...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
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...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]]).
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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
...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)</
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...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
of algebraic systems (cf. [[Algebraic system|Algebraic system]]) $ A _ {i} $
is non-empty and is a subsystem of each of the systems $ A _ {i} , A _ {j} $.
4 KB (674 words) - 16:10, 1 April 2020
...quivalent (cf. [[Equivalence of dynamical systems|Equivalence of dynamical systems]]) near the origin to the normal form
...quivalent (cf. [[Equivalence of dynamical systems|Equivalence of dynamical systems]]) near the origin to the suspension of the normal form (a3) by the standar
6 KB (872 words) - 08:12, 6 June 2020
...cal systems, provided with the $C^1$-topology [[#References|[1]]], . Thus, systems whose trajectories display a behaviour which is more complex and more sensi
...<TR><TD valign="top">[4b]</TD> <TD valign="top"> A.B. Katok, "Dynamical systems with hyperbolic structure" ''Transl. Amer. Math. Soc.'' , '''116''' (1981
8 KB (1,203 words) - 17:01, 23 March 2023
...= 12 : ~/encyclopedia/old_files/data/K110/K.1100100 Kharitonov polynomial theory
...motivation for this theory derives from the issue of robust stability for systems of linear time-invariant differential equations. For a system of linear dif
5 KB (695 words) - 22:14, 5 June 2020
In 1844 W. Woolhouse stated the existence problem for Steiner
systems, and P. Kirkman solved it in 1847 for <img align="absmiddle" border="0" src
...diaofmath.org/legacyimages/s/s087/s087670/s08767034.png" /> (using Steiner
systems).
32 KB (4,451 words) - 15:16, 19 March 2018
The theory of eliminating unknowns from systems of algebraic equations. More precisely, suppose one is given a system of eq
...$x_1,\dots,x_k$, one also consider the homogeneous problem in elimination theory (the inhomogeneous problem is trivial in this case): Find the projection on
8 KB (1,363 words) - 16:04, 13 January 2021
...yclopedia/old_files/data/P071/P.0701560 Parametric resonance, mathematical theory of
The branch of the theory of ordinary differential equations that studies the phenomenon of parametri
10 KB (1,385 words) - 09:16, 13 May 2022
...orks), which is oriented to the qualitative analysis and synthesis of such systems (discovering deadlocks or conflict situations and bottlenecks, computer-aid
...ine is the use of Petri nets as the basis of models for discrete dynamical systems in information technology, economics, digital engineering, etc.
6 KB (897 words) - 19:23, 16 August 2016
...e concept of local coordinates. This, for example, is the situation in the theory of manifolds.
...coordinates "from without" , preserving as it were the "purity" of the theory, have not justified themselves (e.g. the synthetic constructions of [[Proje
7 KB (980 words) - 17:09, 11 April 2014
...group]]) are a special case of BCI-algebras. One may take different axiom systems for BCI-algebras, and one such system says that a BCI-algebra is a non-empt
...top"> C.S. Hoo, "Fuzzy ideals of BCI and MV-algebras" ''Fuzzy Sets and Systems'' , '''62''' (1994) pp. 111–114</TD></TR>
5 KB (790 words) - 02:33, 15 February 2024
...o another is called local Lorentz invariance. Some branches of the general theory of relativity also examine quantities determined by giving a [[congruence]]
<TR><TD valign="top">[1]</TD> <TD valign="top"> V.A. Fok, "Einstein's theory and physical relativity" , Moscow (1967) (In Russian)</TD></TR>
1 KB (179 words) - 08:28, 29 April 2016
...D.V. Anosov (see , [[#References|[2]]]), and they are often called Anosov systems.
...f a $Y$-system of period at most $T$ increases exponentially with $T$. $Y$-systems possess strong ergodic properties with respect to the wide class of so-call
9 KB (1,321 words) - 07:59, 21 June 2014
...space of the representation). Frequently, representation theory means the theory of linear representations. If $V$ is finite-dimensional, then its dimension
...to abstract (or algebraic) representation theory there is a representation theory of topological objects, e.g., topological groups or Banach algebras (cf. [[
4 KB (513 words) - 11:58, 9 November 2014
Problems of finding solutions of (systems of) partial differential equations of hyperbolic type that satisfy specific
...and systems|Mixed and boundary value problems for parabolic equations and systems]]).
31 KB (4,290 words) - 01:21, 14 February 2013
...eorems for loss systems are completely analogous to stability theorems for systems with an infinite number of channels. Suppose sequences $ \{ \tau _ {j} ^
...ds of investigation of loss systems may also be effective for the study of systems with heavy traffic or with a large number of service channels.
