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transversality
This term is a specialized technical descriptor used primarily in differential topology and optimal control theory. It describes a specific geometric relationship where intersections are not tangential but instead occur with a degree of openness or independence, ensuring that the intersection is stable under small perturbations.
In the context of optimization, it functions as a boundary constraint. It dictates how a system must terminate when the final state is not fixed, acting as a necessary condition for optimality in the calculus of variations. Because it is an abstract mathematical property, it is treated as an uncountable mass noun.
Meanings
Examples
The professor explained that transversality ensures the intersection of two manifolds is stable under small perturbations.
The professor explained that transversality ensures the intersection of two manifolds is stable under small perturbations.
We need to check the transversality condition to determine the optimal terminal state of the trajectory.
We need to check the transversality condition to determine the optimal terminal state of the trajectory.
If the two surfaces lack transversality, they might be tangent at the point of intersection.
I wonder if this specific mapping satisfies the requirements for transversality in a higher dimension.
I wonder if this specific mapping satisfies the requirements for transversality in a higher dimension.
The proof relies heavily on the transversality theorem to show that generic intersections are smooth.
The proof relies heavily on the transversality theorem to show that generic intersections are smooth.
Does the transversality condition apply when the final time is fixed or free?
The researcher analyzed the transversality of the intersection to calculate the intersection number.
The researcher analyzed the transversality of the intersection to calculate the intersection number.
Without transversality, the resulting intersection might not be a manifold itself.