...the normal curvature of a surface in a given direction $l$ in terms of the principal curvatures $k_1$ and $k_2$:
...between the direction $l$ and the principal direction corresponding to the principal curvature $k_1$.
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...l space of a [[bundle]], transforming the latter into a [[Bundle#principal|principal $G$-bundle]].
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...of the indicatrix of the curvature (cf. [[Dupin indicatrix]]). If $t$ is a principal direction, the relation (Rodrigues' formula)
...l direction. The normal curvature in a principal direction is known as a [[principal curvature]].
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...principal [[ultrafilter]]. The [[Fréchet filter]] is an example of a non-principal filter.
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...discarding all the terms not containing derivatives of maximal order. The principal part of the differential operator
...differentiations with respect to the various arguments. For instance, the principal part of the differential operator $D_1-D_2^2+\alpha D_2$ is sometimes defin
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...by K.M. Peterson as examples of surfaces allowing of a deformation over a principal base.
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...onversely (Finikov's theorem), the only surface with an infinite number of principal bases is a right helicoid [[#References|[2]]].
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''Dirichlet principal character''
where $D$ is a given natural number. Principal characters serve to define the concepts of primitive and imprimitive charac
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#REDIRECT [[Principal translation]]
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#REDIRECT [[Principal ideal ring]]
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...ly for principal ideals) which is Bezout is a principal ideal ring. As for principal ideal rings, a module of finite type over a Bezout ring is a direct sum of
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...equal to zero, one of the principal curvatures (cf. [[Principal curvature|Principal curvature]]) vanishes, and the coefficients of the [[Second fundamental for
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...that generates a prime ideal). An atomic [[Bezout ring|Bezout ring]] is a principal ideal ring.
...one simply speaks of rings which satisfy the ascending chain condition for principal ideals or the divisor chain condition.
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$#C+1 = 14 : ~/encyclopedia/old_files/data/M062/M.0602310 Mapping, principal net of a,
The directions tangential to the lines of the principal net of the mapping at the point $ x \in G $
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...states that $b^G$ is the principal block of $RG$ if and only if $b$ is the principal block of $RH$.
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$#C+1 = 26 : ~/encyclopedia/old_files/data/P074/P.0704790 Principal type, partial differential operator of,
whose principal part $ P( D) $(
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...ity $A^2=A$ is valid. All [[regular semi-group]]s are semi-simple. If each principal factor of a semi-group is either completely $0$-simple or completely simple
Any semi-group consists, as it were, of its principal factors. This explains, in particular, the important role played by ideally
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...and vice versa. If $\sigma$ is a principal base for a [[Deformation over a principal base|deformation]] of $X$, then $Y$ is a [[Bianchi surface]].
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...f it is closed with respect to all translations (or with respect to merely principal translations).
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...ase|deformation over a principal base]] and to classify them. Thus, if the principal base contains two families of geodesic lines, the functions $U$ and $V$ are
...ss $B_1$ is characterized by the fact that only one family of lines of the principal base are geodesics (one of the functions $U,V$ is constant); conoids may se
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...element of $S$, and similarly the left [[annihilator]] of any element is a principal left ideal on an idempotent element of $S$.
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The left principal ideal $L(\alpha)$ of a ring $A$ contains, in addition to the element $\alph
the right principal ideal $R(\alpha)$ contains all the elements
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...ves as the base of two different deformations $F'$ and $F''$, then it is a principal base of deformation.
...normal curvature of $F$ in the direction of one of the two families of the principal base $\sigma$ at an arbitrary point $M\in F$, while $\kappa'$, $\kappa''$,
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''
principal series of representations''
...images/c/c025/c025750/c0257507.png" />, then the non-degenerate continuous
principal series of representations of <img align="absmiddle" border="0" src="https:/
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A [[Principal G-object|principal $ G $-
then a principal $ G $-
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...y less than $\mathfrak{a}$. The Fréchet filter is not [[principal filter|principal]].
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$#C+1 = 81 : ~/encyclopedia/old_files/data/P074/P.0704740 Principal ideal ring
...[[Associative rings and algebras]]) in which all right and left ideals are principal, i.e. have the form $ aR $
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$#C+1 = 99 : ~/encyclopedia/old_files/data/P074/P.0704710 Principal \BMI G\EMI\AAhobject
...ndle|principal fibre bundle]] in topology, a [[Principal homogeneous space|principal homogeneous space]] in algebraic geometry, etc. Let $ G $
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[[Principal fibre bundle|principal fibre bundle]] the existence of a section implies its triviality. A
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...ere $k_i$ is the principal curvature at a point on the hypersurface in the principal direction $i$.
2 KB (369 words) - 21:48, 1 January 2019
...ipal ideal ring is factorial. A Dedekind ring is factorial only if it is a principal ideal ring. If $S$ is a multiplicative system in a factorial ring $A$, then
3 KB (480 words) - 21:45, 3 January 2021
...a Voss net is isotropic. Every Voss net on a two-dimensional surface is a principal base of a deformation of the surface. Only the helicoid carries an infinite
...lign="top">[2]</TD> <TD valign="top"> S.P. Finikov, "Deformation over a principal base and related problems in geometry", Vereinigt. Wiss.-Techn. Verl. 176 S
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...ngruence (cf. [[Congruence of lines|Congruence of lines]]) with indefinite principal surfaces.
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$#C+1 = 75 : ~/encyclopedia/old_files/data/P074/P.0704690 Principal fibre bundle
The significance of principal fibre bundles lies in the fact that they make it possible to construct asso
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...scending chains of normal subgroups have finite length. If a group has two principal series, then they are isomorphic, i.e. they have the same length and there
The terminology "principal series" is almost never used in the West. Instead one uses chief series. T
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$#C+1 = 45 : ~/encyclopedia/old_files/data/P074/P.0704660 Principal curvature
...direction, i.e. in a direction in which it assumes an extremal value. The principal curvatures $ k _ {1} $
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$#C+1 = 9 : ~/encyclopedia/old_files/data/P074/P.0704750 Principal normal
is the parametric equation of the curve and the value $ t _ {0} $
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The principal linear part of increment of $Q$ under its transformation induced by the loc
...nsor, density, etc.), then the Lie differential $\delta_X Q$ describes the principal linear part of variation with time of $Q$ from the point of view of an obse
1 KB (181 words) - 19:31, 28 December 2014
...with the problem of [[Deformation over a principal base|deformation over a principal base]]. For example, there is a Peterson correspondence between a surface a
...common parametrization, then their third fundamental forms are equal. The principal net for the asymptotic nets (cf. [[Asymptotic net|Asymptotic net]]) of $
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Here $Q(\xi_1,\dots,\xi_n,x_1,\dots,x_n)$ is the principal symbol of the linear partial differential operator, the dot indicates diffe
are valid for at least one value of $s$, then it follows that they are valid for all values of $s$.