10 KB (1,560 words) - 08:02, 14 January 2024
...is as follows: The laws of physics take an identical form in all inertial systems.
...the light source, form the basis of special [[Relativity theory|relativity theory]].
1 KB (221 words) - 17:11, 7 February 2011
...such systems plays an important role in the physical interpretation of the theory.
4 KB (532 words) - 19:41, 5 June 2020
...braic geometry|algebraic geometry]] (commutative algebra, polynomial ideal theory) can be reduced by structurally easy algorithms to the construction of Grö
...Gröbner bases are routinely available in all modern mathematical software systems, as for example Mathematica and Maple.
6 KB (1,000 words) - 17:05, 7 July 2014
...ne to establish whether or not any pair of recursively-presented algebraic systems (cf. [[Algebraic system|Algebraic system]]) in a given class are isomorphic
...isomorphism problem is unsolvable for many important classes of algebraic systems. The unsolvability of the partial isomorphism problem for a finitely-presen
3 KB (386 words) - 22:13, 5 June 2020
...which — the simple theory of types — is described below. The terminology "theory of types" has no rigidly fixed meaning. It denotes formal theories which a
...ble $ y ^ \rho $ must belong (this is the main point of type-theoretic systems) to a higher level in the hierarchy of types than the types $ \sigma _{1}
8 KB (1,261 words) - 20:30, 13 December 2019
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
[[Category:Dynamical systems and ergodic theory]]
1,020 bytes (154 words) - 19:16, 25 October 2014
The modern theory of linear inequalities has been constructed on the basis of this generaliza
...onomic technology and economic planning reduce to the solution of specific systems of linear inequalities; this has significantly determined the modern trend
14 KB (2,052 words) - 10:49, 21 March 2022
...t is hard to say whether field theory, the theory of finite groups and the theory of finite-dimensional Lie algebras should be regarded as general algebra.
...n 1916. Initially this reorientation concerned group theory, and then ring theory. The results of this reorientation are reflected in the monograph of B.L. v
5 KB (725 words) - 21:52, 30 March 2012
...s governing the structure, composition, dynamics, and evolution of stellar systems.
...acteristic problem in stellar kinematics is the solution of overdetermined systems of conditional equations, each equation being derived for an individual sta
9 KB (1,236 words) - 08:23, 6 June 2020
...concerning the qualitative behaviour of linear equations and second-order systems (see [[#References|[3]]], [[#References|[4]]]).
...ontrol|optimal control]], filtration, stabilization of controllable linear systems, etc. (see [[Control system|Control system]], and [[#References|[6]]], [[#R
8 KB (1,202 words) - 14:55, 7 June 2020
..., expressing the most general conditions for the equilibrium of mechanical systems interconnected by ideal constraints.
Condition (*) contains all the equations and equilibrium laws of systems with ideal constraints; it is therefore justifiable to say that the entire
5 KB (805 words) - 08:28, 6 June 2020
...ding of rules a derivation is defined in the Post canonical system. In the theory of calculi one uses the following definition of an enumerable set of words
...t specializations of the notion of a Post canonical system: 1) Post normal systems (all rules have the form
4 KB (601 words) - 14:35, 17 March 2023
Triorthogonal systems of surfaces are given by the systems of coordinate surfaces in an orthogonal curvilinear coordinate system of sp
...an ellipsoid. Spherical transformations are automorphisms of triorthogonal systems of surfaces in Euclidean space.
3 KB (517 words) - 19:56, 12 July 2014
...ced as an algebraic system which is a generalization both of the algebraic systems appearing in the metasymplectic geometry developed by H. Freudenthal and of
...tem]], it is useful in obtaining all Lie algebras, without the use of root systems and Cartan matrices.
8 KB (1,154 words) - 19:44, 27 February 2021
[[Category:Ergodic theory]]
In ergodic theory, properties related to mixing are considered: multiple mixing and weak mixi
5 KB (696 words) - 08:01, 6 June 2020
...of dynamical processes with memory requires the further development of the theory of functional equations with abstract Volterra operators. Numerous applicat
...equations involving classical integral operators of Volterra type when the theory is framed in abstract spaces, see [[#References|[a2]]], [[#References|[a7]]
5 KB (745 words) - 15:29, 13 July 2014
...valign="top"> A.M. Krall, "$M(\lambda)$ theory for singular Hamiltonian systems with one singular point" ''SIAM J. Math. Anal.'' , '''20''' (1989) pp. 6
3 KB (586 words) - 10:02, 15 June 2015
The theory of $ L $-
...the work of A. Lindenmayer, [[#References|[a1]]]. The original aim of this theory was to provide mathematical models for the development of simple filamentou
13 KB (1,968 words) - 18:43, 26 March 2023
...$ is a metric space, then $k\in\N$). The quotient spaces of many dynamical systems do not satisfy any of the separation axioms, even if $S$ does. For example,
...ion in topological dynamics" , ''Sem. Differential Equations and Dynamical Systems II'' , ''Lect. notes in math.'' , '''144''' , Springer (1970) pp. 79–89
4 KB (578 words) - 06:37, 22 April 2012
...i for any fixed $t\in\mathbf R$. These manifolds are widely encountered in systems of type \eqref{*} describing oscillatory processes.
