The geometric notion meaning that at some point two curves (or a curve and a surface) hav
See also [[Tangent vector|Tangent vector]].
461 bytes (75 words) - 17:23, 7 February 2011
$#C+1 = 32 : ~/encyclopedia/old_files/data/V096/V.0906340 Vector,
''geometric''
4 KB (734 words) - 08:28, 6 June 2020
that associates with a differentiable vector field $ X $
and a differentiable geometric object $ Q $
9 KB (1,337 words) - 22:16, 5 June 2020
''of an operator (transformation) $A$ of a vector space $L$ over a field $k$''
An element $\lambda\in k$ such that there is a non-zero vector $x\in L$ satisfying the condition
2 KB (373 words) - 09:18, 12 December 2013
...is called Euclidean if each of its fibres has the structure of a Euclidean vector space with a scalar product $ \langle , \rangle $
is a smooth function on the base. A linear connection on a Euclidean vector bundle is called a Euclidean connection if for any parallel displacement of
2 KB (251 words) - 19:38, 5 June 2020
$#C+1 = 17 : ~/encyclopedia/old_files/data/D030/D.0300210 Darboux vector
The vector $ \pmb\delta $
2 KB (352 words) - 17:32, 5 June 2020
...ocity vector|Phase velocity vector]]) is defined, is non-zero and is not a vector tangent to the curve. The concept was introduced by H. Poincaré [[#Referen
...="top">[2]</TD> <TD valign="top"> S. Lefshetz, "Differential equations: geometric theory" , Interscience (1962)</TD></TR></table>
2 KB (281 words) - 18:48, 5 April 2020
...concentration of its probability distribution around a certain prescribed vector in terms of the second-order moments. Let $ X $
be a random vector assuming values $ x = ( x _ {1}, \dots, x _ {n} ) ^ {T} $
3 KB (375 words) - 16:43, 20 January 2022
Let a geometric object $ \Omega $(
cf. [[Geometric objects, theory of|Geometric objects, theory of]]) be given in a differentiable manifold $ X _ {n} $,
5 KB (725 words) - 19:35, 5 June 2020
...f $K$ along any non-zero integer vector. Here the "width" of $K$ along a vector $v$ in $\mathbf{R}^n$ is
The width of $K$ with respect to $\mathbf{Z}^n$ is greater or equal than the geometric width of $K$, which is the minimum width of $K$ along all unit-length vecto
1 KB (242 words) - 21:16, 8 April 2018
Adjoint norms (cf. [[Norm|Norm]]) in certain vector spaces dual to each other.
vector $ \alpha $,
4 KB (548 words) - 07:59, 6 June 2020
...cyclopediaofmath.org/legacyimages/c/c025/c025470/c02547012.png" /> for any
vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath
...02547060.png" /> (and this is equivalent). It is sometimes called the Reeb
vector field of <img align="absmiddle" border="0" src="https://www.encyclopediaofm
11 KB (1,538 words) - 17:29, 7 February 2011
...beginning of the 20th century. It differs from local geometry, in which a geometric object (field, mapping) is studied in sufficiently small domains only, as i
...ative results, as well as quantitative relations in the large, for regular geometric structures on multi-dimensional singularity-free manifolds.
5 KB (702 words) - 16:57, 15 April 2012
is a differential-geometric structure on $ M $
...ant differentiation|covariant differentiation]], which associates with two vector fields $ X , Y $
6 KB (870 words) - 22:17, 5 June 2020
...nifold similar to a [[Sasakian manifold]]. However, by its topological and geometric properties, such a manifold is closely related to a product manifold (unlik
...$(M,g)$ endowed with an endomorphism $\phi$ of the tangent bundle $TM$, a vector field $\xi$ and a $1$-form $\eta$ which satisfy the conditions
3 KB (562 words) - 15:54, 16 March 2024
...lidean space]] of signature $(1,3)$, suggested by H. Minkowski (1908) as a geometric model of space-time in the special theory of relativity (see ). Correspondi
...e is a time-like vector. The tangent vector to a light ray is an isotropic vector.
5 KB (692 words) - 19:18, 12 April 2014
A geometric object described in a coordinate system $ x = ( x ^ {1}, \dots, x ^ {n}
the tensor density is a tensor (cf. [[Tensor on a vector space|Tensor on a vector space]]). Concepts such as type, valency, covariance, contravariance, etc.
2 KB (350 words) - 10:34, 5 March 2022
...ping|Transversal mapping]]); a concept in linear algebra, differential and geometric topology.
a) Two vector subspaces $ A, B $
5 KB (780 words) - 08:26, 6 June 2020
...stable vector bundles (of fixed rank and degree; cf. also [[Vector bundle|Vector bundle]]) over a given [[Riemann surface|Riemann surface]] $X$ of genus $g\
...#References|[a5]]]; and quantized Hitchin systems with applications to the geometric Langlands program [[#References|[a2]]].
5 KB (752 words) - 15:33, 4 October 2014
''of a set $M$ in an $n$-dimensional vector space''
In general vector spaces, where a hyperplane can be defined as a domain of constant value of
2 KB (331 words) - 04:59, 16 September 2014
A field of geometric quantities on a homogeneous space $ M = G / H $
is a [[Vector bundle|vector bundle]]. In this case, $ H $
8 KB (1,274 words) - 22:13, 5 June 2020
in a finite-dimensional vector space $ V $
and any non-zero vector $ v \in V $
6 KB (818 words) - 12:00, 16 December 2019
...[[Non-linear operator|Non-linear operator]]) between infinite-dimensional vector spaces and also certain classes of non-linear spaces and their mappings. Th
1) Differential calculus of non-linear mappings between Banach, topological vector and certain more general spaces, including theorems on the local inversion
4 KB (490 words) - 17:11, 7 February 2011
...ined as a certain set of objects, which are called its points; they can be geometric objects, functions, states of a physical system, etc. By considering such a
2) The "space of events", playing an important part in the geometric interpretation of relativity theory. Every event is characterized by its po
4 KB (580 words) - 14:27, 28 August 2014
...-geometric structures (cf. [[Differential-geometric structure|Differential-geometric structure]]) on a smooth [[Manifold|manifold]] $ M $
and for each tangent vector $ X _ {x} \in T _ {x} ( M) $
9 KB (1,365 words) - 18:18, 19 January 2021
...ction then one defines a vector field $X_f$ on $S$, called the hamiltonian vector field associated to $f$, by setting
...sely, if $\mathfrak{g}$ is a finite dimensional Lie algebra then its dual vector space $V:=\mathfrak{g}^*$ carries a linear Poisson bracket which is given
6 KB (1,109 words) - 13:50, 12 December 2013
$#C+1 = 46 : ~/encyclopedia/old_files/data/K055/K.0505420 Killing vector,
''more precisely, Killing vector field or infinitesimal motion''
5 KB (766 words) - 22:14, 5 June 2020
...^*$ of the problem of minimizing $f$, the convergence rate being that of a geometric progression with quotient $q<1$.
...this system is equivalent to finding a vector $x^*$ in an $n$-dimensional vector space minimizing the functional
3 KB (519 words) - 19:10, 6 April 2012
is constructed such that every vector $ b _ {i} $ ($ i = 1, \dots, k $)
are obtained from the condition of orthogonality of the vector $ b _ {i+ 1} $
4 KB (542 words) - 08:33, 13 May 2022
several metric concepts of vector algebra can be applied to trivectors. The measure of a trivector $ [ \mat
can be identified with a vector of $ A $,
4 KB (571 words) - 08:58, 10 April 2023
of a space (as a rule, a vector space) $ X $
into a vector space $ Y $
12 KB (1,902 words) - 08:03, 6 June 2020
A [[triple system]] is a [[Vector space|vector space]] $V$ over a field $K$ together with a $K$-[[trilinear mapping]] $V \
A vector space $V$ with triple product $[\,,\,,\,]$ is said to be a Lie triple syste
3 KB (559 words) - 16:43, 15 March 2023
of the real vector space $ \mathbf R ^ {n} $.
of the vector $ x ( p) \in \mathbf R ^ {n} $
7 KB (1,075 words) - 16:43, 4 June 2020
...ry constant. The integral curve is often identified with the solution. The geometric meaning of the integral curves of a scalar equation
the tangent of the angle of inclination of the vector with the $ x $-
3 KB (388 words) - 22:12, 5 June 2020
...Delta : \dim F = i\}\vert$, and call $(f_{-1},f_0,\ldots,f_{d-1})$ the $f$-vector of $\Delta$. Then
...dots,h_d)$, called the $h$-vector of $\Delta$, may be derived from the $f$-vector of $\Delta$ (and vice versa) by the equation
3 KB (544 words) - 20:16, 8 November 2023
The vector bundle consisting of tangent vectors to the ambient manifold that are norma
by the vector bundle $ T _ {Y} $.