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...on the principal diagonal of $L$ will then all be equal, and their common value is the number of spanning trees of $G$. Spanning trees are employed, for ex
The number of spanning trees is a special value of the [[Tutte polynomial]] of the graph, $T_G(1,1)$.
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...th boundary, then they can be extended to the entire space $E^n$ so that a principal fundamental solution will exist for the extended operator.
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is called a principal fibre bundle with structure group $ G $.
is called a fibre bundle with structure group associated with the principal fibre bundle $ \xi $,
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by any of the following unit vectors: the tangent, the principal normal or the binormal of this curve. Let $ \mathbf r = \mathbf r ( s) $
that of the spherical indicatrix of the principal normal by the equation
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A figure consisting of the tangent, the [[Principal normal|principal normal]] and the [[Binormal|binormal]] of a space curve, and the three plan
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...d no subsidiary points; elements of the fourth kind contain a continuum of principal points and infinitely many subsidiary points.
...oebe [[#References|[2]]]. It is based on equivalence classes of paths. The principal theorem in the theory of prime ends is the theorem of Carathéodory: Under
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...this parabolic subgroup; this part is called the corresponding principal (principal degenerate, if the parabolic subgroup is not a Borel subgroup) series repre
3 KB (393 words) - 08:13, 6 June 2020
...riterion]], which reduces all calculations in it to the calculation of the principal minors of only even (or only odd) orders of a Hurwitz matrix.
...[Routh–Hurwitz criterion|Routh–Hurwitz criterion]]); let $\Delta_i$ be its principal minor of order $i$, $i=1,\dotsc,n$.
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...a [[periodic semi-group]]), then $\mathcal D=\mathcal J$. The inclusion of principal left ideals defines in a natural manner a partial order relation on the set
3 KB (507 words) - 16:59, 25 November 2023
...utation. Every isotope (i.e. isotopic magma) of a magma is isomorphic to a principal isotope of the magma. Every magma that is isotopic to a [[quasi-group]] is
1 KB (239 words) - 19:21, 7 January 2016
...trix is an ellipse, the Gaussian curvature of the surface is positive, the principal curvatures of the surface are of the same sign, and for the coefficients of
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...], known as the ''[[Fréchet filter]]'' on $X$. It is contained in any non-principal ultrafilter on $X$.
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...$M^n$ if a certain section is distinguished in the quotient bundle of the principal bundle of frames of order $r$ on $M^n$ by a Lie subgroup $G\subset D_n^r$.
2 KB (310 words) - 12:01, 5 July 2014
...las of the locally trivial bundle. For example, a [[Principal fibre bundle|principal fibre bundle]] with a locally compact space and a Lie group $ G $
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A semi-group in which any two right [[principal ideal]]s intersect is ''left reversible'': $\forall a,b, \in S\ \exists x,y
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associated with a certain principal bundle $ ( X , p , M ^ {n} ) $
is the structure Lie group of the principal bundle $ ( X , p , M ^ {n} ) $
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of the surface in the [[Principal direction|principal direction]]:
are the principal curvatures.
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Half of the sum of the principal curvatures (cf. [[Principal curvature|Principal curvature]]) $ k _ {1} $
are the principal curvatures of the hypersurface, calculated at a point $ A \in \Phi ^ {n}
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...ure Lie group acts simply transitively and analytically. In other words, a principal analytic fibration is a quadruple $ (P,\ B,\ G,\ \pi ) $
dimensional fibre determines a principal analytic fibration with base $ B $
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...]). A submodule $A$ of a module $B$ is called ''inessential'' (small or co-principal) if for any submodule $A'$ the equation $A + A' = B$ implies $A' = B$. Any
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...ff spaces). For calculating the homology of a join (with coefficients in a principal ideal domain), an analogue of the [[Künneth formula|Künneth formula]] is
...the basis of Milnor's construction of a universal [[Principal fibre bundle|principal fibre bundle]].
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[[Principal series|Principal series]]) and a subnormal one a composition series (cf.
2 KB (317 words) - 16:43, 27 November 2013
...striction lines intersect the rays in their boundary points are called the principal surfaces. The set of boundary points of a ray is called the boundary surfac
...me surface; an isotropic congruence, that is, a congruence with indefinite principal surfaces.
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...the minimum condition for inverse sub-semi-groups if and only if it has a principal series (see [[Ideal series|Ideal series]] of a semi-group) each factor of w
...s $M_R$ and $M_H$. At the same time, a semi-simple (see [[Principal factor|Principal factor]] of a semi-group) semi-group with the condition $M_L$ or $M_R$ sati
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...(called a ''[[Fréchet filter]]''). A Fréchet filter is an example of a non-principal filter. 4) The system of subsets containing some fixed point of a set is al
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...differential neighbourhood of this hypersurface; it is essential that the principal quadratic form of the hypersurface does not degenerate. The affine normal a
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...et of elements divisible by $b$ forms an ideal, $bA$, of the ring $A$ (the principal ideal generated by the element $b$), which contains $b$ if $A$ is a ring wi
...re $u$ is an invertible element. Two associated elements generate the same principal ideal. The [[unit divisor]]s coincide, by definition, with invertible eleme
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A square [[Matrix|matrix]] for which all entries below (or above) the principal diagonal are zero. In the first case the matrix is called an upper triangul
successive principal minors are different from zero can be written as a product of a lower trian
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...ghbourhood of the boundary between the illuminated and shadowed zones, the principal term of the asymptotic expansion of $ u _ {K} $
that is correct in the principal terms and remains correct as $ | x | \rightarrow \infty $.
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The principal result concerning induction is the Green indecomposable theorem, described
...local ring]] and a principal ideal domain (cf. also [[Principal ideal ring|Principal ideal ring]]). An integral domain $ S $
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The set of all such frames forms a principal fibre bundle over $ M $,
...p|symplectic group]]. A symplectic connection is just a connection in this principal fibre bundle. There are manifolds $ M $
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...s, its factors are isomorphic to the chief factors (cf. [[Principal factor|Principal factor]]) of $S$.
