...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
...the sense of Lebesgue, but does exist in the sense of the Cauchy principal value, then the integral (2) is called an $ m $-
where the integral is understood in the sense of the Cauchy principal value, generates a continuous integral operator $ \phi \mapsto A \phi $
9 KB (1,369 words) - 22:12, 5 June 2020
...nite discontinuity and the integral logarithm is taken to be the principal value
...also said to define the modified logarithmic integral, and is the boundary value of $ \mathop{\rm li} ( x + i \eta ) \pm \pi i $,
4 KB (515 words) - 18:28, 25 February 2021
An example of a Noetherian ring is any principal ideal ring, i.e. a ring in which every ideal has one generator.
3 KB (539 words) - 18:51, 3 April 2024
The curvature of the curve is equal to the absolute value of the vector $ d ^ {2} \gamma ( t)/dt ^ {2} $,
and the direction of this vector is just the direction of the principal normal to the curve. For the curve $ \gamma $
26 KB (4,098 words) - 17:31, 5 June 2020
of a principal fibre bundle, they must guarantee the invariance of the distribution with r
3 KB (524 words) - 17:36, 13 January 2021
...imensional associative algebra $A$ over a field $F$, while $\Tr(a)$ is the principal trace of the element $a\in A$, which is defined as follows: Let $e_1,\dots,
...rincipal polynomial. For a given element $\a\in A$ the coefficients of the principal polynomial are independent of the basis chosen; for this reason the bilinea
16 KB (2,947 words) - 08:53, 9 December 2016
where the integral is taken to mean the principal value at $ \infty $.
3 KB (473 words) - 20:16, 16 January 2024
tends to become the identity mapping, and that the principal part of its deviation from the identity mapping be defined, with respect to
i.e. when passing to an arbitrary element of the principal fibre bundle $ P $
15 KB (2,221 words) - 10:58, 17 March 2023
have odd degree and are transgressive in any principal fibre bundle of the group $ G $.
3 KB (536 words) - 09:05, 8 April 2023
...heory of linear and bilinear forms in an infinite number of variables. The principal idea of the theory is as follows. Let there be given a complete orthonormal
3 KB (478 words) - 22:10, 5 June 2020
One of the principal uses of the notion of $\mathcal K$-convergence is in formulating versions o
3 KB (545 words) - 12:08, 3 August 2014
with 1 on the principal diagonal is equal to its dimension $ n ( n - 1 ) / 2 $ ,
...of a Lie algebra]]) and is equal to the minimum multiplicity of the eigen value $ \lambda = 1 $
4 KB (566 words) - 15:02, 17 December 2019
is called a Fourier multiplier. The principal problem about Fourier multipliers is to determine conditions on $ \{ \lam
4 KB (568 words) - 08:02, 6 June 2020
the "principal" plane) by rotating them about the lines $ p _ {12} $
4 KB (533 words) - 11:38, 26 March 2023
the mapping in the family must tend to the identity mapping, and the principal part of its deviation from the identity mapping must be determined relative
that is, when passing to an arbitrary element of the principal fibre space $ \Pi $
10 KB (1,455 words) - 06:59, 6 March 2024
is a principal ideal domain [[#References|[4]]], in particular for Lie rings without opera
3 KB (505 words) - 18:57, 5 April 2023
...athematically) speaking, the starting object is a [[Principal fibre bundle|principal fibre bundle]] $ P $
9 KB (1,227 words) - 19:41, 5 June 2020
...ides of these formulas are understood in the sense of the Cauchy principal value and are so-called singular integrals. By taking, under these conditions, $
The Sokhotskii formulas play a basic role in solving boundary value problems of function theory and in the theory of singular integral equation
9 KB (1,249 words) - 14:55, 7 June 2020
...ary differential operator with variable coefficients $a_\a(x)$, then its ''principal symbol'' is the function of $2n$ variables $S(x,p)=\sum_{|\a|=d}a_\a(x)p^\a
3 KB (512 words) - 06:16, 13 June 2022
is the so-called principal scalar invariant.
3 KB (476 words) - 10:58, 29 May 2020
which can be regarded as the principal linear part of the "increment" of $ U $(
The value of covariant differentiation is that it provides a convenient analytic appa
15 KB (2,257 words) - 17:31, 5 June 2020
...various objects of the program that are defined in the program itself. The principal such objects are variables, labels (which name different parts of the progr
...ons in the program; basic operators are the operators of assignment of the value of an expression to a variable, the operator of control transfer (unconditi
12 KB (1,804 words) - 16:56, 7 February 2011
...f objects in this category. The categories of modules over a ring form the principal range of application of this concept.
11 KB (1,715 words) - 22:10, 5 June 2020
...on, under which its net of curvature lines is preserved and therefore is a principal base of the deformation. The umbilical points (cf. [[Umbilical point|Umbili
4 KB (581 words) - 18:20, 28 March 2023
...ations (1), the equations of the tangent at the point corresponding to the value $ t _ {0} $
...the [[Binormal|binormal]]. The figure constructed out of the tangent, the principal normal and the binormal, and also given by the three planes containing pair
33 KB (5,039 words) - 11:46, 26 March 2023
The coefficient $\lambda$ is called an ''[[Eigen value|eigen value]]'' of $A$.
...r subspace. It is called the eigen space of $A$ corresponding to the eigen value $\lambda$ and it coincides with the kernel $\ker(A-\lambda I)$ of the opera
4 KB (571 words) - 12:53, 3 March 2018
...is idempotently generated if and only if the same is true for each of its principal factors [[#References|[10]]]. In an idempotently-generated regular semi-gro
...ular, all inverse semi-groups. A semi-group is orthodox if and only if its principal factors are. There are structure theorems for orthodox semi-groups (see [[#
10 KB (1,515 words) - 08:10, 6 June 2020
...finement of the first. A normal series which cannot be refined is called a principal series (or chief series); while a subnormal series which cannot be refined
4 KB (657 words) - 09:59, 3 January 2021
as an integrability condition for the principal variational equations from which Gauss' equations and the [[Peterson–Coda
4 KB (574 words) - 08:11, 6 June 2020
...{\mathfrak{m}} / i(K_{\mathfrak{m},1})$, where $i$ is the map from $K$ to principal ideals in the ideal group. A coset of $i(K_{\mathfrak{m},1})$ is a ''ray
3 KB (527 words) - 17:00, 23 November 2023
...liptic image. The above integral of the first kind is single-valued on the principal covering surface $ F $ and is a uniformizing variable on $ F $ .