4 KB (554 words) - 06:02, 18 April 2023
is the general linear group of some finite-dimensional vector space $ U $ ,
In the modern geometric theory of invariants, by a comitant one often means any equivariant morphis
3 KB (511 words) - 13:16, 7 April 2023
...eferences|[5]]] consists in the fact that the transfer for the fibres of a vector bundle defined by a horizontal distribution ceases to have a linear charact
...g non-linear connections arose from the need to study various differential-geometric structures of higher orders (such as, for example, a [[Kawaguchi space|Kawa
7 KB (1,089 words) - 12:15, 18 February 2022
...regate of methods of finding various versions of [[Geometric approximation|geometric approximation]] to the solution of the corresponding problem.
If one adopts a geometric approximation for the wave equation and equates to zero the coefficients of
7 KB (1,036 words) - 05:39, 19 March 2022
2) the scalar product is distributive with respect to vector addition:
of a vector $ \mathbf a $
9 KB (1,270 words) - 08:54, 13 January 2024
...ure on a manifold, a geometric quantity, a geometric object, or a field of geometric objects, is a section of a bundle associated with the principal bundle of c
Intuitively, a geometric quantity can be considered as a quantity whose value depends not only on th
11 KB (1,705 words) - 14:55, 7 June 2020
...iteria. On the one hand, a non-Euclidean space may be a finite-dimensional vector space with a scalar product expressible in Cartesian coordinates as
...spaces may also be classified from the point of view of their differential-geometric properties as Riemannian spaces of constant curvature (this includes the ca
2 KB (256 words) - 08:02, 6 June 2020
...connection it is possible to proceed directly from a generalization of the geometric foundation of three-dimensional geometry, from certain axiom systems or fro
...more than three dimensions came gradually, primarily on the grounds of the geometric representation of powers: $ a ^ {2} $
12 KB (1,820 words) - 02:24, 21 January 2022
is the sheaf of holomorphic sections of the [[Negative vector bundle|negative vector bundle]] $ L $
for any [[Positive vector bundle|positive vector bundle]] of rank 1 (here $ K _ {X} $
5 KB (719 words) - 22:14, 5 June 2020
* A singular point of vector field of a special type (with all eigenvalues of the linear part being to o
==Node of a vector field==
5 KB (753 words) - 09:03, 12 December 2013
...y by points in the plane and operations on them correspond to the simplest geometric transformations of the plane. It is not possible to "organize" a number s
of the quaternion and its vector part
11 KB (1,563 words) - 14:54, 7 June 2020
is the cell obtained by shifting by a vector $ v $
and any vector $ v $,
5 KB (743 words) - 07:47, 13 May 2022
...) = A$, $x = p ( y )$. So, every vector field $X$ on $M$ is lifted into a vector field $\Gamma X$ on $Y$. The parallel transport on $Y$ along a curve $\gamm
2) As the connection form $T Y \rightarrow V Y$, transforming every vector of $T _ { y } Y$ into its first component with respect to the direct sum de
6 KB (926 words) - 16:46, 1 July 2020
or, in vector notation, $ \dot{w} = f ( w) $,
defined ( "generated" ) by a smooth vector field $ f ( w) $
3 KB (541 words) - 19:39, 5 June 2020
A [[Differential-geometric structure|differential-geometric structure]] on a smooth manifold $ M $,
is fixed, such that the vertex defined by the vector $ e _ {0} $
11 KB (1,486 words) - 17:46, 4 June 2020
be a basis of left-invariant vector fields on $ G $
also constitute a basis of left-invariant vector fields for the Lie subgroup $ H $.
15 KB (2,363 words) - 08:15, 18 August 2022
...he image $\{ p ( t ) : 0 \leq t \leq 1 \}$ of the interval $[0,1]$ under a vector-valued polynomial $p$. The coefficients $a _ { j }$ in that form readily pr
...td> <td valign="top"> G. Farin, "Curves and surfaces for computer aided geometric design" , Acad. Press (1993) (Edition: Third)</td></tr><tr><td valign="to
4 KB (598 words) - 16:55, 1 July 2020
...terior differentiation was developed by Ü. Lumiste [[#References|[a7]]]. A geometric interpretation of the Faddeev–Popov ghost $c$ and anti-ghost $\overline {
It should be noted that all geometric interpretations of the Faddeev–Popov ghosts mentioned above lay aside the
8 KB (1,181 words) - 20:50, 24 January 2021
...athcal{D} ^ { j }$, where $\mathcal{D} ^ { j }$ denotes the [[Vector space|vector space]] of compactly supported differential forms of degree $j$ on $V$ (cf.
See also [[Geometric measure theory|Geometric measure theory]].
5 KB (733 words) - 17:00, 1 July 2020
...lopediaofmath.org/legacyimages/b/b015/b015280/b0152801.png" />-dimensional
vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath
...kes it possible to consider the barycentric coordinates of the points of a
geometric complex with respect to all of its vertices. Barycentric coordinates are us
6 KB (811 words) - 17:11, 7 February 2011
...relations between them, on which the definitions of figures are based and geometric propositions are proved, was recognized in the works of even the ancient ge
...the clearness of the drawing and the physical feasibility of the necessary geometric constructions, and were not strictly logically deduced from the axioms and
25 KB (3,631 words) - 19:39, 5 June 2020
$#C+1 = 125 : ~/encyclopedia/old_files/data/V096/V.0906490 Vector measure
or, more generally, a topological vector space). A vector measure $ F $
10 KB (1,436 words) - 09:13, 6 January 2024
...), and linear, bilinear and quadratic functions (functionals and forms) on vector spaces.
...inear transformation, and a linear, bilinear and multilinear function on a vector space.
9 KB (1,394 words) - 08:15, 9 January 2024
One of the basic geometric concepts. A straight line is usually implicitly defined by the axioms of ge
determine the coordinates of the normal vector of this straight line.
2 KB (335 words) - 08:23, 6 June 2020
...of measures, subject to some regularity conditions. Such translation by a vector space of "arrows" is called an affine structure, and it is taken to be th
in terms of its displacement vector $ l ( x, \theta ) $
10 KB (1,420 words) - 08:23, 6 June 2020
...control system defined on a smooth manifold by a pair of smooth admissible vector fields has the strong accessibility property if this pair belongs to some o
...polysystems and control theory" D.Q. Mayne (ed.) R.W. Brockett (ed.) , ''Geometric Methods in System Theory. Proc. NATO Advanced Study Institute, London, Augu
2 KB (329 words) - 15:33, 20 November 2014
...heory]] of the energy function on $\Omega G$ [[#References|[a6]]]. Certain geometric aspects of Birkhoff stratification may be described in terms of non-commuta
...top"> B. Bojarski, "On the stability of Hilbert problem for holomorphic vector" ''Bull. Acad. Sci. Georgian SSR'' , '''21''' (1958) pp. 391–398</td><
4 KB (630 words) - 17:01, 1 July 2020
A generalization of [[Vector analysis|vector analysis]], a part of [[Tensor calculus|tensor calculus]] studying differen
...re general geometric objects than tensor fields, such as tensor densities, vector-valued differential forms, etc. is considered as a part of tensor analysis.
11 KB (1,607 words) - 08:25, 6 June 2020
...respect to area, volume, etc.). The concept of a derivative is extended to vector-valued point functions in an abstract space (see [[Differentiation of a map
For a geometric and mechanical interpretation of the derivative, the simplest rules of diff
4 KB (596 words) - 11:47, 5 July 2016
A topological space associated with a vector (or sphere) bundle or spherical fibration.
be a vector bundle over a CW-complex $ X $.
6 KB (876 words) - 08:25, 6 June 2020
...the [[Parallel displacement(2)|parallel displacement]] of the plane by the vector $ ( a , b ) $,
and parallel displacement by a vector $ ( a , b ) $.
