...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
...pplied disciplines, such as theoretical programming, graph theory, systems theory, and mathematical linguistics.
...dences between a family of sets form an ordered [[Category with involution|category with involution]]. Multiplication and involution enable one to express the
8 KB (1,265 words) - 17:31, 5 June 2020
[[Category:Ergodic theorems, spectral theory, Markov operators]]
3 KB (445 words) - 22:08, 7 July 2014
[[Category:Classical measure theory]]
A central theorem in classical measure theory, sometimes called
3 KB (451 words) - 17:37, 18 August 2012
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
statistics relates to the theory of correlation and regression, the bivariate
10 KB (1,403 words) - 16:21, 10 March 2024
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...op">[a1]</td> <td valign="top"> W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , PWN/Springer (1990) pp. Sect. 7.3 (Edition: Seco
3 KB (449 words) - 17:00, 1 July 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
A [[Lagrangian|Lagrangian]] in the theory of gauge fields on an oriented [[Manifold|manifold]] $M$ of dimension $3$.
4 KB (628 words) - 16:58, 1 July 2020
...inite or countable unions of them, nowhere-dense sets or sets of the first category, sets of measure zero. A set $\mathfrak{A} \subset \mathfrak{O}$ is regarde
...sense, a "large" subset in cases c) and d) may mean a subset of the second category in a non-empty open subset of the space $\mathfrak{O}$ or a subset of posit
12 KB (1,758 words) - 00:29, 13 January 2017
...hapt. 9</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> , ''Graph theory. Coverings, imbeddings, tournaments'' , Moscow (1974) pp. 82–159 (In R
...heory" L.W. Beineke (ed.) R.J. Wilson (ed.) , ''Selected topics in graph theory'' , Acad. Press (1978) pp. Chapt. 2</TD></TR>
4 KB (638 words) - 06:59, 6 July 2023
[[Category:Analysis]]
|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
2 KB (369 words) - 11:26, 13 December 2014
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...eferences|[a8]]] (see also [[Spectral theory of compact operators|Spectral theory of compact operators]]).
4 KB (673 words) - 15:30, 1 July 2020
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Both the extensions and applications of the theory of absolutely monotonic functions derive from two major theorems. The first
4 KB (582 words) - 12:38, 23 February 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
..."| Daykin, C.D., Pentikäinen, T. and Pesonen, M. (1994). ''Practical Risk Theory for Actuaries''. Chapman & Hall, London.
9 KB (1,305 words) - 09:48, 28 October 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
In this work he took pains to elaborate an organic theory of the state
10 KB (1,460 words) - 14:50, 18 March 2023
[[Category:Classical measure theory]]
...on the euclidean space. This theorem is used often in [[Geometric measure theory]] and credited to Besicovitch.
4 KB (532 words) - 13:29, 30 November 2012
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
the data to the theory, and graphing the results. His work foreshadows the
9 KB (1,380 words) - 08:31, 9 March 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory in areas as diverse as non-parametric inference and the properties
10 KB (1,466 words) - 08:31, 9 March 2024
Historically, the earliest theory of a [[Cohomology of algebras|cohomology of algebras]].
in the category of $ G $-
16 KB (2,427 words) - 09:48, 26 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...valign="top">[2]</td> <td valign="top"> R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968)</td></tr></table>
3 KB (457 words) - 17:02, 1 July 2020
...orphism; the category of analytic groups of this type is equivalent to the category of finite-dimensional Lie algebras over $ k $.
.../TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> C. Chevalley, "Theory of Lie groups" , '''1''' , Princeton Univ. Press (1946)</TD></TR></table>
9 KB (1,338 words) - 22:16, 5 June 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...inuous spectrum of the unperturbed operator. In [[Spectral theory|spectral theory]] one introduces therefore another subset of $\sigma ( T )$ which is more s
4 KB (564 words) - 17:00, 1 July 2020
<TR><TD valign="top">[1]</TD> <TD valign="top"> , ''Mathematical theory of logical deduction'' , Moscow (1967) (In Russian; translated from Engli
<TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Schutte, "Proof theory" , Springer (1977)</TD></TR>
4 KB (541 words) - 19:23, 26 October 2014
...use of the existence of ultrafilters (cf. [[Ultrafilter|Ultrafilter]]) the theory of filters is preferably used in general topology. By axiomatizing the conc
By a concrete category one means a [[Category|category]] $ {\mathcal C} $
57 KB (8,236 words) - 19:41, 20 January 2021
[[Category:Number theory]]
in 1936 for the purposes of algebraic number theory. The new language
4 KB (757 words) - 21:21, 22 November 2014
[[Category:Limit theorems]]
...This part may be considered as the first serious study ever of probability theory. The book was published in 1713 by N. Bernoulli (a nephew of Jacob Bernoull
4 KB (524 words) - 10:58, 29 May 2020
...n="top">[1]</TD> <TD valign="top"> I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian)</TD></TR>
4 KB (673 words) - 17:01, 26 October 2014
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...f knots, links or three-dimensional manifolds (cf. also [[Knot theory|Knot theory]]; [[Link|Link]]; [[Three-dimensional manifold|Three-dimensional manifold]]
19 KB (2,895 words) - 17:45, 1 July 2020
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There are several (different) notions of regularity in group theory. Most are not intrinsic to a group itself, but pertain to a group acting on
4 KB (716 words) - 07:52, 25 November 2023
...[[#References|[a6]]]) showed that it is consistent with the axioms of set theory that the set $ D $
...ets are of the second category. The key lemma in their proof is the Banach category theorem. H.E. White [[#References|[a4]]] extended the Bradford–Goffman th
8 KB (1,178 words) - 08:25, 26 March 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
Neyman, made fundamental contributions to the theory of hypothesis testing.
11 KB (1,680 words) - 16:32, 24 March 2023
...ory of quantum groups is defined in [[#References|[a1]]] to be dual to the category of Hopf algebras. This is natural for the following reason. There is the fo
is an anti-equivalence between the category of "spaces" and the category of commutative associative unital algebras, perhaps with some additional st
18 KB (2,674 words) - 19:09, 16 December 2019
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...m of L.A. Nazarova [[#References|[a12]]], given a connected quiver $Q$ the category $\operatorname{rep}_K( Q )$ is of tame representation type (see [[#Referenc
18 KB (2,636 words) - 06:50, 15 February 2024
...the theory of automata in constructing transition diagrams, etc. In graph theory itself, an orientation can be introduced in solving certain problems concer
For the terminology in graph theory see also [[Graph]]. A contour is better known as a directed cycle. A compre
5 KB (790 words) - 08:14, 26 November 2023
...ndependent over the field $\mathbb{Q}$, play an important role in [[number theory]], with applications to [[Diophantine equations]].
...of which non-trivial lower bounds for $|L|$ are established belong to the theory of transcendental numbers. In the case $n=2$ a number of inequalities, true
5 KB (776 words) - 08:31, 23 November 2023
[[Category:Classical measure theory]]
A classical theorem in measure theory first established by J. Radon and O.M. Nikodým, which has the following st
3 KB (484 words) - 12:09, 14 December 2012
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
early advocate and practitioner of the `mathematical theory of
10 KB (1,550 words) - 13:04, 18 March 2023
...y spectral sequences of objects of an arbitrary [[Abelian category|Abelian category]] (e.g., bimodules, rings, algebras, co-algebras, Hopf algebras, etc.).
...H $-spaces and $ H $-mappings, then this is a spectral sequence in the category of bigraded Hopf algebras.
14 KB (2,075 words) - 09:23, 18 February 2022
[[Category:Classical measure theory]]
The term might refer to different objects in classical measure theory.
4 KB (584 words) - 09:27, 7 December 2012
[[Category:Field theory and polynomials]]
3 KB (478 words) - 15:26, 30 December 2018
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
...also, quite apart from his many and striking contributions to probability theory.
10 KB (1,435 words) - 08:56, 25 March 2023
...]</TD> <TD valign="top"> N.N. Luzin, "Proof of a theorem in deformation theory" ''Izv. Akad. Nauk. SSSR Otd. Tekhn. Nauk'' , '''10''' (1939) pp. 65–8
[[Category:Geometry]]
5 KB (766 words) - 12:40, 2 November 2014
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory of knowledge which he developed in the specific context of industrial
11 KB (1,624 words) - 12:48, 25 March 2023
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...e theorem are known. The first one [[#References|[a6]]] uses a deformation theory of morphisms over a field of positive characteristic and applies only in th
11 KB (1,780 words) - 14:48, 3 February 2024
...gn="top">[3]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French)</TD></TR>
[[Category:Order, lattices, ordered algebraic structures]]
3 KB (509 words) - 06:25, 28 March 2023
[[Category:Analysis]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC
3 KB (384 words) - 14:28, 14 December 2012
[[Category:Limit theorems]]
Poisson's theorem is a limit theorem in probability theory which is a particular case of the [[Law of large numbers|law of large numbe
4 KB (651 words) - 08:06, 6 June 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
...extensive bibliography. The modern `classic' is B. de Finetti (1974/5) ''Theory of Probability''. John Wiley, London, in 2 volumes, translated from the It
10 KB (1,658 words) - 13:01, 18 March 2023
...TD> <TD valign="top"> P. Erdös, R. Rado, "A partition calculus in set theory" ''Bull. Amer. Math. Soc.'' , '''62''' (1956) pp. 427–489</TD></TR></t
...theorems has grown into a separate branch of combinatorics, called Ramsey theory.
4 KB (709 words) - 17:33, 1 November 2014
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
the probability calculus, by means of his dispersion theory.
10 KB (1,459 words) - 19:56, 5 March 2024
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...="top"> F. Beukers, "Some congruences for the Apéry numbers" ''J. Number Theory'' , '''21''' (1985) pp. 141–155</td></tr>
3 KB (525 words) - 20:51, 23 January 2024
The theory of cluster sets is a branch of function theory in which boundary properties of functions are studied in terms of topologic
...time one basically studied three, geometrically most simple, cases in the theory of cluster sets: a) $ z _ {0} $
25 KB (3,728 words) - 09:43, 26 March 2023
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''BDF theory''
10 KB (1,450 words) - 17:44, 1 July 2020
It turns out that in uniqueness problems the concept of the Baire category of a set is also very important. In fact, there is a Luzin–Privalov bound
is metrically dense and of the second Baire category in $ \sigma $,
10 KB (1,496 words) - 08:27, 6 June 2020
...zation of the Hopf–Steenrod invariant in terms of a generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]).
...the category of finite CW-complexes and taking values in a certain Abelian category $ A $.
12 KB (1,812 words) - 22:11, 5 June 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
A conjecture in [[Homotopy|homotopy]] theory usually referring to a theorem about the contractibility, or homotopy equiv
3 KB (522 words) - 16:57, 1 July 2020
[[Category:Analysis]]
|valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften.
2 KB (382 words) - 23:44, 27 June 2014
...erous applications in algebraic geometry, class field theory and cobordism theory.
is a group object in the category of connected affine formal schemes over $ k $ (
17 KB (2,537 words) - 22:38, 15 December 2019
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
The theory of
10 KB (1,489 words) - 20:37, 22 September 2016
...Arakelov, Yu.G. Zarkhin and L. Szpiro; essential in Faltings' proof is the theory of heights on the moduli space of Abelian varieties. For a good introductio
[[Category:Algebraic geometry]]
7 KB (1,068 words) - 08:01, 6 June 2020
...R><TD valign="top">[2]</TD> <TD valign="top"> J.C. Oxtoby, "Measure and category" , Springer (1971) {{MR|0393403}} {{ZBL|0217.09201}} </TD></TR><TR><TD va
...ive set|Gödel constructive set]]; [[Descriptive set theory|Descriptive set theory]].
