...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
a) Consider all possible recursive schemes — i.e. systems of equations which define $n$-place partial recursive functions. Two scheme
...solvable relations of algorithmic equivalence which would permit complete systems of equivalent transformations that are convenient with regard to their appl
4 KB (663 words) - 12:42, 19 August 2014
of Abelian groups, there exists a Bredon cohomology theory $ \{ H ^ {n} _ {G} ( -,M ) \} _ {n \in \mathbf N } $
are projective objects in the [[Category|category]] of coefficient systems and there is a [[Spectral sequence|spectral sequence]]
5 KB (732 words) - 08:45, 26 March 2023
...g range of computer applications. The term is often understood to mean the theory of numerical methods and algorithms for the solution of typical mathematica
...als with the simplification of the man-computer interaction, including the theory and practice of [[Programming|programming]] computer problems and of [[Auto
13 KB (1,892 words) - 20:11, 31 December 2018
...a certain class. The term "absolute stability" assumes given a class of systems and an indication of the sense in which stability and uniformity are to be
Numerous studies of particular non-linear systems have led to the understanding that in the first place one should take in co
16 KB (2,300 words) - 08:22, 6 June 2020
...of both the geometric and the non-geometric theory of Jordan–Banach triple systems can be found in [[#References|[a25]]].
...and usually denoted by $V _ { y } ^ { \sigma }$. Important notions in the theory of Jordan pairs, such as the [[Jacobson radical|Jacobson radical]], are def
14 KB (2,076 words) - 17:00, 1 July 2020
A method used in non-linear oscillation theory to study oscillatory processes; it is based on an averaging principle, that
...the exact one (see [[#References|[1]]], [[#References|[2]]]). The rigorous theory of the method, with a comprehensive explanation of the essence of the gener
8 KB (1,178 words) - 22:15, 5 June 2020
...tems of linear inequalities and equalities. There exists a well worked-out theory of quadratic programming, and numerical methods have been developed for the
..."top">[a1]</TD> <TD valign="top"> M. Minoux, "Mathematical programming: theory and algorithms" , Wiley (1986)</TD></TR></table>
3 KB (372 words) - 08:07, 22 November 2014
In generic discrete-time dynamical systems defined by iterations of diffeomorphisms, orbits which are homoclinic to a
...TD valign="top"> S. Wiggins, "Introduction to applied non-linear dynamical systems and chaos" , Springer (1990) {{MR|1056699}} {{ZBL|}} </TD></TR></table>
13 KB (1,770 words) - 22:10, 5 June 2020
The mathematical investigation of three-particle systems became possible after L.D. Faddeev in 1960 proposed and studied an integral
..."Mathematical aspects of the three-body problem in the quantum scattering theory" , Israel Program Sci. Transl. (1965) (Translated from Russian)</TD></TR>
4 KB (564 words) - 07:56, 25 April 2022
...deas have found many applications in, e.g.: [[Scheduling theory|scheduling theory]]; circuit and network design; architectural design; control (cf. [[Control
...arametric approach for selecting the most reliable population" ''Queueing Systems'' , '''24''' (1996) pp. 169–176</td></tr></table>
2 KB (296 words) - 16:58, 1 July 2020
...are recursively axiomatized, but not necessarily finitely axiomatized. The theory of axiomatized classes reveals regularities common to all classes of object
...n="top">[2]</TD> <TD valign="top"> A.I. Mal'tsev, "Some problems in the theory of classes of models" , ''Proc. 4-th All-Union Math. Congress (1961)'' , ''
4 KB (658 words) - 10:26, 27 April 2020
...istical physics are basically related to two directions of the statistical theory: to equilibrium statistical mechanics, the problems of which are related to
solves the problems of the equilibrium theory (all equilibrium characteristics, such as internal energy, heat capacity, s
14 KB (2,025 words) - 08:23, 6 June 2020
...stituents of systems for the transmission of information considered in the theory of information transmission. It is used in the mathematical description of
...defined above, are possible; these correspond to more general and complex systems of information transmission (see, for example, [[Channel with feedback|Chan
10 KB (1,309 words) - 17:45, 4 June 2020
...[[Auto-oscillation|Auto-oscillation]]) of one of the simplest oscillating systems (the van der Pol oscillator). In particular, equation (1) serves — after
...="top"> J.J. Stoker, "Nonlinear vibrations in mechanical and electrical systems" , Interscience (1950)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="
6 KB (857 words) - 08:41, 13 May 2022
...golyubov in the justification of the dispersion relations in quantum field theory ([[#References|[1]]], Appendix A). The modern formulation is as follows. Le
...m field theory, in the theory of partial differential equations and in the theory of boundary values of holomorphic functions (especially of functions of sev
11 KB (1,586 words) - 07:17, 13 June 2022
...mptions of [[Relativity theory|relativity theory]]. The space-time of this theory is the four-dimensional [[Pseudo-Euclidean space|pseudo-Euclidean space]] $
...lations in special relativity theory. The space-time of special relativity theory is also called Minkowski space-time, or Lorentz space-time.
6 KB (840 words) - 13:26, 15 August 2014
''for linear systems of ordinary differential equations''
...TR><TR><TD valign="top">[5]</TD> <TD valign="top"> N.A. Izobov, "Linear systems of ordinary differential equations" ''J. Soviet Math.'' , '''5''' : 1 (1
4 KB (578 words) - 08:11, 13 July 2022
[[Category:General theory of linear operators]]
...{Cite|N}}, having in mind in the first instance its application in ergodic theory, when in a [[Measure space|measure space]] $ ( X , \mu ) $
4 KB (632 words) - 17:38, 6 January 2024
...r series associated with the expansion \eqref{7} is played by the spectral theory of linear operators.
...on, etc.). In this way also a whole series of relations is obtained in the theory of special functions.
4 KB (653 words) - 15:15, 14 February 2020
1) In the theory of operators the initial object $ \mathfrak D ( \lambda ) $
...t the solution of the non-linear problem is not unique. The methods of the theory of [[Branching of solutions|branching of solutions]] of non-linear equation
11 KB (1,608 words) - 10:59, 29 May 2020
...of the classical [[Lyapunov function|Lyapunov function]] in the theory of systems of differential equations.
...p"> V.V. Kalashnikov, "Qualitative analysis of the behaviour of complex systems by the method of test functions" , Moscow (1978) (In Russian)</TD></TR></
5 KB (718 words) - 04:11, 6 June 2020
...1]</TD> <TD valign="top"> V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated fro
..."top"> V.V. Nemytskii, "Topological problems in the theory of dynamical systems" ''AMS Transl. Series 1'' , '''5''' (1954) pp. 414–497 ''Uspekhi Mat.
2 KB (291 words) - 08:45, 29 August 2014
...he term "closed", Steklov studied the relevant aspects of various specific systems of orthogonal functions — in particular, the fundamental solutions of the
...onding system of orthogonal polynomials; in particular, he proved that the systems of Hermite and Laguerre polynomials are closed. One of Steklov's sufficient
7 KB (1,066 words) - 15:42, 14 February 2020
...development of technology and, particularly, of multi-processor computing systems, which allow many independent statistical experiments to be simulated simul
...nomena, etc. (see [[#References|[7]]]). In solving complex problems in the theory of rarefied gases, an effective technique is to modify the method of direct
7 KB (1,015 words) - 17:27, 7 February 2011
...opedia/old_files/data/R081/R.0801170 Reliability and inspection of control systems,
''problems in the reliability of control systems''
23 KB (3,205 words) - 08:10, 6 June 2020
...tion introduced by W. Hamilton (1834) to describe the motion of mechanical systems. It is used, beginning with the work of C.G.J. Jacobi (1837), in the classi
...ions for the action function (cf. [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]]) can be written in terms of a Hamilton function.
5 KB (753 words) - 19:43, 5 June 2020
==Oscillatory properties of solutions of linear Hamiltonian systems.==
==Linear Hamiltonian systems with periodic coefficients.==
21 KB (3,088 words) - 11:56, 8 March 2022
...y–Hamilton theorem can be also extended to singular two-dimensional linear systems described by Roesser-type or Fomasini–Marchesini-type models [[#Reference
...heorems may be also extended to two-dimensional continuous-discrete linear systems [[#References|[a5]]].
15 KB (2,237 words) - 17:01, 1 July 2020
...>[6]</TD> <TD valign="top"> S.P. Strelkov, "Introduction to oscillation theory" , Moscow (1964) (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD
...a1]</TD> <TD valign="top"> A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian)</TD></T
6 KB (850 words) - 19:39, 5 June 2020
The mathematical models describing relaxation oscillations are autonomous systems (cf. [[Autonomous system|Autonomous system]]) of ordinary differential equa
Periodic solutions of the type of relaxation oscillations of non-autonomous systems of ordinary differential equations have also been studied (see e.g. [[#Refe
13 KB (1,854 words) - 12:59, 13 January 2024
...</TD> <TD valign="top"> P. Bohl, "Ueber die Beweging eines mechanischen Systems in der Nähe einer Gleichgewichtslage" ''J. Reine Angew. Math.'' , '''127'
...valign="top">[a1]</TD> <TD valign="top"> V.I. Istrăţescu, "Fixed point theory" , Reidel (1981)</TD></TR>
4 KB (586 words) - 11:27, 17 March 2023
...sidues $0,\ldots,m-1$ or the absolutely smallest residues are the complete systems of residues which are most frequently used.
