..."non-classical" constraints of inequality type, in the theory of optimal control, a necessary condition for a strong extremum of the functional (*) is given
...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
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
$#C+1 = 288 : ~/encyclopedia/old_files/data/A014/A.0104090 Automatic control theory
...al apparatus of differential equations and especially an apparatus for the theory of steady motion [[#References|[2]]]; engineers had discovered the idea of
43 KB (6,618 words) - 07:32, 26 March 2023
...the theory of power series, linear operators, random processes, and in the theory of extremal and approximation problems.
...owever, their applications in harmonic analysis are connected not with the theory of Fourier series, but with Fourier transforms.
37 KB (5,073 words) - 18:20, 1 December 2014
...n of a [[Control system|control system]] and of special classes of control systems (contact schemes, diagrams or functional elements), of transition diagrams
Synthesis of electrical networks, or realizability theory, refers to the problem of finding an electrical network with (partly) given
10 KB (1,474 words) - 14:32, 7 June 2020
...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
...f. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]).
Consider the following problem in optimal control: Find a trajectory $ x( t) $
23 KB (3,462 words) - 08:27, 6 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
is only implicit in it — in terms of the control parameter $ \alpha ( t) = \mathop{\rm arg} f _ {t} ( z _ {t} ) $,
...n="top">[6]</TD> <TD valign="top"> V.I. Popov, "Quantization of control systems" ''Soviet Math. Dokl.'' , '''13''' : 6 (1972) pp. 1668–1672 ''Dokl.
8 KB (1,168 words) - 20:13, 12 January 2024
...e one of the possible ways of action, e.g. in solving numerous problems of control and planning of production processes as well as in problems of design and l
At the basis of the theory of convex, and, in particular, linear and quadratic, programming lie the [[
12 KB (2,034 words) - 09:14, 9 June 2016
...ta processing, [[Dynamic programming|dynamic programming]], discrete-event systems, computer science, discrete mathematics, [[Mathematical logic|mathematical
...e leads to a unified approach to various optimization problems and optimal control problems with discrete or continuous state space as well to the correspondi
18 KB (2,598 words) - 22:11, 5 June 2020
...ons occur simultaneously, and the elaboration and fulfillment of plans and control solutions at various levels, including the creation of conditions for the a
...vity of economic systems at various levels; factor models of productivity; systems of demand functions for groups of consumers and specific functions of consu
14 KB (2,040 words) - 19:36, 5 June 2020
...mes (cf. [[Games, theory of|Games, theory of]]). The first studies in this theory appeared in the mid-1950s.
==Formulations of problems in the theory of differential games.==
35 KB (5,293 words) - 20:32, 19 December 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
...estimating the step error is to obtain formulas of Runge–Kutta type with a control term which approximates the principal term of the step error of the method
...p. These methods are suited for the solution of boundary value problems of systems of differential equations. Finite difference analogues of these methods are
32 KB (4,604 words) - 16:55, 7 February 2011
...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
...ions of general Padé approximation problems. In [[#References|[a19]]] this theory is first generalized and then applied to the inversion of (block) Hankel ma
...he theory of Hankel matrices to engineering problems of system and control theory, see [[#References|[a14]]] and [[#References|[a7]]].