4 KB (623 words) - 08:03, 6 June 2020
...is way many classical problems in [[Analytic number theory|analytic number theory]] can be solved (the distribution of the fractional parts of a wide class o
...ences|[5]]]. For the application of Vinogradov's method in analytic number theory see [[#References|[1]]], [[#References|[2]]], , [[#References|[5]]], [[#Ref
9 KB (1,606 words) - 09:42, 9 November 2014
...ar problems are considered for these points. (In any event, the last-named theory is closely connected with the calculus of variations in the large and has c
...ormation in terms of some numerical invariant — the Lyusternik–Shnirel'man category $ \mathop{\rm cat} M $—
14 KB (2,052 words) - 08:27, 6 June 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
mathematical theory and in later life preferred to describe himself
9 KB (1,384 words) - 13:08, 18 March 2023
[[Category:Classical measure theory]]
..."top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958) {{MR|0117523}}
4 KB (498 words) - 22:40, 31 December 2017
[[Category:Special matrices]] [[Category:Markov chains]]
|valign="top"|{{Ref|G}}|| F.R. Gantmacher, "The theory of matrices" , '''1''' , Chelsea, reprint (1977) (Translated from Russian)
6 KB (982 words) - 09:11, 1 October 2023
in the sense of the Pontryagin theory of characters, when the compact group $ G ^ {*} $
...compact) [[#References|[6]]] and in addition the two Milnor axioms on the category of compact metric spaces [[#References|[7]]].
10 KB (1,511 words) - 16:36, 13 January 2024
on the category of $ \Lambda $-
modules with values in the same category, that is covariant in $ A $
16 KB (2,248 words) - 22:15, 5 June 2020
...f a two-person zero-sum game in normal form (cf. [[Games, theory of|Games, theory of]]) amounts to defining sets of strategies $A$ and $B$ of players I and I
...ively) for both players (cf. [[Strategy (in game theory)|Strategy (in game theory)]]). The common value of both parts of equation \eqref{1prm} is called the
5 KB (827 words) - 15:45, 14 February 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
.... This non-separability causes well-known problems of measurability in the theory of weak convergence of measures on the space. To overcome this inconvenienc
8 KB (1,131 words) - 14:43, 27 January 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
Theory and its Applications,'', Vol.1,
8 KB (1,312 words) - 15:46, 25 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...ations on an unbounded domain (cf. also [[Perturbation theory|Perturbation theory]]; [[Perturbation of a linear system|Perturbation of a linear system]]; [[L
9 KB (1,289 words) - 07:19, 25 January 2024
..."On the difficulty of finding reliable witnesses" , ''Algorithmic Number Theory, First Internat. Symp., ANTS-I'' , ''Lecture Notes in Computer Science'' ,
[[Category:Number theory]]
3 KB (516 words) - 17:58, 8 November 2014
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...of topological degree $k$. Another application to gauge theory and to the theory of completely integrable systems is given by a construction of K. Uhlenbeck
5 KB (698 words) - 08:13, 15 February 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
the theory of Brownian movement concided with those of Einstein. The
11 KB (1,571 words) - 07:00, 15 March 2024
...tic ideas (cf. [[Model (in logic)|Model (in logic)]]; [[Model theory|Model theory]]).
...es in the extended language true on all members of $T^*$. The union of the theory of $T^*$ and the finitely satisfiable set from $L$ is itself finitely satis
6 KB (950 words) - 12:20, 30 December 2015
...s is in many respects analogous to the role of associative algebras in the theory of alternative algebras (cf. also [[Alternative rings and algebras|Alternat
A generalization of the theory of finite-dimensional Jordan algebras is the theory of Jordan algebras with the minimum condition for quadratic (inner) ideals
13 KB (1,800 words) - 19:51, 15 March 2023
...></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> F. Harary, "Graph theory" , Addison-Wesley (1969) pp. Chapt. 9</TD></TR></table>
...n="top">[a1]</TD> <TD valign="top"> R.J. Wilson, "Introduction to graph theory" , Longman (1985)</TD></TR></table>
5 KB (780 words) - 20:06, 15 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...</td> <td valign="top"> W.B. Jones, O. Njåstad, W.J. Thron, "Moment theory, orthogonal polynomials, quadrature and continued fractions associated with
3 KB (454 words) - 16:59, 1 July 2020
...gory|category]] of Magari algebras with homomorphisms is equivalent to the category $ \mathbf S $
If one departs from the Zermelo–Fraenkel [[Axiomatic set theory|axiomatic set theory]] $ { \mathop{\rm ZF} } $,
13 KB (1,762 words) - 11:49, 12 January 2021
...D></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P.J. Cohen, "Set theory and the continuum hypothesis" , Benjamin (1966)</TD></TR></table>
is replaced by an arbitrary small category $ {\mathcal C} $)
10 KB (1,489 words) - 19:17, 12 January 2021
[[Category:Markov processes]]
...probability theory and its applications"|"An introduction to probability theory and its applications"]], '''1–2''', Wiley (1966)
4 KB (466 words) - 07:59, 6 June 2020
[[Category:Limit theorems]]
...sion of the [[Central limit theorem|Central Limit Theorem]] of probability theory: If $ S_{n} $ denotes the number of “successes” in $ n $ [[Bernoulli tr
5 KB (692 words) - 19:33, 7 July 2016
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
groups (cf. [[K-theory| $ K $-
25 KB (3,559 words) - 19:33, 7 February 2024
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> F. Harary, "Graph theory" , Addison-Wesley (1969) pp. Chapt. 9</TD></TR><TR><TD valign="top">[2]</
...TD valign="top">[a1]</TD> <TD valign="top"> F. Harary, L. Moser, "The theory of round robin tournaments" ''Amer. Math. Monthly'' , '''73''' (1966) pp
6 KB (931 words) - 20:20, 15 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...in case $f$ is continuous, thus providing a wonderfully short, probability-theory based, constructive proof of the Weierstrass approximation theorem (cf. [[W
4 KB (598 words) - 16:55, 1 July 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...blems of classical [[Differential geometry|differential geometry]]. In the theory of solitons (cf. [[Soliton|Soliton]]) it enters via Lax pairs for non-linea
3 KB (495 words) - 17:56, 2 November 2023
There is also an algebraic description of the category of algebras $ R $,
...s turns out to be dual to the category of compact topological groups. This theory admits a generalization to the case of homogeneous spaces of compact topolo
10 KB (1,483 words) - 17:06, 13 June 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory of the vibrating string, and with [[DAlembert|D'Alembert]]
10 KB (1,551 words) - 07:42, 9 March 2024
|valign="top"|{{Ref|Gr}}||valign="top"| G. Grätzer, "Lattice theory", Freeman (1971)
[[Category:General algebraic systems]]
3 KB (484 words) - 20:54, 28 November 2014
[[Category:Distribution theory]]
4 KB (522 words) - 13:11, 6 January 2024
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...s a starting point for further mathematical research. Nowadays (2000), the theory of pricing of so-called derivative contracts and related subjects has grown
8 KB (1,291 words) - 18:53, 7 February 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory on the propagation of cholera through the London water
9 KB (1,428 words) - 18:38, 24 March 2023
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...\mathbf{x} )$ as a quantum field (cf. [[Quantum field theory|Quantum field theory]]), see e.g. [[#References|[a6]]], [[#References|[a7]]].
5 KB (713 words) - 17:44, 1 July 2020
...emi-groups are the free objects (cf. [[Free algebra|Free algebra]]) in the category of all semi-groups. For a semi-group $F$ the following conditions are equiv
...[Coding, alphabetical]], [[#References|[4]]]–[[#References|[6]]]), and the theory of formal languages and formal grammars (cf. [[Grammar, formal]], see also
5 KB (786 words) - 20:17, 8 December 2015
[[Category:Linear and multilinear algebra; matrix theory]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. C
4 KB (615 words) - 16:20, 24 November 2012
...formulated. The method of coverings plays an important part in [[dimension theory]].
The method of inverse spectra (cf. [[System (in a category)]]; [[Spectrum of spaces]]), closely related to the method of coverings and
10 KB (1,522 words) - 19:35, 25 March 2023
..."top">[3]</TD> <TD valign="top"> A.I. Shirshov, "On a hypothesis in the theory of Lie algebras" ''Sibirsk. Mat. Zh.'' , '''3''' : 2 (1962) pp. 297–3
...D> <TD valign="top"> A.M. Pitts, "Amalgamation and interpolation in the category of Heyting algebras" ''J. Pure Appl. Alg.'' , '''29''' (1983) pp. 155–
4 KB (674 words) - 16:10, 1 April 2020
An intensively studied subject in the
theory of graphs is the colouring of planar graphs (cf. [[Graph colouring|Graph co
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> F. Harary, "Graph
theory" , Addison-Wesley (1969) pp. Chapt. 9</TD></TR></table>
5 KB (774 words) - 16:51, 16 March 2023
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...aracters. There is also a close relationship to [[Clifford theory|Clifford theory]] (of simple modules).
4 KB (692 words) - 17:46, 1 July 2020
...or]]); it is a part of general linear algebra. It deals with the structure theory of projective modules and their automorphism groups. To put it more simply,
...vector bundle]]. These objects may be studied with the aid of the homotopy theory of vector bundles and of topological
14 KB (2,405 words) - 22:14, 10 January 2015
[[Category:Limit theorems]]
...yshev inequality in probability theory|Chebyshev inequality in probability theory]]; these provide the so-called exponential bounds for the probability of la
4 KB (604 words) - 20:08, 7 January 2024
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...nces|[a5]]] has simplified a discussion in the complicated Tomita–Takesaki theory.
6 KB (960 words) - 16:45, 1 July 2020
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...ts. The Weil bundles are particular examples of the bundle functors on the category $\mathcal{M} f$ in the sense of [[#References|[a3]]] and their restrictions
12 KB (1,876 words) - 06:30, 15 February 2024
theory.
of the concept of an algebraic variety by relating it to the theory of
5 KB (706 words) - 13:35, 17 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...td valign="top">[a4]</td> <td valign="top"> I.M. Gel'fand, V.S. Redakh, "A theory of noncommutative determinants and characteristic functions of graphs I" '
3 KB (498 words) - 13:58, 20 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...plicative number theory (cf. also [[Analytic number theory|Analytic number theory]]). Most zeta-functions (cf. also [[Zeta-function|Zeta-function]]) and $L$-
5 KB (706 words) - 17:03, 1 July 2020
[[Category:Classical measure theory]]
|valign="top"|{{Ref|C}}|| Donald L. Cohn, "Measure theory", Birkhäuser (1993). {{MR|1454121}} {{ZBL|0860.28001}}
5 KB (655 words) - 20:33, 18 February 2012
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
dark ages of the theory of probability.
10 KB (1,512 words) - 07:24, 25 March 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
French mathematics from 1815 to 1840. His contribution to probability theory is
11 KB (1,710 words) - 19:08, 6 March 2024
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An essential part of Sperner theory consists of the study of other partially ordered sets having analogous prop
4 KB (654 words) - 20:17, 25 January 2024
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of mathematics, which encompassed path-breaking work in [[probability theory]]. The
12 KB (1,774 words) - 20:00, 7 March 2024
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...="top">[a2]</td> <td valign="top"> J.A. Kelingos, "Contributions to the theory of quasiconformal mappings" , Diss. Univ. Michigan (1963)</td></tr><tr><td
7 KB (1,145 words) - 07:25, 25 January 2024
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...><td valign="top">[a3]</td> <td valign="top"> W.G. Lister, "A structure theory of Lie triple systems" ''Trans. Amer. Math. Soc.'' , '''72''' (1952) pp.
3 KB (559 words) - 16:43, 15 March 2023
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In modern knot theory, Listing's ideas are widely used: colouring of regions (e.g. to define the
3 KB (496 words) - 07:37, 18 March 2023
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Pareto valued theory only insofar as it stood the test of reality;
9 KB (1,349 words) - 08:05, 10 March 2024
[[Category:Number theory]]
3 KB (485 words) - 19:43, 5 June 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
statistical theory. With complimentary graduate programs, the two departments
10 KB (1,490 words) - 07:08, 21 March 2023
[[Category:Classical measure theory]]
...und use not only in set theory and function theory but also in probability theory, see {{Cite|Hal}}, {{Cite|Ko}}.
5 KB (718 words) - 17:50, 5 May 2024
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...em). Minimal unitary dilations and functions of them, defined via spectral theory, allow one to construct a functional calculus for contractive operators. Th
6 KB (897 words) - 20:02, 27 February 2021
[[Category:Stochastic processes]]
is stationary in the usual sense. Thus, in the case of discrete time, the theory of stochastic processes with stationary increments reduces easily to that o
7 KB (994 words) - 08:23, 6 June 2020
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...unication networks; see [[#References|[a2]]] for difference sets in design theory.
