Financial Economics I (HAG106G)

The aim is to provide a theoretical as well as practical overview in financial economics. The efficient markets and the portfolio theory are covered as well as the Markowitz model. Risk, and risk assessment under uncertainty and using the utility function are introduced. Students will get practice in value assessment methods, CAPM, as well as fixed income analysis. Stock valuation and fundamentals of derivatives calculations such as the B&S model are covered. 
Projects are based on understanding of concepts introduced in the course and their usage.  In addition projects are based on Excel usage.

Macroeconomics I (HAG103G)

The course aims to give the students an insight into the main theories, concepts, topics, and principles of macroeconomics and macroeconomic activity. The course stresses both the analytical content and applied usefulness of the topics covered and how they relate to various current economic issues at home and abroad. A sound knowledge of macroeconomics prepares students for various other economics courses, and for life.

Mathematical Analysis IA (STÆ101G)

Main emphasis is on the differential and integral calculus of functions of a single variable. The systems of real and complex numbers. Least upper bound and greatest lower bound. Natural numbers and induction. Mappings and functions. Sequences and limits. Series and convergence tests. Conditionally convergent series. Limits and continuous functions. Trigonometric functions. Differentiation. Extreme values. The mean value theorem and polynomial approximation. Integration. The fundamental theorem of calculus. Logarithmic and exponential functions, hyperbolic and inverse trigonometric functions. Methods for finding antiderivatives. Real power series. First-order differential equations. Complex valued functions and second-order differential equations.

Mathematical Analysis I (STÆ104G)

This is a foundational course in single variable calculus. The prerequisites are high school courses on algebra, trigonometry. derivatives, and integrals. The course aims to create a foundation for understanding of subjects such as natural and physical sciences, engineering, economics, and computer science. Topics of the course include the following:

• Real numbers.
• Limits and continuous functions.
• Differentiable functions, rules for derivatives, derivatives of higher order, applications of differential calculus (extremal value problems, linear approximation).
• Transcendental functions.
• Mean value theorem, theorems of l'Hôpital and Taylor.
• Integration, the definite integral and rules/techniques of integration, primitives, improper integrals.
• Fundamental theorem of calculus.
• Applications of integral calculus: Arc length, area, volume, centroids.
• Ordinary differential equations: First-order separable and homogeneous differential equations, first-order linear equations, second-order linear equations with constant coefficients.
• Sequences and series, convergence tests.
• Power series, Taylor series.

Linear Algebra A (STÆ106G)

Basics of linear algebra over the reals with emphasis on the theoretical side. 

Subject matter: Systems of linear equations, matrices, Gauss-Jordan reduction.  Vector spaces and their subspaces.  Linearly independent sets, bases and dimension.  Linear maps, range space and nullspace. 
The dot product, length and angle measures.  Volumes in higher dimensions and the cross product in threedimensional space.  Flats, parametric descriptions and descriptions by equations.  Orthogonal projections and orthonormal bases.  Gram-Schmidt orthogonalization.  Determinants and inverses of matrices.  Eigenvalues, eigenvectors and diagonalization.

Computer Science 1a (TÖL105G)

Programming in Python (for computations in engineering and science): Main commands and statements (computations, control statements, in- and output), definition and execution of functions, datatypes (numbers, matrices, strings, logical values, records), operations and built-in functions, array and matrix computation, file processing, statistics, graphics. Object-oriented programming: classes, objects, constructors and methods. Concepts associated with design and construction of program systems: Programming environment and practices, design and documentation of function and subroutine libraries, debugging and testing of programmes.

Operations Research (IÐN401G)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover linear programming and the simplex algorithm, as well as related analytical topics. It will also introduce special types of mathematical models, including transportation, assignment, network, and integer programming models. The student will become familiar with a modeling language for linear programming.

Financial Economics II (HAG208G)

The aim of this course is threefold. First, to introduce the fundamentals of financial accounting in order for the students being able to read and understand corporate financial statements. Second, teach the students to analyse and calculate the main important multiples from financial statements as well as being able to interpret their meaning to potential users of this information. Third, the students should be able to conduct fair value estimates of the corporate entities using information from their financial accounts.

Computers, operating systems and digital literacy basics (TÖL205G)

In this course, we study several concepts related to digital literacy. The goal of the course is to introduce the students to a broad range of topics without necessarily diving deep into each one.

The Unix operating system is introduced. The file system organization, often used command-line programs, the window system, command-line environment, and shell scripting. We cover editors and data wrangling in the shell. We present version control systems (git), debugging methods, and methods to build software. Common concepts in the field of cryptography are introduced as well as concepts related to virtualization and containers.

Probability and Statistics (STÆ203G)

Basic concepts in probability and statistics based on univariate calculus. 

Topics: 
Sample space, events, probability, equal probability, independent events, conditional probability, Bayes rule, random variables, distribution, density, joint distribution, independent random variables, condistional distribution, mean, variance, covariance, correlation, law of large numbers, Bernoulli, binomial, Poisson, uniform, exponential and normal random variables. Central limit theorem. Poisson process. Random sample, statistics, the distribution of the sample mean and the sample variance. Point estimate, maximum likelihood estimator, mean square error, bias. Interval estimates and hypotheses testing form normal, binomial and exponential samples. Simple linear regression. Goodness of fit tests, test of independence.

Mathematical Analysis II (STÆ205G)

Open and closed sets. Mappings, limits and continuity. Differentiable mappings, partial derivatives and the chain rule. Jacobi matrices. Gradients and directional derivatives. Mixed partial derivatives. Curves. Vector fields and flow. Cylindrical and spherical coordinates. Taylor polynomials. Extreme values and the classification of stationary points. Extreme value problems with constraints. Implicit functions and local inverses. Line integrals, primitive functions and exact differential equations. Double integrals. Improper integrals. Green's theorem. Simply connected domains. Change of variables in double integrals. Multiple integrals. Change of variables in multiple integrals. Surface integrals. Integration of vector fields. The theorems of Stokes and Gauss.

Introduction to Probability Theory (STÆ210G)

This is an extension of the course "Probability and Statistics" STÆ203G. The basic concepts of probability are considered in more detail with emphasis on definitions and proofs. The course is a preparation for the two M-courses in probability and the two M-courses in statistics that are taught alternately every other year.

Topics beyond those discussed in the probability part of STÆ203G:

Kolmogorov's definition. Proofs of propositions on compound events and conditional probability. Proofs for discrete and continuous variables of propositions on expectation, variance, covariance, correlation, and conditional expectation and variance. Proofs of propositions for Bernoulli, binomial, Poisson, geometric, uniform, exponential, and gamma variables. Proof of the tail-summing proposition for expectation and the application to the geometric variable. Proof of the proposition on memoryless and exponential variables. Derivation of the distribution of sums of independent variables such as binomial, Poisson, normal, and gamma variables. Probability and moment generating functions.

Applied Linear Statistical Models (STÆ312M)

The course focuses on simple and multiple linear regression as well as analysis of variance (ANOVA), analysis of covariance (ANCOVA) and binomial regression. The course is a natural continuation of a typical introductory course in statistics taught in various departments of the university.

We will discuss methods for estimating parameters in linear models, how to construct confidence intervals and test hypotheses for the parameters, which assumptions need to hold for applying the models and what to do when they are not met.

Students will work on projects using the statistical software R.

Stochastic Processes (STÆ415M)

Introduction to stochastic processes with main emphasis on Markov chains.

Subject matter: Hitting time, classification of states, irreducibility, period, recurrence (positive and null), transience, regeneration, coupling, stationarity, time-reversibility, coupling from the past, branching processes, queues, martingales, Brownian motion.

Mathematical Analysis III (STÆ302G)

Functions of a complex variable. Analytic functions. The exponential function, logarithms and roots. Cauchy's Integral Theorem and Cauchy's Integral Formula. Uniform convergence. Power series. Laurent series. Residue integration method. Application of complex function theory to fluid flows. Ordinary differential equations and systems of ordinary differential equations. Linear differential equations with constant coefficients. Systems of linear differential equations. The matrix exponential function. Various methods for obtaining a particular solution. Green's functions for initial value problems. Flows and the phase plane. Nonlinear systems of ordinary differential equations in the plane, equilibrium points, stability and linear approximations. Series solutions and the method of Frobenius. Use of Laplace transforms in solving differential equations.

Data Base Theory and Practice (TÖL303G)

Databases and database management systems. Physical data organization. Data modelling using the Entity-Relationship model and the Relational model. Relational algebra and calculus.  The SQL query language. Design theory for relational data bases, functional dependencies, decomposition of relational schemes, normal forms. Query optimization. Concurrency control techniques and crash recovery. Database security and authorization. Data warehousing.

Non-Life Insurance Mathematics (STÆ414G)

The course will give an overview of some important elements of non-life insurance and reinsurance. Models for claim numbers, the Poisson, mixed Poisson and renewal process. Stochastic models for non-life insurance risks, in particular the compound Poisson, compound mixed Poisson and renewal models. The Cramer-Lundberg and the renewal model as basic risk models. Methods for approximating the distribution of risk models. Small and large claim distributions and their properties. Premium calculation principles for the total claim amount of a portfolio. Experience rating: calculation of the premium in a policy. Reinsurance treaties and their properties. Bayesian methods in a non-life insurance context, in particular the Bayes and linear Bayes estimators for calculating the premium in a policy.

Generalized Linear Models (STÆ421M)

Generalized linear regression models. Exponential dispersion models. Poisson processes and tests for overdispersion. Survival regression models. Nonlinear effects and basis expansions. Parametric, semiparametric and nonparametric likelihood methods. Partial likelihood methods. Generalized linear regression analysis in R.

Life Insurance Mathematics (STÆ413G)

Payment flows; mortality theory; overview of the main forms of insurance; the principle of equivalence; prospective reserves and differential equations for these; cost; general Markov chains in life insurance with applications to disability insurance and multi-life insurance; profits and bonuses; market rate products.

Theoretical Numerical Analysis (STÆ412G)

This is an extension of the course "Numerical Analysis" STÆ405G. The material of Numerical Analysis (STÆ405G) is studied in more detail and more theoretically with emphasis on proofs.

Sets and Metric Spaces (STÆ202G)

Elements of set theory: Sets. Mappings. Relations, equivalence relations, orderings. Finite, infinite, countable and uncountable sets. Equipotent sets. Construction of the number systems. Metric spaces: Open sets and closed sets, convergent sequences and Cauchy sequences, cluster points of sets and limit points of sequences. Continuous mappings, convergence, uniform continuity. Complete metric spaces. Uniform convergence and interchange of limits. The Banach fixed point theorem; existence theorem about solutions of first-order differential equations. Completion of metric spaces. Compact metric spaces. Connected sets. Infinite series, in particular function series.