...n="top">[1]</TD> <TD valign="top"> I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)</TD></TR></table>
2 KB (318 words) - 13:55, 11 August 2014
...cience concerned with the construction of mathematical models of dynamical systems from measured input/output data. The constructed models are mostly of finit
...tworks form an example of common non-linear black-box models for dynamical systems.
7 KB (1,131 words) - 14:45, 16 December 2023
Astronomy is a complex science which deals with celestial objects and their systems from various aspects, which may be very remote from one another. This is wh
...determine an inertial reference coordinate system in space. The coordinate systems which are traditionally employed in astronomy, geodesy and other fields of
8 KB (1,204 words) - 18:48, 5 April 2020
...rticularly useful in the study of the "structural" properties of control systems.
...<TD valign="top">[3]</TD> <TD valign="top"> A. Nakasima, M. Hanzawa, "Theory of equivalent transformation of simple partial parts in relay circuits" ''
18 KB (2,608 words) - 17:21, 7 February 2011
...0]</TD> <TD valign="top"> D.A. Neumann, "Central sequences in dynamical systems" ''Amer. J. Math.'' , '''100''' : 1 (1978) pp. 1–18</TD></TR></table>
4 KB (617 words) - 16:07, 19 August 2014
...over the field of complex numbers which contains many classical polynomial systems. The Appell polynomials were introduced by P.E. Appell [[#References|[1]]].
of Appell polynomials is defined as the set of all possible systems of polynomials $ \{ A _ {n} (z) \} $
11 KB (1,595 words) - 17:13, 2 January 2021
The continuation method is used in the solution of systems of non-linear algebraic and transcendental equations (cf. [[#References|[1]
....F. Davidenko, "On applying the method of variation of parameters to the theory of non-linear functional equations" ''Ukrain. Mat. Zh.'' , '''7''' : 1 (
10 KB (1,328 words) - 17:04, 7 February 2011
...es and complete direct sums (see [[Algebraic systems, variety of|Algebraic systems, variety of]]). Since the totality of polynomial identities that are satisf
...>[a2]</TD> <TD valign="top"> L.H. Rowen, "Polynomial identities in ring theory" , Acad. Press (1980) pp. Chapt. 7</TD></TR>
4 KB (701 words) - 17:01, 23 November 2023
...operty of a scattering matrix follows from the basic principles of quantum theory: it must be unitary.
...$S$-matrix" is applicable in a wider context than only quantum-mechanical systems.
2 KB (387 words) - 22:42, 31 December 2017
1) The classical computability theory initiated by the work of K. Gödel, A. Tarski, A. Church, E. Post, A. Turin
...al languages and automata|Formal languages and automata]]; [[L-systems|$L$-systems]].)
13 KB (2,014 words) - 20:35, 31 December 2018
...ite) ordinal numbers. One considers also more general normal and subnormal systems, the terms of which are indexed by the elements of an ordered set.
|valign="top"|{{Ref|Ha}}||valign="top"| M. Hall jr., "The theory of groups", Macmillan (1959) pp. Sect. 8.4 {{MR|0103215}} {{ZBL|0084.02
2 KB (317 words) - 16:43, 27 November 2013
...rks, telephone networks, water supply systems, electrical circuits, spring systems, and so on. This has been done under various assumptions on routing schemes
There is no fixed definition of Braess's paradox in all systems, but there is a common theme. One assumes some measure of performance, $t$,
6 KB (983 words) - 17:44, 1 July 2020
...are two such fundamental theories: the singular and the spectral homology theory. The former is based on mappings of polyhedra into given spaces and is most
...is defined on the base of groups of chains; subsequent development of this theory led to the study of ordered, rather than of oriented, simplexes by S. Eilen
9 KB (1,350 words) - 11:39, 25 October 2014
...~/encyclopedia/old_files/data/O068/O.0608430 Optimal control, mathematical theory of
...|mathematical programming]] and game theory (cf. [[Games, theory of|Games, theory of]]).
21 KB (3,161 words) - 08:04, 6 June 2020
are obtained in a less general form than for single-server systems, and are connected with a condition on the existence of so-called renewal e
must be satisfied. If for such systems, in triangular arrays, the distributions of $ \tau _ {j} ^ {(n)} e $
18 KB (2,750 words) - 19:29, 16 January 2024
...le$, where $\Omega$ denotes the signature of $T$ (cf. [[Model theory|Model theory]]; [[Structure(2)|Structure]]). If $T$ is substructure complete, then it is
...n for the existence of a common zero of two polynomials $f$, $g$), and the theory of real closed fields (where the [[Sturm theorem|Sturm theorem]] gives a qu
3 KB (435 words) - 09:40, 26 November 2016
There are many Toda systems spawned by Toda's nearest neighbour linking of anharmonic oscillators on th
...Konstant, "The solution to a generalized Toda lattice and representation theory" ''Adv. Math.'' , '''34''' (1979) pp. 195–338</td></tr></table>
8 KB (1,147 words) - 17:45, 1 July 2020
...n the presence of "non-classical" constraints of inequality type, in the theory of optimal control, a necessary condition for a strong extremum of the func
...of canonical transformations and [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]]. Transition to the Hamiltonian form is also useful in the quantization of
6 KB (811 words) - 22:15, 5 June 2020
...In combinatorics, a permanent can be interpreted as follows: The number of systems of distinct repesentatives for a given family of subsets of a finite set is
The most familiar problem in the theory of permanents was van der Waerden's conjecture: The permanent of a [[doubly
8 KB (1,228 words) - 19:13, 17 March 2023
[[Category:General theory of linear operators]]
...HP}}). Means can be understood in various senses in the same way as in the theory of summation of series. The most frequently used means are the Cesàro mean
5 KB (685 words) - 16:32, 6 January 2024
...f [[Orbit stability|orbit stability]] of closed trajectories of autonomous systems of differential equations one has to apply necessary and sufficient conditi
...">[10]</TD> <TD valign="top"> E. Jury, "Inners and stability of dynamic systems" , Wiley (1974)</TD></TR></table>
9 KB (1,283 words) - 14:55, 7 June 2020
...h are upper steps over the series of calculi that give the language of the theory).
...of these calculi one develops the basic notions and methods of the general theory of calculi and finds far-reaching generalizations of calculi; see, e.g., [[
8 KB (1,204 words) - 18:51, 13 November 2014
...t Runge–Kutta methods in particular, applied to initial value problems for
systems of non-linear ordinary differential equations <img align="absmiddle" border
...integrate a stiff problem. Thus there is a need for a special convergence
theory applicable in the presence of stiffness.
11 KB (1,517 words) - 17:23, 7 February 2011
is called the order of equation (1) (below the general theory of linear ordinary differential equations is presented; for equations of th
...]] of linear differential systems it suffices to consider only homogeneous systems.
16 KB (2,441 words) - 17:36, 6 January 2024
...a1]]] and U. Holst [[#References|[a2]]] apply the stochastic approximation theory of H. Robbins and S. Monro [[#References|[a3]]] (cf. also [[Stochastic appr
2 KB (281 words) - 17:22, 7 February 2011
...el is not satisfied, the model is said to be non-regular. Models of formal systems obtained by adjoining to restricted predicate calculus some set $ T $
...en structures for a language, which may or may not satisfy the axioms of a theory under consideration, and models, which necessarily do so; thus the term "r
2 KB (279 words) - 08:01, 6 June 2020
and if for some (hence for all) coordinate systems around $ s _ {i} ( x _ {0} ) $
...gh. This prolongation theorem has important applications in the Lie–Cartan theory of infinite-dimensional Lie groups. The theorem has been extended to cover
9 KB (1,406 words) - 21:26, 3 January 2021
also called regular) play an important role in algebraic group theory. A torus that is not regular is called singular. For reductive groups $ G
can be given in terms of root systems. Thus, if $ T $
2 KB (286 words) - 15:07, 17 December 2019
...sphere [[#References|[a3]]]. It is also of fundamental importance for the theory of singular integral equations [[#References|[a4]]], Riemann–Hilbert prob
...n the theory of completely integrable models [[#References|[a11]]] and $K$-theory [[#References|[a12]]].
