A collection of interconnected notes that grow and evolve over time. Unlike a blog, these notes are continuously refined rather than published as finished pieces. Click any link to explore, and hover to preview.
742 notes · 2698 links
▸
Math
717
▸
Algebra
52
▸
Algebraic Structures
12
▸
Linear Algebra
40
▸
Eigen Stuff
2
▸
Linear Maps
10
▸
Matrices
15
▸
Vector Spaces
10
Closure (Algebra)
Jordan Canonical Form
Set operations
▸
Analysis
131
▸
Functional Analysis
34
▸
Differentiation
3
▸
Metric Spaces
26
Integral Operator
▸
Measure Theory
83
▸
Borel Sets And Functions
7
▸
Convexity
7
▸
Distribution Functions Lebesgue Stieltjes
11
▸
Lebesgue Integration
7
▸
Lp Spaces
7
▸
Measurability
40
▸
Algebras
9
▸
Construction Of Measures
7
▸
Functions
5
▸
Measures
11
▸
Riemann Integration
3
A Summary of MATH 891
▸
Real Analysis
13
Reproducing Kernel Hilbert Space
▸
Convex Optimization
22
▸
Optimization Problems
8
▸
Special Sets
12
Generalized Inequality
Quasiconvex
▸
General Math
9
▸
Geometry
12
▸
Number Theory
29
▸
Elementary Number Theory
16
▸
Finite Fields
8
▸
Quadratic Residues
5
▸
Probability
325
▸
Bayesian Analysis
13
▸
Monte Carlo
3
▸
Variational Inference
5
▸
Stein Variational Inference
4
Variational Inference
Bayesian Statistics
Likelihood
Maximal Likelihood Estimator
Parametric Family
Prior
▸
ℹInformation Theory
134
▸
ℹCoding
93
▸
ℹLossless Coding
23
▸
ℹGeneral
12
▸
ℹUniversal Coding
8
Kac's Lemma (Stationary Ergodic)
Lempel-Ziv Coding
Shannon-Fano-Elias Code
▸
ℹLossy Coding
70
▸
ℹChannel Coding
15
Achievable
Binary Erasure Channel
Binary Symmetric Channel
Binary Symmetric Erasure Channel
Channel Coding Operation
Discrete Communication Channel
Discrete Memoryless Channel
Fixed-Length Codes for Discrete Channels
Information Capacity
Lossless Joint Source-Channel Coding Theorem
Operational Channel Capacity
Rate
Shannon's Channel Coding Theorem for the DMC
Source-Channel Fixed-Length Code
Symmetric Channel
▸
ℹQuantization
35
▸
ℹPredictive Quantization
10
▸
ℹScalar Quantization
15
Bucklew and Wise
Centroid Condition
Companding Quantization
Encoder Decoder Structure of Quantization
High-Resolution Conditions
Lloyd-Max Algorithm
Nearest Neighbour Condition
Optimal Compressor
Optimal Quantizer
Performance Analysis of Quantizers
Quantizer Cell
Regular Quantizer
Scalar Quantizer
Signal to Noise Ratio
Uniform Quantizer
▸
ℹRate Distortion Theory
9
▸
ℹTransform Coding
11
Arithmetic-Geometric Mean Inequality
Gain of Transform Coding
High Resolution Optimality of KLT
Karhunen-Loeve Transform
Lagrange Multiplier
Lagrange Multipliers Technique
Optimal Bit Allocation
Optimal Distortion
Simplification of the Norm
Transform Coding Distortion
Transform Coding with Scalar Quantization
▸
ℹContinuous Information Theory
19
Average Cost Constraint
Average Power Constraint
Block Codes for the Gaussian Channel
Capacity of Parallel Gaussian Channels
Continuous Memoryless Source
Differential Cross-Entropy
Differential Divergence
Differential Entropy
Discrete-Time Continuous Memoryless Channels
Discrete-Time Memoryless Gaussian Channel (AWGN Channel)
Estimation Error and differential entropy
Fading Channel
Gaussian Noise Minimizes Capacity of Additive-Noise Channel
Information Capacity with Input Cost
Joint Differential Entropy of Multivariate Gaussian
Optimal Power Allotment
Shannon's Channel Coding Theorem for AWGN
Upper Bound on Channel Capacity
Volume
▸
ℹElementary Information Theory
22
Asymptotic Equipartition Property
Convexity or Concavity of Information Measure
Cross-Entropy
Data Processing Inequality
Discrete Memoryless Source
Divergence
Entropy Rate
Entropy
Fundamental Inequality
Jensen-Shannon Divergence
Joint Entropy
Log-Sum Inequality
Mutual Information
Renyi Divergence
Renyi Entropy
Self-Information
Source Entropy
Source
State
Stationary & Ergodic Source
Stationary Distribution