Mathematical Analysis IV (STÆ401G)

Aim: To introduce the student to Fourier analysis and partial differential equations and their applications.
Subject matter: Fourier series and orthonormal systems of functions, boundary-value problems for ordinary differential equations, the eigenvalue problem for Sturm-Liouville operators, Fourier transform. The wave equation, diffusion equation and Laplace's equation solved on various domains in one, two and three dimensions by methods based on the first part of the course, separation of variables, fundamental solution, Green's functions and the method of images.

Numerical Analysis (STÆ405G)

Fundamental concepts on approximation and error estimates. Solutions of systems of linear and non-linear equations. PLU decomposition. Interpolating polynomials, spline interpolation and regression. Numerical differentiation and integration. Extrapolation. Numerical solutions of initial value problems of systems of ordinary differential equations. Multistep methods. Numerical solutions to boundary value problems for ordinary differential equations.

Grades are given for programning projects and in total they amount to 30% of the final grade. The student has to receive the minimum grade of 5 for both the projects and the final exam.

Bayesian Data Analysis (STÆ529M)

Goal: To train students in applying methods of Bayesian statistics for analysis of data. Topics: Theory of Bayesian inference, prior distributions, data distributions and posterior distributions. Bayesian inference  for parameters of univariate and multivariate distributions: binomial; normal; Poisson; exponential; multivariate normal; multinomial. Model checking and model comparison: Bayesian p-values; deviance information criterion (DIC). Bayesian computation: Markov chain Monte Carlo (MCMC) methods; the Gibbs sampler; the Metropolis-Hastings algorithm; convergence diagnostistics. Linear models: normal linear models; hierarchical linear models; generalized linear models. Emphasis on data analysis using software, e.g. Matlab and R.

Theoretical Statistics (STÆ313M)

Likelihood, Sufficient Statistic, Sufficiency Principle, Nuisance Parameter, Conditioning Principle, Invariance Principle, Likelihood Theory. Hypothesis Testing, Simple and Composite Hypothesis, The Neyman-Pearson Lemma, Power, UMP-Test, Invariant Tests. Permutation Tests, Rank Tests. Interval Estimation, Confidence Interval, Confidence, Confidence Region. Point Estimation, Bias, Mean Square Error. Assignments are returned using LaTeX and consitute 20% of the final grade.

Theory of linear models (STÆ310M)

Simple and multiple linear regression, analysis of variance and covariance, inference, variances and covariances of estimators, influence and diagnostic analyses using residual and influence measures, simultaneous inference. General linear models as projections with ANOVA as special case, simultaneous inference of estimable functions. R is used in assignments. Solutions to assignments are returned in LaTeX and PDF format.

In addition selected topics will be visited, e.g. generalized linear models (GLMs), nonlinear regression and/or random/mixed effects models and/or bootstrap methods etc.

Students will present solutions to individually assigned
projects/exercises, each of which is handed in earlier through a web-page.

This course is taught in semesters of even-numbered years.

Financial Instruments (VIÐ503G)

This course starts with looking at interest rate markets and how the zero coupon curve is derived. Valuation of different kind of bonds is covered along with the characteristics and risk factors of the major listed bonds, with special emphasis on the Icelandic market. Next the valuation of derivatives is covered along with the main characteristics. Special emphasis is placed on futures/forwards, swaps and options.  The reasons behind derivatives trading are covered and what the main risk factors are.

Algebra (STÆ303G)

Groups, examples and basic concepts. Symmetry groups. Homomorphisms and normal subgroups. Rings, examples and basic concepts. Integral domains. Ring homomorphisms and ideals. Polynomial rings and factorization of polynomials. Special topics.

Mathematical Seminar (STÆ402G)

This course is intended for students who have completed at least 120 ECTS credits. Students who have not completed 120 ECTS credits and are interested in taking the course must obtain the approval of the supervisor prior to signing up for the course.

Each student prepares and studies a selected well-defined topic of mathematics or statistics and will be assigned a mentor related to that topic. Topics vary from year to year. A list of possible topics is released at the start of or prior to the course and students can also suggest topics (provided that a mentor can be found). Students write a thesis on their selected topic and prepare and give a lecture on the topic at a student conference. During the course, students provide each other with constructive critique both regarding the thesis writing and the preparation of the lecture. In addition to presenting their own projects at the student conference, students take active part, listen to their fellow course members and ask questions.

Research Project (STÆ262L)

Research Project

Business Law B - Introduction to Financial Law (VIÐ601G)

The course reviews legislation and legal issues that concern the financial markets, corporate finance and operations. Legal environment of financial compaines will be reviewed, securities law, liability for experts, a chapter in the penal code act regarding wealth deeds and legal issues related to acquisitions and sales of corporations, due diligence, etc. The course will also review contracts and documents in the financial market, including loan-, purchase- and shareholder agreements.

Applied data analysis (MAS202M)

The course focuses on statistical analysis using the R environment. It is assumed that students have basic knowledge of statistics and the statistical software R. Students will learn to apply a broad range of statistical methods in R (such as classification methods, resampling methods, linear model selection and tree-based methods). The course on 12 weeks and will be on "flipped" form. This means that no lectures will be given but students will read some material and watch videos before attending classes. Students will then work on assignments during the classes.

Graph Theory (STÆ520M)

Graphs, homomorphisms and isomorphisms of graphs. Subgraphs, spanning subgraphs. Paths, connected graphs. Directed graphs. Bipartite graphs. Euler graphs and Hamilton graphs; the theorems of Chvátal, Pósa, Ore and Dirac. Tournaments. Trees, spanning trees, the matrix-tree theorem, Cayley's theorem. Weighted graphs, the algorithms of Kruskal and Dijkstra. Networks, the max-flow-min-cut theorem, the algorithm of Ford and Fulkerson, Menger's theorem. Matchings, Berge's theorem, Hall's marriage theorem, the König-Egerváry theorem, the Kuhn-Munkres algorithm. Inseparable and two-connected graphs. Planar graphs, Euler's formula, Kuratowski's theorem, dual graphs. Embeddings of graphs in surfaces, the Ringel-Youngs-Mayer theorem. Colourings, Heawood's coloring theorem, Brooks's theorem, chromatic polynomial; edge colourings, Vizing's theorem.

R Programming (MAS102M)

Students will perform traditional statistical analysis on real data sets. Special focus will be on regression methods, including multiple regression analysis. Students will apply sophisticated methods of graphical representation and automatic reporting. Students will hand in a projects where they apply the above mentioned methods on real datasets in order to answer research questions

Financial Markets (VIÐ505G)

Financial institutions are a pillar of civilized society, supporting people in their productive ventures and managing the economic risks they take on. The workings of these institutions are important to comprehend, if we are to predict their actions today and their evolution in the coming information age. The course strives to offer understanding of the theory of finance and its relation to the history, strengths and imperfections of such institutions as banking, insurance, securities, futures, and other derivatives markets, and the future of these institutions over the next century. The Icelandic Banking System collapse offers myriad of examples and cases that provide a fruitful ground for learning. A frequent reference will be made to those throughout the course.

Numerical Methods for Partial Differential Equations (STÆ537M, TÖL303G)

The aim of the course is to study numerical methods to solve partial differential equations and their implementation.

Data Base Theory and Practice (STÆ537M, TÖL303G)

Databases and database management systems. Physical data organization. Data modelling using the Entity-Relationship model and the Relational model. Relational algebra and calculus.  The SQL query language. Design theory for relational data bases, functional dependencies, decomposition of relational schemes, normal forms. Query optimization. Concurrency control techniques and crash recovery. Database security and authorization. Data warehousing.

Individual Taxation (VIÐ501G, VIÐ505M)

The course covers the principles of Icelandic tax law concerning tax liability and taxable income, including which items are tax deductible. A special emphasis will be placed on the filing of sources of income for individuals and the self-employed through solving problems and cases. The filing of tax returns for individuals, couples, and businesses will be introduced. The determination of benefits and tax credit will be discussed. The fundamental principles of tax law will be covered, along with re-assessment of taxes and the consequences of fraudulent filing. An overview will be given of the key principles of the laws on value added tax and the social insurance fee. Upon completion of the course a student shall be able to file tax returns for individuals and small businesses as well as appeal tax assessments that he/she deems incorrect.

Financial Statements A (VIÐ501G, VIÐ505M)

This course is designed for students on the F- and R-line (finance and accounting). The purpose with the course is that the students obtains knowledge and understanding on matters that management of companies needs to have to prepare financial statements in accordance with generally accepted accounting principles. In the course students, will learn about generally accepted accounting principles according to international accountings standards (IFRS) and icelandic GAAP. Among topics: Financial accounting and accounting standards, income statement, balance sheet and cash flow. Revenue recognition and cost accounting, inventories, accounts receivables, PPE, intangible assets, income tax, impairment test, accounting for financial instruments, liabilities and equity. Students will need to solve assignments during the course.

Engineering Economics (IÐN502G)

The objective of the course is that students get the skills to:

1.    Understand the main concepts in accounting, cost theory and investment theory.

2.    Be able to use methods of measuring the economic feasibility of technical projects.

3.    Be able to develop computer models to assess the profitability of investments, the value of companies and pricing of bonds

Among topics included are accounting, cost theory, cash flow analysis, investment theory, measures of profitability including net present value and internal rate of return, and the building of profitability models. The course ends with a group assignment where the students exercise the development of computer models for feasibility assessment of projects.

Introduction to Financial Accounting (VIÐ103G, VIÐ105G, VIÐ302G)

This course is intended to do the student able to read corporate financial statements. Fundamentals of financial accounting and financial reporting are introduced. The double entry model explained through the accounting equation. Presentation of the conceptual framework for accounting: assumptions, principles and concepts.  The logical relationship between individual chapters in financial statements is in foreground. Whose things have influence on shareholders equity? Main methods of financial statement analysis are presented, especially ratio analysis. Extensive exercises are covered in separate group sessions.