7 KB (989 words) - 16:56, 1 July 2020
...ular points) play an important role in the theory of completely-integrable systems (cf. [[Soliton|Soliton]]). Indeed, the Painlevé property has been proposed
...some exact results on singularity structure and integrability of dynamical systems" ''Acta Appl. Math.'' , '''8''' (1987) pp. 75–104</TD></TR><TR><TD val
5 KB (673 words) - 19:15, 1 May 2024
systems
|valign="top"|{{Ref|Ka}}||valign="top"| T. Kaczorek, "Theory of control and systems", PWN (1993) (In Polish)
4 KB (607 words) - 02:33, 14 September 2022
...<TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Hall, "On the Sylow systems of a soluble group" ''Proc. London Math. Soc.'' , '''43''' (1937) pp. 31
2 KB (287 words) - 08:24, 6 June 2020
of the systems with the same matrix $ A _ {k-1} $
...[Gauss method|Gauss method]], one of the fastest direct methods of solving systems.
10 KB (1,498 words) - 08:08, 21 March 2022
...er triple system is a '''Kirkman triple system''' $\mathrm{KTS}(n)$. Such systems exist if and only if $n \equiv 3 \pmod 6$. The [[Kirkman schoolgirls probl
* Thomas Beth, Dieter Jungnickel, Hanfried Lenz, "Design theory", Cambridge University Press (1986) {{ZBL|0602.05001}}
2 KB (243 words) - 20:10, 8 November 2023
...ction(2)|Contraction]]). He was the first to use the index $\alpha$ in the theory of fixed points [[#References|[a2]]]. Darbo's fixed-point theorem is a gene
...[a3]]]. It is also used to study the controllability problem for dynamical systems represented by implicit differential equations [[#References|[a4]]].
4 KB (600 words) - 00:06, 15 February 2024
The origins of model theory go back to the 1920's and 1930's, when the following two fundamental theore
...n a first-order language is true in every system of an increasing chain of systems, then it is true in the union of the chain (see [[#References|[1]]], [[#Ref
17 KB (2,788 words) - 15:19, 16 December 2020
...[3]</TD> <TD valign="top"> A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian)</TD></T
5 KB (785 words) - 08:05, 6 June 2020
...<TD valign="top"> V.V. Nemytskii, "On the orbit theory of general dynamic systems" ''Mat. Sb.'' , '''23 (65)''' : 2 (1948) pp. 161–186 (In Russian)</TD></T
be involutive. All this generalizes to systems of equations $ \omega ^ {i} = 0 $,
5 KB (702 words) - 17:45, 4 June 2020
...s generalizations and modifications which reflect various features of real systems. In the case of a finite automaton $ (A, S, B, \phi , \psi ) $
..., algebraic theory of|Automata, algebraic theory of]]); problems in coding theory generate the concepts of self-regulating automata, reversible automata, etc
18 KB (2,794 words) - 10:01, 25 April 2020
...es|[a4]]], [[#References|[a6]]]. The most general context to formulate the theory is that of reductively filtered Lie algebras [[#References|[a11]]].
...Bifurcation|bifurcation]] theory. Using Lyapunov–Schmidt reduction and the theory of [[Singularities of differentiable mappings|singularities of differentiab
10 KB (1,362 words) - 08:22, 26 March 2023
...the [[Frobenius theorem on Pfaffian systems|Frobenius theorem on Pfaffian systems]], there are a number of other results which (sometimes) go by the name Fro
ii) The following result in finite group theory. Let $ H $
5 KB (790 words) - 19:40, 5 June 2020
...f natural parallelism occurring in computer-modelling and computer-control systems and processes; and b) a parallelization of information processing in multi-
...and subprograms; and independent problems and programs in multi-processor systems.
11 KB (1,714 words) - 03:17, 5 June 2016
The basic concepts of the theory of partially ordered groups are those of an order homomorphism (cf. [[Order
..."top">[2]</TD> <TD valign="top"> L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)</TD></TR></table>
2 KB (362 words) - 08:05, 6 June 2020
...f a planar vector field $v(x)$ which is [[Local normal forms for dynamical systems#topological equivalence|topologically equivalent]] to the linear center.
...">[a3]</TD> <TD valign="top"> V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Springer (1983) (Translated from Ru
5 KB (706 words) - 09:05, 12 December 2013
...ects participating in the quantum inverse scattering theory for integrable systems. (Thus, the classical Yang–Baxter equation is the classical limit of the
...ntegrable systems II" V.I. Arnold (ed.) S.P. Novikov (ed.) , ''Dynamical Systems VII'' , Springer (1994) pp. 116–259</TD></TR><TR><TD valign="top">[a9]<
7 KB (1,071 words) - 05:50, 18 May 2022
...pursuer (hunter) $P$ and evader (prey) $E$, whose motions are described by systems of differential equations:
...ng a pursuit game means finding an equilibrium (cf. [[Saddle point in game theory]]).
3 KB (423 words) - 20:17, 11 January 2017
[[Category:Dynamical systems with hyperbolic behavior]]
...ather indefinite, as is the term "strange attractor". For smooth dynamical systems two types of strange attractors which are preserved by small perturbations
7 KB (982 words) - 12:14, 18 August 2014
...or increasing the speed of convergence in the iterative solution of linear systems $ Ax = b $.
...2)|linear algebra]] is the solution of finite-dimensional linear algebraic systems
12 KB (1,681 words) - 06:15, 28 March 2023
...become larger and larger with the expansion of electronic mail and similar systems.
...key cryptosystems (as opposed to what has been called above "classical" systems) is due to W. Diffie and M. Hellman, [[#References|[a3]]]: the knowledge of
12 KB (1,852 words) - 17:31, 5 June 2020
...e connection of such planes with quasi-groups, that the development of the theory of quasi-groups properly began.
One of the most important concepts in the theory of quasi-groups is that of an [[Isotopy|isotopy]]. Two quasi-groups $ Q $
16 KB (2,475 words) - 19:16, 18 January 2024
in connection with his theory of the [[Hypergeometric series|hypergeometric series]], but had been consid
...></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G.J. Heckman, "Root systems and hypergeometric functions II" ''Compositio Math.'' , '''64''' (1987) pp.
8 KB (1,050 words) - 22:11, 5 June 2020
...ariety of algebraic systems (cf. [[Algebraic systems, variety of|Algebraic systems, variety of]]), a variety of groups can also be defined by the property of
...roups has the power of the continuum. For examples of infinite independent systems of identities see [[#References|[9]]]. A product of finitely-based varietie
9 KB (1,421 words) - 08:28, 6 June 2020
...mation theory [[#References|[a11]]], physics [[#References|[a14]]], coding theory (cf. also [[Coding and decoding|Coding and decoding]]) [[#References|[a12]]
...y of ovals and hyperovals, the theory of spreads and blocking sets and the theory of combinatorial designs [[#References|[a16]]] (cf. also [[Block design|Blo
7 KB (925 words) - 19:43, 9 November 2014
...stem of identities, or laws (see [[Algebraic systems, variety of|Algebraic systems, variety of]]). Every variety of semi-groups is either periodic, i.e. it co
...are just nine of them. The set of all varieties of semi-groups defined by systems of identities of the type $ w = 0 $
8 KB (1,124 words) - 09:17, 2 April 2023
...s of]]) with singled out equivalent (with respect to the identity mapping) systems $\Sigma$ of generators of the fundamental group $\pi_1(X)$, and in which th
...TD valign="top">[a1]</TD> <TD valign="top"> F.P. Gardiner, "Teichmüller theory and quadratic differentials" , Wiley (1987)</TD></TR>
3 KB (395 words) - 20:20, 22 September 2017
...g has been extended to time-periodic solutions of weakly coupled parabolic systems in [[#References|[a8]]].
...ynamics of Continuous, Discrete and Impulsive Systems (An Internat. J. for Theory and Applications)'' (to appear)</TD></TR><TR><TD valign="top">[a20]</TD> <
8 KB (1,256 words) - 14:43, 16 December 2023
...ich is due to V.V. Nemytskii, was generalized by M.V. Bebutov to dynamical systems defined on an arbitrary metric space; here the condition "xk∞ as k∞"
...1]</TD> <TD valign="top"> V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated fro
2 KB (282 words) - 08:12, 6 June 2020
...special types of uniform asymptotic stability, is extensively used in the theory of stability [[#References|[2]]], [[#References|[3]]], [[#References|[4]]].
...]</TD> <TD valign="top"> N. Rouche, P. Habets, M. Laloy, "Stability theory by Liapunov's direct method" , Springer (1977)</TD></TR></table>
3 KB (365 words) - 18:48, 5 April 2020
...lled Pappus' theorem; it is a special case of the theorem of Pascal in the theory of conic sections (namely, when the conic degenerates to a pair of straight
...theorem in research connected with establishing the independence of axiom systems and logical connections between propositions.
