Mathematical Studies in Joensuu

Most of the undergraduate courses below can be taken as a reading course in English, while the lectures may still be in Finnish. You have to agree personally with the instructor responsible for a particular course about taking the course. Note that the instructors vary from year to year. The credit unit cr below corresponds to the ECTS credit unit. An average student should be able to take 30/60 cr per semester/year by working 40 hours per week during the semester(s). Please contact Dr. Janne Heittokangas for more details.

Basic Studies

  • Analysis I (9 cr) 3316131
  • Basic Calculus a (4 cr) 3315222
  • Basic Calculus b (4 cr) 3315223
  • Introduction to Mathematics (8 cr) 3315231

Intermediate Studies

  • Algebra (8 cr) 3316261
  • Analysis II (9 cr) 3316221
  • Analysis III (9 cr) 3316231
  • Differential Equations a (4 cr) 3316242
  • Differential Equations b (4 cr) 3316243
  • Elementary Topology (4 cr) 3316272
  • Linear Algebra a (5 cr) 3316122
  • Linear Algebra b (4 cr) 3316123
  • Probability Theory a (4 cr) 3316252

Advanced Studies

  • Analysis IV (8 cr) 3317111
  • Analysis Laboratory (2 cr) 3316429
  • Applied Analysis (8 cr) 3316331
  • Calculus of Variations (4 cr) 3317333
  • Communications in Mathematics (3 cr) 3315129
  • Complex Analysis Ia (4 cr) 3317142
  • Complex Analysis Ib (4 cr) 3317143
  • Complex Analysis II (8 cr) 3317171
  • Discrete Mathematics (8 cr) 3316341
  • Dynamical Systems (4 cr) 3317342
  • Finite Element Method (8 cr) 3317361
  • Functional Analysis (8 cr) 3317151
  • Geometry (8 cr) 3317131
  • History of Mathematics (4 cr) 3317222
  • Mathematical Computing in Schools (4 cr) 3317232 (attendance required)
  • Mathematical Modelling (5 cr) 3316362
  • Matrices (8 cr) 3317311
  • Measure and Integration Theory a (4 cr) 3317122
  • Measure and Integration Theory b (4 cr) 3317123
  • Modelling and Optimization (4 cr) 3317392 (material in Finnish only)
  • Number Theory (8 cr) 3317181
  • Numerical Analysis (8 cr) 3316321
  • Numerical Linear Algebra (8 cr) 3317351
  • Optimization (4 cr) 3317332
  • Partial Differential Equations (8 cr) 3317321
  • Partial Differential Equations in Mathematical Modelling (4 cr) 3317393 (material in Finnish only)
  • Probability Theory b (4 cr) 3316253
  • School Mathematics (4 cr) 3316352 (attendance required)
  • Soft Computing (4 cr) 3317394 (material in Finnish only)
  • Statistical Models (4 cr) 3317395 (material in Finnish only)
  • Stochastic Models (4 cr) 3317396 (material in Finnish only)
  • Topology (8 cr) 3317161

