...closed if it is bounded by the two plane domains interior to the curves of intersection of planes $\pi_1$ and $\pi_2$ with it.
353 bytes (61 words) - 16:40, 11 April 2014
$#C+1 = 17 : ~/encyclopedia/old_files/data/I052/I.0502010 Intersection index (in algebraic geometry)
The number of points in the intersection of $ n $
2 KB (289 words) - 06:42, 29 December 2021
...ll these sets (the set of elements common to all $A_\alpha$) is called the intersection of these sets.
The intersection of these sets is denoted by $\bigcap A_\alpha$.
453 bytes (76 words) - 19:46, 8 November 2014
- the [[centroid]] (''i.e.'' the centre of mass), the common intersection point of the three medians (see [[Median (of a triangle)]]);
- the incentre, the common intersection point of the three bisectrices (see [[Bisectrix|Bisectrix]]) and hence the
2 KB (333 words) - 08:29, 23 November 2023
...$A_i = \{ n \in \mathbf{Z} : n > i \}$ is centred; any family in which the intersection of all members is not empty is centred. Every finite centred family of sets
...s compact if and only if every centred family of closed sets has non-empty intersection. Centred families of closed sets in a topological space are used for the co
2 KB (253 words) - 21:48, 17 December 2015
in the same plane defined as follows: 1) the intersection of $ G ^ {*} $
is either empty or is the entire circle, depending on whether the intersection of $ G $
2 KB (241 words) - 16:44, 4 June 2020
...point $H$ of intersection of the altitudes of a triangle, the point $S$ of intersection of its [[Median (of a triangle)|median]]s (the [[centroid]]), and the centr
730 bytes (113 words) - 20:16, 16 January 2016
In a topological space $X$ a subset which is the countable intersection of open sets. See [[Borel set]] and also [[F-sigma]].
193 bytes (29 words) - 18:22, 18 August 2012
...ion of all modular maximal right ideals (cf. [[Modular ideal]]); it is the intersection of all modular maximal left ideals; it contains all quasi-regular one-sided
...adical $J(A)$ is the intersection of all right maximal ideals and also the intersection of all left maximal ideals. Nakayama's lemma says that if $M$ is a finitely
3 KB (444 words) - 06:49, 21 October 2017
...$ denote the operations of [[union of sets|union]], [[intersection of sets|intersection]], [[Difference of two sets|difference]], and [[complementation]] of sets,
...{P}(X)$ of a set $X$ (the set of subsets of $X$), in contrast to union and intersection. This ring is the same as the ring of $\mathbb{Z}/2\mathbb{Z}$-valued funct
2 KB (273 words) - 08:47, 29 April 2023
The ellipse of least surface area obtained as the intersection of a one-sheet [[Hyperboloid|hyperboloid]] with a plane perpendicular to it
145 bytes (22 words) - 17:06, 7 February 2011
An ideal $I$ of a ring $R$ which cannot be expressed as the intersection of a right [[fractional ideal]] $r(I,A)$ and an ideal $B$, each strictly la
...or left and right fractional ideals, and that every ideal decomposes as an intersection of finitely many indecomposable ideals. Then for every ideal $Q$ there exis
2 KB (297 words) - 19:23, 5 October 2017
...rsection of the bisectrix of the angle $C$ with $AB$, and the point $L$ of intersection of the bisectrix of the external angle $C$ with $AB$ forms a [[harmonic qua
1 KB (240 words) - 14:00, 12 November 2023
The intersection $M$ of all subspaces containing $A$. The set $M$ is also called the subspac
...l of a set $A$ is called the ''[[linear closure]]'' of $A$; it is also the intersection of all closed subspaces containing $A$.
