Engineering

Isogenies and Endomorphism Rings

Isogenies as maps of curves respecting the group law, and the two possible endomorphism rings over a field of characteristic zero.

Kernel and Image of a General Matrix

Computing kernel and image bases for rectangular matrices over a field, and why the integer analogue is a different problem.

Kernel of a Linear Transformation

The kernel of a linear transformation: definition, why it is always a subspace, its role as the preimage of zero, and why a trivial kernel means injectivity.

Engineering Mathematics13 Aug 2026 · 7 min read

Kernels, Images and the Isomorphism Theorems

The kernel and image of a homomorphism, and the first isomorphism theorem relating them to a quotient.

L-Functions and the Birch-Swinnerton-Dyer Conjecture

The L-function of an elliptic curve, the Birch-Swinnerton-Dyer conjecture, and what can and cannot be computed about rank.

Language Recognition and Complexity Classes

Randomised complexity classes RP, co-RP, BPP and ZPP, and where primality testing sits among them.

Lattice Definitions and Quadratic Forms

Lattices as discrete subgroups, the Gram matrix, and the correspondence between lattices with a basis and positive definite quadratic forms.

Lattice Determinant and the Hadamard Bound

The lattice determinant as a basis-independent invariant, the Hadamard inequality, and using the orthogonality defect to measure basis quality.

Lattices, Complex Tori and the Weierstrass p-Function

The identification of elliptic curves over the complex numbers with complex tori, and the Weierstrass function that realises it.

Learning Pathways in Computational Number Theory

Suggested routes through the collection for cryptography, computer algebra, coding theory and pure mathematics readers.

Engineering Mathematics13 Aug 2026 · 7 min read

Learning Pathways Through Computational Number Theory

Four routes through this 182-page collection, chosen by what you need to build rather than by chapter order.

Legendre, Jacobi and Kronecker Symbol Computation

The three quadratic symbols, their differing meanings, and the reciprocity-based algorithm that computes them without factoring.

Lehmer's Accelerated GCD Computation

Lehmer's method: running many GCD steps on single-precision leading digits before touching the full multiprecision operands.

Linear Algebra in Practice: The Trail Mix Blending Problem

A full blending and production-planning case study: mass balances as linear equations, unique versus infinite solution sets, feasibility bounds and profit optimisation.

Engineering Mathematics13 Aug 2026 · 25 min read

Linear Algebra Notation Reference

A consolidated reference for linear algebra notation: matrices, entries, vectors, the four subspaces, spans, inner products, transformations and matrix representations.

Engineering Mathematics13 Aug 2026 · 15 min read

Linear Algebra Theorem Map — Applications, Practice and Reference

How the theorems of linear algebra depend on one another: from row operations and RREF through span, bases and dimension to rank, determinants and eigenvalues.

Linear Algebra Theorem Map — Foundations and Core Equivalences

How the theorems of linear algebra depend on one another: from row operations and RREF through span, bases and dimension to rank, determinants and eigenvalues.

Linear Combinations of Vectors

Linear combinations of column vectors: definition, the equivalence between solving a linear system and writing a constant vector as a combination of columns.

Linear Dependence and Spans

Why a linearly dependent set always contains a vector expressible from the others, and how removing it shrinks a spanning set without changing the span it generates.

Linear Independence and Bases

Linear independence, spanning sets and bases, and the conditions under which a basis exists.

Engineering Mathematics13 Aug 2026 · 24 min read

Linear Independence and Nonsingular Matrices

Why a square matrix is nonsingular exactly when its columns are linearly independent, the chained proof, and the growing list of nonsingular equivalences.

Engineering Mathematics13 Aug 2026 · 20 min read

Linear Independence in Abstract Vector Spaces

Relations of linear dependence and linear independence in any vector space: the definitions, the reduction to a homogeneous system, and worked tests in matrix and polynomial spa…

Engineering Mathematics13 Aug 2026 · 24 min read

Linear Independence of Vectors

Relations of linear dependence, the definition of a linearly independent set of column vectors, and the homogeneous-system and rank tests that decide it.

