...
order other than the unit element (see [[
Order|
Order]] of an element of a
group).
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.G. Kurosh, "The
theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian
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''dihedron group''
...In a finite group, two different elements of order 2 generate a dihedral group.
1 KB (202 words) - 16:24, 19 October 2014
''Hamiltonian group''
...rticular, any Hamiltonian group is periodic (cf. [[Periodic group|Periodic group]]).
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...umber) is metacyclic. Polycyclic groups (cf. [[Polycyclic group|Polycyclic group]]) are a generalization of metacyclic groups.
...the more special class of groups whose derived group and derived quotient group are both cyclic.
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...cept $n=4$, this group is simple; this fact plays an important role in the theory of solvability of algebraic equations by radicals.
Note that $A_5$ is the non-Abelian simple group of smallest possible order.
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''torsion-free group''
...with respect to two different prime numbers $p$, then it is a torsion-free group.
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...rder 8 is the smallest finite group that is not a T-group. A group is a T-group if and only if it is equal to its own [[Wielandt subgroup]].
* Derek Robinson, "A Course in the Theory of Groups", Graduate Texts in Mathematics '''80''' Springer (1996) {{ISBN|0
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...e of subgroups of a group is a [[distributive lattice]] if and only if the group is locally cyclic.
* Marshall Hall jr, ''The Theory of Groups'', reprinted American Mathematical Society (1976)[1959] {{ISBN|0-
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''equi-affine group''
The subgroup of the general [[affine group]] consisting of the affine transformations of the $n$-dimensional affine sp
1 KB (158 words) - 22:38, 2 November 2014
...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 following th
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...$ and its centralizer $K$ in $G$; indeed $K$ is isomorphic to the quotient group $G/B$.
...form space]] with respect to the [[uniformity]] implied by the topological group structure.
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$#C+1 = 31 : ~/encyclopedia/old_files/data/P071/P.0701710 Partially ordered group
A [[Group|group]] $ G $
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''cyclic semi-group''
...le$, then $a,\dots,a^{h+d-1}$ are distinct elements and, consequently, the order of $A$ is $h+d-1$; the set
2 KB (405 words) - 19:33, 21 November 2014
''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.
3 KB (443 words) - 18:25, 26 October 2014
...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]]
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...Every pure linear sub-semi-group $P$ of an arbitrary group defines a right order, namely $x<y$ if and only if $yx^{-1}\in P$.
...ll subgroups in $S(G)$ become convex. In a locally nilpotent right-ordered group the system of convex subgroups is solvable.
4 KB (585 words) - 06:17, 28 March 2023
...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]]).
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A metabelian $2$-group (cf. [[Meta-Abelian group|Meta-Abelian group]]) of order 8, defined by generators $x,y$ and relations
The quaternion group can be isomorphically imbedded in the multiplicative group of the algebra of quaternions (cf. [[Quaternion|Quaternion]]; the imbedding
2 KB (350 words) - 14:38, 2 August 2014
''$p$-component of a group element of finite order''
...or $p$-component of $x$ and $z$ is the $p'$-part or $p'$-component. If the order of $x$ is $r=p^{\alpha}s$, $(p,s)=1$, $bp^{\alpha}+cs=1$, then $y=x^{sc}$,
1 KB (201 words) - 20:11, 25 March 2024
A [[P-group| $ p $-
group]] $ G $
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[[Semi-group|semi-group]]. In particular, if $S$ is a group, one obtains a
...p $S$ is semi-simple if and only if all linear representations of the semi-group $S$ over the algebra $\Phi$ are reducible.
2 KB (344 words) - 19:41, 8 December 2015
...7 : ~/encyclopedia/old_files/data/P071/P.0701010 \BMI \Gpi\EMI\AAhsolvable group
A generalization of the concept of a [[Solvable group|solvable group]]. Let $ \pi $
3 KB (420 words) - 08:06, 6 June 2020
A ''torsion group'' (also called ''periodic group'')
[[Group|group]] in which every element has finite order. Any torsion
1 KB (170 words) - 21:20, 29 April 2012
...on, that is, it can be imbedded in a unique locally nilpotent torsion-free group $G^*$ such that all equations of the form $x^n=g$ are solvable in $G^*$, wh
<TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian
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...://www.encyclopediaofmath.org/legacyimages/c/c110/c110400/c1104002.png" />-
group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org
...ges/c/c110/c110400/c11040010.png" /> is partially ordered with the induced
order: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/
15 KB (2,061 words) - 17:13, 7 February 2011
''totally ordered group''
A [[Po-group| $ po $-
4 KB (587 words) - 08:03, 6 June 2020
...eal numbers in the interval $(0,1)$ and the symbol $\infty$ with the usual order and with the operations:
The former case occurs if and only if $S$ is a [[semi-group with cancellation]].
3 KB (413 words) - 09:17, 2 April 2023
...egacyimages/h/h110/h110290/h11029025.png" />, which is identified with the
group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
....encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029051.png" />-Hopf
order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
13 KB (1,801 words) - 19:17, 12 January 2018
...rs from the theory of groups (i.e. $2$-groups). Thus, if $n\geq 3$, an $n$-group has no analogue of the unit element.
Let $\Gamma(\circ)$ be a group with multiplication operation $\circ$; let $n\geq 3$ be an arbitrary intege
2 KB (307 words) - 21:09, 21 November 2014
...with the operation of multiplication in the skew-field. The multiplicative group of a field is Abelian.
...lic group or is a finitely-solvable group and has an extension to a linear group over a skew-field. Some cases are known, e.g., [[#References|[a2]]].
1 KB (219 words) - 20:48, 21 December 2014
''in a group $G$''
...he order of a subgroup $H$ by its index $\left|G:H\right|$ is equal to the order of $G$ (Lagrange's theorem). This relationship applies to finite groups $G$
1 KB (186 words) - 11:18, 20 April 2012
...ng all elements of the groupoid (in any order); these symbols (in the same order) are also listed in front of the first column. If the groupoid has an ident
Cayley tables were first used by A. Cayley in 1854 for groups (cf. [[Group|Group]]).
2 KB (300 words) - 16:43, 4 June 2020
''of a finite group $ G $''
see [[Sylow subgroup|Sylow subgroup]]). A group $ G $
2 KB (376 words) - 08:03, 6 June 2020
be a [[Finite group|finite group]] and $ \pi $
a subset of the prime numbers that divide the order $ n $
2 KB (287 words) - 08:24, 6 June 2020
$#C+1 = 18 : ~/encyclopedia/old_files/data/P074/P.0704570 Primitive group of permutations,
''primitive permutation group''
3 KB (484 words) - 08:07, 6 June 2020
...g/legacyimages/f/f120/f120130/f1201301.png" /> denote an arbitrary [[
Group|
group]] (finite or infinite) and let <img align="absmiddle" border="0" src="https
...opediaofmath.org/legacyimages/f/f120/f120130/f12013038.png" /> is a finite
group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/
16 KB (2,143 words) - 17:10, 7 February 2011
The ''complexification of a Lie group $G$ over $\R$'' is
a complex Lie group $G_\C$ containing $G$ as a real Lie subgroup such that the Lie algebra $\de
3 KB (501 words) - 15:05, 13 April 2024
''right-ordered group''
A [[Group|group]] $ G $
4 KB (580 words) - 06:18, 28 March 2023
...n group theory. Most are not intrinsic to a group itself, but pertain to a group acting on something.
==Regular group of permutations.==
4 KB (716 words) - 07:52, 25 November 2023
$#C+1 = 13 : ~/encyclopedia/old_files/data/L058/L.0508600 Lie group, Banach
endowed with a group structure and an analytic Banach manifold structure (cf. [[Banach analytic
3 KB (427 words) - 22:16, 5 June 2020
The [[Exponent of a group|exponent]] of the multiplicative group of integers modulo $n$, denoted $\lambda(n)$; the least positive integer $\
Since the multiplicative group modulo an odd prime power $p^a$ is cyclic, in these cases we have $\lambda(
1 KB (186 words) - 16:57, 25 November 2023
...ad. A Baer group is maximal of elation, respectively homology, type if the group acts transitively on the non-fixed points on each line $L$ of the spread wh
...gn="top">[a2]</TD> <TD valign="top"> V. Jha, N.L. Johnson, "Structure theory for point-Baer and line-Baer collineation groups in affine planes" , ''Proc
2 KB (305 words) - 17:24, 2 September 2017
A norm on a skew-field the group of values of which is isomorphic to the group of integers $ \mathbf Z $.
is also sometimes understood as the norm having as group of values the $ r $-
1 KB (165 words) - 19:36, 5 June 2020
...cally finite group]]), finiteness of rank (cf. [[Rank of a group|Rank of a group]]), and finiteness of conjugacy classes (cf. [[Conjugate elements|Conjugate
...t groups (cf. [[Nilpotent group|Nilpotent group]]) it is equivalent to the group itself being finitely generated.
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...subgroups are conjugate in $G$. Any $\pi_1$-subgroup of a $\pi$-separable group $G$ is contained in some $\pi_1$-Hall subgroup of $G$ (see [[#References|[2
...and if all prime divisors of $k$ are in $\pi$, then $G$ has a subgroup of order $k$ and all these subgroups are conjugate in $G$. If $\pi$ consists of all
2 KB (379 words) - 09:33, 27 October 2014
...23 : ~/encyclopedia/old_files/data/M064/M.0604790 Monomial substitutions, group of
The subgroup of the group $ \mathop{\rm GL} ( m , \mathbf Z [ H ] ) $
2 KB (302 words) - 08:01, 6 June 2020
...ollowing generalization of Frobenius' theorem: If $G$ is a finite group of order $g$ and $C$ is a [[conjugacy class]] of $G$ of cardinality $h$, then the nu
...mple, $x^4=1$ has exactly $4$ solutions in the [[Symmetric group|symmetric group]] on three letters, but obviously the solutions do not form a subgroup of $
4 KB (650 words) - 20:59, 29 November 2014
...on, a [[Totally ordered set|totally ordered set]] with respect to a binary order relation $\leq$ and satisfies the following axiom: For any elements $x,y,z\
...$ satisfying conditions 1)–4), then $G$ can be made into a totally ordered group with $P$ as set of positive elements.
3 KB (495 words) - 19:18, 17 August 2014
...e group generated by all right translations of a finite distributive quasi-group is solvable [[#References|[4]]].
...ign="top">[3]</TD> <TD valign="top"> V.D. Belousov, "Foundations of the theory of quasi-groups and loops" , Moscow (1967) (In Russian)</TD></TR>
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...oup algebra]]; [[Cross product|Cross product]]), with $|G|=q$. An $R$-Hopf order in $KG$ is a rank-$q$ $R$-Hopf algebra $H$ (cf. [[Hopf algebra|Hopf algebra
...(g)$. Then the $R$-Hopf order in $KG$ determined by $\xi$ (called a Larson order) is of the form
5 KB (919 words) - 04:20, 15 February 2024
...|nil semi-group]], or $S$ is a subdirect product of a group and a nil semi-group.
...nown (including necessary and sufficient ones) under which a periodic semi-group is a band of torsion classes; this clearly occurs for commutative semi-grou
5 KB (775 words) - 17:22, 17 October 2014
...lity in the lattice of all subgroups of a finite group (cf [[Supersolvable group]]).
<TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Hall, Jr., "The theory of groups" , Macmillan (1964) {{ZBL|0116.25403}}</TD></TR>
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''Černikov group''
...also Chernikov; further, an extension of a Chernikov group by a Chernikov group is again Chernikov.
4 KB (589 words) - 18:19, 15 January 2017
''group algebra of a group $G$ over a field $K$''
...a; multiplication of basis elements in the group algebra is induced by the group multiplication. The algebra $KG$ is isomorphic to the algebra of functions
5 KB (872 words) - 22:00, 29 April 2012
A matrix group $G$ over an arbitrary fixed field $K$, all elements of which may be reduced
...mpletely reducible. Every normal subgroup of a completely-reducible matrix group is itself completely reducible.
2 KB (303 words) - 16:55, 10 April 2016
...e poly-nilpotent length is less than the solvable length. A poly-nilpotent group of length 2 is called meta-nilpotent.
...ncreasing) poly-nilpotent series of length $l$ whose factors in increasing order have nilpotent classes not exceeding the numbers $c_1,\dots,c_l$, respectiv
2 KB (229 words) - 16:12, 4 October 2014
The automorphism group of a
group of all automorphisms of the field $L$ leaving the elements of
3 KB (494 words) - 21:56, 5 March 2012
$#C+1 = 10 : ~/encyclopedia/old_files/data/S084/S.0804920 Shmidt group
...lpotent group]]). A Shmidt group is a [[Solvable group|solvable group]] of order $ p ^ \alpha q ^ \beta $,
2 KB (346 words) - 18:45, 11 April 2023
...G. Fubini in 1916 (a generalization of this concept to the geometry of any group of transformations was obtained by E. Cartan in 1920) using the concept of
be the group of transformations of a space $ E $.
3 KB (529 words) - 08:08, 6 June 2020
A group having a [[Normal series|normal series]]
...group itself), and their lengths are equal to the nilpotency class of the group.
3 KB (500 words) - 16:51, 30 December 2018
''(in group theory)''
rank of a [[Finite group|finite group]] $ G $
4 KB (568 words) - 14:10, 31 December 2020
...dered set]], a subset which is totally ordered with respect to the induced order. The [[rank of a partially ordered set]] is the maximal cardinality of a c
...a group $G$ is an element of the tensor product of a chain complex by the group $G$.
1 KB (171 words) - 15:33, 18 March 2023
''idempotent semi-group''
...study of many properties of idempotent semi-groups. Every idempotent semi-group is locally finite.
4 KB (534 words) - 16:11, 12 April 2014
A group of observations in a sample that have the same value. Let $ X _ {1} \dots
of the order statistics (cf. [[Order statistic|Order statistic]])
4 KB (604 words) - 08:25, 6 June 2020
''of an Abelian group or semigroup $M$''
...rac27 n$. The set of odd numbers in $\{1,\ldots,n\}$ is a sum-free set of order $\frac12 n$.
660 bytes (110 words) - 16:47, 23 November 2023
...a so-called Frobenius complement in $G$; the group $G$ is then a Frobenius group by definition. It was proved by G. Frobenius in 1901, see [[#References|[a3
...$ is a subgroup of $G$ is still the only existing proof; it uses character theory! The normal subgroup $N$ is called the Frobenius kernel of $G$.
9 KB (1,463 words) - 07:41, 27 January 2024
$#C+1 = 182 : ~/encyclopedia/old_files/data/O070/O.0700040 Order
The order of an algebraic curve $ F ( x , y ) = 0 $,
13 KB (2,001 words) - 02:12, 1 March 2022
$#C+1 = 80 : ~/encyclopedia/old_files/data/L057/L.0507670 Lattice\AAhordered group,
'' $ l $-group''
5 KB (738 words) - 08:45, 8 October 2023
of a group $ G $
is said to be a coset of the group $ G $
2 KB (298 words) - 19:36, 5 June 2020
...in the [[Group algebra|group algebra]] $F G$ of some [[Finite group|finite group]] $G$. Since the Schur indices for $G$ are trivial in prime characteristic
...is group]]), in the sense of E. Noether, where the factor sets have finite order.
4 KB (692 words) - 17:46, 1 July 2020
...ension $K/k$ is isomorphic to the $p$-adic Lie group $\mathbb{Z}_{p}$, the group of $p$-adic integers.
...e. These results have been applied in algebraic number theory and in group theory.
