## Archive for December, 2019

### Sard’s theorem

December 27, 2019

I have recently been reading Smoller’s book ‘Shock Waves and Reaction-Diffusion Equations’ as background for a course on reaction-diffusion equations I am giving. In this context I came across the subject of Sard’s theorem. This is a result which I had more or less forgotten about although I was very familiar with it while writing my PhD thesis more than thirty years ago. I read about it at that time in Hirsch’s book ‘Differential Topology’, which was an important reference for my thesis. Now I had the idea that this might be something which could be useful for my present research, without having an explicit application in mind. It is a technique which has a different flavour from those I usually apply. The theorem concerns a (sufficiently) smooth mapping between $n$-dimensional manifolds. It is a local result so that it is a enough to concentrate on the case where the domain of the mapping is a suitable subset of Euclidean space and the range is the same space. We define a regular value of $f$ to be a point $y$ such that the derivative of $f$ is invertible at each point $x$ with $f(x)=y$. A singular value is a point of the range which is not a regular value. The statement of Sard’s theorem is that the set $Z$ of singular values has measure zero. By covering the domain with a countable family of cubes we can reduce the proof to the case of a cube. Next we write the cube as the union of $N^n$ cubes, by dividing each side of the original cube into $N$ equal parts. We need to estimate the contribution to the measure of $Z$  from each of the small cubes. Suppose that $y_0$ is a singular value, so that there is a point $x_0$ where the derivative of $f$ is not invertible with $f(x_0)=y_0$. Consider now the contribution to the measure of the image from the cube in which $x_0$ lies. On that cube $f$ can be approximated by its first order Taylor polynomial at $x_0$. The image is contained in the product of a subset of a hyperplane whose volume is of the order $N^{-(n-1)}$ and an interval whose length is of the order $\epsilon N^{-1}$ for an $\epsilon$ which we can choose as small as desired. Adding over the at most $N^n$ cubes which contribute gives a bound for the measure of the set of singular values of order $\epsilon$. Since $\epsilon$ was arbitrary this completes the proof. In words we can describe this argument as follows. The volume of the image of a region which intersects the set of singular points under a suitable linear mapping is small compared to the volume of the region itself and the volume of the image under the nonlinear mapping can be bounded by the corresponding quantity for the linear mapping up to an error which is small compared to the volume defined by the linear mapping.

### Ernst Jünger

December 25, 2019

The book of Jünger which ignited my enthusiasm for his writing is ‘Afrikanische Spiele’. This is a work of fiction but it is based rather closely on Jünger’s own experiences. He was often bored in school and preferred to read adventure stories. For him Africa was the land of adventure and he wanted to go there. He ran away from school and travelled to Verdun, where he enlisted in the Foreign Legion. He was then stationed in Algeria. This was not his real aim and so he deserted and tried to travel further. He was caught and put into prison in solitary confinement. A doctor in the place he was stationed wrote to his father. Since in fact Jünger was not old enough to have joined the foreign legion and had only managed to do so by lying about his age his father was able to get him discharged and took him home. Shortly after he returned the First World War began and Jünger enlisted immediately and had his opportunity for adventure, as related in ‘In Stahlgewittern’. One thing which attracts me to Jüngers writing, in ‘Afrikanische Spiele’ and elsewhere, is the style. At the same time, the content is often remarkable. Here is a striking example. The hero of the book has taken some money with him when he left home. He feels the danger that he might give up and not dare to carry out his plan. To avoid this he takes all the money he has and puts in down a drain in Verdun. In this way he removes the chance of turning back. This reminds me a little of the story of how Nansen became the first to cross the Greenland icecap. He chose the direction of crossing in such a way that failure would have meant almost certain death.

I will mention a passage in ‘Gärten und Strassen’ which particularly struck me and which Banine mentions in her book. In this book Jünger described his experiences during the German invasion of France in the Second World War. This time, in contrast with what happened in the First World War, he was hardly involved in the fighting at all. At one time he was the commanding officer in the town of Laon. Laon has a magnificent gothic cathedral, which I have visited myself. He describes his experience of looking at this cathedral, for whose safety he was responsible at that time, and feeling that this huge building was like a small vulnerable creature. He was successful in preventing treasures from the cathedral being stolen or destroyed, helped by the fact that those who might have done so did not realize how valuable these things were.

Despite my admiration for Jünger’s writings there is one thing which I do not like and which I feel I have to mention. This is a tendency to esotericism which he shows from time to time and which I just try to ignore. Despite this I am sure that I will continue to read Jünger with pleasure in the future.

### Monotone systems revisited

December 4, 2019

There are some topics in mathematics and physics which are a lasting source of dissatisfaction for me since I feel that I have not properly understood them despite having made considerable efforts to do so. In the case of physics the reason is often that the physicists who understand the subject are not able to explain it in a way which provides what a mathematician sees as a comprehensible account. In mathematics the problem is a different one. Mathematicians frequently have a tendency (often justified) to discuss things on a level which is as general as possible. This leads to theorems which are loaded down with detail and where the many technical conditions make it difficult to see the wood for the trees. When confronted with such things I sometimes feel exhausted and give up. I prefer an account which builds up ideas step by step from simple beginnings. Here I return to a subject which I have written about more than once in this blog before but where the sense of dissatisfaction remains. I hope to reduce it here.