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which forms one of the principal subjects in algebra of these days — the study of factorization of ideals.
Principal fractional ideals, as well as invertible fractional ideals, are divisorial
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...are periodic, their inverses are many-valued. The single-valued branches (principal branches) of these functions are denoted by $ { \mathop{\rm arc} \sin }
the principal branches are distinguished by a heavy line.
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is collinear with the [[Principal normal|principal normal]] at each point of $ l $,
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...illustration of the behaviour of specific functions. For the relief of the principal functions see [[#References|[2]]].
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is the [[Principal fibre bundle|principal fibre bundle]] with structure group $ \mathop{\rm GL} _ {n} ( \mathbf R
2 KB (385 words) - 19:39, 5 June 2020
The product of the principal curvatures (cf. [[Principal curvature|Principal curvature]]) of a regular surface at a given point.
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...s contained in some nilpotent ideal, for example, in the [[Principal ideal|principal ideal]] generated by $a$. In a non-commutative ring there may by nilpotent
3 KB (470 words) - 23:58, 24 November 2018
A line on a surface at each point of which the tangent has one of the principal directions. The curvature lines are defined by the equation
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...ns is established by the principal ideal theorem: The height of a non-zero principal ideal is one (cf. [[Krull ring|Krull ring]]). A more general result — Kru
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in absolute value, where
Gauss calculated the precise value of $ S $:
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...ial recursive functions and for Turing machines. In turn, the existence of principal computable enumerations is enabled by the ability of computable functions t
of its arguments. The existence of principal computable enumerations (see [[#References|[1]]], Chapt. 1, Sect. 2) and pa
6 KB (919 words) - 08:26, 6 June 2020
...by W. Krull [[#References|[1]]], who called them rings of finite discrete principal order. They are the most natural class of rings in which there is a divisor
...l ring is a factorial ring if and only if every prime ideal of height 1 is principal.
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...] (e.g., $\mathbb{C}$ or $\mathbb{R}$ or $\mathbb{Q}$). It is known that a principal ideal domain is also a unique factorization domain. If a greatest common
4 KB (673 words) - 17:01, 26 October 2014
...the [[Osculating plane|osculating plane]] is called the [[Principal normal|principal normal]]; the one perpendicular to the osculating plane is called the [[Bin
3 KB (436 words) - 17:15, 30 July 2014
...<TD valign="top"> B. Speh, D.A., jr. Vogan, "Reducibility of general principal series representations" ''Acta Math.'' , '''145''' (1980) pp. 227–299<
3 KB (404 words) - 11:26, 26 March 2023
The principal part of the [[Schwarz symmetric derivative]] of order $n$.
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A $2$-form $\Omega$ on a principal fibre bundle $P$ with structure Lie group $G$, taking values in the Lie alg
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Two space curves $L$ and $L^*$ with common principal normals. Let $k_1$ and $k_2$ be the curvature and the torsion of $L$ respec
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...negative (that is, all its minors are non-negative) if and only if all its principal minors and all elements $b_i$ and $c_i$ ($i=1,\ldots,n-1$) are non-negative
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to take the value $+\infty$. If $f$ is Riemann- (or Lebesgue-) integrable on every interval $
...the integration proceeds, the rule for integrating inequalities, the mean-value theorems, integration by parts, change of variable, and the [[Newton-Leibni
7 KB (1,076 words) - 12:30, 7 July 2013
...s are parabolas. The sections by the planes $Oyz$ and $Oxz$ are called the principal parabolas. The axis of symmetry of an elliptic paraboloid is called its axi
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The principal difficulty in solving problems in non-linear programming is their multi-ext
1 KB (173 words) - 08:52, 25 April 2016
for which only the [[Principal series|principal series]] representations arise. All this allows one to derive the spectrum
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Boundary value problems for elliptic partial differential equations in the finite (interio
The principal difference between exterior and interior boundary value problems consists in the fact that in the former case the boundary conditio
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idele. Such an idele is said to be a principal idele. The subgroup
consisting of all principal ideles is a discrete subgroup of the idele
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...x $z$-plane $\mathbb{C}$ having poles at $a_n$, and only there, with given principal parts (1) of the [[Laurent series|Laurent series]] corresponding to the poi
...morphic function $f(z)$ in $\mathbb{C}$ with poles $a_n$ and corresponding principal parts $g_n(z)$ of the Laurent expansion of $f(z)$ in a neighbourhood of $a_
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..."gauge fields" ) on a principal $G$-bundle (cf. also [[Principal G-object|Principal $G$-object]]) with base space $M$ for a compact connected [[Lie group|Lie g
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principal (respectively, $ \wedge $-
principal) if $ ( a \lor ( b \cdot e ) ) : e = ( a : e ) \lor b $(
8 KB (1,218 words) - 08:02, 6 June 2020
principal and free unknowns. For any values of the free unknowns there
are uniquely determined values of the principal unknowns, which
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The group of principal homogeneous spaces (cf. [[Principal homogeneous space|Principal homogeneous space]]) over an Abelian variety. It was shown by A. Weil [[#Re
of principal homogeneous spaces over $ A $,
7 KB (1,109 words) - 16:59, 1 July 2020
...most refined unitary invariants of a pure semi-normal operator $T$ is the principal function $g_T\in L_{\text{compact}}^{1}(\mathbf{C},\;d\text{area})$, which
...nt under Hilbert–Schmidt perturbations of $T$. The entire behaviour of the principal function qualifies it as the correct two-dimensional analogue of Krein's sp
6 KB (970 words) - 07:33, 18 March 2023
...tic of a connection on a fibre bundle. The holonomy group is defined for a principal fibre bundle $P$ with a Lie structure group $G$ and (second countable) base
'''Reduction theorem'''. Let $P(B,G)$ be a principal fibre bundle satisfying the second axiom of countability; let $\Phi$ be the
3 KB (540 words) - 15:11, 19 November 2023
...fashion one defines the curvature tensor for an arbitrary connection on a principal fibre space with structure Lie group $G$ in terms of a decomposition of the
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...f invariants and invariantly associated structures without restricting the principal fibres of frames. The process of canonization of a frame is included in the
...on can be naturally generalized to classes of fibre spaces associated with principal fibre bundles. Let $ \pi : P ( M, H) \rightarrow M $
7 KB (1,000 words) - 01:15, 19 March 2022
...; as a local one-dimensional Krull ring; as a local Noetherian ring with a principal maximal ideal; as a Noetherian valuation ring; or as a valuation ring with
is a semi-local principal ideal ring; if $ \mathfrak n _ {1} \dots \mathfrak n _ {s} $
5 KB (800 words) - 19:36, 5 June 2020
...ifying the Faddeev–Popov ghost $c$ with a Lie algebra-valued $1$-form on a principal bundle, was proposed by Y. Ne'eman and J. Thierry-Mieg [[#References|[a8]]]
...ld]] with underlying infinite-dimensional manifold of all connections of a principal fibre bundle. In this approach, the Faddeev–Popov ghosts play the role of
8 KB (1,181 words) - 20:50, 24 January 2021
...n the class of commutative rings these turn out to be exactly the Artinian principal ideal rings (see [[#References|[1]]], [[#References|[3]]]). There is also a
1 KB (206 words) - 17:29, 30 April 2014
The principal linear part of growth of the arc length $ M M ^ { \prime } $
in the opposite case. The doubled principal linear part $ 2 \delta $
6 KB (990 words) - 19:40, 5 June 2020
...try; if all distortion indices are different, it is called a trimetry. The principal theorem on axonometries is the Pohlke–Schwartz theorem: Any tetrahedron c
1 KB (194 words) - 16:56, 7 February 2011
...hich is compatible with the isomorphism of the corresponding fibres of the principal $ G $-bundle. Here it is assumed that the structure in question is not id
be a smooth principal $ G $-bundle with structure Lie group $ G $
7 KB (1,089 words) - 12:15, 18 February 2022
...he order of integration in iterated improper integrals of Cauchy principal value type (cf. [[Improper integral|Improper integral]]).