2 KB (344 words) - 17:16, 31 March 2018
...nominator formula had a suitable generalization to the case of affine root systems. For the affine root system of type $ A _ {1} ^ {(1)} $,
...> <TD valign="top"> E. Looyenga, "Invariant theory for generalized root systems" ''Invent. Math.'' , '''61''' (1980) pp. 1–32</TD></TR><TR><TD valign=
7 KB (992 words) - 09:26, 26 March 2023
...Haar condition is called a [[Chebyshev system|Chebyshev system]]. For such systems the [[Chebyshev theorem|Chebyshev theorem]] and the [[De la Vallée-Poussin
...valign="top">[2]</TD> <TD valign="top"> N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)</TD></TR></
3 KB (488 words) - 14:57, 14 February 2020
...P.D. Lax and B. Wendroff [[#References|[a6]]] for solving, approximately, systems of hyperbolic conservation laws; in one space dimension these read:
==Extension to non-linear systems.==
16 KB (2,341 words) - 19:00, 26 March 2023
...any applications to quantificational logic [[#References|[a6]]]. A related theory of polyadic algebras, due to P.R. Halmos [[#References|[a5]]], emphasizes o
...and A. Tarski [[#References|[a7]]], who extended the Stone representation theory that embeds a Boolean algebra $ \mathbf B $
8 KB (1,220 words) - 19:12, 11 December 2020
...) could not be measured, which resulted in quantitative conclusions of the theory not in proper correspondence with observations, so Copernicus was forced to
...osed, in the second half of the 17th century, a dynamic model of the solar systems based on the law of universal gravitation (cf. [[Newton laws of mechanics|N
8 KB (1,318 words) - 17:29, 7 February 2011
[[Category:Classical measure theory]]
In correspondence with the usual tendency in measure theory to ignore sets of measure zero, there is (and is primarily used) a "modulo
8 KB (1,175 words) - 05:58, 15 May 2012
...(as well as their properties) naturally generalize to the case of infinite systems of nested sub-semi-groups. In particular, an ascending ideal series in a se
...p">[2]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc. (1961–1967)</TD></TR></
2 KB (370 words) - 17:11, 14 February 2020
...s have important applications in statistics, signal processing and systems theory. For such matrices there are different algorithms (N. Levison, I. Schur and
...ing operators may be analyzed in terms of methods from mathematical system theory (see [[#References|[a1]]]).
6 KB (899 words) - 20:27, 17 January 2024
...tum theory, both in theoretical and in simulation studies (of quantum spin systems, e.g).
2 KB (344 words) - 21:49, 3 December 2017
Bessel's inequality, Parseval's equality for complete systems and also certain other basic properties of orthogonal expansions essentiall
...</TD> <TD valign="top"> A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–196
2 KB (317 words) - 12:37, 6 January 2024
...sociative calculus" was introduced by A.A. Markov, who also developed the theory of associative calculi [[#References|[2]]].
...ariant under isomorphisms; they are the properties of abstract associative systems. Markov [[#References|[2]]], who based himself on his own studies on the re
9 KB (1,391 words) - 18:48, 5 April 2020
...fian problem was solved by E. Cartan in the analytic case in his theory of systems in involution (cf. [[Involutional system|Involutional system]]). The formul
Cartan's theorem has been generalized to arbitrary differential systems given by ideals in the algebra of differential forms on a manifold (the Car
17 KB (2,624 words) - 19:27, 9 January 2024
...ncyclopedia/old_files/data/U095/U.0905650 Universal behaviour in dynamical systems
...he transition from simple to chaotic dynamics in one-dimensional dynamical systems (cf. also [[Routes to chaos|Routes to chaos]]). By the example of the famil
12 KB (1,685 words) - 08:27, 6 June 2020
...>[3]</TD> <TD valign="top"> A.A. Feinstein, "Foundations of information theory" , McGraw-Hill (1958)</TD></TR></table>
...ign="top">[a2]</TD> <TD valign="top"> A. Rényi, "A diary on information theory" , Akad. Kiado & Wiley (1987)</TD></TR></table>
3 KB (447 words) - 22:12, 5 June 2020
...not always crucial whether or not it is normalized. None the less, if the systems are normalized, a clearer formulation is possible for certain theorems on t
...</TD> <TD valign="top"> A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , '''1–2''' , Pitman (1981) (Tran
2 KB (321 words) - 18:58, 7 September 2017
...) are also called regular equations (systems). This class of equations and systems was introduced by J.L. Fuchs .
...<TD valign="top">[4]</TD> <TD valign="top"> E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–1
12 KB (1,732 words) - 17:50, 5 May 2024
...valign="top"> N.P. Bhatia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970) pp. 30–36</TD></TR></table>
2 KB (330 words) - 08:09, 21 June 2014
...is calculi S1–S5 of [[#References|[1]]] are also known as the Lewis survey systems, [[#References|[a3]]].
.../TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Wójcicki, "Theory of logical calculi" , Kluwer (1988) pp. 154ff</TD></TR></table>
2 KB (356 words) - 16:08, 17 April 2014
A class of dynamical systems (cf. [[Dynamical system|Dynamical system]]). An example is the flow generat
...ri with irrational windings are also frequently encountered in Hamiltonian systems sufficiently close to integrable ones (this problem is closely connected wi
12 KB (1,841 words) - 18:33, 5 June 2020
...ical physics reduce to linear hyperbolic partial differential equations or systems of equations.
...tions and systems, which are sometimes called fully hyperbolic systems, or systems, hyperbolic in the narrow sense. A system (1) is called a strictly hyperbol
20 KB (2,792 words) - 08:50, 7 January 2022
...stationary processes has important applications in the theory of dynamical systems.
Of the many possible generalizations of the concept of entropy in information theory one of the most important is the following. Let $ \xi $
7 KB (1,067 words) - 13:06, 6 January 2024
is zero, and the same applies for a class of systems of univalent functions $ \{ {f _ {k} (z) } : {z \in B _ {k} } \} _ {k=0}
of systems $ \{ {f _ {k} (z) } : {f _ {k} (0) = a _ {k} , z \in \Delta } \} _ {k=1
9 KB (1,232 words) - 07:25, 26 March 2023
systems (to which they are related: there is a construction associating an automorp
...I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian)</TD></TR></table>
3 KB (367 words) - 19:38, 5 June 2020
The main area where Young measures have recently been used is optimization theory. Optimization problems where a local, integral cost functional is to be min
...tness method applied to systems of conservation laws" J.M. Ball (ed.) , ''Systems of Nonlinear Partial Differential Equations'' , Reidel (1983) {{MR|072552
7 KB (981 words) - 13:50, 2 July 2020
...of topology. After their introduction it became possible to speak about a theory of approximation of complicated topological and algebraic-topological objec
At the foundation of the entire theory of approximation of topological spaces by polyhedra, or more precisely by s
14 KB (2,029 words) - 19:53, 3 February 2021
...$ near a saddle point looks like a saddle. See also [[Saddle point in game theory]].
...lign="top"> M.W. Hirsch, S. Smale, "Differential equations, dynamical systems, and linear algebra" , Acad. Press (1974) pp. 190ff {{MR|0486784}} {{ZBL
2 KB (346 words) - 18:52, 26 May 2017
====Different systems of coordinates====
...The celebrated dissertation of B. Riemann on the fundamentals of function theory dates to 1851 (see {{Cite|Ri}}).
7 KB (1,063 words) - 12:51, 22 March 2023
$#C+1 = 104 : ~/encyclopedia/old_files/data/R080/R.0800280 Recursive model theory,
''recursively presented model theory''
14 KB (1,981 words) - 08:10, 6 June 2020
In local coordinate systems $( x ^ { i } )$ on $M$ and $( y ^ { \alpha } )$ on $N$, one has
existence theory for harmonic mappings in prescribed homotopy classes (with existence and no
10 KB (1,396 words) - 16:55, 1 July 2020
$#C+1 = 174 : ~/encyclopedia/old_files/data/Q076/Q.0706240 Qualitative theory of differential equations
...the modern theory of stability of motion (see [[Stability theory|Stability theory]]).
29 KB (4,392 words) - 08:08, 6 June 2020
...in a general way with theoretical ideas in the theory of smooth dynamical systems, and in [[#References|[5]]] the results of [[#References|[1]]] were interpr
...12]]], [[#References|[13]]]) and in the verification of which for specific systems, including (*), one must recourse to numerical integration. Correspondingly
10 KB (1,485 words) - 04:11, 6 June 2020
...nsion play an important part in [[Bifurcation|bifurcation]] theory and the theory of [[Singularities of differentiable mappings|singularities of differentiab
...simple examples (like the one above) can easily be analyzed by elimination theory, where the ground field may be completely arbitrary (usually algebraically
12 KB (1,758 words) - 00:29, 13 January 2017
...ic to a group, is also isomorphic to this group. For this reason, in group theory the concept of isotopy is not used: For groups isotopy and isomorphism coin
...valign="top">[3]</TD> <TD valign="top"> O. Boruvka, "Foundations of the theory of groupoids and groups" , Wiley (1976) (Translated from German)</TD></TR
3 KB (454 words) - 19:12, 7 January 2016
In the more abstract theory of dynamical systems (cf. [[Dynamical system|Dynamical system]]), a trajectory usually means the
2 KB (357 words) - 16:50, 15 September 2014
...y of the structure of connected compact Lie groups is a basic topic in the theory of Lie groups.
...nnected compact Lie groups play an important role in the general structure theory of compact Lie groups.
