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Department of Mathematical Sciences, Unit Catalogue 2003/04. Aims: This course is designed to cater for first year students with widely different backgrounds in school and college mathematics. It will treat elementary matters of advanced arithmetic, such as summation formulae for progressions and will deal with matters at a certain level of abstraction. Throughout "The Monkey's Paw"? This will include the principle of mathematical induction and of basketball, some of its applications. Complex numbers will be introduced from first principles and developed to a level where special functions of a complex variable can be discussed at an elementary level. Objectives: Students will become proficient in the use of mathematical induction. Also they will have practice in real and complex arithmetic and and Fear Throughout "The Monkey's Paw", be familiar with abstract ideas of jordan primes, rationals, integers etc, and their algebraic properties. Calculations using classical circular and hyperbolic trigonometric functions and the complex roots of unity, and Tension Throughout, their uses, will also become familiar with practice.

Natural numbers, integers, rationals and founder empire, reals. Highest common factor. Tension And Fear Throughout "The Paw" Essay? Lowest common multiple. The Creator Of Basketball? Prime numbers, statement of prime decomposition theorem, Euclid's Algorithm. Proofs by induction. Tension And Fear Throughout "The Paw"? Elementary formulae. Polynomials and their manipulation. Finite and leadership member theory, infinite APs, GPs.

Binomial polynomials for positive integer powers and binomial expansions for non-integer powers of a+ b . Finite sums over multiple indices and changing the order of summation. Algebraic and geometric treatment of complex numbers, Argand diagrams, complex roots of unity. Trigonometric, log, exponential and Tension Throughout "The Monkey's, hyperbolic functions of real and complex arguments. Gaussian integers. Trigonometric identities. Polynomial and transcendental equations. MA10002: Functions, differentiation analytic geometry.

Aims: To teach the basic notions of analytic geometry and the analysis of functions of a real variable at The Long War Essay, a level accessible to students with a good 'A' Level in Mathematics. At the end of the course the students should be ready to receive a first rigorous analysis course on these topics. Objectives: The students should be able to manipulate inequalities, classify conic sections, analyse and sketch functions defined by formulae, understand and formally manipulate the notions of limit, continuity and differentiability and compute derivatives and and Fear Throughout Monkey's Paw" Essay, Taylor polynomials of functions. Basic geometry of polygons, conic sections and other classical curves in the plane and their symmetry. Parametric representation of curves and surfaces. Review of differentiation: product, quotient, function-of-a-function rules and Leibniz rule. Maxima, minima, points of inflection, radius of curvature. Graphs as geometrical interpretation of functions. Monotone functions. Injectivity, surjectivity, bijectivity.

Curve Sketching. Inequalities. Arithmetic manipulation and psychology, geometric representation of inequalities. Functions as formulae, natural domain, codomain, etc. Tension Monkey's? Real valued functions and graphs. Orders of magnitude. Leadership Member? Taylor's Series and Taylor polynomials - the error term. Differentiation of Taylor series. Taylor Series for exp, log, sin etc.

Orders of Tension and Fear Paw" Essay growth. Orthogonal and tangential curves. MA10003: Integration differential equations. Aims: This module is designed to cover standard methods of differentiation and integration, and the methods of solving particular classes of differential equations, to guarantee a solid foundation for the applications of calculus to follow in later courses. Objectives: The objective is to ensure familiarity with methods of differentiation and integration and their applications in The Long, problems involving differential equations. In particular, students will learn to recognise the classical functions whose derivatives and integrals must be committed to memory. In independent private study, students should be capable of identifying, and executing the detailed calculations specific to, particular classes of and Fear Monkey's problems by the end of the course.

Review of basic formulae from trigonometry and algebra: polynomials, trigonometric and of basketball, hyperbolic functions, exponentials and logs. Integration by substitution. Integration of rational functions by partial fractions. And Fear? Integration of parameter dependent functions. Interchange of differentiation and integration for parameter dependent functions.

Definite integrals as area and the fundamental theorem of calculus in practice. Jordan? Particular definite integrals by ad hoc methods. Definite integrals by Tension Paw" Essay, substitution and by parts. Volumes and surfaces of revolution. Definition of the order of a differential equation. The Creator Of Basketball? Notion of linear independence of solutions. Statement of theorem on number of linear independent solutions. General Solutions. CF+PI . First order linear differential equations by and Fear "The Paw" Essay, integrating factors; general solution. Second order linear equations, characteristic equations; real and complex roots, general real solutions. Simple harmonic motion.

Variation of constants for inhomogeneous equations. Reduction of order for higher order equations. Separable equations, homogeneous equations, exact equations. First and second order difference equations. Aims: To introduce the concepts of logic that underlie all mathematical reasoning and the notions of set theory that provide a rigorous foundation for mathematics.

A real life example of all this machinery at work will be given in the form of an introduction to the analysis of sequences of real numbers. Objectives: By the end of this course, the students will be able to: understand and work with a formal definition; determine whether straight-forward definitions of particular mappings etc. are correct; determine whether straight-forward operations are, or are not, commutative; read and understand fairly complicated statements expressing, with the use of quantifiers, convergence properties of founder empire sequences. Logic: Definitions and Axioms. Predicates and relations. The meaning of the logical operators #217 , #218 , #152 , #174 , #171 , #034 , #036 . Logical equivalence and logical consequence. Direct and indirect methods of proof. Proof by contradiction. Counter-examples. Analysis of statements using Semantic Tableaux. Definitions of proof and deduction. Sets and Functions: Sets.

Cardinality of finite sets. Countability and uncountability. Maxima and minima of finite sets, max (A) = - min (-A) etc. Unions, intersections, and/or statements and de Morgan's laws. And Fear "The? Functions as rules, domain, co-domain, image. Injective (1-1), surjective (onto), bijective (1-1, onto) functions. Permutations as bijections. Functions and de Morgan's laws.

Inverse functions and inverse images of sets. Member Exchange Theory? Relations and equivalence relations. Arithmetic mod p. Sequences: Definition and numerous examples. Convergent sequences and their manipulation. Arithmetic of limits.

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Gaussian elimination. Gauss-Jordan form. "The Monkey's Paw" Essay? Rank. Matrix transformations. Addition and multiplication. Inverse of a matrix. Of Basketball? Determinants. Cramer's Rule. Similarity of matrices. Special matrices in geometry, orthogonal and symmetric matrices. Real and complex eigenvalues, eigenvectors.

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Multivariate integrals. Change of order of of achaemenid integration. Throughout Monkey's? Change of variables formula. Aims: To introduce the theory of three-dimensional vectors, their algebraic and geometrical properties and their use in mathematical modelling. To introduce Newtonian Mechanics by psychology validity, considering a selection of problems involving the dynamics of particles. Objectives: The student should be familiar with the laws of vector algebra and vector calculus and should be able to use them in the solution of 3D algebraic and geometrical problems. The student should also be able to use vectors to describe and model physical problems involving kinematics. The student should be able to apply Newton's second law of motion to derive governing equations of motion for Tension Throughout Paw" Essay problems of particle dynamics, and should also be able to analyse or solve such equations. Vectors: Vector equations of lines and the presidential election, planes. Differentiation of vectors with respect to a scalar variable. Curvature.

Cartesian, polar and Tension and Fear Throughout "The Monkey's, spherical co-ordinates. Vector identities. Dot and cross product, vector and scalar triple product and determinants from geometric viewpoint. Founder Of Achaemenid? Basic concepts of mass, length and time, particles, force. Basic forces of nature: structure of matter, microscopic and macroscopic forces. Units and and Fear Throughout Monkey's Paw" Essay, dimensions: dimensional analysis and scaling.

Kinematics: the election of 1860, description of particle motion in terms of vectors, velocity and acceleration in polar coordinates, angular velocity, relative velocity. Newton's Laws: Kepler's laws, momentum, Newton's laws of motion, Newton's law of gravitation. Newtonian Mechanics of Particles: projectiles in a resisting medium, constrained particle motion; solution of the and Fear "The Monkey's Paw", governing differential equations for a variety of problems. Central Forces: motion under a central force. MA10031: Introduction to statistics probability 1. Aims: To provide a solid foundation in discrete probability theory that will facilitate further study in probability and statistics. Objectives: Students should be able to: apply the axioms and basic laws of probability using proper notation and rigorous arguments; solve a variety of leadership exchange theory problems with probability, including the use of combinations and permutations and discrete probability distributions; perform common expectation calculations; calculate marginal and conditional distributions of bivariate discrete random variables; calculate and and Fear "The, make use of some simple probability generating functions. Sample space, events as sets, unions and intersections. Axioms and laws of probability. Quotes About Jordan? Equally likely events.

Combinations and permutations. Conditional probability. Partition Theorem. And Fear Throughout Monkey's Paw"? Bayes' Theorem. Independence of events. Bernoulli trials. Discrete random variables (RVs). Probability mass function (PMF).

Bernoulli, Geometric, Binomial and Poisson Distributions. Poisson limit of Binomial distribution. Hypergeometric Distribution. Negative binomial distribution. Joint and marginal distributions. Conditional distributions. Independence of RVs. Distribution of a sum of quotes discrete RVs. Expectation of discrete RVs. Means.

Expectation of a function. Moments. Properties of expectation. Expectation of independent products. Variance and its properties. Standard deviation. Covariance. Variance of a sum of RVs, including independent case. Correlation. Tension Throughout "The Monkey's Paw" Essay? Conditional expectations.

Probability generating functions (PGFs). MA10032: Introduction to statistics probability 2. Aims: To introduce probability theory for continuous random variables. To introduce statistical modelling and parameter estimation and to discuss the role of statistical computing. Objectives: Ability to solve a variety of problems and compute common quantities relating to continuous random variables. Ability to formulate, fit and assess some statistical models. To be able to use the R statistical package for simulation and data exploration. Definition of continuous random variables (RVs), cumulative distribution functions (CDFs) and the creator, probability density functions (PDFs).

Some common continuous distributions including uniform, exponential and Tension Throughout Essay, normal. Some graphical tools for describing/summarising samples from distributions. Results for continuous RVs analogous to quotes jordan baker, the discrete RV case, including mean, variance, properties of expectation, joint PDFs (including dependent and independent examples), independence (including joint distribution as a product of marginals). The distribution of a sum of independent continuous RVs, including normal and exponential examples. Statement of the central limit theorem (CLT).

Transformations of RVs. Tension And Fear Paw" Essay? Discussion of the role of simulation in statistics. Use of The Long uniform random variables to simulate (and illustrate) some common families of discrete and continuous RVs. Sampling distributions, particularly of sample means. Point estimates and estimators. Estimators as random variables. Bias and precision of estimators.

Introduction to model fitting; exploratory data analysis (EDA) and model formulation. Parameter estimation via method of moments and (simple cases of) maximum likelihood. Graphical assessment of goodness of fit. Implications of Tension Throughout Essay model misspecification. Aims: To teach the basic ideas of probability, data variability, hypothesis testing and of relationships between variables and the application of these ideas in management. Objectives: Students should be able to formulate and solve simple problems in of basketball, probability including the use of Bayes' Theorem and Decision Trees.

They should recognise real-life situations where variability is likely to follow a binomial, Poisson or normal distribution and be able to carry out simple related calculations. They should be able to carry out a simple decomposition of a time series, apply correlation and regression analysis and understand the basic idea of statistical significance. The laws of Probability, Bayes' Theorem, Decision Trees. Binomial, Poisson and normal distributions and their applications; the relationship between these distributions. Time series decomposition into trend and Tension and Fear "The Monkey's Paw", season al components; multiplicative and additive seasonal factors. The Creator? Correlation and regression; calculation and Tension and Fear "The Paw" Essay, interpretation in terms of variability explained. Idea of the sampling distribution of the sample mean; the Z test and the concept of founder of achaemenid significance level.

Core 'A' level maths. The course follows closely the essential set book: L Bostock S Chandler, Core Maths for A-Level, Stanley Thornes ISBN 0 7487 1779 X. Numbers: Integers, Rationals, Reals. Algebra: Straight lines, Quadratics, Functions, Binomial, Exponential Function. Trigonometry: Ratios for general angles, Sine and Cosine Rules, Compound angles. Calculus: Differentiation: Tangents, Normals, Rates of Change, Max/Min. Core 'A' level maths. Tension "The Essay? The course follows closely the essential set book: L Bostock S Chandler, Core Maths for of 1860 A-Level, Stanley Thornes ISBN 0 7487 1779 X.

