...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.
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$#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 $\bigcap A_\alpha$.
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- 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
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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
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In a topological space $X$ a subset which is the countable intersection of open sets. See [[Borel set]] and also [[F-sigma]].
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...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
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...$ 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
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The ellipse of least surface area obtained as the intersection of a one-sheet [[Hyperboloid|hyperboloid]] with a plane perpendicular to it
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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
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...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
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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 $A$.
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...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
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...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.
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...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).
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...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;
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The point of intersection of the lines joining the vertices of a triangle to the points where the sid
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$#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
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...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
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...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,
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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
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The [[Closure of a set|closure]] of the [[linear hull]] of $A$; the intersection of all closed linear subspaces of $T$ containing $A$.
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...ction, respectively union, of a finite number, and the union, respectively intersection, of any number of elements of $\mathfrak G$, respectively $\mathfrak F$, is
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...n $R$ is the smallest transitive relation containing $R$: equivalently the intersection of all transitive relations containing $R$ (there exists at least one such,
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The point of intersection of the straight lines joining the vertices of a triangle to the points at w
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...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 $
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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
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...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.
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...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
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...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 $,
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...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
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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
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...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
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...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
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A point $a\in A$ such that the intersection of some [[Neighbourhood|neighbourhood]] of $a$ with $A$ consists of the poi
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...utions is, by definition, the [[Intersection index (in algebraic geometry)|intersection index (in algebraic geometry)]] of the hypersurfaces \eqref{*} at the respe
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...[[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.
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A subset of $X$ that is the intersection of an [[open set]] and a [[closed set]] in $X$: equivalently, a subset that
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Two closed curves obtained as the intersection of two cylinders the axes of which intersect at right angles. The parametri
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...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
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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
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...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 $V_1$.
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$#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 $b$.
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...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
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Thus, one of the tangents at the node has intersection multiplicity at least $4$ with the curve at that point.
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The intersection of all flats (translates of subspaces) of $V$ containing $M$.
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...]]) 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.
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...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
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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
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...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
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The curve of intersection of the surfaces of a sphere of radius $R$ and a certain circular cylinder o
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...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
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such that the following (compactness-type) intersection property holds:
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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
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...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
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...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
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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} $,
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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
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...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.
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...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
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If the partial order of $R$ is an intersection of total orders, then $R$ is a vector ring, and $R$ itself, provided with v
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...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
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...numbers $2,3,5$. The icosahedral space can be defined analytically as the intersection of the surface
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...and is closed under the set-theoretic operations of finite union, finite intersection and taking complements, i.e. such that
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...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
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...tween the bases, $S$ and $S'$ are their areas and $S''$ is the area of the intersection that has equal distance to both bases.
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...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
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...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.
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...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.
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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 $-
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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.
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with the finite intersection property: for every finite subset $ \{ C _ {1} \dots C _ {n} \} \subset
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if and only if its intersection with each $ Y _ \alpha $
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An intersection of the set with an interval in the case of a set on a line, and with an ope
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...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
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the so-called nil radical of the ring; it coincides with the intersection of all prime ideals of $ A $.
i.e. the intersection of all prime ideals, consists of precisely the strongly-nilpotent elements.
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The
intersection of all (proper) maximal subgroups of <img align="absmiddle" border="0" src=
...iaofmath.org/legacyimages/f/f120/f120130/f12013071.png" /> is equal to the
intersection of all normal subgroups <img align="absmiddle" border="0" src="https://www.
16 KB (2,143 words) - 17:10, 7 February 2011
...in [[#References|[a2]]] H. Whitney also shows that the degree and the self-intersection number have different parity.
...are equal (see [[#References|[a1]]]). In this case the degree and the self-intersection number have different parity as well.
3 KB (505 words) - 16:52, 1 July 2020
...axis of two intersecting circles is the line passing through the points of intersection, and the radical axis of two touching circles is their common tangent. For
1 KB (221 words) - 06:14, 16 April 2023
...en (closed) relative to $E$, it is necessary and sufficient that it is the intersection of $E$ and a certain open (closed) set.
1 KB (176 words) - 13:57, 9 November 2014
...sects all the elements of some subsystem $\lambda'$ of $\lambda$, then the intersection of the elements of the system is not empty. Any finite system of closed sub
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...bgroup corresponds to that of a [[pure subgroup]] of an Abelian group. The intersection of isolated subgroups in an $R$-group is an isolated subgroup. A normal sub
1 KB (209 words) - 05:49, 20 June 2023
A congruence on an algebra $\mathbf{A}$ which is expressible as the intersection of all congruences on $\mathbf{A}$ whose factor algebras belong to some fix
1 KB (167 words) - 19:23, 12 December 2015
...al system of surfaces|triorthogonal system of surfaces]], then the line of intersection of any two surfaces of different families will be a curvature line for each
2 KB (232 words) - 17:13, 7 February 2011
...m one another by translation over $k\pi$ along the $x$-axis. The points of intersection with the $x$-axis are $(k\pi,0)$. These are also the points of inflection,
1 KB (213 words) - 18:15, 18 September 2014
...of all (equivalence classes of) representations whose kernels contain the intersection of the kernels of all the representations of this subset. For a commutative
1 KB (204 words) - 16:35, 15 May 2014
is the real line of their intersection, while the quadric $ Q _ {1} $
plane of intersection does not intersect the $ ( n - m - 1 ) $-
6 KB (872 words) - 08:09, 6 June 2020
...ne half-cone. 3) The intersecting plane meets both half-cones (Fig.c); the intersection is a [[Hyperbola|hyperbola]] — it consists of two congruent non-closed pa
...is an ellipse, a parabola or a hyperbola, respectively, if and only if its intersection with the line at infinity (cf. [[Projective plane|Projective plane]]) consi
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...closed intervals: Every family of nested closed intervals has a non-empty intersection ([[Cantor axiom|Cantor's axiom]]); in terms of upper or lower bounds of set
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...M$, with smooth image $f(L)\subset E^3$, such that for any $p\in f(L)$ the intersection of $M$ with the plane $\pi$ through $p$ and perpendicular to $f(L)$ is a [[
1 KB (168 words) - 16:30, 15 April 2014
of intersection of the conic with secants through $ P $(
is the intersection of the polars of these sets. The polar of the intersection of weakly-closed convex balanced sets $ A _ \alpha $
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...finite number of closed half-spaces. An infinite convex polyhedron is the intersection of a finite number of closed half-spaces containing at least one ray; the s
...nvex polyhedron is a special case of a [[Convex set|convex set]]. Being an intersection of half-spaces, a convex polyhedron is described by a system of linear ineq
11 KB (1,564 words) - 17:11, 7 February 2011
A plane algebraic curve of order four that is the line of intersection between the surface of a torus and a plane parallel to its axis (see Figure
1 KB (204 words) - 08:45, 12 November 2023
If a family of connected subsets has a non-empty intersection, then the union of the sets of this family is a connected set. For every po
The quasi-component of a point is the intersection of all open-closed subsets that contain this point. The component of a poin
8 KB (1,209 words) - 06:00, 22 April 2023
that are the poles of the $ ( n - 2 ) $-plane of their intersection with respect to the quadrics cut out on these planes by the absolute cone.
...int of intersection of the given lines, harmonically divided the points of intersection of the lines with the absolute lines.
7 KB (998 words) - 09:31, 21 March 2022
...ying in $O(a,\epsilon)$ and containing this pair of points has a non-empty intersection with $\Phi$. K. Menger and P.S. Urysohn proved that a closed set $\Phi$ lyi
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...et forms a [[Boolean algebra]] with the operations of [[union of sets]], [[intersection of sets]] and [[relative complement]].
1 KB (194 words) - 16:56, 25 November 2023
...of cosets is finite, $H$ is called a subgroup of finite index in $G$. The intersection of a finite number of subgroups of finite index itself has finite index (Po
1 KB (186 words) - 11:18, 20 April 2012
the union and intersection are defined: $ \Gamma = \cup _ {\alpha \in A } \Gamma _ \alpha $
...act valued) mappings is upper semi-continuous (convex-compact valued). The intersection and Cartesian product of any family of convex-compact valued mappings is co
7 KB (994 words) - 08:01, 6 June 2020
...bseteq\mathfrak P$ is closed if $C$ contains every ideal that contains the intersection of all ideals from $C$ (see [[Zariski topology]]). The structure space of a
1 KB (215 words) - 20:41, 1 October 2016
...The linking coefficient is equal to the [[Intersection index (in homology)|intersection index (in homology)]] of any $k$-chain $C^k$ such that $\partial C^k=\alpha
3 KB (548 words) - 20:58, 1 January 2019
The intersection of any family of linear varieties is again a linear variety.
1 KB (197 words) - 09:03, 26 November 2023
...first and the second pair of opposite angles is equal; or 4) the point of intersection of the diagonals bisects each of them. Various types of parallelograms are:
2 KB (265 words) - 17:11, 7 February 2011
i.e. to an intersection point of $ \overline{ {W ^ {n} }}\; $
1 KB (204 words) - 06:28, 30 May 2020
is the index of self-intersection for the class of divisors corresponding to $ D $ (see [[Riemann–Roch th
The intersection index defines an integer-valued bilinear form on $ N( X) $
10 KB (1,443 words) - 15:43, 1 March 2022
...rules with an empty right-hand side, also with respect to iteration). The intersection of a context-free language with a regular language is again a context-free
4 KB (627 words) - 19:42, 5 June 2020
...\pm5a,\ldots,$ with asymptotes $x=\pm2a,\pm4a,\pm6a,\ldots$. The points of intersection with the straight line $y=2a/\pi$ are points of inflection.
1 KB (230 words) - 05:54, 15 April 2023
...is defined in terms of intersections of cycles (cf. [[Intersection theory|Intersection theory]]).
|valign="top"|{{Ref|Fu2}}||valign="top"| W. Fulton, "Intersection theory", Springer (1984) {{MR|0735435}} {{MR|0732620}} {{ZBL|0541.14005}}
4 KB (714 words) - 21:54, 24 April 2012
...all nilpotent groups (cf. [[Nilpotent group|Nilpotent group]]) and if its intersection with the class of finite groups is the class of all finite nilpotent groups
2 KB (245 words) - 17:21, 7 February 2011
...X$ such that a sequence $\{F_i\}$ of closed subsets of $X$ has a non-empty intersection whenever $F_i\supset F_{i+1}$ for all $i$ and each $F_i$ is a subset of som
1 KB (208 words) - 20:11, 12 July 2014
The intersection of all [[closed set]]s of $X$ containing the set $A$.