2 KB (327 words) - 11:17, 9 April 2023
...tiplicative group]] of residue classes modulo $p$ divides the order of the group. Fermat's little theorem was generalized by L. Euler to the case modulo an
...n="top">[1]</TD> <TD valign="top"> I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)</TD></TR></table>
2 KB (257 words) - 18:02, 8 November 2014
...-group $S$ is a [[total order]], then $S$ is called a totally ordered semi-group (cf. also [[Totally ordered set]]). If the relation $\le$ on $S$ defines a
..., is a variety (cf. also [[Variety of groups]]). On a lattice-ordered semi-group the identities
11 KB (1,676 words) - 14:07, 17 March 2020
...al order on the set $E$ of idempotent elements, called the natural partial order on $E$. Two idempotents $u$ and $v$ of a ring are said to be orthogonal if
...ing as an idempotent element of the semi-group of unary operations. In the theory of $R$-modules, the affine operations are those of the form
1 KB (226 words) - 12:30, 14 February 2020
...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
...comes down to finding an algebraic equation over $k$ with the given Galois group. Such equations exist for the symmetric groups, and also for the alternatin
4 KB (586 words) - 04:07, 25 February 2022
...d by identifying 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 $G$
...tiplier of a difference set turns out to be useful: An automorphism of the group $G$ is a multiplier of a $(v,k,\lambda)$-difference set $D$ in $G$ if it is
5 KB (832 words) - 10:37, 15 November 2014
''lattice-ordered group''
A partially ordered group $ \{ G; \cdot, \cle \} $(
9 KB (1,403 words) - 22:15, 5 June 2020
...ound in many textbooks on combinatorics and many elementary books on group theory (see, for example, [[#References|[a5]]], Chap. 9).
.... There are several lemmas and theorems in group theory and representation theory to which the name of William Burnside is correctly attached (for example: B
3 KB (516 words) - 22:29, 1 December 2014
$#C+1 = 37 : ~/encyclopedia/old_files/data/W097/W.0907770 Whitehead group
An Abelian group associated with an associative ring in the following manner. It was introdu
4 KB (564 words) - 08:29, 6 June 2020
$#C+1 = 51 : ~/encyclopedia/old_files/data/M110/M.1100180 Modular group algebra
a [[Group|group]]. The [[Group algebra|group algebra]] $ FG $
5 KB (726 words) - 08:01, 6 June 2020
$#C+1 = 51 : ~/encyclopedia/old_files/data/F040/F.0400290 Finite group scheme
A group scheme that is finite and flat over the ground scheme. If $ G $
5 KB (753 words) - 09:02, 8 April 2023
The notion of a ''group scheme'' is a generalization of the concept of an
[[Algebraic group|algebraic group]]. Let ${\rm Sch}/S$ be the category of
5 KB (831 words) - 21:59, 5 March 2012
...le p,+\rangle$ can be turned into a [[partially ordered set]] (the partial order $\leq$ is defined by the relation $a\leq b$ if and only if $a+b=b$) in whic
...ent (cf. also [[Band of semi-groups]]) (which is a decomposition of a semi-group into sub-semi-groups forming a band). Thus, an upper (lower) semi-lattice d
2 KB (272 words) - 12:05, 23 November 2023
$#C+1 = 112 : ~/encyclopedia/old_files/data/F040/F.0400280 Finite group, representation of a
A homomorphism of a finite group $ G $
10 KB (1,488 words) - 19:39, 5 June 2020
...$K / k$ that leave all elements of $k$ invariant is called the ''[[Galois group]]'' of
...rm{Gal}}\Gal(K/k)$. The study of these groups is a major part of [[Galois theory]].
2 KB (423 words) - 13:52, 25 November 2023
A group $G$ provided with the structure of an
mappings (morphisms) of algebraic varieties. An algebraic group is
4 KB (665 words) - 18:32, 10 December 2015
cf. [[Geometric objects, theory of|Geometric objects, theory of]]) be given in a differentiable manifold $ X _ {n} $,
of this manifold is known as a differential invariant of order $ r $
5 KB (725 words) - 19:35, 5 June 2020
is the structure Lie group of the principal bundle $ ( X , p , M ^ {n} ) $
or, in another terminology, the representation space of the Lie group $ \mathfrak G $.
6 KB (860 words) - 17:33, 5 June 2020
...cending and descending chains of normal subgroups have finite length. If a group has two principal series, then they are isomorphic, i.e. they have the same
...</TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Hall jr., "The theory of groups" , Macmillan (1959) pp. 124</TD></TR><TR><TD valign="top">[a3]<
2 KB (254 words) - 16:51, 30 December 2018
from an (Abelian) [[Semi-group|semi-group]] $ H $
to subsets of an (Abelian) semi-group $ G $
2 KB (318 words) - 16:09, 1 April 2020
be a [[Group|group]] of order $ v $
difference set of order $ n = k - \lambda $
5 KB (803 words) - 06:20, 26 March 2023
be a [[Finite group|finite group]], $ K $
is a splitting field for a finite group $ G $
3 KB (533 words) - 08:12, 6 June 2020
...logical group with respect to this topology and therefore is not a compact group.
...$ \mathbf R $ , respectively) and, more generally, any compact real Lie group.
8 KB (1,117 words) - 20:03, 27 February 2021
...a [[Galois extension|Galois extension]] of $\mathbf Q$ with a given finite group $G$ (see [[#References|[5]]]). The problem is also closely connected with t
...of $K^G$ in the case of an Abelian group $G$ is closely connected with the theory of algebraic tori (cf. [[Algebraic torus|Algebraic torus]]) (see [[#Referen
4 KB (603 words) - 17:59, 23 November 2014
...lement $\zeta$ generates the [[cyclic group]] $\mu_m$ of roots of unity of order $m$.
...t $\zeta^k$ is also a primitive root. The number of all primitive roots of order $m$ is equal to the value of the [[Euler function]] $\phi(m)$ if $\mathrm{h
3 KB (496 words) - 07:46, 20 December 2014
A [[Semi-group|semi-group]] in which every element is regular (see [[Regular element|Regular element]
An arbitrary regular semi-group $ S $
10 KB (1,515 words) - 08:10, 6 June 2020
...rak{gl}(V)$ is naturally identified with the set of all square matrices of order $n$ over $K$ and is denoted by $\mathfrak{gl}(n,K)$. Any linear Lie algebra
...he Lie algebra of the analytic group $G$ (cf. [[Lie algebra of an analytic group]]) is naturally identified with a Lie subalgebra of $\mathfrak{gl}(V)$, tha
3 KB (554 words) - 19:21, 24 February 2017
group $ K _ {2} ( {\mathcal O} _ {F} ) $,
is an [[Abelian group|Abelian group]] of finite order.
5 KB (731 words) - 08:21, 26 March 2023
''inverse semi-group''
...nverse semi-group [[#References|[7]]]. Every congruence on an inverse semi-group is determined by the classes containing idempotents.
7 KB (986 words) - 19:33, 2 January 2018
$#C+1 = 56 : ~/encyclopedia/old_files/data/P075/P.0705050 Profinite group
A topological group that is the [[Projective limit|projective limit]] of an inverse system of f
5 KB (778 words) - 19:48, 21 January 2021
...consists of all such elements. For example, every finite commutative semi-group is a homogroup.
...is, in this case, a [[Completely-simple semi-group|completely-simple semi-group]].
6 KB (961 words) - 18:14, 7 May 2016
A [[Quasi-group|quasi-group]] with an identity, that is, with an element $ e $
...[[Isotopy|Isotopy]]) to a loop. Therefore, one of the main problems in the theory of quasi-groups is to describe the loops to which the quasi-groups of a giv
8 KB (1,291 words) - 06:59, 30 March 2024
...group]] has Fitting length $1$, whereas any finite solvable non-nilpotent group will have Fitting length at least $2$; see any standard reference such as [
...h, to the number of elements needed to generate the Sylow subgroups of the group, to the derived length of the Sylow subgroups or to their nilpotent class.
5 KB (855 words) - 17:03, 1 July 2020
...n of a valued field]]; [[Ramification theory of valued fields|Ramification theory of valued fields]].
...general valuation theory and in the [[Model theory of valued fields|model theory of valued fields]], being Henselian has turned out to be more appropriate t
3 KB (405 words) - 19:33, 28 April 2014
...modulo $\mathfrak{p}^n$ in $R_{\mathfrak{p}}$ in the usual sense of ring theory. For a real place $\mathfrak{r}$ we define $x$ and $y$ to be congruent mo
==Ray class group==
3 KB (527 words) - 17:00, 23 November 2023
...motopic to zero, or, in other words, whose [[Fundamental group|fundamental group]] is not trivial. This means that there are closed paths in $ D $
The order of connectivity of a plane domain $ D $
4 KB (515 words) - 08:02, 6 June 2020
A '''reductive group''' is a
[[Linear algebraic group|linear algebraic group]] $G$ (over an
5 KB (811 words) - 03:39, 4 January 2022
groups (cf. [[L-group| $ l $-
group]]) that is distinguished within the class $ {\mathcal L} $
5 KB (767 words) - 22:15, 5 June 2020
...omorphism group $\mathrm{P}\Gamma\mathrm{U}(3,q^2)$ with associated simple group $\mathrm{PSU}(3,q^2)$ (when $q>2$); see [[#References|[a3]]]. This type is
...in a projective plane in such a way that $G$ is induced by a collineation group of the plane.
3 KB (435 words) - 19:51, 7 August 2016
...[[Adèle|Adèle]]) of a connected [[Linear algebraic group|linear algebraic group]] $G$ defined over a global field $K$ with respect to the [[Tamagawa measur
...ions of $K$). The finiteness of the Tamagawa number follows from reduction theory (see {{Cite|Pl}}).
5 KB (716 words) - 21:31, 4 March 2012
Let $x$ be an element of $G$ whose
order is a power of $p$. The $p$-section of $G$ associated to $x$ is the set of a
...H$. Brauer's second main theorem states that for all elements $y \in H$ of
order prime to $p$, $\chi _ { f } ( x y ) = 0$. Thus, the values of <img align="a
3 KB (540 words) - 17:43, 1 July 2020
$#C+1 = 91 : ~/encyclopedia/old_files/data/C026/C.0206980 Coxeter group
A group with a distinguished system of generators $ \{ {r _ {i} } : {i \in I } \}
15 KB (2,070 words) - 14:45, 1 November 2023
$#C+1 = 82 : ~/encyclopedia/old_files/data/S085/S.0805210 Simple finite group,
''finite simple group''
15 KB (1,883 words) - 08:13, 6 June 2020
A ''normal series of a group $G$ is
...|Subgroup series]]). If each term of the series is normal not in the whole group but only in the preceding term, then the series is called subnormal. Apart
2 KB (317 words) - 16:43, 27 November 2013
then the order on $H$ is called ''strict'', and $H$ is a strictly partially ordered groupo
...est (cf. [[Naturally ordered groupoid]]; [[Ordered semi-group]]; [[Ordered group]]).
3 KB (429 words) - 20:52, 16 March 2023
...yclopedia/old_files/data/G045/G.0405230 Group algebra of a locally compact group
...algebra with [[Involution|involution]] formed by certain functions on the group with multiplication in it defined as convolution. Let the Banach space $
6 KB (888 words) - 10:16, 8 May 2022
.... The theory of HNN-groups is central to geometric and combinatorial group theory and should be viewed in parallel with amalgamated products (cf. also [[Amal
The easiest way to define an HNN-group is in terms of presentations of groups.
6 KB (1,025 words) - 08:55, 29 March 2024
The semi-group on a dual [[Banach space|Banach space]] $ X ^ {*} $
semi-group on $ X $(
7 KB (962 words) - 20:04, 4 April 2020
$#C+1 = 59 : ~/encyclopedia/old_files/data/A014/A.0104000 Automata, algebraic theory of
...f automata was used to obtain a proof of the solvability of certain second-order arithmetical theories, and also to obtain a new, simpler solution of the [[
7 KB (1,008 words) - 18:48, 5 April 2020
$#C+1 = 104 : ~/encyclopedia/old_files/data/I050/I.0500350 Imprimitive group
A group $ G $
6 KB (896 words) - 07:41, 26 February 2022
...d even for the same system of generators. For example, the cyclic group of order two with generator $ a $
...with non-trivial identities have been described (see also [[Group calculus|Group calculus]]).
4 KB (557 words) - 15:18, 7 March 2022
...terms of sums of powers. The characters of representations of a symmetric group (cf. [[Representation of the symmetric groups|Representation of the symmetr
of the symmetric group $ S _ {m} $.
4 KB (519 words) - 16:18, 5 February 2022
...n abstract group (cf. [[Nilpotent group|Nilpotent group]]). An Abelian Lie group is nilpotent. If $ F = \{ V _{i} \} $ is a [[Flag|flag]] in a finite-dim
...( n , k ) $ corresponding to $ N (F \ ) $ consists of all matrices of order $ n = \mathop{\rm dim}\nolimits \ V $ of the form mentioned above.
5 KB (803 words) - 18:12, 12 December 2019
''on a semi-group''
...e inclusion of principal left ideals defines in a natural manner a partial order relation on the set of $\mathcal L$-classes; similar considerations are val
3 KB (507 words) - 16:59, 25 November 2023
...t is hard to say whether field theory, the theory of finite groups and the theory of finite-dimensional Lie algebras should be regarded as general algebra.
If a universal algebra is provided with an order or a topology compatible with the operations, then one has a partially orde
5 KB (725 words) - 21:52, 30 March 2012
...l proof appeared in [[#References|[a7]]], a proof in the context of finite group cohomology appears in [[#References|[a1]]].
...xtbf{Z}/p)^I$ for some indexing set $I$, where $\textbf{Z}/p$ is cyclic of order $p$). Then Serre's theorem asserts that there exist non-trivial $\mod p$ co
6 KB (868 words) - 22:16, 5 February 2021
...he equi-affine transformations, i.e. with respect to the affine unimodular group. For a plane curve $ \mathbf r = \mathbf r (t) $
...he affine parameter of a space curve in the geometry of the general affine group or any one of its subgroups.
2 KB (382 words) - 16:09, 1 April 2020
...group of divisors of a Krull ring is canonically isomorphic to the ordered group $ \mathbf Z ^ {(} I) $.
...align="top">[a1]</TD> <TD valign="top"> R.M. Fossum, "The divisor class group of a Krull domain" , Springer (1973)</TD></TR></table>
3 KB (424 words) - 22:15, 5 June 2020
...ain propositional calculi. It turns out that Abelian groups (cf. [[Abelian group]]) are a special case of BCI-algebras. One may take different axiom systems
A [[partial order]] $\leq$ may be defined by $x \leq y$ if and only if $x \ast y = 0$. A very
5 KB (790 words) - 02:33, 15 February 2024
...ct group|compact group]] that is a finite-dimensional real [[Lie group|Lie group]]. Compact Lie groups can be characterized as finite-dimensional locally co
...y of the structure of connected compact Lie groups is a basic topic in the theory of Lie groups.
9 KB (1,377 words) - 04:00, 9 May 2022
...uces to the discovery of quantities that are invariant with respect to the group of motions of $ E ^ {3} $.
are invariant. The differential neighbourhood of order $ n $
4 KB (613 words) - 22:17, 5 June 2020
is a group, known as the group of all automorphisms of the system $ \mathbf A $;
The subgroups of the group $ \mathop{\rm Aut} ( \mathbf A ) $
10 KB (1,505 words) - 06:43, 26 March 2023
...70 : ~/encyclopedia/old_files/data/P071/P.0701030 \BMI p\EMI\AAhdivisible group,
''Barsotti–Tate group''
6 KB (885 words) - 08:04, 6 June 2020
[[Group|group]] with respect to this multiplication, called the
[[Symmetric group|symmetric group]]. Any subgroup of a symmetric group is called a
7 KB (1,262 words) - 20:15, 27 September 2016
...of the parameter $z$. The operator $I_\alpha^a$ is linear and has the semi-group property:
...fferentiation: If $I_\alpha f=F$, then $f$ is the fractional derivative of order $\alpha$ of $F$. If $0<\alpha<1$, Marchaut's formula applies:
4 KB (614 words) - 15:22, 14 February 2020
$#C+1 = 67 : ~/encyclopedia/old_files/data/M064/M.0604440 Modular group
The group $ \Gamma $
7 KB (1,031 words) - 18:33, 13 January 2024
...$\Q\bigl(\sqrt{-3}\bigr)$ (and generator $\bigl(1+\sqrt{-3}\bigr)/2$), and order 2 (and generator $-1$) for all other imaginary quadratic fields.
