...inear mapping]] $V \times V \times V \rightarrow V$. They are used in the theory of [[Non-associative rings and algebras|non-associative algebras]] and appe
678 bytes (104 words) - 18:13, 19 March 2018
...aces all intuitive notions of complex systems. Typical examples of complex systems are: the nervous system, the brain, a computer, a control system in human s
...ir names, such as systems analysis, system-technology, and general systems theory, among others.
4 KB (532 words) - 17:22, 7 February 2011
...etween Post production systems and Post canonical systems. Post production systems were used by Post and A.A. Markov (1947) to construct the first examples of
...</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Markov, "Theory of algorithms" , Israel Program Sci. Transl. (1961) (Translated from Russ
1 KB (172 words) - 16:25, 1 May 2014
...eneral relativity theory, such reference systems are called local Galilean systems (Galilean frames). The existence of a local Galilean system implies that th
...ems are a corollary of the principles of relativity. In special relativity theory an inertial system is usually defined as a [[Galilean coordinate system|Gal
3 KB (381 words) - 17:51, 13 August 2023
...2]</TD> <TD valign="top"> V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated fro
...valign="top"> N.P. Bhatia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970) pp. 30–36</TD></TR></table>
1 KB (231 words) - 10:45, 15 April 2014
...of the language with free variables $x_1,\dotsc,x_n$ is equivalent in the theory $T$ to one of the formulas of $F_n$. The collection of axioms:
...\aleph_0$. The following general theorem holds: If a first-order countable theory $T$ is categorical in some uncountable cardinality, then it is categorical
4 KB (681 words) - 13:30, 14 February 2020
...ties of algebraic systems are defined by quasi-identities (cf. [[Algebraic systems, quasi-variety of]]). An identity is a special case of a quasi-identity.
...ommonly called ''Horn sentences'' or ''Horn clauses'': see [[Horn clauses, theory of]].
1 KB (193 words) - 07:40, 21 October 2016
...e cosets of a subgroup. Cf. [[#References|[a1]]] for some uses of Schreier systems, such as a proof of the Nielsen–Schreier theorem that subgroups of free g
...lign="top"> W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)
1 KB (211 words) - 12:17, 19 August 2014
An isomorphism is a correspondence (relation) between objects or systems of objects
algebraic systems (initially, with groups) and was extended in a
3 KB (465 words) - 22:01, 5 March 2012
...tributed computing (more generally, architectures occurring in multi-agent systems).
...ation protocol on the power, the complexity, and the properties of grammar systems and of the generated languages.
5 KB (767 words) - 19:10, 5 August 2014
...ame requirement. The concept of a deducible expression in effective formal systems is, generally speaking, not effective.
...some branch of meaningful mathematics. Historically, this class of formal systems arose in connection with the program of D. Hilbert of providing a foundatio
3 KB (392 words) - 12:21, 19 August 2014
...20 : ~/encyclopedia/old_files/data/E110/E.1100090 Equivalence of dynamical systems
Two autonomous systems of ordinary differential equations (cf. [[Autonomous system|Autonomous syst
3 KB (414 words) - 19:37, 5 June 2020
...the point of view of mathematical logic, as models of logics, that is, as systems of propositions with logical operations on them; and from the point of view
...ems of many-valued logic. Among the most important problems for functional systems are those of completeness, of the complexity of expressing some functions i
7 KB (1,087 words) - 19:40, 5 June 2020
Examples of Markov function systems are:
...valign="top">[1]</TD> <TD valign="top"> N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)</TD></TR></
1 KB (202 words) - 06:03, 5 August 2014
...veloping numerical methods for some class of problems. For example, in the theory of [[Quadrature|quadrature]] the problem of calculating integrals of functi
...g of methods for the solution of the [[Cauchy problem|Cauchy problem]] for systems of ordinary differential equations historically was done by investigating t
2 KB (281 words) - 08:01, 6 June 2020
...<TD valign="top">[4]</TD> <TD valign="top"> T. Berger, "Rate distortion theory" , Prentice-Hall (1971)</TD></TR></table>
...J. Körner, "Information theory. Coding theorems for discrete memoryless systems" , Akad. Kiado (1981)</TD></TR></table>
2 KB (247 words) - 18:10, 18 September 2014
...ety of areas: to the theory of non-linear oscillations, to physics, to the theory of automatic control, to astrodynamics, and to others. Averaging methods ha
1) Standard systems in the sense of N.N. Bogolyubov
5 KB (679 words) - 03:42, 21 March 2022
<TR><TD valign="top">[1]</TD> <TD valign="top"> G. Birkhoff, "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc. (1967)</TD></TR>
..."top">[3]</TD> <TD valign="top"> L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) {{ZBL|0137.02001}}</TD></TR>
1 KB (193 words) - 16:45, 4 September 2016
...utomatic control systems (cf. [[Automatic control theory|Automatic control theory]]) if there is a delay in the control mechanism.