9 KB (1,377 words) - 04:00, 9 May 2022
...the development of an analytical method for modeling and designing hybrid systems is to employ a max-plus algebra model of the manufacturing work-cell and ex
...edom cannot be independently actuated, one refers to this class of robotic systems as underactuated manipulators. Denoting by $q_1$ the actuated generalized c
12 KB (1,887 words) - 16:55, 1 July 2020
...hy problem]] for the [[BBGKY hierarchy|BBGKY hierarchy]] for many-particle systems interacting via short-range potentials in the Boltzmann–Grad limit can be
...C. Cercignani, "On the Boltzmann equation for rigid spheres" ''Transp. Theory Stat. Phys.'' , '''2''' (1972) pp. 211–225</TD></TR><TR><TD valign="top
3 KB (408 words) - 18:51, 24 March 2012
$#C+1 = 111 : ~/encyclopedia/old_files/data/R081/R.0801130 Relativity theory
...theory of]]. Relativity theory is also often called Einstein's relativity theory, after A. Einstein who created it (see [[#References|[1]]], [[#References|[
22 KB (3,285 words) - 08:10, 6 June 2020
...ence it plays an important role in the theory of reductions of Hamiltonian systems.
...trong tendency to have a convex image, and is important for representation theory, see [[#References|[a2]]] and [[#References|[a8]]]. There is also a recent
6 KB (935 words) - 10:58, 2 July 2020
For each of the systems $ J $,
...omain (see [[#References|[3]]]). The propositional fragment of each of the systems $ J $,
10 KB (1,489 words) - 19:17, 12 January 2021
...D valign="top"> A.B. Katok, Ya.G. Sinai, A.M. Stepin, "Theory of dynamical systems and general transformation groups with invariant measure" ''J. Soviet Math.
...periodic points under homomorphisms of dynamical systems" ''Math. Systems Theory'' , '''6''' (1972) pp. 26–36 {{MR|0301718}} {{ZBL|0229.54035}} </TD></TR>
10 KB (1,453 words) - 17:20, 27 October 2014
Mathematical models in combustion theory consist of a number of coupled partial differential equations (cf. [[Differ
...f the main problems is the non-linearity due to the reaction kinetics. The systems of partial differential equations describing combustion can be studied usin
3 KB (400 words) - 17:25, 7 February 2011
...in, "Oscillation matrices and kernels and small vibrations of mechanical systems" , Dept. Commerce USA. Joint Publ. Service (1961) (Translated from Russia
...op">[a2]</TD> <TD valign="top"> F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , '''2''' , Chelsea, reprint (1959) pp. Chapt. XIII, §9 (T
2 KB (349 words) - 16:31, 4 November 2014
...equalities commonly provide bounds on the strength of correlations between systems which are no longer interacting but have interacted in their past.
...e briefly described. A source provides an ensemble of identically prepared systems, one after another, and, as part of the preparation, splits each system int
7 KB (958 words) - 10:58, 29 May 2020
...systems, which in terms of local coordinates is equivalent to the study of systems of ordinary differential equations, usually involves analytical methods.
theory [[#References|[1]]], which is the analogue of $ K $-
7 KB (1,066 words) - 06:29, 26 March 2023
...ces|[a21]]], [[#References|[a22]]] for general surveys of linear filtering theory.
...metrization, and this has led to extensive research on characterizing such systems and exploring the connection, uncovered by R.W. Brockett and J.M.C. Clark [
15 KB (2,302 words) - 08:03, 21 January 2024
are the time, be the inertial reference systems of the source of an electromagnetic wave and of the observer. Further, let
...[1]</TD> <TD valign="top"> L.D. Landau, E.M. Lifshitz, "The classical theory of fields" , Addison-Wesley (1951) pp. Chapt. 6 (Translated from Russian
3 KB (399 words) - 19:36, 5 June 2020
...icit-function theorem in non-linear functional analysis and on the general theory of linear problems of corresponding type.
...ed) problems for non-linear parabolic, or hyperbolic, equations this local theory makes it possible to establish the solvability of the problem on a sufficie
30 KB (4,331 words) - 16:42, 20 January 2022
.../TR><TR><TD valign="top">[8]</TD> <TD valign="top"> G.R. Kempf, "Linear systems on homogeneous spaces" ''Ann. of Math.'' , '''103''' (1976) pp. 557–59
7 KB (1,158 words) - 17:45, 4 June 2020
If one introduces two orthogonal coordinate systems with origin at $ 0 $,
...t [[Non-holonomic systems|non-holonomic systems]]. Conversely, Hamiltonian systems are also encountered in several problems in physics.)
27 KB (4,058 words) - 19:36, 5 June 2020
...n of the input-output behaviour of non-linear systems, especially bilinear systems); cf. the collection [[#References|[a1]]] for a first idea.
6 KB (1,093 words) - 08:26, 16 March 2023
...nikov's theorem, there exist uncountably many non-periodic trajectories in systems that display a homoclinic orbit bi-asymptotic to a saddle-focus at $ O $,
...ntal evidences for Shil'nikov homoclinic chaos in non-equilibrium chemical systems are obtained in [[#References|[a7]]], [[#References|[a8]]].
15 KB (2,108 words) - 07:39, 26 March 2023
...atisfy certain compatibility conditions. Generally, almost all known Euler systems satisfy the condition ES) described below. Let $K$ be a number field. Fix a
...tion connected with the corresponding arithmetic object. In this way Euler systems establish (the sought for) relations between arithmetic objects and corresp
19 KB (2,901 words) - 17:41, 25 November 2023
is the length of the boundary layer. All systems of type (1) for which the conditions a) and b) are satisfied simultaneously
...differential systems, lead to an increase in stiffness. Stiff differential systems require special methods of solution.
30 KB (4,292 words) - 05:40, 24 February 2022
...use of methods of mathematical logic led to the consideration of algebraic systems.
The expression "universal algebra" is often used in the sense of "the theory of universal algebras" .
11 KB (1,662 words) - 04:57, 24 February 2022
...riteria relevant to the exactness of reproducibility of information in the theory of information transmission are usually treated statistically, by isolating
...<TD valign="top">[2]</TD> <TD valign="top"> T. Berger, "Rate distortion theory" , Prentice-Hall (1971)</TD></TR></table>
4 KB (536 words) - 22:12, 5 June 2020
...ce in which the modulus is a prime number. A distinguishing feature of the theory of congruences modulo a prime number is the fact that the residue classes m
...prime fields and algebraic-geometric methods, as well as methods of number theory, can be used to study them.
7 KB (1,033 words) - 17:46, 4 June 2020
One of the methods of [[Ergodic theory|ergodic theory]]. Any automorphism $ T $
...aces {{Cite|Ko}}. They served in the construction of a number of dynamical systems with unexpected metric properties {{Cite|KS}}, {{Cite|ACS}}, {{Cite|S}}, or
8 KB (1,151 words) - 12:12, 21 March 2022
...="top"> Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Tra
...ations of white noise as the limit of "wide bandwidth" noise in physical systems and [[#References|[a2]]] for the relationship between differential equation
7 KB (1,011 words) - 08:29, 6 June 2020
...gn="top">[3]</TD> <TD valign="top"> , ''Computational methods in transport theory'' , Moscow (1969) (In Russian)</TD></TR></table>
...a3]</TD> <TD valign="top"> K.M. Case, P.F. Zweifel, "Linear transport theory" , Addison-Wesley (1967)</TD></TR></table>
4 KB (524 words) - 19:35, 5 June 2020
The first significant results in the theory of equations (4) were obtained in [[#References|[11]]], where an effective
...dity of the majority of results listed above has also been established for systems of equations of type (4); however, in contrast to the case of a single equa
32 KB (4,357 words) - 17:14, 7 February 2011
..., G.G. Stokes proposed the existence of periodic wave-trains in non-linear systems. In the case of waves on deep water, the first two terms in the asymptotic
...<TD valign="top"> H.C. Yuen, B.M. Lake, "Nonlinear deep water waves: Theory and experiment" ''Phys. Fluids'' , '''18''' (1975) pp. 956–960</TD></T
7 KB (969 words) - 10:58, 29 May 2020
...lopedia/old_files/data/B017/B.0107390 Boundary value problems in potential theory
...al theory are primarily boundary value problems for elliptic equations and systems (cf. [[Boundary value problem, elliptic equations|Boundary value problem, e
5 KB (673 words) - 06:29, 30 May 2020
...autonomous systems along a periodic (almost periodic) solution are linear systems of differential equations with periodic coefficients (respectively, with al
is invariant relative to the dynamical systems induced by them; then (cf. [[#References|[7]]]):
15 KB (2,177 words) - 16:07, 5 February 2022
...connection between knot and braid theory on the one hand and quantum field theory and statistical mechanics on the other, cf. e.g. [[#References|[a9]]].
...Yang–Baxter equation has much to do with completely-integrable Hamiltonian systems (cf. [[Hamiltonian system|Hamiltonian system]]; [[Soliton|Soliton]]), cf. [
10 KB (1,364 words) - 08:29, 6 June 2020
...rgodic theory. In the latter one considers various properties of dynamical systems in a measure space $ ( X , \mathfrak B , \mu ) $
...ce of invariant measures with various interesting properties for dynamical systems, depending on the type of the latter.
