Date-first mathematical archive

INM: Informal Notes on Mathematics

A dated archive of mathematical notes reconstructed into an English TeX book.

158 dated notes15 thematic volumes in the book versionPDF + TeX for each note

Read full PDF Pre-TeX archive

Showing 158 notes, sorted by date. The full PDF is volume-organized; this website preserves the dated archive order.

2026

Jun

2026

N-20260621Complex Analysis12 sections

The Heat Equation, Diffusion, and Similarity Solutions

Contents preview

Diffusion versus wave motion
Heat kernel on the whole line
Similarity solution on a semi-infinite rod
Fourier series for a finite rod
Maximum principle
Energy decay on an interval
The maximum principle with proof idea
Smoothing and the heat semigroup

PDF TeX

Jun

2026

N-20260620Complex Analysis10 sections

The Second-Order Wave Equation and d'Alembert's Formula

Contents preview

The equation and its physical meaning
Factorization into transport operators
Initial-value problem on the whole line
Finite strings and Fourier series
Energy conservation
Inhomogeneous wave equation and Duhamel's principle
Boundary conditions on a finite interval
Deriving the Fourier coefficients

PDF TeX

Jun

2026

N-20260619Complex Analysis10 sections

First-Order Partial Differential Equations and Characteristics

Contents preview

What a first-order PDE asks for
The unforced transport equation
Example: translating a parabola
Forced transport
What can go wrong
General linear first-order equations
Quasilinear equations
Crossing characteristics and shocks

PDF TeX

Jun

2026

N-20260618Complex Analysis17 sections

First-Order and Higher-Order Differential Equations

Contents preview

Why the differential-equation part is one note
Basic concepts
Separable equations
Homogeneous first-order equations
First-order linear equations
Bernoulli equations
Exact equations
Reducible higher-order equations

PDF TeX

Jun

2026

N-20260617Complex Analysis20 sections

Fourier Series and Orthogonal Expansion of Functions

Contents preview

Why Fourier series is a separate theme
Orthogonality of trigonometric functions
Fourier coefficients
Dirichlet convergence theorem
Even and odd functions
Half-range expansions
Concrete coefficient models
Dirichlet kernel: what partial sums really do

PDF TeX

Jun

2026

N-20260616Complex Analysis14 sections

Numerical Series, Power Series, Taylor Series, and Improper Integrals

Contents preview

Why these topics belong together
Numerical series
Positive-term series
Ratio and root tests
Alternating and arbitrary-sign series
Function series and uniform convergence
Power series
Taylor series

PDF TeX

Jun

2026

N-20260615Vector Calculus20 sections

Surface Integrals, Gauss Theorem, and Stokes Theorem

Contents preview

Why this note is split this way
Four similar-looking integrals
First-kind surface integrals
Template surfaces
Example pattern: spherical cap
Second-kind surface integrals as flux
Model flux computations: closed, capped, singular, and sheeted
Gauss theorem

PDF TeX

Jun

2026

N-20260604Linear Algebra12 sections

Solving Linear Systems: Elimination, Iteration, and Energy

Contents preview

One equation, three ways of thinking
Triangular systems are the basic currency
Gaussian elimination is a factorization in disguise
Why pivoting is part of the algorithm
Structure can make elimination cheaper
When a sequence of guesses is better than a factorization
Jacobi and Gauss–Seidel seen on the same example
Residuals are visible; errors are not

PDF TeX

Jun

2026

N-20260603Linear Algebra10 sections

Generalized Inverses, Projections, and Least-Squares Geometry

Contents preview

What an inverse was supposed to do
The four fundamental subspaces reveal the answer
Why the four Penrose equations appear
SVD turns the definition into a calculation
One rank-one matrix contains the whole geometry
Exact solutions, least squares, and minimum norm
Why inverse-like algebra can fail
Updating a pseudoinverse one column at a time

PDF TeX

Jun

2026

N-20260602Linear Algebra10 sections

Matrix Decompositions: LU, Cholesky, Rank Factorization, and SVD

Contents preview

One matrix, several kinds of hidden structure
Elimination remembers its work
A complete LU calculation
When the first pivot betrays us
Positive definiteness buys a square root
A Cholesky solve from beginning to end
Rank factorization removes redundant coordinates
SVD chooses orthogonal coordinates on both sides

PDF TeX

Apr

2026

N-20260417Vector Calculus13 sections

Green's Theorem, Conservative Fields, and the Planar Stokes Principle

Contents preview

Green's theorem
Proof idea: rectangles and cancellation
Conservative fields and path independence
The punctured-plane warning
Area by Green's theorem
Differential forms notation
Closed forms, exact forms, and topology
Connection with de Rham cohomology

PDF TeX

Apr

2026

N-20260416Vector Calculus14 sections

Line Integrals and Surface Integrals of the First and Second Kind

Contents preview

Two meanings of integration on curves
Line integrals with respect to arc length
Line integrals of vector fields
Surface integrals of scalar fields
Flux integrals
First kind versus second kind
Parametrization invariance
Differential forms viewpoint

PDF TeX

Apr

2026

N-20260408Vector Calculus13 sections

Volumes, Spatial Regions, and Order of Integration

Contents preview

Volume as an integral
Describing spatial regions
A reconstruction workflow
Changing order of integration
Cross-sections
Top minus bottom
Cylindrical and spherical descriptions
Absolute extrema over solids

PDF TeX

Apr

2026

N-20260406Vector Calculus14 sections

Triple Integrals: Solids, Cylindrical Coordinates, and Spherical Coordinates

Contents preview

Triple integrals as weighted volume
Rectangular-coordinate example
Cylindrical coordinates
Spherical coordinates
Coordinate-choice table
Mass, center of mass, and moments
Symmetry
Three ways to understand volume factors

PDF TeX

Apr

2026

N-20260405Vector Calculus13 sections

Computing Double Integrals: Slicing, Polar Coordinates, and Change of Variables

Contents preview

The real task: describe the region
Changing the order of integration
Polar coordinates
Jacobian as local area scaling
Why the absolute value appears
A linear change example
Choosing coordinates
Slicing as pushforward of area

PDF TeX

Apr

2026

N-20260403Vector Calculus13 sections

Double Integrals: Riemann Sums, Area, Symmetry, and Average Value

Contents preview

From area sums to double integrals
Geometric meanings
Basic properties
Existence of the integral
Symmetry before computation
Mean value theorem for double integrals
A worked average-value example
Relation with product measure

PDF TeX

Mar

2026

N-20260325Multivariable Calculus7 sections

Extrema of Two-Variable Functions and the Hessian Test

Contents preview

Local extrema
First-order necessary condition
Quadratic model and the Hessian
Second derivative test
A picture of the Hessian test
Constrained extrema
Endpoint and boundary warnings

PDF TeX

Mar

2026

N-20260318Linear Algebra14 sections

Matrix Computation III: Canonical Forms, Spectral Localization, and Eigenvalue Iteration

Contents preview

Before computing an eigenvalue, decide what can change
The polynomial matrix $\lambda I-A$ contains more than the characteristic polynomial
Invariant factors are a module decomposition
From invariant factors to Jordan blocks
Jordan chains turn algebraic multiplicity into dynamics
Gershgorin reads spectral information directly from rows
Separated discs count eigenvalues, not merely contain them
Diagonal scaling can reveal localization hidden by coordinates

PDF TeX

Mar

2026

N-20260314Multivariable Calculus8 sections

A Problem-Solving Guide for Multivariable Limits and Local Geometry

Contents preview

Purpose of this guide
Classifying points relative to a set
Limit templates
Degree comparison for rational expressions
Differentiability checklist
Tangent and normal templates
Local extrema template
Common failure modes

PDF TeX

Mar

2026

N-20260313Multivariable Calculus8 sections

Multivariable Chain Rule, Total Differentials, and Implicit Differentiation

Contents preview

The chain rule is composition of linear maps
One input variable, two intermediate variables
Two input variables
Total differentials
Implicit differentiation in one equation
Implicit differentiation for systems
A shape table
Singular linearization marks the boundary of implicit solvability

PDF TeX

Mar

2026

N-20260312Linear Algebra11 sections

Matrix Computation II: Spectral Radius, Stability, and Conditioning

Contents preview

One step and many steps are different questions
Spectral radius is not a norm
Gelfand's formula extracts the exponential rate
Jordan blocks explain the boundary
Neumann series turns repeated feedback into an inverse
Condition number is the geometry of inversion
A tiny data change can reverse the solution
Forward error, residual, and backward error

PDF TeX

Mar

2026

N-20260311Linear Algebra11 sections

Matrix Computation I: Norms, Amplification, and Numerical Size

Contents preview

A matrix can be small entrywise and large as an operator
Different norms ask different questions
Hölder duality explains the familiar inequalities
An induced norm measures worst-case amplification
A worked comparison of three operator norms
The spectral norm comes from singular values
Unitary transformations preserve Euclidean geometry
Schatten norms place spectral and Frobenius size in one family