1 KB (198 words) - 16:57, 9 November 2014
to the point of intersection of the altitudes of the triangle. The Simson line is named after R. Simson,
2 KB (236 words) - 08:14, 6 June 2020
is the notation for the intersection of all subcomplexes of $ X $
the intersection $ F \cap \overline{t}\; $
7 KB (1,077 words) - 06:29, 30 May 2020
that is, the intersection of the truth domains of the predicates $R(x)$ and $P(x)$ is non-empty.
1 KB (212 words) - 07:30, 23 August 2014
...th remainder of dimension $\leq k$, then $X$ has an open base in which the intersection of the boundaries of any $k+1$ disjoint sets is compact; etc. If every conn
1 KB (228 words) - 14:25, 10 April 2014
...ideal is a finite irredundant intersection of [[primary ideal]]s. Here an intersection of primary ideals $ \mathfrak a = \cap \mathfrak a _ {i} $
contains an intersection of the others and if the $ \mathfrak a _ {i} $
5 KB (786 words) - 22:15, 5 June 2020
...Zariski topology]]) subsets. A locally closed subset is, by definition, an intersection of an open and a closed subset. The constructible subsets form a Boolean al
1 KB (202 words) - 17:38, 14 April 2018
...that a cocyclic group has a non-trivial minimal subgroup $M$, or that the intersection $M$ of all non-trivial subgroups of $G$ is non-trivial. Each element of $M
1 KB (202 words) - 14:02, 12 November 2023
is the intersection of all subspaces containing $ s $.
In a projective space the operations of addition and intersection of spaces are defined. The sum $ P _ {m} + P _ {k} $
10 KB (1,510 words) - 18:38, 13 January 2024
...ory of a set|Category of a set]]). A space $X$ is Baire if and only if the intersection of each countable family of dense open sets in $X$ is dense (cf. also [[Den
...is finer than $\rho$. A topology $\tau$ has the Slobodnik property if the intersection of each countable family of $\tau$-open $\rho$-dense sets in $X$ is $\rho$-
4 KB (610 words) - 08:48, 18 February 2024
* a $k\times k$ [[Matrix|matrix]] $B$ whose entries are located at the intersection of $k$ distinct columns and $k$ distinct rows of $A$; however a more common
1 KB (233 words) - 10:05, 20 December 2015
...s. The signature of this form is called the signature of the manifold. The intersection form is the most important invariant of a four-dimensional manifold. Two cl
...d every even, integer-valued, symmetric unimodular form is realized as the intersection form of a simply-connected four-dimensional topological manifold. In partic
6 KB (831 words) - 19:39, 5 June 2020
...has a feathering in every Hausdorff compactification. If a set $X$ is the intersection of a sequence $U_1,U_2,\ldots$ of sets open in a space $Y$, then the system
...eans that the set $\Delta = \{(x,x) : x \in X\}$ can be represented as the intersection of a countable family of open sets in $X \times X$. These results and other
6 KB (962 words) - 20:32, 23 September 2017
i.e. the intersection of all prime ideals of $ R $.
1 KB (240 words) - 08:02, 6 June 2020
These six operations are: union, concatenation, Kleene closure ${+}$, intersection with regular languages, morphisms, and inverse morphisms.
...ages]]) if it is closed under non-erasing morphisms, inverse morphisms and intersection with regular languages. A trio closed under union is called a semi-AFL. A s
4 KB (661 words) - 09:20, 2 April 2018
...folds in terms of the intersection pairing (cf. also [[Intersection theory|Intersection theory]]).
...e (with the base point representing $T_1$) and each edge represents a self-intersection. It is not presently known whether different signed trees can correspond to
7 KB (1,047 words) - 15:29, 4 October 2014
...of [[complementation]] in [[set theory]] on [[union of sets|union]] and [[intersection of sets]]; the analogous relationship between [[negation]] in [[proposition
1 KB (199 words) - 14:15, 12 November 2023
...l space $X$ is nowhere dense if, for every nonempty open $U\subset X$, the intersection $U\cap A$ is ''not'' dense in $U$. Common equivalent definitions are:
1 KB (187 words) - 19:07, 7 December 2023
...mbus, then the added term is equal to the product of the difference at the intersection of the path and the column by the coefficient associated with the side of t
...mbus, then the added term is equal to the product of the difference at the intersection of the path and the column by the half-sum of the coefficients associated w
7 KB (939 words) - 19:39, 5 June 2020
...hese hyperplanes which are the poles of their $ ( n - 2) $-hyperplane of intersection relative to the quadrics cut out by the absolute cone on the hyperplanes. I
...er with their point of intersection, harmonically separate their points of intersection with the absolute lines.
7 KB (1,046 words) - 09:03, 13 May 2022
the intersection index $ D ^ {r} \cdot Y $
cf. [[Intersection index (in algebraic geometry)|Intersection index (in algebraic geometry)]]). For other criteria of ampleness see [[#Re
6 KB (950 words) - 12:37, 29 December 2021
a planar curve obtained by the intersection of a circular cone with a plane not passing through the vertex of the cone
...cond axis. The axes of the ellipse are its axes of symmetry. The points of intersection of the ellipse with the axes of symmetry are called its vertices. The major
4 KB (640 words) - 21:38, 24 April 2012
...be an open cone in a real [[topological vector space]] $k$, let $V$ be the intersection of $K$ with a bounded open ball with centre at $0\in k$, and let $U\neq\emp
1 KB (235 words) - 10:40, 26 May 2016
In terms of normal sheaves one can express the self-intersection $ Y \cdot Y $
that is locally a complete intersection, then $ {\mathcal N} _ {Y/X} ^ {*} $
5 KB (799 words) - 08:03, 6 June 2020
sphere if and only if the determinant of the matrix of the integral bilinear intersection $ ( - 1 ) ^ {n} $-
is parallelizable, the diagonal of the intersection matrix of $ 2k $-
9 KB (1,361 words) - 12:24, 10 April 2023
...d surface $M^n$ lying in $N^{n+1}$, the [[Intersection index (in homology)|intersection index (in homology)]] in $N^{n+1}$ of any closed curve $\alpha$ in $M^n$ sa
4 KB (714 words) - 16:10, 5 August 2014
...ed over species in analogy to operations defined over sets, such as union, intersection and others, but since they are to be understood in the specific intuitionis
2 KB (295 words) - 16:47, 19 January 2024
...ytic and algebraic tools. Thus, for instance, the problem of the number of intersection points between a straight line and a circle is reduced to the analytic prob
...[Hyperbola|hyperbola]] and a [[Parabola|parabola]], which are the lines of intersection between a circular cone and planes which do not pass through its vertex (cf
8 KB (1,255 words) - 18:47, 5 April 2020
...e to $K$. If $L_1$ and $L_2$ are normal extensions of $K$, then so are the intersection $L_1 \cap L_2$ and the composite $L_1 \cdot L_2$. However, when $L/K'$ and
2 KB (288 words) - 18:30, 11 April 2016
A family of sets open in $X$ and such that each point of $X$ is the intersection of all elements in the family containing it. A pseudo-basis exists only in
2 KB (280 words) - 07:51, 23 August 2014
is a point in the intersection of the hyperplanes $ \pi _ {1} \dots \pi _ {n + 1 } $
is the maximum number of points of intersection of $ X $
7 KB (1,079 words) - 16:43, 4 June 2020
...A normal space is called perfectly normal if every closed set in it is the intersection of countably many open sets. Every perfectly-normal space is a hereditarily
.../n0676609.png" />-normal spaces in which every closed canonical set is the intersection of countably many open canonical sets are called perfectly $\kappa$-normal.
5 KB (733 words) - 20:41, 19 December 2014
...kew lines have a unique common perpendicular. The equation (as the line of intersection of two planes) of this common perpendicular to the lines $ \mathbf r = \m
2 KB (283 words) - 08:56, 8 April 2023
[[Intersection theory|intersection theory]] provides a satisfactory modern framework. Enumerative geometry dea
{{Cite|Fu2}} for a complete reference on intersection theory; for historical surveys and a discussion of enumerative geometry, se
8 KB (1,263 words) - 08:49, 30 March 2012
...metry axis of a hyperbolic paraboloid is said to be its axis; the point of intersection of a hyperbolic paraboloid with the axis is known as the apex. If $p=q$, th
2 KB (252 words) - 11:12, 25 May 2016
of closed sets has a non-empty intersection; and 3) every open covering of cardinality $ m \in [ a , b ] $
4 KB (577 words) - 08:32, 16 June 2022
...)$, which has good functoriality properties and is equipped with a graded intersection product, at least after tensoring it by $\mathbf{Q}$.
$p \geq 0$, give rise to arithmetic intersection numbers, which are real numbers when their geometric counterparts are integ
8 KB (1,219 words) - 21:00, 13 July 2020
...ch together with each polytope contains all its faces and is such that the intersection between the polytopes is either empty or is a face of each of them. An exam
2 KB (301 words) - 17:51, 11 April 2014
...c notions: $X$ is a submultiset of $Y$ if $\epsilon_X \le \epsilon_Y$, the intersection of $X$ and $Y$ corresponds to $\min(\epsilon_X, \epsilon_Y)$, the union to
2 KB (265 words) - 19:01, 9 November 2023
is equal to the intersection of all maximal modular right (left) ideals of $ R $;
is also equal to the intersection of the kernels of all irreducible right (left) representations of $ R $(
5 KB (764 words) - 08:09, 6 June 2020
be the point of intersection of $ u $
be the point of intersection of $ u $
7 KB (1,026 words) - 08:08, 6 June 2020
The intersection of a family of congruences $\pi_i$, $i \in I$, in the lattice of relations
2 KB (277 words) - 22:08, 12 November 2016
that have a non-empty intersection. There is an obvious simplicial mapping $ \omega _ { {\mathcal A} ^ \pr
set is a finite intersection of closures of open sets. A $ T _ \lambda $-
6 KB (823 words) - 08:09, 6 June 2020
space corresponding to these points according to (a2) have a common point of intersection, say $ ( \theta _ {s} , \rho _ {s} ) $.
...r subsets of picture points can be found by searching coincident points of intersection in the parameter space. In practice this is done by quantizing both paramet
5 KB (768 words) - 22:11, 5 June 2020
...been chosen and a segment of unit length has been specified. The point of intersection ($0$) of the coordinate axes is said to be the coordinate origin. One of th
2 KB (278 words) - 22:36, 16 March 2014
...it is further required that some finite subfamily of the $C_i$ have empty intersection, one obtains the concept of an $S$-weakly infinite-dimensional space.
