Notes for Mechanics class for Spring 2023. Chapter 1 Main Concepts Scalars, Vectors and Matrices Derivatives of Position Chapter 2 Main Concepts Newton’s Laws Definitions Conservation Laws Chapter 3 Main Concepts Chapter 4 Main Concepts Nonlinear Oscillations Phase diagrams Plane Pendulum Chapter 5 Main Concepts Chapter 6 Main Concepts Chapter 7 Main Concepts Chapter 8 Main Concepts Chapter 9 Main Concepts Chapter 10 Main Concepts Chapter 11 Main Concepts Euler Equations Force Field Chapter 1 Main Concepts Scalars, Vectors and Matrices A scalar is some quantity that is invariant under any coordinate transformation A vector is an object in $\mathbb{R^{n}}$ that is invariant under rotations Vectors are closed under addition and scalar multiplication The Dot product is defined as $\vec{A}\cdot \vec{B} = \Sigma_{i} A_{i}B_{i}$ or equivalently $\vec{A}\cdot \vec{B} = |A| |B| \cos(\vec{A},\vec{B})$ $|A| = \sqrt{\Sigma_{i} A_{i}^{2}}$ This is invariant under rotation (since it is a scalar) Abelian The Cross Product is defined as $C = A \times B$ where $C_{i} = \Sigma_{j,k} \epsilon_{ijk}A_{j}B_{k}$ whre $\epsilon_{ijk}$ is the Levi-Civata tensor (0 if indices are the same, 1 if an even permutation of 1,2,3 and -1 if an odd permutation of 1,2,3) Alternatively $|C| = |A||B|\sin\theta$ Non-Abelian $A\times(B\times C) = (A\cdot C)B-(A\cdot B)C$ $A\cdot (B\times C) = B\cdot (C\times A) = C \cdot (A\times B) = ABC$ Matrices are objects in $\mathbb{R^{n\times m}}$ A simple 2D rotation is given by (can be derived by rotating the primed coordinate system around the orign by some angle and adding up lengths) $x_1^{’} = x_1 \cos \theta +x_2 \sin \theta$ $x_2^{’} = -x_1 \sin \theta +x_2 \cos \theta$ In general $x_{i} = \Sigma_{j=1}^{3} \lambda_{ji} x_{j}^{’}$ where the $\lambda$ are the directional cosines defined by $\lambda_{ij} = \cos(x_{i}^{’},x_j)$ ie....

Compilation of notes for Fundamentals of Computer Systems class for Spring 2023. Problem Solving Tips Administrative Stuff Exams Grades Lecture 0 Lecture 1 CPU Models Addition in different bases Definitions Modular Arithmetic Integer format with word size restriction Unsigned binary numbers Binary Addition Algorithm (BAA) Negative numbers Signed Magnitude Representation 1’s Complement Problems with Signed Magnitude and 1’s Complement 2’s complement Intuition behind 2’s complement Lecture 2 Why 2’s complement? Easy subtraction Detecting Overflow Floating Points IEEE standard for a 32 bit word Doubles Underflow Boolean Algebra Boolean Algebra Identities Distributive Law Proof DeMorgan’s Theorem Consensus Theorem circuit Representation of Boolean Algebra Coverting Circuits to Booleans NAND and NOR XOR Duals Lecture 3 Sum of Products (SoP) Product of Sums (PoS) Convert SoP to PoS Minterms Maxterms Karnaugh Map K-map Notation Summary of Simplification with k-maps 2-bit multiplier Don’t Care Conditions Drawing Circuits Lecture 4 Standard Circuits Enabler Decoder Circuit Decoder With Enable MUX (Multiplexer) Representing Functions with Decoders and MUXes MUX trick Shifter Circuit Barrel Shift left w/ Wraparound L-R Shift Circuit with Rollout Unsigned Adder Circuit Half-Adder Full-Adder Signed Adder/Subtractor (2’s-C) Ripple Carry Adder Optimizing Ripple Carry (Carry Lookahead) Merge with known 0’s Code Converter Contraction Example 1 Lecture 5 Latch Intuition SR Latch SR Latch with Control D Latch with control Latches Can’t be Clocked Flip Flops JK Flip Flop (JKFF) Flip Flop Table Trigger Types Sequential Circuit State Machines Registers Register MUXing Shift Register Ripple Counter PLA (Programmable Logic Devices) MIPS Programming Parts of a Program Computer Hardware CPU Memory Clock ISA (Instruction Set Architecture) Programming in MIPS Instruction Types Memory Pointers $pc sp Constants Sign-extend Pseudoinstructions Things to Remember Assembly Code ALU Details of Memory Single Memory Cell Coincident Selection Extending Memory Why is memory not clocked?...

Compilation of notes for Quantum I class for Spring 2023. The Basics The Current Density Operator (probability current) Erenhfest’s Principle Uncertainty Princple Time Independent SE Interference of Stationary States Solving TISE $V(x)=V_{0}$ Infinite Square Well Properties of Eigenfunctions Finite Square Well Dirac Delta Function Free Particle Fourier Transform Definition Scattering States SHO Commutation relation of $\hat{x}$ and $\hat{p}$ Ladder Operators Hermetian Operators Momentum Eigenvalues Fourier transforms Generalized Uncertainty Relationship Dirac Notation Solution to Spherically Symmetric Schrodinger’s Equation Angular Momentum Spherical Coordinates Spin Electron in Magnetic Field Addition of Angular Momenta The Basics The Wavefunction Written in 1 spatial dimension as $\Psi (x,t)$....

