...t a formal axiomatic theory, obtained within a definite [[Meta-theory|meta-theory]].
...imbedded (in a structure-preserving way) into (a power of) the particular category under consideration.
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[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
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...n every normal epimorphism is a cokernel. In an [[Abelian category|Abelian category]] every epimorphism is normal. The concept of a normal epimorphism is dual
[[Category:Category theory; homological algebra]]
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...ace $X$ with values in a category $\def\cK{ {\mathcal K}}\cK$'' (e.g., the category of sets, groups, modules, rings, etc.)
[[Category|category]] of open sets of $X$ and their natural inclusion mappings into $\cK$. Depe
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...nother. Two categories are equivalent if and only if their [[Skeleton of a category|skeletons]] are isomorphic.
...ts (cf. the editorial comments to [[Category]] for the notion of a Kleisli category of a triple).
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A [[subcategory]] $\mathfrak C$ of a [[category]] $\mathfrak K$ such that for any objects $A$ and $B$ from $\mathfrak C$ on
...lass of its objects. Conversely, any subclass of the class of objects of a category $\mathfrak K$ uniquely defines a full subcategory, for which it serves as t
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A category $\mathfrak K$ in which subcategories of epimorphisms $\mathfrak E$ and of m
...mathfrak E\cap\mathfrak M$ coincides with the class of isomorphisms in the category $\mathfrak R$.
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''of a family of objects in a category''
...t of a family of objects in a category|product of a family of objects in a category]].
6 KB (867 words) - 13:57, 26 December 2017
''in a category''
...t). An equivalent definition of a monomorphism is: For any object $X$ of a category $\mathfrak{K}$ the mapping of sets induced by $\mu$,
2 KB (279 words) - 05:35, 12 January 2017
''of a category''
...ects and the class of morphisms, respectively. The class of morphisms of a category $\mathfrak{K}$ is usually denoted by $\operatorname{Mor} \mathfrak{K}$.
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...rns out to be the kernel of its cokernel. In an [[Abelian category|Abelian category]] every monomorphism is normal. The concept of a normal monomorphism is dua
...isomorphism of $G$ onto a normal subgroup of $H$. However, in an additive category the concepts of normal monomorphism and regular monomorphism coincide.
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''category of sequences''
...relation. Then $\mathbb{Z}$ can be considered as a [[Small category|small category]] with integers as objects and all possible pairs $(i,j)$, where $i,j \in \
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...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...[[Closed monoidal category|closed monoidal category]] (cf. also [[Category|Category]]). A [[Functor|functor]] $( - ) ^ { * } : \cal C ^ { \operatorname{op} } \
3 KB (375 words) - 17:46, 1 July 2020
...f a category|Null object of a category]]). An axiomatic description of the category of groups was given by P. Leroux [[#References|[3]]].
...oup object|Group object]]) in $K$ and the homomorphisms between them; this category has some of the properties of $K$; in particular, it is complete if $K$ is
3 KB (379 words) - 05:17, 12 January 2017
[[Category:Topology]]
...set|nowhere dense sets]] in $X$, otherwise $E$ is said to be of the second category (cp. with Chapter 9 of {{Cite|Ox}}).
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$#C+1 = 28 : ~/encyclopedia/old_files/data/Q076/Q.0706870 Quotient category
be an arbitrary [[Category|category]], and suppose that an equivalence relation $ \sim $
2 KB (279 words) - 08:09, 6 June 2020
''terminal object, of a category''
...ight null object of $\mathfrak{K}$. A left null or ''initial object'' of a category is defined in the dual way.
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An [[Abelian category]] with a set of generators (cf. [[Generator of a category]]) and satisfying the following axiom: There exist [[coproduct]]s (sums) of
...y]]) are Grothendieck categories. A full subcategory $\mathfrak{S}$ of the category ${}_R \mathfrak{M}$ of left $R$-modules is known as a ''localizing subcateg
2 KB (366 words) - 19:42, 30 October 2016
A category $\mathfrak C$ in which for any two objects $X$ and $Y$ an Abelian group str
...null object (zero object, cf. [[Null object of a category|Null object of a category]]) as well as the product $X\times Y$ of any two objects $X$ and $Y$.
3 KB (490 words) - 23:53, 10 December 2018
A [[category]] $\mathfrak{C}$ such that the following axioms are satisfied:
These conditions are equivalent to the following: $\mathfrak{C}$ is a category with given products such that the functors
2 KB (374 words) - 20:31, 27 December 2017
...e subsets often do not); see [[#References|[a1]]], for example. In lattice theory, least upper bounds of directed subsets again play a distinctive part; see
[[Category:Order, lattices, ordered algebraic structures]]
2 KB (292 words) - 06:36, 14 October 2014
...ral transformations, the category of modules over $\Gamma$ is an [[Abelian category]], so one can do [[homological algebra]] with these objects.
...ilarly, equivariant local cohomology can be described using modules over a category depending on the space in question.
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[[Category:Linear and multilinear algebra; matrix theory]]
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...re all sets belonging to $U$, with morphisms and composition as above. The category of sets may be denoted by $\mathfrak S$, ENS, $\mathsf{Set}$ or Me.
...t every epimorphism is split is equivalent to the [[axiom of choice]]. The category of sets has a unique [[Bicategory(2)|bicategory]] (factorization) structure
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[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
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A [[category]] with an additional structure, thanks to which the internal Hom-functor ca
A category $\mathfrak{M}$ is said to be closed if a [[bifunctor]] $\otimes: \mathfrak{
3 KB (412 words) - 20:13, 22 December 2017
...}}(Y,X)$ defines a contravariant functor $h_X$ from $\mathcal{C}$ into the category of sets. For any object $F$ of $\hat{\mathcal{C}}$ there exists a natural b
...ieck functor it is possible to define algebraic structures on objects of a category (cf. [[Group object]]; [[Group scheme]]).
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[[Category:Number theory]]
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$#C+1 = 35 : ~/encyclopedia/old_files/data/M064/M.0604480 Modules, category of
The [[Category|category]] mod- $ R $
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[[Category:Number theory]]
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[[Category:Classical measure theory]]
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[[Category:Classical measure theory]]
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...pological vector space which is not a set of the [[Category of a set|first category]] is ultra-barrelled. If a [[locally convex space]] is ultra-barrelled, it
...gn="top">[1]</TD> <TD valign="top"> R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)</TD></TR>
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...system with multiple inputs and multiple outputs; see [[Automatic control theory]].
[[Category:Control theory and optimization]]
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...an 4 cannot, in general, be solved by radicals (see [[Galois theory|Galois theory]]).
...Many questions of the theory of radicals have been studied within category theory. See also [[Radical of a group|Radical of a group]]; [[Radical in a class o
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...lexes or simplicial decompositions. Simplicial spaces are the objects of a category whose morphisms $X\to Y$ are mappings such that every simplex of the triang
...gical spaces (cf. [[Simplicial object in a category|Simplicial object in a category]]).
2 KB (252 words) - 16:30, 9 April 2014
[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
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...s, the exponential law makes the [[category of sets]] a [[Cartesian-closed category]].
* Benjamin C. Pierce, ''Basic Category Theory for Computer Scientists'', MIT Press (1991) {{ISBN|0262660717}}
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[[Category:Group theory and generalizations]]
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...kernel of a homomorphism of groups, rings, etc. Let $\mathfrak{K}$ be a [[category]] with zero or [[null morphism]]s. A morphism $\mu : K \to A$ is called a k
...$ contains a null object (cf. [[Null object of a category|Null object of a category]]).
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[[Category:Classical measure theory]]
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...reflective if it contains a reflection (cf. [[Reflection of an object of a category]]) for every object of $\mathfrak{K}$. Equivalently, $\mathfrak{C}$ is refl
...$\mathfrak{C}$. Thus, a reflective subcategory of a complete (cocomplete) category is complete (cocomplete).
4 KB (670 words) - 09:05, 26 November 2023
The ''fibre product of objects in a category'' is
...t|(inverse or projective) limit]]. Let $\def\fK{ {\mathfrak K}}\fK$ be a [[category]] and let $\def\a{\alpha}\a : A\to C$ and $\def\b{\beta}\b : B\to C$ be giv
3 KB (575 words) - 10:30, 23 November 2013
...mappings of sets. A [[Morphism|morphism]] $\pi : A \to B$ in a [[Category|category]] $\mathfrak{N}$ is called an epimorphism if $\alpha \, \pi = \beta \, \pi$
...ct of two epimorphisms is an epimorphism. Therefore, all epimorphisms of a category $\mathfrak{N}$ form a subcategory of $\mathfrak{N}$ (denoted by $\operatorn
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See also [[Duality|Duality]] in the theory of [[topological vector space]]s.
[[Category:Linear and multilinear algebra; matrix theory]]
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A [[category]] $\mathcal{C}$ is monoidal if it consists of the following data:
1) a category $\mathcal{C}$;
4 KB (612 words) - 14:59, 6 April 2023
...oherent sheaf is similarly defined on a [[Topologized category|topologized category]] with a sheaf of rings.
gives rise to an equivalence of the category of quasi-coherent sheaves of $ {\mathcal A} $-
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...of several arguments, defined on categories, taking values in a [[Category|category]] and giving a one-place [[Functor|functor]] in each argument. More precise
be given. Construct the Cartesian product category $ \mathfrak K = \overline{\mathfrak K}\; _ {1} \times \dots \times \overl
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...between ($\mathcal{U}$-) categories, and in order to admit other "large" category-theoretic constructions.
...olland (1977) ((especially the article of D.A. Martin on Descriptive set theory))</TD></TR>
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[[Category:Linear and multilinear algebra; matrix theory]]
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''of a category''
...concept of a [[Bicategory(2)|bicategory]]. Suppose that in the [[Category|category]] $ \mathfrak K $
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* Biggs, Norman ''Algebraic graph theory'' 2nd ed. Cambridge University Press (1994) {{ISBN|0-521-45897-8}} {{ZBL|07
[[Category:Graph theory]]
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...mathrm{id}_Y$. In a wider sense, a section of any morphism in an arbitrary category is a right-inverse morphism.
[[Category:Set theory]]
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in an [[Abelian category|Abelian category]] $ C $
...ith enough injective objects (e.g., a [[Grothendieck category|Grothendieck category]] has this property). In such categories an object is injective if and only
4 KB (643 words) - 22:12, 5 June 2020
A [[category]] in which any morphism that is both a [[monomorphism]] and an [[epimorphis
* P.T. Johnstone,"Topos theory", Academic Press (1977) {{MR|0470019}} {{ZBL|0368.18001}}
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[[Category:Group theory and generalizations]]
[[Category:Geometry]]
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''of an object in a category''
be some class of epimorphisms in a [[Category|category]] $ \mathfrak K $
3 KB (460 words) - 08:09, 6 June 2020
of an [[Abelian category|Abelian category]] $ \mathfrak A $
In this context, local smallness of a category is the condition: A collection of representatives of the isomorphism classe
3 KB (469 words) - 16:39, 17 March 2023
''of a category''
of a [[Category|category]] $ C $
6 KB (864 words) - 19:42, 5 June 2020
...le with the category structure. More precisely, a covariant functor from a category $ \mathfrak K $
into a category $ \mathfrak C $
8 KB (1,241 words) - 19:40, 5 June 2020
...= 77 : ~/encyclopedia/old_files/data/S085/S.0805380 Simplicial object in a category
from the category $ \Delta $,
7 KB (966 words) - 21:39, 10 June 2020
...clopedia/old_files/data/P075/P.0705030 Product of a family of objects in a category
be an indexed family of objects in the category $ \mathfrak K $.
3 KB (545 words) - 08:07, 6 June 2020
...well-chosen levels $p$ are used in mathematical statistics and probability theory to characterize the dispersion (scatter) of probability distributions. E.g.
...ign="top">[1]</TD> <TD valign="top"> G.U. Yale, "An introduction to the theory of statistics" , Griffin (1916)</TD></TR></table>
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...similar way. The gradations by the integers play an important role in the theory of projective algebraic varieties or schemes.
...</TD> <TD valign="top"> C. Nâstâsescu, F. van Oystaeyen, "Graded ring theory" , North-Holland (1982)</TD></TR></table>
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...= 1 \}$ except, possibly, for a [[First category (set of)|set of the first category]] on $\Gamma$, are either Plessner points or Meier points. By definition, a
...set, of the [[Plessner theorem]], which is formulated in terms of measure theory. A sharpening of Meier's theorem is given in [[#References|[3]]].
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...induced fibration in topology, and extension of the ring of scalars in the
theory of modules.
...ncyclopediaofmath.org/legacyimages/b/b015/b015310/b0153108.png" />) to the
category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
11 KB (1,513 words) - 17:08, 7 February 2011
...Riesz spaces and arbitrary Riesz homomorphisms is dually equivalent to the
category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...a distinguished strong unit and unit-preserving morphisms and the familiar
category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
8 KB (1,097 words) - 22:47, 29 November 2014
...tegory]], and as a rule, the morphisms (cf. [[Morphism|Morphism]]) in this
category are the mappings preserving the relations of <img align="absmiddle" border=
<TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Zykov, "The
theory of finite graphs" , '''1''' , Novosibirsk (1969) (In Russian)</TD></TR>
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[[Category:Number theory]]
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...separated by a spatial distance and a time duration. In special relativity theory the square of an interval is
In general relativity theory one considers the interval between two infinitesimally-close events:
1 KB (224 words) - 14:51, 30 December 2018
$#C+1 = 90 : ~/encyclopedia/old_files/data/A010/A.0100200 Abelian category
...for an abstract construction of homological algebra [[#References|[4]]]. A category $ \mathfrak A $
10 KB (1,515 words) - 18:19, 31 March 2020
...izable in the category of schemes and require its extensions. However, the category of algebraic spaces is closed under these constructions, which renders the
...e category of schemes becomes identical with a complete subcategory of the category of algebraic spaces.