7 KB (1,100 words) - 03:33, 21 March 2022
are the Fourier coefficients of the vector $ f $
The geometric meaning of Bessel's inequality is that the orthogonal projection of an elem
4 KB (620 words) - 04:37, 9 May 2022
is less than one. If in some norm, compatible with the norm of a vector $ x $,
then the simple-iteration method converges at the rate of a geometric series and the estimate
4 KB (568 words) - 20:21, 10 January 2024
...oncept of a CR-manifold (CR-structure) has been defined having in mind the geometric structure induced on a real hypersurface of $ \mathbf C ^ {n} $,
is involutive, i.e., for any complex vector fields $ U $
6 KB (924 words) - 11:07, 26 March 2023
...e _ { k } D _ { k }$ is $\mu$-ample (cf. also [[Ample vector bundle|Ample vector bundle]]).
...B )$ is a Cartier divisor, and let $d$ be the maximum of the dimensions of geometric fibres of $f$. Express $\lambda / r = p / q$ for relatively prime positive
6 KB (873 words) - 09:30, 3 February 2024
...n+1}$. When $\sigma = 1$, $f$ is called a strict contact transformation. A vector field $X$ on a contact manifold is called a contact (or strict contact) inf
<TR><TD valign="top">[1]</TD> <TD valign="top"> P.K. Rashevskii, "Geometric theory of partial differential equations", Moscow-Leningrad (1947) (In Ru
3 KB (384 words) - 07:34, 13 February 2024
dimensional vector spaces $ T ^ {n} $
...of general type present a [[Differential-geometric structure|differential-geometric structure]] of higher order.
6 KB (908 words) - 19:48, 12 January 2024
be a class of objects in algebraic geometry (varieties, schemes, vector bundles, etc.) on which an equivalence relation $ R $
...continuously depend on parameters; 2) the assignment and study of algebro-geometric structures on the parameter sets. The second part forms the matter of modul
16 KB (2,402 words) - 11:49, 16 December 2019
...ration is to the construction of divergence-free vector fields and related geometric problems.
2 KB (330 words) - 17:00, 1 July 2020
...n the appropriate setting, where non-smooth competitors are allowed (cf. [[Geometric measure theory]]); the setting of Bombieri, De Giorgi and Giusti is that of
2 KB (319 words) - 09:01, 4 November 2013
dimensional vector-columns with these numbers as coordinates is found in the columns of this s
see [[Orthogonal Latin squares|Orthogonal Latin squares]]. In geometric terms, an $ \mathop{\rm OA} ( \lambda n ^ {2} , k, n, 2, \lambda ) $
3 KB (395 words) - 08:04, 6 June 2020
can be regarded as symplectic vector fields on $ M $,
are the values of the vector fields $ X, Y \in \mathfrak g $
7 KB (1,006 words) - 08:24, 6 June 2020
A [[Differential-geometric structure|differential-geometric structure]] on a smooth manifold $ M $,
smooth vector fields, linearly independent at each point $ x $
15 KB (2,221 words) - 10:58, 17 March 2023
...41 : ~/encyclopedia/old_files/data/M063/M.0603250 Measure in a topological vector space
...eral problem encountered in the construction of a measure in a topological vector space is that of extending a [[Pre-measure|pre-measure]] to a measure. Let
7 KB (1,017 words) - 08:00, 6 June 2020
...er curves and Bernstein polynomials is that Bézier curves are a parametric vector-valued representation based on control points, whereas Bernstein polynomial
The usefulness of Bézier curves resides in their many geometric and analytical properties. There are elegant and efficient algorithms for e
5 KB (759 words) - 09:49, 27 March 2023
The position vector $ \mathbf r $
...of a [[Homogeneous space|homogeneous space]]. In an arbitrary differential-geometric homogeneous space $ G / H $
4 KB (613 words) - 22:17, 5 June 2020
...visors, imbeddings of varieties in a projective space, differential forms, vector fields and automorphisms, deformations of varieties and subvarieties, in on
...<TD valign="top">[4]</TD> <TD valign="top"> K. Kodaira, "On a differential-geometric method in the theory of stacks" ''Proc. Nat. Acad. Sci. USA'' , '''39''' (1
4 KB (573 words) - 18:20, 17 April 2017
...of all straight lines $l$ in $\mathbf{R} ^ { 2 }$ as follows: let the unit vector $\theta$ represent the direction of $l$ and let $p$ be its signed distance
...="top">[a3]</td> <td valign="top"> L.A. Santaló, "Integral geometry and geometric probability" , ''Encycl. Math. Appl.'' , Addison-Wesley (1976)</td></tr><t
4 KB (682 words) - 06:33, 10 May 2022
...ath.org/legacyimages/m/m062/m062160/m06216079.png" />. The arithmetic-mean
geometric-mean inequality corresponds to the special case <img align="absmiddle" bord
...2/m062160/m06216098.png" />, extended with zeros if necessary to make up a
vector of length <img align="absmiddle" border="0" src="https://www.encyclopediaof
34 KB (4,631 words) - 18:28, 30 November 2016
of a finite-dimensional [[Vector space|vector space]] $ V $,
...001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E. Artin, "Geometric algebra" , Interscience (1957) {{MR|1529733}} {{MR|0082463}} {{ZBL|0077.021
4 KB (558 words) - 16:18, 6 June 2020
A [[Differential-geometric structure|differential-geometric structure]] on a smooth manifold $ M $;
is fixed, with the vertex determined by the vector $ e _ {0} $
10 KB (1,455 words) - 06:59, 6 March 2024
possesses a global time-like vector field, then $ {} ^ {1} V _ {n} $
...a world-line having non-space-like tangents only. The coupling between the geometric model and the physical data is given by Einstein's field equations (cf. [[E
6 KB (888 words) - 08:08, 6 June 2020
A space with a [[differential-geometric structure]] whose points can be provided with coordinates from some algebra
...a module over the algebra, whose definition can be obtained from that of a vector space over a skew-field by replacing the skew-field by an associative algeb
8 KB (1,172 words) - 16:55, 7 June 2020
The [[Geometric genus|geometric genus]] $ p _ {g} $
...{n} \simeq P ( {\mathcal L} _ {n} ) $ (projectivization of two-dimensional vector bundles over the projective line $ \mathbf P ^ {1} $),
5 KB (707 words) - 16:26, 2 March 2022
...and taking values in the extended real line (or, more generally, a normed vector space), with respect to which general sets in the domain of definition $\ma
...efinitions apply to [[Signed measure|signed measures]] or [[Vector measure|vector measures]] $\mu$: in such cases the assumptions above are required to hold
5 KB (751 words) - 09:47, 16 August 2013
these define a vector in $ \mathbf R ^ {2m } $
and can thus be associated with a column vector $ \mathbf u \in \mathbf R ^ {2m \times 1 } $.
11 KB (1,540 words) - 22:11, 5 June 2020
...invariant way the notions of a derivative and a differential for fields of geometric objects on manifolds, such as vectors, tensors, forms, etc. The basic conce
be a smooth vector field, $ X _ {p} \neq 0 $,
15 KB (2,257 words) - 17:31, 5 June 2020
''of a vector space''
be a [[Vector space|vector space]] over a field $ k $
8 KB (1,152 words) - 08:29, 6 June 2020
...gent vector bundles (cf. also [[Lie algebra|Lie algebra]]; [[Vector bundle|Vector bundle]]; [[Tangent bundle|Tangent bundle]]). For a comprehensive treatment
...r space $\Gamma ( A )$ of smooth global sections of $A$, and a morphism of vector bundles $q _ { A } : A \rightarrow T M$, where $T M$ is the tangent bundle
12 KB (1,916 words) - 16:55, 1 July 2020
...al manifolds in non-commutative algebras. The most remarkable examples are geometric quantization (B. Kostant and J.M. Souriau [[#References|[a11]]], [[#Referen
...non-Lagrangian partial differential equations did fail, until some recent geometric studies on the quantization of partial differential equations [[#References
16 KB (2,356 words) - 19:53, 13 January 2018
...$V \mapsto \operatorname { Hom } _ { k } ( V , k )$ for finite-dimensional vector spaces over a field $k$.
...- 1 - k } ( S ^ { n } \backslash X )$, and this forms the basic intuitive geometric idea behind Spanier–Whitehead duality.
