“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

Concept Index

Concept-index

Every concept this site knows about — 748 of them, harvested from the page tree, the skill trees, the hand-curated prerequisite graph, the glossary, and the claims. 22 are gaps: concepts the site assumes but has not written yet. The intended use: search for the thing you don't understand, follow its pages — or, if it's a gap, that's the next thing to build. Regenerate with npx tsx scripts/build_concept_index.mjs after adding content.

748 of 748 concepts covered (100%) — a page or a note attached. 58 carry notes; 0 have neither yet. The goal: every concept resolves to at least a short list of necessarily-true statements, even the ones with no page.

show only the 0 with no page and no note

#

↳ Balanced Binary Tree page
↳ Diameter of Binary Tree page
↳ Kadane with Indices page
↳ Kth Smallest in BST page
↳ Maximum Product Subarray page
↳ Minimum Depth of Binary Tree page
↳ Range Sum / Count in BST page
↳ Recover BST page
↳ Shortest Subarray Sum ≥ K page
↳ Sliding Window Minimum page
↳ Sort a Linked List page
"Fermion sign problem" 7 notes — The 2D Hubbard model — the minimal model of the cuprate superconductors — remains unsolved: quantum Monte Carlo hits the fermion sign problem away from half filling, where weights go negative and noise explodes.
"Frustration" 1 note — Frustration (a non-bipartite lattice) turns the sign problem on even at half filling — the reason frustrated magnets and the doped 2D Hubbard model resist quantum Monte Carlo.
"Genetic programming" 1 note — Genetic programming evolves a population of expression trees by tournament selection, subtree crossover, and mutation, scoring each candidate by its fit to the data.
"Kepler's third law" 1 note — Given planetary periods and semi-major axes, symbolic regression recovers Kepler's third law $T \\propto a^{3/2}$ ($T^2=a^3$) from the numbers alone — the elbow of the Pareto front at $R^2=1$.
"Mott insulator" 2 notes — A Mott insulator is a half-filled band that band theory calls a metal but that insulates from electron repulsion — a metal–insulator transition driven by interaction, with no change in filling and no broken symmetry.
"Mott transition" 5 notes — Two-site DMFT yields the closed form $Z = 1 - (U/U_c)^2$ with $U_c = 3D$ (Bethe lattice) — within about 4% of the full-DMFT (NRG) value $\\approx 2.9D$.
0/1 Knapsack Problem page
1D Eigenvalue Problems page
1D Finite Elements skill — Assemble a stiffness matrix; solve the Galerkin system.
1D Mesh skill — Partition a domain into elements with basis functions.
1s Integrals page
1s Program page
2D finite difference glossary — Discretization of the 2D Laplacian on a regular grid via the five-point stencil (u_{i±1,j} + u_{i,j±1} − 4 u_{ij})/h². Yields a sparse, banded linear system for Poisson and Laplace equations.
2D Finite Difference Method page
2D Finite Elements skill — Triangular elements and gradient assembly in 2D.
2D Ising Model (Metropolis MC) pageprereq node — The model system and the single-spin-flip sampler every refinement below starts from.
3Sum page
6502 Emulator pageskill — Emulate an 8-bit CPU to see an instruction set run.

A

A C Compiler in Python skill — Build a growing subset of C targeting x86-64.
A-Star Search page
A* search glossary — Best-first graph search that orders nodes by f(n) = g(n) + h(n), where g is path cost so far and h is an admissible heuristic estimating remaining cost. Optimal and complete with admissible h.
ABC skill — Bayes by Monte Carlo: accept when output matches.
ABC: Bayes by Monte Carlo page
Absorption spectrum 1 note — The absorption wavelength follows from the excitation energy by $\\lambda_{\\max} = 1240 / E$ with $E$ in eV; the bright band of a conjugated molecule is its HOMO$\\to$LUMO $\\pi\\to\\pi^*$ transition.
Adomian Decomposition pageskillglossary ↗ — A semi-analytic series method for nonlinear functional equations, expanding the solution and the nonlinearity in special Adomian polynomials. Yields recursion relations that can be summed or Padé-accelerated.
Algorithm Overview — Which to Use When page
Almgren-Chriss Optimal Execution page
ALU page
American Options & Longstaff-Schwartz page
Amortized analysis glossary — A technique for averaging the cost of a sequence of operations on a data structure, even when individual operations are occasionally expensive. Common examples: dynamic arrays and union–find.
An Orchestrator: AI-Feynman + a Solver Portfolio page 5 notes
Anderson acceleration glossary — A convergence accelerator that combines a few past iterates of a fixed-point map by minimizing the residual of their linear combination. Widely used to speed up SCF and Picard iterations.
Annihilation operator glossary — The second-quantized operator â_p that removes a fermion from spin orbital p, satisfying anticommutation with creation operators. Acts as zero on the vacuum.
Antisymmetry glossary — The requirement that a many-fermion wavefunction change sign under the exchange of any two particle labels. Imposed via Slater determinants in molecular and atomic problems.
APW Method page
Architecture page
Area of Circle page
Arithmetic Operations page
Arithmetic Units page
Arnoldi Iteration pageprereq nodeglossary ↗ — A Krylov-subspace method that builds an orthonormal basis for span{v, Av, A^2 v, ...} and projects A onto it as an upper Hessenberg matrix. Used for eigenvalues of large nonsymmetric matrices.
Array page
Arrhenius rate laws gap 3 notes — Every rate constant in the furnace is D₀·exp(−Eₐ/kT); temperature is the master knob.
Associated Legendre polynomials glossary — Solutions P_l^m(x) of the associated Legendre equation; combined with e^{imφ} they form spherical harmonics. They appear in any problem with axial symmetry.
Atomic basis functions prereq nodeskill — Gaussian primitives and contractions — the finite function set every matrix below is expressed in.
Atomic orbital glossary — A one-electron wavefunction associated with a single atomic center, used as a building block in LCAO expansions of molecular orbitals. Common forms are Slater-type and Gaussian-type orbitals.
Attention From Scratch page
Augmented plane wave method glossary — A basis for solid-state electronic-structure calculations that uses plane waves in the interstitial region and atom-centered radial solutions of the Schrödinger equation inside spheres around each nucleus, matched at the boundary.
Augmented Plane Waves skill — Muffin-tin spheres matched to plane waves.
Autocallable Notes page

B

Backprop a 2-Layer MLP page
Band structure glossary — The set of energy bands E_n(k) of electrons in a periodic potential, plotted along high-symmetry directions in the Brillouin zone. Determines metallicity, gaps, and effective masses.
Bandwidth Experiments page
Basis set glossary — A finite collection of one-electron functions used to expand molecular orbitals or many-electron wavefunctions. The basis controls accuracy and cost; common families include STO-nG, Pople, and correlation-consistent sets.
Bayes' Rule pageprereq nodeskill — The posterior is the distribution you actually want to sample from.
Beam search 1 note — Greedy decoding commits to the highest-probability token at each step and often paints itself into a corner; beam search keeps several candidate skeletons alive and fits constants to each. Widening the beam moved recovery from 3/20 to 11/20 in the from-scratch SymGPT model.
Bell Numbers page
Bellman-Ford Algorithm pageglossary ↗ — Single-source shortest paths in O(VE) that handles negative edge weights and detects negative cycles. Slower than Dijkstra but more general.
Bernoulli Differential Equation page
Bernoulli Equations skill — Reduce a nonlinear ODE to a separable one.
Best Time to Buy and Sell Stock page
Bi-Orthogonal Lanczos pageprereq node — Lanczos for non-symmetric operators: two coupled sequences kept bi-orthogonal to each other.
Big-O notation glossary — Asymptotic upper bound f(n) = O(g(n)) meaning f grows at most as fast as a constant multiple of g for large n. Standard tool for stating algorithm time and space complexity.
Binary Search pageskillglossary ↗ — A divide-and-conquer search that locates a target in a sorted array in O(log n) by repeatedly halving the search interval. Precondition: the array must be sorted.
Binary Tree Level Order Traversal page
Binary Tree Max Path Sum page
Binder cumulant glossary — The dimensionless ratio U_4 = 1 − ⟨m^4⟩/(3⟨m^2⟩²) used in finite-size scaling of phase transitions. Its crossing of curves for different system sizes locates the critical temperature.
Bisection pageskill ↗ — Robust root-finding by halving a bracketing interval.
Bisection method glossary — A root-finding algorithm that brackets a sign change of a continuous function and repeatedly halves the interval. Convergence is linear but guaranteed.
Black-Scholes skill — Risk-neutral pricing and the closed-form option formula.
Black-Scholes Model pageglossary ↗ — A continuous-time model of European option pricing in which the underlying follows geometric Brownian motion. Its closed-form solution gives the no-arbitrage price as a function of spot, strike, rate, time, and volatility.
BLAS glossary — Basic Linear Algebra Subprograms: a tiered API (Level 1–3) for vector–vector, matrix–vector, and matrix–matrix kernels like DAXPY and DGEMM. Performance-tuned BLAS underlies almost all numerical software.
Bloch's Theorem skillglossary ↗ — States that eigenfunctions of a Hamiltonian periodic in a Bravais lattice can be written as e^{ik·r} u_k(r) with u_k periodic. It gives the band-structure framework of solid-state physics.
Block Davidson method glossary — A generalization of the Davidson algorithm that expands the subspace by several correction vectors per iteration, targeting multiple eigenpairs simultaneously. Standard for low-lying eigenstates of large CI Hamiltonians.
BMI Calculator page
Bond dissociation glossary — The process of pulling a chemical bond apart toward separated fragments. Restricted Hartree–Fock fails qualitatively in this limit, motivating multireference and FCI treatments.
Borel Resummation skill — Transform, Padé, integrate to resum a divergent series.
Borel-Padé Resummation page
Born–Oppenheimer approximation glossary — Separates electronic and nuclear motion by treating nuclei as fixed point charges while solving the electronic Schrödinger equation. It is the foundation of nearly all molecular electronic-structure theory.
Boundary matching glossary — In APW/LAPW, the requirement that wavefunctions and their derivatives agree across the muffin-tin boundary. Sets the linear coefficients that build the augmented basis function.
Boyer-Moore Algorithm pageglossary ↗ — A string-matching algorithm that scans the pattern right-to-left and uses bad-character and good-suffix heuristics to skip large portions of text. Often the fastest in practice for long patterns.
Boys function glossary — The auxiliary function F_0(t) = ½√(π/t) erf(√t) that appears whenever Gaussian primitives are combined to evaluate Coulomb-type integrals. It is the workhorse of analytic two-electron integral evaluation.
Branch Experiments page
Branch Prediction skill — Speculation and the cost of mispredicts.
Branch Prediction Basics page
Breadth-First Search pageskillglossary ↗ — Graph traversal that visits all vertices at the current distance before going deeper, using a FIFO queue. Yields shortest paths in unweighted graphs.
Brent's method glossary — A robust hybrid root-finder combining bisection, secant, and inverse quadratic interpolation. Guarantees convergence on a bracketed root with near-quadratic speed when smooth.
Brillouin zone glossary — The Wigner–Seitz cell of the reciprocal lattice; the natural domain for crystal momentum k. Bloch states with k differing by a reciprocal-lattice vector are equivalent, so band structures are plotted only over one zone.
Brute-Force Hartree-Fock (C++) page ↗
Bubble Sort pageskill — Swap adjacent pairs until the array is sorted.
Bucket Sort page
Build Your Own SCF (6 projects) page

