text: 4 YONATAN GUTMAN, FREDDIE MANNERS AND PE´TER P. VARJU´ We say that a system (T, X) is pronil of degree s, if it is the inverse limit of nilsystems of degree → s. We call a factor X Y of X a pronilfactor of degree s if Y is pronil of degree s, and say Y is the maximal pronilfactor if every other pronilfactor of X factors through Y . { } ∈ { } The discrete cube of dimensio(cid:80)n d is the set 0, 1 d. We write (cid:126)0 = (0, . . . , 0) 0, 1 d and ∈ { } use the notation (cid:126)1 in a similar manner. For a vertex ω = (ω , . . . , ω ) 0, 1 d and a vector n = 1 d (n , . . . , n ) ∈ Zd, we write (cid:104)ω, n(cid:105) = d ω n . If G is a group and g = (g , . . . , g ) ∈ Gd, then we 1 d · · · i=1 i i 1 d write gω = gω1 gωd. 1 d ∈ Let (T, X) be a system. Following Host, Kra and Maass, we say that the pair of points x, y X is { } { } ⊆ regionally proximal of order s if there are sequences of points x , y X and a sequence of i i { } ⊆ Z integer vectors n s such that lim x = x, lim y = y and i i i (cid:104) (cid:105) (cid:104) (cid:105) lim dist(T ω,ni x , T ω,ni y ) = 0 i i ∈ { } \{ } for all ω 0, 1 s (cid:126)0 . We denote this relation by RPs (X). For a nilsystem (T, X) of degree s, T RPs (X) is trivial. This fact is non-trivial, we return to it in Section after introducing some additional T concepts. The following theorem of Host, Kra and Maass characterizes the maximal pronilfactor of (T, X) using the higher order regional proximal relation. 1 Theorem 1.2 ([HKM10, Theorem 1.3]). Let (T, X) be a minimal distal system. Then RPs (X) is a T closed equivalence relation and (T, X/ RPs (X)) is the maximal pronilfactor of (T, X) of degree s. T Host, Kra and Maass also proved that a minimal system (T, X) is distal provided the relation RPs (X) is trivial. Shao and Ye [SY12, Theorem 3.5] proved that RPs (X) is a closed equivalence T T relation for minimal (but not necessarily distal) (T, X) and they combined this with [HKM10] to deduce that Theorem 1.2 holds without the assumption on distality. The deﬁnition of RPs , and those of nilsystems and pronilsystems, generalize straightforwardly to T the context of a dynamical system (H, X) where H is any abelian group (that is, H acts continuously Z on the space X), the previous discussion corresponding to the case H = . In [GGY16], Gutman, Glasner and Ye further generalize the deﬁnition of the regionally proximal relation to the case of an action by an arbitrary, possibly non-abelian group. For abelian group actions, Z in particular for -actions, the new deﬁnition coincides with the old one. We will review this new deﬁnition and discuss it in detail in the next subsection, as well as stating a generalization of Theorem 1.2 to these more general group actions, which is one of the main goals of this paper. 1The case s = 2 had been proven previously in [HM07, Theorem 2]. ##LTLine##

text: YONATAN

text: GUTMAN,

text: FREDDIE

text: MANNERS

text: AND

text: PE´TER

text: P.

text: VARJU´

text: We

text: say

text: that

text: a

text: system

text: (T,

text: X)

text: is

text: pronil

text: of

text: degree

text: s,

text: if

text: it

text: is

text: the

text: inverse

text: limit

text: of

text: nilsystems

text: of

text: degree

text: s.

text: We

text: call

text: a

text: factor

text: X

text: Y

text: of

text: X

text: a

text: pronilfactor

text: of

text: degree

text: s

text: if

text: Y

text: is

text: pronil

text: of

text: degree

text: s,

text: and

text: say

text: Y

text: is

text: the

text: maximal

text: pronilfactor

text: if

text: every

text: other

text: pronilfactor

text: of

text: X

text: factors

text: through

text: Y

text: The

text: discrete

text: cube

text: of

text: d

text: is

text: the

text: set

text: d.

text: We

text: write

text: d

text: and

text: use

text: the

text: notation

text: in

text: a

text: similar

text: manner.

text: For

text: a

text: vertex

text: ω

text: (ω

text: ω

text: d

text: and

text: a

text: vector

text: n

text: d

text: (n

text: n

text: Zd,

text: we

text: write

text: d

text: ω

text: n

text: If

text: G

text: is

text: a

text: group

text: and

text: g

text: (g

text: g

text: Gd,

text: then

text: we

text: d

text: i

text: i

text: d

text: write

text: gω

text: gωd.

text: d

text: Let

text: (T,

text: X)

text: be

text: a

text: system.

text: Following

text: Host,

text: Kra

text: and

text: Maass,

text: we

text: say

text: that

text: the

text: pair

text: of

text: points

text: x,

text: y

text: X

text: is

text: regionally

text: proximal

text: of

text: order

text: s

text: if

text: there

text: are

text: sequences

text: of

text: points

text: x

text: y

text: X

text: and

text: a

text: sequence

text: of

text: i

text: i

text: Z

text: integer

text: vectors

text: n

text: s

text: such

text: that

text: lim

text: x

text: x,

text: lim

text: y

text: y

text: and

text: i

text: i

text: i

text: lim

text: dist(T

text: ω,ni

text: x

text: T

text: ω,ni

text: y

text: i

text: i

text: for

text: all

text: ω

text: s

text: We

text: denote

text: this

text: relation

text: by

text: RPs

text: (X).

text: For

text: a

text: nilsystem

text: (T,

text: X)

text: of

text: degree

text: s,

text: T

text: RPs

text: (X)

text: is

text: trivial.

text: This

text: fact

text: is

text: non-trivial,

text: we

text: return

text: to

text: it

text: in

text: Section

text: after

text: introducing

text: some

text: additional

text: T

text: concepts.

text: The

text: following

text: theorem

text: of

text: Host,

text: Kra

text: and

text: Maass

text: characterizes

text: the

text: maximal

text: pronilfactor

text: of

text: (T,

text: X)

text: using

text: the

text: higher

text: order

text: regional

text: proximal

text: relation.

text: Theorem

text: Theorem

text: Let

text: (T,

text: X)

text: be

text: a

text: minimal

text: distal

text: system.

text: Then

text: RPs

text: (X)

text: is

text: a

text: T

text: closed

text: equivalence

text: relation

text: and

text: (T,

text: X/

text: RPs

text: (X))

text: is

text: the

text: maximal

text: pronilfactor

text: of

text: (T,

text: X)

text: of

text: degree

text: s.

text: T

text: Host,

text: Kra

text: and

text: Maass

text: also

text: proved

text: that

text: a

text: minimal

text: system

text: (T,

text: X)

text: is

text: distal

text: provided

text: the

text: relation

text: RPs

text: (X)

text: is

text: trivial.

text: Shao

text: and

text: Ye

text: Theorem

text: proved

text: that

text: RPs

text: (X)

text: is

text: a

text: closed

text: equivalence

text: T

text: T

text: relation

text: for

text: minimal

text: (but

text: not

text: necessarily

text: distal)

text: (T,

text: X)

text: and

text: they

text: combined

text: this

text: with

text: to

text: deduce

text: that

text: Theorem

text: holds

text: without

text: the

text: assumption

text: on

text: distality.

text: The

text: deﬁnition

text: of

text: RPs

text: and

text: those

text: of

text: nilsystems

text: and

text: pronilsystems,

text: generalize

text: straightforwardly

text: to

text: T

text: the

text: context

text: of

text: a

text: dynamical

text: system

text: (H,

text: X)

text: where

text: H

text: is

text: any

text: abelian

text: group

text: (that

text: is,

text: H

text: acts

text: continuously

text: Z

text: on

text: the

text: space

text: X),

text: the

text: previous

text: discussion

text: corresponding

text: to

text: the

text: case

text: H

text: In

text: Gutman,

text: Glasner

text: and

text: Ye

text: further

text: generalize

text: the

text: deﬁnition

text: of

text: the

text: regionally

text: proximal

text: relation

text: to

text: the

text: case

text: of

text: an

text: action

text: by

text: an

text: arbitrary,

text: possibly

text: non-abelian

text: group.

text: For

text: abelian

text: group

text: actions,

text: Z

text: in

text: particular

text: for

text: -actions,

text: the

text: new

text: deﬁnition

text: coincides

text: with

text: the

text: old

text: one.

text: We

text: will

text: review

text: this

text: new

text: deﬁnition

text: and

text: discuss

text: it

text: in

text: detail

text: in

text: the

text: next

text: subsection,

text: as

text: well

text: as

text: stating

text: a

text: generalization

text: of

text: Theorem

text: to

text: these

text: more

text: general

text: group

text: actions,

text: which

text: is

text: one

text: of

text: the

text: main

text: goals

text: of

text: this

text: paper.

text: case

text: s

text: had

text: been

text: proven

text: previously

text: in

text: Theorem

text: ##LTLine##


text: 32 YONATAN GUTMAN, FREDDIE MANNERS AND PE´TER P. VARJU´ structure, i.e. we denote by Cs+1(U ) the set of (s + 1)-cubes all of whose vertices lie in U . We deﬁne 0 0 → the function ρ : Cs+1(U ) A by ρ(c) = D(σ(c)), where D is the discrepancy function deﬁned above. 0 Since σ and D are continuous, ρ is also continuous. By uniform continuity, there is a num- ber δ > 0 such that dist(ρ(c ), ρ(c )) < τ whenever dist(c , c ) = max (c (ω), c (ω)) < δ , and 0 1 2 1 2 ω 1 2 0 ≤ ≤ dist(σ(x ), σ(x )) τ/C whenever dist(x , x ) 4δ . Here C is the implicit constant from Theorem 1 2 1 2 0 1.25. ≤ We now set δ to be small enough such that δ δ and such that the ball of radius δ around 0 ⊆ ⊆ each vertex of c is contained inside U . We construct open sets U B and U B such that 0 0 1 1 2 2 − ⊆ ⊆ U = ψ 1(U ) and c U U . To this end, let Z be the complement of the union of the δ-balls 1 2 0 1 0 around the vertices of c . Then ψ(Z) is a compact set, which does not contain any vertex of ψ(c ), 0 0 − ∈ { } since diam(ψ 1(ψ(c (ω)))) < δ for all ω 0, 1 k+1. We let U be the complement of ψ(Z) in B . 0 2 2 − Then U = ψ 1(U ) contains c and it is disjoint form Z hence it is contained in the δ neighborhood 1 2 0 ⊆ − ≤ ∈ of the vertices of c . This implies U U . Moreover, diam(σ(ψ 1(b))) τ/C for all b U by the 0 1 0 2 assumptions on ψ, δ and δ. 0 ∈ We check that ρ restricted to Cs+1(U ) is a cocycle. Let c , c , c Cs(U ) be two cubes such that 1 1 2 3 1 [c , c ], [c , c ] and [c , c ] are all cubes. Then we know that [ρ([c , c ])] .σ([c , c ]) are cubes in X for 1 2 1 3 2 3 i j (cid:126)0 i j i, j = 1, 2, 3. It follows then that ([ρ([c , c ]) + ρ([c , c ])] + [ρ([c , c ])] ).σ([c , c ]) (4) 1 2 2 3 (cid:126)0 2 3 (0,...,0,1) 1 2 is also a cube. Indeed, this conﬁguration is obtained from [ρ([c , c ])] .σ([c , c ]) by acting on two 1 2 (cid:126)0 1 2 adjacent vertices with the same element of the top structure group. Using that nilspaces have the gluing property ([GMV16, Proposition 6.2]) for the cube (4) and [ρ([c , c ])] .σ([c , c ]), we obtain that 2 3 (cid:126)0 2 3 [ρ([c , c ]) + ρ([c , c ])] .σ([c , c ]) 1 2 2 3 (cid:126)0 1 3 is also a cube. By the deﬁnition of discrepancy this proves ρ([c , c ]) = ρ([c , c ]) + ρ([c , c ]). 1 3 1 2 2 3 → We check that Theorem 1.25 applies to the ﬁbration ψ : U U . Both U and U are considered 1 2 1 2 with the induced cubespace structure and they are compact, ergodic and have the glueing property. − − Moreover as B is an ergodic nilspace of degree (s 1), ψ is a ﬁbration of degree (s 1). For two cubes 1 ∈ ≤ c , c Cs+1(U ) with ψ(c ) = ψ(c ), we have dist(c , c ) < δ δ , hence dist(ρ(c ), ρ(c )) < τ that 1 2 1 1 2 1 2 0 1 2 (cid:101) is assumed to be smaller than the number denoted by δ in Theorem 1.25. → ◦ ≤ We obtain a function f : U A from Theorem 1.25 such that ρ = ∂s+1f+ρ ψ and dist(f(x), f(y)) 1 · ⊆ C τ/C = τ whenever ψ(x) = ψ(y). This implies that for any two cubes c , c U with ψ(c ) = ψ(c ) (cid:101) (cid:101) 1 2 1 1 2 we have − − − − ∂s+1f(c ) D(σ(c )) = ρ(ψ(c )) = ρ(ψ(c )) = ∂s+1f(c ) D(σ(c )). 1 1 1 2 2 2 (cid:55)→ Thus b f(b).σ(b) is a straight section on U by Lemma 2.11. 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: YONATAN

text: GUTMAN,

text: FREDDIE

text: MANNERS

text: AND

text: PE´TER

text: P.

