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The same as an inverse semi-group (cf. [[Inversion semi-group|Inversion semi-group]]). [[Category:Group theory and generalizations]]

''dihedron group'' ...In a finite group, two different elements of order 2 generate a dihedral group.

...sentation of $G$ on the set of right cosets of $H$ in $G$ (cf [[Coset in a group]]). Then its kernel is the core of $H$ in $G$. .../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

...[Group|Group]]), where the [[Frattini-subgroup(2)|Frattini subgroup]] of a group plays an important role. Due to the many close connections which Lie algebr ...and algebras]]) is the intersection of its maximal subalgebras. Unlike the group case, where the Frattini subgroup is always a [[Normal subgroup|normal subg

''equi-affine group'' The subgroup of the general [[affine group]] consisting of the affine transformations of the $n$-dimensional affine sp

...between the elements of any finite generating set of a finitely-presented group contains a finite set of defining relations in these generators. <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

...perators on $E$. The space $E$ is called the representation space of $\pi$ and the operators $\pi(x)$, $x\in X$, are called the operators of the represent ...valign="top">[1]</TD> <TD valign="top"> A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)</TD></TR>

...gebras was discovered by C. Chevalley [[#References|[2]]] (cf. [[Chevalley group]]). In particular, Chevalley's method makes it possible to obtain Dickson g [[Category:Group theory and generalizations]]

...of $G$ coincides with the rank of $\widehat G$ (cf. [[Rank of an algebraic group]]). ...">[a2]</TD> <TD valign="top"> V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974) {{MR|0376938}} {{ZBL|0371.220

...mi-group]] with unit (i.e. [[Monoid]]) satisfying the [[cancellation law]] and in which any non-invertible element $a$ is decomposable into a product of i a = b_1 \cdots b_k\ \ \text{and}\ \ a = c_1 \cdots c_l

...). As an operation on a class of groups, the verbal product is associative and, within the corresponding [[variety of groups]], it is also free. ...litar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966) {{ZBL|0138.25604}}. pp. 412</TD><

...r groups are said to be cyclic (they are isomorphic to either the additive group $\mathbf Z$ of integers, or the additive groups $\mathbf Z_n$ of residue cl ...groups that are simple (cf. [[Finitely-presented group|Finitely-presented group]]).

''Lie group of type $(E)$'' A real finite-dimensional [[Lie group|Lie group]] $G$ for which the [[Exponential mapping|exponential mapping]] $\exp\colon

''triangular Lie group'' ...f. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]) are real for any element $g$.

...as an increasing ideal series (see [[Ideal series|Ideal series]] of a semi-group) such that for any two adjacent terms $A_\alpha,A_{\alpha+1}$, ...for right (or left) ideals. If all nilpotent sub-semi-groups of a nil semi-group $S$ are finite, then so is $S$.

''of a group $A$ by a group $B$'' ...ruct a semi-direct product one should also know which automorphisms of the group $B$ are induced by conjugation by elements of $A$. More precisely, if $G =

''soluble group'' ...] of a group). The term "solvable group" arose in [[Galois theory|Galois theory]] in connection with the solvability of algebraic equations by radicals.

...of L. Sylow, who proved a number of theorems on such subgroups in a finite group (see [[Sylow theorems|Sylow theorems]]). ...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

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...en the derived group of $G$ is locally finite and, in fact, locally normal and Chernikov.

...$ with degree prime to $p$ (cf. also [[Character of a group|Character of a group]]). The simplest form of the McKay–Alperin conjectures asserts that ...]]] first suggested this might be true when $G$ is a [[Simple group|simple group]]. J.L. Alperin [[#References|[a1]]] observed that it is probably true for

...heory of Lie groups and algebraic groups (cf. [[Lie group, semi-simple|Lie group, semi-simple]]). ...simple. There are examples of simple groups that are not absolutely simple and of simple groups that are not strictly simple.