855 bytes (145 words) - 08:49, 26 November 2023
...all maximal modular right ideals of an associative ring coincides with the intersection of all maximal left modular ideals and is the [[Jacobson radical]] of the r
867 bytes (141 words) - 16:16, 11 September 2016
...of these is the point of intersection of $l$ and $l'$, then the points of intersection of $AB'$ and $A'B$, $BC'$ and $B'C$, $AC'$ and $A'C$ are collinear.
1,003 bytes (158 words) - 12:52, 10 August 2014
...posite to this vertex at a point on the line passing through the points of intersection of the remaining pairs of non-adjacent sides of the pentagon (see Fig. b).
...t $C$ and $D$ with the sides $AD$ and $BC$, respectively, and the point of intersection of $AB$ and $CD$ are collinear (see Fig. c).
3 KB (409 words) - 21:10, 11 April 2014
...d $L^{i+1} = L L^i$, $i \ge 0$, with $\lambda$ denoting the empty string), intersection with [[regular language]]s, morphisms (non-erasing in the context-sensitive
1) union, Kleene ${+}$, non-erasing morphisms, inverse morphisms, and intersection with regular languages is closed under concatenation;
4 KB (577 words) - 21:45, 1 April 2018
The point of intersection of the lines joining the vertices of a triangle to the points where the sid
343 bytes (55 words) - 17:25, 7 February 2011
$#C+1 = 57 : ~/encyclopedia/old_files/data/I052/I.0502020 Intersection index (in homology)
...erizing the algebraic (i.e. including orientation) number of points in the intersection of two subsets of complementary dimensions (in [[General position|general p
4 KB (597 words) - 22:13, 5 June 2020
...f a given ring is a [[lattice]], $S(R)$, with respect to the operations of intersection and join of subrings. The set of ideals (cf. [[Ideal]]) of this ring forms
2 KB (297 words) - 17:54, 3 January 2016
...und the circles $c$ and $c'$ while preserving the distance $TT'$, i.e. the intersection line will be an ellipse ($MF'+MF=TT'$, $MF'=MT'$ and $MF=MT$). In the case
1 KB (207 words) - 11:21, 26 March 2023
...sets in $X$ with empty intersection contains a finite subfamily with empty intersection; 2) every [[Ultrafilter|ultrafilter]] in $X$ is convergent; and 3) every op
1 KB (185 words) - 20:48, 16 October 2014
...dges that have no common vertex are called opposite; the points $P,Q,R$ of intersection of the opposite edges are called diagonal points.
If $S$ and $T$ are the points of intersection of the line $PQ$ with the lines $AD$ and $BC$, then the four points $P,Q,S,
1 KB (226 words) - 16:52, 8 April 2023
An ideal $I$ in a ring which cannot be expressed as the intersection of two strictly larger ideals: that is, $I = J \cap K \Rightarrow I=J \ \te
332 bytes (55 words) - 18:11, 14 November 2023
The [[Closure of a set|closure]] of the [[linear hull]] of $A$; the intersection of all closed linear subspaces of $T$ containing $A$.
190 bytes (34 words) - 22:26, 10 January 2016
...ction, respectively union, of a finite number, and the union, respectively intersection, of any number of elements of $\mathfrak G$, respectively $\mathfrak F$, is
1 KB (214 words) - 06:39, 13 October 2014
...n $R$ is the smallest transitive relation containing $R$: equivalently the intersection of all transitive relations containing $R$ (there exists at least one such,
1 KB (245 words) - 19:34, 17 November 2023
The point of intersection of the straight lines joining the vertices of a triangle to the points at w
659 bytes (97 words) - 13:57, 8 April 2023
...with as simplices the finite non-empty subsets of $\alpha$ with non-empty intersection. In particular, the vertices of $K(\alpha)$ are the non-empty elements of $
255 bytes (37 words) - 12:24, 12 April 2014
A point $x$ such that every neighbourhood of it has non-empty intersection with $A$. The set of all proximate points forms the closure $[A]$, or $\bar
286 bytes (52 words) - 17:15, 11 April 2014
...ions containing the zero and identity operations and which is closed under intersection of projections (i.e., taking the greatest lower bound) and union of project
1 KB (219 words) - 14:33, 24 September 2014
...nvex polygon is the [[convex hull]] of a finite set of points and also the intersection of a finite number of closed half-spaces.
