Module: Analysis 2

Module Specifications.

Current Academic Year 2024 - 2025

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Date posted: September 2024

Module Title	Analysis 2
Module Code	MS231 (ITS) / MTH1043 (Banner)
Faculty	Science & Health	School	Mathematical Sciences
Module Co-ordinator	Martin Friesen
Module Teachers	-
NFQ level	8	Credit Rating	7.5
Pre-requisite	Not Available
Co-requisite	Not Available
Compatibles	Not Available
Incompatibles	Not Available

Repeat examination

Description

This module covers the essential parts of classical and modern Analysis as required for future courses on Probability and Statistics, Differential Equations, and Partial Differential Equations. The topics covered by this course include but are not restricted to: Metric spaces, Continuous functions, Convolutions, and Fourier Analysis

Learning Outcomes

1. Learn important mathematical concepts as used in subsequent courses in Probability Theory, Stochastic Processes, Differential Equations, and Partial Differential Equations
2. Apply theoretical concepts to particular modelling and computational problems related with real-world applications

*Type*	*Hours*	*Description*
Workload	Full-time hours per semester
Lecture	36	Students will attend lectures where new material will be presented and explained. Also attention will be drawn to various supporting material and tutorials as the course progresses.
Tutorial	12	Students will show their solutions to homework questions and will receive help with and feed-back on these solutions.
Independent Study	78	Corresponding to each lecture students will devote approximately one additional hour of independent study to the material discussed in that lecture or to work on support material when attention is drawn to such in lectures. Before each tutorial students will devote approximately three and a half hours to solving homework problems which are to be discussed in that tutorial.
Total Workload: 126

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

Metric and Normed spaces
metrics, norms, inner products, convergence, completeness, open and closed sets, compactness

Continuous functions
completeness, pointwise and uniform convergence, Arzela-Ascoli Theorem, Hoelder and Lipschitz continuity, Banachs fixed-point theorem, applications

Convolutions
elementary properties, Dirac approximation and mollifiers, (applications)

Harmonic Analysis
periodic functions, Fourier coefficients, representation by Fourier series, Plancherel identity, abstract Fourier decomposition in Hilbert spaces

Integration theory
Riemann-Stieltjes integration, integration of distribution functions, Young integral

Assessment Breakdown
Continuous Assessment	20%	Examination Weight	80%

Type	Description	% of total	Assessment Date
Course Work Breakdown
Written Exam	n/a	20%	n/a

Reassessment Requirement Type
Resit arrangements are explained by the following categories: Resit category 1: A resit is available for both* components of the module. Resit category 2: No resit is available for a 100% continuous assessment module. Resit category 3: No resit is available for the continuous assessment component where there is a continuous assessment and examination element.
* ‘Both’ is used in the context of the module having a Continuous Assessment/Examination split; where the module is 100% continuous assessment, there will also be a resit of the assessment
This module is category 3

Indicative Reading List

Other Resources

None