Mathematics

Statistics

2025-08-22

This page pairs well with Probability.

Statistical Inference

Definition (Random Sample & Model)

Let $X=(X_1,\ldots,X_n)$ be i.i.d. from a parametric family $\{F_\theta:\theta\in\Theta\subset\mathbb{R}^p\}$. The parameter $\theta$ is unknown; inference uses the randomness of $X$ to learn about $\theta$.

Probability

2025-08-22

expectation random‑variables distribution frequentist bayesian

This page pairs well with Statistics.

Elements of Probability Theory

Definition (Random Experiment, Sample Space, Events)

A random experiment has uncertain outcomes. The sample space $S$ is the set of all possible outcomes. An event $E$ is a subset of $S$. The certain event is $S$; the impossible event is $\varnothing$.

Real Analysis

2025-07-20

limits continuity topology

I am finding Real Analysis to be more difficult than any other mathematics that I have studied before. I can seem to verify the truth of statements because they seem right; but I am having a difficult time producing rigorous and correct proofs.

It seems that High-School children (on the internet) are able to self-study Fomin with success. Bitterly, we remind ourselves:

"Comparison is the thief of Joy"—Theodore Roosevelt (probably)

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Calculus

newton leibniz change differential integral

You should see gathered listings from this directory below:

Complex Analysis

imaginary residues conformal‑mappings

Topics Analytic Functions & Cauchy-Riemann Equations Contour Integration & Residue Theorem Laurent Series & Singularities Conformal Mapping Important Theorems Cauchy’s Integral Theorem Cauchy’s Integral Formula Residue Theorem Rouché’s Theorem Maximum Modulus Principle

Discrete Mathematics

counting combinatorics logic number‑theory

Topics Set Theory & Boolean Algebra Logic & Proof Techniques Combinatorics & Counting Graph Theory Number Theory (Divisibility, Modular Arithmetic) Recurrence Relations Finite Automata & Formal Languages Discrete Probability

Important Theorems De Morgan’s Laws (Logic & Boolean Algebra) Pigeonhole Principle (Combinatorics) Inclusion-Exclusion Principle (Counting) Euler’s Formula for Graphs ( Handshaking Lemma ( Chinese Remainder Theorem (Number Theory) Fermat’s Little Theorem ( RSA Cryptosystem & Modular Inverses Master Theorem (Recurrence Relations)

Functions

elementary non‑elementary discontinuous algebraic transcendental

Icons

svg log exp complex

All of the site favicons that I use have been generated by contour plots of the complex logarithm and complex exponential functions.

Experiments

Real

Imaginary

Absolute

Linear Algebra

matrices vectors spaces eigen norms

Linear Algebra

Topics Vector Spaces & Linear Independence Matrix Operations & Determinants Eigenvalues & Eigenvectors Linear Transformations Orthogonality & Inner Products Singular Value Decomposition (SVD) Important Theorems Rank-Nullity Theorem Invertible Matrix Theorem Spectral Theorem (Diagonalization of Symmetric Matrices) Cayley-Hamilton Theorem Gram-Schmidt Process Perron-Frobenius Theorem (Positive Matrices)

Optimisation

constrained unconstrained convex

Topics Convexity & Optimization Techniques Gradient Descent & Newton’s Method Lagrangian & Duality Theory Integer & Combinatorial Optimization Linear Programming & Simplex Method Important Theorems KKT Conditions (Karush-Kuhn-Tucker) Lagrange Multipliers Strong & Weak Duality (Linear Programming) Farkas' Lemma Von Neumann’s Minimax Theorem

ἀγεωμέτρητοσ μηδεὶσ εἰσίτω

let no-one ignorant of geometry enter here