10 KB (1,584 words) - 22:13, 5 June 2020
...n theory: denumerable classes). In some cases, influenced by axiomatic set theory (see 2)), the term "class" is used to emphasize the fact that the given c
...standpoint of von Neumann). See also [[Axiomatic set theory|Axiomatic set theory]].
10 KB (1,586 words) - 09:40, 26 March 2023
...eter has been worked-out sufficiently well. Originally it was proposed for systems of algebraic and transcendental equations, integral equations, ordinary and
...d problems in linear algebra, or for proving the solvability of non-linear systems connected with variational problems and the construction of solutions of th
12 KB (1,799 words) - 08:05, 6 June 2020
...sc $|z| \le 1$, with at least one lying on the boundary (cf. also [[Galois theory]]). One should compare this definition with that of a [[Pisot number]]. The
...alysis]], dynamical systems theory (cf. also [[Dynamical system]]) and the theory of quasi-crystals, cf. also [[Pisot number]].
3 KB (504 words) - 18:43, 14 April 2023
The term "oscillator" is used in relation to (mechanical or physical) systems with a finite number of degrees of freedom whose motion is oscillatory (e.g
...gn="top">[1]</TD> <TD valign="top"> L.I. Mandel'shtam, "Lectures on the theory of oscillations" , Moscow (1972) (In Russian)</TD></TR><TR><TD valign="to
2 KB (359 words) - 11:41, 3 August 2014
...elf-consistent (compatible) systems forms the subject of the compatibility theory of partial differential equations.
...top">[10]</TD> <TD valign="top"> N.N. Yanenko, B.L. Rozhdestvenskii, "Systems of quasilinear equations and their applications to gas dynamics" , Amer. Ma
15 KB (2,114 words) - 17:33, 5 June 2020
In the theory of foliations one can also introduce a Poincaré return map (see [[#Referen
...erentiable dynamical systems, see [[#References|[a4]]], Sect. VI.2. In the theory of foliations one can recover the generalization of the Poincaré return ma
9 KB (1,357 words) - 08:06, 6 June 2020
...In a wider sense the Lebesgue constants are defined for other orthonormal systems (cf. [[Orthogonal system|Orthogonal system]]) as the quantities
...y an important role in questions of convergence of Fourier series in these systems. The Lebesgue constants were introduced by H. Lebesgue (1909). See also [[L
7 KB (982 words) - 20:31, 15 November 2023
...D valign="top"> G.I. Marchuk, V.I. Lebedev, "Numerical methods in the theory of neutron transport" , Harwood (1986) (Translated from Russian)</TD></TR
..."top">[a5]</TD> <TD valign="top"> W.C. Rheinboldt, "Methods for solving systems of nonlinear equations" , SIAM (1970)</TD></TR><TR><TD valign="top">[a6]</
8 KB (1,124 words) - 11:46, 17 June 2020
...p) may be found in [[#References|[a1]]]. The matrix-valued version of this theory, which is due to [[#References|[a2]]] (see also [[#References|[a3]]]), is m
...employing a state-space method which is connected to mathematical systems theory (see [[#References|[a4]]], [[#References|[a5]]] and [[Integral equation of
8 KB (1,085 words) - 08:29, 6 June 2020
...(from their own specific standpoints). By revealing the general aspects in systems of such different nature, cybernetics provides a general and moreover new m
In many specific cybernetic systems, time is regarded as a parameter taking only a discrete set of values (usua
35 KB (5,424 words) - 11:17, 26 March 2023
$#C+1 = 79 : ~/encyclopedia/old_files/data/R081/R.0801180 Reliability theory
...ost, etc.) are random. Other widely used methods are those of optimization theory, mathematical logic, etc.
16 KB (2,542 words) - 08:10, 6 June 2020
...tistical aggregate]]), which is defined as a set of a very large number of systems, all dynamically identical with the system under consideration. The ensembl
...ng the equilibrium behaviour of a wide range of both classical and quantum systems [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]].
9 KB (1,357 words) - 16:59, 1 July 2020
...[Lie transformation group|Lie transformation group]]) that leave invariant systems of differential equations. As a result of Lie's work [[#References|[a2]]],
...rrors, if done with pencil and paper. The availability of computer algebra systems (such as Mathematica or Maple) has changed all that. There now exist many s
23 KB (3,178 words) - 22:16, 5 June 2020
For lacunae in function theory see e.g. [[Hadamard theorem|Hadamard theorem]] on gaps; [[Fabry theorem|Fab
A lacuna in the theory of partial differential equations is a subdomain $ D $
6 KB (898 words) - 14:30, 8 January 2022
..."top">[7]</TD> <TD valign="top"> B.L. Rozhdestvenskii, N.N. Yanenko, "Systems of quasilinear equations and their applications to gas dynamics" , Amer. Ma
7 KB (987 words) - 14:03, 24 December 2020
''in graph theory''
...aphs, trees are suitable models for the study of various problems in graph theory. Any tree with $n$
6 KB (1,006 words) - 06:20, 30 March 2023
A logico-mathematical calculus which formalizes elementary number theory. The language of the most common kind of formal arithmetic contains the con
...ivation of the theorems presented in standard courses on elementary number theory. At the time of writing (the 1980's) it would appear that no significant nu
12 KB (1,840 words) - 19:36, 9 February 2024
...as well as from theoretical interest. See also [[Queueing theory|Queueing theory]].
...analysis 1: A foundation of performance evaluation: Vacation and priority systems" , North-Holland (1991) pp. Chap. 3</td></tr>
21 KB (3,138 words) - 20:48, 11 February 2024
[[Category:Distribution theory]]
...ins with continuous time. Similarly, one uses Markov chains in reliability theory, where the fault-free operating times of the individual devices can often b
3 KB (485 words) - 19:23, 10 April 2024
A special topic in scheduling theory, in which unit-execution-time (UET) systems are studied. In such a system, all jobs (tasks) or its operations have an e
3 KB (392 words) - 17:16, 7 February 2011
$#C+1 = 481 : ~/encyclopedia/old_files/data/P110/P.1100120 Pesin theory
...olic behaviour, or simply the theory of non-uniformly hyperbolic dynamical systems.
39 KB (5,692 words) - 14:54, 7 June 2020
The method of degenerate kernels may be applied to systems of integral equations of the type (1), to multi-dimensional equations with
...a3]</TD> <TD valign="top"> I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="
3 KB (401 words) - 17:32, 5 June 2020
The theory of the Chebyshev iteration methods (2), (3) can be extended to partial eige
...D valign="top"> G.I. Marchuk, V.I. Lebedev, "Numerical methods in the theory of neutron transport" , Harwood (1986) (Translated from Russian)</TD></TR
16 KB (2,300 words) - 19:29, 17 January 2024
at least one bounded solution on $ [ 0 , \infty ) $[[#References|[1]]]. The theory of dichotomy [[#References|[2]]], transferred to equations in Banach spaces
...tiplicative ergodic theorem. Characteristic Lyapunov numbers for dynamical systems" ''Trans. Moscow Math. Soc.'' , '''19''' (1969) pp. 197–232 ''Trudy M
3 KB (367 words) - 17:33, 5 June 2020
...="top">[a4]</TD> <TD valign="top"> V.M. Kopytov, N.Ya. Medvedev, "The theory of lattice-ordered groups" , Kluwer Acad. Publ. (1994) (In Russian)</TD><
15 KB (2,061 words) - 17:13, 7 February 2011
[[Category:Ergodic theorems, spectral theory, Markov operators]]
One of the most important theorems in [[Ergodic theory|ergodic theory]]. For an endomorphism $ T $ of a $ \sigma $-finite measure space $ (X,\Sig
3 KB (460 words) - 05:51, 29 November 2016
...system is usually made up of elements which are themselves simple control systems. Formerly synthesis meant that the structure of the elements, a rule for co
Every class of control systems gives rise in a natural way to a specific class of functions. A synthesis p
27 KB (4,142 words) - 14:55, 7 June 2020
..."top">[1]</TD> <TD valign="top"> L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) {{ZBL|0137.02001}}</TD></TR>
...">[a2]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc. (1961–1967)</TD></TR>
3 KB (413 words) - 09:17, 2 April 2023
The first studies of systems such as (1) were concerned with the initial value problem
...are also encountered in various domains of physics and technology: in the theory of non-linear oscillations, hydrodynamics, celestial mechanics, quantum mec
18 KB (2,607 words) - 18:34, 5 June 2020
...he solution of the high-speed problem for the system (11) and other linear systems (see [[#References|[3]]] and [[#References|[8]]]).
...n="top">[12]</TD> <TD valign="top"> T.K. Sirazetdinov, "Optimization of systems with distributed parameters" (1977) (In Russian)</TD></TR><TR><TD valign=
22 KB (3,158 words) - 19:38, 5 June 2020
The principal problem in the theory of congruence equations is the number of solutions of a given congruence. I
...f algebraic geometry are also used in the study of congruence equations or systems of such congruences.