337 bytes (55 words) - 20:12, 9 November 2014
...r contact with $S$ at $x$ one can select the quadrics in which the line of intersection with $S$ has a singular point $x$ with three coincident tangents. On the su
2 KB (240 words) - 08:52, 8 April 2023
...h (possibly empty), and a subset $F\subset X$ is closed if and only if its intersection with every simplex is closed. Every simplicial space is a [[Cellular space|
2 KB (252 words) - 16:30, 9 April 2014
is a vector group if and only if its partial order is an intersection of total orders on $ G $.
where this intersection is taken over all combinations of signs $ \epsilon _ {i} = \pm 1 $,
1 KB (226 words) - 08:28, 6 June 2020
...solutely convex if and only if for any elements $g\not\in H$, $a\in H$ the intersection $S(g)\cap S(ga)$ is non-empty, where $S(x)$ is the minimal invariant sub-se
2 KB (249 words) - 16:16, 12 April 2014
The minimal [[Convex set|convex set]] containing $M$; it is the intersection of all convex sets containing $M$. The convex hull of a set $M$ is denoted
The closure of the convex hull is called the closed convex hull. It is the intersection of all closed half-spaces containing $M$ or is identical with $E^n$. The pa
2 KB (274 words) - 20:10, 9 November 2014
...of a set $M$ in a topological space $X$ is a point $x\in X$ such that the intersection of $M$ with any neighbourhood of $x$ has the same cardinality as the entire
461 bytes (70 words) - 19:37, 15 April 2018
...the intersection points of the opposite sides, and let $C$ and $D$ be the intersection points of the diagonals $SQ$ and $PR$ of $PQRS$ with the straight line $AB$
2 KB (278 words) - 13:42, 29 April 2014
...mension theory and the theory of multiplicities (and [[Intersection theory|intersection theory]]), cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a
6 KB (934 words) - 22:15, 5 June 2020
A point $a\in A$ such that the intersection of some [[Neighbourhood|neighbourhood]] of $a$ with $A$ consists of the poi
608 bytes (90 words) - 08:27, 23 November 2023
...utions is, by definition, the [[Intersection index (in algebraic geometry)|intersection index (in algebraic geometry)]] of the hypersurfaces \eqref{*} at the respe
2 KB (268 words) - 15:02, 14 February 2020
...[[injective hull]] $E$ of the module $M/N$ (cf. [[Injective module]]) the intersection of the kernels of the homomorphisms from $E$ into $E_1$ is trivial. Another
3 KB (455 words) - 19:51, 5 October 2017
...the complex [[projective plane]] is an asymmetric variety, since the self-intersection of the complex straight line is $+1$ or $-1$, depending on the orientation.
591 bytes (78 words) - 06:09, 23 April 2023
A subset of $X$ that is the intersection of an [[open set]] and a [[closed set]] in $X$: equivalently, a subset that
510 bytes (75 words) - 18:35, 19 November 2016
Two closed curves obtained as the intersection of two cylinders the axes of which intersect at right angles. The parametri
379 bytes (73 words) - 09:40, 5 August 2014
...mension $k\times l$, where $1<k<m$, $1<l<n$, formed by the elements at the intersection of $k$ specified rows and $l$ specified columns of $A$ with retention of th
442 bytes (80 words) - 19:36, 11 December 2015
A submodule $E$ of $M$ is essential it has a non-trival intersection with every non-trivial submodule of $M$: that is, $E \cap L = 0$ implies $L
480 bytes (76 words) - 14:23, 12 November 2023
...each point $M$ in $\pi$ is put into correspondence with the point $M_1$ of intersection of the straight line $SM$ with $\pi_1$ (if $SM$ is not parallel to $\pi_1$,
...subspace of minimal dimension containing $U$ and $T$ and let $U_1$ be the intersection of $W$ and $V_1$.
