## Mathematical Studies

MS1001 Mathematical Studies I

Credit Weighting: 15

Teaching Period(s): Teaching Periods 1 and 2.

No. of Students: Min 20, Max 200.

Pre-requisite(s): None

Co-requisite(s): None

Teaching Methods: 72 x 1hr(s) Lectures; 20 x 1hr(s) Tutorials.

Module Co-ordinator: Dr Ben McKay, Department of Mathematics.

Lecturer(s): Mr Alan McCarthy, School of Mathematical Sciences.

Module Objective: To provide an introduction to Mathematics.

Module Content: An introduction to Mathematics, including a study of the following topics: rational, real and complex numbers, elementary algebra, sequences and series, graphs, coordinate geometry, counting, trigonometry, linear algebra, vectors, exponential and logarithm functions, differentiation.

Learning Outcomes: On successful completion of this module, students should be able to:
· Distinguish between and use various kinds of numbers: rational, real and complex;
· Manipulate and simplify algebraic expressions;
· Work with functions and graphs;
· Recall and use the general properties of linear, quadratic, trigonometric, exponential and logarithm functions, and solve problems involving these functions;
· Solve problems using counting techniques;
· Solve problems in geometry and trigonometry, using vectors as necessary;
· Use differentiation to solve extremal problems;
· Solve problems involving sequences and series.

Assessment: Total Marks 300: End of Year Written Examination 240 marks (120 marks for each of Section A and Section B); Continuous Assessment 60 marks (30 marks each for the continuous assessment associated with Section A and Section B. Format of continuous assessment to be in-class test(s) and/or homework assignment(s): students will be given written notification of the format and breakdown of marks for continuous assessment at the first lecture).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40% Students must obtain not less than 40% neither in the combined mark for Section A (continuous assessment and end of year written examination) nor in the combined mark for Section B (continuous assessment and end of year written examination). For students who do not satisfy this requirement, the lower of the two marks, scaled relative to the total marks available for the module, will be returned.

End of Year Written Examination Profile: 1 x 3 hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 3 hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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MS2001 Differential Calculus

Credit Weighting: 5

Teaching Period(s): Teaching Period 1.

No. of Students: Min 20, Max 100.

Pre-requisite(s): MS1001

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Dr Ben McKay, Department of Mathematics.

Lecturer(s): Mr J.P. McCarthy, School of Mathematical Sciences.

Module Objective: To provide an introduction to techniques and applications of differential calculus.

Module Content: Limits, continuity and derivatives of functions of one variable. Applications.

Learning Outcomes: On successful completion of this module, students should be able to:
· Solve inequalities involving real numbers and the modulus function, obtain upper and lower estimates for expressions;
· Reason with both the intuitive idea and the formal definition of the limit of a real function f of one variable, compute limits using a variety of techniques, apply the concept of divergence towards infinity;
· Reason with both the intuitive idea and the formal definition of continuity of a real function f, determine if a given function is continuous and if a discontinuity is removable, apply the Extremal Value Theorem and the Intermediate Value Theorem;
· Reason with both the intuitive idea of the derivative of a real function in one variable and its formal definition, calculate the derivative of a wide variety of functions, derive properties of the function from properties of the derivative via the Mean Value Theorem;
· Find local extrema of a function f by investigating its first and second derivative, solve extremal problems in one variable.

Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s),students will be notified of the format and breakdown of marks for continuous assessment at the first lecture.).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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MS2002 Integral Calculus and Differential Equations

Credit Weighting: 5

Teaching Period(s): Teaching Period 2.

No. of Students: Min 20, Max 100.

Pre-requisite(s): MS2001

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Dr Ben McKay, Department of Mathematics.

Lecturer(s): Dr Aoife Hennessy, Department of Computer Science.

Module Objective: To provide an introduction to Integral Calculus and ordinary differential equations.

Module Content: Techniques and applications of integration of functions of one variable; solution of ordinary differential equations.

Learning Outcomes: On successful completion of this module, students should be able to:
· Reduce simple indefinite integrals to standard form and evaluate them by means of integration by substitution, integration by parts, partial fractions, completion of the square;
· Apply the Fundamental Theorem of Calculus to evaluate definite integrals;
· Apply methods of integration to evaluate plane areas, volumes of rotation and arc length;
· Derive simple properties of the natural logarithm and exponential from properties of definite integrals;
· Apply the trapezoidal rule and Simpson's rule to find approximate values of definite integrals;
· Use differential equations to set up mathematical models of simple growth and decay problems related to physical, sociological and biological phenomena;
· Recognize and solve the following differntial equations: equations of type variables separable, the logistic equation and first-order linear equations.

Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture.).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): None.

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward (whether passed or failed).

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MS2003 Linear Algebra

Credit Weighting: 5

Teaching Period(s): Teaching Period 1.

No. of Students: Min 20, Max 100.

Pre-requisite(s): MS1001

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Dr Ben McKay, Department of Mathematics.

Lecturer(s): Dr Martin Kilian, Department of Mathematics.

Module Objective: To provide an introduction to the theory and applications of matrices.

Module Content: Solutions of systems of linear equations; vector spaces, matrix theory, inverses, eigenvalues and eigenvectors.

Learning Outcomes: On successful completion of this module, students should be able to:
· Solve systems of linear equations by elimination.
· Carry out matrix arithmetic, and invert matrices.
· Find determinants. Use them to decide on solvability and invertibility.
· Determine if vectors belong to subspaces, and in particular to spans. Use this to find bases of subspaces.
· Find eigenvalues, and bases of eigenvectors for eigenspaces.
· Diagonalize symmetric matrices by orthogonal matrices.
· Give examples of matrices with or without various properties such as invertibility, othogonality, symmetry, nature of eigenvalues, and determine whether matrices have these properties.

Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture.).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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MS2005 Discrete Mathematics

Credit Weighting: 5

Teaching Period(s): Teaching Period 1.

No. of Students: Min 20, Max 100.

Pre-requisite(s): MS1001

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Dr Ben McKay, Department of Mathematics.

Lecturer(s): Dr Aoife Hennessy, Department of Computer Science.

Module Objective: To provide an overview of combinatorics, graphs, trees and applications.

Module Content: Induction, recurrence relations. Combinatorics: permutations, combinations, the pigeonhole principle. Discrete probability on finite sample spaces; conditioning and independence. Graph theory: Euler and Hamiltonian paths and cycles, weighted graphs, applications.

Learning Outcomes: On successful completion of this module, students should be able to:
· Devise and carry out proofs by induction.
· Solve first order recurrence relations by iteration, and solve second order linear recurrence relations with constant coefficients.
· Enumerate sets using combinatorial techniques such as permutation and combinations.
· Utilise the pigeonhole principle.
· Derive combinatorial identities.
· Calculate probabilities of events from finite probability spaces, applying conditioning and independence arguments.
· Identify standard properties of given graphs such as the existence of Euler cycles.
· Apply algorithms to calculate quantities associated to graphs, such as the shortest distance between two vertices in a weighted graph, and to produce examples of Euler and Hamilton cycles.

Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture.).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward (whether passed or failed).

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MS2012 Elementary Number Theory

Credit Weighting: 5

Teaching Period(s): Teaching Period 2.

No. of Students: Min 20, Max 100.

Pre-requisite(s): MS1001

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Dr Ben McKay, Department of Mathematics.

Lecturer(s): Dr Andrei Mustata, Department of Mathematics.

Module Objective: To provide an introduction to the study of properties of integers.

Module Content: Divisibility, prime numbers, congruences, cryptography, Pythagorean triples.

Learning Outcomes: On successful completion of this module, students should be able to:
· Detail the basic concepts and theorems of the integer number system including Proof by Induction, primes and the Division Algorithm;
· Derive elementary properties of rational numbers and Gauss' Theorem;
· Apply techniques which have been developed in the lecture to solve problems;
· Explain the basic concepts and main theorems of the theory of congruences;
· Solve problems concerning the Chinese Remainder Theorem, Fermat's Theorem and Euler's Theorem;
· Apply the basic concepts to basic coding theory;
· Develop the RSA algorithm.

Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture.).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): None.

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward (whether passed or failed).

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MS2013 Geometry

Credit Weighting: 5

Teaching Period(s): Teaching Period 2.

No. of Students: Min 20, Max 100.

Pre-requisite(s): MS1001

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Dr Ben McKay, Department of Mathematics.

Lecturer(s): Dr Anca Mustata, Department of Mathematics.

Module Objective: To provide an introduction to Euclidean geometry in dimensions two and three, and to familiarize students with the concept of proof.

Module Content: Reasoning and proof; the fundamental principles and results of planar Euclidean geometry, straightedge constructions, area and length. Elementary solid geometry.