text: VARJU´

text: structure,

text: i.e.

text: we

text: denote

text: by

text: the

text: set

text: of

text: (s

text: all

text: of

text: whose

text: vertices

text: lie

text: in

text: U

text: We

text: deﬁne

text: the

text: function

text: ρ

text: A

text: by

text: ρ(c)

text: D(σ(c)),

text: where

text: D

text: is

text: the

text: discrepancy

text: function

text: deﬁned

text: above.

text: Since

text: σ

text: and

text: D

text: are

text: continuous,

text: ρ

text: is

text: also

text: continuous.

text: By

text: uniform

text: continuity,

text: there

text: is

text: a

text: num-

text: ber

text: δ

text: such

text: that

text: dist(ρ(c

text: ρ(c

text: τ

text: whenever

text: dist(c

text: c

text: max

text: (c

text: (ω),

text: c

text: (ω))

text: δ

text: and

text: ω

text: dist(σ(x

text: σ(x

text: τ/C

text: whenever

text: dist(x

text: x

text: 4δ

text: Here

text: C

text: is

text: the

text: implicit

text: constant

text: from

text: Theorem

text: We

text: now

text: set

text: δ

text: to

text: be

text: small

text: enough

text: such

text: that

text: δ

text: δ

text: and

text: such

text: that

text: the

text: ball

text: of

text: radius

text: δ

text: around

text: each

text: vertex

text: of

text: c

text: is

text: contained

text: inside

text: U

text: We

text: construct

text: open

text: sets

text: U

text: B

text: and

text: U

text: B

text: such

text: that

text: U

text: ψ

text: and

text: c

text: U

text: U

text: To

text: this

text: end,

text: let

text: Z

text: be

text: the

text: complement

text: of

text: the

text: union

text: of

text: the

text: δ-balls

text: around

text: the

text: vertices

text: of

text: c

text: Then

text: ψ(Z)

text: is

text: a

text: compact

text: set,

text: which

text: does

text: not

text: contain

text: any

text: vertex

text: of

text: ψ(c

text: since

text: diam(ψ

text: (ω))))

text: δ

text: for

text: all

text: ω

text: We

text: let

text: U

text: be

text: the

text: complement

text: of

text: ψ(Z)

text: in

text: B

text: Then

text: U

text: ψ

text: contains

text: c

text: and

text: it

text: is

text: disjoint

text: form

text: Z

text: hence

text: it

text: is

text: contained

text: in

text: the

text: δ

text: neighborhood

text: of

text: the

text: vertices

text: of

text: c

text: This

text: implies

text: U

text: U

text: Moreover,

text: diam(σ(ψ

text: τ/C

text: for

text: all

text: b

text: U

text: by

text: the

text: assumptions

text: on

text: ψ,

text: δ

text: and

text: δ.

text: We

text: check

text: that

text: ρ

text: restricted

text: to

text: is

text: a

text: cocycle.

text: Let

text: c

text: c

text: c

text: Cs(U

text: be

text: two

text: cubes

text: such

text: that

text: [c

text: c

text: [c

text: c

text: and

text: [c

text: c

text: are

text: all

text: cubes.

text: Then

text: we

text: know

text: that

text: [ρ([c

text: c

text: .σ([c

text: c

text: are

text: cubes

text: in

text: X

text: for

text: i

text: j

text: i

text: j

text: i,

text: j

text: It

text: follows

text: then

text: that

text: ([ρ([c

text: c

text: ρ([c

text: c

text: [ρ([c

text: c

text: ).σ([c

text: c

text: is

text: also

text: a

text: cube.

text: Indeed,

text: this

text: conﬁguration

text: is

text: obtained

text: from

text: [ρ([c

text: c

text: .σ([c

text: c

text: by

text: acting

text: on

text: two

text: adjacent

text: vertices

text: with

text: the

text: same

text: element

text: of

text: the

text: top

text: structure

text: group.

text: Using

text: that

text: nilspaces

text: have

text: the

text: gluing

text: property

text: Proposition

text: for

text: the

text: cube

text: and

text: [ρ([c

text: c

text: .σ([c

text: c

text: we

text: obtain

text: that

text: [ρ([c

text: c

text: ρ([c

text: c

text: .σ([c

text: c

text: is

text: also

text: a

text: cube.

text: By

text: the

text: deﬁnition

text: of

text: discrepancy

text: this

text: proves

text: ρ([c

text: c

text: ρ([c

text: c

text: ρ([c

text: c

text: We

text: check

text: that

text: Theorem

text: applies

text: to

text: the

text: ﬁbration

text: ψ

text: U

text: U

text: Both

text: U

text: and

text: U

text: are

text: considered

text: with

text: the

text: induced

text: cubespace

text: structure

text: and

text: they

text: are

text: compact,

text: ergodic

text: and

text: have

text: the

text: glueing

text: property.

text: Moreover

text: as

text: B

text: is

text: an

text: ergodic

text: nilspace

text: of

text: degree

text: (s

text: ψ

text: is

text: a

text: ﬁbration

text: of

text: degree

text: (s

text: For

text: two

text: cubes

text: c

text: c

text: with

text: ψ(c

text: ψ(c

text: we

text: have

text: dist(c

text: c

text: δ

text: δ

text: hence

text: dist(ρ(c

text: ρ(c

text: τ

text: that

text: is

text: assumed

text: to

text: be

text: smaller

text: than

text: the

text: number

text: denoted

text: by

text: δ

text: in

text: Theorem

text: We

text: obtain

text: a

text: function

text: f

text: U

text: A

text: from

text: Theorem

text: such

text: that

text: ρ

text: ψ

text: and

text: dist(f(x),

text: f(y))

text: C

text: τ/C

text: τ

text: whenever

text: ψ(x)

text: ψ(y).

text: This

text: implies

text: that

text: for

text: any

text: two

text: cubes

text: c

text: c

text: U

text: with

text: ψ(c

text: ψ(c

text: we

text: have

text: D(σ(c

text: ρ(ψ(c

text: ρ(ψ(c

text: D(σ(c

text: Thus

text: b

text: f(b).σ(b)

text: is

text: a

text: straight

text: section

text: on

text: U

text: by

text: Lemma

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: INVERSE LIMIT REPRESENTATIONS 35 ≥ ∈ We ﬁrst consider the case (cid:96) s + 1. In this case, λ has a unique completion c . Let c C(cid:96)(X) be 1 2 ≈ ∈ { } \{ } ≈ such that c = ϕ(c ). Then c (ω) c (ω) for all ω 0, 1 (cid:96) (cid:126)1 hence c c by Lemma 2.14. Thus 2 1 2 1 2 ϕ(c ) = ϕ(c ) = c as required. 1 2 ≤ ∈ Second, we assume that (cid:96) s. We use that ψ is a ﬁbration and ﬁnd a cube c C(cid:96)(B ) that is 0 1 (cid:48) (cid:48) a completion of π(λ) such that ψ(c ) = π (c). (Here we used that π is a morphism, a fact that we 0 proved above.) We set c to be the unique conﬁguration such that π(c ) = c and ϕ(c ) = c. Since 1 1 0 1 the dimension of c is at most s, any π lift of it is a cube, so c is a cube in particular. 0 1 The fact that ϕ is a ﬁbration, implies by [GMV16, Remark 7.9] that Y is a nilspace and by the (cid:48) (cid:48) universal property that π is a ﬁbration. We leave it to the reader to verify that π is the s-th canonical (cid:3) projection, ϕ is horizontal and ψ is its shadow. − ≥ ∈ We take M suﬃciently large so that diam(ψ 1(x)) < δ for all m M and x B , where δ is n m n m suﬃciently small such that Proposition 2.13 and Proposition 2.9 can both be applied to X∞(n). Then the existence of the nilspace X(n) and the ﬁbration ϕ(n) follows from Proposition 2.13 and it remains m m ◦ to verify the claim about the inverse images of points under the maps ϕ(n) α(n). m (cid:101) ≥ ≥ ≥ ≥ ∈ (cid:101) We take indices n n and m m M such that m M . Let x X(n2) be a point. Let ∈ ◦ − 2 ∈1 ∞ 2 1 n1 2 n2 m2 x (ϕ(mn22) α(n2)) 1(x) X∞( ) be an arbitrary point and take y = ϕ(mn11)(α(n1)(x)). We set out to prove that ◦ − ⊆ ◦ − Z := (ϕ(n2) α(n2)) 1(x) Z := (ϕ(n1) α(n1)) 1(y). 2 m2 1 m1 ⊆ ∈ − We ﬁrst show that α(n1)(Z2) X∞(n1) is a straight ψm2-class. Let b D := ψm12(π(x)) be an arbitrary point. We show that α(n1)(Z2) contains a unique point in the ﬁbre of π above b. Since − (ϕ(n2)) 1(x) is a straight ψ -class, it follows that the points of Z in the ﬁbre of π above b is a single m2 m2 ⊆ 2 (cid:101) Kn2 orbit. This projects to a single point under α(n1) as Kn2 Kn1. (cid:101) { } → { } → Let c : 0, 1 s+1 D be a cube. We show that there is a cube c : 0, 1 s+1 α(n1)(Z2) with − { } → − π(c) = c. Since (ϕ(n2)) 1(x) is a straight ψ -class, we can ﬁnd a cube c : 0, 1 s+1 (ϕ(n2)) 1(x) m2 (cid:101) m2 { } →1 m2 with π(c1) = c. Since α(n2) is a ﬁbration, there is a cube c2 : 0, 1 s+1 Z2 with α(n2)(c2) = c1 and hence π(c2) = c. Thus c := α(n1)(c2) satisﬁes the requirements. This shows that α(n1)(Z2) is indeed a straight ψ -class. m2 We note that α(n1)(Z1) is a straight ψm1-class. Then ∩ − α(n1)(Z ) π 1(cid:101)(D) 1 is a straight ψm2-class and it contains the point α(n1)(x), which is also contained in α(n1)(Z2). By Proposition 2.9 we have hence ⊆ α(n1)(Z ) α(n1)(Z ) 2 1 ⊆ and then Z Z . This completes the proof of Proposition 2.5. 2 1

text: INVERSE

text: LIMIT

text: REPRESENTATIONS

text: We

text: ﬁrst

text: consider

text: the

text: case

text: s

text: In

text: this

text: case,

text: λ

text: has

text: a

text: unique

text: completion

text: c

text: Let

text: c

text: be

text: such

text: that

text: c

text: ϕ(c

text: Then

text: c

text: (ω)

text: c

text: (ω)

text: for

text: all

text: ω

text: hence

text: c

text: c

text: by

text: Lemma

text: Thus

text: ϕ(c

text: ϕ(c

text: c

text: as

text: required.

text: Second,

text: we

text: assume

text: that

text: s.

text: We

text: use

text: that

text: ψ

text: is

text: a

text: ﬁbration

text: and

text: ﬁnd

text: a

text: cube

text: c

text: that

text: is

text: a

text: completion

text: of

text: π(λ)

text: such

text: that

text: ψ(c

text: π

text: (c).

text: (Here

text: we

text: used

text: that

text: π

text: is

text: a

text: morphism,

text: a

text: fact

text: that

text: we

text: proved

text: above.)

text: We

text: set

text: c

text: to

text: be

text: the

text: unique

text: conﬁguration

text: such

text: that

text: π(c

text: c

text: and

text: ϕ(c

text: c.

text: Since

text: the

text: dimension

text: of

text: c

text: is

text: at

text: most

text: s,

text: any

text: π

text: lift

text: of

text: it

text: is

text: a

text: cube,

text: so

text: c

text: is

text: a

text: cube

text: in

text: particular.

text: The

text: fact

text: that

text: ϕ

text: is

text: a

text: ﬁbration,

text: implies

text: by

text: Remark

text: that

text: Y

text: is

text: a

text: nilspace

text: and

text: by

text: the

text: universal

text: property

text: that

text: π

text: is

text: a

text: ﬁbration.

text: We

text: leave

text: it

text: to

text: the

text: reader

text: to

text: verify

text: that

text: π

text: is

text: the

text: s-th

text: canonical

text: projection,

text: ϕ

text: is

text: horizontal

text: and

text: ψ

text: is

text: its

text: shadow.

