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A Questions From a Paper I Read

  1. Apr 2, 2016 #1
    I am considering taking a PhD position in stochastic stability and control within the mechanical engineering department at a university which has offered me a lucrative RA. In discussing my topic with my potential advisor he gave me a paper to read this week. After I read the paper it was clear that I need to develop a background in pure mathematics (I have taken an introduction to proofs course as well as additional math classes in probability and nonlinear differential equations). From the paper I made a list of unfamiliar concepts. I am going to list them below. I am asking for help in identifying the subjects which these concepts span and books which might cover said topics. And as a bonus, if someone could give me a shoet explanation of any of the concepts or how they are related that would be very helpful. The list is below:

    1) SO(3) (rotation group SO(3)) - the set of all rotation in R3?
    2) Tangent Bundle (used in the context of Tangent bundle on SO(3)) - set of all tangent vectors to SO(3)?
    3) unwinding phenomena - in the context of continuous feedback control, im guess this is a type of instability?
    4) frobenius norm - I read a chapter about various norms, but what makes this one special vs an L2 norm (or any other norm for that matter), for example. My understanding is that norms are a measure of how similar two matricies are, but how is this measure significant?
    5) Ito formula - I know this is from stochastic calculus. Wikipedia says this is the "stochastic chain rule"
    6) LMI theory - I believe this is "linear matrix inequalities theory", but what does the theory state?
    7) Lie Algebra and/or Lie groups - I dont know anything about Lie algebra
    8) wong zakai correction - all I know is that this is related to SDEs
    9) class κ function - I know this is used to check stability of a function in control theory, but other than that I am clueless sadly
    10) Gain matrix - again, from control theory, but I know nothing about it
    11) martingale/supermartingale - from stochastic calculus. I know this is basically a model for which the probability of a current event cannot be determined from the outcomes of past events
    12) manifolds - basically a region which can be approximated as euclidean over small dimensions, but is not euclidean globally. I am not sure how these are used in applied math or control theory
    13) morse-lyapunov function - I know what a lyapunov function is (used for stability analysis in nonlinear DEs)
    14) LaSalles theorem - Ive never seen this before but wikipedia says it is another tool in stability analysis
    15) Lie Group Variational Integrator - no idea, never seen before

    Obviously I have a lot to learn. From this list I can see that these topics cover control theory and stochastic calculus, but it is my understanding that before learning stochastic calculus one should learn functional analysis or variational calculus. My background is in aeronautics, specifically fluid mechanics. In providing this list my hope is that someone who has a better mathematical background than myself can point me to materials or subjects so I may get up to speed on my topic. I also know nothing about lie algebra and am not sure where that fits-in in terms of stochastic stability and control.
  2. jcsd
  3. Apr 3, 2016 #2
    With all due respect, but this is not a useful thread. The problem is that this is a huge amount of topics. I can recommend you books for most of them, but they'll take years to complete. So if you want me to do that, then I'll do that. But I think it's far more helpful to go to your (future) advisor and ask him for some books that will contain most of this. We will likely suggest books which approach this from a mathy point of view. This might not be what you need. Your advisor might suggest you books from an engineering point of view, they'll take less time to complete and are probably going to cover more useful material than the math books.
  4. Apr 3, 2016 #3
    I was planning on doing that as well, and I'm not trying to be a mathematical expert, I just need a good starting point. I was hoping there might be a topic that provides a broad foundation covering most of my list.
  5. Apr 3, 2016 #4
    And one thing I cant stand sbout pure math is its inaccesability. I feel like the pure mathematicians go out of their way to make their work as unreadable as possible to new comers.
  6. Apr 3, 2016 #5
    It's really difficult to know how to recommend something if I don't know to what degree you need this. In any case, I think the book by Hubbard & Hubbard "Vector calculus, linear algebra and differential forms: a unified approach" might help you. (Get one of the later editions, they are better and contain more topics. There is a 5th edition now). What does this cover? It covers mainly multivariable calculus which you are familiar with. But it covers matrix norms and their uses in solving numerical problems, it covers manifolds (or at least some kind of predecessor to manifolds). It computes various derivatives of matrix-maps, which is essential for Lie theory.

    As for martingales, the following is nice: http://www.cambridge.org/be/academi...-stochastic-processes/probability-martingales
  7. Apr 3, 2016 #6
    I understand that is how you feel, but it's not true. If you find something inaccessible then you just miss the necessary prereqs. Sometimes it is impossible to describe something in easy terms.
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