3 KB (432 words) - 16:46, 1 July 2020
...the [[Buffon problem|Buffon problem]] (see also [[Geometric probabilities|Geometric probabilities]]; [[Stochastic geometry|Stochastic geometry]]). The uniform
defined as a vector from the origin to a random point on the surface of the unit sphere, is uni
6 KB (911 words) - 12:00, 22 December 2019
...ngent bundle]]) of $X$. A line bundle $L$ on $X$ (cf. also [[Vector bundle|Vector bundle]]) is said to be nef if the degree of the restriction of $L$ to any
...rem. In the late 19th century, the numbers $h ^ { i } ( L )$ intervened in geometric arguments in much the same way as they intervene today, e.g., [[#References
8 KB (1,253 words) - 15:30, 1 July 2020
...eaf of automorphisms of the trivial vector bundle of rank $ n $. Algebraic vector bundles of rank $ 1 $ are said to be '''line bundles'''; they correspond to
...one can associate the projective bundle $ \mathbf{P}(E) $, just like to a vector space one can associate a [[Projective scheme|projective space]].
14 KB (2,169 words) - 08:12, 14 December 2016
...r bundles, endomorphisms, etc.), with the study of their various algebraic-geometric properties and with compactification techniques for moduli spaces (see [[Mo
...2}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford, "Geometric invariant theory" , Springer (1965) {{MR|0214602}} {{ZBL|0147.39304}} </TD>
5 KB (777 words) - 22:00, 7 July 2014
geometrically means that each infinitely-small vector emanating from the point $ a $
...all vector the direction of which forms with the direction of the original vector the angle $ \alpha $.
5 KB (741 words) - 07:34, 10 April 2023
A singular point $x=0$ of a smooth vector field $v(x)=Ax+\cdots$ on the plane,
two real eigenvalues of different sign. Such vector field has two smooth invariant curves
8 KB (1,340 words) - 09:03, 12 December 2013
The totality of geometric properties of surfaces and curves in a [[Pseudo-Riemannian space|pseudo-Rie
...$-axis of an orthogonal coordinate system in which the scalar square of a vector $ \mathbf x $
3 KB (440 words) - 09:31, 22 June 2022
...t can be easily achieved using the formula above for each component of the vector function.
|valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen
2 KB (382 words) - 23:44, 27 June 2014
Since the vector fields $ d _ {n} = - z ^ {n+} 1 ( d / dz) $(
...extension (which is, in fact, universal) of the Lie algebra of holomorphic vector fields on the punctured complex plane having finite Laurent series. For thi
10 KB (1,539 words) - 11:16, 23 March 2023
...y. These include fibrations and bundles over a space, and the differential-geometric and topological concepts connected with it: connections, $G$-structures, ch
...the development of topology itself by making possible a classification of vector bundles and, subsequently, by producing a method of studying the topologica
9 KB (1,298 words) - 14:59, 30 August 2014
Interest arises in metric spaces of
vector densities with groups of similarities and isometries containing infinite-di
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Artin, "
Geometric algebra" , Interscience (1957) pp. Chapt. II</TD></TR><TR><TD valign="top
3 KB (509 words) - 14:19, 12 April 2014
...ition for maximal minimization of the norm of the discrepancy (or residual vector) $ \xi ^ {k+1} = A u ^ {k+1} - f $;
of (1) with the speed of a geometric progression with multiplier $ q = ( M - m ) / ( M + m ) < 1 $.
3 KB (484 words) - 17:03, 23 December 2020
be a [[Vector bundle|vector bundle]] over a finite [[Cellular space|cellular space]] $ X $,
...ciate with algebraic invariants (homotopy classes of mappings) descriptive geometric forms (certain equivalence classes of manifolds that are pre-images under t
5 KB (763 words) - 14:33, 27 April 2024
Vector interpretation. A complex number $z=x+iy$ can be identified with the
vector $(x,y)$ with coordinates $x$ and $y$ starting from the origin (see
14 KB (2,281 words) - 09:55, 26 March 2023
...and $V$ are subspaces in the lattice of subspaces of a finite-dimensional vector space over a finite field of order $q$ and $U \subseteq V$, then
...Rota has proved the following sign theorem: If $X$ is a rank-$k$ flat in a geometric lattice, then $( - 1 ) ^ { k } \mu ( 0 , X )$ is positive. Indeed, $( - 1 )
11 KB (1,758 words) - 09:24, 10 November 2023
is the set of all $m$-dimensional subspaces $W$ of an $n$-dimensional vector space $V$ over a field $k$ satisfying the Schubert conditions: $\dim(W\cap
...were considered by H. Schubert in connection with enumeration problems for geometric objects with given incidence properties. Hilbert's 15th problem concerns a
3 KB (391 words) - 16:26, 9 December 2023
...it is equivalent to Grothendieck's theorem on decomposition of holomorphic vector bundles over the Riemann sphere [[#References|[a3]]]. It is also of fundame
...he classical theory of Riemann–Hilbert problems [[#References|[a10]]]. The geometric approach to Birkhoff factorization, developed in [[#References|[a8]]], has
7 KB (989 words) - 16:56, 1 July 2020
...fel [[#References|[1]]] in connection with systems of linearly independent vector fields on smooth manifolds. First started in [[#References|[1]]], studies o
...align="top"> V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian)</TD></TR><TR><TD va
7 KB (1,080 words) - 12:38, 8 October 2023
...mathematics and physics; e.g. the coordinates of an element (vector) in a vector space are called its components, the coordinates in a product of sets are t
...s in particular to sets whose definitions involve numbers (e.g. metric and vector spaces), and therefore to a very broad range of mathematical objects of pra
7 KB (980 words) - 17:09, 11 April 2014
...olynomials $( p _ { m } ( x ) ) _ { m \geq 1 }$ are attached in a spectral geometric way.
So, let $E$ denote a symplectic vector space of dimension $m = 2 n$ (cf. also [[Symplectic space|Symplectic space]
5 KB (742 words) - 07:27, 25 January 2024
..."Quotient spaces modulo reductive groupes and applications to moduli of vector bundles on algebraic curves" , ''Proc. Internat. Congress Mathematicians (N
3 KB (502 words) - 18:27, 12 December 2019
...\beta ) \rightarrow M$ be smooth vector bundles (cf. also [[Vector bundle|Vector bundle]]) and let $D : \Gamma ( \alpha ) \rightarrow \Gamma ( \beta )$ be a
the vector bundles $C _ { k }$ have the form $C _ { k } = \Lambda ^ { k } T ^ { * } M
8 KB (1,101 words) - 17:44, 1 July 2020
...complex geometry on their set of complex points. The goal is to provide a geometric framework for the study of Diophantine problems in higher dimension (cf. al
...[[Hermitian metric|Hermitian metric]] $h$ on the corresponding holomorphic vector bundle on the complex-analytic manifold $X ( \mathbf{C} )$, one can define
8 KB (1,219 words) - 21:00, 13 July 2020
A [[Differential-geometric structure|differential-geometric structure]] on a smooth fibre bundle with a Lie structure group that genera
with a common non-zero tangent vector $ X \in T _ {x _ {0} } B $
17 KB (2,530 words) - 19:28, 17 January 2024
dimensional vector $ \mathbf x = ( x _ {1} \dots x _ {p} ) ^ \prime $
...ibute being investigated; and the multivariate statistical analysis of the geometric structure of the set of multi-dimensional observations being investigated.
