User:Caliburn/Job List

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Calculus

  • Change of coordinates for polar, spherical and cylindrical coordinates

Real Analysis

  • Proper treatment of indefinite integrals. Do we want to talk about indefinite integrals as cosets in a particular function quotient space? This treatment is rare since indefinite integrals disappear before courses on real analysis.
  • Multivariable analysis - many gaps. To name a few, directional derivative, Frechet derivative, derivatives of functions $\R^n \to \R^m$, interplay with Jacobian, derivative of matrix functions (including $A \mapsto A^{-1}$ and $A \mapsto A^n$, $n \ge 0$), chain rules, inverse function theorem, implicit function theorem, submanifolds of $\R^n$.
    • Generalised integration by substitution with Jacobian

Advanced

  • Rescue pages on the $\delta$ "function" and re-interpret things in terms of distributions. It is worth mentioning abuses of notation present in engineering and physics.
  • Interplay between distributional/weak derivatives and classical derivatives
  • Sobolev spaces
  • Bochner spaces
  • The Definition:Cauchy Principal Value is set up, but isn't used anywhere. Applications include the Hilbert transform, solving the distributional equation $T \cdot x = 1$ and so on.


Complex Analysis

Need more geometry - inc stuff about winding numbers. Improve Residue Theorem. Would need someone stronger in algebraic topology than myself.


Measure Theory

  • A full treatment on the extended real numbers, including limits in $\overline \R$, Bolzano-Weierstrass in $\overline \R$, etc.
  • $L^p$ spaces are not properly set up. This includes arithmetic (which is "obvious" addition, pointwise multiplication, scalar multiplication, reciprocal of equivalence classes, and showing these are well-defined) and integration.
    • Duality in $\ell^p$ and $L^p$ spaces: $\paren {\ell^1}^\ast \cong \ell^\infty$ and $\paren {L^1}^\ast \cong L^\infty$ via an isometric isomorphism. Generally $\paren {L^p}^\ast \cong L^q$ isometrically if positive real numbers $p, q$ have $1/p + 1/q = 1$.
    • Most interchanges of sums/limits, integrals/limits, sums/sums, etc. are not properly justified. (I was guilty of handwaving this years ago) The site isn't quite ready for this - first we will need to cover the interplay between Lebesgue and Riemann integration (on my to-do list), and translate the Monotone Convergence Theorem, Dominated Convergence Theorem, Fubini/Tonelli's Theorem into appropriate results for the Riemann and Lebesgue integral. Then it'd be a matter of going through Special:WhatLinksHere/Fubini's Theorem, Special:WhatLinksHere/Tonelli's Theorem, and so on and fixing the instances. If the technical details are unwieldly and distract from the flow of the page, include this as a lemma.
    • Some results on indefinite integrals involve interchanges, so they would have to be changed to definite integrals leaving one bound arbitrary.
  • Product $\sigma$-algebras in the finite non-binary (more than $2$ $\sigma$-algebras) and countable case. There might be an uncountable case too.


Functional Analysis

  • Many definitions are unnecessarily specific, referring to Hilbert spaces instead of general inner product spaces, Banach spaces or normed vector spaces. I've started some of them in my sandbox. Feel free to finish them and bring them into definitionspace.
  • Dual spaces and dual norms - set up
  • Flesh out/tighten stuff, I will try to cover this stuff over the summer:
    • Hahn-Banach Theorem - complex case still needs doing, also stuff to do with dual norms
    • Second dual, reflexivity - nothing here yet it looks like
    • Weak convergence, Banach-Alaoglu Theorem - fill in redlinks
    • Spectral theory - absent, need bounded and unbounded cases


Probability Theory

Most proofs are only done in the context of classical probability. Also what we should be calling absolutely continuous r.v.s are called continuous. Some random points:

  • Independence
  • Characteristic functions - Lévy's Continuity Theorem and friends
  • Convergence theorems for random variables
    • Uniform integrability
  • Strong Law of Large Numbers, also infinite variance case of Weak Law of Large Numbers
  • Conditional expectation
  • Finish setup of Radon-Nikodym derivatives so we can sort out probability density functions
  • Martingales
  • Foundations for stochastic calculus
  • Convolution of density functions
  • Uniqueness for CDF and PDF


Elementary

  • Let $\theta \sim \ContinuousUniform {-\dfrac \pi 2} {\dfrac \pi 2}$. Let $X$ be the $x$-intercept of the line through $\tuple {x_0, \gamma}$ that makes an angle of $\theta$ with the vertical line $x = x_0$. Then $X \sim \Cauchy {x_0} {\gamma}$.
  • Let $T$ be the time period between consecutive events where these events happen at an average rate of $\lambda$. Then $T \sim \Exponential \beta$ with $\beta = 1/\lambda$.


Analytic Number Theory

  • Build towards rewriting Prime Number Theorem. Tried it and got bored, sorry. If I do an analytic number theory course in my masters year I'll revisit.


Also see