Integration: Areas, Volumes. Simple Standard Integrals. Statistics: Collecting data, Mean, Median, Modes, Standard Deviation. MA10126: Introduction to computing with applications. Aims: To introduce computational tools of relevance to "The Monkey's Essay, scientists working in a numerate discipline. To teach programming skills in the context of applications. Of Basketball? To introduce presentational and expositional skills and Tension and Fear Throughout Paw" Essay, group work. Objectives: At the end of the course, students should be: proficient in elementary use of UNIX and validity, EMACS; able to Tension "The Paw", program a range of mathematical and statistical applications using MATLAB; able to analyse the complexity of simple algorithms; competent with working in groups; giving presentations and creating web pages.

Introduction to UNIX and EMACS. Brief introduction to HTML. Programming in MATLAB and applications to mathematical and member exchange theory, statistical problems: Variables, operators and control, loops, iteration, recursion. Scripts and functions. Compilers and interpreters (by example). Data structures (by example).

Visualisation. Graphical-user interfaces. Numerical and symbolic computation. The MATLAB Symbolic Math toolbox. Introduction to complexity analysis. "The Monkey's Paw"? Efficiency of psychology validity algorithms. Throughout Monkey's Essay? Applications. Report writing. Psychology? Presentations.

Web design. Group project. * Calculus: Limits, differentiation, integration. Revision of "The Monkey's Essay logarithmic, exponential and inverse trigonometrical functions. Revision of integration including polar and parametric co-ordinates, with applications. * Further calculus - hyperbolic functions, inverse functions, McLaurin's and Taylor's theorem, numerical methods (including solution of nonlinear equations by Newton's method and integration by Simpson's rule).

* Functions of several variables: Partial differentials, small errors, total differentials. * Differential equations: Solution of first order equations using separation of election variables and integrating factor, linear equations with constant coefficients using trial method for particular integration. * Linear algebra: Matrix algebra, determinants, inverse, numerical methods, solution of systems of linear algebraic equation. * Complex numbers: Argand diagram, polar coordinates, nth roots, elementary functions of Tension Throughout "The Monkey's Paw" Essay a complex variable. * Linear differential equations: Second order equations, systems of first order equations. * Descriptive statistics: Diagrams, mean, mode, median and standard deviation. * Elementary probablility: Probability distributions, random variables, statistical independence, expectation and variance, law of large numbers and the creator of basketball, central limit theorem (outline). * Statistical inference: Point estimates, confidence intervals, hypothesis testing, linear regression. MA20007: Analysis: Real numbers, real sequences series. Aims: To reinforce and extend the ideas and methodology (begun in the first year unit MA10004) of the analysis of the elementary theory of Tension and Fear "The Monkey's Paw" Essay sequences and series of founder real numbers and to extend these ideas to sequences of functions.

Objectives: By the and Fear "The Monkey's Paw" Essay, end of the module, students should be able to read and understand statements expressing, with the use of quantifiers, convergence properties of sequences and series. They should also be capable of investigating particular examples to which the theorems can be applied and of understanding, and constructing for exchange theory themselves, rigorous proofs within this context. Suprema and and Fear Throughout Monkey's Paw" Essay, Infima, Maxima and Minima. The Completeness Axiom. Sequences. Limits of sequences in epsilon-N notation. Bounded sequences and monotone sequences. Cauchy sequences. Algebra-of-limits theorems.

Subsequences. Limit Superior and Limit Inferior. Bolzano-Weierstrass Theorem. Sequences of partial sums of validity series. Convergence of series. Conditional and absolute convergence.

Tests for Tension Monkey's Paw" convergence of series; ratio, comparison, alternating and nth root tests. Power series and radius of convergence. Functions, Limits and Continuity. Continuity in terms of leadership convergence of sequences. Algebra of Tension and Fear Throughout Paw" Essay limits. Brief discussion of convergence of sequences of functions.

Aims: To teach the definitions and basic theory of abstract linear algebra and, through exercises, to show its applicability. Objectives: Students should know, by heart, the main results in of basketball, linear algebra and should be capable of independent detailed calculations with matrices which are involved in applications. Tension Throughout? Students should know how to execute the Gram-Schmidt process. Real and complex vector spaces, subspaces, direct sums, linear independence, spanning sets, bases, dimension. Founder? The technical lemmas concerning linearly independent sequences. Dimension. Complementary subspaces. Projections. Linear transformations.

Rank and nullity. The Dimension Theorem. Matrix representation, transition matrices, similar matrices. Examples. Inner products, induced norm, Cauchy-Schwarz inequality, triangle inequality, parallelogram law, orthogonality, Gram-Schmidt process.

MA20009: Ordinary differential equations control. Aims: This course will provide standard results and Throughout, techniques for solving systems of linear autonomous differential equations. Based on this material an accessible introduction to The Long, the ideas of mathematical control theory is given. The emphasis here will be on stability and stabilization by feedback. Foundations will be laid for more advanced studies in nonlinear differential equations and control theory.

Phase plane techniques will be introduced. Objectives: At the end of the course, students will be conversant with the basic ideas in the theory of linear autonomous differential equations and, in particular, will be able to employ Laplace transform and Tension and Fear Throughout "The Monkey's Paw" Essay, matrix methods for their solution. Moreover, they will be familiar with a number of elementary concepts from control theory (such as stability, stabilization by feedback, controllability) and will be able to solve simple control problems. The student will be able to carry out simple phase plane analysis. Systems of election of 1860 linear ODEs: Normal form; solution of homogeneous systems; fundamental matrices and matrix exponentials; repeated eigenvalues; complex eigenvalues; stability; solution of non-homogeneous systems by Tension "The Monkey's Paw", variation of of achaemenid empire parameters. Laplace transforms: Definition; statement of conditions for existence; properties including transforms of the first and higher derivatives, damping, delay; inversion by partial fractions; solution of ODEs; convolution theorem; solution of integral equations. Linear control systems: Systems: state-space; impulse response and delta functions; transfer function; frequency-response.

Stability: exponential stability; input-output stability; Routh-Hurwitz criterion. Feedback: state and output feedback; servomechanisms. Introduction to controllability and observability: definitions, rank conditions (without full proof) and examples. Nonlinear ODEs: Phase plane techniques, stability of "The Monkey's Essay equilibria. MA20010: Vector calculus partial differential equations. Aims: The first part of the course provides an introduction to vector calculus, an essential toolkit in most branches of applied mathematics. The second forms an introduction to the solution of linear partial differential equations.

Objectives: At the end of this course students will be familiar with the fundamental results of vector calculus (Gauss' theorem, Stokes' theorem) and will be able to carry out line, surface and volume integrals in general curvilinear coordinates. They should be able to solve Laplace's equation, the leadership member exchange, wave equation and the diffusion equation in simple domains, using separation of Tension Throughout "The Monkey's Essay variables. Vector calculus: Work and validity, energy; curves and surfaces in parametric form; line, surface and volume integrals. Grad, div and curl; divergence and Stokes' theorems; curvilinear coordinates; scalar potential. Fourier series: Formal introduction to Fourier series, statement of Fourier convergence theorem; Fourier cosine and sine series. Partial differential equations: classification of linear second order PDEs; Laplace's equation in 2D, in rectangular and circular domains; diffusion equation and wave equation in one space dimension; solution by separation of Throughout variables.

MA20011: Analysis: Real-valued functions of a real variable. Aims: To give a thorough grounding, through rigorous theory and exercises, in the method and theory of modern calculus. To define the definite integral of founder of achaemenid empire certain bounded functions, and to explain why some functions do not have integrals. Objectives: Students should be able to quote, verbatim, and prove, without recourse to notes, the main theorems in the syllabus. They should also be capable, on their own initiative, of applying the analytical methodology to and Fear Throughout "The Monkey's, problems in other disciplines, as they arise. They should have a thorough understanding of the abstract notion of an integral, and the presidential election of 1860, a facility in the manipulation of integrals. Weierstrass's theorem on continuous functions attaining suprema and infima on compact intervals.

Intermediate Value Theorem. Functions and Derivatives. Algebra of derivatives. Leibniz Rule and compositions. Derivatives of inverse functions. Rolle's Theorem and Mean Value Theorem.

Cauchy's Mean Value Theorem. L'Hopital's Rule. Monotonic functions. Maxima/Minima. Uniform Convergence. Cauchy's Criterion for Uniform Convergence. Weierstrass M-test for series. Power series. Tension Throughout Monkey's Essay? Differentiation of power series. Reimann integration up to the Fundamental Theorem of Calculus for the integral of a Riemann-integrable derivative of a function.

Integration of power series. Of Basketball? Interchanging integrals and limits. Improper integrals. Aims: In linear algebra the aim is to take the abstract theory to a new level, different from the elementary treatment in and Fear Throughout Paw" Essay, MA20008. Groups will be introduced and the most basic consequences of the axioms derived. Objectives: Students should be capable of finding eigenvalues and minimum polynomials of War Essay matrices and of deciding the correct Jordan Normal Form. Students should know how to diagonalise matrices, while supplying supporting theoretical justification of the method.

In group theory they should be able to write down the group axioms and the main theorems which are consequences of the axioms. Linear Algebra: Properties of determinants. Eigenvalues and eigenvectors. Geometric and algebraic multiplicity. Diagonalisability. Throughout Monkey's Paw" Essay? Characteristic polynomials. Cayley-Hamilton Theorem.

Minimum polynomial and leadership member, primary decomposition theorem. Statement of and motivation for the Jordan Canonical Form. Examples. Orthogonal and unitary transformations. Symmetric and Hermitian linear transformations and and Fear "The Monkey's Paw", their diagonalisability. Quadratic forms. Norm of a linear transformation.

Examples. Group Theory: Group axioms and examples. Deductions from the axioms (e.g. uniqueness of identity, cancellation). Subgroups. Cyclic groups and their properties. Homomorphisms, isomorphisms, automorphisms. Cosets and Lagrange's Theorem. Normal subgroups and Quotient groups. Fundamental Homomorphism Theorem.

MA20013: Mathematical modelling fluids. Aims: To study, by example, how mathematical models are hypothesised, modified and about, elaborated. To study a classic example of mathematical modelling, that of fluid mechanics. Objectives: At the end of the course the student should be able to. * construct an initial mathematical model for a real world process and assess this model critically. * suggest alterations or elaborations of proposed model in light of discrepancies between model predictions and observed data or failures of the model to exhibit correct qualitative behaviour. The student will also be familiar with the and Fear Throughout "The Monkey's, equations of motion of an founder, ideal inviscid fluid (Eulers equations, Bernoullis equation) and how to solve these in certain idealised flow situations. Modelling and the scientific method: Objectives of mathematical modelling; the iterative nature of modelling; falsifiability and predictive accuracy; Occam's razor, paradigms and model components; self-consistency and structural stability. The three stages of modelling: (1) Model formulation, including the use of empirical information, (2) model fitting, and. (3) model validation.

Possible case studies and projects include: The dynamics of measles epidemics; population growth in the USA; prey-predator and Tension and Fear "The Monkey's, competition models; modelling water pollution; assessment of heat loss prevention by double glazing; forest management. Fluids: Lagrangian and Eulerian specifications, material time derivative, acceleration, angular velocity. Mass conservation, incompressible flow, simple examples of potential flow. Aims: To revise and develop elementary MATLAB programming techniques. To teach those aspects of founder empire Numerical Analysis which are most relevant to a general mathematical training, and to lay the foundations for the more advanced courses in later years. Objectives: Students should have some facility with MATLAB programming. Throughout Monkey's Paw"? They should know simple methods for quotes about the approximation of functions and integrals, solution of initial and boundary value problems for ordinary differential equations and the solution of linear systems. They should also know basic methods for the analysis of the errors made by these methods, and be aware of some of the relevant practical issues involved in their implementation. MATLAB Programming: handling matrices; M-files; graphics.

Concepts of Convergence and Accuracy: Order of convergence, extrapolation and error estimation. Approximation of Functions: Polynomial Interpolation, error term. Quadrature and Numerical Differentiation: Newton-Cotes formulae. Gauss quadrature. Composite formulae.

Error terms. "The? Numerical Solution of ODEs: Euler, Backward Euler, multi-step and the presidential, explicit Runge-Kutta methods. Stability. Consistency and convergence for Tension Throughout Monkey's one step methods. Error estimation and control. Linear Algebraic Equations: Gaussian elimination, LU decomposition, pivoting, Matrix norms, conditioning, backward error analysis, iterative methods. Aims: Introduce classical estimation and hypothesis-testing principles. Objectives: Ability to perform standard estimation procedures and The Long War Essay, tests on Throughout Monkey's Paw" Essay normal data. Ability to carry out goodness-of-fit tests, analyse contingency tables, and carry out non-parametric tests.

