Engineering Mathematics

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.

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…

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…

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.

Projective Classes of Epimorphisms

Projective classes, relative projectives, allowable epimorphisms, relative exactness, and the examples that motivate relative homological algebra.

Projective Modules over a Principal Ideal Domain

Structure of projective and finitely generated modules over a PID, the structure theorem, torsion and free parts, and the resulting short projective resolutions.

Pullbacks, Pushouts and General Limits

Pullbacks and pushouts, their construction in module categories, limits and colimits over a diagram, and exactness of filtered colimits.

Quadratic Fields and Binary Quadratic Forms

Quadratic fields, their discriminants and integral bases, prime decomposition by the Kronecker symbol, binary quadratic forms, reduction, and the correspondence between form cla…

Quadratic Residues and Quadratic Reciprocity

Quadratic residues modulo a prime and a composite, Euler's criterion, the Legendre and Jacobi symbols, the law of quadratic reciprocity and its supplements, and the quadratic re…

Quasigroups, Loops and Latin Squares

Quasigroups and loops as equationally defined algebras, the correspondence with Latin squares, why division must be in the type for quasigroups to form a variety, and the congru…

Quasiprimal Algebras and Pixley's Theorem

Quasiprimal algebras, Pixley's characterisation via inner isomorphisms, the relationship to primality, and the varieties quasiprimal algebras generate.

Real Quadratic Fields and the Infrastructure Method

Regulators and fundamental units of real quadratic fields, the continued fraction algorithm, the exponential cost problem, and Shanks's infrastructure with the distance function.

Recent Developments: the 1981 Frontier

The source's Recent Developments chapter: the commutator programme, the classification of varieties by congruence conditions, decidability questions, Boolean constructions, stru…

Reduced Products and Filtered Products

Reduced products over a filter, the congruence they induce, the direct product and ultraproduct as extreme cases, and the Horn sentence preservation theorem.

Relative Derived Functors

Construction of relative derived functors from relative resolutions, their properties, the comparison with absolute derived functors, and worked examples.

Representing Algebraic Numbers

The standard, matrix, conjugate-vector and minimal-polynomial representations of an algebraic number, their relative costs, and how to convert between them.

Resolutions for Group Cohomology

The bar resolution and its normalised version, cocycles and coboundaries in explicit terms, resolutions from presentations, and the trade-off between explicitness and size.

Rings, Ideals and Quotient Rings

Commutative rings with unity, units and zero divisors, integral domains and fields, ideals and quotient rings, ring homomorphisms and the isomorphism theorems, with the CRT as a…

Root Finding over the Complex Numbers

Numerical root finding for polynomials over C: conditioning, the Newton and Aberth methods, splitting-circle approaches, and the precision management needed to support exact num…

Satellites

Left and right satellites of an additive functor, their construction, the connection with derived functors, and their use when the category lacks enough projectives.

Satisfaction, Truth and Elementary Equivalence

The satisfaction relation defined by recursion, truth for sentences, theories and models, elementary equivalence, and why elementarily equivalent structures can be non-isomorphic.

Semantic Embeddings and Undecidability

Semantic embeddings as a technique for transferring undecidability, the interpretation of graphs and semigroups into varieties, the decidability results that stand in contrast, …

Semisimple Lie Algebras and the Whitehead Lemmas

The Killing form and Cartan's criterion, the Casimir element, the two Whitehead lemmas, and their consequences including Weyl's complete reducibility theorem and Levi's theorem.

Semisimple Varieties

Semisimple varieties, the relationship to discriminator varieties, residual smallness, and the structure available when subdirect decomposition terminates in simple algebras.