翻訳と辞書 Words near each other ・ Grade skipping ・ Grade-taking ・ Gradec ・ Gradec (Valandovo) ・ Gradec Pokupski ・ Gradec, Albania ・ Gradec, Gjerbës ・ Gradec, Krško ・ Gradec, Litija ・ Gradec, Pivka ・ Gradec, Qendër Skrapar ・ Gradec, Shkodër ・ Gradec, Zagreb ・ Gradec, Zagreb County ・ Gradec, Črnomelj ・ Graded (mathematics) ・ Graded absolutism ・ Graded bedding ・ Graded category ・ Graded English Medium School ・ Graded exercise therapy ・ Graded Lie algebra ・ Graded manifold ・ Graded numerical sequence ・ Graded poset ・ Graded potential ・ Graded reader ・ Graded ring ・ Graded Salience Hypothesis ・ Graded shoreline Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Graded (mathematics) ： ウィキペディア英語版 In mathematics, the term “graded” has a number of meanings, mostly related:In abstract algebra, it refers to a family of concepts:* An algebraic structure X is said to be I-graded for an index set I if it has a gradation or grading, i.e. a decomposition into a direct sum X = \oplus_ X_i of structures; the elements of X_i are said to be “homogeneous of degree ''i''”.** The index set I is most commonly \mathbb or \mathbb, and may be required to have extra structure depending on the type of X.** Grading by \mathbb_2 (i.e. \mathbb/2\mathbb) is also important.** The trivial (\mathbb- or \mathbb-) gradation has X_0 = X, X_i = 0 for i \neq 0 and a suitable trivial structure 0.** An algebraic structure is said to be doubly graded if the index set is a direct product of sets; the pairs may be called “bidegrees” (e.g. see spectral sequence).* A I-graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum V = \oplus_ V_i of spaces.** A graded linear map is a map between graded vector spaces respecting their gradations.* A graded ring is a ring that is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_, with i taken from some monoid, usually \mathbb or \mathbb, or semigroup (for a ring without identity).** The associated graded ring of a commutative ring R with respect to a proper ideal I is \operatorname_I R = \oplus_.* A graded module is left module M over a graded ring which is a direct sum \oplus_ M_i of modules satisfying R_i M_j \subseteq M_.** The associated graded module of an R-module M with respect to a proper ideal I is \operatorname_I M = \oplus_ M.** A differential graded moduledifferential graded category -->, differential graded \mathbb-module or DG-module is a graded module M with a differential d\colon M \to M \colon M_i \to M_ making M a chain complex, i.e. d \circ d=0 .* A graded algebra is an algebra A over a ring R that is graded as a ring; if R is graded we also require A_iR_j \subseteq A_ \supseteq R_iA_j.** The graded Leibniz rule for a map d\colon A \to A on a graded algebra A specifies that d(a \cdot b) = (da) \cdot b + (-1)^a \cdot (db) |a| denotes the “parity” of a ? -->.** A differential graded algebra, DG-algebra or DGAlgebra is a graded algebra which is a differential graded module whose differential obeys the graded Leibniz rule.** A homogeneous derivation on a graded algebra ''A'' is a homogeneous linear map of grade ''d'' = |''D''| on ''A'' such that D(ab)=D(a)b+\varepsilon^aD(b), \varepsilon = \pm 1 acting on homogeneous elements of ''A''.** A graded derivation Graded derivation redirects to Differential algebra, which says nothing about it --> is a sum of homogeneous derivations with the same \varepsilon.** A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see differential graded algebra).** A superalgebra is a \mathbb_2-graded algebra.*** A graded-commutative superalgebra satisfies the “supercommutative” law yx = (-1)^xy.\, for homogeneous ''x'',''y'', where |a| represents the “parity” of a, i.e. 0 or 1 depending on the component in which it lies.** CDGA may refer to the category of augmented differential graded commutative algebras.* A graded Lie algebra is a Lie algebra which is graded as a vector space by a gradation compatible with its Lie bracket.** A graded Lie superalgebra is a graded Lie algebra with the requirement for anticommutativity of its Lie bracket relaxed.** A supergraded Lie superalgebra is a graded Lie superalgebra with an additional super \mathbb_2-gradation.** A Differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map (): L_i \otimes L_j \to L_ and a differential d: L_i \to L_ satisfying () = (-1)^(), for any homogeneous elements ''x'', ''y'' in ''L'', the “graded Jacobi identity” and the graded Leibniz rule.* The Graded Brauer group is a synonym for the Brauer–Wall group BW(F) classifying finite-dimensional graded central division algebras over the field ''F''.* An \mathcal-graded category for a category \mathcal is a category \mathcal together with a functor F:\mathcal \rightarrow \mathcal.** A differential graded category or DG category is a category whose morphism sets form differential graded \mathbb-modules.* Graded manifold – extension of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra, including sections on** Graded function** Graded vector fields** Graded exterior forms** Graded differential geometry** Graded differential calculusIn other areas of mathematics:* Functionally graded elements are used in finite element analysis.