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[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
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The ''Grothendieck group of an additive category''
...ng property. More exactly, let $C$ be a small [[Additive category|additive category]] with set of objects $\mathrm{Ob}(C)$ and let $G$ be an Abelian group. A m
4 KB (701 words) - 06:11, 26 March 2023
...n="top">[a3]</TD> <TD valign="top"> J.A. Bondy, U.S.R. Murthy, "Graph theory with applications" , Macmillan (1976)</TD></TR></table>
[[Category:Graph theory]]
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A construction that arose originally in set theory and topology, and then found numerous applications in many areas of mathema
be a functor from a [[Small category|small category]] $ \mathfrak D $
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[[Category:TeX done]]
...gn="top">[a2]</TD> <TD valign="top"> P. Odifreddi, "Classical recursion theory" , North-Holland (1989) pp. Chapt. II; esp. pp. 199ff</TD></TR><TR><TD va
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In the Zermelo–Fraenkel axiom system for [[Set theory|set theory]], the sum-set axiom expresses that the union of a set of sets is a set.
If the sets $A_\alpha$ are disjoint, then in the category $\mathbf{Set}$ the union of the objects $A_\alpha$ is the sum of these obje
2 KB (282 words) - 19:46, 8 November 2014
...ng the multiplication of which is commutative (cf. [[Commutativity]]). The theory of associative-commutative rings with a unit is called [[commutative algebr
[[Category:Algebra]]
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...ry the term "pullback" is also used, cf. [[Fibre product of objects in a category]].
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[[Category:Linear and multilinear algebra; matrix theory]]
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...theory of Kleinian groups (cf. [[Kleinian group|Kleinian group]]) and the theory of dynamical systems (cf. e.g. [[Limit set of a trajectory|Limit set of a t
[[Category:General topology]]
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...example, the terminology "class" . And then, speaking more formally, set theory deals with objects called classes (cf. [[Class|Class]]), for which there is
...and allows one to consider, for example, such "large" collections as the category of all sets, groups, topological spaces, etc.
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A construction that first appeared in set theory, and then became widely used in algebra, topology and other areas of mathem
of a category $ \mathfrak K $
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[[Category:Topology]]
...set of a Baire space is itself a Baire space. By the [[Baire theorem|Baire category theorem]], any [[Complete metric space|complete metric space]] is a Baire s
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[[Category:Field theory and polynomials]]
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[[Category:Branching processes]]
* {{Ref|H}} Th.E. Harris, "The theory of branching processes", Springer (1963)
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The name derives from the representation theory of groups: a permutation (respectively, $ R $-
monic) and epimorphisms; hence if the domain category $ \mathfrak C $
2 KB (249 words) - 19:41, 20 January 2021
are one-place covariant functors from a category $ \mathfrak K $
into a category $ \mathfrak C $.
4 KB (489 words) - 16:56, 23 November 2023
''in the theory of functions of a complex variable''
...align="top">[a3]</TD> <TD valign="top"> J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390</TD></TR></t
2 KB (260 words) - 17:48, 1 November 2014
...e basic terms in classical statistics and [[Probability theory|probability theory]]. In the axiomatic approach it is defined as any decomposition of the spac
[[Category:Probability and statistics]]
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...oincide. In this sense, the theory of discrete spaces is equivalent to the theory of [[partially ordered set]]s. If $(P,{\sqsubseteq})$ is a [[pre-order]]ed
[[Category:General topology]]
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[[Category:Classical measure theory]]
...re-measurable function. In all classical particular cases in which measure theory is developed in topological spaces, e.g. in Euclidean spaces, the concept o
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...on–Bourbaki theorem is especially useful for generalizations of the Galois theory of finite, normal and separable field extensions.
...ubfields of $P$ containing $F$ and the subgroups of $G$ (cf. also [[Galois theory]]). The elements of $G$ are linear operators on the [[vector space]] $P$ ov
3 KB (470 words) - 19:10, 9 November 2016
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...ch nowadays (as of 2000) plays a very important role in the representation theory of algebras. One can also consider the dual notion of cotilting modules.
3 KB (459 words) - 16:55, 1 July 2020
''situation, $n$-tuple of strategies, in the theory of non-cooperative games''
...ult of a choice by all coalitions of action (see [[Games, theory of|Games, theory of]]) of their strategies with regard to the connections between the strate
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...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...all category|small category]], $\mathcal{A}$ an [[Abelian category|Abelian category]] with exact infinite products, and $M : \mathcal{C} \rightarrow \mathcal{A
9 KB (1,283 words) - 20:55, 8 February 2024
...="top">[a1]</TD> <TD valign="top"> I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)</TD></TR></table>
[[Category:Number theory]]
667 bytes (98 words) - 18:57, 18 October 2014
$#C+1 = 27 : ~/encyclopedia/old_files/data/V096/V.0906280 Variety in a category
is a [[well-powered category]], that is, the admissible subobjects of any object form a set, then every
3 KB (532 words) - 08:28, 6 June 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...n subset is of the second category in itself (cf. also [[Category of a set|Category of a set]]). A space $X$ is Baire if and only if the intersection of each c
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[[Category:Number theory]]
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[[Category:Topology]]
===Baire category===
4 KB (704 words) - 11:07, 6 September 2013
.../TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.R. Scott, "Group theory" , Dover, reprint (1987) (Original: Prentice-Hall, 1964)</TD></TR></table
[[Category:Group theory and generalizations]]
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''in a category''
A concept in the theory of categories, instances of which are a [[Principal fibre bundle|principal
6 KB (908 words) - 16:33, 7 June 2020
...D valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958)</TD></TR>
[[Category:Functional analysis]]
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[[Category:Graph theory]]
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defined in [[#References|[a1]]], on the [[Category|category]] $ G $-
to the category $ { \mathop{\rm Ab} } $
5 KB (732 words) - 08:45, 26 March 2023
[[Category:Set theory]]
453 bytes (76 words) - 19:46, 8 November 2014
* {{Ref|a1}} G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers", Oxford Univ. Press (1979) {{ZBL|0423.10001}}
[[Category:Number theory]]
659 bytes (104 words) - 15:21, 10 April 2023
A concept in category theory. Other names are [[Triple|triple]], monad and functor-algebra.
Let $\mathfrak{S}$ be a [[category]]. A standard construction is a functor $T : \mathfrak{S} \to \mathfrak{S}$
2 KB (317 words) - 09:03, 10 April 2023
''imputation (in the theory of games)''
...valign="top">[a1]</TD> <TD valign="top"> A. Rapoport, "$N$-person game theory: Concepts and applications" , Univ. Michigan Press (1970) pp. 92; 97–10
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[[Category:Classical measure theory]]
In probability theory, the Jordan decomposition of a probability measure $\mu$ is given as $\mu =
532 bytes (81 words) - 10:44, 17 September 2012
{{MSC|03E04}} ''in set theory''
{{MSC|28A}} ''in measure theory''
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be a covariant functor in one argument from a category $ \mathfrak K $
into a category $ \mathfrak C $.
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[[Category:Linear and multilinear algebra; matrix theory]]
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...structures which form categories similar to an [[Abelian category|Abelian category]]. In the non-Abelian case the direct sum is usually called the discrete di
In category theory, the concept dual to that of a product, i.e. that of a [[Coproduct|coproduc
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$#C+1 = 56 : ~/encyclopedia/old_files/data/S091/S.0901930 System (in a category),
''direct and inverse system in a category $ C $''
4 KB (556 words) - 08:24, 6 June 2020
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...h|graph]], i.e. $O G$ and $A G$ are finite sets. A diagram in a [[Category|category]] $\mathcal{C}$ is defined as a diagram $G \rightarrow U \mathcal{C}$, wher
6 KB (937 words) - 19:26, 11 November 2023
...heory of algebraic groups greatly resembles the role played by tori in the theory of Lie groups. The study of algebraic tori defined over algebraic number fi
2 KB (308 words) - 16:57, 23 March 2012
defines a functor from the category of topological (pointed) spaces into itself.
...spaces (cf. [[Loop space|Loop space]]). For any [[Homology theory|homology theory]] $ h _ {*} $(
3 KB (443 words) - 08:24, 6 June 2020
...alition that is possible under the rules of a strategic game (see [[Games, theory of]]). In games in normal form (see [[Non-cooperative game]]), the direct d
...">[a1]</TD> <TD valign="top"> J. Szép, F. Forgó, "Introduction to the theory of games" , Reidel (1985) pp. Sect. 9.1</TD></TR>
1 KB (161 words) - 16:52, 18 October 2014
...[#References|[a1]]]. A groupoid may conveniently be defined as a (small) [[category]] in which every morphism is an isomorphism; equivalently, it is a set $G$
...ategories, play an important role in many areas of application of category theory, including algebra [[#References|[a2]]], differential geometry [[#Reference
2 KB (288 words) - 20:56, 16 March 2023
<TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Fraïssé, ''Theory of Relations'', Studies in Logic and the Foundations of Mathematics, Elsevi
<TR><TD valign="top">[a2]</TD> <TD valign="top"> P. R. Halmos, ''Naive Set Theory'', Springer (1960, repr. 1974) {{ISBN|0-387-90092-6}} {{ZBL|0287.04001}}</T
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...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...numbers) takes the basic structure of the natural numbers object $N$ in a category $\mathcal{C}$ to consist of the zero element $o : 1 \rightarrow N$ (where $
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...tegory]] and which makes it possible to define the homotopy groups of this category, the homology and cohomology groups with values in an Abelian group, etc.
into the category of sets. An arbitrary covering $ U \rightarrow X $
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...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...mportant example of this strategy is given by the [[Tilting theory|tilting theory]] and the tilting functors, as now described.
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[[Category:Classical measure theory]]
551 bytes (87 words) - 14:22, 23 September 2012
An arbitrary monoid can also be considered as a [[category]] with one object. This allows one to associate with a monoid $ M $
...e definition of a monoid in so-called monoidal categories. Suppose given a category $ \mathfrak M $
4 KB (566 words) - 09:54, 16 March 2023
<TR><TD valign="top">[1]</TD> <TD valign="top"> F. Hausdorff, "Set theory" , Chelsea, reprint (1978) (Translated from German)</TD></TR>
[[Category:General topology]]
831 bytes (114 words) - 22:21, 7 November 2014
of objects of an Abelian category $ \mathfrak A $
...of other homology and cohomology theories. Cf. [[Homology theory|Homology theory]]; [[Cohomology|Cohomology]]; [[Cohomology sequence|Cohomology sequence]];
2 KB (323 words) - 20:24, 17 January 2024
[[Algebraic group|algebraic group]]. Let ${\rm Sch}/S$ be the category of
[[Group object|group object]] of this category is known as a group
5 KB (831 words) - 21:59, 5 March 2012
...zation in categories]]). Inverting all the primes yields rational homotopy theory.
...de Rham theory [[#References|[a2]]]. A third discussion using the de Rham theory on the loop space and employing both of the first two descriptions was done
3 KB (409 words) - 16:41, 17 March 2023
[[Category|category]] is an invertible
for all $\a_1,\a_2,\dots$ in $A$ and all $i\in I$, $j\in J$. Thus, in every category of
3 KB (465 words) - 22:01, 5 March 2012
...categories of modules over rings (cf. [[Grothendieck category|Grothendieck category]]). Let $ \mathfrak A $
be an [[Abelian category|Abelian category]]. A full subcategory $ {\mathfrak A ^ \prime } $
10 KB (1,375 words) - 22:17, 5 June 2020
$#C+1 = 79 : ~/encyclopedia/old_files/data/M065/M.0605040 Motives, theory of
...the various [[Cohomology|cohomology]] theories of algebraic varieties. The theory of motives systematically generalizes the idea of using the [[Jacobi variet
8 KB (1,244 words) - 10:52, 16 March 2023
''in analytic number theory''
<TR><TD valign="top">[a2]</TD> <TD valign="top"> E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951)</TD></TR>
1 KB (160 words) - 20:17, 18 October 2014
<TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Hall, Jr., "The theory of groups" , Macmillan (1964) {{ZBL|0116.25403}}</TD></TR>
[[Category:Order, lattices, ordered algebraic structures]]
779 bytes (109 words) - 19:05, 1 May 2024
...chemes. Let $X$ be a scheme. The étale topology on $X$ is the name for the category $X_{\text{et}}$ of étale $X$-schemes the objects of which are étale morph
...thcal{F}(U)$ such that $F_i^*(s) = S_i$. Many standard concepts of [[sheaf theory]] carry over to étale sheaves (that is, sheaves on $X_{\text{et}}$). For e
3 KB (538 words) - 20:07, 3 June 2017
...o considered it as a geometrical illustration of his investigations in the theory of pencils of [[binary quadratic form]]s.