...s $ B _{1} ^{k} \dots B _ {\nu _{k}} ^{k} $ are the coefficients of the principal part of the Laurent expansion of $ f(z) $ in a neighbourhood of $ b _{
11 KB (1,535 words) - 13:55, 18 May 2023
One of the principal possible (along with the [[Schrödinger representation|Schrödinger represe
The average value of the operator $ A $
4 KB (623 words) - 11:53, 25 January 2022
closed fields. One of the principal problems which stimulated the
classifying principal homogeneous spaces of group schemes. Galois
17 KB (2,785 words) - 22:37, 23 December 2014
...s of this theory is based. The analysis of such abstractions is one of the principal tasks of the foundations of mathematics. A careful consideration of the pro
4 KB (591 words) - 17:01, 7 February 2011
...p|Lie group]] of holomorphic automorphisms [[#References|[a7]]]. Here, the principal new feature is the fact that Toeplitz operators $T _ { f }$ (say, on the Ha
...burn, C. Sutherland, "Representations of crossed products by coactions and principal bundles" ''Trans. Amer. Math. Soc.'' , '''299''' (1987) pp. 747–784 {{MR|
8 KB (1,186 words) - 16:46, 1 July 2020
Finding the differential or, in other words, the principal linear part (of increment) of the mapping. The finding of the differential,
...a given point are said to be mutually tangent mappings at this point. The value of the approximating function on an element $ h \in X $,
6 KB (918 words) - 19:35, 5 June 2020
A differential-geometric object on a smooth principal fibre bundle $ P $
4 KB (598 words) - 17:46, 4 June 2020
...ations (cf. [[Algebraic equation|Algebraic equation]]). Solving them was a principal task of algebra in the 16th century and 17th century. If $f(x)$ is a transc
4 KB (692 words) - 13:38, 31 July 2014
...1]</td> <td valign="top"> H. Cartan, "Cohomologie réelle d'un espace fibré principal différentiable" , ''Sém. H. Cartan 1949/50, Exp. 19–20'' (1950)</td></t
4 KB (532 words) - 18:56, 29 February 2024
...onnections on a manifold|Connections on a manifold]]; [[Principal G-object|Principal $G$-object]]) and a section $\phi$ of the associated adjoint bundle of Lie
...} ( 2 )$, where the charge is a positive integer $k$. The absolute minimum value of the functional is then $8 \pi k$ and the coefficient $m$ in the asymptot
11 KB (1,671 words) - 16:59, 1 July 2020
the product of the principal curvatures of $ M ^ {k} $
is one of the principal surfaces of this tensor. This is always the case if $ V ^ {3} $
10 KB (1,530 words) - 08:22, 6 June 2020
but also its principal submatrices of order $ n $
operations. The condition that the principal submatrices are non-singular can be relaxed, while still needing only $ O
11 KB (1,584 words) - 22:13, 5 June 2020
''for a matrix eigen value problem''
Methods for finding the eigen values and eigen vectors (or a principal basis) of a matrix, omitting the preliminary calculation of characteristic
19 KB (2,830 words) - 19:42, 27 February 2021
...n of the existence of convex hypersurfaces with given elementary symmetric principal curvature functions $ \phi _ \nu ( \xi ) $
4 KB (575 words) - 08:00, 6 June 2020
...ined correspondingly. Firs may be regarded as generalizing the notion of a principal ideal domain.
4 KB (594 words) - 12:28, 12 February 2022
takes the value $ x $
...theorem on central isomorphism of direct decompositions of groups having a principal series (cf. [[Krull–Remak–Schmidt theorem|Krull–Remak–Schmidt theor
4 KB (680 words) - 19:35, 5 June 2020
...mpletion of the rational field $\QQ$ with respect to the $p$-adic absolute value $|\cdot|_p$, $p < \infty$, and view $\RR$ as the completion with respect to
...njecture states that the representation $(\pi)_\infty$ of $GL(2,\RR)$ is a principal series $\pi(i t_1, i t_2)$ with trivial central character and
11 KB (1,656 words) - 06:13, 1 April 2023
with zero mean value over the unit sphere $ S ^ {n - 1 } = \{ {x } : {x \in \mathbf R ^ {n} ,
denotes the principal value of the integral. In the one-dimensional case the Calderón–Zygmund operat
5 KB (725 words) - 09:06, 26 March 2023
The principal properties of $ G $
lies in a minimal principal ideal $ ( \Delta _ {i} ) $;
12 KB (1,694 words) - 06:42, 26 March 2023
from the [[Binormal|binormal]] vector to the [[Principal normal|principal normal]] vector.
to point in the direction of a principal vector of the [[Ricci tensor|Ricci tensor]] of $ V ^ {3} $.
16 KB (2,421 words) - 14:56, 7 June 2020
Any principal ideal domain is a Dedekind ring. If $ R $
4 KB (637 words) - 17:32, 5 June 2020
==Singular value factorization.==
An $(n\times n)$-matrix $A$ (over a field) such that the leading principal minors are non-zero,
9 KB (1,294 words) - 22:47, 28 February 2015
...ages/m/m064/m064180/m0641802.png" /> is called a mixed or initial boundary value problem if the desired solution, as well as the boundary conditions, must a
...dary value problems for parabolic equations and systems|Mixed and boundary value problems for parabolic equations and systems]]).
31 KB (4,290 words) - 01:21, 14 February 2013
...ediaofmath.org/legacyimages/d/d032/d032080/d03208010.png" />, while if the
value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
...be solved into
principal terms and terms which are small as compared with
principal terms are often employed in practical work. Small parameter methods (cf. [[
32 KB (4,604 words) - 16:55, 7 February 2011
...yed by the Hurwitz matrix, and that of the first column by the sequence of principal minors $ \Delta _ {i} $,
4 KB (698 words) - 15:52, 10 April 2023
is a commutative principal ideal ring, $ V $
4 KB (640 words) - 03:29, 21 March 2022
algebraic varieties consisting of all principal homogeneous spaces
3 KB (543 words) - 15:48, 17 February 2012
...be encountered in the oldest mathematical texts, which indicates that the principal properties of these operations were known even in early antiquity. In parti
The principal subject dealt with by the algebra of the 18th century and 19th century were
17 KB (2,478 words) - 16:09, 1 April 2020
The principal question which arises when the systems (1)–(3) are changed is the constru
5 KB (679 words) - 03:42, 21 March 2022
...$ be the adèle group and let $G_{\mathbf Q}\subset G_A$ be the subgroup of principal adèles. Then $G_{\mathbf Q}$ is a discrete subgroup in $G_A$; moreover, $G
4 KB (664 words) - 12:18, 30 December 2018
The principal qualitative variational principle is the [[Lindelöf principle|Lindelöf pr
The principal quantitative variational principle obtained by M.A. Lavrent'ev [[#Reference
6 KB (783 words) - 18:43, 11 October 2023
the principal value of $ \mathop{\rm ln} \phi _ \xi ( t) $
5 KB (661 words) - 12:22, 17 January 2021
...nnected with the rejection and renewal of the elements of some system. The principal concepts in renewal theory are those of a renewal process and renewal equat
4 KB (599 words) - 08:10, 6 June 2020
...ction if and only if $\eta ( W ) d g ( W ) \in i \mathbf{R}$, and $W$ is a principal curvature direction if and only if $\eta ( W ) d g ( W ) \in {\bf{R}}$.