Learning Outcomes: On successful completion of this module, students should be able to:
· Prove elementary theorems of planar and solid Euclidean geometry.
· Explain the concepts of axiom and proof.
· Carry out intuitive geometric reasoning.

Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture.).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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MS3001 Introduction to Abstract Algebra

Credit Weighting: 5

Teaching Period(s): Teaching Period 1.

No. of Students: Min 20, Max 100.

Pre-requisite(s): None

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Dr Ben McKay, Department of Mathematics.

Lecturer(s): Vahid Yazdanpanah, School of Mathematical Sciences.

Module Objective: To provide an overview of concepts of abstract algebra.

Module Content: Sets, relations, functions and equivalence relations. Elementary theory of groups.

Learning Outcomes: On successful completion of this module, students should be able to:
· Correctly use logical implications, negations, equivalences, in proving simple mathematical statements.
· Perform operations with sets and display their results in Venn diagrams.
· Discriminate when a relation is reflexive, symmetric, or transitive.
· Determine when a function is injective, surjective or bijective.
· Perform operations with permutations.
· Use the axiom system for groups in determining group structures and their properties.

Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture.).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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Credit Weighting: 5

Teaching Period(s): Teaching Period 1.

No. of Students: Min 20, Max 100.

Pre-requisite(s): MS2001; MS2002

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Professor Bernard Hanzon, School of Mathematical Sciences.

Lecturer(s): Mr David Goulding, School of Mathematical Sciences.

Module Objective: To develop advanced calculus techniques then apply them to quantitative problems in business and management.

Module Content: Multivariable functions, partial derivatives, optimisation problems and Lagrange multipliers.

Learning Outcomes: On successful completion of this module, students should be able to:
· Use the chain rule to compute partial derivatives of functions of several variables;
· Compute equations of tangent planes to two-dimensional surfaces;
· Use partial derivatives to solve problems related to economic concepts such as partial elasticity, production and utility;
· Solve unconstrained optimisation problems for functions of two variables and apply this knowledge to optimisation problems in economics;
· Use the methods of Elimination of Variables and Lagrange Multipliers to solve constrained optimisation problems, including problems from economics.

Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): None.

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward (whether passed or failed).

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MS3005 Transformation Geometry

Credit Weighting: 5

Teaching Period(s): Teaching Period 2.

No. of Students: Min 20, Max 100.

Pre-requisite(s): MS2003

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Dr Ben McKay, Department of Mathematics.

Lecturer(s): Vahid Yazdanpanah, School of Mathematical Sciences.

Module Objective: To provide analysis of axioms and theorems of geometry.

Module Content: Geometrical and algebraic properties of translations, rotations, axial symmetries, isometries, affine transformations and similarity transformations. Applications to ellipses.

Learning Outcomes: On successful completion of this module, students should be able to:
· State the basic concepts of the planar geometry;
· Solve geometric problems using methods from linear algebra;
· Perform computations involving 2x2-systems;
· Formulate the basic properties of conic sections;
· Apply the theory of various transformations of the plane.

Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture.).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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MS3006 History of Mathematics

Credit Weighting: 5

Teaching Period(s): Teaching Period 2.

No. of Students: Min 20, Max 100.

Pre-requisite(s): None

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures.

Module Co-ordinator: Dr Ben McKay, Department of Mathematics.

Lecturer(s): Staff, Department of Mathematics; Ms Cáit Ní Shúilleabháin, School of Mathematical Sciences.

Module Objective: To provide an overview of major developments in Mathematics.

Module Content: The development of Geometry, Algebra and Calculus from ancient times to present day.

Learning Outcomes: On successful completion of this module, students should be able to:
· Describe the development of mathematics from ancient times to the present day.
· Place mathematical events in chronological order.
· Identify at least twenty of the outstanding personalities in the history of mathematics and be able to list their contributions to the subject.
· Describe the social and political environments in which mathematics developed.
· Describe the scientific,economic and military contexts which stimulated mathematicians to promote their subject.
· Explain many mathematical concepts not encountered in undergraduate courses.

Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture.).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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MS3011 Dynamical Systems

Credit Weighting: 5

Teaching Period(s): Teaching Period 2.

No. of Students: Min 20, Max 100.

Pre-requisite(s): MS2001, MS2002, MS2003

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Dr Ben McKay, Department of Mathematics.

Lecturer(s): Mr J.P. McCarthy, School of Mathematical Sciences.