...5]</TD> <TD valign="top"> Yu.A. Mitropol'skii, "Problems of the asymptotic theory of nonstationary vibrations" , D. Davey (1965) (Translated from Russian)</T
4 KB (624 words) - 17:09, 14 February 2020
...fficient conditions for imbeddability, formulated in the language of graph theory (cf., for example, [[#References|[5]]]), are known.
...>[1]</TD> <TD valign="top"> A.I. Mal'tsev, "On inclusion of associative systems in a group" ''Mat. Sb.'' , '''6 (48)''' : 2 (1939) pp. 331–336 (In R
3 KB (456 words) - 19:17, 23 December 2014
...g five codimension-two bifurcations of equilibria in ordinary differential
systems [[#References|[a1]]], [[#References|[a5]]]:<table border="0" cellpadding="0
...olds in the parameter space of (a1). For example, in generic two-parameter
systems, codimension-two bifurcations happen at isolated points in the parameter pl
12 KB (1,485 words) - 13:18, 8 February 2020
...cal models of hysteresis and is convenient for the analysis of closed-loop systems with hysteresis non-linearities is presently (1996) under construction [[#R
In this theory, a hysteresis non-linearity is treated as a transducer, with an input, an o
4 KB (584 words) - 22:11, 5 June 2020
...duction, $L$-systems constituted a significant part of [[formal language]] theory, allowing one to compare parallel rewriting to a more classical sequential
...d with these notions, in particular with D$0$L-sequences, which made D$0$L-systems mathematically very fruitful. The most famous problem is the D$0$L-sequence
10 KB (1,542 words) - 11:21, 26 March 2023
...edback system ensures stabilization of any process belonging to a class of systems. However, a precise and universal definition of adaptive control is still e
...1970s and early 1980s, when rigorous proofs for the stability of adaptive systems were presented. However, very soon a drawback was discovered. The existing
6 KB (870 words) - 06:53, 26 March 2023
...orse–Smale systems were introduced by S. Smale, who considered Morse–Smale systems on a closed $ M $,
...ogical invariants determine the topological equivalence of two Morse–Smale systems. (In the two-dimensional case this question has been solved for a broader c
10 KB (1,474 words) - 20:08, 12 January 2024
is the set of all systems (1), equipped with the structure of a metric space given by the distance
Systems (1) with unstable exponents have been found (cf. [[#References|[2]]], [[#Re
8 KB (1,100 words) - 11:27, 21 June 2020
$#C+1 = 71 : ~/encyclopedia/old_files/data/E035/E.0305760 Entropy theory of a dynamical system
[[Category:Ergodic theory]]
7 KB (1,040 words) - 19:23, 13 January 2024
...ons, construction of projections, and also in studies on the properties of systems of functions. See also [[Gram matrix|Gram matrix]].
...top">[3]</TD> <TD valign="top"> F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , '''1''' , Chelsea, reprint (1977) (Translated from Russian
5 KB (733 words) - 19:42, 5 June 2020
Oscillations in physical systems described by non-linear systems of ordinary differential equations
...also with the use of methods of optimal control of non-linear oscillating systems. The basic general problems of non-linear oscillations are: the search for
14 KB (1,929 words) - 14:54, 7 June 2020
...tional proposition (see [[Configuration|Configuration]]). Non-Desarguesian systems exist, in particular, on certain surfaces and in general on certain Riemann
...o, K. Kuppuswamy Rao, "A class of non-desarguesian planes" ''J. Comb. Theory Ser. A'' , '''19''' (1975) pp. 247–255</TD></TR></table>
3 KB (510 words) - 15:05, 9 April 2014
...nature [[#References|[2]]]; for an account of the theory of distal dynamic systems and their generalizations, as well as the relevant literature, see [[#Refer
...theorem, applicable to distal morphisms between compact minimal dynamical systems: see [[#References|[a1]]], (15.4) or [[#References|[3]]], (3.14.22) for the
5 KB (686 words) - 11:55, 26 July 2014
A morphism in a category of algebraic systems (cf. [[Algebraic system|Algebraic system]]). It is a mapping of an algebrai
...imes applied to morphisms in categories other than categories of algebraic systems (homomorphisms of graphs, sheaves, Lie groups).
