Provide a comprehensive overview of recent trends in biomathematics
A primer on mathematical models in biology / Lee A. Segel, Leah Edelstein-Keshet. Philadelphia, Pa. : Society for Industrial and Applied Mathematics, 2013
Mathematical biology : 1. : An introduction / J. D. Murray. New York : Springer, 2002
Mathematical biology : 2. : Spatial models and biomedical applications / J. D. Murray. New York : Springer, 2003
Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering / Steven H. Strogatz. Cambridge : Perseus Books, 1994
2-MAT-111 Dynamical Systems OR 2-MAT-112 Partial Differential Equations (1) OR 2-MAT-121 Partial Differential Equations (2)
individual work
Continuous assessment: individual work
Exam: final exam and project
Assessment grade scale: A 90%, B 80%, C 70%, D 60%, E 50%
Principles of mathematical modeling, modeling goals, model building, model simulations, parameter selection, non-dimensionalization, model robustness, results analysis. Biochemical kinetics, enzymatic reactions, cooperativity, quasi-stationary approximation. Epidemiological models. Dynamics on neural and other cell membranes, Hodgkin-Huxley model, Fitzhugh-Nagumo model.
50/50
Students will become familiar with the theory and techniques used in current research in mathematical biology and in mathematical models in the natural and social sciences in general. At the same time, students try to work on a separate project in this area. They will also gain new knowledge from population models, chemical kinetics and cell biology.
examination
D. J. Acheson: Elementary Fluid Dynamics, Oxford, Clarendon Press, 1990
G. K. Batchelor: An Introduction to Fluid Dynamics, Cambridge University Press, 2000
P. Guba: Dynamika tekutín, skriptá, FMFI UK, 2021
Interim assessment during the semester has a weight of 30% (homeworks 20%, bonus exercises 10%). The two semester exam papers have a total weight of 70% (the first paper taken in the middle of the semester, the second paper taken at the end of the semester). The student must obtain at least half of the points from each semester exam paper. The final evaluation can be adjusted by an oral exam (theoretical questions, written preparation).
Grading: A (100-91), B (90-81), C (80-71), D (70-61), E (60-51), FX (50-0)
English
Equations of motion for ideal fluids. Vorticity. Irrotational flow. Vorticity equation.
Equations of motion for viscous fluids. Examples of simple viscous flows. Flows with circular streamlines.
Convection and diffusion of vorticity. Gravity waves. Dispersion and group velocity. Surface tension effects and capillary waves.
Internal gravity waves. Waves with finite amplitude. Hydraulic shocks and solitary waves. Kelvin--Helmholtz instability. Thermal convection. Centrifugal instability. Theorem on the stability of shear flow. General theorem on the stability of viscous flow. Uniqueness of steady viscous flow. Transition to turbulence.
30/70
Teach the students to derive and analyse basic equations describing fluid flow.
E. J. Hinch: Perturbation Methods, Cambridge University Press, 1991
J. Kevorkian, J. D. Cole: Multiple Scale and Singular Perturbation Methods, Springer, 1996
Interim assessment during the semester has a weight of 30% (homeworks 20%, bonus exercises 10%). The two semester exam papers have a total weight of 70% (the first paper taken in the middle of the semester, the second paper taken at the end of the semester). The student must obtain at least half of the points from each semester exam paper. The final evaluation can be adjusted by an oral exam (theoretical questions, written preparation).
Grading: A (100-91), B (90-81), C (80-71), D (70-61), E (60-51), FX (50-0)
English
Algebraic equations: Iterative method. Algebraic equations: Expansion method. Singular perturbations and rescaling. Logarithmic Poincare's expansions. Convergence and asymptoticity. Asymptotic approximation of integrals. Watson's lemma. The steepest descent method. Regular perturbation problems in differential equations. Singular perturbation problems in differential equations. Method of matched asymptotic expansions. Multiple scale method. WKBJ method. Poincare-Lindstedt method. Radius of convergence and Domb-Sykes plots.
30/70
To give an overview of basic asymptotic methods for solving algebraic and differential problems in applied mathematics.
Scientific research and elaboration of the results in a written form.
The student completes the corresponding part of the research related to the dissertation and achieves original results.
Scientific research and elaboration of the results in a written form.
The student completes the corresponding part of the research related to the dissertation and achieves original results.
Scientific research and elaboration of the results in written form.