text: We

text: take

text: M

text: suﬃciently

text: large

text: so

text: that

text: diam(ψ

text: δ

text: for

text: all

text: m

text: M

text: and

text: x

text: B

text: where

text: δ

text: is

text: n

text: m

text: n

text: m

text: suﬃciently

text: small

text: such

text: that

text: Proposition

text: and

text: Proposition

text: can

text: both

text: be

text: applied

text: to

text: X∞(n).

text: Then

text: the

text: existence

text: of

text: the

text: nilspace

text: X(n)

text: and

text: the

text: ﬁbration

text: ϕ(n)

text: follows

text: from

text: Proposition

text: and

text: it

text: remains

text: m

text: m

text: to

text: verify

text: the

text: claim

text: about

text: the

text: inverse

text: images

text: of

text: points

text: under

text: the

text: maps

text: ϕ(n)

text: α(n).

text: m

text: We

text: take

text: indices

text: n

text: n

text: and

text: m

text: m

text: M

text: such

text: that

text: m

text: M

text: Let

text: x

text: be

text: a

text: point.

text: Let

text: ∈1

text: x

text: X∞(

text: be

text: an

text: arbitrary

text: point

text: and

text: take

text: y

text: We

text: set

text: out

text: to

text: prove

text: that

text: Z

text: Z

text: We

text: ﬁrst

text: show

text: that

text: is

text: a

text: straight

text: Let

text: b

text: D

text: be

text: an

text: arbitrary

text: point.

text: We

text: show

text: that

text: contains

text: a

text: unique

text: point

text: in

text: the

text: ﬁbre

text: of

text: π

text: above

text: b.

text: Since

text: is

text: a

text: straight

text: ψ

text: -class,

text: it

text: follows

text: that

text: the

text: points

text: of

text: Z

text: in

text: the

text: ﬁbre

text: of

text: π

text: above

text: b

text: is

text: a

text: single

text: orbit.

text: This

text: projects

text: to

text: a

text: single

text: point

text: under

text: as

text: Let

text: c

text: D

text: be

text: a

text: cube.

text: We

text: show

text: that

text: there

text: is

text: a

text: cube

text: c

text: with

text: π(c)

text: c.

text: Since

text: is

text: a

text: straight

text: ψ

text: -class,

text: we

text: can

text: ﬁnd

text: a

text: cube

text: c

text: →1

text: with

text: c.

text: Since

text: is

text: a

text: ﬁbration,

text: there

text: is

text: a

text: cube

text: with

text: and

text: hence

text: c.

text: Thus

text: c

text: satisﬁes

text: the

text: requirements.

text: This

text: shows

text: that

text: is

text: indeed

text: a

text: straight

text: ψ

text: -class.

text: We

text: note

text: that

text: is

text: a

text: straight

text: Then

text: π

text: is

text: a

text: straight

text: and

text: it

text: contains

text: the

text: point

text: which

text: is

text: also

text: contained

text: in

text: By

text: Proposition

text: we

text: have

text: hence

text: and

text: then

text: Z

text: Z

text: This

text: completes

text: the

text: proof

text: of

text: Proposition


text: 44 YONATAN GUTMAN, FREDDIE MANNERS AND PE´TER P. VARJU´ Lemma 4.4. Let (H, Y ) be a pronil system of degree s. Then RPs (Y ) is trivial. H Proof. It is enough to show that RPs (Y ) is trivial for a nilsystem (H, Y ) of degree s. In this case, H the dynamical cubes form a Host–Kra nilspace, which is a nilspace of degree at most s, as shown in ∼ (cid:3) [GMV16, Proposition 2.6]. Hence RPs (Y ) = is indeed trivial. H s Let (H, X) be a minimal system such that RPs (X) is trivial and suppose that H has a dense H subgroup generated by a compact set. This means that we are in the setting of the proof of Theorem 1.4. We have, therefore, a sequence of { } { → } Lie-ﬁbred nilspaces X together with an inverse system of ﬁbrations ϕ : X X such that n m,n n m X = X∞ = limX . We have already proved that the action of H factors through the ﬁbrations ϕ ∞, ←− n n, → ◦ i.e. we have continuous homomorphisms α : H Aut (X ) such that α = ϕ α for m < n. n 1 n m m,n n (cid:104) (cid:105) Moreover, by Theorem 4.2, the group G = Aut (X ), α (H) act transitively on X . n 1 n n n ◦ → We saw in the proof of Theorem 1.28 that there is a continuous homomorphism ψ : Aut (X ) m,n 1 n ◦ Aut (X ) such that 1 m ψ (g).ϕ (x) = ϕ (g.x) m,n m,n m,n ∈ ◦ ∈ → for all g Aut (X ) and x X . We can extend ψ to a homomorphism G G by taking 1 n n m,n n m ψ (α (h)) = α (h). We leave it to the reader to verify that this extension satisﬁes the properties m,n n m claimed in Theorem 1.29. Finally, we note that the proof is valid in the slightly more general setting, when X is an arbitrary → compact ergodic nilspace and H acts on X via a continuous homomorphism H Aut (X). (The action 1 has to be continuous and minimal, and H has to contain a compactly generated dense subgroup.) That is to say, the proof does not require that the cubespace structure on X is deﬁned using the dynamical construction; it may be larger. Appendix A. The purpose of this appendix is to prove the following result stated in [GMV16b]. Theorem A.1. Let X be a compact, ergodic nilspace of degree s. Suppose X is locally connected and has ﬁnite Lebesgue covering dimension, and that Cn(X) is connected for all n. Then X is isomorphic to a nilmanifold G/Γ. That is, there exists a ﬁltered connected Lie group G•, ⊆ ↔ a discrete co-compact subgroup Γ G, and a homeomorphism ϕ : X G/Γ that identiﬁes the cubes Ck(X) with the Host–Kra cubes HKk(G•)/Γ.

text: YONATAN

text: GUTMAN,

text: FREDDIE

text: MANNERS

text: AND

text: PE´TER

text: P.

text: VARJU´

text: Lemma

text: Let

text: (H,

text: Y

text: be

text: a

text: pronil

text: system

text: of

text: degree

text: s.

text: Then

text: RPs

text: (Y

text: is

text: trivial.

text: H

text: Proof.

text: It

text: is

text: enough

text: to

text: show

text: that

text: RPs

text: (Y

text: is

text: trivial

text: for

text: a

text: nilsystem

text: (H,

text: Y

text: of

text: degree

text: s.

text: In

text: this

text: case,

text: H

text: the

text: dynamical

text: cubes

text: form

text: a

text: Host–Kra

text: nilspace,

text: which

text: is

text: a

text: nilspace

text: of

text: degree

text: at

text: most

text: s,

text: as

text: shown

text: in

text: Proposition

text: Hence

text: RPs

text: (Y

text: is

text: indeed

text: trivial.

text: H

text: s

text: Let

text: (H,

text: X)

text: be

text: a

text: minimal

text: system

text: such

text: that

text: RPs

text: (X)

text: is

text: trivial

text: and

text: suppose

text: that

text: H

text: has

text: a

text: dense

text: H

text: subgroup

text: generated

text: by

text: a

text: compact

text: set.

text: This

text: means

text: that

text: we

text: are

text: in

text: the

text: setting

text: of

text: the

text: proof

text: of

text: Theorem

text: We

text: have,

text: therefore,

text: a

text: sequence

text: of

text: Lie-ﬁbred

text: nilspaces

text: X

text: together

text: with

text: an

text: inverse

text: system

text: of

text: ﬁbrations

text: ϕ

text: X

text: X

text: such

text: that

text: n

text: m,n

text: n

text: m

text: X

text: X∞

text: limX

text: We

text: have

text: already

text: proved

text: that

text: the

text: action

text: of

text: H

text: factors

text: through

text: the

text: ﬁbrations

text: ϕ

text: n

text: n,

text: i.e.

text: we

text: have

text: continuous

text: homomorphisms

text: α

text: H

text: Aut

text: (X

text: such

text: that

text: α

text: ϕ

text: α

text: for

text: m

text: n.

text: n

text: n

text: m

text: m,n

text: n

text: Moreover,

text: by

text: Theorem

text: the

text: group

text: G

text: Aut

text: (X

text: α

text: (H)

text: act

text: transitively

text: on

text: X

text: n

text: n

text: n

text: n

text: We

text: saw

text: in

text: the

text: proof

text: of

text: Theorem

text: that

text: there

text: is

text: a

text: continuous

text: homomorphism

text: ψ

text: Aut

text: (X

text: m,n

text: n

text: Aut

text: (X

text: such

text: that

text: m

text: ψ

text: (g).ϕ

text: (x)

text: ϕ

text: (g.x)

text: m,n

text: m,n

text: m,n

text: for

text: all

text: g

text: Aut

text: (X

text: and

text: x

text: X

text: We

text: can

text: extend

text: ψ

text: to

text: a

text: homomorphism

text: G

text: G

text: by

text: taking

text: n

text: n

text: m,n

text: n

text: m

text: ψ

text: (α

text: (h))

text: α

text: (h).

text: We

text: leave

text: it

text: to

text: the

text: reader

text: to

text: verify

text: that

text: this

text: extension

text: satisﬁes

text: the

text: properties

text: m,n

text: n

text: m

text: claimed

text: in

text: Theorem

text: Finally,

text: we

text: note

text: that

text: the

text: proof

text: is

text: valid

text: in

text: the

text: slightly

text: more

text: general

text: setting,

text: when

text: X

text: is

text: an

text: arbitrary

text: compact

text: ergodic

text: nilspace

text: and

text: H

text: acts

text: on

text: X

text: via

text: a

text: continuous

text: homomorphism

text: H

text: Aut

text: (X).

text: (The

text: action

text: has

text: to

text: be

text: continuous

text: and

text: minimal,

text: and

text: H

text: has

text: to

text: contain

text: a

text: compactly

text: generated

text: dense

text: subgroup.)

text: That

text: is

text: to

text: say,

text: the

text: proof

text: does

text: not

text: require

text: that

text: the

text: cubespace

text: structure

text: on

text: X

text: is

text: deﬁned

text: using

text: the

text: dynamical

text: construction;

text: it

text: may

text: be

text: larger.

text: Appendix

text: A.

text: The

text: purpose

text: of

text: this

text: appendix

text: is

text: to

text: prove

text: the

text: following

text: result

text: stated

text: in

text: Theorem

text: Let

text: X

text: be

text: a

text: compact,

text: ergodic

text: nilspace

text: of

text: degree

text: s.

text: Suppose

text: X

text: is

text: locally

text: connected

text: and

text: has

text: ﬁnite

text: Lebesgue

text: covering

text: dimension,

text: and

text: that

text: Cn(X)

text: is

text: connected

text: for

text: all

text: n.

text: Then

text: X

text: is

text: isomorphic

text: to

text: a

text: nilmanifold

text: G/Γ.

text: That

text: is,

text: there

text: exists

text: a

text: ﬁltered

text: connected

text: Lie

text: group

text: G•,

text: a

text: discrete

text: co-compact

text: subgroup

text: Γ

text: G,

text: and

text: a

text: homeomorphism

text: ϕ

text: X

text: G/Γ

text: that

text: identiﬁes

text: the

text: cubes

text: Ck(X)

text: with

text: the

text: Host–Kra

text: cubes

text: HKk(G•)/Γ.


text: 52 YONATAN GUTMAN, FREDDIE MANNERS AND PE´TER P. VARJU´

text: YONATAN

text: GUTMAN,

text: FREDDIE

text: MANNERS

text: AND

text: PE´TER

text: P.