5 KB (755 words) - 22:11, 5 June 2020
...S. Tanaka in [[#References|[a7]]] to generalize the set difference in set theory, and by Y. Imai and Iséki in [[#References|[a5]]] as the algebras of certa
...]</TD> <TD valign="top"> K. Iséki, S. Tanaka, "An introduction to the theory of BCK-algebras" ''Math. Japon.'' , '''23''' (1978) pp. 1–26</TD></TR>
5 KB (892 words) - 02:43, 15 February 2024
For dynamical systems in $ \mathbf R ^ {n} $,
...1]</TD> <TD valign="top"> V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated fro
5 KB (812 words) - 04:31, 14 September 2022
...hains and systems of equations for the Green functions from the chains and systems of equations for the correlation functions (cf. [[Correlation function in s
...top">[4]</TD> <TD valign="top"> S.V. Tyablikov, "Methods of the quantum theory of magnetism" , Plenum (1967) (Translated from Russian)</TD></TR></table>
3 KB (452 words) - 11:18, 25 April 2020
...quivalent (cf. [[Equivalence of dynamical systems|Equivalence of dynamical systems]]) near the origin to the normal form
...quivalent (cf. [[Equivalence of dynamical systems|Equivalence of dynamical systems]]) near the origin to the suspension of the normal form (a3) by the standar
9 KB (1,234 words) - 22:11, 5 June 2020
...anomaly is deeply related to non-commutative [[Geometry|geometry]] and the theory of anti-commutative algebras (cf. [[Anti-commutative algebra|Anti-commutati
....g., octonionic stereo-synthesis) in real-time interactive binocular video-systems.
1 KB (214 words) - 11:13, 5 October 2014
...on equations or delay equations; to find periodic solutions of Hamiltonian systems; to give a rigorous computer-assisted proof of chaos in Lorenz equations; t
...rigorous numerical computations of the Conley index for concrete dynamical systems, since it allows one to incorporate interval arithmetic. Parametrized versi
9 KB (1,428 words) - 16:58, 1 July 2020
...sed subspace of a space is $H$-closed, then the space itself is compact. A theory has been developed for $H$-closed extensions of Hausdorff spaces.
...top"> S.D. Iliadis, S.V. Fomin, "The method of centred systems in the theory of topological spaces" ''Russ. Math. Surveys'' , '''21''' : 4 (1966) pp
2 KB (228 words) - 09:09, 6 August 2014
...introduced in studying various properties of central or normal series and
systems of subgroups (see [[#References|[1]]], [[#References|[2]]]).
<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 (245 words) - 17:21, 7 February 2011
...The equations of quantum mechanics may be put into the form of Hamiltonian systems in which $ p _ {i} ( t) $
...linear operators which satisfy certain commutation relations. Hamiltonian systems (in the usual "finite-dimensional" sense of the word) play an important r
13 KB (1,837 words) - 07:56, 21 January 2024
...al problem]] in the [[Optimal control, mathematical theory of|mathematical theory of optimal control]]. It was first formulated in 1956 by L.S. Pontryagin <r
..._1-t_0$). This statement admits a natural generalization to non-autonomous systems, problems with variable end-points and problems with restricted phase coord
8 KB (1,238 words) - 17:21, 7 June 2016
...bers or even by elements of an ordered set. They are often called subgroup systems (cf.
An important part in group theory is played by subnormal, normal and central series. A subgroup series (1) is
4 KB (657 words) - 09:59, 3 January 2021
$#C+1 = 57 : ~/encyclopedia/old_files/data/O070/O.0700520 Oscillations, theory of
...One makes a distinction between the theory of linear oscillations and the theory of non-linear oscillations.
16 KB (2,396 words) - 08:04, 6 June 2020
''metric theory of dynamical systems''
[[Category:Ergodic theory]]
13 KB (2,012 words) - 22:03, 6 April 2012
...n. Therefore, in general, the maximum principle does not hold for discrete systems.
For systems which are linear in the phase variables,
7 KB (959 words) - 03:47, 4 March 2022
...[1]]]). The Lyapunov transformation is widely used in the theory of linear systems of ordinary differential equations. In many cases the requirement
1 KB (177 words) - 15:55, 1 May 2023
...ical with discrete analysis. Such structures may include certain algebraic systems, infinite graphs, certain types of calculating media (e.g. homogeneous stru
...mathematics|computational mathematics]]; [[Probability theory|probability theory]], and certain other disciplines the subject of which is discrete.