...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 Let $G$ be a finite group of order $p^ms$, where $p$ is a prime number not dividing $s$. Then the fol

...development of the Galilean relativity principle forms part of the general theory of relativity. .../TD> <TD valign="top"> V.A. [V.A. Fok] Fock, "The theory of space, time and gravitation" , Macmillan (1964) (Translated from Russian)</TD></TR></tabl

$#C+1 = 13 : ~/encyclopedia/old_files/data/L058/L.0508600 Lie group, Banach Please remove this comment and the {{TEX|auto}} line below,

...tric|Hyperbolic metric]]), $M_3$ is the Euclidean Hantzche–Wendt manifold, and $M_2$ is the [[Lens space|lens space]] $L(5,2)$ (see [[#References|[a2]]]). ...the three-dimensional sphere $S^3$, discovered by H.M. Hilden, M.T. Lozano and J.M. Montesinos [[#References|[a3]]]. In fact,

...For any elements $x,y,z\in G$, from $x\leq y$ it follows that $xz\leq yz$ and $zx\leq zy$. ...$ satisfying conditions 1)–4), then $G$ can be made into a totally ordered group with $P$ as set of positive elements.

...ons coincide, and in this case one simply speaks of the decomposition of a group with respect to a normal subgroup. [[Category:Group theory and generalizations]]

...|nil semi-group]], or $S$ is a subdirect product of a group and a nil semi-group. ...band of torsion classes; this clearly occurs for commutative semi-groups, and it is true for periodic semi-groups having two idempotents [[#References|[3

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...s the presentation (cf. also [[Finitely-presented group|Finitely-presented group]]; [[Presentation|Presentation]]):

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...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

...(E. Noether, W. Krull), whence follow the analogous theorems for invariant and fully-invariant series. ...normal series of elements in a lattice. 4) Generalizations of the Schreier and Jordan–Hölder theorems have been obtained for normal categories (see [[#

''group algebra of a group $G$ over a field $K$'' the [[Associative rings and algebras|associative algebra]] over $K$ whose elements are all possible fin

Please remove this comment and the {{TEX|auto}} line below, equipped with a transitive transformation group $ G = \{ g \} $

...thbf Z$, and $((x+m))=x-1/2$ if $m\in\mathbf Z$, $0<x<1$. For integers $c$ and $d$, with $c>0$, the Dedekind sum $S(d,c)$ is the rational number defined b ...dekind eta-function]] under substitutions from the [[Modular group|modular group]]. This interpretation leads naturally to the reciprocity relation for Dede

Please remove this comment and the {{TEX|auto}} line below, ...eometry that studies properties of geometrical forms, in particular curves and surfaces, "in the small" . In other words, the structure of a geometrical

Please remove this comment and the {{TEX|auto}} line below, of a locally compact group $ G $

Please remove this comment and the {{TEX|auto}} line below, with a unique non-trivial reduced homology group:

...hisms of [[C*-algebra|$C^*$-algebra]]s from the point of view of [[ergodic theory]]. ...]) for approximately-finite factors (cf. [[Factor]]) of type $\mathrm{II}$ and type $\mathrm{III}_\lambda$, $0 < \lambda < 1$ (see [[#References|[2]]]).

...,\ldots,a_{2m-1}$ is a sequence of elements from the [[Cyclic group|cyclic group]] $\mathbb{Z}_m$, then there exists a set $I\subseteq \{1,\ldots,2m-1\}$ of zeros and $ m - 1 $

...dentifying the elements of the block design with the elements of the group and the blocks with the sets $\{d_1g,\dots,d_kg\}$, where $g$ runs over $G$). ...clic difference set a multiplier is a number $t$ relatively prime with $v$ and with the property that

...ect to this operation, one can form the [[semi-direct product]] $W$ of $B$ and $A^B$, that is, the set of all pairs $(b,\phi)$, where $b \in B$, $\phi \in ...the wreath product (direct wreath product, discrete wreath product) of $A$ and $B$; it is denoted by $A \mathop{wr} B$ (or $A \wr B$). Both wreath product

''in inverse Galois theory'' ...also [[Galois group|Galois group]]) is a free [[Profinite group|profinite group]] of countable rank. Here, $\textbf{Q}^{\text{ab}}$ is the maximal Abelian

The divisor class group under algebraic equivalence on a non-singular projective variety. ...ted by $\mathrm{NS}(X)$. The Néron–Severi theorem asserts that the Abelian group $\mathrm{NS}(X)$ is finitely generated.