text: VARJU´


text: INVERSE LIMIT REPRESENTATIONS 7 { } → { } Deﬁnition 1.5. A map ϕ = (ϕ , . . . , ϕ ) : 0, 1 (cid:96) 0, 1 k is termed a morphism of discrete cubes 1 k − ≤ ≤ if each coordinate function ϕ (ω , . . . , ω ) is equal to either 0, 1, ω or 1 ω for some 1 i m. j 1 l i i ≥ ⊆ { } Deﬁnition 1.6. Let X be a metric space and for each integer (cid:96) 0 let C(cid:96) X 0,1 (cid:96) be a closed set. ◦ ∈ ∈ We say that (X, C(cid:96)) is a cubespace if C0 = X and c ϕ C(cid:96) for any c Ck and any morphism of { } → { } discrete cubes ϕ : 0, 1 (cid:96) 0, 1 k. We refer to this property as cube invariance. ∈ ∈ { } We call the elements of C(cid:96) the (cid:96)-cubes of X. Given c C(cid:96), we call the elements c(ω) for ω 0, 1 (cid:96) the vertices of c. To simplify our notation, we simply write X to refer to the full cubespace structure (X, C(cid:96)), and we write C(cid:96)(X) to refer unambiguously to the cubes C(cid:96) associated to X. The cube-invariance property encodes certain fairly natural operations that produce new cubes from old ones. For instance: • cube-invariance for ϕ(ω , . . . , ω − ) = (ω , . . . , ω − , 0) encodes the fact that a face of an (cid:96)-cube 1 (cid:96) 1 1 (cid:96) 1 − − of dimension ((cid:96) 1) is again an ((cid:96) 1)-cube; • − similarly, cube-invariance for ϕ(ω , . . . , ω ) = (1 ω , . . . , ω ) states that reﬂecting an (cid:96)-cube 1 (cid:96) 1 (cid:96) in one of the coordinate axes yields another (cid:96)-cube; • − using f(ω , . . . , ω ) = (ω , . . . , ω − ) shows that “duplicating” an ((cid:96) 1)-cube creates an (cid:96)-cube; 1 (cid:96) 1 (cid:96) 1 and • applying ϕ(ω , ω , ω ) = (ω , ω , ω ) shows that permuting the coordinates of a cube yields 1 2 3 2 1 3 another cube. As well as considering cubespaces, we will also need to discuss maps between them that respect the cubespace structure. The natural deﬁnition is as follows. → Deﬁnition 1.7. Let X and Y be two cubespaces and let ϕ : X Y be a continuous map. Then we say that ϕ is a cubespace morphism, if { ◦ ∈ } ⊆ ϕ c : c Ck(X) Ck(Y ). Cubespaces also admit a natural notion of sub-objects. ⊆ Deﬁnition 1.8. If X and Y are two cubespaces, we say that Y is a subcubespace of X if Y X ⊆ ⊆ and C(cid:96)(Y ) C(cid:96)(X) for all (cid:96). If Z X is a closed set, the subcubespace of X induced by Z is deﬁned ∩ { } by C(cid:96)(Z) = C(cid:96)(X) Z 0,1 (cid:96) for all (cid:96). We now introduce a further technical deﬁnition.

text: INVERSE

text: LIMIT

text: REPRESENTATIONS

text: Deﬁnition

text: A

text: map

text: ϕ

text: (ϕ

text: ϕ

text: k

text: is

text: termed

text: a

text: morphism

text: of

text: discrete

text: cubes

text: k

text: if

text: each

text: coordinate

text: function

text: ϕ

text: (ω

text: ω

text: is

text: equal

text: to

text: either

text: ω

text: or

text: ω

text: for

text: some

text: i

text: m.

text: j

text: l

text: i

text: i

text: Deﬁnition

text: Let

text: X

text: be

text: a

text: metric

text: space

text: and

text: for

text: each

text: integer

text: let

text: X

text: be

text: a

text: closed

text: set.

text: We

text: say

text: that

text: (X,

text: is

text: a

text: cubespace

text: if

text: X

text: and

text: c

text: ϕ

text: for

text: any

text: c

text: Ck

text: and

text: any

text: morphism

text: of

text: discrete

text: cubes

text: ϕ

text: k.

text: We

text: refer

text: to

text: this

text: property

text: as

text: cube

text: invariance.

text: We

text: call

text: the

text: elements

text: of

text: the

text: of

text: X.

text: Given

text: c

text: we

text: call

text: the

text: elements

text: c(ω)

text: for

text: ω

text: the

text: vertices

text: of

text: c.

text: To

text: simplify

text: our

text: notation,

text: we

text: simply

text: write

text: X

text: to

text: refer

text: to

text: the

text: full

text: cubespace

text: structure

text: (X,

text: and

text: we

text: write

text: to

text: refer

text: unambiguously

text: to

text: the

text: cubes

text: associated

text: to

text: X.

text: The

text: cube-invariance

text: property

text: encodes

text: certain

text: fairly

text: natural

text: operations

text: that

text: produce

text: new

text: cubes

text: from

text: old

text: ones.

text: For

text: instance:

text: cube-invariance

text: for

text: ϕ(ω

text: ω

text: (ω

text: ω

text: encodes

text: the

text: fact

text: that

text: a

text: face

text: of

text: an

text: of

text: dimension

text: is

text: again

text: an

text: similarly,

text: cube-invariance

text: for

text: ϕ(ω

text: ω

text: ω

text: ω

text: states

text: that

text: reﬂecting

text: an

text: in

text: one

text: of

text: the

text: coordinate

text: axes

text: yields

text: another

text: using

text: f(ω

text: ω

text: (ω

text: ω

text: shows

text: that

text: “duplicating”

text: an

text: creates

text: an

text: and

text: applying

text: ϕ(ω

text: ω

text: ω

text: (ω

text: ω

text: ω

text: shows

text: that

text: permuting

text: the

text: coordinates

text: of

text: a

text: cube

text: yields

text: another

text: cube.

text: As

text: well

text: as

text: considering

text: cubespaces,

text: we

text: will

text: also

text: need

text: to

text: discuss

text: maps

text: between

text: them

text: that

text: respect

text: the

text: cubespace

text: structure.

text: The

text: natural

text: deﬁnition

text: is

text: as

text: follows.

text: Deﬁnition

text: Let

text: X

text: and

text: Y

text: be

text: two

text: cubespaces

text: and

text: let

text: ϕ

text: X

text: Y

text: be

text: a

text: continuous

text: map.

text: Then

text: we

text: say

text: that

text: ϕ

text: is

text: a

text: cubespace

text: morphism,

text: if

text: ϕ

text: c

text: c

text: Ck(X)

text: Ck(Y

text: Cubespaces

text: also

text: admit

text: a

text: natural

text: notion

text: of

text: sub-objects.

text: Deﬁnition

text: If

text: X

text: and

text: Y

text: are

text: two

text: cubespaces,

text: we

text: say

text: that

text: Y

text: is

text: a

text: subcubespace

text: of

text: X

text: if

text: Y

text: X

text: and

text: for

text: all

text: If

text: Z

text: X

text: is

text: a

text: closed

text: set,

text: the

text: subcubespace

text: of

text: X

text: induced

text: by

text: Z

text: is

text: deﬁned

text: by

text: Z

text: for

text: all

text: We

text: now

text: introduce

text: a

text: further

text: technical

text: deﬁnition.


text: Anomalous localized resonance phenomena in the nonmagnetic, ﬁnite-frequency regime ∗ † Daniel Onofrei Andrew E. Thaler May 29, 2020 Abstract The phenomenon of anomalous localized resonance (ALR) is observed at the interface between mate- rials with positive and negative material parameters and is characterized by the fact that, when a given source is placed near the interface, the electric and magnetic ﬁelds start to have very fast and large oscillations around the interface as the absorption in the materials becomes very small while they remain smooth and regular away from the interface. In this paper, we discuss the phenomenon of anomalous localized resonance (ALR) in the context of an − − inﬁnite slab of homogeneous, nonmagnetic material (µ = 1) with permittivity (cid:15) = 1 iδ for some small (cid:28) s loss δ 1 surrounded by positive, nonmagnetic, homogeneous media. We explicitly characterize the limit value of the product between frequency and the width of slab beyond which the ALR phenomenon does not occur and analyze the situation when the phenomenon is observed. In addition, we also construct sources for which the ALR phenomenon never appears. 1 Introduction In the following, we discuss the anomalous localized resonance phenomenon (ALR) appearing at the interface between materials with positive and negative material parameters in the ﬁnite frequency regime. We consider the particular slab geometry described by (see Figure 1) R R R C ≡ {(x, y) ∈ 2 : x < 0}; S ≡ {(x, y) ∈ 2 : 0 < x < a}; M ≡ {(x, y) ∈ 2 : x > a}; (1.1) where a > 0 denotes the width of the slab and the sets C, S, M represent the regions to the left of the slab, within the slab, and to the right of the slab, respectively. We also deﬁne d ≡ min{x : (x, y) ∈ suppf} and d ≡ max{x : (x, y) ∈ suppf}. 0 1 In this geometry, we assume that all materials are homogeneous and nonmagnetic (i.e., with magnetic permeability µ = 1); the electrical permittivity is given by 1 for x < 0, ε ≡ −1 − iδ for 0 < x < a, (1.2) 1 for x > a for some δ ∈ (0, 1). We consider the following partial diﬀerential equation (PDE) in 2D: ∗ Department of Mathematics, University of Houston, Houston, TX 77204 † Institute for Mathematics and its Applications, University of Minnesota, College of Science and Engineering, Minneapolis, MN 55455. The research of AET was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation through NSF Award 0931945 and in part by AFOSR under the 2013 YIP Award FA9550-13-1-0078. 1 ##LTLine##

text: Anomalous

label: title

text: localized

text: resonance

text: phenomena

text: in

text: the

text: nonmagnetic,

text: ﬁnite-frequency

text: regime

label: author

text: Daniel

text: Onofrei

text: Andrew

text: E.

text: Thaler

text: May

text: Abstract

text: The

label: abstract

text: phenomenon

text: of

text: anomalous

text: localized

text: resonance

text: (ALR)

text: is

text: observed

text: at

text: the

text: interface

text: between

text: mate-

text: rials

text: with

text: positive

text: and

text: negative

text: material

text: parameters

text: and

text: is

text: characterized

text: by

text: the

text: fact

text: that,

text: when

text: a

text: given

text: source

text: is

text: placed

text: near

text: the

text: interface,

text: the

text: electric

text: and

text: magnetic

text: ﬁelds

text: start

text: to

text: have

text: very

text: fast

text: and

text: large

text: oscillations

text: around

text: the

text: interface

text: as

text: the

text: absorption

text: in

text: the

text: materials

text: becomes

text: very

text: small

text: while

text: they

text: remain

text: smooth

text: and

text: regular

text: away

text: from

text: the

text: interface.

text: In

text: this

text: paper,

text: we

text: discuss

text: the

text: phenomenon

text: of

text: anomalous

text: localized

text: resonance

text: (ALR)

text: in

text: the

text: context

text: of

text: an

text: inﬁnite

text: slab

text: of

text: homogeneous,

text: nonmagnetic

text: material

text: (µ

text: with

text: permittivity

text: iδ

text: for

text: some

text: small

text: s

text: loss

text: δ

text: surrounded

text: by

text: positive,

text: nonmagnetic,

text: homogeneous

text: media.

text: We

text: explicitly

text: characterize

text: the

text: limit

text: value

text: of

text: the

text: product

text: between

text: frequency

text: and

text: the

text: width

text: of

text: slab

text: beyond

text: which

text: the

text: ALR

text: phenomenon

text: does

text: not

text: occur

text: and

text: analyze

text: the

text: situation

text: when

text: the

text: phenomenon

text: is

text: observed.

text: In

text: addition,

text: we

text: also

text: construct

text: sources

text: for

text: which

text: the

text: ALR

text: phenomenon

text: never

text: appears.

text: Introduction

text: In

text: the

text: following,

text: we

text: discuss

text: the

text: anomalous

text: localized

text: resonance

text: phenomenon

text: (ALR)

text: appearing

text: at

text: the

text: interface

text: between

text: materials

text: with

text: positive

text: and

text: negative

text: material

text: parameters

text: in

text: the

text: ﬁnite

text: frequency

text: regime.

text: We

text: consider

text: the

text: particular

text: slab

text: geometry

text: described

text: by

text: (see

text: Figure

text: R

text: R

text: R

text: C

text: {(x,

text: y)

text: x

text: S

text: {(x,

text: y)

text: x

text: a};

text: M

text: {(x,

text: y)

text: x

text: a};

text: where

text: a

text: denotes

text: the

text: width

text: of

text: the

text: slab

text: and

text: the

text: sets

text: C,

text: S,

text: M

text: represent

text: the

text: regions

text: to

text: the

text: left

text: of

text: the

text: slab,

text: within

text: the

text: slab,

text: and

text: to

text: the

text: right

text: of

text: the

text: slab,

text: respectively.

text: We

text: also

text: deﬁne

text: d

text: min{x

text: (x,

text: y)

text: suppf}

text: and

text: d

text: max{x

text: (x,

text: y)

text: suppf}.