Microeconomics I (VIÐ103G, VIÐ105G, VIÐ302G)

The aim of the course is to teach students the basic principles of economic thinking and main theories and concepts in microeconomics. The topics covered include: Markets, specialisation and trade. Supply, demand, elascticity and government policies. Efficiency and welfare. The Icelandic tax system and the effects of taxation on market activity. Externalities, public goods and common resources. Firm behaviour and the organisation of industry. Consumer choice. Labour market, earnings and discrimination. Assymetric information, political economy, behavioural economics.

Business Law A (VIÐ103G, VIÐ105G, VIÐ302G)

This course deals with law and regulation applicable to commercial transactions and business organizations. The purpose of the course is to prepare students for the legal challenges they can expect to encounter as entrepreneurs and managers of private businesses . Topics covered include contracts, torts, negotiable instruments, security and guarantees, and bankruptcy. Laws applicable to business organization will also be studied and the fundamentals of securities laws.

Various approaches to teaching mathematics in upper secondary schools (SNU503M)

In this course, students learn to plan mathematics teaching in upper secondary school using various approaches to provide access for all. An emphasis will be put on exploring different teaching environments and teaching methods that build on research on the teaching and learning of mathematics. In the course, the aims of learning mathematics both in Iceland and its neighboring countries will be discussed based on curricular and governmental documents. Students will read about and get a chance to try out various ways to assess and analyze students’ mathematical achievements. The course format includes lectures, project work, presentations, topic studies connected to practice, and critical topic discussion. An emphasis will be put on students’ discussion about challenges and their search for solutions to problems related to the teaching and learning of mathematics.

Cryptocurrency (STÆ532M)

The course will start by introducing the basic concepts of electronic currencies, such as wallets, addresses and transactions. The students will get to know encoding, transactions, blocks and blockchains. The cryptocurrency Smileycoin will be used as an example throughout the course.
Students will compile their own wallets from source and dive deeply enough into the underlying algorithms to be able to put together their own transactions from the Linux command line and read typical wallet code written in C++.
Students will learn how to call the wallet from other software, e.g. to analyse the flow of funds.
Students will learn how to implement several additions to the traditional use of electronic currency such as encoded messages, running software to react to payments etc.
Students will set up their own examples of addition and study how to set up atomic swaps between different currencies, using the Smileycoin for announcements.

Homework will be individualised, selected from different formats (a) solutions based on the wallet on the command line, (2) documents to form handouts or other material in the tutor-web, (3) short programs (APIs) which respond to transactions being send to particular addresses or to a
particular wallet, (4) programs which talk to exchanged and/or (5) new user interfaces which improve or add to the functionality of a wallet.

All the material and assignments will be in English. Returned assignments will become a part of the open tutor-web educational system.

The course may be taught as a reading course or self-study, but the exact implementation depends on participation.

Measure and Integration Theory (STÆ402M, STÆ418M, VIÐ204G)

Measure spaces, measures, outer measures. The Lebesgue measure on Rn. Measurable functions, the monotone convergence theorem, Fatou’s Lemma. Integrable functions, Lebesgue’s  dominated convergence theorem and applications. Inequalities of Hölder and Minkowski, Lp-spaces, simple facts about Banach and Hilbert spaces. Fourier series. Product of measure spaces, theorems of Tonelli and Fubini. Complex measures. Jordan decomposition and Lebesgue decomposition of measures, Radon-Níkodým theorem. Continuous linear functionals on Lp-spaces. Image measures, transformation formula for the Lebesgue measure on Rn.

Introduction to Measure-Theoretic Probability (STÆ402M, STÆ418M, VIÐ204G)

Probability based on measure-theory.

Subject matter: Probability, extension theorems, independence, expectation. The Borel-Cantelli theorem and the Kolmogorov 0-1 law. Inequalities and the weak and strong laws of large numbers. Convergence pointwise, in probability, with probability one, in distribtution, and in total variation. Coupling methods. The central limit theorem. Conditional probability and expectation.

Management Accounting (STÆ402M, STÆ418M, VIÐ204G)

Introduction to management accounting. Most important cost terms will be presented and cost-volume-profit analysis. Different accounting systems around manufacturing costs and allocation of indirect costs. The difference between absorption costing and variable costing. Budgeting, standard costing and variance analysis. Performance evaluation of different departments and products and cost allocation. After this course the students should understand well the importance of management accounting for decision making in business.

Financial Accounting (VIÐ401G, VIÐ402G, VIÐ604G)

This course is a continuation of the introductory course. The main emphasis here is on the preparation of financial statement, fx. allowance for doubtful accounts, depreciation af property, plant and equipment, goodwill and other intangible assets, inventories valuation, fair value of securities and equities, deferred taxes etc. Preparation of cash-flow statement. In this context the Icelandic legal regulation of accounting and International Financial Accounting Standards (IFRS/IAS) are being dealt with. Calculation of income tax will be presented. Extensive exercises are covered in separate group sessions. After this course students should be capable of preparing financial statement for a comparatively simple company.

Finance II (VIÐ401G, VIÐ402G, VIÐ604G)

Good corporate governance and skilled financial management are the key ingredients for a successfully run corporation.  Finance II builds on the course Finance I, and has its main focus on the corporation and how it is being run from financial management point of view.  The course covers topics in corporate governance, how incentives are embedded in the operation of the firm and what economic and financial outcomes are to be expected from the incentive structure.  The main focus of the course is financial management; the firm’s capital structure, short and long term financing, capital budgeting, dividend policies, short term financial planning as well as financial distress.

Portfolio Management (VIÐ401G, VIÐ402G, VIÐ604G)

The theory behind decisions of investors and corporations regarding building and managing asset and liability portfolios. Risk management of corporations will also be covered.

The course is taught in English

Microeconomics II (HAG201G, MAS202M)

Intermediate microeconomic theory. Basic factors of price theory, uncertainty, including analysis of demand, costs of production and supply relationships, and price and output determination under various market structures, market failures and public choice.

Applied data analysis (HAG201G, MAS202M)

The course focuses on statistical analysis using the R environment. It is assumed that students have basic knowledge of statistics and the statistical software R. Students will learn to apply a broad range of statistical methods in R (such as classification methods, resampling methods, linear model selection and tree-based methods). The course on 12 weeks and will be on "flipped" form. This means that no lectures will be given but students will read some material and watch videos before attending classes. Students will then work on assignments during the classes.

Financial Statements B (VIÐ604M)

This course is a continuation of Financial Statements A, which is taught in the fall semester. It is expected that students of this course are fully familiar with the content of the course Financial Statements A.
The course will cover the principles in accounting under both IFRS and Icelandic law. Topics: cash flow, income tax, earnings per share, financial instruments, finance leases, assets held for sale and discontinued operations, investment properties, provision, information in the financial statements and related parties.
Assignments are part of the course, and students will need submit them.

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Statistical Consulting (LÝÐ201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Statistical Consulting (LÝÐ201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Mathematics for Physicists II (EÐL408G)

Python tools related to data analysis and manipulation of graphs. Differential equations and their use in the description of physical systems. Partial differential equations and boundary value problems. Special functions and their relation to important problems in physics. We will emphasize applications and problem solving.

Mathematics for Physicists I (STÆ211G)

Order of magnitude estimates, scaling relations, and dimensional analysis. Python tools related to data analysis and plotting. Mathematical concepts such as vectors, matrices, differential operators in three dimensions, coordinate transformations, partial differential equations and Fourier series and their relation to undergraduate courses in physics and engineering. We will emphasize applications and problem solving.

Portfolio Management (VIÐ604G)

The theory behind decisions of investors and corporations regarding building and managing asset and liability portfolios. Risk management of corporations will also be covered.

The course is taught in English

Physics 1 R (EÐL107G)

Introduce students to methods and fundamental laws of mechanics, waves and thermodynamics, to the extent that they can apply their knowledge to solve problems. 

Concepts, units, scales and dimensions.  Vectors. Kinematics of particles. Particle dynamics, inertia, forces and Newton's laws. Friction. Work and energy, conservation of energy. Momentum, collisions. Systems of particles, center of mass. Rotation of a rigid body.  Angular momentum and moment of inertia. Statics. Gravity. Solids and fluids, Bernoulli's equation. Oscillations: Simple, damped and forced. Waves. Sound.  Temperature. Ideal gas. Heat and the first law of thermodynamics. Kinetic theory of gases. Entropy and the second law of thermodynamics.

Note that the textbook is accessible to students via Canvas free of charge.

Physics 1 R Lab (EÐL108G)

There are 4 lab sessions with experiments mainly from mechanics, with emphasis on teaching students methods of data collection and data processing. Student hand in a lab report on each experiment. They also hand in a final report from one of these that is intended to look more like a journal article.

Mathematical Analysis IA (STÆ101G)

Main emphasis is on the differential and integral calculus of functions of a single variable. The systems of real and complex numbers. Least upper bound and greatest lower bound. Natural numbers and induction. Mappings and functions. Sequences and limits. Series and convergence tests. Conditionally convergent series. Limits and continuous functions. Trigonometric functions. Differentiation. Extreme values. The mean value theorem and polynomial approximation. Integration. The fundamental theorem of calculus. Logarithmic and exponential functions, hyperbolic and inverse trigonometric functions. Methods for finding antiderivatives. Real power series. First-order differential equations. Complex valued functions and second-order differential equations.

Mathematical Analysis I (STÆ104G)

This is a foundational course in single variable calculus. The prerequisites are high school courses on algebra, trigonometry. derivatives, and integrals. The course aims to create a foundation for understanding of subjects such as natural and physical sciences, engineering, economics, and computer science. Topics of the course include the following:

• Real numbers.
• Limits and continuous functions.
• Differentiable functions, rules for derivatives, derivatives of higher order, applications of differential calculus (extremal value problems, linear approximation).
• Transcendental functions.
• Mean value theorem, theorems of l'Hôpital and Taylor.
• Integration, the definite integral and rules/techniques of integration, primitives, improper integrals.
• Fundamental theorem of calculus.
• Applications of integral calculus: Arc length, area, volume, centroids.
• Ordinary differential equations: First-order separable and homogeneous differential equations, first-order linear equations, second-order linear equations with constant coefficients.
• Sequences and series, convergence tests.
• Power series, Taylor series.

Linear Algebra A (STÆ106G)

Basics of linear algebra over the reals with emphasis on the theoretical side. 

Subject matter: Systems of linear equations, matrices, Gauss-Jordan reduction.  Vector spaces and their subspaces.  Linearly independent sets, bases and dimension.  Linear maps, range space and nullspace. 
The dot product, length and angle measures.  Volumes in higher dimensions and the cross product in threedimensional space.  Flats, parametric descriptions and descriptions by equations.  Orthogonal projections and orthonormal bases.  Gram-Schmidt orthogonalization.  Determinants and inverses of matrices.  Eigenvalues, eigenvectors and diagonalization.

