Hidden oscillation | Unsolved problems in geometry | Real algebraic geometry | Dynamical systems

Hilbert's sixteenth problem

Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. The original problem was posed as the Problem of the topology of algebraic curves and surfaces (Problem der Topologie algebraischer Kurven und Flächen). Actually the problem consists of two similar problems in different branches of mathematics: * An investigation of the relative positions of the branches of real algebraic curves of degree n (and similarly for algebraic surfaces). * The determination of the upper bound for the number of limit cycles in two-dimensional polynomial vector fields of degree n and an investigation of their relative positions. The first problem is yet unsolved for n = 8. Therefore, this problem is what usually is meant when talking about Hilbert's sixteenth problem in real algebraic geometry. The second problem also remains unsolved: no upper bound for the number of limit cycles is known for any n > 1, and this is what usually is meant by Hilbert's sixteenth problem in the field of dynamical systems. The Spanish Royal Society for Mathematics published an explanation of Hilbert's sixteenth problem. (Wikipedia).

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C56 Continuation of previous problem

Adding a bit more depth to the previous problem.

From playlist Differential Equations

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B25 Example problem solving for a Bernoulli equation

See how to solve a Bernoulli equation.

From playlist Differential Equations

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The theorem of Frobenius allows us to calculate a solution around a regular singular point.

From playlist Differential Equations

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C74 Example problem

A first example problem solving a linear, second-order, homogeneous, ODE with variable coefficients around a regular singular point.

From playlist Differential Equations

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How to Solve a Bernoulli Differential Equation

How to Solve a Bernoulli Differential Equation

From playlist Differential Equations

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Solve a Bernoulli Differential Equation (Part 1)

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From playlist Bernoulli Differential Equations

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From playlist Combinatorial Algebraic Geometry: Tropical and Real (HYBRID)

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Solve a Bernoulli Differential Equation (Part 2)

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From playlist Bernoulli Differential Equations

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Algebraic and Convex Geometry of Sums of Squares on Varieties (Lecture 2) by Greg Blekherman

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From playlist Combinatorial Algebraic Geometry: Tropical and Real (HYBRID)

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A06 Example problem including the Wronskian

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From playlist A Second Course in Differential Equations

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From playlist Oxford Mathematics Public Lectures

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B06 Example problem calculating the error

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From playlist A Second Course in Differential Equations

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From playlist Mathematics

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Algebraic and Convex Geometry of Sums of Squares on Varieties (Lecture 4) by Greg Blekherman

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From playlist Combinatorial Algebraic Geometry: Tropical and Real (HYBRID)

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