Course Descriptions of Selected Courses

  • Analysis IV (8 cr)
    Description: A quick review on basic facts in linear algebra and metric spaces. An introduction to Lebesgue integral and measure theory, classical function spaces, normed spaces, inner product spaces, and to linear operators.
    Textbook: B. B. Rynne and M. A. Youngson, Linear Functional Analysis (2002).
  • Applied Analysis (8cr)
    Description: Fourier series and integrals; the Laplace transform and z-transform; continuous and discrete Fourier transformation; applications.
    Textbooks: Kreyszig, Advanced engineering mathematics (1988), Chapters 5 and 10; Jeffrey, Linear algebra and ordinary differential equations (1990), pp. 710-750; Strang, Introduction to applied mathematics (1986), pp. 290-299.
  • Complex Analysis Ia & Ib (8 cr)
    Description: Topics include the algebra and point representation of complex numbers, analytic functions, Cauchy-Riemann equations, complex integration, Cauchy's theorem, the residue theorem, the maximum modulus theorem, Laurent series, the fundamental theorem of algebra, and Moebius transformations.
    Textbook: E. Saff and A. Snider, Fundamentals of Complex Analysis with Applications to Engineering and Science (2003), Chapters 1-7 (key concepts only).
  • Complex Analysis II (8 cr)
    Description: For students who desire a rigorous introduction to the theory of functions of a complex variable. The contents of this course may vary from year to year. Topics usually include some or all of the following: spherical and hyperbolic metrics, normal convergence, some classical results (such as Hurwitz Theorem, Rouche Theorem and Schwarz Lemma), local boundedness and equicontinuity, elliptic functions, and introduction to Nevanlinna theory.
  • Discrete Mathematics (8 cr)
    Description: Introduction to discrete mathematical structures and methods: logic and sets, reasoning and proof, relations and functions, elements of number theory and combinatorics, graphs and algorithms, sequences and recurrence relations.
    Textbook: Robert J. McEliece, Robert B. Ash and Carol Ash, Introduction to discrete mathematics (1989), Chapters 1-5 and 10.
  • Elementary Topology (4 cr)
    Description: Topology of metric spaces and a short introduction to general topological spaces. Topics include the following basic concepts of a metric space: open and closed sets; closure of a set; interior, exterior and boundary points of a set; neighborhood of a point; convergence of a point sequence; Cauchy sequence; continuous function and equivalence of metrics. Complete metric spaces and general topological spaces are briefly discussed at the end of the course.
    Textbooks: I. Kaplansky, Set Theory and Metric Spaces (1977), p. 67-94; P. Long, Introduction to General Topology (1971), Chapters 3 and 9.
  • Functional Analysis (8 cr)
    Description: The course covers the basics of continuous linear operators in normed spaces and in inner product spaces. One of the key notions is the invertibility of the operator. Topics include the open mapping theorem, Riesz-Frechet theorem, Fourier series in Hilbert spaces, the adjoint of an operator, and the notion of compact operator. A student should have taken Analysis IV or equivalent before taking this course.
    Textbook: B. B. Rynne and M. A. Youngson, Linear Functional Analysis (2002).
  • Geometry (8 cr)
    Description: The course is intended mainly for students becoming math teachers. The course presents both the analytic and axiomatic view to the Euclidean geometry. The analytic point of view covers geometric mappings, e.g., isometries, affine mappings, inversions and the related invariances. The classical geometry related to triangles, circles, and lines are studied from both the axiomatic and the analytic point of view.
    Textbooks: D. A. Brannan, M. F. Esplen and J. J. Gray, Geometry (1999); P. J. Ryan, Euclidean and non-Euclidean geometry: an analytic approach (1986).
  • Matrices (8 cr)
    Description: A review of linear algebra; eigenvalues and singular values; spectral decomposition; vector and matrix norms; the singular value decomposition; condition of a linear system; LU-, LDL^T- and Cholesky factorizations; Householder transformations and QR-factorization; linear least squares.
    Textbook: Gill & Murray & Wright, Numerical linear algebra and optimization, vol. 1 (1991), Chapters 1-4 and 6.
  • Measure and Integration Theory a (4 cr)
    Description: The basics of Lebesgue integration theory in Euclidean spaces is covered. Topics include outer measure, measurable set and functions, and the key properties of Lebesgue integral, e.g. the standard convergence theorems such as Fatou's lemma and the dominated convergence theorem.
    Textbooks: E. DiBenedetto, Real Analysis (2002); H. L. Royden, Real Analysis (1988).
  • Measure and Integration Theory b (4 cr)
    Description: The Lebesgue integration theory presented in "Measure and Integration Theory a" is extended and applied to the theory of real analysis. Topics include basic covering results, the fundamental theorem of analysis for absolutely continuous functions, functions of bounded variation, L^p-spaces, and the convolution approximation.
    Textbooks: E. DiBenedetto, Real Analysis (2002); H. L. Royden, Real Analysis (1988).
  • Number Theory (8 cr)
    Description: The course covers the basics of classical number theory as well as some modern applications to the cryptology. The notion of congruence is intensively used throughout the course. Topics include greatest common divisor, Chinese remainder theorem, Fermat's little theorem as well as Euler's theorem, multiplicative functions, pseudoprimes, RSA, primitive roots, and quadratic residues.
    Textbook: K. H. Rosen, Elementary Number Theory and it's Applications (1993).
  • Numerical Analysis (8 cr)
    Description: Error analysis; the numerical approximation of definite integrals; numerical solution of nonlinear equations, including Newton's method; approximation by polynomials and by splines; the numerical solution of ordinary differential equations.
    Textbook: Eldén & Wittmeyer-Koch, Numerical analysis: an introduction (1990).
  • Topology (8 cr)
    Description:An introduction to general topology, topological spaces and continuous functions, compact and connected sets, quotient spaces, homotopy. A student should have taken "Elementary Topology" or equivalent before taking this course.
    Textbook: P. Long, Introduction to General Topology (1971).