Please remove this comment and the {{TEX|auto}} line below, [[Category:General theory of linear operators]]

...es|[2]]]; for an account of the theory of distal dynamic systems and their generalizations, as well as the relevant literature, see [[#References|[3]]]. ...tructure theorem) see [[#References|[3]]], (3.15.42), [[#References|[a3]]] and [[#References|[a5]]].

...pecially useful for generalizations of the Galois theory of finite, normal and separable field extensions. ...F$ is generated by $P$ and $G$; indeed, it is the [[cross product]] of $P$ and $G$.

$#C+1 = 86 : ~/encyclopedia/old_files/data/C023/C.0203180 Cohomotopy group Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, be a [[Group|group]] of order $ v $

The group $\def\Cr#1{{\rm Cr}(\mathbb{P}_k^#1)} \Cr{n}$ of birational automorphisms o over a field $k$; or, equivalently, the group of Cremona

The special linear group ...ant homomorphism $\det_n$. The structure of $\SL(n,R)$ depends on $R$, $n$ and the type of determinant defined on $\GL(n,R)$. There are three main types o

Please remove this comment and the {{TEX|auto}} line below, \textrm{ and } \ \mathfrak Z _ {r} ( \mathfrak Z _ {l} ( H) ) = H

Please remove this comment and the {{TEX|auto}} line below, [[Category:General theory of linear operators]]

Please remove this comment and the {{TEX|auto}} line below, Generalizations of the hypergeometric functions $ { {} _ {p} F _ {q} } $

...3 : ~/encyclopedia/old_files/data/G110/G.1100050 Gamma\AAhinvariant in the theory of Abelian groups, Please remove this comment and the {{TEX|auto}} line below,

...mal subgroups other than $ G $ . A connected Lie group is semi-simple if and only if it splits into a locally direct product of simple non-Abelian norma ...Lie algebras (cf. [[Lie algebra, semi-simple|Lie algebra, semi-simple]]), and also to the global classification of the Lie groups $ G $ that correspon

...while the latter is based on mappings of arbitrary spaces into polyhedra, and is especially useful whenever the application in fact involves such mapping ...tudy of ordered, rather than of oriented, simplexes by S. Eilenberg (1944) and to cubic homology theories in which cubes, rather than simplices, are emplo

A basic result in the theory of polynomial identity rings (PI-rings). A ring $R$ is a PI-ring (cf. also ...over a commutative ring satisfies the standard polynomial of degree $2n$, and no polynomial of lower degree.

Please remove this comment and the {{TEX|auto}} line below, ...discrete topology. Any Hausdorff space can be made into a topological semi-group, e.g. by giving it a left-singular or zero multiplication.

Please remove this comment and the {{TEX|auto}} line below, A [[Semi-group|semi-group]] in which every element is regular (see [[Regular element|Regular element]

...(or Euclidean) space; a connection was also established between this fact and the property of an $r$-dimensional manifold (in an $n$-dimensional space) b ...groups by their dual cohomology groups of the same dimension over the dual group of coefficients.

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...py equivalence, of certain types of mapping spaces. These results are vast generalizations of two different but related conjectures made by D. Sullivan in 1972.

...087670/s08767013.png" /> is a [[balanced incomplete block design]] (BIBD), and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.o In 1844 W. Woolhouse stated the existence problem for Steiner systems, and P. Kirkman solved it in 1847 for <img align="absmiddle" border="0" src="htt

$#C+1 = 64 : ~/encyclopedia/old_files/data/R080/R.0800520 Reflection group Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, are two algebraically-irreducible representations of some group or algebra in two vector spaces $ X $

Please remove this comment and the {{TEX|auto}} line below, ...s of analytic function theory|boundary value problems of analytic function theory]]. It can be stated in the simplest case as follows. Let $ L $

Please remove this comment and the {{TEX|auto}} line below, ...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

...6 : ~/encyclopedia/old_files/data/G110/G.1100060 Gamma\AAhinvariant in the theory of modular forms, Please remove this comment and the {{TEX|auto}} line below,

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...unding the complexity of decompositions as graphs of groups for a discrete group.