C

C compiler in Python — overview page
Cache Coherence page
Cache Conflicts page
Cache Overview page
Cache Size Detection page
Caches skill — L1/L2/L3, cache lines, temporal & spatial locality.
Caffarel ZVMC for STO integrals page
Caffarel ZVMC Reproduction page
Can distance covariance sharpen PLOM? page
Capstone Lab page
Casida for Paints: Color from Conjugation page 8 notes
Casida Hermitian rewrite page
Casida's Equation pageprereq nodeskill — Linear-response electronic excitations.
Cesàro summation glossary — A regularization that assigns a value to a divergent series via the limit of the average of its partial sums. Used in Fourier analysis (Fejér summation) and to relate Picard iterates to Adomian series.
Change of variables in 1D page
Charge gap 1 note — The Mott gap is a many-body quantity, the cost to move charge; it opens with $U$ and grows linearly, approaching $U - W$ in the atomic limit.
Chebyshev Collocation pageskill — Solve BVPs with a spectral differentiation matrix.
Chebyshev Nodes skill — Collocation points that defeat Runge oscillation.
Chebyshev polynomials glossary — The orthogonal polynomials on [-1, 1] with weight 1/sqrt(1 - x^2), defined by T_n(cos theta) = cos(n theta). Their roots are the optimal nodes for polynomial interpolation and underlie spectral collocation methods.
Chebyshev polynomials & nodes prereq node — The orthogonal basis on [−1, 1] the spectral density is expanded in.
Chebyshev Root Distribution page
Class and Object page
Classes & Objects skill — Bundle data with methods; object-oriented design.
Climbing Stairs page
Coin Change page
Coin Change Problem page
Collocation method glossary — A spectral approach that enforces a differential equation exactly at chosen collocation points using a global polynomial expansion. Chebyshev–Lobatto points give spectrally accurate, well-conditioned schemes.
Color 3 notes — Predicting color factors as colorimetry $\\circ$ excited-state QM: the colorimetry is solved, but cheap TDDFT is the bottleneck. A $\\sim$1 eV error in an excitation energy flips a hue to its complement (B3LYP puts indigo's band near 400 nm, predicting yellow, when it is really 605 nm and blue).
Composing flows page
Condensed-Matter Observables skill — Band gaps, densities of states, spectroscopic probes.
Condition number prereq nodeskillglossary ↗ — For a matrix A, κ(A) = ‖A‖ · ‖A^{-1}‖, measuring how much output errors can be amplified relative to input errors. Small κ means well-conditioned; large κ means ill-conditioned.
Condition Number & Stability page
Conditional Statement page
Conditionals skill — Branch with if/else on runtime conditions.
Configuration interaction glossary — A post-Hartree–Fock method that expands the wavefunction as a linear combination of Slater determinants and diagonalizes the many-body Hamiltonian in that basis. CISD truncates at single and double excitations; FCI keeps all of them.
Conjugate Gradient pageprereq nodeskill — Solve SPD systems by stepping along conjugate directions — Krylov power without storing the basis.
Conjugation 3 notes — Carotenoids ($\\beta$-carotene, lycopene) are long polyenes ($\\sim$11 conjugated C=C bonds) whose $\\pi\\to\\pi^*$ band has red-shifted into the visible — the reason carrots are orange and tomatoes red.
Conservation Laws & Riemann Problems page
Container With Most Water page
Contains Duplicate page
Contraction coefficient glossary — The fixed coefficient c_i multiplying each primitive Gaussian when forming a contracted basis function. Together with exponents it defines a contracted Gaussian-type orbital.
Control Unit page
Convergence skill — O(h²) error for smooth solutions on good meshes.
Core Hamiltonian glossary — The one-electron part of the Fock matrix, summing kinetic energy and electron–nucleus attraction integrals over basis functions. It is independent of the density and computed once per geometry.
Correction vector glossary — In Davidson’s method, the preconditioned residual t = (μI − D)^{-1} r used to expand the search subspace each iteration. Choice of preconditioner determines convergence rate.
Correlation & Where HF Fails skill — Dynamic vs static correlation, the stretched-H₂ failure, and the UHF trade.
Could AAA-fit rationals replace Slater orbitals? page
Coulomb operator glossary — The Hartree–Fock mean-field operator Ĵ_j(1) representing the classical electrostatic repulsion of electron 1 from the charge cloud of orbital j. Its matrix element with another orbital is a two-electron integral.
Counting Sort page
Coupled cluster glossary — A post-Hartree–Fock method that parameterizes the wavefunction as e^T |Φ_HF⟩ with a cluster excitation operator T. Truncations like CCSD and CCSD(T) provide highly accurate, size-extensive correlation energies.
Course Schedule page
Covariance (C++) page
Covariance Matrix (C++) pageglossary ↗ — The symmetric positive-semidefinite matrix Σ_{ij} = E[(X_i − μ_i)(X_j − μ_j)] of pairwise covariances. Eigenvectors give principal components.
CPU Mental Model skill — A working model of modern CPU performance tradeoffs.
Creation operator glossary — The second-quantized operator â^†_p that adds a fermion to spin orbital p, with anticommutation rules enforcing antisymmetry. Together with annihilation operators it builds every many-body operator.
Critical exponents glossary — Universal power-law exponents (β, γ, ν, …) governing the divergence of thermodynamic quantities near a continuous phase transition. They are organized into universality classes.
Critical Exponents from Finite-Size Scaling pageprereq node
Cross-Entropy + Softmax Gradient page
Crystal lattice glossary — A periodic arrangement of points in space generated by integer combinations of a set of primitive vectors. Together with a basis of atoms, it defines a crystal structure.
Crystal structure glossary — A periodic arrangement of atoms described by a Bravais lattice plus a basis. Determines the symmetry properties exploited by Bloch states and band-structure calculations.
Crystals page
Crystals & Reciprocal Space skill — Lattice vectors, periodicity, the Brillouin zone.

D

Datapath page
Davidson Method pageprereq nodeglossary ↗ — A subspace iteration for the lowest few eigenpairs of a large Hermitian matrix, augmenting the search subspace with preconditioned residual vectors. The method of choice for large-scale CI and TDDFT eigenproblems.
DAXPY glossary — BLAS Level-1 kernel performing y ← α x + y in double precision. Canonical example of a streaming, bandwidth-bound numerical loop.
DAXPY - A BLAS Example page
Definitions and Flashcards page
Delta (Greek) glossary — The partial derivative ∂V/∂S of an option value with respect to the underlying spot. For a European call under Black–Scholes, delta = N(d_1).
Density as the Variable skill — Trade the wavefunction for the electron density.
Density functional theory (Kohn-Sham) prereq nodeglossary ↗ — A reformulation of the many-electron problem in terms of the electron density rather than the wavefunction, justified by the Hohenberg–Kohn theorems. In practice the density is obtained by solving the Kohn–Sham single-particle equations with an approximate exchange–correlation functional.
Density Matrices & Populations skill — P as the one-electron oracle: Mulliken charges, dipoles, and what is (and is not) observable.
Density matrix glossary — In Hartree–Fock theory, P_{μν} = 2 Σ_i C_{μi} C^*_{νi} summed over occupied orbitals, encoding the one-electron density in the AO basis. SCF iterates the density to self-consistency.
Density of states gap 4 notes — How many states sit at each energy — the spectral fingerprint KPM reconstructs.
Depth-First Search pageskillglossary ↗ — Graph traversal that explores as far as possible along each branch before backtracking, using a stack (or recursion). Underpins topological sort, SCC decomposition, and cycle detection.
Description length / Occam scoring gap 4 notes — Shorter formulas enumerate earlier — searching short-to-long IS the simplicity prior.
Differentiation matrix glossary — A dense matrix D such that (Df)_i approximates f'(x_i) at collocation nodes. The Chebyshev differentiation matrix gives spectral accuracy for smooth functions.
DIIS glossary — Direct Inversion in the Iterative Subspace: Pulay’s method for accelerating SCF by extrapolating the Fock matrix from past iterates that minimize an error vector. Standard tool in quantum chemistry.
Dijkstra's Algorithm pageskillglossary ↗ — Single-source shortest paths in a weighted graph with non-negative edge weights, using a priority queue to greedily extract the closest unsettled vertex. Runs in O((V + E) log V) with a binary heap.
Dimer Energies pageprereq nodeskill — Geometry and binding energy from a Kohn-Sham calculation — the first real observable DFT hands you.
Divergence & Resurgence skill — Factorially divergent series hide nonperturbative structure.
Divide-and-conquer glossary — An algorithmic paradigm that splits a problem into smaller subproblems, solves each recursively, and combines results. Powers merge sort, FFT, Strassen multiplication, and many spectral solvers.
DMRG & matrix product states prereq node
DMRG and Matrix Product States page
Dopant Diffusion pageprereq node — Places the dopant profile whose crossing of the background sets the junction.
Dot Product (C++) page
Dropout From Scratch page
Dynamic programming glossary — An algorithmic technique for problems with overlapping subproblems and optimal substructure: solve each subproblem once and cache the result. The basis of the standard solutions to knapsack, edit distance, and longest common subsequence.
Dynamical mean-field theory 6 notes — Single-site DMFT's local–self-energy approximation fails in 2D, where the pseudogap and $d$-wave pairing live in the momentum dependence $\\Sigma(\\mathbf{k},\\omega)$ it discards — the reason cluster extensions (DCA, cellular DMFT) exist.

