```markdown
### Self-concordant calculus

#### Properties
- Preserved under positive scaling \(\alpha \geq 1\), and sum.
- Preserved under composition with affine function.
- If \(g\) is convex with \(\text{dom } g = \mathbb{R}_{++}\) and \(|g'''(x)| \leq 3g''(x)/x\), then 
  \[
  f(x) = \log(-g(x)) - \log x
  \]
  is self-concordant.

#### Examples
Properties can be used to show that the following are self-concordant:
- \(f(x) = -\sum_{i=1}^m \log(b_i - a_i^T x)\) on \(\{x \mid a_i^T x < b_i, \, i = 1, \ldots, m\}\).
- \(f(X) = -\log \det X\) on \(S_{++}^n\).
- \(f(x) = -\log(y^2 - x^T x)\) on \(\{(x, y) \mid \|x\|_2 < y\}\).

*Source: A. d'Aspremont. Convex Optimization M2.*
```