## Lecture Notes

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Lecture Course Contents

153 First Year Analysis,
A course given to first year students in UMIST up to the merger in 2006.

Properties of the real number system. Inequalities.
Bounded and unbounded sets. Least upper bound and greatest lower bound.
Sequences of real numbers. Subsequences. Limit of a sequence. Convergence and divergence. Convergent sequences are bounded. A monotonic increasing sequence which is bounded above is convergent.
Sums, products and quotients of convergent sequences. The sequences {nk} and {rn}. Sandwich rule.
Series of real numbers as sequences of partial sums. Sums of Geometric series. Divergence of series.
Comparison and ratio tests for series of positive terms.
Series with positive and negative terms. Alternating series test. Relative and absolute convergence. Convergence of power series. Radius of convergence. Cauchy's root test. Functions defined by power series (trigonometric and exponential functions).

MATH10101 Sets, Numbers and Functions.
Last 6 weeks of a first year course.

Numbers of injections and bijections. Numbers of subsets and Binomial Numbers, Pascal's Triangle, Binomial Theorem.
Division Theorem, Greatest Common Divisor, Euclid's Algorithm, Bezout's Lemma,
Linear Diophantine Equations, Congruences, Modular Arithmetic, Solving Linear Congruences, Multiplicative inverses, Pairs of congruences, Triplets of congruences, Method of Successive Squaring, non-linear Diophantine equations,
Congruence Classes, Multiplication Tables, Invertible Elements, Reduced Systems of Classes. Partitions, Relations, generalizing Congruence Classes, from relations to partitions, from partitions to relations.
Prime Numbers, Sieve of Eratosthenes, Infinitude of Primes, Conjectures about Primes, Euler's Theorem, Fermat's Little Theorem. Applications of Euler's and Fermat's Theorem.
Permutations, Bijections, two row notation, Composition. Cycles, factoring, orders. Groups.

1K1 Logic, Set Theory and Matrices
A service course given to Computation students up to 2005.

Propositions, Connectives (and, or, not), Truth Tables, The Boolean Laws of Logics. Equivalence of forms, tautology and contradiction.
Conditional and Biconditional, contrapositive, converse.
Definition of a valid argument. Two ways to use a truth table to show an argument is Valid.
Natural Deduction: A Rule of Assumption, MPP, MTT, DN.
Natural Deduction: Eliminating the "and", Eliminating the "or", Introducing "or", Introducing "and".
Natural Deduction: Conditional Proof, Proof by Contradiction.
Subset, equality, denoting a set. Formal languages.
Complement, union, intersection, difference, symmetric difference. The Boolean Laws for Sets.
Universal and existential quantifiers. Symbolising English sentences. Negating quantified sentences. Proving validity of quantified arguments.
Cartesian product, Power set. Cardinalities of a union of sets, of a produxt of sets and of a power set.
Relations. Digraphs. Reflexive, symmetric, transitive relations. Equivalence Relations.
Describing a function. Onto and one-to-one functions.
Composition and inverses.
Defintion, addition and multiplication. Identity and inverses. Solving systems of linear equations. Gaussian Elimination.

MATH10242 Sequences and Series,
First year course up to 2021.