* A graded poset is a poset P with a rank function \rho\colon P \to \mathbb compatible with the ordering (i.e. \rho(x) ) such that y covers x \implies \rho(y)=\rho(x)+1 . In mathematics, the term “graded” has a number of meanings, mostly related: In abstract algebra, it refers to a family of concepts: * An algebraic structure $X$ is said to be $I$-graded for an index set $I$ if it has a gradation or grading, i.e. a decomposition into a direct sum $X\; =\; \backslash oplus\_\; X\_i$ of structures; the elements of $X\_i$ are said to be “homogeneous of degree ''i''”. * * The index set I is most commonly $\backslash mathbb$ or $\backslash mathbb$, and may be required to have extra structure depending on the type of $X$. * * Grading by $\backslash mathbb\_2$ (i.e. $\backslash mathbb/2\backslash mathbb$) is also important. * * The trivial ($\backslash mathbb$- or $\backslash mathbb$-) gradation has $X\_0\; =\; X,\; X\_i\; =\; 0$ for $i\; \backslash neq\; 0$ and a suitable trivial structure $0$. * * An algebraic structure is said to be doubly graded if the index set is a direct product of sets; the pairs may be called “bidegrees” (e.g. see spectral sequence). * A $I$-graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum $V\; =\; \backslash oplus\_\; V\_i$ of spaces. * * A graded linear map is a map between graded vector spaces respecting their gradations. * A graded ring is a ring that is a direct sum of abelian groups $R\_i$ such that $R\_i\; R\_j\; \backslash subseteq\; R\_$, with $i$ taken from some monoid, usually $\backslash mathbb$ or $\backslash mathbb$, or semigroup (for a ring without identity). * * The associated graded ring of a commutative ring $R$ with respect to a proper ideal $I$ is $\backslash operatorname\_I\; R\; =\; \backslash oplus\_$. * A graded module is left module $M$ over a graded ring which is a direct sum $\backslash oplus\_\; M\_i$ of modules satisfying $R\_i\; M\_j\; \backslash subseteq\; M\_$. * * The associated graded module of an $R$-module $M$ with respect to a proper ideal $I$ is $\backslash operatorname\_I\; M\; =\; \backslash oplus\_\; M$. * * A differential graded module, differential graded $\backslash mathbb$-module or DG-module is a graded module $M$ with a differential $d\backslash colon\; M\; \backslash to\; M\; \backslash colon\; M\_i\; \backslash to\; M\_$ making $M$ a chain complex, i.e. $d\; \backslash circ\; d=0$ . * A graded algebra is an algebra $A$ over a ring $R$ that is graded as a ring; if $R$ is graded we also require $A\_iR\_j\; \backslash subseteq\; A\_\; \backslash supseteq\; R\_iA\_j$. * * The graded Leibniz rule for a map $d\backslash colon\; A\; \backslash to\; A$ on a graded algebra $A$ specifies that $d(a\; \backslash cdot\; b)\; =\; (da)\; \backslash cdot\; b\; +\; (-1)^a\; \backslash cdot\; (db)$ . * * A differential graded algebra, DG-algebra or DGAlgebra is a graded algebra which is a differential graded module whose differential obeys the graded Leibniz rule. * * A homogeneous derivation on a graded algebra ''A'' is a homogeneous linear map of grade ''d'' = |''D''| on ''A'' such that $D(ab)=D(a)b+\backslash varepsilon^aD(b),\; \backslash varepsilon\; =\; \backslash pm\; 1$ acting on homogeneous elements of ''A''. * * A graded derivation is a sum of homogeneous derivations with the same $\backslash varepsilon$. * * A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see differential graded algebra). * * A superalgebra is a $\backslash mathbb\_2$-graded algebra. * * * A graded-commutative superalgebra satisfies the “supercommutative” law $yx\; =\; (-1)^xy.\backslash ,$ for homogeneous ''x'',''y'', where $|a|$ represents the “parity” of $a$, i.e. 0 or 1 depending on the component in which it lies. * * CDGA may refer to the category of augmented differential graded commutative algebras. * A graded Lie algebra is a Lie algebra which is graded as a vector space by a gradation compatible with its Lie bracket. * * A graded Lie superalgebra is a graded Lie algebra with the requirement for anticommutativity of its Lie bracket relaxed. * * A supergraded Lie superalgebra is a graded Lie superalgebra with an additional super $\backslash mathbb\_2$-gradation. * * A Differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map $():\; L\_i\; \backslash otimes\; L\_j\; \backslash to\; L\_$ and a differential $d:\; L\_i\; \backslash to\; L\_$ satisfying $()\; =\; (-1)^(),$ for any homogeneous elements ''x'', ''y'' in ''L'', the “graded Jacobi identity” and the graded Leibniz rule. * The Graded Brauer group is a synonym for the Brauer–Wall group $BW(F)$ classifying finite-dimensional graded central division algebras over the field ''F''. * An $\backslash mathcal$-graded category for a category $\backslash mathcal$ is a category $\backslash mathcal$ together with a functor $F:\backslash mathcal\; \backslash rightarrow\; \backslash mathcal$. * * A differential graded category or DG category is a category whose morphism sets form differential graded $\backslash mathbb$-modules. * Graded manifold – extension of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra, including sections on * * Graded function * * Graded vector fields * * Graded exterior forms * * Graded differential geometry * * Graded differential calculus In other areas of mathematics: * Functionally graded elements are used in finite element analysis. * A graded poset is a poset $P$ with a rank function $\backslash rho\backslash colon\; P\; \backslash to\; \backslash mathbb$ compatible with the ordering (i.e. ) such that $y$ covers $x\; \backslash implies\; \backslash rho(y)=\backslash rho(x)+1$ . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「In mathematics, the term “graded” has a number of meanings, mostly related:In abstract algebra, it refers to a family of concepts:* An algebraic structure X is said to be I-graded for an index set I if it has a gradation or grading, i.e. a decomposition into a direct sum X = \oplus_ X_i of structures; the elements of X_i are said to be “homogeneous of degree ''i''”.** The index set I is most commonly \mathbb or \mathbb, and may be required to have extra structure depending on the type of X.** Grading by \mathbb_2 (i.e. \mathbb/2\mathbb) is also important.** The trivial (\mathbb- or \mathbb-) gradation has X_0 = X, X_i = 0 for i \neq 0 and a suitable trivial structure 0.** An algebraic structure is said to be doubly graded if the index set is a direct product of sets; the pairs may be called “bidegrees” (e.g. see spectral sequence).* A I-graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum V = \oplus_ V_i of spaces.** A graded linear map is a map between graded vector spaces respecting their gradations.* A graded ring is a ring that is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_, with i taken from some monoid, usually \mathbb or \mathbb, or semigroup (for a ring without identity).** The associated graded ring of a commutative ring R with respect to a proper ideal I is \operatorname_I R = \oplus_.* A graded module is left module M over a graded ring which is a direct sum \oplus_ M_i of modules satisfying R_i M_j \subseteq M_.** The associated graded module of an R-module M with respect to a proper ideal I is \operatorname_I M = \oplus_ M.** A differential graded moduledifferential graded category -->, differential graded \mathbb-module or DG-module is a graded module M with a differential d\colon M \to M \colon M_i \to M_ making M a chain complex, i.e. d \circ d=0 .* A graded algebra is an algebra A over a ring R that is graded as a ring; if R is graded we also require A_iR_j \subseteq A_ \supseteq R_iA_j.** The graded Leibniz rule for a map d\colon A \to A on a graded algebra A specifies that d(a \cdot b) = (da) \cdot b + (-1)^a \cdot (db) |a| denotes the “parity” of a ? -->.** A differential graded algebra, DG-algebra or DGAlgebra is a graded algebra which is a differential graded module whose differential obeys the graded Leibniz rule.** A homogeneous derivation on a graded algebra ''A'' is a homogeneous linear map of grade ''d'' = |''D''| on ''A'' such that D(ab)=D(a)b+\varepsilon^aD(b), \varepsilon = \pm 1 acting on homogeneous elements of ''A''.** A graded derivation Graded derivation redirects to Differential algebra, which says nothing about it --> is a sum of homogeneous derivations with the same \varepsilon.** A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see differential graded algebra).** A superalgebra is a \mathbb_2-graded algebra.*** A graded-commutative superalgebra satisfies the “supercommutative” law yx = (-1)^xy.\, for homogeneous ''x'',''y'', where |a| represents the “parity” of a, i.e. 0 or 1 depending on the component in which it lies.** CDGA may refer to the category of augmented differential graded commutative algebras.* A graded Lie algebra is a Lie algebra which is graded as a vector space by a gradation compatible with its Lie bracket.** A graded Lie superalgebra is a graded Lie algebra with the requirement for anticommutativity of its Lie bracket relaxed.** A supergraded Lie superalgebra is a graded Lie superalgebra with an additional super \mathbb_2-gradation.** A Differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map (): L_i \otimes L_j \to L_ and a differential d: L_i \to L_ satisfying () = (-1)^(), for any homogeneous elements ''x'', ''y'' in ''L'', the “graded Jacobi identity” and the graded Leibniz rule.* The Graded Brauer group is a synonym for the Brauer–Wall group BW(F) classifying finite-dimensional graded central division algebras over the field ''F''.* An \mathcal-graded category for a category \mathcal is a category \mathcal together with a functor F:\mathcal \rightarrow \mathcal.** A differential graded category or DG category is a category whose morphism sets form differential graded \mathbb-modules.* Graded manifold – extension of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra, including sections on** Graded function** Graded vector fields** Graded exterior forms** Graded differential geometry** Graded differential calculusIn other areas of mathematics:* Functionally graded elements are used in finite element analysis.* A graded poset is a poset P with a rank function \rho\colon P \to \mathbb compatible with the ordering (i.e. \rho(x) ) such that y covers x \implies \rho(y)=\rho(x)+1 .」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.