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text: VARIE´TE´S

text: DE

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text: STRATIFIE´ES

text: F

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text: X.,

text: -Repr´esentations

text: semi-stables,

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text: p

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text: Estimation

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text: dimensions

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text: de

text: Kisin,

text: J.

text: Reine

text: und

text: Angew.

text: Math.,

text: DOI

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text: Caruso

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text: David

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text: M´ezard

text: A.,

text: Calculs

text: d’anneaux

text: de

text: d´eformations

text: potentiellement

text: Barsotti-Tate,

text: Pr´epublication

text: [CL]

text: Caruso

text: X.,

text: Liu

text: T.,

text: Quasi

text: semistable

text: representations,

text: Bull.

text: Soc.

text: Math.

text: France

text: [Da]

text: David

text: A.,

text: Calculs

text: de

text: multiplicit´es

text: dans

text: la

text: conjecture

text: de

text: Breuil–M´ezard,

text: Pr´epublication

text: [EG]

text: Emerton

text: M.,

text: Gee

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text: A

text: geometric

text: perspective

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text: the

text: Breuil–M´ezard

text: conjecture,

text: J.

text: Math.

text: Jussieu

text: Fontaine

text: J.-M.,

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text: potentiellement

text: semi-stables,

text: Ast´erisque

text: Fontaine

text: J.-M.,

text: Repr´esentations

text: p-adiques

text: des

text: corps

text: locaux

text: I,

text: The

text: Grothendieck

text: Fest-

text: schrift,

text: Vol.

text: II,

text: Progr.

text: Math.,

text: vol.

text: Birkh¨auser

text: Boston,

text: Boston,

text: MA,

text: Hellmann

text: E.,

text: On

text: the

text: structure

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text: group

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text: Math.

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text: [Im]

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text: Kisin

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text: schemes

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text: modularity,

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text: M.,

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text: Amer.

text: Math.

text: Soc.

text: Kisin

text: M.,

text: The

text: structure

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text: semi-stable

text: deformation

text: rings,

text: Actes

text: du

text: Congr`es

text: International

text: des

text: Math´ematiciens

text: vol.

text: II,

text: Q

text: [Ma]

text: Mazur

text: B.,

text: Deforming

text: Galois

text: representations,

text: Galois

text: groups

text: over

text: (Berkeley

text: CA,

text: Math.

text: Sci.

text: Res.

text: Inst.

text: Publ.

text: [PR]

text: Pappas

text: G.,

text: Rapoport

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text: coeﬃcient

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text: [Win]

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text: des

text: normes

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text: corps

text: lo-

text: caux

text: applications,

text: Ann.

text: Sci.

text: ´ecole

text: Norm.

text: Sup.

text: X.

text: Caruso,

text: IRMAR,

text: Universit´e

text: de

text: Rennes

text: UMR

text: Campus

text: de

text: Beaulieu,

text: Rennes

text: Cedex,

text: France

text: E-mail

text: xavier.caruso@normalesup.org

text: A.

text: David,

text: Laboratoire

text: de

text: Math´ematiques

text: de

text: Besanc¸on,

text: UMR

text: Universit´e

text: de

text: Franche-Comt´e,

text: route

text: de

text: Gray,

text: Besanc¸on

text: Cedex,

text: France

text: E-mail

text: Agnes.David@math.cnrs.fr

text: A.

text: Me´zard,

text: Institut

text: de

text: Math´ematiques

text: de

text: Jussieu

text: Paris

text: Rive-Gauche,

text: UMR

text: LabEx

text: SMP

text: Universit´e

text: Pierre

text: et

text: Marie

text: Curie,

text: Paris,

text: France

text: E-mail

text: ariane.mezard@upmc.fr

text: ##LTLine##


text: 30 B. BURGSTALLER References ∗ [1] P. Baum, A. Connes, and N. Higson. Classifying space for proper actions and K-theory of group C - ∗ algebras. Doran, Robert S. (ed.), C -Algebras: 1943-1993. Providence, RI: American Mathematical Society. Contemp. Math. 167, 241-291 (1994). [2] B. Burgstaller. A Green–Julg isomorphims for inverse semigroups. preprint arXiv:1405.1607. [3] B. Burgstaller. An elementary Green imprimitivity theorem for inverse semigroups. preprint arXiv:1405.1619. [4] B. Burgstaller. Equivariant KK-theory of r-discrete groupoids and inverse semigroups. Rocky Mt. J. Math., to appear. preprint arXiv:1211.5006. [5] B. Burgstaller. The universal property of inverse semigroup equivariant KK-theory. preprint arXiv:1405.1613. [6] B. Burgstaller. Equivariant KK-theory for semimultiplicative sets. New York J. Math., 15:505–531, 2009. [7] B. Burgstaller. A descent homomorphism for semimultiplicative sets. Rocky Mt. J. Math., 44(3):809– 851, 2014. [8] J. Chabert and S. Echterhoﬀ. Permanence properties of the Baum-Connes conjecture. Doc. Math., J. DMV, 6:127–183, 2001. [9] J. Cuntz. A new look at KK-theory. K-theory, 1:31–51, 1987. [10] P. Green. The local structure of twisted covariance algebras. Acta Math., 140:191–250, 1978. [11] N. Higson. A characterization of KK-theory. Pac. J. Math., 126(2):253–276, 1987. [12] K. K. Jensen and K. Thomsen. Elements of KK-theory. Mathematics: Theory and Applications. Boston, MA: Birkh¨auser. viii, 202 p. , 1991. [13] P. Julg. K-th´eorie ´equivariante et produits croises. C. R. Acad. Sci., Paris, S´er. I, 292:629–632, 1981. [14] G.G. Kasparov. The operator K-functor and extensions of C*-algebras. Math. USSR, Izv., 16:513–572, 1981. [15] G.G. Kasparov. Equivariant KK-theory and the Novikov conjecture. Invent. Math., 91(1):147–201, 1988. ∗ [16] G.G. Kasparov. K-theory, group C -algebras, and higher signatures (conspectus). In Novikov conjec- tures, index theorems and rigidity. Vol. 1., pages 101–146. Cambridge University Press, 1995. ∗ [17] M. Khoshkam and G. Skandalis. Crossed products of C -algebras by groupoids and inverse semigroups. J. Oper. Theory, 51(2):255–279, 2004. [18] H. Krause. Localization theory for triangulated categories. Holm, Thorsten (ed.) et al., Triangulated categories. Based on a workshop, Leeds, UK, August 2006. Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Series 375, 161-235 (2010)., 2010. [19] P.-Y. Le Gall. Equivariant Kasparov theory and groupoids. I. (Th´eorie de Kasparov ´equivariante et groupo¨ıdes. I.). K-Theory, 16(4):361–390, 1999. [20] R. Meyer. Equivariant Kasparov theory and generalized homomorphisms. K-Theory, 21(3):201–228, 2000.

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text: ATTEMPTS TO DEFINE A BAUM–CONNES MAP 31 [21] R. Meyer and R. Nest. The Baum-Connes conjecture via localisation of categories. Topology, 45(2):209– 259, 2006. ∗ [22] J.A. Mingo and W.J. Phillips. Equivariant triviality theorems for Hilbert C -modules. Proc. Am. Math. Soc., 91:225–230, 1984. [23] A. Neeman. The Grothendieck duality theorem via Bousﬁeld’s techniques and Brown representability. J. Am. Math. Soc., 9(1):205–236, 1996. [24] A. Neeman. Triangulated categories. Annals of Mathematics Studies. 148. Princeton, NJ: Princeton University Press. vii, 449 p., 2001. [25] J. Rosenberg. The Ku¨nneth theorem in equivariant K-theory for actions of a cyclic group of order 2. Algebr. Geom. Topol., 13(2):1225–1241, 2013. ∗ [26] N. Sieben. C -crossed products by partial actions and actions of inverse semigroups. J. Aust. Math. Soc., Ser. A, 63(1):32–46, 1997. [27] K. Thomsen. The universal property of equivariant KK-theory. J. Reine Angew. Math., 504:55–71, 1998. [28] J. L. Tu. The Novikov conjecture for hyperbolic foliations. (La conjecture de Novikov pour les feuil- letages hyperboliques.). K-theory, 12(2):129–184, 1999. Email address: bernhardburgstaller@yahoo.de

