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Longer titles found: Endomorphism ring (view), Frobenius endomorphism (view)

searching for Endomorphism 154 found (392 total)

alternate case: endomorphism

Lattice (order) (5,663 words) [view diff] exact match in snippet view article

a lattice endomorphism is a lattice homomorphism from a lattice to itself, and a lattice automorphism is a bijective lattice endomorphism. Lattices and

In linear algebra, a nilpotent matrix is a square matrix N such that N k = 0 {\displaystyle N^{k}=0\,} for some positive integer k {\displaystyle k} .

In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal (from the upper left to the

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic

an endomorphism whose square is the endomorphism αφ associated to the Frobenius endomorphism φ of the field F. Roughly speaking, this endomorphism απ

however, that many varieties have no interesting self-maps. Let f be an endomorphism of C P n {\displaystyle \mathbf {CP} ^{n}} , meaning a morphism of algebraic

another by an endomorphism-valued one-form. From this perspective, the connection one-form A {\displaystyle A} is precisely the endomorphism-valued one-form

heuristics involving the distribution of eigenvalues of the Frobenius endomorphism. As of 2019, this conjecture is open. More generally, if K is any global

fundamental to observe that there exists only one linear continuous endomorphism X {\displaystyle X} on the space of tempered distributions such that

ad⁡(X)(Y)=[X,Y]{\displaystyle \operatorname {ad} (X)(Y)=[X,Y]}, is a nilpotent endomorphism on g{\displaystyle {\mathfrak {g}}}; i.e., ad⁡(X)k=0{\displaystyle \operatorname

mathematics, an Ockham algebra is a bounded distributive lattice with a dual endomorphism, that is, an operation ~ satisfying ~(x ∧ y) = ~x ∨ ~y, ~(x ∨ y) = ~x

We can then define endomorphism operads in this category, as follows. Let V be a finite-dimensional vector space The endomorphism operad E n d V = { E

maximal abelian subalgebra consisting of elements x such that the adjoint endomorphism ad ⁡ ( x ) : g → g {\displaystyle \operatorname {ad} (x):{\mathfrak {g}}\to

simply be calculated using the characteristic polynomial of the Frobenius endomorphism. This is not the case, for example, in elliptic curve cryptography when

⊗ O = E for any E. Example: E ⊗ E ∗ is canonically isomorphic to the endomorphism bundle End(E), where E ∗ is the dual bundle of E. Example: A line bundle

direct sum of objects having local endomorphism rings. Equivalently, C has split idempotents and the endomorphism ring of every object is semiperfect

V⊗n{\displaystyle V^{\otimes n}} by permutation of indices, the image of the endomorphism determined by that element corresponds to an irreducible representation

This is true more generally for a module M over a ring R, with the endomorphism algebra End(M) (algebra of linear operators on M) replacing the algebra

Endomorphism preserving the inner product

finitely generated modules. For example, if f : M → M is a surjective R-endomorphism of a finitely generated module M, then f is also injective, and hence

adjoint (adjoint of a linear operator) in functional analysis Adjoint endomorphism of a Lie algebra Adjoint representation of a Lie group Adjoint functors

domain [-1,1]. The range of the function is [0,2]. Thus, f is not an endomorphism. Consider the function f ( x ) = x + 1 , {\displaystyle f(x)=x+1,} which

classification of three-folds. Mori completed his Ph.D. titled "The Endomorphism Rings of Some Abelian Varieties" under Masayoshi Nagata at Kyoto University

whose conjugate transpose equals its inverse Unitary transformation – Endomorphism preserving the inner product Halmos 1982, Sect. 127, page 69 Conway 1990

{\displaystyle A(\Gamma )} to a power > 1 {\displaystyle >1} defines and endomorphism of A ( Γ ) {\displaystyle A(\Gamma )} which is injective but not surjective

TM\otimes \mathbb {C} \to M} , the endomorphism J {\displaystyle J} may be extended complex-linearly to an endomorphism J : T M ⊗ C → T M ⊗ C {\displaystyle

important application of rigidity is in the definition of the trace of an endomorphism of a rigid object. The trace can be defined for any pivotal category