text: In

text: this

text: geometry,

text: we

text: assume

text: that

text: all

text: materials

text: are

text: homogeneous

text: and

text: nonmagnetic

text: (i.e.,

text: with

text: magnetic

text: permeability

text: µ

text: the

text: electrical

text: permittivity

text: is

text: given

text: by

text: 1

text: for

text: x

text: ε

text: −1

text: iδ

text: for

text: x

text: a,

text: for

text: x

text: a

text: for

text: some

text: δ

text: We

text: consider

text: the

text: following

text: partial

text: diﬀerential

text: equation

text: (PDE)

text: in

text: Department

text: of

text: Mathematics,

text: University

text: of

text: Houston,

text: Houston,

text: TX

text: Institute

text: for

text: Mathematics

text: and

text: its

text: Applications,

text: University

text: of

text: Minnesota,

text: College

text: of

text: Science

text: and

text: Engineering,

text: Minneapolis,

text: MN

text: The

text: research

text: of

text: AET

text: was

text: supported

text: in

text: part

text: by

text: the

text: Institute

text: for

text: Mathematics

text: and

text: its

text: Applications

text: with

text: funds

text: provided

text: by

text: the

text: National

text: Science

text: Foundation

text: through

text: NSF

text: Award

text: and

text: in

text: part

text: by

text: AFOSR

text: under

text: the

text: YIP

text: Award

text: ##LTLine##


text: we present an heuristic argument, coupled with numerical experiments, to illustrate why we believe the slab lens under consideration exhibits ALR in the long-wavelength regime. 4.1 Blow-up of E (a) δ The key result of this section is Lemma 2: |g (p; γ)| has two real roots when γ < γ∗, namely 1 < p1 < p2. 0 γ γ Because both roots are larger than 1, the main contribution to the blow-up of E (a) comes from the integral δ over the interval 1 ≤ p < ∞. Indeed, the following lemma shows that we do not need to worry about the integral over the interval 0 ≤ p ≤ 1. (cid:90) (cid:90) Lemma 4. Suppose 0 < γ ≤ γ∗ and f ∈ L2(M) with compact support. Then there is a positive constant C and a δ > 0 such that γ γ 1 1 |I |2 L (p; γ) dp = p M (p; γ) dp ≤ C δ |g |2 δ γ 0 0 δ for all 0 < δ ≤ δ . γ Remark 5. We emphasize that Lemma 4 also holds for those sources for which the bound in (3.8) holds (e.g., dipole sources) — see Remark 4. Proof. First, we note that M (p; γ) is continuous for δ ∈ [0, 1], p ∈ [0, 1], and γ ∈ [0, γ∗], so it is bounded by δ a constant independent of δ, p, and γ. Additionally, |I |2 is also bounded by a constant, thanks to Lemma 3. p All that remains for us to show is that |g (p; γ)| is bounded away from 0. δ We deﬁne the function Ξ (γ) ≡ max ||g (p; γ)| − |g (p; γ)|| . (4.1) δ ∈ δ 0 p [0,1] Because |g (p; γ)| and |g (p; γ)| are both continuous for 0 ≤ p ≤ 1, the above maximum is attained, say ∗ δ 0 at p = p (γ). This means that δ ∗ ∗ Ξ (γ) = ||g(p (γ); γ)| − |g (p (γ); γ)||. δ δ 0 δ ∞ ∗ Now let {δ } be a sequence converging to 0 as n → ∞. Because p (γ) is a bounded sequence, it has n n=1 ∗ δn a convergent subsequence p (γ). Along this subsequence, δnk ∗ ∗ Ξ (γ) = ||g (p (γ); γ)| − |g (p (γ); γ)|| → 0 as k → ∞ δnk δnk δnk 0 δnk by Lemma 1. In other words, every sequence Ξ (γ) has a subsequence that converges to 0, which implies δn that every sequence Ξ converges to 0. Because the original sequence δ was arbitrary, this implies that δn n lim Ξ (γ) = 0. → δ δ 0+ In combination with (4.1), this implies that |g (p; γ)| converges to |g (p; γ)| uniformly in p for 0 ≤ p ≤ 1. δ 0 Thus for every (cid:15) > 0 there is a δ > 0 such that γ |g (p; γ)| ≥ |g (p; γ)| − (cid:15) δ 0 for all p ∈ [0, 1] and all 0 < δ ≤ δ . If we take γ 1 (cid:15) = min |g (p; γ)|, 2 0≤p≤1 0 then 1 |g (p; γ)| ≥ |g (p; γ)| ≥ C > 0 δ 2 0 γ for all p ∈ [0, 1] (the last t(cid:90)wo inequalities hold because t(cid:90)he roots of |g0| are larger than 1 by Lemma 2). Combining this result with the ﬁrst paragraph of the proof gives us the bound 1 |I |2 1 1 p M (p; γ) dp ≤ C dp ≤ C |g (p; γ)|2 δ |g (p; γ)|2 γ 0 δ 0 0 for some constant C > 0. γ 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: we

text: present

text: an

text: heuristic

text: argument,

text: coupled

text: with

text: numerical

text: experiments,

text: to

text: illustrate

text: why

text: we

text: believe

text: the

text: slab

text: lens

text: under

text: consideration

text: exhibits

text: ALR

text: in

text: the

text: long-wavelength

text: regime.

label: section

text: Blow-up

text: of

text: E

text: (a)

text: δ

text: The

text: key

text: result

text: of

text: this

text: section

text: is

text: Lemma

text: |g

text: (p;

text: γ)|

text: has

text: two

text: real

text: roots

text: when

text: γ

text: γ∗,

text: namely

text: γ

text: γ

text: Because

text: both

text: roots

text: are

text: larger

text: than

text: the

text: main

text: contribution

text: to

text: the

text: blow-up

text: of

text: E

text: (a)

text: comes

text: from

text: the

text: integral

text: δ

text: over

text: the

text: interval

text: p

text: Indeed,

text: the

text: following

text: lemma

text: shows

text: that

text: we

text: do

text: not

text: need

text: to

text: worry

text: about

text: the

text: integral

text: over

text: the

text: interval

text: p

text: Lemma

text: Suppose

text: γ

text: γ∗

text: and

text: f

text: with

text: compact

text: support.

text: Then

text: there

text: is

text: a

text: positive

text: constant

text: C

text: and

text: a

text: δ

text: such

text: that

text: γ

text: γ

text: |I

text: L

text: (p;

text: γ)

text: dp

text: p

text: M

text: (p;

text: γ)

text: dp

text: C

text: δ

text: |g

text: δ

text: γ

text: δ

text: for

text: all

text: δ

text: δ

text: γ

text: Remark

text: We

text: emphasize

text: that

text: Lemma

text: also

text: holds

text: for

text: those

text: sources

text: for

text: which

text: the

text: bound

text: in

text: holds

text: (e.g.,

text: dipole

text: sources)

text: see

text: Remark

text: Proof.

text: First,

text: we

text: note

text: that

text: M

text: (p;

text: γ)

text: is

text: continuous

text: for

text: δ

text: p

text: and

text: γ

text: γ∗],

text: so

text: it

text: is

text: bounded

text: by

text: δ

text: a

text: constant

text: independent

text: of

text: δ,

text: p,

text: and

text: γ.

text: Additionally,

text: |I

text: is

text: also

text: bounded

text: by

text: a

text: constant,

text: thanks

text: to

text: Lemma

text: p

text: All

text: that

text: remains

text: for

text: us

text: to

text: show

text: is

text: that

text: |g

text: (p;

text: γ)|

text: is

text: bounded

text: away

text: from

text: δ

text: We

text: deﬁne

text: the

text: function

text: Ξ

text: (γ)

text: max

text: ||g

text: (p;

text: γ)|

text: |g

text: (p;

text: γ)||

text: δ

text: δ

text: p

text: Because

text: |g

text: (p;

text: γ)|

text: and

text: |g

text: (p;

text: γ)|

text: are

text: both

text: continuous

text: for

text: p

text: the

text: above

text: maximum

text: is

text: attained,

text: say

text: δ

text: at

text: p

text: p

text: (γ).

text: This

text: means

text: that

text: δ

text: Ξ

text: (γ)

text: ||g(p

text: (γ);

text: γ)|

text: |g

text: (p

text: (γ);

text: γ)||.

text: δ

text: δ

text: δ

text: Now

text: let

text: {δ

text: be

text: a

text: sequence

text: converging

text: to

text: as

text: n

text: Because

text: p

text: (γ)

text: is

text: a

text: bounded

text: sequence,

text: it

text: has

text: n

text: δn

text: a

text: convergent

text: subsequence

text: p

text: (γ).

text: Along

text: this

text: subsequence,

text: δnk

text: Ξ

text: (γ)

text: ||g

text: (p

text: (γ);

text: γ)|

text: |g

text: (p

text: (γ);

text: γ)||

text: as

text: k

text: δnk

text: δnk

text: δnk

text: δnk

text: by

text: Lemma

text: In

text: other

text: words,

text: every

text: sequence

text: Ξ

text: (γ)

text: has

text: a

text: subsequence

text: that

text: converges

text: to

text: which

text: implies

text: δn

text: that

text: every

text: sequence

text: Ξ

text: converges

text: to

text: Because

text: the

text: original

text: sequence

text: δ

text: was

text: arbitrary,

text: this

text: implies

text: that

text: δn

text: n

text: lim

text: Ξ

text: (γ)

text: δ

text: δ

text: In

text: combination

text: with

text: this

text: implies

text: that

text: |g

text: (p;

text: γ)|

text: converges

text: to

text: |g

text: (p;

text: γ)|

text: uniformly

text: in

text: p

text: for

text: p

text: δ

text: Thus

text: for

text: every

text: there

text: is

text: a

text: δ

text: such

text: that

text: γ

text: |g

text: (p;

text: γ)|

text: |g

text: (p;

text: γ)|

text: δ

text: for

text: all

text: p

text: and

text: all

text: δ

text: δ

text: If

text: we

text: take

text: γ

text: min

text: |g

text: (p;

text: γ)|,

text: then

text: |g

text: (p;

text: γ)|

text: |g

text: (p;

text: γ)|

text: C

text: δ

text: γ

text: for

text: all

text: p

text: (the

text: last

text: inequalities

text: hold

text: because

text: roots

text: of

text: are

text: larger

text: than

text: by

text: Lemma

text: Combining

text: this

text: result

text: with

text: the

text: ﬁrst

text: paragraph

text: of

text: the

text: proof

text: gives

text: us

text: the

text: bound

text: |I

text: p

text: M

text: (p;

text: γ)

text: dp

text: C

text: dp

text: C

text: |g

text: (p;

text: δ

text: |g

text: (p;

text: γ

text: δ

text: for

text: some

text: constant

text: C

text: γ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4.2 Sources for which ALR does not occur When 0 < γ < γ∗, the conjectures from the previous section suggest that the zeros of g (p; γ) are 0 responsible for forcing E (a) to blow up in the limit as δ → 0+. This begs the question of whether one δ can design a (realistic) source in the ﬁnite frequency regime (with 0 < γ < γ∗) that eﬀectively cancels the poles that show up in the limit δ → 0+. In other words, we would like to design a source such that |I | = 0 p exactly at the zeros of g (p; γ); heuristically, in the limit as δ → 0+, this would force the integrand in (1.24) 0 to remain bounded at the zeros of g (p; γ) and annihilate the anomalous localized resonance that occurs in 0 (cid:90) this limit. Recall from (1.19) that (cid:98) d1 − I = f(s, k p)e k0νms ds. (4.4) p 0 d0 (cid:98) (cid:98) Lemma 2 implies that g (p; γ) has two roots 1 < p1 < p2. Using this and (4.4), we see that an “ALR- 0 γ γ (cid:98) busting” source f(x, y) can be constructed by choosing f such that f(s, k p1) = f(s, k p2) = 0 for all 0 γ 0 γ s ∈ [d , d ] (which implies I = I = 0). We do not want to just choose any f satisfying this property, 0 1 p1 p2 γ γ however; we restrict ourselves to those sources f ∈ L2(M) with compact support. In summary, we make the following conjecture. Conjecture 4. Suppose f ∈ L2(M) has(cid:98)compact sup(cid:98)port and f(x, k p1) = f(x, k p2) = 0, (cid:98) 0 γ 0 γ where 1 < p1 < p2 are the zeros of g (p; γ) from Lemma 2 and k pj are zeros of order at least 1 for f(x, k p). γ γ 0 0 γ 0 Then there is a δ > 0 and a constant C > 0 such that E (a) ≤ C for all 0 < δ ≤ δ . γ γ δ γ γ There are many sources that satisfy the hypotheses of this theorem. We will present 2 examples here. First, consider (cid:98) f(x, q) = −iχ (x) sinc(α q) sin(α q), (4.5) (d0,d1) 1 2 (cid:40) where sinc(x) = sin(x)/x, 1 for d < x < d , χ (x) = 0 1 (d0,d1) 0 otherwise, π π (cid:98) α1 ≡ k p1 , and α2 ≡ k p2 . (cid:98) 0 γ 0 γ (cid:98) R R Then f(x, ·) ∈ L2( ) and, hence, f(x, ·) ∈ L2( ) by the Plancherel Theorem; moreover, f(x, k p1) = 0 γ f(x, k p2) = 0, where the zeros are order 1. Finally, by direct calculations we have 0 γ 1 f(x, y) = χ (x) · [H(−y − α − α ) − H(−y − α + α ) + H(−y + α + α ) − H(−y − α + α )] , (d0,d1) 4α 1 2 1 2 1 2 2 1 1 (4.6) where H(z) is the Heaviside step function; this f ∈ L2(M) has compact support and thus satisﬁes the hypotheses of Conjecture 4. We may(cid:98)also take f(x, q) = χ (x)J (β q)J (β q), (4.7) (d0,d1) 0 0 1 1 where J and J are the Bessel functions of the ﬁrst kind of orders(cid:98)0 and 1, respectively, and β and 0 1 0 β are such that J (β k p1) = J (β k p2) = 0 (we note that these zeros are also of order 1). Because the 1 0 0 0 γ 1 1 0 γ − R Bessel functions of the ﬁrst kind are O(q 1/2) as q → ∞ [23], we have f(x, ·) ∈ L2( ). By the convolution theorem for Fourier transforms, 21 ##LTLine## ##LTLine## ##LTLine##