Please remove this comment and the {{TEX|auto}} line below, ...lowing mathematical concepts (structures) occur in the definitions of Bohr and Bochner almost-periodicity: 1) the space of continuous functions defined on

Please remove this comment and the {{TEX|auto}} line below, ...-theory. In the narrow sense, $ K $-theory is the generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]) g

Please remove this comment and the {{TEX|auto}} line below, ...4]]] on the other. Another approach has been undertaken by Yu. Berezanskii and collaborators [[#References|[a1]]].

Please remove this comment and the {{TEX|auto}} line below, which is acted upon transitively by the pseudo-unitary group $ \mathop{\rm SU} _ {n,1} $,

$#C+1 = 215 : ~/encyclopedia/old_files/data/C022/C.0202370 Class field theory Please remove this comment and the {{TEX|auto}} line below,

There exist different generalizations of Schauder's theorem: the Markov–Kakutani theorem, Tikhonov's principle, ...">[4]</TD> <TD valign="top"> R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston (1965) {{MR|0221256}} {{ZBL|

...rals are equal, their common value is the Burkill integral of $F$ over $R$ and is denoted by $\int_RF$. If $F$ is integrable over $R$, then $F$ is integra ...ally generalized to set functions with values in a commutative topological group. The Burkill integral is less general than the subsequently introduced [[Ko

Please remove this comment and the {{TEX|auto}} line below, that is homogeneous with respect to the group of linear transformations $ \alpha \in \mathop{\rm GL} _ {n} ( \mathbf R

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...fference set|Difference set]] (Vol. 3). For an extensive discussion of the theory of Abelian difference sets, see also [[#References|[a1]]], Chap. VI.

...in a certain sense, a measure of their deviation from being true. Similar generalizations of the structure theorems of linear algebra appear in topology. A vector sp ...jects may be studied with the aid of the homotopy theory of vector bundles and of topological

Please remove this comment and the {{TEX|auto}} line below, ...nifold]] of constant negative curvature, or, more generally, of a discrete group $ G $

...ncyclopedia/old_files/data/A011/A.0101980 Almost\AAhperiodic function on a group Please remove this comment and the {{TEX|auto}} line below,

...ms is retained — associativity; this is the explanation of the term "semi-group" . Semi-groups are called ''monoids'' if they have, in addition, an identit ...ntial geometry (semi-groups of partial transformations), and the algebraic theory of automata (semi-groups of automata).

Please remove this comment and the {{TEX|auto}} line below, ...es whose geometry was initially developed for the case of three dimensions and only later was generalized to a dimension $ n > 3 $;

Please remove this comment and the {{TEX|auto}} line below, and $ y a = b $

Please remove this comment and the {{TEX|auto}} line below, ''of a [[Topological group|topological group]]''

...e ring $R$ that is a topological space in which the operations of addition and multiplication, as well as the mapping $R\times A\to A$ ($(r,a)\to ra$), ar ...ctures, i.e. groups, semi-groups, rings, lattices, vector spaces, modules, and others, equipped with topologies in which the algebraic operations consider

Please remove this comment and the {{TEX|auto}} line below, ...inear mappings), and linear, bilinear and quadratic functions (functionals and forms) on vector spaces.

Please remove this comment and the {{TEX|auto}} line below, ...nite automata are supplemented by studies of their various generalizations and modifications which reflect various features of real systems. In the case o

...t alphabet of pattern symbols or "variables". Elements of the [[Free semi-group|free semigroup]] of non-empty words $E^+$ are ''patterns''. For a pattern ...$p$ is $k$-unavoidable if it is unavoidable on every alphabet of size $k$ and correspondingly $k$-avoidable if it is avoidable on an alphabet of size $k$

Please remove this comment and the {{TEX|auto}} line below, and where

Please remove this comment and the {{TEX|auto}} line below, in which each row and each column are permutations of the elements of a finite set $ S $

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...ere used in [[#References|[a7]]] to investigate groups given by generators and relations.

$#C+1 = 125 : ~/encyclopedia/old_files/data/K055/K.0505520 Kleinian group Please remove this comment and the {{TEX|auto}} line below,

...that is invariant under some [[Discrete group of transformations|discrete group of transformations]] $ \Gamma $ of analytic transformations of a given c ...al complex space $ \mathbf C ^{n} $ that are invariant under a discrete group $ \Gamma $ of automorphisms of this domain.