PDF TeX

Mar

2026

N-20260306Multivariable Calculus8 sections

Multivariable Limits, Continuity, Differentiability, and the Total Derivative

Contents preview

What changes from one variable to many
How to prove and disprove limits
Continuity
Partial and directional derivatives
Differentiability as linear approximation
Tangent planes and normal vectors
Mixed partial derivatives
The main lesson

PDF TeX

Mar

2026

N-20260305Linear Algebra12 sections

Orthogonal Projection, Least Squares, and the Geometry of Approximation

Contents preview

The nearest point is found by a right angle
Projection onto one direction
A basis turns the geometric condition into equations
The projection matrix remembers the subspace, not the basis
Least squares is projection written as a fitting problem
A complete line-fitting example
The Pythagorean proof of optimality
Orthonormal coordinates remove the Gram matrix

PDF TeX

Mar

2026

N-20260304Multivariable Calculus8 sections

Point-Set Language in $\mathbb R^n$ : Neighborhoods, Boundaries, and Compactness

Contents preview

Why point-set language appears
Balls and neighborhoods
Interior, exterior, and boundary
Open and closed sets
Accumulation points and closure
Boundedness and compactness
Connectedness and path-connectedness
What this vocabulary controls

PDF TeX

Mar

2026

N-20260303Linear Algebra8 sections

Two-Dimensional Linear Maps, Spectral Geometry, and Singular Value Decomposition

Contents preview

Concept map of the chapter
Five model transformations
A spectral example: a biological neural network
SVD as a sequence of geometric operations
SVD and latent structure in data
Rank-one blocks and image compression
A taxonomy of matrix properties
Choose the invariant appropriate to the matrix question

PDF TeX

Mar

2026

N-20260302Machine Learning15 sections

Pure Mathematics as a Language for Machine Learning and Data Analysis

Contents preview

Backpropagation as cotangent pullback
Geometric data analysis
Manifold hypothesis and embeddings
Optimal transport and stochastic dynamics
Symmetry, equivariance, and category-like thinking
Further direction
Backpropagation and differential geometry
Manifold hypothesis and representation learning

PDF TeX

Jan

2026

N-20260106Calculus16 sections

Calculus Review III: Convexity, Integral Inequalities, Wallis Formula, and More

Contents preview

Convexity and the Hermite–Hadamard inequality
Kantorovich-type integral inequality
Integral mean value with odd symmetry
Symmetry of integrals about the midpoint
Midpoint and trapezoidal error terms
First special integral
Positivity of a Fresnel-type integral
A symmetry integral with arcsine

PDF TeX

Jan

2026

N-20260105Linear Algebra14 sections

Spectral Calculus and Positive Matrix Geometry: From Basis Changes to Lyapunov Equations

Contents preview

Coordinates change, but the operator does not
A complete transition-matrix calculation
Polynomial relations turn matrix questions into scalar questions
The minimal polynomial decides diagonalizability
Spectral mapping also controls inverses and adjugates
The symmetric spectral theorem turns spectrum into geometry
Functional calculus applies scalar functions along eigendirections
Rayleigh quotients define the Loewner order

PDF TeX

Jan

2026

N-20260103Linear Algebra12 sections

Rank, Projections, and Long-Term Matrix Dynamics

Contents preview

The first useful question is often which relation the matrix satisfies
Rank factorization exposes the quotient–image structure
The four fundamental subspaces are paired by orthogonality
At corank one, the adjugate becomes a rank-one null-space certificate
Idempotents are projections, but not always orthogonal projections
Nilpotence turns infinite-looking formulas into finite sums
The minimal polynomial reduces every high power
Polynomial Chinese remainders produce primary projectors

PDF TeX

2025

Dec

2025

N-20251217Topology and Homotopy15 sections

A Nontrivial Loop at Zero: FinSet K-Theory and Stable Homotopy

Contents preview

From the meme to the lecture's explanation
The lecture starts from ordinary equations
The Hopf fibration as the first nontrivial map
Homotopy groups and the first stable class
From natural numbers to finite sets
The space of finite sets
Why the first reading was close but needed correction
Finite sets and stable spheres

PDF TeX

Dec

2025

N-20251210Calculus23 sections

Calculus Review II: Monotonicity, Integration Methods, Applications, and More

Contents preview

Strict monotonicity from the derivative
First derivative test for extrema
Second derivative test
Concavity and inflection points
Asymptotes
General procedure for sketching a curve
Basic antiderivative table
Substitution

PDF TeX

Dec

2025

N-20251202Linear Algebra12 sections

Inner-Product Geometry and Spectral Decomposition: From QR to Quadratic Forms

Contents preview

Diagonalization asks whether the space splits into invariant lines
The minimal polynomial detects the missing eigendirections
An inner product turns decomposition into best approximation
Gram–Schmidt factors a basis into orthogonal directions and coefficients
Householder reflections reveal why stable QR avoids repeated subtraction
Orthogonal maps preserve geometry, and reflections generate them
The spectral theorem follows from one Rayleigh extremum and induction
Spectral projectors turn diagonalization into functional calculus

PDF TeX

Nov

2025

N-20251128Combinatorics and Models9 sections

Self-Avoiding Walks III: Lace Expansion, Functional Integrals, and the Critical Dimension Four

Contents preview

The high-dimensional philosophy
The two-point function
The lace expansion as corrected random walk
Bubble diagrams and diagrammatic estimates
Differential inequalities
Functional integral representations
Dimension four and logarithmic corrections
How the three methods fit together

PDF TeX

Nov

2025

N-20251127Combinatorics and Models9 sections

Self-Avoiding Walks II: The Hexagonal Lattice, Holomorphic Observables, and the Exact Connective Constant

Contents preview

Why the hexagonal lattice is special
A domain version of the counting problem
The parafermionic observable
From local cancellation to a global identity
The upper bound on the connective constant
The lower bound on the connective constant
Relation with SLE
Loop models and the same mechanism

PDF TeX

Nov

2025

N-20251126Combinatorics and Models7 sections

Self-Avoiding Walks I: Connective Constants, Bridges, Polygons, and Critical Exponents

Contents preview

What the model is trying to measure
Submultiplicativity and the connective constant
Bridges and why they help
Polygons
Critical exponents
How dimension changes the story
The broader lesson

PDF TeX

Nov

2025

N-20251114Arithmetic Geometry12 sections

Teichmueller Theory Without One Universe: Classical Deformation, Hodge Theaters, and Identification

Contents preview

Why the word Teichmueller matters
Classical Teichmueller theory: fixed topology, variable complex structure
Distortion is measured in one universe
Why the arithmetic analogy cannot be literal
Hodge theaters as arithmetic stages
Etale-like and Frobenius-like directions
The theta-link is the point where one universe breaks
The log-link does not restore naive equality

PDF TeX

Nov

2025

N-20251113Arithmetic Geometry12 sections

When Addition Starts Floating: Theta-Links, Log-Links, and Arithmetic Deformation

Contents preview

What has to be explained
The nonarchimedean theta-function
The local theta-link as a non-ring-theoretic map
The bridge supplied by Kummer theory
Two symmetries and the bookkeeping of labels
Conjugate synchronization and the basepoint problem
The log-link: recovering additive structure from multiplicative structure
The log-theta-lattice

PDF TeX

Nov

2025

N-20251112Calculus12 sections

Calculus Review I: Limits, Completeness, Asymptotic Scales, and Local Approximation

Contents preview

Cauchy–Schwarz and Minkowski are error-control mechanisms
Limits are statements about eventual membership in neighborhoods
Sequence limits are unique and respect algebraic operations
Heine's criterion converts function limits into sequence limits
Asymptotic notation compares scales, not fixed small numbers
Squeeze and monotone convergence use order structure
The Cauchy criterion characterizes convergence in complete spaces
Continuity at each point is weaker than uniform continuity

PDF TeX

Nov

2025

N-20251110Linear Algebra9 sections

Determinants and Rank: From Alternating Volume to Elimination and Rank Strata

Contents preview

Why one scalar can encode both volume and independence
Cofactors and minors are coordinates of exterior powers
Vandermonde determinants measure uniqueness of interpolation
Finite differences expose hidden polynomial dependence
Block elimination produces the Schur complement
Low-rank updates reduce a large determinant to a small one
Recursive determinants are second-order dynamical systems
Cramer's rule lives only on the invertible open set

PDF TeX

Nov

2025

N-20251109Categories and Logic11 sections

Kernels, Cokernels, Images, and Coimages in Abelian Categories

Contents preview

The setting: maps with zero maps
Kernel: the universal object killed by a map
Cokernel: the universal quotient killing a map
Image: the part of the codomain actually reached
Coimage: the effective part of the domain
The standard factorization
A vector-space sanity check
A module example: multiplication by an integer

PDF TeX

Nov

2025

N-20251108Linear Algebra21 sections

Notes on MML: Vectors, the Minus-One Trick, Linear Maps, Basis Change, Kernel, and Image