[[Category:Algebraic geometry]]
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...xiomatic theory of factorization structures $( E , M )$ for morphisms of a category $\frak A$. Here, $E$ and $M$ are classes of $\frak A$-morphisms (the requir
...further assumptions on $E$ and $M$ are required to make the factorization theory useful. A careful analysis has revealed that the crucial requirement that c
11 KB (1,634 words) - 01:54, 13 February 2024
[[Category:Number theory]]
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A (graded) [[Ring|ring]] object in the [[Category|category]] of (graded) co-commutative co-algebras (cf. [[Co-algebra|Co-algebra]]). S
...wn as commutative Hopf algebras with conjugation. Since they belong to the category, they have a coproduct:
7 KB (1,053 words) - 22:11, 5 June 2020
A concept in [[Category|category]] theory. Let $ {\mathcal G} : {\mathcal C} \rightarrow {\mathcal D} $
...mutative associative algebras, Lie algebras, vector spaces, groups) to the category of sets and for a set $ X $,
4 KB (632 words) - 08:27, 6 June 2020
...d in the same way. The differentiable (or analytic) super-manifolds form a category whose morphisms are the morphisms of ringed spaces that are even on the str
...me there are more morphisms in the category of super-manifolds than in the category of vector bundles.
5 KB (723 words) - 08:24, 6 June 2020
The theory of minimal models began with the work of D. Quillen {{Cite|Qu}}. A simply-c
...the homotopy category of simply-connected rational spaces and the homotopy category of connected differential graded Lie algebras (cf. also
7 KB (1,128 words) - 07:52, 9 December 2023
Localization of the additive category $ A ( k) $
with respect to isogeny determines an Abelian category $ M ( k) $,
5 KB (774 words) - 09:04, 8 April 2023
...(cohomology) theory. An axiomatic homology theory is defined on a certain category of pairs $ ( X, A) $
in the category under consideration;
5 KB (779 words) - 19:51, 16 January 2024
...lign="top"> W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)
[[Category:Group theory and generalizations]]
883 bytes (142 words) - 19:33, 11 December 2015
...valign="top">[1]</TD> <TD valign="top"> A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)</TD></TR>
...<TD valign="top"> C.W. Curtis, I. Reiner, "Methods of representation theory" , '''1–2''' , Wiley (Interscience) (1981–1987)</TD></TR>
1 KB (197 words) - 19:18, 25 October 2014
''topologized category.''
...[[Sheaf|sheaf]] on the category. The motivating example has as underlying category the lattice $ {\mathcal O} ( X) $
8 KB (1,216 words) - 18:08, 14 November 2023
...gn="top"> P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Engelking, "Dimension theory" , North-Holland & PWN (1978)</TD></TR>
1 KB (160 words) - 19:51, 3 February 2021
[[Category:Number theory]]
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[[Category:Differential geometry]]
[[Category:Partial differential equations]]
2 KB (319 words) - 09:01, 4 November 2013
<TR><TD valign="top">[a3]</TD> <TD valign="top"> A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1986) {{MR|0855239}} {{ZBL|0
[[Category:Lie theory and generalizations]]
1 KB (177 words) - 19:00, 7 April 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...of manifolds only. This led to an analogous concept of bundle functor on a category over manifolds, [[#References|[a1]]].
4 KB (674 words) - 17:02, 1 July 2020
...ng every piecewise-smooth curve in the base space of a bundle in the given category, which is compatible with the isomorphism of the corresponding fibres of th
...le, a [[Kawaguchi space|Kawaguchi space]]). The foundations of the general theory of non-linear connections are fairly well developed and applications of som
7 KB (1,089 words) - 12:15, 18 February 2022
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...ry) for more exotic spaces in the same way as one did in ordinary homotopy theory for CW-spaces (cf. also [[Cohomology]]; [[Cohomology of a complex]]; [[CW-c
8 KB (1,246 words) - 19:51, 24 December 2023
[[Category:Field theory and polynomials]]
[[Category:History and biography]]
2 KB (312 words) - 19:38, 15 March 2023
...D> <TD valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Sect. 17.7</TD></TR>
[[Category:Number theory]]
1 KB (164 words) - 18:33, 18 October 2014
...re of interest from the point of the view of the recursive analogue of the theory of [[cardinal number]]s.
In [[recursive set theory]] and its applications one also uses certain special subclasses of the clas
1 KB (195 words) - 11:08, 17 March 2023
...ne variety|affine variety]], which plays the role of a local object in the theory of schemes. Let $ A $ be a commutative ring with a unit. An affine scheme c
...first introduced by A. Grothendieck ([[#References|[1]]]), who created the theory of schemes. A scheme is a ringed space that is locally isomorphic to an aff
6 KB (884 words) - 09:29, 13 December 2016
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
..."top">[a1]</td> <td valign="top"> F. Smarandache, "A function in number theory" ''Smarandache Function J.'' , '''1''' (1990) pp. 3–65</td></tr></tabl
903 bytes (140 words) - 16:55, 1 July 2020
...al local theorem, thus making a major contribution to [[Model theory|model theory]]. Later, by improving the method itself, he proved [[#References|[3]]] a l
3 KB (490 words) - 22:23, 26 October 2014
...heory; therefore it can be viewed as the appropriate extension of homotopy theory to general spaces.
...a of operators, or boundaries of certain groups. In all these areas, shape theory has proved useful. In particular, it has applications in the study of cell-
9 KB (1,368 words) - 18:46, 16 March 2024
...ences|[a4]]] that every locally compact locale is spatial, and in fact the category of locally compact locales is equivalent to that of locally compact sober s
...these differences often have the effect of making the former a pleasanter category to work in than the latter. Three examples of this: the property of paracom
7 KB (1,068 words) - 19:43, 23 December 2023
[[Category:Measure-preserving transformations]]
One of the methods of [[Ergodic theory|ergodic theory]]. Any automorphism $ T $
8 KB (1,151 words) - 12:12, 21 March 2022
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian
.../TD> <TD valign="top"> R.C. Lyndon, P.E. Schupp, "Combinatorial group theory" , Springer (1977)</TD></TR>
2 KB (242 words) - 18:26, 26 October 2014
...leason's theorem that extremally-disconnected spaces are projective in the category of Stone spaces. For more details see [[#References|[a1]]].
<TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Koppelberg, "General theory of Boolean algebras" J.D. Monk (ed.) R. Bonnet (ed.) , ''Handbook of Bool
4 KB (532 words) - 17:23, 12 November 2023
[[Category:Ergodic theory]]
...w|continuous flow]]. Measurable flows are used in [[Ergodic theory|ergodic theory]].
956 bytes (174 words) - 16:21, 19 August 2014
[[Category:Markov processes]]
...e|P}}. The model of Brownian motion which is the most important one in the theory of random processes is the so-called [[Wiener process|Wiener process]], and
886 bytes (126 words) - 06:24, 11 May 2012
...valign="top"> I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)</TD></TR>
[[Category:Number theory]]
1 KB (175 words) - 13:22, 14 February 2020
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translate
[[Category:Special functions]]
2 KB (283 words) - 12:43, 14 February 2020
...in defining the fundamental classes of homology), in homological dimension theory, etc.
...><TD valign="top">[4]</TD> <TD valign="top"> E.G. Sklyarenko, "Homology theory and the exactness axiom" ''Russian Math. Surveys'' , '''24''' : 5 (1969)
4 KB (561 words) - 11:33, 22 August 2014
...category]] in a completely similar way, [[#References|[a1]]]. E.g., in the category of sheaves of Abelian groups on a topological space an injective resolution
an injective sheaf. In sheaf theory one often uses resolutions by flabby or soft sheaves (cf. [[Flabby sheaf|Fl
4 KB (611 words) - 01:36, 5 March 2022
[[Category:Linear and multilinear algebra; matrix theory]]
930 bytes (131 words) - 16:56, 30 November 2014
...es|[a2]]], [[#References|[a3]]] for a discussion of the concept of a sound theory and related mathematical principles such as consistency, $\omega$-consisten
<TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Wojcicki, "Theory of logical calculi" , Kluwer (1988) pp. Chapt. 2</TD></TR>
1 KB (207 words) - 19:21, 17 October 2014
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}}
category.
...$L$-fuzzy continuous mappings, and the composition and identities from the
category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
13 KB (1,886 words) - 17:28, 25 November 2023
...ient to use a more general definition: An elementary topos is a [[Category|category]] $\mathcal C$ with products and final object, with a contravariant functor
A topological category is better known as a [[Site|site]].
10 KB (1,557 words) - 09:21, 1 May 2021
[[Category:Dynamical systems and ergodic theory]]
1,020 bytes (154 words) - 19:16, 25 October 2014
...ndition $\Delta_{T_1}=1$, where $T_1$ is the trivial knot (cf. also [[Knot theory]]). For $z=\sqrt t-1/\sqrt t$ one gets the original Alexander polynomial (d
[[Category:Algebraic topology]]
1 KB (155 words) - 07:11, 24 March 2024
A representing object for a generalized cohomology theory. The notion was introduced in [[#References|[1]]] (cf. also [[Generalized c
denotes [[Suspension|suspension]]. Spectra of spaces form a category; a morphism of a spectrum $ \mathbf M $
9 KB (1,281 words) - 14:40, 21 March 2022
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...Auslander and I. Reiten in 1974–1975 and have become a central tool in the theory of representations of finite-dimensional algebras (cf. also [[Representatio
7 KB (1,122 words) - 18:14, 3 April 2024
[[Category:Group theory and generalizations]]
1 KB (153 words) - 16:17, 21 December 2014
...The Koebe function is an extremal function in a number of problems in the theory of univalent functions (cf. [[Bieberbach conjecture]]; [[Univalent function
<TR><TD valign="top">[3]</TD> <TD valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26'''
1 KB (188 words) - 19:36, 5 April 2023
...t space|locally compact]]. Boundedly-compact sets have applications in the theory of approximation in Banach spaces; they have the property that an [[element
...gn="top">[2]</TD> <TD valign="top"> R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)</TD></TR>
1 KB (200 words) - 17:07, 17 October 2014
[[Category:Distribution theory]]
...ibution occur in the study of the fluctuations of random walks, in renewal theory (cf. [[Arcsine law|Arcsine law]]), and are used in mathematical statistics
2 KB (226 words) - 12:50, 11 October 2014
...://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b1202302.png" />-
category''
...nces|[a14]]], [[#References|[a19]]], [[#References|[a22]]] is a [[
Category|
category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org
24 KB (3,338 words) - 17:29, 7 February 2011
...yond", ''Sieve Methods, Exponential Sums, and Their Applications in Number Theory'',
[[Category:Number theory]]
1 KB (220 words) - 10:32, 30 March 2024
<TR><TD valign="top">[2]</TD> <TD valign="top"> V.V. Fedorov, "Theory of optimal experiments" , Acad. Press (1972) (Translated from Russian)</T
...ign="top">[4]</TD> <TD valign="top"> D. Finney, "An introduction to the theory of experimental design" , Univ. Chicago Press (1960)</TD></TR>
2 KB (225 words) - 16:50, 19 October 2014
|valign="top"|{{Ref|Fa2}}||valign="top"| C. Faith, "Algebra" , '''II. Ring theory''' , Springer (1976)
[[Category:Associative rings and algebras]]
948 bytes (141 words) - 19:06, 25 October 2014
...In other words, a co-algebra is the dual concept (in the sense of category theory) to the concept of an associative algebra over a ring $k$.
...extsf{Coalg}_k$ denote, respectively, the category of $k$-algebras and the category of $k$-co-algebras, [[#References|[a2]]]; cf. also [[Hopf algebra]]. But if
4 KB (657 words) - 08:10, 26 November 2023
...$ consists of one prime number $p$, have fundamental significance in group theory. The name is given in honour of L. Sylow, who proved a number of theorems o
...f its Sylow $2$-subgroups. In the theory of infinite groups, except in the theory of locally finite groups, the role of Sylow subgroups is less important, si
2 KB (324 words) - 19:15, 4 April 2023
...R><TD valign="top">[1]</TD> <TD valign="top"> G.G. Hall, "Applied group theory" , Longman (1967)</TD></TR></table>
[[Category:Group theory and generalizations]]
1 KB (202 words) - 16:24, 19 October 2014
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
''braided monoidal category, quasi-tensor category''
17 KB (2,633 words) - 17:42, 1 July 2020
...op"> I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Rus
[[Category:Markov processes]]
1 KB (196 words) - 18:24, 17 October 2014
* Biggs, Norman ''Algebraic graph theory'' 2nd ed. Cambridge University Press (1994) {{ISBN|0-521-45897-8}} {{ZBL|07
[[Category:Graph theory]]
1 KB (180 words) - 17:39, 11 November 2023
[[Category:Limit theorems]]
[[Category:Distribution theory]]
3 KB (488 words) - 20:56, 1 January 2019
The Luzin–Privalov theorems in the theory of functions of a complex variable are classical results of N.N. Luzin and
...a=\{z:\left|z\right|=1\}$ that is metrically dense and of the second Baire category (cf. [[Baire classes]]) on $\sigma$. Then the set $W$ of its radial bo
3 KB (424 words) - 21:56, 24 July 2012
[[Category:Graph theory]]
[[Category:Algorithms]]
3 KB (440 words) - 21:26, 15 November 2014
A term in [[model theory]] denoting the domain of variation of individual (object) variables of a gi
...ign="top">[2]</TD> <TD valign="top"> C.C. Chang, H.J. Keisler, "Model theory" , North-Holland (1973)</TD></TR>
1 KB (219 words) - 10:43, 18 October 2016
''monad, on a category $ \mathfrak R $''
A [[Monoid|monoid]] in the [[Category|category]] of all endomorphism functors on $ \mathfrak R $.
