## Archive for the ‘books’ Category

### The shooting method for ordinary differential equations

November 25, 2022

A book which has been sitting on a shelf in my office for some years without getting much attention is ‘Classical Methods in Ordinary Differential Equations’ by Hastings and McLeod. Recently I read the first two chapters of the book and I found the material both very pleasant to read and enlightening. A central theme of those chapters is the shooting method. I have previously used the method more than once in my own research. I used it together with Juan Velázquez to investigate the existence of self-similar solutions of the Einstein-Vlasov system as discussed here. I used it together with Pia Brechmann to prove the existence of unbounded oscillatory solutions of the Selkov system for glycolysis as discussed here. This method is associated with the idea of a certain type of numerical experiment. Suppose we consider a first order ODE with initial datum $x(0)=\alpha$. Suppose that for some value $\alpha_1$ the solution tends to $+\infty$ for $t\to\infty$ and for another value $\alpha_2>\alpha_1$ the solution tends to $-\infty$. Then we might expect that in between there is a value $\alpha^*$ for which the solution is bounded. We could try to home in on the value of $\alpha^*$ by a bisection method. What I am interested in here is a corresponding analytical procedure which sometimes provides existence theorems.

In the book the procedure is explained in topological terms. We consider a connected parameter space and a property $P$. Let $A$ be the subset where $P$ does not hold. If we can show that $A=A_1\cup A_2$ where $A_1$ and $A_2$ are non-empty and open and $A=A_1\cap A_2=\emptyset$ then $A$ is disconnected and so cannot be the whole of the parameter space. Hence there is at least one point in the complement of $A$ and there property $P$ holds. The most common case is where the parameter space is an interval in the real numbers. For some authors this is the only case where the term ‘shooting method’ is used. In the book it is used in a more general sense, which might be called multi-parameter shooting. The book discusses a number of cases where this type of method can be used to get an existence theorem. The first example is to show that $x'=-x^3+\sin t$ has a periodic solution. In fact this is related to the Brouwer fixed point theorem specialised to dimension one (which of course is elementary to prove). The next example is to show that $x'=x^3+\sin t$ has a periodic solution. After that this is generalised to the case where $\sin t$ is replaced by an arbitrary bounded continuous function on $[0,\infty)$ and we look for a bounded solution. The next example is a kind of forced pendulum equation $x''+\sin x=f(t)$ and the aim is to find a solution which is at the origin at two given times. In the second chapter a wide variety of examples is presented, including those just mentioned, and used to illustrate a number of general points. The key point in a given application is to find a good choice for the subsets. There is also a discussion of two-parameter shooting and its relation to the topology of the plane. This has a very different flavour from the arguments I am familiar with. It is related to Wazewski’s theorem (which I never looked at before) and the Conley index. The latter is a subject which has crossed my path a few times in various guises but where I never really developed a warm relationship. I did spend some time looking at Conley’s book. I found it nicely written but so intense as to require more commitment than I was prepared to make at that time. Perhaps the book of Hastings and McLeod can provide me with an easier way to move in that direction.

### The autobiography of Andrew Carnegie

November 4, 2022

Carnegie later got involved in the production of iron and things constructed out of it, such as rails and bridges. He got an advantage over his competitors by employing a chemist. The iron ore from some mines was unpopular and correspondingly relatively cheap. There had been problems with smelting it. A chemical analysis revealed the source of the problem – that ore contained too much iron for the smelting process to work well. The solution was to modify the process (with the help of scientific considerations) and then it was possible to buy the high quality ore at a cheap price while others continued to buy low quality ore at a high price. Previously nobody really knew what they were buying. Carnegie believed in the value of real knowledge. Carnegie did not like the stock exchange and emphasized that except for once at the beginning of his career he never speculated. It was always his policy to buy and sell things on the basis of their real value. Carnegie was no friend of unions and often fought them hard. On the other hand he was, or claimed to be, a friend of the working man. His idea was not to give people money just like that but to give them the opportunity to improve their own situation. In later years he gave a huge amount of money, about 300 million dollars in total for various causes. He gave money for libraries (more than two hundred), for scientific research, for church organs and of course for the Carnegie Hall. These were all things which he believed would do people good.