Contents preview

Vectors beyond arrows
The minus-one trick
Example of the trick
Linear mappings
Complex numbers as a real vector space
Finite-dimensional isomorphism theorem
Basis change
Equivalence and similarity

PDF TeX

Oct

2025

N-20251011Machine Learning11 sections

The Reparameterization Trick and Auto-Encoding Variational Bayes

Contents preview

The gradient problem
Score-function estimator and pathwise estimator
Gaussian reparameterization
The stochastic ELBO estimator
Why implementations output log variance
Training as ELBO maximization
Decoder likelihoods and reconstruction losses
Generation after training

PDF TeX

Oct

2025

N-20251010Machine Learning11 sections

Variational Inference: KL Divergence, Jensen, and the ELBO

Contents preview

The approximate posterior as an optimization problem
The ELBO identity
Jensen's inequality derivation
Reconstruction term and regularization term
The ELBO gap
Closed-form KL for diagonal Gaussians
Negative ELBO as a loss
Empirical lower bounds

PDF TeX

Oct

2025

N-20251009Machine Learning10 sections

Latent Variable Models: Priors, Decoders, and Intractable Posteriors

Contents preview

From autoencoders to latent variable models
The generative story
Graphical model notation
Marginal likelihood
The posterior problem
Why naive Monte Carlo is not enough
The recognition model
Amortized inference

PDF TeX

Oct

2025

N-20251008Machine Learning9 sections

Probability Language for Variational Autoencoders

Contents preview

Why probability enters generative modeling
Random variables, distributions, and densities
Joint, marginal, and conditional distributions
Bayes' rule as inversion of a generative story
Gaussian distributions
Expectation and Monte Carlo approximation
KL divergence
Jensen's inequality

PDF TeX

Sep

2025

N-20250926Machine Learning16 sections

Backpropagation in One MLP Layer: From a Three-Dimensional Tensor to an Outer Product

Contents preview

The setting
The theoretical tensor
Computing the sparse derivative
Contraction by the chain rule
Why the tensor disappears
Contravariant reading of the same calculation
Outer-product expression
Numerical example

PDF TeX

Jul

2025

N-20250707Curves and Structures8 sections

Nikonov's Paper at One Crossing: Region Probes, Partial Tribrackets, and What the Seminar Was About

Contents preview

Following the region probes
The local dictionary
A region is not just a blank patch
At a crossing, four regions want to talk to each other
Why the operation is partial
Probe clearing and color rules
What about crossings themselves?
The paper in one playful sentence

PDF TeX

Jul

2025

N-20250705Curves and Structures7 sections

Nikonov's Seminar Prelude: Knots, Diagrams, Probes, and the Space Between the Lines

Contents preview

First orientation
The topological object and its planar shadow
Some basic knot-theory language
The pieces of a diagram
Nikonov's bridge: diagram elements as probes
Invariants, coinvariants, and labels
From probes to colorings

PDF TeX

Mar

2025

N-20250301Curves and Structures12 sections

Projective Schemes and Gluing: Building Projective Space from Affine Windows

Contents preview

Projective geometry returns with functions attached
The projective line as the model
Graded rings and homogeneous data
Projective space as Proj
The projective plane through three windows
A conic across charts
Projective closure with scheme memory
What Proj adds

PDF TeX

Feb

2025

N-20250211Classical Geometry17 sections

Transformations of Conics and the Erlangen View of Geometry

Contents preview

The organizing question
Affine transformations
Conics in matrix form
Projective geometry
Cross-ratio
Projective duality and conics
Inversion
Möbius transformations

PDF TeX

Jan

2025

N-20250118Curves and Structures12 sections

Schemes Gently: A Guided Tour of Spectrum, Nilpotents, and Local Rings

Contents preview

A new kind of point
The affine line spectrum
Closed sets and distinguished opens
Localization is zooming in
The structure sheaf
Nilpotents: invisible but geometric
The arithmetic curve
Maximal ideals are not enough

PDF TeX

2024

Oct

2024

N-20241026Curves and Structures14 sections

Zariski Geometry: When Functions Become the Space

Contents preview

Starting from functions, not pictures
Coordinate rings: a space through its functions
Nullstellensatz: the dictionary becomes reliable
The Zariski topology is coarse on purpose
Distinguished opens and localization
Sheaves: local data with a gluing rule
Stalks: looking through a microscope
A local example: two axes crossing

PDF TeX

Sep

2024

N-20240914Curves and Structures11 sections

Divisors on Curves: The Bookkeeping of Zeros, Poles, and Functions

Contents preview

A ledger for meromorphic functions
Principal divisors: the ledger of one function
Reading a divisor as permission
Line bundles: permission becomes geometry
The canonical divisor
Riemann–Roch without intimidation
Hurwitz as topology with branch points
A divisor game on the sphere

PDF TeX

Jul

2024

N-20240727Topology and Homotopy7 sections

Hopf Fibration, Homotopy, and the Infinite-Stability Reading

Contents preview

From the meme to infinite stability
The analytic reflex
The Eilenberg swindle
Where the Hopf fibration enters
Reading the ladder
The infinite-stability reading
The finite clues

PDF TeX

Jun

2024

N-20240608Curves and Structures14 sections

Plane Curves as Algebraic Creatures: Equations, Rings, and Singularities

Contents preview

The curve is not only the drawing
The function ring of a curve
When factorization becomes geometry
Smooth points and singular points
Tangent cones: the first nonzero shadow
The cusp as a ring
Intersections: one visible point may count twice
Projective closure repairs missing intersections

PDF TeX

May

2024

N-20240518Classical Geometry11 sections

Projective Geometry as the Art of Removing Exceptions

Contents preview

The annoyance of parallel lines
Homogeneous coordinates are ratios
A coordinate scene: where parallel lines meet
Duality: points and lines speak the same language
Transformations and what survives
A first invariant: the cross-ratio
Conics as one projective family
The projective plane as all lines through the origin

PDF TeX

Apr

2024

N-20240427Vector Calculus12 sections

Differential Forms and Stokes: The Geometry of Boundary Measurements

Contents preview

A theorem looking for its natural language
One-forms: measuring motion
The pullback experiment
Two-forms: measuring oriented area
Exterior derivative as one operator
A familiar theorem reappears
Orientation is not decoration
Why arc length is a different kind of geometry

PDF TeX

Apr

2024

N-20240413Multivariable Calculus12 sections

Local Linear Geometry: A Mapmaker's View of Derivatives and Tangent Spaces

Contents preview

The mapmaker's problem
Derivative first, partial derivatives second
The chain rule is not a formula trick
A small laboratory: polar coordinates
Tangent planes from equations
A manifold is a space with local Euclidean windows
Two pictures of tangent vectors
Regular values: one clean machine for making manifolds

PDF TeX

Mar

2024

N-20240330Combinatorics and Models10 sections

From Extremal Triangles to Euler Characteristic and Degree

Contents preview

Maximum-area triangles live on the convex hull
Two maximal triangles must alternate around the boundary
Cyclic lifts turn alternation into the bound $n$
Euler's formula follows by adding non-tree edges one at a time
Incidence counting leaves only five regular polyhedra
Polygon gluings compute Euler characteristics of quotient surfaces
A chain-complex cancellation proves the homological Euler formula
Degree on the circle is winding made algebraic

PDF TeX

Mar

2024

N-20240323Combinatorics and Models20 sections

Combinatorics: Identities, Generating Functions, Catalan Numbers, and More

Contents preview

Binomial identities
Good subsets
Generating functions
A probability example with repeated trials
Catalan numbers
Convex sets and convex hulls
Separating convex sets
Planar graphs and Euler-type counting

PDF TeX

Feb

2024

N-20240211Categories and Logic16 sections

Topos Theory: Categorical Preliminaries, Internal Logic, and the Geometry of Sets

Contents preview

Why topoi enter the story
Categories
Universal properties and limits
Exponentials and cartesian closure
Subobjects and the subobject classifier
Elementary topoi
A toy presheaf topos
Grothendieck topoi and sheaves

PDF TeX

2023

Aug

2023

N-20230801Curves and Structures11 sections

From Self-Duality to Spectral Curves: A Computable Route through Hitchin's Equations

Contents preview

Why the story begins in four dimensions
Dimensional reduction produces three coupled equations
Gauge symmetry explains why the solution space is a quotient
The complex equation turns the pair into a Higgs bundle
A local rank-two ansatz turns the matrix equation into one scalar PDE
A solution also produces a family of flat connections
The Hitchin map records global eigenvalue data
A global rank-two model requires a theta characteristic

PDF TeX

Jul

2023

N-20230710Sets and Foundations12 sections

Foundations of Mathematics: Set Theory, Category Theory, and Type Theory

Contents preview

The question of foundations
Set theory
Category theory
Type theory
Homotopy type theory and univalence
Comparative analysis
Modern applications
Toward a unified perspective

