Varphi Josephson junction

Template:Disambiguation In mathematics, the term “graded” has a number of related meanings:

• 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 $X={\oplus }_{i\in I}{X}_i$ of structures; the elements of $X_i$ are said to be “homogenous of degree i”.
  • The index set I is most commonly $\mathbb{\mathbb{N}}$ or $\mathbb{\mathbb{Z}}$, and may be required to have extra structure depending on the type of $X$.
  • The trivial ($\mathbb{\mathbb{Z}}$- or $\mathbb{\mathbb{N}}$-) gradation has $X_0=X$, $X_i=0$ for Failed to parse (syntax error): {\displaystyle i ≠ 0} and a suitable trivial structure $0$.
  • An algebraic structure is said to be doubly graded if the index set is a direct product; the pairs may be called as “bidegrees” (e.g. see spectral sequence).
• A I-graded vector space or graded linear space for a set I is thus a vector space with a decomposition into a direct sum $V={\oplus }_{i\in I}{V}_i$ 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 $R_i{R}_j\subseteq R_{i+j}$, with $i$ taken from some monoid, usually $\mathbb{\mathbb{N}}$ or $\mathbb{\mathbb{Z}}$, or semigroup (for a ring without identity).
  • The associated graded ring of a commutative ring $R$ with respect to a proper ideal $I$ is ${\mathrm{gr}}_{I}R={\oplus }_{n\in \mathbb{\mathbb{N}}}{I}^n/I^{n+1}$.
• A graded module is left module $M$ over a graded ring which is a direct sum ${\oplus }_{i\in I}{M}_i$ of modules satisfying $R_i{M}_j\subseteq M_{i+j}$.
  • The associated graded module of an $R$-module $M$ with respect to a proper ideal $I$ is ${\mathrm{gr}}_{I}M={\oplus }_{n\in \mathbb{\mathbb{N}}}{I}^{n}M/I^{n+1}M$.
  • A differential graded module, differential graded $\mathbb{\mathbb{Z}}$-module or DG-module is a graded module $M$ with a differential $d:M\to M:M_i\to M_{i+1}$ 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 Failed to parse (syntax error): {\displaystyle A_iR_j \subseteq A_{i+j} ⊇ R_iA_j} .
  • The graded Leibniz rule for a map $d:A\to A$ on a graded algebra $A$ specifies that $d(a\cdot b)=(da)\cdot b+(-1{)}^{\left|a\right|}a\cdot (db)$ .
  • 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 DGA is an augmented DG-algebra, or differential graded augmented algebra, (see differential graded algebra).
  • A superalgebra is a a Z₂-graded algebra.
    • A graded-commutative superalgebra satisfies the “supercommutative” law $yx=(-1{)}^{\left|x\right|\left|y\right|}xy.$ for homogenous x,y, where $\left|a\right|$ represents the “parity” of $a$, i.e. 0 or 1 depending on the component it lies in.
  • 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 Z/2Z-gradation.
  • A Differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map $[,]:L_i\otimes L_j\to L_{i+j}$ and a differential $d:L_i\to L_{i-1}$ satisfying $[x,y]=(-1{)}^{\left|x\right|\left|y\right|+1}[y,x],$ 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 $\mathcal{𝒜}$-graded category for a category $\mathcal{𝒜}$ is a category $\mathcal{𝒞}$ together with a functor $F:\mathcal{𝒞}\to \mathcal{𝒜}$.
  • A differential graded category or DG category is a category whose morphism sets form differential graded $\mathbb{\mathbb{Z}}$-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
• Graded derivation , but see also the section Graded derivations in Derivation (abstract algebra)
• Functionally graded elements are elements used in finite element analysis.
• A graded poset is a poset P with a rank function Failed to parse (syntax error): {\displaystyle ρ\colon P \to N} compatible with the ordering (so ρ(x)<ρ(y) ⇐ x < y) such that y covers x ⇒ $\rho (y)=\rho (x)+1$ .