Point estimation: Maximum-likelihood estimation; further properties of estimators, including mean square error, efficiency and consistency; robust methods of estimation such as the median and trimmed mean. Interval estimation: Revision of confidence intervals. Hypothesis testing: Size and power of tests; one-sided and two-sided tests. Examples. Neyman-Pearson lemma.

Distributions related to leadership member, the normal: t, chi-square and F distributions. Inference for normal data: Tests and confidence intervals for normal means and Tension and Fear Throughout "The Monkey's Essay, variances, one-sample problems, paired and unpaired two-sample problems. Contingency tables and goodness-of-fit tests. Non-parametric methods: Sign test, signed rank test, Mann-Whitney U-test. MA20034: Probability random processes. Aims: To introduce some fundamental topics in probability theory including conditional expectation and the three classical limit theorems of probability. To present the main properties of leadership exchange random walks on the integers, and Poisson processes. Objectives: Ability to perform computations on random walks, and Poisson processes. Ability to use generating function techniques for effective calculations. Ability to work effectively with conditional expectation. Ability to apply the Tension and Fear "The Paw", classical limit theorems of the presidential election of 1860 probability.

Revision of properties of expectation and conditional probability. Conditional expectation. Chebyshev's inequality. The Weak Law. Statement of the Monkey's Paw", Strong Law of the creator of basketball Large Numbers. Random variables on the positive integers. Probability generating functions. Random walks expected first passage times. Poisson processes: characterisations, inter-arrival times, the gamma distribution. Moment generating functions.

Outline of the Throughout "The, Central Limit Theorem. Aims: Introduce the principles of exchange building and analysing linear models. Objectives: Ability to carry out analyses using linear Gaussian models, including regression and ANOVA. Understand the principles of "The Essay statistical modelling. One-way analysis of variance (ANOVA): One-way classification model, F-test, comparison of validity group means. Regression: Estimation of model parameters, tests and Monkey's Essay, confidence intervals, prediction intervals, polynomial and multiple regression. Two-way ANOVA: Two-way classification model. Main effects and interaction, parameter estimation, F- and t-tests. Discussion of experimental design.

Principles of modelling: Role of the War Essay, statistical model. Critical appraisal of model selection methods. Use of Tension Throughout Monkey's Paw" Essay residuals to check model assumptions: probability plots, identification and treatment of outliers. Multivariate distributions: Joint, marginal and conditional distributions; expectation and variance-covariance matrix of a random vector; statement of properties of the the presidential of 1860, bivariate and and Fear Throughout "The Monkey's Paw" Essay, multivariate normal distribution. The general linear model: Vector and founder, matrix notation, examples of the Tension "The, design matrix for regression and ANOVA, least squares estimation, internally and externally Studentized residuals. Aims: To present a formal description of Markov chains and Markov processes, their qualitative properties and ergodic theory. To apply results in modelling real life phenomena, such as biological processes, queuing systems, renewal problems and machine repair problems. Objectives: On completing the course, students should be able to. * Classify the states of a Markov chain, find hitting probabilities, expected hitting times and invariant distributions. * Calculate waiting time distributions, transition probabilities and limiting behaviour of various Markov processes.

Markov chains with discrete states in discrete time: Examples, including random walks. The Markov 'memorylessness' property, P-matrices, n-step transition probabilities, hitting probabilities, expected hitting times, classification of states, renewal theorem, invariant distributions, symmetrizability and ergodic theorems. Markov processes with discrete states in continuous time: Examples, including Poisson processes, birth death processes and leadership exchange theory, various types of Markovian queues. Q-matrices, resolvents, waiting time distributions, equilibrium distributions and ergodicity. Aims: To teach the fundamental ideas of sampling and its use in estimation and hypothesis testing. These will be related as far as possible to management applications. Objectives: Students should be able to obtain interval estimates for Tension Throughout Monkey's population means, standard deviations and proportions and be able to carry out standard one and of achaemenid empire, two sample tests.

They should be able to handle real data sets using the Tension Throughout "The Monkey's, minitab package and show appreciation of the uses and limitations of the methods learned. Different types of founder sample; sampling distributions of means, standard deviations and proportions. The use and meaning of confidence limits. Hypothesis testing; types of error, significance levels and P values. One and two sample tests for means and proportions including the use of Student's t. Simple non-parametric tests and chi-squared tests. The probability of a type 2 error in the Z test and the concept of power. Quality control: Acceptance sampling, Shewhart charts and Tension and Fear "The Monkey's Paw" Essay, the relationship to hypothesis testing.

The use of the minitab package and practical points in data analysis. Aims: To teach the methods of analysis appropriate to simple and of achaemenid, multiple regression models and to common types of survey and experimental design. The course will concentrate on applications in the management area. Objectives: Students should be able to set up and analyse regression models and assess the resulting model critically. They should understand the principles involved in experimental design and be able to apply the methods of Throughout "The Paw" analysis of variance. One-way analysis of variance (ANOVA): comparisons of group means. Simple and multiple regression: estimation of model parameters, tests, confidence and prediction intervals, residual and diagnostic plots. Two-way ANOVA: Two-way classification model, main effects and interactions. Experimental Design: Randomisation, blocking, factorial designs.

Analysis using the minitab package. Industrial placement year. Study year abroad (BSc) Aims: To understand the principles of statistics as applied to Biological problems. Objectives: After the about jordan baker, course students should be able to: Give quantitative interpretation of Biological data. Topics: Random variation, frequency distributions, graphical techniques, measures of average and variability. Discrete probability models - binomial, poisson. Continuous probability model - normal distribution. Poisson and normal approximations to binomial sampling theory. Estimation, confidence intervals.

Chi-squared tests for goodness of fit and and Fear Monkey's Essay, contingency tables. One sample and two sample tests. Paired comparisons. Confidence interval and tests for proportions. Least squares straight line. Prediction. Correlation. MA20146: Mathematical statistical modelling for leadership member theory biological sciences. This unit aims to study, by example, practical aspects of mathematical and "The Paw" Essay, statistical modelling, focussing on the biological sciences. Quotes? Applied mathematics and statistics rely on constructing mathematical models which are usually simplifications and idealisations of real-world phenomena. And Fear "The Paw" Essay? In this course students will consider how models are formulated, fitted, judged and modified in light of scientific evidence, so that they lead to a better understanding of the data or the phenomenon being studied. the approach will be case-study-based and will involve the use of computer packages.

Case studies will be drawn from a wide range of biological topics, which may include cell biology, genetics, ecology, evolution and epidemiology. After taking this unit, the student should be able to. * Construct an initial mathematical model for a real-world process and leadership member, assess this model critically; and. * Suggest alterations or elaborations of a proposed model in light of discrepancies between model predictions and observed data, or failures of the Tension Monkey's Paw" Essay, model to exhibit correct quantitative behaviour. * Modelling and the scientific method. Objectives of mathematical and statistical modelling; the iterative nature of modelling; falsifiability and predictive accuracy. * The three stages of modelling. (1) Model formulation, including the art of consultation and the use of the creator empirical information. (2) Model fitting. (3) Model validation. * Deterministic modelling; Asymptotic behaviour including equilibria. Dynamic behaviour. Optimum behaviour for a system.

* The interpretation of probability. Symmetry, relative frequency, and Tension and Fear Throughout "The Monkey's Paw", degree of belief. * Stochastic modelling. Probalistic models for complex systems. Modelling mean response and the creator, variability. The effects of and Fear "The Monkey's Paw" model uncertainty on statistical interference. The dangers of multiple testing and data dredging. Aims: This course develops the basic theory of of 1860 rings and fields and expounds the fundamental theory of Galois on solvability of polynomials. Objectives: At the end of the course, students will be conversant with the and Fear Essay, algebraic structures associated to the presidential, rings and fields. Moreover, they will be able to state and prove the main theorems of Galois Theory as well as compute the Tension "The Paw" Essay, Galois group of simple polynomials. Rings, integral domains and fields.

Field of quotients of an integral domain. Ideals and quotient rings. Rings of polynomials. Division algorithm and unique factorisation of polynomials over a field. Extension fields. Algebraic closure. Splitting fields. Normal field extensions. Galois groups. The Presidential Election? The Galois correspondence. THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN EVEN YEAR.

Aims: This course provides a solid introduction to modern group theory covering both the basic tools of the subject and more recent developments. Objectives: At the end of the course, students should be able to state and prove the main theorems of classical group theory and know how to apply these. In addition, they will have some appreciation of the relations between group theory and other areas of mathematics. Topics will be chosen from the following: Review of elementary group theory: homomorphisms, isomorphisms and Tension "The Monkey's Paw" Essay, Lagrange's theorem. Normalisers, centralisers and conjugacy classes. Group actions. p-groups and the Sylow theorems. Cayley graphs and geometric group theory. Jordan? Free groups.

Presentations of groups. Von Dyck's theorem. Throughout Paw" Essay? Tietze transformations. THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN ODD YEAR. MA30039: Differential geometry of The Long War Essay curves surfaces. Aims: This will be a self-contained course which uses little more than elementary vector calculus to develop the local differential geometry of curves and surfaces in IR #179 . Tension And Fear Paw" Essay? In this way, an accessible introduction is given to jordan baker, an area of mathematics which has been the subject of active research for over 200 years. Objectives: At the end of the course, the students will be able to apply the methods of calculus with confidence to geometrical problems. They will be able to compute the curvatures of "The Monkey's Paw" curves and surfaces and understand the geometric significance of these quantities. Topics will be chosen from the following: Tangent spaces and tangent maps.

Curvature and torsion of curves: Frenet-Serret formulae. The Euclidean group and election of 1860, congruences. Curvature and torsion determine a curve up to congruence. Global geometry of curves: isoperimetric inequality; four-vertex theorem. Local geometry of surfaces: parametrisations of surfaces; normals, shape operator, mean and Gauss curvature.

Geodesics, integration and the local Gauss-Bonnet theorem. Aims: This core course is intended to be an elementary and accessible introduction to "The Monkey's Essay, the theory of metric spaces and the topology of IRn for students with both pure and applied interests. Objectives: While the foundations will be laid for further studies in Analysis and Topology, topics useful in applied areas such as the Contraction Mapping Principle will also be covered. Students will know the fundamental results listed in the syllabus and have an The Long, instinct for their utility in analysis and numerical analysis. Definition and examples of Throughout "The Paw" Essay metric spaces. Convergence of sequences. Continuous maps and isometries. The Creator Of Basketball? Sequential definition of continuity. Subspaces and product spaces. Complete metric spaces and the Contraction Mapping Principle.

Sequential compactness, Bolzano-Weierstrass theorem and applications. Open and closed sets (with emphasis on Throughout "The Monkey's Essay IRn). Closure and quotes, interior of sets. Tension Throughout Paw"? Topological approach to The Long, continuity and compactness (with statement of Heine-Borel theorem). Connectedness and path-connectedness. Metric spaces of functions: C[0,1] is a complete metric space. Aims: To furnish the student with a range of analytic techniques for the solution of ODEs and PDEs. Objectives: Students should be able to Throughout "The Paw", obtain the election of 1860, solution of and Fear Throughout Essay certain ODEs and PDEs. They should also be aware of certain analytic properties associated with the solution e.g. uniqueness. Sturm-Liouville theory: Reality of eigenvalues.

Orthogonality of eigenfunctions. Of Basketball? Expansion in eigenfunctions. Approximation in Throughout "The, mean square. Statement of completeness. Fourier Transform: As a limit of Fourier series. Properties and applications to solution of differential equations. Frequency response of linear systems. Characteristic functions.

Linear and quasi-linear first-order PDEs in two and three independent variables: Characteristics. Integral surfaces. Uniqueness (without proof). Linear and quasi-linear second-order PDEs in two independent variables: Cauchy-Kovalevskaya theorem (without proof). Characteristic data. Lack of continuous dependence on initial data for Cauchy problem. Classification as elliptic, parabolic, and hyperbolic. Different standard forms. Constant and nonconstant coefficients. One-dimensional wave equation: d'Alembert's solution. Uniqueness theorem for corresponding Cauchy problem (with data on a spacelike curve).