...oup of order 2 generated by $-1$ and $\{\epsilon\}$ is the infinite cyclic group generated by a fundamental unit $\epsilon$. For example, for $\Q\bigl(\sqrt
5 KB (867 words) - 17:41, 12 November 2023
...ric $( 111,11,1 )$-design, i.e. a [[Projective plane|projective plane]] of order $10$, shows; see [[#References|[a5]]].
...$; such a design is equivalent to a [[Hadamard matrix|Hadamard matrix]] of order $4 n$, which is conjectured to exist for all values of $n$. Similarly, a sy
7 KB (1,152 words) - 16:45, 1 July 2020
...S. Tanaka in [[#References|[a7]]] to generalize the set difference in set theory, and by Y. Imai and Iséki in [[#References|[a5]]] as the algebras of certa
6) $0 \ast x = 0$ for all $x$. A [[Partial order|partial order]] $\le$ can then be defined by putting $x \le y$ if and only if $x \ast y =
5 KB (892 words) - 02:43, 15 February 2024
...by means of radicals. It is customary to write the operation in an Abelian group in additive notation, i.e. to use the plus sign ($+$) for that operation, c
...is an arbitrary prime number, are Abelian (cf. [[Group-of-type-p^infinity|Group of type $p^\infty$]]).
11 KB (1,810 words) - 22:12, 29 August 2015
...p|rank of an algebraic group]] and the [[Rank of a Lie group|rank of a Lie group]] in a special way.
...or space. However, there exists another, unrelated, concept of rank in the theory of Lie algebras (see [[Rank of a Lie algebra|Rank of a Lie algebra]]).
4 KB (631 words) - 10:02, 15 April 2014
A square matrix of order $ n $
...p]]; the converse is also true: the multiplication table of a finite quasi-group is a Latin square. For a Latin square $ A = \| a _ {ij} \| $
11 KB (1,681 words) - 11:23, 17 March 2023
...ent of $\mathrm{GF}(p^n)^*$ is a root of the polynomial $X^{p^n-1}-1$. The group $\mathrm{GF}(p^n)^*$ is cyclic, and its generators are the primitive roots
...\mathrm{Gal}\left({\mathrm{GF}(p^m)/\mathrm{GF}(p^n)}\right)$ is cyclic of order $m/n$. The automorphism $\tau$ may be taken as the generator of $\mathrm{Ga
4 KB (749 words) - 18:32, 2 November 2014
A ''Chevalley group'' is
a linear algebraic group over some field, related to a semi-simple
6 KB (1,002 words) - 19:25, 3 November 2013
A permutation group is
...ons. In other words, a permutation group is a pair $(G,X)$, where $G$ is a group and $X$ is a set, and each $\def\g{\gamma}\g\in G$ corresponds to a transfo
9 KB (1,519 words) - 07:41, 30 November 2014
...e order $|G|$ of any [[Finite group|finite group]] $G$ is divisible by the order $|H|$ of any subgroup $H$ of it. The theorem was actually proved by J.L. La
.... Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR></table>
4 KB (562 words) - 21:05, 11 October 2014
...sarily associative) which is a [[Partially ordered group|partially ordered group]] under addition and in which for any elements $ a , b , c \in R $
Every ring is an ordered ring for the trivial order. As examples of ordered rings one may take an [[Ordered field|ordered field
7 KB (1,019 words) - 08:04, 6 June 2020
...: ~/encyclopedia/old_files/data/L058/L.0508490 Lie algebra of an analytic group,
''Lie algebra of a Lie group $ G $
9 KB (1,338 words) - 22:16, 5 June 2020
$#C+1 = 117 : ~/encyclopedia/old_files/data/R110/R.1100020 Ramification theory of valued fields
...of [[Commutative algebra|commutative algebra]] and [[Number theory|number theory]] in which certain distinguished intermediate fields of algebraic extension
7 KB (1,108 words) - 08:09, 6 June 2020
...nts of the real plane second-order curves (cf. [[Second-order curve|Second-order curve]]). Thus, let $ M $
these mappings are also called invariants of real plane second-order non-splitting curves. The values of these invariants on a specific curve en
8 KB (1,242 words) - 07:04, 6 May 2022
...ogic]]. The Morley rank is an ordinal-valued dimension associated to first-order formulas with parameters from a model $ M $
of a complete first-order theory $ T $.
4 KB (551 words) - 08:01, 6 June 2020
One of the classical groups, defined as the group of automorphisms of a skew-symmetric bilinear form $\Phi$ on a left $K$-mod
[[Classical group|Classical group]]). In the case when $E=K^{2m}$ and the matrix of $\Phi$ with respect to th
6 KB (1,078 words) - 14:22, 3 November 2013
...tation]]s of $G$ over $K$ are precisely the irreducible modules over the [[group algebra]] $R = k[G]$.
...ncept of an irreducible module is fundamental in the theories of rings and group representations. By means of it one defines the [[composition sequence]] an
2 KB (391 words) - 18:11, 18 March 2018
of a [[Linear algebraic group|linear algebraic group]] $ G $
is realized as a closed subgroup of the group $ \mathop{\rm GL}\nolimits (V) $
6 KB (816 words) - 16:46, 17 December 2019
cf. [[Geometric objects, theory of|Geometric objects, theory of]]) a new geometric object $ {\mathcal L} _ {X} Q $,
with respect to the one-parameter (local) transformation group $ \phi _ {t} $
9 KB (1,337 words) - 22:16, 5 June 2020
...tion with the problem of deciding for a well-formed formula from the first-order [[predicate calculus]] whether the formula is valid. Generally speaking, wh
...of an effective computational procedure is suitably formalized, as in the theory of algorithms.
3 KB (490 words) - 22:33, 10 January 2016
...oup (cf. also [[Lie group, semi-simple|Lie group, semi-simple]]), the loop group $\Omega G = \{ \gamma : S ^ { 1 } \rightarrow G : \gamma ( 1 ) = 1 \}$; her
...ently, $\Omega G$ is a [[Homogeneous space|homogeneous space]] of the loop group $L G _ { \text{C} } = \{ \gamma : S ^ { 1 } \rightarrow G _ { \text{C} } \}
5 KB (698 words) - 08:13, 15 February 2024
A triple $(W,G,F)$, where $W$ is a topological space, $G$ is a topological group and $F$ is a continuous function $G\times W\to W$ defining a left action of
...action. In this case the arguments of $F$ are usually written in the other order (expressing $F$ as a mapping $W\times G\to W$), and \eqref{1} is replaced b
4 KB (690 words) - 15:52, 14 February 2020
$#C+1 = 22 : ~/encyclopedia/old_files/data/F040/F.0400270 Finite group
...cally, many concepts in abstract group theory have had their origin in the theory of finite groups.
12 KB (1,742 words) - 09:10, 4 July 2020
...e Brauer group of $k$ and is denoted by $\Br(k)$. The zero element of this group is the class of full matrix algebras, while the element inverse to the clas
...ined for number fields in connection with the development of [[class field theory]]. The general form of the reciprocity law is formulated in terms of Brauer
7 KB (1,232 words) - 12:12, 30 December 2015
...ell as under restriction and glueing of mappings. More precisely, a pseudo-group of transformations $ \Gamma $
...7]]]) or even of an arbitrary set. As a group of transformations, a pseudo-group of transformations determines an equivalence relation on $ M $;
13 KB (1,866 words) - 18:13, 20 January 2022
...nym for "elementary subsystem" (cf. [[Model theory]]). If $L$ is a first-order language of signature $\Omega$ and $A$ is an algebraic system of signature
A first-order language $L$ is uniquely determined by its signature $\Omega=(\Omega_f,\Ome
3 KB (527 words) - 07:17, 12 February 2024
[[Algebraic group|Algebraic group]]) of the
[[General linear group|general linear group]] of all automorphisms of
3 KB (458 words) - 04:43, 4 January 2022
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.
4 KB (687 words) - 05:47, 15 April 2023
all transition functions between them belonging to a given pseudo-group $ \Gamma $
The pseudo-group $ \Gamma $
11 KB (1,659 words) - 08:01, 21 January 2024
$#C+1 = 100 : ~/encyclopedia/old_files/data/S091/S.0901670 Symmetric group
The group of all permutations (self-bijections) of a set $ X $
7 KB (1,003 words) - 08:24, 6 June 2020
associated to a [[P-divisible group| $ p $-
divisible group]] $ G $
4 KB (651 words) - 08:25, 6 June 2020
can be considered as first-order differentiation on the algebra of differential forms on $ M $
is considered as a differentiation of order zero.
6 KB (880 words) - 16:10, 1 April 2020
...transformations of the homogeneous space. The fundamentals of the general theory of Kawaguchi spaces were developed along the formal lines of a generalized
of order $ q = 2 p - 1 $,
6 KB (908 words) - 19:48, 12 January 2024
The special linear group
''of degree (order) $\def\SL{\textrm{SL}}\def\GL{\textrm{GL}} n$ over a ring $R$''
6 KB (988 words) - 22:16, 10 January 2015
group (in particular, for groups, Abelian groups, vector spaces, and rings) one c
group $ G $
4 KB (680 words) - 19:35, 5 June 2020
...anced design ($\operatorname{PBD}(\nu,K,\lambda)$ or $(K,\lambda)$-PBD) of order $\nu$ with block sizes from $K$ is a pair $(V,\mathcal{B})$ where $V$ is a
...$0$ and $1$ are considered to be flats of all pairwise balanced designs of order $v > 1$. Such flats are trivial.
8 KB (1,452 words) - 12:17, 13 February 2024
...ntially the character group of the dyadic group. (This connection made the theory of Walsh functions a special case of the general study of [[Harmonic analys
4) the Walsh–Dirichlet kernels of order $2^n$, $D_{2^n}(x)=\sum_{k=0}^{2^n-1}w_k(x)$, are non-negative on $[0,1)$.
3 KB (467 words) - 09:42, 27 November 2018
...unctions|Walsh functions]] (either in Walsh's original order or in Paley's order). A Walsh–Fourier series is a Walsh series whose coefficients satisfy
The theory of Walsh series is distinguished from that of other orthonormal series (cf.
4 KB (668 words) - 08:28, 6 June 2020
is a square matrix of order $ m $
is called the Jordan block of order $ m $
5 KB (708 words) - 11:11, 17 December 2019
...fference set|Difference set]] (Vol. 3). For an extensive discussion of the theory of Abelian difference sets, see also [[#References|[a1]]], Chap. VI.
...ference sets comprise the Paley difference sets, the families using higher-order residues and also the twin prime power series, see [[#References|[a1]]], Se
9 KB (1,331 words) - 19:36, 13 February 2024
Let $G$ be the [[Group|group]] $\operatorname {SL} _ { n } ( F )$ ($n \geq 3$, $F$ any [[Field|field]]).
...entral extensions uniquely (cf. also [[Extension of a group|Extension of a group]]). It follows that every [[Projective representation|projective representa
8 KB (1,339 words) - 16:56, 1 July 2020
...algebraic groups. As in the analytic case, the Lie algebra of an algebraic group $ G $
group, $ A = K [G] $
7 KB (1,070 words) - 21:44, 16 December 2019
In the most general sense, ''Galois theory''
is a theory dealing with mathematical
11 KB (1,965 words) - 04:47, 16 January 2022
$#C+1 = 75 : ~/encyclopedia/old_files/data/F042/F.0402210 Fundamental group,
''Poincaré group''
6 KB (829 words) - 05:44, 13 April 2023
...on-trivial connected normal subgroups other than $ G $ . A connected Lie group is semi-simple if and only if it splits into a locally direct product of si
...bra, semi-simple]]), and if $ G $ is a group without centre (an adjoint group), then $ \Gamma (G) $ is the lattice $$
11 KB (1,458 words) - 18:15, 12 December 2019
...representation theory (see [[Representation of a group|Representation of a group]]; [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#Ref
is played by the group algebra $ RG $,
6 KB (990 words) - 10:59, 29 May 2020
...ly define the relevant homology (cohomology) theory. An axiomatic homology theory is defined on a certain category of pairs $ ( X, A) $
an Abelian group (or module over some ring) $ H _ {q} ( X, A) $
5 KB (779 words) - 19:51, 16 January 2024
of order $ n $
of order $ n $
17 KB (2,485 words) - 12:08, 21 March 2022
...al{Z} _ { m + 1 } ^ { \pi }$ to $\mathcal{Z} _ { m} ^{\pi }$. The real $K$-theory groups of $\mathbf{R} \pi$ are given by:
...a6]]] defined a $K$-theory-valued invariant $\alpha$ taking values in this group which generalizes the $\hat{A}$-genus. It was conjectured that this might p
6 KB (850 words) - 14:01, 25 November 2023
of a group $ G $
of the same group $ G $
5 KB (737 words) - 10:19, 17 December 2019
$#C+1 = 244 : ~/encyclopedia/old_files/data/G044/G.0404270 Geometric objects, theory of
...ential forms makes it possible to introduce differential criteria into the theory of geometric objects, which convert it to an effective tool in differential
21 KB (3,190 words) - 19:41, 5 June 2020
$#C+1 = 18 : ~/encyclopedia/old_files/data/G045/G.0405240 Group calculus
An [[Associative calculus|associative calculus]] in which the natural group requirement of the existence of an inverse operation is effectively fulfill
9 KB (1,334 words) - 19:42, 5 June 2020
that is homogeneous with respect to the group of linear transformations $ \alpha \in \mathop{\rm GL} _ {n} ( \mathbf R
The automorphism group of $ K _ {n} $
9 KB (1,244 words) - 22:10, 5 June 2020
...cteristic $p$, and values of irreducible complex characters on elements of order divisible by $p$, are controlled to a considerable extent by the $p$-local
...common multiple of the orders of the elements of $G$. The [[Group algebra|group algebra]] $R G$ has a unique decomposition as a direct sum of indecomposabl
9 KB (1,465 words) - 17:46, 1 July 2020
...3 : ~/encyclopedia/old_files/data/G110/G.1100050 Gamma\AAhinvariant in the theory of Abelian groups,
...ian group]]). By a theorem of S. Shelah (see [[#References|[a7]]]), such a group is free if it is of [[singular cardinal]]ity, so the invariant is defined f
6 KB (845 words) - 19:41, 5 June 2020
...rst-order sentences true in a given structure forms a complete first-order theory $ T $.
For example, the models of the theory of the complex field $ ( \mathbf C ,\ +,\ \cdot ,\ 0,\ 1) $
11 KB (1,671 words) - 11:38, 22 December 2019
...to the case $d = 2$, a design $A _ { 1 } ( s )$ is just an affine plane of order $s$, see also [[Plane|Plane]].
...on plane (cf. [[Plane|Plane]]; [[Primitive group of permutations|Primitive group of permutations]]). Detailed studies of translation planes may be found in
5 KB (820 words) - 11:00, 17 March 2023
for some group $ A $.
together with such structures as the group of automorphisms, the semi-group of endomorphisms, the lattice of congruences, etc.). The problems involved
7 KB (1,037 words) - 08:24, 6 June 2020
...re non-isomorphic real forms of the Lie algebra of all complex matrices of order $n$ (which also has other real forms).
|valign="top"|{{Ref|Na}}||valign="top"| M.A. Naimark, "Theory of
3 KB (431 words) - 17:08, 22 November 2013
...the plane is the number of points on a circle minus one. A Möbius plane of order $ n $
circles pass. The following model of the Möbius plane of order $ n = p ^ {h} $
5 KB (757 words) - 08:02, 6 June 2020
with [[Galois group]] $ G $,
the Galois group of which is a proper quotient group of $ G $.
7 KB (1,042 words) - 19:53, 16 January 2024
of the set. A quasi-group with a unit is called a [[Loop|loop]].
...e connection of such planes with quasi-groups, that the development of the theory of quasi-groups properly began.