2 KB (275 words) - 17:01, 5 October 2017
...e set of points $M$ for which $OM=CB$, where $B$ and $C$ are the points of intersection of the line $OM$ with a circle and the tangent $AB$ to the circle at the po
2 KB (300 words) - 09:39, 26 March 2023
...nt conics belonging to one plane. The two other focal points are points of intersection of a conic with a characteristic of its plane.
and the four points of intersection of the conic with an adjacent conic in the same plane. The geometric proper
8 KB (1,191 words) - 06:39, 9 April 2023
...$M_2$, $M_3$, $M_4$ and $M_1'$, $M_2'$, $M_3'$, $M_4'$ are obtained by the intersection of the same quadruple of straight lines $m_1$, $m_2$, $m_3$, $m_4$, then
2 KB (296 words) - 09:34, 13 April 2014
Let $u$ be the axis coinciding with the line of intersection of the planes $0xy$ and $0x'y'$, oriented so that the three lines $0z$, $0z
2 KB (331 words) - 14:12, 13 November 2014
The intersection of connected subgroups of an exponential Lie group is connected. The centra
2 KB (253 words) - 22:20, 14 November 2014
then the entry at the intersection of the $ i $-
2 KB (300 words) - 16:43, 4 June 2020
...s \cup \tilde U_k$, where $\tilde U_i$ is the largest open set in $bX$ the intersection of which with $X$ is the set $U_i$ ($X$ is assumed to be [[Completely-regul
2 KB (294 words) - 16:43, 24 September 2017
...of $\nu$-primitive $A$-modules. The radical $J_{1/2}(A)$ is defined as the intersection of the maximal right module ideals. All four radicals are different, and
5 KB (767 words) - 11:23, 27 October 2014
The set-theoretical intersection of this family, that is, $ \cap F $.
2 KB (275 words) - 04:11, 6 June 2020
...set coincides with itself. This is equivalent to the requirement that the intersection of all neighbourhoods of a point $ x \in X $
2 KB (275 words) - 08:25, 6 June 2020
with index of self-intersection $ ( C ^ {2} ) _ {X} > 0 $.
which is a smooth rational curve with index of self-intersection $ ( S _ {n} ^ {2} ) _ {F} = - n $.
5 KB (707 words) - 16:26, 2 March 2022
...es the other. For example, each of the operations [[union of sets]] and [[intersection of sets]] distributes over the other.
2 KB (301 words) - 19:27, 24 January 2016
...mical systems) for a set $E$ in a topological space $X$ which contains the intersection of a countable number of open dense subsets in $X$. The terminology is in g
It is more common to call (a set containing) an intersection of countably many dense open sets [[Residual set|residual]] or comeager, fo
4 KB (704 words) - 11:07, 6 September 2013
...e. Think of the Boolean algebra of all subsets of a given set under union, intersection and complement to interpret these formulas.
2 KB (335 words) - 15:09, 13 August 2014
...roblem of additive type may also be regarded as the problem of finding the intersection of arithmetical sums of sets. For instance, the set $ M $
2 KB (277 words) - 19:35, 5 June 2020
...sic set has the structure of a closed subscheme $B$ in $X$, defined as the intersection of all divisors of the movable part of the linear system. The removal of th
2 KB (288 words) - 04:27, 15 February 2024
The polygon resulting from the intersection of the pyramid by the plane is called the upper base, while the base of the
2 KB (400 words) - 21:32, 11 April 2014
<TR><TD valign="top">[2]</TD> <TD valign="top"> C. Chevalley, "Intersection of algebraic and algebroid varieties" ''Trans. Amer. Math. Soc.'' , '''57'
2 KB (317 words) - 17:54, 10 January 2016
...ation is the intersection of their images and, conversely, the image of an intersection is the sum of the images. In particular, the image of a point is a hyperpla
6 KB (878 words) - 14:59, 2 May 2023
* A singular (non-smooth) point of an algebraic curve which is a transversal intersection of smooth branches;
...e node (or ''nodal point'', also ''nodal singularity'') is a point of self-intersection of smooth branches of the curve.
5 KB (753 words) - 09:03, 12 December 2013
...centre $O$ of the sphere, then a so-called great circle is obtained as the intersection. A unique great circle can be drawn through any two points $A$ and $B$ on t
...gle $A'BC'$ between the tangents of the corresponding arcs at the point of intersection $B$ or by the dihedral angle formed by the planes $OBA$ and $OBC$.
8 KB (1,389 words) - 15:53, 19 April 2014
...1}$ and $\overrightarrow{M_0M_2}$, where $M_1$ and $M_2$ are the points of intersection of the circle and an arbitrary straight line passing through $M_0$. In part
2 KB (298 words) - 06:25, 12 October 2023
...andt subgroup $\omega ( G )$ of a [[Group|group]] $G$ is defined to be the intersection of the normalizers of all the subnormal subgroups of $G$ (cf. also [[Subnor
...nilpotent (cf. [[Nilpotent group|Nilpotent group]]), $\omega ( G )$ is the intersection of the normalizers of all the subgroups of $G$. The latter is called the no
11 KB (1,625 words) - 07:40, 10 February 2024
one takes the intersection with $ D $
2 KB (293 words) - 18:47, 5 April 2020
Any union and any intersection of subcomplexes of $ X $
2 KB (333 words) - 16:43, 4 June 2020
intersection and a countable union of ${\mathcal A}$-sets is an ${\mathcal A}$-set. Any
2 KB (327 words) - 14:21, 15 August 2023
...three points $A,B,C$ and $A_1,B_1,C_1$ are given, other than the point of intersection of the lines; if $CB_1$ is parallel to $BC_1$ and $CA_1$ is parallel to $AC
2 KB (344 words) - 17:16, 31 March 2018
...fmath.org/legacyimages/n/n066/n066970/n066970132.png" /> is defined as the
intersection of all elements of the form <img align="absmiddle" border="0" src="https://
21 KB (2,970 words) - 17:28, 7 February 2011
be two domains in the plane with non-empty intersection and such that the solution to the Dirichlet problem for the Laplace equatio
...2]]]) can be applied to finding a solution to the Dirichlet problem in the intersection of two domains $ A $
6 KB (852 words) - 08:12, 6 June 2020
...ssociative rings and algebras|Non-associative rings and algebras]]) is the intersection of its maximal subalgebras. Unlike the group case, where the Frattini subgr
2 KB (323 words) - 13:52, 25 April 2014
...d lying in different faces, or, in other words, by the angle formed by the intersection of the dihedral angle and a plane normal to the edge.
2 KB (324 words) - 09:48, 26 May 2016
...int operators on a separable Hilbert space such that $A+B$, defined on the intersection of the domains of $A$ and $B$, is essentially self-adjoint. And in the form
2 KB (344 words) - 21:49, 3 December 2017
...wo arbitrary points all points of the segment connecting these points. The intersection of any family of convex sets is itself a convex set.
...A closed convex set is the intersection of its supporting half-spaces. The intersection of a finite number of closed half-spaces is a convex polyhedron. The [[face
9 KB (1,558 words) - 17:30, 23 October 2017
...erically has a certain property if the property holds outside (a countable intersection of) open dense set(s). For instance, a polynomial has generically no double
2 KB (324 words) - 12:46, 4 September 2014
should have non-empty intersection with $ E _ {f} $
2 KB (311 words) - 19:43, 5 June 2020
...a triangle is F. Morley's theorem (1899), stating that the three points of intersection of the adjacent trisectors of the angles of an arbitrary triangle form an e
2 KB (321 words) - 13:05, 7 December 2014
A theorem on the product of the simple ratios in which the points of intersection of an algebraic curve with the sides of a triangle divide these sides. Supp
2 KB (347 words) - 10:08, 4 June 2020
where the right-hand side is the [[Intersection index (in homology)|intersection index]] of the classes $ z _ {1} $
5 KB (839 words) - 06:50, 28 April 2024
...is called the complement of the set $A$ (in $X$). The operations of union, intersection and taking the complement are connected by the so-called [[de Morgan laws]]
8 KB (1,307 words) - 19:50, 30 November 2014
...f polynomial ideals. In particular, he developed the [[Intersection theory|intersection theory]] on a non-singular projective algebraic variety. The results of the
...ly projective) over an arbitrary ground field, the theory of divisors, the intersection theory for such varieties, and the general theory of Abelian varieties, whi
8 KB (1,165 words) - 16:08, 1 April 2020
...r $k$. Two parabolic subgroups of a group $G$ are called opposite if their intersection is a Levi subgroup of each of them. A closed subgroup of a group $G$ is a p
...containing $B$. Each parabolic subgroup coincides with its normalizer. The intersection of any two parabolic subgroups contains a subgroup of $G$ that is conjugate
5 KB (932 words) - 11:43, 17 December 2019
...omain|integral domain]]. In this case every maximal ideal is prime and the intersection of all prime ideals is the radical of the null ideal (i.e. is the set of ni
2 KB (393 words) - 08:07, 6 June 2020
...is a continuum, the components of a Hausdorff compactum are continua, the intersection of a decreasing sequence of continua is a continuum. No continuum can be d
2 KB (336 words) - 15:31, 23 July 2012
...\Delta<0$, the double point is said to be a nodal point or a point of self-intersection; e.g. for the curve
2 KB (417 words) - 11:59, 26 March 2023
are called the faces of the analytic polyhedron. The intersection of any $ k $
2 KB (335 words) - 18:47, 5 April 2020
The property of a subgroup to be subnormal is transitive. An intersection of subnormal subgroups is again a subnormal subgroup. The subgroup generate
3 KB (420 words) - 10:03, 3 January 2021
...he second by the angular coordinates $\alpha$ and $\beta$ of the points of intersection of the chord with the circumference of the disc; in the third case by the C
2 KB (374 words) - 10:24, 16 March 2023
...Hamilton group contains a subgroup isomorphic to the quaternion group. The intersection of all non-trivial subgroups of the quaternion group (and also of any gener
2 KB (350 words) - 14:38, 2 August 2014
be a contravariant functor of global [[Intersection theory|intersection theory]] from $ V ( k) $
does not, in some sense, depend on the intersection theory of $ C $,
8 KB (1,244 words) - 10:52, 16 March 2023
...f it coincides with its socle. The socle of $M$ can also be defined as the intersection of all the essential submodules of $M$. The socle is the largest semi-simpl
2 KB (371 words) - 15:04, 19 November 2023
..., $S(g)$ does not contain the unit of $G$ and if for any $x,y\in S(g)$ the intersection $S(x)\cap S(y)$ is non-empty.
