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The birational point of view can afford to be careless about subsets of codimension 1. To have a moduli space as a scheme is on one side a question about characterising schemes as representable functors (as the Grothendieck school would see it); but geometrically it is more like a compactification question, as the stability criteria revealed.
The stability of fixed points of a system of constant coefficient linear differential equations of first order can be analyzed using the eigenvalues of the corresponding matrix. An autonomous system ′ =, where x(t) ∈ R n and A is an n×n matrix with real entries, has a constant solution =
This is a fixed point of the action of the 1-PS , and so the line over in the affine space + is preserved by the action of . An action of the multiplicative group C ∗ {\displaystyle \mathbb {C} ^{*}} on a one dimensional vector space comes with a weight , an integer we label μ ( x , λ ) {\displaystyle \mu (x,\lambda )} , with the property that
The appearance or the disappearance of a periodic orbit through a local change in the stability properties of a fixed point is known as the Hopf bifurcation. The following theorem works for fixed points with one pair of conjugate nonzero purely imaginary eigenvalues. It tells the conditions under which this bifurcation phenomenon occurs.
Note that subcritical and supercritical describe the stability of the outer lines of the pitchfork (dashed or solid, respectively) and are not dependent on which direction the pitchfork faces. For example, the negative of the first ODE above, x ˙ = x 3 − r x {\displaystyle {\dot {x}}=x^{3}-rx} , faces the same direction as the first picture ...
DJT stock surged after Donald Trump said he would not sell his shares in the company, the home of Trump's social media platform, Truth Social.
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically, for functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set.
The death of an Oregon house cat and a pet food recall are raising questions about the ongoing outbreak of bird flu and how people can protect their pets. Bird flu has been spreading for years in ...