4 KB (595 words) - 16:57, 1 July 2020
The true order of the value of $\theta$ is unknown (1988). According to one hypothesis,
Regarding the value of $ \Delta _{k} (x) $,
11 KB (1,578 words) - 11:58, 26 March 2023
whose principal linear parts belong to the group $ \mathop{\rm GL} ( n , K ) $.
determines a principal subbundle $ \pi _ {k} : B ^ {k} \rightarrow M $
11 KB (1,659 words) - 08:01, 21 January 2024
...the [[Differential|differential]] of a function of one variable. It is the principal linear part of the increment of the functional in a certain direction; it i
5 KB (687 words) - 08:27, 6 June 2020
The canonical connection can also be viewed as a connection on the principal $ \mathop{\rm GL} _ {n} ( \mathbf C ) $-bundle $ \widetilde \pi : P
4 KB (681 words) - 03:41, 21 March 2022
One defines the principal curvature, principal direction in the direction of $ \xi $,
Using elementary symmetric functions, one can construct various principal-curvature functions, such as, for example, the mean curvature
25 KB (3,789 words) - 19:41, 5 June 2020
...such as: regular semi-groups with a finite number of idempotents in every principal factor [[#References|[2]]], Clifford inverse semi-groups, and completely $0
5 KB (643 words) - 16:42, 19 September 2014
as $n \rightarrow \infty$. The value of the infinite product is the limit
An infinite product has the rearrangement property (i.e. its value is independent of the order of the factors) if and only if it is absolutely
5 KB (741 words) - 17:20, 24 April 2016
...rticular cases are '''[[Vector bundle]]''', '''[[Tangent bundle]]''', '''[[Principal fibre bundle]], $\dots$
To define a vector bundle, one has in addition to the principal definition assume the following:
32 KB (5,299 words) - 12:27, 12 December 2020
is a non-principal character mod $ D $,
4 KB (617 words) - 13:30, 14 May 2022
In particular, a Gram determinant is equal to zero if any of its principal minors (which is also a Gram determinant) is zero.
5 KB (733 words) - 19:42, 5 June 2020
...omorphic'' to $F_a$. The diffeomorphism maps every point $z\in F_a$ to the value $s_z(b)\in F_b$, where $s_z(\cdot)$ is the uniquely defined horizontal sect
==Connections on principal and vector bundles==
16 KB (2,769 words) - 08:44, 12 December 2013
The principal property by which branching processes differ from Markov processes is that
is indecomposable and aperiodic, it has a simple positive eigen value $ \lambda $
4 KB (636 words) - 06:29, 30 May 2020
the mean-value property
In the theory of harmonic functions an important role is played by the principal fundamental solutions of the Laplace equation:
21 KB (3,095 words) - 19:43, 5 June 2020
...hen solving [[Boundary value problems of analytic function theory|boundary value problems of analytic function theory]] (see also [[#References|[3]]]) and w
...in these formulas are singular integrals and exist in the Cauchy principal-value sense (see [[#References|[3]]], and also [[Hilbert singular integral|Hilber
5 KB (698 words) - 18:20, 26 January 2022
...scale of Gevrey classes $G ^ { S }$ and, if possible, to find the critical value(s) of $s$, i.e. those for which a change of behaviour occurs. In particular
...rator|Symbol of an operator]]; [[Principal part of a differential operator|Principal part of a differential operator]]). The operator is called $G ^ { S }$-hypo
21 KB (3,112 words) - 08:01, 6 February 2024
The smallest possible value of $ m $
is called the principal symbol of a classical pseudo-differential operator of order $ m $.
29 KB (4,077 words) - 04:02, 4 January 2022
called the value group of the valuation $ v $.
...he so-called canonical valuation for its field of fractions, for which the value group is $ K ^ {*} / U $,
14 KB (2,135 words) - 08:27, 6 June 2020
...integral exponential function is understood in the sense of the principal value of this integral:
here the value of $ \mathop{\rm ln} ( - z) $
6 KB (811 words) - 19:20, 11 January 2024
...s do not change from one trial to another. Bernoulli trials are one of the principal schemes considered in probability theory.
4 KB (636 words) - 13:12, 6 February 2020
for $\mu \neq 0$; he also gave other examples involving principal fibre bundles.
5 KB (681 words) - 17:43, 1 July 2020
...s an amalgam of four Abelian groups that is not imbeddable in a group. The principal problem concerning amalgams of groups is, generally speaking, as follows. L
5 KB (883 words) - 16:10, 1 April 2020
on a principal fibre bundle $ P $
5 KB (683 words) - 14:33, 21 March 2022
are called the principal open sets. They form a basis for the topology on $ \mathop{\rm Spec} A
5 KB (778 words) - 13:13, 6 January 2024
...Suppose also that $0 < \epsilon < 1$ and that the $\log \alpha_i$ are the principal values of the logarithms. If there are rational integers $b_1,\ldots,b_n$,
5 KB (776 words) - 08:31, 23 November 2023
A method for studying boundary value problems in mathematical physics by reducing them to integral equations; th
is a principal [[Fundamental solution|fundamental solution]] of $ L $,
4 KB (653 words) - 08:07, 6 June 2020
consisting of Lie algebras in which all principal factors have order $ \leq m $,
5 KB (790 words) - 22:16, 5 June 2020
is called the principal symbol of the operator $ A $.
