to the formula 
.
things become more complicated. If 
is a function of two variables then there is no condition on the derivatives of 
at the origin which will ensure the existence of a function 
. This can be seen by choosing 
. All derivatives of this function at the origin are zero. However there is no smooth function 
. For setting 
in a relation of this form gives a contradiction. There are true analogues of the theorem in one dimension for 
with 
. This can be formulated in algebraic language as follows. In the ring of germs of smooth functions at the origin the ideal of functions which vanish at the origin is generated by the monomials 
. The proof is very similar to that in the one-dimensional case. More generally, if for a given 
the function 
vanish at the origin there exist functions 
such that 
. Here 
is a multi-index.
, to play a special role. It is now assumed that the restriction 
is divisible by 
, i.e. 
for a smooth function 
. The conclusion is that under this assumption it is possible to multiply 
in such a way that the result is a polynomial of degree 
. Another related result is the Mather division theorem. In that case we start with any smooth function 
such that 
and 
with coefficients which are smooth functions of 
in the second and rearrange the result. For the converse first multiply by a positive function to make 
. The Malgrange preparation theorem is an analogue for smooth functions of the Weierstrass preparation theorem for analytic functions of complex variables. The big difference is that the analytic theorem includes a uniqueness statement while the smooth one does not. It seems that the Malgrange theorem cannot simply be got by approximating the smooth functions by analytic ones and using the Weierstrass theorem. Instead it is proved by decomposing the Fourier transform using a partition of unity.
, where 
and 
denote the first and second order derivatives of 
, respectively. The boundary condition is an inhomogeneous Dirichlet condition 
. Now replace 
to get the equation 
. Suppose that this equation has a unique solution for a given function 
between suitable function spaces by 
. A solution of the problem we want to solve is then a fixed point of 
. Since the equation which defines 
. A priori estimates for the elliptic problem then give the bound needed to apply the theorem to this problem and obtain an existence theorem for the boundary value problem for the nonlinear equation.
be the open unit ball in a Banach space 
and let 
into 
is contained in 
. They involve both scaling the argument and modifying the parameter. A fixed point is obtained for each 
and then 
. It is supposed that any fixed point of the latter with 
satisfies the inequality 
. Now 
which satisfies 
for 
and 
for 
. Let 
be the closed ball of radius 
about the origin. Then 
at this point then it is also a fixed point of 
then 
is proportional to 
with a factor 
. Applying the assumption of the theorem we see that the norm of 
is no greater than 
is less than one and there nevertheless exists a positive steady state. The bifurcation which occurs in the proof is a transcritical bifurcation. Intuitively this means that as a parameter is varied a steady state moves across another steady state which is fixed. There is a sign involved in a bifurcation of this type. One value of the sign corresponds to a backward bifurcation while the other corresponds to the forward bifurcations more familiar in epidemiology. The steady state which arises in a backward bifurcation is unstable. In simulations of models with backward bifurcations it is often seen that the steady state can be continued to larger values of the parameter until it reaches a point where a fold bifurcation takes place. There it changes from unstable to stable. This phenomenon cannot be captured by the analysis of the transcritical bifurcation since in general it occurs far away from the bifurcation point. We have now found an extension of that analysis which allows the fold bifurcation to be treated. The idea is to replace the transcritical bifurcation by one whose codimension is one higher. This is similar to passing from a fold to a cusp. The transcritical bifurcation could be given the intuitive name ‘moving hyperbolic steady state’ and by analogy with this we give the bifurcation studied in the paper the name ‘moving fold’. In this type of bifurcation two positive steady states arise, one stable and one unstable. This is a tool which may be used to prove the existence of more than one positive steady state in epidemiological models. The general theorem we prove applies to both the cases 
and 
but up to now we have only found interesting examples for the first of these cases. Our original model for hepatitis C is one such example. We were not able to decide whether in that example there exist parameters which would allow the second case of the theorem to be applied. We proved the relative statement that if parameters with this property exist there actually exist three positive steady states. An interesting side effect of this work is that we found a limiting case of that model (where two parameters are set to zero) where there can exist a continuum of steady states.
Hydrobates 