10 KB (1,435 words) - 07:01, 29 March 2024
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...">[a2]</td> <td valign="top"> M. Marcus, H. Minc, "A survey of matrix theory and matrix inequalities" , Dover (1992) pp. Sect. I.3</td></tr></table>
4 KB (549 words) - 15:30, 1 July 2020
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...ave absolute value strictly less than $1$ (cf. also [[Galois theory|Galois theory]]). That is, $\theta$ satisfies a polynomial equation of the form $P ( x )
5 KB (723 words) - 19:03, 23 January 2024
...1]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "The algebraic theory of semi-groups" , '''1''' , Amer. Math. Soc. (1961)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> L.N. Shevrin, "On the theory of periodic semigroups" ''Soviet Math. Izv. Vyz.'' , '''18''' : 5 (1974)
5 KB (775 words) - 17:22, 17 October 2014
[[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
...ntial geometry (semi-groups of partial transformations), and the algebraic theory of automata (semi-groups of automata).
...ersection or union; etc. The following example is important in the general theory and in some applications. Let $X$ be an arbitrary set, and let an operation
17 KB (2,435 words) - 09:18, 2 April 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
modern statistical theory). When Pearson was working out the mathematical
11 KB (1,612 words) - 18:41, 22 March 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
...work on Markov chains, the concept of Markovian dependence pervades modern theory and application of random processes. His textbook influenced the developmen
13 KB (1,874 words) - 18:35, 8 March 2024
If a partially ordered set is regarded as a [[Small category|small category]], then it is a lattice if and only if has products and coproducts of pairs
The class of all lattices forms a category if homomorphisms are taken as morphisms.
14 KB (2,066 words) - 15:31, 11 November 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...valign="top"> C. Brezinski, M. Redivo Zaglia, "Extrapolation methods. Theory and practice" , North-Holland (1991)</td></tr><tr><td valign="top">[a2]</t
9 KB (1,390 words) - 20:20, 25 January 2024
...ar space]]) and are particularly important in [[Dimension theory|dimension theory]]. Every closed subspace of a normal space is normal (normality is heredita
<TR><TD valign="top">[2]</TD> <TD valign="top"> V.I. Zaitsev, "On the theory of Tikhonov spaces" ''Vestnik Moskov. Univ. Mat. Mekh.'' : 3 (1967) pp.
5 KB (733 words) - 20:41, 19 December 2014
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
of one of the few treatises of probability theory in the 19th century. He
11 KB (1,779 words) - 16:32, 4 March 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
integral part of statistics and his theory of errors remained a major focus of
11 KB (1,665 words) - 15:52, 25 March 2023
...[[morphism]]s $\alpha_i\colon A\rightarrow A_i$, $i\in I$, of an arbitrary category $\mathfrak{A}$ with common initial object $A$ is called a morphism cone wit
The theory of cones in [[Banach space]]s is more thoroughly developed <ref group="comm
13 KB (2,188 words) - 13:05, 10 June 2016
[[Category:Nonparametric inference]]
|valign="top"|{{Ref|M}}|| R. von Mises, "Mathematical theory of probability and statistics" (1964) (Translated from German)
4 KB (572 words) - 11:08, 26 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
Two types of applications arise in classical [[Potential theory|potential theory]] (see [[#References|[a4]]], [[#References|[a6]]], [[#References|[a7]]]).
7 KB (966 words) - 16:11, 11 February 2024
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...op">[a5]</td> <td valign="top"> R. Sikorski, "On Mikusiński's algebraic theory of differential equations" ''Studia Math.'' , '''16''' (1958) pp. 230–
4 KB (576 words) - 10:55, 2 July 2020
Basic problems studied in the representation theory of associative algebras are that of obtaining necessary and sufficient cond
...[9]</TD> <TD valign="top"> P. Donovan, M.R. Freislich, "The representation theory of finite graphs and associated algebras" , Carleton Univ. (1974) {{MR|0357
22 KB (3,137 words) - 20:01, 15 March 2023
...TD valign="top">[4]</TD> <TD valign="top"> N.I. Akhiezer, "Elements of the theory of elliptic functions" , Amer. Math. Soc. (1990) (Translated from Russian)
[[Category:Functions of a complex variable]]
4 KB (565 words) - 19:36, 5 June 2020
[[Category:Analysis]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC
4 KB (631 words) - 08:46, 18 November 2012
[[Category:Classical measure theory]]
..."top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958) {{MR|0117523}}
5 KB (751 words) - 09:47, 16 August 2013
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
first - and one on interpolation theory (Thiele, 1909).
11 KB (1,576 words) - 19:39, 7 October 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory of statistical experimentation and sample surveys, but especially
12 KB (1,801 words) - 16:39, 14 August 2023
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...ner derivations are a tool in the study of differential identities, Galois theory with derivations, and enveloping algebra smash products. To start with, [[#
4 KB (655 words) - 16:46, 1 July 2020
[[Category:Analysis]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Pres
3 KB (493 words) - 09:25, 16 August 2013
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
where in the feverish atmosphere generated by the Einstein theory
12 KB (1,840 words) - 20:30, 6 March 2024
[[Category:Classical measure theory]]
|valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory", v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
4 KB (595 words) - 17:27, 18 October 2012
...t the same time classes of locally convex spaces that are important in the theory and applications) are normed spaces, countably-normed spaces and Fréchet s
...e with its dual or [[Adjoint space|adjoint space]]. The foundation of this theory of [[Duality|duality]] for locally convex spaces is the [[Hahn–Banach the
18 KB (2,674 words) - 22:17, 5 June 2020
...TD> <TD valign="top"> A.A. Markov, N.M. [N.M. Nagornyi] Nagorny, "The theory of algorithms" , Reidel (1988) (Translated from Russian) {{ZBL|0663.03023
[[Category:Logic and foundations]]
4 KB (587 words) - 20:53, 8 December 2015
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
..."top">[a2]</td> <td valign="top"> D.E. Edmunds, W.D. Ewans, "Spectral theory and differential operators" , Clarendon Press (1987) {{MR|0929030}} {{ZBL
4 KB (579 words) - 19:34, 7 February 2024
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...ns (i.e., of functions all of whose level sets are convex) and for duality theory in micro-economics (duality between direct and indirect utility functions).
6 KB (838 words) - 17:44, 1 July 2020
...es all [[Steenrod–Eilenberg axioms|Steenrod–Eilenberg axioms]], and on the category of paracompact spaces it is uniquely determined by those axioms together wi
...for paracompact spaces it is isomorphic to the cohomology defined in sheaf theory.
5 KB (775 words) - 08:13, 5 May 2022
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...also arise naturally in the diffusion approximation in stochastic network theory.
4 KB (565 words) - 16:56, 1 July 2020
this gives an isomorphism between the category of representations of $ L $
and the category of modules over $ U ( L) $.
10 KB (1,344 words) - 08:11, 6 June 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
community. As a result, his work, especially his studies in statistical theory
11 KB (1,620 words) - 06:53, 8 October 2023
[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
11 KB (1,769 words) - 05:54, 7 April 2018
[[Category:Topology]]
|valign="top"|{{Ref|He}}|| L. Helms, "Potential theory". Universitext. Springer-Verlag London, Ltd., London, 2009. {{MR|2526019}}
8 KB (1,260 words) - 15:29, 5 January 2017
...classification theory of unipotent commutative algebraic groups and in the theory of commutative formal groups (amongst other things) [[#References|[a3]]].
[[Category:TeX done]]
4 KB (709 words) - 13:46, 20 March 2023
From the point of view of the theory of invariants, a Jordan matrix is a canonical representative in the orbits
...n="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 (708 words) - 11:11, 17 December 2019
[[Category:Distribution theory]]
...o probability theory and its applications"|"An introduction to probability theory and its applications"]], '''1–2''', Wiley (1957–1971)
6 KB (864 words) - 13:51, 12 December 2013
[[Category:Classical measure theory]]
A concept of classical measure theory.
5 KB (745 words) - 17:18, 18 August 2012
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...|group algebra]] $\mathbf{C} [ \Gamma ]$ with the algebraic [[K-theory|$K$-theory]] index of an invariant (pseudo-) differential operator on a covering $\til
5 KB (698 words) - 17:02, 1 July 2020
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...ation has a visual interpretation in the framework of [[Morse theory|Morse theory]] of the energy function on $\Omega G$ [[#References|[a6]]]. Certain geomet
4 KB (630 words) - 17:01, 1 July 2020
[[Category:Distribution theory]]
...B.V. Gnedenko, [[Gnedenko, "A course in the theory of probability"|"The theory of probability"]], Chelsea, reprint (1962) (Translated from Russian)
5 KB (776 words) - 10:59, 29 May 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...a3]]] with his theory of capacities (cf. also [[Capacity|Capacity]]). This theory had influences on many parts of mathematics and different areas of sciences
7 KB (1,001 words) - 16:57, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
of equinoxes, and lunar theory. We owe to the young D'Alembert
11 KB (1,657 words) - 20:48, 21 December 2023
[[Category:Markov processes]]
|valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1979) (Translated from Russi
5 KB (778 words) - 22:14, 5 June 2020
The theory of modular representations of the symmetric groups has also been developed
...TD valign="top">[5]</TD> <TD valign="top"> G.D. James, "The representation theory of the symmetric groups" , Springer (1978) {{MR|0513828}} {{ZBL|0393.20009}
18 KB (2,500 words) - 13:04, 18 February 2022
...gebraic tori]]. A generalization of algebraic tori in the framework of the theory of group schemes is the following
...group scheme over $S$ can equivalently be defined as a group object in the category of $S$-schemes.
4 KB (629 words) - 20:08, 15 December 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
his excursions into pure theory a very small part of his writings. Her
11 KB (1,692 words) - 13:48, 4 November 2023
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...l has many significant connections with invariants in [[Graph theory|graph theory]]. Of central importance is the relationship between $t ( M )$ and the [[ch
11 KB (1,736 words) - 06:27, 15 February 2024
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...2)</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> D. Gabor, "Theory of communication" ''J. IEE'' , '''93''' (1946) pp. 429–457</td></tr><t
4 KB (543 words) - 17:00, 1 July 2020
from the category of $ \mathfrak G $-
modules into the category of $ K $-
21 KB (3,027 words) - 17:45, 4 June 2020
...ses the concept of a random graph makes it possible to utilize probability theory as a tool in order to obtain asymptotic solutions of enumeration problems.
...TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> B. Bollobas, "Graph theory and combinatorics" , Acad. Press (1984)</TD></TR></table>
6 KB (999 words) - 20:05, 15 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
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...> <td valign="top"> R.V. Kadison, J.R. Ringrose, "Fundamentals of the theory of operator algebras" , '''1–2''' , Acad. Press (1983)</td></tr><tr><td
4 KB (651 words) - 15:30, 1 July 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...op">[a5]</td> <td valign="top"> F.J. MacWilliams, N.J.A. Sloane, "The theory of error-correcting codes" , North-Holland (1977)</td></tr></table>
4 KB (592 words) - 16:52, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
popularize British mathematical statistics. He was a pioneer in the theory of
11 KB (1,766 words) - 18:41, 6 March 2024
[[Category:Partial differential equations]]
The Korn inequality has several applications in the theory of nonlinear elasticity (and was in fact originally derived by Korn in line
4 KB (541 words) - 11:59, 3 November 2013
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
'''Summary.''' Geary contributed to areas of statistical theory,
12 KB (1,874 words) - 10:53, 25 February 2024
[[Category:Ordinary differential equations]]
First integrals of motions are particularly studied in the theory of [[Hamiltonian system|Hamiltonian systems]]. For physically relevant case
3 KB (563 words) - 18:44, 15 January 2015
[[Category:Field theory and polynomials]]
4 KB (749 words) - 18:32, 2 November 2014
...is called a '''$ B $-morphism'''. Fibre spaces with their morphisms form a category — one that contains fibre spaces over $ B $ with their $ B $-morphisms as
...ign="top">[a1]</TD> <TD valign="top"> A. Dold, “Partitions of unity in the theory of fibrations”, ''Ann. of Math.'', '''78''' (1963), pp. 223–255.</TD></
5 KB (754 words) - 01:34, 10 December 2016
...lynomial is also related to various other polynomials encountered in graph theory. These include the [[chromatic polynomial]] (see [[#References|[a13]]]), th
...TD> <TD valign="top"> E.J. Farrell, "A graph-theoretic approach to Rook theory" ''Caribb. J. Math.'' , '''7''' (1988) pp. 1–47</TD></TR>
6 KB (1,005 words) - 20:14, 15 March 2023
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...4]</td> <td valign="top"> D. Gottlieb, M.Y. Hussaini, S.A. Orszag, "Theory and application of spectral methods" R.G. Voigt (ed.) D. Gottlieb (ed.)