8 KB (1,148 words) - 17:46, 4 June 2020
...ntially the character group of the dyadic group. (This connection made the theory of Walsh functions a special case of the general study of [[Harmonic analys
...alsh system: J.J. Price [[#References|[a2]]] proved that among orthonormal systems whose functions $f_n$ alternate sign on finer and finer partitions of $[0,1
3 KB (467 words) - 09:42, 27 November 2018
...ns in mathematical analysis are all equations of the type (1), as are also systems of algebraic equations, both finite and infinite, finite-difference equatio
...tems of compatible functional equations. Moreover, functional equations or systems of functional equations can contain a greater number of essential, essentia
19 KB (2,803 words) - 19:40, 5 June 2020
In ergodic theory, Bernoulli automorphisms (or, more exactly, the cascades generated by itera
...I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian)</TD></TR></table>
4 KB (566 words) - 10:58, 29 May 2020
.... Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR></table>
4 KB (599 words) - 08:24, 6 June 2020
| This article ''Queueing Theory'' was adapted from an original article by U. Narayan Bhat, which appeared i
<center>'''Queueing Theory'''</center>
20 KB (3,169 words) - 11:35, 3 April 2016
...who applied it to various problems in [[Approximation theory|approximation theory]] (e.g., Kolmogorov widths, $\varepsilon$-entropy, etc.). Subsequently the
...has also been applied to other series expansions, e.g., multiple wavelets systems.
3 KB (472 words) - 01:04, 15 February 2024
[[Category:Ergodic theorems, spectral theory, Markov operators]]
|valign="top"|{{Ref|VY}}|| A.M. Vershik, S.A. Yuzvinskii, "Dynamical systems with invariant measure" ''Progress in Math.'' , '''8''' (1970) pp. 151–21
2 KB (340 words) - 08:12, 9 January 2024
...exttt{false}$. In the strict case $q = \texttt{false}$ is excluded. A Horn theory is a set of Horn clauses.
...si-varieties in [[Universal algebra]] (cf. also [[Quasi-variety]]; [[Model theory]]). Model-theoretic properties of Horn theories which allow arbitrary quant
7 KB (1,111 words) - 06:59, 21 October 2016
[[Intersection theory|intersection theory]] provides a satisfactory modern framework. Enumerative geometry deals with
{{Cite|Fu2}} for a complete reference on intersection theory; for historical surveys and a discussion of enumerative geometry, see
8 KB (1,263 words) - 08:49, 30 March 2012
...<TD valign="top"> Ya.Z. Tsypkin, "Foundations of the theory of learning systems" , Acad. Press (1973) (Translated from Russian)</TD></TR></table>
...ark, "Stochastic approximation methods for constrained and unconstrained systems" , Springer (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top
9 KB (1,349 words) - 08:23, 6 June 2020
...bjects, such as complex spaces, topological manifolds, algebras, etc. Such systems occur in may parts of mathematics both as tools and as objects to be invest
parameter systems of smooth curves $ \{ \sigma ^ {1} \dots \sigma ^ {n} \} $
20 KB (3,056 words) - 08:25, 6 June 2020
...same set of strategies (cf. [[Strategy (in game
theory)|Strategy (in game
theory)]]) and the same pay-off function. (Important texts in this area are [[#Ref
...quilibrium (cf. also [[Nash theorem (in game
theory)|Nash theorem (in game
theory)]]). It follows that <img align="absmiddle" border="0" src="https://www.enc
28 KB (3,866 words) - 06:43, 27 September 2016
...], in dependence on which they take different names. For example, with the theory of uniqueness for lacunary trigonometric series there is associated the con
3 KB (452 words) - 12:33, 6 January 2024
...g [[dynamic programming]]), computer science, automata and formal language theory, numerical methods, [[parallel programming]], etc. (cf. also [[Idempotent a
...Green, "Minimax algebra" , ''Lecture Notes in Economics and Mathematical Systems'' , '''166''' , Springer (1979) {{ISBN|3-540-09113-0}} {{ZBL|0399.90052}}<
7 KB (1,026 words) - 20:42, 16 November 2023
...ther fixed or periodic (cf. [[Poincaré–Bendixson theory|Poincaré–Bendixson theory]]).
...[1]]], [[#References|[3]]]). A generalization of this theorem to dynamical systems given on a space of infinite measure is the Hopf recurrence theorem (cf. [[
4 KB (535 words) - 16:59, 23 March 2023
[[Category:Ergodic theory]]
...see {{Cite|Fo}}, {{Cite|B}}). There are also similar results for dynamical systems (at present only cascades and flows) in appropriate measurable spaces (cf.
9 KB (1,352 words) - 10:41, 9 November 2014
...r–Planck equation is relevant: for charged particles and for gravitational systems. When dealing with a self-consistent force field $F$, one obtains in both c
...<TD valign="top"> T. Padmanabhan, "Statistical mechanics of gravitating systems" ''Phys. Rep.'' , '''188''' (1990) pp. 285–362</TD></TR></table>
6 KB (900 words) - 08:31, 22 August 2014
...pplied disciplines, such as theoretical programming, graph theory, systems theory, and mathematical linguistics.
...><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian) {{MR|}} {{ZBL|0266.08001}} </T
8 KB (1,265 words) - 17:31, 5 June 2020
...sification of reduced root systems (respectively, irreducible reduced root systems).
Simple Lie algebras that correspond to root systems of types $ A $ – $ D $ are said to be classical and have the followin
25 KB (3,293 words) - 19:24, 26 March 2023
...ulated as the basis of the theory, while the remaining propositions of the theory are obtained as logical consequences of these axioms.
...ass of specific problems which eventually established [[Proof theory|proof theory]] as one of the main chapters of modern mathematical logic.
17 KB (2,707 words) - 10:02, 25 April 2020
...relationship between the signs is. In this way language resembles abstract systems studied in mathematics, and it can therefore be studied using mathematical
...es|[5]]]), although there is as yet (1984) no sufficiently-complete formal theory of phonemes. Also, in the area of distinctive signs, the minimal meaningful
8 KB (1,194 words) - 08:24, 6 June 2020
...tic equation|Quadratic equation]]; [[Cubic equation|Cubic equation]]). The theory of the solution of quadratic equations was first expounded in the book Arit
...e solvability of algebraic equations by radicals in [[Galois theory|Galois theory]] can be stated as follows. Let $ f(x) $
18 KB (2,778 words) - 16:09, 1 April 2020
for arbitrary complete orthonormal systems $ \{ \phi _ {j} \} $,
...align="top">[a3]</TD> <TD valign="top"> N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman (1981) (Tran
3 KB (501 words) - 22:10, 5 June 2020
...mentary flow|elementary flow]] describing the demand flow in many queueing systems. The distributions of the random variables $ \tau _ {n} - \tau _ {n-} 1 $
...p"|{{Ref|GSY}}|| I.I. Gikhman, A.V. Skorokhod, M.I. Yadrenko, "Probability theory and mathematical statistics" , Kiev (1979) (In Russian) {{MR|2026607}} {{ZB
3 KB (480 words) - 08:06, 6 June 2020
...lign="top">[a3]</TD> <TD valign="top"> E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. 323</TD></TR
3 KB (398 words) - 18:47, 5 April 2020
...s by many mathematicians and experts in celestial mechanics on Hamiltonian systems which are close to being completely integrable, the term "adiabatic invari
...escribed in fact belong to the field of asymptotic methods of perturbation theory (if the problem is posed in a more general manner, a few rigorous results c
7 KB (1,019 words) - 16:09, 1 April 2020
.... Neither does a general decision procedure exist for finding solutions of systems of such Diophantine equations in the ring of integers of any number field o
More precisely, Skolem asked whether the [[Elementary theory|elementary theory]] of the ring of all algebraic integers $ {\widetilde{\mathbf Z} } $
12 KB (1,775 words) - 11:58, 4 April 2020
...sufficient number of particles but also form quasi-homogeneous statistical systems.
...</TD> <TD valign="top"> C. Cercignani, "Mathematical methods in kinetic theory" , Plenum (1969)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">
4 KB (525 words) - 20:13, 12 October 2014
...e [[Integrals in involution|integrals in involution]], and the Hamiltonian systems that they generate (known as higher Korteweg–de Vries equations) are comp
...the latter constitute completely-integrable finite-dimensional Hamiltonian systems. Any periodic potential can be approximated by a finite-gap potential. Let
14 KB (2,037 words) - 20:24, 16 January 2024
...es, a theory of areas, etc. The theory of areas of polygons underlying the theory of measurements of areas in a non-Archimedean plane is based on the concept
4 KB (583 words) - 15:04, 9 April 2014
...ined for number fields in connection with the development of [[class field theory]]. The general form of the reciprocity law is formulated in terms of Brauer
...dditive group of integers. This fact is of importance in local class-field theory.