2 KB (365 words) - 20:04, 12 July 2014
...the ideals in a regular commutative Banach algebra: A closed ideal is the intersection of maximal ones ( "it admits spectral synthesis" ) if and only if its annih
as cospectrum, namely the intersection of all closed maximal ideals corresponding to the points of $ E $.
9 KB (1,382 words) - 08:22, 6 June 2020
...(see [[#References|[1]]]). One of them reduces to the requirement that an intersection of $ \widetilde{X} $
2 KB (374 words) - 16:44, 4 June 2020
is a normal complete intersection of $ X $(
...ld in étale cohomology [[#References|[a4]]] and in [[Intersection homology|intersection homology]] [[#References|[a5]]], [[#References|[a6]]]. For the proof of the
11 KB (1,584 words) - 11:51, 8 April 2023
which may be traversed such that the points of self-intersection are visited only twice. For a curve to be unicursal it is necessary and suf
2 KB (342 words) - 13:37, 7 June 2020
The converse is true for free groups, but not in general: The intersection of two verbal subgroups may not be a verbal subgroup.
2 KB (363 words) - 11:34, 12 January 2021
the self-intersection index of the anti-canonical divisor $ (- K _ {X} ^ {3} ) \leq 64 $.
then the self-intersection index $ d = H ^ {3} $
6 KB (869 words) - 13:13, 26 March 2023
...tification]] $\overline{\mathbf R^n}$); and 2) the images of any non-empty intersection $V_x\cap V_y$ (under the two homeomorphisms to $\mathbf R^n$ or $\overline{
2 KB (357 words) - 11:31, 27 October 2014
...int $A$ distinct from the centre of a given pencil of circles the point of intersection $A'$ of the circles of the pencil passing through $A$.
3 KB (469 words) - 18:16, 17 December 2017
...]] $I$ of a [[ring]] $R$ (or of a submodule $N$ of a [[module]] $M$) as an intersection of primary ideals (primary submodules, cf. [[Primary ideal]]). The primary
2 KB (386 words) - 20:06, 5 October 2017
for the wave equation is determined only by the initial data at the intersection of the initial manifold with the characteristic cone of $ ( M, t) $
depends only on the initial data at the intersection of the initial manifold with the characteristic conoid if and only if the [
6 KB (856 words) - 12:26, 24 March 2024
...gebraic variety $V\subset P^g$ of dimension $n$ is the number of points of intersection with a generic hyperplane of dimension $g-n$ in $P^g$. Thus, the degree of
3 KB (391 words) - 13:15, 7 April 2023
...family of non-empty closed sets of a topological space $X$ has a non-empty intersection, is one of the definitions of compactness of $X$ (see [[#References|[1]]],
8 KB (1,230 words) - 20:30, 21 January 2021
can be uniquely represented as the intersection of homogeneous prime ideals:
3 KB (364 words) - 08:09, 13 July 2022
...ational boundary component $F_{i}$ of $X ^ { * }$, and $\Gamma_{i}$ is the intersection of $\Gamma$ with the stabilizer of $F_{i}$. In addition, $V$ and each $V _
...rsection cohomology [[#References|[a4]]] (cf. also [[Intersection homology|Intersection homology]]) of the Baily–Borel compactification coincides with its $L^{2}
13 KB (1,983 words) - 17:00, 1 July 2020
...mined by the class of all primary rings; the least special radical; or the intersection of the prime ideals of the ring;
...mitive rings; the intersection of all primitive ideals of the ring; or the intersection of all modular maximal right (left) ideals. It is a quasi-regular ideal con
16 KB (2,540 words) - 08:09, 6 June 2020
The intersection of any set of normal subgroups is normal, and the subgroup generated by any
2 KB (406 words) - 22:11, 5 March 2012
so that its image is the intersection (generally incomplete) of the quadrics containing it [[#References|[1]]].
...></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.N. Tyurin, "On the intersection of quadrics" ''Russian Math. Surveys'' , '''30''' : 6 (1975) pp. 51–106 '
11 KB (1,607 words) - 08:08, 6 June 2020
...s greatest element, that is, the identity with respect to the operation of intersection.
2 KB (403 words) - 19:25, 3 April 2016
...ll of the space $\mathbf R^k$ with centre at the origin and $D_+^k$ is the intersection of this ball with the half-space $x_k\geq0$.
2 KB (383 words) - 08:32, 19 April 2014
...[6]]]. The local components of a height constructed there play the role of intersection indices in arithmetic.
...$ over a number field $K$. Heights also play an important role in Arakelov intersection theory, which via moduli spaces of algebraic curves has also become importa
6 KB (893 words) - 08:43, 1 May 2023
...riangle, the centroid of the three corners, each with weight unity, is the intersection point of the three medians of the triangle (see [[Plane trigonometry|Plane
3 KB (410 words) - 16:43, 4 June 2020
system. The intersection of all elements of a chain is called its kernel, and the union of all kerne
3 KB (440 words) - 18:48, 5 April 2020
and their infimum is the intersection $ X \cap Y $.
An intersection of subalgebras may be empty, and therefore for certain types of algebras (e
7 KB (1,037 words) - 08:24, 6 June 2020
...the homotopy groups (cf. [[Homotopy group|Homotopy group]]); and also the intersection ring of a manifold, soon to be replaced by the cohomology ring of Alexander
3 KB (442 words) - 19:40, 25 March 2023
Any space in which the intersection of any countable family of dense open subsets is dense (cp. with Section 9
2 KB (353 words) - 17:21, 18 August 2012
...o surfaces of different families form a right angle at each point of their intersection. It is assumed that the surfaces of the triorthogonal system are regular, a
3 KB (517 words) - 19:56, 12 July 2014
as the intersection of all ideals that contain $ X $.
The intersection, and for semi-groups also the union, of left (two-sided) ideals is again a
13 KB (2,163 words) - 22:11, 5 June 2020
Every lattice ordering of a group $G$ is the intersection of some of its right-orderings (cf. [[Lattice-ordered group|Lattice-ordered
4 KB (585 words) - 06:17, 28 March 2023
where $C$ is the point of intersection of the characteristics. When one replaces the differential relations \eqref
up to the intersection with $AB$ at $D$. The value of the entropy $S$ at $D$ is determined by mean
6 KB (1,008 words) - 15:54, 14 February 2020
operation is stronger than the operation of countable union and countable intersection, and is idempotent. With respect to $ {\mathcal A} $-
3 KB (432 words) - 16:59, 1 July 2020
is bounded. The intersection $ K = \cap _ {n = 1 } ^ \infty R _ {n} $
3 KB (439 words) - 22:14, 5 June 2020
...is case the left-hand side of (1) can be determined in the same way as the intersection index $ \Gamma _ {f} \Delta $
is the diagonal (cf. [[Intersection index (in algebraic geometry)|Intersection index (in algebraic geometry)]]).
10 KB (1,491 words) - 19:04, 26 March 2023
...note that while the meet of a family of closed sets is their set-theoretic intersection, the join of a family of closed sets is the closure of their set-theoretic
3 KB (459 words) - 12:03, 23 November 2023
in $\PP^n$ its image with respect to the Veronese mapping $v_m$ is the intersection of the Veronese variety $v_m(\PP^n)$ with the hyperplane
2 KB (408 words) - 23:35, 22 October 2018
...c curve]]), transcendental curves can have an infinite number of points of intersection with a straight line and an infinite number of points of inflection. On tra
3 KB (434 words) - 13:49, 9 April 2023
...I} \sim_i$ {{Cite|EM85}}. As an immediate consequence, the set-theoretic intersection of a family $\{\sim_i\}_{i\in I}$ of $\Sigma$-congruences is again a $\Sigm
8 KB (1,264 words) - 16:16, 18 February 2013
...at a subset of $ \operatorname{Irr}(A) $ is a Borel set if and only if its intersection with each $ {\operatorname{Irr}_{n}}(A) $ belongs to the Borel structure on
3 KB (478 words) - 04:32, 4 December 2016
The intersection of cycles $ \gamma _ {i} \gamma _ {j} = q _ {ij} $
and the intersection matrices $ Q $
6 KB (967 words) - 08:26, 6 June 2020
...t (of the intersection of their domain of definition) which has nontrivial intersection with $V$.
6 KB (1,048 words) - 21:19, 14 January 2021
...ofmath.org/legacyimages/b/b015/b015260/b0152607.png" />, continuous in the
intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
10 KB (1,381 words) - 11:57, 27 September 2012
Under a projective transformation the sum and intersection of subspaces are preserved, points are mapped to points, and independence o
3 KB (417 words) - 08:08, 6 June 2020
...)_{\lambda\in\Lambda}$ is called a separating family of congruences if the intersection of all the $q_\lambda$ is the diagonal congruence (the equality relation) o
3 KB (536 words) - 22:24, 26 October 2014
...tion over the Jacobian (cf. [[Jacobi variety]]) of an algebraic curve. The intersection form of one-dimensional cycles in the homology of an algebraic curve $X$
3 KB (414 words) - 18:28, 17 April 2024
is a non-empty compact space; 2) the intersection of any [[Centred family of sets|centred system]] of closed sets in $ X $
is non-empty; 3) the intersection of any maximal centred system of closed sets in $ X $
26 KB (3,890 words) - 16:31, 5 June 2020
then the curve of intersection of any plane $ x = \textrm{ const } $
6 KB (901 words) - 02:55, 4 March 2022
...mal element, if this exists. A [[complete lattice]] always has a zero: the intersection of all elements.
