Engineering Mathematics

Applications of the Künneth Formulas

Applications of the Kunneth and universal coefficient theorems to products of spaces, group cohomology of direct products, and the ring structure on cohomology.

Applications of the LLL Algorithm

Practical applications of lattice reduction: integer kernel and image computation, integer relation detection, recovering minimal polynomials from numerical approximations, and …

Asymptotic Notation and Machine Models

Big-O, Omega and Theta notation, the RAM model, bit complexity versus operation counts, input size measured in bits, polynomial versus subexponential versus exponential time, an…

Baby-Step Giant-Step and Class Group Structure

The baby-step giant-step method for discrete logarithms and group order, its application to class groups when an approximation to the order is available, and determination of gr…

Engineering Mathematics13 Aug 2026 · 6 min read

Boolean Algebras: Axioms and Structure

Boolean algebras as an equational class, the two-element algebra as the unique subdirectly irreducible member, atoms and atomlessness, and why the variety is arithmetical.

Engineering Mathematics13 Aug 2026 · 6 min read

Boolean Powers

Boolean powers A[B]*, their construction as locally constant functions on a Stone space, the identities they preserve, and filtered Boolean powers as the refinement that carries…

Boolean Products

Boolean products as subdirect products over a Boolean space with a patching condition, the relationship to sheaves, the equaliser condition, and the classes of algebras admittin…

Engineering Mathematics13 Aug 2026 · 6 min read

Boolean Rings and the Boolean Algebra-Ring Correspondence

Boolean rings, the mutual translation with Boolean algebras, term equivalence as the precise relationship, and why the ring picture makes ideals and the prime spectrum available.

Categories and Morphisms

Categories, objects and morphisms, monomorphisms and epimorphisms defined by cancellation, isomorphisms, and why the arrow-theoretic definitions differ from the element-based ones.

Chain Complexes and Homology

Chain and cochain complexes, cycles and boundaries, homology as a functor, and the abelian category of complexes.

Chain Homotopy and the Comparison Theorem

Chain homotopy between chain maps, homotopy equivalence, the comparison theorem for projective resolutions, and independence of derived functors from the chosen resolution.

Change of Rings

Restriction and extension of scalars, induced and coinduced modules, the change-of-rings theorems, and the associated spectral sequences.

Class Group and Unit Computation in General Number Fields

The general sub-exponential algorithm for class groups and units: factor base selection, ideal reduction, relation collection, the relation matrix and its kernel, and verificati…

Class Groups, Units and the Regulator

The ideal class group, Dirichlet's unit theorem, the regulator, Minkowski's bound, and the analytic class number formula used to verify computed values.

Class Numbers of Imaginary Quadratic Fields

Computing class numbers of imaginary quadratic fields by enumeration of reduced forms, by analytic class number formulas, and by modular form methods, with the Gauss class numbe…

Class Operators H, S, P and the Definition of a Variety

The class operators I, S, H, P and P_S, the inclusions and idempotency relations among their composites, and the identification of HSP as the variety-generating closure operator.

Classical Factoring: Trial Division, Fermat and Lehman

Trial division and wheel factorisation, Fermat's difference of squares method, Lehman's improvement, and the role of these methods as a preprocessing stage.

Classical Primality Proofs: Pocklington and Lehmer

The Pocklington-Lehmer N−1 test, partial factorisation requirements, the N+1 test with Lucas sequences, and combined methods.

Closure Operators and Galois Connections

Closure operators and closure systems, the correspondence with complete lattices, the finitary case and algebraic lattices, and Galois connections as the standard source of clos…

Cofree Modules and Essential Extensions

Cofree modules via Hom from the ring, the adjunction producing enough injectives, essential extensions, and the existence and uniqueness of injective hulls.

Cohomology of Finite Cyclic Groups

The periodic free resolution for a finite cyclic group, the resulting periodic cohomology, norm and difference maps, and Tate cohomology.

Cohomology of Products and Coproducts of Groups

Cohomology of a direct product via Kunneth, cohomology of a free product as a direct sum, Mayer-Vietoris for amalgamated products, and the contrast between the two constructions.

Complete Lattices and Algebraic Lattices

Complete lattices, algebraic lattices, compact elements, and the theorem that a lattice is algebraic exactly when it is the lattice of closed sets of a finitary closure operator.

Completions of Filtrations and lim1

Inverse limits and their failure of exactness, the derived functor lim^1, the Mittag-Leffler condition, the Milnor sequence, and completion of filtered objects.

