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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 t
    4 KB (490 words) - 17:11, 7 February 2011
  • [[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 the
    1 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 and
    9 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. 2
    2 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
  • 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 g
    9 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 $[-T,T]$, by
    5 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 problem
    14 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 transf
    30 KB (4,462 words) - 07:59, 6 June 2020
  • ...of order $k$ reveals that $\varphi$ is completely determined by its values on the coordinate functions centred at this point $x _ { 0 }$. Thus, ...thbf{R} ^ { m }$ (cf. [[Weil algebra|Weil algebra]]) extends to all smooth manifolds and smooth mappings
    12 KB (1,876 words) - 06:30, 15 February 2024
  • cf. [[Tensor analysis|Tensor analysis]]) on a [[Manifold|manifold]] $ M $ ...iant under [[Parallel displacement(2)|parallel displacement]] along curves on $ M $.
    8 KB (1,160 words) - 08:05, 6 June 2020
  • denotes the [[Lie algebra|Lie algebra]] of vector fields on $ M $), ...Riemannian curvature tensor, the Ricci operator, and the scalar curvature on $ M $,
    15 KB (2,270 words) - 08:28, 26 March 2023
  • ...ww.encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634601.png" /> (or on a Riemann surface <img align="absmiddle" border="0" src="https://www.encycl ...aofmath.org/legacyimages/m/m063/m063460/m06346020.png" />. It follows that on a non-compact Riemann surface <img align="absmiddle" border="0" src="https:
    44 KB (5,974 words) - 22:47, 29 November 2014
  • A Hermitian metric on a complex vector space $ V $ is a positive-definite [[Hermitian form|Hermitian form]] on $ V $.
    4 KB (681 words) - 03:41, 21 March 2022
  • A generalization of the concept of a [[Divisor|divisor]] of positive degree on a [[Riemann surface|Riemann surface]]. A holomorphic [[Vector bundle|vector if on $ E $
    7 KB (1,004 words) - 08:07, 6 June 2020
  • which is called a Morse–Smale diffeomorphism in this case) on a compact (usually closed) differentiable $ m $- ...ed system). This guarantees the existence of stable and unstable invariant manifolds $ W ^ {s} $
    10 KB (1,474 words) - 20:08, 12 January 2024
  • Mathematical background on dynamical systems can be found in [[#References|[a3]]], [[#References|[a7]] ...has a strictly positive part. In the remaining cases the stability depends on the non-linear terms in the Taylor expansion of $G$ (cf. also [[Stability t
    13 KB (1,928 words) - 17:00, 1 July 2020
  • ...rties and applications. Moreover, these integral formulas typically depend on the domain under consideration, and they intimately reflect complex-analyti ...ves iteration of the one-variable formula on product domains. For example, on a poly-disc $P = \{ ( z _ { 1 } , \dots , z _ { n } ) : | z _ { j } - a _ {
    15 KB (2,167 words) - 16:10, 11 February 2024
  • ...85 : ~/encyclopedia/old_files/data/D032/D.0302180 Differential geometry of manifolds ...simal structures (cf. [[Infinitesimal structure|Infinitesimal structure]]) on a manifold and their connection with the structure of the manifold and its
    30 KB (4,323 words) - 19:35, 5 June 2020
  • A closed smooth curve on a [[Riemannian manifold|Riemannian manifold]] $ M $ not constant on any subinterval, is a closed geodesic if, for sufficiently small $ \epsil
    13 KB (1,911 words) - 17:44, 4 June 2020
  • of functions bounded on the interval $ ( 0, 1) $ ...orems of the second type, estimates of the norms of traces of functions on manifolds of smaller dimension in terms of their weighted norms are given.