The student completes the corresponding part of the research related to the dissertation and achieves original results.
Achieving new scientific results in the area of student’s PhD study. Writing a scientific manuscript and its publication.
A publication in a journal covered by Mathematical Reviews, Zentralblatt MATH, Referativnyi Zhurnal, INSPEC; in fully peer reviewed proceedings of an international conference held in an international language, with a large international participation, reviewed by title in Mathematical Reviews, Zentralblatt MATH, Referativnyi Zhurnal; in proceedings of an internationally renowned organization (IFAC) or publisher (Springer, Elsevier) if it is not of category A; a monograph or a chapter in monograph in an international language published by an international publisher if it is not of category A. The number of credits will be 0.P times 25 if the student’s participation on the publication is P%.
Achieving new scientific results in the area of student’s PhD study. Writing a scientific manuscript and its publication.
A publication reviewed, but not by title, in Mathematical Reviews, ZbMATH, Referativnyi Zhurnal; in peer reviewed proceedings of an international conference with an international organizing committee, if it is not of categories A or B; a monograph or a chapter in monograph published by a Slovak publishing house in an international language, exceptionally also in Slovak, reviewed in Mathematical Reviews, ZbMATH, Referativnyi Zhurnal, if it contains new original results by the student. The number of credits will be 0.P times 20 if the student’s participation on the publication is P%.
Achieving new scientific results in the area of student’s PhD study. Writing a scientific manuscript and its publication.
A scientific publication by the student that cannot be classified as of type A, B, or C. The number of credits will be 0.P times 10 if the student’s participation on the publication is P%.
Reading of the Bc-degree thesis in question. Writing a referee report on it.
The student writes a referee report on a Bc-degree thesis.
Preparations for producing a new teaching aid or text. Production of a new teaching aid or text.
The student produces a new teaching aid or text.
Exam: Oral
0/100
Exam: Oral
0/100
The aim of this lecture, based on principles and methods of the traditional geometry and the computational synthetic geometry, is to deepen the knowledge of the students useful in technical applications of geometry and machine vision. The lecture will also supply the necessary geometric basis for the theory of computer vision and image recognition. The students will also find inspiration for creating new algorithms or improving effectiveness of the traditional algorithms in computer graphics.
R. Hartley, A Zisserman: Multiple View Geometry, Cambridge University Press 2002
O. Faugeras, Q – T. Luong ( T. Papadopoulo) Geometry of Multiple Images, The MIT Press, Cambridge, Massachusetts, London 2001
Final exam (oral exam with written preparation 100%). Rough evaluation scale: A 90%, B 80%, C 70%, D 60%, E 50%
Synthetic constructions of curves, surfaces and solids for technical applications. Viewing methods and traditional methods of the realistic imaging (illumination, shadowing, isophotes). The use of computers and professional software in constructions and photo-realistic imaging. Geometric photogrammetry and its use in creating virtual cities. Single view geometry and camera models. Mathematical and geometric principles of multiple view geometries. Two-view and epipolar geometry. Projections used in cartography or related fields.
The aim of this lecture, based on principles and methods of the traditional geometry and the computational synthetic geometry, is to deepen the knowledge of the students useful in technical applications of geometry and machine vision. The lecture will also supply the necessary geometric basis for the theory of computer vision and image recognition. The students will also find inspiration for creating new algorithms or improving effectiveness of the traditional algorithms in computer graphics.
examination
1. S. Bosch: Algebraic Geometry and Commutative Algebra, Springer, 2013
2. M. F. Atiyah, I. G. MacDonald: Introduction to commutative algebra: Advanced book program.Oxford : Westview, 1969
3. D. Eisenbud: Commutative algebra with a view toward algebraic geometry. New York: Springer,2004
4. H.Matsumura: Commutative Ring Theory, Cambridge University Press, Cambridge, 1989
Homeworks (written solutions of problems and their consulting 50%), final exam (oral exam with written preparation 50%). Rough evaluation scale: A 90%, B 80%, C 70%, D 60%, E 50%
English
Commutatiove rings and modules. Theory of noetherian rings. Integral extensions of rings. Extensions and restrictions of coefficients. Homological methods: Ext and Tor.
Homeworks/ final exam: 50/50
PhD. student masters selected methods of modern commutative algebra with emphasis on application in other areas, especially in algebraic geometry.
