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...s the closure of an open set; in other words, it is the closure of its own interior $ \langle M\rangle $: A set which is the interior of a closed set is called a canonical open set or $ \kappa o $-

algebra (for example, the Borel $ \sigma $- algebra) and containing exactly one representative from each orbit. However, if $

...ological space $X$ with the Baire property is a [[Algebra of sets|$\sigma$-algebra]]. ...first category (an open set is called ''regular'' if it coincides with the interior of its closure).

The elements of the Lie algebra $ \mathfrak g $ is the operation of interior multiplication by $ X $

...simple polygonal curve, so that non-adjacent sides have no common points (interior or end), then the polygon is called simple. If the boundary of a two-dimens ...he point is in the exterior domain, if it is odd, then the point is in the interior.

...ion algebra $\mathcal{A} ( X )$ defined on $X$; cf. [[Algebra of functions|Algebra of functions]]) if each $f \in \mathcal{A} ( X )$ vanishing on $E$ can be a Ditkin sets were first studied for the [[Fourier-algebra(2)|Fourier algebra]] $A ( \widehat { G } ) \cong L ^ { 1 } ( G )$, with the norm defined by $\

...f functions|Algebra of functions]]). The algebra $A$ is called a Dirichlet algebra if $A + \overline{A}$ is uniformly dense in $C ( X )$. Dirichlet algebras w ...f D }$. The algebra $A ( \mathbf{D} )$ is a typical example of a Dirichlet algebra on the unit circle $\partial \mathbf{D}$. For $A ( \mathbf{D} )$, the measu

[[Pseudo-Boolean algebra|Pseudo-Boolean algebra]]) or open sets in topological spaces (for intuitionistic logic), or elemen where $\text{ Int }(X)$ denotes the interior of the set $X$.

of maximal ideals of a [[Commutative Banach algebra|commutative Banach algebra]] $ A $ is a commutative Banach algebra containing $ A $

$#C+1 = 179 : ~/encyclopedia/old_files/data/A011/A.0101370 Algebra of functions, ''function algebra''

...t is called an ''[[ultrafilter]]'' (a maximal proper filter in any Boolean algebra is also called an ultrafilter). ...urhood]]s of any point $x \in E$ (the subsets of $E$ containing $x$ in the interior) form a filter.

$#C+1 = 86 : ~/encyclopedia/old_files/data/P075/P.0705610 Pseudo\AAhBoolean algebra Every pseudo-Boolean algebra is a [[Distributive lattice|distributive lattice]] with largest element 1 (

form on the Lie algebra $ T _ {e} G $ of interior multiplication by $ \Omega $;

...been successfully used in recent proofs of "deep" results in topological algebra (cf. also [[Separate and joint continuity|Separate and joint continuity]]),

[[Banach algebra|Banach algebra]] $A$ over the field of complex numbers, with an involution $x \rightarrow 1) The algebra $C_0(X)$ of continuous complex-valued functions on a locally compact Hausd

of parallel fields forms a subalgebra of the algebra of all tensor fields on $ M $ ...t under contraction of tensor fields and permutation of their indices. The algebra $ \Pi ( M, \nabla ) $

and analytic at interior points of $ E $). is analytic at interior points of $ E $

...an operator usually takes values in an algebra (in particular, an operator algebra) that is simpler than the original one. has symbols of two types: interior and boundary. The interior symbol $ \sigma ^ {0} ( \mathfrak A ) $

is decaying fast enough at infinity. ii) The interior problem, i.e. determining $ f $ in the interior of $ K $,

multiplication) as well as interior (cup-multiplication). This is equivalent to the statement that the mapping ..., "A general algebraic approach to Steenrod operations" , ''The Steenrod Algebra and Its Applications'' , ''Lect. notes in math.'' , '''168''' , Springer (

...— the [[Stone–Čech compactification|Stone–Čech compactification]] — is the algebra $M ( A )$ of multipliers of $A$, defined by R.C. Busby in 1967 [[#Reference ...{ * }$-algebra of double centralizers of $A$ and the concrete $C ^ { * }$-algebra $M ( A )$. This, in particular, shows that $M ( A )$ is independent of the

ii) considering the interior $\Theta ( \mu )$ of the convex set of those $\theta \in E ^ { * }$ such tha ...xponential family by the mean. The domain of the means is contained in the interior $C _ { F }$ of the convex hull of the support of $F$. When $C _ { F } = M _

...nts of the form (2) are given not only at the end points, but also at some interior points of $ a \leq x \leq b $. ...1]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)<

...is denoted by $\beta\omega$. Here, $\mathcal{P}(\omega)$ is the power set algebra of $\omega$ and $\text{fin}$ denotes its ideal of finite sets. The points i ...ality of the continuum]]) in which non-empty $G_\delta$ sets have infinite interior (for short, a Parovichenko space). This theorem had wide applications both

and the operator of interior multiplication $ i _ {X} $( ...ion" ''Progress in Math.'' , '''6''' (1970) pp. 229–269 ''Itogi. Nauk. Algebra Topol. Geom. 1965'' (1967) pp. 429–465</TD></TR><TR><TD valign="top">[3

