...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