Linear Regression You have a set of data $(x_{i},y_{i},\sigma_[i])$, where x is the independent variable, y is the dependent variable, and $\sigma$ is the uncertainty in the dependent variable You want to fit a line to the data ($\hat{y} = mx+b$) You also want the uncertainties of the parameters Why would you want to do this? To know the parameters of the model To extrapolate/interpolate values of the dataset that you don’t have How you do this depends on your viewpoint a frequentist approach would try to maximize the likelihood estimate (information theory) Suppose that you have a probability density for the data $y_{i}$ ranging from 1 to N Treat this density as a function of the parameters....

Logistics Thermodynamics 0th Law 1st Law Response Functions 2nd Law Entropy 3rd Law Thermodynamic Potentials Enthalpy Helmhotz Free Energy Gibbs Free Energy Grand Potential Probability Definitions Important Probability Distributions Kinetic Theory of Gases Liouville’s Theorem Bogoliubov-Born-Green-Kirkwood-Yvon Hierarchy (BBGKY) Boltzmann Equation H-Theorem Classical Statistical Mechanics Microcanonical Ensemble Two State System Example 0th Law 1st Law 2nd Law Ideal Gas Mixing Entropy The Canonical Ensemble Grand Canonical Ensemble Conserved Ensemble Notes Interacting Particles Cumulant Expansion Mean Field Theory of Condensation Critical Points Quantum Statistical Mechanics Dilute Polyatomic Gases Quantized Vibrational Modes Quantized Rotational Modes Blackbody Radiation Formalism Identical Particles Grand Canonical Formulation Degenerate Fermi Gas $f_{m}^{\eta}(z)$ Denegerate Bose Gas Logistics Book(s) Landau and Lifshitz Kardar Pathria Thermodynamics 0th Law Thermodynamic equilibrium means that each object in thermal contact with each other have the same temperature Ideal gases, as you cool them down, all converge to a single point, which defines absolute zero....

The Basics Hibert Space Review In an infinitely dimensional space, we have some abstract vector $\Psi$. We can choose coordinate axes such that all values can be taken by the position x, so that we can describe $\Psi$ as as set of components $\Psi(x)$ Could have also chosen p instead for our coordinate axes $(v,w) \geq 0$, where (a,b) denotes an inner product This is linear in the right argument, and anti-linear in the first argument (ie....

Happy New Year! The semester is over, and I passed all my classes, and that’s all that matters! First, I finished integrating my POVFan, and it works! (see below) It doesn’t look like a continuous image, since the frame rate of my camera is much higher than the angular frequency of the motor. But in person, you see a continuous image (the “framerate” of your eyes can’t keep up with the rotation speed of the blade)....

The bad news: It’s midterm season ATM, so life could be better. I got my first roach. Don’t know how I haven’t encountered one in the 5 years I’ve been in New York, but here we are. Dealt with it immediately though. The good news: With regards to the Lox Rust compiler, I managed to implement functions and closure (closures are such a pain…). I’m now starting down the barrel of writing a garbage collector for the VM, which would require a fundamental restructuring of the codebase....

IEEE Floating Point Standard Arrays Numerics Random Numbers LCM (Linear Congruent Method) Testing RNG Nonuniform Distributions Non-Analytic Nonuniform Distributions Interpolation Linear Interpolation Fitting Basis Coefficients Polynomial fits Fourier Basis Integration Trapezoid Rule Simpson’s Rule Rescaling Integrals Differentiation Scaling Linear Algebra LU SVD Eigensystems Uses Principal Component Analysis (PCA) Dimensional Reduction Root-Finding Minimization Quadratic Fit Golden Section Search Multi-Dimensional FFTs ODEs PDEs Initial Condition Problems Boundary Condition Problems MCMC (Markov Chain Monte Carlo) Markov Chain Metropolis Hastings Burn-in and Proposal Distribution IEEE Floating Point Standard $\pm 1+ 2^{e-127}\Sigma_{i=0}^{22} f_{i} 2^{-(i-23)}$ e is the exponent number $f_{i}$ represents the floating bits Some edge cases: e = 0, $f \neq 0$ are called subnormal numbers, where “1” above is replaced with 0 and $e-127$ is replaced by $e-126$ e=0 and f=0 is a signed zero e=255, f=0 is a signed $\inf$ e=255, $f\neq = 0$ is NaN Arrays TL;DR if you want to go fast, then use numpy arrays, since they are more contiguous in memory (and hence cache friendly) compared to python lists You also don’t have to worry about type checking since Python needs to check if operations There are also some built-in functions that numpy provides which are faster than Python built-ins Numerics TL;DR Keep numbers from being too small and too large and you’re good $E[\delta (f(a))^{2}] \approx E[[\frac{\partial f}{\partial a}]^{2} \delta a^{2}]$ Assuming similar errors for each operations, you find that the round-off error scales as $\sqrt{N}$ Making finer time steps yields a better approximation, but the round-off error degrades this....

