The age of innocence

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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Persistence and permanence of dynamical systems

In mathematical models for chemical or biological systems the unknowns, which might be concentrations or populations, are naturally positive quantities. It is of interest to know whether in the course of time these quantities may approach zero in some sense. For example in the case of ecological models this is related to the question whether some species might become extinct. The notions of persistence and permanence are attempts to formulate precise concepts which could be helpful in discussing these questions. Unfortunately the use of these terms in the literature is not consistent.

I first consider the case of a dynamical system with continuous time defined by a system of ODE . The system is defined on the closure of a set which is either the positive orthant in Euclidean space (the subset where all coordinates are non-negative) or the intersection of that set with an affine subspace. The second case is included to allow the imposition of fixed values of some conserved quantities. Let be the boundary of , where in the second case we mean the boundary when is considered as a subset of the affine subspace. Suppose that for any there is a unique solution with . If let , where is the solution with initial datum and is the Euclidean distance. Let . A first attempt to define persistence (PD1) says that it means that for all . Similarly, a first attempt to define uniform persistence (PD2) is to say that it means that there is some such that for all . The system may be called weakly persistent (PD3) if for all but this concept will not be considered further in what follows. A source of difficulties in dealing with definitions of this type is that there can be a mixing between what happens at zero and what happens at infinity. In a system where all solutions are bounded PD1 is equivalent to the condition that no positive solution has an -limit point in . Given the kind of examples I am interested in I prefer to only define persistence for systems where all solutions are bounded and then use the definition formulated in terms of -limit points. In that context it is equivalent to PD1. The term permanence is often used instead of uniform persistence. I prefer the former term and I prefer to define it only for systems where the solutions are uniformly bounded at late times, i.e. there exists a constant such that all components of all solutions are bounded by for times greater than some , where might depend on the solution considered. Then permanence is equivalent to the condition that there is a compact set such that for any positive solution for . A criterion for permanence which is sometimes useful will now be stated without proof. If a system whose solutions are uniformly bounded at infinity has the property that then it is permanent. Here is the -limit set of , i.e. the union of the -limit sets of solutions starting at points of . If there is a point in there is a solution through that point on an interval which is non-negative. For some systems points like this can be ruled out directly and this gives a way of proving that they are permanent.

Let be the flow of the dynamical system, i.e. is the value at time of the solution with . The time-one map of the system is defined as . Its powers define a discrete semi-dynamical system with flow , where is a non-negative integer. For a discrete system of this type there is an obvious way to define -limit points, persistence and permanence in analogy to what was done in the case with continuous time. For the last two we make suitable boundedness assumptions, as before. It is then clear that if the system of ODE is persistent or permanent its time-one map has the corresponding property. What is more interesting is that the converse statements also hold. Suppose the the time-one map is persistent. For a given let be the closure of the set of points . This is a compact set and because of persistence it lies in . Let . This is also a compact subset of . If is sufficiently large then where is the largest integer smaller that . This implies that and proves that the system of ODE is persistent. A similar argument shows that if the time-one map is permanent the system of ODE is permanent. We only have to start with the compact set provided by the permanence of the time-one map.