~shreyasminocha/uni-notes

4567b591ba891dad5e3b320054e12a5682a9d0a6 — Shreyas Minocha 6 months ago 10268a3
Update
3 files changed, 46 insertions(+), 0 deletions(-)

M comp/323.md
M math/365.md
M phil/360.md
M comp/323.md => comp/323.md +15 -0
@@ 209,3 209,18 @@ Shank's Algorithm: $O(\sqrt{N} \log N)$
If we have an algorithm that takes $O(S_{q^e})$ steps to solve the DLP $g^x = h$ in a group $G$ for any element with $order(g) = q^e$ where $q$ is prime. We can solve

### Miller-Rabin Test

### Polynomials

**Fundamental Theorem of Algebra**: every polynomial of degree $n$ has exactly $n$ roots (possibly multiple and complex).

**Bezout's Theorem**: suppose two polynomials $F(x, y)$ and $G(x, y)$ of degrees $m$ and $n$ respectively have no trivial factors. Then, $\{F(x, y) = 0\} \cap \{G(x, y) = 0\}$ has exactly $mn$ points.

Homogenization: $P(x, y, w) = \sum_{0 \leq i + j \leq n} a_{i, j} x^i y^j w^{n - i - j}$

Dehomogenization: $P(x, y, 1)$

-   $w = 0$: points at infinity
-   $w = 1$: affine points

Note: $(0, 0, 0)$ is not a point in the projective space.

M math/365.md => math/365.md +8 -0
@@ 293,3 293,11 @@ infinite continued fractions are irrational
all irrational numbers have infinite continued fraction expressions

distinct infinite continued fractions correspond to distinct irrational numbers

## Quadratic Forms

A binary integral quadratic form is a polynomial $Q(x, y) = ax^2 + bxy + cy^2$. if $n = Q(x, y)$ has a solution, we say that $n$ is represented by $Q$.

representation problem: given a quadratic form $Q$, which integers are represented by $Q$?

pairs with gcd greater than one can be reduced to pairs with gcd = 1. pairs with gcd = 1 are called primitive pairs.

M phil/360.md => phil/360.md +23 -0
@@ 443,3 443,26 @@ objectivity vs realism. objectivity is a feature of our beliefs. realism: our be
Mackie denies realism; Nagel accepts realism, but distinguishes it from objectivity

to Nagel, there are degrees of objectivity.

### _Perfectionism_ by Thomas Hurka

### _Psychological Egoism_ by Joel Feinberg

by helping others they'd be helping themselves. guilt-alleviation.

-   moral education

    rewards/punishments motivate us for self-interest reasons

-   self-deception
-   desire as motivation

    "we only do things that we desire"

    subject vs object of desire. conflating aim and consequences.

    lincoln and the pigs. why would you not have piece of mind?

-   helping others makes us happy

    e.g. warm glow of thinking of yourself as a good person by helping vs just taking a pill