{\mathfrak {g}}_{1}} . Via π {\displaystyle \pi } , it acts on V as a linear endomorphism (namely, π ( c ) = ∑ i π ( e i ) ∘ π ( e i ) : V → V {\displaystyle \pi

radicial S-endomorphism of a scheme X of finite type over S is bijective (10.4.11), and that if X/S is of finite presentation, and the endomorphism is a monomorphism

vector field, which can be defined for any vector bundle, the canonical endomorphism J:TTM→TTM;JξX:=vlξ⁡(πTM)∗X,X∈TξTM{\displaystyle J:TTM\to TTM;\qquad J_{\xi

A\subset \operatorname {End} (V)} be the associative subalgebra of the endomorphism algebra of V generated by π ( g ) {\displaystyle \pi ({\mathfrak {g}})}

Eleanor Green Dawley (1966). Abelian groups and their endomorphism rings and the quasi-endomorphism of torsion free abelian groups. Syracuse, NY: Syracuse

a compact closed category. Any endomorphism f of a dualizable object admits a trace, which is a certain endomorphism of the monoidal unit of C. This

itself is the identity endomorphism of the space of 2-forms, Ω2(M), under the identification (using the metric) of the endomorphism ring End(Ω2(M)) with

diagram). As a simple example, the diagram of a single object with an endomorphism ( f : X → X {\displaystyle f\colon X\to X} ), or with two parallel arrows

Singular: dim(H1(O)) = 1 and is acted on non-trivially by the Frobenius endomorphism. This implies K = 0, and Picτ is μ2. The surface is a quotient of a K3

as multiplication and τ {\displaystyle \tau } acting as the Frobenius endomorphism of L {\displaystyle L} . The ring L { τ } {\displaystyle L\{\tau \}}

the group scheme α p {\displaystyle \alpha _{p}} (the kernel of the endomorphism of the additive group induced by taking pth powers) is its own Cartier

{\text{End}}_{R}(A)} via the map sending a ⊗ b {\displaystyle a\otimes b} to the endomorphism x ↦ a x b {\displaystyle x\mapsto axb} of A {\displaystyle A} . Over

application g ↦ g ω {\displaystyle g\mapsto g^{\omega }} is then an endomorphism of G. From this, it results that a Ω-group can also be viewed as a group

algebra is an element x such that the adjoint endomorphism a d x {\displaystyle ad_{x}} is a nilpotent endomorphism. 4.  A nilpotent cone normalizer The normalizer

multiplication). For convenience, we write ρ a {\displaystyle \rho _{a}} for the endomorphism ρ ( a ) {\displaystyle \rho (a)} . To be an algebra homomorphism, ρ {\displaystyle

sufficiently large: each block is a full matrix algebra over F, the endomorphism ring of the vector space underlying the associated simple module. To

finite-dimensional unital associative algebra A has a natural homomorphism to its own endomorphism ring End(A). A bilinear form can be defined on A in the sense of the

Reid, PhD., Jones wrote her thesis entitled "Abelian Groups and Their Endomorphism Rings and the Quasi-Endomorphisms of Torsion Free Abelian Groups." In

fixed base form a Sturmian word is a transcendental number.: 64, 85  An endomorphism of the free monoid B∗ on a 2-letter alphabet B is Sturmian if it maps

exceptional Jordan algebras. A derivation of a Jordan algebra A is an endomorphism D of A such that D(xy) = D(x)y+xD(y). The derivations form a Lie algebra

Equivalently, it can be defined as a subbundle H {\displaystyle H} of the endomorphism bundle End ⁡ ( T M ) {\displaystyle \operatorname {End} (TM)} such that

{g}}}a_{i}(x)t^{i}} be the characteristic polynomial of the adjoint endomorphism ad ⁡ ( x ) : y ↦ [ x , y ] {\displaystyle \operatorname {ad} (x):y\mapsto

fashion as the TM-valued one-form θ on M corresponding to the identity endomorphism of the tangent bundle under the duality isomorphism End(TM) ≈ TM ⊗ T∗M

from V to W. ... Definition: The following terms are also employed: (2) endomorphism for linear operator ... (6) automorphism for bijective linear operator

density theorem Double commutant theorem Note that for a field, the endomorphism algebra of a one-dimensional vector space (a line) is canonically equal

the Lefschetz number of an endomorphism of an elliptic complex in terms of a sum over the fixed points of the endomorphism. As special cases their formula

space in German. Fixed-point space – Topological space such that every endomorphism has a fixed point, a Hausdorff space X such that every continuous function

Hopf algebra of symmetric functions the Verschiebung Vn is the algebra endomorphism that takes the complete symmetric function hr to hr/n if n divides r

unique (1-dimensional) formal group law F such that e(x) = px + xp is an endomorphism of F, in other words e ( F ( x , y ) ) = F ( e ( x ) , e ( y ) ) .  