16 KB (2,475 words) - 19:16, 18 January 2024
The representation of a Lie group $ G $
cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]). The coadjoint representation acts on the dual $ \mathfrak g ^ {*} $
4 KB (606 words) - 17:45, 4 June 2020
equipped with a transitive transformation group $ G = \{ g \} $
is the group of all parallel translations. Sometimes the term "homogeneous random field
5 KB (684 words) - 08:09, 6 June 2020
...<TR><TD valign="top">[1]</TD> <TD valign="top"> M. Hall, "Combinatorial theory" , Wiley (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">
mutually [[Orthogonal Latin squares|orthogonal Latin squares]] of order $ t $.
5 KB (737 words) - 06:33, 22 February 2022
...proved a still stronger assertion: Every two normal series of an arbitrary group have isomorphic refinements (Schreier's theorem). The Jordan–Hölder theo
...relation of normality or an operation of multiplication were introduced in order to define a normal series of elements in a lattice. 4) Generalizations of t
4 KB (660 words) - 20:00, 11 April 2014
...[2]]]. Discrete systems of this type characteristically display long-range order at temperatures below a transition point — that is, a general regularity
...ordering phenomena is obtained by theories of the type of molecular field theory. Despite the mathematical simplicity of the model, an exact general solutio
4 KB (585 words) - 11:53, 26 March 2023
....G. Thompson [[#References|[2]]], who showed that all finite groups of odd order are solvable.
...D valign="top"> W. Feit, J.G. Thompson, "Solvability of groups of odd order" ''Pacific J. Math.'' , '''13''' (1964) pp. 775–1029</TD></TR></table>
15 KB (2,211 words) - 08:50, 26 March 2023
[[Algebraic group|algebraic group]] that is a complete
variety is regular; the group law on an Abelian variety is
8 KB (1,216 words) - 20:39, 5 March 2012
is the rank of the group $ E $
...raic number|Algebraic number]]; [[Algebraic number theory|Algebraic number theory]]) and $ v $
3 KB (430 words) - 10:48, 20 January 2024
of a Lie group $ G $
of a Lie group $ G $,
8 KB (1,274 words) - 22:13, 5 June 2020
An orthogonal group is a group of all linear transformations of an $n$-dimensional
...\varphi}\phi$ such that $Q(\phi(v))=Q(v)$ for all $v\in V$). An orthogonal group is a
14 KB (2,418 words) - 18:20, 7 November 2014
...ps so that each term of the form depends linearly on the variables in each group, then the form is called a ''[[multilinear form]]''. Every form can be obta
...]s. The theory of quadratic forms is closely connected with that of second-order curves and surfaces (see also [[Hermitian form]]).
1 KB (237 words) - 07:31, 19 January 2016
that does not change under any of the transformations of a given Lie group $ G $
of transformations. The group $ G $
11 KB (1,730 words) - 19:04, 9 January 2024
Cauchy's theorem in group theory: If the order of a finite group $ G $
contains an element of order $ p $.
5 KB (824 words) - 15:35, 4 June 2020
A classical group is the group of automorphisms of some
...ly satisfy extra conditions. There is no precise definition of a classical group. It is supposed that $f$ is either the null form or is a non-degenerate ref
12 KB (1,991 words) - 18:05, 29 November 2014
The additive group of the ring $ W( k) $
is said to be the Witt group of the field $ k $
9 KB (1,359 words) - 08:29, 6 June 2020
...atively answered in 1966 (cf. [[#References|[2]]]). A square matrix $A$ of order $n\times n$ over an associative ring $R$ is said to be non-full if it can b
be square matrices of order $n\times n$ over $R$ in which all columns (except, possibly, the first colu
5 KB (851 words) - 14:57, 30 July 2014
with structure group $ \mathop{\rm Spin} _ {n} $(
see [[Spinor group|Spinor group]]), covering some principal fibre bundle $ \pi : P \rightarrow M $
6 KB (923 words) - 17:42, 6 January 2024
...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).
17 KB (2,435 words) - 09:18, 2 April 2023
i) the value group $vK$ is a [[P-divisible group|$p$-divisible group]];
...the latter by a theorem of Kaplansky (which has a certain analogue in the theory of [[real closed field]]s: If $K$ has no non-trivial immediate algebraic ex
3 KB (523 words) - 19:54, 1 January 2015
is a [[Galois extension|Galois extension]] with Galois group $ G ( K/k) $,
is precisely the order of the inertia subgroup $ T ( \mathfrak P _ {i} ) $
3 KB (473 words) - 08:09, 6 June 2020
...stimator|equivariant estimator]] for the shift parameter with respect to a group of real shifts, having minimal risk with respect to a quadratic loss functi
be the group of real shifts operating in the realization space $ \mathbf R ^ {1} = (-
4 KB (549 words) - 19:05, 17 January 2024
A principal subbundle with structure group $ G $
bundle of all co-frames of order $ k $
17 KB (2,677 words) - 19:40, 5 June 2020
A homomorphism from the [[Stable homotopy group|stable homotopy group]] of the spectrum of $ \mathop{\rm SO} $
to the stable homotopy group of the spectrum of the sphere $ S ^ {0} $,
9 KB (1,201 words) - 20:31, 16 January 2024
...ups) that asks whether every [[Finitely-generated group|finitely-generated group]] of exponent $n$ is finite, or, equivalently, whether the free Burnside gr
...the Burnside groups $B ( m , 6 )$ of exponent $6$ are finite and have the order given by the formula $| B ( m , 6 ) | = 2 ^ { \alpha } 3 ^ { C _ { \beta }
14 KB (2,317 words) - 06:45, 16 March 2023
...nd let $W$ be the subgroup of permutation matrices. In the [[Group algebra|group algebra]] $k [ G ]$ of $G$ over any field $k$ of characteristic $0$ or $p$,
...o $p$, then $\chi ( x )$ equals, up to a sign which can be determined, the order of a Sylow $p$-subgroup of the centralizer of $x$; otherwise it equals $0$.
9 KB (1,369 words) - 13:31, 25 November 2023
For lacunae in function theory see e.g. [[Hadamard theorem|Hadamard theorem]] on gaps; [[Fabry theorem|Fab
For lacunae in geometry see [[Group of motions|Group of motions]]; [[Lacunary space|Lacunary space]].
6 KB (898 words) - 14:30, 8 January 2022
...unding the complexity of decompositions as graphs of groups for a discrete group.
...of Grushko's theorem [[#References|[a6]]], which asserts that if a [[Group|group]] $G$ is generated by a subset of cardinality $d$ and decomposes as a non-t
9 KB (1,405 words) - 15:30, 1 July 2020
Here the factors on the right-hand side are to be taken in the natural order for (a1), while in the case of (a2) the product is first taken over $ m $
...pplication in (combinatorial) group theory, algebraic topology and quantum theory, cf., e.g., [[#References|[a2]]]–[[#References|[a4]]]. For convergence re
3 KB (471 words) - 05:21, 19 January 2022
...s/m/m062/m062160/m06216014.png" />. This defines a [[Partial
order|partial
order]] which occurs under various names in various parts of mathematics: majorit
...mages/m/m062/m062160/m06216047.png" /> the corresponding [[Weyl
group|Weyl
group]]. Consider the adjoint action of <img align="absmiddle" border="0" src="ht
34 KB (4,631 words) - 18:28, 30 November 2016
''of order $ m $''
one obtains a Young diagram of order $ m - p $.
6 KB (623 words) - 19:39, 9 November 2023
...they can be applied to sets whose elements are of an indefinite nature. In order to define a structure, relations are given in which the elements of the set
..."top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French)</TD></TR></tabl
11 KB (1,705 words) - 14:55, 7 June 2020
The field of velocities of a (local) one-parameter group of motions on a Riemannian manifold $ M $.
...eld is complete, that is, it is the field of velocities of a one-parameter group of motions. The set $ i ( M) $
5 KB (766 words) - 22:14, 5 June 2020
Historically the first branch of linear algebra was the theory of linear algebraic equations (cf. [[Linear equation|Linear equation]]). In
...elli theorem|Kronecker–Capelli theorem]]). The construction of the general theory of systems of linear equations was thus completed at the end of the 19th ce
9 KB (1,394 words) - 08:15, 9 January 2024
...,\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
...the definition of the Erdös–Ginzburg–Ziv constant for an arbitrary Abelian group, as follows. If $ G $
10 KB (1,573 words) - 17:25, 28 January 2020
...(or posets; cf. also [[Partially ordered set|Partially ordered set]]). The theory of Möbius inversion matured in the classic paper [[#References|[a4]]] of G
...$ is the $x y$-entry in the inverse of the incidence matrix of the partial order relation on $P$.
11 KB (1,758 words) - 09:24, 10 November 2023
defines the square matrix of order $ 2n + 2 $
...equal to one. The symplectic transformations form a group, which is a Lie group.
7 KB (1,005 words) - 14:55, 7 June 2020
...eristic of $k$ (or if the characteristic is 0), then the group $U_n$ is of order $n$ and its generators are known as primitive $n$-th roots of unity. The nu
...symbol|norm-residue symbol]]; etc.). In field theory and algebraic number theory an important position is occupied by fields obtained by adjunction of roots
4 KB (680 words) - 13:40, 30 December 2018
The set of all orthogonal matrices of order $ n $
forms a subgroup of the [[General linear group|general linear group]] $ \mathop{\rm GL} _ {n} ( R) $.
3 KB (437 words) - 14:54, 7 June 2020
''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
11 KB (1,686 words) - 18:54, 6 February 2021
is the multiplicative group of the field of real numbers. The number $ \tau ( C ^ \prime ) $
is a finite group, then the element $ \omega \in \mathop{\rm Wh} ( \pi ) $
3 KB (446 words) - 08:10, 6 June 2020
be a [[Finite group|finite group]]. A representation of $ G $
is a group [[Homomorphism|homomorphism]] $ X : G \rightarrow { { \mathop{\rm GL} } (
11 KB (1,659 words) - 08:43, 26 March 2023
Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing r
...covered just recently (relative to the patterns' age)-- in particular, the theory of automorphism groups of finite geometries.
5 KB (802 words) - 02:45, 19 April 2023
...fied, up to an isomorphism, with the elements of the [[Brauer group|Brauer group]] $ B( F ) $
...ex numbers, and the [[Quaternion|quaternion]] algebra. For this reason the group $ B( \mathbf R ) $
5 KB (706 words) - 09:14, 28 June 2022
be its [[Galois group|Galois group]] and $ \mathbf Z [ G ] $
the group ring of $ G $ (cf. also [[Group algebra|Group algebra]]; [[Cross product|Cross product]]) over the rational integers $
7 KB (1,035 words) - 05:58, 19 March 2022
is a totally ordered Abelian group, the adjoined element $ \infty $
called the value group of the valuation $ v $.
14 KB (2,135 words) - 08:27, 6 June 2020
A cellular algebra $W$ of order $n$ and rank $r$ is a matrix subalgebra of the full matrix algebra $\mathbf
...algebras is again a cellular algebra. For each set of matrices of the same order $n$ it is possible to determine a minimal cellular algebra containing this
7 KB (1,000 words) - 16:48, 14 January 2021
of the same curvature by factorization with respect to a discrete group $ \Gamma $
are conjugate in the group of all motions of $ \widetilde{M} {} ^ {n} $.
18 KB (2,658 words) - 19:36, 18 January 2024
...ly ordered, subject to the condition that $[ab] > b$, while preserving the order of the basic commutators of lengths less than $n$. Any set of basic commuta
<TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Hall jr., "The theory of groups" , Macmillan (1959)</TD></TR>
3 KB (539 words) - 08:07, 13 February 2024
$#C+1 = 108 : ~/encyclopedia/old_files/data/C027/C.0207190 Crystallographic group
A discrete group of motions of an $ n $-
14 KB (1,983 words) - 18:41, 26 March 2023
is called a finite projective plane of order $ n $
exists is unanswered (1990). The existence of a finite projective plane whose order is a power of a prime number has been proved (cf. [[#References|[4]]]). The
11 KB (1,726 words) - 06:13, 23 April 2023
...od have required refinement, and they have been formulated in terms of the theory of bundles.
be the $ r $-dimensional [[Lie group|Lie group]] of its transformations ( $ G $
15 KB (2,363 words) - 08:15, 18 August 2022
...e of stellar subdivisions with centres in all open simplices of $K$ in the order of decreasing dimensions. For an arbitrary closed subcomplex $K$ of a compl
...ign="top">[2]</TD> <TD valign="top"> P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1965)</TD
3 KB (522 words) - 14:11, 19 April 2014
...l quasi-triangular Hopf algebra, co-quasi-triangular Hopf algebra, quantum group''
...oping algebras $U _ { q } ( \mathfrak { g } )$ or their associated quantum group coordinate rings $G_q$.
14 KB (2,048 words) - 18:24, 3 August 2020
$#C+1 = 125 : ~/encyclopedia/old_files/data/K055/K.0505520 Kleinian group
of the group of all fractional-linear mappings (cf. [[Fractional-linear mapping|Fraction
13 KB (1,863 words) - 19:27, 17 January 2024
...isomorphic to the fundamental groups (cf. [[Fundamental group|Fundamental group]]) $ G ( K) = \pi _ {1} ( M ( k) ) $
dimensional homology group $ H _ {2} ( G ; \mathbf Z ) $
12 KB (1,807 words) - 18:41, 13 January 2024
...] $ K $ . The choice of a basis in $ V $ realizes a linear group as a group of non-singular square $ ( n \times n ) $ -matrices over $ K $ . In thi
...p of $ ( n \times n ) $ -matrices or linear group of order $ n $ . The theory of linear groups is most developed when $ K $ is commutative, that is,
16 KB (2,362 words) - 18:01, 12 December 2019
The [[projective plane]] $\mathrm{P}(2,2)$ of order $2$, consisting of $7$ points and $7$ lines each containing $3$ points, in
A Steiner triple system gives rise to a [[quasi-group]] structure on the underlying set, defined by the binary operation $x \cdot
2 KB (243 words) - 20:10, 8 November 2023
...erous applications in algebraic geometry, class field theory and cobordism theory.
A formal group over a field $ k $
17 KB (2,537 words) - 22:38, 15 December 2019
...ntary theory]]; [[Field|Field]]; [[Valuation|Valuation]]). The basic first-order language is that of rings (or fields) together with a unary [[relation symb
...lly closed field that is complete under a valuation with Archimedean value group. The decidability is one ingredient in the proof that Hilbert's 10th proble
12 KB (1,846 words) - 09:28, 26 November 2016
The classical theory of differential-geometric objects was revisited from the functorial point o
...he form of a natural transformation $F \rightarrow G$. Moreover, the $k$th order natural operators of $F$ into $G$ (cf. [[Natural operator in differential g
4 KB (674 words) - 17:02, 1 July 2020
...le, a [[Kawaguchi space|Kawaguchi space]]). The foundations of the general theory of non-linear connections are fairly well developed and applications of som
be a smooth principal $ G $-bundle with structure Lie group $ G $
7 KB (1,089 words) - 12:15, 18 February 2022
...ionic structure is isomorphic to the [[General linear group|general linear group]] $ \mathop{\rm GL} ( m , \mathbf H ) $
The group of all automorphisms of a quaternionic structure is isomorphic to the direc
8 KB (1,179 words) - 15:10, 19 January 2021
...[[Fourier integral|Fourier integral]]). Classical harmonic analysis — the theory of Fourier series and Fourier integrals — underwent a rapid development,
...locally compact Abelian groups (cf. [[Character of a group|Character of a group]]), posed the problem of the natural limits of the main results of classica
66 KB (9,085 words) - 17:28, 31 March 2020
...l basis for $E/F$, that is, a basis consisting of an orbit of the [[Galois group]] $G = \mathrm{Gal}(E/F)$. Thus, an element $z \in E$ generates a normal ba
...basis consisting of primitive elements (elements of maximal multiplicative order, cf. [[Primitive element of a Galois field]]), see [[#References|[a4]]]
2 KB (296 words) - 19:12, 24 November 2023
...discrete topology. Any Hausdorff space can be made into a topological semi-group, e.g. by giving it a left-singular or zero multiplication.