E

Edit Distance pageskill ↗ — Dynamic programming over string transformations.
Egg Dropping Problem page
Eigenvalue glossary — A scalar λ such that for some operator A and non-zero vector v, Av = λv. The vector v is called an eigenvector.
Electron affinity glossary — The energy released when an extra electron is added to a neutral atom or molecule. Approximated by minus the LUMO energy in single-determinant theory.
Electron correlation gapglossary 4 notes — The energy difference E_exact − E_HF, capturing the correlated motion of electrons that mean-field Hartree–Fock cannot describe. Recovering it is the central task of post-HF methods.
Electron–nucleus integral glossary — A one-electron matrix element ⟨φ_μ| -Z/|r-R_N| |φ_ν⟩ giving the attraction of an electron to a nucleus. With Gaussian primitives it reduces to the Boys F_0 function.
Entanglement entropy gap 4 notes — How much a bipartition of a quantum state knows about itself — the Schmidt spectrum across a cut.
ERI tensor glossary — The four-index tensor of electron-repulsion integrals (μν|λσ) used to assemble the Coulomb and exchange contributions of the Fock matrix. Storing it scales as N^4 in basis size and dominates HF cost.
Etude: Local Functional Derivatives page
Euclidean Algorithm page
Euler Method skillglossary ↗ — The simplest numerical integrator for ordinary differential equations, advancing y by h f(t, y) at each step. It has first-order global accuracy and is the textbook starting point for more sophisticated schemes.
Euler Method - Exponential (Fortran) page
Euler Method - Harmonic Oscillator (Fortran) page
Euler-Maruyama Method (Fortran) pageglossary ↗ — The stochastic analog of explicit Euler for SDEs, advancing dY = a dt + b dW by a · Δt + b · √Δt · ξ with ξ standard normal. First-order strong scheme commonly used in finance simulations.
Exchange operator glossary — The non-local operator K̂_j arising in Hartree–Fock from antisymmetry of the wavefunction; it has no classical analog and lowers the energy of same-spin electron pairs. Together with Ĵ it builds the mean-field part of the Fock operator.
Exchange–correlation functional glossary — The unknown density functional E_xc[ρ] that contains all many-body effects beyond the classical Coulomb energy in Kohn–Sham DFT. Approximations include LDA, GGA (e.g., PBE), and hybrid functionals.
Excited state glossary — Any eigenstate of a Hamiltonian with energy above the ground state. Block Davidson and CI methods target several low-lying excited states at once.
Execution Model pageskill — Pipelining and out-of-order execution.
Exotic Options (Digital, Barrier, Asian, Lookback) page
Expanding in the Eigenbasis pageprereq node
Explicit Euler Method page
Exponential Search page

F

False Sharing Experiment page
FCI for H₂ prereq node — FCI worked end to end on H₂ — the smallest system where correlation is visible.
Feature Bases skill — When a linear model stops being linear.
Feature selection 1 note — Sure Independence Screening ranks every candidate feature by the magnitude of its correlation with the target and keeps only the top few dozen; it is dimensionality reduction with a high-probability guarantee that the truly relevant features survive.
Feedback & Stability skill — Feedback control, dynamical systems, and stability.
Fejér summation glossary — The Cesàro mean of partial Fourier sums; smooths out the Gibbs phenomenon and converges uniformly for continuous periodic functions. A standard tool of classical Fourier analysis.
Fermion glossary — A particle with half-integer spin obeying Fermi–Dirac statistics and the Pauli exclusion principle. Electrons, protons, and neutrons are fermions.
Fermion-to-qubit mapping (Jordan-Wigner) gap 4 notes — Rewrites the electronic Hamiltonian as a sum of Pauli strings a quantum computer can measure.
Fibonacci and the DP Family page
Fibonacci Sequence page
File I/O page
Filling Orbitals: RHF, UHF, and the Ansatz Ladder page
Find Median from Data Stream page
Finite difference method glossary — A discretization that replaces derivatives by differences of values at neighboring grid points, turning a differential equation into a linear or nonlinear algebraic system. The simplest way to put a PDE on a computer.
Finite Difference Methods page
Finite Differences skill — Replace derivatives with stencils on a grid.
Finite element method glossary — A discretization that expands the solution of a PDE in piecewise-polynomial basis functions over a mesh and enforces Galerkin or collocation conditions. Standard tool for elliptic and parabolic PDEs in complex geometries.
Finite-difference stencil glossary — A pattern of grid points used to approximate a differential operator by a weighted sum of function values. Stencils encode the order, accuracy, and bandwidth of the resulting discrete operator.
Finite-size scaling 1 note — A finite box rounds the transition: below a crossover $\\Delta\\varphi^\\ast \\sim 1/N$ the power law is masked, so the asymptotic exponent must be read from an *intermediate* offset window — not from the offsets closest to $\\varphi_J$, which are the most contaminated.
FIRE minimization 1 note — FIRE finds the energy minimum by running molecular dynamics and zeroing the velocity whenever it points against the force — momentum with a restart rule, reaching a minimum without ever forming a Hessian.
Fixed-Point Iteration pageglossary ↗ — An iteration of the form x_{k+1} = g(x_k) that converges, when stable, to a solution of x = g(x). The SCF loop and Picard iteration are concrete examples.
Fixed-point theorem glossary — Result guaranteeing existence (and sometimes uniqueness) of x with g(x) = x. Banach’s contraction-mapping theorem also yields convergence of fixed-point iteration.
Floating-Point Arithmetic page
Floyd-Warshall Algorithm pageglossary ↗ — All-pairs shortest paths via a triple loop in O(V^3), based on dynamic programming over intermediate vertices. Works with negative edges if there are no negative cycles.
Fock matrix glossary — The matrix representation of the Fock operator in an atomic-orbital basis, equal to the core Hamiltonian plus density-dependent Coulomb and exchange contributions. Diagonalizing it yields molecular orbital energies and coefficients.
Fock operator glossary — The one-electron operator that defines the Hartree–Fock problem: core Hamiltonian plus mean-field Coulomb and exchange contributions from every other occupied orbital.
For Loop page
Formula Discovery Without Search (Meijer G) pageprereq node
Fourier Coefficients (C++) page
Fourier Series pageskillglossary ↗ — An expansion of a periodic function as a sum of sines and cosines with frequencies that are integer multiples of the fundamental. The coefficients are inner products of the function against the basis elements.
Full Configuration Interaction pageprereq nodeskillglossary ↗ — The exact diagonalization of the electronic Hamiltonian within a given one-electron basis: the wavefunction is expanded in every Slater determinant that can be built from the orbitals. Cost grows factorially with system size.
Full-adder page
Function page
Function Pointer page
Function Pointers (Advanced) page
Functional Derivatives pageprereq node — Both DFT and the response kernel are defined by derivatives of functionals.
Functions skill — Package logic into reusable functions with parameters.

G

Gamma (Greek) glossary — The second derivative ∂²V/∂S² of an option value with respect to spot, equivalently ∂Δ/∂S. Measures the convexity of the option position.
Gamma Knife Dose Simulation page
GARCH and Realized Volatility page
Gauss–Seidel iteration glossary — A stationary iterative solver for Ax = b that uses the most recent updates within a sweep, accelerating Jacobi convergence at the cost of inherent sequentiality. Converges for diagonally dominant or SPD systems.
Gaussian Elimination pageprereq nodeskillglossary ↗ — The standard direct method for solving a linear system, in which row operations reduce the matrix to upper triangular form and the solution is obtained by back substitution. Partial pivoting is normally used for numerical stability.
Gaussian exponent glossary — The α parameter controlling the radial decay of a Gaussian primitive e^{-α r²}. Small exponents describe diffuse functions, large exponents describe tight core functions.
Gaussian product theorem prereq nodeglossary ↗ — States that the product of two Gaussian functions centered at different points is itself a Gaussian centered at a weighted midpoint, scaled by an overlap factor. This identity makes integrals over Gaussian primitives analytically tractable.
Gaussian Product Theorem = Parallel Axis Theorem page
Gaussian-type orbital glossary — A basis function of the form exp(-alpha r^2) times a polynomial in the Cartesian coordinates. Two-electron integrals over Gaussians are analytic, which is why nearly all production quantum-chemistry codes use them.
Generalized eigenvalue problem glossary — The matrix equation A v = λ B v, with B typically symmetric positive-definite. The Roothaan–Hall equations and APW/LAPW secular equations both take this form.
Generate Parentheses page
Geometric Brownian motion glossary — The stochastic process dS = mu S dt + sigma S dW. It is the standard model for asset prices because its logarithm is a Brownian motion with drift and it stays strictly positive.
Geometric frustration 1 note — Geometric frustration (e.g. a triangular lattice) prevents all antiferromagnetic bonds from being satisfied at once, leaving a degenerate classical manifold — the doorway to quantum spin liquids.
Geometric Series (Fortran) page
Geometry Optimization (by diff) page
Getline from Console page
Gibbs phenomenon glossary — The fixed ~9% overshoot near a jump discontinuity that persists when truncating a Fourier series, regardless of how many terms are kept. Cesàro/Fejér summation suppresses it.
Givens rotation glossary — A 2×2 plane rotation used to selectively zero entries of a matrix. Alternative to Householder reflections for QR factorization, especially in sparse settings.
GPU-Accelerated Ising (PyTorch/MPS) page
Gradient Descent pageprereq node — Following −∇ downhill; HMC needs the same gradient of the log-density to move.
Gram–Schmidt process glossary — A procedure for orthonormalizing a set of vectors by sequentially subtracting projections onto previously processed vectors. Modified Gram–Schmidt is the numerically stable variant used in Arnoldi.
Greedy algorithm glossary — A strategy that builds a solution by always making the locally optimal choice, hoping it yields a global optimum. Works for problems with matroid or exchange-argument structure (e.g., MST).
Greeks & Hedging skill — Delta/gamma/vega and replication by rebalancing.
Greeks and Delta Hedging page
Green's Functions pageskill — Solve inhomogeneous PDEs by convolving with a kernel.
Ground state glossary — The lowest-energy eigenstate of a Hamiltonian. Variational methods deliver upper bounds on its energy.
Group Anagrams page