2 KB (359 words) - 22:17, 11 April 2014
$#C+1 = 58 : ~/encyclopedia/old_files/data/I052/I.0502000 Intersection homology
...pecial stress on [[Poincaré duality|Poincaré duality]] in its homological (intersection) form: Let $ M $
7 KB (1,043 words) - 22:13, 5 June 2020
...ity on two straight lines $a$ and $b$ and let $G$ and $K$ be the points of intersection of these lines with the absolute of the space. Then the angle $\phi$ betwee
...the direction vectors of the isotropic lines passing through the point of intersection of the lines $a$ and $b$.
2 KB (294 words) - 09:19, 7 August 2014
...ential submodule. The sum of all inessential submodules coincides with the intersection of all maximal submodules. A left ideal $I$ belongs to the Jacobson radical
2 KB (361 words) - 19:20, 2 October 2016
Thus, one of the tangents at the node has intersection multiplicity at least $4$ with the curve at that point.
727 bytes (113 words) - 20:45, 28 December 2014
The intersection of all flats (translates of subspaces) of $V$ containing $M$.
532 bytes (83 words) - 19:07, 7 December 2023
...]]) consisting of cosets with respect to closed submodules has a non-empty intersection. Every linearly-compact module is a complete topological group.
...every finite intersection of elements of $\{V_\alpha\}$ is non-empty, the intersection $\bigcap V_\alpha$ is non-empty. Such a linear subvariety is closed.
4 KB (667 words) - 21:41, 21 November 2018
...to be ''essential'' if whenever $H$ is a non-trivial subgroup of $G$, the intersection of $S$ and $H$ is non-trivial: here "non-trivial" means "containing an ele
594 bytes (85 words) - 14:23, 12 November 2023
A plane curve obtained as the intersection of a circular cone with a plane not passing through the vertex of the cone
...rix is called the parameter. A parabola is a symmetric curve; the point of intersection of a parabola with its axis of symmetry is called the vertex of the parabol
2 KB (364 words) - 08:04, 6 June 2020
...which there exists a family of cardinality $\tau$ of sets open in $X$ with intersection $A$. It is usually denoted by $\psi(A,X)$. The pseudo-character $\psi(A,X)$
...s the smallest infinite cardinal number $\tau$ such that each point is the intersection of a family of cardinality $\leq\tau$ of sets which are open in $X$. Spaces
2 KB (346 words) - 13:09, 27 September 2014
The curve of intersection of the surfaces of a sphere of radius $R$ and a certain circular cylinder o
593 bytes (95 words) - 19:52, 20 September 2017
...perations of [[Union of sets|union]] $R_1\cup R_2$, [[Intersection of sets|intersection]] $R_1\cap R_2$, and negation or [[Relative complement|complementation]] $R
...the Cartesian product $A \times B$. As before we may speak of the union, intersection or negation of $R$ as a relation on $A \times B$. The transpose $R^t$ is n
4 KB (625 words) - 08:47, 26 November 2023
such that the following (compactness-type) intersection property holds:
549 bytes (88 words) - 12:15, 12 December 2013
The foci of a second-order curve can be defined as the points of intersection of the tangents to that curve from the [[circular points]] of the plane. Th
829 bytes (132 words) - 05:43, 9 April 2023
...e ring]] with unit element in which any [[Prime ideal|prime ideal]] is the intersection of the [[maximal ideal]]s containing it, i.e. a ring any integral quotient
...a (non-commutative) ring $A$ is a Jacobson ring if every prime ideal is an intersection of [[primitive ideal]]s or, equivalently, if every prime factor ring $A/\ma
3 KB (415 words) - 20:32, 19 January 2016
