My book summary for the book by Ernest Nagel and James Newman.

Quantum Mechanics Differences from Classical Mechanics States have different logical structure than CM. States and measurements are different unlike CM. e.g., Position and Momemntum can be determined by experiments in CM. Spins Particles have properties attached to it. e.g., mass, electric charge. Even a specific particle is not completely specified by its position. Attached to electron is an extra degree of freedom, called spin. Spin is as quantum mechanical as it can and we should not try to visualize it.

Notation $ F_i({q}) $ is the force on ith particle. $ {q} $ is the set of cordinates of all particles. If there are N particles in the system, Configration space: 3N dimensional space, records positions. State space: 6N dimensional space, records position and velocity. Phase space: 6N dimesnional space, records position and momentum. Newton’s second law gives us 6N equations. $$ \begin{align*} \dot{p}_i &= F_i({q}) \ \dot{q}_i &= \frac{p_i}{m_i} \end{align*} $$

API bf.add(x): adds x in the data structure x in bf: tests if x is in the data structure. Why not use Set or Dict? Bloom filters are more space efficients. They take memory lesser than the keys themselves. Cons can’t store associated data. does not support deletions. It is probabilistic data structure. That means, x in bf might have false positives. There are no false negatives. Applications Spell checkers: (40 years ago) Add the dictionary into the filter.

Lecture 2 Aug 19, 1991 Moskow: Coup d’etat against Mikhail Gorbichov (came power in Feb 1985) When came to power in 1985, west thought what would become of Soviet Union? Gorbachev was younger, talked like a western politician, allowed criticism of the regime. His policies were more open. It seemed like it was starting to change. In 1989 while revolution in Eastern Europe, Russia didn’t intervene; this was opposite to Russia’s earlier policies.

Convex Optimization Note: These notes are not mathematically rigorous. These are meant for quick reference, please read the Convex Optimization book from Boyd and Vandenberghe for more rigorous treatment. Chapter 2: Convex Sets Sets Affine Sets Affine combination: $\sum_{i=1}^{i=N} \theta_i x_i$ such that $\sum_{i=1}^{i=N} \theta_i = 1$. Draw a line passing through any two points, if the whole line is in the set, the set is affine. Every affine set C can be written as $C = V + x_0$ for any $x_0 \in C$, where V is a subspace.

Definitions Interior Point $x \in C$ A point is an interior point if we can find a ball around x which is completely inside C. Open Set A set is open, if all points are interior points. Closed Set A set is closed, if its complement is open. Closure Take the set’s complement. Find its interior. Take the complement of it. You got a closure. A point is in a closure, if for every $\epsilon$, you can find a point from the original set within $\epsilon$ distance.