Computer Science 1a (TÖL105G)

Programming in Python (for computations in engineering and science): Main commands and statements (computations, control statements, in- and output), definition and execution of functions, datatypes (numbers, matrices, strings, logical values, records), operations and built-in functions, array and matrix computation, file processing, statistics, graphics. Object-oriented programming: classes, objects, constructors and methods. Concepts associated with design and construction of program systems: Programming environment and practices, design and documentation of function and subroutine libraries, debugging and testing of programmes.

Computer Science 2 (TÖL203G)

The course will cover various data structures, algorithms and abstract data types. Among the data structures covered are lists, stacks, queues, priority queues, trees, binary trees, binary search trees and heaps along with related algorithms. Various search and sort algorithms will be covered. Algorithms will be analysed for their space and time complexity. There will be small programming assignments in Java using the given data structures and algorithms.

Computers, operating systems and digital literacy basics (TÖL205G)

In this course, we study several concepts related to digital literacy. The goal of the course is to introduce the students to a broad range of topics without necessarily diving deep into each one.

The Unix operating system is introduced. The file system organization, often used command-line programs, the window system, command-line environment, and shell scripting. We cover editors and data wrangling in the shell. We present version control systems (git), debugging methods, and methods to build software. Common concepts in the field of cryptography are introduced as well as concepts related to virtualization and containers.

Probability and Statistics (STÆ203G)

Basic concepts in probability and statistics based on univariate calculus. 

Topics: 
Sample space, events, probability, equal probability, independent events, conditional probability, Bayes rule, random variables, distribution, density, joint distribution, independent random variables, condistional distribution, mean, variance, covariance, correlation, law of large numbers, Bernoulli, binomial, Poisson, uniform, exponential and normal random variables. Central limit theorem. Poisson process. Random sample, statistics, the distribution of the sample mean and the sample variance. Point estimate, maximum likelihood estimator, mean square error, bias. Interval estimates and hypotheses testing form normal, binomial and exponential samples. Simple linear regression. Goodness of fit tests, test of independence.

Mathematical Analysis II (STÆ205G)

Open and closed sets. Mappings, limits and continuity. Differentiable mappings, partial derivatives and the chain rule. Jacobi matrices. Gradients and directional derivatives. Mixed partial derivatives. Curves. Vector fields and flow. Cylindrical and spherical coordinates. Taylor polynomials. Extreme values and the classification of stationary points. Extreme value problems with constraints. Implicit functions and local inverses. Line integrals, primitive functions and exact differential equations. Double integrals. Improper integrals. Green's theorem. Simply connected domains. Change of variables in double integrals. Multiple integrals. Change of variables in multiple integrals. Surface integrals. Integration of vector fields. The theorems of Stokes and Gauss.

Introduction to Probability Theory (STÆ210G)

This is an extension of the course "Probability and Statistics" STÆ203G. The basic concepts of probability are considered in more detail with emphasis on definitions and proofs. The course is a preparation for the two M-courses in probability and the two M-courses in statistics that are taught alternately every other year.

Topics beyond those discussed in the probability part of STÆ203G:

Kolmogorov's definition. Proofs of propositions on compound events and conditional probability. Proofs for discrete and continuous variables of propositions on expectation, variance, covariance, correlation, and conditional expectation and variance. Proofs of propositions for Bernoulli, binomial, Poisson, geometric, uniform, exponential, and gamma variables. Proof of the tail-summing proposition for expectation and the application to the geometric variable. Proof of the proposition on memoryless and exponential variables. Derivation of the distribution of sums of independent variables such as binomial, Poisson, normal, and gamma variables. Probability and moment generating functions.

Introduction to Quantum Mechanics (EÐL306G)

The course is devoted to theoretical foundations of wave and quantum mechanics. The main concepts characterizing classical waves, such as wave equation, plane waves, wavepackets and phase and group velocity are discussed and then, after the introduction of the concept of particle-wave dualism are used to describe the properties of the de Broglie material waves corresponding to quantum particles. Dynamic and stationary Schrodinger equations are introduced, and their solutions for a set of physically important particular cases, including quantum tunneling, quantum potential well, quantum harmonic oscillator and Coulomb potential are analyzed in all necessary detail. The last part of the course is devoted to the quantum description of spin.

Machine Learning (REI505M)

An overview of some of the main concepts, techniques and algorithms in machine learning. Supervised learning and unsupervised learning. Data preprocessing and data visualization. Model evaluation and model selection. Linear regression, nearest neighbors, support vector machines, decision trees and ensemble methods. Deep learning. Cluster analysis and the k-means algorithm. The students implement simple algorithms in Python and learn how to use specialized software packages. At the end of the course the students work on a practical machine learning project.

Engineering Hydraulics 1 (UMV502G)

The course gives the students basic knowledge in fluid mechanics. Theoretical background for fluids and fluid flow is presented. The fundamental equations of fluid mechanics are derived and used to solve problems. The students perform laboratory experiments.

Fluid Mechanics (VÉL502G)

Properties of liquids and gases. Pressure and force fields in liquids at rest, pressure gauges. Equations of motion, continuity, momentum and energy. Bernoulli equation of motion. Dimensional analysis and dynamic similarity. Two dimensional flow, non-viscous fluids, boundary layers theory, laminar and turbulent flow, fluid friction and form drag. Flow of compressible fluids, velocity of sound. Mach number, sound waves, nozzle shape for supersonic speed. Open channel flow. Several experiments are conducted.

Classical Mechanics (EÐL302G)

Newtonian dynamics of a particle in various coordinate systems. Harmonic, damped and forced oscillations of a pendulum. Nonlinear oscillations and chaos. Gravitation and tidal forces. Calculus of variations. Lagrangian and Hamiltonian dynamics, generalized coordinates and constraints. Central force motion and planetary orbits. Dynamics of a system of particles, collisions in a center-of-mass coordinate system and in a lab system. Motion in a non-inertial reference frame, Coriolis and centrifugal forces. Motion relative to the Earth. Mechanics of rigid bodies, inertia tensors and principal axes of inertia. Eulerian angles, and Euler's equations for a rigid body. Precession, motion of a symmetric top and stability of rigid body rotations. Coupled oscillations, eigenfrequencies and normal modes.

Numerical Linear Algebra (STÆ511M)

Iterative methods for linear systems of equations.  Decompositions of matrices: QR, Cholesky, Jordan, Schur, spectral and singular value decomposition (SVD) and their applications.  Discrete Fourier transform (DFT) and the fast Fourier transform (FFT).  Discrete cosine transform (DCT) in two-dimensions and its application for the compression of images (JPEG) and audio (MP3, AAC).  Sparse matrices and their representation.

Special emphasis will be on the application and implementation of the methods studied.

Numerical Methods for Partial Differential Equations (STÆ537M)

The aim of the course is to study numerical methods to solve partial differential equations and their implementation.

Mathematical Analysis III (STÆ302G)

Functions of a complex variable. Analytic functions. The exponential function, logarithms and roots. Cauchy's Integral Theorem and Cauchy's Integral Formula. Uniform convergence. Power series. Laurent series. Residue integration method. Application of complex function theory to fluid flows. Ordinary differential equations and systems of ordinary differential equations. Linear differential equations with constant coefficients. Systems of linear differential equations. The matrix exponential function. Various methods for obtaining a particular solution. Green's functions for initial value problems. Flows and the phase plane. Nonlinear systems of ordinary differential equations in the plane, equilibrium points, stability and linear approximations. Series solutions and the method of Frobenius. Use of Laplace transforms in solving differential equations.

Data Base Theory and Practice (TÖL303G)

Databases and database management systems. Physical data organization. Data modelling using the Entity-Relationship model and the Relational model. Relational algebra and calculus.  The SQL query language. Design theory for relational data bases, functional dependencies, decomposition of relational schemes, normal forms. Query optimization. Concurrency control techniques and crash recovery. Database security and authorization. Data warehousing.

Electromagnetism 1 (EÐL401G)

The equations of Laplace and Poisson. Magnetostatics. Induction.  Maxwell's equations. Energy of the electromagnetic field. Poynting's theorem. Electromagnetic waves. Plane waves in dielectric and conducting media, reflection and refraction.  Electromagnetic radiation and scattering. Damping.

Thermodynamics 1 (EÐL402G)

Basic concepts of thermodynamic systems, the zeroth law of thermodynamics. Work, internal energy, heat, enthalpy, the first law of thermodynamics for closed and open systems. Ideal and real gases, equations of state. The second law of thermodynamics, entropy, available energy. Thermodynamic cycles and heat engines, cooling engines and heat pumps. Thermodynamic potentials, Maxwell relations. Mixture of ideal gases. Properties for water and steam. Chemical potentials, chemical reactions of ideal gases, the third law of thermodynamics.

Introduction to Systems Biology (LVF601M)

Systems biology is an interdisciplinary field that studies the biological phenomena that emerge from multiple interacting biological elements. Understanding how biological systems change across time is a particular focus of systems biology. In this course, we will prioritize aspects of systems biology relevant to human health and disease.

This course provides an introduction to 1) basic principles in modelling molecular networks, both gene regulatory and metabolic networks; 2) cellular phenomena that support homeostasis like tissue morphogenesis and microbiome resilience, and 3) analysis of molecular patterns found in genomics data at population scale relevant to human disease such as patient classification and biomarker discovery. In this manner, the course covers the three major scales in systems biology: molecules, cells and organisms.

The course activities include reading and interpreting scientific papers, implementation of computational algorithms, working on a research project and presentation of scientific results.

Lectures will comprise of both (1) presentations on foundational concepts and (2) hands-on sessions using Python as the programming language. The course will be taught in English.

Theoretical Numerical Analysis (STÆ412G)

This is an extension of the course "Numerical Analysis" STÆ405G. The material of Numerical Analysis (STÆ405G) is studied in more detail and more theoretically with emphasis on proofs.

Sets and Metric Spaces (STÆ202G)

Elements of set theory: Sets. Mappings. Relations, equivalence relations, orderings. Finite, infinite, countable and uncountable sets. Equipotent sets. Construction of the number systems. Metric spaces: Open sets and closed sets, convergent sequences and Cauchy sequences, cluster points of sets and limit points of sequences. Continuous mappings, convergence, uniform continuity. Complete metric spaces. Uniform convergence and interchange of limits. The Banach fixed point theorem; existence theorem about solutions of first-order differential equations. Completion of metric spaces. Compact metric spaces. Connected sets. Infinite series, in particular function series.