1. S. Bosch: Algebraic Geometry and Commutative Algebra, Springer, 2013
2. D. Eisenbud: Commutative algebra with a view toward algebraic geometry. New York: Springer,2004
3. D. Eisenbud, J. Harris: The Geometry of Schemes, Springer, 2000
4. U. Görtz, T. Wedhorn: Algebraic Geometry I: Schemes, Springer, 2020
Homeworks (written solutions of problems and their consulting 50%), final exam (oral exam with written preparation 50%). Rough evaluation scale: A 90%, B 80%, C 70%, D 60%, E 50%
English
Affine schemes and their basic constructions. Techniques of global schemes.
Homeworks / Final evaluation: 50/50
PhD. student masters basic notions and methods of modern algebraic geometry in order to read contemporary literature in the area.
This is a follow-up to the master course in algebraic topology, an elaboration of homology and cohomology theories at advanced levels.
Topology and geometry / Glen E. Bredon. New York : Springer, 1993
Algebraic topology / Edwin H. Spanier. New York : Springer, 1966
Algebraic topology / Allen Hatcher. New York : Cambridge University Press, 2001
Exam: written and oral
Indicative evaluation scale: A 90%, B 80%, C 70%, D 60%, E 50%
Homology and cohomology theories, the universal coefficient theorems and Kuenneth fomulae, products, Steenrod cohomology operations, duality.
This is a follow-up to the master course in algebraic topology, an elaboration of homology and cohomology theories at advanced levels.
examination
An elaboration of higher homotopy theory and basic facts from the theory of spectral sequences.
Topology and geometry / Glen E. Bredon. New York : Springer, 1993
Algebraic topology / Edwin H. Spanier. New York : Springer, 1966
Algebraic topology / Allen Hatcher. New York : Cambridge University Press, 2001
Exam: written and oral
Indicative evaluation scale: A 90%, B 80%, C 70%, D 60%, E 50%
Higher homotopy groups, the Hurewicz homomorphism, homotopy properties of CW-complexes, homology and cohomology spectral sequences of fibrations and their applications.
Elaboration of higher homotopy theory and basic facts from theory of spectral sequences.
examination
An elaboration of the advanced theory of smooth manifolds and vector bundles.
Topology and geometry / Glen E. Bredon. New York : Springer, 1993
Differential topology / Morris W. Hirsch. New York : Springer, 1997
Exam: written and oral
Indicative evaluation scale: A 90%, B 80%, C 70%, D 60%, E 50%
The Morse-Sard theorem, transversality, vector bundles, immersions and embeddings of manifolds in Euclidean spaces, the degree of a map, the Euler characteristic, basic facts from Morse theory.
An elaboration of the advanced theory of smooth manifolds and vector bundles.
examination
Topology and geometry / Glen E. Bredon. New York : Springer, 1993
Diferenciálna geometria a Lieove grupy pre fyzikov / Marián Fecko. Bratislava : Iris, 2004
Differential geometry and Lie groups for physicists / Marián Fecko. Cambridge : Cambridge University Press, 2006
Carter, Roger, Segal, Graeme, and MacDonald, Ian, Lectures on Lie Groups and Lie Algebras, Cambridge University Press, 1995.
Exam: written and/or oral examin
Indicative grading scheme: A: 90%, B: 80%, C: 70%, D: 60% , E:50%
English, Slovak
The notion of Lie group, examples. The action of a Lie group on a manifold. Homogeneous spaces. Classical Lie groups. Invariant vector fields. Basic information on representations. Fundamentals of the theory of Lie algebras.
0/100
Students will learn the basic concepts, methods and results of the theory of Lie groups and algebras and will be able to apply them in solving problems in geometry and topology.
To acquaint the students with basic notions, methods and results of the different-ial geometry on manifolds.
Foundations of differential geometry : Volume 2 / Shoshichi Kobayashi, Katsumi Nomizu. New York : John Wiley, 1969
A comprehensive introduction to differential geometry : volume 1 / Michael Spivak. Berkeley : Publish or Perish, 1979
A comprehensive introduction to differential geometry : Volume 2 / Michael Spivak. Berkeley : Publish or Perish, 1979
A comprehensive introduction to differential geometry : Volume 3 / Michael Spivak. Berkeley : Publish or Perish, 1979
A comprehensive introduction to differential geometry : Volume 4 / Michael Spivak. Berkeley : Publish or Perish, 1979
A comprehensive introduction to differential geometry : Volume 5 / Michael Spivak. Berkeley : Publish or Perish, 1979
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry I. Interscience Publishers N. York, 1996.