PDF TeX

Mar

2023

N-20230316Functions and Inequalities9 sections

Fixed Points, Convergent Iteration, and Compactness Methods

Contents preview

A fixed point does not by itself explain an iteration
Contractions make successive points geometrically closer
Completeness supplies the point that the estimates approach
The proof gives two practical stopping bounds
The equation $x=\cos x$ is a complete example
Picard iteration moves the theorem to a function space
Topological existence does not provide Banach's algorithm
Schauder replaces finite-dimensional compactness by a compact map

PDF TeX

Mar

2023

N-20230314Categories and Logic8 sections

Free Constructions, Adjunctions, Concrete Categories, and Group Actions

Contents preview

A free object solves an extension problem
The universal property is a natural Hom-set bijection
Unit and counit encode the same adjunction
Full and faithful describe maps on arrows, not objects
A concrete category includes a chosen faithful underlying-set functor
A group is a one-object category
Natural transformations are equivariant maps
An adjunction generates a monad by adding and flattening structure

PDF TeX

Mar

2023

N-20230313Categories and Logic10 sections

Category Theory Continued: Posets, Groups as One-Object Categories, Products, and Monomorphisms

Contents preview

What this note continues
Posets are categories
A small poset example
Isomorphisms in a poset category
Groups as one-object categories
Why this example matters
Products of categories
The opposite category

PDF TeX

Mar

2023

N-20230311Vector Calculus11 sections

From Oscillation to Stable Limits: Riemann and Lebesgue Integration

Contents preview

The difficult question is whether integration survives a limit
Riemann integration controls oscillation on intervals
Dense oscillation defeats every interval partition
Measure spaces decide which sets may be counted
Simple functions build the nonnegative integral from below
Monotone convergence is built into the definition
Fatou and dominated convergence control general limits
Escape and concentration explain why domination is needed

PDF TeX

Mar

2023

N-20230310Calculus16 sections

Derivatives, Tangents, Extremal Problems, Young–Holder–Minkowski, and Jensen

Contents preview

The derivative from secants to tangents
Derivative notation and local definition
Rules of differentiation
Elementary derivative formulas
Extrema, Fermat's lemma, and critical points
A derivative proof of a basic inequality
Young's inequality
Holder's inequality

PDF TeX

Feb

2023

N-20230225Sets and Foundations9 sections

Completed Decimals, Hyperfinite Truncations, and Nonstandard Analysis

Contents preview

One notation can hide three different objects
The real decimal is a completed limit
An ultrapower turns sequences into numbers
Infinitesimal closeness and standard part
Transfer explains why finite algebra still works
Internal sets are the domain of transfer
Derivatives become standard parts of infinitesimal quotients
Hyperfinite sums recover the Riemann integral

PDF TeX

Feb

2023

N-20230223Classical Geometry12 sections

Trigonometric Systems, Multiple-Angle Formulas, and a Determinant Recurrence

Contents preview

Correcting a trigonometric mistake
A trigonometric system
A rational-parametrization idea
A direct algebraic route
Multiple-angle formulas
The determinant recurrence
From technique to structure
Trigonometric systems as algebraic systems

PDF TeX

Feb

2023

N-20230220Classical Geometry10 sections

Special Proofs of Trigonometric Addition Formulas

Contents preview

Why prove the same identity more than once?
A chord-length proof of the cosine difference formula
Signed area proves the sine difference formula
The rotation proof begins from a group law
An analytic Euler proof with no hidden circularity
Where the circular Euler proof enters
Tangent lives on a projective line
A dependency graph for the proofs

PDF TeX

Feb

2023

N-20230217Linear Algebra12 sections

Tensors in Nonorthogonal Coordinates: Dual Bases, Metrics, and Contraction

Contents preview

Why coordinates become interesting when they are not orthogonal
Tensor products package bilinear dependence
Vectors and covectors are different kinds of objects
A nonorthogonal basis exposes the dual basis
The metric is the Gram matrix of the frame
Raising and lowering indices is the metric acting as a map
Tensor type records what the object consumes and produces
Contraction creates invariants by closing one input-output pair

PDF TeX

Feb

2023

N-20230209Linear Algebra11 sections

Linear Maps, Matrix Representations, and Tensor Invariance

Contents preview

From vector identities to linear representations
Coordinates as isomorphisms
The basis-change formula
A worked basis-change example
Which quantities are invariant?
Covectors and the inverse-transpose rule
Bilinear forms are not linear operators
Symmetric and skew parts

PDF TeX

Feb

2023

N-20230205Multivariable Calculus9 sections

Vector Identities from Exterior Algebra: Wedge, Contraction, and the Hodge Star

Contents preview

Why the vector triple product is easy to misremember
The scalar triple product is a volume form
The wedge product records an oriented plane before choosing a normal
The metric turns vectors into covectors
The Hodge star converts complementary dimensions
Lagrange's identity is an inner product of bivectors
Contraction removes one vector from an alternating form
The Levi–Civita symbol is the coordinate form of volume

PDF TeX

Feb

2023

N-20230201Vector Calculus12 sections

Newton's Shell Theorem: Solid Angle, Potential, and Gauss Law

Contents preview

First orientation
Statement of Newton's shell theorem
Why symmetry alone is not enough
Solid-angle proof in full detail
Potential proof: doing the integral
The direct force integral
Gauss-law viewpoint
Harmonic potential

PDF TeX

Jan

2023

N-20230127Number Theory Basics10 sections

Cyclotomic Pullbacks: Exact Orders, Galois Conjugacy, and Polynomial Divisibility

Contents preview

A plausible generalization fails for a precise reason
Taking powers changes order by a greatest common divisor
The correct divisibility criterion
The exact common factor is stronger information
Why one primitive root forces an entire rational factor
Cyclotomic polynomials separate roots by exact order
Substitution pulls cyclotomic factors back along a power map
Möbius inversion reconstructs the exact-order factor

PDF TeX

Jan

2023

N-20230123Vector Calculus15 sections

Definite Integrals, Line Integrals, Work, Green-Type Thinking, and Vector Operations

Contents preview

Indefinite and definite integrals
Area by slicing
Work as an integral
Line integrals
Green-type intuition
Polar coordinates
Spherical coordinates
Dot product and cross product

PDF TeX

Jan

2023

N-20230119Classical Geometry17 sections

Plane Geometry I: Ceva, Menelaus, Ptolemy, Simson Line, and Euler Line

Contents preview

Why this geometry chapter matters
Ceva's theorem
Menelaus' theorem
Ptolemy's theorem
Generalized Ptolemy inequality
Simson line
Euler line and Euler formula
Why the pattern matters

PDF TeX

Jan

2023

N-20230118Combinatorics and Models10 sections

Local Information, Global Structure: Pigeonhole, Ramsey, Radon, and Helly

Contents preview

Pigeonhole is an averaging theorem with a design problem hidden inside
Partial sums turn a divisibility problem into a finite-state collision
Repeated pigeonhole becomes sequential compactness on an interval
Ramsey theory forces a pattern, not merely a repeated label
Intervals give the one-dimensional Helly theorem
Radon's theorem is an affine dependence split into two convex combinations
Helly's theorem follows by applying Radon to carefully chosen witnesses
Carathéodory reduces a large convex representation to $d+1$ points

PDF TeX

Jan

2023

N-20230117Functions and Inequalities18 sections

More Function Exercises: Periodicity, Piecewise Definitions, Images, and Functional Equations

Contents preview

Continuing the function chapter
Piecewise functions
Periodicity from functional equations
Graph transformations
Image of an interval
Functional equations and substitution
Counting functions between finite sets
A compact bridge

PDF TeX

Jan

2023

N-20230116Functions and Inequalities11 sections

Functions through Fibers, Iteration, and Functional Equations

Contents preview

A function includes its source and target
Images and preimages obey different algebraic laws
Direct and inverse image form a Galois connection
Fibers measure information loss
Counting injections, surjections, and bijections
A finite function is a directed dynamical system
Iteration equations become restrictions on cycle lengths
Left inverses, right inverses, and what they prove

PDF TeX

Jan

2023

N-20230107Functions and Inequalities11 sections

Function Graphs, Symmetry Centers, and Transformations

Contents preview

A note on graph problems
Centers of symmetry
Symmetry about a line
Absolute value transformations
Zero counting by graphs
How to find a center of symmetry in practice
Absolute value as folding
Parameter motion and intersection counting

PDF TeX

2022

Dec

2022

N-20221229Topology and Homotopy19 sections

Algebraic Topology and Algebra: Products, Wedge Sums, CW Complexes, Cosets, and Surface Gluings

Contents preview

From quotient pictures to algebraic invariants
Disjoint union
Wedge sum
CW complexes: building spaces by cells
The torus as a quotient of a square
Cosets as algebraic quotients
The Klein bottle
Polygon schemata for surfaces

PDF TeX

Dec

2022

N-20221228Topology and Homotopy15 sections

Quotient Topology and Product Topology

Contents preview

Interval modulo endpoints
More quotient spaces
Definition of quotient topology
Product topology
Universal property of quotient topology
Endpoint identification in detail
Product topology via projections
Quotient and product as opposite directions