14 KB (2,048 words) - 09:29, 3 July 2021
Quite generally, a torsion theory for an Abelian category $ {\mathcal C} $
from a torsion theory for the category $ R \textrm{ - Mod } $
4 KB (674 words) - 07:14, 12 July 2022
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...X , Y )$, where $\operatorname { rad }$ is the [[Jacobson radical]] of the category of all finite-dimensional $A$-modules. The Auslander–Reiten quiver is a t
9 KB (1,240 words) - 08:04, 25 November 2023
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...t is, it will correspond to a commutative diagram in $HoTop$, the homotopy category of spaces and homotopy classes of mappings, but more is true. The homotopie
9 KB (1,449 words) - 00:50, 15 February 2024
...ther fixed or periodic (cf. [[Poincaré–Bendixson theory|Poincaré–Bendixson theory]]).
...oints are Poisson stable, with the exception of a certain set of the first category of measure zero (cf. [[#References|[1]]], [[#References|[3]]]). A generaliz
4 KB (535 words) - 16:59, 23 March 2023
...n="top">[2]</TD> <TD valign="top"> H. Davenport, "Multiplicative number theory" , Springer (1980)</TD></TR>
[[Category:Number theory]]
1 KB (211 words) - 20:25, 31 October 2014
be finite. The category $ \mathop{\rm mod} kQ $
modules is equivalent to the category of representations of $ Q $(
12 KB (1,755 words) - 19:50, 21 March 2023
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...y|homology theory]] $h_* ^ { S }$ supposed to be defined on the [[Category|category]] of pairs of compact metric (i.e., metrizable) spaces $\bf K$, satisfying
8 KB (1,134 words) - 17:46, 1 July 2020
...n="top">[a2]</TD> <TD valign="top"> C. Kuratowski, "Introduction to set theory and topology" , Pergamon (1961) pp. 128ff (Translated from French)</TD><
[[Category:General topology]]
1 KB (176 words) - 13:57, 9 November 2014
...<TD valign="top"> A.S. Smogorzhevskii, E.S. Stolova, "Handbook of the theory of planar curves of the third order" , Moscow (1961) (In Russian)</TD></T
[[Category:Geometry]]
1 KB (203 words) - 14:02, 9 April 2023
[[Category:Classical measure theory]]
|valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften.
5 KB (753 words) - 09:41, 16 August 2013
...dense in the $w$-plane. The strengthened version of Fatou's theorem in the theory of boundary properties of analytic functions asserts that if $\gamma$ is a
...TR><TD valign="top">[3]</TD> <TD valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26'''
2 KB (264 words) - 21:15, 1 November 2014
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...</td> <td valign="top"> M.A. Krasnoselskij, "Topological methods in the theory of nonlinear integral equations" , Pergamon (1964)</td></tr><tr><td valign
3 KB (554 words) - 07:22, 13 February 2024
[[Category:Distribution theory]]
1 KB (202 words) - 19:41, 5 June 2020
...determined by $\phi$. A more symmetric terminology, borrowed from category theory, calls $\phi$ a left adjoint and $\phi'$ a right adjoint (see [[Adjoint fun
...">[a1]</TD> <TD valign="top"> T.S. Blyth, M.F. Janowitz, "Residuation theory" , Pergamon (1972)</TD></TR>
1 KB (218 words) - 20:53, 8 January 2016
...a deterministic polynomial-time algorithm for primality (cf. [[Complexity theory]]). In addition, they can be used to select keys for public-key cryptograph
...>[a3]</TD> <TD valign="top"> E. Bach, J. Shallit, "Algorithmic number theory" , '''1: Efficient Algorithms''' , MIT (1996)</TD></TR>
3 KB (485 words) - 22:01, 25 October 2014
In [[graph theory]], the '''Tutte matrix''' $A$ of a [[graph]] $G = (V,E)$ is a matrix used t
[[Category:Graph theory]]
1 KB (226 words) - 07:28, 14 November 2023
...between two rings $R$ and $S$ it is necessary and sufficient that in the [[category]] of left $R$-modules there is a finitely-generated projective generator $U
<TR><TD valign="top">[2]</TD> <TD valign="top"> H. Bass, "Algebraic $K$-theory" , Benjamin (1968)</TD></TR>
3 KB (499 words) - 20:17, 26 November 2016
...lign="top">[a1]</TD> <TD valign="top"> R. Gilmer, "Multiplicative ideal theory" , M. Dekker (1972)</TD></TR></table>
[[Category:Associative rings and algebras]]
1 KB (190 words) - 19:52, 2 November 2014
...the analogues of ordinary homotopy and cohomotopy groups in the suspension category — for the $ S $-
...xtra-ordinary (generalized) homology and cohomology theories. A suspension category, or $ S $-
11 KB (1,442 words) - 13:45, 8 June 2020
[[Category:Group theory and generalizations]]
1 KB (187 words) - 21:05, 15 November 2017
[[Category:Number theory]]
1 KB (201 words) - 20:18, 14 October 2014
...n="top">[1]</TD> <TD valign="top"> I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)</TD></TR></table>
...D> <TD valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979)</TD></TR></table>
2 KB (257 words) - 18:02, 8 November 2014
..."top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French)</TD></TR></tabl
...ry]]; [[Sets, category of|Sets, category of]]). More precisely, a concrete category is a pair $ ( {\mathcal C}, U) $
11 KB (1,705 words) - 14:55, 7 June 2020
...e sense of [[Many-one reducibility|$m$-reducibility]] (cf. [[Recursive set theory]]) between [[solvable set]]s and [[creative set]]s. The latter are the larg
<TR><TD valign="top">[3]</TD> <TD valign="top"> H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967)
2 KB (250 words) - 20:46, 23 November 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...l–Katona theorem is probably the most important one in finite extremal set theory.
6 KB (871 words) - 14:52, 11 November 2023
in the derived category $ D _ {c} ^ {b} ( X _ {0} ) $.
in the category of regular holonomic $ D _ {X} $-
5 KB (728 words) - 08:27, 6 June 2020
[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
4 KB (579 words) - 20:31, 18 February 2012
...out to be creative (assuming a natural enumeration of all formulas of the theory); in particular, this is the case for Peano arithmetic and, in general, for
<TR><TD valign="top">[1]</TD> <TD valign="top"> H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967)</
2 KB (295 words) - 06:52, 28 September 2016
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
An immediate consequence is a basic estimate in [[Galois theory|Galois theory]]: If $E$, $F$ are field extensions of a field $K$ and the degree $[ E : K
2 KB (275 words) - 16:55, 1 July 2020
..."+" denotes the functor from the category of topological spaces into the category of pointed spaces $ X ^ {+} = ( X \cup x _ {0} , x _ {0} ) $.
is a generalized cohomology theory in which the Chern classes $ \sigma _ {i} $
5 KB (632 words) - 11:51, 21 March 2022
is often regarded as a functor defined only on the category of CW-complexes of dimension at most $ 2 n - 2 $,
with values in the category of Abelian groups. For CW-complexes $ X $
7 KB (1,086 words) - 17:45, 4 June 2020
...cursively enumerable. Many sets playing an important role in recursive set
theory and its applications are productive (e.g. the set of all Gödel numbers of
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Rogers jr., "
Theory of recursive functions and effective computability" , McGraw-Hill (1967)
2 KB (288 words) - 17:33, 15 November 2014
...actness, so that the use of cohomology proved to be more effective. In the category of metrizable compacta, where the groups $H_p(X,A;G)$ and $H^p(X,A;G^*)$ ar
Homological dimension theory originated from a theorem by P.S. Aleksandrov: The relation $\dim X\leq n$,
5 KB (741 words) - 06:05, 22 November 2018
[[Category:Linear and multilinear algebra; matrix theory]]
1 KB (191 words) - 18:12, 7 February 2017
[[Category:Number theory]]
1 KB (198 words) - 05:56, 15 April 2023
[[Category:Measure-theoretic ergodic theory]]
For further information and references see [[Ergodic theory|Ergodic theory]].
2 KB (260 words) - 08:00, 6 June 2020
...eory|proof theory]]. In order to render the concept of a derivation in the theory effective, additional restrictions must be imposed to ensure that the premi
[[Category:Logic and foundations]]
2 KB (271 words) - 16:50, 2 November 2014
...deal of all functions that vanish off meager sets (cf. [[Category of a set|Category of a set]]). (In either case, the mapping $ T : L \rightarrow M $
...ign="top">[a3]</TD> <TD valign="top"> B.Z. Vulikh, "Introduction to the theory of partially ordered spaces" , Wolters–Noordhoff (1967) (In Russian)</T
3 KB (509 words) - 17:32, 5 June 2020
[[Category:Ergodic theory]]
A property considered in [[Ergodic theory|ergodic theory]]. Originally it was defined for a [[Cascade|cascade]] $\{T^k\}$ or a [[Flo
1 KB (242 words) - 21:18, 8 November 2014
...n="top">[a1]</TD> <TD valign="top"> J.H. van Lint, "Introduction to coding theory" , Springer (1982) {{MR|1540511}} {{ZBL|0485.94015}} </TD></TR>
...of a code via the Delsarte–MacWilliams inequalities" ''IEEE Trans. Inform. Theory'' , '''23''' (1977) pp. 157–166 {{MR|439403}} {{ZBL|}} </TD></TR>
5 KB (779 words) - 07:49, 26 January 2018
...been rarely examined. The major importance of transference theorems in the theory of Diophantine approximation can be explained by a transference theorem in
[[Category:Number theory]]
2 KB (255 words) - 20:56, 25 October 2014
[[Category:Linear and multilinear algebra; matrix theory]]
1 KB (200 words) - 18:13, 26 October 2014
In the theory of finite, or [[Galois field]]s, a ''primitive polynomial'' is a polynomial
[[Category:Field theory and polynomials]]
2 KB (271 words) - 19:29, 2 November 2014
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...lso essential for the [[Closed category|closed category]] structure on the category of crossed complexes. However, the reduced case, i.e. when $C _ { 0 }$ is a
13 KB (1,937 words) - 13:10, 24 December 2020
...some group into account. More precisely, an equivariant cohomology in the category of $ G $-
taking values in the category of Abelian groups) and natural transformations
5 KB (772 words) - 11:12, 20 January 2021
<TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) pp. Chapt.
[[Category:Functions of a complex variable]]
1 KB (199 words) - 17:44, 1 November 2014
...for finding optimal (or $\epsilon$-optimal) strategies of the players. The theory of duels has military as well as economic applications (competition for mar
...n="top">[1]</TD> <TD valign="top"> S. Karlin, "Mathematical methods and theory in games, programming and economics" , Addison-Wesley (1959)</TD></TR></ta
2 KB (294 words) - 12:24, 9 November 2014
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> F. Harary, "Graph theory" , Addison-Wesley (1969) pp. Chapt. 9</TD></TR></table>
...ces|[a3]]]. A good reference for this and other algebraic aspects of graph theory is [[#References|[a1]]]. A related reference is [[#References|[a2]]].
2 KB (330 words) - 20:09, 15 March 2023
<TR><TD valign="top">[a1]</TD> <TD valign="top"> Józef Słomiński, "The theory of abstract algebras with infinitary operations" ''Rozprawy Mat.'' , '''18
[[Category:General algebraic systems]]
1 KB (216 words) - 05:53, 15 April 2023
[[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 (1957–196
2 KB (254 words) - 12:03, 14 December 2012
is a covariant functor from the category of groups into the category of Abelian groups. If $ f : \Pi \rightarrow \Pi $
...//www.encyclopediaofmath.org/legacyimages/w/w097/w097770/w09777031.png" />-theory" , Princeton Univ. Press (1971)</TD></TR></table>
4 KB (564 words) - 08:29, 6 June 2020
...gue of the Riemann hypothesis for elliptic curves by H. Hasse in 1933. The theory of algebraic curves over an arbitrary field of constants, which was develop
...ideals. In particular, he developed the [[Intersection theory|intersection theory]] on a non-singular projective algebraic variety. The results of the studie
8 KB (1,165 words) - 16:08, 1 April 2020
...ry. Gabriel topologies correspond to Serre localizing subcategories of the category of left modules over the ring.
...yen, A. Verschoren, "Reflectors and localization. Application to sheaf theory" , M. Dekker (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="to
2 KB (315 words) - 19:21, 14 August 2014
In algebraic number theory the term integer is also used to denote elements of an algebraic [[number f
...>[a1]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1975) (Translated from Russian) (German translation: Bir
2 KB (283 words) - 17:19, 30 November 2014
<TR><TD valign="top">[3]</TD> <TD valign="top"> G.F. Kangro, "Theory of summability of sequences and series" ''J. Soviet Math.'' , '''5''' (19
...align="top">[4]</TD> <TD valign="top"> S.A. Baron, "Introduction to the theory of summability of series" , Tartu (1966) (In Russian)</TD></TR>
2 KB (276 words) - 20:05, 9 November 2014
A class of special functors from the category of pairs of spaces into the category of graded Abelian groups.