Final exam (oral exam with written preparation 100%). Rough evaluation scale: A 90%, B 80%, C 70%, D 60%, E 50%
Differentiable manifolds, smooth mappings, tangent vectors. Tensors and tensor fields. Covariant derivative, affine connection. Geodesics. Riemannian metric, the Levi-Civita connection. Homogeneous spaces, invariant connections.
The students will master basic ideas and results of the global differential geometry of manifolds.
examination
Knots / Gerhard Burde, Heiner Zieschang, Michael Heusener. New York : Walter de Gruyter, 2013
Algebraic topology / Allen Hatcher. New York : Cambridge University Press, 2001
Algebraic topology / Edwin H. Spanier. New York : Springer, 1966
Knots, Links, Braids and 3-manifolds: An Introduction to the New Invariants in Low-Dimensional Topology / V. V. Prasolov, A. B. Sossinsky: AMS, Providence, 1996
Exam: written and oral exam
Indicative grading scheme: A: 90%, B: 80%, C: 70%, D: 60% , E:50%
English, Slovak
The contents of the course will be adjusted, to some extent, to the interests of the students. The topics covered in the course will include mainly the following:
Basic notions, methods and results of knot and link theory; results from the topology of 3-dimensional manifolds: examples of manifolds, lens spaces, surgery, torus decompositions;
results on 4-dimensional manifolds: examples of manifolds, algebro-topological invariants, complex surfaces, the Kirby calculus.
0/100
The students will master classical results, as well as selected outcomes of the modern low-dimensional topological research.
Symplectic geometry, grupoids, and integrable systems / Pierre Dazord, Alan Weinstein. New York : Springer, 1991
Algebraic topology / Edwin H. Spanier. New York : Springer, 1966
Introduction to Symplectic Topology / Dusa McDuff, Dietmar Salamon: Oxford Science Publications, 1998
Exam: written and/or oral exam
Indicative grading scheme: A: 90%, B: 80%, C: 70%, D: 60% , E:50%
English, Slovak
The contents of the course will be adjusted, to some extent, to the interests of the students. The topics covered in the course will include mainly the following:
basic notions, methods and results of symplectic topology, tools and analytical background of theory of J-holomorphic curves, the Gromov-Witten invariants, applications of the gauge theory in the topology of 3-dimensional and 4-dimensional manifolds.
0/100
Students will master classical results, as well as selected outcomes of the current research in symplectic topology, topological applications of gauge theories etc.
Examination (A: 90%, B: 80%, C: 70%, D: 60% , E:50% )
Geometric incidence structures, projective and affine planes. Coordinatization of projective and affine planes and the relations between geometric and corresponding coordinate algebraic structures. Finite geometries and geometries over the field. Ordered geometry. Absolute geometry as the starting point for introduction of hyperbolic non-euclidean geometry. Riemann's non-euclidean geometry. Fundamental principles for construction of four-dimensional geometry. Selected chapters of Euclidean geometry (special properties of triangle, isometries and similarities in Euclidean plane and space, two-dimensional crystallography, geometry of circles and spheres, Platonic solids, golden section, phyllotaxis).
The PhD students learn some interesting but little known properties of geometric planar and spatial objects, as well as the fundamental principles and methods in the axiomatic approach to classical geometry related to some geometric and algebraic structures and formation of non-euclidean geometries. They will be then able to use the knowledge obtained to solve problems in geometry and topology.
A comprehensive introduction to differential geometry : volume 1 / Michael Spivak. Berkeley : Publish or Perish, 1979
A comprehensive introduction to differential geometry : Volume 2 / Michael Spivak. Berkeley : Publish or Perish, 1979
A comprehensive introduction to differential geometry : Volume 3 / Michael Spivak. Berkeley : Publish or Perish, 1979
A comprehensive introduction to differential geometry : Volume 4 / Michael Spivak. Berkeley : Publish or Perish, 1979
A comprehensive introduction to differential geometry : Volume 5 / Michael Spivak. Berkeley : Publish or Perish, 1979
Lectures on classical differential geometry / Dirk J. Struik. Cambridge : Addison-Wesley Press, 1950
Final exam (oral exam with written preparation 100%). Rough evaluation scale: A 90%, B 80%, C 70%, D 60%, E 50%
Curves: Analytic expression. Tangent lines and osculating planes. Natural parametrization. Curvature and torsion, the Frenet formulas. Natural equations of curves.