Module Objective: To provide an overview of dynamical systems and their application.

Module Content: Discrete time systems, fixed point and stability analysis, complex dynamics, applications.

Learning Outcomes: On successful completion of this module, students should be able to:
· Compute the iterates of a real or complex valued function of a single variable and the orbits of points,
· Determine the fixed and periodic points of such functions and the nature of these points,
· Investigate the dynamics of families of functions of a real variable,
· Determine the bifurcation points of such families of functions, in particular the families of logistic maps and tent maps,
· Sketch the Julia sets of elementary quadratic maps of a complex variable.

Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s), students will be notified of the format and breakdown of marks for continuous assessment at the first lecture.).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward. whether passed or failed.

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MS3012 Introduction to Statistics

Credit Weighting: 5

Teaching Period(s): Teaching Period 1.

No. of Students: Min 20, Max 100.

Pre-requisite(s): MS1001

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials; 10 x 1hr(s) Practicals.

Module Co-ordinator: Prof Finbarr O'Sullivan, Department of Statistics.

Lecturer(s): Staff, Department of Statistics.

Module Objective: To provide an introduction to fundamental statistical techniques.

Module Content: Understanding data, one variable summaries, introduction to SPSS, multi-variable summaries, probability, statistical studies, the Poisson distribution, the Binomial distribution, the Normal distribution, standard errors, confidence intervals, significance tests and regression analysis.

Learning Outcomes: On successful completion of this module, students should be able to:
· Calculate and interpret descriptive statistics such as the mean, median, standard deviation, quartiles, percentiles, etc.
· Draw and interpret graphical summaries of data e.g. histograms, box plots, stem and leaf plots.
· Calculate probabilities for discrete probability distributions e.g. Binomial distribution and Poisson distribution using probability mass function or statistical tables
· Calculate probabilities for the Normal distribution using the approximation to the Standard Normal distribution.
· Carry out hypothesis tests for one mean and one proportion and make conclusions based on the p-value for the test.
· Compute descriptive statistics and construct graphs using SPSS.
· Model the relationships between variables using Regression Analysis.

Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (SPSS computer project - 15 marks, homework - 10 marks).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. Marks in passed element(s) of Continuous Assessment are carried forward, Failed element(s) of Continuous Assessment must be repeated (as specified by Module Coordinator.).

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MS5005 Discrete Mathematics

Credit Weighting: 5

Teaching Period(s): Teaching Period 1.

No. of Students: Min 5, Max 30.

Pre-requisite(s): MS1001 or equivalent

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Dr Ben McKay, Department of Mathematics.

Lecturer(s): Staff, Department of Mathematics.

Module Objective: To provide an overview of combinatorics, graphs, trees and applications.

Module Content: Induction, recurrence relations. Combinatorics: permutations, combinations, the pigeonhole principle. Discrete probability on finite sample spaces; conditioning and independence. Graph theory: Euler and Hamiltonian paths and cycles, weighted graphs, applications. The students will perform self-directed literature/library assignment. This literature/library assignment is designed to teach students how key discoveries were made, the people behind such discoveries and the scientific landscape at the time the ground breaking research was carried out. The literature project will involve independent research from the available literature in the Boole Library, the department and other literature sources. Students will be encouraged to present their own ideas and interpretations of the literature reviewed and to draw conclusions.

Learning Outcomes: On successful completion of this module, students should be able to:
· Devise and carry out proofs by induction.
· Solve first order recurrence relations by iteration, and solve second order linear recurrence relations with constant coefficients.
· Enumerate sets using combinatorial techniques such as permutation and combinations.
· Utilise the pigeonhole principle
· Derive combinatorial identities.
· Calculate probabilities of events from finite probability spaces, applying conditioning and independence arguments.
· Identify standard properties of given graphs such as the existence of Euler cycles.
· Apply algorithms to calculate quantities associated to graphs, such as the shortest distance between two vertices in a weighted graph.
· Interpret, synthesise, and critically assess the current scientific literature on the topic of this module in a literature review format. Students will learn how key discoveries are made in science and the thinking at the time that enabled these discoveries to be made.

Assessment: Total Marks 100: End of Year Written Examination 50 marks; Continuous Assessment 50 marks (In-class test 15 marks, assigned homework 10 marks, literature assignment 25 marks).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. Marks in passed element(s) of Continuous Assessment are carried forward, Failed element(s) of Continuous Assessment must be repeated.

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