14 KB (2,126 words) - 16:45, 1 July 2020
...ounced by R. Thom [[#References|[1]]] in terms of their gradient dynamical systems and the analogous list of critical points of codimension $ \leq 4 $(
...t in an individual object, but in a collection of them, depending on some "control" parameters. Degenerate singularities which are removable for each fixed va
11 KB (1,481 words) - 18:07, 1 June 2023
...a unique definition for each definite difference scheme. For an effective control of the viscosity to be possible, it is advisable to examine classes of diff
..."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
...tural numbers. Based on an elaborate system of verbal calculation, written systems of calculation arose and slowly the methods of doing the four arithmetic op
...mathematical theory. From arithmetic gradually grew [[Number theory|number theory]]. The systematic study of magnitudes and measurements was created. The pro
22 KB (3,256 words) - 21:54, 30 March 2012
...e one of the most important types of control systems (cf. [[Control system|Control system]]). Substantially, a finite automaton may be described as a system w
...Depending on the particular point of view, such system may include formal systems (cf. [[Formal system|Formal system]]), real automata, living organisms, etc
19 KB (2,906 words) - 07:33, 26 March 2023
A test for a given table, with control target $ \mathfrak N $
...plexity. The construction of minimal tests is a central problem of testing theory. The reasons for this can be demonstrated by the example of an absolute tes
22 KB (3,351 words) - 08:25, 6 June 2020
$#C+1 = 56 : ~/encyclopedia/old_files/data/Q076/Q.0706850 Queueing theory
The branch of probability theory in which one studies mathematical models of various kinds of real queues (c
12 KB (1,824 words) - 08:09, 6 June 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
...ve. This is the Vitali–Hahn–Saks theorem. Another striking result from the theory of vector measures is the so-called Nikodým boundedness theorem: For a col
...4]]]); see [[#References|[a5]]] for the role of vector measures in control theory.
10 KB (1,436 words) - 09:13, 6 January 2024
...1]</TD> <TD valign="top"> D.R. Brillinger, "Time series. Data analysis and theory" , Holt, Rinehart & Winston (1975) {{MR|0443257}} {{ZBL|0321.62004}} </
...valign="top">[a4]</TD> <TD valign="top"> L. Ljung, "System identification theory for the user" , Prentice-Hall (1987) {{MR|1157156}} {{ZBL|0615.93004}} </TD
5 KB (740 words) - 19:58, 16 January 2024
...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
==Oscillatory properties of solutions of linear Hamiltonian systems.==
...–(4) are involved in a number of problems in variational calculus, optimum control, studies on the properties of the spectrum of the corresponding differentia
21 KB (3,088 words) - 11:56, 8 March 2022
...f. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]) in fact arose in this manner.
...t this equation is solvable, a special apparatus had to be developed — the theory of the [[Riemann integral|Riemann integral]].
33 KB (4,933 words) - 01:50, 23 January 2022
...( \theta ) = H ^ { 2 } \ominus \theta H ^ { 2 }$. There also has evolved a theory of so-called $C _ { 0 }$ operators, for which the operator $T ( \theta )$ (
...nnections with engineering applications, such as signal processing, robust control and system identification, have led to new questions, different points of v
12 KB (1,802 words) - 17:01, 1 July 2020
...Tucker conditions are applicable to a wide range of problems, from optimal control problems in engineering [[#References|[a5]]], [[#References|[a6]]], [[#Refe
..."> M.S. Bazaraa, H.D. Sherali, C.M. Shetty, "Nonlinear programming: Theory and algorithms" , Wiley (1993) (Edition: Second)</TD></TR>
16 KB (2,514 words) - 17:28, 23 October 2017
...of mathematics it is stimulating research in pure mathematics (e.g. coding theory, $*$-algebras, quantum topology) as well as requiring and providing many op
...ach which led to the basic definitions of computer science and information theory as formulated by A. Church, A.M. Turing, C. Shannon and others in the first
21 KB (3,030 words) - 16:52, 1 July 2020
...<TD valign="top"> Ya.Z. Tsypkin, "Foundations of the theory of learning systems" , Acad. Press (1973) (Translated from Russian)</TD></TR></table>
...ung, "Analysis of recursive stochastic algorithms" ''IEEE Trans. Autom. Control'' , '''AC-22''' (1977) pp. 551–575</TD></TR></table>
9 KB (1,349 words) - 08:23, 6 June 2020
...near functions on sets of an $n$-dimensional [[vector space]] specified by systems of linear inequalities and equations; linear programming is one of the bran
The basis for studying properties of linear programming problems is the theory of duality. The dual problem to the problem \eqref{eq:1}–\eqref{eq:3} is
12 KB (1,882 words) - 22:25, 1 September 2016
...ally, the Lyapunov equation has many applications in stability and control theory; efficient numerical algorithms for solving it are available.