...gn="top">[7]</TD> <TD valign="top"> G.H. Hardy, "The theory of Cauchy's principal values" ''Proc. London Math. Soc.'' , '''7''' : 2 (1909) pp. 181–208<
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i.e. the principal ideal generated by $ e $
such that the principal band generated by $ e $
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Example: a [[principal direction]].
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...ed on the same straight line of the congruence are equal and negative. The principal surfaces of a $B$-congruence cut out conjugate line systems on its focal su
1 KB (216 words) - 16:33, 19 October 2014
must converge to the identity mapping, and the principal part of its deviation from the latter must be defined, relative to the fiel
...ing condition (1), that is, under a passage to an arbitrary element of the principal fibre bundle $ \Pi $
11 KB (1,486 words) - 17:46, 4 June 2020
A principal bundle with structure group $ G $
be the [[Grassmann manifold|Grassmann manifold]]; it is the base of the principal $ \textrm{ O } _ {n} $-
9 KB (1,417 words) - 20:39, 23 December 2023
...matrices with principal minors unequal to zero. Generally, a choice of the principal element is adopted. In this scheme, the bordering rows and columns used are
has the maximum absolute value. The calculated matrix will then differ from $ A ^ {-1} $
10 KB (1,498 words) - 08:08, 21 March 2022
is a connection in the principal bundle $ B ( M) $
on the principal frame bundle $ B ( M) $,
6 KB (870 words) - 22:17, 5 June 2020
...ucture group $G$, and $\Gamma$ is $G$-invariant, then $\Gamma$ is called a principal connection. These connections have been used most frequently. On the other
6 KB (926 words) - 16:46, 1 July 2020
If the row indices and column indices are the same, then the minor is called principal, and if they are the first $k$ rows and columns, then it is called a corner
1 KB (233 words) - 10:05, 20 December 2015
...is not equal to $0$ at the point $x_0$, then the Taylor polynomial is the principal part of the [[Taylor formula]].
1 KB (233 words) - 17:01, 16 March 2013
A trio (semi-AFL, AFL) $\mathcal{F}$ is said to be ''principal'' if there is a language $L$ such that $\mathcal{F}$ is the least trio (sem
All four families in the Chomsky hierarchy are principal; for instance, from the Chomsky–Schützenberger characterization of conte
4 KB (661 words) - 09:20, 2 April 2018
can be interpreted as the set of isomorphism classes of topological principal bundles over $ X $
...lian cohomology for a site; for an interpretation see [[Principal G-object|Principal $ G $-
12 KB (1,712 words) - 09:30, 20 December 2019
is a regular ring; b) every principal left ideal of $ R $
is generated by an idempotent; c) the principal left ideals of $ R $
14 KB (2,228 words) - 08:10, 6 June 2020
...ries. The series (1) is assumed to converge if and only if its regular and principal parts converge. Properties of Laurent series: 1) if the domain of convergen
while the principal part is
8 KB (1,203 words) - 10:36, 20 January 2024
A linear differential or pseudo-differential operator with an invertible principal symbol (see [[Symbol of an operator|Symbol of an operator]]).
with principal symbol $ \sigma _ {A} ( x , \xi ) $.
12 KB (1,764 words) - 05:06, 24 February 2022
to the surface and two mutually orthogonal principal unit tangent vectors $ \mathbf r _ {1} $
1 KB (219 words) - 17:32, 5 June 2020
a) the principal moments of inertia $ \lambda = ( \lambda _ {1} , \lambda _ {2} , \lambda
c) the principal angular momentum is perpendicular to the direction of gravity, i.e., $ \l
4 KB (547 words) - 19:42, 5 June 2020
...m (see [[Jordan matrix]]). For $n>1$ the ring $k[x_1,\ldots,x_n]$ is not a principal ideal ring.
...led a zero of a system of polynomials $F \subset k[x_1,\ldots,x_n]$ if the value of each polynomial from $F$ at this point is $0$. For a ring of polynomials
6 KB (1,011 words) - 06:01, 30 September 2023
...f classical mechanics]], postulated by H. Hertz [[#References|[1]]] as the principal law of a mechanics developed by himself. In it, unlike in Newtonian mechani
2 KB (244 words) - 16:54, 7 February 2011
$H$ is said to be the principal congruence subgroup corresponding to $\fp$. The concept of a congruence sub
1 KB (233 words) - 20:51, 28 November 2014
of degree one is called a principal polarization.