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• Part I Sequences
• 2 Convergence . . . . . . . . . . . . . .. . . . . . . . . . . . . .16
• 2.1 What is a Sequence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
• 2.2 The Triangle Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
• 2.3 The Definition of Convergence . . . . . . . . . . . . . . . . . . . . . . . . 19
• 2.4 The Completeness Property for ℝ . . . . . . . . . . . . . . . . . . . . . . 25
• 2.5 Some General Theorems about Convergence . . . . . . . . . . . . . . . . 29
• 2.6 Exponentiation - a digression . . . . . . . . . . . . . . . . . . . . . . . . 31
• 3 The Calculation of Limits . . . . . . . . . . . . . .. . . . . . . . . . . . . .34
• 3.1 The Sandwich Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
• 3.2 The Algebra of Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
• 4 Some Special Sequences . . . . . . . . . . . . . .. . . . . . . . . . . . . .43
• 4.1 Basic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
• 4.2 New Sequences from Old . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
• 4.3 Newton’s Method for Finding Roots of Equations - optional . . . . . . . 56
• 5 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59
• 5.1 Sequences that Tend to Infinity . . . . . . . . . . . . . . . . . . . . . . . 59
• 6 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64
• 6.1 The Subsequence Test for Non-Convergence . . . . . . . . . . . . . . . . . . . . . 64
• 6.2 Cauchy Sequences and the Bolzano-Weierstrass Theorem . . . . . . . . . . . . . . . 68
• 6.2.1 Proofs for the section - optional . . . . . . . . . . . . . . . . . . . . . . . 69
• 7 L'Hôpital's Rule . . . . . . . . . . . . . .. . . . . . . . . . . 74
• Part II Series
• 8 Introduction to Series . . . . . . . . . . . . . .. . . . . . . . . . . . . .79
• 8.1 The Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
• 9 Series with Non-Negative Terms . . . . . . . . . . . . . . . . . . . . . . . . . .85
• 9.1 The Basic Theory of Series with Non-Negative Terms . . . . . . . . . . . . .. .85
• 9.2 The Integral Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
• 10 Series with Positive and Negative Terms . . . . . . . . . . . . . .. . . . . . . . 98
• 10.1 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
• 10.2 Absolute Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
• 11 Power Series . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .106
• 11.1 The Radius of Convergence of a Power Series . . . . . . . . . . . . . . . . 107
• 11.2 The n-th Root Test . . . . . . . . . . . . . . . . . . . . . . . . 107
• 12 Further Results on Power Series - further reading . . . . . . . . . . . . . .114
• 12.1 More General Taylor Series. . . . . . . . . . . . . . . . . . . . . . . . . . 114
• 12.2 Rearranging Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
• 12.3 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

MATH20101 Real Analysis,
Second year course up to 2022.

Webpage

Limit of a real-valued function at a finite point, one-sided limits, limits of a function at infinity.
Divergence. Limits Rules, including the Sum, Product and Quotient rules and the Sandwich Rule.
Special Limits; the exponential and trigonometric functions.
Definition of continuity and continuous functions. Continuity Rules, including the Sum, Product and Quotient rules and the Composite rule.
The Intermediate Value Theorem, the Boundedness Theorem and the Inverse Function Theorem.
Definition of differentiation Rules for differentiation, including the Sum, Product and Quotient rules, the Chain Rule and the Inverse Rule.
Derivative Results including: Rolle's Theorem, the Mean Value Theorem, Cauchy's Mean Value Theorem and L'Hôpital's Rule.
Taylor Polynomials and Taylor's Theorem with Cauchy's and Lagrange's forms of the error. Taylor Series and a number of standard examples.
Definition of integration with worked examples. Fundamental Theorem of Integration.

MATH20132 Calculus of Several Variables,
Second year course up to 2022.

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Functions of Several Variables: Notation; Limits of vector-valued functions; Scalar-valued examples; Sandwich Rule; Directional limits; Limits along curves; Continuity of vector-valued functions; Continuity Laws; Composition Laws.
Appendix Uniqueness of Limit; Example of Limit; Directional limits; Composition Results.

Differentiation of functions of several variables: Directional derivatives; Partial Derivatives; Differentiable implies Continuous; Vector-valued Linear Functions; Fréchet Derivative. Fréchet Differentiable implies Continuous; Derivative exists implies directional derivative exists. Jacobian Matrix; Special cases: functions of one variable and scalar valued functions. When is a function differentiable? C1 functions are Fréchet differentiable. Examples; Product and Quotient functions; The Chain Rule; Rules for differentiation; Special cases; the Inverse function.
Appendix Partial and directional derivatives; Directional derivative implies directional continuity; Derivative is unique; Mean Value Theorem; Differentiable does not implies C1; Earlier example revisited; Rules of Differentiation; Chain Rule.

Surfaces and the Implicit Function Theorem: A surface as a graph; a surface as an Image set, i.e. given parametrically; a surface as a Level set, i.e. given Implicitly; a graph is a level set; converses.
Linear Algebra, Vector Subspaces & planes,
Level sets are graphs (locally): Implicit Function Theorem; Parametric sets are graphs (locally): Inverse Function Theorem; Manifolds.
Tangent Spaces and Tangent Planes; Tangent Space for a graph; Tangent Space for a Level Set; Tangent Space for a Parametric set;
Appendix on Surfaces Full Rank; Jacobian matrix of a graph written as an image set; Jacobian matrix of a graph written as a level set; ℝn or ℝr x ℝn-r?; Surface of a unit ball in ℝ3; Permuting the coordinate functions in a parametric set; Permuting the variables in a level set;
Background Linear Algebra; Dimension; Surfaces as level sets f-1(0). Which f?;
Proof of the Inverse Function Theorem.
Appendix on Tangent Spaces Tangent Spaces for a surface in R3 with illustrations.