is constant and the principal symbol $ \widetilde{b} _ {0} $
24 KB (3,360 words) - 19:39, 5 June 2020
be a principal $ G $-
bundle (cf. also [[Principal G-object|Principal $ G $-
16 KB (2,293 words) - 19:37, 5 June 2020
...ith the tangent bundle is the frame bundle of the manifold $M$, which is a principal bundle. The bundle $\tau^*(M)$ dual to the tangent bundle $\tau (M)$, calle
5 KB (893 words) - 11:45, 20 May 2017
...in arithmetical progressions have their own specific features. One of the principal achievements in this field, due to C.L. Siegel (1935), is as follows: Let $
...s does not apply to the problem of the asymptotic behaviour of the average value of $\tau(n)$ or, to put it in a somewhat different manner, the problem of t
27 KB (4,516 words) - 18:38, 18 October 2014
The principal use of equivariant cohomology is in equivariant obstruction theory and in s
5 KB (772 words) - 11:12, 20 January 2021
One of the principal objects of study in algebraic geometry. The
5 KB (706 words) - 13:35, 17 March 2023
..., a compact generalized Hopf manifold arises as the total space of a flat, principal $ S ^ {1} $
5 KB (697 words) - 22:11, 5 June 2020
is an abbreviation for the value $ \mathop{\rm exp} ( w) $
in (*) one obtains the principal value. An interesting example is obtained if $ z = a= i $:
6 KB (860 words) - 17:32, 6 January 2024
...d using the boundary mapping on $K _ { 1 }$ applied to $\sigma ( D )$, the principal symbol of $D$ (cf. also [[Symbol of an operator|Symbol of an operator]]). T
5 KB (698 words) - 17:02, 1 July 2020
In modern studies principal stress is laid on regular hypercomplex functions analytic according to Fuet
In this case the analogues of power series are easier to construct, but the value of the integral depends on the path. For a commutative algebra $ \mathfra
5 KB (724 words) - 22:11, 5 June 2020
that acts according to the rule: the value of the section $ d _ {m} ( u) $
The symbol (principal system) of a linear differential operator $ A : E \rightarrow F $
25 KB (3,768 words) - 09:07, 14 June 2022
...of $M$ (cf. also [[Minor|Minor]]) comes from a spanning tree of $G$ having value $\pm 1$. In the case of the complete graph $K _ { \nu }$ (with some orienta
...$th row of $L^-$ (respectively, $i$th column of $L ^ { + }$). In fact, the principal minor of $L$ obtained by deleting rows and columns indexed by $v _ { i_1 }
10 KB (1,636 words) - 08:14, 15 February 2024
...nen-Lo\`eve Representation, Longitudinal Data, Nonparametrics, Panel Data, Principal Component, Registration, Regression,
...pansions, penalized splines, or local polynomial smoothing, and functional principal
29 KB (4,005 words) - 20:28, 13 March 2024
...imple near-rings if and only if: a) it satisfies the minimum condition for principal ideals; b) $A$ does not contain ideals with zero multiplication; and c) any
5 KB (767 words) - 11:23, 27 October 2014
...e $M$, denoted by $T P / G$, and sometimes called the Atiyah bundle of the principal bundle $P$. This vector bundle is a Lie algebroid, with bracket induced by
12 KB (1,916 words) - 16:55, 1 July 2020
is a polarized Abelian surface. A principal polarization is a polarization of degree $ 1 $.
is twice a principal polarization on $ A $.
18 KB (2,511 words) - 06:25, 26 March 2023
...ed to express all the specific characteristics of statistical systems. The principal difficulties involved in the study of the functions (3) or (5) are caused b
5 KB (743 words) - 10:59, 29 May 2020
...ristics are related with the use of computer resources by the program, the principal resources being the execution time and the amount of memory used.
5 KB (775 words) - 17:05, 7 February 2011
...motions forms part of the condensation segment with sequence number $k$. A principal problem is the distribution of lacunas and condensation segments of possibl
6 KB (940 words) - 09:43, 11 August 2014
...e domain of definition, a fairly complete theory of the principal boundary value problems is available. See [[Differential equation, partial, discontinuous
...[5]</TD> <TD valign="top"> O.A. Oleinik, "Solution of fundamental boundary value problem for second-order equations with discontinuous coefficients" ''Dokl.
5 KB (714 words) - 17:33, 5 June 2020
...der the natural ways of combination of algorithms. This fact is one of the principal arguments in favour of the basic assumption about algorithms (the [[Church
6 KB (805 words) - 17:59, 1 April 2020
...ist quite a number of convergence criteria. The following criteria are the principal ones.
...n series with non-negative terms, it is often reasonable to single out the principal part of its $ n $-
29 KB (4,393 words) - 19:21, 27 January 2020
is the principal part (in some sense) of the vector function $ f $,
5 KB (748 words) - 08:05, 6 June 2020
...ctions: $x\land y$, $x\lor y$, $Cx$, $x\to y$, etc. When this is done, the value of any complex proposition, constructed with the aid of the fundamental log
...uments can be defined by the subset of the corners at which it assumes the value "1". This subset, when written out as a matrix whose rows are selections of
5 KB (836 words) - 12:39, 14 February 2020
on the other hand, a conformal connection is the connection in the reduced principal fibre bundle that is defined by the given conformal structure.
6 KB (864 words) - 17:46, 4 June 2020
consisting of the principal ideals. The divisor class group is Abelian and is usually denoted by $ C
5 KB (820 words) - 19:36, 5 June 2020
...e. The fundamental reference for this subject is [[#References|[a1]]]. The principal methods for proving the $h$-principle are removal of singularities, continu
5 KB (700 words) - 00:42, 15 February 2024
...^ { \prime }$). In particular, $T _ { A }$ is completely determined by its value on $\mathbf{R}$, which is the Weil algebra $A$ itself. Even its algebra str
...by the functorial action. Similarly, they behave nicely on vector bundles, principal bundles, etc.
12 KB (1,876 words) - 06:30, 15 February 2024
...simal deformations has numerous applications in mathematics and mechanics. Principal applications include problems of isometric imbedding by the method of exten
6 KB (807 words) - 16:00, 12 April 2023
...is called a left $A$-module if there is a mapping $A\times M \to M$ whose value on a pair $(a, m)$, for $a \in A$, $m \in M$, written $am$, satisfies the a
...is called a left $G$-module if there is a mapping $G\times M \to M$ whose value at a pair $(g, m)$, where $g \in G$, $m \in M$, is written as $gm$, and whe
23 KB (3,918 words) - 04:31, 23 July 2018
is a [[Principal homogeneous space|principal homogeneous space]] over $ {\mathcal J} _ {B} ( X) / B $;
15 KB (2,046 words) - 05:46, 13 June 2022
is a principal ideal, i.e. a set of the form $ \{ {x \in X } : {x \leq u } \} $;
If the weights of all non-zero principal ideals are identical, then the Boolean algebra is called uniform; such alge
14 KB (2,077 words) - 21:17, 17 January 2021
There are two principal methods for constructing operators $ L _ {h} $
...ow not only provide discretizations of differential equations and boundary value problems, but also the solution of these problems. References [[#References
11 KB (1,674 words) - 14:30, 7 April 2020
...tation and electromagnetism on the basis of general relativity theory. The principal problems of geometro-dynamics at this stage may be stated in a simplified m
5 KB (712 words) - 19:41, 5 June 2020
The principal difficulty in problems involving a heavy fluid is that the boundary conditi
...precludes any analytic approximation. Moreover, these problems display the principal drawback of equations of elliptic type — the propagation of the effect of
25 KB (3,871 words) - 20:01, 4 January 2024
...troduction of variable magnitudes into mathematics by R. Descartes was the principal factor in the creation of differential calculus. Differential and integral