PDF TeX

Dec

2022

N-20221227Categories and Logic16 sections

Category Theory Continued and the Question Topology

Contents preview

More examples of categories
Subcategories and functors
Forgetful functors
A topology generated by a distance-like question
Posets as categories, with an example
Functorial thinking
Generated topology
Why category theory appears beside topology

PDF TeX

Dec

2022

N-20221223Topology and Homotopy10 sections

Paths to Groups: Homotopy, Coverings, Fundamental Groups, and Presentations

Contents preview

Path concatenation is reversible but not strictly associative
Homotopy relative endpoints is a congruence for concatenation
The fundamental groupoid remembers every basepoint at once
The exponential covering turns loops into integer endpoints
The covering computation gives $\pi_1(S^1)\cong\mathbb Z$
The punctured plane retracts onto the circle
Continuous maps induce homomorphisms
Van Kampen assembles groups from overlapping pieces

PDF TeX

Dec

2022

N-20221222Topology and Homotopy8 sections

Open-Set Calculus: Continuity, Closure, Nets, and Connected Components

Contents preview

Continuity is the stability of tests under preimage
The Sierpiński space classifies open subsets
Closure is an operator from which the topology can be recovered
Subspace topology is the weakest topology making inclusion continuous
Nets recover closure in every topological space
Connectedness is the absence of a discrete-valued separator
Paths give connectedness, but not every connected space is path-connected
Local path-connectedness repairs the gap between components

PDF TeX

Dec

2022

N-20221221Topology and Homotopy10 sections

Completion, Uniform Structure, and Quotient Topology

Contents preview

A Cauchy sequence can describe a point missing from the space
The completion metric is a limit of mutual distances
The original space sits isometrically and densely inside its completion
A diagonal approximation proves that the completed space is complete
Uniformly continuous maps extend because they preserve Cauchy information
The universal property makes completion unique
Homeomorphism forgets the uniform scale of closeness
A continuous bijection needs extra hypotheses to be a homeomorphism

PDF TeX

Dec

2022

N-20221218Topology and Homotopy17 sections

Basic Topology I: Metric Spaces, Boundedness, Norms, Open Sets, and Completeness

Contents preview

Why metrics come first
Boundedness: diameter and radius
Metric spaces
Sequences and continuity
Standard metrics
Norms and inner products
Open and closed balls
Open sets, closed sets, and limit points

PDF TeX

Dec

2022

N-20221217Functions and Inequalities22 sections

Function Exercises: Domains, Ranges, Dirichlet Function, Parameters, and Functional Equations

Contents preview

Structure of the exercise sheet
Domain problems
Floor functions
Range problems
Dirichlet function
Graph transformations and zero counting
Logarithmic parameter equation
Functional equations with period-like recurrence

PDF TeX

Dec

2022

N-20221214Curves and Structures18 sections

Tarski's Circle-Squaring Problem: From Polygon Dissection to Borel Equidecomposition

Contents preview

The route of the note
Equidecomposability and dissection congruence
Hilbert's third problem and the Dehn invariant
Measure-theoretic background
Tarski's circle-squaring problem
Scissors congruence and why nice pieces fail
Laczkovich's positive theorem
First idea: work in the torus

PDF TeX

Dec

2022

N-20221211Calculus11 sections

Derivatives: Local Linearization, Computation Graphs, and Higher-Order Data

Contents preview

A tangent line is a local model, not only a slope
The algebra of small remainders generates the rules
Powers, exponentials, and logarithms
The inverse rule and where it fails
A small derivative library and its dependency map
Forward mode: propagate a perturbation
Reverse mode: pull sensitivity backward
Second-order automatic differentiation is jet composition

PDF TeX

Dec

2022

N-20221210Combinatorics and Models15 sections

Binomial Theorem, Absolute-Value Estimates, and Set Exercises

Contents preview

Binomial theorem
Absolute value of binomial sums
A summation exercise
Set exercises
Counting subsets with properties
Set-builder notation
Binomial coefficients count the same choices in two languages
Binomial identities as counting in two languages

PDF TeX

Dec

2022

N-20221209Combinatorics and Models18 sections

Set-Theoretic and Combinatorial Exercises after the First Lecture

Contents preview

What this page is doing
A floor-function example
A binomial counting identity
Solving a system by discriminants
A prime-number exercise
A nested-set exercise
A recurrence of sets
An intersection-counting problem

PDF TeX

Dec

2022

N-20221207Combinatorics and Models17 sections

Set Problems, Extremal Counting, and Venn-Diagram Reasoning

Contents preview

A set construction problem
What this problem teaches
A Venn-diagram necessary-and-sufficient problem
Difference identities
Coordinate and algebraic exercises
A note on proof writing
Extremal set problems: why constructions matter
Venn diagrams versus proofs

PDF TeX

Dec

2022

N-20221205Combinatorics and Models11 sections

Counting by Logic, Projection, Inversion, and Generating Series

Contents preview

Set operations are logical operations written geometrically
Addition counts disjoint cases; multiplication counts successive choices
The binomial theorem counts choices inside a product
Double counting is counting one incidence set through two projections
Inclusion–exclusion is a pointwise product expansion
Boolean Möbius inversion explains the alternating signs structurally
Ordinary generating functions turn decomposition into multiplication
Generating functions solve recurrences and coin-change problems

PDF TeX

Nov

2022

N-20221126Number Theory Basics18 sections

Congruence Classes, Complete Residue Systems, and Euler's Function

Contents preview

Congruence
Basic properties
Residue classes
Complete and reduced residue systems
Example: last digit
Example: reducing a long expression
A polynomial congruence by induction
Where the idea leads

PDF TeX

Nov

2022

N-20221120Arithmetic Geometry9 sections

Elliptic Curves from Smooth Cubics to Finite Groups

Contents preview

A cubic becomes an elliptic curve only after two additions
The discriminant is exactly the obstruction to smoothness
A line determines the third point, and Vieta determines its coordinates
Tangency, inverses, and the point at infinity complete the rule
A rational example shows both addition and doubling
Divisors explain why the chord rule is associative
The same formulas work over finite fields
One point generates the entire eight-element example

PDF TeX

Nov

2022

N-20221119Number Theory Basics17 sections

Prime Numbers, Composite Numbers, Divisor Functions, Congruences, and Number-Theoretic Exercises

Contents preview

Prime and composite numbers
Euclid's theorem
Prime divisibility
Fundamental theorem of arithmetic
Perfect numbers and square numbers
Congruence exercises
Broader perspective
Prime exponents and valuations

PDF TeX

Nov

2022

N-20221107Calculus9 sections

Trigonometric Limits, Derivatives, and Inverse Trigonometric Examples

Contents preview

Radians are built into the derivative
The circle squeeze proves the first basic limit
The second limit and the correct use of small-angle equivalence
Sine and cosine emerge from their addition laws
The other four derivatives are generated, not memorized
Inverse functions require branches
Endpoint blow-up is a failure of local invertibility
Three examples that use different ideas

PDF TeX

Nov

2022

N-20221105Functions and Inequalities10 sections

Function Exercises and Inequality Estimates

Contents preview

A short note about elementary methods
Periodicity from a transformed argument
A quadratic minimum
A logarithmic comparison
A parameterized polynomial
The conceptual turn
Periodicity from transformed arguments
Quadratic and logarithmic estimates

PDF TeX

Oct

2022

N-20221031Number Theory Basics13 sections

Congruences and Integer Linear Algebra: From Residue Classes to Smith Normal Form

Contents preview

A remainder is a coordinate on a quotient
A linear congruence is an image problem
A complete congruence calculation
The Chinese remainder theorem glues local coordinates
A three-modulus reconstruction
When moduli are not coprime
Integer row reduction needs a stricter notion of invertibility
Smith normal form diagonalizes divisibility

PDF TeX

Oct

2022

N-20221030Number Theory Basics9 sections

Integer Arithmetic: Euclidean Descent, Bézout Identities, and the Euler Function

Contents preview

Descent is the algorithmic form of well-ordering
The gcd is preserved when a remainder replaces a number
A complete Euclidean calculation
Bézout's identity is an ideal statement
Coprimality is exactly invertibility modulo $n$
Prime divisibility makes factorization rigid
Euler's function counts the unit group
Euler's theorem is Lagrange's theorem in disguise

PDF TeX

Oct

2022

N-20221013Number Theory Basics10 sections

Oriented Boundary Sums, Polynomial Remainders, and Congruence Classes

Contents preview

Three short computations hide three different kinds of structure
A determinant is the signed area of two position vectors
The shoelace formula is a boundary integral evaluated edge by edge
Translation and orientation provide immediate consistency checks
A complete polygon calculation
Evaluation removes every multiple of $x-a$
Remainders modulo several linear factors are interpolation data
The polynomial Chinese remainder theorem explains uniqueness

