Engineering

Gram–Schmidt Orthogonalisation

The Gram-Schmidt procedure, the mu coefficients and their role in reduction conditions, the numerical instability of the classical algorithm, and exact integral alternatives.

Group Extensions and H2

Group extensions with abelian kernel, factor sets and their equivalence, the classification by H^2, central extensions, and the obstruction interpretation.

H2, Hopf's Formula and the Schur Multiplier

Hopf's formula for H_2 in terms of a free presentation, the Schur multiplier, universal central extensions, and the connection to the lower central series.

Hensel Lifting and Factorisation over the Integers

Hensel's lemma and quadratic lifting, Mignotte's coefficient bounds, the exponential recombination problem, and the LLL-based polynomial-time solution.

Hilbert's Syzygy Theorem

Hilbert's chain-of-syzygies theorem, the Koszul resolution, global dimension of polynomial rings, and the historical and modern significance.

Homological Algebra and Algebraic Topology

The topological origins of homological algebra, singular and cellular chains, the Eilenberg-Zilber and Kunneth theorems in topology, classifying spaces, and the dictionary betwe…

Homological Algebra: Discipline Overview

Structural overview of homological algebra: the failure of exactness, resolutions and derived functors, the central invariants Ext and Tor, and how the machinery specialises to …

Homology of Small Categories

Homology of a small category with coefficients in a functor, the nerve and its geometric realisation, comparison with group homology, and the associated spectral sequences.

Homomorphisms and the Isomorphism Theorems

Homomorphisms, kernels, the first, second and third isomorphism theorems in arbitrary type, the correspondence theorem, and the congruence extension property.

Identities and Birkhoff's HSP Theorem

Identities, satisfaction, the operators M and Id as a Galois connection, and Birkhoff's HSP theorem identifying varieties with equationally definable classes, with proof.

Injective Modules and Dualization

Injective modules, the extension property, Baer's criterion, divisible groups, and the existence of enough injectives.

Injective Modules over a Principal Ideal Domain

Injective modules over a PID characterised by divisibility, the classification of injective abelian groups, and the resulting short injective resolutions.

Integer Square Roots and Perfect Power Detection

Computing exact integer square roots by Newton's method, fast rejection of non-squares by modular filters, and detection of perfect powers and prime powers.

Kan Extensions

Left and right Kan extensions, their construction by colimits and limits, the adjunction characterisation, and their role in homology of small categories.

Lattice Homomorphisms, Isomorphisms and Sublattices

Lattice homomorphisms, isomorphisms and sublattices: why every lattice homomorphism is monotone but not conversely, how to recognise a genuine sublattice, and the role of embedd…

Lattices and Quadratic Forms

Lattices as discrete subgroups, bases and unimodular change of basis, the Gram matrix and determinant, successive minima, and the correspondence with positive definite quadratic…

Legendre, Jacobi and Kronecker Symbols

Quadratic residues, Euler's criterion, the Legendre symbol, its Jacobi and Kronecker extensions, and the reciprocity-based algorithm that evaluates them in logarithmic time.

Lie Algebra Extensions and H2

Extensions of Lie algebras with abelian kernel, the classification by H^2, central extensions, and examples including the Heisenberg and affine algebras.

Lie Algebras and the Universal Enveloping Algebra

Lie algebras, representations as modules, the universal enveloping algebra and its universal property, the Poincare-Birkhoff-Witt theorem, and the augmentation ideal.

Linear Algebra Algorithms over Fields and Rings

Exact Gaussian elimination, fraction-free elimination, determinant and characteristic polynomial algorithms, and kernel and image computation over fields and over ℤ.

Engineering Mathematics13 Aug 2026 · 3 min read

Linearly Generated Sequences and Sparse Linear Systems

Linearly generated sequences and their minimal polynomials, the Berlekamp–Massey and extended-Euclidean methods of computing them, and Wiedemann's algorithm for solving huge spa…

Low-Dimensional Group Cohomology

Explicit descriptions of H^0, H^1 and H_1 for group cohomology, with trivial and non-trivial coefficients, and their interpretations.

Mal'cev Conditions I: Congruence Permutability

Mal'cev conditions in general, the ternary Mal'cev term characterising congruence permutability, and why term conditions are the right way to classify varieties.

Mal'cev Conditions II: Congruence Distributivity and Jonsson Terms

Congruence-distributive varieties, the Jónsson term characterisation, Jónsson's lemma, and congruence modularity with Day terms.

