mushetty.me

A place to compile all my notes for various classes. This exists since reviewing my notes has started to become a pain.

Trying to figure out which notebook holds what
Attempting to decipher my messy handwriting
The stacks of 3 subject notebooks takes up a lot of space

All of the above is mitigated if I publish my notes online. That way, I can quickly find what I want, the notes don’t take up physical space, and I don’t need to read my god-awful handwriting. I can also access my notes from anywhere.

Feel free to browse. Any errata is of my own accord, and many of these are works in progress. The notes are also quite messy because I type them in class.

OperatingSystems

Notes following this MIT course and is based on the xv6 toy operating system. Operating System Interfaces Processes and Memory Forking Exec Operating System Interfaces Operating systems serve a number of purposes Abstracts away lower level hardware, so that programs don’t have to care about what disk you are writing to Allows multiple programs to run “at the same time” Provide well defined interface for programs to interact with each other For the interface, we want a simple interface that is easy to implement, but allows high sophisticated operations to happen UNIX philosophy of composing many simple programs together helps keep this problem trackable Can divide operating system into kernel space and user space kernel space is the program which provides services to other programs....

Quantum II

Symmetries Hibert Space Review $(v,w) \geq 0$, where (a,b) denotes an inner product This is linear in the right argument, and anti-linear in the first argument (ie. take complex conjugates) (v,v) = 0 implies v = 0 $v = \Sigma_{i} e_{i} (e_{i}, v)$ always holds in Hilbert spaces (ie. can always decompose into an eigenbasis, where $(e_{i}, e_{j}) = \delta_{ij}$) This can be extended to non-orthonormal bases (ie. $(e_{i}, e_{j}) = G_{ij}$), which implies that $(v,w) = \Sigma_{ij} (v_{i}e_{i})G_{ij}^{-1} (e_{j}, w)$ You can add Hilbert spaces of different dimensions together The new Hilbert space has the equivalence relationship of $<v_{1}+v_{2},w_{1}+w_{2}> = <v_{1},w_{1}> + <v_{2},w_{2}>$ The total dimensionality of the new Hilbert space is just the sum of the previous two You can also multiply two Hilbert spaces together The equivalence relation which must hold for the product is that $\lambda <v,w> = <\lambda v,w> = <v,\lambda w>$ You can decompose this product Hilbert space as a sum of tensor products of the eigenbases of each prior Hilbert space (ie....

ComputationalPhysics

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....

ExperimentalTechniques

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....

FPGA/Verilog Basics

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 Boolean Algebra Theorems Let all variables be boolean in nature (ie....

Art of Electronics

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...

CQM

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....

Databases

An introduction to databases. Logistics Intro Data Models Database Design Phases Bad Design ER Model Drawing ER Diagrams SQL (Structed Query Language) SQL Schema SQL Query Anatomy SELECT and FROM WHERE AS keyword String Operations LIKE ORDER BY Set Operations UNION INTERSECT EXPECT Null Values Aggregation GROUP BY HAVING Nested Subqueries Subquery tests for WHERE Subquery test for FROM DELETE, INSERT, UPDATE DELETE INSERT UPDATE JOINS Natural/Inner Joins Outer Joins Views Integrity Constraints Authorization Privileges Roles Object-relational Databases Complex Types User Defined Types Inheritance Textual Data Big Data Streaming Data Physical Storage Types Interfaces Magnetic Disks Flash RAID RAID Levels Indexing Ordered Indices B+ Tree Hash Indices Transactions Scheduling Query Processing Parsing Optimization Evaluation Relational Algebra Building Execution graph Query Cost Relational Algebra Implementations Selection Sorting Joins Duplicate elimination Projection Set Operations Implementation Outer Join Aggregations Evaluating Expressions Pipelining Logistics Course link...

Solid State Physics

An introduction to solid state physics. Logistics Heat Capacity of Solids Einstein Model of Heat Capacity of Solids Debye Model of Heat Capacity of Solids Electrical Properties (Drude Model) Constant E&M Force Thermal Conductivity Sommerfield Model Fermi-Dirac Statistics Sommerfield Model (cont.) LCAO (Linear Combination of Atomic Orbitals)/ Tight Binding Theory Chemistry Review Shell Theory Ionic Bonds Covalent Bonds LCAO Basics Vibrations 2 Types of Springs Crystal Structure Reciprocal Space Logistics Location: 307 Pupin Time: 2:40-3:55 PM Tuesdays and Thursdays Textbook: The Oxford Solid State Basics Grading Scheme 30% problem sets (5 in total....

Plasma Physics

An introduction to plasma physics. Logistics Grading Scheme MHD Equillibrium Consequences of MHD Equillibrium Theta Pinch Z Pinch Screw Pinch Toroidal Geometry Definitions Forces Grad Shafranov Equation Stream Functions Grad Shafranov Derivation Sketch Aspect Ratio Expansion Logistics Location: Mudd 825 Time: 1:10-2:40 Book: MHD Stability of Tokamaks by Zohm Grading Scheme Projects: 60% Homework: 40% MHD Equillibrium Suprisingly useful since particles in plasma undergo gyration, which enhances number of collisions and causes discrete particle to behave more fluid-like Setting time derivatives in MHD equations to zero This is a good assumption, since deviations from this result in motion with timescales on the order of microseconds (called the Alfven time $\tau_{A}$) Hence, if you are physically observe the plasma, good chance it is in EQ $\vec{v} = 0$ is the magneto-static limit (or that $v« v_{a}$) where $v_{a}$ is the Alfven speed (which in turn is defined as size of detector divided by $\tau_{a}$) In space, the magneto-static approximation is normally not valid Define the plasma beta as $\beta = \frac{p}{\frac{B^{2}}{2\mu}}$ or the thermal energy divided by the magnetic field energy Equation of state $P = n_{e}T_{e}+n_{i}T_{i}$ Momentum equation: $\rho(\frac{\partial}{\partial t}\vec{v}+\vec{v}\nabla\cdot \vec{v}) = \vec{J}\times B =\nabla P$ Maxwell’s equations $\nabla \times \vec{B} = \mu_{0}\vec{J}$ $\nabla \cdot B = 0$ $v \times B +E =0 \rightarrow E = 0$ $\frac{\partial }{\partial t} B = -\nabla \times E \rightarrow \frac{\partial }{\partial t} B =0$ Continuity $\frac{\partial}{\partial t}\rho + \nabla \cdot v = 0$ 0=0 Energy $\frac{\partial}{\partial t}P+\vec{v}\cdot\nabla p -c_{s}^{2}(\frac{\partial}{\partial t}\rho + \vec{v}\cdot \nabla \rho) = 0$ To Summarize, for MHD equilibrium, we have that...