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Contents
01Semantic Segmentation: Resolving a Domain and Typing its Pieces02A High School Student's Introduction to Physics03Linear Algebra I: Vector Spaces and Linear Maps04Quaternions Are the Rotors of Space05Mathematical Pharmacology
Lead Article
No. 001
Geometry

Semantic Segmentation: Resolving a Domain and Typing its Pieces

Every segmentation method resolves a domain into pieces and then assigns each piece a type. Reading the field through those two operations puts pixels, point clouds, meshes and shape spaces on one axis, recovers the primal-dual mesh pair on which discrete exterior calculus is built, and gives a precise reason why a message-passing network may place its dual vertices where a discrete Hodge star may not.

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PhysicsNo. 002

A High School Student's Introduction to Physics

Physics built from arithmetic, assuming nothing beyond addition and subtraction. The number systems and the equals sign, then units, then the affine space of positions, then the geometric product forced by the demand that a vector square to its own length. The calculus is constructed on the way, with the limit stated in full and the derivative defined by Caratheodory's slope function. In the geometric algebra of the plane: the kinematic formulas derived as a corollary of integrating a constant, centripetal acceleration by differentiating a rotor twice, the harmonic oscillator's cosine proved rather than announced, angular momentum as a conserved bivector with Kepler's equal areas as its consequence, and work and energy as the fundamental theorem along a path.

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AlgebraNo. 003

Linear Algebra I: Vector Spaces and Linear Maps

A proof-based reconstruction of linear algebra from the axioms: vector spaces, bases and dimension by Steinitz exchange, linear maps and rank-nullity, matrices as coordinate representations, the four fundamental subspaces, and the affine solution set of a linear system. The application is stoichiometry: balancing a reaction is a search for integer vectors in a null space, conservation laws are a left null space, and an ICE table is an affine set cut by a nonlinear equilibrium variety. Part I of a six-part lecture series converging on the singular value decomposition and geometric algebra.

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AlgebraNo. 004

Quaternions Are the Rotors of Space

Hamilton's quaternions reconstructed on their own terms and then recognised as the even subalgebra of the geometric algebra of space: the imaginaries i, j, k are the basis bivectors, a unit quaternion is a rotor, and the half-angle, the two-to-one cover, and the absence of gimbal lock become plain facts about rotors. A companion to the classical-mechanics article, closing with the native rotation expressions this settles in code.

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MedicineNo. 005

Mathematical Pharmacology

Pharmacokinetics, pharmacodynamics, and quantitative systems pharmacology reconstructed as one state-space object: controllability, observability, BIBO stability, and Metzler positivity built once at arbitrary compartment count, specialised to the reversible/dissipative (GENERIC) structure of compartmental kinetics, generalised to tensor QSP networks, and closed by a computable answer to when adding a compartment buys real structure versus an unidentifiable direction.

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AlgebraNo. 006

Classical Mechanics from Zero, in Two Languages

Classical mechanics constructed from nothing but an inertial frame, in matrix and linear algebra and in geometric algebra side by side: rotations and rigid-body dynamics without Euler angles, one vector derivative replacing grad, div, and curl, and the Lagrangian and Hamiltonian formalisms in both languages.

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AnalysisNo. 007

Configuration-Space Curvature and the Navier–Stokes Singular Set

The curvature of SDiff⁡(T3)\operatorname{SDiff}(\mathbb{T}^3) is the pressure Hessian. Read against the Caffarelli–Kohn–Nirenberg theory, this places the Navier–Stokes singular set where curvature concentrates or the strain–enstrophy imbalance q=∣S∣2−12∣ω∣2q = |S|^2 - \tfrac{1}{2}|\omega|^2 vanishes. In the scale-critical L3L^3 class the balanced regime is excluded; beyond it the Liouville problem for bounded ancient pressureless flows remains open.

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SystemsNo. 008

Causal Models of Concurrency

From labelled transition systems and bisimulation through Mazurkiewicz traces and event structures to happens-before, logical clocks, and the observability coefficient: a mathematical account of what it means for a concurrent system to be causally intelligible.

