The Codimension of a singularity

In summary: For example, the codimension of a critical point is not defined until very late in the book. Instead, the author uses phrases like "the codimension of a singularity is not defined until very late in the book," which makes it difficult to follow. Additionally, the author is very vague in some places. For example, he talks about a mapping H(x,p,\cdot) being maxima at "K distinct points" but does not explain what "distinct points" mean.
  • #1
Kreizhn
743
1
This may seem like a foolish question, but I can't seem to find the answer anywhere. Also, please forgive the question if it is ambiguous but the context in which it arises is not clear to me:

There is a mapping [itex] H(x,p,\cdot): \mathbb R \to \mathbb R [/itex] with x,p fixed, which attains its maxima at K distinct points [itex] u_k, k \in\left\{1,\ldots, K\right\} [/itex]. Each point [itex] u_k [/itex] is a critical point with a singularity of codimension [itex] c_k [/itex].

What is the codimension of a singularity?

I believe the author plans on later generalizing this for a mapping [itex] H:T^*M\times\mathbb R \to \mathbb R [/itex] for smooth mfld M, so if you could explain it in that context it would be helpful.
 
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  • #2
Perhaps the fact that the manifolds are R and that the word singularity is used is what is throwing me off.

I know that a regular point means that the pushforward is surjective. So is the codimension of a critical point the dimension of the relative complement of the image of the pushforward?
 
  • #3
surely the author defines his own terms.
 
  • #4
If they did, it was very subtly mentioned. I've been reading this book from the beginning and have not seen any mention of it. I shall go back and look closer.

The thing is, it does not seem to be an obscure term. I quick search of google scholar, for example, yields many papers that talk about "Codimension-n singularities" where n seems to be most often one, two, or three. Unfortunately, the papers often seem to define the codimension based on some obscure sets or assume that the reader already has knowledge of singularity codimension. For this reason I was hoping that perhaps I was just unaware of existing terminology.
 
  • #5
Also, while I have found this book to be generally very valuable, I have found it to be very poorly written. It is

"Singular Trajectories and their Role in Control Theory" by Bonnard and Chyba

The book is great for people who already have a working background knowledge in the field, but there is a dearth of definitions.
 

1. What is the codimension of a singularity?

The codimension of a singularity is a measure of the number of independent variables that must be varied in order to eliminate the singularity. It is the difference between the dimension of the space in which the singularity occurs and the dimension of the singularity itself.

2. How is the codimension of a singularity determined?

The codimension of a singularity is determined by analyzing the equations that describe the singularity. It is usually equal to the number of equations that must be satisfied in order to eliminate the singularity.

3. What is the significance of the codimension of a singularity?

The codimension of a singularity is important because it can provide information about the behavior of the system near the singularity. It can also help determine the type of singularity and how it may affect the overall system dynamics.

4. Can the codimension of a singularity change?

Yes, the codimension of a singularity can change if the equations that describe the singularity change. This can occur if parameters in the system are varied or if the system undergoes a phase transition.

5. How does the codimension of a singularity relate to other measures of singularity?

The codimension of a singularity is related to other measures of singularity, such as the multiplicity and the degree of the singularity. However, it provides a more quantitative measure of the singularity and can be used to classify different types of singularities.

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