...independent branches of topological semi-groups have emerged: the general theory of compact semi-groups (cf. [[Compactness|Compactness]]); homotopy properti
16 KB (2,287 words) - 14:36, 19 March 2023
as a group; and 4) the line as a topological space.
one considers an arbitrary (not necessarily topological) group $ G $
13 KB (1,809 words) - 19:41, 5 June 2020
The group of principal homogeneous spaces (cf. [[Principal homogeneous space|Principa
defined over k, has a group structure. The group $ { \mathop{\rm WC} } ( A, k) $
7 KB (1,109 words) - 16:59, 1 July 2020
of finite index in the [[Modular group|modular group]] $ \Gamma $;
(see [[Modular group|Modular group]]). The least such $ N $
8 KB (1,144 words) - 05:57, 13 February 2024
A non-empty set on which some [[Order (on a set)|order relation]] is given.
...es of partially-ordered sets. 1) The set of natural numbers with the usual order relation. 2) The set of natural numbers, where $ a \leq b $
21 KB (3,255 words) - 11:54, 19 March 2023
freely acted upon by a free Abelian group $ J ^ {a} $
of the [[Fundamental group|fundamental group]] $ \pi _ {1} (M) = G $
12 KB (1,694 words) - 06:42, 26 March 2023
== $ K $-theory and classification.==
By the [[K-theory| $ K $-
12 KB (1,766 words) - 06:39, 26 March 2023
be the discrete group of automorphisms of $ D $.
...e multipliers to obtain a theta-series. A Poincaré series, associated to a group $ \Gamma $,
5 KB (778 words) - 08:25, 6 June 2020
The most important examples of general-purpose systems are (in alphabetical order) AXIOM, Derive, Maple, Mathematica, and REDUCE. Their design features diffe
...d SYMDE (in MACSYMA), CRACK and ODESOLVE (in REDUCE), PDELIE (Lie symmetry group methods package in MACSYMA); for tensor calculus: MathTensor and Ricci (in
9 KB (1,292 words) - 20:51, 18 September 2016
...can either be reduced to lower-order ones or be completely integrated via group-theoretic techniques.
...wn ones, and PDEs can be classified into equivalence classes. Furthermore, group-invariant solutions obtained via Lie's approach may provide insight into th
23 KB (3,178 words) - 22:16, 5 June 2020
...aces that are invariant under transformations of the [[Affine group|affine group]] or its subgroups. The differential geometry of equi-affine space has been
...e curve. Constant equi-affine curvature characterizes curves of the second order. A natural equation $ k = f(s) $
8 KB (1,146 words) - 19:51, 4 April 2020
...ally become invalid, and the Grothendieck group $K_0(R)$ and the Whitehead group $K_1(R)$ are, in a certain sense, a measure of their deviation from being t
...vector bundle]]. These objects may be studied with the aid of the homotopy theory of vector bundles and of topological
14 KB (2,405 words) - 22:14, 10 January 2015
...gree 1 on the subgroup $H$ (cf. [[Character of a group]]) of the symmetric group $S_n$ (one obtains the [[determinant]] for $H=S_n$, $\chi =\pm 1$, in accor
The permanent is used in linear algebra, probability theory and combinatorics. In combinatorics, a permanent can be interpreted as foll
8 KB (1,228 words) - 19:13, 17 March 2023
of the [[Fundamental group|fundamental group]] of the domain, and the mapping is taken to be a [[Cofibration|cofibration
...Quillen [[#References|[a15]]] in order to define the higher [[Algebraic K-theory|algebraic $ K $-
7 KB (1,010 words) - 16:47, 17 March 2023
...) in which $<$ is interpreted as a [[total order]] (cf. [[Order (on a set)|Order (on a set)]]). Then $M$ is called $o$-minimal if every parametrically defin
An [[elementary theory]] is called $o$-minimal if every model of it is $o$-minimal.
5 KB (799 words) - 09:27, 26 November 2016
The most important class of formal systems is that of formal first-order theories (see [[#References|[4]]]) formalizing some branch of meaningful ma
...riety of branches of mathematics, for example in group theory and category theory.
3 KB (392 words) - 12:21, 19 August 2014
...mation" ) arose in the classical theory of electromagnetic fields. In this theory the four-dimensional electromagnetic vector potential $ A _ {n} ( x) $,
remain unchanged; this is the so-called gauge invariance of field theory. If the function $ f $
9 KB (1,227 words) - 19:41, 5 June 2020
...ncyclopedia/old_files/data/A011/A.0101980 Almost\AAhperiodic function on a group
be an (abstract) group. A bounded complex-valued function $ f(x) $,
12 KB (1,716 words) - 11:05, 10 May 2020
...s of analytic function theory|boundary value problems of analytic function theory]]. It can be stated in the simplest case as follows. Let $ L $
...index in the class of solutions with an admissible infinity at the end of order less than one (an integrable infinity) is equal to one plus the index in th
10 KB (1,509 words) - 08:11, 6 June 2020
[[Category:Distribution theory]]
...eory. It is used most effectively in proving limit theorems of probability theory. For example, the proof of the central limit theorem for independent identi
8 KB (1,162 words) - 19:58, 19 January 2024
analytic boundary components, then, in order that they be conformally equivalent, it is necessary that $ 6g- 6+ 3n $
...blem for Riemann surfaces, which studies the nature of these parameters in order to introduce them, if possible, in such a way that they would define a comp
14 KB (1,973 words) - 08:11, 6 June 2020
...function]]; on the other, a finite [[Sporadic simple group|sporadic simple group]]. Moonshine is the explanation and generalization of this unlikely connect
...divides $o ( g ) \operatorname { gcd } ( 24 , o ( g ) )$ ($o ( g )$ is the order of $g$).
12 KB (1,765 words) - 09:46, 10 November 2023
The multiplicative group $ R ^ \times $
by the group of so-called principal units $ 1 + pR $.
7 KB (1,070 words) - 19:40, 5 June 2020
is called the order of $ f ( z) $.
There are no elliptic functions of order less than 2.
9 KB (1,292 words) - 19:08, 20 January 2022
$#C+1 = 126 : ~/encyclopedia/old_files/data/F041/F.0401890 Fuchsian group
...omorphic transformations (cf. [[Discrete group of transformations|Discrete group of transformations]]) of an (open) disc $ K $
14 KB (2,109 words) - 20:03, 15 March 2023
The order of a point $ p \in \pi ( k) $
A point of order two is called a double point and a point of a order $ > 1 $
10 KB (1,621 words) - 18:58, 10 April 2023
...encyclopedia/old_files/data/R081/R.0801420 Representation of a topological group
A mapping of the group $ G $
20 KB (2,775 words) - 16:40, 31 March 2020
...inear representation|Linear representation]]) of the [[Spinor group|spinor group]] $ \mathop{\rm Spin} _ {n} ( Q) $,
see [[Spinor group|Spinor group]]; $ Q $
8 KB (1,127 words) - 18:58, 19 January 2024
A theorem reducing the description of the action of a transformation group on some neighbourhood $U$ of a given [[Orbit|orbit]] to that of the [[Stabi
Let $G$ be a topological [[Transformation group|transformation group]] of a [[Hausdorff space|Hausdorff space]] $X$. A subspace $S$ of $X$ is ca
11 KB (1,819 words) - 15:30, 1 July 2020
...k_i$, called the partial indices of $f$, are uniquely determined up to the order and define an interesting decomposition of the space of matrix-functions (c
...sphere [[#References|[a3]]]. It is also of fundamental importance for the theory of singular integral equations [[#References|[a4]]], Riemann–Hilbert prob
7 KB (989 words) - 16:56, 1 July 2020
of first-order closed formulas of the signature $ \Omega $
...both it and its complement are closed with respect to ultra-products. The theory of axiomatizable classes of algebraic systems deals with the connection bet
8 KB (1,165 words) - 16:10, 1 April 2020
...commutative group of operators on a Banach space and, more generally, any group with relatively compact trajectories. If $ \mathop{\rm dim} X < \infty
...dition is not sufficient, even for normal operators on a Hilbert space. In order that a [[Normal operator|normal operator]] $ A $
9 KB (1,382 words) - 08:22, 6 June 2020
in order to include arbitrary positive operators in this correspondence one has to i
The theory of positive-definite kernels extends the theory of positive-definite functions (cf. [[Positive-definite function|Positive-d
3 KB (383 words) - 08:07, 6 June 2020
relative to a [[Group|group]] $ G $
is a homogeneous space of the group $ G $
13 KB (2,049 words) - 19:35, 17 January 2024
...in order to study obstruction theory for $G$-spaces. If $G$ is a [[finite group]], let $\mathcal{O}_G$ be the category of orbits $G/H$ and $G$-mappings bet
...D valign="top"> T. tom Dieck, "Transformation groups and representation theory" , ''Lecture Notes Math.'' , '''766''' , Springer (1979)</TD></TR>
3 KB (473 words) - 18:52, 28 October 2016
...[[Non-Euclidean geometries|non-Euclidean geometries]], i.e. a geometrical theory based on axioms whose requirements are different from the requirements of t
==Group I. Axioms of incidence.==
19 KB (3,123 words) - 23:52, 14 December 2020
The ''determinant of a square matrix $A = (a_{ij})$ of order $n$ over a commutative associative ring $R$ with unit 1'' is
...with 1, $\def\Mn{\textrm{M}_n(R)}\Mn$ is the set of all square matrices of order $n$ over $R$ and $E_n$ is the identity matrix over $R$. Let $A\in\Mn$, whil
11 KB (1,876 words) - 20:27, 30 November 2016
$#C+1 = 142 : ~/encyclopedia/old_files/data/D032/D.0302500 Dimension theory
in such a way that the multiplicity (or order) of this covering is $ n + 1 $,
12 KB (1,894 words) - 19:35, 5 June 2020
...ve algebras are the group algebras $k G$ for finite groups $G$ (cf. also [[Group algebra]]). An arbitrary finite-dimensional algebra $A$ is said to be repre
...hbf{E} _ { n }$ ($n = 6,7,8$) and $G$ is an infinite [[Cyclic group|cyclic group]] of automorphisms of the translation quiver $\mathbf{Z} \overset{\righthar
9 KB (1,240 words) - 08:04, 25 November 2023
===Dirichlet's theorem in the theory of Diophantine approximations===
A theorem describing the structure of the multiplicative group of units of an algebraic number field; obtained by P.G.L. Dirichlet [[#Refe
7 KB (1,065 words) - 14:05, 17 March 2020
...mathfrak g $ is the Lie algebra of all complex square matrices of a fixed order, then the subalgebra of all diagonal matrices is a Cartan subalgebra in $
...cisely, they can be transformed into another by operators of the algebraic group $ D $ of automorphisms of $ \mathfrak g $ whose Lie algebra is the co
7 KB (925 words) - 17:15, 17 January 2020
Historically, the earliest theory of a [[Cohomology of algebras|cohomology of algebras]].
is a group and $ A $
16 KB (2,427 words) - 09:48, 26 March 2023
are "indexed" by elements of some abstract [[Semi-group|semi-group]] $ \mathfrak A $
is known as a representation of the semi-group $ \mathfrak A $.
34 KB (5,024 words) - 09:12, 21 January 2024
cf. [[Remainder of an integer|Remainder of an integer]]). In order to express the congruence of the numbers $ a $
and in order to express the incongruency of $ a $
20 KB (3,011 words) - 09:59, 26 March 2023
...e projective; the property of points to be on one line (collinearity), the order of algebraic curves, etc., are such properties.
...ns, a "richer" projective geometry, this group of axioms is completed by order axioms and the axiom of continuity (for real projective space), the [[Pappu
7 KB (1,119 words) - 13:21, 12 December 2013
the order of any element of $ \Sigma $
runs through the symmetric group $ S _{n} $ .
30 KB (4,468 words) - 18:44, 17 December 2019
The branch of number theory with the basic aim of studying properties of algebraic integers in algebrai
...e group of Pell units). The question arises: What is the structure of this group?
28 KB (4,440 words) - 22:00, 11 December 2019
[[Algebraic curve|algebraic curve]] of genus 1. The theory of elliptic
geometry. But historically the theory of elliptic curves arose as a
19 KB (3,251 words) - 20:37, 19 September 2017
...essive performance (composition) of transformations, etc. The concept of a group is historically one of the first examples of abstract algebraic systems and
A group is a non-empty set $G$ with one binary operation that satisfies the followi
21 KB (3,246 words) - 17:24, 9 October 2016
Given a [[Group|group]] $G$ and a subset $S \subset G$ which does not contain the identity of $G$
...References|[a20]]] that an arbitrary graph $\Gamma$ is a Cayley graph of a group $G$ if and only if $\operatorname{Aut} \Gamma$ contains a regular subgroup
10 KB (1,435 words) - 07:01, 29 March 2024
...of lowest degree for which there exist moduli (cf. [[Moduli theory|Moduli theory]]). Every smooth cubic curve $ X $
into an Abelian group with neutral element $ x _ {0} $.
10 KB (1,376 words) - 11:12, 26 March 2023
Thus, the complete theory of areas of polygons can be constructed on the basis of the theorem on the
The theory of volumes in $ \mathbf R ^ {3} $
11 KB (1,725 words) - 19:37, 5 June 2020
A generalization of the [[Fundamental group|fundamental group]], proposed by W. Hurewicz [[#References|[1]]] in the context of problems o
...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
33 KB (4,910 words) - 10:04, 15 December 2019
...classification theory of unipotent commutative algebraic groups and in the theory of commutative formal groups (amongst other things) [[#References|[a3]]].
...f{N})$ is free with basis $\mathbf{N}^*$, the free monoid (see [[Free semi-group]]) of all words in the alphabet $\mathbf{N}$ with the duality pairing $\mat
4 KB (709 words) - 13:46, 20 March 2023
...) \right\}$. The theory of cosine functions, which is very similar to the theory of semi-groups, was originated by S. Kurera [[#References|[a2]]] and was de
...\Omega )$. Since the equation is of first order, one can apply semi-group theory (see [[#References|[a4]]], [[#References|[a5]]]). Indeed, the operator
10 KB (1,449 words) - 17:45, 1 July 2020
...countable models of $T$, up to isomorphism (cf. also [[Model theory|Model theory]]); $n(T)\leq2^{\aleph_{0}}$. In 1961, R. Vaught [[#References|[a17]]] aske
...#References|[a10]]] proved, using [[Descriptive set theory|descriptive set theory]], that if $n(T)>\aleph_0$, then $n(T)=\aleph_1$ or $2^{\aleph_{0}}$ (actua
7 KB (1,023 words) - 19:13, 15 November 2023
...ectively), i.e. the sphere is a (hyper-)quadric or a surface of the second order of special form.The position of any point in space relative to a sphere is
...$O(n+1)$ — that maps one pair onto the other); this group is the complete group of isometries of $S^n$; finally, a sphere is a [[Homogeneous space|homogene
9 KB (1,500 words) - 21:06, 8 November 2014
be a set of generators for a finitely-generated group $ G $.
...below. The subject of growth functions for algebras and groups studies the order of growth of functions like $ f _ {G} ( n) $
19 KB (2,908 words) - 20:20, 12 January 2024
of this form is said to be the degree (order) of the hypersurface. A closed subscheme $ W $
of order $ m $(
6 KB (927 words) - 22:11, 5 June 2020
is said to be the order of the differential expression and the order of the differential operator defined by this expression.
the order of $ D $
17 KB (2,519 words) - 19:35, 5 June 2020
...f an evolution operator (1972, [[#References|[a5]]]), concerning the first-order evolution problem with a time-dependent $m$-accretive operator in a general
...of lines in order to show the existence and the representation of the semi-group generated by an $m$-accretive operator $T$ in terms of a product formula. T
7 KB (1,062 words) - 17:03, 1 July 2020
...[[Lie-group, p-adic|Lie group, $ p $ -adic]]; [[Analytic group|Analytic group]]).