...://www.encyclopediaofmath.org/legacyimages/m/m062/m062160/m0621601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l ...www.encyclopediaofmath.org/legacyimages/m/m062/m062160/m0621608.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofma

Please remove this comment and the {{TEX|auto}} line below, are the fields of components of the velocity, of the pressure and of the temperature of a turbulent fluid flow (see [[#References|[1]]]). An

Please remove this comment and the {{TEX|auto}} line below, that is graded by means of an Abelian group $ A $,

Please remove this comment and the {{TEX|auto}} line below, and $ \delta $

...among the widely available systems is REDUCE, which runs on many platforms and is still being further developed. ...1983. Maple was released in 1986, while Scratchpad II (later called AXIOM) and Mathematica appeared by 1988.

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...ntribution from 1929 [[#References|[a7]]] (see also [[#References|[a8]]]), and Wolff's boundary version of the [[Schwarz lemma|Schwarz lemma]] from 1926 [

...$A$ is called a left polygon over a monoid $R$ if for any $\lambda \in R$ and $a \in A$ the product $\lambda a \in A$ is defined, such that and

...of them generalizes the theorem to the case of proper morphisms of schemes and asserts that the direct image of a [[Coherent sheaf|coherent sheaf]] under and on properties of its singular points (see [[#References|[5]]], [[#Reference

Please remove this comment and the {{TEX|auto}} line below, of a disc and a closed interval, each point $ ( x , 0 ) $

Please remove this comment and the {{TEX|auto}} line below, and $ x _ {1} $

...lem was provided in the middle of the 20th century by B.L. van der Waerden and K. Schütte. ...f mathematics (topology, functional analysis, geometry in the large, graph theory, etc.) are extensively used. One of the central groups of problems in combi

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...ups) that asks whether every [[Finitely-generated group|finitely-generated group]] of exponent $n$ is finite, or, equivalently, whether the free Burnside gr

.... It has been completely proved thanks to the fundamental work of A. Wiles and R. Taylor {{Cite|TaWi}}, and its further refinements

Please remove this comment and the {{TEX|auto}} line below, ...For example, the [[Korteweg–de Vries equation|Korteweg–de Vries equation]] and the non-linear [[Schrödinger equation|Schrödinger equation]] for inverse

$#C+1 = 281 : ~/encyclopedia/old_files/data/B017/B.0107470 Braid theory Please remove this comment and the {{TEX|auto}} line below,

...discrete topological space, a rational straight line, analytic manifolds, and Lie groups over fields with ultra-metric absolute values. ...er extreme conditions inside particles, and possibly also in astrophysical and cosmological singularities — the spatial characteristics may manifest the

Please remove this comment and the {{TEX|auto}} line below, ...y has been constructed for various spaces, primarily Euclidean, projective and homogeneous spaces.

Please remove this comment and the {{TEX|auto}} line below, ...aces that are invariant under transformations of the [[Affine group|affine group]] or its subgroups. The differential geometry of equi-affine space has been

Please remove this comment and the {{TEX|auto}} line below, ...e value|absolute value]]; [[Norm|norm]], etc. All of them, in essence, are generalizations of the idea of the [[Absolute value|absolute value]] of a real or complex n

$#C+1 = 112 : ~/encyclopedia/old_files/data/E035/E.0305900 Equal content and equal shape, figures of Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, is a bijection and the mapping $ T ^ {-} 1 $

Please remove this comment and the {{TEX|auto}} line below, the integrals (2) and (3) exist in the sense of the principal value for almost all $ x $.

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...the study of many classes of partial and functional differential equations and has solved problems that had been left open by the previous classical metho

Please remove this comment and the {{TEX|auto}} line below, and $ \mathbf T $

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...tive variety. The Kawamata–Viehweg vanishing theorem states that for a nef and big line bundle on a complex projective manifold $X$,

Please remove this comment and the {{TEX|auto}} line below, ...of a geometry over an algebra. A recent survey of the development of this theory is [[#References|[a3]]].