6 KB (866 words) - 18:56, 24 January 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
queueing theory.
12 KB (1,866 words) - 14:20, 10 March 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
20 papers in analytic number theory over the next seven years.
13 KB (2,000 words) - 18:58, 5 March 2024
[[Category:Classical measure theory]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC
5 KB (734 words) - 11:57, 2 May 2014
[[Category:Classical measure theory]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CR
4 KB (703 words) - 11:32, 14 January 2013
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...given. In treating some problems in vector-valued measure and integration theory, B.J. Pettis needed to use this result but noted that no proof was supplied
5 KB (714 words) - 15:30, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
completed with a dissertation on probability theory as a basis for theoretical
12 KB (1,787 words) - 07:49, 18 August 2023
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According to Einstein's theory of general relativity (cf. [[Relativity theory]]), [[space-time]] can be modeled by a $4$-dimensional Lorentzian manifold
4 KB (606 words) - 19:47, 24 November 2023
[[Category:Distribution theory]]
...ion|normal distribution]], the uniform distribution appears in probability theory as an exact distribution in some problems and as a limit in others.
6 KB (911 words) - 12:00, 22 December 2019
[[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
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Balayaged functions were introduced in classical [[Potential theory|potential theory]] by M. Brelot and play an important role. Given any set $A \subset \mathbf
6 KB (870 words) - 16:57, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
achieved practical progress, if not the desired confirmation of his theory in
12 KB (1,727 words) - 10:18, 10 March 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
...ef|6}}||valign="top"| Kendall, M.G. and Stuart, A. (1973). ''The Advanced Theory of Statistics''. Vol. 2, 3rd edition. Griffin, London.
11 KB (1,719 words) - 20:18, 12 March 2016
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...mplexity theory, algorithmic analysis, number theory, analysis and measure theory have all been influential. See also [[#References|[a2]]], [[#References|[a4
6 KB (871 words) - 17:45, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
to the theory of functions. His later years, until his early death
12 KB (1,787 words) - 12:11, 28 October 2023
The
theory of graphs studies methods for determining the connectivity of graphs, condi
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> F. Harary, "Graph
theory" , Addison-Wesley (1969) pp. Chapt. 9</TD></TR><TR><TD valign="top">[2]</
7 KB (1,143 words) - 07:00, 6 July 2023
...perty for the $2$-primary component of the class group follows from Gauss' theory of genera.
The theory of complex multiplication (see {{Cite|CaFr}}) enables one to construct Abel
5 KB (867 words) - 17:41, 12 November 2023
$#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
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
almost alone in France, represents the theory of probabilities, which he has cultivated
12 KB (1,845 words) - 19:59, 7 March 2024
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...tomography. In [[Tomography|tomography]], or at least in tomography as the theory was initially constructed in the early 1980s, ridge functions were basic. H
5 KB (780 words) - 16:45, 1 July 2020
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...is not right perfect, then it is consistent with ZFC (cf. [[Set theory|Set theory]]; [[Zermelo axiom|Zermelo axiom]]) that there are no p-test modules, [[#Re
5 KB (732 words) - 17:46, 1 July 2020
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...regularity theory here as $C ^ { \infty } ( \Omega )$ does in distribution theory (for example, $u \in \mathcal{G} ( \Omega )$ and $\Delta u \in \mathcal{G}
7 KB (996 words) - 20:13, 4 February 2024
Thus, the problem of homotopy types on the category of $ \mathop{\rm CW}\nolimits $ -
in the category of simplicial sets (with respect to homotopy groups in the sense of Kan, se
31 KB (4,636 words) - 12:07, 15 December 2019
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory of (linear) sets which are now called after him, to break the confines
14 KB (2,139 words) - 18:37, 8 March 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
`theory of collectives' which is built on the frequency interpretation of
16 KB (2,419 words) - 16:35, 14 August 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
..., [[#References|[a7]]]) in the area of finite transformation groups. Smith theory is now (2000) best understood via cohomological methods, following an appro
6 KB (1,018 words) - 16:56, 1 July 2020
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..."top">[a1]</td> <td valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number Theory" , Acad. Press (1966) (In Russian)</td></tr>
5 KB (751 words) - 13:28, 25 November 2023
[[Category:Analysis]]
...p"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory", '''1''', Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}}
4 KB (655 words) - 12:01, 14 December 2012
The representation theory of finite-dimensional (associative) algebras is a very active branch of mat
[[Category:Associative rings and algebras]]
4 KB (607 words) - 20:54, 15 March 2023
is a set of the first category in $ X _ {0} $.
The basic problem in the theory of operator semi-groups is to establish relations between properties of a s
5 KB (786 words) - 19:41, 5 June 2020
...t spaces, is a reflective category (cf. [[Reflective subcategory|Reflexive category]]). Spaces with compact completions (i.e. pre-compact spaces) are character
...ons. Such a product, though identical with the product in the sense of the category of proximity spaces, is geometrically inconvenient and mainly serves in the
20 KB (2,990 words) - 16:25, 15 October 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory. His investigations in this field inspired many workers in
13 KB (2,023 words) - 18:43, 24 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...with respect to $\lambda$. This theorem is closely related to integration theory [[#References|[a8]]], [[#References|[a3]]]. Namely, if $\{ f _ { n } \}$ is
4 KB (614 words) - 07:39, 24 January 2024
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...receded by a Gaussian smoothing step (see [[Scale-space theory|Scale-space theory]]; [[Edge detection|Edge detection]]).
5 KB (686 words) - 16:45, 1 July 2020
[[Category:Distribution theory]]
4 KB (578 words) - 11:31, 21 March 2022
defined on the category of $ A $-
...D valign="top">[1]</TD> <TD valign="top"> D. Gorenstein, "An arithmetic theory of adjoint plane curves" ''Trans. Amer. Math. Soc.'' , '''72''' (1952) p
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...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
the atomic theory of matter. For detailed discussions of his
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...(in either case) requires only elementary (albeit ingenious) [[probability theory]] and takes less than a page.
[[Category:Combinatorics]]
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...ces|[a1]]]. This theorem is the cornerstone of one-relator [[Group|group]] theory.
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A morphism in a category of algebraic systems (cf. [[Algebraic system|Algebraic system]]). It is a m
...ign="top">[2]</TD> <TD valign="top"> C.C. Chang, H.J. Keisler, "Model theory" , North-Holland (1973)</TD></TR></table>
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[[Category:Branching processes]]
..."top"|{{Ref|S2}}|| B.A. Sewastjanow, "Age-dependent branching processes" ''Theory Probab. Appl.'' , '''9''' : 4 (1964) pp. 521–537 ''Teor. Veroyatnost. i P
4 KB (535 words) - 06:29, 30 May 2020
...clopedia/old_files/data/C021/C.0201890 Chebyshev inequality in probability theory,
[[Category:Distribution theory]]
6 KB (876 words) - 16:43, 4 June 2020
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...i, G.A. Kamenskii, L.E. El'sgol'ts, "Mathematical foundations of the theory of control systems" , Moscow (1969) (In Russian)</td></tr><tr><td valign=
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...gn="top">[a17]</td> <td valign="top"> D.B. West, "Introduction to graph theory" , Prentice-Hall (1996)</td></tr></table>
10 KB (1,636 words) - 08:14, 15 February 2024
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...e first of four Zolotarev problems in [[Approximation theory|approximation theory]]. Zolotarev's second problem asks for the monic polynomial $p$ of degree $
7 KB (974 words) - 10:10, 11 November 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
Bortkiewicz consolidated Lexis's theory based on his famous
13 KB (1,926 words) - 18:33, 17 April 2024
[[Category:Markov processes]]
|valign="top"|{{Ref|GS}}|| I.I. Gikhman, A.V. Skorokhod, "Introduction to the theory of random processes" , Saunders (1969) (Translated from Russian) {{MR|02476
9 KB (1,271 words) - 19:35, 5 June 2020
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...embodies the phase factor. The aim of [[Quantum field theory|quantum field theory]] is to treat other elementary particles by quantizing appropriate classica
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...uperalgebra|Superalgebra]]), first appeared in [[Deformation|deformation]] theory, in a geometric context, in the work of A. Frölicher and A. Nijenhuis in t
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[[Category:Markov processes]]
A family of measures used in the theory of Markov processes for determining the distribution at future instants fro
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In the
theory of linear operators the two special cases <img align="absmiddle" border="0"
...rators (morphisms) form, together with the composition law, the [[
Category|
category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org
67 KB (9,247 words) - 17:12, 29 October 2017
The inverse assertion, although true even in $C^\infty$-smooth category, has no practical meaning since a center-type singularity may have only ''f
...">[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
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...was introduced by G. de Rham in [[#References|[a6]]], to obtain a homology theory including both forms and chains, but a precise definition, see [[#Reference
5 KB (733 words) - 17:00, 1 July 2020
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...f Einstein's theory of relativity (cf. also [[Relativity theory|Relativity theory]]).
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...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
Complètes'', Volume VIII). In this work, he applied the theory of
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...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
catalyst that enabled probability theory to develop beyond mere combinatorial
15 KB (2,384 words) - 06:52, 6 March 2024
...>[a2]</TD> <TD valign="top"> L.H. Rowen, "Polynomial identities in ring theory" , Acad. Press (1980) pp. Chapt. 7</TD></TR>
[[Category:Associative rings and algebras]]
4 KB (701 words) - 17:01, 23 November 2023
...erences|[a8]]] is usually viewed as the starting point of modern portfolio theory. Markowitz considered static problems and called a portfolio mean-variance
[[Category:Game theory, economics, social and behavioral sciences]]
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modules can be reduced to corresponding problems in the theory of rings [[#References|[2]]].
...ed for the description of the tangent space to an arbitrary functor in the category of schemes [[#References|[1]]], [[#References|[3]]].
6 KB (895 words) - 19:36, 5 June 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
noteworthy contributions to probability theory, most
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[[Category:Classical measure theory]]
|valign="top"|{{Ref|Hal}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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...[#References|[a2]]], which contains an excellent history of stratification theory).
5 KB (830 words) - 17:44, 1 July 2020
...TD> <TD valign="top"> O. Zariski, "The problem of minimal models in the theory of algebraic surfaces" ''Amer. J. Math.'' , '''80''' (1958) pp. 146–18
[[Category:Algebraic geometry]]
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[[Category:Harmonic analysis]]
''real-variable theory of, real-variable $\mathcal{H}^p$ theory''
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6) $K$-theory. $K _ { 1 } ( {\cal O} _ { n } ) = 0$; $K _ { 0 } ( \mathcal{O} _ { n } ) =
11 KB (1,565 words) - 07:20, 25 January 2024
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...ives, normals and tangent cones are used in non-smooth analysis, a body of theory concerned with the calculus of functions and sets that do not admit linear
5 KB (716 words) - 17:03, 1 July 2020
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...al operators (mappings) differ in the $C ^ { \infty }$ and in the analytic category, it is natural to investigate the behaviour of such operators (mappings) in
21 KB (3,112 words) - 08:01, 6 February 2024
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...<td valign="top"> P. Erdös, M. Kac, "On certain limit theorems of the theory of probability" ''Bull. Amer. Math. Soc.'' , '''52''' (1946) pp. 292–3
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...="top">[5]</TD> <TD valign="top"> K. Ueno, "Introduction to classification theory of algebraic varieties and compact complex spaces" , ''Lect. notes in math.