    9 KB (1,435 words) - 08:13, 13 January 2024
  • ...] - \infty , 0 ] )$ is bounded and the $f_j$ do not vanish simultaneously on $\partial K$. Letting $f = ( f _ { 1 } , \dots , f _ { n } )$, Kronecker sh ...d manifolds of the same finite dimension. He used it to prove the theorems on invariance of dimension and invariance of domain (cf. also [[Brouwer theore
    12 KB (1,815 words) - 17:42, 1 July 2020
  • on $ X $; ...ing analytic set in a local model (cf. [[Analytic set|Analytic set]]). The global dimension is defined by the formula:
    22 KB (3,277 words) - 01:53, 19 January 2022
  • ...s of topological spaces and extremal properties of functions (functionals) on them. Morse theory is a branch of [[Variational calculus in the large|varia on a smooth manifold $ M $(
    21 KB (3,095 words) - 08:01, 6 June 2020
  • ...n-diffusion systems [[#References|[a19]]], and fluctuations in fluid films on inclines [[#References|[a30]]]. Indeed, (a1) generically describes the dyna ...e that as suggested by the presence of chaotic solutions and by a Painlevé analysis [[#References|[a7]]] (cf. also [[Painlevé test|Painlevé test]]), the Kura
    21 KB (3,050 words) - 17:43, 1 July 2020
  • with coefficients depending on variables $ x $, The type of a Monge–Ampère equation depends on the sign of the expression
    17 KB (2,601 words) - 08:01, 6 June 2020
  • '''On a ''general'' complex quintic threefold, there are only finitely many smoot .../math> is the projectivization of the kernel of the natural restriction of global sections
    14 KB (2,200 words) - 18:27, 23 October 2017
  • ...the general framework of the theory of connections. As a device of tensor analysis, covariant differentiation is widely used in theoretical physics, particula Let an affine connection be given on an $ n $-
    15 KB (2,257 words) - 17:31, 5 June 2020
  • ...ransformation) to general position" , the precise meaning of which depends on the context. Usually the set $\mathfrak{O}$ of all objects considered has a ...eft({\lambda_1, \dots, \lambda_n}\right)$, depending sufficiently smoothly on $n$ (scalar) parameters, and when all possible such families form a Baire s
    12 KB (1,758 words) - 00:29, 13 January 2017
  • ...analysis that a bounded and [[Holomorphic function|holomorphic function]] on the whole plane (cf. also [[Entire function|entire function]]) must be a co ...ypercone in $\mathbb R^n$ must be planar for $n\leq 7$. His proof is based on an inequality for the norm of the second fundamental form of minimal surfac
    10 KB (1,518 words) - 21:22, 14 January 2021
  • ...uctive representations of commutative algebras characterizing differential manifolds in non-commutative algebras. The most remarkable examples are geometric qua ..., as the quotient with respect to gauge groups produces finite-dimensional manifolds [[#References|[a2]]], [[#References|[a34]]], [[#References|[a35]]], [[#Refe
    16 KB (2,356 words) - 19:53, 13 January 2018
  • ...yellow;">L. Euler was the first to employ charts (curvilinear coordinates) on surfaces in geometric research. B. Riemann took up the notion of a chart as The specification of an atlas defines on $ M $
    7 KB (1,075 words) - 16:43, 4 June 2020
  • ...{ M } \langle F _ { A } \wedge * F _ { A } \rangle$ to any connection $A$ on $\xi $; here $*$ refers to the Hodge star operator (cf. also [[Laplace oper ...ections, the space of gauge equivalence classes of Yang–Mills connections. On this space, the problem of gauge fixing is that of choosing continuously a
    7 KB (1,013 words) - 16:45, 1 July 2020
  • ...see, e.g., [[Metric space|Metric space]]; [[Functional analysis|Functional analysis]]). ...y developed in Ancient Greece. The result was an accumulation of knowledge on metric relationships governing triangles, determinations of areas and volum
    20 KB (2,829 words) - 18:41, 11 December 2020
  • see [[#References|[a30]]]. From now on $ \mathop{\rm char} ( K) = 0 $. is generated by two elements. The global [[Homological dimension|homological dimension]] of $ A _ {n} ( K) $
    28 KB (4,182 words) - 19:30, 19 January 2024
  • ...t as a topological isomorphism. Thus, one can define the Fourier transform on $\mathcal{Q}$ by duality; it also gives an isomorphism. Since there is a co ...x open cone with vertex at the origin and $F _ { j } ( z )$ is holomorphic on the wedge $\mathbf{R} ^ { n } + i \Gamma _ { j }$ satisfying the infra-expo
    26 KB (3,903 words) - 17:01, 1 July 2020
  • be a differential or pseudo-differential (as a rule, matrix) operator on a domain $ X \subset \mathbf R ^ {n} $ is a matrix-valued function on $ X \times ( \mathbf R ^ {n} \setminus 0) $
    12 KB (1,764 words) - 05:06, 24 February 2022
  • be the volume element on $ \Lambda $. lying on $ \Lambda $.