9 KB (1,264 words) - 16:56, 15 April 2012
One of the most important invariants in [[Ergodic theory|ergodic theory]]. Basic is the concept of the entropy $ h ( S) $
...see [[#References|[3]]], [[#References|[4]]], and [[Ergodic theory|Ergodic theory]]).
11 KB (1,535 words) - 17:57, 29 October 2020
...and finitistically-provable assertion that if the Zermelo–Fraenkel formal theory is consistent, then it remains so after adding the axiom of choice.
...n syntax and semantics is not so essential. One uses so-called semi-formal systems, where the notion of a deduction depends on certain semantic stipulations.
2 KB (328 words) - 17:25, 7 February 2011
...ning sufficient conditions for the existence of periodic solutions of such systems is of considerable interest.
...D valign="top">[1]</TD> <TD valign="top"> J.W. [Lord Rayleigh] Strutt, "Theory of sound" , '''1''' , Dover, reprint (1945)</TD></TR><TR><TD valign="top">
3 KB (475 words) - 08:10, 6 June 2020
...apping turned into a fibration) is the space $K(\pi_n, n)$. Such Postnikov systems are called cellular. The limit of a cellular Postnikov system is a CW-compl
The fundamental theorem in the theory of Postnikov systems states (see
11 KB (1,963 words) - 07:35, 8 August 2018
$#C+1 = 49 : ~/encyclopedia/old_files/data/A010/A.0100710 Additive theory of ideals
...uniqueness" theorem must hold. The fundamental principles of the additive theory of ideals were introduced in the 1920s and the 1930s by E. Noether [[#Refer
7 KB (1,035 words) - 20:23, 4 April 2020
...localizations (i.e. its rings of fractions with respect to multiplicative systems $S$, cf. [[Localization in a commutative algebra]]) are again Bezout rings.
...lign="top">[a1]</TD> <TD valign="top"> R. Gilmer, "Multiplicative ideal theory" , M. Dekker (1972)</TD></TR></table>
1 KB (190 words) - 19:52, 2 November 2014
...below. During the enormous activity on mathematical aspects of integrable systems and [[Soliton|soliton]] equations, starting in the late 1960{}s, some proto
...es|[a30]]], and the modified Korteweg–de Vries equation for representation theory [[#References|[a6]]], [[#References|[a15]]].
13 KB (1,901 words) - 09:09, 26 March 2023
...n the areas of flexible manufacturing, communication networks and logistic systems.
...dresses the synthesis of controllers (i.e. supervisors) for discrete event systems to satisfy a set of qualitative specifications on the admissible orderings
14 KB (2,165 words) - 19:35, 5 June 2020
...n the second half of the 19th century, S. Lie and his school developed the theory of an important class of topological groups (groups of differentiable trans
...gical algebraic systems; and questions of duality of topological algebraic systems.
9 KB (1,142 words) - 17:15, 15 November 2023
...f. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]), in which the control $ u= u( t) $
...(or for complete controllability) are known in computable form for linear systems
20 KB (2,978 words) - 08:04, 6 June 2020
hold. For linear systems with constant or periodic coefficients,
but there exist systems for which the corresponding inequalities are strict (see [[Uniform stabilit
8 KB (1,194 words) - 08:14, 6 June 2020
systems $ \gamma = ( V,T,A,R ) $
systems with finite sets of axioms and splicing rules, with the splicing operation
5 KB (695 words) - 19:55, 17 December 2020
...algebraic system is one of the basic mathematical concepts and its general theory has been developed in depth. This was done in the 1950s, and the work took
Two algebraic systems $ \mathbf A ,\ \mathbf A^ \prime $
30 KB (4,711 words) - 18:22, 30 January 2020
The theoretical basis of the development of automatic translation is the theory of formal grammars (cf. [[Grammar, formal|Grammar, formal]]). An automatic
...lation are due to the relatively backward state-of-the-art of the semantic theory of languages, which could be used for an exact formulation of the rules of
7 KB (1,008 words) - 17:03, 7 February 2011
...ed and is also used, with various changes, for the approximate solution of systems of equations whose coefficients are also known approximately.
...em|Kronecker–Capelli theorem]]). The construction of the general theory of systems of linear equations was thus completed at the end of the 19th century.
9 KB (1,394 words) - 08:15, 9 January 2024
...cal systems with discrete states and changes of states called events; such systems arise e.g. in digital network communication protocols.
