...ar spaces and their mappings. The basic divisions of non-linear functional analysis are the following.
...e study of spaces that are locally linear and of Banach manifolds — global analysis.
4 KB (490 words) - 17:11, 7 February 2011
The comparison of algorithms and the analysis of numerical problems in a Bayesian setting, cf. also [[Bayesian approach|B
...worst-case sense over the class $P$. Alternatively, in Bayesian numerical analysis, one puts an [[A priori distribution|a priori distribution]] $\mu$ on the i
6 KB (908 words) - 18:44, 21 March 2024
...orithms have been constructed for the realization of a random search for a global extremum of a function in several variables (see [[#References|[5]]]). Thes
...gn="top">[1]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from
5 KB (649 words) - 17:24, 7 February 2011
[[Category:Global analysis]]
1 KB (203 words) - 16:38, 17 November 2012
...10/b1100108.png" />) are of no use for an appropriate characterization and analysis of methods which are able to efficiently integrate a stiff problem. Thus th
...zing parameter goes back to [[#References|[a7]]], where it was used in the analysis of multi-step methods. The point is that stiffness is often compatible with
11 KB (1,517 words) - 17:23, 7 February 2011
...t by means of local semantics of statements of the program (so-called flow analysis of the program); checking certain properties of the information collected (
...s are divided into local, the economy part is not larger than a statement; global, the economy part is the entire program; and quasi-local, for which the eco
5 KB (775 words) - 17:05, 7 February 2011
In other words, the problem is to construct a global meromorphic function with locally specified polar singularities.
satisfying the compatibility condition there corresponds a uniquely defined global section of the sheaf $ {\mathcal M} / {\mathcal O} $,
16 KB (2,209 words) - 11:03, 26 March 2023
...braic and analytic geometry, etc. is a frequently used method to construct global objects such as varieties, schemes, differentiable manifolds, vector bundle
and morphisms of schemes between them. Cf. [[Scheme|Scheme]]. Here also global separation properties must be added to obtain a scheme. For vector bundles
4 KB (636 words) - 09:22, 15 January 2024
A very useful fact in analysis is that $C^1$ maps $f$ such that $\left. df\right|_{x_0}$ is invertible at
...the differential is invertible at ''every point'' does not guarantee the ''global invertibility'' of the map. Indeed, a famous example is the exponential map
10 KB (1,719 words) - 16:56, 30 November 2014
...TD valign="top">[a1]</TD> <TD valign="top"> S.S. Chern, "Studies in global analysis and geometry" , ''Studies in Mathematics'' , '''4''' , Math. Assoc. America
2 KB (325 words) - 19:23, 26 March 2024
...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
...so cannot be a local minimum point of the modulus of $f(z)$. An equivalent global formulation of the maximum-modulus principle is that, under the same condit
|valign="top"|{{Ref|Ah}}||valign="top"| L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241 {{ZBL|0395.30001}}
4 KB (614 words) - 06:23, 12 October 2023
<TR><TD valign="top">[1]</TD> <TD valign="top"> A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated
...TD valign="top">[a2]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)</TD></TR>
3 KB (413 words) - 13:42, 17 March 2023
a finite number of global sections $ s _{1} \dots s _{N} $
by its global sections.)
9 KB (1,307 words) - 20:04, 27 February 2021
==Local and global convergence theory.==
...f$ is decreased as the iteration progresses. There are several variants of global convergence theorems for BFGS and related methods, [[#References|[a9]]], [[
13 KB (1,868 words) - 07:21, 13 February 2024
...D valign="top">[2]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)</TD></TR></table>
3 KB (490 words) - 17:46, 4 June 2020
==Local and global theory.==
...see [[Deformation|Deformation]] 1) and 2)). The fundamental methods of the global theory are those of the theory of representable functors and geometric inva
16 KB (2,402 words) - 11:49, 16 December 2019
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
...solution in the space of sequences of bounded functions), and non-standard analysis methods.