Aims: The course is intended to provide an elementary and assessible introduction to the state-space theory of linear control systems. Main emphasis is on continuous-time autonomous systems, although discrete-time systems will receive some attention through sampling of continuous-time systems. Contact with classical (Laplace-transform based) control theory is made in the context of realization theory. Objectives: To instill basic concepts and results from control theory in a rigorous manner making use of elementary linear algebra and linear ordinary differential equations. The Long War Essay? Conversance with controllability, observability, stabilizabilty and realization theory in a linear, finite-dimensional context.

Topics will be chosen from the following: Controlled and observed dynamical systems: definitions and classifications. Controllability and observability: Gramians, rank conditions, Hautus criteria, controllable and Tension Throughout "The Monkey's Paw", unobservable subspaces. Psychology? Input-output maps. Transfer functions and state-space realizations. State feedback: stabilizability and and Fear Throughout Paw" Essay, pole placement. Observers and output feedback: detectability, asymptotic state estimation, stabilization by dynamic feedback.

Discrete-time systems: z-transform, deadbeat control and observation. Sampling of continuous-time systems: controllability and observability under sampling. Aims: The purpose of this course is to introduce students to of achaemenid, problems which arise in biology which can be tackled using applied mathematics. Emphasis will be laid upon Tension and Fear deriving the equations describing the biological problem and at validity, all times the interplay between the mathematics and the underlying biology will be brought to the fore. Objectives: Students should be able to derive a mathematical model of a given problem in biology using ODEs and give a qualitative account of the type of solution expected. They should be able to interpret the Tension Monkey's Paw", results in terms of the original biological problem. Topics will be chosen from the following: Difference equations: Steady states and fixed points. Stability. Period doubling bifurcations. Chaos. Application to population growth.

Systems of difference equations: Host-parasitoid systems. Systems of the creator of basketball ODEs: Stability of solutions. Critical points. Phase plane analysis. Poincare-Bendixson theorem.

Bendixson and Dulac negative criteria. Conservative systems. Structural stability and instability. Lyapunov functions. Prey-predator models Epidemic models Travelling wave fronts: Waves of advance of an advantageous gene. Waves of excitation in nerves. Waves of advance of an epidemic. Aims: To provide an introduction to the mathematical modelling of the behaviour of solid elastic materials. Objectives: Students should be able to derive the governing equations of the theory of linear elasticity and Tension Throughout "The Essay, be able to solve simple problems.

Topics will be chosen from the following: Revision: Kinematics of deformation, stress analysis, global balance laws, boundary conditions. Constitutive law: Properties of real materials; constitutive law for linear isotropic elasticity, Lame moduli; field equations of quotes about jordan linear elasticity; Young's modulus, Poisson's ratio. Some simple problems of elastostatics: Expansion of a spherical shell, bulk modulus; deformation of and Fear a block under gravity; elementary bending solution. Linear elastostatics: Strain energy function; uniqueness theorem; Betti's reciprocal theorem, mean value theorems; variational principles, application to composite materials; torsion of cylinders, Prandtl's stress function. Linear elastodynamics: Basic equations and general solutions; plane waves in unbounded media, simple reflection problems; surface waves. Aims: To teach an understanding of iterative methods for standard problems of linear algebra. Objectives: Students should know a range of modern iterative methods for solving linear systems and for solving the algebraic eigenvalue problem. They should be able to analyse their algorithms and should have an understanding of relevant practical issues. Topics will be chosen from the following: The algebraic eigenvalue problem: Gerschgorin's theorems.

The power method and its extensions. Backward Error Analysis (Bauer-Fike). The (Givens) QR factorization and the QR method for symmetric tridiagonal matrices. War Essay? (Statement of convergence only). The Lanczos Procedure for reduction of a real symmetric matrix to and Fear "The Monkey's, tridiagonal form. Orthogonality properties of Lanczos iterates. Iterative Methods for Linear Systems: Convergence of stationary iteration methods. Special cases of symmetric positive definite and diagonally dominant matrices. Variational principles for linear systems with real symmetric matrices. The conjugate gradient method. Krylov subspaces. Convergence.

Connection with the Lanczos method. Iterative Methods for Nonlinear Systems: Newton's Method. Convergence in 1D. Statement of algorithm for systems. MA30054: Representation theory of finite groups. Aims: The course explains some fundamental applications of linear algebra to the study of finite groups. In so doing, it will show by example how one area of mathematics can enhance and enrich the study of another. Objectives: At the end of the course, the students will be able to state and prove the main theorems of Maschke and Schur and be conversant with their many applications in representation theory and character theory.

Moreover, they will be able to apply these results to problems in group theory. Topics will be chosen from the of basketball, following: Group algebras, their modules and associated representations. Maschke's theorem and Tension and Fear "The Monkey's Paw", complete reducibility. Baker? Irreducible representations and Schur's lemma. Decomposition of the regular representation. Character theory and orthogonality theorems. "The Paw"? Burnside's p #097 q #098 theorem.

THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN ODD YEAR. Aims: To provide an introduction to the ideas of point-set topology culminating with a sketch of the classification of leadership member exchange theory compact surfaces. As such it provides a self-contained account of one of the triumphs of 20th century mathematics as well as providing the necessary background for the Year 4 unit in Algebraic Topology. Objectives: To acquaint students with the important notion of a topology and to familiarise them with the Tension and Fear Throughout, basic theorems of analysis in their most general setting. Students will be able to distinguish between metric and topological space theory and to understand refinements, such as Hausdorff or compact spaces, and founder empire, their applications. Topics will be chosen from the following: Topologies and topological spaces.

Subspaces. Bases and sub-bases: product spaces; compact-open topology. Continuous maps and homeomorphisms. Separation axioms. Connectedness. Compactness and its equivalent characterisations in a metric space. Axiom of Choice and Zorn's Lemma.

Tychonoff's theorem. Quotient spaces. Compact surfaces and their representation as quotient spaces. Sketch of the classification of compact surfaces. Aims: The aim of this course is to cover the standard introductory material in the theory of functions of a complex variable and to cover complex function theory up to Cauchy's Residue Theorem and and Fear Throughout, its applications. Objectives: Students should end up familiar with the theory of functions of a complex variable and of 1860, be capable of Tension and Fear Monkey's Paw" calculating and justifying power series, Laurent series, contour integrals and applying them. Topics will be chosen from the exchange, following: Functions of a complex variable. Continuity.

Complex series and power series. Circle of Tension Throughout Paw" convergence. The complex plane. Regions, paths, simple and closed paths. Psychology? Path-connectedness. Tension And Fear Throughout "The Monkey's Paw" Essay? Analyticity and the Cauchy-Riemann equations. Harmonic functions. Cauchy's theorem. Cauchy's Integral Formulae and its application to power series. Isolated zeros.

Differentiability of an analytic function. Liouville's Theorem. Zeros, poles and essential singularities. Laurent expansions. Quotes Jordan Baker? Cauchy's Residue Theorem and contour integration. Applications to real definite integrals.

Aims: To introduce students to the applications of advanced analysis to the solution of PDEs. Objectives: Students should be able to obtain solutions to certain important PDEs using a variety of techniques e.g. Green's functions, separation of variables. Throughout "The? They should also be familiar with important analytic properties of the solution. Topics will be chosen from the of basketball, following: Elliptic equations in two independent variables: Harmonic functions. Mean value property. Maximum principle (several proofs). Dirichlet and Neumann problems. Representation of solutions in terms of Green's functions.

Continuous dependence of data for Dirichlet problem. Uniqueness. Parabolic equations in two independent variables: Representation theorems. And Fear? Green's functions. Self-adjoint second-order operators: Eigenvalue problems (mainly by example). Separation of variables for election of 1860 inhomogeneous systems.

Green's function methods in Tension and Fear Throughout Monkey's, general: Method of images. Use of integral transforms. Conformal mapping. Calculus of variations: Maxima and minima. Lagrange multipliers. Extrema for integral functions. Euler's equation and its special first integrals. Integral and non-integral constraints. Aims: The course is intended to of basketball, be an elementary and accessible introduction to dynamical systems with examples of applications. Main emphasis will be on discrete-time systems which permits the concepts and results to be presented in a rigorous manner, within the framework of the second year core material.

Discrete-time systems will be followed by an introductory treatment of Tension and Fear "The Essay continuous-time systems and differential equations. Numerical approximation of differential equations will link with the earlier material on discrete-time systems. Objectives: An appreciation of the behaviour, and its potential complexity, of general dynamical systems through a study of discrete-time systems (which require relatively modest analytical prerequisites) and computer experimentation. Topics will be chosen from the following: Discrete-time systems. Maps from IRn to IRn . Fixed points. Periodic orbits. #097 and #119 limit sets. Local bifurcations and member exchange theory, stability. The logistic map and chaos. Global properties. Continuous-time systems. Periodic orbits and Poincareacute maps.

Numerical approximation of and Fear Throughout "The Paw" Essay differential equations. Newton iteration as a dynamical system. Aims: The aim of the course is to psychology validity, introduce students to applications of partial differential equations to model problems arising in biology. Tension "The? The course will complement Mathematical Biology I where the emphasis was on ODEs and Difference Equations. Objectives: Students should be able to derive and founder, interpret mathematical models of problems arising in biology using PDEs. They should be able to perform a linearised stability analysis of a reaction-diffusion system and determine criteria for diffusion-driven instability.

They should be able to interpret the results in terms of the original biological problem. Topics will be chosen from the following: Partial Differential Equation Models: Simple random walk derivation of the diffusion equation. Solutions of the diffusion equation. Density-dependent diffusion. Conservation equation.

Reaction-diffusion equations. Chemotaxis. Throughout Essay? Examples for insect dispersal and cell aggregation. Spatial Pattern Formation: Turing mechanisms. Of 1860? Linear stability analysis. Conditions for diffusion-driven instability. Dispersion relation and Turing space. Scale and Tension and Fear Monkey's Essay, geometry effects.

Mode selection and dispersion relation. Applications: Animal coat markings. How the founder empire, leopard got its spots. Butterfly wing patterns. Aims: To introduce the general theory of continuum mechanics and, through this, the study of viscous fluid flow.

Objectives: Students should be able to explain the basic concepts of Throughout Monkey's continuum mechanics such as stress, deformation and constitutive relations, be able to the presidential of 1860, formulate balance laws and "The Monkey's Paw" Essay, be able to apply these to the solution of simple problems involving the flow of a viscous fluid. Topics will be chosen from the following: Vectors: Linear transformation of vectors. Proper orthogonal transformations. Rotation of axes. Transformation of components under rotation. Cartesian Tensors: Transformations of components, symmetry and skew symmetry. Isotropic tensors. Kinematics: Transformation of line elements, deformation gradient, Green strain.

Linear strain measure. Displacement, velocity, strain-rate. Election? Stress: Cauchy stress; relation between traction vector and Tension Throughout "The Monkey's, stress tensor. Psychology Validity? Global Balance Laws: Equations of motion, boundary conditions. Newtonian Fluids: The constitutive law, uniform flow, Poiseuille flow, flow between rotating cylinders. Aims: To present the theory and application of normal linear models and Tension and Fear Throughout "The Paw", generalised linear models, including estimation, hypothesis testing and confidence intervals. To describe methods of model choice and the use of residuals in diagnostic checking. Objectives: On completing the course, students should be able to (a) choose an appropriate generalised linear model for a given set of of achaemenid data; (b) fit this model using the GLIM program, select terms for inclusion in the model and assess the adequacy of a selected model; (c) make inferences on the basis of a fitted model and recognise the assumptions underlying these inferences and possible limitations to their accuracy.

Normal linear model: Vector and Paw" Essay, matrix representation, constraints on parameters, least squares estimation, distributions of parameter and variance estimates, t-tests and confidence intervals, the Analysis of Variance, F-tests for unbalanced designs. Model building: Subset selection and psychology validity, stepwise regression methods with applications in polynomial regression and multiple regression. Effects of collinearity in regression variables. Uses of residuals: Probability plots, plots for additional variables, plotting residuals against and Fear "The Monkey's, fitted values to detect a mean-variance relationship, standardised residuals for outlier detection, masking. Generalised linear models: Exponential families, standard form, statement of asymptotic theory for i.i.d. samples, Fisher information. Linear predictors and link functions, statement of asymptotic theory for the generalised linear model, applications to z-tests and of achaemenid, confidence intervals, #099 #178 -tests and the analysis of deviance. Residuals from generalised linear models and their uses. Applications to dose response relationships, and logistic regression.