27 KB (3,850 words) - 14:44, 7 June 2020
...} ( M , P )$ is the [[Lie algebra|Lie algebra]] of all locally Hamiltonian vector fields $Y \in \mathfrak { X } ( M )$ satisfying $\mathcal{L} _ { Y } P = 0$
The Hamiltonian vector field mapping can be subsumed into the following exact sequence of Lie alge
6 KB (935 words) - 10:58, 2 July 2020
be the [[Geometric genus|geometric genus]] and let $ q $
...="top">[2]</TD> <TD valign="top"> F.A. Bogomolov, "Holomorphic tensors and vector bundles on projective varieties" ''Math. USSR-Izv.'' , '''13''' : 3 (1979)
7 KB (986 words) - 19:41, 5 June 2020
denotes a column vector, the transposed of which is the row vector $ {} ^ {\textrm{ t } } \mathbf u = ( u _ {0} , u _ {1} , u _ {2} , u _ {3
4) The [[Vector field|vector field]] $ X $
8 KB (1,111 words) - 22:15, 5 June 2020
...^ { - } )$ (i.e. a pair $V = ( V ^ { + } , V ^ { - } )$ of real or complex vector spaces together with trilinear mappings $V ^ { \pm } \times V ^ { \mp } \ti
...{ * }$-triple theory. A general account of both the geometric and the non-geometric theory of Jordan–Banach triple systems can be found in [[#References|[a25
14 KB (2,076 words) - 17:00, 1 July 2020
...\mu$ (cf. [[Measure in a topological vector space|Measure in a topological vector space]]) defined on the [[Algebra of sets|σ-algebra]] $\mathcal{B} (X)$ of
|valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure theory", Springer-Verlag (1979). {{MR|0257325}} {{ZBL|0874.49001}}
4 KB (638 words) - 19:35, 1 January 2021
A term denoting a geometric structure that describes the spatial and temporal relations in those physic
...s forms an isotropic cone, whose vectors (cf. [[Isotropic vector|Isotropic vector]]) have scalar square zero and correspond to the motion of light and other
6 KB (840 words) - 13:26, 15 August 2014
Another direction of research is the discovery and study of invariant geometric objects on a homogeneous space (see [[Invariant object|Invariant object]] o
on the space of various geometric objects on $ M $ (
20 KB (2,996 words) - 08:42, 16 December 2019
The branch of geometry dealing with the differential-geometric properties of curves and surfaces that are invariant under transformations
determines a curve up to an equi-affine transformation. The vector $ n = {d ^ {2} \mathbf r } / {d s ^ {2} } $
8 KB (1,146 words) - 19:51, 4 April 2020
A topological or differential geometric construction generalizing the idea of [[parallel translation]] in affine sp
===(Genuine) parallel translation in vector and affine spaces===
16 KB (2,769 words) - 08:44, 12 December 2013
[[Vector bundle|Vector bundle]];
In the case of a finite-dimensional vector space $\R^n$, a coordinate system is given by a basis, and two bases are po
18 KB (2,980 words) - 17:05, 13 June 2020
with tangent vector $ a _ {z} ( q) $,
...pond specific singularities of the quadratic differential (poles), and the geometric properties of the solution are related in a suitable fashion to the structu
7 KB (1,091 words) - 19:19, 20 January 2022
...lectromagnetic fields. In this theory the four-dimensional electromagnetic vector potential $ A _ {n} ( x) $,
...mplified. Thus, the Lorentz condition, which is linear with respect to the vector potential $ A _ {n} $,
9 KB (1,227 words) - 19:41, 5 June 2020
is the mass of the $ r $-
vector $ \alpha ( p) $.
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Whitney, "
Geometric integration theory" , Princeton Univ. Press (1957)</TD></TR></table>
5 KB (645 words) - 21:34, 20 February 2021
...etric figures on a sphere, in the same way as planimetry is concerned with geometric figures in a plane.
...ing, as the notation implies, in $x$. Let $x_y$ be the unit length tangent vector to $xy$ at $x$ and let $x_z$ be analogously defined. Then the angle between
8 KB (1,389 words) - 15:53, 19 April 2014
on a finite-dimensional vector space $ V $
...84)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E. Artin, "Geometric algebra" , Interscience (1957)</TD></TR></table>
4 KB (640 words) - 03:29, 21 March 2022
form a [[geometric lattice]] (cf. [[Semi-Dedekind lattice|Semi-Dedekind lattice]]). A subset
of a vector space $ V $
4 KB (651 words) - 17:45, 4 June 2020
...sections and construction problems. Relatively rigorous logical proofs of geometric theorems also appeared during that period. The [[Elements-of-Euclid| $ Ele
...tly used in geometry and in other fields of mathematics ([[Vector calculus|Vector calculus]]; [[Tensor calculus|Tensor calculus]]; the method of differential
20 KB (2,829 words) - 18:41, 11 December 2020
vector field $ v $
the set of fixed geometric points of the morphism $ F ^ { n } $
10 KB (1,491 words) - 19:04, 26 March 2023
...nster $\bf M$ as a symmetry group of a certain infinite-dimensional graded vector space with natural additional structure. The additional structure can be ex
...ng" of Riemann spheres with punctures was initiated by Frenkel. A precise geometric formulation of the notion of vertex operator algebra in terms of partial op
20 KB (2,919 words) - 00:57, 15 February 2024
...es provide the standard tangent vectors (cf. also [[Tangent vector|Tangent vector]]) as $1$-jets of smooth curves (compare with the above definition via homo
...defined simply by the functorial action. Similarly, they behave nicely on vector bundles, principal bundles, etc.
12 KB (1,876 words) - 06:30, 15 February 2024
...ns occur as the equations describing the group orbit of the highest weight vector. A similar description holds for these combined differential-difference equ
...l } ( t ) ) _ { l \in \mathbf Z }$. The group orbit of the highest weight vector can then be characterized by a set of so-called Hirota bilinear relations f
15 KB (2,265 words) - 17:00, 1 July 2020
is the $ \mathcal F( M) $-module of smooth [[vector field|vector fields]] on $ M $.
...ned above, one may also study exterior differential forms with values in a vector space $ V $
27 KB (4,062 words) - 01:31, 7 May 2022
...meaning that it is the composite of the spherical mapping (or unit normal vector) $N : M \rightarrow S ^ { 2 }$ and the [[Stereographic projection|stereogra
4 KB (595 words) - 16:57, 1 July 2020
====Cone in a real vector space (by M.I. Voitsekhovskii)====
A cone in a real vector space $E$ is a set $K\in E$ such that $\lambda K\in K$ for any $\lambda >0$
13 KB (2,188 words) - 13:05, 10 June 2016
is a geometric concept which is a generalization of the concept of a
defined as a curve for which the tangent vector field is parallel
11 KB (1,778 words) - 21:59, 5 March 2012
...ef|Bic}}|| K. Bichteler, "Integration theory (with special attention to vector measures)" , ''Lect. notes in math.'' , '''315''' , Springer (1973)
|valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure theory", Springer-Verlag (1979). {{MR|0257325}} {{ZBL|0874.49001}}
4 KB (498 words) - 22:40, 31 December 2017
a [[Vector field|vector field]] on $ X $,
.... This chaoticity is strongly suggested by numerical results combined with geometric arguments: what is lacking is a tedious numerical verification. This equati
8 KB (1,192 words) - 16:43, 4 June 2020
An inequality concerning the derivatives of vector functions $f:\mathbb R^n\to \mathbb R^n$. Assuming that $f$ is continuosly
..."A new approach to counterexamples to $L^1$ estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions", ''A
4 KB (541 words) - 11:59, 3 November 2013
...cs and geometric measure theory is no exception: techniques and ideas from geometric measure theory have been found useful in the study of partial differential
...slant, [[#References|[a9]]] discusses applications of some of the ideas of geometric measure theory in the theory of Sobolev spaces and functions of bounded var
21 KB (3,319 words) - 17:46, 1 July 2020
...eories]]) generated by the category of vector bundles (cf. [[Vector bundle|Vector bundle]]).
...ory, which consists of the fact that the space of continuous sections of a vector bundle can be regarded as a module over the algebra of continuous functions
17 KB (2,459 words) - 07:32, 26 February 2022
...pletion. It has been established that for the rings of algebraic geometry (geometric rings) passage to the completion preserves a number of important properties
...nd correspond in the geometric representation to the notion of a family of vector spaces. The usual constructions of linear algebra such as the direct sum, a
16 KB (2,400 words) - 17:45, 4 June 2020
a non-degenerate vector field $ \Phi $
is put in correspondence with a non-zero vector $ \Phi ( x) $.
14 KB (2,172 words) - 19:39, 5 June 2020
be a finite-dimensional vector space over an ordered field $ k $,
...001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E. Artin, "Geometric algebra" , Interscience (1957) {{MR|1529733}} {{MR|0082463}} {{ZBL|0077.021
5 KB (659 words) - 06:06, 23 April 2024
<TD valign="top"> I.P. Egorov, "Motions in generalized differential-geometric spaces" ''Progress in Math.'' , '''6''' (1970) pp. 171–228 ''Itogi Nauk.