...ces of summable functions (or kernel operators, in the quantum case): time-global solutions for general classes of an interaction potential;
10 KB (1,427 words) - 07:38, 7 February 2024
The same analysis has been generalized to the case of a bounded domain in [[#References|[a1]]
...D></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J.A. Carrillo, "Global weak solutions of the absorption and reflection-type initial-boundary value
6 KB (900 words) - 08:31, 22 August 2014
``long-run", or ``global" statistical dependence in the Earth sciences.
peculiar method of analysis that follows very
9 KB (1,398 words) - 20:37, 22 September 2016
...r equipments, and computer networks), which is oriented to the qualitative analysis and synthesis of such systems (discovering deadlocks or conflict situations
...conducted along two lines. The mathematical theory is advanced by a formal analysis of their properties. The most interesting problems include recognizing dead
6 KB (897 words) - 19:23, 16 August 2016
...theory is based on the implicit-function theorem in non-linear functional analysis and on the general theory of linear problems of corresponding type.
The global theory of non-linear problems is less completely developed, and then only f
30 KB (4,331 words) - 16:42, 20 January 2022
.... Subsequently, fundamental results were obtained by methods of functional analysis and by algebraic methods, concerning the homotopy invariance of classes and
...ying the topological invariants, provided by $K$-theory. Multi-dimensional global problems of the calculus of variations on manifolds proved to be more diffi
9 KB (1,298 words) - 14:59, 30 August 2014
...ture of the boundary conditions or any supplementary conditions). Such a "global" character of variational calculus in the large proper is stressed by the
...#References|[12]]]). Variational calculus in the large is also employed in global [[Differential geometry|differential geometry]] [[#References|[13]]].
14 KB (2,052 words) - 08:27, 6 June 2020
..., "Anosov diffeomorphisms" S.-S. Chern (ed.) S. Smale (ed.) , ''Global analysis'' , ''Proc. Symp. Pure Math.'' , '''14''' , Amer. Math. Soc. (1970) pp. 6
...300</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Shub, "Global stability of dynamical systems" , Springer (1986)</TD></TR><TR><TD valign=
9 KB (1,321 words) - 07:59, 21 June 2014
...nnected the theory of variational inequalities to [[Convex analysis|convex analysis]], especially to the notion of subdifferentiability, and to the theory of m
...R><TD valign="top">[a10]</TD> <TD valign="top"> V.K. Le, K. Schmitt, "Global bifurcation in variational inequalities" , Springer (1997)</TD></TR><TR><T
5 KB (737 words) - 20:35, 18 March 2024
...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}.
Global derivatives are maps from $C^1 (M)$ to $C^0 (M)$ satisfying the (analog of)
5 KB (757 words) - 10:34, 11 December 2013
...D valign="top">[4]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)</TD></TR><TR><TD valign="top">[5]<
4 KB (543 words) - 22:15, 5 June 2020
...integral formulas is one of the most important tools in classical complex analysis (cf. also [[Boundary value problems of analytic function theory|Boundary va
In applications involving the construction of global holomorphic functions satisfying special properties, and in order to solve
15 KB (2,167 words) - 16:10, 11 February 2024
...p">[5]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973)</TD></TR></table>
6 KB (880 words) - 16:10, 1 April 2020
...e manifold gives, for a sufficiently smooth manifold, the largest possible global degree of smoothness of the function which is obtained as a result of exten
...ns of several variables and imbedding theorems" S.M. Nikol'skii (ed.) , ''Analysis III'' , ''Encycl. Math. Sci.'' , '''26''' , Springer (1990) pp. 1–140
9 KB (1,435 words) - 08:13, 13 January 2024
...D valign="top">[3]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)</TD></TR><TR><TD valign="top">[4]<
4 KB (681 words) - 03:41, 21 March 2022
If the problem satisfies the global contractivity condition
...having the same sign. This result can be used in the asymptotic stability analysis of Runge–Kutta methods, see [[#References|[a5]]].