Proof of the Implicit Function Theorem: by Induction.

Extremal Values: Extremal Values and Lagrange multipliers;
Appendix Examples including the GM-AM inequality & the Cauchy-Schwarz inequality along with their extensions.

Differential forms and their integration: Differential 1-forms; Exact 1-forms; higher-order derivatives; Closed 1-forms; line integrals; Differential 2-forms; Surface integrals; Products of 1-forms; Derivatives of 1-forms; Stoke's Theorem (statement of); Green's Theorem (statement of). Differentiable k-forms; products and derivatives of k-forms; integration of a k-form over a manifold.
Appendix Projections are linearly independent; Second derivatives need not be equal; motivation for the definition of a 2-form; Proof of Stoke's Theorem; Vector fields; A basis for k-forms.

Higher Derivatives: The Hessian Matrix; Taylor's Theorem; Surfaces; examples of local minima, maxima and saddle points; Tests for definiteness or otherwise.

Appendix Further worked example; Tests for definiteness or otherwise of matrices.

341 Measure Theory
A course given to third year students in UMIST up to 2002.

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Classes of subsets: Topology, Rings and Fields. Cardinality, Topological Space results.
Outer measure and Measurable sets, Lebesgue Measurable sets, Non-measurable sets, Sets of measure zero.
Measurable functions: Sequences of functions.
Simple functions. An interesting simple function.
Integration: Integration of non-negative simple functions. Chebychev's Theorem and an application to Normal Numbers.
Interchanging integrals with other operations. Extended version of Monotonic Convergence Theorem with yet another proof that ∑(1/n2)=π2/6.
Integration of measurable functions. Comparison of the Riemann and Lebesgue integrals, Measure Preserving Transformations including Poincare's Recurrence Theorem. Spaces of integrable functions, showing that these spaces are complete.
Product Spaces and Fubini's Theorem. Completion of product spaces.

MATH6\41022 Analytic Number Theory.

Webpage

Two proofs of the infinitude of primes. Definition of the Riemann zeta function, infinite products.

Elementary Prime Number Theory. von Mangoldt's Function; Partial Summation; Replacing sums by integrals.
Chebyshev's bounds for ψ(x), π(x) and θ(x), Asymptotic relations between ψ(x) & θ(x) and π(x) & θ(x).
Merten's results on weighted sums over primes; extended with improved error terms; Merten's Theorem on Euler products
The Statement of the Prime Number Theorem, π(x)~ x/logx iff, ψ(x)~ x; the logarithmic integral.

Arithmetic functions. Cauchy Products; Convolutions; Dirichlet Series; multiplicative and additive functions.
Multiplicative Functions. Möbius function; Möbius inversion; square-free and k-free numbers; Liouville's function.
Factorising arithmetic functions; Dirichlet Series as products and quotients of the Riemann zeta function; factoring Q2 , 2 ω,
d 2
Euler’s phi function. Inverses of Arithmetic functions. The Dirichlet Series for multiplicative functions has an Euler Product. Alternative method of factorising an arithmetic function.

Sums of Convolutions. Convolution method for 1*g; Various summations -- of Q2 , 2 ω, d(n2)d 2(n) and d 3(n).
Sums of Convolutions. Dirichlet's hyperbolic method.
Table of Arithmetic Functions, their associated Dirichlet Series, and Summations;
Convolution Method, the general case. Asymptotic Result on Summation of dk(n). Extra terms in Summation results.

Sums of Additive Functions. Turán's inequality; Hardy-Ramanujan Theorem. Appendix Turán-Kubilius inequality; statement of the Erdos-Kac Theorem.

The Prime Number Theorem. Introduction
Step 1 Analytic Properties of the Riemann zeta function.
Step 2 Relating the prime counting function, ψ(x), to the Riemann zeta function.
Step 3 ζ (σ +it) ≠ 0 for σ ≥ 1.
Step 4 Bounds on the Riemann zeta function.
i.   Upper bounds for σ ≥ 1 - η(t) for some function η(t) > 0,
ii.   Lower bounds for σ ≥ 1- φ(t) for some function φ(t) > 0.
Divergence of ζ (1+it).
Step 5 Moving the line of integration.
Step 6 Final deduction of the Prime Number Theorem.

Actual, though unproved in this course, & conjectured results on the zeros of ζ(s).