3 KB (471 words) - 21:06, 3 January 2015
...every non-empty family of principal ideals has a maximal element, and the intersection of any two principal ideals is principal. Every principal ideal ring is fac
3 KB (480 words) - 21:45, 3 January 2021
...ro. The Lie algebra of an intersection of Lie subgroups coincides with the intersection of their Lie algebras. The Lie algebra of the kernel of a homomorphism $
9 KB (1,338 words) - 22:16, 5 June 2020
it coincides with the intersection of all maximal left (also, all right) ideals.
contains the intersection $ I $
7 KB (1,160 words) - 08:00, 6 June 2020
...gth of a segment of a helical line between two of its successive points of intersection with the same generator is called the turn of the helical line, while the l
3 KB (537 words) - 15:42, 29 September 2014
planes into themselves. The lines of intersection of these planes with the absolute of $ {} ^ {3} S _ {5} $
3 KB (461 words) - 19:40, 5 June 2020
<TR><TD valign="top">[5]</TD> <TD valign="top"> Y. Nakai, "Note on the intersection of an algebraic variety with the generic hyperplane" ''Mem. Coll. Sci. Univ
3 KB (451 words) - 11:57, 23 April 2017
A planar curve which is obtained as the intersection of a circular cone with a plane not passing through the apex of the cone an
3 KB (527 words) - 16:34, 19 September 2014
is the intersection of a suitable set of right orders $ P _ \alpha $
is the intersection of all total orders of $ G $
9 KB (1,403 words) - 22:15, 5 June 2020
2) every countable intersection of neighbourhoods of $x$ contains a neighbourhood of $x$. The latter is com
3 KB (483 words) - 05:16, 21 November 2017
and 2) if $x,y$ are different points in $R$, then the intersection $l(x,y)$ of the straight line $l$ through $x$ and $y$ with $R$ is either al
3 KB (497 words) - 16:16, 1 May 2014
one considers the intersection of this group with an order $ {\mathcal O} $
3 KB (430 words) - 10:48, 20 January 2024
...er, a coincidence number, a fixed-point index, a vector field index, or an intersection number. For any mapping $ p : X \rightarrow Y $,
3 KB (442 words) - 10:33, 29 May 2020
the intersection of any two successive members of which is an $ ( l -1) $ -
6 KB (908 words) - 21:26, 16 December 2019
...'left [[Reversible semi-group|reversible]]'': that for all $a,b \in S$ the intersection of right [[ideal]]s $aS \cap bS$ is non-empty.
3 KB (456 words) - 19:17, 23 December 2014
...crossing number $\text{cr} ( G )$ of a graph $G$ is the minimum number of intersection points among the curves representing edges, over all possible drawings of t
iii) intersection points among the curves representing edges are crossing points, i.e. the cu
6 KB (1,021 words) - 17:46, 1 July 2020
...sets satisfying three conditions: 1) the empty set is not included; 2) the intersection of two subsets in the system again belongs to it; and 3) for any subset, ei
...r is also called fixed in that point. An ultrafilter is called free if the intersection of all its elements is the empty set, in other words, if it is not fixed in
10 KB (1,605 words) - 16:57, 13 January 2024
is the intersection of all prime ideals containing $ A $,
3 KB (505 words) - 08:09, 6 June 2020
A subgroup system that is closed with respect to union and intersection is called complete. A complete subgroup system is called subnormal if for a
4 KB (599 words) - 08:24, 6 June 2020
obtained as the point of intersection of the ray drawn from the north pole of the sphere, $ P( 0, 0, 1) $,
3 KB (453 words) - 13:59, 17 March 2023
...clopediaofmath.org/legacyimages/m/m063/m063660/m06366025.png" />: take the
intersection of the considered subsets with <img align="absmiddle" border="0" src="https
8 KB (1,175 words) - 05:58, 15 May 2012
...ic group coincides with the [[connected component of the identity]] in the intersection of all [[Borel subgroup]]s of $G$; it is the smallest closed normal subgrou
3 KB (517 words) - 11:55, 23 November 2013
...direct sum of irreducible unitary finite-dimensional representations. The intersection of the kernels of the continuous homomorphisms of a topological group $ G
coincides with the intersection of the kernels of the irreducible unitary finite-dimensional representation
9 KB (1,290 words) - 19:39, 5 June 2020
...is in the West almost exclusively used for a subset of the real line whose intersection with every nowhere-dense set is countable (see [[Luzin space|Luzin space]])
3 KB (535 words) - 04:11, 6 June 2020
...sitively closed as far as the sets of subjects and objects have a non-void intersection. The ultimate goal of such a formal specification is the verifiability of a
3 KB (469 words) - 17:22, 7 February 2011
into any domain will also yield an analytic curve. If the set of intersection points of two analytic curves is infinite, these analytic curves coincide.
3 KB (480 words) - 18:31, 5 April 2020
..._i$ of $X$ such that $F_{i+1}\subset F_i$, $i=1,2,\ldots,$ has a non-empty intersection (a generalization of Cantor's principle of nested intervals).
3 KB (481 words) - 18:40, 16 April 2014
A perfect differential ideal may be represented as the intersection of a finite number of prime differential ideals. To such a representation c
the intersection of all $ \Sigma _{i} $
30 KB (4,468 words) - 18:44, 17 December 2019
...coprime, then the minimal surface has $(m-1)(m+n)$ different lines of self-intersection issuing from the singular point in different directions. There is a relatio
3 KB (495 words) - 13:00, 3 October 2014
...zeros of vector fields, fixed points of diffeomorphisms, and the points of intersection of a pair of submanifolds of complementary dimension, symplectic cohomology
...of $L _ { + }$ and $L_{-}$ in $M$, respectively (cf. [[Intersection theory|Intersection theory]]). When $\phi$ is symplectically isotopic to $\operatorname{id}$, $
15 KB (2,211 words) - 07:18, 15 February 2024
to its intersection with the plane $ \Pi ^ \prime $
3 KB (526 words) - 08:08, 6 June 2020
...tive Noetherian ring every ideal has a representation as an incontractible intersection of finitely many primary ideals. Although such a representation is not uniq
3 KB (539 words) - 18:51, 3 April 2024
...cf. also [[Divisor|Divisor]]; [[Intersection index (in algebraic geometry)|Intersection index (in algebraic geometry)]]). The set $N _ { 1 } ( X / S )$ of all the
11 KB (1,780 words) - 14:48, 3 February 2024
is the intersection form of the 3-dimensional homology space on the manifold $ M ^ {6} $,
3 KB (435 words) - 17:53, 20 January 2022
see [[Jordan decomposition|Jordan decomposition]]) and their intersection coincides with the set of all semi-simple elements of the centre of $ G $
and their intersection is the centre of $ G $ .
9 KB (1,402 words) - 11:29, 17 December 2019
...\in S},(f^{A_i})_{f\in F})_{i\in I}$ of $\Sigma$-subalgebras $A_i$, their intersection $A=((s^{A})_{s\in S}, (f^{A})_{f\in F})$ is a $\Sigma$-subalgebra given by
13 KB (2,332 words) - 14:56, 21 April 2013
..._0$-spaces for which all singletons are closed. A $T_0$-space in which the intersection of any number of open sets is open is called a discrete space (in the broad
3 KB (527 words) - 07:33, 24 November 2023
...ing the two conditions: 1) the empty set does not belong to it; and 2) the intersection of two subsets belonging to it contains some third subset belonging to it.
3 KB (602 words) - 17:00, 25 September 2017
in terms of elementary functions. The intersection of $ V _ {3} $
The intersection of $ V _ {3} $
9 KB (1,366 words) - 17:45, 4 June 2020
<TR><TD valign="top">[3]</TD> <TD valign="top"> G.P. Kukin, "Intersection of subalgebras of a free Lie algebra" ''Algebra and Logic'' , '''16''' (1
3 KB (564 words) - 19:53, 15 March 2023
denote, respectively, intersection, difference and projection along the $ ( n+ 1 ) $-
4 KB (529 words) - 22:11, 5 June 2020
Regarding the operations of intersection and union of varieties, defined by the formula
by the application of a finite number of operations of intersection and multiplication [[#References|[1]]].
9 KB (1,421 words) - 08:28, 6 June 2020
remaining within the intersection of the domain $ D $
if the intersection of $ E $
10 KB (1,496 words) - 08:27, 6 June 2020
...ts, which is called a play. By definition, player II wins this play if the intersection $\cap \bar B_i$ of the closures of $B_i$ in $Y$ is contained in $X$. Otherw
3 KB (538 words) - 19:20, 1 January 2021
...ut directly in terms of the cohomological product (or, dually, in terms of intersection of cycles) without referring to the category.
3 KB (507 words) - 09:35, 19 November 2017
...an Abelian variety (cf. [[Picard scheme|Picard scheme]]). The operation of intersection of cycles makes it possible to define a multiplication in the quotient grou
cf. [[Intersection theory|Intersection theory]]).
12 KB (1,874 words) - 06:08, 17 April 2024
is the intersection form of one-dimensional homologies of the surface $ M ^ {2} $.
4 KB (529 words) - 17:40, 5 April 2023
is the intersection of $ E $
be the intersection of $ E $
11 KB (1,648 words) - 16:53, 20 January 2024
...ion. The product of the radius of a parallel by the cosine of the angle of intersection of the surface of rotation with the parallel is constant along a geodesic (
4 KB (581 words) - 18:20, 28 March 2023
...ider for instance a $C^1$ embedding $\gamma: [0,1]\to \mathbb R^2$ and the intersection $V$ of the usual [[Non-measurable set|Vitali set]] of $\mathbb R$ with $[0,
...is called {{Anchor|Purely unrectifiable}} ''purely unrectifiable'' if its intersection with any $k$-dimensional rectifiable set is an $\mathcal{H}^k$-null set.
12 KB (1,962 words) - 17:00, 13 June 2020
...element is non-empty. Each thread is contained in some maximal thread. The intersection of all sets which are elements of a maximal thread $ \xi $
are the points of intersection of the straight line $ AB $
10 KB (1,505 words) - 04:04, 9 May 2022
...ty intersection with the interior domain and if the vertex belongs to this intersection. (See Fig. d.) The sum of the interior angles of a simple $n$-gon is equal
...the boundary or only to the right of it; the domain of a polygon with self-intersection may be on different sides of it.