Surfaces: Analytic expression. Curves on surfaces. Tangent planes and normals. Developable surfaces. The first fundamental form of a surface. Mappings between surfaces. The second fundamental form of a surface. The normal curvature of a surface. Directions on a surface. Principal directions and principal curvatures. The Gaussian curvature of a surface.
Geometry on surfaces: Geodesics, geodetic coordinates. Surfaces with constant curvature. Non-euclidean geometry.
The PhD students master important notions, methods and results of the classical differential geometry of curves and surfaces. They will be able to apply their knowledge to problems in geometry and topology.
Solid modeling by computers : From theory to applications / edited Mary S. Pickett, John W. Boyse. New York : Plenum Press, 1984
Level set methods and dynamic implicit surfaces / Stanley Osher, Ronald Fedkiw. New York : Springer, 2003
Michael Mortenson : Geometric Modeling. John Wiley & Sons, Inc. New York, NY, 3rd Edition 2006
Final exam (oral exam with written preparation 100%). Rough evaluation scale: A 90%, B 80%, C 70%, D 60%, E 50%
Mathematical methods in geometric modeling. Extended methods in paramteric modeling (Coons body construction, bodies as special hypersurfaces in E4). Geometric modeling using splines, especially in rational form. Recursive subdivision and refinement in modeling. Fundamentals of implicit modeling of curves and surfaces. F-rep. Implicitization. Selected elimination methods. Polar forms. Shape modification tools and fitting. Mesh in geometric modeling. Effective methods of tesselation for smooth surfaces and boundaries.
0/100
The students are supposed to acquire modern trends in parameteric, implicit and subdivision modeling of curves, surfaces and volumes. They should be able to solve tasks connected with constructive modelling requirements of technical praxis as well as contemporary problems of modeling in computer graphics, entertainment industry and mass media.
Real-time rendering / Tomas Akenine-Möller, Eric Haines, Naty Hoffman. Wellesley : A. K. Peters, 2008
Study materials in Slovak and selected recent papers and monographs in Visual Computing research.
individual work 100%, examination 0%
Slovak and English
The course content can be (partially) individualized with regard to the dissertation topic.
Methodology of science, methodology of mathematical modeling and structured engineering design in computer graphics and image processing. Mathematical models relevant to the field of Visual Computing. Digital Processing of Visual Information with regard to the application of projective and affine geometry. Theory of HCI (Human Computer Interface).
(Applications.) CAD / CAM. Real-time rendering, visual effects and computer games. Advanced techniques of special modeling and 3D computer animation. International standardization (philosophy of the MPEG and SEDRIS standards, standardization for geodata and biodata). Medical informatics.
The PhD students will acquire the necessary general and specific methodological theoretical knowledge on trends and applications of computer graphics and they will be able to apply them to solve relevant theoretical and practical problems.
Image processing, analysis, and machine vision / Milan Sonka, Vaclav Hlavac, Roger Boyle. [Stamford] : Cengage Learning, 2008
Počítačové videnie / Elena Šikudová, Zuzana Černeková, Wanda Benešová, Zuzana Haladová, Júlia Kučerová. Praha: Wikina 2013.
https://vgg.fiit.stuba.sk/kniha/Pocitacove%20Videnie%20-%20Detekcia%20a%20Rozpoznavanie%20Objektov.pdf
Study materials in Slovak and selected recent papers and monographs in Visual Computing research.
individual work
Examination 100% (A: 90%, B: 80%, C: 70%, D: 60% , E:50% )
1. Digital image and its properties (overview)
2. The image preprocessing (advanced method)
3. Segmentation (advanced method)
4. Pattern recognition (advanced method)
5. Image understanding
6. 3D vision, geometry and radiometry
7. Uses of 3D vision
8. Motion Analysis
9. Case studies (selected topics)
The PhD students receive an overview of recent advanced methods in computer vision, which reflect the current state of knowledge in the field (preprocessing, segmentation, image recognition and understanding), with an emphasis on the most challenging ideas in 3D vision. They will be able to apply the acquired knowledge in solving theoretical and practical problems of computer vision.
The Visual Display of Quantitative Information / Edward R.Tufte. Graphics Press 2001.