[[#References|[3]]], while in the case of surfaces of principal type
4 KB (644 words) - 13:06, 17 April 2023
is known as the principal character and is denoted by $ \chi _ {0} $:
The principal properties of Dirichlet characters are expressed by the formulas
10 KB (1,462 words) - 11:49, 26 March 2023
...duced the concepts of composition and chief series (cf. [[Principal series|Principal series]]) for such groups and proved that the indices of two series of the
4 KB (660 words) - 20:00, 11 April 2014
One of the methods for studying boundary value problems for differential equations with variable coefficients by means of
is called the principal part of $ L $,
7 KB (986 words) - 14:54, 7 June 2020
...ation of principal analytic fibrations (cf. [[Principal analytic fibration|Principal analytic fibration]]) with base $ X $
6 KB (951 words) - 06:45, 22 February 2022
...[[Taylor formula|Taylor formula]]), by means of which one singles out the principal part of a function. Thus, in the case of an indeterminacy of the type $0/0$
2 KB (420 words) - 18:01, 3 August 2014
be the principal radii of curvature of the surfaces $ u ^ {i} = \textrm{ const } $,
corresponding to the principal direction $ H _ {k} du ^ {k} $,
4 KB (653 words) - 07:55, 25 April 2022
...r a [[factorial ring]]. An element of a ring is prime if and only if the [[principal ideal]] generated by this element is a [[prime ideal]].
1 KB (252 words) - 16:49, 13 August 2023
in which the integrals are understood in the sense of the principal value.
understood in the sense of its principal value, is also called the Hilbert transform of $ f $.
5 KB (696 words) - 22:17, 28 January 2020
and its deviation from the latter depends in its principal part only on $ x $
that defines the principal part of the deviation of the isomorphism $ \Gamma L _ {t} $
9 KB (1,365 words) - 18:18, 19 January 2021
...element set $\{1,2\}$ and taking the value $1$ to elements of $X$, and the value $2$ to elements of $Y$. This identification leads to a general definition o
...alue of a predicate $P(f_1,\ldots,f_k)$ is true if for every $i \in I$ the value of $P(f_1(i),\ldots,f_k(i))$ is true. Moreover, if in all $X_i$ an equation
4 KB (769 words) - 20:29, 7 February 2017
the set of classes of isomorphic real-analytic principal fibre bundles with a real structure Lie group $ G $,
...ne-to-one correspondence with the set of classes of isomorphic topological principal fibre bundles with the same structure group $ G $.
5 KB (671 words) - 16:15, 11 February 2022
...pletely regular. If $a$ is a regular element of a semi-group $S$, then the principal right (left) ideal in $S$ generated by $a$ is generated by some idempotent;
2 KB (246 words) - 08:24, 12 April 2014
An integrally-closed Noetherian domain is an integral domain all principal ideals of which are equi-dimensional, [[#References|[a1]]], p. 196.
2 KB (246 words) - 19:37, 5 June 2020
called the principal category; $ f $
assigns the principal category. The category $ [ \Phi \setminus \Psi ] $(
6 KB (946 words) - 19:42, 5 June 2020
...titions can be performed without introducing an enumeration parameter. The principal programming systems include Al'fa (for M-20 type machines), Algibr (for the
2 KB (260 words) - 17:20, 7 February 2011
A principal fibre bundle $ \widetilde \pi : \widetilde{P} \rightarrow M $
see [[Spinor group|Spinor group]]), covering some principal fibre bundle $ \pi : P \rightarrow M $
6 KB (923 words) - 17:42, 6 January 2024
is a [[Principal ideal|principal ideal]], and the residue field $ R / P $
5 KB (680 words) - 22:17, 5 June 2020
is the principal value of $ \mathop{\rm log} z $.
2 KB (247 words) - 08:03, 6 June 2020
...nd $a_0>0$, to have negative real parts. It consists of the following: All principal minors $\Delta_i$, $i=1,\dotsc,n$, of the Hurwitz matrix $H$ are positive (
2 KB (262 words) - 12:37, 14 February 2020
...of lower-triangular (upper-triangular) matrices with unit elements on the principal diagonal, and $ G _ {0} $
whose principal minors are non-zero, then the decomposition $ G _ {0} = NH N ^ {*} $
5 KB (778 words) - 19:41, 5 June 2020
called a principal crossed homomorphism, or cocycle cohomologous to $ e $.
2 KB (288 words) - 08:28, 18 February 2022
where the integral in (2) must be understood in the sense of the principal value at $ \infty $
the integrals (2) and (3) exist in the sense of the principal value for almost all $ x $.
5 KB (752 words) - 13:19, 20 March 2023
defines the so-called principal connex, expressing the incidence of the point $ x $
2 KB (252 words) - 17:46, 4 June 2020
as a discrete subgroup (it is called the group of principal idèles), while the quotient group $ C _ {L} = I _ {L} / L ^ {*} $,
is principal. All divisors of $ K $
17 KB (2,620 words) - 07:48, 13 February 2024
...ch pair of circles one can choose from the components of their pencils two principal spheres. The latter are defined by the property that for the pencils in ter
...circle). The invariants of a pair of circles are expressed in terms of the principal angles as follows:
14 KB (2,020 words) - 12:57, 13 January 2024
contain identical elements. Cayley tables which are symmetric about the principal diagonal represent commutative binary operations; in particular, this is th
2 KB (300 words) - 16:43, 4 June 2020
...ed by arithmetization (or Gödel numbering); this has now become one of the principal methods of proof theory (meta-mathematics); it is described below.
...rmulas, finite sequences of formulas, etc.) by natural numbers so that the principal properties of these objects (to be an [[Axiom|axiom]] or a logical derivati
5 KB (732 words) - 18:30, 18 November 2018
that the principal results on the arithmetic of quadratic forms over
subgroup in $G_A$, called the subgroup of principal adèles. If $\infty$ is
6 KB (1,071 words) - 06:53, 28 December 2021
is a non-principal character, one has
is a [[principal character]] $ \mathop{\rm mod} d $,
15 KB (2,181 words) - 11:50, 26 March 2023
...(a)\in D_0$ of an element $a\in A^*$ is denoted by $(a)$ and is called the principal divisor of the element $a$. One says that $a\in A^*$ is divisible by the di
...Then for any $c \in K^*$, $K^* = K\setminus 0$, it is possible to define a principal divisor $(c) \in D$. If $c = a/b$ where $a,b\in A^*$, then $(c) = (a)/(b)$.