denote the corresponding increment of the value of the function. If there exists a (finite or infinite) limit
37 KB (5,252 words) - 11:45, 26 March 2023
$#C+1 = 113 : ~/encyclopedia/old_files/data/B017/B.0107350 Boundary value problem, elliptic equations
The classical boundary value problems are special cases of the following problem: Find a solution to equ
13 KB (1,836 words) - 06:29, 30 May 2020
...[#References|[4]]], [[#References|[6]]]. It is based on solving a boundary value problem with an oblique derivative for the Laplace equation. The field of t
is expressed in terms of its value $ W _ {0} $
11 KB (1,690 words) - 19:41, 5 June 2020
...roblem for the [[Laplace equation|Laplace equation]] in plane domains. The principal idea of the Schwarz alternating method as applied to the simplest case of t
Schwarz' alternating method is also used to solve boundary value problems of a more general nature for general equations of elliptic type (i
6 KB (852 words) - 08:12, 6 June 2020
form a principal fibre bundle over $ M ^ {2n} $
6 KB (837 words) - 08:24, 6 June 2020
...ms often have no solution for all values of $b$ so that, if an approximate value of $b$ is specified, it must be borne in mind that a formal solution to the
The principal task of the theory of programming may be said to be the achievement of more
13 KB (1,892 words) - 20:11, 31 December 2018
A norm is also called an absolute value or a multiplicative valuation. Norms may (more generally) be considered on
the ordinary [[Absolute value|absolute value]] or modulus of the number $ x \in \mathbf R $,
6 KB (1,003 words) - 21:35, 13 January 2021
...numerical characteristics and metric properties of Boolean functions. The principal parts of this theory are concerned with properties of "almost-all" Boolean
be the minimal value of the order of the neighbourhood in which $ S _ {k} ( \mathfrak A , f) $
14 KB (2,056 words) - 06:28, 30 May 2020
...ntial boundary value problem by difference boundary value problems]]). The principal idea of the method consists in regarding the solution $ u _ {h} ( x) $
which is an approximate value of the sought solution $ u ( x) $—
7 KB (1,037 words) - 17:08, 13 January 2024
...(cf. [[Automata, experiments with|Automata, experiments with]]). Here, the principal problem is to obtain certain information about the structure of an automato
7 KB (975 words) - 18:49, 5 April 2020
with principal part $ P _ {m} $
6 KB (850 words) - 22:11, 5 June 2020
The principal contribution of Arveson in this connection is a result that generalizes a t
The principal application of Arveson's theorem is to this very general set up: Suppose $A
14 KB (2,151 words) - 17:43, 1 July 2020
an eigen vector of $A$ and $\lambda$ is called an eigen value of $A$. An
element $\lambda\in K$ is an eigen value of a matrix $A$ if and only if it is a
18 KB (3,377 words) - 17:54, 2 November 2013
..., where the integral on the right-hand side is taken to mean the principal value (6). The Fourier integral is also studied when $ f $
7 KB (974 words) - 19:39, 5 June 2020
...systems [[#References|[a3]]] and [[#References|[a4]]]; Hitchin systems for principal $G$-bundles [[#References|[a5]]]; and quantized Hitchin systems with applic
5 KB (752 words) - 15:33, 4 October 2014
The principal subject of the Galois theory of rings are the correspondences:
6 KB (975 words) - 04:16, 19 March 2022
...may be, the method is used throughout Book 12 of Euclid's Elements as the principal deductive tool. Euclid's chain of reasoning may be written in modern form a
are equal to each other and to a constant value $ k $,
22 KB (3,357 words) - 17:34, 1 January 2021
The square of the principal linear part of the increment of the length of the arc $ MM ^ \prime $
6 KB (917 words) - 09:59, 12 October 2023
denotes the Cauchy principal value, $ \delta ( \nu - \mu ) $
...ign="top"> W. Greenberg, C. van der Mee, V. Protopopescu, "Boundary value problems in abstract kinetic theory" , Birkhäuser (1987)</TD></TR><TR><TD
6 KB (887 words) - 20:31, 25 February 2021
...additive problems in which the integrals over "major" arcs, which yield a "principal" part of $J_k(N)$, can be investigated fairly completely, while the integra
5 KB (868 words) - 21:24, 18 November 2016
The principal divisors form a subgroup $ P(X) $
...[[Vector bundle, algebraic|Vector bundle, algebraic]]). The degree of the principal divisors on a smooth projective curve is zero, and thus all divisors in one
22 KB (3,307 words) - 17:02, 17 December 2019
...n of the problem may be described entirely in terms of the fully specified principal submatrices, the set of all solutions can be parametrized by a fractional-l
...ngular patterns, namely for the case when the set of given entries forms a principal submatrix, a full-width submatrix, a full-length submatrix, or an off-diago
14 KB (2,045 words) - 15:19, 28 February 2022
...onstitute the basis for various computational schemes for solving boundary value problems, including the finite element and collocation methods (cf. [[Galer
...roblems for equations of elliptic type, as a rule, the energy space of the principal part of the corresponding differential operator is chosen as $ H $.
7 KB (1,019 words) - 11:15, 21 March 2022
...e corresponding quotient semi-lattice is isomorphic to the semi-lattice of principal ideals of $ S $.
7 KB (947 words) - 17:44, 4 June 2020
''complete eigen value problem''
...n values but also to construct a basis composed of eigen vectors, or root (principal) vectors, for the matrix.
15 KB (2,340 words) - 08:54, 21 January 2024
...ven though such a computation is limited by the necessity of selecting the value of, for example, a procedure out of a given finite population of "routine
...ain an assignment statement, or any sequence of statements that produces a value. Together with the possibility of computing names and procedures, and also
6 KB (897 words) - 20:29, 5 December 2023
...a semi-lattice, see [[Idempotents, semi-group of]]); each left or right [[principal ideal]] of $S$ has a unique generating idempotent. Every group is an invers
7 KB (986 words) - 19:33, 2 January 2018
is the principal ideal generated by $ \prod _ {i = 1 } ^ {m} g _ {b _ {i} } ( \mathfrak
7 KB (1,035 words) - 05:58, 19 March 2022
by choosing the directions of the coordinate axes along the so-called principal axes (axes of symmetry) of the conic section. If $ A $
...ts the sum of whose distances from two given points (the foci) has a given value. Another planimetrical definition of conic sections can be given, encompass
6 KB (955 words) - 10:00, 26 March 2023
...lign="top"> D.V. Alekseevskii, A.M. Vinogradov, V.V. Lychagin, "The principal ideas and methods of differential geometry" , ''Encycl. Math. Sci.'' , '''2
7 KB (1,006 words) - 08:24, 6 June 2020
...thematical cybernetics and discrete mathematics, and reflect the following principal features of real and abstract control systems (cf. [[Control system|Control
...ssociated with real functional systems splits into the concepts of a truth-value functional system and a sequential functional system. In the first case, th
7 KB (1,087 words) - 19:40, 5 June 2020
The principal theorem of the local theory of deformations, proved by M. Kuranishi [[#Refe
...s to the ones above. In particular, for any [[Principal analytic fibration|principal analytic fibration]] (bundle)
41 KB (5,916 words) - 11:24, 26 March 2023
...ndations of classical [[Differential geometry|differential geometry]]. Its principal parts are the theories of curves and surfaces, and they were intensively de
==The subject of geometry, its principal branches and connection with other branches of mathematics.==
20 KB (2,829 words) - 18:41, 11 December 2020
...eal constant coefficients (cf. [[Principal part of a differential operator|Principal part of a differential operator]]).