7 KB (1,232 words) - 12:12, 30 December 2015
...f. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]), several approaches to the establishment of sufficient conditions for
...or optimality are permissible for the more general cases of non-autonomous systems, problems with Mayer- and Bolza-type functions (see [[Bolza problem|Bolza p
10 KB (1,504 words) - 08:04, 6 June 2020
The representation theory of quivers has been developed in order to deal effectively with certain typ
In general, the representation theory of the $n$-subspace quiver
12 KB (1,755 words) - 19:50, 21 March 2023
A term used in the qualitative theory of differential equations, used for several closely
...tructure which, nevertheless, can be described in terms of [[Sector in the theory of ordinary differential equations|sectors]] of three different type (ellip
8 KB (1,340 words) - 09:03, 12 December 2013
in the sense of every normalized invariant measure of shift dynamical systems (cf. [[Shift dynamical system|Shift dynamical system]]), $ S = \mathop{\
...TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.A. Izobov, "Linear systems of ordinary differential equations" ''J. Soviet Math.'' , '''5''' : 1 (1
8 KB (1,097 words) - 16:43, 4 June 2020
...x of $D$ (cf. also [[Index formulas|Index formulas]]; [[Index theory|Index theory]]). For elliptic Lie equations the index can be expressed in terms of chara
...td valign="top">[a6]</td> <td valign="top"> D. Spencer, "Overdetermined systems of linear partial differential operators" ''Bull. Amer. Math. Soc.'' , '''
8 KB (1,101 words) - 17:44, 1 July 2020
<TR><TD valign="top">[3]</TD> <TD valign="top"> A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)</TD></TR>
A further meaning of the word "filter" occurs in the theory of (partially observed) stochastic processes, cf. [[Stochastic processes, f
3 KB (602 words) - 17:00, 25 September 2017
Solution of linear systems of equations. Given a matrix $A \in \mathbf{R} ^ { n \times n }$ and a vect
...sparse matrices or compute eigenvalues of sparse matrices or both. Linear systems are either solved by direct sparse methods or by iterative methods. A surve
10 KB (1,478 words) - 17:02, 1 July 2020
...p, local]]) and its Lie algebra. Lie's theorems are the foundations of the theory developed in the 19th century by S. Lie and his school (see ).
...</TR><TR><TD valign="top">[6]</TD> <TD valign="top"> N.G. Chebotarev, "The theory of Lie groups" , Moscow-Leningrad (1940) (In Russian) {{MR|}} {{ZBL|}} </TD
8 KB (1,148 words) - 10:21, 16 December 2019
in a complex domain is connected to the theory of univalent functions. See [[#References|[a8]]], [[#References|[a5]]] and
...njugacy has also been studied for certain second-order linear differential systems of higher dimension [[#References|[a1]]], [[#References|[a9]]]. In the hist
8 KB (1,195 words) - 07:36, 23 January 2024
...with a free boundary in the plane, extensive use is made of methods of the theory of functions of a complex variable.
...ign="top"> A. Fasano (ed.) M. Primicerio (ed.) , ''Free boundary problems: theory and application'' , '''1–2''' , Pitman (1983) {{MR|0714899}} {{ZBL|}} </T
8 KB (1,055 words) - 17:33, 5 June 2020
...I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian)</TD></TR>
.../TD> <TD valign="top"> S. Lefschetz, "Differential equations: geometric theory" , Interscience (1957)</TD></TR>
3 KB (502 words) - 07:59, 21 March 2023
For systems in which $ \{ \tau _ {j} ^ {s} - \tau _ {j} ^ {e} \} \in G _ {I} $
Then, for the systems considered,
24 KB (3,620 words) - 16:42, 30 December 2020
.../TD> <TD valign="top"> N.N. Moiseev, "Elements of the theory of optimal systems" , Moscow (1975) (In Russian)</TD></TR></table>
3 KB (447 words) - 10:39, 31 August 2014
...ry relations, semi-groups of subsets (or subsystems of different algebraic systems, for example ideals in rings and semi-groups), etc. Every ordered semi-grou
...formed by the ordered groups (cf. [[Ordered group|Ordered group]]); their theory forms an independent part of algebra. In distinction to ordered groups, the
11 KB (1,676 words) - 14:07, 17 March 2020
...the Nielsen transformations over the system of free generators lead to new systems of free generators, and any system of free generators can be obtained from
...lign="top"> W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)
4 KB (572 words) - 19:40, 5 June 2020
...[[Model
theory|model
theory]] studying the [[Elementary
theory|elementary
theory]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org
...Khovanskii's finiteness theorem [[#References|[a5]]], it follows that this
theory is [[O-minimal|<img align="absmiddle" border="0" src="https://www.encyclope
9 KB (1,282 words) - 17:26, 22 February 2013
by virtue of the systems (6), (10) until the control $ u $
...ons and the necessary conditions for the optimality of moving and singular systems" ''USSR Comp. Math. Math. Phys.'' , '''7''' : 2 (1967) pp. 22–54 ''Z
17 KB (2,522 words) - 20:03, 12 January 2024
One of the problems in the theory of information transmission (cf. [[Information, transmission of|Information
...<TD valign="top">[4]</TD> <TD valign="top"> T. Berger, "Rate distortion theory" , Prentice-Hall (1971)</TD></TR></table>
8 KB (1,081 words) - 22:12, 5 June 2020
$#C+1 = 184 : ~/encyclopedia/old_files/data/A014/A.0104310 Axiomatic set theory
...ry aiming at the construction of some fragment of informal ( "naive" ) set theory.
22 KB (3,585 words) - 17:07, 25 April 2020
This problem has been greatly generalized in the theory of stationary stochastic processes (cf. [[Stationary stochastic process|Sta
For systems of stochastic differential equations the problem of interpolation of some c
6 KB (777 words) - 08:23, 6 June 2020
The Vapnik–Chervonenkis dimension plays an important role in learning theory, especially in probably approximately correct ([[PAC]]) learning. Thus, lea
...Krizanc, B. Ruf, J. Urrutia, G. Wöginger, "The VC-dimension of set systems defined by graphs" ''Discr. Appl. Math.'' , '''77''' : 3 (1997) pp. 237
3 KB (443 words) - 12:31, 11 December 2016
...p"> B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytskii, "The theory of Lyapunov exponents and its applications to problems of stability" , Mosc
...ign="top"> F. Verhulst, "Nonlinear differential equations and dynamical systems" , Springer (1989)</TD></TR></table>
3 KB (437 words) - 07:39, 26 February 2022
...th the difference that the transition from geometry to algebra in homology theory is irreversible).
...as, the theory of finite-dimensional algebras, the theory of rings and the theory of quadratic forms.
12 KB (1,885 words) - 23:48, 23 April 2017
...ns, for finitely-presented groups (or algebras), i.e. ones given by finite systems of generators and defining relations, it is possible, using a finite number
...lign="top"> W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)
4 KB (557 words) - 15:18, 7 March 2022
...f. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]), consisting of a synthesis of an optimal control (a feedback synthesi
...tended to problems of optimal synthesis control for discrete (multi-stage) systems, where the corresponding Bellman equation is a finite-difference equation (
32 KB (4,621 words) - 08:04, 6 June 2020
In the theory of rings of differential operators, e.g. Weyl algebras (cf. [[Weyl algebra|
...align="top">[a2]</TD> <TD valign="top"> P. Schapira, "Microdifferential systems in the complex domain" , Springer (1985)</TD></TR></table>
3 KB (480 words) - 16:44, 13 January 2024
...2]</TD> <TD valign="top"> V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated fro
3 KB (470 words) - 08:25, 6 June 2020
...atical linguistics is expanding; its methods have found application in the theory of programming.
...ministic effective systems (algorithms) but to construct non-deterministic systems (calculi), which allow either for a given object at some level to enumerate
14 KB (2,122 words) - 17:14, 3 November 2014
...t in the subsets of the system. The concept of a block design arose in the theory of design (planning) of (statistical) experiments in the 1920s and 1930s, b
The theory of block designs considers problems on the existence and classification as
9 KB (1,344 words) - 16:08, 6 February 2020
...em of the solvability of equations by radicals (see [[Galois theory|Galois theory]]). Jordan introduced the concepts of composition and chief series (cf. [[P
...the Jordan–Hölder theorem have been considered in the language of lattice theory and partially ordered sets. A generalization of Schreier's theorem has been
4 KB (660 words) - 20:00, 11 April 2014
are atlases of their local coordinate systems ( $ \phi _ \alpha : U _ \alpha \rightarrow D _ \alpha \subset \mat
...<TR><TD valign="top">[a2]</TD> <TD valign="top"> S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)</TD></TR></table>
4 KB (590 words) - 11:46, 30 December 2020
...stigation of these models was an important stage in the development of the theory of many-valued logics.
...m the point of view of algebra, models of many-valued logics are algebraic systems (cf. [[Algebraic system|Algebraic system]]) having both applied and theoret
34 KB (5,105 words) - 19:20, 16 January 2024