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text: ATTEMPTS TO DEFINE A BAUM–CONNES MAP 5 ∈ C (cid:111) − − generated by G and all projections p G of the form p = e (1 e ) . . . (1 e ) for 0 1 n ∈ ≥ ∈ ˜ e E and n 0. Note that every element of G is of the form gp with g G and p as i before. ˜ Every G-action α on a G-algebra (or G-Hilbert module) extends to a G-action by linearity, − − that is, α = α α (α α ) . . . (α α ), where p is as before (see [3, Lemma 2.1]). We gp g e0 1 e1 1 en ˜ sometimes extend G-actions to G-actions in this way implicitly without saying. We shall ⊆ ˜ also consider discrete groupoids H G, and we may regard them as inverse semigroups ∪ { } ⊆ ˜ H 0 G with zero element in order to consistently redeﬁne the known notion of H- ∪ { } equivariant KK-theory KKH via the inverse semigroup H 0 , where 0 is understood to act always as zero. Provided is here however that the H-algebras are deﬁned in the groupoid sense, that is, that they are also C (H(0))-algebras, see [15, Deﬁnition 1.5]. (Cf. also [4].) 0 ⊆ ⊆ ˜ Let G L G be a subinverse semigroup. Then we have (3) KKG(A, B) = KKL(A, B) = KKG˜(A, B) via the identity map on cycles when using the above mentioned extension of G-actions ˜ ˜ for all G-algebras A and B. (A G-Hilbert B-module inherits the linearly extended G-action from B by compatibility.) Denote by X or X the totally disconnected, locally compact G ∗ ∗ Hausdorﬀ space such that C (X) is the universal commutative C -algebra C (E) generated 0 by the commuting projections E. (Actually X is compact since E is unital.) C (X) is 0 ∈ ∈ endowed with the G-action g(1 ) = 1 ∗ for e E and g G. Every G-algebra A may e geg → M be regarded as a C (X)-algebra (see Kasparov [15, Section 1.5]) by π : C (X) Z( (A)) 0 0 ⊗ with π(1 )(a) = e(a) since E has a unit. Write A X B for the balanced tensor product e ⊗ ⊗ − ⊗ ∈ (A B divided by all elements of the form e(a) b a e(b) where e E), see Le Gall [19] or [15, Section 1.6]. ⊆ ˜ Deﬁnition 2.5. The groupoid H G associated to a given ﬁnite subinverse semi- (cid:102) (cid:48) ⊆ { ∈ | ∈ (cid:48) ∈ ˜ group H G is deﬁned to be the ﬁnite groupoid H = hp G h H , p (cid:48) ∗ ≥ } E(H ) is a minimal projection, h h p . (cid:48) (cid:48) Observe that KKH (A, B) = KKH(A, B) for all H -algebras or H-algebras A and B by ∗ ∗ (cid:48) the equivalence of C (cid:48) and C , and KKH and KKH, respectively, see [4]. (Our notion H(cid:92) H (cid:48) (cid:48) KKH coincides with KKH of [4].) All subinverse semigroups of G are assumed to contain

text: ATTEMPTS

text: TO

text: DEFINE

text: A

text: BAUM–CONNES

text: MAP

text: C

text: generated

text: by

text: G

text: and

text: all

text: projections

text: p

text: G

text: of

text: the

text: form

text: p

text: e

text: e

text: e

text: for

text: n

text: e

text: E

text: and

text: n

text: Note

text: that

text: every

text: element

text: of

text: G

text: is

text: of

text: the

text: form

text: gp

text: with

text: g

text: G

text: and

text: p

text: as

text: i

text: before.

text: Every

text: G-action

text: α

text: on

text: a

text: G-algebra

text: (or

text: G-Hilbert

text: module)

text: extends

text: to

text: a

text: G-action

text: by

text: linearity,

text: that

text: is,

text: α

text: α

text: α

text: (α

text: α

text: (α

text: α

text: where

text: p

text: is

text: as

text: before

text: (see

text: Lemma

text: We

text: gp

text: g

text: en

text: sometimes

text: extend

text: G-actions

text: to

text: G-actions

text: in

text: this

text: way

text: implicitly

text: without

text: saying.

text: We

text: shall

text: also

text: consider

text: discrete

text: groupoids

text: H

text: G,

text: and

text: we

text: may

text: regard

text: them

text: as

text: inverse

text: semigroups

text: H

text: G

text: with

text: zero

text: element

text: in

text: order

text: to

text: consistently

text: redeﬁne

text: the

text: known

text: notion

text: of

text: H-

text: equivariant

text: KK-theory

text: KKH

text: via

text: the

text: inverse

text: semigroup

text: H

text: where

text: is

text: understood

text: to

text: act

text: always

text: as

text: zero.

text: Provided

text: is

text: here

text: however

text: that

text: the

text: H-algebras

text: are

text: deﬁned

text: in

text: the

text: groupoid

text: sense,

text: that

text: is,

text: that

text: they

text: are

text: also

text: C

text: see

text: Deﬁnition

text: (Cf.

text: also

text: Let

text: G

text: L

text: G

text: be

text: a

text: subinverse

text: semigroup.

text: Then

text: we

text: have

text: KKG(A,

text: B)

text: KKL(A,

text: B)

text: KKG˜(A,

text: B)

text: via

text: the

text: identity

text: map

text: on

text: cycles

text: when

text: using

text: the

text: above

text: mentioned

text: extension

text: of

text: G-actions

text: for

text: all

text: G-algebras

text: A

text: and

text: B.

text: (A

text: G-Hilbert

text: B-module

text: inherits

text: the

text: linearly

text: extended

text: G-action

text: from

text: B

text: by

text: compatibility.)

text: Denote

text: by

text: X

text: or

text: X

text: the

text: totally

text: disconnected,

text: locally

text: compact

text: G

text: Hausdorﬀ

text: space

text: such

text: that

text: C

text: (X)

text: is

text: the

text: universal

text: commutative

text: C

text: -algebra

text: C

text: (E)

text: generated

text: by

text: the

text: commuting

text: projections

text: E.

text: (Actually

text: X

text: is

text: compact

text: since

text: E

text: is

text: unital.)

text: C

text: (X)

text: is

text: endowed

text: with

text: the

text: G-action

text: for

text: e

text: E

text: and

text: g

text: G.

text: Every

text: G-algebra

text: A

text: may

text: e

text: geg

text: M

text: be

text: regarded

text: as

text: a

text: C

text: (X)-algebra

text: (see

text: Kasparov

text: Section

text: by

text: π

text: C

text: (X)

text: Z(

text: (A))

text: with

text: )(a)

text: e(a)

text: since

text: E

text: has

text: a

text: unit.

text: Write

text: A

text: X

text: B

text: for

text: the

text: balanced

text: tensor

text: product

text: e

text: (A

text: B

text: divided

text: by

text: all

text: elements

text: of

text: the

text: form

text: e(a)

text: b

text: a

text: e(b)

text: where

text: e

text: E),

text: see

text: Le

text: Gall

text: or

text: Section

text: Deﬁnition

text: The

text: groupoid

text: H

text: G

text: associated

text: to

text: a

text: given

text: ﬁnite

text: subinverse

text: semi-

text: group

text: H

text: G

text: is

text: deﬁned

text: to

text: be

text: the

text: ﬁnite

text: groupoid

text: H

text: hp

text: G

text: h

text: H

text: p

text: E(H

text: is

text: a

text: minimal

text: projection,

text: h

text: h

text: p

text: Observe

text: that

text: KKH

text: (A,

text: B)

text: KKH(A,

text: B)

text: for

text: all

text: H

text: -algebras

text: or

text: H-algebras

text: A

text: and

text: B

text: by

text: the

text: equivalence

text: of

text: C

text: and

text: C

text: and

text: KKH

text: and

text: KKH,

text: respectively,

text: see

text: (Our

text: notion

text: H

text: KKH

text: coincides

text: with

text: KKH

text: of

text: All

text: subinverse

text: semigroups

text: of

text: G

text: are

text: assumed

text: to

text: contain


text: 8 B. BURGSTALLER ∗ ∈ C -algebras C (G ) and C (G /H) are endowed with the G-action g(h) := [gh G ] gh, 0 H 0 H H ∈ ∈ ∈ ∗ ≥ ∗ where g G and h G (of course, gh G is equivalent to g g hh ). Deﬁne H H ⊆ ⊗ C (G /H, B) C (G /H) B to be the G-invariant ideal which is the closure of the 0 H 0 H ⊗ ∗ ∈ ∈ linear span of all elements of the form g gg (b) (g G , b B). Similarly, denote by H ∈ L ⊗ ⊗ ⊗ ⊗ ⊗ ∗ p Z( (IndG(A) B)) (center) the central projection p(g a b) := g a gg (b) for H ∈ ∈ ∗ ∈ g G , a g g(A) and b B. We have a direct sum decomposition H (cid:0) (cid:1) (cid:0) (cid:1) ∼ ⊗ ⊗ ⊕ − ⊗ (5) IndG(A) B = p IndG(A) B (1 p) IndG(A) B , H H H → ⊗ and we denote the ﬁrst summand (and ideal) by IndG(A) B. H ⊆ ˜ Lemma 4.4 (Cf. line (17) in [21]). Let B be a G-algebra and H G a ﬁnite subgroupoid. (cid:88) ∗ Then there is a G-equivariant -isomorphism −→ ⊗ Θ : IndGResH(B) C (G /H, B), Θ(f) = g g(f(g)) H G 0 H ∈ g GH/H ∈ ⊆ ⊗ for all f IndGResH(B) C (G ) B. (The sum is understood that we choose for every H G 0 H ∈ equivalence class in G /H exactly one arbitrary representative g G .) H H (cid:3) Proof. The proof is straightforward. ⊆ ˜ Lemma 4.5 (Cf. line (16) in [21]). Let H G be a ﬁnite subgroupoid, A a H-algebra and (cid:0) (cid:1) ∗ B a G-algebra. Then there is a G-equivariant -isomorphism → ⊗ −→ ⊗ ⊗ ⊗ ⊗ ⊗ Θ : IndG A XH ResH(B) IndG(A) B, Θ(g a b) = g a g(b) H G H ∈ ∈ ∗ ∈ ∗ for all g G , a g g(A) and b g g(B). H ⊗ Proof. The tensor product A XH ResH(B) denotes the balanced groupoid tensor product G and is endowed with the diagonal H-action. In other words, we may regard A and ResH(B) G ∪ { } as H 0 -inverse semigroup algebras and take the usual diagonal inv(cid:0)erse semigroup ac(cid:1)tion ⊗ ∪{ } for the tensor product A XH∪{0} ResH 0 (B). ⊗ ∗ ⊗ G˜ ⊗ ∗ ⊗ ∗ ⊗ ⊆ Note that we have gt t (a b) = gt t (a) t (b) in IndG A XH ResH(B) H G ⊗ ⊗ ∈ ∈ ∈ ∈ ∈ C (G ) A B for all g G , t H, a A and b B with gt G , so we can achieve the 0 H H H ∗ required format in the argument of Θ when setting t := g g. Surjectivity of Θ is obvious. ∗ → ∗ (cid:3) That Θ is isometric is also clear as the transition g gB gB by Θ is a -isomorphism. From now on we restrict ourselves to trivially graded G-algebras.