Stationary
▸
Probability Theory
77
▸
Convergence
16
Almost Sure Convergence
Central Limit Theorem
Convergence in Distribution
Convergence in Expectation
Existence of Sequences of Independent rvs
In Probability Convergence
Law of Large Numbers
Norm-like Function
Pointwise Convergence
Portmanteau's Theorem
Prokhorov's Theorem
Relatively Sequentially Compact
Scheffé's Theorem
Skohorod's Theorem
Theorems on Convergence
Weak convergence
▸
Expectation Variance
13
▸
Fundamentals Of Probability
27
Almost surely
Axioms of Probability
Bayes' Rule
Bernoulli Trial
Borel-Cantelli Lemma
Coin Tossing Probability Space
Conditional Independence
Conditional Probability of Events
Continuity of Probability
Countable Sub-additivity of Events
De Finetti's Theorem
Distribution
Event Space
Existence of Uniform Measure
Extension Theorem
Independent
Infinitely often
Kolmogorov 0-1 Law
Law of Total Probability
Probability Measure
Probability Space
Random Variable
Random Vector
Sample Space
Tight
iid
σ(X)
▸
Probability Functions
6
▸
Types Of Rvs
10
Summary of MATH 895
▸
Stochastic Differential Equations
24
▸
Ito Calculus
19
Bichteler-Dellacherie Theorem
Cross Variation
Doléans Measure
Itô Isometry
Itô Representation Theorem
Itô Stochastic Integral is a Local Martingale
Itô Stochastic Integral
Itô's Formula
Multidimensional Itô Formula
Mutual Variation of Itô Stochastic Integrals
Mutual Variation
Predicable Processes
Predictable sets
Quadratic Variation
Semimartingale Integral
Semimartingale
Solution to SDE
Stochastic Interval
Stopping Time Integral
▸
Mathematical Finance
5
▸
Stochastic Processes
77
▸
Markov Chains
36
▸
Stability
17
(n-μ)-small
Atom
Convergence to Equilibrium
Dobrushin's Ergodic Coefficient
Existence of Invariant measure
Foster-Lyapunov Theorems
Harris Recurrent
Invariant Distribution ↔ Positive Recurrence
Invariant probability measure
Kac's Lemma
Null Recurrence
Petite Set
Positive Harris Recurrent
Positive Recurrent
Recurrent
Transient
Uniquely Ergodic
Accessible
Aperiodic
Chapman Kolmogorov Equation
Communication
Continuous Time Markov Chain
Detailed Balance
Equivalent Bounds
Ergodic Theorem
Indistinguishable
Invariant Distribution for CTMC
Invariant Distribution
Invariant Measure
Irreducible
Joint Markov Chain
Markov Property
Markov chain
Q-Matrix
Stationary
Transition Kernel
▸
Martingales
14
Continuous Martingale with Finite Variation is Constant
Càdlàg
Doob's Forward Convergence Theorem
Doob's Maximal Inequalities
Doob's Optional Sampling Theorem
Doob's Upcrossing Inequality
Local Martingale
Lp Martingale
Lévy's Convergence Theorems
Martingale Convergence Theorem
Martingale Equivalence for Stopping Times
Martingale
Stopped Process is also R.C. Martingale
Supermartingale
▸
Special Processes
5
▸
Times
8
Adapted
Filtration
Indistinguishable
Ionescu Tulcea Theorem
Kolmogorov Extension Theorem
Modification
Non-Explosive
Same Finite-Dimensional Distribution
Stochastic Process
Stochastic Realization
▸
Systems Control
94
▸
Linear Control
41
▸
Calculus Of Variations
7
▸
Controllability Observability
8
▸
Existence Uniqueness
2
▸
Feedback
7
▸
General
4
▸
Realization Theory
10
▸
🚧Safe Control
8
▸
Stochastic Control
20
▸
Optimal Stochastic Control
13
Average Cost problem
Bellman's Optimality Principle
Blackwell's Irrelevant Information Theorem
Controlled Markov Chain
Discounted Infinite Horizon Optimization
Finite Horizon Optimization
Linear Quadratic Problem
Markov Decision Process
Measurable Selection Conditions
Optimal
Policy
Strategic measure
Value Iteration Algorithm
▸
Partially Observed
5
▸
Stochastic Learning
2
▸
Stochastic Teams
18
▸
Systems
7
▸
Topology
43
▸
Compactness
7
▸
Spaces
10
▸
Topology Examples
6
Closed
Closure
Heine-Borel Theorem
Interior Point
Metrizable
Neighbourhood
Open
Topological Space
Topology
▸
Engineering
12
▸
Machine Learning
13