Mathematical Analysis IV (STÆ401G)

Aim: To introduce the student to Fourier analysis and partial differential equations and their applications.
Subject matter: Fourier series and orthonormal systems of functions, boundary-value problems for ordinary differential equations, the eigenvalue problem for Sturm-Liouville operators, Fourier transform. The wave equation, diffusion equation and Laplace's equation solved on various domains in one, two and three dimensions by methods based on the first part of the course, separation of variables, fundamental solution, Green's functions and the method of images.

Numerical Analysis (STÆ405G)

Fundamental concepts on approximation and error estimates. Solutions of systems of linear and non-linear equations. PLU decomposition. Interpolating polynomials, spline interpolation and regression. Numerical differentiation and integration. Extrapolation. Numerical solutions of initial value problems of systems of ordinary differential equations. Multistep methods. Numerical solutions to boundary value problems for ordinary differential equations.

Grades are given for programning projects and in total they amount to 30% of the final grade. The student has to receive the minimum grade of 5 for both the projects and the final exam.

Algebra (STÆ303G)

Groups, examples and basic concepts. Symmetry groups. Homomorphisms and normal subgroups. Rings, examples and basic concepts. Integral domains. Ring homomorphisms and ideals. Polynomial rings and factorization of polynomials. Special topics.

Operations Research (IÐN401G)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover linear programming and the simplex algorithm, as well as related analytical topics. It will also introduce special types of mathematical models, including transportation, assignment, network, and integer programming models. The student will become familiar with a modeling language for linear programming.

Mathematical Seminar (STÆ402G)

This course is intended for students who have completed at least 120 ECTS credits. Students who have not completed 120 ECTS credits and are interested in taking the course must obtain the approval of the supervisor prior to signing up for the course.

Each student prepares and studies a selected well-defined topic of mathematics or statistics and will be assigned a mentor related to that topic. Topics vary from year to year. A list of possible topics is released at the start of or prior to the course and students can also suggest topics (provided that a mentor can be found). Students write a thesis on their selected topic and prepare and give a lecture on the topic at a student conference. During the course, students provide each other with constructive critique both regarding the thesis writing and the preparation of the lecture. In addition to presenting their own projects at the student conference, students take active part, listen to their fellow course members and ask questions.

Machine Learning (REI505M)

An overview of some of the main concepts, techniques and algorithms in machine learning. Supervised learning and unsupervised learning. Data preprocessing and data visualization. Model evaluation and model selection. Linear regression, nearest neighbors, support vector machines, decision trees and ensemble methods. Deep learning. Cluster analysis and the k-means algorithm. The students implement simple algorithms in Python and learn how to use specialized software packages. At the end of the course the students work on a practical machine learning project.

Theory of linear models (STÆ310M)

Simple and multiple linear regression, analysis of variance and covariance, inference, variances and covariances of estimators, influence and diagnostic analyses using residual and influence measures, simultaneous inference. General linear models as projections with ANOVA as special case, simultaneous inference of estimable functions. R is used in assignments. Solutions to assignments are returned in LaTeX and PDF format.

In addition selected topics will be visited, e.g. generalized linear models (GLMs), nonlinear regression and/or random/mixed effects models and/or bootstrap methods etc.

Students will present solutions to individually assigned
projects/exercises, each of which is handed in earlier through a web-page.

This course is taught in semesters of even-numbered years.

Applied Linear Statistical Models (STÆ312M)

The course focuses on simple and multiple linear regression as well as analysis of variance (ANOVA), analysis of covariance (ANCOVA) and binomial regression. The course is a natural continuation of a typical introductory course in statistics taught in various departments of the university.

We will discuss methods for estimating parameters in linear models, how to construct confidence intervals and test hypotheses for the parameters, which assumptions need to hold for applying the models and what to do when they are not met.

Students will work on projects using the statistical software R.

Stochastic Processes (STÆ415M)

Introduction to stochastic processes with main emphasis on Markov chains.

Subject matter: Hitting time, classification of states, irreducibility, period, recurrence (positive and null), transience, regeneration, coupling, stationarity, time-reversibility, coupling from the past, branching processes, queues, martingales, Brownian motion.

Geometry (STÆ508M)

The subject of the course is the foundations of geometry. Affine geometry: Axioms and models. Affine planes and their isomorphisms. Parallel transformations, translations, dilatations. Translation planes and vector addition. Desargue's theorem and coordinates over division rings. The theorem of Pappus. Finite affine planes and the theorem of Wedderburn. Isomorphisms and automorphisms of Desarguesian affine planes. Affine spaces of three and higher dimensions. Projective geometry: Projective planes. Duality principle. The connection between affine and projective planes. The Bruck-Ryser theorem. Automorphisms. The theorems of Desargues and Pappus and coordinates in projective planes. Classical geometry: Incidence axioms, order axioms, congruence axioms. Neutral geometry and neutral planes. Angles and congruence theorems. Different continuity axioms. Euclidean geometry, the theorem of Pappus and coordinates over pythagorean and euclidean fields. Ruler and compass constructions. Hyperbolic geometry, Hilbert's axiom. The end calculus of Hilbert and coordinates in hyperbolic geometry.

Graph Theory (STÆ520M)

Graphs, homomorphisms and isomorphisms of graphs. Subgraphs, spanning subgraphs. Paths, connected graphs. Directed graphs. Bipartite graphs. Euler graphs and Hamilton graphs; the theorems of Chvátal, Pósa, Ore and Dirac. Tournaments. Trees, spanning trees, the matrix-tree theorem, Cayley's theorem. Weighted graphs, the algorithms of Kruskal and Dijkstra. Networks, the max-flow-min-cut theorem, the algorithm of Ford and Fulkerson, Menger's theorem. Matchings, Berge's theorem, Hall's marriage theorem, the König-Egerváry theorem, the Kuhn-Munkres algorithm. Inseparable and two-connected graphs. Planar graphs, Euler's formula, Kuratowski's theorem, dual graphs. Embeddings of graphs in surfaces, the Ringel-Youngs-Mayer theorem. Colourings, Heawood's coloring theorem, Brooks's theorem, chromatic polynomial; edge colourings, Vizing's theorem.

Distributions (STÆ523M)

Fundamentals of distribution theory with applications to partial differential equations

Subject matter: Test funcitons, distributions, differnetiation of distributions, convergence of sequences of distributions, Taylor expansions in several variables, localization, distributions with compact support, multiplication by functions, transpostition: pullback and push-forward of distributions, convolution of distributions, fundamental solutions, Fourier transformation, Fourier series, and fundamental solutions and Fourier transforms.

Introduction to Logic (STÆ528M)

Logical deductions and proofs. Propositional calculus, connectives, truth functions and tautologies. Formal languages, axioms, inference rules. Quantifiers. First-order logic. Interpretations. The compactness theorem. The Lövenheim-Skolem theorem. Computability, recursive functions. Gödel's theorem.

Bayesian Data Analysis (STÆ529M)

Goal: To train students in applying methods of Bayesian statistics for analysis of data. Topics: Theory of Bayesian inference, prior distributions, data distributions and posterior distributions. Bayesian inference  for parameters of univariate and multivariate distributions: binomial; normal; Poisson; exponential; multivariate normal; multinomial. Model checking and model comparison: Bayesian p-values; deviance information criterion (DIC). Bayesian computation: Markov chain Monte Carlo (MCMC) methods; the Gibbs sampler; the Metropolis-Hastings algorithm; convergence diagnostistics. Linear models: normal linear models; hierarchical linear models; generalized linear models. Emphasis on data analysis using software, e.g. Matlab and R.

Formal Languages and Computability (TÖL301G)

Finite state machines, regular languages and grammars, push-down automata, context-free languages and grammars, Turing machines, general languages and grammars, and their basic properties. Recursive and recursively enumerable languages, reduction between languages, connection to decision problems and proving unsolvability of such problems. The complexity classes P and NP, and NP-completeness. Examples of various models of computation.

Data Base Theory and Practice (TÖL303G)

Databases and database management systems. Physical data organization. Data modelling using the Entity-Relationship model and the Relational model. Relational algebra and calculus.  The SQL query language. Design theory for relational data bases, functional dependencies, decomposition of relational schemes, normal forms. Query optimization. Concurrency control techniques and crash recovery. Database security and authorization. Data warehousing.

Various approaches to teaching mathematics in upper secondary schools (SNU503M)

In this course, students learn to plan mathematics teaching in upper secondary school using various approaches to provide access for all. An emphasis will be put on exploring different teaching environments and teaching methods that build on research on the teaching and learning of mathematics. In the course, the aims of learning mathematics both in Iceland and its neighboring countries will be discussed based on curricular and governmental documents. Students will read about and get a chance to try out various ways to assess and analyze students’ mathematical achievements. The course format includes lectures, project work, presentations, topic studies connected to practice, and critical topic discussion. An emphasis will be put on students’ discussion about challenges and their search for solutions to problems related to the teaching and learning of mathematics.

Cryptocurrency (STÆ532M)

The course will start by introducing the basic concepts of electronic currencies, such as wallets, addresses and transactions. The students will get to know encoding, transactions, blocks and blockchains. The cryptocurrency Smileycoin will be used as an example throughout the course.
Students will compile their own wallets from source and dive deeply enough into the underlying algorithms to be able to put together their own transactions from the Linux command line and read typical wallet code written in C++.
Students will learn how to call the wallet from other software, e.g. to analyse the flow of funds.
Students will learn how to implement several additions to the traditional use of electronic currency such as encoded messages, running software to react to payments etc.
Students will set up their own examples of addition and study how to set up atomic swaps between different currencies, using the Smileycoin for announcements.

Homework will be individualised, selected from different formats (a) solutions based on the wallet on the command line, (2) documents to form handouts or other material in the tutor-web, (3) short programs (APIs) which respond to transactions being send to particular addresses or to a
particular wallet, (4) programs which talk to exchanged and/or (5) new user interfaces which improve or add to the functionality of a wallet.

All the material and assignments will be in English. Returned assignments will become a part of the open tutor-web educational system.

The course may be taught as a reading course or self-study, but the exact implementation depends on participation.