1 KB (198 words) - 17:10, 7 February 2011
...y]]), and the system of the theory of types (cf. [[Types, theory of|Types, theory of]]).
407 bytes (61 words) - 17:28, 7 February 2011
One of the invariants in [[Ergodic theory|ergodic theory]], the construction of which recalls the construction of the cohomology of
...single non-trivial example. The use of "homological" concepts in ergodic theory stems from the fact that in many real cases it may be important to know (an
4 KB (528 words) - 22:10, 5 June 2020
$#C+1 = 63 : ~/encyclopedia/old_files/data/A011/A.0101680 Algebraic systems, quasi\AAhvariety of
A class of algebraic systems ( $ \Omega $-
6 KB (881 words) - 16:10, 1 April 2020
...an 4 cannot, in general, be solved by radicals (see [[Galois theory|Galois theory]]).
...Many questions of the theory of radicals have been studied within category theory. See also [[Radical of a group|Radical of a group]]; [[Radical in a class o
2 KB (254 words) - 16:33, 19 April 2014
Many dynamical systems (cf. [[Dynamical system|Dynamical system]]) are described by difference equ
or by autonomous systems of differential equations (cf. [[Autonomous system|Autonomous system]]) $
6 KB (764 words) - 08:03, 6 June 2020
A shift operator is used, in particular, in the theory of dynamical systems (see [[Shift dynamical system|Shift dynamical system]]; [[Bernoulli automor
...e often easy to analyze. They are of great importance in dynamical systems theory, owing to the Smale–Birkhoff theorem: A discrete-time [[Dynamical system|
3 KB (392 words) - 08:13, 6 June 2020
...s is given in [[#References|[a1]]]. A general consideration of supertriple systems is given in [[#References|[a2]]] and [[#References|[a5]]].
...]</td> <td valign="top"> W.G. Lister, "A structure theory of Lie triple systems" ''Trans. Amer. Math. Soc.'' , '''72''' (1952) pp. 217–242</td></tr><t
3 KB (559 words) - 16:43, 15 March 2023
$#C+1 = 91 : ~/encyclopedia/old_files/data/A011/A.0101670 Algebraic systems, class of
A class of algebraic systems of the same type. All systems of a given type are assumed to be written in a given signature $ \Omega $
8 KB (1,165 words) - 16:10, 1 April 2020
is described by several systems of differential equations
...etails see [[#References|[3]]]). The participation of several differential systems $ ( S _ {i} ) $
4 KB (557 words) - 08:26, 6 June 2020
''in control theory''
...ates have this property. The accessibility property is typical for control systems. Namely, every control system defined on a smooth manifold by a pair of smo
2 KB (329 words) - 15:33, 20 November 2014
...ems theory. For systems in the sense of logics, see [[Formal_system|formal systems]].
...cussed at the interplay between them (e.g. cybernetic models of biological systems). A more fundamental definition of a system was required encompassing the
8 KB (1,362 words) - 17:44, 25 September 2012
...6]]]. L. Neustadt has shown the significance of these operators in control theory [[#References|[a6]]].
...on abstract Volterra operators and associated equations. However, a solid theory of this kind of operator does not yet exist.
6 KB (863 words) - 15:34, 13 July 2014
...TD> <TD valign="top"> A.N. Sharkovskii, V.A. Dobrynskii, , ''Dynamical systems and problems of stability of solutions of differential equations'' , Kiev
...dy of hyperbolic sets (cf. [[Hyperbolic set|Hyperbolic set]]) in dynamical systems; see Chapt. 2 in [[#References|[a2]]].
