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Results tagged with ca.classical-analysis-and-odes
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user 497175
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
2
votes
1
answer
75
views
How many strict local minima can a quintic polynomial in two real variables have?
A quadratic or cubic polynomial (in two variables) can have at most one strict local minimum. A quartic polynomial can have up to five strict local minima [1]. So, how many strict local minima can a …
- 291
7
votes
1
answer
771
views
Can a cubic polynomial in two real variables have three saddle points?
Is there a cubic polynomial $c(x,y)$ with exactly 3 saddle point critical points?
In other words, can a cubic polynomial in two variables have three critical points, all of which are saddle points? …
- 291
8
votes
1
answer
596
views
How many saddle points can a quartic polynomial in two real variables have? All 9?
By Bézout's theorem a quartic polynomial $p(x,y)$ can have at most 9 isolated critical points. Can all of them be saddle points?
In case of a cubic polynomial there is a mechanical way to answer thi …
- 291
0
votes
0
answers
87
views
Polynomial / quadratic autonomous system of ODEs – proving monotonicity / convexity
Problem:
Consider the autonomous ODE system
\begin{align*}
\dot{x} &= (1-x) (z-xy)\\
\dot{y} &= \tfrac 1 2 y^2 - (a+xy)(1-y) \\
\dot{z} &= \tfrac 1 2 z^2 - \tfrac 1 2 y^2 + (a+xy)z
\end{align*}
for $ …
- 291