...l calculus" D. Franke (ed.) F. Kraus (ed.) , ''Design Methods of Control Systems'' , '''2''' , Pergamon (1991) pp. 723–728</TD></TR></table>
8 KB (1,166 words) - 06:28, 30 May 2020
...hening of the pointwise ergodic theorem (cf. also [[Ergodic theory|Ergodic theory]]) announced in [[#References|[a21]]] and stating that if $( X , \mathcal{F
...neral phenomena in which sampling is "good" for an uncountable number of systems. Since [[#References|[a21]]], several proofs of the "Wiener–Wintner theo
9 KB (1,431 words) - 17:03, 1 July 2020
...th Archimedean factors, and so totally ordered groups have solvable normal systems (see [[Subgroup system|Subgroup system]]).
Specific for the theory of totally ordered groups are questions connected with the extension of par
3 KB (495 words) - 19:18, 17 August 2014
The role of the Lyapunov characteristic exponent in the theory of Lyapunov stability is based on the following assertion: If $ \lambda _
...stem (1) has constant or periodic coefficients, and also for certain other systems); in other words, the functionals $ \lambda _ {i} ( A) $
12 KB (1,746 words) - 15:37, 1 May 2023
...with many CAD systems using them, and many important developments in their theory.
Therefore, some CAD systems use Bézier curves and surfaces, some employ the $ B $-
5 KB (759 words) - 09:49, 27 March 2023
...systems of modal logic have been considered, interrelations between these systems have been revealed, and their interpretations have been studied.
...modal logic which have been studied are based on classical logic; however, systems based on intuitionistic logic have also been discussed (see, for example, [
13 KB (1,978 words) - 20:34, 14 January 2021
...te polynomials]]; and [[Laguerre polynomials|Laguerre polynomials]]. These systems of [[Orthogonal polynomials|orthogonal polynomials]] have the following pro
...systems of orthogonal polynomials mentioned satisfy these properties; for systems obtained from these three by linear transformations of the independent vari
8 KB (1,132 words) - 14:19, 17 March 2023
runs through some uncountable set of values. Systems of the type (2) contain an uncountable set of functions $ \{ x _ \alpha
...[#References|[1]]] is the author of the first publication on the theory of systems of differential equations of the type (1). His main result was an existence
8 KB (1,114 words) - 17:33, 5 June 2020
...l–Katona theorem is probably the most important one in finite extremal set theory.
<tr><td valign="top">[a1]</td> <td valign="top"> K. Engel, "Sperner theory" , Encyclopedia of Mathematics and its Applications '''65''', Cambridge Uni
6 KB (871 words) - 14:52, 11 November 2023
...and retrieval of data), system access in the case of multi-user computing systems, etc.
...of programs, program verification, mathematical models, and general system theory, has been developed in this connection.
2 KB (210 words) - 17:28, 7 February 2011
Systems of linear inequalities appear in several forms; the following examples are
...e name "transposition theorem" . The relation between the primal and dual systems is sometimes given as a "theorem of alternatives" , listing alternatives,
12 KB (1,986 words) - 19:13, 31 March 2017
...In addition, they can be used to select keys for public-key cryptographic systems (cf. [[Cryptography]]; [[Cryptology]]).
...>[a3]</TD> <TD valign="top"> E. Bach, J. Shallit, "Algorithmic number theory" , '''1: Efficient Algorithms''' , MIT (1996)</TD></TR>
3 KB (485 words) - 22:01, 25 October 2014
...assical (elementary) projective and analytic geometry one speaks of linear systems of curves, surfaces, quadrics, etc. These are families of curves, surfaces,
...trol and systems theory: for linear input/output systems, linear dynamical systems or linear control system.
6 KB (855 words) - 22:17, 5 June 2020
which turn up in the theory below are supposed to be real-valued functions of the variables $ x _ {s}
Chetaev's theorem on perturbations of stable motions of Hamiltonian systems. This Chetaev theorem is a theorem on the properties of the Poincaré varia
7 KB (1,104 words) - 13:23, 6 January 2024
...absolute space and absolute time. They remain valid in inertial reference systems. The conclusions as to the motion of material bodies are arrived at in dyna
...ns motions of bodies which are interconnected with each other. Dynamics of systems include dynamics of a solid, dynamics of a system with a variable mass, dyn
11 KB (1,677 words) - 20:41, 21 December 2016
...ion of an [[Inverse matrix|inverse matrix]]. As for the solution of linear systems, methods for numerical inversion can be subdivided into direct and iterativ
...se matrix. The accuracy of the computed solution, as in the case of linear systems, depends on the rate of growth of the matrix entries in the intermediate st
11 KB (1,584 words) - 22:13, 5 June 2020
...studying the rotation of the Earth. Methods in astrometry are based on the theory and the practice of geometrical measurements on the celestial sphere and th
...sphere. Spherical trigonometry makes it possible to use various coordinate systems on the celestial sphere and to determine numerous relations between the ang
4 KB (554 words) - 17:13, 7 February 2011
...mber of degrees of freedom. They are also called dynamical or conservative systems (cf. [[Dynamical system|Dynamical system]]).