...<td valign="top"> A. Halanay, D. Wexler, "Qualitative theory of pulse systems" , Nauka (1971) (In Russian)</td></tr><tr><td valign="top">[a5]</td> <td
5 KB (876 words) - 16:52, 1 July 2020
...after [[#References|[a3]]] have witnessed an explosive development of the theory of functional-differential equations and their applications (see, e.g. [[#R
...he boundary value problem, the existence of periodic solutions, stability, control, etc. For example, the Cauchy problem (also called the initial problem or b
15 KB (2,182 words) - 12:41, 23 February 2024
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 predicting some compon
6 KB (908 words) - 08:23, 6 June 2020
[[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
...st interested in algorithms of this kind. Fairly effective are interactive systems combining the advantages of the human eye assessments with the computer's a
...R><TD valign="top">[6]</TD> <TD valign="top"> I.V. Sobolev, "Production control in the timber industry" , Petrozavodsk (1976) (In Russian)</TD></TR><TR><
8 KB (1,179 words) - 17:25, 27 October 2014
...-matrix. Together these transformations make up the feedback group (of the
control system <img align="absmiddle" border="0" src="https://www.encyclopediaofmat
...top">[a10]</TD> <TD valign="top"> A. Mead, E. Ruch, A. Schönhofer, "
Theory of chirality functions, generalized for molecules with chiral ligands" ''T
34 KB (4,631 words) - 18:28, 30 November 2016
''theory of programming''
...nvestigating models of programs. The formal combinatorial methods form the theory of program schemes, which studies properties of a program that are invarian
21 KB (3,241 words) - 08:25, 6 June 2020
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
...as well as from theoretical interest. See also [[Queueing theory|Queueing theory]].
....g., a user data field) is served in pre-emptive resume manner because the control information has already been acquired, and the last portion (e.g., a messag
21 KB (3,138 words) - 20:48, 11 February 2024
''in the theory of differential equations''
...tements have been obtained on the basis of the comparison principle in the theory of the stability of motion (see [[#References|[4]]]–[[#References|[6]]]).
8 KB (1,142 words) - 17:45, 4 June 2020
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
...esis problems for antennas and other physical systems); and various other control problems described by differential equations (in particular, differential
...sign of antennas and other systems or constructions, problems of optimal control and many others) are also called synthesis problems.
33 KB (5,246 words) - 00:23, 25 April 2012
$#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
A representing object for a generalized cohomology theory. The notion was introduced in [[#References|[1]]] (cf. also [[Generalized c
...construct the homotopy category of spectra [[#References|[2]]]. Postnikov systems (cf. [[Postnikov system|Postnikov system]]) of spectra of spaces have also
9 KB (1,281 words) - 14:40, 21 March 2022
...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
$#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
...]]). Context-free languages are also sometimes defined by so-called normal systems of equations in languages, which are another form of notation of context-fr
...valign="top">[1]</TD> <TD valign="top"> S. Ginsburg, "The mathematical theory of context-free languages" , McGraw-Hill (1966)</TD></TR></table>
14 KB (2,055 words) - 07:50, 18 November 2023
$#C+1 = 144 : ~/encyclopedia/old_files/data/B017/B.0107310 Boundary\AAhlayer theory
In the theory of systems of ordinary differential equations
24 KB (3,570 words) - 08:40, 26 March 2023
...te noise. The precise description of such a filtering problem requires the theory of stochastic differential equations (e.g., [[#References|[a4]]]; see also
...n="top"> R.E. Mortensen, "Optimal control of continuous-time stochastic systems" ''PhD Diss. Univ. California, Berkeley'' (1966)</td></tr><tr><td valign=
9 KB (1,283 words) - 17:44, 1 July 2020
...which the true classification is a priori known. In developing the general theory of recognition algorithms the most complete results have been obtained with
...top">[3]</TD> <TD valign="top"> V.N. Vapnik, A.Ya. Chervonenkis, "The theory of pattern recognition" , Moscow (1974) (In Russian)</TD></TR><TR><TD val
8 KB (1,189 words) - 13:43, 17 April 2014
...earch are considered in the theory of games (cf. [[Games, theory of|Games, theory of]]). If an information state contains several physical states, and if the
...ion problems, as well as problems with preference relations, relate to the theory of games; they are classified according to their game-theoretic properties.