Taken from Experimental Techniques in Modern High-Energy Physics: by Hanagaki, Tanaka, Tomoto, Yamazaki Why? Two basic categories of HEP experiments Find any new particle by converting collision energy to particles Done through finding resonances in the final state Investigate the nature of a particular interaction Done through precise measurements of other physical observables In theory, you work with a small number of states to keep problems tractable. Experimentally, you can get thousands of final states which you have to parse through The number of particles goes as $ln \sqrt{s}$ where s is the square of the center of mass energy of the collision This the angular density to go up, which create “jets” (ie....

Notes adapted from MIT Digital Electronics Lab Also from here Boolean Algebra Theorems Truth Tables Specific Functions 1-Bit Functions 2-bit Functions XOR Overview Verilog Basics Variables Parameters Operators and Loops Enums Syntax Oddities Combinational Logic Continuous Assignment Procedural Assignment Glitches! Gotchas Nesting Modules Combinational Versus Sequential Logic Latches Characteristic Equations Registers Verilog for Sequential Blocks Sequential Always Blocking and Non-Blocking Assignment External Signals (Metastability) FSMs FSM State Encoding Converting to Verilog Clocks PLLS The Treachery of Clocks Skew Jitter Proper usage Timing Timing Parameters Slack Video Black and White TV Color TV VGA HDMI Memory History of Memory Key parameters of memory FPGA SRAM Fabric Case Studies Off FPGA Memory EPROM Family DRAM Buffering DRAM FSM Modularization Pipelining Math Complexity Division CORDIC AXI AXI Signals Signed Number DSP48 CRCs Polynomial Arithmetic Boolean Algebra Theorems Let all variables be boolean in nature (ie....

Notes taken while reading “The Art of Electronics”. I realized that my electronics knowledge is a bit spotty and more theoretically inclined, hence why I’m doing this. The Basics, Because You Have to Start Somewhere Resistors Resistor Rule’s of Thumb Voltage Dividers Thevenin Equivalence Misc Use Cases Capacitors Common Applications RC Time Constants Frequency Filters Differentiators Integrators Inductors Applications Voltage Converters Transformers Diodes Applications Rectifiers Dual Rail Power Supply Voltage Multipliers Biasing Logic Gates Clamps Limiters Logarithm Inductive Kickback Transistors (BJTs) BJTs (Bipolar Junction Transistors) Transistor Switches and Saturation Switching From High Pulse Generator Schmitt Trigger Impedences of Sources and Loads Emitter Follower Emitter Follower Biasing Case Study Canceling Offset Transistor Current Sources Common-emitter Amplifier Unity Gain Phase Splitter Phase Shifter Transconductance Ebers-Moll equation Rules of thumb Emitter Follower Redux Bias Stability Current Mirrors The Basics, Because You Have to Start Somewhere There are two quantities that we care about with circuits: Voltage and Current...

Logistics Principle of “Least” Action Symmetries Gauge Invariance Example Reduce to Quadratures Hamiltonian Mechanics Phase Portraits Poisson Brackets Properties Jacobi Proof Sketch Poisson’s theorem Canonical Transformations Why We Care Liouville’s Theorem Canonical Invariants Entropy Tangent Infinitesimal CTs Hamiltonian Noether’s Theorem Hamilton Jacobi Equation Action Variables Multiperiodic motion Planetary motion Liouville’s Integrability Theorem Cannonical Peturbation Theory Adiabatic Motion Quantum Mach Zender Hilbert Space Linear Algebra Review Time Evolution Heisenberg Formulation Compatible Observables Symmetries Uncertainty Relationship Phase Space Interpretation Entanglement And Mixed States Mixed States Von Neumann Entropy Partial Trace (“Tracing Out”) Infinite Spaces Translation Operator Scattering Probability Current Delta Function Path Integral Free Particle Saddle Point Approximation Path Integral Proof WKB Logistics Matthew Klebon klebon@nyu....