text: B.

text: BURGSTALLER

text: C

text: -algebras

text: C

text: (G

text: and

text: C

text: (G

text: /H)

text: are

text: endowed

text: with

text: the

text: G-action

text: g(h)

text: [gh

text: G

text: gh,

text: H

text: H

text: H

text: where

text: g

text: G

text: and

text: h

text: G

text: (of

text: course,

text: gh

text: G

text: is

text: equivalent

text: to

text: g

text: g

text: hh

text: Deﬁne

text: H

text: H

text: C

text: (G

text: /H,

text: B)

text: C

text: (G

text: /H)

text: B

text: to

text: be

text: the

text: G-invariant

text: ideal

text: which

text: is

text: the

text: closure

text: of

text: the

text: H

text: H

text: linear

text: span

text: of

text: all

text: elements

text: of

text: the

text: form

text: g

text: gg

text: (b)

text: (g

text: G

text: b

text: B).

text: Similarly,

text: denote

text: by

text: H

text: L

text: p

text: Z(

text: (IndG(A)

text: B))

text: (center)

text: the

text: central

text: projection

text: p(g

text: a

text: b)

text: g

text: a

text: gg

text: (b)

text: for

text: H

text: g

text: G

text: a

text: g

text: g(A)

text: and

text: b

text: B.

text: We

text: have

text: a

text: direct

text: sum

text: decomposition

text: H

text: IndG(A)

text: B

text: p

text: IndG(A)

text: B

text: p)

text: IndG(A)

text: B

text: H

text: H

text: H

text: and

text: we

text: denote

text: the

text: ﬁrst

text: summand

text: (and

text: ideal)

text: by

text: IndG(A)

text: B.

text: H

text: Lemma

text: (Cf.

text: line

text: in

text: Let

text: B

text: be

text: a

text: G-algebra

text: and

text: H

text: G

text: a

text: ﬁnite

text: subgroupoid.

text: Then

text: there

text: is

text: a

text: G-equivariant

text: -isomorphism

text: Θ

text: IndGResH(B)

text: C

text: (G

text: /H,

text: B),

text: Θ(f)

text: g

text: g(f(g))

text: H

text: G

text: H

text: g

text: GH/H

text: for

text: all

text: f

text: IndGResH(B)

text: C

text: (G

text: B.

text: (The

text: sum

text: is

text: understood

text: that

text: we

text: choose

text: for

text: every

text: H

text: G

text: H

text: equivalence

text: class

text: in

text: G

text: /H

text: exactly

text: one

text: arbitrary

text: representative

text: g

text: G

text: H

text: H

text: Proof.

text: The

text: proof

text: is

text: straightforward.

text: Lemma

text: (Cf.

text: line

text: in

text: Let

text: H

text: G

text: be

text: a

text: ﬁnite

text: subgroupoid,

text: A

text: a

text: H-algebra

text: and

text: B

text: a

text: G-algebra.

text: Then

text: there

text: is

text: a

text: G-equivariant

text: -isomorphism

text: Θ

text: IndG

text: A

text: XH

text: ResH(B)

text: IndG(A)

text: B,

text: Θ(g

text: a

text: b)

text: g

text: a

text: g(b)

text: H

text: G

text: H

text: for

text: all

text: g

text: G

text: a

text: g

text: g(A)

text: and

text: b

text: g

text: g(B).

text: H

text: Proof.

text: The

text: tensor

text: product

text: A

text: XH

text: ResH(B)

text: denotes

text: the

text: balanced

text: groupoid

text: tensor

text: product

text: G

text: and

text: is

text: endowed

text: with

text: the

text: diagonal

text: H-action.

text: In

text: other

text: words,

text: we

text: may

text: regard

text: A

text: and

text: ResH(B)

text: G

text: as

text: H

text: -inverse

text: semigroup

text: algebras

text: and

text: take

text: the

text: usual

text: diagonal

text: semigroup

text: for

text: the

text: tensor

text: product

text: A

text: ResH

text: (B).

text: G˜

text: Note

text: that

text: we

text: have

text: gt

text: t

text: (a

text: b)

text: gt

text: t

text: (a)

text: t

text: (b)

text: in

text: IndG

text: A

text: XH

text: ResH(B)

text: H

text: G

text: C

text: (G

text: A

text: B

text: for

text: all

text: g

text: G

text: t

text: H,

text: a

text: A

text: and

text: b

text: B

text: with

text: gt

text: G

text: so

text: we

text: can

text: achieve

text: the

text: H

text: H

text: H

text: required

text: format

text: in

text: the

text: argument

text: of

text: Θ

text: when

text: setting

text: t

text: g

text: g.

text: Surjectivity

text: of

text: Θ

text: is

text: obvious.

text: That

text: Θ

text: is

text: isometric

text: is

text: also

text: clear

text: as

text: the

text: transition

text: g

text: gB

text: gB

text: by

text: Θ

text: is

text: a

text: -isomorphism.

text: From

text: now

text: on

text: we

text: restrict

text: ourselves

text: to

text: trivially

text: graded

text: G-algebras.


text: 2 protocol that effectively coordinates the users’ transmissions under the above described information structure. In the design of transmission protocols, there are two major performance metrics of interest: throughput and delay. The throughput region of a protocol is the set of arrival rates for which the users’ queues are stable (see detailed deﬁnition in Section ) under the protocol. The delay performance of a protocol is the average waiting time of a packet in the communication system. An efﬁcient transmission protocol should achieve the maximum throughput region and incur low transmission delay. In this paper, we propose a common information (see [2, 3]) based multiple access protocol (CIMA) that uses the common channel feedback to coordinate users. In CIMA, each user constructs upper bounds on the lengths of the queues of all users, including itself, based on previous transmission strategies and the common feedback. Since the upper bounds are common knowledge, users can coordinate their transmission through these common upper bounds to avoid collision. We prove that without knowledge of any statistics, CIMA achieves the full throughput region of the collision channel. We also prove that the CIMA protocol incurs low transmission delay; the delay is upper-bounded by a linear function of the number of users. There is a rich literature on multiple access communications. Below we present a survey of this literature. Related Work There are primarily two classes of protocols for the situation where the alphabet of the feedback channel is { } { } 0, 1, e = no transmission, successful transmission, collision : collision-free and contention-based protocols. Time-division-multiple-access (TDMA) [4] and adaptive TDMA [5, 6] are collision-free protocols. In adaptive TDMA protocols the (common) information provided by the feedback is used to adaptively coordinate users to avoid collision. Adaptation resolves the problems due to asymmetric arrivals, and collision avoidance results in higher throughput and lower delay than TDMA. However, there is no theoretical analysis of adaptive TDMA protocols. Backoff-type protocols and Aloha protocols [4] allow for contention/collision. Due to collision, most contention based protocols can not achieve full throughput. However, polynomial back-off protocols, presented and analyzed in [7], achieve full throughput. Nevertheless, polynomial back-off protocols have exponential delay performance in simulation. Several types of multiple access protocols were proposed when the common information among the users is more { } than 0, 1, e . The authors of [8–10] proposed decentralized random access protocols that achieve full throughput when each user knows the maximum queue length in the system or all other users’ transmission results. When channel sensing is allowed, carrier sense multiple access (CSMA) protocols, proposed in [11–16], achieve full throughput when the channel sensing portion of time is not taken into account in the throughput calculation. A survey of CSMA protocols is presented in [17]. In terms of delay performance, the CSMA protocols proposed in [15, 16] achieve delay that is linear in the number of users. Multiple access protocols for adversarial queueing models were presented in [18, 19]. In [18, 19] it is proved that these protocols achieve full throughput and have linear delay in the number of users. Other models for multiple access have also been proposed in the literature. In [20], channel switching policies that achieve high throughput for multiple access have been considered within the context of the slotted Aloha protocol and the IEEE 802.11 WLANs protocol. The stability region of the multi-packet reception multiple access channel May 27, 2020 DRAFT

text: protocol

text: that

text: effectively

text: coordinates

text: the

text: users’

text: transmissions

text: under

text: the

text: above

text: described

text: information

text: structure.

text: In

text: the

text: design

text: of

text: transmission

text: protocols,

text: there

text: are

text: two

text: major

text: performance

text: metrics

text: of

text: interest:

text: throughput

text: and

text: delay.

text: The

text: throughput

text: region

text: of

text: a

text: protocol

text: is

text: the

text: set

text: of

text: arrival

text: rates

text: for

text: which

text: the

text: users’

text: queues

text: are

text: stable

text: (see

text: detailed

text: deﬁnition

text: in

text: Section

text: under

text: the

text: protocol.

text: The

text: delay

text: performance

text: of

text: a

text: protocol

text: is

text: the

text: average

text: waiting

text: time

text: of

text: a

text: packet

text: in

text: the

text: communication

text: system.