H

H2 Atom and FCI page
H2+ Orbital page
Half-adder page
Hamiltonian glossary — The energy operator of a quantum system; its eigenvalues are the allowed energies. For an N-electron molecule it sums kinetic, electron–nucleus, and electron–electron contributions.
Hamiltonian mechanics gap 4 notes — Phase space, energy conservation, and the symplectic flow that leapfrog approximates.
Hamiltonian Monte Carlo pageprereq node
Hardware Prefetch page
Harmonic Oscillator (FD) pageprereq nodeskillglossary ↗ — A particle in a quadratic potential V(x) = ½ k x², the canonical exactly solvable quantum system with equally spaced energy levels ℏω(n + ½). Used as a test problem for finite-difference and spectral methods.
Hartree product glossary — A simple product of one-particle wavefunctions, ignoring antisymmetry. Historical predecessor to the Slater determinant.
Hartree-Fock prereq node — Defines the occupied/virtual orbital picture that excitations live in.
Hartree-Fock Facts page
Hartree-Fock Flashcards page
Hartree-Fock Method pageglossary 8 notes ↗ — A mean-field approximation in which the many-electron wavefunction is a single Slater determinant of orbitals, optimized to minimize the energy.
Hartree-Fock on a grid prereq node — The same HF equations on a real-space grid instead of a basis — stripped of the integral machinery.
Hartree-Fock-Bogoliubov pageskill — Pairing correlations in a self-consistent mean field.
Heap Sort page
Heisenberg model glossary 2 notes — A spin model with isotropic nearest-neighbor exchange interactions Σ S_i · S_j. Captures magnetic ordering and emerges as the strong-coupling limit of the Hubbard model.
Hello World pageskill ↗ — Your first program: compile, run, print a line.
Heston Stochastic Volatility page
Hodgkin-Huxley skill — Voltage-gated channels drive the action potential.
Hohenberg–Kohn theorems glossary — Two theorems showing that the ground-state electron density uniquely determines the external potential (up to a constant) and that the total energy is a variational functional of the density. They are the formal foundation of DFT.
HOMO glossary — The highest occupied molecular orbital. Its energy controls a system’s ionization potential and many low-energy chemical responses.
House Robber page
Householder QR prereq node — The stable orthogonal factorization A = QR behind least squares and the QR eigenvalue algorithm.
Householder QR Factorization pageglossary ↗ — A factorization A = QR built by applying Householder reflectors that successively zero out subdiagonal entries. It is the standard numerically stable way to compute QR for dense matrices.
Hubbard dimer glossary — The two-site Hubbard model with on-site repulsion U and hopping t. The smallest interacting lattice problem that exhibits FCI-like correlation effects analytically.
Hubbard Dimer and FCI page
Hubbard dimer FCI prereq node — The two-site Hubbard model solved exactly — correlation in its most stripped-down form.
Hubbard model glossary — A lattice model of interacting fermions with a nearest-neighbor hopping term t and an on-site Coulomb repulsion U. It is the simplest setting that captures the competition between kinetic delocalization and electronic correlation.
Huffman Coding pageglossary ↗ — A greedy algorithm that builds an optimal prefix code for a given symbol-frequency distribution by repeatedly merging the two least-frequent subtrees. Used for lossless compression.
Hyperasymptotics pageskill — Exponentially small corrections refine the asymptotic series.

I

IEEE Benchmark Test Systems page
ILP Experiments page
Implement Trie page
Implicit Euler Method pageglossary ↗ — A backward time-stepping scheme y_{n+1} = y_n + h f(t_{n+1}, y_{n+1}) that requires solving an algebraic equation at each step. It is unconditionally stable, making it suitable for stiff problems.
Implicit Methods skill — Unconditionally stable stepping for stiff systems.
Implied Volatility pageskillglossary ↗ — The volatility input that, when fed into the Black–Scholes formula, reproduces the market price of an option. It is found numerically by inverting the pricing formula, typically with Newton's method on vega.
Infinite Square Well pageglossary ↗ — Synonym for the particle-in-a-box potential: zero inside an interval [0, L] and infinite outside. Often used to benchmark numerical eigensolvers.
Input/Output page
Insertion Sort page
Instruction Latency page
Integral Recurrences: OS vs MD (overview) page
Interactive Algorithm Example page
Interpolation Search page
Intractable Likelihoods skill — When the likelihood hides in latent simulator paths.
Ionization potential glossary — The minimum energy required to remove an electron from a neutral atom or molecule in its ground state. Approximated by minus the HOMO energy via Koopmans’ theorem.
Ising Model pageglossary ↗ — A lattice model of classical spins taking values plus or minus one with nearest-neighbor coupling. It is the prototypical example of a system with a continuous phase transition and is usually simulated by Metropolis Monte Carlo.
Isostaticity 2 notes — At jamming the packing is *isostatic*: exactly $Z_{iso} = 2d$ contacts per particle — the Maxwell minimum that constrains every degree of freedom, with none to spare.
Itô's lemma glossary — The chain rule for stochastic calculus: for f(t, X) with dX = μ dt + σ dW one has df = (f_t + μ f_x + ½ σ² f_{xx}) dt + σ f_x dW. Used to derive the Black–Scholes PDE.

J

Jamming pageprereq node 8 notes — The hard-sphere fluid at its density limit: where a packing first becomes rigid, found by energy minimization (FIRE).
Jellium glossary — A model of an interacting electron gas in a uniform positive background charge. Reference system for many DFT functionals and the homogeneous-electron-gas limit.
Jump Search page

K

K-Means From Scratch page
Kahan summation glossary — A compensated-summation algorithm that accumulates the rounding error of each addition in a separate variable and adds it back on the next step. Reduces the error of summing N floating-point numbers from O(N eps) to essentially O(eps).
KCL, KVL, and Ohm's Law page
Kelly Criterion skill — The bet size that maximizes log-wealth growth.
Kelly Criterion & Position Sizing page
Kernel Polynomial Method (density of states) pageprereq node
Kinetic energy integral glossary — A one-electron matrix element T_{μν} = ⟨φ_μ| -½∇² |φ_ν⟩ contributing to the core Hamiltonian. For Gaussian primitives it has a clean closed form (Szabo–Ostlund Appendix A).
KMP Algorithm pageglossary ↗ — Knuth–Morris–Pratt string-matching algorithm running in O(n + m) by precomputing a longest-prefix-suffix table that lets it skip ahead on mismatch. Optimal for text-with-pattern problems with repetitive structure.
Knapsack Problem page
Kohn-Sham Orbitals pageprereq node — Excitations are occupied → virtual transitions among these orbitals.
Kohn–Sham equations glossary — A reformulation of DFT that maps the interacting many-electron problem onto a fictitious system of non-interacting electrons sharing the same density. The single-particle equations close via an exchange–correlation potential.
Koopmans' theorem glossary — In Hartree–Fock theory, the negative of an occupied orbital energy approximates the ionization potential, and the negative of a virtual orbital energy approximates the electron affinity. The approximation neglects orbital relaxation upon electron removal or addition.
Kruskal's Algorithm pageglossary ↗ — A greedy algorithm that builds a minimum spanning tree by adding the next-cheapest edge that does not form a cycle. Uses union–find for cycle detection.
Krylov / Arnoldi skill — Orthonormal bases of Krylov subspaces for large problems.
Krylov subspace glossary — The span K_m(A, v) = span{v, Av, A²v, …, A^{m−1}v}. Iterative methods like Arnoldi, Lanczos, and Davidson search for eigen- or linear-system solutions inside Krylov subspaces.
Krylov subspaces gap 4 notes — span{b, Ab, A²b, …}: everything you can reach with matrix-vector products alone — the arena of every large-scale iterative method.
Kth Largest Element page