The essential contents of the theory of complex autonomous systems — unlike in the real case — is found in the case of an analytic $ f(x
13 KB (1,960 words) - 07:35, 26 March 2023
...commonly called strange or wild. It is widely believed that typically the systems are either stochastic or deterministic (or a combination of them), but ther
...<td valign="top"> I.P. Cornfeld, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982)</td></tr><tr><td valign="top">[a5]</td> <td valign="top
5 KB (722 words) - 16:46, 1 July 2020
A term in [[Ergodic theory|ergodic theory]] and [[Topological dynamics|topological dynamics]] used in phrases like:
In ergodic theory, the concept "transformation with a quasi-discrete spectrum" is in fact c
7 KB (978 words) - 05:56, 19 March 2022
...tions, which has to be borne in mind in [[Perturbation theory|perturbation theory]].
...s relate to the [[Qualitative theory of differential equations|qualitative theory of differential equations]]. The latter, in particular, establishes the spe
9 KB (1,244 words) - 08:05, 6 June 2020
...he limiting (when $t\to\pm\infty$) behaviour of trajectories of autonomous systems of two differential equations of the first order:
...l possible to use the basic "technical" premises of the Poincaré–Bendixson theory: the [[Jordan theorem|Jordan theorem]] and the [[Poincaré return map|Poinc
8 KB (1,265 words) - 15:06, 14 February 2020
<TR><TD valign="top">[a1]</TD> <TD valign="top"> Józef Słomiński, "The theory of abstract algebras with infinitary operations" ''Rozprawy Mat.'' , '''18
[[Category:General algebraic systems]]
1 KB (216 words) - 05:53, 15 April 2023
...shock (cf. [[Shock waves, mathematical theory of|Shock waves, mathematical theory of]]) followed by a rarefaction wave. The solution method was simplified by
4 KB (526 words) - 06:29, 30 May 2020
systems.
...quivalent (cf. [[Equivalence of dynamical systems|Equivalence of dynamical systems]]) near the origin to the system
10 KB (1,376 words) - 16:43, 4 June 2020
...to grammar forms has been developed also for parallel rewriting. (See [[L-systems| $ L $-
systems]], [[#References|[a5]]].)
6 KB (867 words) - 14:14, 31 December 2020
[[Category:Measure-theoretic ergodic theory]]
...ng transformations arise, for example, in the study of classical dynamical systems (cf. (measurable) [[Cascade|Cascade]]; [[Measurable flow|Measurable flow]])
2 KB (260 words) - 08:00, 6 June 2020
...etc.). The following have had a decisive influence on the creation of the theory of orthogonal series:
...em of moments (mid-19th century), which led to his creation of the general theory of [[Orthogonal polynomials|orthogonal polynomials]];
18 KB (2,612 words) - 13:13, 18 February 2022
[[Category:Ergodic theory]]
A property considered in [[Ergodic theory|ergodic theory]]. Originally it was defined for a [[Cascade|cascade]] $\{T^k\}$ or a [[Flo
1 KB (242 words) - 21:18, 8 November 2014
...in [[#References|[a2]]], the theory of abstract families of languages, AFL-theory, is a framework for systematically studying the closure properties of famil
...xamples of anti-AFLs can be found, for example, in the area of Lindenmayer systems [[#References|[a3]]].
4 KB (661 words) - 09:20, 2 April 2018
...of symbolic dynamics in the narrow sense to the investigation of dynamical systems which themselves are defined completely independently of $\Omega$ and $\sig
...2]</TD> <TD valign="top"> R. Bowen, "Equilibrium states and the ergodic theory of Anosov diffeomorphisms" , ''Lect. notes in math.'' , '''470''' , Springe
15 KB (2,197 words) - 08:48, 29 April 2023
...g semantics of constructive mathematics by the method of stepwise semantic systems.