text: An

text: efﬁcient

text: transmission

text: protocol

text: should

text: achieve

text: the

text: maximum

text: throughput

text: region

text: and

text: incur

text: low

text: transmission

text: delay.

text: In

text: this

text: paper,

text: we

text: propose

text: a

text: common

text: information

text: (see

text: based

text: multiple

text: access

text: protocol

text: (CIMA)

text: that

text: uses

text: the

text: common

text: channel

text: feedback

text: to

text: coordinate

text: users.

text: In

text: CIMA,

text: each

text: user

text: constructs

text: upper

text: bounds

text: on

text: the

text: lengths

text: of

text: the

text: queues

text: of

text: all

text: users,

text: including

text: itself,

text: based

text: on

text: previous

text: transmission

text: strategies

text: and

text: the

text: common

text: feedback.

text: Since

text: the

text: upper

text: bounds

text: are

text: common

text: knowledge,

text: users

text: can

text: coordinate

text: their

text: transmission

text: through

text: these

text: common

text: upper

text: bounds

text: to

text: avoid

text: collision.

text: We

text: prove

text: that

text: without

text: knowledge

text: of

text: any

text: statistics,

text: CIMA

text: achieves

text: the

text: full

text: throughput

text: region

text: of

text: the

text: collision

text: channel.

text: We

text: also

text: prove

text: that

text: the

text: CIMA

text: protocol

text: incurs

text: low

text: transmission

text: delay;

text: the

text: delay

text: is

text: upper-bounded

text: by

text: a

text: linear

text: function

text: of

text: the

text: number

text: of

text: users.

text: There

text: is

text: a

text: rich

text: literature

text: on

text: multiple

text: access

text: communications.

text: Below

text: we

text: present

text: a

text: survey

text: of

text: this

text: literature.

text: Related

text: Work

text: There

text: are

text: primarily

text: two

text: classes

text: of

text: protocols

text: for

text: the

text: situation

text: where

text: the

text: alphabet

text: of

text: the

text: feedback

text: channel

text: is

text: e

text: no

text: transmission,

text: successful

text: transmission,

text: collision

text: collision-free

text: and

text: contention-based

text: protocols.

text: Time-division-multiple-access

text: (TDMA)

text: and

text: adaptive

text: TDMA

text: are

text: collision-free

text: protocols.

text: In

text: adaptive

text: TDMA

text: protocols

text: the

text: (common)

text: information

text: provided

text: by

text: the

text: feedback

text: is

text: used

text: to

text: adaptively

text: coordinate

text: users

text: to

text: avoid

text: collision.

text: Adaptation

text: resolves

text: the

text: problems

text: due

text: to

text: asymmetric

text: arrivals,

text: and

text: collision

text: avoidance

text: results

text: in

text: higher

text: throughput

text: and

text: lower

text: delay

text: than

text: TDMA.

text: However,

text: there

text: is

text: no

text: theoretical

text: analysis

text: of

text: adaptive

text: TDMA

text: protocols.

text: Backoff-type

text: protocols

text: and

text: Aloha

text: protocols

text: allow

text: for

text: contention/collision.

text: Due

text: to

text: collision,

text: most

text: contention

text: based

text: protocols

text: can

text: not

text: achieve

text: full

text: throughput.

text: However,

text: polynomial

text: back-off

text: protocols,

text: presented

text: and

text: analyzed

text: in

text: achieve

text: full

text: throughput.

text: Nevertheless,

text: polynomial

text: back-off

text: protocols

text: have

text: exponential

text: delay

text: performance

text: in

text: simulation.

text: Several

text: types

text: of

text: multiple

text: access

text: protocols

text: were

text: proposed

text: when

text: the

text: common

text: information

text: among

text: the

text: users

text: is

text: more

text: than

text: e

text: The

text: authors

text: of

text: proposed

text: decentralized

text: random

text: access

text: protocols

text: that

text: achieve

text: full

text: throughput

text: when

text: each

text: user

text: knows

text: the

text: maximum

text: queue

text: length

text: in

text: the

text: system

text: or

text: all

text: other

text: users’

text: transmission

text: results.

text: When

text: channel

text: sensing

text: is

text: allowed,

text: carrier

text: sense

text: multiple

text: access

text: (CSMA)

text: protocols,

text: proposed

text: in

text: achieve

text: full

text: throughput

text: when

text: the

text: channel

text: sensing

text: portion

text: of

text: time

text: is

text: not

text: taken

text: into

text: account

text: in

text: the

text: throughput

text: calculation.

text: A

text: survey

text: of

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text: 14 | P(cid:80)λ,CIMA(qk+1 f0:k) | Pλ,CIMA(q , f f − ) = k+1(cid:48) k 0:k| 1 . (23) q(cid:48) Pλ,CIMA(qk+1, fk f0:k−1) k+1 Let v = v(bCIMA). Consider the numerator in (23). There are two cases: f = 1 and f = 0. t k k When f = 1, we have k (cid:88) | Pλ,CIMA(q , F = 1 f − ) k+1 k 0:k 1 | = (cid:88) Pλ,CIMA(q , q , F = 1 f − ) k+1 k k 0:k 1 qk = Pλ,CIMA(q , q , Qv > 0|f − ) (24) (cid:88) k+1 k k 0:k 1 qk  | = Pλ,CIMA(q f − )  k 0:k 1 (cid:89) qk,qkv>0  − − Pλ(Av= qv qv + 1) Pλ(An = qn qn) (25) (cid:88) k (cid:89)k+1 k (cid:54) k k+1 k  n=v N | = Pλ,CIMA(qn f − )  k (cid:89)0:k 1 qk,qkv>0 n=1  − − Pλ(Av = qv qv + 1) Pλ(An = qn qn) (26) (cid:89) (cid:88)k k+1 k (cid:54) k k+1 k  n=v  − | = (cid:88) Pλ(An = qn qn)Pλ,CIMA(qn f − ) k k+1 k k 0:k 1 (cid:54) n=v qn k − | Pλ(Av = qv qv + 1)Pλ,CIMA(qv f − ). (27) k k+1 k k 0:k 1 qv>0 k Equation (24) holds because F = 1 if and only if Qv(BtCIMA) > 0. Equation (25) is true because of the system k k dynamics (1) and the fact that An, n = 1, 2, . . . , N are mutually independent and independent of all variables before k k. Equation (26) follows from the induction hypothesis for (7). Equation (27) is true because each term in (26) depends only on each qn for n = 1, 2, . . . , N. k Using (27), the denominator in (23) becomes May 27, 2020 DRAFT ##LTLine##