...><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Ueno, "Classification theory of algebraic varieties and compact complex spaces" , Springer (1975) {{MR|0
5 KB (747 words) - 06:29, 30 May 2020
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...by W.F. Stinespring, using a very interesting non-commutative integration theory [[#References|[a6]]]. The case of a general locally compact group $G$ was i
4 KB (622 words) - 16:58, 1 July 2020
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...nches of pure and applied mathematics. In [[Probability theory|probability theory]] it means the following theorem: Let $A _ { 1 } , \ldots , A _ { n }$ be e
11 KB (1,700 words) - 06:06, 15 February 2024
...problems concerning flows admit an efficient solution by methods of graph theory.
...revealed the common basis of a number of results obtained before in graph theory and combinatorial analysis. It turned out that the following theorems can b
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[[Category:Analysis]]
...nd smoothness are extensively used in [[Approximation theory|approximation theory]] and Fourier analysis (cf. [[Harmonic analysis|Harmonic analysis]]).
6 KB (907 words) - 21:19, 14 January 2021
[[Category:Classical measure theory]]
...="top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958) {{MR|0117523}}
5 KB (762 words) - 09:52, 16 August 2013
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...l genus of the knot, to the genus of the knot (cf. also [[Knot theory|Knot theory]]), to the planar genus of the knot (from the Seifert construction), to the
6 KB (918 words) - 16:58, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
Asymptotic Theory for Bootstrap Methods in Statistics
14 KB (2,104 words) - 19:26, 4 March 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
of tangency a `contract curve' in his theory of economic exchange
12 KB (1,871 words) - 14:51, 18 March 2023
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...align="top">[a1]</td> <td valign="top"> J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984)</td></tr></table>
5 KB (743 words) - 17:01, 1 July 2020
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A Liouville inequality is one embodying the principle in number theory that algebraic numbers cannot be very well approximated by rational numbers
4 KB (619 words) - 16:56, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
century probability theory. His major probabilistic work, the ''
16 KB (2,329 words) - 12:09, 28 October 2023
...c semi-automaton (cf. [[Automata, algebraic theory of|Automaton, algebraic theory of]]) with set $ S $
...ized in categorical terms, as those for which the forgetful functor to the category of sets preserves coproducts (that is, such that the coproduct of a family
6 KB (889 words) - 08:27, 6 June 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
...arquis de|Laplace]] had convinced him of the essential role of probability theory.
13 KB (1,945 words) - 10:52, 25 February 2024
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...erization of matrix variate elliptically contoured distributions" , ''Adv. Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz'' , Wiley (19
7 KB (992 words) - 07:26, 28 January 2024
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...ails on the use of $r _ { S }$ in hypothesis testing, and for large-sample theory, see [[#References|[a1]]].
7 KB (1,033 words) - 18:56, 22 January 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
architect of the statistical theory of extreme values.
12 KB (1,919 words) - 18:58, 7 March 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
...s false, can then be optimized. A definitive account of the Neyman-Pearson theory
22 KB (3,172 words) - 20:20, 12 March 2016
$#C+1 = 214 : ~/encyclopedia/old_files/data/M065/M.0605010 Morse theory
...ory of categories (cf. [[Category (in the sense of Lyusternik–Shnirel'man)|Category (in the sense of Lyusternik–Shnirel'man)]]).
21 KB (3,095 words) - 08:01, 6 June 2020
[[Category:Markov processes]]
|valign="top"|{{Ref|Dy}}|| E.B. Dynkin, "Foundations of the theory of Markov processes" , Springer (1961) (Translated from Russian) {{MR|01318
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...variation known as E-theory. The foremost application, however, is to the theory of extensions: An extension of $C ^ { * }$-algebras $A$ and $B$ is any $C ^
17 KB (2,644 words) - 17:46, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
was Huber's (1964, 1981) minimax theory, where the statistical problem
13 KB (1,889 words) - 19:48, 13 March 2024
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...in non-commuting variables, which have applications in system and control theory [[#References|[a3]]]. It turns out that the previous characterization in th
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[[Category:Distribution theory]]
...bution as an exact probability distribution is discussed more fully in the theory of random processes (see [[Poisson process|Poisson process]]), where the Po
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...tor from the category of open subsets of $X$ and their inclusions into the category of groups (rings, etc.) and their homomorphisms. The mappings $F_V^U$ are c
...one can consider them from a categorical point of view and apply to sheaf theory the constructions of
26 KB (4,342 words) - 15:06, 15 July 2014
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...l geometry]] and also in Einstein's general [[Relativity theory|relativity theory]]. In the latter it has important physical interpretations and its algebrai
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[[Category:Set functions and measures on topological spaces]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Pr
5 KB (764 words) - 09:39, 16 August 2013
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...f the original mechanical problem, hence it plays an important role in the theory of reductions of Hamiltonian systems.
6 KB (935 words) - 10:58, 2 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
all mathematicians interested in the theory of probability at the beginning
13 KB (1,939 words) - 18:16, 13 August 2023
[[Category:Real functions]]
...levant for absolutely convergent improper integrals, since one can use the theory of Lebesgue. Indeed, consider any Lebesgue measurable set $\Omega$ and a fu
4 KB (598 words) - 21:45, 9 July 2013
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...Even very general density topologies, or also fine topologies of potential theory, have the Luzin–Menshov property, hence they have the Zahorski property a
5 KB (822 words) - 16:46, 1 July 2020
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...ce of a dynamical system (cf. [[Automatic control theory|Automatic control theory]]) is appraised in terms of the $ H ^ \infty $-
7 KB (1,101 words) - 16:08, 27 January 2024
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...of exponential sum encountered in [[Analytic number theory|analytic number theory]]. In the first type, one starts with polynomials $g ( X ) , h ( X ) \in \m
10 KB (1,528 words) - 07:11, 24 January 2024
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...n response [[#References|[a1]]] (cf. also [[Scale-space theory|Scale-space theory]]). However, such edge curves satisfying
6 KB (889 words) - 15:31, 1 July 2020
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The theory of kernel operators (cf. also [[Kernel of an integral operator|Kernel of an
5 KB (722 words) - 15:30, 1 July 2020
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...ally, the Lyapunov equation has many applications in stability and control theory; efficient numerical algorithms for solving it are available.
5 KB (876 words) - 16:52, 1 July 2020
...n="top">[a3]</TD> <TD valign="top"> J.H. van Lint, "Introduction to coding theory" , Springer (1992)</TD></TR>
[[Category:Information and communication, circuits]]
5 KB (676 words) - 17:25, 12 November 2023
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...nacci numbers, groups and manifolds" , ''Combinatorial and Geometric Group Theory (1993)'' , ''Lecture Notes'' , '''204''' , London Math. Soc. (1995) pp. 2
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.... The theory of HNN-groups is central to geometric and combinatorial group theory and should be viewed in parallel with amalgamated products (cf. also [[Amal
6 KB (1,025 words) - 08:55, 29 March 2024
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...ties: positive entropy (cf. [[Entropy theory of a dynamical system|Entropy theory of a dynamical system]]) and [[Ergodicity|ergodicity]]; [[Mixing|mixing]] o
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[[Category:Distribution theory]]
...istribution is known only in a few cases. One of the basic problems in the theory of stable distributions is the description of their domains of attraction (
6 KB (799 words) - 18:06, 14 October 2023
...cular, the points of any locale are exactly its points in the sense of the theory of locales.
...W. Pelletier, "A quantisation of the calculus of relations" , ''Category Theory 1991, CMS Conf. Proc.'' , '''13''' , Amer. Math. Soc. (1992) pp. 345–36
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The theory of such transformations is still under construction. As can be shown, all t
5 KB (720 words) - 17:01, 1 July 2020
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...valign="top"> D. Goss, "On a Fermat equation arising in the arithmetic theory of function fields" ''Math. Ann.'' , '''261''' (1982) pp. 269–286</td>
4 KB (702 words) - 16:45, 1 July 2020
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...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
4 KB (600 words) - 00:06, 15 February 2024
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There are, however, still many open problems in the theory. Apart from difficulties of rather technical nature, the whole scale of the
6 KB (849 words) - 16:08, 27 January 2024
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''Small Sample Theory of the Mean''
16 KB (2,352 words) - 07:17, 19 March 2023
...system, or an inverse spectrum (see [[System (in a category)|System (in a category)]]). Indeed, every complex $ N _ \alpha $
...of topology. After their introduction it became possible to speak about a theory of approximation of complicated topological and algebraic-topological objec
14 KB (2,029 words) - 19:53, 3 February 2021
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
...mack, W.O. and McKendrick, A.G. (1927). A contribution to the mathematical theory of epidemics I. ''Proc.Roy.Soc. A'' '''115''', 700-721. (Parts I,II,and III
13 KB (2,053 words) - 06:46, 9 March 2024
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...s of analytic function theory|Boundary value problems of analytic function theory]]): Given a [[Holomorphic function|holomorphic function]] $f$ in $D$ that i
7 KB (996 words) - 09:31, 3 February 2024
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...f : X \rightarrow Y$; it has applications, for example, in the fixed-point theory for mappings. There are variants of this theorem depending on the choice of
16 KB (2,462 words) - 06:56, 15 February 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
Raymond T. Birge, an influential paper "On the Statistical Theory of
14 KB (2,166 words) - 13:19, 18 March 2023
[[Category:Classical measure theory]]
|valign="top"|{{Ref|Tay}}|| A.E. Taylor, "General theory of functions and integration" , Blaisdell (1965) {{MR|0178100}} {{ZBL|013
5 KB (691 words) - 09:52, 6 July 2013
...|{{Ref|KoFo}}||valign="top"| A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis", '''1–2''', Graylock (1957–1961)
[[Category:Linear and multilinear algebra; matrix theory]]
5 KB (903 words) - 21:31, 3 January 2021
...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
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<tr><td valign="top">[a6]</td> <td valign="top"> E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Oxford Univ. Press (1986) ((revised by D.
5 KB (666 words) - 18:50, 26 January 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
problem" for which his name persists in probability theory.
13 KB (2,017 words) - 09:45, 15 August 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
generalized law of error, via index numbers, the theory of banking,
14 KB (2,144 words) - 13:23, 18 March 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
...' B. Petrov & F. Csáki (eds), ''2nd International Symposium on Information Theory'', Académiai Kiadó, Budapest, pp. 267--281.
12 KB (2,037 words) - 08:10, 10 March 2024
[[Category:Classical measure theory]]
A much deeper theory is available for [[standard Borel space|standard]], [[analytic Borel space|
15 KB (2,605 words) - 07:10, 23 September 2012
[[Category:Stochastic processes]]
|valign="top"|{{Ref|L}}|| M. Loève, "Probability theory" , Princeton Univ. Press (1963) {{MR|0203748}} {{ZBL|0108.14202}}
6 KB (836 words) - 14:55, 7 June 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theoretical hypotheses. As a consequence the modern theory of
13 KB (2,014 words) - 07:23, 25 March 2023
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...ents, method of (in probability theory)|Moments, method of (in probability theory)]]). For example, if $\psi$ is a probability distribution on a set $I \subs
6 KB (888 words) - 18:32, 21 December 2020
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...also [[Knot and link diagrams|Knot and link diagrams]]; [[Knot theory|Knot theory]]).
7 KB (1,078 words) - 15:30, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
thereby anticipating a theory developed much later. The multivariate
13 KB (1,905 words) - 13:07, 18 March 2023
[[Category:TeX done]]
...orphic function (see , ) is one of the basic theorems in analytic function theory, giving for meromorphic functions an analogue of the expansion of a rationa
6 KB (980 words) - 18:47, 24 May 2017
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...<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
The construction of Thom spaces is natural on the category of bundles: Any morphism of (vector) bundles $ f: \xi \rightarrow \eta $
For a multiplicative generalized cohomology theory $ E $(
6 KB (876 words) - 08:25, 6 June 2020
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...e sheets are geodesic lines. There are natural correspondences between the theory of measured foliations, that of measured laminations, and that of measured
5 KB (803 words) - 16:56, 1 July 2020
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...xt of Galois fields than in algebra in general, see [[Galois theory|Galois theory]] and [[Primitive polynomial|Primitive polynomial]].