16 KB (2,473 words) - 09:45, 7 August 2014
$#C+1 = 55 : ~/encyclopedia/old_files/data/S083/S.0803330 Scheduling theory
...scheduling theory include management, production, transportation, computer systems, construction, etc.
21 KB (3,117 words) - 10:04, 18 February 2021
plans and contributions to Statistical Systems of other countries as
control for changes in weather on the surface of the earth was
13 KB (1,963 words) - 16:40, 19 March 2023
...ical applications, such as domain decomposition methods for solving linear systems of equations, and certain multi-grid schemes used for solving elliptic part
...inert, "Error bounds for the method of alternating projections" ''Math. Control Signals Syst.'' , '''1''' (1988) pp. 43–59</td></tr><tr><td valign="top
8 KB (1,270 words) - 16:56, 1 July 2020
...tion of]]; [[Boolean functions, metric theory of|Boolean functions, metric theory of]]) $ | M | > 2 ^ {2 ^ {n} } $.
...cheduling theory]]), extremal problems on graphs (cf. [[Graph theory|Graph theory]]), and problems on the minimal representation of Boolean functions and fun
10 KB (1,391 words) - 12:30, 17 February 2021
...fundamental work of C. Shannon [[#References|[1]]] published in 1948. The theory of alphabetical coding is a branch of applied mathematics that consists in
...exity of a circuit realization by automata goes beyond the scope of coding theory.) The so-called prefix codes (that is, codes for which no word of $ V $
9 KB (1,453 words) - 18:06, 15 November 2023
...pparatus. A delicate question is how to express the type of socio-economic systems which are modelled, taking account of the social structure. It often happen
...y if the cost of the outputs is equal to the cost of the inputs. The given theory of efficient production methods and their characterization using $ \pi $
37 KB (5,562 words) - 11:11, 7 January 2024
[[Category:Linear and multilinear algebra; matrix theory]]
led to the matrix calculus — the subject matter of the theory of
18 KB (3,377 words) - 17:54, 2 November 2013
...ingle species models, age structure is considered in many multi-population systems. In the domain of demography, two-sex models are of special interest and in
...ons; however, in recent years, a functional-analytic approach based on the theory of semi-groups of operators (cf. also [[Semi-group of operators|Semi-group
11 KB (1,634 words) - 13:07, 10 February 2024
...um hypothesis is independent of the Zermelo–Frankel axioms. See also [[Set theory]].
...alism]], [[Formal system]], [[Axiomatic method]], [[Consistency]], [[Proof theory]], and [[Unsolvability]]. For the history of mathematics associated with th
29 KB (4,109 words) - 19:54, 18 March 2018
...e traditional logical problems by algebraic methods. With the birth of the theory of sets (in the 1870s), propositions and logical operations on them became
...logic, which is a part of the problem of synthesis of control (switching) systems. The minimization of functions of the algebra of logic consists in construc
27 KB (3,984 words) - 15:58, 22 December 2019
...earning involves sequences of actions and the task consists of acquiring a control strategy for choosing appropriate actions to achieve a goal, this mode is s
...uncertainty as to the correct answer (cf. [[Probability theory|Probability theory]]). They calculate the probability distribution for the unknown variables,
21 KB (3,152 words) - 17:43, 1 July 2020
$#C+1 = 20 : ~/encyclopedia/old_files/data/G045/G.0405010 Graph theory
...Methods were developed for solving a number of extremal problems in graph theory; one such problem is the construction of the maximum flow across a network
13 KB (1,976 words) - 20:08, 15 March 2023
Unlike formal languages, natural languages are dynamic, by nature. Rule systems and vocabularies of natural languages continuously change over time, the le
...ned unification is $\cal N P$-complete (cf. [[Complexity theory|Complexity theory]]). On the other hand, carefully crafted "mildly context-sensitive" gramm
18 KB (2,627 words) - 17:03, 1 July 2020
...e number of computations, so efficient approximation techniques attempt to control the number of calculations for both computational and accuracy effectivenes
See also [[Errors, theory of|Errors, theory of]]; [[Error|Error]].