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. Crossed complexes are a variant of chain complexes of modules over integral group rings but strengthened in two ways:

A set together with a given transitive [[group action]]. More precisely, $ M $ is a homogeneous space with group $ G $

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...e unit sphere in $E$ (in case $E$ has finite dimension, $G$ is semi-simple and also has finite dimension — the induced [[Unitary representation|unitary

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...orm. For a substantial recent (as of 2000) account of automatic continuity theory for Banach algebras, see [[#References|[a4]]]; all the terms that are used

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...e {p} } \leq \epsilon \right\}$, for $a \in \widehat { K } _ { \text{p} }$ and $\epsilon \in \mathbf{R}$, $\epsilon &gt; 0$. The $\text{p}$-topology has c

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...s on the unit disc) in terms of Schur parameters (see [[#References|[a9]]] and [[#References|[a16]]]) has proved more useful for engineering applications.

Please remove this comment and the {{TEX|auto}} line below, ...can either be reduced to lower-order ones or be completely integrated via group-theoretic techniques.

Please remove this comment and the {{TEX|auto}} line below, ...ometry in which the basic concepts of geometry, the relations between them and related questions are studied.

Please remove this comment and the {{TEX|auto}} line below, and $ U _ {2} $:

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. .... They are also intimately connected with representations of the symmetric and general linear groups (cf. also [[Representation of the symmetric groups|Re

Please remove this comment and the {{TEX|auto}} line below, ...the form of a soap film stretched on a wire framework [[#References|[1]]], and the problem has come to be called the Plateau problem.

Please remove this comment and the {{TEX|auto}} line below, and $ X $ (cf. [[Chern class|Chern class]]). It can be used to calculate the

Please remove this comment and the {{TEX|auto}} line below, ..., was the first to use the letters of the alphabet to denote the constants and the variables in a problem. Most of the present-day symbols of algebra were

...://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b1202306.png" /> and arrows <img align="absmiddle" border="0" src="https://www.encyclopediaofmat and has vertical and horizontal composition operations

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...of homomorphisms: Let $G_1$ be a [[Group|group]] and let $G_2$ be a metric group with a [[Metric|metric]] $d ( \cdot , \cdot )$. Given $\varepsilon > 0$, do

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...homotopies coming from subcollections or subcomposites of the $n$ mappings and these various homotopies at different levels are related by coherence condi

...all png images have been 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

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...e theories. Various equivalent definitions of the notion of vertex algebra and of the variant notion of vertex operator algebra are given below.

Please remove this comment and the {{TEX|auto}} line below, ...s) is well established in the literature since the activity of V.P. Maslov and his collaborators (see e.g. [[#References|[a1]]], [[#References|[a2]]], [[#

Please remove this comment and the {{TEX|auto}} line below, ...the concept of an [[Analytic manifold|analytic manifold]]. A local model (and, at the same time, the most important example) of an analytic space over a

Please remove this comment and the {{TEX|auto}} line below, Problems on the selections and arrangements of elements of a finite set, often having as their origin some

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...[[Algebraic topology|algebraic topology]] for the necessary constructions and definitions).

$#C+1 = 157 : ~/encyclopedia/old_files/data/R077/R.0707130 Radical of rings and algebras Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, be an analytic submanifold of complex codimension one and let $ \omega ( x) $

Please remove this comment and the {{TEX|auto}} line below, ...wo equally important sides of the same subject, not merely yielded elegant and conclusive results, but resulted in a mutual enrichment of the two topics [

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. A formula giving an integral representation of the identity operator and having a number of realizations depending on the initial setting. Historica

...e consists of a topological space $X$ (the underlying space of the scheme) and a ==Basic concepts and properties.==

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...oth a bracket and an associative multiplication defined on the same space, and to impose a "Leibniz rule" relating both operations, stating that the adj

...a successive construction of topology on the basis of manifolds, mappings and differential forms date back to the end of 19th century (H. Poincaré), but ...classification of imbeddings of one manifold into another and its various generalizations.