16 KB (2,805 words) - 02:18, 6 January 2022
...cos\theta+\beta\sin\theta$, $q=-\nu\sin\theta+\beta\cos\theta$, $v$ is the principal normal, $\beta$ is the binormal, $x$ is the torsion of the curve $\Gamma$,
2 KB (281 words) - 15:05, 10 April 2023
...d all be combined in geometrical form: Consider a [[Principal fibre bundle|principal fibre bundle]] $\xi : P \rightarrow M$ over a smooth oriented [[Riemannian
...is sometimes referred to as the Gribov ambiguity. Suitably normalized, the value of the absolute minimum of the Yang–Mills functional just amounts to the
7 KB (1,013 words) - 16:45, 1 July 2020
...Quadratic reciprocity law|Quadratic reciprocity law]]). In addition to the principal reciprocity law of Gauss for quadratic residues, which may be expressed as
2 KB (295 words) - 17:43, 19 December 2014
...curvature and the torsion of the curve, $\theta$ is the angle between the principal normal of the curve and the normal to the surface, and $s$ is the natural p
2 KB (294 words) - 09:19, 7 August 2014
represents the representative fraction called the principal scale of the map. The fundamental characteristic of a map projection at any
is called the linear deformation. The numerical value of the principal scale $ M $
12 KB (1,890 words) - 09:14, 26 March 2023
One of the principal possible equivalent representations (together with the [[Schrödinger repre
...itary transformations of the type \eqref{1} and, consequently, the average value should not depend on the selection of the representation. Differentiation o
2 KB (324 words) - 15:31, 14 February 2020
The principal results in the classification of algebraic surfaces were obtained by geomet
The principal discrete birational invariants of smooth complete algebraic varieties over
6 KB (844 words) - 04:10, 1 August 2014
...tivity of all flat left modules is equivalent to the minimum condition for principal right ideals (cf. [[Perfect ring|Perfect ring]]).
2 KB (327 words) - 07:32, 13 December 2016
...ctions on a manifold|Connections on a manifold]]; [[Principal fibre bundle|Principal fibre bundle]]; or [[#References|[a3]]] for a review). In natural units $\h
...). The existence of a monopole indicates non-triviality of a corresponding principal bundle. The monopole of charge $\hbar \nmid 2 e$ is the connection in the [
10 KB (1,478 words) - 17:30, 25 November 2023
...elds can be characterized as rings over which all modules are free. Over a principal ideal domain a submodule of a free module is free. Near to free modules are
2 KB (351 words) - 12:24, 14 February 2020
...s_1,\dots,s_n$ there exists an $x\in A$ such that the decomposition of the principal ideal $(x)$ into a product of prime ideals has the form
2 KB (355 words) - 13:31, 11 July 2018
...s the local [[Maxwell distribution|Maxwell distribution]], while the limit value of the $H$-function as $t\to\infty$ is equal to the entropy of the ideal ga
From the point of view of statistical mechanics, the principal significance of the Boltzmann $H$-theorem consists in the mathematical expr
2 KB (326 words) - 17:14, 30 December 2018
are the principal curvatures of the surface and $ \alpha $
2 KB (307 words) - 18:47, 11 April 2023
.... An ideal generated by a single element is said to be a [[Principal ideal|principal ideal]].
...ns on their ideals or on the lattice of ideals (see [[Principal ideal ring|Principal ideal ring]]; [[Artinian ring|Artinian ring]]; [[Noetherian ring|Noetherian
13 KB (2,163 words) - 22:11, 5 June 2020
...et all its boundary points) yields a convex set of the same dimension. The principal subject of the theory of convex sets is the study of [[convex body|convex b
...for some unique convex body. Specifying the support function is one of the principal methods of specifying a convex body.
9 KB (1,558 words) - 17:30, 23 October 2017
The kernel of this mapping is called the principal congruence subgroup modulo $ I $.
...subgroup|congruence subgroup]] is a subgroup of finite index containing a principal congruence subgroup. H. Bass, J. Milnor and Serre proved that in $ SL _ {
5 KB (775 words) - 07:59, 26 March 2023
...ts (points, straight lines, planes, second-order curves and surfaces). The principal means of study in analytic geometry are the method of coordinates and the m
The principal technique in the study and classification of such curves is to select a Car
8 KB (1,255 words) - 18:47, 5 April 2020
tangent to the special principal null congruence with rotation (of non-gradient type), and $ \eta _ {\mu \
2 KB (312 words) - 22:14, 5 June 2020
A principal subbundle with structure group $ G $
of the principal bundle of co-frames on the manifold. More exactly, let $ \pi _ {k} : M _
17 KB (2,677 words) - 19:40, 5 June 2020
...onnected problems: 1) a demonstration of the solvability of, say, boundary value problems taking into account changes in their formulation which are a conse
...dysh was the first to note the dependence of the formulation of a boundary value problem on the nature of the degeneration of an elliptic equation on the bo
7 KB (1,059 words) - 17:32, 5 June 2020
admits a [[Principal series|principal series]], then any two decompositions of $ G $
5 KB (786 words) - 22:15, 5 June 2020
...e (curve)]]); 4) the universal Milnor bundle (cf. [[Principal fibre bundle|Principal fibre bundle]]).
6 KB (949 words) - 07:39, 14 June 2022
and is expressed in terms of the principal normal $ \mathbf n $
2 KB (352 words) - 17:32, 5 June 2020
....encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346040.png" />, the principal part of a Laurent expansion (cf. [[Laurent series|Laurent series]])
...no solution — supplementary conditions concerning the compatibility of the principal parts must be imposed.