29 KB (4,373 words) - 17:21, 2 January 2021
see [[#References|[8]]]). With practically no changes, the theory of Smith normal forms goes over to the case when $ K $
...icular, this holds for the theory of finitely-generated Abelian groups and theory of the Jordan normal form (see below).
51 KB (7,267 words) - 07:39, 14 January 2024
...of assignment of the value of an expression to a variable, the operator of control transfer (unconditional or conditional jump, or goto), the operator of proc
...tion of an intermediate form, are employed in optimizing translators or in systems with mixed generation for various types of machines; in certain very comple
12 KB (1,804 words) - 16:56, 7 February 2011
...extremum is guaranteed only for convex and related unimodal functions. The theory of finding a global extremum is still (1989) in the initial phase of develo
...><TD valign="top">[15]</TD> <TD valign="top"> , ''The current state of the theory of Operations Research'' , Moscow (1979) (In Russian)</TD></TR><TR><TD va
13 KB (1,911 words) - 08:00, 6 June 2020
The classification adopted in the theory of linear differential operators refers mainly to linear differential opera
...rential operator of|Principal type, partial differential operator of]]). A theory of local solvability and smoothness of solutions has been developed for suc
25 KB (3,768 words) - 09:07, 14 June 2022
but is also suitable for systems subject to uncertainties of the form $ \Delta A = \epsilon | A | $
th eigenvalue alone. From first-order perturbation theory one finds, [[#References|[a1]]],
30 KB (4,205 words) - 18:47, 5 April 2020
...ng processes, Krein spaces, the geometry of complex Banach spaces, control theory, and optimization. These semi-groups can be considered natural non-linear a
...|[a18]]]. He used this approach to study several important problems in the theory of linear operators on indefinite metric spaces. Note that, generally speak
24 KB (3,989 words) - 20:19, 11 January 2021
...tationary stochastic process is widely used in applications of probability theory in various areas of natural science and technology, since these processes a
In the mathematical theory of stationary stochastic processes, an important role is played by the mome
24 KB (3,614 words) - 08:52, 21 January 2024
$#C+1 = 33 : ~/encyclopedia/old_files/data/A011/A.0101910 Algorithms, theory of
...ers, cf. [[Turing machine|Turing machine]]). Subsequent development of the theory of algorithms is due to the studies of Kleene, Post [[#References|[6]]], [[
19 KB (2,846 words) - 16:10, 1 April 2020
...4th ACM STOC Symp.'' (1982) pp. 147–152 (Revised version: Inform. & Control 68 (1986), 86-104)</td></tr><tr><td valign="top">[a25]</td> <td valign="top
22 KB (3,250 words) - 17:43, 1 July 2020
...r, this is not achievable anymore in a perfect way. The effects of quantum theory may begin to have an influence on the outcome of the calculation.
Quantum theory would not be probabilistical without measurements. The probabilistic nature
27 KB (4,213 words) - 18:53, 26 April 2014
..."arithmetization of analysis" refers to 19th century efforts to create a "theory of real numbers ... using set-theoretic constructions, starting from the na
# the creation of the theory of real numbers.
91 KB (14,186 words) - 16:58, 23 November 2023