boundary conditions, and these have been completely described. Boundary value problems take a simple form when the expression $ L _ {0} ^ \prime $
32 KB (4,393 words) - 08:22, 6 June 2020
The principal problems of the theory of harmonic spaces include the theory of solvability
7 KB (1,087 words) - 19:43, 5 June 2020
[[Principal homogeneous space|principal homogeneous space]] over
one chooses the one for which the absolute value of the discriminant
19 KB (3,251 words) - 20:37, 19 September 2017
...conformal invariant of weight $k = - 1$ whenever $n \geq 4$. Although the principal feature of $W ( g )$ is that $W ( g ) = 0$ if and only if the Riemannian ma
..., g \}$ of $( M , g )$, and $\Phi \{ M , g \} \in S ^ { 1 }$ is a critical value if and only if $\{ M , g \}$ is conformally flat. S.S. Chern [[#References|
40 KB (5,895 words) - 17:45, 1 July 2020
is a sheaf of principal ideals. For a connected non-singular algebraic variety this condition means
6 KB (927 words) - 22:11, 5 June 2020
...], [[#References|[9]]]). Here the domains of basic resonance form those of principal resonance, and together with the domains of combined resonance there appear
10 KB (1,385 words) - 09:16, 13 May 2022
...a coordinate system; cf. also [[Principal part of a differential operator|Principal part of a differential operator]]; [[Symbol of an operator|Symbol of an ope
...tangent bundle $T M$ of $M$. If $P$ is an elliptic operator as above, its principal symbol $a = \sigma ( P )$ defines a [[K-theory|$K$-theory]] class $[ a ]$ w
35 KB (5,243 words) - 17:45, 1 July 2020
A branch of modern algebra. Its principal task is to represent any [[Ideal|ideal]] of a ring (or of another algebraic
7 KB (1,035 words) - 20:23, 4 April 2020
have an envelope is called a principal surface of $ \Gamma _ {t} $.
8 KB (1,191 words) - 06:39, 9 April 2023
...in "block index" form $C _ { A B }$ where each capital index can take a value $1$–$6$ and represents a pair of skew indices according to the scheme:
...|[a5]]]. Associated with each Petrov type at at point $p$ in $M$ are $4$ "principal" null directions (some possibly coincident). These directions are instrume
6 KB (975 words) - 15:02, 30 March 2023
the group of principal divisors $P(X)$, i.e. divisors of rational
6 KB (948 words) - 11:41, 31 January 2022
...cter of the surface is clearly illustrated in this figure, which shows the principal sections of the surface at an arbitrary point $ O $.
be their normal curvatures, i.e. the principal curvatures at the point $ O $(
48 KB (7,189 words) - 08:02, 6 June 2020
...ll syntactic problems in a given sentence may be considered as solved. The principal difficulties in the creation of completely automatic systems of high-qualit
7 KB (1,008 words) - 17:03, 7 February 2011
...ces|[a5]]] for an example in the case of double cosets with respect to the principal congruence subgroup
6 KB (1,050 words) - 00:38, 15 February 2024
...eneral meaning it is considered nowadays as the analysis of connections in principal fibre spaces or fibre spaces associated to them. There is a way of defining
7 KB (1,108 words) - 19:43, 13 August 2023
equivariant mapping of the principal $ \mathop{\rm GL} ^ {k} ( n) $-
The value of the Lie derivative $ {\mathcal L} _ {X} Q $
9 KB (1,337 words) - 22:16, 5 June 2020
...obtained. See [[Boundary value problem, complex-variable methods|Boundary value problem, complex-variable methods]].
and retains their principal features. For this reason solutions of equations of the type (7) are known
30 KB (4,345 words) - 09:02, 21 January 2024
...ues), from its behaviour with respect to a specific cone in the space (the principal eigenvalues of a positive operator), or from the scalar product (the spectr
...blems is explained by the fact that one can often successfully isolate the principal part and perturbation part in the same terms (by redistributing the coeffic
16 KB (2,424 words) - 08:22, 6 June 2020
The principal trends in the theory of Hilbert spaces with an indefinite metric are the sa
Beginning with the study of L.S. Pontryagin [[#References|[1]]], the principal problem of the theory is the existence of semi-definite invariant subspaces
17 KB (2,626 words) - 20:43, 16 January 2024
and the integral is understood in the sense of the Cauchy principal value (see [[#References|[1]]]–[[#References|[4]]]).
...TR><TD valign="top">[7]</TD> <TD valign="top"> T.D. Dzhuraev, "Boundary value problems for equations of mixed and mixed-composite type" , Tashkent (1979
7 KB (1,014 words) - 08:26, 6 June 2020
in the principal fibration of frames over $ M $
9 KB (1,280 words) - 08:10, 6 June 2020
...e concept of an analytic space was introduced in the 1950's and became the principal subject of the theory of analytic functions; many fundamental results obtai
7 KB (1,028 words) - 13:31, 25 April 2014
The principal problem in the theory of differential inequalities is to describe, starting
The theory of boundary value problems was also studied for differential inequalities. The inequality $
8 KB (1,230 words) - 19:35, 5 June 2020
.... [[Algebraic curve|Algebraic curve]]; [[Modular form|Modular form]]). The principal congruence subgroup $ \Gamma ( N) $
7 KB (1,031 words) - 18:33, 13 January 2024
...area, the surface integral of the [[elementary symmetric function]] of its principal curvatures (in the case of a $C^{2}$-smooth body), and also the correspo
of the principal radii of curvature, regarded as a function of the normal to the sphere $
12 KB (1,811 words) - 05:48, 26 April 2023
is the [[Cauchy operator|Cauchy operator]] (i.e. the fundamental solution or principal solution) of the system
8 KB (1,194 words) - 08:14, 6 June 2020
...h’s method to non-compact manifolds and analytic imbeddings, and also by a principal refinement of the process of extension with respect to a parameter, it has
7 KB (1,113 words) - 16:28, 16 January 2017
...ll Heyting algebras (see [[Brouwer lattice]]); however, for the classes of principal ideal domains, of Brouwer lattices, and of normal distributive lattices, th
7 KB (1,160 words) - 08:00, 6 June 2020
...e exterior algebra of a module is employed in the theory of modules over a principal ideal ring [[#References|[5]]].