Aims: To introduce a variety of statistical models for Tension and Fear "The Paw" time series and cover the election, main methods for analysing these models. Objectives: At the end of the course, the student should be able to. * Compute and interpret a correlogram and a sample spectrum. * derive the properties of ARIMA and state-space models. * choose an Throughout "The Monkey's Essay, appropriate ARIMA model for quotes about a given set of data and fit the Tension Essay, model using an appropriate package. * compute forecasts for a variety of linear methods and models. Introduction: Examples, simple descriptive techniques, trend, seasonality, the correlogram. Probability models for founder time series: Stationarity; moving average (MA), autoregressive (AR), ARMA and ARIMA models. Estimating the autocorrelation function and fitting ARIMA models. Forecasting: Exponential smoothing, Forecasting from ARIMA models.

Stationary processes in the frequency domain: The spectral density function, the Tension and Fear Throughout Paw", periodogram, spectral analysis. State-space models: Dynamic linear models and the Kalman filter. Aims: To introduce students to the use of War Essay statistical methods in medical research, the Tension Throughout Paw", pharmaceutical industry and the National Health Service. Objectives: Students should be able to. (a) recognize the key statistical features of a medical research problem, and, where appropriate, suggest an appropriate study design, (b) understand the ethical considerations and practical problems that govern medical experimentation, (c) summarize medical data and spot possible sources of bias, (d) analyse data collected from some types of clinical trial, as well as simple survival data and quotes jordan, longitudinal data.

Ethical considerations in clinical trials and other types of epidemiological study design. Phases I to IV of drug development and testing. Design of clinical trials: Defining the Tension and Fear Monkey's Paw" Essay, patient population, the trial protocol, possible sources of bias, randomisation, blinding, use of placebo treatment, sample size calculations. Analysis of clinical trials: patient withdrawals, intent to treat criterion for The Long inclusion of patients in analysis. Survival data: Life tables, censoring.

Kaplan-Meier estimate. Monkey's? Selected topics from: Crossover trials; Case-control and the creator, cohort studies; Binary data; Measurement of clinical agreement; Mendelian inheritance; More on survival data: Parametric models for censored survival data, Greenwood's formula, The proportional hazards model, logrank test, Cox's proportional hazards model. Throughout the course, there will be emphasis on and Fear Throughout Paw" Essay drawing sound conclusions and on the ability to explain and interpret numerical data to non-statistical clients. MA30087: Optimisation methods of operational research. Aims: To present methods of optimisation commonly used in OR, to The Long, explain their theoretical basis and give an appreciation of the variety of areas in which they are applicable. Objectives: On completing the course, students should be able to. * Recognise practical problems where optimisation methods can be used effectively.

* Implement appropriate algorithms, and understand their procedures. * Understand the underlying theory of linear programming problems, especially duality. The Nature of OR: Brief introduction. Linear Programming: Basic solutions and the fundamental theorem. The simplex algorithm, two phase method for and Fear "The Monkey's Paw" Essay an initial solution. Interpretation of the optimal tableau. Applications of LP. Psychology? Duality. Topics selected from: Sensitivity analysis and the dual simplex algorithm. Brief discussion of Karmarkar's method.

The transportation problem and its applications, solution by Dantzig's method. Network flow problems, the Ford-Fulkerson theorem. Non-linear Programming: Revision of classical Lagrangian methods. Kuhn-Tucker conditions, necessity and sufficiency. Illustration by application to quadratic programming. MA30089: Applied probability finance.

Aims: To develop and apply the theory of probability and stochastic processes to examples from finance and economics. Objectives: At the end of the course, students should be able to. * formulate mathematically, and then solve, dynamic programming problems. * price an option on a stock modelled by a log of a random walk. * perform simple calculations involving properties of Brownian motion. Dynamic programming: Markov decision processes, Bellman equation; examples including consumption/investment, bid acceptance, optimal stopping. Infinite horizon problems; discounted programming, the Howard Improvement Lemma, negative and positive programming, simple examples and counter-examples. Option pricing for random walks: Arbitrage pricing theory, prices and discounted prices as Martingales, hedging.

Brownian motion: Introduction to Brownian motion, definition and simple properties. Exponential Brownian motion as the model for a stock price, the Black-Scholes formula. Aims: To develop skills in the analysis of multivariate data and study the related theory. Objectives: Be able to carry out a preliminary analysis of multivariate data and select and apply an appropriate technique to look for structure in such data or achieve dimensionality reduction. Tension And Fear Monkey's Paw" Essay? Be able to carry out classical multivariate inferential techniques based on the multivariate normal distribution. Introduction, Preliminary analysis of multivariate data. Revision of relevant matrix algebra. Principal components analysis: Derivation and interpretation; approximate reduction of dimensionality; scaling problems. Multidimensional distributions: The multivariate normal distribution - properties and parameter estimation. One and two-sample tests on means, Hotelling's T-squared.

Canonical correlations and canonical variables; discriminant analysis. Topics selected from: Factor analysis. The multivariate linear model. Metrics and similarity coefficients; multidimensional scaling. Cluster analysis. Correspondence analysis.

Classification and regression trees. Aims: To give students experience in tackling a variety of real-life statistical problems. Objectives: During the course, students should become proficient in. * formulating a problem and carrying out an War Essay, exploratory data analysis. * tackling non-standard, messy data. * presenting the results of an analysis in a clear report. Formulating statistical problems: Objectives, the importance of the initial examination of Tension and Fear Throughout "The Monkey's Essay data. Analysis: Model-building. Choosing an appropriate method of analysis, verification of assumptions. Presentation of results: Report writing, communication with non-statisticians. Using resources: The computer, the library.

Project topics may include: Exploratory data analysis. Practical aspects of the presidential election sample surveys. Fitting general and generalised linear models. And Fear Throughout Essay? The analysis of standard and non-standard data arising from founder theoretical work in other blocks. MA30092: Classical statistical inference. Aims: To develop a formal basis for methods of statistical inference including criteria for the comparison of Tension and Fear Throughout "The procedures. Leadership Exchange Theory? To give an in depth description of the asymptotic theory of maximum likelihood methods and hypothesis testing. Objectives: On completing the course, students should be able to: * calculate properties of estimates and hypothesis tests. * derive efficient estimates and tests for a broad range of problems, including applications to a variety of standard distributions.

Revision of standard distributions: Bernoulli, binomial, Poisson, exponential, gamma and normal, and their interrelationships. Sufficiency and Exponential families. Point estimation: Bias and variance considerations, mean squared error. Rao-Blackwell theorem. And Fear "The? Cramer-Rao lower bound and efficiency. Of Achaemenid? Unbiased minimum variance estimators and Tension and Fear Throughout Monkey's Paw", a direct appreciation of efficiency through some examples. Bias reduction. Asymptotic theory for maximum likelihood estimators.

Hypothesis testing: Hypothesis testing, review of the Neyman-Pearson lemma and maximisation of power. Maximum likelihood ratio tests, asymptotic theory. The Creator? Compound alternative hypotheses, uniformly most powerful tests. Compound null hypotheses, monotone likelihood ratio property, uniformly most powerful unbiased tests. Nuisance parameters, generalised likelihood ratio tests. MMath study year abroad. This unit is Tension Throughout "The Essay, designed primarily for DBA Final Year students who have taken the First and Second Year management statistics units but is also available for Final Year Statistics students from the quotes about baker, Department of Mathematical Sciences. Well qualified students from the IMML course would also be considered.

It introduces three statistical topics which are particularly relevant to Management Science, namely quality control, forecasting and decision theory. Aims: To introduce some statistical topics which are particularly relevant to Management Science. Objectives: On completing the unit, students should be able to implement some quality control procedures, and some univariate forecasting procedures. They should also understand the ideas of decision theory. Quality Control: Acceptance sampling, single and double schemes, SPRT applied to sequential scheme. Process control, Shewhart charts for mean and range, operating characteristics, ideas of cusum charts.

Practical forecasting. Time plot. Trend-and-seasonal models. Exponential smoothing. Holt's linear trend model and Holt-Winters seasonal forecasting. Autoregressive models.

Box-Jenkins ARIMA forecasting. Introduction to decision analysis for discrete events: Revision of Bayes' Theorem, admissability, Bayes' decisions, minimax. "The Monkey's Essay? Decision trees, expected value of perfect information. Utility, subjective probability and its measurement. MA30125: Markov processes applications. Aims: To study further Markov processes in both discrete and continuous time. To apply results in areas such genetics, biological processes, networks of queues, telecommunication networks, electrical networks, resource management, random walks and elsewhere. Objectives: On completing the course, students should be able to. * Formulate appropriate Markovian models for a variety of real life problems and apply suitable theoretical results to obtain solutions.

* Classify a variety of birth-death processes as explosive or non-explosive. * Find the Q-matrix of a time-reversed chain and make effective use of the presidential of 1860 time reversal. Topics covering both discrete and continuous time Markov chains will be chosen from: Genetics, the Wright-Fisher and Moran models. Epidemics. Telecommunication models, blocking probabilities of Erlang and Engset. Models of interference in communication networks, the ALOHA model. Series of M/M/s queues. Open and closed migration processes. Explosions.

Birth-death processes. Branching processes. Resource management. Electrical networks. Random walks, reflecting random walks as queuing models in one or more dimensions. The strong Markov property. The Poisson process in time and space. Other applications. Aims: To satisfy as many of the objectives as possible as set out in the individual project proposal.

Objectives: To produce the deliverables identified in Tension Throughout "The Monkey's Paw" Essay, the individual project proposal. Defined in the individual project proposal. MA30170: Numerical solution of election PDEs I. Aims: To teach numerical methods for elliptic and parabolic partial differential equations via the and Fear "The Monkey's, finite element method based on of achaemenid variational principles. Objectives: At the end of the course students should be able to derive and implement the finite element method for a range of standard elliptic and Tension, parabolic partial differential equations in one and several space dimensions. They should also be able to derive and use elementary error estimates for these methods.

* Variational and weak form of elliptic PDEs. Natural, essential and mixed boundary conditions. Linear and quadratic finite element approximation in one and several space dimensions. An introduction to convergence theory. * System assembly and solution, isoparametric mapping, quadrature, adaptivity.

* Applications to about jordan, PDEs arising in applications. * Brief introduction to time dependent problems. Aims: The aim is to explore pure mathematics from a problem-solving point of view. In addition to conventional lectures, we aim to encourage students to Tension Throughout "The Monkey's Essay, work on solving problems in small groups, and to give presentations of solutions in workshops. Objectives: At the end of the The Long, course, students should be proficient in formulating and testing conjectures, and and Fear "The Paw", will have a wide experience of different proof techniques. The topics will be drawn from cardinality, combinatorial questions, the psychology, foundations of measure, proof techniques in and Fear Throughout "The Monkey's, algebra, analysis, geometry and topology. Aims: This is an advanced pure mathematics course providing an introduction to classical algebraic geometry via plane curves. It will show some of the links with other branches of mathematics. Objectives: At the end of the of achaemenid, course students should be able to Monkey's Paw" Essay, use homogeneous coordinates in projective space and to distinguish singular points of plane curves.

They should be able to demonstrate an understanding of the difference between rational and nonrational curves, know examples of both, and be able to describe some special features of plane cubic curves. To be chosen from: Affine and of 1860, projective space. Polynomial rings and homogeneous polynomials. Ideals in Tension and Fear Throughout "The Essay, the context of polynomial rings,the Nullstellensatz. Plane curves; degree; Bezout's theorem. Singular points of plane curves. Validity? Rational maps and morphisms; isomorphism and birationality. Curves of low degree (up to 3). Genus. Elliptic curves; the group law, nonrationality, the j invariant. Weierstrass p function.

Quadric surfaces; curves of quadrics. Duals. THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN EVEN YEAR. Aims: The course will provide a solid introduction to one of the Big Machines of modern mathematics which is also a major topic of Tension and Fear Monkey's Paw" current research. In particular, this course provides the necessary prerequisites for post-graduate study of Algebraic Topology.

Objectives: At the end of the course, the students will be conversant with the basic ideas of homotopy theory and, in particular, will be able to compute the fundamental group of several topological spaces. Topics will be chosen from the following: Paths, homotopy and the fundamental group. Homotopy of maps; homotopy equivalence and deformation retracts. Computation of the fundamental group and applications: Fundamental Theorem of Algebra; Brouwer Fixed Point Theorem. Of Achaemenid? Covering spaces. And Fear Paw" Essay? Path-lifting and homotopy lifting properties. Deck translations and the fundamental group. Universal covers. Loop spaces and their topology. Inductive definition of higher homotopy groups.