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text: Topic_6 Topic_19 Topic_27 Topic_47 word prob. word prob. word prob. word prob. food 0.027 drug 0.031 medical 0.033 dog 0.011 restaurant 0.008 drugs 0.019 hospital 0.024 garden 0.009 eat 0.008 cancer 0.019 care 0.019 tree 0.009 LDA more 0.005 study 0.011 patients 0.018 dogs 0.009 chicken 0.005 patients 0.011 doctors 0.016 plants 0.008 cooking 0.005 treatment 0.009 health 0.013 trees 0.008 eating 0.005 fda 0.009 doctor 0.009 animal 0.007 one 0.005 heart 0.008 patient 0.009 plant 0.007 good 0.005 risk 0.008 surgery 0.008 animals 0.006 foods 0.005 more 0.007 center 0.008 zoo 0.006 word/topic cos. word/topic cos. word/topic cos. word/topic cos. cheeseburgers 0.564 topic_62 0.618 topic_19 0.519 dogwood 0.498 meatless 0.535 aricept 0.531 topic_62 0.478 dogwoods 0.494 smoothies 0.534 topic_27 0.519 neonatal 0.466 topic_33 0.485 Topic2Vec topic_95 0.533 memantine 0.514 topic_13 0.457 bark 0.484 meatloaf 0.530 enbrel 0.512 anesthesiologists 0.445 fescue 0.483 tastier 0.530 gabapentin 0.511 anesthesia 0.439 aphids 0.478 topic_52 0.527 colorectal 0.509 reconstructive 0.437 mulched 0.478 cheeseburger 0.525 prilosec 0.507 comatose 0.437 azaleas 0.477 concoctions 0.522 placebos 0.507 hysterectomy 0.433 shrub 0.475 vegetarians 0.515 intravenously 0.504 ventilator 0.432 camellias 0.472 Topic_53 Topic_67 Topic_79 Topic_93 word prob. word prob. word prob. word prob. government 0.022 www 0.028 computer 0.016 russia 0.028 africa 0.015 com 0.023 technology 0.010 russian 0.027 people 0.015 hotel 0.018 phone 0.009 putin 0.017 LDA african 0.011 travel 0.015 software 0.009 soviet 0.013 country 0.009 trip 0.011 digital 0.008 moscow 0.012 international 0.008 night 0.010 apple 0.008 president 0.010 darfur 0.007 per 0.009 use 0.007 country 0.007 sudan 0.007 day 0.008 system 0.006 former 0.007 south 0.007 tour 0.008 microsoft 0.006 state 0.007 human 0.007 cruise 0.007 up 0.006 union 0.006 word/topic cos. word/topic cos. word/topic cos. word/topic cos. mozambique 0.428 fairmont 0.569 wirelessly 0.584 topic_88 0.469 uganda 0.423 motorcoach 0.553 handhelds 0.573 boris 0.435 ghana 0.419 stateroom 0.547 desktops 0.572 leonid 0.411 addis 0.417 uniworld 0.540 pda 0.566 dmitry 0.404 Topic2Vec darfur 0.412 maarten 0.533 smartphone 0.566 vladimir 0.397 burundi 0.408 tourcrafters 0.529 megabyte 0.562 mikhail 0.397 lanka 0.407 wyndham 0.528 macbook 0.556 dmitri 0.396 congo 0.406 cunard 0.527 handheld 0.549 alexei 0.394 ababa 0.403 safaris 0.522 treo 0.549 eduard 0.392 darfurians 0.402 trafalgar 0.518 modems 0.548 kasparov 0.391 Figure 2: Nearest words and topics for each topic. Words are listed with conditional probabilities in LDA while words and topics are listed with calculated cosine similarity in Topic2Vec. 4.1 Dataset L (cid:88) (cid:88) − (D) Skip gram We use the English Gigaword Fifth Edition2 as 1 M | our training data for learning fundamental word (3b) = (log p(w w ) M − ≤ ≤ (cid:54) i+c i and topic representations. We randomly extract i=1 k c k,c=0 | part of documents and construct our training set + log p(w z )). i+c i described as follows: we chose 100,000 docu- Topic2Vec aims at learning topic representa- ments, where each consists of more than 1,000 tions along with word representations. Consid- characters from subfolder ltw eng (Los Angeles ering the simplicity and efﬁcient solution, we Times) containing 411,032 documents. Besides, just follow the optimization scheme that used in we eliminate those words that occur less than 5 Word2Vec (Mikolov et al., 2013a). To approxi- times and the stop words. In the end, training set mately maximize the probability of the softmax, contains about 42 million words and the vocabu- we use Negative Sampling without Hierarchical lary size is 102,644. Softmax (Mikolov et al., 2013b). Stochastic gra- 4.2 Evaluation Methods dient descent (SGD) and back-propagation algo- In experiments, we run Topic2Vec in Skip-gram rithm are used to optimize our model. By the way, and learn topic representations together with word complexity of our Topic2Vec is linear with size of representations. And then we evaluate topic repre- dataset, same with Word2Vec. sentations via comparing Topic2Vec with LDA in 4 Experiments 2https://catalog.ldc.upenn.edu/LDC2011T07 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: [14] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hin- ton, “Imagenet classiﬁcation with deep convolutional neural networks,” in NIPS, 2012, pp. 1097–1105. [15] Jeff Donahue, Yangqing Jia, Oriol Vinyals, Judy Hoff- man, Ning Zhang, Eric Tzeng, and Trevor Darrell, “De- caf: A deep convolutional activation feature for generic visual recognition,” arXiv preprint arXiv:1310.1531, 2013. [16] Chao Dong, Chen Change Loy, Kaiming He, and Xi- aoou Tang, “Learning a deep convolutional network for image super-resolution,” in ECCV, pp. 184–199. Springer, 2014. [17] Matthew D Zeiler, Graham W Taylor, and Rob Fergus, “Adaptive deconvolutional networks for mid and high level feature learning,” in ICCV, 2011, pp. 2018–2025. [18] Matthew D Zeiler and Rob Fergus, “Visualizing and understanding convolutional networks,” in ECCV, pp. 818–833. Springer, 2014. [19] Anita Roth-Nebelsick, Dieter Uhl, Volker Mosbrugger, and Hans Kerp, “Evolution and function of leaf venation architecture: a review,” Annals of Botany, vol. 87, no. 5, pp. 553–566, 2001. [20] Héctor Candela, Antonio Martınez-Laborda, and José Luis Micol, “Venation pattern formation in arabidop- sis thaliana vegetative leaves,” Developmental biology, vol. 205, no. 1, pp. 205–216, 1999. [21] Adam Runions, Martin Fuhrer, Brendan Lane, Pavol Federl, Anne-Gaëlle Rolland-Lagan, and Przemyslaw Prusinkiewicz, “Modeling and visualization of leaf ve- nation patterns,” ACM Transactions on Graphics, vol. 24, no. 3, pp. 702–711, 2005. [22] Robert J Mullen, Dorothy Monekosso, Sarah Barman, Paolo Remagnino, and Paul Wilkin, “Artiﬁcial ants to extract leaf outlines and primary venation patterns,” in Ant Colony Optimization and Swarm Intelligence, pp. 251–258. Springer, 2008.

text: Alex

text: Krizhevsky,

text: Ilya

text: Sutskever,

text: and

text: Geoffrey

text: E

text: Hin-

text: ton,

text: “Imagenet

text: classiﬁcation

text: with

text: deep

text: convolutional

text: neural

text: networks,”

text: in

text: NIPS,

text: pp.

text: Jeff

text: Donahue,

text: Yangqing

text: Jia,

text: Oriol

text: Vinyals,

text: Judy

text: Hoff-

text: man,

text: Ning

text: Zhang,

text: Eric

text: Tzeng,

text: and

text: Trevor

text: Darrell,

text: “De-

text: caf:

text: A

text: deep

text: convolutional

text: activation

text: feature

text: for

text: generic

text: visual

text: recognition,”

text: arXiv

text: preprint

text: Chao

text: Dong,

text: Chen

text: Change

text: Loy,

text: Kaiming

text: He,

text: and

text: Xi-

text: aoou

text: Tang,

text: “Learning

text: a

text: deep

text: convolutional

text: network

text: for

text: image

text: super-resolution,”

text: in

text: ECCV,

text: pp.

text: Springer,

text: Matthew

text: D

text: Zeiler,

text: Graham

text: W

text: Taylor,

text: and

text: Rob

text: Fergus,

text: “Adaptive

text: deconvolutional

text: networks

text: for

text: mid

text: and

text: high

text: level

text: feature

text: learning,”

text: in

text: ICCV,

text: pp.

text: Matthew

text: D

text: Zeiler

text: and

text: Rob

text: Fergus,

text: “Visualizing

text: and

text: understanding

text: convolutional

text: networks,”

text: in

text: ECCV,

text: pp.

text: Springer,

text: Anita

text: Roth-Nebelsick,

text: Dieter

text: Uhl,

text: Volker

text: Mosbrugger,

text: and

text: Hans

text: Kerp,

text: “Evolution

text: and

text: function

text: of

text: leaf

text: venation

text: architecture:

text: a

text: review,”

text: Annals

text: of

text: Botany,

text: vol.

text: no.

text: pp.

text: Héctor

text: Candela,

text: Antonio

text: Martınez-Laborda,

text: and

text: José

text: Luis

text: Micol,

text: “Venation

text: pattern

text: formation

text: in

text: arabidop-

text: sis

text: thaliana

text: vegetative

text: leaves,”

text: Developmental

text: biology,

text: vol.

text: no.

text: pp.

text: Adam

text: Runions,

text: Martin

text: Fuhrer,

text: Brendan

text: Lane,

text: Pavol

text: Federl,

text: Anne-Gaëlle

text: Rolland-Lagan,

text: and

text: Przemyslaw

text: Prusinkiewicz,

text: “Modeling

text: and

text: visualization

text: of

text: leaf

text: ve-

text: nation

text: patterns,”

text: ACM

text: Transactions

text: on

text: Graphics,

text: vol.

text: no.

text: pp.

text: Robert

text: J

text: Mullen,

text: Dorothy

text: Monekosso,

text: Sarah

text: Barman,

text: Paolo

text: Remagnino,

text: and

text: Paul

text: Wilkin,

text: “Artiﬁcial

text: ants

text: to

text: extract

text: leaf

text: outlines

text: and

text: primary

text: venation

text: patterns,”