2 KB (271 words) - 16:50, 2 November 2014
..., arose from the study of the long-term statistical behaviour of dynamical systems (cf. also [[Measure-preserving transformation|Measure-preserving transforma
...re-preserving transformations: the entropy, which they took from Shannon's theory of information (''cf.'' also [[Entropy of a measurable decomposition]]; [[S
8 KB (1,391 words) - 13:02, 12 December 2013
...intention of providing an alternative foundation for mathematics. Curry's theory is divided into two parts: pure combinatory logic ( $ { \mathop{\rm CL} }
In the presence of the rule of extensionality, the theory $ { \mathop{\rm CL} } $
11 KB (1,625 words) - 22:11, 5 June 2020
...e subsets often do not); see [[#References|[a1]]], for example. In lattice theory, least upper bounds of directed subsets again play a distinctive part; see
2 KB (292 words) - 06:36, 14 October 2014
...ontrol theory|automatic control theory]]. A systematic construction of the theory of ordinary differential equations with deviated arguments was begun in 194
...h a deviation of the argument, permitting the construction of a meaningful theory, has been introduced. Several properties of such equations are directly ana
27 KB (3,819 words) - 16:54, 7 February 2011
...chip by T. Veltman is an example). The earliest among the widely available systems is REDUCE, which runs on many platforms and is still being further develope
Among the objects handled by these systems are polynomials over effective rings (e.g., the integers, the rational numb
9 KB (1,292 words) - 20:51, 18 September 2016
...n problems in number theory have also turned out to be connected with root systems [[#References|[6]]].
==General properties of root systems.==
22 KB (3,351 words) - 19:14, 21 December 2019
...nsformational grammars; some of them are intended for processing component systems, others for processing hierarchy trees. As an example, one can quote the so
grammars, which are finite systems of elementary transformations of the form $ t _ {1} \Rightarrow t _ {2} \
4 KB (600 words) - 19:42, 5 June 2020
...lement (zero) subsystems. The direct sum, or (discrete) direct product, of systems $ X _ {i} $,
In category theory, the concept dual to that of a product, i.e. that of a [[Coproduct|coproduc
4 KB (680 words) - 19:35, 5 June 2020
...gue inequality and relations similar to it are often used in approximation theory to obtain estimates of best approximations from below. The inequality was e
2 KB (236 words) - 09:15, 1 August 2014
...ferences|[a1]]], but their most natural roots lie in the study of chemical systems: the components of the vector $u$ may then represent concentrations of chem
...of systems, the research impetus in this field comes more from viewing the systems as models of specific natural phenomena, rather than from interest in them
12 KB (1,852 words) - 13:56, 26 July 2014
...heory; therefore it can be viewed as the appropriate extension of homotopy theory to general spaces.
...a of operators, or boundaries of certain groups. In all these areas, shape theory has proved useful. In particular, it has applications in the study of cell-
9 KB (1,368 words) - 18:46, 16 March 2024
...ts. Actually, Wold introduced (a1) as a joint representation for AR and MA systems (cf. also [[Mixed autoregressive moving-average process|Mixed autoregressiv
...esponding to stable and miniphase ARMA or (finite-dimensional) state space systems is by far the most important one. In this case there is a wide class of ide
9 KB (1,297 words) - 09:30, 3 February 2024
...ions of the Routh–Hurwitz criterion in the stability theory of oscillating systems, a stable polynomial. There are other criteria for the stability of polynom
...top">[2]</TD> <TD valign="top"> F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , '''1''' , Chelsea, reprint (1977) (Translated from Russian
2 KB (262 words) - 12:37, 14 February 2020
...c theory]] analogous to the [[Metric entropy|metric entropy]] of dynamical systems (introduced in [[#References|[1]]]). For an open covering $ \mathfrak A $
...TD valign="top">[16]</TD> <TD valign="top"> C. Grilleneberger, "Ergodic theory on compact spaces" , Springer (1976)</TD></TR></table>
10 KB (1,429 words) - 16:40, 29 October 2023
...ction-angle coordinates in the theory of completely-integrable Hamiltonian systems. Each such system (with finite degrees of freedom) can be transformed into
1 KB (239 words) - 14:39, 28 August 2014
...n) functions describing all possible states of a many-particle system. For systems of finitely many particles the BBGKY hierarchy is equivalent to the [[Liouv
...ution function|Distribution function]]). The BBGKY hierarchy for classical systems reads as follows:
10 KB (1,427 words) - 07:38, 7 February 2024
''software for dynamical systems''
Mathematical background on dynamical systems can be found in [[#References|[a3]]], [[#References|[a7]]] or [[#References
13 KB (1,928 words) - 17:00, 1 July 2020
A branch of topology dealing with the topological problems of the theory of differentiable manifolds and differentiable mappings, in particular diff
...y algebraic methods, concerning the homotopy invariance of classes and the theory of Hermitian forms over cochains with an involution. Of major importance ar
9 KB (1,298 words) - 14:59, 30 August 2014
...f. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]), consisting in the determination of the minimum time
...e-optimal control can also be studied for non-autonomous systems, i.e. for systems whose right-hand side $ f $
7 KB (997 words) - 08:25, 6 June 2020