<TD valign="top"> S. Bochner, "Vector fields and Ricci curvature" ''Bull. Amer. Math. Soc.'' , '''52''' (1946) pp
19 KB (2,034 words) - 04:04, 15 February 2024
An area can be introduced on the basis of an approximating, integral-geometric, or functional approach or else axiomatically. The most common concepts are
===Integral-geometric areas.===
21 KB (3,223 words) - 18:48, 5 April 2020
{{Cite|Ka2}}. In the non-symmetrizable case more sophisticated geometric methods are required
...$L(\L)$, which is determined by the property that there exists a non-zero vector $v_\L\in L(\L)$ such that
15 KB (2,514 words) - 20:27, 15 November 2017
that is, a vector field
then the vector field $ \partial / \partial x ^ {1} $
16 KB (2,336 words) - 08:02, 6 June 2020
coincides with its geometric multiplicity (i.e. the defect of the matrix $ A - \lambda E $).
is the initial vector in the method. Such a matrix is usually very poorly conditioned, as is evid
15 KB (2,340 words) - 08:54, 21 January 2024
vector corresponding to the $ p $-
the vector group of homogeneous polynomials in $ 2 m $
8 KB (1,110 words) - 16:10, 1 April 2020
and suppose that the vector field $ a ( x) = ( a _ {1} ( x) \dots a _ {n} ( x) ) $
are positive integers and the vector functions $ f ( x , y , z ) $,
9 KB (1,342 words) - 19:08, 9 January 2024
...up of all non-singular linear transformations of a finite-dimensional real vector space $ V $,
Analogues of these results have been obtained in the geometric theory of invariants (cf. [[Invariants, theory of|Invariants, theory of]])
13 KB (2,049 words) - 19:35, 17 January 2024
on a finite-dimensional vector space $ V $
...tified with the set of regular sections of the one-dimensional homogeneous vector bundle over $ G / B $
9 KB (1,346 words) - 10:57, 20 December 2019
...wise motion through $n$ quadrants. (An equivalent condition is: The radius vector $P(i\omega)$, as $\omega$ increases from $0$ to $+\infty$, never vanishes a
...to the [[Routh–Hurwitz criterion|Routh–Hurwitz criterion]]; however, it is geometric in character and does not require the verification of determinant inequalit
5 KB (692 words) - 15:30, 1 July 2020
...d a unique highest weight vector. The $ B $-weight of the highest weight vector is $ - w _{0} ( \chi ) $,
...a16]]]. They used the notion of Frobenius splitting to establish the basic geometric vanishing properties of line bundles on generalized flag varieties and gene
10 KB (1,486 words) - 08:19, 4 March 2022
...*$ is well-known to be a vector space. In fact, $P^*$ is isomorphic to the vector space of [[formal power series]] $\mathcal{F}$ via the mapping
In particular, for linear functionals $L$ and $M$, the geometric series $M,ML,ML^2,\ldots$ makes sense, and so one may define a sequence $s_
7 KB (1,142 words) - 20:34, 9 December 2015
...d|field]] of characteristic not two. A Euclidean space is a [[Vector space|vector space]] $X$ over $F$ equipped with a symmetric [[Bilinear form|bilinear for
...amental problem is to characterize classes of Euclidean spaces by means of geometric structures, i.e. structures of abstract points and lines equipped with suit
12 KB (1,960 words) - 17:44, 1 July 2020
...$\Phi$ a non-degenerate skew-symmetric bilinear form on an $n$-dimensional vector space $E$ over $K$. If $n$ is even, then the symplectic group associated wi
vector space $\def\H{ {\mathbb H}}\H^m$ of dimension $m=p+q$ over the division rin
6 KB (1,078 words) - 14:22, 3 November 2013
can be naturally imbedded in the Lie algebra of $ r $-jets of vector fields on $ M $
consists of the $ ( r+ 1) $-jets of vector fields on $ M $
13 KB (1,866 words) - 18:13, 20 January 2022
''vector lattice''
A real partially ordered vector space $ X $(
14 KB (2,252 words) - 08:09, 21 January 2024
...eferences|[a8]]], one wishes to solve the problem of finding the intrinsic geometric properties (i.e., the basic differential invariants) of (a1) under non-sing
Define the KCC-covariant differential of a contravariant vector field $\xi ^ { i } ( x )$ on $\Omega$ by
10 KB (1,467 words) - 17:47, 1 July 2020
being the components of the normal vector to this surface. Accordingly, equation (1) defines a connection between the
of the normal vector to the integral surface, and at each point $ ( x , u ) $
15 KB (2,114 words) - 17:33, 5 June 2020
...he space $C ^ { \infty } ( E )$ of smooth sections of some [[Vector bundle|vector bundle]] $E$ over $M$ equipped with a positive-definite Riemannian inner pr
...troduces certain functions of the eigenvalues which can be used to extract geometric information from the spectrum. Some useful such functions having interestin
14 KB (2,008 words) - 16:55, 1 July 2020
...a subset of the real line), the map $A$ takes the form $A (t) = a t$: the vector $a$ is then the [[Approximate derivative|approximate derivative]] of $f$ at
|valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissen
5 KB (734 words) - 11:57, 2 May 2014
The algebro-geometric solution was expressed by I.M. Krichever [[#References|[a5]]] in terms of t
...f $\partial$. From this point of view, the curve associated to the algebro-geometric solutions is the spectral curve of the ring of differential operators in $x
8 KB (1,099 words) - 08:01, 20 February 2023
The approximate limit of a function taking values in a finite-dimensional vector space can be defined using its coordinate functions and the definition abov
|valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissen
5 KB (745 words) - 17:18, 18 August 2012
...sence, to each [[Lagrangian|Lagrangian]] $L$, the Euler operator assigns a geometric object $\mathcal{E} ( L )$ whose components $\mathcal{E} ^ { a } ( L )$, $a
...s its corresponding flow. Then the prolongation $Z^k$ of $Z$ to a vertical vector field on $E ^ { k }$ has flow $\phi _ { t } ^ { k }$ (cf. also [[Prolongati
17 KB (2,575 words) - 17:45, 1 July 2020
be two real vector spaces in separate duality with respect to a bilinear form $ \langle \cd
The following result is a generalization of the geometric [[Hahn–Banach theorem|Hahn–Banach theorem]] on the bipolar of a set: th
6 KB (921 words) - 19:36, 5 June 2020
...[[#References|[3]]], [[#References|[4]]]). Poincaré made extensive use of geometric methods, regarding the solutions of systems of differential equations as cu
The geometric approach of Poincaré was developed in the 1920s by George Birkhoff, who di
29 KB (4,392 words) - 08:08, 6 June 2020
[[Vector space|vector space]] $V$ over a field $k$ which preserve a fixed non-singular
...an anisotropic symmetric bilinear form such that $f(v,v)\in L_1^2$ for any vector $v$.
14 KB (2,418 words) - 18:20, 7 November 2014
or, in vector notation,
==Geometric interpretation of an autonomous system.==
13 KB (1,960 words) - 07:35, 26 March 2023
...]]) are the most general spaces figuring in functional analysis. These are vector (linear) spaces $ X $
is a metric space, then one has a metric vector space.
36 KB (5,132 words) - 08:10, 30 January 2022
dimensional vector space over $ K = \mathbf R , \mathbf C $
be the pseudo-group of local transformations of a vector space $ V $
11 KB (1,659 words) - 08:01, 21 January 2024
...of infinitesimal motions (or Killing vectors, cf. [[Killing vector|Killing vector]]), which are defined as velocity fields of one-parameter subgroups of $
in the direction of the vector $ X \in T _ {p} M $.
30 KB (4,323 words) - 19:35, 5 June 2020
The problem of the geometric characterization of domains in a given [[Analytic space|analytic space]] th
in an infinite-dimensional complex topological vector space $ E $.
6 KB (828 words) - 22:16, 5 June 2020
A generalization of representation functions are representation sections of a vector $ G $-bundle $ E $
...R><TD valign="top">[a1]</TD> <TD valign="top"> S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4</TD></TR></table>
6 KB (891 words) - 12:31, 18 February 2022
Methods of differential inequalities and Lyapunov vector functions have been applied to establish theorems on asymptotic $ y $-
..."top">[a2]</TD> <TD valign="top"> S. Lefshetz, "Differential equations: geometric theory" , Dover, reprint (1977) pp. 78, 83</TD></TR></table>
8 KB (1,186 words) - 08:22, 6 June 2020
...ics, Bayesian Kriging, Transformed-Gaussian Kriging, Correlation Function, Geometric Anisotropy, Random Field, Generalized Linear Geostatistical Model
...oldsymbol{\theta}=\left(\vartheta_1,\vartheta_2,\kappa\right)$ denotes the vector of correlation parameters. The parameter $\kappa$ controls the smoothness o
25 KB (3,705 words) - 19:33, 5 October 2023
...[[#References|[a2]]].) Let $A$ be a given matrix and $\mathbf{b}$ a given vector.
for some matrix $B$ and vector $\mathbf{b}$.