TM∖{0} satisfies JH = V and [V, H] = H, where J and V are the canonical endomorphism and the canonical vector field on TM∖{0}. Hence, by definition, H is

Andrew V; Rotger, Victor (2012). "Sato-Tate distributions and Galois endomorphism modules in genus 2". Compositio Mathematica. 149 (5): 1390–1442. arXiv:1110

(X\oplus Y)=\operatorname {Ber} (X)\mathrm {Ber} (Y)} The determinant of an endomorphism of a free module M can be defined as the induced action on the 1-dimensional

endomorphisms is a ring with the multiplication given by composition. The endomorphism ring EndR(M) acts on M by left multiplication defined by f.x = f(x).

involution 1.  An involution of a Banach algebra A is an isometric endomorphism A → A , x ↦ x ∗ {\displaystyle A\to A,\,x\mapsto x^{*}} that is conjugate-linear

F_{\nabla }\in \Omega ^{2}(\operatorname {End} (E))} with values in the endomorphism bundle, defined by F ∇ ( v 1 , v 2 ) = ∇ v 1 ∇ v 2 − ∇ v 2 ∇ v 1 − ∇

ρ {\displaystyle w\cdot u=w(u+\rho )-\rho } . Δα for α a root is the endomorphism of the Z-module Z[P] defined by Δ α ( u ) = u − s α ⋅ u 1 − e − α {\displaystyle

Every injective group is algebraically compact. Ulm factors of cotorsion groups are algebraically compact. On endomorphism rings of Abelian groups v t e

{\displaystyle R} -module and f : M → M {\displaystyle f:M\to M} is a surjective endomorphism, then f {\displaystyle f} is an isomorphism. Over a local ring, one can

and Lambek) of a Boolean ring R is a Boolean ring, since every partial endomorphism is idempotent. Every prime ideal P in a Boolean ring R is maximal: the

∇, unit η, and counit ε. The convolution is a product defined on the endomorphism algebra End(X) as follows. Let φ, ψ ∈ End(X), that is, φ, ψ: X → X are

Every element x of g {\displaystyle {\mathfrak {g}}} defines the adjoint endomorphism ad(x) (also written as adx) of g {\displaystyle {\mathfrak {g}}} with

ideal ( x , y ) {\displaystyle (x,y)} . Let x {\displaystyle x} be an endomorphism between a finite-dimensional vector space V. Then the subalgebra A ⊂

abelian Lie algebra V with a 1-dimensional Lie algebra acting on V as an endomorphism b such that b is not nilpotent and Tr(b2)=0. In characteristic 0, every

operator playing a central role in the theory takes values in a space of endomorphism-valued formal distributions. Let R {\displaystyle R} be an algebra over

generated spaces. Fixed-point space – Topological space such that every endomorphism has a fixed point, a Hausdorff space where every continuous function

(V)} may be first viewed as an endomorphism of T 1 1 ( V ) {\displaystyle T_{1}^{1}(V)} and then viewed as an endomorphism of E n d ( V ) {\displaystyle

Konieczny, Janusz (2007), "A method of finding automorphism groups of endomorphism monoids of relational systems", Discrete Mathematics, 307: 1609–1620

product of two G-sets induces a symmetric monoidal structure on A(G); The endomorphism ring of the point (that is the G-set with only one element) is the Burnside

the ring W of Witt vectors of k, together with a σ-linear injective endomorphism of M. An F-isocrystal is defined in the same way, except that M is a

surjective map from a module in the family F to M factors through X Any endomorphism of X commuting with the map to M is an automorphism. In general an F-cover

of Ω to the generating function of Bernoulli numbers and the adjoint endomorphism of Ω, Ω ′ = ad Ω exp ⁡ ( ad Ω ) − 1 A , {\displaystyle \Omega '={\frac

