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Analysis and Algebra
Cursusdoel
After completing this course, students are able to:
- Find and classify equilibria of systems of ordinary differential equations, and sketch their phase portraits.
- Analyze systems of ordinary differential equations in terms of phase portraits, stability properties, dependence on parameters, and attractors.
- Apply these techniques to gain insights into topics in natural and social sciences.
- Read and write formal proofs in group theory.
- Recognize and work with group-theoretic structures and concepts, such as subgroup, order, permutations, isomorphism, direct product, and Lagrange’s Theorem, in a variety of different contexts.
- Justify and apply key results about analytic functions, complex differentiation and integration.
- Use complex methods, such as path integrals in the complex plane, to solve problems from other areas such as real-valued calculus, physics, and engineering.
| Description of assignment | Assesses which learning goals? |
| Midterm Exam Final Exam Assignments pre-midterm Assignments post-midterm Active participation |
1-2, 4-7 1-2, 4-7 1-7 1-7 1-7 |
Vakinhoudelijk
This course replaces the former Math labs UCSCIMATL5, UCSCIMATL3, and UCSCIMATL6.
The course will cover two or three of the following main topics.
Dynamical systems. Many systems across the natural sciences and beyond are characterized by local and instantaneous rules, such as mechanical forces between bodies or the interaction of individuals in a population, but it is often interesting and desirable to study, through mathematical analysis, more global and qualitative aspects of the behavior of such systems. Techniques to this end include systems of ordinary differential equations, phase plane analysis, stability analysis, linearization, limit cycles, Poincaré-Benedixson Theorem.
Group theory. Abstraction and formalization are powerful tools in modern mathematics. By studying at an abstract level certain algebraic structures and symmetries, group theory provides a powerful conceptual lens and a language to express structural relations that have proved ubiquitous in mathematics and physics. Group theory is also a natural setting in which to learn styles of proof-writing and abstract thought characteristic of much of modern mathematics.
Complex analysis. Generalizing the ideas of ordinary calculus or real analysis to the domain of complex numbers is both mathematically rich and fruitful in applications. Topics include the calculus of complex-valued functions and power series, geometric properties of analytic functions, the Cauchy-Riemann equations, topological properties of integration in the complex plane, Cauchy’s Theorem, Cauchy’s Formula.
Which of the above topics are covered may vary from year to year.
The course will cover two or three of the following main topics.
Dynamical systems. Many systems across the natural sciences and beyond are characterized by local and instantaneous rules, such as mechanical forces between bodies or the interaction of individuals in a population, but it is often interesting and desirable to study, through mathematical analysis, more global and qualitative aspects of the behavior of such systems. Techniques to this end include systems of ordinary differential equations, phase plane analysis, stability analysis, linearization, limit cycles, Poincaré-Benedixson Theorem.
Group theory. Abstraction and formalization are powerful tools in modern mathematics. By studying at an abstract level certain algebraic structures and symmetries, group theory provides a powerful conceptual lens and a language to express structural relations that have proved ubiquitous in mathematics and physics. Group theory is also a natural setting in which to learn styles of proof-writing and abstract thought characteristic of much of modern mathematics.
Complex analysis. Generalizing the ideas of ordinary calculus or real analysis to the domain of complex numbers is both mathematically rich and fruitful in applications. Topics include the calculus of complex-valued functions and power series, geometric properties of analytic functions, the Cauchy-Riemann equations, topological properties of integration in the complex plane, Cauchy’s Theorem, Cauchy’s Formula.
Which of the above topics are covered may vary from year to year.
Format
Lectures, exercise classes, project work, presentations
Lectures, exercise classes, project work, presentations
Werkvormen
UCU SCI 2 course
Toetsing
Active participation
Verplicht | Weging 10% | ECTS 0,75
Assignments
Verplicht | Weging 20% | ECTS 1,5
Presentation
Verplicht | Weging 20% | ECTS 1,5
*midterm FEEDBACK*
Niet verplicht
Midterm exam
Verplicht | Weging 25% | ECTS 1,88
Final exam
Verplicht | Weging 25% | ECTS 1,88
Ingangseisen en voorkennis
Ingangseisen
Er moet voldaan zijn aan de cursus:
Voorkennis
Er is geen informatie over ingangseisen bekend.
Voertalen
- Engels
Competenties
-
Interdisciplinariteit
Cursusmomenten
Gerelateerde studies
Tentamens
Er is geen tentamenrooster beschikbaar voor deze cursus
Verplicht materiaal
| Materiaal | Omschrijving |
|---|---|
| BOEK | Steven H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd ed., CRC Press, 2015, ISBN 978-0813349107 |
Aanbevolen materiaal
Er is geen informatie over de aanbevolen literatuur bekend
Coördinator
| dr. V.N.E. Blasjö | v.n.e.blasjo@uu.nl |
Docenten
| dr. M. Ruijgrok | M.Ruijgrok@uu.nl |
Inschrijving
Let op: deze cursus is niet toegankelijk voor studenten van andere faculteiten, bijvakkers mogen zich dus niet inschrijven.
Naar OSIRIS-inschrijvingen
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