Carnegie was committed to the goal of world peace. He had a lot of influence with powerful politicians and it seems that in at least one case he used it to prevent the US becoming involved in a war. He got into contact with Kaiser Wilhelm II. It turned out that both of them were admirers of Robert the Bruce. He had great hopes for the Kaiser as someone who could help to bring peace and he must have been bitterly disappointed in 1914 when things went a very different way. Then he transferred his hopes to President Wilson. At that point his autobiography breaks off. Here I have only been able to present a few selected things from a fascinating book which I thoroughly recommend. I find Carnegie an admirable character.

### Andrew Carnegie and me

October 10, 2022

At the moment I am rereading the autobiography of Andrew Carnegie. For me he is a leading example of a ‘good capitalist’. I see some parallels between my early life and that of Carnegie and I have been thinking about similarities and differences. We were both born in Scotland to parents who were not rich. His father was a weaver. At some stage advances in technology meant that his form of industry was no longer viable. The family had difficulty earning their living and decided to emigrate to America. While still in Scotland Carnegie’s mother started a business to supplement the income of her husband.

My father was a farmer. When he began the size of the farm (70 acres) was sufficient. (I recently noticed the coincidence that the father of Robert Burns, Scotland’s national poet, also had a farm of 70 acres.) Later there was a trend where the size of farms increased and the machinery used to work them became more advanced. In order to do this the farmers, who were mostly not rich, had to borrow money to buy more land and better machines. The banks were eager to lend them that money. Of course this meant a certain risk but many of the people concerned were prepared to take that risk. My father, on the other hand, never borrowed any money in his life and so he missed taking part in this development. This meant that under the new conditions the farm was too small (and in fact some of it was not very good land – it was too wet) to support our family (my parents, my grandmother and myself) very easily. My mother did everything she could to supplement the income of my father. In particular she took in bed and breakfast guests during the summer. I should point out that we were not poor. We did not lack anything essential, living in part from our own produce such as milk, butter, cheese, eggs, potatoes and meat. My grandmother kept a pig and hens. The school I attended, Kirkwall Grammar School, was the school for all children in the area – there was no alternative. The parents of many of the other children I went to school with were better off financially than my parents. As a sign of this, I mention an exchange between our school and one in Canada. Many of the other pupils took part in that. My parents could not afford to finance it for me. At a time when many people were getting their first colour TV we still had a very old black and white device where with time ‘black’ and ‘white’ were becoming ever more similar. I did not feel disadvantaged but I just mention these things to avoid anyone claiming that I grew up in particularly fortunate economic circumstances.

Both Carnegie and I benefitted from the good educational system in Scotland. School was already free and compulsory in his time. My university education was mostly financed by the state, although I did win a couple of bursaries in competitions which helped to make my life more comfortable. In my time parents had to pay a part of the expenses for their childrens’ university education, depending on their incomes. My parents did not have to pay anything. Some of the people I studied with should have got a contribution from their parents but did not get as much as they should have. Thus I actually had an advantage compared to them. Carnegie’s father was involved in politics and had quite a few connections. My parents had nothing like that. It might be thought that since my parents did not have very much money or connections and since there were very few books in our house I started life with some major disadvantages. I would never make this complaint since I know that my parents gave me some things which were much more important than that and which helped me to build a good life. I grew up in a family where I felt secure. My parents taught me to behave in certain ways, not by command but by their example. They taught me the qualities of honesty, reliability, hard work and humility. Carnegie received the same gifts from his parents.

Let me now come back to the question of books. As a child I was hungry for them. We had a good school library which included some unusual things which I suppose not all parents would have been happy about if they had known the library as well as I did. For instance there was a copy of the ‘Malleus Maleficarum’. What was important for my future was that there were current and back issues of Scientific American and New Scientist. There was also a public library from which I benefitted a lot. Apparently this was the first public library in Scotland, founded in 1683. In the beginning it was by subscription. It became free due to a gift of money from Andrew Carnegie in 1889. He spent a huge amount of time and effort in supporting public libraries in many places. In 1903 Carnegie also gave the money to construct a building for the library and that is the building it was still in when I was using it. He visited Kirkwall to open the library in 1909. Thus it can be said that I personally received a gift of huge value from Carnegie, the privilege of using that library. His activity in this area was his way of returning the gift which he received as a young boy when someone in Pittsburgh opened his private library to working boys.

