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A Convolution, singularity, kernel, principle values, linear stability analysis, quadrature points

  1. May 21, 2017 #1
    I'm reading a book on vortex methods and I came across the above mentioned terms, however, I don't understand what they mean in mathematical terms. The book seems to be quite valuable with its content and therefore I would like to understand what the author is trying to say using the above mentioned terms in his mathematics. Can someone please tell me what branch of mathematics or what book of mathematics I should read in order to understand these terms? If it helps, I would like to point out that I have a decent understanding of college level calculus.

    Thanks in advance!
     
  2. jcsd
  3. May 21, 2017 #2

    fresh_42

    Staff: Mentor

    This is a rather broad question and depends very likely on the context of vortex theory, as I assume you don't want to hear a standard answer on what a singularity is. Could you narrow it down by some context, examples or further explanations? Otherwise it's almost impossible to answer without complaints about what is meant where and by whom.
     
  4. May 21, 2017 #3
    Yes, I agree it's a rather broad question. Sorry about that.
    I would actually like to learn about singularities in a strict mathematical sense. So, if I have to learn about convolution, singularity and kernels in particular where should I start looking?

    I did google about them a bit, found some information in wikipedia. But I think it would be much better if I knew the basics of whatever it is I have to know to understand convolution, singularity and kernel. That way I would be able to understand the context of application of these concepts.
     
  5. May 21, 2017 #4

    fresh_42

    Staff: Mentor

    Well, all these belong to calculus in a way. E.g. I've found all terms in the book from Hewitt and Stromberg
    https://www.amazon.com/Abstract-Analysis-Graduate-Texts-Mathematics/dp/0387901388
    but this isn't quite easy to read as it is mainly based on measure theory, whereas usual college courses proceed along the lines real analysis - vector analysis - complex analysis and maybe followed by function theory and functional analysis. In addition to understand vortex theory, even some basics on differential geometry and differential equations might be needed. So in order to deal with vortex theory in special, it might be more promising to look out for individual papers, that deal with certain questions. Google often leads to lecture notes on certain topics, that can be read in a reasonable amount of time. Many universities provide such notes on the internet. But as a tip: it's better to search via Google rather than on the universities' homepages, as you normally cannot get through to the individual papers by starting on their homepages.
     
  6. May 21, 2017 #5
    Thanks a lot for the information, fresh_42. I'll try to look up lecture notes that are made available online. I think that's the easier way to learn too.
     
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