A generalized cohomology theory is a pair $ ( h ^ {*} , \delta ) $,
23 KB (3,297 words) - 19:41, 5 June 2020
...rem [[#References|[1]]]: The transversal mappings form a set of the second category in the set of all continuous mappings $ M \rightarrow Z $.
...oximation theorem holds; see [[#References|[3]]]. Also, in the topological category every continuous mapping is homotopic to a transversal one; this was proved
5 KB (763 words) - 14:33, 27 April 2024
[[Category:Analysis]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Pr
1 KB (197 words) - 23:32, 27 June 2014
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...ame ambient isotopy class of oriented links (cf. also [[Braid theory|Braid theory]]), then one can transform one braid to another by a sequence of Markov mov
2 KB (295 words) - 08:04, 19 March 2023
[[Category:Linear and multilinear algebra; matrix theory]]
1 KB (233 words) - 10:05, 20 December 2015
...of endomorphisms of a projective generator (injective co-generator) of the category of left $ R $-
is faithful if and only if it is a generator of the category of left $ R $-
7 KB (1,087 words) - 08:09, 6 June 2020
[[Category:Group theory and generalizations]]
1 KB (215 words) - 20:26, 4 December 2014
[[Category:Number theory]]
1 KB (197 words) - 13:12, 25 November 2023
...force for the development of [[algebraic number theory]] and [[class field theory]]. A far-reaching generalization of the quadratic reciprocity law is known
[[Category:Number theory]]
2 KB (295 words) - 17:43, 19 December 2014
A general method in number theory which generalizes the principle of sifting [[composite number]]s from the [
The most advanced branch of the general theory of the sieve method and its applications is that of the linear sieve (when
4 KB (636 words) - 19:42, 17 November 2023
...A$ is similarly transformed into an $F$-polygon (cf. [[Automata, algebraic theory of]]).
...may consider a [[wreath product]] construction important in the algebraic theory of automata. The free product (or co-product) of polygons coincides with th
6 KB (1,055 words) - 05:59, 22 April 2023
[[Category:Analysis]]
...of one variable, proposed by Pierpont in {{Cite|Pi}}. However the modern theory of functions of bounded variation uses a different generalization (see
2 KB (261 words) - 12:31, 27 September 2012
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...theorem|Brunn–Minkowski theorem]] and is the basis for the Brunn–Minkowski theory of convex bodies (i.e., compact convex sets).
4 KB (596 words) - 15:30, 1 July 2020
''co-universal square, pull-back square, in a category''
...[1]</TD> <TD valign="top"> I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968)</TD></TR></table>
2 KB (209 words) - 11:42, 8 February 2020
which is a covariant functor on the category of pairs $ ( X, A) $
...invariants. The natural connections with homotopy theory make the singular theory indispensable to homotopical topology.
8 KB (1,130 words) - 08:14, 6 June 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...opology" . The exposition in [[#References|[a3]]] suggested the equivalent category of Hausdorff spaces and [[compactly continuous map]]s (mappings continuous
10 KB (1,545 words) - 18:18, 20 January 2021
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...up|permutation group]]. A classical and beautiful application of character theory is provided in elucidating the structure of Frobenius groups. Namely, let $
9 KB (1,463 words) - 07:41, 27 January 2024
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...opf algebra|Hopf algebra]], or bi-algebra, in a [[Braided category|braided category]]. In physical terms, this is a generalization of both the notion of a quan
21 KB (3,130 words) - 17:42, 1 July 2020
<TR><TD valign="top">[2]</TD> <TD valign="top"> F. Harary, "Graph theory" , Addison-Wesley (1969) pp. Chapt. 9</TD></TR>
[[Category:Graph theory]]
2 KB (303 words) - 20:09, 15 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
''Riesz theory of compact operators''
2 KB (301 words) - 17:00, 1 July 2020
A special method in [[Analytic number theory|analytic number theory]] that uses non-commutative arithmetic and reduces questions on the [[Unifo
.... Malyshev, "A new version of the ergodic method of Yu.V. Linnik in number theory" ''J. Soviet Math.'' , '''11''' (1978) pp. 346–352 ''Zap. Nauchn. Sem. Le
3 KB (340 words) - 17:57, 19 October 2014
...o the inequalities (see [[Saddle point in game theory|Saddle point in game theory]]):
...op"> E. Vilkas, "Axiomatic definition of the value of a matrix game" ''Theory Probabl. Appl.'' , '''8''' (1963) pp. 304–307 ''Teor. Veroyatnost. i P
2 KB (344 words) - 15:57, 14 February 2020
...</TD> <TD valign="top"> A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–196
<TR><TD valign="top">[2]</TD> <TD valign="top"> P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}} </TD></TR>
2 KB (289 words) - 20:54, 25 October 2014
[[Category:Set theory]]
1 KB (230 words) - 21:19, 18 December 2014
...pt. 9</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Ore, "Theory of graphs" , Amer. Math. Soc. (1962)</TD></TR><TR><TD valign="top">[3]</TD
...e above (which is now the four-colour theorem) originally belonged to this category (see [[Graph, planar|Graph, planar]]). Recent surveys on edge colourings ar
5 KB (799 words) - 20:12, 15 March 2023
...matical theory is untrue leads to a contradiction in the theory; since the theory is consistent, this proves that "not A" is untrue, i.e. in accordance wit
[[Category:Special functions]]
2 KB (374 words) - 21:07, 1 November 2014
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...deas have found many applications in, e.g.: [[Scheduling theory|scheduling theory]]; circuit and network design; architectural design; control (cf. [[Control
2 KB (296 words) - 16:58, 1 July 2020
...researches by economists in the 18th century; the basis of modern utility theory was laid in the 1940s by J. von Neumann and O. Morgenstern [[#References|[1
Utility theory also deals with stochastic ordering and ordering of the sums or differences
5 KB (854 words) - 09:14, 7 April 2018
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...} \rightarrow 0$ with $T _ { 1 }$ and $T_2$ in $\operatorname{add} T$, the category of finite direct sums of direct summands of $T$. Here, $\operatorname {p.di
8 KB (1,215 words) - 19:50, 24 December 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...ar potential theory, and many of the results and concepts of the classical theory have been extended to the non-linear setting, sometimes in unexpected ways.
5 KB (766 words) - 20:14, 19 November 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...denotes the projective dimension of $T$ and $\operatorname{add} T$ is the category of finite direct sums of direct summands of $T$ (see [[Tilting module|Tilti
16 KB (2,221 words) - 09:47, 11 November 2023
...align="top">[a2]</TD> <TD valign="top"> P. Odifreddi, "Classical recursion theory" , North-Holland (1989) pp. Chapt. II; esp. pp. 199ff {{MR|0982269}} {{ZBL|
[[Category:Logic and foundations]]
2 KB (233 words) - 17:26, 14 October 2014
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
he also wrote a widely read text-book on probability theory.
6 KB (913 words) - 18:45, 4 March 2024
[[Category:Branching processes]]
|valign="top"|{{Ref|BH}}|| R. Bellman, T.E. Harris, "On the theory of age-dependent stochastic branching processes" ''Proc. Nat. Acad. Sci. U
1 KB (204 words) - 19:33, 28 October 2014
...omorphic or topologically equivalent. They are isomorphic objects in the [[category]] of topological spaces and continuous mappings. A homeomorphism must not b
...tiable function|differentiability]], as a result of the development of set theory and the axiomatic method. This problem, which was explicitly stated for the
11 KB (1,634 words) - 17:03, 23 March 2023
In the theory of coverings (cf. [[Covering (of a set)]]) one also has a notion of irreduc
[[Category:General topology]]
2 KB (241 words) - 08:04, 3 January 2016
...ies may be characterized as those equipped with a forgetful functor to the category of sets which is monadic (cf. [[Triple|Triple]]) and preserves filtered col
...exactly those which are Abelian categories (cf. [[Abelian category|Abelian category]]). Note that the second of these classes of varieties is closed under Mori
10 KB (1,593 words) - 01:28, 17 January 2017
...gn="top">[1]</TD> <TD valign="top"> R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)</TD></TR><TR><TD v
[[Category:Geometry]]
2 KB (274 words) - 20:10, 9 November 2014
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...o J.H.C. Whitehead [[#References|[a9]]] and plays an important role in the theory of algebraic models of homotopy types of spaces.
9 KB (1,326 words) - 16:58, 1 July 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...is residual in some interval of $\mathbf{R}$, cf. also [[Category of a set|Category of a set]]; [[Baire classes|Baire classes]]), then player $A$ has a winning
10 KB (1,579 words) - 17:01, 1 July 2020
A [[Cohomology|cohomology]] theory with values in a [[Sheaf|sheaf]] and with supports contained in a given sub
defines a left-exact functor from the category of sheaves of Abelian groups on $ X $
10 KB (1,477 words) - 22:17, 5 June 2020
(In category theory, this property is used to define ''monomorphisms''.)
1 KB (234 words) - 12:11, 12 December 2013
A partially exact homology theory (cf. [[Homology theory|Homology theory]]) which satisfies the following axiom of compact support: For each element
which is induced by the inclusion. If the homology theory $ H $
4 KB (549 words) - 22:11, 5 June 2020
...<TD valign="top"> K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968) {{MR|0249348}} {{ZBL|0169.37502}}</TD></TR>
...D> <TD valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 {{MR|0568909}
2 KB (318 words) - 09:27, 10 November 2023
...S. Tannaka in [[#References|[a6]]] to generalize the set difference in set theory, and by Y. Imai and Iséki in [[#References|[a3]]] as the algebras of certa
...hat the category of Abelian groups is equivalent to the subcategory of the category of BCI-algebras formed by the $p$-semi-simple BCI-algebras. Here, a [[Homom
5 KB (790 words) - 02:33, 15 February 2024
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...ratorname{Map}( X , Y )$ is a fundamental problem of [[Homotopy|homotopy]] theory. The set of path components, $\pi_0 \; \operatorname { Map } ( X , Y ) = [
8 KB (1,265 words) - 06:16, 15 February 2024
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...ly found wide applicability in a variety of areas of [[Homotopy|homotopy]] theory, most notably in the stable homotopy groups of spheres ([[#References|[a9]]
7 KB (996 words) - 11:15, 20 January 2021
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
of queueing theory for application to telephone traffic and was a precursor
7 KB (990 words) - 13:25, 18 March 2023
[[Category:Number theory]]
2 KB (266 words) - 18:19, 18 October 2014
...top">[1]</TD> <TD valign="top"> S. Karlin, "Mathematical methods in the theory of games, programming and economics" , Addison-Wesley (1959)</TD></TR></ta
[[Category:Game theory, economics, social and behavioral sciences]]
2 KB (322 words) - 12:24, 9 November 2014
<TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1''' , Chelsea (1960) (Translated from Russian)</TD></TR>
[[Category:Group theory and generalizations]]
2 KB (359 words) - 16:54, 23 November 2023
...(cf. [[#References|[7]]], [[#References|[8]]]). The earlier history of the theory of universal algebras goes back to the 19th century. The active study in th
The expression "universal algebra" is often used in the sense of "the theory of universal algebras" .
11 KB (1,662 words) - 04:57, 24 February 2022
...n terms of intersections of cycles (cf. [[Intersection theory|Intersection theory]]).
The Chow ring is the domain of values for the classical theory of Chern classes of vector bundles (cf. {{Cite|Ha}}). More precisely, if $E
4 KB (714 words) - 21:54, 24 April 2012
[[Category:Distribution theory]]
...probability theory and its applications"|"An introduction to probability theory and its applications"]], '''2''', Wiley (1971)
6 KB (843 words) - 11:58, 20 October 2012
Luzin's theorem in the theory of functions of a complex variable (the local principle of finite area) is
Luzin's theorems in descriptive set theory are, by convention, split into three parts. The first and main part is dire
7 KB (1,075 words) - 19:29, 1 January 2021
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
<center>Department of Probability Theory, Statistics and
20 KB (3,391 words) - 12:09, 28 October 2023
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...nt of view for clarifying and unifying various aspects of pointed homotopy theory, but is often heuristic rather than strictly categorical.
13 KB (1,969 words) - 08:10, 15 February 2024
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...uction of the shape category $\operatorname{Sh}$ (cf. [[Shape theory|Shape theory]]). A metric compactum (cf. [[Metric space|Metric space]]; [[Compact space|
4 KB (699 words) - 15:31, 1 July 2020
The branch of the theory of manifolds (cf. [[Manifold|Manifold]]) concerned with the study of relati
is the category of differentiable (smooth) manifolds; $ \mathop{\rm PL} $
15 KB (2,155 words) - 14:56, 7 June 2020
...up, proved by L. Sylow [[#References|[1]]] and playing a major role in the theory of finite groups. Sometimes the union of all three theorems is called Sylow
<TR><TD valign="top">[2]</TD> <TD valign="top"> M. Hall, "Group theory" , Macmillan (1959)</TD></TR>
2 KB (372 words) - 19:17, 4 April 2023
...uch functions is a Cartesian-closed category (cf. [[Closed category|Closed category]]).