I start with a system of ordinary differential equations $\dot x_i=f_i(x)$. It should be defined on the $n$-dimensional Euclidean space or on one of its orthants. (An orthant is the subset of Euclidean space defined by making a choice of the signs of its components. It generalises a quadrant in the two-dimensional case.) The system is said to be cooperative if $\frac{\partial f_i}{\partial x_j}>0$ for all $i\ne j$. The name comes from the fact that the equations for the population dynamics of a set of species has this property if each species benefits the others. Suppose we now have two solutions $x$ and $\bar x$ of the system and that $x_i(t_0)\le\bar x_i(t_0)$ for all $i$ at some time time $t_0$. We may abbreviate this relation by $x(t_0)\le\bar x(t_0)$. Here we see a partial order on Euclidean space defined by the ordering of the components. A theorem of Müller and Kamke says that if the initial data for two solutions of a cooperative system at time $t_0$ satisfies this relation then $x(t)\le\bar x(t)$ for all $t\ge t_0$. Another way of saying this is that the time-$t$ flow of the system is preserves the partial order. A system of ODE with this property is called monotone. Thus the Müller-Kamke theorem says that a cooperative system is monotone.

The differential condition for monotonicity can be integrated. If $x$ and $\bar x$ are two points in Euclidean space with $x_i=\bar x_i$ for a certain $i$ and $x_j\le\bar x_j$ for $j\ne i$ then $f_i(x)\le f_i(\bar x)$. To see this we join $x$ to $\bar x$ by a piecewise linear curve where the coordinates other than the $i$th are increased successively from $x_j$ to $\bar x_j$. On each segment of this curve the value of $f_i$ does not decrease, as a consequence of the fundamental theorem of calculus. Hence its value at the end of the entire path is at least as big as its value at the beginning. We now want to prove that a certain inequality holds at all times $t\ge t_0$. In order to do this we would like to consider the first time $t_*>t_0$ where the inequality fails and get a contradiction. Unfortunately there might be no such time – in principle the condition might fail immediately. To get around this we deform the system for the solution $\bar x$ to $\frac{d\bar x_i}{dt}=f_i(\bar x)+\epsilon$. If we can prove the result for the deformed system the result for the initial system follows by continuous dependence of the solution on $\epsilon$. For the deformed system let $t_*$ be the supremum of the times where the desired inequality holds. If the inequality does not hold globally then the system is still defined at $t=t_*$. For $t=t_*$ we have $x_i=\bar x_i$ for some $i$ and we can assume w.l.o.g. that $x_j<\bar x_j$ for some $j$ since otherwise the two solutions would be equal and the result trivial. The integrated form of the cooperativity condition implies that at $t_*$ the right hand side of the evolution equation for $\bar x_i-x_i$ is positive. On the other hand the fact that it just reached zero coming from positive values implies that the right hand side of the evolution equation is non-positive and we get a contradiction.

A key source of information about monotone dynamical systems is the book of Hal Smith with this title. I have repeatedly looked at this book but always got bogged down quite quickly. Now I realise that for my purposes it would have been much better if I had started with chapter 3. The Müller-Kamke theorem is discussed in section 3.1. The range of application of this theorem can be extended considerably by the following trick, discussed in section 3.5. Suppose that we define $y_i=(-1)^{m_i}x_i$ where each of the $m_i$ are zero or one. This transforms the signs of $Df$ in a certain way and so cooperativity of the system for $y$ corresponds to a certain sign pattern for the entries of $Df$. A first important condition is that each off-diagonal element of $Df(x)$ should be either non-negative or non-positive. Next, the sign of $\frac{\partial f_i}{\partial x_j}\frac{\partial f_j}{\partial x_i}$ is not changed be the transformation and must thus be non-negative. In the context of population models this can be interpreted as saying that there is no pair of species which are in a predator-prey relationship. Given that these two conditions are satisfied we consider a labelled graph where the nodes are the numbers from $1$ to $n$ and there is an edge between two nodes if at least one of the corresponding partial derivatives is non-zero at some point. The edge is then labelled with the sign of this non-zero value. A loop in the graph can be assigned the sign which is the product of those of its edges. It turns out that a system can be transformed to a cooperative system in the way indicated if and only if the graph contains no negative loops. I will call a system of this type ‘cooperative up to sign reversal’. The system can be transformed by a permutation of the variables into one where $Df$ has diagonal blocks with non-negative entries and off-diagonal elements with non-positive entries.

If all elements of $Df$ are required to be non-positive we get the class of competitive systems. It should be noted that being competitive leads to less restrictions on the dynamics of a system (towards the future) than being cooperative. We can define a class of systems which are competitive up to sign reversal. An example of such a system is the basic model of virus dynamics. In that system the unknowns are the populations of uninfected cells $x$, infected cells $y$ and virus particles $v$. The transformation $y\mapsto -y$ makes it into a competitive system. In various models of virus dynamics including the immune response the target cells of the virus and the immune cells are in a predator-prey relationship and so these systems can be neither cooperative up to sign or competitive up to sign.