7 KB (1,013 words) - 19:38, 5 June 2020
...ong to $D$, then the cardinality of the ultraproduct with respect to a non-principal ultrafilter $D$ on a countable set $I$ is equal to that of the continuum. F
...-saturated systems is given by ultraproducts. For example, if $D$ is a non-principal ultrafilter on a countable set $I$, then $\prod_{i\in I}A_i/D$ is an $\alep
17 KB (2,788 words) - 15:19, 16 December 2020
...which mathematical methods of investigation are extensively employed. The principal task of theoretical astrophysics is the interpretation of results of observ
One of the principal problems in astrophysics is the study of the structure and the evolution of
22 KB (3,445 words) - 09:41, 15 August 2023
...odifications obtained by changing components or the mode of operation. The principal concept — a finite automaton — originated in the mid-20th century in co
...uring the operation. This leads to the concept of a growing automaton. The principal modifications and subclasses of finite automata, together with their most i
18 KB (2,794 words) - 10:01, 25 April 2020
i.e. the value of $ f (z) $
...solution of [[Boundary value problems of analytic function theory|boundary value problems of analytic function theory]], of singular integral equations conn
26 KB (3,804 words) - 08:11, 13 February 2022
...olongements d'une variété différentiable. I. Calcul des jets, prolongement principal. II. L'espace des jets d'ordre $r$ de $V _ { n }$ dans $V _ { m }$. III. Tr
7 KB (1,051 words) - 06:27, 15 February 2024
...c functions (cf. [[Characteristic function|Characteristic function]]) is a principal instrument for studying convergence of probability distributions.
...are the [[Mathematical expectation|mathematical expectation]] (the average value), the [[Dispersion|dispersion]] (or variance), the [[Median (in statistics)
8 KB (1,003 words) - 21:59, 21 November 2018
intersect the level surface of the principal function transversally (cf. [[Transversality condition|Transversality condi
8 KB (1,116 words) - 19:43, 5 June 2020
where the improper integral is understood in the sense of the principal value. For (5) he established the validity of three theorems, which are nowadays
...gn="top">[6]</TD> <TD valign="top"> I.I. Danilyuk, "Nonregular boundary value problems on the plane" , Moscow (1975) (In Russian)</TD></TR><TR><TD vali
8 KB (1,144 words) - 08:02, 6 June 2020
the value of the tensor field $ A $
structure]], that is, the principal subbundle $ P( A) $
8 KB (1,160 words) - 08:05, 6 June 2020
is rarely known in advance, and the principal meaning of (2) is the possibility that it offers of comparing the merits of
In many methods, the principal term of the error increases according to a similar law, while the computati
18 KB (2,636 words) - 08:00, 13 May 2022
occupy the principal diagonal. In affine spaces over algebras one can specify a Hermitian metric
8 KB (1,172 words) - 16:55, 7 June 2020
and the related homogeneous boundary value problem satisfies
...s|[a7]]]. Another interesting boundary value problem is the focal boundary value problem $ y ^ {( i ) } ( x _ {j} ) = 0 $,
8 KB (1,195 words) - 07:36, 23 January 2024
...s, the physical processes taking place in them and their evolution. One of principal problems of astrophysics is the structure and evolution of stars. The theor
8 KB (1,204 words) - 18:48, 5 April 2020
...od may be stated as follows. The Stokes approximation is considered as the principal term of the asymptotic expansion for small Reynolds numbers Re in a neighbo
The boundary-layer equations yield the principal term in a suitably constructed asymptotic expansion of the Navier–Stokes
23 KB (3,690 words) - 20:00, 4 January 2024
as $n \to \infty$, with $\lambda_V(Q)$ the principal eigenvalue of the positive matrix $Q_V$. By the ergodic theorem,
with $ \lambda_V(a)$ the principal eigenvalue of the matrix $(a(i,j)+\delta(i,j)V(j))_{i,j}$.
13 KB (2,280 words) - 10:48, 28 October 2023
are the congruence subgroups containing a principal congruence subgroup $ \Gamma (N) $
8 KB (1,144 words) - 05:57, 13 February 2024
is an integer. The maximum value of the magnitude
or the degree of (1). The maximum value of the magnitudes $ i _ {s} $,
8 KB (1,148 words) - 17:46, 4 June 2020
The two principal surface forms are also constructed: the quadratic form
8 KB (1,146 words) - 19:51, 4 April 2020
...}); in this case the $P_1,\dots,P_s$ form a complete set of non-isomorphic principal indecomposable left $A$-modules, that is, indecomposable $A$-modules into w
6 KB (1,055 words) - 08:14, 22 May 2012
The order of an elliptic function is the number of times it takes each value in its period parallelogram, cf. [[Elliptic function|Elliptic function]].
For each possible value $ \alpha $,
13 KB (2,001 words) - 02:12, 1 March 2022
to find the line for which the functional (1) assumes an extremal value. The value of the derivative $ \dot{x} $
The principal method for finding necessary conditions is the method of variations. Constr
28 KB (4,182 words) - 07:56, 16 April 2023
The principal idea of the Runge–Kutta method was proposed by C. Runge [[#References|[1]
...hods, the Runge–Kutta method, as other one-step methods, only requires the value at the last time point of the approximate solution and allows one to carry
9 KB (1,315 words) - 20:04, 15 January 2024
...ndary from inside a domain by a hypersurface for which a certain number of principal curvatures is positive. In the theory of functions of several complex varia
15 KB (2,101 words) - 17:13, 7 February 2011
...in applications of the theory of functions, e.g. in the theory of boundary value problems for partial differential equations.
The principal problem in the study of weighted spaces is to obtain [[Imbedding theorems|i
9 KB (1,435 words) - 08:13, 13 January 2024
...K$ is the conjugation operator defined on $L [ 0,2 \pi ]$ by the principal value integral
7 KB (1,076 words) - 07:40, 7 February 2024
is called unmixed. In a ring of polynomials, the ideals of the principal series are unmixed or, in modern terminology, a ring of polynomials is a [[
...generalized by C. Chevalley and A.I. Uzkov. Krull had proved a theorem on principal ideals, initiating the dimension theory of Noetherian rings, and also a the
16 KB (2,400 words) - 17:45, 4 June 2020
b) Whether a model of type (1) is unique (identifying it). The principal difficulty in computing and interpreting a model consists in the fact that
8 KB (1,152 words) - 19:38, 5 June 2020
...contains a comprehensive biography, a complete bibliography and Erlang's principal works in English translation).
