...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>
730 bytes (104 words) - 12:00, 27 January 2018
...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
...economics and cybernetics. In the framework of combinatorial mathematics, systems of distinct representatives play an important role in questions of choice a
In view of the wide use of systems of distinct representatives, algorithms for their practical determination (
6 KB (900 words) - 08:25, 6 June 2020
...investigated degenerate equilibrium positions are those of two-dimensional systems, for which several methods for studying the behaviour of trajectories in a
...valign="top"> A.A. Bryuno, "Stepwise asymptotic solutions of non-linear systems" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''29''' (1965) pp. 329–364 (In
3 KB (453 words) - 17:32, 5 June 2020
The design, development, tuning, and operation of computer and communication systems heavily rely on mathematical techniques which are usually indicated as perf
...f the daily operation and capacity planning of computer- and communication systems also requires techniques from such areas as combinatorial optimization (sch
5 KB (732 words) - 08:05, 6 June 2020
Just as for traditional dynamical systems the phase space of a flow usually is provided with a certain structure with
In applications one usually encounters flows described by autonomous systems (cf. [[Autonomous system|Autonomous system]]) of ordinary differential equa
3 KB (541 words) - 19:39, 5 June 2020
...> <TD valign="top"> D. Ruelle, "Small random perturbations of dynamical systems and the definition of attractors" ''Comm. Math. Phys.'' , '''82''' (1981)
2 KB (382 words) - 12:08, 18 August 2014
...the channel coding problem, [[Steiner system]]s, $t$-designs, and in the theory of finite groups. The most important special case is the sphere packing in
491 bytes (69 words) - 01:32, 11 February 2012
...eorem for polynomials of best uniform approximation is valid for Chebyshev systems (see also [[Haar condition|Haar condition]]; [[Chebyshev set|Chebyshev set]
...="top">[5]</TD> <TD valign="top"> S. Karlin, V. Studden, "Tchebycheff systems with applications in analysis and statistics" , Interscience (1966)</TD></
3 KB (491 words) - 10:46, 15 August 2014
...nts; all Gentzen formal systems are sometimes treated as natural deduction systems, since they reflect to some extent the usual methods of handling logical co
Natural deduction systems comprise rules for the introduction of logical symbols and their discharge.
10 KB (1,428 words) - 19:41, 5 June 2020
An important particular case of a dissipative system are the so-called systems with convergence, for which all solutions $ x ( t ; t _ {0} , x _ {0} )
...ution on the entire axis which is asymptotically stable in the large. Such systems have been thoroughly studied (see, for example, [[#References|[1]]]).
2 KB (276 words) - 19:36, 5 June 2020
...ultitude of algebraically completely integrable systems in the 1970s. Such systems are given by a Lax-pair equation: $L=[M,L]$ with $(n\times n)$-matrices $L$
...tems for principal $G$-bundles [[#References|[a5]]]; and quantized Hitchin systems with applications to the geometric Langlands program [[#References|[a2]]].
5 KB (752 words) - 15:33, 4 October 2014
of algebraic systems of the same signature''
...plica for any algebraic system of the same signature. A class of algebraic systems of a fixed signature is replica full if and only if it contains a one-eleme
2 KB (359 words) - 08:11, 6 June 2020
...projective or affine geometry. Another characteristic example of incidence systems is that of [[block design]]s: for example, [[balanced incomplete block desi
Two incidence systems $S = (A,\mathfrak{B},I)$ and $S' = (A',\mathfrak{B'},I')$ are called isomor
3 KB (488 words) - 19:37, 7 November 2023
...s or rational numbers). The study of such equations forms the topic of the theory of [[Diophantine equations|Diophantine equations]].
608 bytes (91 words) - 17:19, 7 February 2011
...ing]]s, tertiary ideals are the same as [[primary ideal]]s (cf. [[Additive theory of ideals]]; [[Primary decomposition]]).
...ions (of ideals of a ring, of submodules of a module, and others) leads to systems with fractions in which the general notions of $S$-primarity and $S$-primar
2 KB (297 words) - 19:23, 5 October 2017
...objects are arbitrary mathematical proofs is called a [[Proof theory|proof theory]] or a meta-mathematics. An example of the application of a meta-mathematic
...ot be proved by the tools of the relevant theory itself (provided that the theory is in fact consistent).
3 KB (544 words) - 15:41, 30 December 2018
A shift dynamical system is used in the theory of dynamical systems mainly to construct examples (here $ S $
...is everywhere dense, and others), and also in the theory of non-autonomous systems of ordinary differential equations, where $ S $
6 KB (862 words) - 08:13, 6 June 2020
...ätzer, "General lattice theory" , Birkhäuser (1978) (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978)</TD></TR></table>
...are, e.g., investigations concerning the structural stability of dynamical systems up to some tolerance, the Zeeman tolerance stability conjecture, [[#Referen
3 KB (402 words) - 21:38, 12 October 2014
...nalogy is not possible; this gave rise to the development of the theory of systems of hypercomplex numbers.