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SystemsNo. 009

An FFT for every GPU: ferrum-gpu and gpufft

Two Python packages, one idea: GPU FFTs that don't care whose GPU you own. gpufft wraps cuFFT and VkFFT for cross-vendor transforms today (NVIDIA, AMD, Intel, Apple); ferrum-gpu writes the kernels in pure Rust, compiled to PTX by cuda-oxide, within 1.3-3.7× of cuFFT. Both on PyPI.

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PhysicsNo. 010

Active kk-atic Fluids on Riemannian 3-Manifolds

A covariant geometric-calculus formulation of active kk-atic hydrodynamics on Riemannian 3-manifolds: one SU(2)/H^\mathrm{SU}(2)/\hat H order-parameter family, and a non-equilibrium variational principle selecting braided and knotted disclination structures.

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PhysicsNo. 011

Interstellar: a brane-bulk reading

A physics reading of Nolan's Interstellar (2014) in which Kerr geometry, Randall-Sundrum II brane-bulk coupling, and M-theory singularity resolution collapse the film's apparent plot holes into consequences of one maintained parameter: Gargantua's spin.

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GeometryNo. 012

From RVE to Mesh: A Pipeline for Heterogeneous Continua

A single pipeline from microstructure to discrete solver: mean-field homogenisation on a representative volume element produces an SPD(3)\mathrm{SPD}(3)-valued permeability tensor field Keff(x)K^{\mathrm{eff}}(x), which induces a Riemannian metric g=(Keff)−1g = (K^{\mathrm{eff}})^{-1}, whose Hodge star discretises the Laplace-Beltrami operator, and whose scalar curvature R(g)R(g) drives adaptive remeshing.

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AnalysisNo. 013

Analysis on Manifolds V: Stokes’ Theorem

The generalised Stokes theorem ∫Mdω=∫∂Mω\int_M \mathrm{d}\omega = \int_{\partial M} \omega proved in full, recovering FTC, Green’s theorem, the divergence theorem, and classical Stokes as special cases. Hodge decomposition Ωk=im d⊕Hk⊕im d∗\Omega^k = \mathrm{im}\,\mathrm{d} \oplus \mathcal{H}^k \oplus \mathrm{im}\,\mathrm{d}^* with complete Sobolev proof. Harmonic representatives, Betti numbers, and the de Rham isomorphism HdRk(M)≅Hom(Hk(M;Z),R)H^k_{\mathrm{dR}}(M) \cong \mathrm{Hom}(H_k(M;\mathbb{Z}),\mathbb{R}). The conclusion of a five-part lecture series.

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AnalysisNo. 014

Analysis on Manifolds IV: Integration

Forms as antiderivatives, oriented manifolds, manifolds with boundary, partitions of unity, integration of kk-forms over kk-submanifolds, the change-of-variables theorem via pullback, the Riemannian volume form, period integrals, the Mayer-Vietoris sequence, and the full de Rham cohomology H∗(M)H^*(M) of spheres, tori, and surfaces. Part IV of a five-part lecture series on differential forms and the generalised Stokes theorem.

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AnalysisNo. 015

Analysis on Manifolds III: Differential Forms

Smooth manifolds, tangent and cotangent spaces, differential forms as smooth sections of Λk(T∗M)\Lambda^k(T^*M), the exterior derivative, pullback along smooth maps, and the recovery of grad, curl, and div as the exterior derivative in R3\mathbb{R}^3. Part III of a five-part lecture series on differential forms and the generalised Stokes theorem.

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AnalysisNo. 016

Analysis on Manifolds II: Exterior Algebra

The algebraic machinery behind differential forms: dual spaces, multilinear alternating maps, the wedge product, bases and dimension of Λk(V∗)\Lambda^k(V^*), determinants as top forms, the interior product, and the Hodge star. Part II of a five-part lecture series on differential forms and the generalised Stokes theorem.

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ProbabilityNo. 017

Probability and Statistics: A Geometric Foundation

A measure-theoretic construction of probability and statistics, from sigma-algebras through estimation theory and hypothesis testing to the Riemannian geometry of statistical manifolds.

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GeometryNo. 018

Numerical Analysis via Discrete Exterior Calculus

A self-contained reconstruction of numerical analysis through discrete exterior calculus: simplicial complexes, cochains, the discrete Hodge star, and the Hodge Laplacian, applied to quantum mechanics, computational electromagnetics, and fluid dynamics.