3 KB (395 words) - 10:06, 24 August 2014
$#C+1 = 1 : ~/encyclopedia/old_files/data/A014/A.0104080 Automata, theory of
...[[Algorithms, theory of|Algorithms, theory of]]), particularly so with the theory of abstract machines, since automata can be regarded as a special case of s
7 KB (975 words) - 18:49, 5 April 2020
== In dynamical systems ==
4 KB (608 words) - 11:50, 1 May 2023
A property of an axiom system for a given axiomatic theory, defined as follows: Every axiom in the system is independent, i.e. it is n
...endent if and only if there is an [[Interpretation|interpretation]] of the theory in which the axiom is false, while all the other axioms are true. The const
3 KB (478 words) - 17:19, 7 February 2011
...is important to note that if the equations of some of the elements of the systems are unknown, the Nyquist diagram can be constructed experimentally, by feed
...ce been developed for multivariable, infinite-dimensional and sampled-data systems, e.g. [[#References|[5]]], , , .
4 KB (619 words) - 13:06, 10 August 2014
...ase the theory of representations of infinite groups is connected with the theory of representations of the group algebras of these groups.
...> <TD valign="top"> B.I. Plotkin, "Groups of automorphisms of algebraic systems" , Wolters-Noordhoff (1972)</TD></TR></table>
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...c); [[Qualitative theory of differential equations]]; [[Poincaré–Bendixson theory]].
* {{Ref|a1}} O. Hajek, "Dynamical systems in the plane", Acad. Press (1968)
535 bytes (71 words) - 14:12, 8 April 2023
...erent Perron transformations [[#References|[1]]], [[#References|[2]]]. For systems (2) with bounded continuous coefficients, all the Perron transformations ar
...TR><TR><TD valign="top">[4]</TD> <TD valign="top"> N.A. Izobov, "Linear systems of ordinary differential equations" ''J. Soviet Math.'' , '''5''' : 1 (1
3 KB (408 words) - 19:31, 11 January 2024
''in the theory of dynamical systems, discrete-time dynamical system''
...y investigated, although in applications, mostly continuous-time dynamical systems (cf. [[Flow (continuous-time dynamical system)|Flow (continuous-time dynami
3 KB (418 words) - 05:47, 18 May 2022
...f. [[Sector in the theory of ordinary differential equations|Sector in the theory of ordinary differential equations]]) by semi-trajectories (the separatrice
...eontovich, "Methods and means for a qualitative investigation of dynamical systems on the plane" , Moscow (1976) (In Russian)</TD></TR>
4 KB (577 words) - 09:06, 1 October 2023
...been rarely examined. The major importance of transference theorems in the theory of Diophantine approximation can be explained by a transference theorem in
[[Category:Number theory]]
2 KB (255 words) - 20:56, 25 October 2014
...of its origin. A syntactic language is used to study this aspect of formal systems.
...e [[Gödel incompleteness theorem|Gödel incompleteness theorem]] for formal systems is based on this fact.
2 KB (394 words) - 17:28, 7 February 2011
with zero initial conditions. In control theory, relation (1) is represented graphically as:
...on is one of the basic concepts in the theory of linear stationary control systems. It is independent of the control actions imposed on the system and is gove
4 KB (608 words) - 13:08, 18 June 2020
...development of the Galilean relativity principle forms part of the general theory of relativity.
...><TD valign="top">[1]</TD> <TD valign="top"> V.A. [V.A. Fok] Fock, "The theory of space, time and gravitation" , Macmillan (1964) (Translated from Russi
2 KB (272 words) - 17:21, 7 February 2011
...groups (cf. [[Kleinian group|Kleinian group]]) and the theory of dynamical systems (cf. e.g. [[Limit set of a trajectory|Limit set of a trajectory]]).
348 bytes (56 words) - 19:35, 19 October 2014
.... Let $\left\{{\psi_n}\right\}$ and $\left\{{g_n}\right\}$ be two complete systems of functions in $L_2 \! \left({a, b}\right) = L_2$ (i.e. measurable functio
1 KB (233 words) - 05:56, 14 January 2017
...cal systems'' , '''I: Ordinary differential equations and smooth dynamical systems''' , Springer (1988) pp. 159ff (Translated from Russian)</TD></TR></tabl
3 KB (535 words) - 21:47, 30 April 2014
$#C+1 = 15 : ~/encyclopedia/old_files/data/D033/D.0303160 Discrete systems in statistical mechanics
...change is considered to be independent of the others. One of the simplest systems — the Ising model (1925) — is characterized by the Hamiltonian [[#Refer
4 KB (585 words) - 11:53, 26 March 2023
...[3]</TD> <TD valign="top"> A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Pergamon (1966) (Translated from Russian)</TD></TR></ta
...I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian)</TD></TR></table>
2 KB (364 words) - 15:57, 14 February 2020