...lude [[Double and dual numbers|double and dual numbers]], and hypercomplex systems of the form
3 KB (484 words) - 19:37, 28 December 2020
...rem is true: Systems with fixed critical points can only be constituted by systems (1) which, after the introduction of a suitable parameter $ \epsilon $,
...l systems (1) without moving critical singular points, and in the study of systems belonging to these classes (cf. [[Singular point|Singular point]] of a diff
5 KB (748 words) - 08:05, 6 June 2020
...lastic and elastic-plastic systems; often the term stability of deformable systems is used.
...ov stability]]. The central question in the theory of stability of elastic systems is to find a region in the parameter space of the system with its external
9 KB (1,409 words) - 08:22, 6 June 2020
A notion imported from group theory (cf. also [[Group|Group]]), where the [[Frattini-subgroup(2)|Frattini subgr
Related concepts are being studied in very general algebraic systems [[#References|[a2]]].
2 KB (323 words) - 13:52, 25 April 2014
...series]] using [[Summation methods|summation methods]]. The best developed theory of the summation of Fourier series is that which uses the trigonometric sys
Apart from these, the most important in the theory of one-dimensional trigonometric series are the [[Cesàro summation methods
4 KB (614 words) - 11:33, 2 September 2014
...ies of the system itself (see also [[Oscillations, theory of|Oscillations, theory of]]). The term "auto-oscillation" was introduced by A.A. Andronov (see [[#
...mous auto-oscillating systems with one degree of freedom may be defined as systems whose equations of motion have one or more limit cycles in the phase plane
9 KB (1,404 words) - 16:55, 15 April 2012
...) is a solution of the other system (equation), and conversely, where both systems (equations) are considered in one and the same domain.
...o-called elimination of the unknowns (cf. [[Elimination theory|Elimination theory]]).
4 KB (692 words) - 13:38, 31 July 2014
Every link has a closed braid presentation (cf. also [[Braid theory|Braid theory]]; [[Link|Link]]).
...ng the theory of braids, [[#References|[a1]]] (cf. also [[Knot theory|Knot theory]]). Alexander's theorem has its roots in Brunn's result (1897) that every k
946 bytes (146 words) - 08:32, 17 March 2023
$#C+1 = 101 : ~/encyclopedia/old_files/data/E035/E.0305360 Elementary theory
...llection of closed formulas of first-order predicate logic. The elementary theory $ \mathop{\rm Th} ( K) $
9 KB (1,436 words) - 19:37, 5 June 2020
$#C+1 = 115 : ~/encyclopedia/old_files/data/A011/A.0101690 Algebraic systems, variety of
A class of algebraic systems (cf. [[Algebraic systems, class of|Algebraic systems, class of]]) of a fixed signature $ \Omega $,
10 KB (1,438 words) - 16:10, 1 April 2020
...attention to the more detailed internal properties. Therefore, two control systems, having in some sense identical behaviour and identical purposes, are not r
...Buslenko [[#References|[3]]] and others. A complete definition of control systems was established in [[#References|[1]]]. This definition includes all known
12 KB (1,844 words) - 16:56, 15 April 2012
...ajectory for which only one Floquet multiplier has modulus one. In certain systems the whole phase space is a hyperbolic set (cf. [[Y-system| $ Y $-
...and have since played an important role in the theory of smooth dynamical systems, both as objects of studies and as a part in many examples [[#References|[3
5 KB (745 words) - 22:11, 5 June 2020
When describing pure logical systems (propositional and predicate calculus), the terms "constructive" , "intui
...n to belong to constructive logic. The general manner in which most of the systems of constructive logic reflect the specific constructive understanding of th
10 KB (1,524 words) - 20:25, 4 June 2020
Chebyshev points are often chosen as "solutions" of incompatible linear systems of equations and inequalities.
...><TD valign="top">[3]</TD> <TD valign="top"> I.I. Eremin, "Incompatible systems of linear inequalities" ''Dokl. Akad. Nauk SSSR'' , '''138''' : 6 (1961)
2 KB (296 words) - 16:59, 22 November 2018
...of the principles of the elementary constructive theory of numbers. These systems were originally conceived as formalizations of parts of intuitionistic logi
...us) are obtained from the ordinary versions of the corresponding classical systems with a full set ( $ \wedge , \lor , \supset , \neg , \forall , \exists $)
9 KB (1,350 words) - 22:10, 5 June 2020
...op">[1]</TD> <TD valign="top"> V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from
In arbitrary dynamical systems (where the phase space is not necessarily metric) the periodic points are c
5 KB (770 words) - 18:12, 16 December 2020
A concept in the theory of orthogonal systems (cf. [[Orthonormal system|Orthonormal system]]). Let a [[Complete system of
..., therefore one has to single out the special class of Riesz systems, i.e. systems $\{\psi_n\}$ satisfying
4 KB (643 words) - 19:53, 27 February 2021
...al analysis which is known as the [[Riemann sphere|Riemann sphere]] in the theory of functions of a complex variable.
