systems or any other class of $ \Omega $-
systems comprising free systems of any (non-zero) rank. An automorphism $ \phi $
10 KB (1,505 words) - 06:43, 26 March 2023
...f every point. It is sometimes possible to introduce orthogonal coordinate systems in the large. In an orthogonal system, the metric tensor $ g _ {ij} $
The most frequently used orthogonal coordinate systems are: on a plane — [[Cartesian coordinates|Cartesian coordinates]]; [[Elli
11 KB (1,597 words) - 08:58, 4 March 2022
...detail (under the name of potential systems) and gave numerous examples of systems of this type. An Egorov system $ \Sigma $
The solutions of these equations define two other Egorov systems, $ \Sigma _ {1} $
4 KB (653 words) - 07:55, 25 April 2022
...ce|verbal congruence]] (see also [[Algebraic
systems, variety of|Algebraic
systems, variety of]]).
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The
theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian
2 KB (363 words) - 11:34, 12 January 2021
A method in the geometric theory of functions of a complex variable that is used to solve extremal problems
To find the ranges of functionals and systems of functionals on such classes the following theorems are sometimes useful.
6 KB (949 words) - 06:44, 9 May 2022
...agonal, can be solved by a modified form of [[Gaussian elimination]]: such systems appear, for example, in the [[Crank-Nicolson method]] for numerical solutio
* Thomas Muir. ''A treatise on the theory of determinants''. (Dover Publications, 1960 [1933])
897 bytes (116 words) - 18:31, 14 April 2018
...at states conditions for the solvability of the Lur'e equations in control theory:
...<TR><TD valign="top">[6]</TD> <TD valign="top"> D.D. Siljak, "Nonlinear systems. Parameter analysis and design" , Wiley (1969)</TD></TR><TR><TD valign="to
6 KB (854 words) - 12:59, 13 January 2021
...re-ordered sets and lattices, every Boolean ring, and some other algebraic systems are determined up to isomorphism by their endomorphism semi-groups. The sam
<TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Zykov, "The theory of finite graphs" , '''1''' , Novosibirsk (1969) (In Russian)</TD></TR>
3 KB (388 words) - 17:46, 25 October 2014
...ically important estimates for the free energies of various multi-particle systems [[#References|[4]]]. There exist generalizations of the Bogolyubov inequali
...top">[3]</TD> <TD valign="top"> S.V. Tyablikov, "Methods of the quantum theory of magnetism" , Plenum (1967) (Translated from Russian)</TD></TR><TR><TD
6 KB (788 words) - 10:59, 29 May 2020
...the theory of Hamiltonian systems, since many specific properties of such systems are directly related to these integral invariants (cf. [[#References|[4]]],
...tems. Up to a multiplier, all universal integral invariants of Hamiltonian systems can be reduced to the ones indicated (cf. [[#References|[4]]], [[#Reference
6 KB (919 words) - 18:23, 5 April 2023
...al local theorem, thus making a major contribution to [[Model theory|model theory]]. Later, by improving the method itself, he proved [[#References|[3]]] a l
3 KB (490 words) - 22:23, 26 October 2014
A notion in the theory of ordinary linear differential equations with an independent complex varia
...form of the [[Lamé equation|Lamé equation]]. Extensions of the concept to systems of partial differential equations are mentioned in (the editorial comments
5 KB (815 words) - 21:42, 31 July 2015
...[[Derived rule|Derived rule]]; [[Deducible rule|Deducible rule]]), render systems with the symbol $\vdash$ much like a [[Gentzen formal system|Gentzen formal
...f and of studying proofs by mathematical methods (cf. [[Proof theory|Proof theory]]). The concept of a formal derivation proved to be a good approximation to
4 KB (684 words) - 10:29, 11 October 2014
A physical phenomenon that occurs in macroscopic systems and consists in the following. In certain equilibrium states of the system
...</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Ya.G. Sinai, "Theory of phase transitions" , Pergamon (1982) (Translated from Russian)</TD></T
1 KB (187 words) - 17:28, 7 February 2011
Some of the main achievements of automated deduction systems include:
...nd evaluate theories, for example by testing the consistency of a proposed theory extension and by generating ramifications of a proposed extension.
6 KB (814 words) - 19:26, 16 April 2014
may be used to express all the specific characteristics of statistical systems. The principal difficulties involved in the study of the functions (3) or (
This study is essential for physical systems of various kinds and is most advanced for the case of short-range interacti
5 KB (743 words) - 10:59, 29 May 2020
...h transformations is due to the fact that in certain theorems of [[ergodic theory]] automorphisms with "too many" periodic points are considered as trivial
...op"> V.A. Rokhlin, "Selected topics from the metric theory of dynamical systems" ''Amer. Math. Soc. Transl. Series 2'' , '''49''' pp. 171–240 ''Uspekh
1 KB (199 words) - 14:01, 30 April 2023
...d by systems of equalities and inequalities. There exists a quite complete theory of convex programming, and numerous methods have been developed for solving
...D> <TD valign="top"> I.I. Eremin, N.N. Astaf'ev, "Introduction to the theory of linear and convex programming" , Moscow (1976) (In Russian)</TD></TR>
2 KB (236 words) - 12:57, 3 May 2016
The dynamical systems (or models describing deterministic evolution, cf. [[Dynamical system|Dynam
The main examples of chaotic dynamical systems (and dynamical systems which are supposed to be chaotic) are discussed below.
8 KB (1,192 words) - 16:43, 4 June 2020
...the theory of dynamical systems in which one studies topological dynamical systems (cf. [[Dynamical system|Dynamical system]]; [[Topological dynamical system|
...ves arose in the study of more concrete objects — differentiable dynamical systems. Various "repetitiveness" properties are (in increasing order of generality
12 KB (1,685 words) - 22:03, 10 December 2017