PDF TeX

Oct

2022

N-20221012Functions and Inequalities11 sections

Classical Inequalities from Convexity, Duality, and Norm Geometry

Contents preview

A convex graph lies above each supporting tangent
Jensen averages the supporting-line inequality
Smoothing leads from Jensen to majorization
Young's inequality is a tangent bound in dual exponents
Hölder follows by normalizing Young's inequality
Hölder identifies the dual norm exactly
Minkowski is the triangle inequality produced by duality
Cauchy's Engel form is projection geometry in disguise

PDF TeX

Oct

2022

N-20221005Categories and Logic11 sections

Linear Logic: Resources, Dialogue, Narrative, and Geometry

Contents preview

First orientation
From truth to resources
Narrative as a resource pipeline
Tensor, additives, and branching stories
Dialogue as a game
Schopenhauer's stratagem as a ludic interaction
Ludics: loci instead of signifieds
Orthogonality and negation

PDF TeX

Oct

2022

N-20221002Arithmetic Geometry21 sections

Finite Fields and Polynomial Factorization

Contents preview

Why finite fields matter
Fields
Field extensions
Constructing finite fields
Important properties of finite fields
Polynomial factorization over finite fields
Algorithms
Applications

PDF TeX

Sep

2022

N-20220918Number Theory Basics11 sections

Divisibility by Finite Differences and Pairing

Contents preview

A general odd-power divisibility principle
Why finite differences work
An alternating harmonic numerator
The next abstraction
Finite differences as discrete derivatives
The binomial basis and integer-valued polynomials
Pairing as group symmetry
P-adic valuation and lifting intuition

PDF TeX

Sep

2022

N-20220913Number Theory Basics15 sections

Divisibility, Congruences, and the Habit of Proving Necessity

Contents preview

From guessing to proving
Divisibility
Example: reducing the dividend
Example: polynomial divisibility in one variable
Congruences
A large power sum
A wider interpretation
Divisibility as ideal containment

PDF TeX

Sep

2022

N-20220911Classical Geometry10 sections

Analytic Geometry through Normals, Line Pencils, and Quadratic Forms

Contents preview

A line is an affine level set, not a collection of unrelated formulas
Angles and Euclidean distance come from the normal vector
Distance to a hyperplane is governed by the dual norm
Circles are quadratic level sets, and subtraction reveals their common geometry
A fractional range problem is a pencil of lines through one point
Ratios on a graph are radial coordinates
A perpendicular moving-line problem is a circle problem
A general conic is an affine quadratic form

PDF TeX

Aug

2022

N-20220813Complex Analysis10 sections

Contour Integrals: Parametrization, Exactness, Winding, and Residues

Contents preview

A contour integral is a limit of transformed tangent increments
Orientation, concatenation, and reparametrization
The complex integral is the pullback of a one-form
Primitives are equivalent to path independence
Cauchy–Goursat turns holomorphicity into exactness on simple domains
Powers on a circle isolate one Fourier mode
Winding number is the period of $dz/(z-a)$
Closed need not mean exact on a punctured domain

PDF TeX

Aug

2022

N-20220811Functions and Inequalities18 sections

Inequality Exercises: Cauchy, AM–GM, Vectors, and Optimization

Contents preview

How to read this note
A fraction problem with a constraint
A weighted square-root comparison
Maximizing a radical expression
Equality in Cauchy–Schwarz
Triangle-angle inequalities
Two spheres and a linear equation
A product inequality

PDF TeX

Aug

2022

N-20220810Functions and Inequalities11 sections

Basic Inequalities, Cauchy, Triangle Inequality, and Coordinate Geometry

Contents preview

Basic inequalities
AM–GM
Cauchy's inequality
Triangle inequality and its variants
Worked AM–GM examples
A Cauchy example
Coordinate geometry at the end of the note
What the examples reveal: three structures

PDF TeX

Aug

2022

N-20220808Functions and Inequalities16 sections

Set Counting, Modular Patterns, and Mathematical Induction

Contents preview

A set-counting problem
A modular arithmetic construction
Why not every construction is finished
Sequence exercises
Mathematical induction
Examples of induction
Induction structure
Modular patterns

PDF TeX

Aug

2022

N-20220806Functions and Inequalities23 sections

Set Theory Exercises, Closures, Functions, and Algebraic Side Notes

Contents preview

Images, preimages, and finite set examples
Closure under addition
A combinatorial maximal-set problem
Functional and algebraic exercises
Matrix and coordinate side computations
Images and preimages behave differently
Closure operations
The reusable pattern

PDF TeX

Aug

2022

N-20220803Number Theory Basics13 sections

Arithmetic and Geometric Sequence Exercises

Contents preview

A trigonometric opening example
Logarithmic sequences
Recovering terms from partial sums
Sums and recurrence relations
Geometric progression products
A harmonic-looking sequence
A final synthesis
Sequences as discrete dynamical systems

PDF TeX

Aug

2022

N-20220802Classical Geometry10 sections

Trigonometric Functions, Radians, Identities, and Triangle Formulas

Contents preview

An angle is a motion before it is a number
Coordinates on the unit circle
Rotations contain the addition formulas
Complex multiplication and multiple angles
Chebyshev polynomials turn angles into algebra
Triangle formulas are inner-product geometry
The ambiguous triangle is a real geometric bifurcation
The circle group explains why frequencies add

PDF TeX

Aug

2022

N-20220801Functions and Inequalities24 sections

Set Exercises, Functions, and Elementary Problem-Solving

Contents preview

Subsets and characteristic functions
Set laws
Venn diagram exercises
Inclusion–exclusion with literature examples
Set constructions with squares
Floor functions
Functions and monotonicity
Characteristic functions and subsets

PDF TeX

Jul

2022

N-20220728Multivariable Calculus11 sections

What Does ``Product'' Mean? Spatial Vectors, Cross Products, and Universal Products

Contents preview

One word can hide different mathematical types
Coordinates describe a vector only after a basis is chosen
The dot product adds metric information
The cross product packages area and orientation
The scalar triple product measures oriented volume
A cross product can generate a rotation
Why three dimensions are exceptional
Cartesian products package two independent pieces of data

PDF TeX

Jul

2022

N-20220725Linear Algebra10 sections

Permutations, Inversions, and Determinants

Contents preview

Inversions and signs
Definition of determinant
Basic determinant properties
Sarrus and triangular matrices
Cofactor expansion
A structured determinant
The determinant line and exterior powers
Why permutation signs are forced

PDF TeX

Jul

2022

N-20220723Number Theory Basics9 sections

Polygon Motions and Roots of Unity: Modular Hops, Chords, and Vanishing Sums

Contents preview

The ant on a decagon is a modular-arithmetic problem
The greatest common divisor determines the orbit
Roots of unity place the polygon on the complex plane
Power maps have a visible kernel and image
The sum of the vertices is zero
Orthogonality is a vanishing polygon sum
Chord lengths depend only on modular distance
The roots of $z^N-1$ form the polygon

PDF TeX

Jul

2022

N-20220721Functions and Inequalities12 sections

Probability, Conditional Probability, and Substitution Tricks in Inequalities

Contents preview

Random experiments and sample spaces
Classical model and Venn diagrams
Conditional probability
A transition to substitution tricks
A classical symmetric equation
Inequality examples
Jensen, trigonometry, and contest style
The hidden structure

PDF TeX

Jul

2022

N-20220720Combinatorics and Models15 sections

Counting Principles, Binomial Identities, and Mixed Problem Solving

Contents preview

Counting principles
A divisibility example
Permutations and combinations
Binomial theorem
Polynomial coefficients
Further contest examples
Binomial identities by stories
When to use multiplication and when to use addition

PDF TeX

Jul

2022

N-20220719Classical Geometry22 sections

Solid Geometry: Points, Lines, Planes, Parallelism, and Perpendicularity

Contents preview

Why solid geometry feels different
Basic axioms of incidence
Relations between two lines
A line and a plane
Parallel planes and parallel lines
Perpendicularity
Projection theorem
A coordinate viewpoint

PDF TeX

Jul

2022

N-20220718Functions and Inequalities12 sections

Nonlinear Recurrences, Fixed Points, and Metric-Space Exercises

Contents preview

Fixed-point iteration
Möbius recurrences
Example
Second-order linear recurrences
Dedekind cuts and metric exercises
Fixed points of recurrence maps
Monotone bounded sequences
After the examples

PDF TeX

Jul

2022

N-20220717Number Theory Basics13 sections

Recurrences as Orbits: Linear Dynamics, Normalization, and Reversible Transformations

Contents preview

A recurrence generates an orbit
Arithmetic progressions are translation orbits
Geometric progressions are scaling orbits
Affine recurrences become geometric after shifting a fixed point
Variable forcing gives a discrete variation-of-constants formula
Normalization should match the coefficient growth
Second-order recurrences are matrix powers
Generating functions turn shifts into multiplication

PDF TeX

Jul

2022

N-20220716Classical Geometry11 sections

Conic Exercises: Lines, Ellipses, Parabolas, and Tangents

Contents preview

Line and ellipse
Tangents from a point to an ellipse
Line and hyperbola
Parabola tangent through a point
A parabola and a circle
An ellipse with prescribed quadrilateral area
Discriminants distinguish tangency from asymptotic intersection
Tangency via discriminants