Engineering Mathematics13 Aug 2026 · 6 min read

Modular Exponentiation and Powering Algorithms

Left-to-right and right-to-left binary powering, sliding window exponentiation, addition chains, and the generic monoid formulation that makes one routine serve many algebraic s…

Modular Representation Theory

Representation theory in characteristic dividing the group order, failure of semisimplicity, relative projectivity, complexity and support varieties, and the Quillen stratificat…

Modules and Module Homomorphisms

Modules over a ring, submodules and quotients, module homomorphisms, exact sequences, the short five lemma and the snake lemma.

Modules, Vector Spaces and Matrices

Modules over a ring, vector spaces and dimension, linear independence and bases, matrices as linear maps, rank and kernel, and Gaussian elimination over a field with its algorit…

Multiprecision Integer Arithmetic

Representation of multiprecision integers, schoolbook and fast multiplication, division, modular reduction strategies, and how to choose the right base ring for a computation.

Natural Transformations

Natural transformations and naturality squares, natural isomorphisms, functor categories, and the role of naturality in the long exact sequences.

Nilpotent Groups and Homology

The lower central series, homological characterisation of nilpotent quotients, the Stallings and Stammbach theorems, and applications to group presentations.

Numerical Tables: KEVOS Sourcing Policy

The KEVOS two-layer knowledge architecture applied to number theory: durable method in the vault, numeric catalogue data sourced from current authoritative databases such as LMF…

Orders and Ideals in Number Fields

Orders, the maximal order, fractional and integral ideals, unique factorisation of ideals in a Dedekind domain, ideal arithmetic by Hermite normal form, and the two-element repr…

Engineering Mathematics13 Aug 2026 · 6 min read

Orthogonal Latin Squares and the Refutation of Euler's Conjecture

Orthogonal Latin squares, Euler's officer problem, the refutation by Bose, Shrikhande and Parker, and the algebraic reading of orthogonality as a condition on a pair of quasigro…

Pollard's p−1 Method and Its Relatives

The p−1 method with its two stages, smoothness assumptions, the p+1 method using Lucas sequences, and the implications for choosing cryptographic primes.

Engineering Mathematics13 Aug 2026 · 7 min read

Polynomial Arithmetic and Applications

Basic and fast polynomial arithmetic, the Euclidean algorithm over F[X], modular inverses and Chinese remaindering for polynomials, evaluation and interpolation, rational functi…

Polynomial Arithmetic and GCD in Unique Factorisation Domains

Dense and sparse polynomial representation, multiplication algorithms, pseudo-division over a UFD, primitive parts and content, and the subresultant and modular remedies for coe…

Engineering Mathematics13 Aug 2026 · 7 min read

Polynomial Rings and Unique Factorization

The ring F[X] as a Euclidean domain, division with remainder, roots and degree bounds, irreducibility, unique factorization, Gauss's lemma for ℤ[X], and polynomial congruences a…

Posets and the Two Definitions of a Lattice

The order-theoretic and algebraic definitions of a lattice, the equivalence between them, and why universal algebra needs the algebraic form: only an equationally defined class …

Preliminaries: Sets, Classes, Relations and Notation

Sets, proper classes, relations, kernels and the notational conventions universal algebra shares with model theory: why the collection of all groups cannot be a set, how relatio…

Preservation Theorems: Horn, Universal and Positive Sentences

The preservation theorems: Łoś–Tarski for substructures, Lyndon for homomorphisms, Keisler–Galvin for reduced products, Chang–Łoś–Suszko for unions of chains, and their relation…

Primal Algebras

Primal algebras, Foster's theorem that they generate varieties equivalent to Boolean algebras, the representation of every member as a Boolean power, and the relationship to fun…

Primality Testing versus Factoring

The separation between primality testing and factoring, compositeness tests versus primality proofs, probable primes, pseudoprimes, and how to select a testing strategy.

Principal Congruence Formulas

Principal congruence formulas, Mal'cev's chain characterisation, when the chains are uniformly bounded, and the application to bounding the size of subdirectly irreducible algeb…

Engineering Mathematics13 Aug 2026 · 7 min read

Probabilistic Algorithms

Models of randomised computation, one-sided and two-sided error, error reduction by repetition, generating random numbers in a range without bias, rejection sampling, and genera…

Probabilistic Primality Testing

Trial division, the Fermat test and its failure on Carmichael numbers, the Miller–Rabin test with its 1/4 error bound, practical prime generation, perfect power testing and the …

Products, Coproducts and Universal Constructions

Universal properties, products and coproducts in a general category, initial and terminal objects, and uniqueness up to unique isomorphism.

Projective and Injective Resolutions

Construction of projective and injective resolutions, the horseshoe lemma, projective and injective dimension, and global dimension of a ring.