13 KB (2,146 words) - 22:23, 25 April 2012
The intersection $ k $
...esembling the pairing determined by the [[Intersection index (in homology)|intersection index (in homology)]] in $ ( n + 1 ) $-
13 KB (1,934 words) - 19:26, 17 January 2024
is self-polar, since it is the line of intersection of the polar planes $ EAB $
4 KB (553 words) - 02:25, 16 June 2022
If one permits self-
intersection, then there are <img align="absmiddle" border="0" src="https://www.encyclop
15 KB (1,875 words) - 17:26, 7 February 2011
...s of function germs [[#References|[a6]]], [[#References|[a7]]], and on the intersection homology groups of algebraic varieties with coefficients in a polarizable v
...">[a8]</TD> <TD valign="top"> E. Cattani, A. Kaplan, W. Schmid, "$L^2$ and intersection cohomologies for a polarizable variation of Hodge structure" ''Invent. Math
9 KB (1,281 words) - 12:14, 12 January 2021
...superposition or extension of renewal processes, as well as in high-level intersection problems in various random-path schemes, in critical branching processes, e
3 KB (485 words) - 19:23, 10 April 2024
and the points of intersection $ \alpha $
4 KB (546 words) - 08:06, 6 June 2020
it gives their intersection. For these four values, the $ p $-
4 KB (628 words) - 16:09, 1 April 2020
...ir intersection is strictly smaller than the dimension of each set. If the intersection of two irreducible sets in $ U $
The union of a finite number and the intersection of any family of analytic sets in $ U $
12 KB (1,915 words) - 18:47, 5 April 2020
The intersection of all admissible subgroups containing a given subset $ M $
5 KB (837 words) - 08:04, 6 June 2020
for which the intersection $ U _ {1} \cap U _ {2} $
4 KB (560 words) - 17:46, 4 June 2020
...dimensional. A [[totally separated space]], one in which each point is the intersection of all [[Open-closed set|closed-and-open set]]s including it, is totally di
4 KB (646 words) - 13:05, 3 January 2016
that point of the intersection of $ V $
for which such an intersection exists). (The hypersurface $ V $
9 KB (1,357 words) - 08:06, 6 June 2020
and c) the intersection of $ G _ {i} $
4 KB (680 words) - 19:35, 5 June 2020
dimensional Lebesgue measure of the intersection of the set $ M $
5 KB (645 words) - 21:34, 20 February 2021
...hcal{K} ^ { n }$ such that $K = L + M$. Each summand of $K$ is a non-empty intersection of a family of translates of $K$; the converse is true for $n = 2$. $K$ is
4 KB (596 words) - 15:30, 1 July 2020
...ace|complete metric space]] has a non-empty, and in fact everywhere-dense, intersection
3 KB (517 words) - 17:25, 31 December 2016
...re at $\infty$. Now all circles (lines) either touch or have two points of intersection.
4 KB (597 words) - 18:10, 1 June 2023
...{B} $ respectively. If $ D_{A} \cap D_{B} \neq \varnothing $, then on this intersection, we can define the operator $ (\alpha A + \beta B)(x) \stackrel{\text{df}}{
4 KB (675 words) - 04:55, 1 March 2017
...2$ is Borel. And if $T$ is σ-compact (in particular, if $T=\R^n$) then the intersection operation is Borel.
11 KB (1,769 words) - 05:54, 7 April 2018
ii) the intersection of two subspaces is a subspace;
4 KB (654 words) - 08:06, 6 June 2020
is closed relative to union, multiplication, intersection with regular sets, truncated iteration, unabbreviated homomorphisms, and in
4 KB (595 words) - 19:39, 5 June 2020
is an imbedding; 2) the self-intersection of $ \phi $
4 KB (597 words) - 08:26, 6 June 2020
classes of algebraic groups have a trivial intersection: If an
4 KB (665 words) - 18:32, 10 December 2015
...lues of infinite multiplicity, of the Weyl spectrum, which is equal to the intersection of the spectra of all compact perturbations, etc.
5 KB (744 words) - 08:22, 6 June 2020
the intersection
4 KB (674 words) - 16:10, 1 April 2020
The tangent to the line of intersection of any surface $ \omega = \textrm{ const } $
4 KB (653 words) - 07:55, 25 April 2022
...ifold. Then a heteroclinic (homoclinic) point is any point which is in the intersection of the stable manifold of one invariant point and the unstable manifold of
4 KB (662 words) - 21:17, 23 November 2018
is the intersection of the quadrics and cubics in $ P ^ {g-} 1 $
4 KB (609 words) - 08:02, 6 June 2020
the intersection of all congruences $ \theta $
5 KB (767 words) - 14:54, 7 June 2020
...s in $A$ coincides with the intersection of all maximal right ideals; this intersection is called the radical of the algebra $A$ and is denoted by $\rad A$. An ele
14 KB (2,346 words) - 22:48, 29 November 2014
...$P\subset\mathbf{R}^n$ is polyhedral (and necessarily convex) if it is the intersection of finitely many closed half-spaces, say
In each case, the dual system uses the intersection $C^* \cap L^\perp$, where $L^\perp$ is the orthogonal complement of $L$. Fo
12 KB (1,986 words) - 19:13, 31 March 2017
is defined as the intersection of all boundaries $ S $
5 KB (677 words) - 13:07, 7 April 2023
...Signature|signature]] of the manifold, i.e. the signature of the quadratic intersection form defined on $ H _ {n/2} ( M) $,
5 KB (680 words) - 08:07, 6 June 2020
is not determined by a single integer. [[Intersection homology|Intersection homology]] plays an analogous role vis à vis stratified Morse theory as or
then the intersection homology group $ \mathop{\rm IH} _ {i} ( A , B ) $
21 KB (3,095 words) - 08:01, 6 June 2020
is the line of intersection of these two planes, while the quadric $ Q _ {1} $
...tive coordinates of the given planes. In case the $ ( n - 2 ) $-plane of intersection of the two given planes intersects $ T _ {0} $,
8 KB (1,103 words) - 14:58, 2 May 2023
having non-empty intersection with $ A $.
5 KB (692 words) - 10:54, 21 June 2020
is the intersection of all germs of complex-analytic sets containing $ V _ {a} $;
5 KB (671 words) - 16:15, 11 February 2022
...led the solid tangent cone.) In other words, $V(O)$ is the boundary of the intersection of all half-spaces containing $S$ and defined by the [[Supporting hyperplan
4 KB (653 words) - 13:02, 17 June 2016
...^ { + }$ are well-defined covers the whole of $\mathbf{R} ^ { 3 }$. In the intersection of these regions ($0 < \theta < \pi$) the vector potentials $\mathbf{A} ^ {
...n hemisphere including the equator ($\theta < \pi / 2 + \epsilon$). The intersection $H _ { + } \cap H _ { - }$ is parametrized by the azimuthal angle $\phi$. I
10 KB (1,478 words) - 17:30, 25 November 2023
that satisfies the condition that the intersection $ G _ \alpha \cap G _ \beta $
5 KB (883 words) - 16:10, 1 April 2020
The intersection of the sets obtained in this way is the Sierpiński carpet (Fig. b).
the intersection of which is a one-dimensional continuum $ M $,
19 KB (2,971 words) - 19:30, 26 March 2023
...the circle, the product of the distances from that point to both points of intersection of each secant with the circle is a constant number (for the given point);
4 KB (771 words) - 13:05, 27 September 2014
...his procedure indefinitely, one obtains a decreasing sequence of tori. Its intersection is called the $ \mathbf n $-
4 KB (631 words) - 08:14, 6 June 2020
...k } ( R )$ or $\mathfrak{D}$ necessarily forces $R$ to be a local complete intersection. It is not known (as of 2000) whether this is true if $\mathfrak{D}$ is pro
5 KB (705 words) - 16:52, 1 July 2020
the level line has self-intersection, and decomposes into non-intersecting simple closed curves; for sufficientl
6 KB (963 words) - 14:03, 18 May 2023
It is conjectured that the intersection of the image of the moduli space and the Kummer variety of an indecomposabl
is a Jacobian. Similarly for the intersection of the Kummer variety and the tangent space to the moduli space at the poin
11 KB (1,642 words) - 14:55, 7 June 2020
[[Filter|filter]] (that is, for any $V_1,V_2$ the intersection $V_1\cap
has a cluster point (that is, a point lying in the intersection of the
16 KB (2,875 words) - 21:57, 12 October 2014
On the intersection of three neighbourhoods $ u _ \alpha \cap u _ \beta \cap u _ \gamma
6 KB (989 words) - 19:40, 5 June 2020
...ious constructions and theorems, FANR spaces must be pointed. E.g., if the intersection of two pointed FANR spaces is a pointed FANR, then their union is also a po
4 KB (699 words) - 15:31, 1 July 2020
...hematical language. Similarly, in the theory of filters on a given set the intersection of all non-empty elements of the filter determines a non-standard element;
5 KB (677 words) - 23:47, 29 April 2012
A monodromy transformation preserves the intersection form on $ H ^ {n} ( X _ {t} ) $.
relative to the intersection form on $ H ^ {n} ( X _ {0} ) $,
14 KB (1,947 words) - 08:01, 6 June 2020
In the intersection of the coordinate neighbourhoods, $ dx ^ {i ^ \prime} = ( {\partial x ^
on each coordinate neighbourhood which transforms in the intersection of two neighbourhoods according to formula (5). The system $ \Gamma _ {jk
15 KB (2,221 words) - 10:58, 17 March 2023
with a non-empty intersection $ D _ {0} $
...e Dirichlet problem. Schwarz's method may be employed for the union or the intersection of any finite number of domains.
17 KB (2,472 words) - 08:26, 21 March 2022
...if for any complex line $\operatorname{l} \subset \mathbf{C} ^ { n }$ the intersection $E \cap \bf l$ is both connected and simply connected (meaning that its com
...., compact $\mathbf{C}$-convex sets with non-connected interior. Also, the intersection of two $\mathbf{C}$-convex sets is not necessarily $\mathbf{C}$-convex.
16 KB (2,533 words) - 09:46, 18 February 2024
...and 2, 4, 6 lie on two straight lines (are collinear), then the points of intersection of the pairs of lines $(1,2)$ and $(4,5)$, $(2,3)$ and $(5,6)$, $(3,4)$ and
5 KB (796 words) - 21:08, 11 April 2014
the intersection $ \overline{c}\; _ {n} \cap \overline{c}\; _ {n + 1 } $
5 KB (847 words) - 22:16, 5 June 2020
...he calculation. A very simple difference grid is obtained as the points of intersection of straight lines parallel to the axes of a Cartesian coordinate system (a
5 KB (759 words) - 10:59, 16 April 2014
...y is connected with "genericity" and with the "good" properties of the intersection $ L \cap M $,
5 KB (780 words) - 08:26, 6 June 2020
of the zero variation of the intersection of $ E $
7 KB (1,016 words) - 08:27, 6 June 2020
...and is a [[Normal subgroup|normal subgroup]] of the group of motions. The intersection of these two subgroups is the unit of the group of motions. The quotient gr
6 KB (940 words) - 09:43, 11 August 2014
...o be defined using other operations that preserve regularity (for example, intersection, taking complements, etc.), and also by presenting the set of words derivab
5 KB (830 words) - 14:27, 12 November 2023
...ems concern, e.g., finding the convex hull of a set of points, finding the intersection of two convex sets, constructing the Voronoi diagram of a set of points in
5 KB (738 words) - 19:10, 17 December 2015
is the self-intersection index and $ p _ {g} (X) $
5 KB (747 words) - 06:29, 30 May 2020
vanish on the intersection $ M = \cap _ {i} \{ {( \widetilde{x} , \widetilde{p} ) } : {\widetilde{
6 KB (864 words) - 17:45, 4 June 2020
such that the intersection (product) $ D _ {a _ {1} } \cdot \dots \cdot D _ {a _ {n} } $
5 KB (790 words) - 19:40, 5 June 2020
is the intersection index in $ X \times X $
5 KB (732 words) - 08:12, 21 January 2024
...n of almost-periodicity of the solution is related to the structure of the intersection of the spectrum of $ A $
...erating operator of a bounded strongly-continuous semi-group and the above intersection is countable, then for the almost-periodicity of a bounded solution defined
26 KB (3,943 words) - 20:22, 16 January 2024
that have non-empty intersection with $ A $.