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FinanceNo. 019

Accounting in Quantitative Finance and Algorithmic Trading

A graduate-level bridge from double-entry bookkeeping to P&L attribution, the Greeks, risk measures, and tax lot accounting for algorithmic trading.

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AnalysisNo. 020

Analysis on Manifolds I: Analysis on Rn\mathbb{R}^n

A self-contained, proof-based reconstruction of single and multivariable analysis from first principles: topology of Rn\mathbb{R}^n, the derivative as a linear map, the chain rule, and the inverse and implicit function theorems. Part I of a five-part lecture series on differential forms and the generalised Stokes theorem.

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FinanceNo. 021

Performance Measurement Under Uncertainty

A measure-theoretic construction of risk-adjusted return: Sharpe, Sortino, Calmar, Omega, and Rachev as functionals on the space of return distributions, with four novel theorems and empirical verification on live backtest data.

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ProbabilityNo. 022

The Kelly Criterion from Shannon Information Theory

A rigorous derivation of Kelly's growth-rate-optimal betting strategy from Shannon's mutual information, with application to binary prediction markets.

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PhysicsNo. 023

Reservoir Geometry: Riemannian Manifolds in Oil and Gas

Darcy's law recast as geodesic flow on a Riemannian manifold (R,g)(\mathcal{R}, g), pressure diffusion as the Laplace–Beltrami equation, and permeability tensor interpolation via SPD(3)\mathrm{SPD}(3) geodesics, with no_std\texttt{no\_std} Rust for embedded well-site monitoring.

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SystemsNo. 024

Rust, from first principles

Types, ownership, operational semantics, async, and the FFI bridge: a graduate-level treatment of Rust from mathematical first principles.

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PhysicsNo. 025

Maxwell's Equations and Gauge Theory: Electromagnetism as a Principal Bundle

Four languages for one theory: vector calculus, differential forms, spacetime algebra, and principal fiber bundles. From the classical field equations to gauge invariance, the Aharonov-Bohm effect, Yang-Mills theory, and Dirac monopoles.

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AnalysisNo. 026

Navier–Stokes: Derivation in R3\mathbb{R}^3 and on a Riemannian Manifold

An end-to-end derivation of the incompressible Navier–Stokes equations from continuum mechanics axioms, geometric reformulation via differential forms, coordinate-free lift to a Riemannian manifold, the Millennium Prize problem, functional analysis, and geometric algebra.

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FinanceNo. 027

Price as Geometry: Resolution, Coarse-Graining, and the Structure of Market Noise

A rigorous tour through stationary and non-stationary models of price evolution, with geometric analysis at the forefront. From the random walk null and Black-Scholes as flat geometry, through mean reversion as curved Riemannian diffusion, wavelets, geometric harmonics, and information geometry, anchored throughout by empirical evidence from BTC/ETH millisecond data.

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AnalysisNo. 028

Diffusion on Curved Spaces

From the Gaussian heat kernel on Rn\mathbb{R}^n to the Laplace-Beltrami operator on Riemannian manifolds, with the short-time heat kernel expansion and spectral theory.

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GeometryNo. 029

Manifolds: The Language of Modern Geometry

A rigorous construction of smooth manifolds from first principles: charts, tangent spaces, Riemannian metrics, curvature tensors, and geometric flows.

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ProbabilityNo. 030

Kuramoto: How Order Emerges from Chaos

Fireflies, neurons, power grids: all governed by the same equation. A tour through the Kuramoto model, its order parameter, and the phase transition that turns noise into rhythm.

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FinanceNo. 031

BTC/ETH Lead-Lag: Resolution-Dependent Direction Reversal on Binance Spot

Resolution-dependent direction reversal in BTC/ETH lead-lag on Binance spot: ETH leads at 1ms, BTC leads at 100ms, crossover at 15–20ms. January and full year 2025.

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Activity
  1. newSemantic Segmentation: Resolving a Domain and Typing its Pieces
  2. newA High School Student's Introduction to Physics
  3. newLinear Algebra I: Vector Spaces and Linear Maps

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31 Articles · Vol. I