These sources explain mathematical ideas, from proofs to queues to braided hair. This page lists eight sources, oldest first where dated.
On the Early History of the Singular Value Decomposition
The article is a historical survey with modern notation laid over old derivations. It walks through each mathematician's argument and judges its clarity. It ends with later extensions, naming and applications. Evidence is the original papers, paraphrased.
A mathematical formalisation of dimensional analysis
Terence Tao explains how units and dimensions can be built on firm mathematical ground. The post gives a parametric model and an abstract model. The post is argument and worked examples, with no data. The downloaded text lost the equations, so the read is partial.
Waiting time, load factor, and queueing theory: why you need to cut your systems a bit of slack
The post uses queueing theory to explain why a busy system responds slowly. It backs the claim with a short simulation and one exact formula. It then applies the idea to personal workload and to computing. Evidence is a simulated queue, not field data.
How Gödel’s Proof Works
Natalie Wolchover gives a simplified, informal tour of Gödel's 1931 proof. It follows the scheme in a 1958 book by Nagel and Newman. The article uses small worked examples of Gödel numbering. It offers explanation, not new results.
Early years of Biased Random-Key Genetic Algorithms: A systematic review
Four authors review 253 papers that use BRKGA, a genetic algorithm framework. The report counts papers, authors and countries and maps citation and keyword patterns. It describes the literature and does not compare how well the method performs.
A Good Lemma Is Worth a Thousand Theorems (2007)
The source is a modest thread, not the essay. Commenters name favorite lemmas and argue about what makes one important. One side thread turns technical. The linked essay is a short opinion piece with a few added quotations.
Robust Safety Filters for Lipschitz-Bounded Adaptive Closed-Loop Systems with Structured Uncertainties
The paper adds a safety filter to adaptive control of systems with uncertain parameters. It derives a safety condition from error bounds and solves it as a convex program. Its evidence is proofs plus one MATLAB quadrotor simulation compared against four other safety filters.
Mathematical Patterns in African American Hairstyles
Gloria Gilmer reports on the geometry found in hair braiding. The paper was prepared for a teachers' workshop. Evidence is observation and interviews in salons, described briefly. It ends with four classroom activities and a reference list.
Connections to other subjects
These links are this wiki's own reading, not claims the sources make.
- Waiting time, load factor, and queueing theory: why you need to cut your systems a bit of slack and Embracing the parallel coding agent lifestyle, from Harnesses and agent loops. Willison calls review the bottleneck when agents run in parallel. Bernhardsson's queue shows what that costs. Latency climbs fast past about 80% load. So a fully busy reviewer may be the slowest part. Perhaps reviewers should stay half idle on purpose.
- Mathematical Patterns in African American Hairstyles and What Heraldry and Japanese Mon Can Teach Us About Building Visual-Identity Generators, from Images made by rules. Overmyer asks which rules a tradition held and which came later from explainers. Gilmer's braiders use tessellations without seeing them as mathematics. So a rule can live in the hands before anyone writes it down.
- A mathematical formalisation of dimensional analysis and Everyone Says Assembly Is Untyped—Everyone Is Wrong, from Compilers and interpreters. Tao says dimensional consistency catches errors and reveals the form of laws. gingerBill says assembly already has types: operand forms and widths. In both, the structure existed before anyone checked it. A checker only shows what the domain already enforced.
Glossary
- Adaptive control
- A controller that adjusts its own parameters online to cope with an uncertain plant.
- Angle bisector
- A line that splits an angle into two equal parts.
- Axiom
- A basic statement that is assumed true as a starting point.
- Axiom of choice
- An assumption that lets mathematicians pick one item from each of many sets.
- Bibliometric analysis
- The quantitative study of publications, authors and citations.
- Bilinear form
- A formula mixing two sets of variables, each appearing to the first power.
- Co-citation analysis
- A method that links papers that are cited together by other papers.
- Complete
- Able to prove every true statement in its area.
- Consistent
- Never leading to a contradiction.
- Continuum hypothesis
- A question about the sizes of infinity that axioms cannot settle.
- Control barrier function
- A function whose sign tells a controller whether the system is inside a safe region.
- Covector
- A vector-like object that measures vectors, and transforms differently from a vector.
- Coyoneda lemma
- A result in category theory, used in functional programming.
- Cycle time
- The time from starting a piece of work to finishing it.
- Decoder
- A function that turns a chromosome into an actual solution and scores it.
- Deflation
- Solving a problem by removing one solved part to leave a smaller one.
- Dimensional analysis
- Checking and deriving physical relations by tracking units such as length, mass and time.
- Ethnomathematics
- The study of mathematical ideas inside the cultural practices of a people.
- Forward invariance
- A set is forward invariant when a trajectory that starts inside it never leaves.
- Fundamental shape
- The single shape that is repeated to make a tessellation.
- Genetic algorithm
- A search method that evolves a population of candidate solutions by selection and mixing.
- Geometric distribution
- A distribution for the number of trials before the first success.
- Gödel number
- A whole number that stands for a formula or a sequence of formulas.
- Halting problem
- The question of whether a program will eventually stop or run forever.
- Integral equation
- An equation in which the unknown function sits inside an integral.
- Johnson-Lindenstrauss lemma
- A result on shrinking high-dimensional data while roughly keeping distances.
- Lemma
- A smaller proved statement used as a step toward a larger result.
- Lipschitz bound
- A limit on how fast a quantity can change when its input changes.
- Load factor
- The share of a system's capacity that is in use.
- M/M/1
- A queue with one worker and random, exponentially spaced arrivals and service times.
- Metaheuristic
- A general search strategy for finding good answers to hard optimization problems.
- Perturbation theory
- The study of how results change when the input changes slightly.
- Queueing theory
- The mathematics of waiting lines and how long things wait in them.
- Random-key chromosome
- A list of numbers between 0 and 1 that encodes a candidate solution.
- Reference model
- A known model whose behavior the real plant is made to follow.
- Second-order cone program
- A convex optimization problem that can be solved efficiently and reliably.
- Singular value decomposition
- A way to write a matrix as a product of rotations and a diagonal scaling.
- Sobolev inequality
- A bound on a function's size in terms of the size of its derivatives.
- Structure group
- A group of transformations, such as rescalings, that acts on the quantities being studied.
- Tensor product
- A standard way of combining two vector spaces into a larger one.
- Tessellation
- A covering of a flat surface by copies of a shape, with no gaps.
- Theorem
- A major proved statement that a piece of mathematics aims to establish.
- Throughput
- How much work a system completes in a given time.
- Torsor
- A space that looks like a group but has no chosen starting point.