Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Analyzing Critical Points in Finite Solutions

In course

MATH-301: Ordinary differential equations

Ce cours donne une introduction rigoureuse au principaux thèmes de la théorie des équations différentielles ordinaires (EDO). Les EDO sont fondamentales pour l'étude des systèmes dynamiques et des équations aux dérivées partielles. Des applications dans ce ...

Description

This lecture delves into the analysis of critical points in finite solutions, exploring the concept of isolated critical points and their implications. The instructor discusses the properties of solutions, such as being a bridge or a point, and demonstrates how to identify them in various scenarios. Through a series of examples and theoretical explanations, the lecture showcases the importance of understanding critical points in the context of finite solutions.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

François Samer Genoud

2000-2005 Etudes en Physique à l'EPFL dont une année d'échange à Zürich (ETH, 2002-2003). 2005 Diplôme de Physicien EPF (Master) en physique mathématique, sous la direction du Prof. C.-E. Pfister. 2005-2008 Doctorat sous la direction du Prof. C.A. Stuart, sur l'étude de certaines équations de Schrödinger non-linéaires, en particulier de questions liées à l'existence et la stabilité orbitale d'ondes stationnaires. 2009-2010 Chercheur postdoc FNS à OxPDE, Université d'Oxford. 2010-2013 Chercheur postdoc à l'Université Heriot-Watt, Edimbourg. 2013-2014 Chercheur postdoc à l'Université de Vienne. 2015-2017 Professeur Assistant à TU Delft. 2017-2018 Professeur Associé à l'Université libre de Bruxelles. 2018- Directeur du CMS Domaines d'intérêt scientifique: analyse non-linéaire, calcul des variations, équations aux dérivées partielles, physique mathématique et physique théorique en général.

Joachim Krieger

Official source

Related concepts (32)

Isolated point

In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S and there exists a neighborhood of x that does not contain any other points of S. This is equivalent to saying that the singleton {x} is an open set in the topological space S (considered as a subspace of X). Another equivalent formulation is: an element x of S is an isolated point of S if and only if it is not a limit point of S.

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about converges to the function in some neighborhood for every in its domain.

Stationary point

In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name). For a differentiable function of several real variables, a stationary point is a point on the surface of the graph where all its partial derivatives are zero (equivalently, the gradient is zero).

Inflection point

In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.

Singular point of a curve

In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. Algebraic curves in the plane may be defined as the set of points (x, y) satisfying an equation of the form where f is a polynomial function f: \R^2 \to \R. If f is expanded as If the origin (0, 0) is on the curve then a_0 = 0. If b_1 ≠ 0 then the implicit function theorem guarantees there is a smooth function h so that the curve has the form y = h(x) near the origin.

Related lectures (30)

Continuous Functions: Definition MOOC: Analysis I MOOC: Analysis I (part 4) : Limit of a function, continuous functions

Explains the definition of continuous functions and isolated points.

Taylor Series and Function Analysis

Explores Taylor series, function properties, inflection points, and critical points in graphical and mathematical contexts.

Sub-Varieties and Topological Concepts

Covers sub-varieties in different dimensions and topological concepts with examples and applications of theorems.

Shell Components in Meromorphic Parameter Spaces

Explores shell components in transcendental parameter planes and attracting cycles.

Proper Actions and Quotients