7 KB (990 words) - 13:25, 18 March 2023
...ch of mathematics flourished to the greatest extent in Ancient Greece. The principal source is Aritmetika by Diophantus (probably the 3rd century A.D.), which c
...n studied to a lesser extent, because of inherent difficulties. One of the principal methods of study in the representation of numbers by forms of higher degree
18 KB (2,685 words) - 08:57, 18 August 2022
The principal task of the theory of curls is to determine the velocity field of the liqui
7 KB (1,148 words) - 18:18, 21 June 2020
...of the cohomology theory of Lie algebras to the study of the cohomology of principal bundles and homogeneous spaces (see [[#References|[8]]], [[#References|[14]
...lperin, R. Vanstone, "Connections, curvature and cohomology. Cohomology of principal bundles and homogeneous spaces" , '''3''' , Acad. Press (1975) {{MR|0400275
21 KB (3,027 words) - 17:45, 4 June 2020
...arising boundary conditions, \eqref{1} gives rise to an elliptic boundary value problem. The system \eqref{1} is conveniently called the system of equation
...\eqref{2} are of at most second order. The order of the determinant of the principal symbol (characteristic polynomial) in \eqref{2} is 4, while in \eqref{1} it
8 KB (1,200 words) - 17:04, 14 February 2020
...ath.org/legacyimages/m/m064/m064430/m06443044.png" /> it takes each finite
value, except <img align="absmiddle" border="0" src="https://www.encyclopediaofma
....encyclopediaofmath.org/legacyimages/m/m064/m064430/m064430103.png" /> the
value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
39 KB (5,287 words) - 17:07, 7 February 2011
...is method is easily implemented on a computer; its scheme with choice of a principal element makes it possible to solve a system with an arbitrary matrix, and a
it is required to determine an eigen value $ \lambda $
20 KB (3,033 words) - 22:16, 5 June 2020
are the principal diagonal minors of the matrix
9 KB (1,283 words) - 14:55, 7 June 2020
...tralizer]]; [[Normalizer of a subset|Normalizer of a subset]]). One of the principal goals of representation theory (cf. also [[Representation of a group|Repres
9 KB (1,465 words) - 17:46, 1 July 2020
...a theory describing the "deviation" from Hasse's principle in the class of principal homogeneous spaces of Abelian varieties [[#References|[7]]], [[#References|
...an be associated to each Abelian variety (the Tate–Shafarevich group). The principal difficulty of the theory is that methods for calculating the groups $ {\m
24 KB (3,602 words) - 11:48, 26 March 2023
whose principal symbols belong to $ W $,
is a principal ideal, so $ M $
28 KB (4,182 words) - 19:30, 19 January 2024
...rime } = \Gamma$, the integral in (a1) is understood as a Cauchy principal value integral (cf. also [[Cauchy integral|Cauchy integral]]).
8 KB (1,185 words) - 17:00, 1 July 2020
The principal function classes $ W $
13 KB (1,964 words) - 08:11, 6 June 2020
...topology of $4$-manifolds have been obtained by S.K. Donaldson: Consider a principal $G$-bundle $\xi $ on an oriented Riemannian $4$-manifold $M$, for a compact
7 KB (1,126 words) - 17:45, 1 July 2020
principal new directions in analysis of this century
11 KB (1,737 words) - 12:11, 23 November 2023
is negative definite. The principal properties of the function $ \theta $
...of this school were G. Castelnuovo, F. Enriques and F. Severi. One of the principal achievements of the Italian school was the classification of algebraic surf
29 KB (4,414 words) - 17:20, 17 December 2019
Principal integral representations. If $ \mathop{\rm Re} \gamma > \mathop{\rm Re
12 KB (1,576 words) - 01:14, 21 January 2022
...iplicities $r_1 , \ldots , r_n$ respectively (cf. also [[Eigen value|Eigen value]]), i.e.
10 KB (1,502 words) - 22:40, 12 December 2020
...ars to be C.F. Gauss who first stated explicitly the difference, being the principal one in constructive mathematics, between potential infinity and the actual
...s has a certain similarity to the latter. Apart from this, there is also a principal difference both of a general philosophical and a concrete mathematical natu
20 KB (2,913 words) - 07:40, 14 August 2023
...ebraic ovals). However, these non-algebraic conditions apply only to the ''principal Taylor part'' of the vector fields: in the restricted problems where this p
7 KB (1,080 words) - 09:14, 12 December 2013
every finite value exactly $ r $
...t, second and third kind, respectively. An integral of the first kind is a principal uniformizing function for $ F $
9 KB (1,292 words) - 19:08, 20 January 2022
The numerical value of a probability may sometimes be obtained from its "classical" definition:
...licated cases, a statistical approach is needed to determine the numerical value of a probability. For instance, if a marksman has hit the target in 39 out
9 KB (1,350 words) - 18:22, 27 May 2012
In other words, the value $ x _ {r-} 1 ^ {0} $
the value $ x _ {r-} 2 = x _ {r-} 2 ^ {0} $
10 KB (1,474 words) - 19:41, 5 June 2020
one can speak of the value of $ f $
at a point $ x \in X $ (which coincides with the value of the analytic function $ { \mathop{\rm red} } f $
22 KB (3,277 words) - 01:53, 19 January 2022
are called the principal geometric object determining functions (or representation-determining funct
...neous object, or a tensor. A geometric object is linear if and only if the principal functions which define it are of the form
21 KB (3,190 words) - 19:41, 5 June 2020
...t under diffeomorphisms, and using the Weyl quantization one gets that the principal symbol $a _ { m }$ is invariantly defined on the cotangent bundle $T ^ { *
of the principal symbol. If one writes
30 KB (4,520 words) - 19:54, 6 February 2024
principal in the Scioto debacle - a failed scheme projected to settle
between 1786 and 1822 remain his life's principal accomplishment.
15 KB (2,346 words) - 20:38, 22 September 2016
does not completely determine the value of the Abelian integral (1), or, which is the same, the integral (1) turns
and the difficulty of the inversion problem can already be noticed. The principal advances in the theory of inversion of Abelian integrals are due to B. Riem
10 KB (1,594 words) - 06:20, 17 April 2024
...f sets has made it possible to state a proposition on the unsolvability in principal (in an exact sense) of certain mathematical problems and has made it possib
The principal method used for establishing that a formula $ A $
22 KB (3,585 words) - 17:07, 25 April 2020
...es the predicate term and goes along an arc corresponding to the predicate value. When hitting a transformer with operator $ y := \tau $,
one computes the value of $ \tau $
21 KB (3,241 words) - 08:25, 6 June 2020
...the theory of Abelian groups can be applied to the case of modules over a principal ideal ring. Owing to their relative simplicity and to the fact that they ha
11 KB (1,810 words) - 22:12, 29 August 2015
...curve, and the improper integral is to be understood as a Cauchy principal value, i.e.
for some admissible value $ \alpha $
27 KB (3,955 words) - 10:05, 8 May 2022
...ecently Mardešić and collaborators have successfully explained some of the principal peculiarities, using the concept of an approximate inverse limit. The appro
14 KB (2,029 words) - 19:53, 3 February 2021
is the decomposition of the principal divisor $ ( b) $
10 KB (1,442 words) - 08:10, 6 June 2020
A function related to integral representations of solutions of boundary value problems for differential equations.
The Green function of a boundary value problem for a linear differential equation is the [[Fundamental solution|fu
33 KB (4,788 words) - 08:53, 13 May 2022
...set of $k'$-isomorphisms $B\otimes_k k'\to A\otimes_k k'$ is naturally a [[principal homogeneous space]] over $\Aut_k'(A)$ and thus defines an element of $\H^1(
9 KB (1,595 words) - 22:23, 22 November 2013