L

Ladder operators glossary — Pairs of operators that raise and lower the eigenvalue of a chosen observable, classically illustrated by â and â^† for the harmonic oscillator. They underlie second quantization and angular-momentum algebra.
Lagrange multipliers glossary — Auxiliary variables introduced to enforce equality constraints during optimization. In the Hartree–Fock derivation they enforce orbital orthonormality and yield the Fock equations.
Lambda page
Lambdas & Closures skill — Anonymous functions that carry state. Capture by value vs reference, and the closure type behind the syntax.
Lanczos ↔ continued fractions page
Lanczos adapted basis prereq node — Reshaping the Lanczos basis toward the part of the spectrum you actually want.
Lanczos Iteration pageprereq nodeskillglossary ↗ — The symmetric specialization of Arnoldi: a three-term recurrence builds an orthonormal Krylov basis and projects a Hermitian matrix onto a tridiagonal one. Standard for extremal eigenvalues of large sparse Hermitian operators.
Lanczos Learns the Optimal Basis page
Lanczos MOR for Large Circuits page
LAPACK glossary — A standard Fortran library of dense linear algebra routines (factorizations, eigensolvers, least-squares) layered on top of BLAS. The numerical workhorse beneath NumPy, MATLAB, and quantum-chemistry codes.
LAPW skill — Linearized APW with derivative continuity.
LAPW Method page
Largest Rectangle in Histogram page
Lasso Regression (C++) pageglossary ↗ — Linear regression with L1 penalty λ‖w‖_1 that produces sparse coefficient estimates by zeroing out unimportant features. Solves a convex but non-smooth optimization problem.
Layer Normalization page
Leaky Integrate-and-Fire pageskill — An RC threshold-and-reset neuron with a closed-form f-I.
Learning to Sample (SLMC on PySCF) page
Legendre Polynomials pageskillglossary ↗ — The orthogonal polynomials on [-1, 1] with respect to unit weight, obtained by Gram–Schmidt on {1, x, x^2, ...}. They arise as the angular part of solutions to Laplace's equation in spherical coordinates.
Lennard-Jones MD (Velocity Verlet) pageprereq node — Integrating Newton’s equations with leapfrog — the exact integrator HMC borrows to make proposals.
Lexing & Tokens skill — Turn source text into a token stream.
LIF Network Linear-Response (Lanczos+CF) page
LIF Network Response skill — Transfer functions of sparse networks via Lanczos+CF.
Limits, Completeness, and Finitism page
Linear algebra gap 4 notes — Vectors, matrices, and how operators act on them.
Linear combination of atomic orbitals glossary — An ansatz that writes each molecular orbital as a sum of atomic basis functions weighted by coefficients to be determined variationally. It reduces solving the Hartree–Fock equations to a matrix eigenvalue problem.
Linear Regression (Fortran) pageskillglossary ↗↗ — Fits a linear model y ≈ Xβ by minimizing ‖y − Xβ‖². The normal equations X^T X β = X^T y give a closed-form least-squares solution when X has full column rank.
Linear Response Theory pageprereq node — The poles of the response function ARE the excitation energies.
Linear Search pageskill — Scan an array element-by-element for a target.
Linear systems (Ax = b) gap 4 notes — What it means to solve for x, and when a solution exists or is unique.
Linearized augmented plane wave method glossary — An efficient variant of the APW method that linearizes the energy dependence of the muffin-tin radial functions, removing the need to solve a nonlinear secular equation at each k-point. The de facto standard for all-electron DFT in solids.
Liouville-Lanczos Linear-Response TDDFT page
Lock Contention page
Log-return glossary — The continuously compounded return r_t = log(S_t / S_{t−1}). For geometric Brownian motion, log-returns are normally distributed.
Logic Gates page
Logistic Regression pageskill — Sigmoid, cross-entropy, gradient descent.
Longest Common Subsequence page
Longest Consecutive Sequence page
Longest Increasing Subsequence page
Longest Palindromic Subsequence page
Longest Palindromic Substring page
Longest Substring No Repeat page
Loops skill — Repeat work with for/while; master iteration.
Lowest Common Ancestor page
LRU Cache page
LUMO glossary — The lowest unoccupied molecular orbital. Its energy controls a system’s electron affinity and serves as the first acceptor in excitations.

M

m-scheme glossary — A many-body basis built from Slater determinants of single-particle states with definite magnetic quantum number m. Standard in shell-model diagonalization and FCI implementations.
Magic numbers glossary — Numbers of nucleons (2, 8, 20, 28, 50, 82, 126) at which nuclei are unusually stable, explained by closed shells in the nuclear shell model with spin–orbit coupling.
Many-electron wavefunctions prereq node — Antisymmetric many-electron wavefunctions — the Pauli principle written as a determinant.
Map page
Markov chain glossary — A stochastic process whose next state depends only on the current state. Provides the theoretical backbone of Metropolis sampling and many MCMC methods.
Matrix Chain Multiplication page
Matrix eigenvalue problems prereq node — Excitation energies are eigenvalues; you must be fluent in solving Ax = λx.
Matrix product states gap 4 notes — A wavefunction rewritten as a chain of small matrices; the bond dimension caps the entanglement it can carry.
Matrix storage prereq nodeskill — How a matrix actually sits in memory — the layout every algorithm’s indexing and speed assume.
Maximum Depth of Binary Tree page
Maximum Subarray page ↗
MCMC From Scratch pageprereq node — Sampling by a Markov chain corrected with the Metropolis accept/reject test — the correction HMC reuses.
McMurchie-Davidson pageprereq node — Hermite-Gaussian recurrences for the electron-repulsion integrals — one of the two standard schemes.
Mean-field approximation glossary — Replaces interactions among many particles with each particle’s coupling to an average (effective) field built from the others. Hartree–Fock and Kohn–Sham DFT are mean-field theories.
Mean-Variance skill — Markowitz efficient frontier from the covariance matrix.
Mean-Variance Portfolio Optimization page
Measurement Bridge skill — Map lab observables to model outputs.
Measurement Methodology page
Median of Two Sorted Arrays page
Memoization glossary — Caching the results of function calls keyed by arguments to avoid redundant recomputation. The top-down complement to bottom-up dynamic programming.
Memory Bandwidth pageskill — Throughput limits and how to measure them.
Merge Intervals page
Merge K Sorted Lists page
Merge Sort pageskillglossary ↗ — A divide-and-conquer comparison sort that splits the array, sorts each half recursively, and merges them in O(n log n) time. Stable and well-suited to external sorting.
Merge Two Sorted Lists page
Method of Characteristics pageskill — Turn a PDE into ODEs along solution curves.
Method of Lines pageskill — Discretize space, then integrate ODEs in time.
Metropolis algorithm glossary — A Markov-chain Monte Carlo method for sampling from a Boltzmann distribution: propose a local move and accept it with probability min(1, exp(-Delta E / kT)). Detailed balance guarantees the chain converges to the target distribution.
Metropolis–Hastings algorithm glossary — A Markov-chain Monte Carlo method that generates samples from a target distribution by proposing moves and accepting them with probability min(1, π(new)/π(old)) (suitably symmetrized). Underlies the Metropolis algorithm and Ising-model simulations.
Min Stack page
Modular Exponentiation pageglossary ↗ — Computes a^b mod m in O(log b) multiplications by squaring and multiplying based on the binary expansion of b. Foundation of RSA and many cryptographic primitives.
Molecular orbital glossary — A one-electron wavefunction delocalized over a molecule, typically expanded as a linear combination of atomic orbitals (LCAO). Occupied molecular orbitals build up the many-electron Slater determinant.
Møller–Plesset perturbation theory glossary — Rayleigh–Schrödinger perturbation theory built on the Hartree–Fock reference, with the fluctuation potential as perturbation. MP2 is the lowest non-trivial order and a workhorse correlated method.
Monte Carlo Integration pageglossary ↗ — Estimation of an integral by averaging the integrand over random samples of the domain. Error decreases as 1/sqrt(N) independent of dimension, making it the method of choice in high dimensions.
Monte Carlo method glossary — A class of algorithms that estimate quantities by averaging over random samples, with error decaying like 1/√N independently of dimension. Underlies integration, option pricing, and statistical sampling.
Monte Carlo Option Pricing page
Monte Carlo Pi (Fortran) page
Monte Carlo Pricing skill — Average discounted payoffs over simulated paths.
Monte Carlo π glossary — Estimation of π by sampling random points in the unit square and counting those falling inside the inscribed quarter-circle. Pedagogical demonstration of Monte Carlo convergence.
Moshinsky Transform pageskill — Relative vs center-of-mass coordinates for two-body forces.
Move Semantics skill — Stealing instead of copying. std::move as a cast, the move constructor, and the moved-from state.
MP2 skill — Correlation as a perturbation: the workhorse second-order correction.
Muffin-tin potential glossary — An approximation that takes the crystal potential to be spherically symmetric inside non-overlapping atomic spheres and constant in the interstitial region. Foundational for APW and KKR methods.
Multi-Head Attention page
Multipole Expansion skillglossary ↗ — A power-series expansion of a Coulomb-like interaction in spherical harmonics, separating angular and radial dependence. In the nuclear shell model it factors two-body interactions into Slater radial integrals times angular coefficients.
Multipole Expansion in Shell Model page

N

Neumaier summation glossary — A variant of Kahan summation that also corrects when the running sum becomes smaller in magnitude than the next addend. Strictly improves on Kahan in mixed-magnitude inputs.
Neural Posterior Estimation pageskill — Amortize the posterior across observations with a network.
Newton–Raphson method glossary — A root-finding iteration x_{k+1} = x_k − f(x_k)/f'(x_k) that converges quadratically near a simple root. Sensitive to the initial guess and requires the derivative.
Newton's Method (Fortran) pageglossary ↗ — An iterative root finder that updates x by subtracting f(x)/f'(x). It converges quadratically near a simple root but requires a derivative and can diverge from a poor initial guess.
NN-Guided Symmetry Discovery pageprereq node
No-arbitrage glossary — The principle that one cannot construct a riskless portfolio with positive return in an idealized market. Underlies the existence of a risk-neutral measure and the Black–Scholes derivation.
Normalizing Flows pageskill — Invertible maps learn flexible conditional densities.
Nuclear Shell Model pageskillglossary ↗ — A model in which nucleons fill single-particle levels of an effective central potential including a strong spin–orbit term. It explains the observed magic numbers and the systematics of low-lying nuclear states.
Number of Islands page
Numerov method glossary — A fourth-order finite-difference scheme tailored to second-order ODEs of the form y'' = f(x) y, with no first-derivative term. Widely used to integrate the radial Schrödinger equation.
Numerov stencil glossary — The five-point implicit difference relation underlying Numerov’s method, accurate to O(h^6) for equations of the form y'' = g(x) y + s(x). Yields a tridiagonal eigenproblem for the radial Schrödinger equation.
Numerov's Method pageskill ↗ — High-order integrator for y''=g(x)y+s(x).