12 KB (1,986 words) - 19:13, 31 March 2017
...'', '''[[Fibration]]''', '''Skew product''' etc. Particular cases are '''[[Vector bundle]]''', '''[[Tangent bundle]]''', '''[[Principal fibre bundle]], $\dot
...etrized by the ''index set'' which itself has an additional topological or geometric structure (topological space, smooth or holomorphic manifold etc.).
32 KB (5,299 words) - 12:27, 12 December 2020
...GL}}\GL_n(k)$ of non-singular linear transformations of an $V$-dimensional vector space $k$ over a field $k$ are precisely the subgroups $P(\nu)$ consisting
In the case where $k=\R$, the parabolic $\R$-subgroups admit the following geometric interpretation (see ). Let $G_\R$ be a non-compact real semi-simple Lie gro
5 KB (932 words) - 11:43, 17 December 2019
==Geometric interpretation of solutions of a system of linear
the row of coordinates of a vector $x$ of an $n$-dimensional vector
12 KB (2,085 words) - 22:05, 5 March 2012
...eir domains of definition (this means that the local resolvent of the zero vector is equal to zero, which holds, for example, for all operators without eigen
at the vector $ x $
24 KB (3,593 words) - 18:51, 13 January 2024
...In this case $N = \partial M$, $N_x$ is the inwardly pointing unit normal vector and $U ^ { + } \partial M = \{ v \in S N : \langle v , N _ { x } \rangle &g
...="top">[a6]</td> <td valign="top"> L.A. Santaló, "Integral geometry and geometric probability (With a foreword by Mark Kac)" , ''Encyclopedia Math. Appl.'' ,
5 KB (875 words) - 06:04, 5 April 2023
...Black and Scholes, [[#References|[a3]]], stock prices are modelled by the geometric Brownian motion
...ependent tradeable assets is at least one larger than the dimension of the vector of Brownian motions appearing in the model. In particular, in a complete ma
9 KB (1,337 words) - 18:28, 17 October 2014
A branch of [[Conformal geometry|conformal geometry]] in which the geometric quantities that are invariant under conformal transformations are studied b
each point or circle is defined by a vector $ \mathbf x = ( x _ {1} , x _ {2} , x _ {3} , x _ {4} ) $,
6 KB (942 words) - 17:46, 4 June 2020
...pace of (usually vector-valued) functions (or sections of a differentiable vector bundle) on differentiable manifolds or else on a space dual to a space of t
in the space of sections of a vector bundle $ \xi $
17 KB (2,519 words) - 19:35, 5 June 2020
...{{Ref|AeMa}}||valign="top"| A. Aepply, L. Markus, "Integral equivalence of vector fields on manifolds and bifurcation of differential systems" ''Amer. J. Mat
|valign="top"|{{Ref|CaNe}}||valign="top"| C. Camacho, A.L. Neto, "Geometric theory of foliations", Birkhäuser (1985) {{MR|0824240}} {{ZBL|0568.57002}}
6 KB (918 words) - 17:18, 9 January 2013
is the displacement vector at the point $ x= ( x _ {1} , x _ {2} , x _ {3} ) $
...been studied. If the jumps along the characteristic affect only the normal vector component $ \mathop{\rm grad} u $,
20 KB (3,044 words) - 19:36, 5 June 2020
...ian mechanics, the spatial properties of physical processes are defined by geometric properties of three-dimensional Euclidean space, while the time variable ap
...ed in one tensor object in Minkowski space. For example, an energy-impulse vector is formed in the following way: its first component in a Galilean coordinat
22 KB (3,285 words) - 08:10, 6 June 2020
..."top">[3]</TD> <TD valign="top"> S. Lefschetz, "Differential equations: geometric theory" , Interscience (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD v
...Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983)</TD></TR></table>
6 KB (857 words) - 08:41, 13 May 2022
[[Geometric genus|geometric genus]] $ p _ {g} = h ^ {2} ( A, {\mathcal O} _ {A} ) =1 $
b) the flows of the integrable vector fields $ X _ {u _ {i} } $
18 KB (2,511 words) - 06:25, 26 March 2023
is imbedded as an orbit in a vector space $ W $
be a linear geometric structure of type $ W $(
17 KB (2,677 words) - 19:40, 5 June 2020
is the position vector of the directrix and $ m = m ( v) $
is the unit vector of the generator passing through $ \rho ( v) $,
12 KB (1,684 words) - 08:12, 6 June 2020
...uperalgebra]]), first appeared in [[Deformation|deformation]] theory, in a geometric context, in the work of A. Frölicher and A. Nijenhuis in the late 1950s, a
...ors, and those in $V _ { \overline{1} }$, called odd vectors. In the super vector space representing physical particles, the even vectors represent bosons an
18 KB (2,722 words) - 17:47, 1 July 2020
...cy, etc. For instance, if $A$ is a field and $V=W$ is a finite-dimensional vector space over $A$ with basis $e_1,\dots,e_n$, then for the vectors
|valign="top"|{{Ref|Ar}}||valign="top"| E. Artin, "Geometric algebra", Interscience (1957) {{MR|1529733}} {{MR|0082463}} {{ZBL|0077.0
6 KB (1,157 words) - 08:58, 9 December 2016
A geometric object which locally has the structure (topological, smooth, homological, e
...lutions of non-degenerate systems of equations and also as various sets of geometric and other objects allowing local parametrization (see below); for example,
30 KB (4,462 words) - 07:59, 6 June 2020
...diaofmath.org/legacyimages/r/r082/r082150/r08215031.png" /> is the tangent
vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
...w.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215089.png" /> are
vector fields and <img align="absmiddle" border="0" src="https://www.encyclopediao
55 KB (7,564 words) - 16:56, 7 February 2011
...izations of the structure theorems of linear algebra appear in topology. A vector space can be regarded as a special case of a
...dle]]. These objects may be studied with the aid of the homotopy theory of vector bundles and of topological
14 KB (2,405 words) - 22:14, 10 January 2015
form a one-dimensional subspace, spanned by a unit vector $ e $.
...R><TD valign="top">[a4]</TD> <TD valign="top"> S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4</TD></TR></table>
8 KB (1,119 words) - 08:22, 6 June 2020
...(see also [[#References|[a26]]]), ad hoc and without giving any intrinsic geometric interpretation for its meaning or origin, introduced a new tensor as an ana
denotes the [[Lie algebra|Lie algebra]] of vector fields on $ M $),
15 KB (2,270 words) - 08:28, 26 March 2023
...or the later development of science was the accumulation of arithmetic and geometric knowledge in Egypt and Babylon. In Babylon, on the basis of the techniques
...ns which, in contrast to the concepts of a natural number, a fraction or a geometric figure, have no fairly sound support in prescientific ordinary human experi
22 KB (3,256 words) - 21:54, 30 March 2012
Let $M ( k )$ be the vector space of (entire) modular forms of weight $k$, see [[Modular form|Modular f
...\prime }$, where the sum is over all sublattices of $L$ of index $n$. This geometric definition, [[#References|[a4]]], makes (a1) easier to understand.
6 KB (1,050 words) - 00:38, 15 February 2024
In vector form, the absolute null polarity can be written in the form $ \mathbf u =
...56}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E. Artin, "Geometric algebra" , Interscience (1957) pp. Chapt. II {{MR|1529733}} {{MR|0082463}}
7 KB (1,005 words) - 14:55, 7 June 2020
...veloped most profoundly for the case when $K$ is a skew-field and $E$ is a vector space of finite dimension $n$ over $K$. From now on, these conditions will
...consists in the study of special elements in the classical groups and the geometric properties of them, principally the study of transvections, involutions and
12 KB (1,991 words) - 18:05, 29 November 2014
At the basis of the geometric approach to the study of Kleinian groups is the notion of a fundamental dom
...culated by means of the [[Riemann–Roch theorem|Riemann–Roch theorem]]. The geometric structure of such groups $ \Gamma $
13 KB (1,863 words) - 19:27, 17 January 2024
a [[Wiener process|Wiener process]] which generates the observed vector process $ y $
through the state vector process $ x $.
