## ~shreyasminocha/uni-notes

4567b591ba891dad5e3b320054e12a5682a9d0a6 — Shreyas Minocha 6 months ago
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
+
+
+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