    24 KB (3,360 words) - 19:39, 5 June 2020
  • ...tional-differential, as well as more complicated equations in mathematical analysis are all equations of the type (1), as are also systems of algebraic equatio into itself and regard in it an ordinary differential equation on the infinite interval $ 0 \leq \lambda < \infty $:
    19 KB (2,803 words) - 19:40, 5 June 2020
  • A term denoting a number of problems related to research on extremals and global minima of functionals in the $ k $-dimensional volume $ \mathop{\rm vo defined on $ k $-dimensional generalized surfaces that are imbedded in an $ n $-di
    29 KB (4,259 words) - 11:43, 3 March 2022
  • ...ifold. A Lie group is said to be real, complex or $ p $ -adic, depending on the field over which its analytic manifold is considered. Henceforth, as a ...(see [[Lie group, local|Lie group, local]]). Systematic research into the global structure of Lie groups was first begun by E. Cartan and H. Weyl. The first
    26 KB (3,980 words) - 17:06, 13 June 2020
  • ...ich must be credited to B. Riemann) was to introduce an analytic structure on the set of all such structures. The idea was made precise by the following of complex manifolds parametrized by a complex space
    41 KB (5,916 words) - 11:24, 26 March 2023
  • The important role of basic concepts and the relations between them, on which the definitions of figures are based and geometric propositions are p ...Results justified by the use of [[Euclidean geometry|Euclidean geometry]] on the basis of the same principles and concepts as in the $ Elements $
    25 KB (3,631 words) - 19:39, 5 June 2020
  • ...rential geometry the properties of curves and surfaces are usually studied on a small scale, i.e. the study concerns properties of sufficiently small pie ...ometry. Many geometrical concepts were defined prior to their analogues in analysis. For instance, the concept of a tangent is older than that of a derivative,
    33 KB (5,039 words) - 11:46, 26 March 2023
  • ...ry velocity and of the force, while the projection of the phase trajectory on the space $ x _ {i} $( ...ed by a system of differential equations such as (1). On the other hand, a global (i.e. suitable for all states of the dynamical system) and invariant (i.e.
    27 KB (4,058 words) - 19:36, 5 June 2020
  • {{Cite|Ku}} under the name of "ideal divisor" in his studies on cyclotomic fields. ...= \sum n_i\f p_i$, the mapping $a\mapsto \sum n_i$ is a discrete valuation on $K$, and is known as the essential valuation of $K$. The homomorphism $\phi
    16 KB (2,805 words) - 02:18, 6 January 2022
  • The simplest ordinary differential equation is already encountered in analysis: The problem of finding the primitive function of a given continuous functi defined and differentiable on some interval $ I $
    33 KB (4,933 words) - 01:50, 23 January 2022
  • ...local study of singular points of algebraic surfaces. The fundamental work on this subject was done by D. Mumford (1961), who deduced important invariant ...ximum number of linearly independent regular two-dimensional differentials on $ V $(
    26 KB (3,736 words) - 13:08, 8 February 2020
  • ...or studying many problems in contemporary algebra, geometry, topology, and analysis. ...e stalks $\cF_x$ are discrete, the stalk-wise algebraic operations defined on $\cF$ by taking direct limits are continuous and the natural projection $p:
    26 KB (4,342 words) - 15:06, 15 July 2014
  • ...Weierstrass. The "transition to the complex domain" had a decisive effect on this theory. The theory of analytic functions was constructed as the theory ...ther approach, which was systematically developed by Weierstrass, is based on the possibility of representing functions by power series; it is thus conne
    61 KB (9,850 words) - 19:04, 20 January 2022
  • ...). An obvious generalization is the action of a group (or semi-group) $G$ on the space $A^G$ of functions from $G$ to $A$. ...ut not periodic; see [[#References|[a10]]], Chapt. 12. For general results on shift systems, consult [[#References|[a11]]], .
    15 KB (2,197 words) - 08:48, 29 April 2023
  • Duality between the different cohomology spaces on ...gebraically closed field $k$ and let $\mathcal{L}$ be a locally free sheaf on $X$. Serre's duality theorem states that the finite-dimensional cohomology
    64 KB (9,418 words) - 12:44, 8 February 2020
  • ...and solved a number of fundamental problems on the dependence of solutions on parameters (see below). Lyapunov studied the behaviour of solutions in a ne different solutions on the interval $ 0 \leq x \leq \omega $.
    29 KB (4,392 words) - 08:08, 6 June 2020
  • ...[[#References|[a24]]] (all problems except 4, 9, 14; with special emphasis on developments from 1975–1992). Cantor's problem on the [[Continuum, cardinality of the|cardinal number of the continuum]].
    29 KB (4,109 words) - 19:54, 18 March 2018
  • on a curve $ L _{1} : \ z = \phi _{1} (t) $, and on a curve $ L _{2} : \ z = \phi _{2} (t) $,
    66 KB (9,825 words) - 01:45, 23 June 2022