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In differential geometry, the slice theorem states: given a manifold on which a Lie group acts as diffeomorphisms, for any in , the map /, [] extends to an invariant neighborhood of / (viewed as a zero section) in / so that it defines an equivariant diffeomorphism from the neighborhood to its image, which contains the orbit of .
In mathematics, sine and cosine are trigonometric functions of an angle.The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that ...
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
We can also generalize the induced Ramsey's theorem to a multicolor setting. For graphs H 1, H 2, …, H r, define r ind (H 1, H 2, …, H r) to be the minimum number of vertices in a graph G such that any coloring of the edges of G into r colors contain an induced subgraph isomorphic to H i where all edges are colored in the i-th color for ...
The hinge theorem holds in Euclidean spaces and more generally in simply connected non-positively curved space forms.. It can be also extended from plane Euclidean geometry to higher dimension Euclidean spaces (e.g., to tetrahedra and more generally to simplices), as has been done for orthocentric tetrahedra (i.e., tetrahedra in which altitudes are concurrent) [2] and more generally for ...
The most common example of Bloch's theorem is describing electrons in a crystal, especially in characterizing the crystal's electronic properties, such as electronic band structure. However, a Bloch-wave description applies more generally to any wave-like phenomenon in a periodic medium.
Moreover, satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart near in which = + … +. Taking an exterior derivative now shows ω = d θ = d x 1 ∧ d y 1 + … + d x m ∧ d y m . {\displaystyle \omega =\mathrm {d} \theta =\mathrm {d} x_{1}\wedge \mathrm {d} y_{1}+\ldots +\mathrm {d} x_{m ...
Lie's third theorem says that every finite-dimensional real Lie algebra is the Lie algebra of a Lie group. It follows from Lie's third theorem and the preceding result that every finite-dimensional real Lie algebra is the Lie algebra of a unique simply connected Lie group.