...the field $ \mathbf R $ of real numbers (see also [[Linear group|Linear group]]) and its subgroups, closed in the natural Euclidean topology (J. von Neum
26 KB (3,980 words) - 17:06, 13 June 2020
is the Hamiltonian. The second group of equations (1) may be replaced by the equations
...n, then the equations (3) solve the mechanical problem in which the second group of equations (3) defines, in an implicit way, the law of motion of the syst
7 KB (976 words) - 19:30, 11 January 2024
The Legendre polynomial of order $ n $
and satisfies a [[Lipschitz condition|Lipschitz condition]] of order $ \alpha > 1 / 2 $,
5 KB (670 words) - 17:45, 13 January 2024
...eared in the theory of algorithms (cf. [[Algorithms, theory of|Algorithms, theory of]]) practically all at the same time, and they all proved to be essential
The first examples of unsolvable algorithmic problems were noted in the theory of algorithms itself. These include the problem of recognition of belonging
21 KB (3,264 words) - 06:46, 26 March 2023
is the group of permutations of the set consisting of the first $ n $
of square matrices of order $ n $
15 KB (2,252 words) - 08:04, 6 June 2020
...d orientation: If the first vertex does not belong to $\s^{n-1}$, then the order of the others is taken to be positive for $\s^{n-1}$.
...be defined as the bundle $\def\L{\Lambda}\L^n(M)$ of differential forms of order $n$. It has a non-zero section only in the orientable case and then such a
18 KB (2,980 words) - 17:05, 13 June 2020
...e of the criteria in order to arrive at a global judgement. Moreover, in a group of decision makers each member faces the question of how to judge the quali
...new criteria may emerge, old ones may be dropped, and the decision-making group may change. When a family selects a car, for instance, these features of th
18 KB (2,526 words) - 15:30, 1 July 2020
...the context of differential geometry and, lastly, in the context of group theory.
...the order of the points on a straight line is linear, i.e. similar to the order in the set of real numbers; in elliptic geometry, the points of a straight
17 KB (2,582 words) - 08:02, 6 June 2020
...ediaofmath.org/legacyimages/s/s087/s087670/s08767011.png" /> is called the
order of the Steiner system <img align="absmiddle" border="0" src="https://www.en
...087670/s08767027.png" />; 2) the existence of Steiner systems with a given
group of automorphisms; 3) the imbedding of partial Steiner systems (not containi
32 KB (4,451 words) - 15:16, 19 March 2018
In order that a sequence $ \{ f _ {n} \} $
then in order that $ f $
8 KB (1,132 words) - 08:27, 6 June 2020
...nces|[a4]]] are given, stated in historical order. They are centred around group symmetry, relative to unitary representations of Lie groups in Hilbert spac
...) \rtimes {\bf R} ^ { 4 }$ as a covering of the [[Poincaré group|Poincaré group]] of relativity, and a vacuum state vector $\psi_0$ fixed by the representa
10 KB (1,584 words) - 07:43, 27 January 2024
The central problem in the theory of varieties of Lie algebras is to describe bases of identities of a variet
of matrices of order 2 over a field $ k $
5 KB (790 words) - 22:16, 5 June 2020
The second point of interest in the theory of Seifert fibrations is to show that a closed manifold $ M ^ {3} $
...e is defined by the fundamental group. The [[Fundamental group|fundamental group]] $ \pi _ {1} ( M ^ {3} ) $
10 KB (1,350 words) - 12:59, 28 April 2024
...ger-valued quadratic forms [[#References|[5]]]. Certain problems in number theory have also turned out to be connected with root systems [[#References|[6]]].
The finite group $ A (R) $
22 KB (3,351 words) - 19:14, 21 December 2019
is a matrix-function of order $ n \times n $,
...="top">[3]</TD> <TD valign="top"> L.V. [L.V. Ovsyannikov] Ovsiannikov, "Group analysis of differential equations" , Acad. Press (1982) (Translated from
3 KB (441 words) - 17:32, 5 June 2020
...he elements of which are known as (integral) divisors of the ring $A$. The theory of divisors makes it possible to reduce a series of problems connected with
...omorphism $\phi : A^* \to D_0$ be given. The homomorphism $\phi$ defines a theory of divisors of the ring $A$ if it satisfies the following conditions.
16 KB (2,805 words) - 02:18, 6 January 2022
is a properly-discontinuous group of biholomorphic automorphisms of $ D $
is the group of covering homeomorphisms of this covering, i.e. $ D _ {0} /G $
14 KB (2,068 words) - 02:08, 18 July 2022
...H. Conway [[#References|[a2]]]. They find their origin in the area of game theory. Their description can be found in Conway's book [[#References|[a2]]] (1976
...umbers do not form a set but a proper class (cf. [[Types, theory of|Types, theory of]]).
8 KB (1,226 words) - 16:11, 3 July 2016
...t an infinite distance, and by a detailed study of the singular points. In order to study all the points of an affine curve, the curve is imbedded into a pr
of the group $ \mathop{\rm Div}\nolimits \ X $
22 KB (3,307 words) - 17:02, 17 December 2019
...sub-particles such as quarks. The Dirac equation is the foundation of the theory of particles with half-integral spin ($ \dfrac{1}{2} $, $ \dfrac{3}{2} $, $
...complex coefficients that is invariant with respect to the general Lorentz group of transformations:
9 KB (1,306 words) - 18:13, 17 March 2023
...any applications to quantificational logic [[#References|[a6]]]. A related theory of polyadic algebras, due to P.R. Halmos [[#References|[a5]]], emphasizes o
...with G. Frobenius in the 1880s, referring to a collection of elements of a group as a "complex" ).
8 KB (1,220 words) - 19:12, 11 December 2020
...ius transformations (cf. also [[Discrete group of transformations|Discrete group of transformations]]; [[Fractional-linear mapping|Fractional-linear mapping
...rom the cohomology of varieties over $\Q$ (cf. also [[Galois theory|Galois theory]]).
7 KB (1,047 words) - 01:52, 18 July 2022
A set together with a given transitive [[group action]]. More precisely, $ M $
is a homogeneous space with group $ G $
20 KB (2,996 words) - 08:42, 16 December 2019
A coherent algebra $W$ of
order $n$ and rank $r$ is a matrix subalgebra of the full matrix algebra <img ali
...itive) [[#References|[a2]]], [[#References|[a10]]] (cf. also [[Permutation
group]]; [[Centralizer]]). This leads to many important applications of coherent
7 KB (1,092 words) - 17:44, 1 July 2020
The theory of invariant measures (with respect to continuous groups of transformations
...s requires finding an [[Integral invariant|integral invariant]] of the Lie group. The latter can be found as a solution to the system of partial differentia
28 KB (4,292 words) - 08:45, 1 July 2022
A theory intended for the calculation of rounding errors of calculations on numerica
is a semi-group under addition and multiplication. The following equalities hold:
5 KB (677 words) - 22:13, 5 June 2020
...definition of the covariant of a tensor with respect to the general linear group $ \mathop{\rm GL} ( V) $.
The notion of a covariant arose in the classical theory of invariants and is a special case of the notion of a [[Comitant|comitant]
4 KB (627 words) - 17:31, 5 June 2020
...athematics) one builds several classes of formal languages, of which first-order logic and equational logic are especially important. Languages of the first
...a model of $E$, then $\mathcal{A}$ is also a model of the least equational theory including $E$.
9 KB (1,433 words) - 17:00, 1 July 2020
...lead to abstract generalizations and analogues of ordinary analytic number theory, which may then be applied in a unified way to further enumeration question
An arithmetical semi-group is, by definition, a commutative [[Semi-group|semi-group]] $G$ with identity element $1$, which contains a countable subset $P$ such
24 KB (3,738 words) - 07:41, 7 February 2024
the corresponding equality is valid only up to summands of order 2.
...or cohomology with coefficients in an arbitrary finitely-generated Abelian group $ \pi $ (see [[#References|[2]]], [[#References|[3]]]). In final form thi
6 KB (870 words) - 10:04, 11 July 2022
$#C+1 = 1 : ~/encyclopedia/old_files/data/T092/T.0902550 Theory of surfaces
...ifferential geometry|differential geometry]] dealing with surfaces. In the theory of surfaces one examines the shape of a surface, its curvature, the propert
9 KB (1,331 words) - 08:25, 6 June 2020
...n the highest weight. Two of these are classical results in representation theory: Freudenthal's formula and Kostant's formula.
be a partial order relation on $ \mathfrak t ^ {*} $
6 KB (885 words) - 08:02, 6 June 2020
...Symbol of an operator]]). The differential operators $D _ { k }$ are first-order partial differential operators whose symbols are induced by the exterior mu
...x of $D$ (cf. also [[Index formulas|Index formulas]]; [[Index theory|Index theory]]). For elliptic Lie equations the index can be expressed in terms of chara
8 KB (1,101 words) - 17:44, 1 July 2020
form an Abelian group $ \Gamma $
...which is known as the period group (or the period module). A basis of this group is known as a basis system of periods of the Abelian function, or also as a
11 KB (1,602 words) - 16:08, 1 April 2020
...tic equation|Quadratic equation]]; [[Cubic equation|Cubic equation]]). The theory of the solution of quadratic equations was first expounded in the book Arit
...e solvability of algebraic equations by radicals in [[Galois theory|Galois theory]] can be stated as follows. Let $ f(x) $
18 KB (2,778 words) - 16:09, 1 April 2020
...investigate their topological properties by algebraic methods (cf. [[Knot theory]]). Named after H. Seifert [[#References|[1]]], who applied the constructio
are of finite order, then $ \theta ( z _ {1} \otimes z _ {2} ) = 0 $.
5 KB (839 words) - 06:50, 28 April 2024
forms an Abelian group with respect to the composition: $ ( \theta + \psi ) _ {X} = \theta _ {X}
is a group homomorphism.
29 KB (4,197 words) - 09:49, 26 March 2023
...ly found wide applicability in a variety of areas of [[Homotopy|homotopy]] theory, most notably in the stable homotopy groups of spheres ([[#References|[a9]]
...$ modulo $2 p$, the group $[ T ( n ) , X ]$ is naturally isomorphic to the group $D _ { n } H_{*} \Omega ^ { \infty } X$ of homogeneous elements of degree $
7 KB (996 words) - 11:15, 20 January 2021
...er and R.L. Griess in 1973. This group is now known as the monster and has order
...imes dividing the order of $\mathcal{M}$ are also the primes for which the group $\Gamma _ { 0 } ( p ) +$ has genus zero.
21 KB (3,107 words) - 09:12, 10 November 2023
...aracterizing the properties of vacuum, sometimes introduced in the general theory of relativity. Einstein's equations (cf. [[Einstein equations|Einstein equa
...Einstein equations have no such solution is not considered a defect of the theory. There are no reliable indications that the cosmological constant is distin
5 KB (648 words) - 17:31, 5 June 2020
...sm of the homotopy groups in all dimensions (cf. [[Homotopy group|Homotopy group]]). Correspondingly, two spaces $ X $
...lent if and only if their minimal simplicial sets are isomorphic. Thus, in order to solve the problem of weak homotopy type, all that remains to be done is
31 KB (4,636 words) - 12:07, 15 December 2019
...s method, the computation of the characteristic polynomial for a matrix of order $ n $
...th rounding-off errors could not be tested for problems of any substantial order until digital computers became available. Such tests were carried out in th
15 KB (2,340 words) - 08:54, 21 January 2024
Crossed complexes are a variant of chain complexes of modules over integral group rings but strengthened in two ways:
...ces; covering spaces, and in particular Cayley graphs; and the equivariant theory. It is also essential for the [[Closed category|closed category]] structure
13 KB (1,937 words) - 13:10, 24 December 2020
...om it. The integral is defined for a knot $K$ (cf. also [[Knot theory|Knot theory]]) embedded in the three-dimensional space $ \mathbf{R} ^ { 3 } = \mathbf{
...itive orientation of the space $\mathbf{R} ^ { m }$ defined by the natural order of the coordinates $t _ { 1 } , \ldots , t _ { m }$.
10 KB (1,481 words) - 19:05, 23 January 2024
is a Cohen–Macaulay ring; for example, this is true of any semi-group ring $ K [ G \cap \mathbf Z ^ {n} ] $,
is a finite group acting on a Cohen–Macaulay ring $ A $,
7 KB (1,158 words) - 17:45, 4 June 2020
...iples and results concerning the concepts listed above. In particular, the theory of generalized displacement operators has substantial applications in abstr
...ults in this field are also due to him. The systematic construction of the theory of generalized displacement operators was mainly given in the work of B.M.
30 KB (4,254 words) - 17:53, 13 January 2024
...tegral geometry. It was introduced by R. Penrose in the context of twistor theory [[#References|[a4]]] but many mathematicians have introduced transforms whi
...$K$ assumed to have the same rank as $G$ (cf. [[Lie group, semi-simple|Lie group, semi-simple]]). In complex geometry there is also the Andreotti–Norguet
7 KB (1,055 words) - 11:15, 13 February 2024
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
though this condition is not necessary. In order to compute the integral $ I ( f ) $
...als of equal degree have been ordered arbitrarily, e.g. in lexicographical order. In this enumeration $ \phi _ {1} ( x) = 1 $,
17 KB (2,639 words) - 04:38, 29 December 2021
the Pauli matrices form a complete system of second-order matrices by which an arbitrary linear operator (matrix) of dimension 2 can
...he coordinate system by a linear two-valued representation of the rotation group. Under a rotation by an infinitesimal angle $ \theta $
6 KB (785 words) - 01:11, 19 March 2022
''cobordism theory''
...structures in the stable tangent or normal bundle to a manifold. Cobordism theory is dual (in the sense of [[S-duality| $ S $-
42 KB (6,290 words) - 19:33, 17 January 2024
...t in the subsets of the system. The concept of a block design arose in the theory of design (planning) of (statistical) experiments in the 1920s and 1930s, b
The theory of block designs considers problems on the existence and classification as
9 KB (1,344 words) - 16:08, 6 February 2020
[[Abelian group]] with the distributive action of a ring. A module is a generalization of a
Let a ring $A$ be given. An additive Abelian group $M$ is called a left $A$-module if there is a mapping $A\times M \to M$ who
23 KB (3,918 words) - 04:31, 23 July 2018
An object of study in classical homotopy theory. The calculation of the homotopy groups of the spheres, $ \pi _ {i} ( S
==I. General theory.==
33 KB (4,248 words) - 08:17, 21 March 2022
One of the axioms in set theory. It states that for any family $F$ of
axioms of set theory (e.g. in the system ZF).
4 KB (647 words) - 18:30, 4 December 2017
...ences are generated. A configuration is called regular if its automorphism group is transitive on the points and on the lines of the configuration.
requires an auxiliary second-order curve. The configuration $ ( 10 _ {3} ) $
27 KB (3,830 words) - 09:57, 26 March 2023
mentioned above, then an order $ \leq $
...ht annihilators of this semi-group. The semi-group itself is a [[Baer semi-group]], i.e. the right and left annihilators of each of its elements are generat
14 KB (2,066 words) - 15:31, 11 November 2023
...ce). This theorem had wide applications both in topology as well as in the theory of Boolean algebras. The method of proof of this result goes back to W. Rud
...ega^*$ under CH. This result cannot be proved in ZFC alone (cf. also [[Set theory]]). In 1968, K. Kunen [[#References|[a12]]] proved that in a model formed b
11 KB (1,671 words) - 21:29, 19 November 2017
...investigating the class groups (cf. also [[Class field theory|Class field theory]]) of real Abelian extensions of $\mathbf{Q}$ (cf. also [[Extension of a fi
...arithmetic objects such as the class group of a number field or the Selmer group of an elliptic curve. On the other hand, an Euler system encodes values of
19 KB (2,901 words) - 17:41, 25 November 2023
...c quantum equation proved possible in [[Quantum field theory|quantum field theory]] but not in quantum mechanics. In [[#References|[3]]] an interpretation of
...ds; it plays the role of one of the fundamental equations of quantum field theory.
