Engineering

Filtered Differential Objects

Filtrations of chain complexes, the associated graded object, the spectral sequence of a filtered complex, and the interpretation of the pages.

Engineering Mathematics13 Aug 2026 · 11 min read

Filters, Ideals and Ultrafilters

Filters and ideals on a Boolean algebra, principal and free filters, ultrafilters and their three equivalent characterisations, and the role of ultrafilters throughout the rest …

Engineering Mathematics13 Aug 2026 · 7 min read

Finite Fields: Existence, Uniqueness and Structure

Why finite fields have prime-power order, how they are constructed and why they are unique, the subfield lattice, conjugates, norms and traces, and the Frobenius automorphism.

Engineering Mathematics13 Aug 2026 · 11 min read

Finite State Acceptors and Recognisable Languages

Finite state acceptors as unary algebras, recognisable languages, the Myhill–Nerode congruence, the minimal automaton as a quotient, and the closure properties that follow.

Finiteness Conditions on Groups

Cohomological dimension of groups, the Stallings-Swan theorem, finiteness conditions FP_n and F_n, duality groups, and the role of torsion.

First-order Languages and Structures

First-order languages with relation and operation symbols, structures as the semantic objects, the syntax of terms and formulas, free and bound variables, and how algebras sit i…

Engineering Mathematics13 Aug 2026 · 11 min read

Free Algebras and the Universal Mapping Property

Free algebras over a class, their construction as quotients of the term algebra, the universal mapping property, and the fact that F_K(X) lies in the variety generated by K.

Free and Projective Modules

Free modules and bases, projective modules and the lifting property, the equivalence with direct summands of free modules, and projective resolutions.

Fully Invariant Congruences and Equational Theories

Equational theories as fully invariant congruences on the term algebra, the dual isomorphism with the lattice of varieties, and equational bases.

Functionally Complete Algebras

Functionally complete algebras, the polynomial clone, the characterisation via simplicity and the discriminator as a polynomial, and the contrast with primality.

Functors and Their Exactness Properties

Covariant and contravariant functors, composition, additive functors, faithful and full functors, and the exactness hierarchy that governs derived functors.

Generators and Discrete Logarithms in ℤ*p

Finding a generator of ℤ*p, generic discrete logarithm algorithms including baby-step giant-step and Pollard rho, the Pohlig–Hellman reduction, and the Diffie–Hellman key establ…

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.