### Trip to Scotland with some obstacles

August 21, 2022

What conclusions do I draw from this? Firstly, I do not believe that we had specially bad luck but rather that this is the usual state of affairs at the moment. (The luggage of several other members of our group, arriving from different airports with different airlines, also took many days to arrive, in one case even a day longer than ours.) We also experienced a number of other things while in Scotland, such as lifts or coffee machines in hotels which were not working and had been waiting for months to be repaired. Many hotels in Scotland, especially in rural areas, have closed, at least for the season and maybe for ever. For these reasons we had to stay in some cases at hotels much further away from the points we wanted to visit than planned and there were long drives. The situation with logistics is dire. We were not organizing the trip alone. The organization was being done by a company which has many years of experience organizing trips of this kind in Scotland and all over the world. We know from previous experience that this company is very good. Thus things are very difficult even for the experts. If things do not change quickly this type of tourism is threatened. In future I will think very carefully about flying anywhere. This has nothing to do with the frequently discussed environmental issues but simply with the doubt that I will arrive successfully with my luggage and without an excessive amount of stress. If I do fly anywhere then I will be prepared to pay a higher price to get a direct flight. This is then the analogue of my present practise with train trips where I try to minimize the number of connections which have to be reached since the trains cannot be expected to be on time. It seems that these days the most reasonable thing is to expect that everything that can go wrong will go wrong. Travelling has become an adventure again. Will this change soon? I do not expect it will.

### The Prime of Miss Jean Brodie

May 30, 2022

As the title suggests the novel ‘The prime of Miss Jean Brodie’ is dominated by one character. She is strong and fascinating but it is wise to be cautious of having too much admiration for her. This is made clear at the latest by her professed admiration for Mussolini. (The novel is set in the years leading up to the Second World War.) She is a teacher at a private school, her pupils being girls of under the age of twelve. A group of these girls from one year come together to form ‘the Brodie set’. They are brought together not by any similarities between them but by their bond to Jean Brodie. This also keeps them together for the rest of their time at school. She is regarded by most of the teachers at the conventional school as too progressive and the headmistress is keen to find a reason to get rid of her. Eventually she succeeds in doing so. I cannot see Jean Brodie as a model for a teacher. She hatches out schemes to allow her pupils to avoid the work they should be doing and to listen to the stories she tells them. She has very definite ideas, for instance as to the relative importance of subjects: art first, philosophy second, science third. My ordering would be: science first, art second, philosophy third. The book is often very funny but I think there is also a lot in it which is very serious. Apart from the content, the use of language is very impressive. I enjoyed trying to imagine what it means for a jersey to be ‘a dark forbidding green’. There is even a little mathematics, although that is a subject which Jean Brodie has little liking for. Concerning the conflict of Miss Brodie with the Kerr sisters we read, ‘Miss Brodie was easily the equal of both sisters together, she was the square on the hypotenuse of a right-angled triangle and they were only the squares on the other two sides’. For me this is one of those few books which is able to bring some movement into my usual routine. I watched a documentary about Muriel Spark herself, which is also very interesting. Maybe I will soon read another of her novels. ‘Aiding and Abetting’ has also found its way from my distant past onto my bookshelf but I am not sure it will be the next one I read.

