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- ...ar spaces and their mappings. The basic divisions of non-linear functional analysis are the following. ...ch, topological vector and certain more general spaces, including theorems on the local inversion of a differentiable mapping and the implicit-function t4 KB (490 words) - 17:11, 7 February 2011
- |26|28|30|31|32|33|40|41|42|43|44 = [[Category:Analysis]] |46|47|48 = [[Category:Functional analysis and applications]]5 KB (494 words) - 22:21, 4 December 2014
- [[Category:Global analysis]] Let $M$ and $N$ be two [[Differentiable manifold|$C^r$ manifolds]] and $f:M\to N$ a $C^r$ map. If $r> \max \{0, \dim M - \dim N\}$, then the1 KB (203 words) - 16:38, 17 November 2012
- ...hod to construct global objects such as varieties, schemes, differentiable manifolds, vector bundles, sheaves $ \dots $ Consider, for example, the case of differentiable manifolds of dimension $ n $.4 KB (636 words) - 09:22, 15 January 2024
- ...bundles. Attempts at a successive construction of topology on the basis of manifolds, mappings and differential forms date back to the end of 19th century (H. P .... Subsequently, fundamental results were obtained by methods of functional analysis and by algebraic methods, concerning the homotopy invariance of classes and9 KB (1,298 words) - 14:59, 30 August 2014
- ...TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> K. Krzyzewski, "On analytic invariant measures for expanding mappings" ''Colloq. Math.'' , '' ...M. Shub, "Expanding maps" S.-S. Chern (ed.) S. Smale (ed.) , ''Global analysis'' , ''Proc. Symp. Pure Math.'' , '''14''' , Amer. Math. Soc. (1970) pp. 22 KB (337 words) - 19:13, 9 October 2014
- of linear transformations of the tangent spaces on a manifold $ M $ extended by linearity on $ T ^ {\mathbf C } M $)6 KB (880 words) - 16:10, 1 April 2020
- Cartan's theorem on the highest weight vector. Let $ \mathfrak g $ on $ \mathfrak t $9 KB (1,307 words) - 20:04, 27 February 2021
- Cartan's theorem on the highest weight vector. Let $8$ be a complex semi-simple Lie algebra, le ...ighest weight vector of the representation $0$ and the linear function $1$ on $1$ defined by the condition $\Lambda ( h _ { i } ) = k _ { i }$, $\dot { i9 KB (1,346 words) - 13:36, 17 October 2019
- and its inverse, defined on $f(X)$, is also denoted by $f^{-1}$. ...{\rm id}$. A notable example of involution is the [[complex conjugation]] on the [[Complex number|complex plane]]:10 KB (1,719 words) - 16:56, 30 November 2014
- An [[Affine connection|affine connection]] on a Hermitian manifold $ M $ If the affine connection on $ M $3 KB (413 words) - 13:42, 17 March 2023
- $#C+1 = 81 : ~/encyclopedia/old_files/data/I051/I.0501770 Integration on manifolds ...nalogues of differential calculus. There is also an "integral calculus on manifolds" . Let $ \Delta _ {n} = [ 0, 1] ^ {n} \subset \mathbf R ^ {n} $6 KB (827 words) - 22:13, 5 June 2020
- A [[Hermitian metric|Hermitian metric]] on a [[Complex manifold|complex manifold]] whose fundamental form $ \omega $ the [[Fubini–Study metric|Fubini–Study metric]] on the complex projective space $ \mathbf C P ^ {n} $;4 KB (543 words) - 22:15, 5 June 2020
- ...d the maps $f,g U \to \mathbb R$ are differentiable along a [[Vector field on a manifold|vector field]]. The rule becomes then ...b C$ and $f,g: U \to \mathbb C$ are differentiable in the sense of complex analysis (cf. [[Analytic function]]). Then the formula reads as \eqref{e:rule}.5 KB (757 words) - 10:34, 11 December 2013
- A complex structure on a real vector space $ V $ is the structure of a complex vector space on $ V $3 KB (490 words) - 17:46, 4 June 2020
- ...of tori and geodesic flows (cf. [[Geodesic flow|Geodesic flow]]) on closed manifolds of negative curvature are $Y$-systems. There are also other examples of a r The existence of a $Y$-system on a manifold imposes restrictions on the topological properties of the manifold. Little is known about this in g9 KB (1,321 words) - 07:59, 21 June 2014
- A global version of the same statement is the following The global Theorem 2 holds also when $[0,T]$ is replaced by $[-T, 0]$ or $[-T,T]$, by5 KB (851 words) - 11:10, 30 November 2013
- which is meromorphic on the entire manifold $ M $ In other words, the problem is to construct a global meromorphic function with locally specified polar singularities.16 KB (2,209 words) - 11:03, 26 March 2023
- ...ture of the boundary conditions or any supplementary conditions). Such a "global" character of variational calculus in the large proper is stressed by the ...f the number of closed geodesics (cf. [[Closed geodesic|Closed geodesic]]) on a closed Riemannian (and, more generally, a Finsler) manifold. This problem14 KB (2,052 words) - 08:27, 6 June 2020
- ...rements, both of mathematics itself and of other sciences. In mathematics, manifolds arose first of all as sets of solutions of non-degenerate systems of equati ...utions of multi-dimensional variational problems (soap films), as integral manifolds of Pfaffian systems and of dynamical systems, as groups of geometric transf30 KB (4,462 words) - 07:59, 6 June 2020