15 KB (2,302 words) - 08:03, 21 January 2024
...near (that is, additive and homogeneous) function on a real [[Vector space|vector space]] $ L ( \mathbf R ) $
...hedrally-closed systems (see [[#References|[1]]]). In the case of the real vector space $ \mathbf R ^ {n} $,
14 KB (2,052 words) - 10:49, 21 March 2022
...s of the polynomial $p ( \lambda _ { 1 } , \lambda _ { 2 } )$. The algebro-geometric assumptions on the polynomial $p ( \lambda _ { 1 } , \lambda _ { 2 } )$ and
...points at infinity) fit together to form two complex holomorphic rank-$r$ vector bundles $\mathfrak{C}$ and $\tilde{\frak E}$ on a compact Riemann surface $
24 KB (3,136 words) - 20:00, 24 November 2023
...powerful source of algebraic complete integrability over moduli spaces of vector bundles; 7) the bispectral property.
...Lamé equation|Lamé equation]]); moduli spaces of the corresponding algebro-geometric configurations (tangential covers) were described in [[#References|[a31]]]
13 KB (1,901 words) - 09:09, 26 March 2023
...sional Hamiltonian systems, where the ring of functions is replaced by the vector space of functionals.
...ry and prequantization" K. Bleuler (ed.) A. Reetz (ed.) , ''Differential Geometric Methods in Mathematical Physics (Bonn, 1975)'' , ''Lecture Notes in Mathema
8 KB (1,172 words) - 19:18, 17 March 2023
all rows of which coincide with the vector $\pi$ (see also
be estimated by a geometric progression with any exponent $\rho$ that has
6 KB (982 words) - 09:11, 1 October 2023
are topological vector spaces), defined by the formula
of differentiable complex-valued vector functions on $ M $
41 KB (5,908 words) - 22:12, 5 June 2020
A [[Vector space|vector space]] $ H $
is an infinite-dimensional vector space.
31 KB (4,586 words) - 18:43, 13 January 2024
...morphism]]), in particular of integral trajectories and singular points of vector fields on differentiable manifolds (dynamical systems), as well as the vari
...|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957) {{MR|0087148}} {{ZBL|008
17 KB (2,329 words) - 01:38, 30 December 2021
Geometric interpretation of the differential. The equation of the tangent to the grap
...ctor-functions of one or more real variables; and to complex functions and vector-functions of one or more complex variables. In functional analysis they are
15 KB (2,211 words) - 11:44, 26 March 2023
...ogical characteristics of a two-dimensional manifold with its differential-geometric properties is the [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]] [[#Refe
...the sum of the indices (cf. [[Index|Index]]) of the singular points of any vector field on a closed surface is equal to $ \chi ( M) $.
20 KB (3,075 words) - 08:26, 6 June 2020
one associates the axiom vector $ \pi _ {G} = ( v _ {1} \dots v _ {n} ) $,
is a column vector of dimension $ n $
13 KB (1,968 words) - 18:43, 26 March 2023
...e surface $g$, and $u_j(x)$ are the desired components of the displacement vector of a point on the mean surface. The system \eqref{1} is solved under four b
...D></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.V. Pogorelov, "Geometric methods in the non-linear theory of elastic shells" , Moscow (1967) (In R
8 KB (1,200 words) - 17:04, 14 February 2020
or, more generally, a generalized section of a smooth vector bundle), then $ \mathop{\rm WF} ( u) $
of a bicharacteristic (that is, a trajectory of the Hamiltonian vector field on $ T ^ {*} X \setminus 0 $
12 KB (1,704 words) - 18:55, 29 December 2021
...e now available concerning the homology invariants of the singularities of vector fields, frame fields and tensor fields, as well as of the singularities of
...vector bundles are especially important (cf. [[Principal fibre bundle]]; [[Vector bundle]]).
19 KB (2,788 words) - 09:43, 28 October 2023
defines a vector bundle $ \xi _ {r} = \phi ^ {*} \gamma _ {r} $
bordism theory, which admits of a geometric definition using the concept of a so-called [[B-Phi-structure| $ ( B , \ph
42 KB (6,290 words) - 19:33, 17 January 2024
...f M.S. Narasimhan and C.S. Seshadri on moduli spaces of stable holomorphic vector bundles on Riemann surfaces. Among others, Atiyah and Bott derived formulas
...uction of a new kind of gauge theory, phrased in terms of the differential-geometric equation by N. Seiberg and E. Witten, which, in turn, had again originated
7 KB (1,126 words) - 17:45, 1 July 2020
...braic variety if and only if it admits a [[Negative vector bundle|negative vector bundle]]. Kodaira's theorem can be generalized to complex spaces (see [[#Re
...>[7]</td> <td valign="top"> R.E. Greene, H. Wu, "A theorem in complex geometric function theory" , ''Value-distribution Theory'' , '''A''' , M. Dekker (19
12 KB (1,750 words) - 19:20, 27 October 2023
is an integer) of the vector $ x $
for every other vector $ y $
21 KB (3,203 words) - 06:33, 13 June 2022
Conversely, for a finite vector space of dimension $ d $
This geometric structure is called a diagonal space.
11 KB (1,611 words) - 08:03, 6 June 2020
...ubset of $\mathbf{C} ^ { n }$, one denotes by $A ( E )$ the [[Vector space|vector space]] of holomorphic functions on $E$, endowed with the projective (respe
For any vector $a \in \mathbf{C} ^ { n } \backslash \{ 0 \}$, let $( a , \partial )$ denot
16 KB (2,533 words) - 09:46, 18 February 2024
...y, often simply called the trajectory) in the phase space. In this space a vector field can be defined by associating with each point $ w $
the vector $ f ( w) $
27 KB (4,058 words) - 19:36, 5 June 2020
...TD valign="top"> A.N. Šošitaišvili, "Bifurcations of topological type of a vector field near a singular point" , ''Proc. Petrovskii Sem.'' , '''1''' , Moscow
10 KB (1,376 words) - 16:43, 4 June 2020
...ct product]] of lattices of subspaces of vector spaces (cf. [[Vector space|Vector space]]) [[#References|[a22]]].
...been characterized for various classes of rings via the Arguesian law and geometric conditions on the lattice, e.g. for completely primary uniserial rings [[#R
29 KB (4,201 words) - 16:31, 9 December 2023
$n$-dimension vector space, $\Gamma\subset\C^n$ is a lattice (cf.
geometric structures of algebraic varieties. E.g., they were used to
8 KB (1,216 words) - 20:39, 5 March 2012
The simplest control systems are described by an ordinary (vector) differential equation
is the state vector of the system, $ u \{ u _ {1} \dots u _ {r} \} $
43 KB (6,618 words) - 07:32, 26 March 2023
the system (4) takes the vector form:
The vector function
33 KB (4,933 words) - 01:50, 23 January 2022
The branch of mathematics dealing with geometric objects connected with commutative rings: algebraic varieties (cf. [[Algebr
is an integer vector, $ \alpha _{j} $
29 KB (4,414 words) - 17:20, 17 December 2019
be a [[Vector bundle|vector bundle]] on $ X $
can be interpreted as elements of the $ \mathbf C $-vector space $ \mathop{\rm Hom} _ {\mathcal D _ {X} } ( M, {\mathcal O} _ {X}
24 KB (3,511 words) - 07:03, 10 May 2022
...in the sample, a certain part of the information is contained in the rank vector only. One can construct statistical procedures based only on the ranks with
...ped it became clear that the part of the information contained in the rank vector can prove to be significant, in which case these procedures are highly effi
37 KB (5,331 words) - 17:14, 7 February 2011
...pendent concept. The term "function" first appeared in works of Leibniz. Geometric, analytic and kinematic ideas were used to specify a function, but graduall
...te structures. For example, if the ranges of functions belong to a certain vector space, then such functions can be added; if they belong to a ring, then the
34 KB (5,509 words) - 22:06, 28 January 2020
A group of linear transformations of a [[Vector space|vector space]] $ V $ of finite dimension $ n $ over some [[Skew-field|skew-f
...079}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E. Artin, "Geometric algebra" , Interscience (1957) {{MR|1529733}} {{MR|0082463}} {{ZBL|0077.021
16 KB (2,362 words) - 18:01, 12 December 2019
==Geometric interpretation of the derivative.==
...ions in one or several real variables, and to complex-valued functions and vector functions in one or several complex variables. In functional analysis the i
37 KB (5,252 words) - 11:45, 26 March 2023
are non-empty or not. The dimensions of these vector spaces are $ n _ {+} $
Such a formal difference of (equivalence classes) of vector bundles defines an element of the $ K $-
27 KB (3,739 words) - 06:51, 26 March 2023