### Advances in the treatment of lung cancer

May 1, 2022

Now let me come back to the lectures. The first important message is nothing new: most cases of lung cancer are caused by smoking. Incidentally, the secretary I mentioned above smoked a lot when she was young but gave up smoking very many years ago. The message is: if you smoke then from the point of view of lung cancer it is good to stop. However it may not be enough. In the first lecture it was emphasized that the first step these days when treating lung cancer is to do a genetic analysis to look for particular mutations since this can help to decide what treatments have a chance of success. In the case of the secretary the doctors did look for mutations but unfortunately she belonged to the majority where there were no mutations which would have been favourable for her prognosis under a suitable treatment. In the most favourable cases there are possibilities available such as targeted therapies (e.g. kinase inhibitors) and immunotherapies. These lectures are intended to be kept understandable for a general audience and accordingly the speaker did not provide many details. This means that since I have spent time on these things in the past I did not learn very much from that lecture. The contents of the second lecture, on surgical techniques, were quite unfamiliar to me. The main theme was minimally invasive surgery which is used in about 30% of operations for lung cancer in Germany. It is rather restricted to specialized centres due to the special expertise and sophisticated technical equipment required. It was explained how a small potential tumour in the lung can be examined and removed. In general the tumour will be found by imaging techniques and the big problem in a operation is to find it physically. We saw a film where an anaesthetised patient is lying on an operating table while the huge arms of a mobile imaging device do a kind of dance around them. The whole thing looks very futuristic. After this dance the device knows where the tumour is. It then computes the path to be taken by a needle to reach the tumour from outside. A laser projects a red point on the skin where the needle is to be inserted. The surgeon puts the point of the needle there and then rotates it until another red point coincides with the other end. This fixes the correct direction and he can then insert the needle. At the end of the needle there is a microsurgical device which can be steered from a computer. Of course there is also a camera which provides a picture of the situation on the computer screen. The movements of the surgeon’s hands are translated into movements of the device at the end of the needle. These are scaled but also subject to noise filtering. In other words, if the surgeon’s hands shake the computer will filter it out. There is also a further refinement of this where a robot arm connected to the imagining device automatically inserts the needle in the right way. The result of all this technology is that, for instance, a single small metastasis in the lung can be removed very effectively. One of the most interesting things the surgeon said concerned the effects of the pandemic. One effect has been that people have been more reluctant to go to the doctor and that it has taken longer than it otherwise would have for lung cancer patients to go into hospital. The concrete effect of this on the work of the surgeon is that he sees that the tumours he has to treat are on average in a more advanced state than they were than before the pandemic. Putting this together with other facts leads to the following stark conclusion which it is worth to state clearly, even if it is sufficiently well known to anyone who is wiling to listen. The reluctance of people to get vaccinated against COVID-19 has led to a considerable increase in the number of people dying of cancer.

### The age of innocence

February 20, 2022

In a previous post I talked about my search for authors to read who are new to me. Now, following one suggestion there, I have read ‘The age of innocence’ by Edith Wharton. I did appreciate the book a lot. What are the reasons? The ‘psychological subtlety’ I mentioned in my previous post is also very much present in the novel. The society portrayed in the book is that of a small group of rich and elitist people in the New York of the late nineteenth century. In that group people tend to avoid talking about the things which are really important to them. This means that they have to be good at communicating without words. The novel is very successful in presenting the resulting non-verbal communication. Another aspect of the book I appreciate is the use of language. There are often phrases for which I felt the necessity to pause and savour them. The book is also rich in humour and irony.

What are the main themes of the novel? One is a certain society. We see a small number of people who are very rich, know each other well and are very resistent to letting anyone else into their circle. In some ways it is similar to the aristocracy in Europe at that time except that it lacks the long historical tradition and the codification of its rules. It consists mainly of people without special talents or great intelligence. In this society people are known because they are known and because certain other people in the same circle accept them. It reminds me a bit of the ‘celebrities’ of the present day who have similar characteristics including in many cases a lack of any obvious talent other than self-presentation. The main difference is that there are more chances to enter the circle of the celebrities. The two main characters in the book, Newland Archer and Ellen Olenska, are exceptions to the general rule. The Countess Olenska, as she is usually referred to in the book, comes from the small society described there but married a Polish count. She left him due to his bad treatment of her and came back to New York. What this treatment was is never really specified. In fact, it is not only the characters in the book who do not say openly what they mean. The author also presents many things in a way which is more suggestive than specific. The case of what really happened between the Countess Olenska and her husband is a good example but there are many more. In a way I found it a little frustrating to feel that I was waiting for information which never came. On the other hand I find that this allusive style has its own attraction. For me the Countess Olenska, who often ignores the strict rules of the society she is living in and is clearly highly intelligent and thoughtful, is the most attractive character in the novel. Newland Archer is the central character in the book in the sense that the story is told from his point of view, although in the third person. In his own way he questions the standards of the society around him. He often has impulses to act in a positive way but seems incapable of following them. He marries May Welland, a cousin of the Countess Olenska, who is the ideal partner for him in terms of her social standing and physical appearance. May is extremely conventional and not very intelligent. Her husband soon sees her as stupid, although in the end she has enough cunning to trick him and, in a sense, triumph over her clever cousin. There is a great deal of erotic tension between Archer and the Countess but, in the abstract and not only the literal sense, the orgasm never comes. May is treated cruelly in the text. Although it happens without any further comment the fact that she is given a painting of sheep as a present is a good example.