Non-Life Insurance Mathematics (STÆ414G)

The course will give an overview of some important elements of non-life insurance and reinsurance. Models for claim numbers, the Poisson, mixed Poisson and renewal process. Stochastic models for non-life insurance risks, in particular the compound Poisson, compound mixed Poisson and renewal models. The Cramer-Lundberg and the renewal model as basic risk models. Methods for approximating the distribution of risk models. Small and large claim distributions and their properties. Premium calculation principles for the total claim amount of a portfolio. Experience rating: calculation of the premium in a policy. Reinsurance treaties and their properties. Bayesian methods in a non-life insurance context, in particular the Bayes and linear Bayes estimators for calculating the premium in a policy.

Life Insurance Mathematics (STÆ413G)

Payment flows; mortality theory; overview of the main forms of insurance; the principle of equivalence; prospective reserves and differential equations for these; cost; general Markov chains in life insurance with applications to disability insurance and multi-life insurance; profits and bonuses; market rate products.

Generalized Linear Models (STÆ421M)

Generalized linear regression models. Exponential dispersion models. Poisson processes and tests for overdispersion. Survival regression models. Nonlinear effects and basis expansions. Parametric, semiparametric and nonparametric likelihood methods. Partial likelihood methods. Generalized linear regression analysis in R.

Physics 2 R (EÐL206G)

Introduction to electrodynamics in material; from insulators to superconductors.  Charge and electric field. Gauss' law. Electric potential. Capacitors and dielectrics. Electric currents and resistance. Circuits. Magnetic fields. The laws of Ampère and Faraday. Induction. Electric oscillation and alternating currents. Maxwell's equations. Electromagnetic waves. Reflection and refraction. Lenses and mirrors. Wave optics.

Mathematical Physics (EÐL612M)

Continuum mechanics: Stress and strain, equations of motion. Seismic waves. Maxwell's equations and electromagnetic waves. Plane waves, reflection and refraction. Distributions and Fourier transforms. Fundamental solutions of linear partial differential equation. Waves in homogeneous media. Huygens' principle and Ásgeirsson's mean value theorem. Dispersion, phase and group velocities, Kramers-Kronig equations. The method of stationary phase. Surface waves on liquids.

Software Development (HBV401G)

In this course, software engineers and computer scientists take the step from programming-in-the-small (i.e. individual developers creating compact modules that solve clearly defined problems) to programming-in-the-large (i.e. teams of developers building complex systems that satisfy vague customer requirements). To deal with the complexities of such projects, this course introduces key software engineering concepts such as agile and plan-driven software process models, requirements engineering, effort estimation, object-oriented analysis and design, software architecture and test-driven development. These concepts are immediately applied in practice as students team up to develop and integrate component-based systems using the Java programming language.

Introduction to Measure-Theoretic Probability (STÆ418M)

Probability based on measure-theory.

Subject matter: Probability, extension theorems, independence, expectation. The Borel-Cantelli theorem and the Kolmogorov 0-1 law. Inequalities and the weak and strong laws of large numbers. Convergence pointwise, in probability, with probability one, in distribtution, and in total variation. Coupling methods. The central limit theorem. Conditional probability and expectation.

Topology (STÆ419M)

General topology: Topological spaces and continuous maps. Subspaces, product spaces and quotient spaces. Connected spaces and compact spaces. Separation axioms, the lemma of Urysohn and a metrization theorem. Completely regular spaces and compactifications.

Analysis of Algorithms (TÖL403G)

Methodology for the design of algorithms and the analysis of their time conplexity. Analysis of algorithms for sorting, searching, graph theory and matrix computations. Intractable problems, heuristics, and randomized algorithms.

Statistical Consulting (LÝÐ201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Statistical Consulting (LÝÐ201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Mathematics for Physicists II (EÐL408G)

Python tools related to data analysis and manipulation of graphs. Differential equations and their use in the description of physical systems. Partial differential equations and boundary value problems. Special functions and their relation to important problems in physics. We will emphasize applications and problem solving.

Mathematics for Physicists I (STÆ211G)

Order of magnitude estimates, scaling relations, and dimensional analysis. Python tools related to data analysis and plotting. Mathematical concepts such as vectors, matrices, differential operators in three dimensions, coordinate transformations, partial differential equations and Fourier series and their relation to undergraduate courses in physics and engineering. We will emphasize applications and problem solving.

Portfolio Management (VIÐ604G)

The theory behind decisions of investors and corporations regarding building and managing asset and liability portfolios. Risk management of corporations will also be covered.

The course is taught in English

Mathematical Structure (TÖL104G)

Propositions, predicates, inference rules. Set operations and Boolean algebra. Induction and recursion. Basic methods of analysis of algorithms and counting. Simple algorithms in number theory. Relations, their properties and representations. Trees and graphs and related algorithms. Strings, examples of languages, finite automata and grammars.

Computers, operating systems and digital literacy basics (TÖL108G)

In this course, we study several concepts related to digital literacy. The goal of the course is to introduce the students to a broad range of topics without necessarily diving deep into each one.

The Unix operating system is introduced. The file system organization, often used command-line programs, the window system, command-line environment, and shell scripting. We cover editors and data wrangling in the shell. We present version control systems (git), debugging methods, and methods to build software. Common concepts in the field of cryptography are introduced as well as concepts related to virtualization and containers.

Mathematical Analysis IA (STÆ101G)

Main emphasis is on the differential and integral calculus of functions of a single variable. The systems of real and complex numbers. Least upper bound and greatest lower bound. Natural numbers and induction. Mappings and functions. Sequences and limits. Series and convergence tests. Conditionally convergent series. Limits and continuous functions. Trigonometric functions. Differentiation. Extreme values. The mean value theorem and polynomial approximation. Integration. The fundamental theorem of calculus. Logarithmic and exponential functions, hyperbolic and inverse trigonometric functions. Methods for finding antiderivatives. Real power series. First-order differential equations. Complex valued functions and second-order differential equations.

Mathematical Analysis I (STÆ104G)

This is a foundational course in single variable calculus. The prerequisites are high school courses on algebra, trigonometry. derivatives, and integrals. The course aims to create a foundation for understanding of subjects such as natural and physical sciences, engineering, economics, and computer science. Topics of the course include the following:

• Real numbers.
• Limits and continuous functions.
• Differentiable functions, rules for derivatives, derivatives of higher order, applications of differential calculus (extremal value problems, linear approximation).
• Transcendental functions.
• Mean value theorem, theorems of l'Hôpital and Taylor.
• Integration, the definite integral and rules/techniques of integration, primitives, improper integrals.
• Fundamental theorem of calculus.
• Applications of integral calculus: Arc length, area, volume, centroids.
• Ordinary differential equations: First-order separable and homogeneous differential equations, first-order linear equations, second-order linear equations with constant coefficients.
• Sequences and series, convergence tests.
• Power series, Taylor series.

Linear Algebra A (STÆ106G)

Basics of linear algebra over the reals with emphasis on the theoretical side. 

Subject matter: Systems of linear equations, matrices, Gauss-Jordan reduction.  Vector spaces and their subspaces.  Linearly independent sets, bases and dimension.  Linear maps, range space and nullspace. 
The dot product, length and angle measures.  Volumes in higher dimensions and the cross product in threedimensional space.  Flats, parametric descriptions and descriptions by equations.  Orthogonal projections and orthonormal bases.  Gram-Schmidt orthogonalization.  Determinants and inverses of matrices.  Eigenvalues, eigenvectors and diagonalization.

Computer Science 1a (TÖL105G)

Programming in Python (for computations in engineering and science): Main commands and statements (computations, control statements, in- and output), definition and execution of functions, datatypes (numbers, matrices, strings, logical values, records), operations and built-in functions, array and matrix computation, file processing, statistics, graphics. Object-oriented programming: classes, objects, constructors and methods. Concepts associated with design and construction of program systems: Programming environment and practices, design and documentation of function and subroutine libraries, debugging and testing of programmes.

Operations Research (IÐN401G)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover linear programming and the simplex algorithm, as well as related analytical topics. It will also introduce special types of mathematical models, including transportation, assignment, network, and integer programming models. The student will become familiar with a modeling language for linear programming.

Introduction to data science (REI202G)

The course provides an introduction to the methods at the heart of data science and introduces widely used software tools such as numpy, pandas, matplotlib and scikit-learn.

The course consists of 6 modules:

1. Introduction to the Python programming language.
2. Data wrangling and data preprocessing.
3. Exploratory data analysis and visualization.
4. Optimization.
5. Clustering and dimensionality reduction.
6. Regression and classification.

Each module concludes with a student project.

Note that there is an academic overlap with REI201G Mathematics and Scientific Computing and both courses cannot be valid for the same degree.

Computer Science 2 (TÖL203G)

The course will cover various data structures, algorithms and abstract data types. Among the data structures covered are lists, stacks, queues, priority queues, trees, binary trees, binary search trees and heaps along with related algorithms. Various search and sort algorithms will be covered. Algorithms will be analysed for their space and time complexity. There will be small programming assignments in Java using the given data structures and algorithms.

Probability and Statistics (STÆ203G)

Basic concepts in probability and statistics based on univariate calculus. 

Topics: 
Sample space, events, probability, equal probability, independent events, conditional probability, Bayes rule, random variables, distribution, density, joint distribution, independent random variables, condistional distribution, mean, variance, covariance, correlation, law of large numbers, Bernoulli, binomial, Poisson, uniform, exponential and normal random variables. Central limit theorem. Poisson process. Random sample, statistics, the distribution of the sample mean and the sample variance. Point estimate, maximum likelihood estimator, mean square error, bias. Interval estimates and hypotheses testing form normal, binomial and exponential samples. Simple linear regression. Goodness of fit tests, test of independence.

Mathematical Analysis II (STÆ205G)

Open and closed sets. Mappings, limits and continuity. Differentiable mappings, partial derivatives and the chain rule. Jacobi matrices. Gradients and directional derivatives. Mixed partial derivatives. Curves. Vector fields and flow. Cylindrical and spherical coordinates. Taylor polynomials. Extreme values and the classification of stationary points. Extreme value problems with constraints. Implicit functions and local inverses. Line integrals, primitive functions and exact differential equations. Double integrals. Improper integrals. Green's theorem. Simply connected domains. Change of variables in double integrals. Multiple integrals. Change of variables in multiple integrals. Surface integrals. Integration of vector fields. The theorems of Stokes and Gauss.

Introduction to Probability Theory (STÆ210G)

This is an extension of the course "Probability and Statistics" STÆ203G. The basic concepts of probability are considered in more detail with emphasis on definitions and proofs. The course is a preparation for the two M-courses in probability and the two M-courses in statistics that are taught alternately every other year.