The intersection of two nets of circles is a pencil of circles. An elliptic net contains onl
...e of which is parabolic can only be an elliptic or a parabolic pencil. The intersection of two nets one of which is non-degenerate can only be a non-degenerate pen
5 KB (865 words) - 13:07, 16 July 2014
...d by the spherical [[Polygon|polygon]] (see Fig.) which is obtained by the intersection of the faces of the polyhedral angle with a sphere of unit radius with cent
737 bytes (124 words) - 17:11, 7 February 2011
with non-empty intersection, i.e. $ \cap \{ {U } : {U \in \mathfrak B ^ \prime } \} \neq \emptyset
has the finite intersection property (i.e. the intersection of any finite number of $ U _ \alpha $
4 KB (580 words) - 09:08, 26 March 2023
set, but their intersection need not be. A set which is a finite intersection of $ \kappa a $-
2 KB (408 words) - 06:29, 30 May 2020
are coprime, then the intersection of $ V $
are not coprime, this intersection also lies on an unknotted torus $ T ^ {2} \subset S ^ {3} $,
3 KB (418 words) - 07:38, 18 March 2023
intersection and union. The ideal of the intersection $X\cap Y$ is identical
identical with the intersection of their ideals ${\mathfrak A}_X \cap {\mathfrak A}_Y$. Any set ${\bar k}^n
4 KB (616 words) - 21:49, 30 March 2012
The point of intersection of the three altitudes of a triangle, one of the classical [[triangle centr
920 bytes (144 words) - 19:18, 6 November 2016
...any two (or any set of) subgroups of a group $G$ is a subgroup of $G$. The intersection of all subgroups of $G$ containing all elements of a certain non-empty set
3 KB (467 words) - 14:22, 30 August 2014
...pole $O$ intersects the cochleoid; the tangents to the cochleoid at these intersection points pass through the same point.
1,018 bytes (151 words) - 20:46, 5 December 2023
...gated. It is a complete chain, i.e., it is closed with respect to join and
intersection. The system <img align="absmiddle" border="0" src="https://www.encyclopedia
...ncyclopediaofmath.org/legacyimages/c/c110/c110400/c11040090.png" /> is the
intersection of a suitable set of prime subgroups. If <img align="absmiddle" border="0"
15 KB (2,061 words) - 17:13, 7 February 2011
...space|Hausdorff]] [[topological space]] in which a subset is closed if its intersection with any compact subset is closed. Every [[locally compact space|locally c
871 bytes (114 words) - 14:05, 19 November 2023
If the partial order of $R$ is an intersection of total orders, then $R$ is a vector ring, and $R$ itself, provided with v
755 bytes (120 words) - 21:01, 22 December 2014
...following axioms: $X$ itself and the empty set $\emptyset$ are closed; the intersection of any number of closed sets is closed; the union of finitely many closed s
813 bytes (138 words) - 10:36, 16 April 2014
...numbers $2,3,5$. The icosahedral space can be defined analytically as the intersection of the surface
867 bytes (132 words) - 12:19, 10 April 2023
...and is closed under the set-theoretic operations of finite union, finite intersection and taking complements, i.e. such that
789 bytes (133 words) - 18:36, 25 November 2012
...he case $b=(1-a)^{-1}a(a+1)$, $c=-(1-a)^{-1}(a+1)^2$ the projection of the intersection is a [[Cardioid|cardioid]].
3 KB (479 words) - 11:37, 26 March 2023
...simultaneously left-, right- and middle-associative (or, equivalently, the intersection of the left, right and middle kernels of the loop). An element $a$ of a loo
797 bytes (132 words) - 20:04, 29 October 2016
...tween the bases, $S$ and $S'$ are their areas and $S''$ is the area of the intersection that has equal distance to both bases.