14 KB (2,032 words) - 14:35, 19 March 2023
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...presentations of these groups analogous to Brauer's modular representation theory of finite groups (cf. also [[Finite group, representation of a|Finite group
33 KB (5,081 words) - 10:26, 11 November 2023
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...th applications in micro-economics are given in [[#References|[a12]]]. The theory and applications of production functions with multiple inputs and of differ
6 KB (958 words) - 15:30, 1 July 2020
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...Onsager and S. Machlup in [[#References|[a1]]], who used this to develop a theory of fluctuations in (non-) equilibrium thermodynamics. Their work was restri
5 KB (784 words) - 16:52, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
pilot's licence). The lectures on the theory of flight which he gave to
17 KB (2,521 words) - 14:28, 15 August 2023
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...a handlebody" M. Gerstenhaber (ed.) J.D. Stasheff (ed.) , ''Deformation Theory and Quantum Groups with Applications to Mathematical Physics'' , ''Contemp.
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An invariant of oriented links (cf. also [[Knot theory|Knot theory]]).
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...d Pisot sequences" F.Q. Gouvea (ed.) N. Yui (ed.) , ''Advances in Number Theory'' , Oxford Univ. Press (1993) pp. 333–340</td></tr><tr><td valign="top"
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[[Category:Analysis]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC
5 KB (736 words) - 09:19, 17 June 2014
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
also dedicated his book on the mathematical theory of the pendulum clock,
16 KB (2,633 words) - 12:03, 1 November 2023
[[Category:Limit theorems]]
A limit theorem in probability theory which is a refinement of the [[Strong law of large numbers|strong law of la
6 KB (875 words) - 08:56, 21 January 2024
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...lian subspace of $\frak g$. This reduction is important in the Cartan–Weyl theory of the classification of irreducible representations, as well as their char
8 KB (1,184 words) - 16:59, 1 July 2020
...y continuous. Therefore, topological spaces and continuous mappings form a category.
...rstrass theorem]]. These and other investigations led to the creation of a theory of rings of continuous functions, the methods of which turned out to be qui
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...al{Z} _ { m + 1 } ^ { \pi }$ to $\mathcal{Z} _ { m} ^{\pi }$. The real $K$-theory groups of $\mathbf{R} \pi$ are given by:
6 KB (850 words) - 14:01, 25 November 2023
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...sphere [[#References|[a3]]]. It is also of fundamental importance for the theory of singular integral equations [[#References|[a4]]], Riemann–Hilbert prob
7 KB (989 words) - 16:56, 1 July 2020
...umbers by rational numbers is one of the more difficult problems in number theory; attempts to solve it yielded very important results, including the Thue, T
...ies, units of cubic fields), A.A. Markov (cubic fields), Yu.V. Sokhotskii (theory of ideals) and others — also made significant contributions.
10 KB (1,645 words) - 17:08, 14 February 2020
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...td valign="top"> M. Born, E. Wolf, "Principles of optics. Electromagnetic theory of propagation, interference and diffraction of light" , Pergamon (1993)
6 KB (935 words) - 17:30, 4 February 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
of the foundation of the modern theory of sample surveys.
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...<td valign="top">[a3]</td> <td valign="top"> P.A.M. Dirac, "The quantum theory of the electron" ''Proc. Royal Soc. London'' , '''A117''' (1928) pp. 610
7 KB (926 words) - 22:43, 12 December 2020
...at quotient mappings with care and that from the point of view of category theory the class of quotient mappings is not as harmonious and convenient as that
...top"> J. Isbell, "A note on complete closure algebras" ''Math. Systems Theory'' , '''3''' (1969) pp. 310–312</TD>
7 KB (1,119 words) - 10:25, 1 January 2018
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
implying "both'' the formulation of economic theory in a
15 KB (2,119 words) - 18:50, 24 March 2023
''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
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...h a process is called a renewal process (cf. also [[Renewal theory|Renewal theory]]) and objects of interest are the behaviour for large (time) $t$:
10 KB (1,526 words) - 19:03, 23 January 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
Jakob also cultivated the theory of series which were published in five
16 KB (2,550 words) - 08:45, 25 March 2023
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...op">[a9]</td> <td valign="top"> N.E. Hurt, "Exponential sums and coding theory. A review" ''Acta Applic. Math.'' , '''46''' (1997) pp. 49–91</td></tr
6 KB (1,050 words) - 00:38, 15 February 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
large-scale sample surveys and planning. For him, theory grew out
13 KB (1,963 words) - 16:40, 19 March 2023
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of $\sigma : E \rightarrow E$ for even $m = 2 n$, of course, spectral theory suggests a complex contour integral representation of the Euler–Frobenius
5 KB (742 words) - 07:27, 25 January 2024
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''Fermi–Thomas theory''
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[[Category:Classical measure theory]]
A concept in measure theory (see also [[Absolute continuity]]). If $\mu$ and $\nu$ are two measures on
6 KB (957 words) - 07:56, 15 December 2016
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A theory of multi-scale representation of sensory data developed by the image proces
7 KB (994 words) - 15:30, 1 July 2020
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...by W. Specht [[#References|[a5]]]. Their importance in the representation theory for symmetric groups (cf. also [[Representation of the symmetric groups]])
8 KB (1,203 words) - 10:18, 11 November 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
The theory does not only apply to independent variables,
13 KB (2,280 words) - 10:48, 28 October 2023
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..." A.G. Ramm (ed.) P.N. Shivakumar (ed.) A.V. Strauss (ed.) , ''Operator Theory And Its Applications'' , ''Fields Inst. Commun.'' , '''25''' , Amer. Math.
6 KB (833 words) - 19:11, 22 January 2024
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...ptic partial differential equation]]). Some of the early highlights of the theory, related to the time-dependent problem, include: Kato's fundamental existen
7 KB (1,062 words) - 17:03, 1 July 2020
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...><tr><td valign="top">[a7]</td> <td valign="top"> A. Turull, "Character theory and length problems" , ''Finite and Locally Finite Groups (Istanbul, 1994)'
5 KB (855 words) - 17:03, 1 July 2020
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...oblems based on possibility theory. Fuzzy programming based on possibility theory is called possibilistic programming. Until now (2000), a lot of fuzzy progr
7 KB (1,115 words) - 17:44, 1 July 2020
...ratorname{Max}B\to\operatorname{Max}A$, so that the affinoid spaces form a category.
...ined for rigid analytic spaces that are analogous to known theorems in the theory of complex spaces. Thus, there are analogues of Cartan's Theorems A and B (
6 KB (892 words) - 20:07, 22 December 2018
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
entitled "Mathematical Theory of Statistics". This was the first
14 KB (2,125 words) - 19:20, 24 March 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
...(A careful analysis is carried out of a very original contribution on the theory of index numbers known as ``Gini's identities'').
13 KB (1,954 words) - 07:48, 14 August 2023
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...n), subject to some naturality conditions. A very readable account of this theory can be found in [[#References|[a2]]]. One particular application of this fo
5 KB (700 words) - 00:42, 15 February 2024
...he arithmetical nature of zeta values and other questions of transcendence theory over $L$, cf. also
[[L-function|$L$-function]]). The invention and basic theory as well as large parts of the deeper results about Drinfel'd modules are du
19 KB (3,204 words) - 20:11, 14 April 2012
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Szegö's theory involves the solution of this extremum problem and related questions such a
7 KB (1,105 words) - 10:02, 11 November 2023
[[Category:Analysis]]
...ite|Fr}} and it is sometimes called Fréchet variation. However the modern theory of functions of bounded variation uses a different generalization (see [
4 KB (637 words) - 12:29, 10 February 2020
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...ntial theory, the theory of poly-analytic and poly-harmonic functions, the theory of [[Eisenstein series]], etc. (see [[#References|[a7]]]).
7 KB (955 words) - 10:00, 16 March 2023
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In the language of design theory (cf. [[Block design|Block design]] and the links given there), a net of ord
7 KB (1,085 words) - 16:58, 1 July 2020
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...generally, Mahler's method has interesting deep connections with automata theory (cf. also [[Formal languages and automata|Formal languages and automata]]).
6 KB (816 words) - 20:26, 5 December 2023
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...) \leq k$ is $\cal N P$-complete for every $k > 3$ (cf. also [[Complexity theory]]).
7 KB (1,139 words) - 20:32, 5 December 2023
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...equation appears in various areas of mathematics and physics, such as the theory of surfaces [[#References|[a4]]], the propagation of sound [[#References|[a
9 KB (1,341 words) - 20:49, 23 January 2024
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...ent developments involving the Fredholm spectrum include the approximation theory of Hilbert-space operators developed by Apostol, D. Herrero, and Voiculescu
5 KB (812 words) - 17:00, 1 July 2020
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...|[a7]]], [[#References|[a8]]], [[#References|[a10]]]. Prime object of this theory is the statistical experiment $ {\mathcal E} = ( \Omega, {\mathcal F}, {\
9 KB (1,287 words) - 16:56, 1 July 2020
[[Category:Classical measure theory]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC
6 KB (1,027 words) - 15:56, 12 November 2013
...y: for system of linear differential equations, and in control and systems theory: for linear input/output systems, linear dynamical systems or linear contro
[[Category:Algebraic geometry]]
6 KB (855 words) - 22:17, 5 June 2020
[[Category:Distribution theory]]
|valign="top"|{{Ref|Lo}}|| M. Loève, "Probability theory" , v. Nostrand (1963) pp. 178 {{MR|0203748}} {{ZBL|0108.14202}}
7 KB (1,007 words) - 04:11, 6 June 2020
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...gnificant property of Asplund spaces, and with application in optimization theory, was established by D. Preiss [[#References|[a6]]], who showed that every l
6 KB (893 words) - 16:59, 1 July 2020
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..." in string theory [[#References|[a9]]] and its precursor, dual resonance theory; the term refers more specifically to the closely related operators used in
8 KB (1,142 words) - 16:46, 1 July 2020
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One major goal of [[Complexity theory|complexity theory]] is to obtain precise bounds on the number of operations performed by algo
7 KB (1,161 words) - 16:46, 1 July 2020
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<tr><td valign="top">[a2]</td> <td valign="top"> M. Carmeli, "Statistical theory and random matrices" , M. Dekker (1983)</td></tr><tr><td valign="top">[a3]
8 KB (1,193 words) - 17:37, 11 November 2023
[[Category:Distribution theory]]
...eory. It is used most effectively in proving limit theorems of probability theory. For example, the proof of the central limit theorem for independent identi
8 KB (1,162 words) - 19:58, 19 January 2024
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...f the unknowns, or for the number of solutions. The development of Baker's theory (cf. also [[Gel'fond–Baker method|Gel'fond–Baker method]]) and its $p$-
7 KB (1,096 words) - 17:46, 1 July 2020
while preserving the orientation. Depending on the category (Diff, PL or Top) from which the terms "submanifold" and "isotopy" in these
dimensional knots (in each category) is an Abelian semi-group with respect to the operation of connected sum. I
13 KB (1,934 words) - 19:26, 17 January 2024
[[Category:Differential geometry]]
[[Category:Partial differential equations]]
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...utions $P$, $Q$. His work was very influential in [[Ergodic theory|ergodic theory]] in connection with generalizations of the [[Ornstein isomorphism theorem|
6 KB (948 words) - 16:55, 1 July 2020
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...4]</td> <td valign="top"> D. Gottlieb, M.Y. Hussaini, S.A. Orszag, "Theory and application of spectral methods" R.G. Voigt (ed.) D. Gottlieb (ed.)
