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diff --git a/md/research/ou-math.md b/md/research/ou-math.md index bf7dcde..c44b97d 100644 --- a/md/research/ou-math.md +++ b/md/research/ou-math.md @@ -159,6 +159,88 @@ Using Minitab to carry out straightforward data analyses http://www.open.ac.uk/courses/qualifications/details/m208?orig=q31 +#### Unit A1 Sets, functions and vectors +Revises important foundations of pure mathematics and the mathematical +language used to describe them. +#### Unit A2 Number systems +Systems of numbers most widely used in mathematics: the integers, rational +numbers, real numbers, complex numbers and modular or 'clock' arithmetic, +and looks at when and how certain types of equations can be solved in the +system. +#### Unit A3 Mathematical language and proof +Writing of pure mathematics and some of the methods used to construct +proofs, and as a further introduction to abstract mathematical thinking +equivalence relations are introduced. +#### Unit A4 Real functions, graphs and conics +Reminder of the principles underlying the sketching of graphs of functions +and other curves. +#### Unit B1 Symmetry and groups Symmetry of plane figures and solids, and shows how this topic leads to the +definition of a group, which is a set of elements that can be combined with +each other in a way that has four basic properties called group axioms. +#### Unit B2 Subgroups and isomorphisms Subgroups, which are groups that lie inside other groups, and also at cyclic +groups, which are groups whose elements can all be obtained by repeatedly +combining a single element with itself. It also investigates groups that +appear different but have identical structures. +#### Unit B3 Permutations +Functions that rearrange the elements of a set: it shows how these +functions form groups and looks at some of their properties. +#### Unit B4 Lagrange's Theorem and small groups +Fundamental theorem about groups, and uses it to investigate the +structures of groups that have only a few elements, before focusing on +improving skills in understanding theorems and proofs in the context of +group theory. +#### Unit C1 Linear equations and matrices +Why simultaneous equations may have different numbers of solutions, and +also explains the use of matrices. +#### Unit C2 Vector spaces +Generalises the plane and three-dimensional space, providing a common +structure for studying seemingly different problems. +#### Unit C3 Linear transformations +Mappings between vector spaces that preserve many geometric and +algebraic properties. +#### Unit C4 Eigenvectors +Diagonal representation of a linear transformation, and applications to +conics and quadric surfaces. +#### Unit D1 Numbers Real numbers as decimals, rational and irrational numbers, and goes on to +show how to manipulate inequalities between real numbers. +#### Unit D2 Sequences The 'null sequence' approach, used to make rigorous the idea of +convergence of sequences, leading to the definitions of pi and e. +#### Unit D3 Series +Convergence of series of real numbers and the use of series to define the +exponential function. +#### Unit D4 Continuity +Sequential definition of continuity, some key properties of continuous +functions, and their applications. +#### Unit E1 Cosets and normal subgroups +Revision of Units B1-B4 and looks at how a group can be split into 'shifts' of +any one of its subgroups. +#### Unit E2 Quotient groups and conjugacy +How to 'divide' a group by one of its subgroups to obtain another group, and +how in any group some elements and some subgroups are similar to each +other in a particular sense. +#### Unit E3 Homomorphisms +Functions that map groups to other groups in a way that respects at least +some of the structure of the groups. +#### Unit E4 Group actions +How group elements can sometimes be applied to elements of other sets in +natural ways. This leads to a method of counting how many different +objects there are of certain types, such as how many different coloured +cubes can be produced if their faces can be painted any of three different +colours. +#### Unit F1 Limits +The epsilon-delta approach to limits and continuity, and relates these to the +sequential approach to limits of functions. +#### Unit F2 Differentiation +Differentiable functions and gives L'Hopital's rule for evaluating limits. +Integration explains the fundamental theorem of calculus, the Maclaurin +integral test and Stirling's formula. +#### Unit F3 Integration +The fundamental theorem of calculus, the Maclaurin integral test and +Stirling's formula. +#### Unit F4 Power series +Finding power series representations of functions, their properties and +applications. + ### DD209 Running the economy http://www.open.ac.uk/courses/qualifications/details/dd209?orig=q15 @@ -177,6 +259,108 @@ http://www.open.ac.uk/courses/qualifications/details/m249?orig=q36 http://www.open.ac.uk/courses/qualifications/details/m269?orig=r38 ### M303 Further pure mathematics http://www.open.ac.uk/courses/qualifications/details/m303?orig=q31 + +#### Chapter 1 Foundations +Proof by induction, divisibility, linear Diophantine equations +#### Chapter 2 