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The tennis racket theorem or intermediate axis theorem, is a kinetic phenomenon of classical mechanics which describes the movement of a rigid body with three distinct principal moments of inertia. It has also been dubbed the Dzhanibekov effect , after Soviet cosmonaut Vladimir Dzhanibekov , who noticed one of the theorem's logical consequences ...
In 1985 he demonstrated stable and unstable rotation of a T-handle nut from the orbit, subsequently named the Dzhanibekov effect. The effect had been long known from the tennis racket theorem, which says that rotation about an object's intermediate principal axis is unstable while in free fall. In 1985 he was promoted to the rank of major ...
Given a manifold and a Lie algebra valued 1-form over it, we can define a family of p-forms: [3]. In one dimension, the Chern–Simons 1-form is given by []. In three dimensions, the Chern–Simons 3-form is given by
This is a formulation of the Lax–Milgram theorem which relies on properties of the symmetric part of the bilinear form. It is not the most general form. It is not the most general form. Let V {\displaystyle V} be a real Hilbert space and a ( ⋅ , ⋅ ) {\displaystyle a(\cdot ,\cdot )} a bilinear form on V {\displaystyle V} , which is
Theorem [7] — Suppose T is a distribution on U with compact support K. There exists a continuous function f {\displaystyle f} defined on U and a multi-index p such that T = ∂ p f , {\displaystyle T=\partial ^{p}f,} where the derivatives are understood in the sense of distributions.
Importantly, the geometry of the domain on which a ¯-closed differential form is ¯-exact is more restricted than for the Poincaré lemma, since the proof of the Dolbeault–Grothendieck lemma holds on a polydisk (a product of disks in the complex plane, on which the multidimensional Cauchy's integral formula may be applied) and there exist ...
For example, when = and =, the intersection form is trivial and the moduli space has dimension = =. This agrees with existence of the BPST instanton , which is the unique ASD instanton on S 4 {\displaystyle S^{4}} up to a 5 parameter family defining its centre in R 4 {\displaystyle \mathbb {R} ^{4}} and its scale.
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).