{\displaystyle w\mapsto R(u,v)w} is also called the curvature transformation or endomorphism. N.B. There are a few books where the curvature tensor is defined with

element X of a semisimple Lie algebra g is called nilpotent if its adjoint endomorphism ad X: g → g,   ad X(Y) = [X,Y] is nilpotent, that is, (ad X)n = 0 for

σ{\displaystyle \sigma } to τ{\displaystyle \tau }. The boundary operator is the endomorphism ∂{\displaystyle \partial } of the free abelian group generated by X{\displaystyle

the ring W of Witt vectors of k, together with a σ-linear injective endomorphism of M. An F-isocrystal is defined in the same way, except that M is a

point x. Then the almost complex structure j on X induces a vector space endomorphism j of G such that j2=–1 on G/K [x,y] + j[jx,y] + j[x,jy] – [jx,jy] = 0

Topological space such that every endomorphism has a fixed point

cross-ratio. The determinant, trace, eigenvectors, and eigenvalues of a linear endomorphism are invariant under a change of basis. In other words, the spectrum of

doi:10.1090/S0002-9939-1995-1277109-4. A. Facchini: Module theory. Endomorphism rings and direct sum decompositions in some classes of modules. Progress

− X A {\displaystyle \operatorname {ad} _{A}(X)=AX-XA} (the adjoint endomorphism of the Lie algebra gl(n, C) of all n×n matrices with complex entries)

{\displaystyle Z\mapsto R(X,Y)Z} is also called the curvature transformation or endomorphism. Occasionally, the curvature tensor is defined with the opposite sign

a_{j}\rangle =\delta _{ij}.} If A is an m × m matrix, then A induces an endomorphism of W with respect to this basis and the transpose AT induces a transformation

decomposition is equivalently given, up to a sign choice, by a symmetric endomorphism of R4,2 whose square is the identity and whose ±1 eigenspaces are σ and

pointed out that a Grothendieck topology can be entirely described as an endomorphism of the subobject representor, and Tierney showed that the conditions

ring is a commutative ring R {\displaystyle R} together with a ring endomorphism σ : R → R {\displaystyle \sigma \colon R\to R} . Often it is assumed

abelian group M. Every map mr given by mr(x) = xr creates an additive endomorphism of M, that is, an element of E. The map r → mr is a ring homomorphism

B {\displaystyle B} is the centralizer of A {\displaystyle A} in the endomorphism ring End ⁡ ( U ) {\displaystyle \operatorname {End} (U)} . The proof

object in C {\displaystyle {\mathcal {C}}} and that there exists an endomorphism α : Y → Y {\displaystyle \alpha :Y\to Y} that does not have any fixed

of a (smooth projective) algebraic variety V {\displaystyle V} , an endomorphism φ : V → V {\displaystyle \varphi :V\to V} , and a line bundle L → V {\displaystyle

infinite Fibonacci word is a morphic word, generated in {0,1}* by the endomorphism 0 → 01, 1 → 0. The nth element of a Fibonacci word, s n {\displaystyle

always hopfian. Any simple ring is hopfian, since the kernel of any endomorphism is an ideal, which is necessarily zero in a simple ring. In contrast

Travis; Petit, Christophe (2018), "Supersingular isogeny graphs and endomorphism rings: Reductions and solutions" (PDF), in Nielsen, Jesper Buus; Rijmen

Steinberg (1968) gave a useful improvement to the theorem. Suppose that F is an endomorphism of an algebraic group G. The Lang map is the map from G to G taking g

Travis; Petit, Christophe (2018), "Supersingular isogeny graphs and endomorphism rings: Reductions and solutions" (PDF), in Nielsen, Jesper Buus; Rijmen

{B}},m)} be a probability space, and T : X → X a measure-preserving endomorphism of X. The maximal function of f ∈ L1(X,m) is f ∗ ( x ) := sup n ≥ 1 1

cohomology H1(X/W(k)) ⊗ Qp of the special fiber (with the Frobenius endomorphism on this group and the Hodge filtration on this group tensored with K)

manifold to GL(n,C). Such a reduction is uniquely determined by a C∞-linear endomorphism J ∈ End(TM) such that J2 = −1. In this situation, the torsion can be