Topics beyond those discussed in the probability part of STÆ203G:

Kolmogorov's definition. Proofs of propositions on compound events and conditional probability. Proofs for discrete and continuous variables of propositions on expectation, variance, covariance, correlation, and conditional expectation and variance. Proofs of propositions for Bernoulli, binomial, Poisson, geometric, uniform, exponential, and gamma variables. Proof of the tail-summing proposition for expectation and the application to the geometric variable. Proof of the proposition on memoryless and exponential variables. Derivation of the distribution of sums of independent variables such as binomial, Poisson, normal, and gamma variables. Probability and moment generating functions.

Stochastic Processes (STÆ415M)

Introduction to stochastic processes with main emphasis on Markov chains.

Subject matter: Hitting time, classification of states, irreducibility, period, recurrence (positive and null), transience, regeneration, coupling, stationarity, time-reversibility, coupling from the past, branching processes, queues, martingales, Brownian motion.

R Programming (MAS102M)

Students will perform traditional statistical analysis on real data sets. Special focus will be on regression methods, including multiple regression analysis. Students will apply sophisticated methods of graphical representation and automatic reporting. Students will hand in a projects where they apply the above mentioned methods on real datasets in order to answer research questions

Applied Linear Statistical Models (STÆ312M)

The course focuses on simple and multiple linear regression as well as analysis of variance (ANOVA), analysis of covariance (ANCOVA) and binomial regression. The course is a natural continuation of a typical introductory course in statistics taught in various departments of the university.

We will discuss methods for estimating parameters in linear models, how to construct confidence intervals and test hypotheses for the parameters, which assumptions need to hold for applying the models and what to do when they are not met.

Students will work on projects using the statistical software R.

Mathematical Analysis III (STÆ302G)

Functions of a complex variable. Analytic functions. The exponential function, logarithms and roots. Cauchy's Integral Theorem and Cauchy's Integral Formula. Uniform convergence. Power series. Laurent series. Residue integration method. Application of complex function theory to fluid flows. Ordinary differential equations and systems of ordinary differential equations. Linear differential equations with constant coefficients. Systems of linear differential equations. The matrix exponential function. Various methods for obtaining a particular solution. Green's functions for initial value problems. Flows and the phase plane. Nonlinear systems of ordinary differential equations in the plane, equilibrium points, stability and linear approximations. Series solutions and the method of Frobenius. Use of Laplace transforms in solving differential equations.

Data Base Theory and Practice (TÖL303G)

Databases and database management systems. Physical data organization. Data modelling using the Entity-Relationship model and the Relational model. Relational algebra and calculus.  The SQL query language. Design theory for relational data bases, functional dependencies, decomposition of relational schemes, normal forms. Query optimization. Concurrency control techniques and crash recovery. Database security and authorization. Data warehousing.

Applied data analysis (MAS202M)

The course focuses on statistical analysis using the R environment. It is assumed that students have basic knowledge of statistics and the statistical software R. Students will learn to apply a broad range of statistical methods in R (such as classification methods, resampling methods, linear model selection and tree-based methods). The course on 12 weeks and will be on "flipped" form. This means that no lectures will be given but students will read some material and watch videos before attending classes. Students will then work on assignments during the classes.

Theoretical Numerical Analysis (STÆ412G)

This is an extension of the course "Numerical Analysis" STÆ405G. The material of Numerical Analysis (STÆ405G) is studied in more detail and more theoretically with emphasis on proofs.

Sets and Metric Spaces (STÆ202G)

Elements of set theory: Sets. Mappings. Relations, equivalence relations, orderings. Finite, infinite, countable and uncountable sets. Equipotent sets. Construction of the number systems. Metric spaces: Open sets and closed sets, convergent sequences and Cauchy sequences, cluster points of sets and limit points of sequences. Continuous mappings, convergence, uniform continuity. Complete metric spaces. Uniform convergence and interchange of limits. The Banach fixed point theorem; existence theorem about solutions of first-order differential equations. Completion of metric spaces. Compact metric spaces. Connected sets. Infinite series, in particular function series.

Mathematical Analysis IV (STÆ401G)

Aim: To introduce the student to Fourier analysis and partial differential equations and their applications.
Subject matter: Fourier series and orthonormal systems of functions, boundary-value problems for ordinary differential equations, the eigenvalue problem for Sturm-Liouville operators, Fourier transform. The wave equation, diffusion equation and Laplace's equation solved on various domains in one, two and three dimensions by methods based on the first part of the course, separation of variables, fundamental solution, Green's functions and the method of images.

Numerical Analysis (STÆ405G)

Fundamental concepts on approximation and error estimates. Solutions of systems of linear and non-linear equations. PLU decomposition. Interpolating polynomials, spline interpolation and regression. Numerical differentiation and integration. Extrapolation. Numerical solutions of initial value problems of systems of ordinary differential equations. Multistep methods. Numerical solutions to boundary value problems for ordinary differential equations.

Grades are given for programning projects and in total they amount to 30% of the final grade. The student has to receive the minimum grade of 5 for both the projects and the final exam.

Theory of linear models (STÆ310M)

Simple and multiple linear regression, analysis of variance and covariance, inference, variances and covariances of estimators, influence and diagnostic analyses using residual and influence measures, simultaneous inference. General linear models as projections with ANOVA as special case, simultaneous inference of estimable functions. R is used in assignments. Solutions to assignments are returned in LaTeX and PDF format.

In addition selected topics will be visited, e.g. generalized linear models (GLMs), nonlinear regression and/or random/mixed effects models and/or bootstrap methods etc.

Students will present solutions to individually assigned
projects/exercises, each of which is handed in earlier through a web-page.

This course is taught in semesters of even-numbered years.

Theoretical Statistics (STÆ313M)

Likelihood, Sufficient Statistic, Sufficiency Principle, Nuisance Parameter, Conditioning Principle, Invariance Principle, Likelihood Theory. Hypothesis Testing, Simple and Composite Hypothesis, The Neyman-Pearson Lemma, Power, UMP-Test, Invariant Tests. Permutation Tests, Rank Tests. Interval Estimation, Confidence Interval, Confidence, Confidence Region. Point Estimation, Bias, Mean Square Error. Assignments are returned using LaTeX and consitute 20% of the final grade.

Bayesian Data Analysis (STÆ529M)

Goal: To train students in applying methods of Bayesian statistics for analysis of data. Topics: Theory of Bayesian inference, prior distributions, data distributions and posterior distributions. Bayesian inference  for parameters of univariate and multivariate distributions: binomial; normal; Poisson; exponential; multivariate normal; multinomial. Model checking and model comparison: Bayesian p-values; deviance information criterion (DIC). Bayesian computation: Markov chain Monte Carlo (MCMC) methods; the Gibbs sampler; the Metropolis-Hastings algorithm; convergence diagnostistics. Linear models: normal linear models; hierarchical linear models; generalized linear models. Emphasis on data analysis using software, e.g. Matlab and R.

Introduction to deep neural networks (TÖL506M)

In this course we cover deep neural networks and methods related to them. We study networks and methods for image, sound and text analysis. The focus will be on applications and students will present either a project or a recent paper in this field.

Algebra (STÆ303G)

Groups, examples and basic concepts. Symmetry groups. Homomorphisms and normal subgroups. Rings, examples and basic concepts. Integral domains. Ring homomorphisms and ideals. Polynomial rings and factorization of polynomials. Special topics.

Research Project (STÆ262L)

Research Project

Mathematical Seminar (STÆ402G)

This course is intended for students who have completed at least 120 ECTS credits. Students who have not completed 120 ECTS credits and are interested in taking the course must obtain the approval of the supervisor prior to signing up for the course.

Each student prepares and studies a selected well-defined topic of mathematics or statistics and will be assigned a mentor related to that topic. Topics vary from year to year. A list of possible topics is released at the start of or prior to the course and students can also suggest topics (provided that a mentor can be found). Students write a thesis on their selected topic and prepare and give a lecture on the topic at a student conference. During the course, students provide each other with constructive critique both regarding the thesis writing and the preparation of the lecture. In addition to presenting their own projects at the student conference, students take active part, listen to their fellow course members and ask questions.

Physics 1 R (EÐL107G)

Introduce students to methods and fundamental laws of mechanics, waves and thermodynamics, to the extent that they can apply their knowledge to solve problems. 

Concepts, units, scales and dimensions.  Vectors. Kinematics of particles. Particle dynamics, inertia, forces and Newton's laws. Friction. Work and energy, conservation of energy. Momentum, collisions. Systems of particles, center of mass. Rotation of a rigid body.  Angular momentum and moment of inertia. Statics. Gravity. Solids and fluids, Bernoulli's equation. Oscillations: Simple, damped and forced. Waves. Sound.  Temperature. Ideal gas. Heat and the first law of thermodynamics. Kinetic theory of gases. Entropy and the second law of thermodynamics.

Note that the textbook is accessible to students via Canvas free of charge.

Biostatistics II (Clinical Prediction Models ) (LÝÐ301F)

This course is a continuation of Biostatistics I and constitutes a practical guide to statistical analyses of student's own research projects. The course covers the following topics. Estimation of relative risk/odds ratios and adjusted estimation of relative risk/odds ratios, correlation and simple linear regression, multiple linear regression and logistic regression. The course is based on lectures and practical sessions using R for statistical analyses.

R Programming (MAS102M)

Students will perform traditional statistical analysis on real data sets. Special focus will be on regression methods, including multiple regression analysis. Students will apply sophisticated methods of graphical representation and automatic reporting. Students will hand in a projects where they apply the above mentioned methods on real datasets in order to answer research questions

Machine Learning (REI505M)

An overview of some of the main concepts, techniques and algorithms in machine learning. Supervised learning and unsupervised learning. Data preprocessing and data visualization. Model evaluation and model selection. Linear regression, nearest neighbors, support vector machines, decision trees and ensemble methods. Deep learning. Cluster analysis and the k-means algorithm. The students implement simple algorithms in Python and learn how to use specialized software packages. At the end of the course the students work on a practical machine learning project.