text: Sources

text: for

text: which

text: ALR

text: does

text: not

text: occur

text: When

text: γ

text: γ∗,

text: the

text: conjectures

text: from

text: the

text: previous

text: section

text: suggest

text: that

text: the

text: zeros

text: of

text: g

text: (p;

text: γ)

text: are

text: responsible

text: for

text: forcing

text: E

text: (a)

text: to

text: blow

text: up

text: in

text: the

text: limit

text: as

text: δ

text: This

text: begs

text: the

text: question

text: of

text: whether

text: one

text: δ

text: can

text: design

text: a

text: (realistic)

text: source

text: in

text: the

text: ﬁnite

text: frequency

text: regime

text: (with

text: γ

text: γ∗)

text: that

text: eﬀectively

text: cancels

text: the

text: poles

text: that

text: show

text: up

text: in

text: the

text: limit

text: δ

text: In

text: other

text: words,

text: we

text: would

text: like

text: to

text: design

text: a

text: source

text: such

text: that

text: |I

text: p

text: exactly

text: at

text: the

text: zeros

text: of

text: g

text: (p;

text: γ);

text: heuristically,

text: in

text: the

text: limit

text: as

text: δ

text: this

text: would

text: force

text: the

text: integrand

text: in

text: to

text: remain

text: bounded

text: at

text: the

text: zeros

text: of

text: g

text: (p;

text: γ)

text: and

text: annihilate

text: the

text: anomalous

text: localized

text: resonance

text: that

text: occurs

text: in

text: this

text: limit.

text: Recall

text: from

text: that

text: I

text: f(s,

text: k

text: p)e

text: ds.

text: p

text: Lemma

text: implies

text: that

text: g

text: (p;

text: γ)

text: has

text: two

text: roots

text: Using

text: this

text: and

text: we

text: see

text: that

text: an

text: “ALR-

text: γ

text: γ

text: busting”

text: source

text: f(x,

text: y)

text: can

text: be

text: constructed

text: by

text: choosing

text: f

text: such

text: that

text: f(s,

text: k

text: f(s,

text: k

text: for

text: all

text: γ

text: γ

text: s

text: [d

text: d

text: (which

text: implies

text: I

text: I

text: We

text: do

text: not

text: want

text: to

text: just

text: choose

text: any

text: f

text: satisfying

text: this

text: property,

text: γ

text: γ

text: however;

text: we

text: restrict

text: ourselves

text: to

text: those

text: sources

text: f

text: with

text: compact

text: support.

text: In

text: summary,

text: we

text: make

text: the

text: following

text: conjecture.

text: Conjecture

text: Suppose

text: f

text: and

text: f(x,

text: k

text: f(x,

text: k

text: γ

text: γ

text: where

text: are

text: the

text: zeros

text: of

text: g

text: (p;

text: γ)

text: from

text: Lemma

text: and

text: k

text: pj

text: are

text: zeros

text: of

text: order

text: at

text: least

text: for

text: f(x,

text: k

text: p).

text: γ

text: γ

text: γ

text: Then

text: there

text: is

text: a

text: δ

text: and

text: a

text: constant

text: C

text: such

text: that

text: E

text: (a)

text: C

text: for

text: all

text: δ

text: δ

text: γ

text: γ

text: δ

text: γ

text: γ

text: There

text: are

text: many

text: sources

text: that

text: satisfy

text: the

text: hypotheses

text: of

text: this

text: theorem.

text: We

text: will

text: present

text: examples

text: here.

text: First,

text: consider

text: f(x,

text: q)

text: −iχ

text: (x)

text: sinc(α

text: q)

text: sin(α

text: q),

text: where

text: sinc(x)

text: sin(x)/x,

text: for

text: d

text: x

text: d

text: χ

text: (x)

text: otherwise,

text: π

text: π

text: α1

text: k

text: and

text: α2

text: k

text: γ

text: γ

text: R

text: R

text: Then

text: f(x,

text: and,

text: hence,

text: f(x,

text: by

text: the

text: Plancherel

text: Theorem;

text: moreover,

text: f(x,

text: k

text: γ

text: f(x,

text: k

text: where

text: the

text: zeros

text: are

text: order

text: Finally,

text: by

text: direct

text: calculations

text: we

text: have

text: γ

text: f(x,

text: y)

text: χ

text: (x)

text: [H(−y

text: α

text: α

text: H(−y

text: α

text: α

text: H(−y

text: α

text: α

text: H(−y

text: α

text: α

text: 4α

text: where

text: H(z)

text: is

text: the

text: Heaviside

text: step

text: function;

text: this

text: f

text: has

text: compact

text: support

text: and

text: thus

text: satisﬁes

text: the

text: hypotheses

text: of

text: Conjecture

text: We

text: take

text: f(x,

text: q)

text: χ

text: (x)J

text: (β

text: q)J

text: (β

text: q),

text: where

text: J

text: and

text: J

text: are

text: the

text: Bessel

text: functions

text: of

text: the

text: ﬁrst

text: kind

text: of

text: and

text: respectively,

text: and

text: β

text: and

text: β

text: are

text: such

text: that

text: J

text: (β

text: k

text: J

text: (β

text: k

text: (we

text: note

text: that

text: these

text: zeros

text: are

text: also

text: of

text: order

text: Because

text: the

text: γ

text: γ

text: R

text: Bessel

text: functions

text: of

text: the

text: ﬁrst

text: kind

text: are

text: O(q

text: as

text: q

text: we

text: have

text: f(x,

text: By

text: the

text: convolution

text: theorem

text: for

text: Fourier

text: transforms,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4.2.1 Current sources for which ALR does not occur In the monochromatic electromagnetic setting, f must satisfy additional restrictions for it to represent a realistic (divergence-free) current source — see §(cid:32). In particul(cid:33)ar, the function f should be in L2(M) with compact support and be of the form (cid:101) (cid:101) ∂J ∂J f = µ y − x (4.10) 0 ∂x ∂y for a current (cid:101) (cid:101) (cid:101) J = J (x, y)e + J (x, y)e (4.11) x x y y (cid:101) (cid:101) satisfying the continuity equation ∂J ∂J x + y = 0. (4.12) ∂x ∂y We now construct a source f satisfying the hypotheses of Conjecture 4 that is of the form (4.10)–(4.12). For simplicity, we assume that(cid:101)the current from (4.11) ha(cid:101)s the form J (x, y) = r (x)t (y) and J (x, y) = r (x)t (y). (4.13) x 1 1 y 2 2 with r , r , t , t smooth enough. Then the continuity equation (4.12) becomes 1 2 1 2 (cid:48) (cid:48) r (x)t (y) + r (x)t (y) = 0. 1 1 2 2 Taking the Fourier transform of this equatio(cid:98)n with respect(cid:98)to y gives (cid:48) r (x)t (q) + r (x)iqt (q) = 0. (4.14) 1 1 2 2 We further simplify the problem by taking(cid:98) (cid:98) t (q) = iqt (q). (4.15) 1 2 (cid:98) Then (4.14) becomes (cid:48) [r (x) + r (x)] t (q) = 0, 1 2 1 which is satisﬁed for all x and q if (cid:48) r (x) = −r (x). (4.16) 2 1 (cid:2) (cid:3) (cid:2) (cid:3) Then (4.10), (4(cid:98).13), (4.15), and ((cid:98)4.16) imply that (cid:98) (cid:98) (cid:48) (cid:48)(cid:48) f(x, q) = µ r (x)t (q) − r (x)iqt (q) = µ −r (x)t (q) + r (x)q2t (q) (4.17) 0 2 2 1 1 0 1 2 1 2 (cid:98) (cid:98) At this point, f satisﬁes (4.10)–(4.12)(cid:98). (cid:98) (cid:98) R R By the Plancherel Theorem and (4.17), f ∈ L2(M) if and only if t (q) ∈ L2( ) and q2t (q) ∈ L2( ). 2 2 We must also be careful to also choose t (q) in such a way that t (k p1) = t (k p2) = 0 and f(x, y) = (cid:48)(cid:48) (cid:48)(cid:48) 2 2 0 γ 2 0 γ µ [−r (x)t (y) − r (x)t (y)] has compact support in y. 0 1 2 1 2 There are many examples of functions that accomplish these tasks. Unfortunately, the functions in (4.6) and (4.8) lead to current sources that are discontinuous and, hence, not divergence free, so we need to be a bit more careful. To ﬁnd a smooth current with compact support satisfying our requirements, we take 23 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Current

text: sources

text: for

text: which

text: ALR

text: does

text: not

text: occur

text: In

text: the

text: monochromatic

text: electromagnetic

text: setting,

text: f

text: must

text: satisfy

text: additional

text: restrictions

text: for

text: it

text: to

text: represent

text: a

text: realistic

text: (divergence-free)

text: current

text: source

text: see

label: equation

text: In

text: the

text: function

text: f

text: should

text: be

text: in

text: with

text: compact

text: support

text: and

text: be

text: of

text: the

text: form

text: ∂J

text: ∂J

text: f

text: µ

text: y

text: x

text: ∂x

text: ∂y

text: for

text: a

text: current

text: J

text: J

text: (x,

text: y)e

text: J

text: (x,

text: y)e

text: x

text: x

text: y

text: y

text: satisfying

text: the

text: continuity

text: equation

text: ∂J

text: ∂J

text: x

text: y

text: ∂x

text: ∂y

text: We

text: now

text: construct

text: a

text: source

text: f

text: satisfying

text: the

text: hypotheses

text: of

text: Conjecture

text: that

text: is

text: of

text: the

text: form

text: For

text: simplicity,

text: we

text: assume

text: current

text: from

text: the

text: form

text: J

text: (x,

text: y)

text: r

text: (x)t

text: (y)

text: and

text: J

text: (x,

text: y)

text: r

text: (x)t

text: (y).

text: x

text: y

text: with

text: r

text: r

text: t

text: t

text: smooth

text: enough.

text: Then

text: the

text: continuity

text: equation

text: becomes

text: r

text: (x)t

text: (y)

text: r

text: (x)t

text: (y)

text: Taking

text: the

text: Fourier

text: transform

text: of

text: this

text: with

text: y

text: gives

text: r

text: (x)t

text: (q)

text: r

text: (x)iqt

text: (q)

text: We

text: further

text: simplify

text: the

text: problem

text: by

text: t

text: (q)

text: iqt

text: (q).

text: Then

text: becomes

text: [r

text: (x)

text: r

text: (x)]

text: t

text: (q)

text: which

text: is

text: satisﬁed

text: for

text: all

text: x

text: and

text: q

text: if

text: r

text: (x)

text: −r

text: (x).

text: Then

text: and

text: imply

text: that

text: f(x,

text: q)

text: µ

text: r

text: (x)t

text: (q)

text: r

text: (x)iqt

text: (q)

text: µ

text: −r

text: (x)t

text: (q)

text: r

text: (q)

text: At

text: this

text: point,

text: f

text: satisﬁes

text: R

text: R

text: By

text: the

text: Plancherel

text: Theorem

text: and

text: f

text: if

text: and

text: only

text: if

text: t

text: (q)

text: and

text: (q)

text: We

text: must

text: also

text: be

text: careful

text: to

text: also

text: choose

text: t

text: (q)

text: in

text: such

text: a

text: way

text: that

text: t

text: (k

text: t

text: (k

text: and

text: f(x,

text: y)

text: γ

text: γ

text: µ

text: [−r

text: (x)t

text: (y)

text: r

text: (x)t

text: (y)]

text: has

text: compact

text: support

text: in

text: y.

text: There

text: are

text: many

text: examples

text: of

text: functions

text: that

text: accomplish

text: these

text: tasks.

text: Unfortunately,

text: the

text: functions

text: in

text: and

text: lead

text: to

text: current

text: sources

text: that

text: are

text: discontinuous

text: and,

text: hence,

text: not

text: divergence

text: free,

text: so

text: we

text: need

text: to

text: be

text: a

text: bit

text: more

text: careful.