5 KB (775 words) - 08:13, 5 May 2022
...to stress their similarity to the set-theoretical operations of union and intersection. The notation $ \overline{x}\; , x ^ \prime $
coincide with union and intersection, respectively.
14 KB (2,077 words) - 21:17, 17 January 2021
closed under intersection, such that $ x _ {0} \in \cap _ \lambda U _ \lambda $;
6 KB (829 words) - 05:44, 13 April 2023
to be a regular boundary point it is necessary and sufficient that in the intersection $ U _ {0} = U \cap D $
5 KB (799 words) - 09:19, 6 January 2024
...cal signature . Their affine difference appears in the type of the line of intersection with the improper plane: an ellipsoid intersects it in an imaginary oval, a
19 KB (2,446 words) - 17:35, 31 March 2012
denote the real algebraic variety defined as the intersection system
C) For a regular complete intersection, if $ n $
16 KB (2,200 words) - 08:30, 14 January 2024
at that point. Here "tangent" means that the intersection of $ {\mathcal O} $
5 KB (839 words) - 17:04, 16 March 2023
...transformation]]) and of semi-universal deformations of isolated complete intersection singularities [[#References|[a2]]], [[#References|[a3]]]. In the latter cas
5 KB (728 words) - 08:27, 6 June 2020
...D}$ with $\lambda > 1$ and $s > 2$ in which every line (that is, the intersection of all blocks through two given points) meets every non-parallel block is i
5 KB (820 words) - 11:00, 17 March 2023
The intersection of domains of holomorphy is a domain of holomorphy; any [[Biholomorphic map
6 KB (977 words) - 19:36, 5 June 2020
given this imbedding, is the intersection with $ X ^ {n} $
6 KB (839 words) - 11:24, 21 March 2022
...erstands the intersection of a finite number of closed half-spaces if this intersection is bounded. Local finiteness of the family means that each point in $\R^n$
12 KB (1,899 words) - 20:28, 25 April 2012
stands for the intersection $ I \cap T $.
6 KB (836 words) - 14:55, 7 June 2020
...en two intersecting curves having a well-defined tangent in their point of intersection one understands the angle formed by these tangents.
6 KB (1,009 words) - 11:05, 4 December 2016
...f a [[conical surface]], and is sometimes also called a cone. Finally, the intersection of $K$ with a half-space containing $0$ and bounded by a plane not passing
...eq 0$ for $x\in K$); if $M$ is a vector subspace of $E$ having a non-empty intersection with the interior of $K$ and $f$ is a linear form on $M$ that is positive o
13 KB (2,188 words) - 13:05, 10 June 2016
...">[a4]</TD> <TD valign="top"> E. Cattani, A. Kaplan, W. Schmid, "$L^2$ and intersection cohomologies for a polarizable variation of Hodge structure" ''Invent. Math
6 KB (883 words) - 18:39, 2 January 2021
closed under intersection and such that every open set in $ S $
6 KB (936 words) - 19:36, 5 June 2020
...is led to the algebraic problem of representing an ideal in the form of an intersection of ideals of simpler structure. This problem was solved by E. Lasker (see [
...iating the dimension theory of Noetherian rings, and also a theorem on the intersection of the powers of an ideal in a Noetherian ring, which is the basis of the s
16 KB (2,400 words) - 17:45, 4 June 2020
...t and is closed under the set-theoretic operations of finite union, finite intersection and taking complements, i.e. such that
6 KB (884 words) - 09:36, 16 August 2013
has non-empty intersection with $ M $.
7 KB (1,085 words) - 22:12, 5 June 2020
are the points of intersection of the straight line $ XY $
7 KB (1,055 words) - 10:04, 21 March 2022
and the point of intersection $ \mathbf X $
6 KB (942 words) - 17:46, 4 June 2020
...ly of lines $x_2={\rm const}$ (more precisely, the connected components of intersection of the leaves with $U$ have this equation). A foliation is called orientabl
6 KB (918 words) - 17:18, 9 January 2013
...tical relations: of being contained one in one another, of having an empty intersection or a non-empty symmetric difference. Let $ m _ {f} $
has a non-empty intersection with at least one interval corresponding to a conjunction of $ S _ {k-1}
22 KB (3,328 words) - 08:31, 26 March 2023
...All recursively-enumerable sets, together with the operations of union and intersection, form a [[Lattice|lattice]] $ {\mathcal E} $.
...ing finite (recursively-enumerable) sets, each of which also has non-empty intersection with the complement of $ A $.
16 KB (2,323 words) - 08:10, 6 June 2020
if the intersection
and if the intersection of the chordal cluster sets, $ \cap C ( f, \zeta ; h ( \zeta , \phi )) $
25 KB (3,728 words) - 09:43, 26 March 2023
and of any finite flat extension. A complete intersection of a Cohen–Macaulay ring $ A $,
7 KB (1,158 words) - 17:45, 4 June 2020
...zero. The intersection index of divisors is identical with the topological intersection index of cycles. The following equalities are valid: $ \sigma $
by the intersection index equals $ 2p _ {g} (V) + 1 $.
26 KB (3,736 words) - 13:08, 8 February 2020
field $K$, then their intersection $M\cap N$ is a subfield in $K$; also,
6 KB (929 words) - 00:28, 18 May 2013
...while the values of connectives and quantifiers are: for the conjunction — intersection; for the existential quantifier — taking the least upper bound, etc. Clas
7 KB (1,091 words) - 18:50, 14 August 2014
...t to the operations of permutation and identification of arguments, union, intersection, direct product, and projection (the projection of a set $ \mathfrak M $
7 KB (998 words) - 19:35, 5 June 2020
...a free product of two of its subgroups with as amalgamated subgroup their intersection, viz. the subgroup of linear affine transformations and the subgroup of tri
6 KB (923 words) - 18:31, 12 December 2019
...any questions to non-singular varieties, and in studying these one can use intersection theory and the apparatus of differential forms.
7 KB (965 words) - 08:11, 6 June 2020
...The knots are located as follows. The first group of knots consists of the intersection points of the circle of radius $ a $,
...at the origin, with the coordinate axes. The second group consists of the intersection points of the circle of radius $ b $,
17 KB (2,639 words) - 04:38, 29 December 2021
...n a straight line which is perpendicular to the radical axis. The point of intersection of the line of the centres of a pencil and its radical axis is called the c
Any pencil of circles is the intersection of two nets (cf. [[Net|Net]]) of circles.
20 KB (3,056 words) - 08:25, 6 June 2020
...ylinder. The simplest kind of convex body is a convex polyhedron, i.e. the
intersection of a finite number of half-spaces. The surface of a convex polyhedron is co
...yimages/c/c026/c026390/c026390134.png" /> (the supporting half-space). The
intersection of all supporting half-spaces at <img align="absmiddle" border="0" src="htt
34 KB (5,010 words) - 17:24, 7 February 2011
...al operators use essentially three ingredients from set theory, namely set intersection, union, complementation. In the Euclidean case, translation is added to the
6 KB (839 words) - 07:59, 6 June 2020
are the coordinates of the point of intersection of the pair of straight lines and $ \alpha _ {1} $
are the coordinates of the points of intersection of the straight lines, it is equal to
28 KB (4,292 words) - 08:45, 1 July 2022
...roper Lorentz transformations and form the proper Lorentz group $L_+$. The intersection of $L_\uparrow$ and $L_+$ is often simply called the Lorentz group.
6 KB (942 words) - 13:30, 10 August 2014
...athbb R^3$ be an open neighborhood of $x_0$ and $U\subset \mathbb R^2$ the intersection of an open neighborhood of $0\in \mathbb R^2$ with the closed upper half pl
5 KB (824 words) - 09:41, 29 January 2014
of the intersection of the interior of the characteristic cone of a linear hyperbolic system
6 KB (898 words) - 14:30, 8 January 2022
An '''intersection complement''' of $N$ is a submodule $C$ such that $C \cap N = \{0\}$ and $C
7 KB (1,130 words) - 20:30, 15 November 2023
is exceptional if and only if the intersection matrix $ ( A _ {i} A _ {j} ) $
7 KB (990 words) - 19:38, 5 June 2020
...ages/c/c025/c025650/c02565060.png" /> (which is necessarily defined in the
intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
26 KB (3,622 words) - 17:12, 7 February 2011
quadric that is the intersection of an $ n $-
9 KB (1,339 words) - 22:17, 5 June 2020
The intersection of cellular algebras is again a cellular algebra. For each set of matrices
7 KB (1,000 words) - 16:48, 14 January 2021
is positive outside the intersection of $ U $
7 KB (1,087 words) - 19:43, 5 June 2020
...Omega \subset {\bf C} ^ { n }$ be a $\mathbf{C}$-convex domain (i.e. every intersection of $\Omega$ with a complex affine line is connected and simply connected, c
7 KB (968 words) - 16:56, 1 July 2020
Very remarkable is Morley's theorem: The points of intersection of the adjacent trisectors of the angles of any triangle form the vertices
7 KB (1,026 words) - 14:54, 7 June 2020
...ided ideals in which it is contained; any closed left ideal in $A$ is the intersection of the maximal regular left ideals in which it is contained.