...he foundations of ancient mathematics: elementary geometry, number theory, algebra, the general theory of proportion, and a method for the determination of ar ...ght lines lying in the same plane intersect a third, and if the sum of the interior angles on one side of the latter is less than the sum of two right angles,

...undamental frequency; it does not reduce torsional rigidity or the maximal interior conformal radius (see [[#References|[3]]]). ...op">[a5]</TD> <TD valign="top"> M. Marcus, "Finite dimensional multilinear algebra" , '''1''' , M. Dekker (1973) pp. 78ff {{MR|0352112}} {{ZBL|0284.15024}} </

dimensional if and only if it contains interior points with respect to $ E ^ {n} $. containing interior points (with respect to $ E ^ {n} $).

...et of all holomorphic vector fields on $\textbf{D}$ is a [[Lie algebra|Lie algebra]] under the commutator bracket For absolutely convex domains, interior flow invariance conditions can be given in terms of their support functiona

is the operation of interior multiplication (contraction), is called the polar system of the integral el ...been generalized to arbitrary differential systems given by ideals in the algebra of differential forms on a manifold (the Cartan–Kähler theorem).

...manifold with boundary (for example, an operator from the Boutet de Monvel algebra, [[#References|[10]]], [[#References|[11]]]) at a boundary point means inve is an interior point of $ X $)

The search for interior points of the spectrum of a sparse matrix $ A $ ...processes to compute eigen values of a large sparse matrix (not only "the interior points" ). The interesting aspect is that the computed vectors by no means

the finite plane) and the interior of the unit disc $ D = \{ {z} : {| z | < 1} \} $ ( in the case of the interior of the unit disc, $ G $

...and, more generally, to the open unit ball $U$ of a so-called $J ^ { * }$-algebra (see [[#References|[a37]]], [[#References|[a25]]] and the references there) ...se, if $F \in \operatorname{Hol} ( {\cal D} )$ is not the identity, has an interior fixed point and is power convergent, then $c$ is unique. However, this is n

The branch of topology and algebra concerned with braids, the groups formed by their equivalence classes and v the interior of which contains $ \omega $.

...al X $ can be transformed by means of an element of $ \Gamma $ into an interior point of $ X $ or into a pseudo-concave point of the boundary $ \parti ...oup with Lie algebra $ \mathfrak g $ . Identify the universal enveloping algebra $ U \mathfrak g $ of $ \mathfrak g $ with the right-invariant differe

with an interior point in $ B $ ...in the theory of locally convex spaces is played by methods of homological algebra connected with the study of the category of locally convex spaces and their

.... Soviet Math.'' , '''14''' (1980) pp. 1363–1407 ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''15''' (1977) pp. 93–171</TD></TR> c) A strongly pseudo-convex domain can be written as the interior of the intersection of a properly decreasing family of strongly pseudo-conv

and its closure is contained in the interior of the subspace $ A $, ...rticular, it is possibly to give a new proof of the fundamental theorem of algebra.

...formally equivalent either to the Riemann sphere, the complex plane or the interior of the unit disc. In the first case the algebraic function is a rational, i ...align="top">[1]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001

...e length. In this way, the completeness of the surface in the sense of its interior metric is guaranteed. This example shows that an intrinsically-complete sur is the interior domain of an horocycle in the Lobachevskii plane. The universal covering $

...generalizations and translation of geometric concepts into the language of algebra are incorporated, is the system of axioms proposed by H. Weyl (1916). One o ...replaced by calculations after having been translated into the language of algebra.

...ds of the theory of functions of a complex variable on the one hand and of algebra and algebraic geometry on the other hand is characteristic of the whole per are called interior and the other points, that are mapped to the points of the segment

...s it to the second component. After this it is necessary to remove all the interior points of the tape as well as the boundary points at which the gluing has t ...with which it is easier to work (e.g. they can be described by commutative algebra) and which at the same time provide enough information. The most important

is the distance between two points in the interior of the oval, then ...try" ''Progress in Math.'' , '''12''' (1972) pp. 173–214 ''Itogi Nauk. Algebra. Topol. Geom. 1968'' (1970) pp. 157–191</TD></TR></table>

...'J. Soviet Math.'' , '''2''' : 3 (1974) pp. 264–303 ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''10''' (1972) pp. 47–112 {{MR|}} {{ZBL|1068.14059}} < ...closed sets" , ''General topology and its relations to modern analysis and algebra (Proc. Symp. Prague)'' , Acad. Press (1961) pp. 123–132 {{MR|}} {{ZBL|}}