text: in

text: Ant

text: Colony

text: Optimization

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text: Intelligence,

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text: Springer,


text: Shielding superconductors with thin ﬁlms Sam Posen,1 Mark K. Transtrum,2 Gianluigi Catelani,3 Matthias U. Liepe,1 and James P. Sethna4 ∗ 1LEPP, Physics Department, Newman Laboratory, Cornell University, Ithaca, NY 14853-2501 2Department of Physics and Astronomy, Brigham Young University, Provo, UT 84602 3Forschungszentrum Ju¨lich, Peter Gru¨nberg Institut (PGI-2), 52425 Ju¨lich, Germany 4LASSP, Physics Department, Clark Hall, Cornell University, Ithaca, NY 14853-2501 (Dated: May 27, 2020) Determining the optimal arrangement of superconducting layers to withstand large amplitude AC magnetic ﬁelds is important for certain applications such as superconducting radiofrequency cavities. In this paper, we evaluate the shielding potential of the superconducting ﬁlm/insulating ﬁlm/superconductor (SIS’) structure, a conﬁguration that could provide beneﬁts in screening large AC magnetic ﬁelds. After establishing that for high frequency magnetic ﬁelds, ﬂux penetration must be avoided, the superheating ﬁeld of the structure is calculated in the London limit both numerically and, for thin ﬁlms, analytically. For intermediate ﬁlm thicknesses and realistic material parameters we also solve numerically the Ginzburg-Landau equations. It is shown that a small enhancement of the superheating ﬁeld is possible, on the order of a few percent, for the SIS’ structure relative to a bulk superconductor of the ﬁlm material, if the materials and thicknesses are chosen appropriately. PACS numbers: 74.78.Fk, 74.25.Op, 64.60.Ht, 77.55.-g I. INTRODUCTION SIS’ heterolaminates compared to bulk superconductors. The SIS’ structure may provide beneﬁts in other ways for realistic materials with surface defects,8 but considering Can one engineer a better superconducting magnetic those beneﬁts is beyond the scope of the present work. shield? How can one optimally arrange materials to maintain complete ﬂux exclusion from a region, and what The paper is organized as follows: we start our analysis is the maximum external ﬁeld that can be screened? It by arguing in Sec. that for an SIS’ structure, a signiﬁ- has long been known that superconducting ﬁlms of width cant enhancement of the ﬂux penetration ﬁeld could be d smaller than the London magnetic penetration depth achieved only if a∇signiﬁcant gradient in the phase of the λ can remain superconducting at much higher magnetic order parameter φ can be established across the ﬁlm ﬁelds than bulk samples,1 so it has been proposed that shielding the bulk. Since this would result in a level of ﬁlms could be used to shield bulk superconductors.2 In dissipation that is likely unmanageable, we restrict our this paper, we investigate the shielding properties of the analysis to ﬁelds below Bsh, where both ﬁlm and bulk su- ﬁlm/insulator/bulk (SIS’) structure and compare to the perconductor are in a (meta)stable regime, with no phase single superconducting slab. The focus here is on AC gradient across the ﬁlm. In Sec. , numerical calculations rather than DC shielding; the latter has already been are performed in the London limit. The thin ﬁlm regime studied extensively.3–7 is examined in Sec. with an analytical Ginzburg-Landau approach. In Sec. , the results are extended to ﬁlms of Superconducting radio-frequency (SRF) cavities are an intermediate thicknesses via a full numerical Ginzburg- example of an application in which shielding of large- Landau analysis. We summarize our work in Sec. . amplitude high frequency magnetic ﬁelds is required. II. FLUX EXCLUSION This technology underlies particle accelerators used in high-energy physics, nuclear physics, neutron sources, and X-ray light sources. The large AC accelerating elec- The fundamental link between superconducting order tric ﬁeld of these cavities induces a correspondingly large and magnetism is the fact that the free energy and prop- ∇ magnetic ﬁeld. If the magnetic ﬁeld exceeds the ﬂux pen- erties of the system are governed not by the gradients ψ etration ﬁeld of the material, it causes a quench in the of the superconducting order, but by a ‘covariant’ deriva- ∇ − ∗ (cid:126) ∗ cavity. If SIS’ structures could enhance the ﬂux pene- tive Dψ = ( e iA/ )ψ, where e = 2e is the Cooper tration ﬁeld relative to that of a bulk superconductor, it pair charge and A is the magnetic vector potential. If we could allow these cavities to achieve higher accelerating write the complex superconducting order parameter in | | ﬁelds.? In this paper, we examine the superheating ﬁelds terms of two real ﬁelds as ψ = ψ exp(iφ), the covariant B of these structures, where ﬂux penetration would oc- derivative becomes sh cur in defect-free superconductors. In fact, part of the motivation for this work is that there has been signiﬁcant ∇| | | | ∇ − ∗ (cid:126) confusion in the SRF community regarding the maximum Dψ = [ ψ + i ψ ( φ e A/ )] exp(iφ) (1) ∇| | | | ∗ (cid:126) ﬁelds that SIS’ structures can screen; we hope that this = [ ψ + i ψ (m v / )] exp(iφ), (2) s study will clarify the screening mechanism and its limita- tions. Our calculations show modest shielding gains for

text: Shielding

label: title

text: superconductors

text: with

text: thin

text: ﬁlms

text: Sam

label: author

text: Mark

text: K.

text: Gianluigi

text: Matthias

text: U.

text: and

text: James

text: P.

text: Physics

text: Department,

text: Newman

text: Laboratory,

text: Cornell

text: University,

text: Ithaca,

text: NY

text: of

text: Physics

text: and

text: Astronomy,

text: Brigham

text: Young

text: University,

text: Provo,

text: UT

text: Ju¨lich,

text: Peter

text: Gru¨nberg

text: Institut

text: Ju¨lich,

text: Germany

text: Physics

text: Department,

text: Clark

text: Hall,

text: Cornell

text: University,

text: Ithaca,

text: NY

text: (Dated:

text: May

label: date

text: Determining

label: abstract

text: the

text: optimal

text: arrangement

text: of

text: superconducting

text: layers

text: to

text: withstand

text: large

text: amplitude

text: AC

text: magnetic

text: ﬁelds

text: is

text: important

text: for

text: certain

text: applications

text: such

text: as

text: superconducting

text: radiofrequency

text: cavities.

text: In

text: this

text: paper,

text: we

text: evaluate

text: the

text: shielding

text: potential

text: of

text: the

text: superconducting

text: ﬁlm/insulating

text: ﬁlm/superconductor

text: (SIS’)

text: structure,

text: a

text: conﬁguration

text: that

text: could

text: provide

text: beneﬁts

text: in

text: screening

text: large

text: AC

text: magnetic

text: ﬁelds.

text: After

text: establishing

text: that

text: for

text: high

text: frequency

text: magnetic

text: ﬁelds,

text: ﬂux

text: penetration

text: must

text: be

text: avoided,

text: the

text: superheating

text: ﬁeld

text: of

text: the

text: structure

text: is

text: calculated

text: in

text: the

text: London

text: limit

text: both

text: numerically

text: and,

text: for

text: thin

text: ﬁlms,

text: analytically.

text: For

text: intermediate

text: ﬁlm

text: thicknesses

text: and

text: realistic

text: material

text: parameters

text: we

text: also

text: solve

text: numerically

text: the

text: Ginzburg-Landau

text: equations.

text: It

text: is

text: shown

text: that

text: a

text: small

text: enhancement

text: of

text: the

text: superheating

text: ﬁeld

text: is

text: possible,

text: on

text: the

text: order

text: of

text: a

text: few

text: percent,

text: for

text: the

text: SIS’

text: structure

text: relative

text: to

text: a

text: bulk

text: superconductor

text: of

text: the

text: ﬁlm

text: material,

text: if

text: the

text: materials

text: and

text: thicknesses

text: are

text: chosen

text: appropriately.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: SIS’

text: heterolaminates

text: compared

text: to

text: bulk

text: superconductors.

text: The

text: SIS’

text: structure

text: may

text: provide

text: beneﬁts

text: in

text: other

text: ways

text: for

text: realistic

text: materials

text: with

text: surface

text: but

text: considering

text: Can

text: one

text: engineer

text: a

text: better

text: superconducting

text: magnetic

text: those

text: beneﬁts

text: is

text: beyond

text: the

text: scope

text: of

text: the

text: present

text: work.

text: shield?

text: How

text: can

text: one

text: optimally

text: arrange

text: materials

text: to

text: maintain

text: complete

text: ﬂux

text: exclusion

text: from

text: a

text: region,

text: and

text: what

text: The

text: paper

text: is

text: organized

text: as

text: follows:

text: we

text: start

text: our

text: analysis

text: is

text: the

text: maximum

text: external

text: ﬁeld

text: that

text: can

text: be

text: screened?

text: It

text: by

text: arguing

text: in

text: Sec.

text: that

text: for

text: an

text: SIS’

text: structure,

text: a

text: signiﬁ-

text: has

text: long

text: been

text: known

text: that

text: superconducting

text: ﬁlms

text: of

text: width

text: cant

text: enhancement

text: of

text: the

text: ﬂux

text: penetration

text: ﬁeld

text: could

text: be

text: d

text: smaller

text: than

text: the

text: London

text: magnetic

text: penetration

text: depth

text: achieved

text: only

text: if

text: a∇signiﬁcant

text: gradient

text: in

text: the

text: phase

text: of

text: the

text: λ

text: can

text: remain

text: superconducting

text: at

text: much

text: higher

text: magnetic

text: order

text: parameter

text: φ

text: can

text: be

text: established

text: across

text: the

text: ﬁlm

text: ﬁelds

text: than

text: bulk

text: so

text: it

text: has

text: been

text: proposed

text: that

text: shielding

text: the

text: bulk.

text: Since

text: this

text: would

text: result

text: in

text: a

text: level

text: of

text: ﬁlms

text: could

text: be

text: used

text: to

text: shield

text: bulk

text: In

text: dissipation

text: that

text: is

text: likely

text: unmanageable,

text: we

text: restrict

text: our

text: this

text: paper,

text: we

text: investigate

text: the

text: shielding

text: properties

text: of

text: the

text: analysis

text: to

text: ﬁelds

text: below

text: Bsh,

text: where

text: both

text: ﬁlm

text: and

text: bulk

text: su-

text: ﬁlm/insulator/bulk

text: (SIS’)

text: structure

text: and

text: compare

text: to

text: the

text: perconductor

text: are

text: in

text: a

text: (meta)stable

text: regime,

text: with

text: no

text: phase

text: single

text: superconducting

text: slab.

text: The

text: focus

text: here

text: is

text: on

text: AC

text: gradient

text: across

text: the

text: ﬁlm.

text: In

text: Sec.

text: numerical

text: calculations

text: rather

text: than

text: DC

text: shielding;

text: the

text: latter

text: has

text: already

text: been

text: are

text: performed

text: in

text: the

text: London

text: limit.

text: The

text: thin

text: ﬁlm

text: regime

text: studied

text: is

text: examined

text: in

text: Sec.

text: with

text: an

text: analytical

text: Ginzburg-Landau

text: approach.

text: In

text: Sec.

