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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 pons asinorum in Oliver Byrne's edition of the Elements [1]. In geometry, the theorem that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (/ ˈ p ɒ n z ˌ æ s ɪ ˈ n ɔːr ə m / PONZ ass-ih-NOR-əm), Latin for "bridge of asses", or more descriptively as the isosceles triangle theorem.
A relation equal to its converse is a symmetric relation; in the language of dagger categories, it is self-adjoint. Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relations as sets), and actually an involutive quantale .
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It was originally given by Jürgen Moser in 1965 to check when two volume forms are equivalent, [1] but its main applications are in symplectic geometry. It is the standard argument for the modern proof of Darboux's theorem, as well as for the proof of Darboux-Weinstein theorem [2] and other normal form results. [2] [3] [4]
In the mathematical theory of automorphic forms, a converse theorem gives sufficient conditions for a Dirichlet series to be the Mellin transform of a modular form. More generally a converse theorem states that a representation of an algebraic group over the adeles is automorphic whenever the L-functions of various twists of it are well-behaved.
The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions. The converse does hold e.g. if the domain is simply connected; this is Cauchy's integral theorem, stating that the line integral of a holomorphic function along a closed curve is ...
A twist-hinge dissection is one which use a three-dimensional "hinge" which is placed on the edges of pieces rather than their vertices, allowing them to be "flipped" three-dimensionally. [ 10 ] [ 11 ] As of 2002, the question of whether any two polygons must have a common twist-hinged dissection remains unsolved.