O

Obara-Saika pageprereq node — Recurrence relations that build high-angular-momentum integrals from low ones.
One-electron integrals prereq nodeskill — Overlap, kinetic, and nuclear-attraction matrices — the one-body part of the Hamiltonian.
Operational intensity glossary — The ratio of floating-point operations to bytes transferred from main memory for a kernel. The horizontal axis of the roofline diagram.
Operator Splitting pageskill — Alternate the evolution of decomposed operators.
Option Greeks glossary — Sensitivities of an option price to underlying parameters: delta (spot), gamma (delta sensitivity), theta (time), vega (volatility), and rho (rate). Used for hedging and risk management.
Orbital Energies & Koopmans skill — What the ε's mean: ionization potentials from frozen orbitals, and why the errors cancel.
Orthogonal polynomials from a stochastic Sturm-Liouville? page
Orthogonality & projection gap 4 notes — Inner products, orthonormal bases, and projecting onto a subspace — the move behind QR, Arnoldi, and least squares.
Orthonormality glossary — A set of vectors or functions is orthonormal if it is mutually orthogonal and each element has unit norm. Required of spin orbitals in a normalized Slater determinant.
Oscillator Strengths pageprereq node — Turns eigenvectors into the intensities a spectrum actually shows.
Overlap matrix glossary — The Gram matrix of basis functions, S_{μν} = ⟨φ_μ|φ_ν⟩. A non-identity overlap turns the Hartree–Fock equations into a generalized eigenvalue problem requiring symmetric orthogonalization.
Overview page ↗
Ownership & Smart Pointers (by diff) page

P

Pade Approximant page
Padé Approximant skillglossary ↗ — A rational function of fixed numerator and denominator degree whose Taylor expansion matches a given power series to as high an order as possible. Often dramatically extends the useful range of a divergent or slowly convergent series.
Padé Approximants pageskill — Rational approximation reaches past the radius of convergence.
Padé via Lanczos (Feldmann & Freund 1995) page
Pairs Trading and Cointegration page
Parsing & the AST skill — Build an abstract syntax tree from tokens.
Partial differential equation glossary — An equation involving partial derivatives of a multivariable unknown function. Classified into elliptic, parabolic, and hyperbolic, each with characteristic numerical strategies.
Partial pivoting glossary — A row-swap strategy in Gaussian elimination that picks the largest-magnitude entry in the current column as pivot. It controls roundoff growth and is essential for numerical stability.
Particle in a box prereq nodeskillglossary ↗ — An infinite square-well potential with hard walls at 0 and L; eigenfunctions are sines with quantized energies E_n = n²π²ℏ²/(2mL²). The simplest non-trivial bound-state problem in quantum mechanics.
Particle-Hole Excitations pageprereq node — The basis of the Casida matrix is the set of occupied→virtual pairs.
Path-Integral Frame skill — Every SBI method approximates the same latent integral.
Pauli exclusion principle glossary — No two identical fermions can occupy the same quantum state simultaneously. Equivalently, the many-fermion wavefunction must change sign under exchange of any two particles.
PCA From Scratch page
Performance Analysis page
Permutation invariance 1 note — The input $(x_i, y_i)$ points are a set, not a sequence, so each is embedded by an MLP and fed as context with no positional encoding, leaving the model permutation-invariant over the samples. The generated expression tokens do get positional encodings and a causal mask.
Permutations page
Perturbation theory prereq nodeglossary ↗ — A general framework that expands eigenvalues and eigenvectors of H_0 + λV in powers of λ. In quantum chemistry it underlies Møller–Plesset and many response theories.
Perturbation Theory & Resurgence page
Phase transition glossary — A qualitative change in the macroscopic state of a system as a control parameter (e.g., temperature) crosses a critical value. The Ising model is the canonical lattice example.
Photolithography pageprereq node — Draws the openings every other fab step is patterned through.
Picard Iteration pageskillglossary ↗ — A fixed-point iteration that solves x = g(x) by repeatedly applying g to a starting guess. Used to prove existence of solutions to ODEs and as a practical method when g is a contraction.
Pivoting glossary — Reordering rows or columns of a matrix during elimination to avoid small or zero pivots and to improve numerical stability. Partial and complete pivoting are the standard variants.
Plane-wave expansion glossary — Representation of crystal wavefunctions as Σ_G c_G e^{i(k+G)·r} over reciprocal-lattice vectors G. The native basis for periodic DFT codes and the starting point of APW/LAPW.
Playground (run Knot in your browser) page
PMNN algorithm glossary — Power-Method Neural Network: parameterizes an eigenvector candidate by a neural network and trains it to minimize the residual loss until it falls below a threshold. Hybridizes power iteration with gradient-based optimization.
Pointer page
Pointer Chasing page
Pointers & Memory skill — Addresses, dereferencing, and dynamic allocation.
Poisson's equation & electrostatics gap 4 notes — The junction's field and potential are the solution of ψ″ = −ρ/ε with charge that depends on ψ.
Policy Calculation page
Policy-Based Design page
Positronium: Para, Ortho & the Zeeman Effect page
Positronium: para, ortho & Zeeman prereq node
Power Iteration pageprereq nodeskillglossary ↗ — A simple eigenvalue algorithm that repeatedly multiplies a vector by A and renormalizes; the iterate converges to the eigenvector of largest magnitude. Forms the conceptual basis of Arnoldi, Lanczos, and Davidson.
Power Method Neural Network page
Preconditioner glossary — A matrix M ≈ A used to transform Av = b or Av = λv into a better-conditioned problem M^{-1}A v = M^{-1}b. The diagonal of A − μI is the standard Davidson preconditioner.
Prim's Algorithm pageglossary ↗ — A greedy MST algorithm that grows a single tree, repeatedly adding the cheapest edge connecting the tree to a non-tree vertex. Implementations use a priority queue.
Primitive Gaussian glossary — A single uncontracted Gaussian basis function with a specific exponent and angular factor. Multiple primitives are combined linearly with contraction coefficients to form a contracted basis function.
Probabilistic Learning on Manifolds page
Product of Array Except Self page
Put–call parity glossary — The arbitrage identity C − P = S − K e^{-rT} relating European call and put prices on the same underlying and strike. Holds independently of the option pricing model.
PySCF Basics (by diff) page
Python · Step 1 — The smallest C program page
Python · Step 2 — Unary operators page

Q

QR Factorization skillglossary ↗ — The decomposition A = Q R with Q orthogonal (or unitary) and R upper triangular. Building block of least-squares solvers, the QR algorithm for eigenvalues, and Arnoldi.
Quadratic Equation Solver page
Quantities of Interest (Measurement ↔ Model) pageprereq node ↗↗↗↗↗ — From a converged density to what you can measure — energies, charges, dipoles — and what stays unobservable.
Quantum magnetism 2 notes — The classical Néel state is *not* the quantum ground state: the $S_i^+S_j^- + S_i^-S_j^+$ terms flip antiparallel pairs, so the true ground state is an entangled superposition lying below the classical energy.
Quantum Magnetism & Superexchange pageprereq node 7 notes — The large-U face of the Hubbard model: frozen charges whose spins order antiferromagnetically via J = 4t^2/U.
Quantum mechanics (states & operators) prereq node — Wavefunctions, expectation values, and the language of the whole field.
Quasiparticle weight 1 note — The quasiparticle weight $Z$ sets the effective mass $m^*/m = 1/Z$; at the Mott transition $Z \\to 0$ and the mass diverges — the metal localizes without the carriers disappearing.
Qubits, gates & measurement gap 4 notes — The state space VQE prepares its trial wavefunction in, and reads expectation values from.
Quick Sort pageglossary ↗ — A randomized comparison sort that partitions the array around a pivot and recurses; expected O(n log n) time, in-place but not stable. Standard library default in many languages.

R

Rabin-Karp Algorithm pageglossary ↗ — A string-matching algorithm using a rolling hash to compare windows of text against a pattern in expected O(n + m). Naturally generalizes to multiple-pattern search.
Radix Sort page
RAII skill — Resource lifetime tied to object lifetime. Deterministic destructors, LIFO teardown, exception safety for free.
Rayleigh plane-wave expansion glossary — Expansion of a plane wave e^{ik·r} in spherical Bessel functions and spherical harmonics. Used to match plane-wave and atomic-sphere solutions in APW and LAPW.
Rayleigh quotient glossary — The scalar v^T A v / v^T v, equal to an eigenvalue of A when v is the corresponding eigenvector. Stationary values of the quotient are the eigenvalues; minimization gives the lowest one.
Real Space vs Second Quantization page
Reciprocal lattice glossary — The Fourier-dual lattice of a Bravais lattice, with primitive vectors b_i satisfying a_i · b_j = 2π δ_{ij}. Hosts the Brillouin zone and indexes plane-wave momenta.
Regular Expression Matching page
Reservoir Sampling page
Residual glossary — For an approximate eigenpair (λ, x): r = (A − λI) x. For a linear system Ax = b: r = b − Ax. Vanishing residual signals an exact solution.
Restricted Hartree–Fock glossary — A closed-shell Hartree–Fock variant in which alpha and beta electrons are constrained to share the same spatial orbitals. Appropriate for systems with all electrons paired.
Reverse Linked List page
Reverse-Mode Autodiff Engine page
RHF, UHF & the ansatz ladder prereq node — Restricted vs unrestricted spin — when one set of spatial orbitals is too rigid for the physics.
Rho (Greek) glossary — The sensitivity ∂V/∂r of an option value to the risk-free interest rate. Generally smaller in magnitude than the other Greeks for short-dated options.
Ripple-carry Adder page
Risk-neutral measure glossary — A probability measure under which discounted asset prices are martingales, so that option prices equal expected discounted payoffs. Equivalent to no-arbitrage in idealized markets.
RK4 glossary — The classical fourth-order Runge–Kutta method, advancing y' = f(t, y) by averaging four slope estimates per step. The default workhorse for non-stiff initial-value problems.
Roadmap page
Rod Cutting page
Roofline Model pageskillglossary ↗ — A visual performance model that bounds attainable FLOP/s by min(peak compute, bandwidth · operational intensity). Distinguishes compute-bound from memory-bound kernels.
Roothaan-Hall skill — The generalized eigenproblem for molecular orbitals.
Roothaan-Hall equations prereq nodeglossary ↗ — The Hartree–Fock equations expressed in a finite atomic-orbital basis as the generalized eigenvalue problem F C = S C ε. They convert the integro-differential HF problem into a matrix problem.
Row-Major and Column-Major Storage page
Rule of Five / Zero skill — The five special members that move together — and the Rule of Zero that lets you write none of them.
Run a real SCF in the browser prereq node — The full SCF loop compiled to run live — every matrix above assembled and iterated in front of you.
Run a real SCF in your browser (wasm) page
Runge-Kutta skill — Fourth-order accuracy from staged slope evaluations.
Runge-Kutta Method pageglossary ↗ — A family of single-step ODE integrators that combine several evaluations of the right-hand side per step to achieve high order of accuracy. The classical fourth-order variant (RK4) is the standard workhorse.