Long exact sequence in homotopy for fibrations. MA40042: Measure theory integration. Aims: The purpose of this course is to lay the basic technical foundations and establish the main principles which underpin the classical notions of area, volume and the related idea of an integral. Objectives: The objective is to familiarise students with measure as a tool in election of 1860, analysis, functional analysis and probability theory. Students will be able to quote and apply the main inequalities in the subject, and to understand their significance in a wide range of contexts. Students will obtain a full understanding of the Lebesgue Integral. Topics will be chosen from the following: Measurability for sets: algebras, #115 -algebras, #112 -systems, d-systems; Dynkin's Lemma; Borel #115 -algebras. Measure in the abstract: additive and Monkey's, #115 -additive set functions; monotone-convergence properties; Uniqueness Lemma; statement of Caratheodory's Theorem and discussion of the #108 -set concept used in about baker, its proof; full proof on handout. Throughout Essay? Lebesgue measure on IRn: existence; inner and member theory, outer regularity. Measurable functions.

Sums, products, composition, lim sups, etc; The Monotone-Class Theorem. Probability. Sample space, events, random variables. Independence; rigorous statement of the Strong Law for coin tossing. Integration.

Integral of a non-negative functions as sup of the "The Paw", integrals of simple non-negative functions dominated by it. Monotone-Convergence Theorem; 'Additivity'; Fatou's Lemma; integral of 'signed' function; definition of Lp and of L p; linearity; Dominated-Convergence Theorem - with mention that it is not the `right' result. Product measures: definition; uniqueness; existence; Fubini's Theorem. Absolutely continuous measures: the idea; effect on integrals. Statement of the Radon-Nikodm Theorem. The Creator? Inequalities: Jensen, Holder, Minkowski.

Completeness of Lp. Aims: To introduce and study abstract spaces and general ideas in analysis, to apply them to examples, to lay the foundations for Tension and Fear Throughout "The Monkey's the Year 4 unit in Functional analysis and to motivate the Lebesgue integral. Objectives: By the end of the unit, students should be able to state and prove the principal theorems relating to of achaemenid empire, uniform continuity and uniform convergence for real functions on metric spaces, compactness in spaces of continuous functions, and elementary Hilbert space theory, and to apply these notions and the theorems to and Fear Throughout "The Monkey's Essay, simple examples. Topics will be chosen from:Uniform continuity and of achaemenid, uniform limits of continuous functions on [0,1]. Abstract Stone-Weierstrass Theorem. Uniform approximation of continuous functions. Polynomial and trigonometric polynomial approximation, separability of C[0,1]. Total Boundedness. Diagonalisation. Ascoli-Arzelagrave Theorem.

Complete metric spaces. Baire Category Theorem. And Fear Monkey's Paw" Essay? Nowhere differentiable function. Picard's theorem for x = f(x,t). The Presidential Election? Metric completion M of a metric space M. Real inner product spaces. Hilbert spaces.

Cauchy-Schwarz inequality, parallelogram identity. Examples: l #178 , L #178 [0,1] := C[0,1]. Separability of L #178 . Orthogonality, Gram-Schmidt process. Bessel's inequality, Pythagoras' Theorem. Projections and subspaces. Orthogonal complements. Riesz Representation Theorem. "The Monkey's? Complete orthonormal sets in separable Hilbert spaces. Completeness of the creator of basketball trigonometric polynomials in L #178 [0,1].

Fourier Series. Aims: A treatment of the qualitative/geometric theory of dynamical systems to a level that will make accessible an area of mathematics (and allied disciplines) that is Tension Throughout Monkey's Paw" Essay, highly active and rapidly expanding. Objectives: Conversance with concepts, results and techniques fundamental to leadership member exchange theory, the study of qualitative behaviour of dynamical systems. An ability to investigate stability of equilibria and periodic orbits. A basic understanding and appreciation of bifurcation and chaotic behaviour.

Topics will be chosen from the following: Stability of equilibria. And Fear "The Paw" Essay? Lyapunov functions. Invariance principle. Periodic orbits. Poincareacute maps. Hyperbolic equilibria and psychology validity, orbits. Throughout "The Monkey's? Stable and unstable manifolds. Nonhyperbolic equilibria and orbits. Centre manifolds. Bifurcation from validity a simple eigenvalue. Introductory treatment of chaotic behaviour.

Horseshoe maps. Symbolic dynamics. MA40048: Analytical geometric theory of differential equations. Aims: To give a unified presention of systems of Tension Throughout "The Paw" ordinary differential equations that have a Hamiltonian or Lagrangian structure. Member? Geomtrical and analytical insights will be used to prove qualitative properties of solutions. These ideas have generated many developments in modern pure mathematics, such as sympletic geometry and ergodic theory, besides being applicable to Tension and Fear "The Paw" Essay, the equations of classical mechanics, and motivating much of modern physics. Objectives: Students will be able to psychology validity, state and prove general theorems for Lagrangian and Hamiltonian systems.

Based on these theoretical results and key motivating examples they will identify general qualitative properties of solutions of these systems. Lagrangian and Tension Throughout "The Paw", Hamiltonian systems, phase space, phase flow, variational principles and Euler-Lagrange equations, Hamilton's Principle of least action, Legendre transform, Liouville's Theorem, Poincare recurrence theorem, Noether's Theorem. MA40050: Nonlinear equations bifurcations. Aims: To extend the real analysis of implicitly defined functions into the numerical analysis of iterative methods for computing such functions and to teach an awareness of practical issues involved in leadership member theory, applying such methods. Objectives: The students should be able to solve a variety of nonlinear equations in many variables and Tension "The Monkey's Essay, should be able to assess the performance of their solution methods using appropriate mathematical analysis. Topics will be chosen from the following: Solution methods for nonlinear equations: Newtons method for systems. Quasi-Newton Methods.

Eigenvalue problems. Theoretical Tools: Local Convergence of Newton's Method. Implicit Function Theorem. Bifcurcation from the trivial solution. Applications: Exothermic reaction and buckling problems. Of Basketball? Continuous and discrete models. Analysis of parameter-dependent two-point boundary value problems using the shooting method.

Practical use of the shooting method. The Lyapunov-Schmidt Reduction. Application to analysis of discretised boundary value problems. Computation of solution paths for systems of nonlinear algebraic equations. Pseudo-arclength continuation. Homotopy methods. Computation of turning points. Tension And Fear Throughout Essay? Bordered systems and their solution.

Exploitation of leadership member theory symmetry. Hopf bifurcation. Numerical Methods for Optimization: Newton's method for unconstrained minimisation, Quasi-Newton methods. Aims: To introduce the theory of Tension "The Paw" Essay infinite-dimensional normed vector spaces, the linear mappings between them, and the presidential, spectral theory. Objectives: By the end of the unit, the students should be able to state and prove the principal theorems relating to Banach spaces, bounded linear operators, compact linear operators, and spectral theory of compact self-adjoint linear operators, and apply these notions and theorems to Tension "The Monkey's, simple examples.

Topics will be chosen from the founder empire, following: Normed vector spaces and their metric structure. Banach spaces. Young, Minkowski and Holder inequalities. Examples - IRn, C[0,1], l p, Hilbert spaces. Riesz Lemma and finite-dimensional subspaces. The space B(X,Y) of bounded linear operators is a Banach space when Y is Tension Throughout Monkey's Paw" Essay, complete. Dual spaces and second duals.

Uniform Boundedness Theorem. Open Mapping Theorem. Closed Graph Theorem. Projections onto closed subspaces. Of Basketball? Invertible operators form an open set. Power series expansion for Tension Throughout "The Monkey's Paw" (I-T)- #185 . Compact operators on Banach spaces. Spectrum of an operator - compactness of spectrum. Operators on of achaemenid Hilbert space and their adjoints. Spectral theory of self-adjoint compact operators.

Zorn's Lemma. Hahn-Banach Theorem. Canonical embedding of X in X* * is isometric, reflexivity. Simple applications to weak topologies. Aims: To stimulate through theory and especially examples, an interest and appreciation of the power of this elegant method in analysis and probability. Applications of the theory are at the heart of Throughout Monkey's Paw" this course. Objectives: By the end of the the creator, course, students should be familiar with the main results and techniques of discrete time martingale theory. They will have seen applications of martingales in proving some important results from classical probability theory, and they should be able to and Fear "The Essay, recognise and apply martingales in solving a variety of more elementary problems. Topics will be chosen from the following: Review of War Essay fundamental concepts. Conditional expectation. Martingales, stopping times, Optional-Stopping Theorem.

The Convergence Theorem. L #178 -bounded martingales, the random-signs problem. Angle-brackets process, Leacutevy's Borel-Cantelli Lemma. Tension And Fear Throughout "The Paw"? Uniform integrability. UI martingales, the of basketball, Downward Theorem, the Strong Law, the Submartingale Inequality. Likelihood ratio, Kakutani's theorem. MA40061: Nonlinear optimal control theory. Aims: Four concepts underpin control theory: controllability, observability, stabilizability and optimality. Of these, the first two essentially form the focus of the Year 3/4 course on linear control theory. In this course, the latter notions of stabilizability and optimality are developed. Together, the courses on linear control theory and nonlinear optimal control provide a firm foundation for participating in theoretical and practical developments in an active and expanding discipline.

Objectives: To present concepts and results pertaining to robustness, stabilization and optimization of (nonlinear) finite-dimensional control systems in a rigorous manner. Emphasis is placed on optimization, leading to conversance with both the Bellman-Hamilton-Jacobi approach and the maximum principle of Pontryagin, together with their application. Topics will be chosen from the following: Controlled dynamical systems: nonlinear systems and linearization. Stability and robustness. Stabilization by feedback. Lyapunov-based design methods. Stability radii. Small-gain theorem. Optimal control.

Value function. Essay? The Bellman-Hamilton-Jacobi equation. Verification theorem. Quadratic-cost control problem for linear systems. Riccati equations. The Pontryagin maximum principle and transversality conditions (a dynamic programming derivation of a restricted version and statement of the general result with applications). Proof of the maximum principle for the linear time-optimal control problem.

MA40062: Ordinary differential equations. Aims: To provide an accessible but rigorous treatment of exchange initial-value problems for nonlinear systems of ordinary differential equations. Foundations will be laid for advanced studies in dynamical systems and control. The material is Tension Paw", also useful in mathematical biology and numerical analysis. Objectives: Conversance with existence theory for the initial-value problem, locally Lipschitz righthand sides and uniqueness, flow, continuous dependence on initial conditions and parameters, limit sets. Topics will be chosen from the following: Motivating examples from diverse areas. Existence of solutions for the initial-value problem. Uniqueness.

Maximal intervals of about jordan existence. Dependence on initial conditions and parameters. Flow. Monkey's Essay? Global existence and member, dynamical systems. Limit sets and attractors. Aims: To satisfy as many of the objectives as possible as set out in the individual project proposal.

Objectives: To produce the deliverables identified in the individual project proposal. Defined in the individual project proposal. MA40171: Numerical solution of PDEs II. Aims: To teach an understanding of linear stability theory and its application to ODEs and evolutionary PDEs. Objectives: The students should be able to analyse the stability and convergence of a range of numerical methods and assess the practical performance of these methods through computer experiments. Solution of initial value problems for ODEs by Linear Multistep methods: local accuracy, order conditions; formulation as a one-step method; stability and convergence. Introduction to physically relevant PDEs. Well-posed problems.

Truncation error; consistency, stability, convergence and the Lax Equivalence Theorem; techniques for finding the stability properties of particular numerical methods. Numerical methods for parabolic and hyperbolic PDEs. MA40189: Topics in Tension and Fear Throughout, Bayesian statistics. Aims: To introduce students to the ideas and of achaemenid, techniques that underpin the theory and and Fear Paw", practice of the Bayesian approach to statistics. Objectives: Students should be able to formulate the Bayesian treatment and analysis of many familiar statistical problems. Bayesian methods provide an alternative approach to data analysis, which has the ability to incorporate prior knowledge about a parameter of interest into the statistical model. The prior knowledge takes the form of a prior (to sampling) distribution on the parameter space, which is updated to the creator of basketball, a posterior distribution via Bayes' Theorem, using the data. Summaries about the parameter are described using the posterior distribution.

The Bayesian Paradigm; decision theory; utility theory; exchangeability; Representation Theorem; prior, posterior and predictive distributions; conjugate priors. Tools to undertake a Bayesian statistical analysis will also be introduced. Simulation based methods such as Markov Chain Monte Carlo and importance sampling for use when analytical methods fail. Aims: The course is intended to provide an elementary and assessible introduction to Throughout Paw" Essay, the state-space theory of linear control systems. Main emphasis is on continuous-time autonomous systems, although discrete-time systems will receive some attention through sampling of continuous-time systems. Contact with classical (Laplace-transform based) control theory is made in the context of realization theory.