F_{A}\cdot \omega =\lambda (E)\operatorname {Id} _{E}} and the identity endomorphism has trace given by the rank of E {\displaystyle E} , we obtain λ ( E

a frequent co-author. The Igusa–Todorov functions and Igusa–Todorov endomorphism algebras are named for their joint work. Todorov is also the namesake

sporadic simple groups. In general the finite group associated to an endomorphism of a simply connected simple algebraic group is the universal central

rings. If D is a division ring and σ {\displaystyle \sigma } is a ring endomorphism which is not an automorphism, then the skew polynomial ring D [ x , σ

{\displaystyle M} , endowed with a tensor of type (1, 1), i.e. a vector bundle endomorphism (called an almost complex structure) J : T M → T M {\displaystyle J:TM\rightarrow

called an Elkies prime if the characteristic equation of the Frobenius endomorphism, ϕ 2 − t ϕ + q = 0 {\displaystyle \phi ^{2}-t\phi +q=0} , splits over

T x ) {\displaystyle Uf(x)=f(Tx)\,} where T is a measure-preserving endomorphism of X, thought of in applications as representing a time-step of a discrete

S} -module M = M 0 ⊕ M 1 {\displaystyle M=M_{0}\oplus M_{1}} with an endomorphism d = [ 0 d 1 d 0 0 ] {\displaystyle d={\begin{bmatrix}0&d_{1}\\d_{0}&0\end{bmatrix}}}

: x ↦ x p {\displaystyle \varphi :x\mapsto x^{p}} is a GF(p)-linear endomorphism and a field automorphism of GF(q), which fixes every element of the subfield

g {\displaystyle X\in {\mathfrak {g}}} . That is, adX is a nilpotent endomorphism in the usual sense of linear endomorphisms (rather than of Lie algebras)

nonsingular. The injective hull E(RR) is a nonsingular right R-module. The endomorphism ring S = E n d ( E ( R R ) ) {\displaystyle S=\mathrm {End} (E(R_{R}))\

T ∈ Ob ⁡ Π ( G ) {\displaystyle T\in \operatorname {Ob} \Pi (G)} an endomorphism of the space of T and satisfies the conditions of compatibility with

G is cyclic. In particular, the group G is not residually finite. An endomorphism of G is either an automorphism or its image is a cyclic subgroup of G

representation of the Weil group on a vector space V together with a nilpotent endomorphism N of V such that wNw−1=||w||N, or equivalently a representation of the

S for some prime number p, then the taking of pth powers induces an endomorphism of Ga, and the kernel is the group scheme αp. Over an affine base such

can be embedded into its own automorphism group. In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an

this one. An almost complex structure on a real Lie algebra g is an endomorphism I : g → g {\displaystyle I:\;g\rightarrow g} which squares to −Idg. This

bundle and a holomorphic ( 1 , 0 ) {\displaystyle (1,0)} -form, since the endomorphism of a line bundle are trivial. In particular, the Higgs field is uncoupled

, being a (decorated framed) tangle without boundary, represents an endomorphism of the monoidal identity (the empty set in this calculus), or in other

^{2}} have opposite sign and satisfy ab = −1. Since a square of such an endomorphism defines an Inoue surface of type S+, an Inoue surface of type S− has

)1_{W^{+}}\right)} from W + {\displaystyle W^{+}} to the a traceless Hermitian endomorphism of W + {\displaystyle W^{+}} identified with an imaginary self-dual 2-form

Osinga H M. (2008) Bifurcations of the global stable set of a planar endomorphism near a cusp singularity. International Journal of Bifurcation and Chaos

of a holomorphic vector bundle E {\displaystyle E} with a holomorphic endomorphism-valued ( 1 , 0 ) {\displaystyle (1,0)} -form Φ {\displaystyle \Phi }

proper over an Artinian subscheme of S. For each L in Pic(X), define the endomorphism c1(L) of G (called the first Chern class of L) by c 1 ( L ) F = F − L

This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex. Atiyah, M. F.; Bott, R. (1967), "A Lefschetz

that every map from a flat module onto M factors through F, and any endomorphism of F over M is an automorphism. This flat cover conjecture was explicitly

checking. These are equal because they are the kernel and cokernel of the endomorphism F× xn→ F×; formally, |μn| ⋅ |(F×)n| = |F×|. More abstractly, the first

S2CID 119176440 (with Alexander Slávik) 2006, 2012: Approximations and Endomorphism Algebras of Modules, de Gruyter Expositions in Mathematics 41, Vol. 1

exactly the set of zero-divisors on M, that is, an element r such that the endomorphism m ↦ r m , M → M {\displaystyle m\mapsto rm,M\to M} is not injective.