S

SCF in the Browser (Knot → C → wasm) page
SCF iteration prereq nodeskill ↗ — Produces the converged orbitals the response is computed around.
Schrödinger equation glossary — The fundamental equation of non-relativistic quantum mechanics. Its time-independent form, H psi = E psi, is an eigenvalue problem for the Hamiltonian whose solutions are the stationary states and their energies.
Search in Rotated Sorted Array page
Secant method glossary — A derivative-free root-finder that replaces f' in Newton’s method by a finite-difference slope from the last two iterates. Convergence is superlinear (order ~1.618).
Second Quantization skillglossary ↗ — A reformulation of many-body quantum mechanics in terms of creation and annihilation operators acting on Fock space. Antisymmetry of fermions is automatic via the anticommutation relations, and operators take a particle-number-independent form.
Second-quantization formalism glossary — A reformulation of many-body quantum mechanics in terms of creation and annihilation operators acting on Fock space, automatically enforcing fermion antisymmetry. It is the natural language of post-Hartree–Fock methods and quantum field theory.
Secular equation glossary — The determinantal equation det(H − E S) = 0 whose roots are eigenvalues of a generalized eigenvalue problem. Historical name for the matrix eigenproblems arising in quantum mechanics.
Selection Sort page
Self-consistent field glossary — An iterative procedure in which an effective potential is built from a guess wavefunction, the resulting equations are solved for new orbitals, and the cycle repeats until the orbitals stop changing.
Self-Learning MC on a PySCF Surface page
Self-Learning Monte Carlo page
Self-Test Checklist page
SEM · Series overview page
SEM · Step 1 — GLL points and the Lagrange basis page
SEM · Step 2 — Quadrature and the diagonal mass matrix page
Separation of Variables pageskill — u(x,t)=X(x)T(t) splits a PDE into eigenproblems.
Serialize Binary Tree page
Series overview page ↗↗
Shanks / Wynn skill — Nonlinear sequence acceleration removes leading error.
Shanks Transformation & Wynn's ε-Algorithm page
Sharpe Ratio and Backtest Evaluation page
Shell Sort page
Sieve of Eratosthenes pageglossary ↗ — An ancient algorithm that finds all primes up to N by iteratively marking multiples of each newly discovered prime. Runs in O(N log log N).
Sigmoid and Its Gradient page
Simple Addition page
Simple Interest page
Simpson's rule glossary — A Newton–Cotes quadrature rule that fits a parabola through three equally spaced points to integrate over a panel, with error O(h^5) per panel. One step up from the trapezoidal rule.
Single-Particle Models page
Singular Value Decomposition pageprereq node — Every matrix as rotate–stretch–rotate; the most informative factorization there is.
SISSO: Symbolic Regression as Compressed Sensing page 5 notes
Slater determinant glossary — An antisymmetrized product of single-particle spin orbitals; the simplest many-electron wavefunction that satisfies the Pauli exclusion principle.
Slater Determinants skill — Antisymmetric many-electron wavefunctions.
Slater integrals glossary — The radial integrals F^k and G^k that appear after a multipole expansion of 1/r_{12} between two atomic or nuclear orbitals. They store all the radial physics of two-body matrix elements.
Slater Orbitals pageprereq nodeskill — Exponential orbitals with the correct nuclear cusp — the physically right shape a basis set tries to reproduce.
Slater Orbitals: The Integral Problem page
Slater-type orbital glossary — A one-electron basis function of the form r^(n-1) exp(-zeta r) times a spherical harmonic. Slater-type orbitals capture the correct cusp at the nucleus and the correct exponential decay at long range, but are expensive to integrate over.
Slater–Condon rules glossary — Six matrix-element rules for one- and two-electron operators between Slater determinants that differ by zero, one, or two spin orbitals. They reduce determinantal expectation values to sums of orbital integrals.
Sliding Window Maximum page
Smart Pointers skill — unique_ptr, shared_ptr, weak_ptr — RAII and move semantics combined into ownership you can read off the type.
Sparse regression 1 note — The Sparsifying Operator is $\\ell_0$ regularization: for each descriptor dimension $D$ it finds the $D$-feature subset whose linear least-squares fit is best, reporting the smallest $D$ that suffices. The jump from a near-perfect 1D fit to an exact 2D fit signals two independent physical contributions.
Spatial orbital glossary — The position-dependent factor of a spin orbital, shared by an alpha- and a beta-spin electron in restricted theories. Doubly occupying a spatial orbital is the signature of closed-shell wavefunctions.
SPD matrices & quadratic forms gap 3 notes — When xᵀAx > 0, solving Ax = b becomes minimizing a bowl — the picture conjugate gradient descends.
Spectra from Moments page
Spectral accuracy glossary — Convergence faster than any polynomial in 1/N for smooth solutions, characteristic of well-designed spectral and pseudospectral methods. Produced by global, high-order basis expansions.
Spectral Derivative page
Spectral Derivatives skill — Transform-based differentiation, exponentially accurate.
Spectral Functions from Krylov Moments page
Spectral method glossary — A discretization that represents the solution of a differential equation as a global expansion in smooth basis functions (Fourier or Chebyshev), so that derivatives become matrix multiplications. For smooth solutions it converges exponentially in the number of basis functions.
Spectral Methods pageskill — Differentiate via transforms for exponential accuracy.
Spectrum glossary — The set of eigenvalues of an operator or matrix. The ground-state energy is the smallest element of the Hamiltonian’s spectrum.
Spherical harmonics glossary — The orthonormal angular eigenfunctions Y_{lm}(θ, φ) of the squared angular-momentum operator. They give the angular factor of atomic orbitals and Slater-type orbitals.
Spin & Determinants skill — S² eigenfunctions, the 2K singlet-triplet split, and the ⟨S²⟩ contamination diagnostic.
Spin contamination glossary — In unrestricted methods, the deviation of the computed total spin expectation value from its exact eigenvalue, indicating mixing of higher-spin states. Symptomatic of broken-symmetry solutions.
Spin orbital glossary — A product of a spatial orbital and a one-electron spin function (alpha or beta). Slater determinants are antisymmetrized products of spin orbitals.
Spin–orbit coupling glossary — A relativistic interaction between a particle’s spin and its orbital angular momentum, splitting otherwise degenerate levels. It is responsible for fine structure in atoms and shell-model magic numbers in nuclei.
Spline interpolation glossary — Piecewise-polynomial interpolation with continuity of derivatives across knots; cubic splines are the most common variety. Avoids the oscillations of high-degree polynomial interpolation.
Stable Softmax page
Stable Summation pageglossary ↗ — Strategies for summing many floating-point numbers while controlling roundoff growth, including pairwise, Kahan, and Neumaier summation. Crucial when adding numbers of disparate magnitudes.
States, Operators & Measurement pageprereq node — The postulates in matrix form — the grammar every quantum page assumes.
Static correlation glossary — The portion of electron correlation arising from near-degeneracies of multiple electronic configurations, dominant in bond-breaking and biradical situations. Single-determinant methods cannot capture it.
Step 1 — The setup page
Step 1 — The weak form page
Step 10 — Poisson on a Delaunay mesh page
Step 10 — The H2 binding curve page
Step 11 — Afterward page
Step 2 — A 1D mesh from scratch page
Step 2 — The variational principle page
Step 3 — Atomic basis functions page
Step 3 — Solving 1D Poisson page
Step 4 — A 2D triangular mesh page
Step 4 — Many-electron wavefunctions page
Step 5 — Solving 2D Poisson page
Step 5 — The Hartree-Fock equations page
Step 6 — Convergence page
Step 6 — The Roothaan-Hall equations page
Step 7 — The heat equation page
Step 7 — The SCF iteration page
Step 8 — Linear elasticity page
Step 8 — Multi-centre 1-electron integrals page
Step 9 — Delaunay triangulation page
Step 9 — Multi-centre 2-electron integrals page
STO-nG basis glossary — A minimal basis set in which each Slater-type orbital is approximated by a fixed contraction of n Gaussian primitives. STO-3G is the canonical example.
STO-nG Fitting pageprereq node — Fit a Slater orbital with a sum of Gaussians — buy cusp-shaped accuracy at Gaussian integral cost.
Stochastic differential equation glossary — A differential equation containing a stochastic forcing term, typically a Wiener process. Models option pricing, particle diffusion, and noisy dynamical systems.
Stochastic trace estimation gap 4 notes — Tr A = E⟨r|A|r⟩ for random r — a trace from a few matrix–vector products, no diagonal sum.
Stochastic Volatility skill — Coupled SDEs for spot and variance (Heston).
Strassen multiplication glossary — A divide-and-conquer matrix multiplication algorithm running in O(n^{log_2 7}) ≈ O(n^{2.807}) by replacing eight subblock products with seven. First sub-cubic algorithm for matrix multiplication.
Struct page
Subset Sum Problem page
Subsets page
Subtraction page
Superexchange 3 notes — In the large-$U$ Hubbard model, virtual hopping generates an antiferromagnetic spin coupling $J = 4t^2/U$ — magnetism with no magnetic force, only repulsion and the Pauli principle.
Surrogate Models skill — Learn a cheap stand-in for an expensive simulator.
Switch Case page
Symbolic regression 16 notes — Separability is detected from data: fit a smooth surrogate, hold all variables but one at a base point, and read off each 1-D profile. If the profiles ADD back to the data the law is additively separable; if they MULTIPLY back it is multiplicatively separable. The profiles are the subproblems.
Symbolic Regression in the Browser pageprereq node
Symbolic Regression: Rediscovering Laws page 5 notes
Symbolic Rule Engines skill — Derivations as searchable rewrite state-spaces.
Symbolic Rule-Learning Engines page
SymGPT: A Transformer for Symbolic Regression page 5 notes
Symmetric / Hermitian operators gap 3 notes — Real eigenvalues and orthogonal eigenvectors — the structure that makes Lanczos and the SVD work.
Symmetric orthogonalization glossary — The transformation X = S^{-1/2} that orthogonalizes a non-orthogonal basis while staying as close as possible to the original. Used to convert F C = S C ε into a standard eigenvalue problem.