15 KB (2,316 words) - 21:05, 20 April 2012
...he length of the $i$-th black interval and $a_i,\beta_i$ are the angles of intersection of $0x$ with $\partial U$ at the ends $b_i$, for different $i$, are indepen
7 KB (1,051 words) - 09:27, 22 August 2014
which is a ring of sets, i.e. contains the intersection and the union of any two elements in it. The base $ \mathfrak B $
an (open) base such that the intersection of the boundaries of any of its $ n + 1 $
25 KB (3,639 words) - 09:53, 26 March 2023
be the intersection of all completions of $ {\mathcal N} $
7 KB (1,069 words) - 19:40, 5 June 2020
...e defines the middle and right kernels. They always exist in a loop. Their intersection is called the kernel of the loop. Every kernel is an associative subloop, t
8 KB (1,291 words) - 06:59, 30 March 2024
and the disjoint intersection of $ A, B \subseteq S $
are "non-positively correlated" with respect to the disjoint intersection $ A \square B $.
18 KB (2,717 words) - 20:14, 8 December 2023
there exists an arbitrarily small translation of one of them such that their intersection will also be a submanifold (see [[#References|[2]]], [[#References|[4]]], a
9 KB (1,400 words) - 11:09, 26 March 2023
such that the image of the intersection $ E \cap U $
to the intersection of the set $ E $
32 KB (5,224 words) - 19:36, 25 March 2023
in a locally convex space is the intersection of all closed half-spaces containing it. A non-empty closed convex subset
is a [[Baire space|Baire space]] in the induced topology (that is, the intersection of any sequence of open subsets of $ \partial _ {e} K $
18 KB (2,674 words) - 22:17, 5 June 2020
respectively, and the intersection of the $ a $-
8 KB (1,265 words) - 17:31, 5 June 2020
...et|Cantor set]] which puts zero on each gap of the set and $2^{-n}$ on the intersection of the set with the interval of generation $n$ (such measure is also the [[
6 KB (957 words) - 07:56, 15 December 2016
...#References|[2]]]). Another nilpotent ideal has also been considered — the intersection of the kernels of the irreducible finite-dimensional representations (the n
10 KB (1,457 words) - 18:48, 13 January 2024
and if its self-intersection index $ ( Y \cdot Y ) $
8 KB (1,076 words) - 07:58, 21 January 2024
...[Isomorphism]]) to the [[quotient group]] $F(X)/N(R)$, where $N(R)$ is the intersection of all [[normal subgroup]]s of $F(X)$ containing $R$. The subgroup $N(R)$ i
6 KB (1,025 words) - 08:55, 29 March 2024
...ther differently from subspaces: this is most evident in the fact that the intersection of any number of dense sublocales of a given locale is dense. It is this pr
7 KB (1,068 words) - 19:43, 23 December 2023
...gy|cohomology]] of singular spaces, [[Hodge structure|Hodge structure]] on intersection cohomology, singularity theory (cf. [[Singularities of differentiable mappi
An example of a perverse sheaf is the intersection cohomology complex $ I C _ {Y} ^ { \bullet } $,
24 KB (3,511 words) - 07:03, 10 May 2022
topology. (A set is said to be massive (generic) if it is the intersection of a countable number of open dense subsets. A property is called generic i
In the study of the self-intersection of the image of a smooth mapping, the concept of a multi-jet is used. Let
30 KB (4,632 words) - 08:14, 6 June 2020
denotes the self-intersection number. Every Abelian surface admits an ample invertible sheaf and hence is
of two elliptic curves with intersection number $ ( E _ {1} \cdot E _ {2} ) = 1 $,
18 KB (2,511 words) - 06:25, 26 March 2023
...cal{D}$ with $\mu > 1$ and $s > 2$ in which every line (that is, the intersection of all blocks through two given points) has cardinality $s$ is isomorphic t
7 KB (1,085 words) - 16:58, 1 July 2020
...by strings of zeros and ones. Consider, for example, some regular (without intersection of edges) imbedding of a tree $ D $
6 KB (1,006 words) - 06:20, 30 March 2023
non-empty, the following holds in this intersection:
7 KB (1,091 words) - 19:19, 20 January 2022
there is an intersection pairing $ \cap $:
is given by "intersection with the fundamental class" , that is, it has the form $ x \rightarrow z
23 KB (3,297 words) - 19:41, 5 June 2020
..., the author of [[#References|[1]]]) if a set is closed if and only if its intersection with every compact subspace is closed. The $k$-spaces are precisely the quo
8 KB (1,267 words) - 05:59, 22 April 2023
the set of intersection numbers of blocks in a tight $ t $-design contains exactly $ s $
9 KB (1,332 words) - 02:59, 21 March 2022
...structure constants $p _ { i,j } ^ { k }$ of $W$ are sometimes called the intersection numbers of $\mathfrak{M}$. They have the following combinatorial interpreta
7 KB (1,092 words) - 17:44, 1 July 2020
...uages as sets, one may immediately define the Boolean operations of union, intersection, complementation (the complementation of a language $ L $
...dable whether the intersection of two context-free languages is empty. The intersection of a context-free and a regular language is always context-free; and, hence
28 KB (4,202 words) - 08:53, 1 February 2020
is a complete intersection at $ x $
8 KB (1,143 words) - 16:48, 19 February 2022
...f a set $M\subset X$ as a point every neighbourhood of which has non-empty intersection with $M$. This definition implies that every point of the set $M$ itself is
...[M_2]$; and 3) $[\,[M]\,]=[M]$. The closure of an arbitrary set $M$ is the intersection of all closed sets containing $M$; alternatively, $[M]$ is the smallest clo
21 KB (3,431 words) - 05:43, 18 December 2016
...al unitary group. The unitary symplectic group ${\rm USp}_{2m}(\C)$ is the intersection (in ${\rm GL}_{2m}(\C)$) of the
6 KB (1,078 words) - 14:22, 3 November 2013
...collection of group elements that commute with all subgroups, that is, the intersection of the normalizers of all subgroups (cf. [[Normalizer of a subset]]). The n
8 KB (1,284 words) - 21:49, 6 June 2016
...d on the planarity of the Earth's equator and the vernal equinox (i.e. the intersection line of the Earth's equator with the plane of the ecliptic), are not inerti
8 KB (1,204 words) - 18:48, 5 April 2020
The intersection of two distinct dimensional components of an $ n $-
9 KB (1,337 words) - 06:29, 30 May 2020
is the set-theoretic complete intersection of ample divisors on $ X $,
9 KB (1,321 words) - 17:45, 4 June 2020
can be obtained as the intersection of a root system of type $ E _{8} $
can be obtained as the intersection of a root system of type $ E _{8} $
22 KB (3,351 words) - 19:14, 21 December 2019
...7,601,000 six-nodal plane quintics on <math>F</math>, each of which is the intersection of <math>F</math> with a sixfold tangent plane. A proof that the correspond
*[Ka2] S. Katz, ''Lines on complete intersection threefolds with <math>K=0</math>'', Math. Z. '''191''' (1986), 293–29
14 KB (2,200 words) - 18:27, 23 October 2017
...sign $\mathcal{D}$ with $\lambda > 1$ in which every line (that is, the intersection of all blocks through two given points) meets every block is isomorphic to
7 KB (1,152 words) - 16:45, 1 July 2020
elements has empty intersection.
13 KB (2,001 words) - 02:12, 1 March 2022
...cyimages/c/c026/c026400/c0264002.png" />. A set is convex: 1) if it is the intersection of open half-spaces; or 2) if it contains with two arbitrary points the seg
15 KB (2,101 words) - 17:13, 7 February 2011
...e are spaces that are not zero-dimensional and in which every point is the intersection of a family of open-and-closed sets, but no such space can be compact.
8 KB (1,230 words) - 13:00, 3 January 2016
to the point of intersection of the trajectory of (1) through $ \xi $
9 KB (1,244 words) - 08:05, 6 June 2020
...amiltonian system, linear|Hamiltonian system, linear]]) and having a large intersection with the theory of linear systems, the theory of Lyapunov characteristic ex
9 KB (1,252 words) - 10:41, 15 April 2014
...<TD valign="top"> J.L. Colliot-Thélène, J.J. Sansuc, P. Swinnerton-Dyer, "Intersection of two quadrics and Chatelet surfaces" ''J. Reine Angew. Math.'' , '''374''
18 KB (2,685 words) - 08:57, 18 August 2022
...be the rank of the positive part of the intersection form $q$ on $M$; the intersection form being defined by
16 KB (2,663 words) - 10:57, 13 February 2024
A tiny area, known as a curl point, is formed at the intersection of a curl thread with the plane $ XOY $.
7 KB (1,148 words) - 18:18, 21 June 2020
The intersection of all varieties of signature $ \Omega $
10 KB (1,438 words) - 16:10, 1 April 2020
...thod for calculating the orthogonal projection $P _ { U \cap V }$ onto the intersection of two closed subspaces $U$ and $V$ in a [[Hilbert space|Hilbert space]] $H
8 KB (1,270 words) - 16:56, 1 July 2020
...n of definition from the mapping described here. The mappings agree on the intersection of their domains, and both are defined in the important case when $C _ { G
9 KB (1,465 words) - 17:46, 1 July 2020
...ation groups. Let $(G,X)$ and $(H,Y)$ be two permutation groups with empty intersection $X\cap Y$. The sum $(G,X)+(H,Y)$ is defined as the permutation group consis
9 KB (1,519 words) - 07:41, 30 November 2014
...structure of $4$-manifolds. Donaldson first discovered restrictions on the intersection form of a compact $4$-manifold and deduced that certain topological $4$-man
7 KB (1,126 words) - 17:45, 1 July 2020
...n Euclidean, metric or topological space). These operations include union, intersection, taking a complement or projection, etc. Descriptive set theory was created
...tions. The same applies to the operations of countable union and countable intersection. One of the key theorems of the general theory of operations over sets in t
44 KB (6,667 words) - 11:40, 26 March 2023
...cyclopediaofmath.org/legacyimages/h/h046/h046420/h046420365.png" /> is the
intersection of the maximal ideals in which it is contained, one says that <img align="a
66 KB (9,085 words) - 17:28, 31 March 2020
...e set of these vertices constitutes an abstract simplex if and only if the intersection of the corresponding elements of $\gamma$ is not empty. Systems of open cov
10 KB (1,606 words) - 07:49, 17 May 2017
is the intersection of $ G _ {2n,n } ( K ) $
9 KB (1,440 words) - 16:43, 4 June 2020
of an arithmetic nature, i.e., closely related to the Gillet–Soulé arithmetic intersection pairing (on $ X _ {\mathbf Z} $),
and generalizing Arakelov's intersection pairing on arithmetic surfaces.
25 KB (3,559 words) - 19:33, 7 February 2024
...of a space metrizable by a complete metric is the [[Baire property]]: The intersection of any countable family of everywhere-dense open sets is everywhere dense.
9 KB (1,484 words) - 08:00, 6 June 2020