text: the

text: results

text: are

text: extended

text: to

text: ﬁlms

text: of

text: Superconducting

text: radio-frequency

text: (SRF)

text: cavities

text: are

text: an

text: intermediate

text: thicknesses

text: via

text: a

text: full

text: numerical

text: Ginzburg-

text: example

text: of

text: an

text: application

text: in

text: which

text: shielding

text: of

text: large-

text: Landau

text: analysis.

text: We

text: summarize

text: our

text: work

text: in

text: Sec.

text: amplitude

text: high

text: frequency

text: magnetic

text: ﬁelds

text: is

text: required.

text: II.

text: FLUX

text: EXCLUSION

text: This

text: technology

text: underlies

text: particle

text: accelerators

text: used

text: in

text: high-energy

text: physics,

text: nuclear

text: physics,

text: neutron

text: sources,

text: and

text: X-ray

text: light

text: sources.

text: The

text: large

text: AC

text: accelerating

text: elec-

text: The

text: fundamental

text: link

text: between

text: superconducting

text: order

text: tric

text: ﬁeld

text: of

text: these

text: cavities

text: induces

text: a

text: correspondingly

text: large

text: and

text: magnetism

text: is

text: the

text: fact

text: that

text: the

text: free

text: energy

text: and

text: prop-

text: magnetic

text: ﬁeld.

text: If

text: the

text: magnetic

text: ﬁeld

text: exceeds

text: the

text: ﬂux

text: pen-

text: erties

text: of

text: the

text: system

text: are

text: governed

text: not

text: by

text: the

text: gradients

text: ψ

text: etration

text: ﬁeld

text: of

text: the

text: material,

text: it

text: causes

text: a

text: quench

text: in

text: the

text: of

text: the

text: superconducting

text: order,

text: but

text: by

text: a

text: ‘covariant’

text: deriva-

text: cavity.

text: If

text: SIS’

text: structures

text: could

text: enhance

text: the

text: ﬂux

text: pene-

text: tive

text: Dψ

text: e

text: iA/

text: )ψ,

text: where

text: e

text: is

text: the

text: Cooper

text: tration

text: ﬁeld

text: relative

text: to

text: that

text: of

text: a

text: bulk

text: superconductor,

text: it

text: pair

text: charge

text: and

text: A

text: is

text: the

text: magnetic

text: vector

text: potential.

text: If

text: we

text: could

text: allow

text: these

text: cavities

text: to

text: achieve

text: higher

text: accelerating

text: write

text: the

text: complex

text: superconducting

text: order

text: parameter

text: in

text: ﬁelds.?

text: In

text: this

text: paper,

text: we

text: examine

text: the

text: superheating

text: ﬁelds

text: terms

text: of

text: two

text: real

text: ﬁelds

text: as

text: ψ

text: ψ

text: exp(iφ),

text: the

text: covariant

text: B

text: of

text: these

text: structures,

text: where

text: ﬂux

text: penetration

text: would

text: oc-

text: derivative

text: becomes

text: sh

text: cur

text: in

text: defect-free

text: superconductors.

text: In

text: fact,

text: part

text: of

text: the

text: motivation

text: for

text: this

text: work

text: is

text: that

text: there

text: has

text: been

text: signiﬁcant

text: confusion

text: in

text: the

text: SRF

text: community

text: regarding

text: the

text: maximum

text: Dψ

text: ψ

text: i

text: ψ

text: φ

text: e

text: A/

text: exp(iφ)

text: ﬁelds

text: that

text: SIS’

text: structures

text: can

text: screen;

text: we

text: hope

text: that

text: this

text: ψ

text: i

text: ψ

text: (m

text: v

text: exp(iφ),

text: s

text: study

text: will

text: clarify

text: the

text: screening

text: mechanism

text: and

text: its

text: limita-

text: tions.

text: Our

text: calculations

text: show

text: modest

text: shielding

text: gains

text: for


text: 4 Image Meissner-screened external Image these structures compared to the bulk value of the ﬁlm antivortex Vortex field (pushes vortex into SC) antivortex material. However, the range of ﬁlm thicknesses over (attracts (attracts which the increase is appreciable ((cid:38) few %) is relatively vortex) vortex) small, and the gain decreases as the thickness of the in- sulating layer increases. … … Exterior Thin Film SC Insulator Bulk SC FIG. 3: Forces on a vortex in a homolaminate. As the ﬁlm is IV. THIN FILMS IN THE GINZBURG-LANDAU made thinner, the image antivortex to the right of the ﬁlm has APPROACH a stronger pull on the vortex, lowering the barrier to vortex penetration. Calculating B using London theory, as done in the sh previous section, fails to take into account 2D instabil- ities in the order parameter, therefore overestimating reaches that material’s bulk B while B is still relatively sh i B in many circumstances. The problem of calculat- small. However, between these two extremes, there is a sh ing B for bulk samples while taking into account 2D situation in which the ﬁlm provides some screening, so sh instabilities has a long history (see, e.g., Ref. 19) and that B is large but still smaller than B . In this case, a i 0 has mostly been tackled in the Ginzburg-Landau (GL) beneﬁt can be realized – the small penetration depth of framework. Only recently calculations beyond GL the- the material in the bulk causes B to be larger than with i ory were performed;26,27 they showed that while the GL the exponential decay expected for a thick ﬁlm [Eq. (6)]. results cannot be trusted quantitatively at low tempera- This in turn reduces the magnitude of the negative gra- tures, they give a qualitatively correct estimate. There- dient in B , bolstering the barrier to ﬂux penetration M fore, for simplicity we restrict ourselves to GL theory (Eqn. 4). This increase in the barrier is depicted in Fig. 4. G even in the low-temperature regime where its quantita- The dark curves show B , B , and for a Nb Sn thin M V 3 tive predictions are not exact. ﬁlm/insulator/Nb bulk SIS structure with 10 nm thick insulator and d/λ = 0.64 (the peak of the cyan curve The approach we use to ﬁnd B is described in detail f sh in Fig. 2). The light curves show calculations for a bulk in Ref. 19: we ﬁrst extremize the GL free energy, a func- Nb Sn ﬁlm (for this case, the dark shaded region repre- tional of the spatially dependent order parameter ψ and 3 senting the insulator does not apply). In this example, superﬂuid velocity v , and then study its stability against s B = 300 mT. The Gibbs free energy of the SIS’ struc- small perturbation of these functions. In the present case, 0 ture is still sharply peaked, showing a relatively robust the GL free energy is the sum of the contributions for the energy barrier, but that of the bulk ﬁlm is almost ﬂat, bulk and the ﬁlm. The boundary conditions are the usual showing that ﬂux penetration is likely to occur at slightly ones for the order parameter (vanishing of its derivatives higher ﬁelds. at all surfaces); the superﬂuid velocity vanishes deep into the bulk, and its derivative at the external ﬁlm surface is 500 proportional to the applied magnetic ﬁeld. Similarly, the B M ﬁeld between the ﬁlm and the bulk is proportional to the B 400 V derivatives of the superﬂuid velocities at the two surfaces. G = 0/0(BV/2+BM) However, this internal ﬁeld is not externally imposed, but pJ] 300 must be calculated consistently with Maxwell equations; G [ this gives the ﬁnal condition of continuity of the vector T], potential across all surfaces. Hence for a very thin insu- m 200 B [ lating barrier the superﬂuid velocities at the bulk surface and the internal ﬁlm surface coincide, while the ﬁlm su- 100 perﬂuid velocity would be higher for a thicker insulator. For an analytical estimate of the SIS’ superheating 0 0 50 100 150 200 ﬁeld BSIS(cid:48), we consider the simplest possible case of a x [nm] sh (cid:29) strongly type II bulk material (GL parameter κ 1) GL shielded by an insulator of negligible thickness and a (cid:28) (cid:28) strongly type II thin ﬁlm with ξ d λ . With a FIG. 4: Comparison of a SIS’ ﬁlm with near-optimal param- f f eters (dark curves) to a bulk ﬁlm (light curves). The slower thin ﬁlm, the diﬀerence between the internal ﬁeld (at the G decay of B in the large-λ thin ﬁlm inﬂuences , bolstering bulk surface) and the applied ﬁeld (outside the ﬁlm) is M the barrier to ﬂux penetration. Note that the dark shaded small, and the maximum possible ﬁeld is reached when region representing the insulating region of the SIS’ can be the internal ﬁeld coincides with the bulk superheating ignored for the bulk ﬁlm. ﬁeld B . Indeed, within GL theory and at linear order sh,b in d/λ , using the boundary conditions discussed above (cid:48) f The impact of this is a modest increase in BSIS for we ﬁnd [see Appendix ]: sh ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Image

text: Meissner-screened

text: external

text: Image

text: these

text: structures

text: compared

text: to

text: the

text: bulk

text: value

text: of

text: the

text: ﬁlm

text: antivortex

text: Vortex

text: field

text: (pushes

text: vortex

text: into

text: SC)

text: antivortex

text: material.

text: However,

text: the

text: range

text: of

text: ﬁlm

text: thicknesses

text: over

text: (attracts

text: (attracts

text: which

text: the

text: increase

text: is

text: appreciable

text: few

text: is

text: relatively

text: vortex)

text: vortex)

text: small,

text: and

text: the

text: gain

text: decreases

text: as

text: the

text: thickness

text: of

text: the

text: in-

text: sulating

text: layer

text: increases.

text: Exterior

text: Thin

text: Film

text: SC

text: Insulator

text: Bulk

text: SC

text: FIG.

text: Forces

label: caption

text: on

text: a

text: vortex

text: in

text: a

text: homolaminate.