...-linear or smooth (diffeomorphisms); imbeddings of one object into another and also immersions (local imbeddings). (cf. [[Homeomorphism]]; [[Diffeomorphis ...imbeddings (or immersions) with respect to isotopies (regular homotopies), and the classification of general continuous mappings up to homotopy. An import

Please remove this comment and the {{TEX|auto}} line below, ...ear on its domain) is an operator defined by some differential expression, and acting on a space of (usually vector-valued) functions (or sections of a di

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...{ 0 }$ is a [[Polynomial|polynomial]] in $X$ with coefficients in $\bf Z$ and there exists an $x \in \widetilde{\mathbf{Z}}$ such that $f ( x )$ is a uni

Please remove this comment and the {{TEX|auto}} line below, ...ology of a Riemannian manifold. Thus, problems relating to the topological and the metric structure of Riemannian manifolds with given conditions on the c

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. One of the fundamental results in Ramsey theory.

and let $ \mathfrak t $ and the linear function $ \Lambda $

Please remove this comment and the {{TEX|auto}} line below, and is such that the functions $ f - f _ \alpha $

$#C+1 = 20 : ~/encyclopedia/old_files/data/G045/G.0405010 Graph theory Please remove this comment and the {{TEX|auto}} line below,

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. groups (cf. [[K-theory| $ K $-

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...b )$, $( c , d )$ of points $a$, $b$, $c$, $d$ are said to be congruent if and only if $Q ( a - b ) = Q ( c - d )$.

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...theorem|Fermat last theorem]]). For details and generalizations of Iwasawa theory, see [[#References|[a10]]], [[#References|[a7]]], [[#References|[a12]]], [[

Please remove this comment and the {{TEX|auto}} line below, ...non-degenerate systems of equations and also as various sets of geometric and other objects allowing local parametrization (see below); for example, the

...varieties (cf. [[Algebraic variety|Algebraic variety]]) and their various generalizations (schemes, algebraic spaces, etc., cf. [[Scheme|Scheme]]; [[Algebraic space| ...g into a proof as for the formulation of such proofs in their most obvious and most general form.

$#C+1 = 28 : ~/encyclopedia/old_files/data/G045/G.0405050 Gravitation, theory of Please remove this comment and the {{TEX|auto}} line below,

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...[[Symmetric group|Symmetric group]]; [[Character of a group|Character of a group]]), the associated generalized matrix function $d _ { \chi } ^ { G } : \mat

Please remove this comment and the {{TEX|auto}} line below, ...iples and results concerning the concepts listed above. In particular, the theory of generalized displacement operators has substantial applications in abstr

$#C+1 = 214 : ~/encyclopedia/old_files/data/M065/M.0605010 Morse theory Please remove this comment and the {{TEX|auto}} line below,

''metric theory of dynamical systems'' [[Category:Ergodic theory]]

{{Cite|Ch}} on conics and was systematized and used to great effect by H. Schubert in {{Cite|Sc}}. The justification of Schubert's enumerative calculus and the verification of the numbers he obtained was the contents of Hilbert's 1

Please remove this comment and the {{TEX|auto}} line below, ...in applications of probability theory in various areas of natural science and technology, since these processes accurately describe many real phenomena a

It is a well-known and elementary fact in complex analysis that a bounded and [[Holomorphic function|holomorphic function]] on the whole plane (cf. also ...e|mean curvature]] of a graph can be explicitly computed in terms of first and second derivatives of $f$, the Bernstein problem can be stated in the follo

Please remove this comment and the {{TEX|auto}} line below, ...te 1970s. The historical development can be seen from [[#References|[a1]]] and [[#References|[a6]]].

Please remove this comment and the {{TEX|auto}} line below, ...e]]) on a manifold and their connection with the structure of the manifold and its topology.

Please remove this comment and the {{TEX|auto}} line below, and, finally, on the specific nature of the problem to be solved (for example,

...ay one defines the concepts of a subalgebra, an ideal, a quotient algebra, and a homomorphism of Lie algebras. A Lie algebra $L$ is said to be commutative ...tant case is that in which $k$ is a field (especially when $k=\R$ or $\C$) and $L$ is a vector space (of finite or infinite dimension) over $k$.