7 KB (1,028 words) - 17:44, 1 July 2020
...nstant multiplier) quadratic differential form that is invariant under the category of statistical decision rules. Because of this fact it arises in the formul
<TR><TD valign="top">[1]</TD> <TD valign="top"> R.A. Fisher, "Theory of statistical estimation" ''Trans. Cambridge Philos. Soc.'' , '''22''' (
6 KB (792 words) - 07:58, 14 January 2024
...ng the exactness axiom. In the category of locally contractible spaces the theory $ H $,
considered with compact support, is isomorphic to the singular theory (cf. [[Singular homology|Singular homology]]). The groups $ H _ {n} ^ {x
9 KB (1,445 words) - 22:10, 5 June 2020
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...om it. The integral is defined for a knot $K$ (cf. also [[Knot theory|Knot theory]]) embedded in the three-dimensional space $ \mathbf{R} ^ { 3 } = \mathbf{
10 KB (1,481 words) - 19:05, 23 January 2024
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...tion theory of coherent algebras as a generalization of the representation theory of finite permutation groups (cf. also [[Finite group, representation of a|
7 KB (1,092 words) - 17:44, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
and originality in economic theory and policy, made him one of the most
17 KB (2,523 words) - 19:38, 21 March 2023
[[Category:Dynamical systems with hyperbolic behavior]]
|valign="top"|{{Ref|A}}|| V.I. Arnol'd, et al., "Theory of bifurcations" , ''Encycl. Math. Sci.'' , '''5. Dynamical systems''' , Sp
7 KB (982 words) - 12:14, 18 August 2014
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...<tr><td valign="top">[a8]</td> <td valign="top"> V.N. Zasukhin, "On the theory of multidimensional stationary processes" ''Dokl. Akad. Nauk SSSR'' , '''3
9 KB (1,297 words) - 09:30, 3 February 2024
[[Category:Branching processes]]
...A.M. Zubkov, "Life-like periods of a branching process with immigration" ''Theory Probab. Appl.'' , '''17''' : 1 (1972) pp. 174–183 ''Teor. Veroyatnost. i
5 KB (775 words) - 06:29, 30 May 2020
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...also an abstract version (see also [[Potential theory, abstract|Potential theory, abstract]]), dealing with harmonic spaces, which states (see [[#References
8 KB (1,180 words) - 17:00, 1 July 2020
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...a1]</td> <td valign="top"> T. Beth, D. Jungnickel, H. Lenz, "Design theory" , Cambridge Univ. Press (1999) (Edition: Second)</td></tr>
5 KB (820 words) - 11:00, 17 March 2023
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For higher dimensions $n \geq 3$ the theory of quasi-regular mappings is essentially different from the plane case $n =
9 KB (1,309 words) - 17:02, 1 July 2020
...ge creates a basis for formalizing the deductive methods of a mathematical theory under investigation. The concept of a formalized language lies at the botto
...; [[Predicate calculus|Predicate calculus]], and [[Types, theory of|Types, theory of]].
9 KB (1,407 words) - 19:01, 21 November 2018
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...nalysis of algorithmic complexity, cf. also [[Complexity theory|Complexity theory]]. (Tarjan's result that the well-known path compression algorithm for set
7 KB (994 words) - 16:46, 1 July 2020
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...ce then, most mathematical attention has been paid to existence-uniqueness theory for the Wigner equation in $\mathbf{R} ^ { 3 }$ and, more recently, in a cl
7 KB (1,053 words) - 16:59, 1 July 2020
[[Category:Classical measure theory]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC
7 KB (1,028 words) - 17:17, 18 August 2012
...is (see [[#References|[6]]], [[#References|[7]]]) a generalization of the theory of minimal models of surfaces to regular two-dimensional schemes. Minimal m
...[[#References|[a2]]] and [[#References|[a4]]] are good surveys of this new theory.
7 KB (1,005 words) - 21:13, 21 November 2018
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...ntation theory|representation theory]], besides [[Spectral theory|spectral theory]].
8 KB (1,099 words) - 08:01, 20 February 2023
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...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
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One of the fundamental results in Ramsey theory.
10 KB (1,506 words) - 00:47, 15 February 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
A key topic in Markov chain theory is the study of the limiting
16 KB (2,671 words) - 20:19, 12 March 2016
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...te noise. The precise description of such a filtering problem requires the theory of stochastic differential equations (e.g., [[#References|[a4]]]; see also
9 KB (1,283 words) - 17:44, 1 July 2020
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...td></tr><tr><td valign="top">[a11]</td> <td valign="top"> T.N. Thiele, "Theory of observations" , Layton, London (1903) (Danish edition 1897 (Reprinted
5 KB (787 words) - 16:55, 1 July 2020
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==Perturbation theory.==
11 KB (1,780 words) - 16:12, 11 February 2024
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A systematic presentation of the theory of Ehresmann connections (under the name of general connections) can be fou
6 KB (926 words) - 16:46, 1 July 2020
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A symplectic approach in terms of Yang–Mills theory for a bundle on a closed $2$-manifold enabled M.F. Atiyah and R. Bott [[#Re
7 KB (1,126 words) - 17:45, 1 July 2020
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...he introduced the concept of "[[generating function]]s" in [[probability theory]]. Closely related to generating functions is the Z-transform, which may be
9 KB (1,351 words) - 19:32, 2 February 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
If it is bounded, there is a nice theory of ''nonparametric
17 KB (2,589 words) - 12:44, 28 October 2023
[[Category:Markov processes]]
...f measurability varies according to the concrete situation. In the general theory of Markov processes one takes the following definition of a functional. Sup
7 KB (1,069 words) - 19:40, 5 June 2020
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in a complex domain is connected to the theory of univalent functions. See [[#References|[a8]]], [[#References|[a5]]] and
8 KB (1,195 words) - 07:36, 23 January 2024
Alternative skew-fields play a substantial part in the theory of projective planes, since a projective plane is a Moufang plane (i.e. a t
...potency of an alternative ring is completely parallel to the corresponding theory for associative rings. This is a consequence of the following fact: Let $
12 KB (1,930 words) - 19:54, 15 March 2023
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...tial procedures for statistical estimation, hypothesis testing or decision theory, the number of observations taken (or sample size) is not pre-determined, b
8 KB (1,155 words) - 19:33, 2 February 2024
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...M. Gromov, "Hyperbolic groups" S.M. Gersten (ed.) , ''Essays in Group Theory'' , Springer (1987) pp. 75–263</td></tr><tr><td valign="top">[a6]</td>
14 KB (2,317 words) - 06:45, 16 March 2023
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..."top"> K. Murasugi, "Jones polynomial and classical conjectures in knot theory" ''Topology'' , '''26''' : 2 (1987) pp. 187–194</td></tr><tr><td vali
7 KB (1,054 words) - 07:42, 10 February 2024
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A [[Quantum field theory|quantum field theory]] contains massive fields if the [[Hilbert space|Hilbert space]] obtained b
9 KB (1,385 words) - 20:57, 8 February 2024
...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
...ogy, which can be described in terms of pointwise convergence. In homotopy theory an important role is played by the path space of a topological space, i.e.
...categories of topological spaces which are Cartesian closed (cf. [[Closed category]]), i.e., roughly, such that $\text{Mor}(X,Y)$ is again a space of the same
8 KB (1,267 words) - 05:59, 22 April 2023
...}) $, and the category of algebraic vector bundles on $ X $ is dual to the category of locally free sheaves of $ \mathcal{O}_{X} $-modules. Moreover, for an $
H. Bass, “Algebraic $ K $-theory”, Benjamin (1968). {{MR|249491}} {{ZBL|}}</TD></TR>
14 KB (2,169 words) - 08:12, 14 December 2016
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...e closely connected to important fields of geometric analysis, e.g., index theory, geometric quantization and several complex variables.
8 KB (1,186 words) - 16:46, 1 July 2020
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The [[Space-time|space-time]] of general [[Relativity theory|relativity theory]] consists of a connected Hausdorff [[Manifold|manifold]] $M$ of dimension
6 KB (975 words) - 15:02, 30 March 2023
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...tower problem (cf. also [[Tower of fields|Tower of fields]]; [[Class field theory]]), that $D$ is finite. The best explicit upper bound for it, $D \leq 92.4$
6 KB (866 words) - 19:51, 24 November 2023
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<tr><td valign="top">[a9]</td> <td valign="top"> W.M. Singer, "Invariant theory and the lambda algebra" ''Trans. Amer. Math. Soc.'' , '''280''' (1983) p
6 KB (860 words) - 16:28, 12 July 2020
...pecial system of theorems that constitute [[Probability theory|probability theory]] and [[Mathematical statistics|mathematical statistics]].
...d to compute new probabilities, in accordance with the laws of probability theory. For instance, if the marksman can hit the target with a probability of 4/1
9 KB (1,350 words) - 18:22, 27 May 2012
...tively and smoothly. A connection on this fibre bundle is a mapping of the category of piecewise-smooth curves in the base $ B $
into the category of diffeomorphisms of the fibres that associates with a curve $ L = L ( x
17 KB (2,530 words) - 19:28, 17 January 2024
...duced to the constructive selection principle in many constructions of the theory of recursive functions (cf. [[Recursive function|Recursive function]]), whi
...ciple and Markov's rule for theories of choice sequences" , ''ISILC. Proof theory symposium'' , ''Lect. notes in math.'' , '''500''' , Springer (1975) pp.
7 KB (1,033 words) - 17:48, 1 November 2014
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...most important unsolved problems in the [[Metric theory of numbers|metric theory of numbers]] (as of 1998). It was inspired by an effort to replace $f ( q )
10 KB (1,454 words) - 07:42, 4 February 2024
...mutativity]]). They are named after N.H. Abel, who used such groups in the theory of solving algebraic equations by means of radicals. It is customary to wri
...e primary with respect to $p$ (the term $p$-group is used in general group theory). Every torsion group splits uniquely into a direct sum of primary groups t
11 KB (1,810 words) - 22:12, 29 August 2015
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...heory" L.W. Beineke (ed.) R.J. Wilson (ed.) , ''Selected Topics in Graph Theory'' , Acad. Press (1978) pp. 15–50</td></tr><tr><td valign="top">[a9]</td
6 KB (1,021 words) - 17:46, 1 July 2020
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...ign="top">[a6]</td> <td valign="top"> K. Hoffman, H. Rossi, "Function theory from a multiplicative linear functional" ''Trans. Amer. Math. Soc.'' , '''
7 KB (1,114 words) - 19:36, 23 December 2023
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...mportant [[Integral transform|integral transform]] used in the diffraction theory of aberrations (see [[#References|[a1]]], Chap. 9): Let $k , l \in {\bf N}
8 KB (1,192 words) - 08:49, 10 November 2023
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...)$ the usual supremum norm (cf. also [[Approximation theory|Approximation theory]]). The subspace $C ( S ) + C ( T )$ might be more properly written as $C (
8 KB (1,276 words) - 07:35, 8 February 2024
...Lie algebras are a generalization of the Poisson algebras, adapted to the theory of infinite-dimensional Hamiltonian systems, where the ring of functions is
In the category of $ \mathbf Z $-
8 KB (1,172 words) - 19:18, 17 March 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
worth a great deal to shape one's explanatory theory into a form in
15 KB (2,197 words) - 14:46, 18 March 2023
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...exist suggest that $m ( P )$ has intimate connections with [[K-theory|$K$-theory]]. For example, $m ( 1 + x + y ) = L ^ { \prime } ( - 1 , \chi _{- 3} )$, w
7 KB (1,101 words) - 14:50, 27 January 2024
[[Category:Classical measure theory]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC
9 KB (1,397 words) - 10:08, 16 August 2013
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...ces|[a1]]], [[#References|[a4]]], and [[#References|[a8]]]. The underlying theory is typically based on averaging, relating (a7) to an associated differentia
7 KB (1,131 words) - 14:45, 16 December 2023
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...er equation]]). It was based on a combination of time-dependent scattering theory and phase-space analysis. In his first work, Enss solved the two-body scatt
9 KB (1,396 words) - 16:57, 1 July 2020
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...which $\{ x \in X : f ( x ) \neq 0 \}$ is meagre, that is, of first Baire category (cf. also [[Baire classes|Baire classes]]). Then $B ( K ) / M ( K )$ is a c
14 KB (2,118 words) - 16:46, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
than in the estimation theory. We proceed now to their
18 KB (2,622 words) - 20:30, 6 March 2024
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...) \right\}$. The theory of cosine functions, which is very similar to the theory of semi-groups, was originated by S. Kurera [[#References|[a2]]] and was de
10 KB (1,449 words) - 17:45, 1 July 2020
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...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
8 KB (1,101 words) - 17:44, 1 July 2020
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...to study both prime rings satisfying generalized identities and the Galois theory of semi-prime rings (cf. also [[Prime ring|Prime ring]]; [[Rings and algebr
9 KB (1,402 words) - 13:37, 25 November 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
...likelihood principle. In ''Second International Symposium on Information Theory''. eds. Petrov B. and Cs\'{a}ki F., Akadémiai Kiad\'{o}, Budapest, 267--28
17 KB (2,653 words) - 16:21, 10 March 2024
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...a3]</td> <td valign="top"> S. Dafermos, A. Nagurney, "On some traffic theory equilibrium paradoxes" ''Transportation Res. B'' , '''18''' (1984) pp. 1
6 KB (983 words) - 17:44, 1 July 2020
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...rences|[a6]]], G. Hermann set out to show that all standard objects in the theory of polynomial ideals over fields $k$, including the prime ideals associated
10 KB (1,498 words) - 14:47, 3 February 2024
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...K _ { 2 } R$. The motivation comes from [[Algebraic K-theory|algebraic $K$-theory]], where this definition fits in well with earlier definitions of $K _ { 0
8 KB (1,339 words) - 16:56, 1 July 2020
[[Category:Ordinary differential equations]]
...branches of mathematics (differential topology, dinamical systems and Lie theory among others).