PDF TeX

Jul

2022

N-20220715Classical Geometry14 sections

Conics and the Dedekind Construction of Addition

Contents preview

Conic sections
Latus rectum and tangent geometry
Rational and irrational density
Vector spaces
Dedekind cuts and order
Addition of Dedekind cuts
Additive inverses
Conics through focus and directrix

PDF TeX

Jul

2022

N-20220713Sets and Foundations14 sections

Completeness, Supremum, Dedekind Cuts, and Metric Spaces

Contents preview

Large and small positive real numbers
The geometric view of the real line
Supremum property
Nested intervals imply completeness
Dedekind cuts
Metric spaces
Subspaces and density
Supremum as least upper bound

PDF TeX

Jul

2022

N-20220710Sets and Foundations12 sections

Vectors, Ordered Fields, and the First Axioms of Real Numbers

Contents preview

Plane vectors
Field axioms
Ordered fields
Elementary consequences of the order axioms
Nested intervals
Ordered fields and the real line
Vectors as algebraicized geometry
The conceptual payoff

PDF TeX

Jul

2022

N-20220708Classical Geometry17 sections

Directed Angles, Reflections, and the Miquel Point

Contents preview

Reflecting the orthocenter
The incenter–excenter lemma
Miquel point of a triangle
Directed angles reduce case distinctions
Miquel point proof pattern
Further context
Directed angles as projective bookkeeping
Reflections and isogonal structure

PDF TeX

Jun

2022

N-20220628Calculus15 sections

Techniques of Integration: Partial Fractions, Parts, and Substitution

Contents preview

Linearity
Partial fractions
Integration by parts
Substitution
Partial fractions as algebra before calculus
Integration by parts as product rule reversed
A second lens
Partial fractions as algebra of poles

PDF TeX

Jun

2022

N-20220615Classical Geometry19 sections

A Russian-Style Early Olympiad Game: Coordinates, Packing, and Symmetry

Contents preview

The main problem types
Problem 1: two distances under a linear constraint
Problem 2: parabola and distance
Problem 3: packing unit circles
Problem 4: a symmetric Diophantine equation
The larger pattern
Coordinate choice and invariants
Packing and density

PDF TeX

May

2022

N-20220501Multivariable Calculus19 sections

Lagrange Multipliers, Hessians, Bordered Hessians, and Degenerate Extrema

Contents preview

The full problem of constrained extrema
Lagrange multipliers: the basic idea
Example: minimizing distance to a line
Why the method is valid
Hessian matrix and the second derivative test
Sylvester criterion and eigenvalues
Bordered Hessian
Example revisited with a bordered Hessian

PDF TeX

Apr

2022

N-20220417Functions and Inequalities9 sections

Designing Inequality Proofs: Equality Cases, Homogeneity, and Cyclic Structure

Contents preview

Finding the likely answer is only half of a proof
Homogeneity tells us what may be normalized
Cyclic and symmetric expressions carry different information
Equality cases guide the choice of inequality
Denominators often reveal an Engel-form Cauchy step
Smoothing explains why equal variables often extremize
Triangle substitutions encode the boundary conditions
Schur's inequality shows how ordering controls a cyclic sign

PDF TeX

Apr

2022

N-20220412Linear Algebra10 sections

Indices, Dual Maps, Adjoints, Trace, and Singular Values

Contents preview

Indices carry type information before they abbreviate sums
Matrix multiplication is contraction of an output with an input
A change of basis transforms both indices and leaves contractions unchanged
Transpose is the matrix of the dual map
An adjoint is a dual map returned to vectors by an inner product
In nonorthonormal coordinates the Gram matrices enter the adjoint
Rank is the dimension of the image and the size of a minimal factorization
Trace is contraction, and cyclicity is a legal index relabelling

PDF TeX

Apr

2022

N-20220411Linear Algebra18 sections

Invertible Matrices, Adjoints, Row Reduction, and Rotations

Contents preview

Invertibility
Diagonal matrices
Products of inverses
Adjugate formula
Row reduction for inverses
Rotation matrices
Beyond the first calculation
Invertibility as isomorphism

PDF TeX

Apr

2022

N-20220409Calculus11 sections

Composition and Elimination: Higher Derivatives, Determinants, and Schur Complements

Contents preview

Two bookkeeping problems that should not be confused
Repeated derivations produce the higher Leibniz rule
Faà di Bruno records partitions of derivative labels
The fourth derivative as a complete partition calculation
Jets turn higher differentials into truncated Taylor algebra
Parametric differentiation is repeated application of one operator
Determinants inherit a multinomial Leibniz rule
Matrix multiplication is composition with one index hidden

PDF TeX

Apr

2022

N-20220407Linear Algebra12 sections

Exercises on $\mathbb R^n$ , $\mathbb C^n$ , Vector Spaces, and Subspaces

Contents preview

Complex arithmetic
Vector space identities
A non-example with infinity
Subspaces
Bridge to later ideas
How to read basic vector-space exercises
Beyond real and complex scalar structures
Quotient spaces

PDF TeX

Apr

2022

N-20220405Multivariable Calculus10 sections

Fréchet Differentiability, Hessians, and Constrained Tests

Contents preview

Fréchet differentiability and complex differentiability
Fréchet derivative
Visualizing complex maps
Hessian matrices and Taylor expansion
Bordered Hessian
Example
Finite fields and Galois theory
Why complex analysis meets Hessians

PDF TeX

Mar

2022

N-20220323Complex Analysis7 sections

Complex Differentiability, Cauchy–Riemann Equations, and Holomorphic Rigidity

Contents preview

One difference quotient must survive every planar direction
Cauchy–Riemann is sufficient under real differentiability
Wirtinger derivatives isolate the holomorphic part
The Jacobian must be a rotation followed by a scaling
Cauchy's integral formula turns boundary values into derivatives
Holomorphic functions are analytic
Zeros are isolated unless the whole function vanishes

PDF TeX

Mar

2022

N-20220321Complex Analysis10 sections

Motion on the Unit Circle, Roots of Unity, and Finite Fourier Analysis

Contents preview

The differential equation $Z'=iZ$ produces uniform circular motion
The exponential map wraps a line onto a circle
Taking an $n$ -th root means dividing angles and choosing a branch
The roots of unity form a finite cyclic group
Polynomial divisibility becomes evaluation on a finite subgroup
Character orthogonality filters one congruence class
The discrete Fourier transform is the full set of character coordinates
Fourier characters diagonalize every circulant operator

PDF TeX

Mar

2022

N-20220320Calculus14 sections

A Review of Calculus: Mean Value Theorems and Continuity

Contents preview

From the mean value theorem to Cauchy's form
Continuity at a point
How this note fits the previous one
Rolle's theorem as the engine
Cauchy's theorem and L'Hopital's rule
Uniform continuity
Further direction
Mean value theorems as compactness plus local linearity

PDF TeX

Mar

2022

N-20220319Calculus12 sections

A Review of Calculus: Sequences, Limits, Derivatives, and Integrals

Contents preview

Sequences and convergence
Limits of functions
Derivatives
Small-o and big-O
Integration
Why the epsilon definition is shaped this way
Continuity by sequences
Derivative as best linear approximation

PDF TeX

Mar

2022

N-20220315Calculus11 sections

Quadratic Parameters, Tangency, and Constrained Extrema

Contents preview

A scattered algebraic note
A quadratic with parameters
A parabola and a hyperbola
A constrained extremum
Tangency as a double-root condition
Completing the square
A wider frame
Tangency as multiplicity

PDF TeX

Feb

2022

N-20220212Curves and Structures16 sections

Riemann Surfaces as One-Dimensional Complex Manifolds

Contents preview

Structure beyond topology
From smooth structure to complex structure
Definition of a Riemann surface
Examples
Holomorphic maps between Riemann surfaces
Meromorphic functions as maps to the sphere
Degree and multiplicity
Local power form is not global

PDF TeX

Feb

2022

N-20220205Sets and Foundations12 sections

A First Comparison of ZFC, NBG, KM, and New Foundations

Contents preview

Why there are several axiomatizations of set theory
The main systems
ZFC
NBG
Kelley–Morse set theory
New Foundations
How to compare the systems
What remains unclear

PDF TeX

Jan

2022

N-20220131Sets and Foundations12 sections

Rethinking Set Theory Through Functions Rather Than Membership

Contents preview

Why this translation/note matters
The discomfort with ZFC
Three misunderstandings about ETCS
Elements as special functions
The informal ETCS-style principles
Subsets via characteristic functions
Products, exponentials, and inverse images
Choice as right inverses

PDF TeX

Jan

2022

N-20220123Arithmetic Geometry17 sections

From Algebraic Number Theory to the First Shadow of Langlands

Contents preview

Why algebraic number theory comes first
Number fields and algebraic integers
Ideals and factorization
Quadratic fields and class number
First understanding of Langlands
Local and global correspondences
Cross-field intuitions
Numerical experiments