...e inclusions, cf. [[Injective object|Injective object]]) in the [[Category|category]] of $T_0$ topological spaces [[#References|[a1]]]; equivalently, they are
11 KB (1,664 words) - 04:50, 15 February 2024
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...CC-group are locally conjugate. The theory of formations of groups and the theory of Fitting classes in finite solvable groups has also been extended to the
2 KB (350 words) - 16:55, 1 July 2020
A basic concept in the branch of the theory of algorithms called enumeration theory, which investigates general properties of classes of objects numbered by ar
...|category]] theory, which permitted one to look at problems in enumeration theory from a new point of view (see [[#References|[4]]]).
12 KB (1,899 words) - 19:37, 5 June 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory of probability.
7 KB (1,004 words) - 19:40, 8 March 2024
[[Category:Stochastic processes]]
is a particular case of a renewal process (cf. [[Renewal theory|Renewal theory]]). To an elementary flow is related the [[Poisson process|Poisson process]
2 KB (312 words) - 19:37, 5 June 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
<tr><td valign="top">[a1]</td><td valign="top"> R.E. Blahut, "Theory and practice of error control codes", Addison-Wesley (1983)</td></tr>
4 KB (622 words) - 19:04, 17 February 2024
$#C+1 = 296 : ~/encyclopedia/old_files/data/H047/H.0407860 Homology theory
...of homotopy, in which deformations are used for the same purpose. Homology theory was introduced towards the end of the 19th century by H. Poincaré (cf. [[H
23 KB (3,393 words) - 08:51, 25 April 2022
...are two such fundamental theories: the singular and the spectral homology theory. The former is based on mappings of polyhedra into given spaces and is most
...is defined on the base of groups of chains; subsequent development of this theory led to the study of ordered, rather than of oriented, simplexes by S. Eilen
9 KB (1,350 words) - 11:39, 25 October 2014
can be considered as a [[Small category|small category]], whose objects are the elements of $ P $
and is empty otherwise. Conversely, every small category in which $ H ( a , b ) \cup H( b, a) $
21 KB (3,255 words) - 11:54, 19 March 2023
...re (later published as [[#References|[a2]]]). This development is based on category-theoretic rather than set-theoretic foundations, and is compatible with inf
...his definition. Much of "infinitesimal" differential geometry, such as the theory of connections, curvature, etc., can be similarly developed in a purely alg
5 KB (681 words) - 08:24, 6 June 2020
...ng the extensions of concepts and notations in constructing a mathematical theory). The available apparatus of deducible rules serves to bring the conversion
[[Category:Logic and foundations]]
2 KB (266 words) - 19:22, 17 October 2014
...of real or $ p $ -adic analytic groups is a complete subcategory in the category of locally compact topological groups. The question of the extent to which
.../TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> C. Chevalley, "Theory of Lie groups" , '''1''' , Princeton Univ. Press (1946) {{MR|0082628}} {{
6 KB (902 words) - 17:45, 12 December 2019
[[Category:Classical measure theory]]
...gy in real analysis, since the class $\mathcal{A}$ is strongly tied to the theory of Lebesgue integration and differentiation. For example, a bounded functio
6 KB (865 words) - 22:40, 17 August 2013
be an Abelian category. A graded object is a sequence $ K = ( K _ {n} ) _ {n \in \mathbf Z } $
A chain complex in a category $ A $
6 KB (913 words) - 19:44, 16 January 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
Fréchet in particular that we owe the first theory
11 KB (1,737 words) - 12:11, 23 November 2023
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''$n$-category''
18 KB (2,710 words) - 00:41, 15 February 2024
[[Category:Markov processes]]
...probability theory and its applications"|"An introduction to probability theory and its applications"]], '''1–2''', Wiley (1966)
2 KB (256 words) - 18:38, 27 April 2024
...h fibre is linear. The set of vector bundles and their morphisms forms the category $ \mathbf{Bund} $.
...ebraic topology, the theory of linear connections, algebraic geometry, the theory of (pseudo-) differential operators, etc.
11 KB (1,695 words) - 20:37, 23 December 2023
[[Category:Limit theorems]]
...}}|| M. Kac, "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer. (1963) {{MR|1530983}} {{MR|0110114}} {{ZBL|0112.09101
2 KB (236 words) - 10:23, 8 February 2017
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
set]]; [[Descriptive set theory|Descriptive set theory]]). The $ {\mathcal A} $-
3 KB (432 words) - 16:59, 1 July 2020
[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
6 KB (884 words) - 09:36, 16 August 2013
[[Category:Classical measure theory]]
[[Category:classical measure theory]]
4 KB (638 words) - 19:35, 1 January 2021
[[Category:Lie theory and generalizations]]
2 KB (253 words) - 22:20, 14 November 2014
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
a major innovator in demographic analysis and apportionment theory and
6 KB (895 words) - 12:27, 9 March 2024
One of the fundamental questions in the theory of Lie groups is the question of how general a character the example given
...algebra defines an equivalence of the category of local Lie groups and the category of finite-dimensional Lie algebras over $ k $.
10 KB (1,553 words) - 22:16, 5 June 2020
...] of a group). The term "solvable group" arose in [[Galois theory|Galois theory]] in connection with the solvability of algebraic equations by radicals.
.... Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR></table>
3 KB (443 words) - 18:25, 26 October 2014
...s of pairwise non-adjacent edges. Such problems occur, for example, in the
theory of scheduling (partitioning of the edges of a bipartite graph into a minima
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> O. Ore, "
Theory of graphs" , Amer. Math. Soc. (1962)</TD></TR></table>
3 KB (475 words) - 13:16, 25 November 2023
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* {{Ref|a2}} A. Kawauchi, "A survey of knot theory", Birkhäuser (1996) {{ZBL|0861.57001}}
2 KB (289 words) - 07:50, 25 November 2023
====Category of $\Sigma$-Algebras====
...$-algebras together with the $\Sigma$-algebra-morphisms forms a [[Category|category]] {{Cite|W90}}.
13 KB (2,332 words) - 14:56, 21 April 2013
...bundle (cf. [[Vector bundle|Vector bundle]]), definitions are given of the category of analytic vector bundles, and of the concepts of a subbundle, a quotient
defines an equivalence between the category of analytic vector bundles over $ X $
6 KB (951 words) - 06:45, 22 February 2022
...(see, for example, [[#References|[5]]]) are equivalent to problems in the theory of representations connected with the complementary series of adèle groups
In the theory of semi-simple Lie groups the notion of a complementary series representati
3 KB (393 words) - 18:23, 26 October 2014
[[Category:Analysis]]
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Pr
2 KB (249 words) - 14:09, 2 May 2014
...bgame are non-zero (cf. also [[Strategy (in game theory)|Strategy (in game theory)]]). In this case the game is almost certainly completed in a finite number
...Milnor, L.S. Shapley, "On games of survival" , ''Contributions to the theory of games'' , '''3''' , Princeton Univ. Press (1957) pp. 15–45</TD></TR>
2 KB (397 words) - 16:58, 18 October 2014
[[Category:Limit theorems]]
...06024.png" />-statistics — which are especially of interest in statistical theory — were studied by many authors over the last 15 years (as of 1988). An im
6 KB (866 words) - 11:09, 12 May 2012
[[Category:Analysis]]
...tial differential equations]], [[Metric geometry]] and [[Geometric measure theory]].
2 KB (257 words) - 16:49, 9 November 2013
<TR><TD valign="top">[2]</TD> <TD valign="top"> F. Harary, "Graph theory" , Addison-Wesley (1969) pp. Chapt. 9</TD></TR>
...n="top">[a1]</TD> <TD valign="top"> R.J. Wilson, "Introduction to graph theory" , Longman (1972) pp. §27</TD></TR>
2 KB (415 words) - 22:37, 26 October 2014
...1 = 79 : ~/encyclopedia/old_files/data/S087/S.0807320 Statistical decision theory
...n a broader interpretation of the term, statistical decision theory is the theory of choosing an optimal non-deterministic behaviour in incompletely known si
12 KB (1,679 words) - 08:23, 6 June 2020
<TR><TD valign="top">[3]</TD> <TD valign="top"> H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967)
...gn="top">[a4]</TD> <TD valign="top"> P. Odifreddi, "Classical recursion theory" , North-Holland (1989) pp. Chapt. II; esp. pp. 199ff</TD></TR>
2 KB (371 words) - 21:13, 2 November 2014
[[Category:Classical measure theory]]
In correspondence with the usual tendency in measure theory to ignore sets of measure zero, there is (and is primarily used) a "modulo
8 KB (1,175 words) - 05:58, 15 May 2012
...[3]</TD> <TD valign="top"> A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Pergamon (1966) (Translated from Russian)</TD></TR></ta
.... Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian)
2 KB (364 words) - 15:57, 14 February 2020
...p">[a1]</TD> <TD valign="top"> J.L. Alperin, "The main problem of block theory" W.R. Scott (ed.) F. Gross (ed.) , ''Proc. Conf. Finite Groups (Park City
[[Category:Group theory and generalizations]]
2 KB (358 words) - 18:45, 13 October 2014
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iii) Seeking higher-dimensional analogues of methods of combinatorial group theory. Later workers would also see algebraic homotopy as including:
9 KB (1,441 words) - 17:45, 1 July 2020
<TR><TD valign="top">[a2]</TD> <TD valign="top"> M.M. Rao, "Measure theory and integration" , Interscience (1987)</TD></TR>
[[Category:Classical measure theory]]
2 KB (378 words) - 15:57, 8 October 2017
...lled "Cantor's [[diagonal process]]" and plays a significant role in set theory (and elsewhere). Cantor's theorem implies that no two of the sets
...es not exist. This means that one must not include among the axioms of set theory the assertion that for each propositional function (or predicate) $\phi(x)$
8 KB (1,230 words) - 20:30, 21 January 2021
...riety of branches of mathematics, for example in group theory and category theory.
3 KB (392 words) - 12:21, 19 August 2014
[[Category:Limit theorems]]
2 KB (303 words) - 11:49, 23 November 2023
...implex in $K_2$. Simplicial complexes and their simplicial mappings form a category.
...|$ is a covariant functor from the category of simplicial complexes to the category of cellular spaces. A topological space $X$ homeomorphic to the body $|K|$
11 KB (1,773 words) - 09:56, 13 February 2024
[[Category:Classical measure theory]]
A concept of classical measure theory.
3 KB (373 words) - 17:15, 18 August 2012
within the framework of the theory of [[scheme]]s. To define the Picard scheme for an arbitrary $ S $-
in the category $ \mathop{\rm Sch}\nolimits /S $
7 KB (1,008 words) - 07:42, 20 March 2024
...seph Oesterlé]] in 1985. It is connected with other problems of [[number theory]]: for example, the truth of the ABC conjecture would provide a new proof
...ves and the abc-conjecture", ed. Wüstholz, Gisbert; ''A panorama in number theory or The view from Baker's garden'', (2002), pp. 128-147, Cambridge Universit
2 KB (362 words) - 19:28, 14 November 2023
...|Pseudo-prime]]) for every such base $a$. These numbers play a role in the theory of probabilistic primality tests (cf. [[Probabilistic primality test|Probab
...top">[a2]</TD> <TD valign="top"> R.D. Carmichael, "Note on a new number theory function" ''Bull. Amer. Math. Soc.'' , '''16''' (1910) pp. 232–238 (S
3 KB (485 words) - 17:34, 18 October 2014
[[Category:Lie theory and generalizations]]
2 KB (303 words) - 18:20, 12 December 2019
''in the theory of functions of a complex variable''
is of the first Baire category (cf. [[Baire classes|Baire classes]]) on $ \Gamma $.
3 KB (354 words) - 04:11, 6 June 2020
Then the category of sheaves of Abelian groups on $ X _ {et} $
is an Abelian category with a sufficient collection of injective objects. The functor $ \Gamma $
5 KB (746 words) - 11:54, 8 April 2023
[[Category:Field theory and polynomials]]
2 KB (296 words) - 19:12, 24 November 2023
.... Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR></table>
[[Category:Group theory and generalizations]]
2 KB (343 words) - 18:24, 26 October 2014
...mation theory [[#References|[a11]]], physics [[#References|[a14]]], coding theory (cf. also [[Coding and decoding|Coding and decoding]]) [[#References|[a12]]
...y of ovals and hyperovals, the theory of spreads and blocking sets and the theory of combinatorial designs [[#References|[a16]]] (cf. also [[Block design|Blo
7 KB (925 words) - 19:43, 9 November 2014
...In fact, it may be claimed that, at a very basic level, logic and category theory are the same.
...used for presenting the foundations of mathematics, and here too category theory has something to say.
24 KB (4,067 words) - 20:57, 21 December 2017
...ies of certain modules over it — in particular, from the properties of the category of all left (or right) modules over this ring (cf. [[Morita equivalence]];
...ribed by formulas of first-order predicate calculus in the language of the theory of modules [[#References|[4]]].
2 KB (327 words) - 07:32, 13 December 2016
...D valign="top">[2]</TD> <TD valign="top"> L.N. Shevrin, "On the general theory of semi-groups" ''Mat. Sb.'' , '''53''' : 3 (1961) pp. 367–386 (In R
[[Category:Group theory and generalizations]]
2 KB (360 words) - 17:24, 14 October 2014
...ed cohomology theories|Generalized cohomology theories]]) generated by the category of vector bundles (cf. [[Vector bundle|Vector bundle]]).