Virtual Space / Lars Qvortrup et al. Springer-Verlag London 2002 2002.
The Data Science Design Manual / Steven S. Skiena. Cham: Springer Nature 2017.
individual work 100%, examination 0%
Slovak and English
The course content can be (partially) individualized with regard to the dissertation topic.
Methodology of scientific and technical visualization. Information visualisation. Visualization techniques and scenarios for geodata and medical data. Special representation (point-based graphics, implicit surfaces, volume graphics). Selected applications.
Virtual environments by Qvortrup. Cyberspace and virtual navigation, interaction and cooperation. Computer games, visual effects and their programming in real time.
Digital storytelling.
The PhD student receives theoretical and practical knowledge of visualization and virtual environments. If it is appropriate and possible, the students will participate in solving problems in a cooperating company, thus achieving practical applications of their theoretical knowledge.
General topology / Stephen Willard. Mineola : Dover, 1970
Topology / James R. Munkres. Upper Saddle River : Prentice-Hall , 2000
Modern general topology / Jun-Iti Nagata. Amsterdam : North-Holland, 1968
Exam: written and oral
Indicative evaluation scale: A 90%, B 80%, C 70%, D 60%, E 50%
Compact spaces and compactifications, compactly generated spaces, perfect mappings, paracompact spaces, topologies on sets of continuous mappings, compact - open topology, metrizable spaces, metrization theorems, uniform spaces, selected topical themes.
To master selected notions and methods of general topology and their applications in geometry and topology, mathematical analysis and algebra.
Categories for the working mathematician / Saunders Mac Lane. New York : Springer, 1997
Algebra a príbuzné disciplíny / Milan Kolibiar ...[et al.]. Bratislava : Alfa, 1992
Abstract and Concrete Categories/Jiří Adámek, Horst Herrlich, George E. Strecker. http://katmat.math.uni-bremen.de/acc/acc.pdf
Exam: written and oral
Indicative evaluation scale: A 90%, B 80%, C 70%, D 60%, E 50%
Abstract and concrete categories and functors, adjoint functors, existence of adjoints, reflective and coreflective subcategories, factorization structures, topological categories and functors, selected topical themes.
To master selected category-theoretical methods used in topology and algebra.
Bochnak, J.; Coste, M.; Roy, M.-F., Real algebraic geometry. Berlin: Springer. ix, 430 p., (1998)
Basu, S.; Pollack, R.; Roy, M.-F., Algorithms in real algebraic geometry. 2nd ed. Berlin: Springer, 662 p., 2006
Examination (A: 90%, B: 80%, C: 70%, D: 60% , E:50% )
Real closed fields. Semialgebraic sets, their properties and decomposition. Real and complex algebraic sets. Properties, examples, constructions. Real roots of polynomial systems and their localization – methods and algorithms.
The student should master basic qualitative results and computational methods of work with real (semi)algebraic varieties. The student learns typical examples in spaces of dimensions 1, 2, 3, 4 and becomes able to use them in applications based on real semialgebraic objects, e.g. in geometric modeling, visualization etc.
A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations; Springer 1983
Preliminary grading: The student will complete three homework assignments of 15 points.
Exam: oral
Indicative scale of assessment: A 90%, B 80%, C 70%, D 60%, E 50%
English
The Gauss-Weierstras semigroup, C0-semigroups and their generators, the Hille-Yosida theorem. Analytic semigroups and their generators. Generation of semigroups by elliptic operators. Powers of operators. Fractional, interpolation and extrapolation spaces, and properties of semigroups in those spaces. Existence and properties of the solution of a model nonlinear parabolic equation.
45/55
The students will learn the basics of the theory of C0-semigroups
and its use in the analysis of evolution partial differential equations,
primarily equations of parabolic type.
Slovak, English
1. Initial value problem and the method of steps.
2. Existence and uniqueness of solutions of systems with bounded delays.
3. Linear delay differential systems. Variation of parameters.
4. Lyapunov method for uniform stability. Asymptotic stability.
Students learn the basics of the theory of differential equations with delayed arguments and functional differential equations.