Engineering Mathematics13 Aug 2026 · 24 min read

Linear Transformations

The definition of a linear transformation between vector spaces: additivity, homogeneity, the zero-to-zero theorem, and how to tell a linear map from an affine one.

Linear Transformations and Linear Combinations

A linear transformation passes through any linear combination, so its values on a basis determine it everywhere. The superposition theorem and its use in prediction.

Linearly Generated Sequences

Sequences satisfying linear recurrences, their minimal polynomials, and the equivalence with rational generating functions.

LLL for Linearly Dependent Generating Sets

Extending LLL to generating sets that are not independent, and using the resulting zero vectors to extract relations.

LLL Reduction Quality and Proof Sketch

Why LLL terminates in polynomial time and what its output guarantees, via the potential function and the Lovasz condition.

LLL with Deep Insertions

Allowing a vector to move further than one position, the quality gain, and the loss of the polynomial time guarantee.

LLL-Based Hermite Normal Form Computation

Using lattice reduction to control entry growth during Hermite normal form computation, and when this beats the modular approach.

Lucas Sequences and Lucas Pseudoprimes

Lucas sequences, the Lucas probable prime test, and why it complements the strong pseudoprime test rather than duplicating it.

Machine Models and Complexity Theory

The random access machine and Turing machine models, what counts as a primitive operation, and how the choice of model affects stated complexities.

Engineering Mathematics13 Aug 2026 · 7 min read

Markov's and Chebyshev's Inequalities

Markov's and Chebyshev's inequalities, their proofs, and how they bound deviation from the mean.

Mathematical Notation and Standing Conventions

The notational conventions, symbol set and standing assumptions used throughout the KEVOS computational number theory collection.

Matrices and Augmented Matrices

What a matrix is, the entry notation for row and column, and how appending the constants column turns a system of linear equations into a single rectangular array.

Matrices and Linear Maps

The correspondence between matrices and linear maps, change of basis, and why the correspondence depends on a choice.

Matrices and Linear Transformations

Every matrix defines a linear transformation, and every linear transformation on column vectors comes from one unique matrix built from images of the unit vectors.

Matrices, Complex Conjugation and the Adjoint

Entrywise complex conjugation of a matrix, how it respects addition and scalar multiplication, commutation with the transpose, and the adjoint or conjugate transpose.

Matrices: Basic Definitions and Properties

Matrices over a ring, their arithmetic, and the properties that survive when the base ring is not a field.

Matrix and Vector Notation for Linear Systems

Column vectors, the zero vector, coefficient matrix, vector of constants and solution vector: the notation that compresses a whole system of equations into the symbol LS(A,b).

Matrix Equality, Addition and Scalar Multiplication

Entry-by-entry definitions of matrix equality, matrix addition and scalar multiplication on the set of m by n matrices, with worked examples and engineering use.

Matrix Multiplication

Matrix multiplication defined column by column through the matrix-vector product, the inner dimension rule, why the operation is not commutative, and what it means.

Matrix Multiplication, Entry-by-Entry

The entry formula for a matrix product: each entry of AB sums over the shared dimension, why it follows from the column definition, and what it costs.

Matrix Representation and Cost Model

Dense and sparse matrix representations, the cost model for exact linear algebra, and why coefficient growth rather than operation count usually decides performance.

Matrix Representations and Similarity

Changing basis converts one matrix representation of a transformation into another by a similarity transformation, which is why eigenvalues and rank are intrinsic.

Matrix Representations of Invertible Transformations

An invertible linear transformation has an invertible matrix representation, and the representation of its inverse is the matrix inverse with the bases exchanged.

Matrix Representations of Linear Transformations

How a linear transformation between abstract vector spaces becomes a matrix: the definition of a matrix representation, the fundamental theorem, and basis choice.

Matrix Transpose and Symmetric Matrices

Definition of the matrix transpose, symmetric matrices, why symmetric matrices are square, and how the transpose interacts with matrix addition and scalar multiplication.