PDF TeX

2021

Oct

2021

N-20211007Arithmetic Geometry11 sections

From Diophantine Inequalities to Arithmetic Deformation

Contents preview

The question is an inequality, not a slogan
A first chain of rigidity results
From Mordell finiteness to effective inequalities
The Szpiro inequality as a concrete target
Vojta's inequality and the hyperbolic curve viewpoint
Arithmetic fundamental groups
The local elliptic curve model: Tate curves and q-parameters
Normalised degree as log-volume

PDF TeX

Aug

2021

N-20210806Topology and Homotopy9 sections

Group Actions, Homotopy, and a Rough Road to the Hopf Fibration

Contents preview

Purpose of the note
Group actions
Homotopy
Cofibrations, fibrations, and deformation retracts
Spheres and homotopy groups
The Hopf fibration
Further reading and visual models
Group actions as symmetry bookkeeping

PDF TeX

Aug

2021

N-20210804Categories and Logic11 sections

Boolean Algebras, Boolean Rings, and the First Hint of Stone Duality

Contents preview

Boolean algebra beyond computing
A first definition via Heyting algebras
A small amount of algebra
Boolean rings
Examples of Boolean rings
Boolean algebras directly
Moving between the two structures
The logical shadow

PDF TeX

Jul

2021

N-20210721Categories and Logic11 sections

A Gateway to Dualities: Space, Algebra, and the Dualizing Object

Contents preview

Starting point
The table is not a list
The dualizing object
Stone duality as the first complete example
What Stone duality says in one sentence
Gelfand duality as a continuous analogue
Affine schemes and the affine line
Locales and the Sierpi'nski space

PDF TeX

Jun

2021

N-20210621Linear Algebra14 sections

Prop Presentations for Linear Algebra: Diagrams, Matrices, and Relations

Contents preview

First orientation
Props as typed wiring diagrams
The alphabet: copy, discard, add, zero, and scalars
Two colours: a comonoid and a monoid
The bialgebra law
Scalars as ring operations
How a matrix becomes a diagram
Completeness: a proof system for matrices

PDF TeX

Feb

2021

N-20210207Sets and Foundations8 sections

Choice, Well-Ordering, and the Arithmetic of Cardinal Size

Contents preview

Choice enters the story
Partially ordered sets
Well-ordering
Equipollence and cardinal numbers
Cantor's theorem
Comparing cardinals
Infinite cardinals and finite disturbance
Concrete encodings behind the arithmetic

PDF TeX

Feb

2021

N-20210206Sets and Foundations6 sections

Relations, Products, and the First Appearance of Universal Properties

Contents preview

Relations and equivalence classes
Partitions
Indexed products and projections
The first universal property
Integers and induction
Division, gcd, and congruence

PDF TeX

Feb

2021

N-20210204Sets and Foundations7 sections

Sets, Classes, and the First Warnings About Size

Contents preview

Why classes enter before functions
Russell's class as the first warning
Power sets and indexed families
Set identities
Images, inverse images, and functions
One-sided inverses
Set-level bookkeeping behind functions

PDF TeX

2026

The Heat Equation, Diffusion, and Similarity Solutions

The Second-Order Wave Equation and d'Alembert's Formula

First-Order Partial Differential Equations and Characteristics

First-Order and Higher-Order Differential Equations

Fourier Series and Orthogonal Expansion of Functions

Numerical Series, Power Series, Taylor Series, and Improper Integrals

Surface Integrals, Gauss Theorem, and Stokes Theorem

Solving Linear Systems: Elimination, Iteration, and Energy

Generalized Inverses, Projections, and Least-Squares Geometry

Matrix Decompositions: LU, Cholesky, Rank Factorization, and SVD

Green's Theorem, Conservative Fields, and the Planar Stokes Principle

Line Integrals and Surface Integrals of the First and Second Kind

Volumes, Spatial Regions, and Order of Integration

Triple Integrals: Solids, Cylindrical Coordinates, and Spherical Coordinates

Computing Double Integrals: Slicing, Polar Coordinates, and Change of Variables

Double Integrals: Riemann Sums, Area, Symmetry, and Average Value

Extrema of Two-Variable Functions and the Hessian Test

Matrix Computation III: Canonical Forms, Spectral Localization, and Eigenvalue Iteration

A Problem-Solving Guide for Multivariable Limits and Local Geometry

Multivariable Chain Rule, Total Differentials, and Implicit Differentiation

Matrix Computation II: Spectral Radius, Stability, and Conditioning

Matrix Computation I: Norms, Amplification, and Numerical Size

Multivariable Limits, Continuity, Differentiability, and the Total Derivative

Orthogonal Projection, Least Squares, and the Geometry of Approximation

Point-Set Language in Rn\mathbb R^nRn: Neighborhoods, Boundaries, and Compactness

Two-Dimensional Linear Maps, Spectral Geometry, and Singular Value Decomposition

Pure Mathematics as a Language for Machine Learning and Data Analysis

Calculus Review III: Convexity, Integral Inequalities, Wallis Formula, and More

Spectral Calculus and Positive Matrix Geometry: From Basis Changes to Lyapunov Equations

Rank, Projections, and Long-Term Matrix Dynamics

2025

A Nontrivial Loop at Zero: FinSet K-Theory and Stable Homotopy

Calculus Review II: Monotonicity, Integration Methods, Applications, and More

Inner-Product Geometry and Spectral Decomposition: From QR to Quadratic Forms

Self-Avoiding Walks III: Lace Expansion, Functional Integrals, and the Critical Dimension Four

Self-Avoiding Walks II: The Hexagonal Lattice, Holomorphic Observables, and the Exact Connective Constant

Self-Avoiding Walks I: Connective Constants, Bridges, Polygons, and Critical Exponents

Teichmueller Theory Without One Universe: Classical Deformation, Hodge Theaters, and Identification

When Addition Starts Floating: Theta-Links, Log-Links, and Arithmetic Deformation

Calculus Review I: Limits, Completeness, Asymptotic Scales, and Local Approximation

Determinants and Rank: From Alternating Volume to Elimination and Rank Strata

Kernels, Cokernels, Images, and Coimages in Abelian Categories

Notes on MML: Vectors, the Minus-One Trick, Linear Maps, Basis Change, Kernel, and Image

The Reparameterization Trick and Auto-Encoding Variational Bayes

Variational Inference: KL Divergence, Jensen, and the ELBO

Latent Variable Models: Priors, Decoders, and Intractable Posteriors

Probability Language for Variational Autoencoders

Backpropagation in One MLP Layer: From a Three-Dimensional Tensor to an Outer Product

Nikonov's Paper at One Crossing: Region Probes, Partial Tribrackets, and What the Seminar Was About

Nikonov's Seminar Prelude: Knots, Diagrams, Probes, and the Space Between the Lines

Projective Schemes and Gluing: Building Projective Space from Affine Windows

Transformations of Conics and the Erlangen View of Geometry

Schemes Gently: A Guided Tour of Spectrum, Nilpotents, and Local Rings

2024

Zariski Geometry: When Functions Become the Space

Divisors on Curves: The Bookkeeping of Zeros, Poles, and Functions

Hopf Fibration, Homotopy, and the Infinite-Stability Reading

Plane Curves as Algebraic Creatures: Equations, Rings, and Singularities

Projective Geometry as the Art of Removing Exceptions

Differential Forms and Stokes: The Geometry of Boundary Measurements

Local Linear Geometry: A Mapmaker's View of Derivatives and Tangent Spaces

From Extremal Triangles to Euler Characteristic and Degree

Combinatorics: Identities, Generating Functions, Catalan Numbers, and More

Topos Theory: Categorical Preliminaries, Internal Logic, and the Geometry of Sets

2023

From Self-Duality to Spectral Curves: A Computable Route through Hitchin's Equations

Foundations of Mathematics: Set Theory, Category Theory, and Type Theory

Fixed Points, Convergent Iteration, and Compactness Methods

Free Constructions, Adjunctions, Concrete Categories, and Group Actions

Category Theory Continued: Posets, Groups as One-Object Categories, Products, and Monomorphisms

From Oscillation to Stable Limits: Riemann and Lebesgue Integration

Derivatives, Tangents, Extremal Problems, Young–Holder–Minkowski, and Jensen

Completed Decimals, Hyperfinite Truncations, and Nonstandard Analysis

Trigonometric Systems, Multiple-Angle Formulas, and a Determinant Recurrence

Special Proofs of Trigonometric Addition Formulas

Tensors in Nonorthogonal Coordinates: Dual Bases, Metrics, and Contraction

Linear Maps, Matrix Representations, and Tensor Invariance

Vector Identities from Exterior Algebra: Wedge, Contraction, and the Hodge Star

Newton's Shell Theorem: Solid Angle, Potential, and Gauss Law

Point-Set Language in $\mathbb R^n$ : Neighborhoods, Boundaries, and Compactness

Exercises on $\mathbb R^n$ , $\mathbb C^n$ , Vector Spaces, and Subspaces