9 KB (1,275 words) - 17:43, 1 July 2020
...submersions" , ''Lecture Notes'' , '''40''' , Research Inst. Math., Global Analysis Research Center, Seoul Nat. Univ. (1998)</td></tr><tr><td valign="top">[a5
5 KB (681 words) - 17:43, 1 July 2020
...ighest order) are minimized. However, since the relation between the true (global) error and the local error is generally not known, it is questionable wheth
...D valign="top">[a1]</TD> <TD valign="top"> J.C. Butcher, "The numerical analysis of ordinary differential equations. Runge–Kutta and general linear method
7 KB (1,053 words) - 17:13, 14 February 2020
...ions of representation and approximation of functions, and their local and global properties. The modern theory of functions of a real variable typically inv
...n urgent need for a new critical review of the foundations of mathematical analysis, which was carried out at the end of the 19th century and beginning of the
11 KB (1,738 words) - 18:15, 24 March 2018
...ing analytic set in a local model (cf. [[Analytic set|Analytic set]]). The global dimension is defined by the formula:
The theory of analytic spaces has two aspects: the local and the global aspect. Local analytic geometry is concerned with germs of analytic sets in
22 KB (3,277 words) - 01:53, 19 January 2022
...ion is useful when some functions are not differentiable. Using non-smooth analysis, one can replace derivatives by other objects such as subgradients (see, e.
...0\right\}$. Here $f^0(\theta)=0$ for any $\theta$, hence $\theta^*=0$ is a global minimum. The saddle-point condition requires $U_1=U_1(\theta)\geq 0$ such t
16 KB (2,514 words) - 17:28, 23 October 2017
...ed only for convex and related unimodal functions. The theory of finding a global extremum is still (1989) in the initial phase of development (see [[Multi-e
.../TD> <TD valign="top"> Yu.G. Evtushenko, "Numerical methods for finding global extrema (case of a non-uniform mesh)" ''USSR Comp. Math. Math. Phys.'' , '
13 KB (1,911 words) - 08:00, 6 June 2020
Global stability of the trivial solution of a non-linear system of ordinary differ
then one has global exponential stability:
16 KB (2,300 words) - 08:22, 6 June 2020
...ithms and programs for the computer realization of the discrete models, an analysis of the sensitivity of the model to variations of the parameters, an estimat
...ng of [[Time series|time series]] on a network of measurements, space-time analysis and the compatibility of meteorological fields), and also the use of method
15 KB (2,159 words) - 17:08, 7 February 2011
Another global construction of the Weil bundles on all manifolds $M$ is due to A. Morimoto
...lign="top"> P.W. Michor, A. Kriegl, "The convenient setting of global analysis" , ''Math. Surveys Monogr.'' , '''53''' , Amer. Math. Soc. (1997)</td></tr
12 KB (1,876 words) - 06:30, 15 February 2024
...al structure ( "the very same as Rn" ), this idea admits a whole series of global features typical for manifolds: (non-) orientability, homological [[Poincar
...ion|Morse function]]), etc., which are used for the geometric study of the global structure of manifolds, and, roughly speaking, consist of constructing a po
30 KB (4,462 words) - 07:59, 6 June 2020
briefly, Morse theory 1) is divided into two parts: local and global. The local part is related to the idea of a critical point of a smooth func
The basic results in global Morse theory are as follows. Let $ f $
21 KB (3,095 words) - 08:01, 6 June 2020
...face of any rough plate (see [[#References|[2]]]). In the investigation of global atmospheric processes on an Earth scale, the field of ground pressure and o
...fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press (1964) (Translated from Russian)</TD></TR><TR><T
9 KB (1,319 words) - 08:09, 6 June 2020
...="top">[3]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian)</TD></TR></table>
...D></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> S.G. Gindikin, "Analysis on homogeneous domains" ''Russian Math. Surveys'' , '''19''' (1964) pp.
10 KB (1,514 words) - 07:41, 26 March 2023
be a finite-dimensional smooth manifold. Tangent spaces and such provide the global analogues of differential calculus. There is also an "integral calculus on
...nifolds and calculus on manifolds" W. Schiehlen (ed.) W. Wedig (ed.) , ''Analysis and estimation of stochastic mechanical systems'' , Springer (Wien) (1988)
6 KB (827 words) - 22:13, 5 June 2020
...lude numerical integration, optimal recovery (approximation) of functions, global optimization, and solution of integral equations and partial differential e
...average-case setting (see [[Bayesian numerical analysis|Bayesian numerical analysis]]).
12 KB (1,706 words) - 20:29, 9 December 2023
cf. [[Tensor analysis|Tensor analysis]]) on a [[Manifold|manifold]] $ M $
.../TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated f
8 KB (1,160 words) - 08:05, 6 June 2020