text: To

text: ﬁnd

text: a

text: smooth

text: current

text: with

text: compact

text: support

text: satisfying

text: our

text: requirements,

text: we

text: take

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Proof. We have |ν − (1 + iδ)ν |2 ≥ |Re [ν − (1 + iδ)ν ]|2 . (A.10) s m s m (cid:115) (cid:112) If 0 ≤ p ≤ 1, then (cid:112) (p2 + 1) + (p2 + 1)2 + δ2 |Re [ν − (1 + iδ)ν ]| = + δ 1 − p2 ≥ 1. (A.11) s m 2 If 1 ≤ p < ∞, then (cid:115) (cid:112) (cid:112) (cid:112) (cid:112) (p2 + 1) + (p2 + 1)2 + δ2 |Re [ν − (1 + iδ)ν ]| = − p2 − 1 ≥ p2 + 1 − p2 − 1. (A.12) s m 2 Squaring both sides of (A.11) and (A.12) and utilizing (A.10) gives us the desired result. In the next lemma, we provide upper bounds on the second term in (A.8). Lemma 6. Suppose γ > γ∗; then there is a constant 0 < C < 1 such that  (cid:16) (cid:17) (cid:16)(cid:112) (cid:112) (cid:17)  − (cid:48) C for 0 ≤ p ≤ 1, 0 < δ ≤ 0.4, |νs + (1 + iδ)νm|2 e 2γνs ≤ 1 + 3√δ p2 + 1 + p2 − 1 2 e−2γ√p2+1 for 1 ≤ p < ∞, 0 < δ ≤ 1. (A.13) (cid:115) (cid:112) Proof. The triangle inequality, the assumptions that γ > γ∗ and δ >(cid:112)0, and the bound (cid:48) (p2 + 1) + (p2 + 1)2 + δ2 ν = ≥ p2 + 1 (A.14) s (cid:16)2 (cid:112) (cid:17) imply that √ − (cid:48) − |νs + (1 + iδ)νm| e γνs ≤ |νs| + 1 + δ2|νm| e γ∗ p2+1. For 0 ≤ p ≤ 1 and 0 < δ ≤ 0.4, the deﬁn(cid:110)it(cid:2)ions of ν and ν(cid:3) from(cid:112)(1.12)(cid:112)and (1.1(cid:111)5), respectively, imply that s m the above bound becomes − (cid:48) (cid:104)(cid:0) (cid:1) (cid:112) (cid:105) − √ |νs + (1 + iδ)νm| e γνs ≤ (p2 + 1)2 + δ2 1/4 + 1 + δ2 1 − p2 e γ∗ p2+1 (cid:104)(cid:0) (cid:1) (cid:112) −(cid:105) ≤ 4 + δ2 1/4 + 1 + δ2 e γ∗ − ≤ 4 + 0.42 1/4 + 1 + 0.42 e γ∗ ≈ 0.9813. Therefore, if we take C = 0.99, we have − (cid:48) |νs + (1 + iδ)νm|2 e 2γνs ≤ C. Now we consider the case 1 ≤ p < ∞(cid:16). The t(cid:112)riangle inequ(cid:17)ality, the bound in (A.14), and the deﬁnitions of ν and ν for p ≥ 1 (see (1.12) and (1.15), respectively), imply that s m − (cid:48) (cid:110)(cid:2) (cid:3) (cid:112)− √ (cid:112) (cid:111) |νs + (1 + iδ)νm| e γνs ≤ |νs| + 1 + δ2 |νm| e γ p2+1 √ − = (p2 + 1)2 + δ2 1/4 + 1 + δ2 p2 − 1 e γ p2+1. 29 ##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: Proof.

text: We

text: have

text: |ν

text: iδ)ν

text: |Re

text: [ν

text: iδ)ν

text: s

text: m

text: s

text: m

text: If

text: p

text: then

text: δ2

text: |Re

text: [ν

text: iδ)ν

text: δ

text: s

text: m

text: If

text: p

text: then

text: δ2

text: |Re

text: [ν

text: iδ)ν

text: s

text: m

text: Squaring

text: both

text: sides

text: of

text: and

text: and

text: utilizing

text: gives

text: us

text: the

text: desired

text: result.

text: In

text: the

text: next

text: lemma,

text: we

text: provide

text: upper

text: bounds

text: on

text: the

text: second

text: term

text: in

text: Lemma

text: Suppose

text: γ

text: γ∗;

text: then

text: there

text: is

text: a

text: constant

text: C

text: such

text: that

text: C

text: for

text: p

text: δ

text: |νs

text: e

text: 2γνs

text: 3√δ

text: for

text: p

text: δ

text: Proof.

text: The

text: triangle

text: inequality,

text: the

text: assumptions

text: that

text: γ

text: γ∗

text: and

text: δ

text: and

text: the

text: bound

text: δ2

text: ν

text: s

text: imply

text: that

text: |νs

text: iδ)νm|

text: e

text: γνs

text: |νs|

text: e

text: γ∗

text: For

text: p

text: and

text: δ

text: the

text: of

text: ν

text: and

text: respectively,

text: imply

text: that

text: s

text: m

text: the

text: above

text: bound

text: becomes

text: |νs

text: iδ)νm|

text: e

text: γνs

text: δ2

text: δ2

text: e

text: γ∗

text: δ2

text: δ2

text: e

text: γ∗

text: e

text: γ∗

text: Therefore,

text: if

text: we

text: take

text: C

text: we

text: have

text: |νs

text: e

text: 2γνs

text: C.

text: Now

text: we

text: consider

text: the

text: case

text: p

text: The

text: the

text: bound

text: in

text: and

text: the

text: deﬁnitions

text: of

text: ν

text: and

text: ν

text: for

text: p

text: (see

text: and

text: respectively),

text: imply

text: that

text: s

text: m

text: |νs

text: iδ)νm|

text: e

text: γνs

text: |νs|

text: δ2

text: |νm|

text: e

text: γ

text: δ2

text: δ2

text: e

text: γ

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text: Synthesis, characterization and physical properties of layered bismuthide PtBi 2 † C. Q. Xu1, X. Z. Xing2, Xiaofeng Xu1 , L. Q. Che3, Bin Chen4, Xin Lu3, Jianhui Dai1, Z. X. Shi2 1Department of Physics and Hangzhou Key Laboratory of Quantum Matters, Hangzhou Normal University, Hangzhou 310036, China 2Department of Physics and Key Laboratory of MEMS of the Ministry of Education, Southeast University, Nanjing 211189, China 3Center for Correlated Matter and Department of Physics, Zhejiang University, Hangzhou, 310058, China and 4Department of Physics, University of Shanghai for Science & Tehcnology , Shanghai, China (Dated: May 29, 2020) We report details of single crystal growth of stoichiometric bismuthide PtBi whose structure 2 consists of alternate stacking of Pt layer sandwiched by Bi bilayer along the c-axis. The compound crystallizes in space group P-3 with a hexagonal unit cell of a=b=6.553˚A, c=6.165˚A. The magne- tization data show opposite sign for ﬁelds parallel and perpendicular to the Pt layers, respectively. The T-dependent resistivity is typical of a metal and the magnetic response shows clear two types of charge carriers and the validity of the semi-classical Kohler’s rule. Its physical properties was discussed in comparison with recently proposed topological superconductor β-PdBi . 2 I. INTRODUCTION (a) (b) Binary bismuthide β-PdBi has attracted much inter- 2 est recently as a promising candidate of topological su- perconductor (TS).1–6 Topological superconductivity is a new state of matter possessing symmetry-protected sur- face states while the bulk states are fully gapped by su- perconducting pairing.7–10 The Majorana fermions are believed to exist on the surface or vortex core in such FIG. 1: (Color online) Crystal lattice of PtBi . (a) The prim- TSs, which may not only be of scientiﬁc importance, but 2 itive unit cell for hexagonal PtBi . The coordinates of Pt(1) also can lead to a wide-ranging applications in microelec- 2 and Pt(2) are (a, 2a, 0.92c) and (2a, a, 0.08c), respectively. tronic devices and quantum computing. The centrosym- 3 3 3 3 ∼ (b) The structure as seen from a perspective along the c-axis. metric stoichiometric β-PdBi (T 5 K) was claimed to 2 c be topologically nontrivial in view of the observation of the topologically-protected surface modes by spin- and angle-resolved ARPES.4 However, no Andreev bound plane magnetization displays pronounced anisotropy, be- states associated with Majorana fermions are detectable ing diamagnetic with ﬁeld aligned along the plane and through point-contact spectroscopy,6 in sharp contrast paramagnetic when ﬁeld is perpendicular to the plane. to the cases in Cu-intercalated Bi2Se311,12 and In-doped The magnetoresistance (MR) and Hall resistivity mea- SnTe.13 On the other hand, it becomes the common wis- sured on the same sample both show two types of car- dom that spin-orbit interaction (SOI) in heavy elements riers and the former one scales well to the semi-classical is crucial for the topological states. It is therefore heuris- Kohler’s rule.16,17 tic to ask what if we replace Pd by heavier Pt element II. EXPERIMENTAL with enhanced SOI. In this study, we substituted Pt for Pd in PdBi and PtBi single crystals were fabricated via a melt-growth 2 2 found that this new material actually crystallizes in a method. The starting materials of high purity, Bi pow- distinct structure. Unlike β-PdBi which has the tetrag- der(4N) and Pt powder (4N), were mixed thoroughly in 2 onal structure in an I4/mmm space group, PtBi crys- the prescribed molar ratio of Bi:Pt = 2:1 (2 g in to- 2 tallizes in space group P-3 with a hexagonal unit cell tal weight). All these preparations were performed in of a=b=6.553˚A, c=6.165˚A.14 It is also diﬀerent from its a glove box ﬁlled with protective argon gas (both H O 2 homologue PtBi superconductor (T =1 K) with a mono- and O contents were limited below 0.1ppm). The mix- c 2 clinic unit cell.15 The in-plane resistivity of PtBi shows tures were loaded and sealed in an evacuated quartz tube. 2 ◦ metallic behaviors down to 2 K, the lowest tempera- This quartz tube was then heated to 700 C quickly in a ture studied in this work. The intra-plane and inter- sintering furnace and kept at this temperature for 48h, ◦ ◦ before being slowly cooled down to 450 C(3 C/h), and ﬁnally being quenched into cold water. Large pieces of dark-gray plate-like PtBi single crystals of typical 7-8 2 † Corresponding author: xiaofeng.xu@hznu.edu.cn mm in length were harvested. ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Synthesis,

text: characterization

text: and

text: physical

text: properties

text: of

text: layered

text: bismuthide

text: PtBi

text: C.

text: Q.

text: X.

text: Z.

text: Xiaofeng

text: L.

text: Q.

text: Bin

text: Xin

text: Jianhui

text: Z.

text: X.

text: of

text: Physics

text: and

text: Hangzhou

text: Key

text: Laboratory

text: of

text: Quantum

text: Matters,

text: Hangzhou

text: Normal

text: University,

text: Hangzhou

text: China

text: of

text: Physics

text: and

text: Key

text: Laboratory

text: of

text: MEMS

text: of

text: the

text: Ministry

text: of

text: Education,

text: Southeast

text: University,

text: Nanjing

text: China

text: for

text: Correlated

text: Matter

text: and

text: Department

text: of

text: Physics,

text: Zhejiang

text: University,

text: Hangzhou,

text: China

text: and

text: of

text: Physics,

text: University

text: of

text: Shanghai

text: for

text: Science

text: Tehcnology

text: Shanghai,

text: China

text: (Dated:

text: May

label: date

text: We

text: report

text: details

text: of

text: single

text: crystal

text: growth

text: of

text: stoichiometric

text: bismuthide

text: PtBi

text: whose

text: structure

text: consists

text: of

text: alternate

text: stacking

text: of

text: Pt

text: layer

text: sandwiched

text: by

text: Bi

text: bilayer

text: along

text: the

text: c-axis.

text: The

text: compound

text: crystallizes

text: in

text: space

text: group

text: with

text: a

text: hexagonal

text: unit

text: cell

text: of

text: The

text: magne-

text: tization

text: data

text: show

text: opposite

text: sign

text: for

text: ﬁelds

text: parallel

text: and

text: perpendicular

text: to

text: the

text: Pt

text: layers,

text: respectively.

text: The

text: T-dependent

text: resistivity

text: is

text: typical

text: of

text: a

text: metal

text: and

text: the

text: magnetic

text: response

text: shows

text: clear

text: two

text: types

text: of

text: charge

text: carriers

text: and

text: the

text: validity

text: of

text: the

text: semi-classical

text: Kohler’s

text: rule.

text: Its

text: physical

text: properties

text: was

text: discussed

text: in

text: comparison

text: with

text: recently

text: proposed

text: topological

text: superconductor

text: β-PdBi

text: I.

text: INTRODUCTION

text: (a)

text: (b)

text: Binary

text: bismuthide

text: β-PdBi

text: has

text: attracted

text: much

text: inter-

text: est

text: recently

text: as

text: a

text: promising

text: candidate

text: of

text: topological

text: su-

text: perconductor

text: Topological

text: superconductivity

text: is

text: a

text: new

text: state

text: of

text: matter

text: possessing

text: symmetry-protected

text: sur-

text: face

text: states

text: while

text: the

text: bulk

text: states

text: are

text: fully

text: gapped

text: by

text: su-

text: perconducting

text: The

text: Majorana

text: fermions

text: are

text: believed

text: to

text: exist

text: on

text: the

text: surface

text: or

text: vortex

text: core

text: in

text: such

text: FIG.