...A second source of $ K $-theory is the connection with algebraic $ K $-theory, which consists of the fact that the space of continuous sections of a vect
17 KB (2,459 words) - 07:32, 26 February 2022
[[Category:Analysis]]
|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
2 KB (301 words) - 14:05, 10 December 2012
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...1</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> M. Sugeno, "Theory of fuzzy integrals and its applications" ''PhD Thesis Tokyo Inst. Technol.
5 KB (729 words) - 19:04, 23 January 2024
...fschetz formula]] for the number of fixed points. The necessity for such a theory was pointed out by A. Weil [[#References|[1]]], who showed that the rationa
from the category of varieties into the category of finite-dimensional graded anti-commutative $ K $-
5 KB (732 words) - 08:12, 21 January 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory to data.
8 KB (1,114 words) - 06:48, 8 October 2023
is a functor from the category of pointed topological spaces into the category of (non-Abelian) groups. For any path $ \phi $
...><TR><TD valign="top">[4]</TD> <TD valign="top"> J.R. Stallings, "Group theory and three-dimensional manifolds" , Yale Univ. Press (1972)</TD></TR>
6 KB (829 words) - 05:44, 13 April 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...ems of [[Functional analysis|functional analysis]] and [[Measure|measure]] theory.
6 KB (845 words) - 16:55, 1 July 2020
[[Category:Analysis]]
|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
2 KB (271 words) - 08:48, 16 September 2012
[[Category:Analysis]]
|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
2 KB (269 words) - 08:46, 16 September 2012
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...[[#References|[a2]]] for problems in [[Approximation theory|approximation theory]]. They have approximately the same number of harmonics as a hyperbolic cro
3 KB (468 words) - 13:55, 21 January 2024
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...fields) by J. Leray [[#References|[a2]]]. Several important results in the theory of differential equations were proved by using domain invariance as a tool.
2 KB (333 words) - 16:45, 1 July 2020
...align="top">[a2]</TD> <TD valign="top"> Y.N. Moschovakis, "Descriptive set theory" , North-Holland (1980)</TD></TR>
[[Category:Topology]]
2 KB (327 words) - 14:21, 15 August 2023
...ition of a simple group given here differs somewhat from that given in the theory of Lie groups and algebraic groups (cf. [[Lie group, semi-simple|Lie group,
In the theory of infinite groups two notions stronger than simplicity are used, viz. thos
3 KB (510 words) - 17:22, 14 October 2014
...Geometry of numbers|geometry of numbers]] as a separate division of number theory. It was established by H. Minkowski in 1896 (see [[#References|[1]]]). Let
...try of numbers]]). These refinements have applications in algebraic number theory and in Diophantine approximation. For a collection of other conditions whic
3 KB (468 words) - 19:26, 15 November 2014
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
statistician who also contributed to Mendelian theory and time-series
8 KB (1,250 words) - 14:47, 18 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...op">[2]</td> <td valign="top"> M.G. Kendall, A. Stuart, "The advanced theory of statistics" , '''2. Inference and relationship''' , Griffin (1979)</td>
4 KB (556 words) - 18:53, 24 January 2024
...joint is faithful if and only if the co-unit is epic. (See also [[Category|Category]].)
10 KB (1,506 words) - 19:36, 5 June 2020
...]</TD> <TD valign="top"> N. Rouche, P. Habets, M. Laloy, "Stability theory by Liapunov's direct method" , Springer (1977)</TD></TR>
[[Category:Ordinary differential equations]]
2 KB (299 words) - 15:31, 14 February 2020
...orms the basis for the application of universal enveloping algebras in the theory of representations of Lie algebras ([[#References|[3]]], [[#References|[4]]
A.A. Kirillov, “Elements of the theory of representations”, Springer (1976). (Translated from Russian)</TD></TR>
6 KB (970 words) - 18:59, 5 April 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
'''Summary.''' K.Jordan, best known for his work on probability theory and finite
11 KB (1,734 words) - 12:02, 1 November 2023
A simplicial object in the category of sets $ \mathop{\rm Ens} $(
cf. [[Simplicial object in a category|Simplicial object in a category]]), that is, a system of sets ( $ n $-
32 KB (4,905 words) - 09:31, 13 February 2024
[[Category:Markov processes]]
|valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Rus
2 KB (354 words) - 22:14, 5 June 2020
[[Category:Classical measure theory]]
|valign="top"|{{Ref|H}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
3 KB (434 words) - 08:00, 6 June 2020
<TR><TD valign="top">[1]</TD> <TD valign="top"> G. Birkhoff, "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc. (1973)</TD></TR>
...="top">[3]</TD> <TD valign="top"> L.A. Skornyakov, "Elements of lattice theory" , A. Hilger & Hindustan Publ. Comp. (1977) (Translated from Russian)
5 KB (726 words) - 13:16, 25 November 2023
...ion''; but $X$ need not be uniquely determined by $A$ and $B$. Both in the theory of modules and in its applications there is a need to describe all differen
...ective hull]] or envelope of $A$. The notion can be defined in any Abelian category, cf. [[#References|[a1]]]. The dual notion is that of a [[projective coveri
2 KB (354 words) - 21:55, 30 October 2016
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...et|Normalizer of a subset]]). One of the principal goals of representation theory (cf. also [[Representation of a group|Representation of a group]]) is to fi
9 KB (1,465 words) - 17:46, 1 July 2020
...gate questions connected with the Krull–Remak–Schmidt theorem; it uses the category of submodules of direct sums of the modules in question.
...></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian
5 KB (786 words) - 22:15, 5 June 2020
[[Category:Ergodic theory]]
A continuous flow in [[Ergodic theory|ergodic theory]] is a family $\{T^t\}$ (where $t$ ranges over the real axis $\R$) of autom
3 KB (480 words) - 02:51, 15 February 2024
...ed group]]). One has also considered framed tangles and graph tangles. The category of tangles, with boundary points as objects and tangles as morphisms, is im
2 KB (336 words) - 09:53, 16 March 2023
...sites (i.e. topological categories; cf. [[Topologized category|Topologized category]]) in dimensions 0, 1. A unified approach to non-Abelian cohomology can be
...-Abelian complexes of differential forms are also an important tool in the theory of pseudo-group structures on manifolds.
12 KB (1,712 words) - 09:30, 20 December 2019
...top">[3]</TD> <TD valign="top"> P. Turán, "An extremal problem in graph theory" ''Mat. Fiz. Lapok'' , '''48''' (1941) pp. 436–452 ((in Hungarian; Ge
...remal graph theory. A comprehensive account of this point of view of graph theory can be found in [[#References|[a1]]].
4 KB (567 words) - 20:10, 15 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...xiomatic potential theory (see also [[Potential theory, abstract|Potential theory, abstract]]), see [[#References|[a1]]], where it is called the axiom of nat
3 KB (449 words) - 17:00, 1 July 2020
[[Category:Markov processes]]
|valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russi
3 KB (412 words) - 19:53, 1 November 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
v) for a (generalized) homology theory $E _ { * }$ there is a natural isomorphism between $E _ { k } ( X )$ and $E
3 KB (432 words) - 16:46, 1 July 2020
[[Category:Analysis]]
..."|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory", '''1''', Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}}
3 KB (384 words) - 11:58, 14 December 2012
[[Category:Classical measure theory]]
A concept in classical measure theory related to the [[Jordan decomposition]]. Consider a [[Algebra of sets|σ-al
3 KB (403 words) - 10:49, 10 December 2012
Cyclic cohomology was developed as a replacement of the de Rham theory in the context of non-commutative algebras. It was discovered independently
homology (cf. also [[K-theory| $ K $-
9 KB (1,226 words) - 17:31, 5 June 2020
[[Category:Distribution theory]]
2 KB (315 words) - 19:35, 5 June 2020
[[Category:Analysis]]
Hardy in {{Cite|Har}} (see also {{Cite|Ha}}). However the modern theory of functions of bounded variation uses a different generalization (see [[F
4 KB (644 words) - 10:52, 10 December 2012
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
apply theory, if only "shop arithmetic", to real statistical problems.
7 KB (1,078 words) - 09:01, 18 March 2023
[[Category:Distribution theory]]
...probability theory and its applications"|"An introduction to probability theory and its applications"]], '''2''' , Wiley (1966)
4 KB (498 words) - 15:35, 4 June 2020
...e 1950s. Their importance is due to the fact that all problems in homotopy theory can be reduced (cf. [[Homotopy type|Homotopy type]]), to a greater or lesse
from the [[Category|category]] of pointed pairs into the category of pointed sets. This functor is homotopy invariant, i.e. $ f _ \star = g
33 KB (4,910 words) - 10:04, 15 December 2019
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...ategory|category]] of (co-) modules of $H$ is a [[Braided category|braided category]] (cf. also [[Quantum groups|Quantum groups]]). This is arguably the key pr
14 KB (2,048 words) - 18:24, 3 August 2020
[[Category:Limit theorems]]
...probability theory and its applications"|"An introduction to probability theory and its applications"]], '''2''' , Wiley (1966) pp. 210
2 KB (324 words) - 18:48, 5 April 2020
is locally nilpotent [[#References|[4]]]. The category of étale $ p $-
divisible groups is equivalent to the category of $ p $-
6 KB (885 words) - 08:04, 6 June 2020
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...homology complex of a Loday algebra is a [[Co-algebra|co-algebra]] in the category of dual–Loday algebras.
9 KB (1,452 words) - 06:17, 15 February 2024
[[Category:Markov chains]]
|valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''1''' , Springer (1975) (Translated from Russi
2 KB (309 words) - 08:26, 6 June 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...gn="top">[a4]</td> <td valign="top"> G. Targonski, "Topics in iteration theory" , Vandenhoeck and Ruprecht (1981) pp. 82ff.</td></tr><tr><td valign="top
2 KB (377 words) - 16:58, 1 July 2020
[[Category:Analysis]]
C. Arzelà in {{Cite|Ar}} (see also {{Cite|Ha}}, p. 543). However the modern theory of functions of bounded variation uses a different generalization (see [[Fu
2 KB (323 words) - 09:38, 16 August 2013
...f schemes $\Spec k(x)\to X$. An important property is the existence in the category of schemes of direct and fibre products (cf.
[[Fibre product of objects in a category|Fibre product of objects in a category]]), which generalize the concept of the tensor product of rings. The underl
12 KB (2,191 words) - 13:20, 24 November 2013
...e exceptional graph. It is at least arguable that the development of graph theory was in large extent due to the interest in the [[Four-colour problem|four-c
...valign="top">[a3]</td> <td valign="top"> W.T. Tutte, "On the algebraic theory of graph colourings" ''J. Combin. Th.'' , '''1''' (1966) pp. 15–50</td
3 KB (472 words) - 19:41, 15 March 2023
[[Category:Classical measure theory]]
A concept in classical measure theory related to the [[Hahn decomposition]]. Consider a [[Algebra of sets|σ-alg
3 KB (419 words) - 12:56, 2 December 2012
[[Category:Markov chains]]
...probability theory and its applications"|"An introduction to probability theory and its applications"]], '''1''', Wiley (1968)
3 KB (415 words) - 16:08, 1 April 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory of probability. In 1708, the first edition of his ''Essay d'analyse
7 KB (1,099 words) - 19:39, 8 March 2024
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...lead to abstract generalizations and analogues of ordinary analytic number theory, which may then be applied in a unified way to further enumeration question
24 KB (3,738 words) - 07:41, 7 February 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
'''Summary'''. Boole's name in probability theory and statistical
9 KB (1,457 words) - 19:16, 24 March 2023
[[Category:Distribution theory]]
[[Category:Stochastic processes]]
8 KB (1,006 words) - 01:14, 19 January 2022
[[Category:Analysis]]
..."|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory", '''1''', Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}}
2 KB (359 words) - 15:24, 9 September 2012
$#C+1 = 423 : ~/encyclopedia/old_files/data/C020/C.0200740 Category
...closed with respect to successive composition (or product) of mappings. A category $ \mathfrak C $
43 KB (6,447 words) - 09:17, 26 March 2023
...oduct|fibre products]]. Indeed, let $\def\fC{ {\mathfrak C}}\fC$ be such a category, and $S$ an object in $\fC$. An object over $\fC$ is a morphism in $\fC$, $
...k')$, $S = \def\Spec{ {\rm Spec}}\Spec(k)$ and $\fC$ is, for instance, the category of (affine) [[Scheme|schemes]] this corresponds to extending scalars.)
9 KB (1,595 words) - 22:23, 22 November 2013
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
==Rotor in graph theory.==
4 KB (579 words) - 16:57, 1 July 2020
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
publication in 1939 of his other great book, called simply ''Theory of Probability''. The substantially revised third edition
9 KB (1,388 words) - 13:03, 18 March 2023
[[Category:Analysis]]
...point of view. For a thorough discussion of the historical aspects of the theory of functions of bounded variation we refer to Section 3.12 of {{Cite|AFP}
3 KB (386 words) - 10:54, 16 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" , '''3. Design and analysis''' , Griffin (1969)</td></tr>
7 KB (873 words) - 07:45, 27 January 2024
...tic methods are used in the last approach (see [[Errors, theory of|Errors, theory of]]).