8 KB (1,082 words) - 22:14, 5 June 2020
...es the theory of algebraic functions, the theory of Abelian integrals, the theory of special functions, etc. Special functions — [[Bessel functions|Bessel
==Linear theory.==
16 KB (2,410 words) - 11:15, 28 January 2020
...is a thermodynamic (in statistical mechanics) or vacuum (in quantum field theory) average of dynamical quantities in a specially modified averaging procedur
...btained by extra averaging of the quasi-average over the symmetry-breaking group.
11 KB (1,490 words) - 08:09, 6 June 2020
[[Algebraic group|algebraic group]]. If $S$ is a smooth projective
group (the points of $S^{(d)}$ correspond to effective divisors of degree
10 KB (1,582 words) - 22:02, 5 March 2012
...sult. In particular, there is an action of the [[Symmetric
group|symmetric
group]] $S _ { n }$ in <img align="absmiddle" border="0" src="https://www.encyclo
...relevant objects in the tensor product. On the other hand, one is free to
group some of the objects in the tensor product together as a single object and r
17 KB (2,633 words) - 17:42, 1 July 2020
...partial differential equations. The order of (1) is defined as the highest order of a derivative occurring in the equation.
...with respect to its variables corresponding to the derivatives of highest order, then the type of (1) is defined as that of the principal linear part of $
30 KB (4,331 words) - 16:42, 20 January 2022
...of a geometry over an algebra. A recent survey of the development of this theory is [[#References|[a3]]].
be its group of units. To define the projective line $ \mathbf P ( A ) $
9 KB (1,440 words) - 16:43, 4 June 2020
...he exterior electro-magnetic field $( \phi , \mathbf{A} )$. It is a second-order, [[Hyperbolic partial differential equation|hyperbolic partial differential
...\mathbf{x} )$ as a quantum field (cf. [[Quantum field theory|Quantum field theory]]), see e.g. [[#References|[a6]]], [[#References|[a7]]].
5 KB (713 words) - 17:44, 1 July 2020
...ed functions and predicates. For example, the language of elementary group theory is obtained according to this scheme from the language of classical predica
is the identity of the group, $ \mathop{\rm inv} ( x) $
13 KB (1,882 words) - 04:11, 6 June 2020
In order to have a one-factorization, a graph must have an even number of vertices a
...hm was described by J.H. Dinitz and D.R. Stinson [[#References|[a10]]]. In order to use the hill-climbing approach, it is necessary to formulate the search
16 KB (2,342 words) - 09:33, 19 January 2021
...let $\Gamma$ be an arithmetic subgroup (cf. [[Arithmetic group|Arithmetic group]]) of $G ( \mathbf{Q} )$, commensurable with the integer points of $G$. The
In order to describe the structure sheaf of $V ^ { * }$ (cf. also [[Scheme|Scheme]])
13 KB (1,983 words) - 17:00, 1 July 2020
...gn="top">[2]</TD> <TD valign="top"> K. Kuratowski, A. Mostowski, "Set theory" , North-Holland (1968)</TD></TR></table>
Ultrafilters support a considerable body of theory both in general topology and in mathematical logic. For a topologist, they
10 KB (1,605 words) - 16:57, 13 January 2024
...ed subsets, namely lines and planes, with the usual relations: membership, order, congruence (either defined by distance or by motion) and in which all the
is the theory of convex bodies.
12 KB (1,820 words) - 02:24, 21 January 2022
...tiation is widely used in theoretical physics, particularly in the general theory of relativity.
...y differentiation for which the well-known relationships between the first-order partial derivatives and differentials remain valid.
15 KB (2,257 words) - 17:31, 5 June 2020
...is reduction is carried out with respect to the field of definition of the group.
...s action centralizes the natural place permutation action of the symmetric group $\mathfrak{S}_r$ on $E ^{ \otimes r}$. The image of $K G$ in $\operatorname
33 KB (5,081 words) - 10:26, 11 November 2023
A first-order ordinary differential equation of the form
...stions concerning the qualitative behaviour of linear equations and second-order systems (see [[#References|[3]]], [[#References|[4]]]).
8 KB (1,202 words) - 14:55, 7 June 2020
If the base of the cone is a [[disc]], then the cone is called circular <ref group="comment" name="Right circular cone" />. A circular cone is called straight
...er a topological space can be generalized in the framework of [[category]] theory: A set of [[morphism]]s $\alpha_i\colon A\rightarrow A_i$, $i\in I$, of an
13 KB (2,188 words) - 13:05, 10 June 2016
...richness of problems and the variety of ideas used to solve them makes the theory of algebraic surfaces one of the most interesting fields of algebraic geome
...ants, criteria of rationality and ruledness of algebraic surfaces, and the theory of minimal models. Most of these results were critically reviewed and demon
26 KB (3,736 words) - 13:08, 8 February 2020
is the identity matrix of order $ n $,
...e to introduce an addition, converting it into an Abelian group — the Witt group of $ k $(
8 KB (1,152 words) - 08:29, 6 June 2020
...g [[dynamic programming]]), computer science, automata and formal language theory, numerical methods, [[parallel programming]], etc. (cf. also [[Idempotent a
...$, and $a \odot c \le b \odot c$ if $a \le b$. Using this standard partial order it is possible to define in the usual way the notions of upper and lower bo
7 KB (1,026 words) - 20:42, 16 November 2023
The bordering method allows one to solve higher-order systems owing to the effective use of computer memory. This is caused by th
and the coefficients of a $ k $-th order system of equations is necessary, that is, a series of numbers of length $
10 KB (1,498 words) - 08:08, 21 March 2022
with $g ( z )$ belonging to the loop group $C ^ { \infty } ( S ^ { 1 } , \operatorname{SL}_ { 2 } ( {\bf C} ) )$. If t
This phenomenon is due to the fact that there is a natural lattice group that commutes with the commuting flows corresponding to the parameters $t _
15 KB (2,265 words) - 17:00, 1 July 2020
...points and lines equipped with suitable relations on the objects (e.g., of order, congruence, parallelity, continuity). The classical case $F = \mathbf{R}$,
...tween" was not properly defined in mathematics. Pasch now realized that a theory of between-ness was important in the systematic approach to geometry.
12 KB (1,960 words) - 17:44, 1 July 2020
...old_files/data/B017/B.0107400 Boundary value problems of analytic function theory
Hilbert initially reduced this problem to a singular integral equation in order to give an example of the application of such an equation.
17 KB (2,509 words) - 03:12, 13 January 2022
of order $ n $
...> <TD valign="top"> A.F. Nikiforov, V.B. Uvarov, "Fundamentals of the theory of special functions" , Moscow (1974) (In Russian)</TD></TR><TR><TD valig
8 KB (1,132 words) - 14:19, 17 March 2023
A group of theorems in combinatorial theory related to the selection of elements from a set which in some way correspon
...It is equivalent to the Ph. Hall theorem. It is employed, for example, in order to prove that certain matrices are linear combinations of permutation matri
13 KB (1,942 words) - 08:13, 6 June 2020
The notion of an algorithm has to be formalized in order to show that some problem is undecidable. The undecidability of a problem m
...ivity is considered, eight problems arise in this setup: group versus semi-group, general versus Abelian, and equation versus unidirectional relation. The w
7 KB (1,138 words) - 13:14, 10 April 2014
that is graded by means of an Abelian group $ A $,
is an [[Ordered group|ordered group]], then for every filtered Lie algebra (cf. [[Filtered algebra|Filtered alg
20 KB (2,828 words) - 19:11, 26 March 2023
with a first-order polar singularity on $ S $.
and have a first-order polar singularity on $ S $.
11 KB (1,662 words) - 08:11, 6 June 2020
...$M _ { \operatorname{sa} }$ as a partially ordered real vector space with order-unit $1$ (cf. also [[Semi-ordered space|Semi-ordered space]]). The positive
...takes $G_1$ to be the additive group of integers and $G_2$ to be the free group on two generators the corresponding $A W ^ { * }$-factors are isomorphic. T
14 KB (2,118 words) - 16:46, 1 July 2020
...-injective ring with zero Jacobson radical is regular. The group ring of a group $ G $
is finite and the order of every such subgroup is invertible in the original regular ring (see [[#R
14 KB (2,228 words) - 08:10, 6 June 2020
...of associative rings of characteristic zero (that is, rings whose additive group is torsion-free) there exists a positive integer $n=n(\mathfrak M)$ such th
...>[a2]</TD> <TD valign="top"> L.H. Rowen, "Polynomial identities in ring theory" , Acad. Press (1980) pp. Chapt. 7</TD></TR>
4 KB (701 words) - 17:01, 23 November 2023
The term "second curvature" is commonly used in higher-dimensional Frénet theory, where the curve is considered in Euclidean $ n $-
...difference between the end-points of the evolvent, evaluated up to second-order terms, has the components $ \Omega ^ {k} $,
16 KB (2,421 words) - 14:56, 7 June 2020
...[[Symmetric group|Symmetric group]]; [[Character of a group|Character of a group]]), the associated generalized matrix function $d _ { \chi } ^ { G } : \mat
...semi-definite Hermitian matrices $A$. This gives a [[Partial order|partial order]] on the characters and also the immanants. In this context, Schur's inequa
19 KB (2,837 words) - 05:34, 15 February 2024
among which the most important ones are the order, the addition and the multiplication. In this connection the basic properti
...[[Smoothness, modulus of|Smoothness, modulus of]]) (and even moduli in the theory of elasticity (compression, shear)). However, in all these cases it is poss
4 KB (659 words) - 08:01, 6 June 2020
''theory of programming''
...nvestigating models of programs. The formal combinatorial methods form the theory of program schemes, which studies properties of a program that are invarian
21 KB (3,241 words) - 08:25, 6 June 2020
...of motion of an ideal liquid and Einstein's equation of general relativity theory) can be interpreted as equations of geodesics for certain infinite-dimensio
...ne is M. Morse's variational theory of geodesics (cf. [[Morse theory|Morse theory]], [[#References|[20]]]).
30 KB (4,323 words) - 19:35, 5 June 2020
is called the order of the linear differential operator $ A $.
of the first order. Another equivalent definition of a linear differential operator $ A : E
25 KB (3,768 words) - 09:07, 14 June 2022
The questions a) and b) and many others in the theory of singularities are studied along the following lines:
...a number of individual results appeared far earlier ([[Morse theory|Morse theory]] of critical points of functions, Whitney's theorem on singularities of im
30 KB (4,632 words) - 08:14, 6 June 2020
that generates a one-parameter group of transformations preserving $ A $,
In field theory, where $ n = 4 $
16 KB (2,336 words) - 08:02, 6 June 2020
...es, in particular, idempotent integration theory, linear algebra, spectral theory, and functional analysis. Applications include various optimization problem
...amilton–Jacobi equation (cf. also [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]]) can be treated as linear over suitable semi-rings. Maslov's superpositio
18 KB (2,598 words) - 22:11, 5 June 2020
...ots,K_n)$, lying on any of these surfaces, can be executed in parallel. In order to represent by a program the parallel processing of arrays one needs speci
...able but relatively complicated, by an immediate specification of parallel group operations.
11 KB (1,714 words) - 03:17, 5 June 2016
...a certain structure (defined by algebraic operations, a topology, or by an order relation). The general definition of an operator coincides with the definit
...th the weak topology (the concept of semi-continuity is mainly used in the theory of non-linear operators); an operator is said to be strongly continuous if
14 KB (2,265 words) - 17:06, 24 January 2020
...combinatorics and graph theory, linear algebra, topology, and probability theory.
...rict best-to-worst ranking of $X$. When voters are asked to preferentially order the alternatives, $D$ may be the same as $V$. But $D$ may be different from
13 KB (2,002 words) - 09:34, 10 November 2023
...ential-geometric structure]] on a smooth fibre bundle with a Lie structure group that generalizes [[Connections on a manifold|connections on a manifold]], i
on which a Lie group $ G $
17 KB (2,530 words) - 19:28, 17 January 2024
The common name for a group of six special ordinary differential equations of the type
...that do not reduce to other known ones. Arranged in the generally-accepted order, the Painlevé equations have the following form ( $ a, b, c, d \in \mathb
5 KB (673 words) - 19:15, 1 May 2024
structure, where the pseudo-group consists of mappings that are $ l $-
...under the above concept of equivalence, and in essence coincides with the theory of surfaces, particularly when one considers topics related to the geometry
6 KB (923 words) - 22:11, 5 June 2020
...$, cf. [[#References|[a3]]], 35.8. Then evaluation at the Taylor series of order $k$ reveals that $\varphi$ is completely determined by its values on the co
...Weil bundles on Lie groups are again Lie groups (cf. also [[Lie group|Lie group]]) and all structural mappings, like the exponential or adjoint ones, are d
12 KB (1,876 words) - 06:30, 15 February 2024
The variational equation of order $ k $
...linear homogeneous equation corresponding to a variational equation of any order is the same (i.e. independent of $ k $),
15 KB (2,177 words) - 16:07, 5 February 2022
...theorem|Fermat last theorem]]). For details and generalizations of Iwasawa theory, see [[#References|[a10]]], [[#References|[a7]]], [[#References|[a12]]], [[
...k ) \cong \mathbf{Z} _ { p }$, where $\mathbf{Z} _ { p }$ is the additive group of $p$-adic integers. Then there is a sequence of fields
19 KB (2,876 words) - 05:38, 15 February 2024
(observe that in order for \eqref{e:formula_integral} to be well defined, i.e. independent of the
...s holomorphic and $\gamma \subset D$ a closed rectifiable curve [[Homotopy group|homotopic]] to $0$, then \eqref{e:integral_vanishes} holds.
4 KB (659 words) - 13:04, 3 January 2014
This leads to the theory of webs, cf. [[Webs, geometry of|Webs, geometry of]] and [[#References|[a13
is called the degree (or order) of the pencil of lines. A first-order algebraic pencil of lines is given by an equation
20 KB (3,056 words) - 08:25, 6 June 2020
on an arbitrary locally compact Abelian group $ G $,
of integration is the corresponding character group $ \widehat{G} $(
17 KB (2,406 words) - 20:00, 12 January 2024
...gebraic groups and simple finite groups connected with Lie groups), in the theory of differential equations ( $ K $ -
...es (topoi, Abelian categories), and in functional analysis (representation theory). Conversely, the ideas and methods of these disciplines are utilized in al
29 KB (4,414 words) - 17:20, 17 December 2019
are path-connected, then for every coefficient group $ G $
and so the group $ E _ \infty ^ {s,t} = \cap _ {r>s} E _ {r} ^ {s,t} $
14 KB (2,075 words) - 09:23, 18 February 2022
...the properties of the sum \eqref{eq1} to solve certain problems in number theory; a particular case is one of the proofs of the [[Quadratic reciprocity law|
The significance of Gauss sums in number theory only became evident in the 1920s. At that time H. Weyl used general trigono
6 KB (921 words) - 19:40, 29 March 2024
$#C+1 = 111 : ~/encyclopedia/old_files/data/R081/R.0801130 Relativity theory
...theory of]]. Relativity theory is also often called Einstein's relativity theory, after A. Einstein who created it (see [[#References|[1]]], [[#References|[
22 KB (3,285 words) - 08:10, 6 June 2020
defines a representation of the braid group $ B _ {n} $,
...connection between knot and braid theory on the one hand and quantum field theory and statistical mechanics on the other, cf. e.g. [[#References|[a9]]].
10 KB (1,364 words) - 08:29, 6 June 2020
''real-variable theory of, real-variable $\mathcal{H}^p$ theory''
...r $p<1$, the dual of $\mathcal{H}^p$ is the homogeneous Lipschitz space of order $\frac{n-np}{p}$; see {{Cite|FoS}}.