5 KB (851 words) - 11:10, 30 November 2013
...to non-singular varieties, and in studying these one can use intersection theory and the apparatus of differential forms.
[[Category:Algebraic geometry]]
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...ns given above, but in this case there are relatively few examples and the theory is not so well-developed.
6 KB (1,026 words) - 18:09, 5 August 2020
[[Category:Probabilistic number theory]]
...absolutely-normal numbers was established by Borel on the basis of measure theory. The construction of normal numbers in an explicit form was first achieved
6 KB (960 words) - 08:11, 4 November 2023
...unctions of a complex variable, theory of|Functions of a complex variable, theory of]]) and the [[Fourier transform|Fourier transform]] method [[#References|
...stic bodies, etc. [[#References|[a1]]]. The linear elastic two-dimensional theory is the most developed [[#References|[a3]]], including the process of crack
10 KB (1,410 words) - 19:39, 5 June 2020
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...[Qualitative theory of differential equations in Banach spaces|Qualitative theory of differential equations in Banach spaces]]). Here, $A ( t )$ is the infin
13 KB (1,982 words) - 07:42, 4 February 2024
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category.
<table><tr><td valign="top">[a1]</td> <td valign="top"> H. Weyl, "The
theory of groups and quantum mechanics" , Dover (1931)</td></tr><tr><td valign="t
6 KB (955 words) - 17:44, 1 July 2020
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...roup (and the Alexander and Markov theorems, cf. also [[Braid theory|Braid theory]]; [[Alexander theorem on braids|Alexander theorem on braids]]; [[Markov br
18 KB (2,713 words) - 05:14, 15 February 2024
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...[#References|[a11]]]) and the already tight relations between Banach space theory and [[Harmonic analysis|harmonic analysis]] were further solidified.
6 KB (912 words) - 13:22, 28 October 2023
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...r Toeplitz matrices, as well as problems of minimal realizations in system theory [[#References|[a2]]].
8 KB (1,155 words) - 18:48, 26 January 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory of series. This theorem was the background for a quarrel with the Scotch
19 KB (3,099 words) - 16:34, 14 August 2023
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Applications of branch groups to the theory of the discrete Laplace operator on graphs are given in [[#References|[a2]]
8 KB (1,322 words) - 13:08, 10 February 2024
...ject of an algebraic structure. The concept of an ideal first arose in the theory of rings. The name ideal derives from the concept of an [[Ideal number|idea
of a category with null morphisms is a [[Subobject|subobject]] $ ( U, \mu ) $
13 KB (2,163 words) - 22:11, 5 June 2020
...both the $ n $-dimensional case, and, in connection with problems of the theory of functions of a complex variable, the hyperbolic plane; regular partition
==Basic results of the theory of reflection groups.==
12 KB (1,735 words) - 19:59, 15 March 2023
...aces) depending on parameters (cf. [[Analytic space|Analytic space]]). The theory of deformations originated with the problem of classification of all possib
...made a local study of moduli problems, thus laying the foundations of the theory of deformation of complex manifolds and analytic bundles. An analytic defor
41 KB (5,916 words) - 11:24, 26 March 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
<center>'''Queueing Theory'''</center>
20 KB (3,169 words) - 11:35, 3 April 2016
The (infinite-dimensional) representation theory of semi-simple Lie groups over $ \mathbf R $ has been created for a larg
...Sporadic simple group]]). For a survey of the structure and representation theory of the Chevalley groups see [[#References|[a2]]].
11 KB (1,458 words) - 18:15, 12 December 2019
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...t H ^ { 2 } d A$ is known as the Polyakov extrinsic action and in membrane theory it is the Helfrich free energy.
8 KB (1,192 words) - 16:56, 1 July 2020
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..."> S.A. Telyakovskii, "An estimate, useful in problems of approximation theory, of the norm of a function by means of its Fourier coefficients" ''Proc. S
11 KB (1,550 words) - 20:48, 23 January 2024
<TR><TD valign="top">[a2]</TD> <TD valign="top"> T. Björk, "Arbitrage theory in continuous time" , Oxford Univ. Press (1998)</TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top"> R.C. Merton, "Theory of rational option pricing" ''Bell J. Economics and Management Sci.'' , ''
9 KB (1,337 words) - 18:28, 17 October 2014
[[Category:Analysis]]
...|| A.G. Vitushkin, "Analytic capacity of sets in problems of approximation theory"" Russian Math. Surveys , 22 : 6 (1967) pp. 139–200 Uspekhi Mat. Nauk , 2
5 KB (810 words) - 07:23, 3 October 2012
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==Examples in probability theory.==
9 KB (1,390 words) - 07:41, 4 February 2024
[[Category:Analysis]]
|valign="top"|{{Ref|Pa}}|| R.S. Palais, "Morse theory on Hilbert manifolds", ''Topology'' , '''2''' (1963) pp. 299–340 {{MR|015
5 KB (792 words) - 09:32, 28 June 2014
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The study of adaptive algorithms is generally made through the theory of [[Stochastic approximation|stochastic approximation]].
8 KB (1,242 words) - 15:31, 1 July 2020
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..._ { a } ^ { p }$, B. Korenblum has established a theory that parallels the theory of [[Hardy spaces|Hardy spaces]] with regard to Riesz factorization and inv
12 KB (1,859 words) - 14:41, 27 January 2024
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...="top">[a4]</td> <td valign="top"> H.-H. Kuo, "White noise distribution theory" , CRC (1996)</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> N
9 KB (1,318 words) - 17:43, 1 July 2020
[[Category:Classical measure theory]]
...space" {{Cite|P|Sect. 1.4C, p. 15}}. "Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of m
9 KB (1,257 words) - 05:15, 4 September 2013
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...al analysis" S.S. Gupta (ed.) J.O. Berger (ed.) , ''Statistical Decision Theory and Related Topics IV'' , '''1''' , Springer (1988) pp. 163–175</td></t
6 KB (908 words) - 18:44, 21 March 2024
In this context, "homotopy" means homotopy in the category of differential graded algebras (see [[#References|[a2]]] or [[#References|
...lign="top"> S. Halperin, Y. Félix, J.-C. Thomas, "Rational homotopy theory" , Univ. Toronto (1996) (Preprint)</TD></TR><TR><TD valign="top">[a6]</TD
6 KB (936 words) - 16:08, 1 April 2020
[[Category:Analysis]]
In the theory of optimal control and in the [[Pontryagin maximum principle|Pontryagin max
6 KB (985 words) - 17:11, 22 June 2014
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...le coefficients was posed (avoiding singular points, as needed even in the theory of ordinary differential equations). H. Lewy [[#References|[a5]]] found the
8 KB (1,236 words) - 19:09, 26 March 2023
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<tr><td valign="top">[a1]</td> <td valign="top"> N. Aronszajn, "Theory of reproducing kernels" ''Trans. Amer. Math. Soc.'' , '''68''' (1950) pp
11 KB (1,680 words) - 17:31, 4 February 2024
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...(or posets; cf. also [[Partially ordered set|Partially ordered set]]). The theory of Möbius inversion matured in the classic paper [[#References|[a4]]] of G
11 KB (1,758 words) - 09:24, 10 November 2023
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..._ { 1 + \infty}$-algebra appears as a higher symmetry of conformal field theory.
9 KB (1,398 words) - 10:21, 11 November 2023
The theory of mathematical models of physical events; it holds a special position, bot
...n of the foundations of classical mechanics, universal gravitation and the theory of light (cf. [[Newton laws of mechanics|Newton laws of mechanics]]). Subse
13 KB (1,857 words) - 21:26, 8 November 2014
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A [[Quantum field theory|quantum field theory]] is said to contain massless fields if in the [[Hilbert space|Hilbert spac
10 KB (1,467 words) - 13:12, 2 April 2023
...n="top">[1]</TD> <TD valign="top"> F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , '''1''' , Chelsea, reprint (1977) (Translated from Russian)
...</TD> <TD valign="top"> C.E. Fröberg, "Introduction to numerical analysis, theory and applications" , Benjamin/Cummings (1985)</TD></TR>
7 KB (1,025 words) - 22:14, 5 June 2020
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...ults for this type of stopping functionals which are based on Peano kernel theory; for a survey see [[#References|[a8]]].
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''Arakelov theory''
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Much of the development of the theory of group splittings is parallel to the theory of incompressible embedded surfaces in three-dimensional manifolds (cf. als
9 KB (1,405 words) - 15:30, 1 July 2020
[[Category:Stochastic analysis]]
|valign="top"|{{Ref|LS}}|| R.Sh. Liptser, A.N. Shiryayev, "Theory of martingales" , Kluwer (1989) (Translated from Russian) {{MR|1022664}} {{
10 KB (1,420 words) - 19:10, 9 January 2024
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...applications in commutative algebra, algebraic geometry and algebraic $K$-theory; see [[#References|[a5]]] and [[#References|[a6]]].
8 KB (1,239 words) - 17:01, 1 July 2020
The theory of almost-periodic functions on a group depends essentially on the mean-val
In the theory of almost-periodic functions on a group they play the same role as the func
12 KB (1,716 words) - 11:05, 10 May 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
|valign="top"|{{Ref|1}}||valign="top"| Anderson, T. W. (1948). On the theory of testing serial correlation. ''Skandinavisk Aktuarietidskrift'', '''31'''
22 KB (3,535 words) - 16:20, 10 March 2024
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...fference set|Difference set]] (Vol. 3). For an extensive discussion of the theory of Abelian difference sets, see also [[#References|[a1]]], Chap. VI.
9 KB (1,331 words) - 19:36, 13 February 2024
$#C+1 = 130 : ~/encyclopedia/old_files/data/I052/I.0502040 Intersection theory
The theory of intersections of algebraic subvarieties and cycles. Let $ X $
12 KB (1,730 words) - 22:13, 5 June 2020
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...12]]] have produced singular solutions which are in agreement with Moore's theory. Krasny's method [[#References|[a5]]] used a non-linear filter to remove th
8 KB (1,185 words) - 17:00, 1 July 2020
[[Category:Classical measure theory]]
|valign="top"|{{Ref|KF}}|| A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–1961
8 KB (1,363 words) - 05:26, 8 August 2018
[[Category:Distribution theory]]
...tribution is given by the [[Limit theorems|limit theorems]] of probability theory (see also [[Laplace theorem|Laplace theorem]]; [[Lyapunov theorem|Lyapunov
11 KB (1,515 words) - 08:03, 6 June 2020
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...concepts of a triple system and a supertriple system are important in the theory of quarks and Yang–Baxter equations.
8 KB (1,154 words) - 19:44, 27 February 2021
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The main result of KCC-theory is the following assertion: Two systems of the form (a1) on $\Omega$ are eq
10 KB (1,467 words) - 17:47, 1 July 2020