44 KB (5,974 words) - 22:47, 29 November 2014
...$\epsilon > 0$ there exists a $c = c(\epsilon) > 0$ such that for any non-principal real [[Dirichlet character|Dirichlet character]] $\chi$ of modulus $k$,
...L(z,\chi) > c_1(\epsilon) \neq 0$ for $z > 1 - c_1/k^\epsilon$ for any non-principal real Dirichlet character $\chi$. The constants $c(\epsilon)$ and $c_1(\epsi
5 KB (784 words) - 20:40, 18 October 2014
A [[Connection|connection]] in a [[Principal fibre bundle|principal fibre bundle]] over a (pseudo-)Riemannian manifold whose curvature satisfie
be a principal $ G $-bundle over a Riemannian manifold $ M $,
14 KB (2,063 words) - 02:45, 18 July 2022
is a [[Principal ideal ring|principal ideal ring]] and if all $ H _ {n} ^ {x} $
...of the definition of generalized manifolds. Any cohomology manifold over a principal ideal ring is a homology manifold over $ G $,
9 KB (1,445 words) - 22:10, 5 June 2020
The [[Gaussian curvature|Gaussian curvature]] (the product of the principal curvatures) of a regular surface in Euclidean space $ E ^ {3} $
all even elementary symmetric functions of the principal curvatures
6 KB (891 words) - 19:41, 5 June 2020
...l{P}(M)$ is complete. For any $x \in M$ the set $(x^\Delta)^\nabla$ is the principal ideal generated by $x$. Put $i(x) = (x^\Delta)^\nabla$ for all $x \in M$. T
2 KB (347 words) - 14:30, 18 October 2014
...ssical principal curvatures of the surface (cf. also [[Principal curvature|Principal curvature]]) and $d A$ is the area element of the induced metric on $\Sigma
8 KB (1,192 words) - 16:56, 1 July 2020
A procedure is a principal tool for overcoming complexities in programming by way of systematic partit
...tual parameters between brackets, denotes the "command" of computing the value of the function corresponding to these parameters.
3 KB (398 words) - 16:54, 7 February 2011
lie on the principal diagonal). The covariance matrix is a symmetric positive semi-definite matr
3 KB (387 words) - 16:42, 20 January 2024
...symmetric way). Rickart rings are characterized by the projectivity of all principal left (right) ideals. [[Regular ring (in the sense of von Neumann)|Regular (
2 KB (418 words) - 06:38, 18 October 2017
The principal term of the asymptotic expansion (2) is usually called the WKB approximatio
...ng points have been obtained [[#References|[1]]], [[#References|[4]]]. The principal term of the asymptotic expansion is expressed in the form of Bessel functio
6 KB (879 words) - 19:52, 16 January 2024
...surfaces allowing a [[Deformation over a principal base|deformation over a principal base]]; etc.
...ty of the sphere, and also the following result of D. Hilbert: The maximum principal curvature radius of part of a surface of constant curvature 1 is attained o
11 KB (1,579 words) - 17:32, 5 June 2020
The principal relations are:
9 KB (1,307 words) - 13:15, 9 December 2012
in the opposite case). The double principal linear part $ 2 \delta $
3 KB (387 words) - 08:12, 6 June 2020
and it introduces the structure of a principal $ H $-bundle with base $ X _ {n} $
as a new basis. This means that the principal bundle $ G( \pi , M) \rightarrow M $
15 KB (2,363 words) - 08:15, 18 August 2022
such that the principal ideal $ ( p) $
3 KB (388 words) - 08:27, 6 June 2020
\nu f ( x) & \textrm{ if the value
if it exists, is called the principal computable enumeration of $ A $.
12 KB (1,899 words) - 19:37, 5 June 2020
...itian matrix is called non-negative (or positive semi-definite) if all its principal minors are non-negative, and positive definite if they are all positive. No
2 KB (359 words) - 22:10, 5 June 2020
The principal difficulty of the problem is that all the above conditions must be satisfie
The principal results arrived at in the theory of infinitely-small waves made it possible
7 KB (1,020 words) - 20:50, 31 October 2023
...l for the creation of strong methods of solution of the principal boundary value problems. Studies in potential theory concerning properties of different po
...t half of the 20th century, a great stimulus for the generalization of the principal problems and the completion of the existing formulations in potential theor
38 KB (5,597 words) - 10:29, 30 January 2022
The improper integral (in the sense of the Cauchy principal value)
3 KB (368 words) - 22:10, 5 June 2020
...n of the principal analytic fibrations (cf. [[Principal analytic fibration|Principal analytic fibration]]) with as basis a given reduced Stein space $ X $
10 KB (1,513 words) - 08:23, 6 June 2020
from its tangent plane and is the same in absolute value as the curvature of the corresponding [[Normal section|normal section]]. Th
is the unit principal normal vector of the normal section and $ \mathbf N $
3 KB (426 words) - 08:03, 6 June 2020
...Newton in 1669 (see [[#References|[1]]]) to determine the exponents of the principal terms of algebraic functions. The process of finding successively the terms
axis gives another possible value of $ \epsilon $.
7 KB (1,019 words) - 08:02, 6 June 2020
is a self-dual Abelian variety with principal polarization, and $ T _ {G} ^ {2k+1} ( X) $
is a principal torus.
6 KB (953 words) - 12:29, 29 December 2021
...the envelope of a one-parameter family of spheres having the corresponding principal radii of curvature as their radii. Conversely, if a regular curve $\zeta(s)
3 KB (414 words) - 09:10, 26 March 2023
such that the principal ideal generated by $ p $
3 KB (461 words) - 22:12, 5 June 2020
by the group of so-called principal units $ 1 + pR $.
The group of principal units is isomorphic, for $ m = 2 $
7 KB (1,070 words) - 19:40, 5 June 2020
principal achievements of the theory of Weil. In fact, if $m$ is
[[Riemann hypotheses|Riemann hypotheses]]) and is also the principal
8 KB (1,216 words) - 20:39, 5 March 2012
...ytic continuation|analytic continuation]] of functions. If $ f(z) $ is the principal branch of an analytic function, regular at zero and represented by a series
2 KB (377 words) - 02:56, 2 June 2013
is the sum of all principal minors of order $ m $,
...oot is counted according to its multiplicity. See also [[Eigen value|Eigen value]].
3 KB (434 words) - 11:43, 24 December 2020
One of the principal objects of study in the arithmetic theory of linear algebraic groups are ar
[[Arithmetic group]]), and one of the principal technical instruments is the
11 KB (1,671 words) - 18:19, 24 May 2019
...pondence exists. On manifolds an important role is played by the so-called principal symbol, which is defined for certain pseudo-differential operators and is a
...rators their principal symbols are added and multiplied, respectively. The principal symbol does not change when terms of lower order are added to the operator.