6 KB (1,018 words) - 13:36, 26 December 2013
...[[#References|[1]]]). The determination of the number of magic squares of order $ n $
...f order 6. Euler conjectured that there are no orthogonal Latin squares of order $ n = 4 k + 2 $,
12 KB (1,892 words) - 19:25, 13 January 2024
If $ \mathbb{k} $ is Noetherian and the module $ \mathfrak{g} $ has finite order, then the algebra $ U(\mathfrak{g}) $ is left- and right-Noetherian. If $ \
...orms the basis for the application of universal enveloping algebras in the theory of representations of Lie algebras ([[#References|[3]]], [[#References|[4]]
6 KB (970 words) - 18:59, 5 April 2023
...le coefficients was posed (avoiding singular points, as needed even in the theory of ordinary differential equations). H. Lewy [[#References|[a5]]] found the
...unctions of a complex variable, theory of|Functions of a complex variable, theory of]]).
8 KB (1,236 words) - 19:09, 26 March 2023
...d the concept of a "motion" . This concept made it possible to introduce a group approach in geometry to study motions and to put the research methods in an
==Group I.==
25 KB (3,631 words) - 19:39, 5 June 2020
$#C+1 = 184 : ~/encyclopedia/old_files/data/A014/A.0104310 Axiomatic set theory
...ry aiming at the construction of some fragment of informal ( "naive" ) set theory.
22 KB (3,585 words) - 17:07, 25 April 2020
$#C+1 = 311 : ~/encyclopedia/old_files/data/P074/P.0704150 Potential theory, abstract
...see [[#References|[3]]]). The probabilistic approach to abstract potential theory, the origins of which could be found already in the works of P. Lévy, J. D
23 KB (3,517 words) - 08:07, 6 June 2020
$#C+1 = 104 : ~/encyclopedia/old_files/data/R080/R.0800280 Recursive model theory,
''recursively presented model theory''
14 KB (1,981 words) - 08:10, 6 June 2020
...nsional space, which was one of the origins of algebraic topology; [[braid theory]] is a related case.
2) An important role in the history of algebraic topology was played by the theory of homology invariants of the position of various sets in a Euclidean space
19 KB (2,788 words) - 09:43, 28 October 2023
...nly if the [[Galois group]] of the corresponding algebraic problem has the order $2^k$, $k \in \mathbf{N}$. For example, the regular $n$-gon is constructibl
...cts and computational complexity of such algorithms (cf. also [[Complexity theory]] and [[Algorithm, computational complexity of an]]).
5 KB (738 words) - 19:10, 17 December 2015
[[Category:Linear and multilinear algebra; matrix theory]]
$K$. If $m=n$, then (1) is called a square matrix of order $n$. The set
18 KB (3,377 words) - 17:54, 2 November 2013
There are important applications of the Tits form in representation theory. One easily proves that if $Q$ is connected, then $q_Q$ is positive definit
...\operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$ defines a group isomorphism $\underline{\operatorname { dim }} : K _ { 0 } ( Q ) \rightarro
18 KB (2,636 words) - 06:50, 15 February 2024
...ely, for "statement" , "nouns of gender x, number y and case z" , "verb group in the third person" , and "transitive verb in the third person" , while t
...languages generated by a context-sensitive grammar with time complexity of order $ n ^ {2} $,
9 KB (1,320 words) - 16:57, 25 April 2020
...breviation of a system of six ordinary differential equations of the first order. Here, the phase space is six-dimensional Euclidean space, the six componen
...cal system (or its configuration space) is the special orthogonal group of order three SO(3). The phase space $ W ^ {6} $
27 KB (4,058 words) - 19:36, 5 June 2020
A non-autonomous normal system of ordinary differential equations of order $ m $
is said to be stiff if the autonomous system of order $ m + 1 $
30 KB (4,292 words) - 05:40, 24 February 2022
The first group includes, for instance, the [[Method of characteristics|method of character
The methods in the second group yield non-singular difference schemes (cf. [[Difference scheme|Difference s
24 KB (3,317 words) - 22:11, 5 June 2020
...[a24]]]. A lattice is Arguesian if and only if its [[Partial order|partial order]] dual is Arguesian.
...group]]; [[Congruence (in algebra)|Congruence (in algebra)]]) of a [[Group|group]] and any lattice of permuting [[Equivalence|equivalence]] relations [[#Ref
29 KB (4,201 words) - 16:31, 9 December 2023
...ions over arbitrary fields. The methods employed are purely algebraic. The theory of valuations and extensions of fields are especially important. In the alg
...erts the Puiseux series of elements in one cycle into each other in cyclic order, i.e. there is a cyclic permutation of the series and of the corresponding
20 KB (3,036 words) - 07:17, 15 June 2022
...semi-simple) have obtained a fairly complete description in the classical theory: Any semi-simple finite-dimensional associative algebra is a direct sum of
...ereditary for the idempotent Baer radical. The construction of the general theory of radicals was initiated by S. Amitsur
16 KB (2,540 words) - 08:09, 6 June 2020
...ted following O. Heaviside and H.A. Lorentz, is the following set of first-order partial differential equations at any regular point $\mathbf{x}$ (whether o
...somewhat blurred introduction of this term made Hertz say that "Maxwell's theory is none other than Maxwell's equations" . As a consequence of (a1)–(a4) o
13 KB (1,991 words) - 12:50, 17 March 2023
...roup (and the Alexander and Markov theorems, cf. also [[Braid theory|Braid theory]]; [[Alexander theorem on braids|Alexander theorem on braids]]; [[Markov br
...ond approach one considers the Hecke algebra associated to the Artin braid group and constructs on it the Jones–Ocneanu trace, which essentially is invari
18 KB (2,713 words) - 05:14, 15 February 2024
and as structure group the group $ \mathop{\rm SO} _ {n} $,
realizes a quadratic form of order eight, in which every element on the main diagonal equals 2, while the sign
9 KB (1,361 words) - 12:24, 10 April 2023
$#C+1 = 55 : ~/encyclopedia/old_files/data/S083/S.0803330 Scheduling theory
...tive) sets of operations. The area of application of results in scheduling theory include management, production, transportation, computer systems, construct
21 KB (3,117 words) - 10:04, 18 February 2021
...be obtained as a central [[Quotient group|quotient]] of a finite reductive group.
...special linear group. This group is a [[Simple finite group|finite simple group]] except in the following cases:
48 KB (8,458 words) - 18:22, 13 August 2023
...re a good illustration of the above. This point of view was adopted in the theory of Diophantine equations only at a later date, while a systematic study of
...]], [[#References|[10]]]. The deviation is described in terms of a special group $ {\mathop{\amalg\kern-0.30em\amalg}} $,
24 KB (3,602 words) - 11:48, 26 March 2023
...es|[3]]], in proving the existence of the thermodynamic limit, and also in order to obtain physically important estimates for the free energies of various m
...top">[3]</TD> <TD valign="top"> S.V. Tyablikov, "Methods of the quantum theory of magnetism" , Plenum (1967) (Translated from Russian)</TD></TR><TR><TD
6 KB (788 words) - 10:59, 29 May 2020
...ll the singular points of (1) (or (2)) are branch points of it of infinite order.
A second-order Fuchsian equation with singular points $ z _ {1} \dots z _ {k} , \infty $
12 KB (1,732 words) - 17:50, 5 May 2024
...methodology. The role which differential geometry can play in statistical theory has been realized effectively only since the late 1970s. The historical dev
...|Affine connection]]) on the parameter space, together with various higher-order geometrical objects. Observed geometries are more directly relevant to the
11 KB (1,396 words) - 19:32, 5 June 2020
...oretical concepts, namely [[Renormalization group analysis|renormalization group analysis]] [[#References|[a6]]], by which one can understand how non-mean-f
...d valign="top"> D. Stauffer, A. Aharony, "Introduction to percolation theory" , Taylor&Francis (1992)</td></tr></table>
16 KB (2,407 words) - 19:47, 17 February 2024
$#C+1 = 76 : ~/encyclopedia/old_files/data/P074/P.0704170 Potential theory, mixed boundary value problems of
In order to highlight the difference between the new method and the old one, here is
18 KB (2,501 words) - 14:54, 7 June 2020
...s based on the sum of the ranks of the first sample in the series of joint order statistics. One rejects the hypothesis that the distributions are equal if
An extensive group of non-parametric tests is based on the use of empirical distribution funct
37 KB (5,331 words) - 17:14, 7 February 2011
...xt of Galois fields than in algebra in general, see [[Galois theory|Galois theory]] and [[Primitive polynomial|Primitive polynomial]].
...nt role. For any [[Galois extension|Galois extension]] $E / F$ with Galois group $G$, one defines the trace and the norm (over $F$) of an element $z \in E$
14 KB (2,032 words) - 14:35, 19 March 2023
...s defined by the form of the coefficients with even derivatives of minimal order with respect to the spatial variables in the functions to be calculated, un
...possible, using the apparatus of differential approximations, to produce a group classification of difference schemes (see [[#References|[9]]]).
7 KB (987 words) - 14:03, 24 December 2020
...r hand is characteristic of the whole period of further development of the theory of Riemann surfaces, associated with the names of F. Klein, H. Poincaré, P
is one of the $ n $-th order roots of unity, $ \epsilon ^ {n} = 1 $.
34 KB (4,972 words) - 13:13, 6 January 2022
and undergoes sign changes under the two other translations of exact order $ 2 $.
Accordingly, one defines the Euler formal group law $ F ( U,V ) \in K [ [ U,V ] ] $
16 KB (2,293 words) - 19:37, 5 June 2020
...time (also called the dynamics), i.e. the semi-group (more often an entire group) of transformations $ U _ {t} ^ {V} $,
...e experimentally-measured values). For a long time it was believed that in order to prove this postulate, an unknown ergodic hypothesis had to be demonstrat
28 KB (4,019 words) - 10:50, 5 March 2022
...uations|Hamilton equations]]) if the variational principle comprises first-order derivatives only. If the variational principle comprises derivatives of ord
first [[Integrals in involution|integrals in involution]], the order of the Hamiltonian system may be reduced by $ 2k $ (at least in a certain
13 KB (1,837 words) - 07:56, 21 January 2024
The following equations can be reduced to (2): the second-order vector equation
is a $ k $-th order vector, $ R( t) = R( t) ^ {*} $
21 KB (3,088 words) - 11:56, 8 March 2022
$#C+1 = 254 : ~/encyclopedia/old_files/data/K055/K.0505600 Knot theory
In a wider sense the subject of knot theory is the imbedding of a sphere in a manifold (cf. [[Multi-dimensional knot|Mu
37 KB (5,599 words) - 11:39, 10 April 2023
An [[Abelian group]] $E$, written additively, in which a multiplication of the elements by sca
...ew-field; the theory of such vector spaces is much more difficult than the theory of vector spaces over a field.
14 KB (2,558 words) - 11:28, 21 June 2016
$#C+1 = 33 : ~/encyclopedia/old_files/data/A011/A.0101910 Algorithms, theory of
...ers, cf. [[Turing machine|Turing machine]]). Subsequent development of the theory of algorithms is due to the studies of Kleene, Post [[#References|[6]]], [[
19 KB (2,846 words) - 16:10, 1 April 2020
...\dots , x _ { n } | R _ { 1 } , \dots , R _ { n } \rangle$ of the trivial group. $3$-deformations can be translated into a sequence of Andrews–Curtis mov
...the Andrews–Curtis conjecture is: Any balanced presentation of the trivial group can be transformed into the empty presentation by Andrews–Curtis moves.
31 KB (4,667 words) - 17:46, 1 July 2020
is of order at most two; this is due to the fact that for the first Chern class $ c _
be a generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]) i
14 KB (1,987 words) - 19:15, 16 January 2024
...nstance, the Kustaanheimo–Stiefel mapping is inherent to the Cartan spinor theory, since (a1), with $ p = q = - 1 $,
...ty can be made precise by transformation properties of (first- and) second-order elliptic and hyperbolic differential operators. For example, this yields (f
8 KB (1,111 words) - 22:15, 5 June 2020
...dynamics by computational algorithms. Below the fundamental aspects of the theory of numerical methods for solving problems in gas dynamics will be considere
In order to close the relations (*) the magnitudes $ \Sigma _ {j + 1/2 } ^ {*} $
28 KB (3,843 words) - 19:41, 5 June 2020
...ing in the definition of a category presupposes the use of axioms from set theory which distinguish between sets and classes. The most commonly used is the a
...tegory was introduced in 1945 [[#References|[8]]]. The origins of category theory and the initial stimulus for its development came from algebraic topology.
43 KB (6,447 words) - 09:17, 26 March 2023
...|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-
...ng non-self-adjoint operators with finite non-Hermitian ranks and function theory on a compact real [[Riemann surface|Riemann surface]] is based on the notio
24 KB (3,136 words) - 20:00, 24 November 2023
Alternative skew-fields play a substantial part in the theory of projective planes, since a projective plane is a Moufang plane (i.e. a t
...potency of an alternative ring is completely parallel to the corresponding theory for associative rings. This is a consequence of the following fact: Let $
12 KB (1,930 words) - 19:54, 15 March 2023
[[Lie group|Lie group]], see also
[[Lie group, local|Lie group, local]];
25 KB (4,037 words) - 07:06, 23 April 2016
of order $ n $
...ngs to the basis [[#References|[6]]]. The problem of dividing the class of group algebras into tame and wild ones has been completely solved [[#References|[
22 KB (3,137 words) - 20:01, 15 March 2023
...ices may entirely be stored in a computer memory, and for problems of high order, in which the information is usually stored in compact form.
are multiplied in reverse order. If $ \widetilde{Q} _ {k} = Q _ {1} \dots Q _ {k} $
19 KB (2,830 words) - 19:42, 27 February 2021
====Discriminant of an [[algebra (ring theory) | algebra]]====
...m of elements of a field is one of the most important constructions in the theory of field extensions. Let $K$ be a finite
16 KB (2,947 words) - 08:53, 9 December 2016
...ction|Dirichlet $L$-function]]) form the basis of modern [[analytic number theory]]. In addition to Riemann's zeta-function one also distinguishes the genera
...ion]]. This accounts for the important role played by $\zeta(s)$ in number theory. As a function of a real variable, $\zeta(s)$ was introduced in 1737 by L.
45 KB (7,251 words) - 02:20, 29 June 2022
...variable, generally infinite, number of degrees of freedom. Quantum field theory unifies the description of fields and particles, which, in classical physic
The notion of a quantum field plays a central role in the theory. It is convenient to explain how it is introduced by the example of an elec
35 KB (5,270 words) - 23:26, 6 December 2016
$#C+1 = 184 : ~/encyclopedia/old_files/data/H046/H.0406300 Handle theory,
''handle-body theory''
15 KB (2,248 words) - 19:53, 5 November 2023
...logy, and technology; they are usually treated in terms of the statistical theory of detection of systematic differences between the results of direct measur
...ubjective examination of the quality of a number of objects, effected by a group of independent experts. Another example is a statistical study of the produ
23 KB (3,288 words) - 19:36, 5 June 2020
...mathematical objects and their approximation by simpler objects. Thus, in order to calculate the area of a circle, a sequence of areas of regular polygons
...refore any normed space (although by no means every semi-normed space). In order for a sequence to converge in a complete metric space it is necessary and s
22 KB (3,726 words) - 10:31, 2 September 2017
$#C+1 = 37 : ~/encyclopedia/old_files/data/P075/P.0705430 Proof theory
...ented his formalization method, which is one of the basic methods in proof theory.
18 KB (2,822 words) - 08:08, 6 June 2020
$#C+1 = 332 : ~/encyclopedia/old_files/data/Q076/Q.0706250 Qualitative theory of differential equations in Banach spaces
is a [[Strongly-continuous semi-group|strongly-continuous semi-group]] of operators for $ t \geq 0 $
26 KB (3,943 words) - 20:22, 16 January 2024
A basis of the natural numbers of order $ k $
of order 4. In general, the sequence of $ m $-
28 KB (4,564 words) - 07:37, 26 March 2023