T

Temperature Conversion page
Templates skill — Compile-time polymorphism: function and class templates, instantiation, and where the error messages come from.
Templates for Functions page
Tensor contraction glossary — Summation over one or more shared indices of multidimensional arrays, generalizing matrix multiplication. Building the Fock matrix from the ERI tensor and density matrix is a canonical example.
Tensor-product state glossary — A many-body state expressible as a single tensor product |ψ_1⟩⊗|ψ_2⟩⊗… of subsystem states. Hartree products and unentangled wavefunctions are tensor-product states; Slater determinants are not.
Ternary Search page
The area law gap 4 notes — Ground states of local Hamiltonians entangle only across the boundary — the reason a wavefunction compresses at all.
The diffusion equation gap 4 notes — Fick's second law ∂C/∂t = ∂ₓ(D ∂ₓC) governs both oxide growth and dopant redistribution.
The Dipole Operator & Transition Moments pageprereq node — Light couples to molecules through the dipole; it sets which excitations are bright.
The eigenvalue problem (Ax = λx) gap 4 notes — The directions an operator only stretches — every iterative spectral method hunts these.
The Fermion Sign Problem pageprereq node 6 notes — Why the doped 2D Hubbard model is unsolved: fermion antisymmetry makes QMC weights negative and the average sign decays exponentially.
The Fock Matrix skill — Coulomb and exchange as a mean-field operator.
The Fourier transform gap 4 notes — A projection lens is a low-pass filter in spatial-frequency space — imaging is an FFT and a pupil.
The Hard-Sphere Fluid pageprereq node
The Hodgkin-Huxley Model page
The Knot Playground page
The Legendre Transform page
The Likelihood Perspective page
The Limit Order Book page
The Mott Transition pageprereq node 6 notes — Repulsion turns a half-filled band (a band-theory metal) into an insulator — correlation, not filling.
The non-Hermitian (A, B) eigenproblem prereq node — Casida is a coupled (A+B)(A−B) eigenproblem, not a plain symmetric one.
The Path-Integral Frame page
The Pipeline skill — Lex → parse → codegen: the classical compiler stages.
The PN Junction pageprereq node — The device the flow was building — solved by nonlinear Poisson, measured by C-V.
The Radial Distribution Function pageprereq node
The RC Circuit page
The Resolvent & Spectral Function pageprereq node
The RLC Circuit page
The RPA (A, B) Eigenproblem pageprereq node
The Three Classical Equations pageskill — Heat, wave, Laplace: parabolic, hyperbolic, elliptic.
The Wilson-Cowan Model page
Thermal Oxidation (Deal-Grove) pageprereq node — Grows the SiO₂ that masks diffusion and becomes the gate dielectric — the first step of the flow.
Theta (Greek) glossary — The partial derivative ∂V/∂t of an option value with respect to time, capturing time decay. Long options typically have negative theta.
Three Representations of 1/r₁₂ page
Time-dependent density functional theory 1 note — Linear-response TDDFT reduces to Casida's equation, a generalized eigenvalue problem whose eigenvalues are the excitation energies and whose eigenvectors give the transition densities (hence oscillator strengths).
Time-Dependent DFT pageprereq node — Casida is the matrix form of the TDDFT linear-response equations.
Time-Dependent Schrödinger page
Time-independent Schrödinger equation glossary — The eigenvalue equation Ĥψ = Eψ for stationary states of a quantum system. Most of electronic-structure theory consists of approximate strategies for solving it.
TLB Basics page
TLB Experiments page
Top K Frequent Elements page
Topological Sort pageglossary ↗ — A linear ordering of the vertices of a DAG such that every edge points from earlier to later. Computed via Kahn’s algorithm or DFS post-order reversal.
Training a flow on two moons page
Transformer From Scratch (GPT) page
Trapezoidal Rule pageglossary ↗ — A quadrature rule that approximates the integral on each subinterval by the area of a trapezoid. Second-order accurate for smooth functions and exponentially accurate for periodic ones.
Trapping Rain Water page
Trial wavefunction glossary — A parameterized ansatz for an unknown wavefunction whose parameters are tuned to minimize the energy expectation value. It is the input to any variational calculation.
Trie page
Truncated CI & Size Consistency skill — CISD, the size-consistency disease, and the Davidson patch.
Two Sum page
Two-Body Integrals skill — Coulomb interaction and the Gaussian product theorem.
Two-electron integral glossary — A four-index integral (μν|λσ) over a product of two basis functions and the 1/r_{12} kernel, encoding all electron–electron repulsion in a finite basis. The most expensive ingredient in Hartree–Fock and post-HF methods.
Two-electron integrals (ERIs) prereq nodeskill — The coupling matrix in Casida is built from these.
Two-Level Systems & Spin-1/2 pageprereq node — The smallest quantum system that does everything — the 2×2 machinery of mixing, avoided crossings, and Rabi flopping.
Two-Site DMFT pageprereq node 6 notes — Maps the lattice to a self-consistent impurity; the minimal runnable method that captures the Mott transition.

U

UHF & Broken Symmetry skill — Pople-Nesbet, the Coulson-Fischer point, and the spin-contamination trade.
Union-Find Algorithm page
Union–find glossary — Disjoint-set data structure supporting near-constant amortized union and find with path compression and union by rank. Backbone of Kruskal’s MST and connected-component algorithms.
Unrestricted Hartree–Fock glossary — A Hartree–Fock variant that allows alpha and beta electrons to occupy distinct spatial orbitals. Useful for open-shell systems and bond-breaking, but the wavefunction is generally not a spin eigenstate.

V

Valid Anagram page
Valid Parentheses page
Valid Sudoku page
Validate BST page
Value at Risk page
Value Categories skill — lvalues, rvalues, xvalues — what they are and the one test that tells them apart (can you take its address?).
Variable Declaration page
Variables & Arithmetic skill — Declare variables, do arithmetic, meet the basic types.
Variables and Arithmetic page
Variance reduction glossary — Techniques (control variates, importance sampling, antithetic variates, stratification) that lower the variance of a Monte Carlo estimator without changing its mean. Lets you hit a target accuracy with fewer samples.
Variance Swaps page
Variational Perturbation skill — A trial parameter tamed by minimal sensitivity.
Variational Perturbation Theory page
Variational Principle pageprereq nodeskillglossary ↗ — For any normalized trial wavefunction, the expectation value of the Hamiltonian is an upper bound on the true ground-state energy. Minimizing this expectation value over a parameterized family of trial wavefunctions yields the best approximation within that family.
Variational Quantum Eigensolver (H₂) pageprereq node
Variational Rational page
Vector page
Vector & matrix norms gap 4 notes — How you measure the size of a vector or matrix — the language stability and convergence are stated in.
Vectorization skill — SIMD: data parallelism within one core.
Vectorization Experiments page
Vectorization Overview page
Vega glossary — The sensitivity ∂V/∂σ of an option value to volatility. Used as the derivative in Newton-Raphson root-finding for implied volatility.
Volatility smile glossary — The empirical pattern of implied volatility plotted against strike, often showing higher vol for far-from-the-money options. Contradicts the constant-volatility Black–Scholes assumption.
Volatility surface glossary — Implied volatility as a function of strike and time to expiration. Used to price exotic options and calibrate stochastic-volatility models.
Volatility Surfaces and Calibration page

W

Wang-Landau on a PySCF Surface page
Wavefunction glossary — A complex amplitude ψ whose squared modulus gives a probability density for the configuration of a quantum system. Solutions of the Schrödinger equation; central object of quantum mechanics.
Weak Form skill — Recast a PDE as a variational problem over test functions.
Welford Mean & Variance page
When a Linear Model Stops Being Linear page
Where ABC Breaks pageskill — Curse of dimensionality and wasted simulations.
While Loop page
Why Some Likelihoods Can't Be Written Down page
Wiener process glossary — A continuous-time stochastic process W_t with W_0 = 0, independent Gaussian increments W_t − W_s ~ N(0, t − s), and continuous paths. Mathematical model of Brownian motion; building block of SDEs.
Wilson-Cowan skill — Excitatory/inhibitory population-rate dynamics.
WKB / Semiclassics skill — Semiclassical quantization via the classical action.
WKB and Semiclassical Methods page
Word Break page
Word Ladder page
Word Search page
Worked Ladder: Spin Operators in Positronium pageprereq node
Write CSV page

Z

Zone Folding pageglossary ↗ — The mapping of band-structure information from a small Brillouin zone into a larger one (or vice versa) when a unit cell is enlarged or reduced. Essential for interpreting supercell calculations.