Please remove this comment and the {{TEX|auto}} line below, are their partial derivatives, and $ L $

Please remove this comment and the {{TEX|auto}} line below, ''of curves and surfaces''

Please remove this comment and the {{TEX|auto}} line below, ...white noise analysis has developed into a viable framework for stochastic and infinite-dimensional analysis [[#References|[a4]]]–[[#References|[a6]]],

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...|Riemann surface]]), and with the theory of multi-dimensional systems. The theory of operator vessels can be also generalized for the study of tuples of non-

Please remove this comment and the {{TEX|auto}} line below, ...e size, in which the matrices may entirely be stored in a computer memory, and for problems of high order, in which the information is usually stored in c

...od for studying many problems in contemporary algebra, geometry, topology, and analysis. ...orphism. The space $\cF$ together with the stalk-wise algebraic operations and the projection $p$ is called the sheaf of Abelian groups (rings, etc.) over

Please remove this comment and the {{TEX|auto}} line below, is a locally convex space if and only if the topology of $ E $

...ction|Dirichlet $L$-function]]) form the basis of modern [[analytic number theory]]. In addition to Riemann's zeta-function one also distinguishes the genera which converges absolutely and uniformly in any bounded domain of the complex $s$-plane for which $\sigma\

...ting of elements of an arbitrary nature, called points of the given space, and a topological structure, or topology, on this set $X$ (cf. ...g the so-called rational (open) balls, that is, balls for which the radius and the coordinates of the centre are rational numbers. A topology is often def

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...one has to specify to what classes of operators these formulas will apply, and, finally, one has to explain how to use these formulas in applications.

$#C+1 = 70 : ~/encyclopedia/old_files/data/D031/D.0301750 Difference schemes, theory of Please remove this comment and the {{TEX|auto}} line below,

...hat the finite-dimensional cohomology (vector) spaces $H^i(X,\mathcal{L})$ and $H^{n-i}(X,\check{\mathcal{L}}\otimes\omega_X)$ are mutually dual. Here $\o [[sheaf]] of germs of regular differential forms of degree $n$ on $X$, and $\check{\mathcal{L}}=\operatorname{Hom}(\mathcal{L},\mathcal{O}_X)$ is the

...um hypothesis is independent of the Zermelo–Frankel axioms. See also [[Set theory]]. ...y of mathematics associated with this problem, see [[Hilbert 2nd problem]] and [[Hilbert program]].

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...of Fourier hyperfunctions was proposed by M. Sato at the same time as his theory of hyperfunctions, in which the transformed objects can be interpreted natu

Please remove this comment and the {{TEX|auto}} line below, ...wski [[#References|[a25]]] (1922), who axiomatized the idea of "closure" and thus explained "nearness" between a point $ x $

Please remove this comment and the {{TEX|auto}} line below, is continuous; and 2) the mapping $ ( k, x) \rightarrow kx $,

Please remove this comment and the {{TEX|auto}} line below, and there exists a real number $ k $

Please remove this comment and the {{TEX|auto}} line below, Relations between analytic and topological invariants of operators of a certain class. More precisely, ind

Please remove this comment and the {{TEX|auto}} line below, ...e "and" , "or" , "if … then" . In general, models of many-valued logic are generalizations of the [[Algebra of logic|algebra of logic]]. It is important to note that

.... Metrics are important in the study of convergence (of series, functions) and for the solution of questions concerning approximation. The development of the theory of metric spaces has proceeded in the following main directions.

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...|[a5]]], [[#References|[a3]]], [[#References|[a6]]], [[#References|[a16]]] and the original articles cited therein.

Please remove this comment and the {{TEX|auto}} line below, ...as basis and is generated from it by the operation of immediate succession and its iteration. The set of all natural numbers $ >1 $

$#C+1 = 247 : ~/encyclopedia/old_files/data/T092/T.0902810 Tight and taut immersions Please remove this comment and the {{TEX|auto}} line below,

...t of convergence arises, for example, in the study of mathematical objects and their approximation by simpler objects. Thus, in order to calculate the are ...e to calculate this solution with the required accuracy. Both for ordinary and partial differential equations there are various convergent difference meth

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...discrete data, then called the log-linear model; see [[#References|[a6]]], and [[#References|[a10]]].

Please remove this comment and the {{TEX|auto}} line below, ...most general form such a study falls into three parts: 1) the introduction and study of infinite-dimensional spaces as such; 2) the study of the simplest

...te evolution of the process is observed by the controller. A corresponding theory has also been developed in the case of partial observations (incomplete dat ...//www.encyclopediaofmath.org/legacyimages/c/c026/c026010/c02601030.png" /> and control action <img align="absmiddle" border="0" src="https://www.encyclope