967 bytes (161 words) - 16:41, 8 May 2024
...int $x$ a condensation point (of a set $M$) in a topological space if (the intersection of $M$ with) every neighbourhood of $x$ is an uncountable set. (See also [[
861 bytes (136 words) - 09:56, 26 March 2023
...wo Borel subgroups of a group $G$ contains a maximal torus of $G$; if this intersection is a maximal torus, such Borel subgroups are said to be opposite. Opposite
3 KB (423 words) - 17:51, 27 April 2012
$#C+1 = 130 : ~/encyclopedia/old_files/data/I052/I.0502040 Intersection theory
...ent in some sense but that are in general position, and one then takes the intersection of $ Y ^ \prime $
12 KB (1,730 words) - 22:13, 5 June 2020
...here $f$ is the surface area of the triangle $MNP$ and $P$ is the point of intersection of the straight lines $m$ and $n$. The affine distance for two elements tan
730 bytes (122 words) - 15:22, 30 July 2014
...sent any [[Ideal|ideal]] of a ring (or of another algebraic system) as the intersection of a finite number of ideals of special type (primary, tertiary, primal, un
is satisfied. The intersection theorem is valid for primary ideals: The intersection of two primary ideals having the same primary radical $ P $
7 KB (1,035 words) - 20:23, 4 April 2020
...decomposition|primary decomposition]], that is, can be represented as the intersection of finitely-many primary ideals. Similarly, an $A$-module is called a Laske
1 KB (157 words) - 11:43, 29 June 2014
...ins all solvable groups (cf. [[Solvable group|Solvable group]]) and if its intersection with the class of finite groups is the class of all finite solvable groups.
1 KB (163 words) - 17:20, 7 February 2011
...y small in relation to the coverings from the given uniform structure, the intersection of the elements of this system is not empty. On a topological group there a
...n Hausdorff compactification. All such spaces have the Baire property: The intersection of a countable family of non-empty open everywhere-dense sets is always non
5 KB (764 words) - 17:23, 9 December 2013
Helly's theorem on the intersection of convex sets with a common point: Let $ K $
...-empty intersection, then all the elements of this family have a non-empty intersection.
4 KB (649 words) - 22:10, 5 June 2020
...of symmetry of an elliptic paraboloid is called its axis and the point of intersection of the axis with the elliptic paraboloid is its vertex.
892 bytes (150 words) - 13:52, 29 April 2014
such that the intersection of these congruences is the identity congruence and $ B/ \rho _ {i} \sime
is not representable as an intersection of strictly larger congruences). The theorem that every algebra is represen
3 KB (495 words) - 08:24, 6 June 2020
The [[characteristic subgroup]] $\Phi(G)$ of a group $G$ defined as the intersection of all [[maximal subgroup]]s of $G$, if there are any; otherwise $G$ is its
1 KB (169 words) - 20:06, 18 October 2017
...called a cone if it is closed under homomorphism, inverse homomorphism and intersection with regular languages. A cone that is closed under union, catenation and c
free homomorphism, inverse homomorphism, intersection with regular languages, union, catenation, and $ \lambda $-
5 KB (715 words) - 16:08, 1 April 2020
A cusp can also be defined via the so-called intersection number of two plane curves at a point, cf. [[#References|[a1]]], pp. 74-82.
844 bytes (130 words) - 19:23, 1 November 2014
with the finite intersection property: for every finite subset $ \{ C _ {1} \dots C _ {n} \} \subset
977 bytes (144 words) - 08:05, 6 June 2020
if and only if its intersection with each $ Y _ \alpha $
954 bytes (143 words) - 19:36, 5 June 2020
An intersection of the set with an interval in the case of a set on a line, and with an ope
939 bytes (161 words) - 08:07, 6 June 2020
...y them, if $ab=ba$ for any two elements $a \in A$ and $b \in B$ and if the intersection $A \cap B$ lies in the [[Centre of a group|centre]] $\mathcal{Z}(G)$. In pa
857 bytes (151 words) - 21:01, 10 January 2017