text: (Color

text: online)

text: Crystal

text: lattice

text: of

text: PtBi

text: (a)

text: The

text: prim-

text: TSs,

text: which

text: may

text: not

text: only

text: be

text: of

text: scientiﬁc

text: importance,

text: but

text: itive

text: unit

text: cell

text: for

text: hexagonal

text: PtBi

text: The

text: coordinates

text: of

text: also

text: can

text: lead

text: to

text: a

text: wide-ranging

text: applications

text: in

text: microelec-

text: and

text: are

text: (a,

text: and

text: a,

text: respectively.

text: tronic

text: devices

text: and

text: quantum

text: computing.

text: The

text: centrosym-

text: (b)

text: The

text: structure

text: as

text: seen

text: from

text: a

text: perspective

text: along

text: the

text: c-axis.

text: metric

text: stoichiometric

text: β-PdBi

text: (T

text: K)

text: was

text: claimed

text: to

text: c

text: be

text: topologically

text: nontrivial

text: in

text: view

text: of

text: the

text: observation

text: of

text: the

text: topologically-protected

text: surface

text: modes

text: by

text: spin-

text: and

text: angle-resolved

text: However,

text: no

text: Andreev

text: bound

text: plane

text: magnetization

text: displays

text: pronounced

text: anisotropy,

text: be-

text: states

text: associated

text: with

text: Majorana

text: fermions

text: are

text: detectable

text: ing

text: diamagnetic

text: with

text: ﬁeld

text: aligned

text: along

text: the

text: plane

text: and

text: through

text: point-contact

text: in

text: sharp

text: contrast

text: paramagnetic

text: when

text: ﬁeld

text: is

text: perpendicular

text: to

text: the

text: plane.

text: to

text: the

text: cases

text: in

text: Cu-intercalated

text: and

text: In-doped

text: The

text: magnetoresistance

text: (MR)

text: and

text: Hall

text: resistivity

text: mea-

text: On

text: the

text: other

text: hand,

text: it

text: becomes

text: the

text: common

text: wis-

text: sured

text: on

text: the

text: same

text: sample

text: both

text: show

text: two

text: types

text: of

text: car-

text: dom

text: that

text: spin-orbit

text: interaction

text: (SOI)

text: in

text: heavy

text: elements

text: riers

text: and

text: the

text: former

text: one

text: scales

text: well

text: to

text: the

text: semi-classical

text: is

text: crucial

text: for

text: the

text: topological

text: states.

text: It

text: is

text: therefore

text: heuris-

text: Kohler’s

text: tic

text: to

text: ask

text: what

text: if

text: we

text: replace

text: Pd

text: by

text: heavier

text: Pt

text: element

text: II.

text: EXPERIMENTAL

text: with

text: enhanced

text: SOI.

text: In

text: this

text: study,

text: we

text: substituted

text: Pt

text: for

text: Pd

text: in

text: PdBi

text: and

text: PtBi

text: single

text: crystals

text: were

text: fabricated

text: via

text: a

text: melt-growth

text: found

text: that

text: this

text: new

text: material

text: actually

text: crystallizes

text: in

text: a

text: method.

text: The

text: starting

text: materials

text: of

text: high

text: purity,

text: Bi

text: pow-

text: distinct

text: structure.

text: Unlike

text: β-PdBi

text: which

text: has

text: the

text: tetrag-

text: and

text: Pt

text: powder

text: were

text: mixed

text: thoroughly

text: in

text: onal

text: structure

text: in

text: an

text: space

text: group,

text: PtBi

text: crys-

text: the

text: prescribed

text: molar

text: ratio

text: of

text: Bi:Pt

text: g

text: in

text: to-

text: tallizes

text: in

text: space

text: group

text: with

text: a

text: hexagonal

text: unit

text: cell

text: tal

text: weight).

text: All

text: these

text: preparations

text: were

text: performed

text: in

text: of

text: It

text: is

text: also

text: diﬀerent

text: from

text: its

text: a

text: glove

text: box

text: ﬁlled

text: with

text: protective

text: argon

text: gas

text: (both

text: H

text: O

text: homologue

text: PtBi

text: superconductor

text: (T

text: K)

text: with

text: a

text: mono-

text: and

text: O

text: contents

text: were

text: limited

text: below

text: The

text: mix-

text: c

text: clinic

text: unit

text: The

text: in-plane

text: resistivity

text: of

text: PtBi

text: shows

text: tures

text: were

text: loaded

text: and

text: sealed

text: in

text: an

text: evacuated

text: quartz

text: tube.

text: metallic

text: behaviors

text: down

text: to

text: K,

text: the

text: lowest

text: tempera-

text: This

text: quartz

text: tube

text: was

text: then

text: heated

text: to

text: C

text: quickly

text: in

text: a

text: ture

text: studied

text: in

text: this

text: work.

text: The

text: intra-plane

text: and

text: inter-

text: sintering

text: furnace

text: and

text: kept

text: at

text: this

text: temperature

text: for

text: before

text: being

text: slowly

text: cooled

text: down

text: to

text: C/h),

text: and

text: ﬁnally

text: being

text: quenched

text: into

text: cold

text: water.

text: Large

text: pieces

text: of

text: dark-gray

text: plate-like

text: PtBi

text: single

text: crystals

text: of

text: typical

text: Corresponding

text: author:

text: xiaofeng.xu@hznu.edu.cn

text: mm

text: in

text: length

text: were

text: harvested.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 θ˙1(t) = −λ2δ˙(t). (81) R˙ = x˙c +iy˙c = N1 ∑N eiθk. (89) k=1 On substituting θ˙1(t) from (81) and by using (80) in (79), For N = 2, (89) gives we get a differential equation in terms of δ(t) as 1 1 δ˙(t)−κ sinδ(t) = 0, (82) x˙c = 2 [cosθ1 +cosθ2]; y˙c = 2 [sinθ1 +sinθ2]. (90) where, κ = (K1 +K2)/2 > 0. Integrating both sides of (82) Integrating (90), we ge(cid:90)t as following (cid:90) (cid:90) 1 (cid:90) t δ dδ = t κdt, (83) xc(t)−xc0 = 2 0 {cosθ1 +cosθ2}dt (91) δ0 sinδ 0 yc(t)−yc0 = 12 t {sinθ1 +sinθ2}dt, (92) 0 we get δ(t) = 2tan−1(φ0eκt), (84) thewihneitriea,l(lxocc0a,tyico0n) =of(txhce(0c)e,nytcr(o0i)d)., denotes the coordinates of We can compute the above integrals by using the following trigonometric relations: (cid:18) (cid:19) (cid:18) (cid:19) where, δ0 = δ(0) = θ20 − θ10 (using (80)), and φ0 = tan(δ0/2). Now, substituting for(cid:8)δ(t) in (80), we get (cid:9) cosθ1 +cosθ2 = 2cos(cid:18) θ1 +2 θ2(cid:19) cos(cid:18) θ1 −2 θ2(cid:19) (93) θ1(t) = λ2 K2c2 −2tan−1(φ0eκt) . (85) sinθ1 +sinθ2 = 2sin θ1 +2 θ2 cos θ1 −2 θ2 (94) (cid:8) (cid:9) Also, substituting θ1(t) in (39) for N = 2, we get From (85) and (86), we have θ2(t) = λ1 K1c2 +2tan−1(φ0eκt) , (86) (θ1 +θ2)/2 = λ1θ10 +λ2θ20 +(λ1 −λ2)tan−1(φ0eκt) (95) where, λ is deﬁned in (68). These results show how hetero- 1 geneous controller gains affect the agents’ velocity directions. and Subtracting (85) from (86), we get (θ1 −θ2)/2 = tan−1(−φ0eκt). (96) θ2(t)−θ1(t) = 2tan−1(φ0eκt). (87) Th(cid:18)us, (cid:19) (cid:0) (cid:1) In steady-state, that is, as t → ∞, (87) simpliﬁed to (cid:113) θ2f −θ1f = sgn(φ0)π (88) cos θ1 −2 θ2 = cos tan−1(−φ0eκt) = 1+1φ2e2κt . (97) 0 where, θkf = θk(t → ∞),k = 1,2, and sgn(φ0) is the signum Using relations (93) and (94) along with (95) and (97) in function of φ0. Thus, the difference between the velocity (91) and (92), we(cid:90)get (cid:90) directions of agents in phase balancing is, |θ2f − θ1f | = π radians, as desired. t t B. Convergence Point xc(t)−xc0 = f(t)dt; yc(t)−yc0 = g(t)dt. (98) 0 0 The centroid of a group of agents is stabilized to a ﬁxed point when they form a balanced formation. This ﬁxed point where, (cid:0) (cid:1) is called the convergence point of the system. Thus, the (cid:113) cboensvheorwgenncine pthoiisntseiscttihoenctehnattroaiddeosfirtehde cgoronuvperagsentc→e p∞o.inItt wcailnl f(t) = cos (cid:0)λ1θ10 +λ2θ20 +(λ1 −λ2)tan−1(φ0eκt)(cid:1), (99) be achieved by suitably selecting the heterogeneous controller (cid:113)1+φ02e2κt gains K1 and K2 of the two agents. sin λ1θ10 +λ2θ20 +(λ1 −λ2)tan−1(φ0eκt) Let xc(t) and yc(t) are the abscissa and the ordinate of the g(t) = . (100) centroid of the group at any time instant t. Then, the rate of 1+φ2e2κt 0 change of centroid’s position in (4) can be written as ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R˙

text: x˙c

text: +iy˙c

text: ∑N

text: eiθk.

text: On

text: substituting

text: from

text: and

text: by

text: using

text: in

text: For

text: N

text: gives

text: we

text: get

text: a

text: differential

text: equation

text: in

text: terms

text: of

text: δ(t)

text: as

text: δ˙(t)−κ

text: sinδ(t)

text: x˙c

text: y˙c

text: where,

text: κ

text: Integrating

text: both

text: sides

text: of

text: Integrating

text: we

text: as

text: following

text: t

text: δ

text: dδ

text: t

text: κdt,

text: δ0

text: sinδ

text: t

text: we

text: get

text: δ(t)

text: denotes

text: the

text: coordinates

text: of

text: We

text: can

text: compute

text: the

text: above

text: integrals

text: by

text: using

text: the

text: following

text: trigonometric

text: relations:

text: where,

text: δ0

text: (using

text: and

text: φ0

text: Now,

text: substituting

text: in

text: we

text: get

text: θ1

text: θ1

text: −2

text: λ2

text: θ1

text: θ2

text: cos

text: θ1

text: −2

text: θ2

text: Also,

text: substituting

text: in

text: for

text: N

text: we

text: get

text: From

text: and

text: we

text: have

text: λ1

text: where,

text: λ

text: is

text: deﬁned

text: in

text: These

text: results

text: show

text: how

text: hetero-

text: geneous

text: controller

text: gains

text: affect

text: the

text: agents’

text: velocity

text: directions.

text: and

text: Subtracting

text: from

text: we

text: get

text: In

text: steady-state,

text: that

text: is,

text: as

text: t

text: simpliﬁed

text: to

text: −θ1f

text: cos

text: θ1

text: −2

text: θ2

text: cos

text: where,

text: θkf

text: θk(t

text: ∞),k

text: and

text: is

text: the

text: signum

text: Using

text: relations

text: and

text: along

text: with

text: and

text: in

text: function

text: of

text: Thus,

text: the

text: difference

text: between

text: the

text: velocity

text: and

text: directions

text: of

text: agents

text: in

text: phase

text: balancing

text: is,

text: π

text: radians,

text: as

text: desired.

text: t

text: t

text: B.

text: Convergence

text: Point

text: f(t)dt;

text: g(t)dt.

text: The

text: centroid

text: of

text: a

text: group

text: of

text: agents

text: is

text: stabilized

text: to

text: a

text: ﬁxed

text: point

text: when

text: they

text: form

text: a

text: balanced

text: formation.

text: This

text: ﬁxed

text: point

text: where,

text: is

text: called

text: the

text: convergence

text: point

text: of

text: the

text: system.

text: Thus,

text: the

text: cboensvheorwgenncine

text: cgoronuvperagsentc→e

text: p∞o.inItt

text: wcailnl

text: f(t)

text: cos

text: be

text: achieved

text: by

text: suitably

text: selecting

text: the

text: heterogeneous

text: controller

text: gains

text: and

text: of

text: the

text: two

text: agents.

text: sin

text: Let

text: xc(t)

text: and

text: yc(t)

text: are

text: the

text: abscissa

text: and

text: the

text: ordinate

text: of

text: the

text: g(t)

text: centroid

text: of

text: the

text: group

text: at

text: any

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