Chern class c1(O(1)){\displaystyle c_{1}({\mathcal {O}}(1))} as a group endomorphism of the Chow group of P(C⊕1){\displaystyle \mathbb {P} (C\oplus 1)}, the

supersingular elliptic curve over a finite field of characteristic p. The endomorphism ring of this is an order in a quaternion algebra over the rationals,

defined by adX Y = [X,Y] = XY − YX for some fixed X ∈ g. (The adjoint endomorphism encountered above.) Denote with AdA for fixed A ∈ G the linear transformation

{\displaystyle f:\mathbb {D} ^{n}\rightarrow \mathbb {D} ^{n}} is an endomorphism of the unit-ball. For k ≥ 1 {\displaystyle k\geq 1} all its homology

{\displaystyle M_{i},i\in I} are countably generated modules with local endomorphism rings and if N {\displaystyle N} is a countably generated module that

{L}}^{\otimes n}} and endomorphisms of the graded ring S. The identity endomorphism of S corresponds to a homomorphism ε {\displaystyle \varepsilon } . Applying

even. Hence the right hand side equals ω, so in this case the Frobenius endomorphism of Z[i]/(p) is the identity. Kummer had already established that if f

implies that adx{\displaystyle \operatorname {ad} _{x}} is a d-dimensional endomorphism, and so one has the characteristic equation det(tI−adx)=∑n=0dpn(x)tn{\displaystyle

T(|A|)=|P_{1}(A)|} ; this is an external endomorphism of the cardinals just as the T operation on ordinals is an external endomorphism of the ordinals. A set A is

e. the image of Λ ( V n ) {\displaystyle \Lambda (V_{n})} under the endomorphism formed from taking the product of the Clifford representation endomorphisms

called sheet (as in a sheet of paper), or sometimes a page or a term; an endomorphism d r {\displaystyle d_{r}} is called boundary map or differential. Sometimes

homomorphisms. These are useful to determining some information about the endomorphism ring End ( X ) ⊗ Q {\displaystyle {\text{End}}(X)\otimes \mathbb {Q}

series derived from the power series of the exponential map of a linear endomorphism, as in matrix exponentiation. When G is a matrix Lie group, all occurrences

r_{u\sigma v}:k\langle X\rangle \to k\langle X\rangle }, which is an endomorphism of k⟨X⟩{\displaystyle k\langle X\rangle } that fixes all elements of

( a ) : x ↦ a x   {\displaystyle L(a):x\mapsto ax\ } and define a K-endomorphism of A, called the quadratic representation, by Q ( a ) = 2 L ( a ) 2 −

{\displaystyle p} ring to characteristic 0 together with a lift of its Frobenius endomorphism. To make this precise, define a δ {\displaystyle \delta } -ring ( R

assigns to each ξ ∈ T p ∗ M {\displaystyle \xi \in T_{p}^{\ast }M} an endomorphism of the space of symmetric 2-tensors on T p M {\displaystyle T_{p}M} given

Y)Z} is C∞(M)-linear in each variable, it follows that R(x,Y)p is an endomorphism at p. For X and Y linearly independent tangent vectors at p, K=(R(X,Y)Y

or almost clean, and a module is called clean or almost clean if its endomorphism ring is clean or almost clean. CM Abbreviation for Cohen–Macaulay. CoCoA

modules. Young 1.  Alfred Young 2.  The Young symmetrizer is the G-linear endomorphism c λ : V ⊗ n → V ⊗ n {\displaystyle c_{\lambda }:V^{\otimes n}\to V^{\otimes

carries positive operators onto positive operators. For a in E, define an endomorphism of E, called the quadratic representation, by Q ( a ) = 2 L ( a ) 2 −