The two main characters are very different but in a way they mirror each other. I wonder to what extent the Countess Olenska is an image of the author and to what extent Newland Archer might be a better one. For me the most interesting aspect of the novel is what it has to say about love. I found the ending, whose content I will not reveal here, striking.

After I finished this post I remembered a question which I had asked myself and forgotten. This concerns the question of the relation of Marcel Proust, an author I much admire, to Edith Wharton. After all they did live in the same place at the same time. A short search led me to the following interesting answer.

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### Looking for a good read

January 9, 2022

The other book I bought on that day was a collection of writings of Jean Paul, ‘Die wunderbare Gesellschaft in der Neujahrsnacht’. Unfortunately I found it completely inaccessible for me and I only read a small part of it.

### Uğur Şahin, Özlem Türeci and The Vaccine

October 4, 2021

I now come to the story itself. Şahin and Türeci are both Germans whose parents came to Germany from Turkey. They studied medicine and they met during the practical part of their studies. They were both affected by seeing patients dying of cancer while medicine was helpless to prevent it. They decided they wanted to change the situation and have pursued that goal with remarkable consistency since then. They later came to the University of Mainz. They founded a biotechnology company called Ganymed producing monoclonal antibodies which was eventually sold for several hundred million Euros. They then went on to found BioNTech with the aim of using mRNA technology for cancer immunotherapy. An important role was played by money provided by the Strüngmann brothers. They had become billionaires through their company Hexal which sold generic drugs. They were relatively independent of the usual mechanisms of the financial markets and this was a big advantage for BioNTech. (A side remark: I learned from the book that the capital NT in the middle of the company name stands for ‘new technology’.) In early January 2020 Şahin foresaw the importance of COVID-19 and immediately began a project to apply the mRNA technology of BioNTech to develop a vaccine. The book is the story of many of the obstacles which he and Türeci had to overcome to attain this goal. In the US the vaccine is associated with the name Pfizer and it is important to mention at this point what the role of Pfizer was, namely to provide money and logistics. The main ideas came from Şahin and Türeci. Of course no important scientific development is due to one or two people alone and there are many contributions. In this case a central contribution came from Katalin Karikó.

How does the BioNTech vaccine work? The central idea of an mRNA vaccine is as follows. The aim is to introduce certain proteins into the body which are similar to ones found in the virus. The immune response to these proteins will then also act against the virus. What is actually injected is mRNA and that is then translated into the desired proteins by the cellular machinery. To start with the sequences of relevant proteins must be identified and corresponding mRNA molecules produced in vitro based on a DNA template. The RNA does not only contain the code for the protein but also extra elements which influence the way in which it behaves or is treated within a cell. In addition it is coated with some lipids which protect it from degradation by certain enzymes and help it to enter cells. Karikó played a central role in the development of this lipid technology. After the RNA has been injected it has to get into cells. A good target cell type are the dendritic cells which take up material from their surroundings by macropinocytosis. They then produce proteins based on the RNA template, cut them up into small peptides and display these on their surface. They also move to the lymph nodes. There they can present the antigens to T cells, which get activated. For T cells to get activated a second signal is also necessary and it is fortunate that mRNA can provide such a signal – in the language of vaccines it shows a natural adjuvant activity. In many more popular accounts of the role of the immune system in the vaccination against COVID-19 antibodies are the central subject. In fact according to the book many vaccine developers are somewhat fixated on antibodies and underestimate the role of T cells. There Şahin had to do a lot of convincing. It is nevertheless the case that antibodies are very important in this story and there is one point which I do not understand. Antibodies are produced by B cells and in order to do so they must be activated by the antigen. For this to happen the antigen must be visible outside the cells. So how do proteins produced in dendritic cells get exported so that B cells can see them?

I admire Şahin and Türeci very much. This has two aspects. The first is their amazing achievement in producing the vaccine against COVID-19 in record time. However there is also another aspect which I find very important. It is related to what I have learned about these two people from the book and from other sources. It has to do with a human quality which I find very important and which I believe is not appreciated as it should be in our society. This is humility. In their work Şahin and Türeci have been extremely ambitious but it seems to me that in their private life they have remained humble and this makes them an example to be followed.

October 2, 2020