be coordinate systems in $ \sigma $
2 KB (322 words) - 10:58, 29 May 2020
...properties of a given system. For this reason, in the theory of dynamical systems one speaks of a continuous spectrum whenever $U_T$ or $A$ have no eigenfunc
...to a [[Metric isomorphism|metric isomorphism]] {{Cite|CFS}}. An analogous theory has also been developed for transformation groups more general than $\mathb
5 KB (710 words) - 12:58, 16 July 2014
...ideration of difference schemes for integrating one-dimensional hyperbolic systems. If $\tau$ is the grid spacing with respect to $t$, $h$ the grid spacing wi
...="top">[1]</TD> <TD valign="top"> S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian)</
729 bytes (103 words) - 10:39, 10 August 2014
...bers; as written notations of numerical symbols began to appear, so number systems began to vary in the character of their numerical signs and in the principl
...numbers are formed by grouping nodal numbers together are called additive systems. Thus, in ancient Egyptian (hieroglyphic) notation, the numbers 1, 2, 3, 4,
8 KB (1,143 words) - 08:03, 6 June 2020
...inear mapping]] $V \times V \times V \rightarrow V$, is mainly used in the theory of non-associative algebras and appears in the construction of Lie algebras
...d> <td valign="top"> W. Hein, "A construction of Lie algebras by triple systems" ''Trans. Amer. Math. Soc.'' , '''205''' (1975) pp. 79–95</td></tr>
3 KB (498 words) - 19:51, 27 February 2021
...rmational processes in [[#References|[a11]]], and in networks of cognitive systems in [[#References|[a15]]] and [[#References|[a16]]].
...o be used in more complex social processes, involving many self-organizing systems.
9 KB (1,251 words) - 18:24, 14 January 2021
There are systems of other types, where each request must necessarily be served, such as, for
==I. Systems with waiting (systems with a queue, queueing systems).==
15 KB (2,349 words) - 09:05, 21 January 2024
...there is also an analogue of Mikhailov's criterion for non-linear control systems (see [[#References|[6]]]).
...i, L.E. El'sgol'ts, "Mathematical foundations of the theory of control systems" , Moscow (1969) (In Russian)</td></tr><tr><td valign="top">[6]</td> <td
5 KB (692 words) - 15:30, 1 July 2020
...equations may be employed to find approximate solutions of the non-linear systems.
which is typical in the theory of quasi-linear oscillations, the harmonic balance method consists in repla
4 KB (613 words) - 19:43, 5 June 2020
...ion of the initial elements are such that the operation of complex control systems can be described with the aid of Boolean functions. Boolean functions are a
...cube (cf. [[Boolean functions, metric theory of|Boolean functions, metric theory of]]), as well as of the various algebras of Boolean functions (cf. [[Many-
5 KB (836 words) - 12:39, 14 February 2020
...ties: positive entropy (cf. [[Entropy theory of a dynamical system|Entropy theory of a dynamical system]]) and [[Ergodicity|ergodicity]]; [[Mixing|mixing]] o
systems are sometimes called Kolmogorov systems (flows, etc.), after their originator (see ), who used the term "quasi-reg
7 KB (1,037 words) - 16:58, 1 July 2020
...ary to consider statistical states of systems; for classical (non-quantum) systems they are described by probability distributions on the phase space. The rul
...ferentiable [[Dynamical system|dynamical system]] (which includes the main systems considered in analytical mechanics and classical statistical physics), the
6 KB (906 words) - 21:49, 30 April 2014
...ent or a more general type, including the case of non-ideal and quantified systems.
The theory was presented by L. Boltzmann in 1872.
2 KB (326 words) - 17:14, 30 December 2018
...of motion (understood in the same wide form). Thus, stability theory is a theory in the widest sense of this word. Among the different concepts of the stabi
...]]) and having a large intersection with the theory of linear systems, the theory of Lyapunov characteristic exponents (cf. [[Lyapunov characteristic exponen
9 KB (1,252 words) - 10:41, 15 April 2014
...I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian)</TD></TR></table>
698 bytes (110 words) - 15:19, 17 July 2014
...of Chetaev functions have been suggested, in particular for non-autonomous systems (cf. [[#References|[3]]]).
...bility of motion. Applications of Lyapunov's second method to differential systems and equations with delay" , Stanford Univ. Press (1963) (Translated from
2 KB (299 words) - 15:31, 14 February 2020