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As a mathematics graduate with focus on programming we did a whole lot of coding of some mathematical statements (as well as proving them), but yet rarely giving real life examples and applications for given statements.

So for my future work i would like to learn some of the applications (mostly in electronics, programming, physics...) and get some references where i can continue to learn so that - given a problem i can correlate that to something i have learned before.

I am most interested in applications for these fields:

  1. Numerical analysis

    • Interpolation (Hermite, multidimensional, Newton, Spline ...)
    • Approximation ( Least squares, uniform approximation,..)
    • Numerical methods for solving differential equations
    • Numerical methods for finding eigenvalues and eigenvectors
    • Numerical Methods for solving non-linear equations

      ...

  2. Mathematical analysis

    • Integration by curve, surface
    • Integrals with parameters
    • Fourier series,Fourier transformation,Fourier integral
    • Uniform convergence (for sequence of functions, series, integrals)
    • Weierstrass function, Riemann zeta function

      ...

  3. Measure theory

    • Lebesgue measure, Lebesgue integral
    • Radon - Nikodym theorem , derivate
    • Monotone convergence theorem, dominant convergence theorem
    • Lp spaces, norms

      ...

  4. Complex analysis

    • Cauchy integral theorem
    • Picard's theorem
    • Laurent series ...
    • Holomorphic functions

      ...

References, books, websites - anything will do, just as long as there are multiple examples (also the more on the side of programming (algorithms, problem solving) and electronics the better)

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  • $\begingroup$ This is way too vague, for the Fourier series and PDE the application are all the audio/video signal processing and many physical simulation. And if you want to review all this, my advice would be to start reading a book on the Riemann zeta function and in the same time an introduction to the distribution theory. $\endgroup$ – reuns Jul 3 '16 at 13:03
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Probability theory is to me strictly pure mathematics, but with strong applicable properties. If you like measure theory, along with functional/modern analysis then probability theory is the area for you; The axioms of probability theory (Kolmogorov) are strictly measure theoretic. Then how one builds up expected value and so on is also integration theoretical. One integrates with respect to a probability measure, thus making the Lebesgue integral or equivalent necessary. I strongly encourage you to read the table of content in Kallenberg's Foundations of modern probability theory, to get a glimpse of what probability theory might entail, from the analytical stand point. It is sadly a bit advanced though, but I do not expect someone to read it all.

Probability theory is to some degree either very analytical or very combinatorial/graph theoretical, i.e. either you get hooked on stochastic analysis and SDE:s (along with stochastic integration and such), or you get hooked on percolation theory and Markov analysis, the latter having intimate connections with statistical mechanics (Ising's model for instance).

And as a last point, probability theory is computationally challenging, in so far as we need strong programmers as well. For instance, Monte Carlo methods are a very popular numerical method.

I hope you do not mind me diverging from your wish list, but I believe probability theory is worth a notice.

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Let's check out the OP's wish list:

  1. Numerical analysis

    • Interpolation (Hermite, multidimensional, Newton, Spline ...)
    • The webpage Programming in Delphi contains some material (theory & software) about interpolation, with 2nd order splines only.
    • Numerical methods for solving differential equations
    • My own experience in this field has been summarized as : Highlights .
      (Link, because I don't want this answer look like personal advertizing)
      Research concerns the linear partial differential equations for convection / diffusion.
      And a Least Squares (Finite Element) Method is employed with ideal flow.
    Some of my favorite readings: Warning. The mathematics in such practical applications (engineering) books may be very specific, i.e not so "general" as is common in "theoretical" treatments. It may contain a lot of hand waving argumentation as well. Looking through all this is part of the challenge. What's more, the two main competitors in Numerical Analysis, the Finite volume method and Finite element method, have been invented by engineers in the first place, not by mathematicians. The latter have jumped in (especially with FEM) only after the beef has been roasted already.
  2. Mathematical analysis

  3. Measure theory

    • Somebody somehow has to show me the indispensability of Measure theory for practical applications. Of course, in some of the comments, Probability theory is mentioned as the counter example par excellence. Then we have the following MSE question as a reference: Between bayesian and measure theoretic approaches .
  4. Complex analysis

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    $\begingroup$ Isn't Measure theory the basis for Probability theory? I don't think I have to stress the importance of the latter for practical applications $\endgroup$ – Yuriy S Jun 29 '16 at 14:39
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    $\begingroup$ @daniels_pa, I'm a physicist, so I'm not the best person to answer you. Just a remark - the basics of probability theory (which I've learned in the third year as well) are not enough for serious quantum mechanics stuff. $\endgroup$ – Yuriy S Jun 29 '16 at 18:05
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    $\begingroup$ @daniel_pa Strictly speaking the Hermite polynomials are not in $L^2(\mathbb{R})$, but the Hermite functions form an orthonormal basis for $L^2$ and they are made from the polynomials themselves. They are also the eigenfunctions for the time independent Schrodinger eqn for the Harmonic oscillator. $\endgroup$ – Joel Jun 29 '16 at 18:32
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    $\begingroup$ Moreover, given different measures (or "weight functions") you have a different system of orthogonal polynomials. You are correct that this is intimately related to approximation theory. For instance, for gaussian quadrature, the zeros of orthogonal polynomials are used as sample sites. The polynomials in turn depend on a particular measure. $\endgroup$ – Joel Jun 29 '16 at 18:36
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    $\begingroup$ @HandeBruijn Thats the right path! Though would be great if there were books, websites, articles regarding specific fields (in this case fluid motion) using complex analysis, and not just MSE questions. They somehow got to the point where they need math - i am on the opposite side i have the math but have no idea how they got to need it. And thats the part I'm most interested in. Guiding some non physics people (but instead good at math) towards real world problem where they see the value of their knowledge. $\endgroup$ – daniels_pa Jun 30 '16 at 20:20

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