13 KB (1,836 words) - 14:55, 7 June 2020
...entation often is introduced in a different fashion, viz. as a generalized principal series representation (cf. [[Continuous series of representations|Continuou
3 KB (393 words) - 18:23, 26 October 2014
...c manifolds (cf. [[Characteristic manifold|Characteristic manifold]]). The principal problems of this type are the characteristic Cauchy problem (cf. [[Cauchy c
3 KB (535 words) - 22:45, 10 December 2018
...sed by small displacements of the argument are studied. This is one of the principal methods for studying extremal problems (hence the name [[Variational calcul
...rtain neighbourhood defined by the restrictions that are in force. Let the value $t=0$ correspond to $x_0$. As $v$ runs through the set of all parameters, t
3 KB (429 words) - 14:15, 27 August 2014
The following formulas can be used to compute the value of the gradient:
of the argument is to yield the principal linear part of the increment $ f( t) - f( t _ {0} ) $
8 KB (1,190 words) - 19:42, 5 June 2020
...ngs. These correspondences became known as [[Functor|'''functors''']]. The principal advantages of this language — the amount of information conveyed, natural
...40’s, homological algebra had become an independent branch of algebra. The principal domain of application of homological algebra is the category of modules ove
12 KB (1,885 words) - 23:48, 23 April 2017
multivariate $T^2$, principal components analysis, and canomical
mathematical economics. He introduced the idea of principal component
13 KB (1,905 words) - 13:07, 18 March 2023
is the principal branch of an analytic function, regular at the origin and representable by
3 KB (417 words) - 08:17, 6 January 2024
...ian variety (cf. [[Polarized algebraic variety]]) this form determines the principal polarization over the Jacobian $J(X)$
3 KB (414 words) - 18:28, 17 April 2024
with principal symbol $ a _ {m} ( x, \xi ) $,
If the principal symbol $ a _ {m} ( x, \xi ) $
12 KB (1,704 words) - 18:55, 29 December 2021
...kind-complete Riesz space has the projection property and, a fortiori, the principal projection property.
be a Riesz space with the principal projection property, let $ e $
14 KB (2,252 words) - 08:09, 21 January 2024
th entry on the principal diagonal by the valency of the vertex $ v _ {i} $.
The cofactors of all entries on the principal diagonal of $ M $
6 KB (1,006 words) - 06:20, 30 March 2023
The principal term of the asymptotics of $ H ( \phi ( \xi )) $
3 KB (448 words) - 17:33, 5 June 2020
...curvature and the torsion of a cylindrical helical line are constant. The principal normals of a cylindrical helical line intersect the cylinder axis at a righ
...tinguishes between right-handed and left-handed helical lines, i.e. as the value of the coordinate $z$ increases, the rotation of the helical line around th
3 KB (537 words) - 15:42, 29 September 2014
The principal properties of the Bessel kernels $ G _ \alpha (x) $
3 KB (423 words) - 10:58, 29 May 2020
namely the principal fibre bundle of frames on $ M $
which is the principal bundle of frames on $ B $
8 KB (1,110 words) - 16:10, 1 April 2020
The minimal multiplicity of the eigen value $ \lambda = 0 $
...ices; or 5) the algebra of all upper-triangular matrices with zeros on the principal diagonal. For these algebras the ranks are $ n $,
3 KB (460 words) - 08:09, 6 June 2020
...hich, in conjunction with Dehn's lemma and the loop theorem, is one of the principal tools in the topology of three-dimensional manifolds: If $M$ is an oriented
3 KB (466 words) - 08:36, 1 May 2023
...At all points of convergence, the binomial series represents the principal value of the function $ {(1 + z) } ^ \alpha $
3 KB (470 words) - 08:17, 26 March 2023
...reduces the theory of analytic surfaces to that of algebraic surfaces. The principal results on the classification of analytic surfaces were obtained by K. Koda
The principal invariant in the classification of analytic surfaces is the transcendence d
8 KB (1,169 words) - 18:47, 5 April 2020
...re widely spread. These systems are Postnikov systems which consist of the principal fibrations $p_n: X_n \to X_{n-1}$ induced from the standard Serre fibration
A split Postnikov system is a sequence of principal fibrations
11 KB (1,963 words) - 07:35, 8 August 2018
...an infinite set of linearly independent solutions corresponds to the eigen value $ \sqrt {2 / \pi } $
is an eigen value, namely, to every positive number $ \lambda $
3 KB (465 words) - 15:24, 28 February 2022
...bed by these points are called lines of tangency of the ruled surface. The principal surfaces of a complex are the ruled surfaces for which the lines of tangenc
3 KB (539 words) - 17:46, 4 June 2020
...d or principal) and free ultrafilters. An ultrafilter is called trivial or principal if it is the system of all subsets containing a given point; such an ultraf
Then the Ax–Kochen theorem says that for each non-principal ultrafilter $ \Phi $,
10 KB (1,605 words) - 16:57, 13 January 2024
...lic first-order differential operator, used for the notation of one of the principal differential operations of vector analysis. In a rectangular Cartesian coor
4 KB (556 words) - 19:43, 5 June 2020
...and fibre bundles (fibrations, cf. [[Fibration]]) and their sections. The principal types of mappings considered in algebraic topology are arbitrary continuous
...y important notion in algebraic topology is that of a [[deformation]]. The principal types of deformations include: a homotopy, i.e. an arbitrary continuous (sm
19 KB (2,788 words) - 09:43, 28 October 2023
...he direct task is simple, and the solution is obtained by quadratures. The principal difficulty consists in solving the inverse problem. Methods of the theory o
...implest models of medium structure only. However, in a number of cases the principal quantity that is experimentally determined in seismic exploration are the t
13 KB (1,884 words) - 19:41, 5 June 2020
...over an algebraic number field; the finiteness conjecture for the group of principal homogeneous locally trivial spaces; the conjecture of Birch and Swinnerton-
there exists a principal polarized five-dimensional Abelian variety — the intermediate Jacobian $
10 KB (1,376 words) - 11:12, 26 March 2023
...a field of geometric objects, is a section of a bundle associated with the principal bundle of coframes on the manifold $ M $.
Intuitively, a geometric quantity can be considered as a quantity whose value depends not only on the point $ x $
11 KB (1,705 words) - 14:55, 7 June 2020
...has a non-zero [[Socle|socle]]; 3) $R$ satisfies the minimum condition for principal right ideals; 4) $R$ satisfies the minimum condition for finitely-generated
3 KB (491 words) - 19:59, 30 October 2016
...h ideals. Therefore a basic problem in class field theory is to define the principal class in terms of the field $ k $ itself in such a way that its infinite
...ing to Weber is a field $ K / k $ in which only the prime ideals in its principal class $ H _{f} $ , and only these, split completely. Dirichlet's theorem
28 KB (4,440 words) - 22:00, 11 December 2019