Compilation of notes for Machine Learning class for Spring 2023. Administrative Stuff Topic 1 Workflow of Classification Problems (Supervised Learning) Statistical Approach Classifier Maximum Likelihood Estimation Example Convergence Naive Bayes Classifier How do you quantify the quality of a classifier? Approaches to Classification Generative Approach Discriminative Approach Topic 2 Nearest K-Neighbor Importance of Closeness for k-Nearest Neighbors Distances Similarities Issues with the nearest neighbor k-NN Optimality Proof Practical Considerations of k-nearest neighbor Finding the k-th nearest neighbors takes time Metric of Closeness is Sometimes Unclear Derivative w....

Compilation of notes for Astrophysics and Cosmology class for Spring 2023. Special Relativity Spacetime Diagrams Worldline Inertial Frames Postulates of SR Deriving Time Dilation from diagrams Deriving Doppler Shift from Diagram Metrics Metric tensor (SR) Covariant versus contravariant vectors Metric in Spherical Coordinates 4-Momentum Example Classical Cosmology The Cosmological Principle Comoving Coordinates Hubble’s Law Wavelength relation to scale factor Luminosity Distance (L) Non-Euclidean Geometry Metric of a 2-Sphere Metric of the 3-sphere Metric of the Pseudosphere (hyperbolic geometry) Most general homogeneous isotropic metric What is $a(t)?...

Here is a compendium of my notes that I took to review for Machine Learning. This was meant to be a quick refresher on concepts and to put everything in one place. For a less cursory version, see the background section under Resources at the course website. Probability and Statistics Basic Concepts Axioms of Probability Definitions Baye’s Rule Expectation Common Probability Distributions Bernoulli Distribution Binomial Distribution Poisson Distribution Categorical Distribution Multinomial Distribution Guassian(Normal) Distribution Multivariate Gaussian Distribution Laplace distribution Dirac Delta Distribution Mixtures of Distributions Common Functions Logistic sigmoid $\sigma(x)$ Softplus Function $\Zeta(x)$ Information Theory Self-Information Shannon Entropy Kullback-Leibler (KL) divergence Cross-entropy Exponential Distribution Chi-squared Distribution Basics Goodness of Fit (GOF) Independence T-test One Sample Two Sample Paired Unpaired Linear Algebra Types of Objects in Linear Algebra Matrix Operations Central Problem of Linear Algebra Gaussian Elimination Reduced Row Echelon Form (RREF) Reading of Solutions of Ax=b from RREF LU Decomposition LDU Decomposition Identity and Inverses Vector Spaces Subspaces Orthogonal Complements Linear Transformations Diagonalization Change of Basis Linear Dependence and Span Basis and Dimension Norms Orthogonal Bases Gram-Schmidt and Orthogonal Complements Eigendecomposition Singular Value Decomposition (SVD) Moore-Penrose Pseudoinverse QR Decomposition Gram-Schmidt Trace Determinant Kernel, Range, Nullity,Rank Mapping definitions Kernel Rank and Nullity Least Squares Probability and Statistics Basic Concepts Axioms of Probability Probability Measure $P: \mathbb{F} \rightarrow \mathbb{{R}}$ such that $P(A) \geq 0$ for all $A \in \mathbb{F}$ $P(\Omega) = 1$ If $A_{1},A_{2}…$ are disjoint events, then $P(\cup A_{i}) = \Sigma_{i} P(A_{i})$ Definitions Sample space $\Omega$: The set of all possible outcomes Event space $\mathbb{F}:A$ is a subset of $\Omega$ a random variable quantity that has an uncertain value....

The Ising Model is a simple thermodynamic model of magnetic material that can visually demonstrate phase transitions. Attached is a client-side simulation of the Ising model that takes heavy inspiration from Daniel Schroeder. I modified his code to be asyncronous, allowed an arbitrary square lattice size, introduced more parameters that could be fiddled with, and added plots to visualize parameters of interest. Temp: 5 Coupling: 1 Field: 0 Steps Per Cycle: 1000 Size: 100 Start!...

I’ve always wanted to create my own website, so during winter break, I decided to do just that. I could have gone about it the smart way by hosting on a cloud provider like AWS or the like. I decide to do it the hard way and run my own webserver. I technically succeed. Here is what I did. The exact procedure might differ between operating systems, but the overall structure is the same....

The good news: Classes are going well so far. I’m learning about classical mechanics in the framework of Lagrangians and Hamiltonians in greater depth than I have every done before. In previous classes, we just stopped at deriving Hamilton’s equations. So far, we are exploring the theorems associated with phase space in greater detail Electricity and Magnetism gives me an excuse to read Jackson Computational physics covers a smattering of numerical techniques as they are applicable to physics....