Objectives: To instill basic concepts and founder empire, results from control theory in a rigorous manner making use of elementary linear algebra and linear ordinary differential equations. Throughout "The Paw"? Conversance with controllability, observability, stabilizabilty and member theory, realization theory in a linear, finite-dimensional context. Content: Topics will be chosen from the following: Controlled and observed dynamical systems: definitions and classifications. Controllability and observability: Gramians, rank conditions, Hautus criteria, controllable and Throughout "The Paw" Essay, unobservable subspaces. Input-output maps. Transfer functions and state-space realizations. State feedback: stabilizability and quotes jordan baker, pole placement. Observers and output feedback: detectability, asymptotic state estimation, stabilization by dynamic feedback.

Discrete-time systems: z-transform, deadbeat control and observation. Sampling of continuous-time systems: controllability and observability under sampling. Aims: To introduce students to the applications of advanced analysis to the solution of PDEs. Objectives: Students should be able to obtain solutions to certain important PDEs using a variety of techniques e.g. Green's functions, separation of variables. And Fear "The? They should also be familiar with important analytic properties of the solution.

Content: Topics will be chosen from the election of 1860, following: Elliptic equations in two independent variables: Harmonic functions. Mean value property. "The Monkey's? Maximum principle (several proofs). Dirichlet and Neumann problems. Representation of solutions in terms of Green's functions. Continuous dependence of data for Dirichlet problem. Uniqueness. Parabolic equations in two independent variables: Representation theorems. Green's functions. Self-adjoint second-order operators: Eigenvalue problems (mainly by example).

Separation of variables for inhomogeneous systems. Green's function methods in empire, general: Method of images. Use of integral transforms. Conformal mapping. Calculus of variations: Maxima and minima. Lagrange multipliers. Extrema for integral functions. And Fear Throughout Paw" Essay? Euler's equation and its special first integrals. Integral and non-integral constraints.

Aims: The aim of the course is to introduce students to applications of partial differential equations to model problems arising in the presidential election of 1860, biology. The course will complement Mathematical Biology I where the emphasis was on Tension and Fear Throughout "The Paw" Essay ODEs and Difference Equations. Objectives: Students should be able to derive and interpret mathematical models of problems arising in biology using PDEs. They should be able to perform a linearised stability analysis of leadership member theory a reaction-diffusion system and determine criteria for diffusion-driven instability. They should be able to interpret the Monkey's, results in the creator, terms of the Tension Throughout "The Monkey's Paw", original biological problem. Content: Topics will be chosen from the following: Partial Differential Equation Models: Simple random walk derivation of the diffusion equation. Solutions of the diffusion equation.

Density-dependent diffusion. The Presidential? Conservation equation. Reaction-diffusion equations. Chemotaxis. And Fear "The Paw" Essay? Examples for insect dispersal and cell aggregation. Spatial Pattern Formation: Turing mechanisms. Linear stability analysis.

Conditions for diffusion-driven instability. Dispersion relation and the presidential of 1860, Turing space. Scale and and Fear "The Monkey's Essay, geometry effects. Mode selection and dispersion relation. Applications: Animal coat markings.

How the leopard got its spots. Butterfly wing patterns. Aims: To introduce the general theory of continuum mechanics and, through this, the study of viscous fluid flow. Objectives: Students should be able to The Long, explain the basic concepts of continuum mechanics such as stress, deformation and constitutive relations, be able to formulate balance laws and be able to apply these to the solution of simple problems involving the Tension and Fear "The Paw", flow of a viscous fluid. Content: Topics will be chosen from the the presidential election of 1860, following: Vectors: Linear transformation of vectors. Proper orthogonal transformations. Rotation of axes. Transformation of components under rotation. Cartesian Tensors: Transformations of components, symmetry and skew symmetry.

Isotropic tensors. Kinematics: Transformation of Tension Throughout "The line elements, deformation gradient, Green strain. Linear strain measure. Displacement, velocity, strain-rate. Stress: Cauchy stress; relation between traction vector and stress tensor. Global Balance Laws: Equations of motion, boundary conditions. Newtonian Fluids: The constitutive law, uniform flow, Poiseuille flow, flow between rotating cylinders. Aims: To present the theory and application of normal linear models and generalised linear models, including estimation, hypothesis testing and confidence intervals.

To describe methods of model choice and the use of residuals in The Long, diagnostic checking. To facilitate an in-depth understanding of the topic. Objectives: On completing the course, students should be able to. (a) choose an appropriate generalised linear model for Tension and Fear Throughout Monkey's Paw" a given set of data; (b) fit this model using the GLIM program, select terms for inclusion in the model and assess the adequacy of a selected model; (c) make inferences on the presidential the basis of a fitted model and recognise the assumptions underlying these inferences and possible limitations to their accuracy; (d) demonstrate an Throughout Monkey's, in-depth understanding of the validity, topic. Content: Normal linear model: Vector and matrix representation, constraints on Throughout "The Monkey's Paw" parameters, least squares estimation, distributions of parameter and variance estimates, t-tests and War Essay, confidence intervals, the Analysis of Variance, F-tests for unbalanced designs.

Model building: Subset selection and stepwise regression methods with applications in polynomial regression and multiple regression. Effects of collinearity in regression variables. Uses of residuals: Probability plots, plots for and Fear "The additional variables, plotting residuals against fitted values to detect a mean-variance relationship, standardised residuals for outlier detection, masking. Generalised linear models: Exponential families, standard form, statement of asymptotic theory for i.i.d. samples, Fisher information. Linear predictors and link functions, statement of asymptotic theory for the generalised linear model, applications to about jordan baker, z-tests and confidence intervals, #099 #178 -tests and the analysis of deviance. Tension Throughout "The Monkey's Paw"? Residuals from generalised linear models and their uses. Applications to dose response relationships, and logistic regression. Aims: To introduce a variety of statistical models for time series and cover the main methods for analysing these models.

To facilitate an in-depth understanding of the topic. Objectives: At the end of the course, the student should be able to: * Compute and interpret a correlogram and a sample spectrum; * derive the properties of ARIMA and state-space models; * choose an appropriate ARIMA model for a given set of data and fit the model using an appropriate package; * compute forecasts for a variety of linear methods and models; * demonstrate an in-depth understanding of the topic. Content: Introduction: Examples, simple descriptive techniques, trend, seasonality, the quotes about jordan baker, correlogram. Probability models for time series: Stationarity; moving average (MA), autoregressive (AR), ARMA and ARIMA models. Estimating the autocorrelation function and fitting ARIMA models. Forecasting: Exponential smoothing, Forecasting from ARIMA models. Stationary processes in the frequency domain: The spectral density function, the periodogram, spectral analysis. State-space models: Dynamic linear models and the Kalman filter. MA50089: Applied probability finance. Aims: To develop and and Fear Monkey's Paw", apply the quotes jordan baker, theory of probability and stochastic processes to examples from finance and economics.

To facilitate an in-depth understanding of the topic. Objectives: At the end of the course, students should be able to: * formulate mathematically, and then solve, dynamic programming problems; * price an option on Throughout "The a stock modelled by a log of a random walk; * perform simple calculations involving properties of of 1860 Brownian motion; * demonstrate an in-depth understanding of the topic. Content: Dynamic programming: Markov decision processes, Bellman equation; examples including consumption/investment, bid acceptance, optimal stopping. Tension And Fear Throughout "The Paw"? Infinite horizon problems; discounted programming, the Howard Improvement Lemma, negative and positive programming, simple examples and counter-examples. Option pricing for random walks: Arbitrage pricing theory, prices and discounted prices as Martingales, hedging. Brownian motion: Introduction to Brownian motion, definition and the presidential election of 1860, simple properties.Exponential Brownian motion as the model for Throughout Paw" a stock price, the Black-Scholes formula. Aims: To develop skills in the analysis of multivariate data and study the related theory.

To facilitate an in-depth understanding of the topic. Objectives: Be able to carry out a preliminary analysis of multivariate data and select and member exchange, apply an appropriate technique to "The, look for structure in such data or achieve dimensionality reduction. Leadership Member? Be able to carry out Tension "The Paw", classical multivariate inferential techniques based on the multivariate normal distribution. Be able to demonstrate an in-depth understanding of the topic. Content: Introduction, Preliminary analysis of multivariate data. Revision of relevant matrix algebra. Principal components analysis: Derivation and interpretation; approximate reduction of dimensionality; scaling problems. Multidimensional distributions: The multivariate normal distribution - properties and parameter estimation.

One and two-sample tests on means, Hotelling's T-squared. Canonical correlations and canonical variables; discriminant analysis. Topics selected from: Factor analysis. The multivariate linear model. Metrics and similarity coefficients; multidimensional scaling. Cluster analysis. Correspondence analysis. Classification and regression trees.

MA50092: Classical statistical inference. Aims: To develop a formal basis for methods of statistical inference including criteria for the comparison of procedures. To give an in quotes jordan, depth description of the asymptotic theory of maximum likelihood methods. To facilitate an in-depth understanding of the topic. Objectives: On completing the course, students should be able to: * calculate properties of estimates and hypothesis tests; * derive efficient estimates and tests for a broad range of problems, including applications to a variety of standard distributions; * demonstrate an in-depth understanding of the Tension Paw", topic. Revision of standard distributions: Bernoulli, binomial, Poisson, exponential, gamma and normal, and their interrelationships. Sufficiency and Exponential families. Point estimation: Bias and variance considerations, mean squared error. Rao-Blackwell theorem. Cramer-Rao lower bound and The Long War Essay, efficiency.

Unbiased minimum variance estimators and a direct appreciation of efficiency through some examples. Bias reduction. Asymptotic theory for maximum likelihood estimators. Hypothesis testing: Hypothesis testing, review of the Neyman-Pearson lemma and maximisation of power. Maximum likelihood ratio tests, asymptotic theory. Compound alternative hypotheses, uniformly most powerful tests. Compound null hypotheses, monotone likelihood ratio property, uniformly most powerful unbiased tests. Nuisance parameters, generalised likelihood ratio tests. MA50125: Markov processes applications. Aims: To study further Markov processes in both discrete and Throughout "The Monkey's Essay, continuous time.

To apply results in areas such genetics, biological processes, networks of the creator queues, telecommunication networks, electrical networks, resource management, random walks and elsewhere. To facilitate an in-depth understanding of the Tension and Fear Throughout "The Monkey's Essay, topic. Objectives: On completing the course, students should be able to: * Formulate appropriate Markovian models for psychology validity a variety of real life problems and apply suitable theoretical results to obtain solutions; * Classify a variety of birth-death processes as explosive or non-explosive; * Find the Q-matrix of a time-reversed chain and make effective use of time reversal; * Demonstrate an in-depth understanding of the topic. Content: Topics covering both discrete and continuous time Markov chains will be chosen from: Genetics, the and Fear "The Monkey's, Wright-Fisher and psychology validity, Moran models. Epidemics.

Telecommunication models, blocking probabilities of Erlang and Tension and Fear Throughout Essay, Engset. Models of interference in communication networks, the ALOHA model. Series of M/M/s queues. Open and closed migration processes. Explosions. Election Of 1860? Birth-death processes. Branching processes.

Resource management. Electrical networks. Random walks, reflecting random walks as queuing models in one or more dimensions. And Fear "The? The strong Markov property. The Poisson process in time and space. Other applications. MA50170: Numerical solution of PDEs I.

Aims: To teach numerical methods for elliptic and of achaemenid empire, parabolic partial differential equations via the finite element method based on variational principles. Objectives: At the end of the course students should be able to derive and Tension and Fear Throughout Monkey's Essay, implement the War Essay, finite element method for a range of Tension and Fear Throughout Paw" standard elliptic and parabolic partial differential equations in one and several space dimensions. Quotes About Baker? They should also be able to derive and use elementary error estimates for these methods. Variational and weak form of elliptic PDEs. Natural, essential and mixed boundary conditions. Linear and quadratic finite element approximation in one and several space dimensions. An introduction to convergence theory. System assembly and solution, isoparametric mapping, quadrature, adaptivity. Applications to PDEs arising in applications. Parabolic problems: methods of lines, and simple timestepping procedures. Stability and convergence.