[[Category:Numerical analysis and scientific computing]]
3 KB (519 words) - 22:31, 1 November 2014
...th the difference that the transition from geometry to algebra in homology theory is irreversible).
...as, the theory of finite-dimensional algebras, the theory of rings and the theory of quadratic forms.
12 KB (1,885 words) - 23:48, 23 April 2017
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
variance and the theory of interval estimation. In 1931 he started
8 KB (1,167 words) - 15:28, 18 March 2023
...an important role in the study of central extensions of groups and in the theory of projective representations of finite groups [[#References|[1]]]. If $
...ion of the class field tower problem (cf. [[Class field theory|Class field theory]]), the Kurosh problem on nil algebras (cf. [[Nil algebra|Nil algebra]]) an
19 KB (2,870 words) - 09:48, 26 March 2023
An invariant of cohomology type associated with schemes in [[Algebraic K-theory|algebraic $ K $-
theory]]. More precisely, in algebraic $ K $-
8 KB (1,120 words) - 20:06, 31 October 2023
[[Category:Classical measure theory]]
|valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
3 KB (459 words) - 14:45, 19 September 2012
...stablishing an isomorphism between graphs is an important problem in graph theory. There are algorithms for certain classes of graphs with the aid of which i
...es|[a1]]] and [[#References|[a2]]] and also [[Complexity theory|Complexity theory]].
4 KB (648 words) - 17:28, 18 October 2014
[[Category:Distribution theory]]
...ins with continuous time. Similarly, one uses Markov chains in reliability theory, where the fault-free operating times of the individual devices can often b
3 KB (485 words) - 19:23, 10 April 2024
...gn="top"> J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) {{MR|0323842}} {{ZBL|0254.17004}} </TD></TR></table>
[[Category:Nonassociative rings and algebras]]
9 KB (1,348 words) - 08:49, 8 April 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...">[2]</td> <td valign="top"> T.W. Anderson, D.A. Darling, "Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes" ''A
3 KB (396 words) - 20:47, 22 January 2024
...o homology, depend contravariantly, as a rule, on the objects of the basic category on which they are defined. In contrast to homology, connecting homomorphism
...of cohomology is dual to that of homology (see [[Homology theory|Homology theory]]; [[Homology group|Homology group]]; [[Aleksandrov–Čech homology and co
16 KB (2,386 words) - 16:47, 20 January 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory of mathematical expectation, were thus already
8 KB (1,269 words) - 18:56, 7 March 2024
be an appropriate [[Cohomology|cohomology]] theory: either $ H ^ {. } X = H ^ {. } ( X, \mathbf Q ) $
...variant functor from the category of schemes and proper morphisms into the category of Abelian groups. In this case, for a [[Proper morphism|proper morphism]]
10 KB (1,385 words) - 03:10, 2 March 2022
[[Category:Topology]]
...ehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978)) {{MR|1034865}} {{MR|0979016}} {{MR|0031025}} {{ZBL|1175.
5 KB (746 words) - 08:32, 18 August 2013
...d $k$ with given [[Galois group|Galois group]] (cf. [[Galois theory|Galois theory]]), and of stating the conditions which ensure the existence (and non-exist
...ological group]]) makes it possible to solve the inverse problem of Galois theory in one stroke for special classes of fields: finite fields, local fields or
4 KB (586 words) - 04:07, 25 February 2022
[[Category:Classical measure theory]]
...p"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory", '''1''', Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}}
3 KB (399 words) - 17:27, 18 August 2012
...orresponding systems based on constructive logic. There are systems of set theory based on constructive logic in which classical systems can be embedded. The
In the theory of constructions one studies the very rules of construction and proof that
10 KB (1,524 words) - 20:25, 4 June 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...valign="top">[a2]</td> <td valign="top"> D. Spring, "Convex integration theory" , ''Monogr. Math.'' , '''92''' , Birkhäuser (1998) {{MR|1488424}} {{ZBL
2 KB (330 words) - 17:00, 1 July 2020
...see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]]) and the theory of non-cooperative non-atomic games (see [[#References|[2]]]).
...lso analogues of the [[Nash theorem (in game theory)|Nash theorem (in game theory)]], as well as general results concerning the existence of equilibrium situ
4 KB (683 words) - 21:44, 8 November 2014
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...="top">[a1]</td> <td valign="top"> Yu.V. Egorov, "A contribution to the theory of generalized functions" ''Russian Math. Surveys'' , '''45''' : 5 (1990
2 KB (342 words) - 16:45, 1 July 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...inted space|Pointed space]]; [[Continuum|Continuum]]; [[Shape theory|Shape theory]]), which induces isomorphisms of the shape groups $f _ { \# } : \check{\pi
4 KB (621 words) - 16:56, 1 July 2020
...s in algebraic geometry (see [[Period mapping|Period mapping]]) and in the
theory of singularities of smooth mappings (see [[#References|[4]]]).
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W.V.D. Hodge, "The
theory and application of harmonic integrals" , Cambridge Univ. Press (1952) {{MR|
9 KB (1,281 words) - 12:14, 12 January 2021
[[Category:Ergodic theorems, spectral theory, Markov operators]]
One of the most important theorems in [[Ergodic theory|ergodic theory]]. For an endomorphism $ T $ of a $ \sigma $-finite measure space $ (X,\Sig
3 KB (460 words) - 05:51, 29 November 2016
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...ber of critical points of even functionals; this can be used, e.g., in the theory of differential equations.
2 KB (353 words) - 16:57, 1 July 2020
...umbers. I.M. Vinogradov in 1937 discovered a new method in analytic number theory — the method of estimating trigonometric sums involving prime numbers —
...d.) , ''Number theory and applications (Proc. First Conf. Canadian Number Theory Assoc., Banff, April 1988)'' , Kluwer (1989) ((Also: Mathematical Centr
3 KB (514 words) - 21:14, 9 January 2015
...alized Riemann hypothesis in various classical problems in additive number theory. There are also various other improvements of density theorems.
...n="top"> A.F. Lavrik, "A survey of Linnik's large sieve and the density theory of zeros of L-functions" ''Russian Math. Surveys'' , '''35''' : 2 (1980)
3 KB (519 words) - 11:35, 26 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...e|$\operatorname{BMO}$-space]]), the theory of integral operators, and the theory of $\overline { \partial }$-equations (cf. also [[Neumann d-bar problem|Neu
3 KB (431 words) - 20:26, 5 December 2023
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...align="top">[a1]</td> <td valign="top"> H. Matsumura, "Commutative ring theory" , Cambridge Univ. Press (1989)</td></tr><tr><td valign="top">[a2]</td> <t
3 KB (406 words) - 17:45, 1 July 2020
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
2) There is a system of first-order logical axioms for [[Set theory|set theory]] (e.g., ZFC) that serves as a basis for mathematics.
4 KB (514 words) - 16:59, 1 July 2020
...hbb R$ and free of oscillatory discontinuities are intensively used in the theory of stochastic processes (cf. [[Stochastic process|Stochastic process]]), wh
|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby, "Measure and category" , Springer (1971).
2 KB (408 words) - 12:10, 30 November 2013
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...N.G. de Bruijn, "The asymptotic behaviour of a function occurring in the theory of primes" ''J. Indian Math. Soc. (N.S.)'' , '''15''' (1951) pp. 25–32
2 KB (376 words) - 08:55, 10 November 2023
[[Category:Analysis]]
* $\frac{1}{2}$ compatibly with the convention, in use in the theory of [[Function of bounded variation|functions of bounded variation]], of def
3 KB (370 words) - 17:22, 20 November 2016
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...who applied it to various problems in [[Approximation theory|approximation theory]] (e.g., Kolmogorov widths, $\varepsilon$-entropy, etc.). Subsequently the
3 KB (472 words) - 01:04, 15 February 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
theory of Brownian motion,
9 KB (1,320 words) - 06:58, 25 March 2023
function fields. The theory of associative rings and algebras became
century. This theory has many contact points with numerous fields of
11 KB (1,726 words) - 20:09, 15 December 2020
An [[Injective object|injective object]] in the category of (right) modules over an associative [[ring with identity]] $ R $,
Baer's criterion). There are "enough" injective objects in the category of $ R $-
7 KB (1,085 words) - 22:12, 5 June 2020
...D></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Zykov, "The theory of finite graphs" , '''1''' , Novosibirsk (1969) (In Russian)</TD></TR></
...><TD valign="top">[a4]</TD> <TD valign="top"> W.K. Chen, "Applied graph theory" , North-Holland (1971)</TD></TR></table>
4 KB (632 words) - 20:10, 15 March 2023
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...)$ are valid for abstract Hardy spaces [[#References|[a2]]]. Using such a theory, J. Wermer [[#References|[a4]]] showed that if the Gleason part $G ( \phi )
3 KB (474 words) - 19:56, 27 February 2021
Baxter algebras originated in the following problem in fluctuation theory: Find the distribution functions of the maxima $\max\{0, S_1, \ldots, S_n \
...[#References|[a3]]] proved that the standard Baxter algebra is free in the category of Baxter algebras (cf. also [[Free algebra]]).
6 KB (960 words) - 07:40, 18 November 2023
[[Category:Set functions and measures on topological spaces]]
..."top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958) {{MR|0117523}}
3 KB (393 words) - 16:15, 29 November 2012
...of modules were connected with the application of methods and ideas of the theory of categories (cf.
[[Category|Category]]), in particular, methods of
23 KB (3,918 words) - 04:31, 23 July 2018
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...inear mapping]] $V \times V \times V \rightarrow V$, is mainly used in the theory of non-associative algebras and appears in the construction of Lie algebras
3 KB (498 words) - 19:51, 27 February 2021
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
'''Summary.''' Lundberg was a founder of mathematical Risk Theory.
9 KB (1,437 words) - 14:21, 10 March 2024
[[Category:Stochastic processes]]
...p"|{{Ref|GSY}}|| I.I. Gikhman, A.V. Skorokhod, M.I. Yadrenko, "Probability theory and mathematical statistics" , Kiev (1979) (In Russian) {{MR|2026607}} {{ZB
3 KB (480 words) - 08:06, 6 June 2020
...zability. Open mappings with discrete fibres play an important role in the theory of functions of one complex variable: these include all holomorphic functio
...D></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1–2''' , Moscow (1962) (In Rus
3 KB (511 words) - 08:04, 6 June 2020
...">[a2]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc. (1961–1967)</TD></TR>
[[Category:Order, lattices, ordered algebraic structures]]
3 KB (413 words) - 09:17, 2 April 2023
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
|valign="top"|{{Ref|5}}||valign="top"| Durrett, R. (1995). ''Probability: Theory and Examples''. 2nd edition. Duxbury, Belmont.
11 KB (1,725 words) - 19:55, 5 March 2024
[[Category:Stochastic processes]]
|valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russi
3 KB (473 words) - 08:23, 6 June 2020
[[Category:Stochastic processes]]
A notion used in probability theory for random variables having the property of independence of the "future" .
4 KB (526 words) - 07:59, 6 June 2020
[[Category:Ergodic theorems, spectral theory, Markov operators]]
2 KB (340 words) - 08:12, 9 January 2024
[[Category:TeX done]]
...><TR><TD valign="top">[4]</TD> <TD valign="top"> M. Loève, "Probability theory" , Princeton Univ. Press (1963)</TD></TR></table>
4 KB (642 words) - 11:43, 14 June 2017
.../TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1–2''' , Chelsea (1977) (Trans
<TR><TD valign="top">[a3]</TD> <TD valign="top"> R. von Mises, "Theory of flight" , Dover, reprint (1959)</TD></TR>
4 KB (542 words) - 19:58, 4 January 2024
...icense'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
mathematically-based study entitled "The Theory of Limiting Utility" which
9 KB (1,322 words) - 16:31, 4 March 2024
...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...d valign="top">[a3]</td> <td valign="top"> W. Feit, "The representation theory of finite groups" , North-Holland (1982)</td></tr><tr><td valign="top">[a4
3 KB (540 words) - 17:43, 1 July 2020
...tive algebraic groups [[#References|[4]]], [[#References|[5]]], and in the theory of formal groups [[#References|[6]]]. Let $ A $
...functor from the category of commutative rings with unit element into the category of rings. This functor may be represented by the ring of polynomials $ \m
17 KB (2,502 words) - 17:25, 22 December 2019
...ebraic (cf. [[#References|[a4]]]). An illuminating account of the analytic theory may be found in [[#References|[a15]]] (cf. also [[#References|[a2]]], [[#Re
For an adequate setting of the theory of $ \mathcal D $-modules the machinery of derived categories and derived
24 KB (3,511 words) - 07:03, 10 May 2022
[[Category:Markov chains]]
...probability theory and its applications"|"An introduction to probability theory and its applications"]], '''1''' , Wiley (1966)
3 KB (400 words) - 16:12, 30 July 2014
A theory studying continuous families of objects in algebraic geometry.
...ructures on the parameter sets. The second part forms the matter of moduli theory.
16 KB (2,402 words) - 11:49, 16 December 2019
[[Category:Field theory and polynomials]]
2 KB (384 words) - 07:34, 18 December 2014
[[Category:Markov chains]]
|valign="top"|{{Ref|BF}}|| A. Blanc-Lapierre, R. Fortet, "Theory of random functions" , '''1–2''' , Gordon & Breach (1965–1968) (Tra
3 KB (325 words) - 07:59, 6 June 2020