Geometry (STÆ508M)

The subject of the course is the foundations of geometry. Affine geometry: Axioms and models. Affine planes and their isomorphisms. Parallel transformations, translations, dilatations. Translation planes and vector addition. Desargue's theorem and coordinates over division rings. The theorem of Pappus. Finite affine planes and the theorem of Wedderburn. Isomorphisms and automorphisms of Desarguesian affine planes. Affine spaces of three and higher dimensions. Projective geometry: Projective planes. Duality principle. The connection between affine and projective planes. The Bruck-Ryser theorem. Automorphisms. The theorems of Desargues and Pappus and coordinates in projective planes. Classical geometry: Incidence axioms, order axioms, congruence axioms. Neutral geometry and neutral planes. Angles and congruence theorems. Different continuity axioms. Euclidean geometry, the theorem of Pappus and coordinates over pythagorean and euclidean fields. Ruler and compass constructions. Hyperbolic geometry, Hilbert's axiom. The end calculus of Hilbert and coordinates in hyperbolic geometry.

Numerical Linear Algebra (STÆ511M)

Iterative methods for linear systems of equations.  Decompositions of matrices: QR, Cholesky, Jordan, Schur, spectral and singular value decomposition (SVD) and their applications.  Discrete Fourier transform (DFT) and the fast Fourier transform (FFT).  Discrete cosine transform (DCT) in two-dimensions and its application for the compression of images (JPEG) and audio (MP3, AAC).  Sparse matrices and their representation.

Special emphasis will be on the application and implementation of the methods studied.

Graph Theory (STÆ520M)

Graphs, homomorphisms and isomorphisms of graphs. Subgraphs, spanning subgraphs. Paths, connected graphs. Directed graphs. Bipartite graphs. Euler graphs and Hamilton graphs; the theorems of Chvátal, Pósa, Ore and Dirac. Tournaments. Trees, spanning trees, the matrix-tree theorem, Cayley's theorem. Weighted graphs, the algorithms of Kruskal and Dijkstra. Networks, the max-flow-min-cut theorem, the algorithm of Ford and Fulkerson, Menger's theorem. Matchings, Berge's theorem, Hall's marriage theorem, the König-Egerváry theorem, the Kuhn-Munkres algorithm. Inseparable and two-connected graphs. Planar graphs, Euler's formula, Kuratowski's theorem, dual graphs. Embeddings of graphs in surfaces, the Ringel-Youngs-Mayer theorem. Colourings, Heawood's coloring theorem, Brooks's theorem, chromatic polynomial; edge colourings, Vizing's theorem.

Distributions (STÆ523M)

Fundamentals of distribution theory with applications to partial differential equations

Subject matter: Test funcitons, distributions, differnetiation of distributions, convergence of sequences of distributions, Taylor expansions in several variables, localization, distributions with compact support, multiplication by functions, transpostition: pullback and push-forward of distributions, convolution of distributions, fundamental solutions, Fourier transformation, Fourier series, and fundamental solutions and Fourier transforms.

Introduction to Logic (STÆ528M)

Logical deductions and proofs. Propositional calculus, connectives, truth functions and tautologies. Formal languages, axioms, inference rules. Quantifiers. First-order logic. Interpretations. The compactness theorem. The Lövenheim-Skolem theorem. Computability, recursive functions. Gödel's theorem.

Numerical Methods for Partial Differential Equations (STÆ537M)

The aim of the course is to study numerical methods to solve partial differential equations and their implementation.

Formal Languages and Computability (TÖL301G)

Finite state machines, regular languages and grammars, push-down automata, context-free languages and grammars, Turing machines, general languages and grammars, and their basic properties. Recursive and recursively enumerable languages, reduction between languages, connection to decision problems and proving unsolvability of such problems. The complexity classes P and NP, and NP-completeness. Examples of various models of computation.

Various approaches to teaching mathematics in upper secondary schools (SNU503M)

In this course, students learn to plan mathematics teaching in upper secondary school using various approaches to provide access for all. An emphasis will be put on exploring different teaching environments and teaching methods that build on research on the teaching and learning of mathematics. In the course, the aims of learning mathematics both in Iceland and its neighboring countries will be discussed based on curricular and governmental documents. Students will read about and get a chance to try out various ways to assess and analyze students’ mathematical achievements. The course format includes lectures, project work, presentations, topic studies connected to practice, and critical topic discussion. An emphasis will be put on students’ discussion about challenges and their search for solutions to problems related to the teaching and learning of mathematics.

Cryptocurrency (STÆ532M)

The course will start by introducing the basic concepts of electronic currencies, such as wallets, addresses and transactions. The students will get to know encoding, transactions, blocks and blockchains. The cryptocurrency Smileycoin will be used as an example throughout the course.
Students will compile their own wallets from source and dive deeply enough into the underlying algorithms to be able to put together their own transactions from the Linux command line and read typical wallet code written in C++.
Students will learn how to call the wallet from other software, e.g. to analyse the flow of funds.
Students will learn how to implement several additions to the traditional use of electronic currency such as encoded messages, running software to react to payments etc.
Students will set up their own examples of addition and study how to set up atomic swaps between different currencies, using the Smileycoin for announcements.

Homework will be individualised, selected from different formats (a) solutions based on the wallet on the command line, (2) documents to form handouts or other material in the tutor-web, (3) short programs (APIs) which respond to transactions being send to particular addresses or to a
particular wallet, (4) programs which talk to exchanged and/or (5) new user interfaces which improve or add to the functionality of a wallet.

All the material and assignments will be in English. Returned assignments will become a part of the open tutor-web educational system.

The course may be taught as a reading course or self-study, but the exact implementation depends on participation.

Generalized Linear Models (STÆ421M)

Generalized linear regression models. Exponential dispersion models. Poisson processes and tests for overdispersion. Survival regression models. Nonlinear effects and basis expansions. Parametric, semiparametric and nonparametric likelihood methods. Partial likelihood methods. Generalized linear regression analysis in R.

Life Insurance Mathematics (STÆ413G)

Payment flows; mortality theory; overview of the main forms of insurance; the principle of equivalence; prospective reserves and differential equations for these; cost; general Markov chains in life insurance with applications to disability insurance and multi-life insurance; profits and bonuses; market rate products.

Non-Life Insurance Mathematics (STÆ414G)

The course will give an overview of some important elements of non-life insurance and reinsurance. Models for claim numbers, the Poisson, mixed Poisson and renewal process. Stochastic models for non-life insurance risks, in particular the compound Poisson, compound mixed Poisson and renewal models. The Cramer-Lundberg and the renewal model as basic risk models. Methods for approximating the distribution of risk models. Small and large claim distributions and their properties. Premium calculation principles for the total claim amount of a portfolio. Experience rating: calculation of the premium in a policy. Reinsurance treaties and their properties. Bayesian methods in a non-life insurance context, in particular the Bayes and linear Bayes estimators for calculating the premium in a policy.

Mathematical Physics (EÐL612M)

Continuum mechanics: Stress and strain, equations of motion. Seismic waves. Maxwell's equations and electromagnetic waves. Plane waves, reflection and refraction. Distributions and Fourier transforms. Fundamental solutions of linear partial differential equation. Waves in homogeneous media. Huygens' principle and Ásgeirsson's mean value theorem. Dispersion, phase and group velocities, Kramers-Kronig equations. The method of stationary phase. Surface waves on liquids.

Software Development (HBV401G)

In this course, software engineers and computer scientists take the step from programming-in-the-small (i.e. individual developers creating compact modules that solve clearly defined problems) to programming-in-the-large (i.e. teams of developers building complex systems that satisfy vague customer requirements). To deal with the complexities of such projects, this course introduces key software engineering concepts such as agile and plan-driven software process models, requirements engineering, effort estimation, object-oriented analysis and design, software architecture and test-driven development. These concepts are immediately applied in practice as students team up to develop and integrate component-based systems using the Java programming language.

Simulation (IÐN403M)

Simulation techniques and system modelling find application in fields as diverse as physics, chemistry, biology, economics, medicine, computer science, and engineering. The purpose of this course is to introduce fundamental principles and concepts in the general area of systems modelling and simulation. Topics to be covered in this course are discrete event simulation, statistical modelling, and simulation modelling design, experimental design, model testing and interpretation of simulation results. The maximum likelihood estimation of probability distributions base on real data is presented. The course will also introduce the generation of random variates and testing. Fundamental programming of simulation models in C is covered and specialized simulation packages introduced. The students will complete a real world simulation project where the emphasis will be on manufacturing or service systems.

Design & Experimental Execution (IÐN405G)

The purpose of the course is to train an engineering approach to experiments and experimental thinking. Experiments are designed, carried out, data collected and processed using statistical methods. Finally, it discussed how conclusions can be drawn from data / information when using experiments in for example product design and the design and operation of production systems.

Course material: Linear and non-linear regression analysis. Analysis of Variances (ANOVA). Design of experiments. Statistical quality control. Non-parametric tests that can be used in data processing. Use of statistical programs when solving tasks.

Introduction to Measure-Theoretic Probability (STÆ418M)

Probability based on measure-theory.

Subject matter: Probability, extension theorems, independence, expectation. The Borel-Cantelli theorem and the Kolmogorov 0-1 law. Inequalities and the weak and strong laws of large numbers. Convergence pointwise, in probability, with probability one, in distribtution, and in total variation. Coupling methods. The central limit theorem. Conditional probability and expectation.

Topology (STÆ419M)

General topology: Topological spaces and continuous maps. Subspaces, product spaces and quotient spaces. Connected spaces and compact spaces. Separation axioms, the lemma of Urysohn and a metrization theorem. Completely regular spaces and compactifications.

Analysis of Algorithms (TÖL403G)

Methodology for the design of algorithms and the analysis of their time conplexity. Analysis of algorithms for sorting, searching, graph theory and matrix computations. Intractable problems, heuristics, and randomized algorithms.

Statistical Consulting (LÝÐ201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Statistical Consulting (LÝÐ201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Mathematics for Physicists II (EÐL408G)

Python tools related to data analysis and manipulation of graphs. Differential equations and their use in the description of physical systems. Partial differential equations and boundary value problems. Special functions and their relation to important problems in physics. We will emphasize applications and problem solving.

Mathematics for Physicists I (STÆ211G)

Order of magnitude estimates, scaling relations, and dimensional analysis. Python tools related to data analysis and plotting. Mathematical concepts such as vectors, matrices, differential operators in three dimensions, coordinate transformations, partial differential equations and Fourier series and their relation to undergraduate courses in physics and engineering. We will emphasize applications and problem solving.

Portfolio Management (VIÐ604G)

The theory behind decisions of investors and corporations regarding building and managing asset and liability portfolios. Risk management of corporations will also be covered.

The course is taught in English