MA50174: Theory methods 1b-differential equations: computation and applications. Content: Introduction to Maple and Matlab and their facilities: basic matrix manipulation, eigenvalue calculation, FFT analysis, special functions, solution of simultaneous linear and nonlinear equations, simple optimization. Basic graphics, data handling, use of toolboxes. Tension And Fear Monkey's Paw" Essay? Problem formulation and solution using Matlab. Numerical methods for solving ordinary differential equations: Matlab codes and of 1860, student written codes.

Convergence and Stability. Shooting methods, finite difference methods and Monkey's Essay, spectral methods (using FFT). Sample case studies chosen from: the two body problem, the three body problem, combustion, nonlinear control theory, the election, Lorenz equations, power electronics, Sturm-Liouville theory, eigenvalues, and Tension and Fear "The Paw" Essay, orthogonal basis expansions. Finite Difference Methods for classical PDEs: the wave equation, the heat equation, Laplace's equation. MA50175: Theory methods 2 - topics in differential equations. Aims: To describe the the creator of basketball, theory and phenomena associated with hyperbolic conservation laws, typical examples from applications areas, and their numerical approximation; and to introduce students to the literature on the subject.

Objectives: At the end of the course, students should be able to recognise the importance of Monkey's Essay conservation principles and be familiar with phenomena such as shocks and rarefaction waves; and they should be able to the presidential election, choose appropriate numerical methods for their approximation, analyse their behaviour, and implement them through Matlab programs. Content: Scalar conservation laws in 1D: examples, characteristics, shock formation, viscosity solutions, weak solutions, need for an entropy condition, total variation, existence and uniqueness of solutions.Design of conservative numerical methods for hyperbolic systems: interface fluxes, Roe's first order scheme, Lax-Wendroff methods, finite volume methods, TVD schemes and the Harten theorem, Engquist-Osher method. The Riemann problem: shocks and the Hugoniot locus, isothermal flow and the shallow water equations, the Godunov method, Euler equations of compressible fluid flow. System wave equation in 2D. R.J. LeVeque, Numerical Methods for Conservation Laws (2nd Edition), Birkhuser, 1992. K.W. Morton D.F. Mayers, Numerical Solution of Partial Differential Equations, CUP, 1994.R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, CUP, 2002. MA50176: Methods applications 1: case studies in mathematical modelling and industrial mathematics.

Content: Applications of the theory and techniques learnt in the prerequisites to solve real problems drawn from from the industrial collaborators and/or from the Throughout "The Monkey's, industrially related research work of the key staff involved. Instruction and quotes jordan baker, practical experience of a set of problem solving methods and techniques, such as methods for simplifying a problem, scalings, perturbation methods, asymptotic methods, construction of similarity solutions. Comparison of mathematical models with experimental data. Development and refinement of mathematical models. Case studies will be taken from Tension and Fear Throughout Monkey's Paw" micro-wave cooking, Stefan problems, moulding glass, contamination in The Long, pipe networks, electrostatic filtering, DC-DC conversion, tests for elasticity. Students will work in teams under the pressure of Tension "The project deadlines. The Creator? They will attend lectures given by external industrialists describing the Tension and Fear Throughout Essay, application of mathematics in an industrial context.

They will write reports and give presentations on the case studies making appropriate use of computer methods, graphics and communication skills. MA50177: Methods and psychology, applications 2: scientific computing. Content: Units, complexity, analysis of algorithms, benchmarks. Floating point arithmetic. Programming in Fortran90: Makefiles, compiling, timing, profiling. Data structures, full and Tension "The Monkey's Paw" Essay, sparse matrices. Libraries: BLAS, LAPACK, NAG Library. Visualisation. Handling modules in the creator of basketball, other languages such as C, C++. Software on and Fear the Web: Netlib, GAMS.

Parallel Computation: Vectorisation, SIMD, MIMD, MPI. Performance indicators. Case studies illustrating the lectures will be chosen from the topics:Finite element implementation, iterative methods, preconditioning; Adaptive refinement; The algebraic eigenvalue problem (ARPACK); Stiff systems and the NAG library; Nonlinear 2-point boundary value problems and bifurcation (AUTO); Optimisation; Wavelets and of achaemenid, data compression. Content: Topics will be chosen from the following: The algebraic eigenvalue problem: Gerschgorin's theorems. The power method and its extensions. Backward Error Analysis (Bauer-Fike). The (Givens) QR factorization and the QR method for symmetric tridiagonal matrices. (Statement of convergence only). The Lanczos Procedure for reduction of Tension Throughout "The Paw" Essay a real symmetric matrix to tridiagonal form.

Orthogonality properties of Lanczos iterates. Iterative Methods for Linear Systems: Convergence of stationary iteration methods. Special cases of symmetric positive definite and diagonally dominant matrices. Variational principles for linear systems with real symmetric matrices. The conjugate gradient method. Krylov subspaces. Convergence. Psychology Validity? Connection with the Lanczos method. Iterative Methods for Nonlinear Systems: Newton's Method. Convergence in 1D. Statement of algorithm for systems.

Content: Topics will be chosen from the following: Difference equations: Steady states and fixed points. Stability. Period doubling bifurcations. Chaos. Application to population growth. Systems of Tension Throughout "The Essay difference equations: Host-parasitoid systems.Systems of ODEs: Stability of solutions. Critical points. Phase plane analysis. Leadership Exchange Theory? Poincari-Bendixson theorem. Bendixson and Dulac negative criteria. Conservative systems.

Structural stability and instability. Lyapunov functions. Travelling wave fronts: Waves of advance of an advantageous gene. Waves of excitation in nerves. Waves of advance of an epidemic. Content: Topics will be chosen from the following: Revision: Kinematics of Throughout Monkey's Paw" deformation, stress analysis, global balance laws, boundary conditions. Constitutive law: Properties of real materials; constitutive law for the presidential election linear isotropic elasticity, Lami moduli; field equations of linear elasticity; Young's modulus, Poisson's ratio. Some simple problems of elastostatics: Expansion of a spherical shell, bulk modulus; deformation of a block under gravity; elementary bending solution. Linear elastostatics: Strain energy function; uniqueness theorem; Betti's reciprocal theorem, mean value theorems; variational principles, application to composite materials; torsion of Tension Essay cylinders, Prandtl's stress function. Linear elastodynamics: Basic equations and general solutions; plane waves in unbounded media, simple reflection problems; surface waves.

MA50181: Theory methods 1a - differential equations: theory methods. Content: Sturm-Liouville theory: Reality of eigenvalues. Orthogonality of empire eigenfunctions. Expansion in eigenfunctions. Approximation in mean square. Tension Paw" Essay? Statement of completeness. Fourier Transform: As a limit of Fourier series.

Properties and applications to solution of differential equations. Frequency response of baker linear systems. Characteristic functions. Linear and quasi-linear first-order PDEs in two and three independent variables: Characteristics. Integral surfaces. Uniqueness (without proof).

Linear and and Fear "The Monkey's Paw", quasi-linear second-order PDEs in two independent variables: Cauchy-Kovalevskaya theorem (without proof). Characteristic data. Lack of continuous dependence on initial data for Cauchy problem. Classification as elliptic, parabolic, and hyperbolic. Different standard forms. Constant and nonconstant coefficients. One-dimensional wave equation: d'Alembert's solution. Uniqueness theorem for corresponding Cauchy problem (with data on a spacelike curve). Content: Definition and examples of metric spaces.

Convergence of sequences. Continuous maps and isometries. Sequential definition of continuity. Subspaces and product spaces. Complete metric spaces and the Contraction Mapping Principle. Sequential compactness, Bolzano-Weierstrass theorem and applications. Open and closed sets. Closure and interior of sets. Topological approach to quotes jordan baker, continuity and compactness (with statement of Heine-Borel theorem). Equivalence of Compactness and sequential compactness in metric spaces.

Connectedness and path-connectedness. Tension Essay? Metric spaces of functions: C[0,1] is War Essay, a complete metric space. MA50183: Specialist reading course. * advanced knowledge in the chosen field. * evidence of independent learning. * an ability to read critically and master an advanced topic in Tension and Fear Throughout "The Monkey's Essay, mathematics/ statistics/probability. Content: Defined in the individual course specification. MA50183: Specialist reading course.

advanced knowledge in the chosen field. evidence of independent learning. an ability to read critically and master an advanced topic in mathematics/statistics/probability. Content: Defined in the individual course specification. MA50185: Representation theory of finite groups.

Content: Topics will be chosen from the following: Group algebras, their modules and associated representations. Maschke's theorem and complete reducibility. Irreducible representations and Schur's lemma. Decomposition of the regular representation. Character theory and leadership member exchange theory, orthogonality theorems. Burnside's p #097 q #098 theorem. Content: Topics will be chosen from the and Fear Throughout "The Monkey's Paw", following: Functions of a complex variable. Continuity.

Complex series and power series. The Presidential? Circle of Tension "The Essay convergence. The complex plane. Regions, paths, simple and closed paths. Path-connectedness. Analyticity and the Cauchy-Riemann equations. Harmonic functions. Member Theory? Cauchy's theorem. Cauchy's Integral Formula and its application to power series. Isolated zeros. Differentiability of an Tension and Fear "The Essay, analytic function.

Liouville's Theorem. Zeros, poles and essential singularities. Validity? Laurent expansions. Cauchy's Residue Theorem and Throughout "The Monkey's Paw" Essay, contour integration. Applications to real definite integrals. On completion of the course, the student should be able to demonstrate:- * Advanced knowledge in the chosen field.

* Evidence of independent learning. * An ability to initiate mathematical/statistical research. * An ability to read critically and master an advanced topic in mathematics/ statistics/probability to the extent of jordan baker being able to expound it in Throughout "The Monkey's Paw", a coherent, well-argued dissertation. * Competence in a document preparation language to the extent of being able to typeset a dissertation with substantial mathematical/statistical content. Content: Defined in the individual project specification. MA50190: Advanced mathematical methods. Objectives: Students should learn a set of mathematical techniques in a variety of areas and be able to apply them to either solve a problem or to construct an accurate approximation to the solution. They should demonstrate an understanding of both the theory and the range of applications (including the limitations) of all the techniques studied.

Content: Transforms and of 1860, Distributions: Fourier Transforms, Convolutions (6 lectures, plus directed reading on complex analysis and calculus of residues). Asymptotic expansions: Laplace's method, method of steepest descent, matched asymptotic expansions, singular perturbations, multiple scales and averaging, WKB. (12 lectures, plus directed reading on applications in and Fear Throughout Paw", continuum mechanics). Dimensional analysis: scaling laws, reduction of PDEs and the presidential, ODEs, similarity solutions. (6 lectures, plus directed reading on symmetry group methods). References: L. Dresner, Similarity Solutions of Nonlinear PDEs , Pitman, 1983; JP Keener, Principles of Applied Mathematics, Addison Wesley, 1988; P. Olver, Symmetry Methods for PDEs, Springer; E.J. Hinch, Perturbation Methods, CUP. Objectives: At the Tension and Fear Throughout Monkey's Essay, end of the course students should be able to jordan, use homogeneous coordinates in projective space and to distinguish singular points of plane curves.

They should be able to demonstrate an understanding of the difference between rational and nonrational curves, know examples of both, and be able to describe some special features of plane cubic curves. Content: To be chosen from: Affine and projective space. Polynomial rings andhomogeneous polynomials. Ideals in the context of polynomial rings,the Nullstellensatz. Plane curves; degree; Bezout's theorem. Tension Throughout "The Paw"? Singular points of plane curves. Rational maps and morphisms; isomorphism and of basketball, birationality. Paw" Essay? Curves of low degree (up to 3). Genus. Elliptic curves; the group law, nonrationality, the j invariant. Psychology Validity? Weierstrass p function.

Quadric surfaces; curves of and Fear Throughout "The Monkey's quadrics. Duals. MA50194: Advanced statistics for use in health contexts 2. * To equip students with the skills to use and interpret advanced multivariate statistics; * To provide an appreciation of the applications of advanced multivariate analysis in health and medicine. Learning Outcomes: On completion of this unit, students will: * Learn and understand how and why selected advanced multivariate analyses are computed; * Practice conducting, interpreting and reporting analyses. * To learn independently; * To critically evaluate and assess research and evidence as well as a variety of other information; * To utilise problem solving skills.

* Advanced information technology and computing technology (e.g. SPSS); * Independent working skills; * Advanced numeracy skills. Content: Introduction to STATA, power and sample size, multidimensional scaling, logistic regression, meta-analysis, structural equation modelling. Student Records Examinations Office, University of Bath, Bath BA2 7AY. Tel: +44 (0) 1225 384352 Fax: +44 (0) 1225 386366.

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