text: As

text: the

text: ﬁlm

text: is

text: IV.

text: THIN

text: FILMS

text: IN

text: THE

text: GINZBURG-LANDAU

text: made

text: thinner,

text: the

text: image

text: antivortex

text: to

text: the

text: right

text: of

text: the

text: ﬁlm

text: has

text: APPROACH

text: a

text: stronger

text: pull

text: on

text: the

text: vortex,

text: lowering

text: the

text: barrier

text: to

text: vortex

text: penetration.

text: Calculating

text: B

text: using

text: London

text: theory,

text: as

text: done

text: in

text: the

text: sh

text: previous

text: section,

text: fails

text: to

text: take

text: into

text: account

text: instabil-

text: ities

text: in

text: the

text: order

text: parameter,

text: therefore

text: overestimating

text: reaches

text: that

text: material’s

text: bulk

text: B

text: while

text: B

text: is

text: still

text: relatively

text: sh

text: i

text: B

text: in

text: many

text: circumstances.

text: The

text: problem

text: of

text: calculat-

text: small.

text: However,

text: between

text: these

text: two

text: extremes,

text: there

text: is

text: a

text: sh

text: ing

text: B

text: for

text: bulk

text: samples

text: while

text: taking

text: into

text: account

text: situation

text: in

text: which

text: the

text: ﬁlm

text: provides

text: some

text: screening,

text: so

text: sh

text: instabilities

text: has

text: a

text: long

text: history

text: (see,

text: e.g.,

text: Ref.

text: and

text: that

text: B

text: is

text: large

text: but

text: still

text: smaller

text: than

text: B

text: In

text: this

text: case,

text: a

text: i

text: has

text: mostly

text: been

text: tackled

text: in

text: the

text: Ginzburg-Landau

text: (GL)

text: beneﬁt

text: can

text: be

text: realized

text: the

text: small

text: penetration

text: depth

text: of

text: framework.

text: Only

text: recently

text: calculations

text: beyond

text: GL

text: the-

text: the

text: material

text: in

text: the

text: bulk

text: causes

text: B

text: to

text: be

text: larger

text: than

text: with

text: i

text: ory

text: were

text: they

text: showed

text: that

text: while

text: the

text: GL

text: the

text: exponential

text: decay

text: expected

text: for

text: a

text: thick

text: ﬁlm

text: [Eq.

text: results

text: cannot

text: be

text: trusted

text: quantitatively

text: at

text: low

text: tempera-

text: This

text: in

text: turn

text: reduces

text: the

text: magnitude

text: of

text: the

text: negative

text: gra-

text: tures,

text: they

text: give

text: a

text: qualitatively

text: correct

text: estimate.

text: There-

text: dient

text: in

text: B

text: bolstering

text: the

text: barrier

text: to

text: ﬂux

text: penetration

text: M

text: fore,

text: for

text: simplicity

text: we

text: restrict

text: ourselves

text: to

text: GL

text: theory

text: (Eqn.

text: This

text: increase

text: in

text: the

text: barrier

text: is

text: depicted

text: in

text: Fig.

text: G

text: even

text: in

text: the

text: low-temperature

text: regime

text: where

text: its

text: quantita-

text: The

text: dark

text: curves

text: show

text: B

text: B

text: and

text: for

text: a

text: Nb

text: Sn

text: thin

text: M

text: V

text: tive

text: predictions

text: are

text: not

text: exact.

text: ﬁlm/insulator/Nb

text: bulk

text: SIS

text: structure

text: with

text: nm

text: thick

text: insulator

text: and

text: d/λ

text: (the

text: peak

text: of

text: the

text: cyan

text: curve

text: The

text: approach

text: we

text: use

text: to

text: ﬁnd

text: B

text: is

text: described

text: in

text: detail

text: f

text: sh

text: in

text: Fig.

text: The

text: light

text: curves

text: show

text: calculations

text: for

text: a

text: bulk

text: in

text: Ref.

text: we

text: ﬁrst

text: extremize

text: the

text: GL

text: free

text: energy,

text: a

text: func-

text: Nb

text: Sn

text: ﬁlm

text: (for

text: this

text: case,

text: the

text: dark

text: shaded

text: region

text: repre-

text: tional

text: of

text: the

text: spatially

text: dependent

text: order

text: parameter

text: ψ

text: and

text: senting

text: the

text: insulator

text: does

text: not

text: apply).

text: In

text: this

text: example,

text: superﬂuid

text: velocity

text: v

text: and

text: then

text: study

text: its

text: stability

text: against

text: s

text: B

text: mT.

text: The

text: Gibbs

text: free

text: energy

text: of

text: the

text: SIS’

text: struc-

text: small

text: perturbation

text: of

text: these

text: functions.

text: In

text: the

text: present

text: case,

text: ture

text: is

text: still

text: sharply

text: peaked,

text: showing

text: a

text: relatively

text: robust

text: the

text: GL

text: free

text: energy

text: is

text: the

text: sum

text: of

text: the

text: contributions

text: for

text: the

text: energy

text: barrier,

text: but

text: that

text: of

text: the

text: bulk

text: ﬁlm

text: is

text: almost

text: ﬂat,

text: bulk

text: and

text: the

text: ﬁlm.

text: The

text: boundary

text: conditions

text: are

text: the

text: usual

text: showing

text: that

text: ﬂux

text: penetration

text: is

text: likely

text: to

text: occur

text: at

text: slightly

text: ones

text: for

text: the

text: order

text: parameter

text: (vanishing

text: of

text: its

text: derivatives

text: higher

text: ﬁelds.

text: at

text: all

text: surfaces);

text: the

text: superﬂuid

text: velocity

text: vanishes

text: deep

text: into

text: the

text: bulk,

text: and

text: its

text: derivative

text: at

text: the

text: external

text: ﬁlm

text: surface

text: is

text: proportional

text: to

text: the

text: applied

text: magnetic

text: ﬁeld.

text: Similarly,

text: the

text: B

text: M

text: ﬁeld

text: between

text: the

text: ﬁlm

text: and

text: the

text: bulk

text: is

text: proportional

text: to

text: the

text: B

text: V

text: derivatives

text: of

text: the

text: superﬂuid

text: velocities

text: at

text: the

text: two

text: surfaces.

text: G

text: However,

text: this

text: internal

text: ﬁeld

text: is

text: not

text: externally

text: imposed,

text: but

text: pJ]

text: must

text: be

text: calculated

text: consistently

text: with

text: Maxwell

text: equations;

text: G

text: this

text: gives

text: the

text: ﬁnal

text: condition

text: of

text: continuity

text: of

text: the

text: vector

text: T],

text: potential

text: across

text: all

text: surfaces.

text: Hence

text: for

text: a

text: very

text: thin

text: insu-

text: m

text: B

text: lating

text: barrier

text: the

text: superﬂuid

text: velocities

text: at

text: the

text: bulk

text: surface

text: and

text: the

text: internal

text: ﬁlm

text: surface

text: coincide,

text: while

text: the

text: ﬁlm

text: su-

text: perﬂuid

text: velocity

text: would

text: be

text: higher

text: for

text: a

text: thicker

text: insulator.

text: For

text: an

text: analytical

text: estimate

text: of

text: the

text: SIS’

text: superheating

text: ﬁeld

text: we

text: consider

text: the

text: simplest

text: possible

text: case

text: of

text: a

text: x

text: [nm]

text: sh

text: strongly

text: type

text: II

text: bulk

text: material

text: (GL

text: parameter

text: κ

text: GL

text: shielded

text: by

text: an

text: insulator

text: of

text: negligible

text: thickness

text: and

text: a

text: strongly

text: type

text: II

text: thin

text: ﬁlm

text: with

text: ξ

text: d

text: λ

text: With

text: a

text: FIG.

text: Comparison

text: of

text: a

text: SIS’

text: ﬁlm

text: with

text: near-optimal

text: param-

text: f

text: f

text: eters

text: (dark

text: curves)

text: to

text: a

text: bulk

text: ﬁlm

text: (light

text: curves).

text: The

text: slower

text: thin

text: ﬁlm,

text: the

text: diﬀerence

text: between

text: the

text: internal

text: ﬁeld

text: (at

text: the

text: G

text: decay

text: of

text: B

text: in

text: the

text: large-λ

text: thin

text: ﬁlm

text: inﬂuences

text: bolstering

text: bulk

text: surface)

text: and

text: the

text: applied

text: ﬁeld

text: (outside

text: the

text: ﬁlm)

text: is

text: M

text: the

text: barrier

text: to

text: ﬂux

text: penetration.

text: Note

text: that

text: the

text: dark

text: shaded

text: small,

text: and

text: the

text: maximum

text: possible

text: ﬁeld

text: is

text: reached

text: when

text: region

text: representing

text: the

text: insulating

text: region

text: of

text: the

text: SIS’

text: can

text: be

text: the

text: internal

text: ﬁeld

text: coincides

text: with

text: the

text: bulk

text: superheating

text: ignored

text: for

text: the

text: bulk

text: ﬁlm.

text: ﬁeld

text: B

text: Indeed,

text: within

text: GL

text: theory

text: and

text: at

text: linear

text: order

text: sh,b

text: in

text: d/λ

text: using

text: the

text: boundary

text: conditions

text: discussed

text: above

text: f

text: The

text: impact

text: of

text: this

text: is

text: a

text: modest

text: increase

text: in

text: BSIS

text: for

text: we

text: ﬁnd

text: [see

text: Appendix

text: sh

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