Complex Multiplication and Class Fields

Complex multiplication, isogenies, the relation between CM curves and imaginary quadratic orders, Hilbert and Weber class polynomials, and the CM method for curve construction.

Compositeness Tests: Fermat and Miller–Rabin

The Fermat test and its failure on Carmichael numbers, the strong probable prime test of Miller-Rabin, error bounds, deterministic base sets for bounded ranges, and Baillie-PSW.

Computation and Authoritative Sources

Software for computing resolutions, Ext, Tor and group cohomology, together with the authoritative databases and the KEVOS policy on reproducing computed data.

Computing Ext Groups

Techniques for computing Ext: choice of variable, use of the long exact sequences, standard computations over the integers, and Ext for cyclic and finitely generated modules.

Computing Galois Groups of Number Fields

The resolvent method for determining Galois groups, the Frobenius cycle-type approach via Dedekind's theorem, transitive group classification by degree, and test polynomials.

Engineering Mathematics13 Aug 2026 · 7 min read

Computing Modular Square Roots

Algorithms for the Jacobi symbol, testing quadratic residuosity, extracting square roots modulo a prime with Tonelli–Shanks and Cipolla, lifting to prime powers by Hensel's lemm…

Computing the Maximal Order: the Round 2 Algorithm

The Pohst-Zassenhaus theorem, the Dedekind criterion, the radical and the ring of multipliers, and the Round 2 algorithm for computing the ring of integers.

Congruences and Quotient Algebras

Congruences as equivalence relations compatible with the operations, the construction of quotient algebras, and why congruences rather than subobjects are the general quotient d…

Continued Fraction Expansions

Simple continued fractions, convergent recurrences and best-approximation properties, Lagrange's periodicity theorem, and the expansion of a square root used for Pell's equation…

Convergence of Spectral Sequences

Convergence conditions for spectral sequences, first-quadrant and bounded cases, conditional convergence, and the failure modes including non-vanishing lim^1.

Decomposition of Prime Numbers in Number Fields

Ramification indices and residue degrees, the fundamental identity, Dedekind's theorem relating prime decomposition to polynomial factorisation modulo p, and computing valuations.

Definition of Group Homology and Cohomology

Group homology and cohomology as derived functors of coinvariants and invariants, equivalently as Tor and Ext over the group ring, with the basic properties and long exact seque…

Definition of Lie Algebra Cohomology

Lie algebra cohomology as Ext over the enveloping algebra, invariants and coinvariants, the explicit description in degrees 0 and 1, and derivations.

Derivations and the Semidirect Product

Derivations and principal derivations, the semidirect product, complements and their conjugacy, and the interpretation of H^1 as classifying complements.

Derived and Stable Categories

The derived category, localisation at quasi-isomorphisms, triangulated structure, derived functors in the modern sense, and the stable module category.

Derived Functors

Left and right derived functors, their construction from resolutions, the fundamental properties, and the axiomatic characterisation by universality.

Engineering Mathematics13 Aug 2026 · 7 min read

Deterministic Primality Testing: the AKS Algorithm

The AKS primality test — the polynomial identity it is based on, the algorithm and its correctness argument, the role of the auxiliary parameter r, complexity, later improvement…

Direct Products, Factor Congruences and Direct Indecomposability

Direct products, projection homomorphisms, factor congruences as the internal signature of a decomposition, directly indecomposable algebras and the limits of unique factorisation.

Direct Sums, Products and Split Sequences

Direct sums and direct products of modules, their universal properties, the splitting lemma and equivalent characterisations of split short exact sequences.

Directly Representable Varieties

Directly representable varieties, McKenzie's theorem that they are congruence-permutable, the classification of their directly indecomposable members, and the connection to the …

Discrete Probability for Algorithm Analysis

Finite probability distributions, conditional probability and independence, random variables, expectation and variance, Chebyshev and Chernoff bounds, the birthday paradox, hash…

Discriminants and Integral Bases

The field discriminant, integral bases, the index of an equation order, and why computing the maximal order reduces to factoring the polynomial discriminant.

Discriminator Varieties

The ternary discriminator term, discriminator varieties, the Bulman-Fleming–Keimel–Werner representation theorem, and the exceptional package of structural properties that f…

Distributive and Modular Lattices: the M5 and N5 Criteria

Distributive and modular lattices, the self-duality of both conditions, and the forbidden-sublattice theorems that characterise them by the non-embeddability of M5 and N5.