Prime numbers +Prime numbers, Fundamental Theorem of Arithmetic, prime +decomposition of integers, the theta-function, Fibonacci numbers. +#### Chapter 3 Congruence +Definition of congruence, properties of congruence, divisibility +tests, linear congruences, solution of linear congruences, +solving systems of linear congruences. +#### Chapter 4 Fermat's and Wilson's Theorems +Fermat's Little Theorem, Wilson's Theorem, polynomial +congruences, Lagrange's Theorem (for numbers) +#### Chapter 5 Examples of groups +Group axioms, subgroups, cosets, Lagrange's Theorem (for +groups), normal subgroups, quotient groups, conjugate +elements, homomorphism of groups, isomorphism of groups, +first isomorphism theorem, correspondence theorem. +#### Chapter 6 Towards classification +Direct product of groups, internal direct product theorem, +cyclic groups, direct product of cyclic groups, decomposition +of finite cyclic groups, group actions, orbits and stabilisers. +#### Chapter 7 Finite groups +Group presentations, dihedral groups, dicyclic groups, +#### Chapter 8 The Sylow Theorems +Sylow p-subgroup, the Sylow theorems, applications of the +Sylow theorems, prime power subgroups theorem. +#### Chapter 9 Multiplicative functions +Multiplicative functions, Euler's phi-function, reduced set of +residues, Euler's theorem, primitive roots. +#### Chapter 10 Quadratic reciprocity +Solutions of quadratic congruences, quadratic residues, +Euler's criterion, the Legendre symbol, Gauss's Lemma, +quadratic character of 2, the law of quadratic reciprocity, +quadratic character of 3, the Jacobi symbol. +#### Chapter 11 Rings and polynomials +Ring axioms, subrings, units, fields, polynomials over fields, +division algorithm for polynomials, factors of a polynomial, +Euclidean algorithm for polynomials, factorising polynomials, +irreducibility for polynomials, rational root test, Gauss's +lemma, Eisenstein's criterion +#### Chapter 12 Fermat's Last Theorem and unique factorisation +Pythagorean triples; integral domains; associates, +irreducibles and primes in rings; integral domains, norms for +integral domains; Euclidean domains; division algorithm for +Euclidean domains; highest common factors in Euclidean +domains; unique factorisation domains. +#### Chapter 13 Distance and continuity +Sequences in the real line; real null sequences; continuity of +real-valued functions; intermediate value theorem; extreme +value theorem; continuity on the plane; Euclidean distance on +the plane. +#### Chapter 14 Metric spaces and continuity 1 +Continuity of functions from R^n to R^m, Euclidean distance on +R^n, convergent sequences in R^n, metrics, metric spaces, +convergence of sequences in metric spaces, continuity in +metric spaces. +#### Chapter 15 Metric spaces and continuity 2 +Induced metrics, Cantor metric, equivalent metrics, product +metrics, pointwise convergence of functions, uniform +convergence of functions, the max metric on C[0,1]. +#### Chapter 16 Open and closed sets +Open sets, closed sets, dense sets, nowhere dense sets, +closure of a set, interior of a set, boundary of a set, countable +sets, uncountable sets. +##### Chapter 17 Rings and homomorphisms +Fields of fractions, ring isomorphisms, primitive polynomials, +ideals, principal ideals, principal ideal domain, algebra of +ideals, cosets of an ideal, quotient rings, ring +homomorphisms, maximal ideals, prime ideals. +#### Chapter 18 Fields and polynomials +Isomorphism of fields, field extensions, vector spaces over +fields, degree of a field extension, algebraic and +transcendental elements, minimal polynomials, the KLM Theorem for field extensions, finite fields, roots of unity, +splitting fields, splitting polynomials, classification of finite +fields. +#### Chapter 19 Fields and geometry +Subfield generated by a set, field extensions of finite degree, +field of algebraic numbers, transcendental extension, ruler +and compass constructions, constructible number, +impossibility of doubling the cube, squaring the circle and +trisecting the angle pi/3 +#### Chapter 20 Public-key cryptography +RSA cryptosystem, Diffie-Hellman cryptosystem, elliptic +curves, Diffie-Hellman-ElGamal cryptosystem, Menezes-Vanstone cryptosystem. +#### Chapter 21 Connectedness +Homeomorphisms, disconnections, connectedness, +connected components, totally disconnected sets, +connectedness in Euclidean spaces, the intermediate value +theorem, path-connectedness, the topologist's cosine. +#### Chapter 22 Compactness +Sequential compactness, the Heine-Borel theorem, +generalised extreme value theorem, Arzela-Ascoli Theorem, +open covers, compact metric spaces, equivalence of +sequential compactness and compactness in metric spaces. +#### Chapter 23 Completeness +Cauchy sequences, complete metric spaces, the contraction +mapping theorem, completion of a metric space. +#### Chapter 24 Fractals +The Hausdorff metric, self-similar sets, iterated function +schemes, box dimension, open set condition. + ### DD309 Doing economics: people, markets and policy http://www.open.ac.uk/courses/qualifications/details/dd309?orig=q15 ### MST326 Mathematical methods and fluid mechanics |