M. Medveď: Dynamické systémy, Veda, 1988.
M. Medveď: Fundamentals of dynamical systems and bifurcation theory, Philadelphia, Adam Hilger, 1992.
(1-MAT-801 Topology or 1-MAT-150 Mathematical Analysis (2)) and 1-MAT-310 Ordinary Differential Equations (1)
Exam: oral and written exam
Generic characterization of singular points and periodic trajectories of dynamical systems. Invariant manifolds. Reduction to the central manifold. Calculation of normal forms. Single and multiparametric bifurcations close to singular points and periodic trajectories. Homoclinic trajectories and Melnikov functions. Introduction to chaos theory.
0/100
Absolvent will gain a good foundation in the theory of dynamical systems and the theory of bifurcations, which he will be able to apply in solving specific problems in the field of natural and technical sciences.
Methods of nonlinear analysis : Applications to differential equations / Pavel Drábek, Jaroslav Milota. Basel : Birkhäuser, 2007
Nonlinear functional analysis and its applications : II/B: Nonlinear Monotone Operators / Eberhard Zeidler ; Translated by Author and by Leo F. Boron. New York : Springer, 1990
An introduction to nonlinear boundary value problems / Stephen R. Bernfeld, V. Lakshmikantham. New York : Academic Press, 1974
Exam: oral
Indicative rating scale: A 90%, B 80%, C 70%, D 60%, E 50%
Fundamentals of the theory of the degree of mappings, introduction to the theory of monotone operators, nonlinear boundary value problems.
0/100
To show possible applications of nonlinear functional analysis, especially in examining the properties of solutions of differential equations.
The present course is devoted to the study of initial and boundary value problems (IVP and BVP) for ODEs.
Hairer, E., Norsett, S. P., Wanner, G.: Solving Ordinary Differential Equations I Nonstiff Problems.Springer Verlag 1987
Hairer, E.,Wanner, G.: Solving Ordinary Differential Equations II Stiff and Differential – Algebraic Problems. Springer Verlag 1991
Ascher, U. M., Mattheij, R. M. M., Russell, R. D.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM 1995
Dávid, A., Chocholatý, P.: Numerická matematika II (Okrajové úlohy pre obyčajné diferenciálne rovnice) UK Bratislava 1985
test
IVP: one step methods, multistep methods, stability, convergence, nonstiff and stiff problems, explicit RK-methods of higher order, implicit RK-methods, delay differential equations.
BVP:.conditioning of BVPs, initial value methods, finite difference methods, finite element methods, mesh selection, singular perturbations, functional differential equations, solving of nonlinear multipoint BVPs.
oral exam
To gain theoretical basics of modern numerical methods.
K. Rektorys: Variational Methods in Mathematics, Science and Engineering, SNTL, Praha 1974 (in Czech)
J. Nečas: Les Methodes Discrete en Theorie des Equations Elliptiques, Academia, Praha 1967
J. Wloka: Partial Differential Equations, University Press, Cambridge
Exam: oral
Indicative rating scale: A 90%, B 80%, C 70%, D 60%, E 50%
Sobolev spaces, generalized solutions of boundary value elliptic problems, Lax-Milgram theorem, Ritz and Galerkin methods, Fredholm alternative, spectral theory, generalized solutions of parabolic and hyperbolic problems.
0/100
To gain theoretical basics of modern numerical methods.
oral and written exam
To get acquainted with the basic methods of solving hyperbolic conservation systems.
Le Veque: Numerical methods for conservative law, ETH Zurich, Birkhauser-Verlag, Basel, 1992
individual work
Continuous assessment: individual work
Exam: oral
Indicative assessment scale: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
Hyperbolic systems; linear problems and their numerical methods; consistence, convergence and Lax‘s theorem, Lax Vendroffova method, nonlinear hyperbolic problems, weak and entropy solutions, conservative and entropy methods, Riemann problem and its solution, Godunov method, Roas method, nonlinear hyperbolic systems and the methods of their solutions.
To get acquainted with the basic methods of solving hyperbolic conservation systems.
exam
Metóda konečných prvkov / Marián Slodička. Bratislava : Fakulta matematiky, fyziky a informatiky UK, 2001
Continuous assessment: individual work
Exam: oral
Indicative assessment scale: A 90%, B 80%, C 70%, D 60%, E 50%
Slovak, English
Galerkin method, interpolation theory in H-spaces 1D and 2D. Finite element method error estimation. Bases in specific spaces. The first Strang's lemma, nonconformal elements, the second Strang's lemma, the Maltigrid method, algebraic solution, solution of evolutionary problems by the finite element method.
0/100
Acquire methods and practices for the implementation of modern computational procedures.