text: SYNCHRONIZATION FOR AN INFINITE CLASS OF CONTINUED FRACTION TRANSFORMATIONS 23 · · · ··· · ··· · 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 2 1 3 1 3 1 2 1 · · ··· 3 1 3 1 2+3 1 3 1 3 3 1 3 1 2 1 3 1 3 3 1 3 1 2 1 3 1 3 1 3 Figure 10. Naive calculations for f( Θ (313) ). From left to right, ﬁnding: ∞ ≺ ∞ ∞ (cid:31) 1∞ ∞ ≺ ∞ (31) (31312) ; (3131) (313121) ; (313121) (31312131) . The beginning of second copies of words are marked with a dot. · given by r (α) = AkC r (α). That is, the ﬁrst simpliﬁed digit of r (α) is k. In other 1 0 0 words, I = [ζ , ω ). k,1 k,1 k,1 Being a ﬁxed point, r (ω ) = k, k, . . . , is purely periodic with period k. Note 0 k,1 however that this is not a T -admissible expansion, as were it so then r for this value α ∈ I 1 of α would equal r . But, r (α) / ! 0 0 α (2) ow suppose c > 1. By deﬁnition, f(c) = c, that is v = c is full branched. Therefore, for 1 each k, ω is such that r (ω ) is the ﬁxed point of (AkC)c. Of course, (AkC)c has k,c 0 k,c the same ﬁxed point as AkC. That is, ω = ω . Related to this, there are values of α k,c k,1 suﬃciently close to ω so that for each of these α, the ﬁrst c simpliﬁed digits of r (α) k,1 0 are all equal to k. Equivalently, r (α), r (α), . . . , r (α) are all in ∆ (k, 1). And, this 0 1 c α is also to say that d(k, c) gives the ﬁrst c simpliﬁed digits of r (α). The reader should 0 easily ﬁnd α in the complement of I inside of I . k,c k,1 (3) Consider v = 111. We consider each preﬁx in turn. Of course u = 1 is full-branched. We ∞ · · · · · · ∞ next compare (11) = 11 11 11 with 1 ; by our convention for powers, we certainly ∞ ≺ ∞ ∞ ∞ · · · · · · ﬁnd that (11) 1 . We next compare (11) with (111) = 111 111 111 = · · · ∞ ≺ ∞ 121212 ; certainly (11) (111) . Therefore, f(111) = 11. Compare this with Figure 11. (4) Consider v = Θ (313) = 313(1213)13. Arguing just as for the previous case, we ﬁnd ∞ 1 { ∞ ∞ ∞} that (31) is the minimal element of 3 , (31) , (313) . Since 3131 = (31)2, we ∞ ∞ certainly have that (31) = (3131) , and thus this latter is our current candidate for ∞ the maximal length full branched preﬁx of v. We compare it with (31312) . We ﬁnd that these inﬁnite words agree in their ﬁrst four letters, but in the ﬁfth (a “c ”-position) they ≺ ∞ ≺j ∞ diﬀer. Confer Figure 10. Since 31313 31315, we ﬁnd that (3131) (31315) . We ∞ ∞ ∞ thus now compare (3131) with (313121) . Confer Figure 10. we ﬁnd that (313121) ∞ is the smaller. One easily sees that is it also smaller than (3131213) . We now compare ∞ ∞ it with (31312131) . See Figure 10. Again, (313121) is the smaller. One easily sees ∞ ≺ ∞ ∞ that also (313121) v . Therefore, f(v) = (313121) . The following shows that certain phenomena illustrated in the above examples hold in general. ∈ V Lemma 28. If v is of length greater than one, then f(v) has even length. In this case, for ∈ N ∞ each k , a simpliﬁed digit expansion for r (ω ) is d(k, ( f(v) ) ). 0 k,v Proof. Our convention for powers of words shows that any preﬁx u of odd length greater than one, thus having initial letter c and ﬁnal letter some c , has its second power including the 1 ∞i letter c + c . Already u2 is larger than the preﬁx (c d ) . The ﬁrst statement thus holds. j 1 · · · 1 1 For any word u = c d d , we have 1 1 j ∞ · · · ∞ ∞ d(k, u ) = [kc1, (k + 1)d1, , (k + 1)dj] = d(k, u ) . ∞ (cid:3) Thus, since ω is deﬁned to be ( d(k, f(u) ) ) , the result holds. k,v ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SYNCHRONIZATION

text: FOR

text: AN

text: INFINITE

text: CLASS

text: OF

text: CONTINUED

text: FRACTION

text: TRANSFORMATIONS

text: Figure

text: Naive

label: caption

text: calculations

text: for

text: f(

text: Θ

text: From

text: left

text: to

text: right,

text: ﬁnding:

text: 1∞

text: The

text: beginning

text: of

text: second

text: copies

text: of

text: words

text: are

text: marked

text: with

text: a

text: dot.

text: given

text: by

text: r

text: (α)

text: AkC

text: r

text: (α).

text: That

text: is,

text: the

text: ﬁrst

text: simpliﬁed

text: digit

text: of

text: r

text: (α)

text: is

text: k.

text: In

text: other

text: words,

text: I

text: [ζ

text: ω

text: Being

text: a

text: ﬁxed

text: point,

text: r

text: (ω

text: k,

text: k,

text: is

text: purely

text: periodic

text: with

text: period

text: k.

text: Note

text: however

text: that

text: this

text: is

text: not

text: a

text: T

text: -admissible

text: expansion,

text: as

text: were

text: it

text: so

text: then

text: r

text: for

text: this

text: value

text: α

text: I

text: of

text: α

text: would

text: equal

text: r

text: But,

text: r

text: (α)

text: α

text: ow

text: suppose

text: c

text: By

text: deﬁnition,

text: f(c)

text: c,

text: that

text: is

text: v

text: c

text: is

text: full

text: branched.

text: Therefore,

text: for

text: each

text: k,

text: ω

text: is

text: such

text: that

text: r

text: (ω

text: is

text: the

text: ﬁxed

text: point

text: of

text: (AkC)c.

text: Of

text: course,

text: (AkC)c

text: has

text: k,c

text: k,c

text: the

text: same

text: ﬁxed

text: point

text: as

text: AkC.

text: That

text: is,

text: ω

text: ω

text: Related

text: to

text: this,

text: there

text: are

text: values

text: of

text: α

text: k,c

text: suﬃciently

text: close

text: to

text: ω

text: so

text: that

text: for

text: each

text: of

text: these

text: α,

text: the

text: ﬁrst

text: c

text: simpliﬁed

text: digits

text: of

text: r

text: (α)

text: are

text: all

text: equal

text: to

text: k.

text: Equivalently,

text: r

text: (α),

text: r

text: (α),

text: r

text: (α)

text: are

text: all

text: in

text: (k,

text: And,

text: this

text: c

text: α

text: is

text: also

text: to

text: say

text: that

text: d(k,

text: c)

text: gives

text: the

text: ﬁrst

text: c

text: simpliﬁed

text: digits

text: of

text: r

text: (α).

text: The

text: reader

text: should

text: easily

text: ﬁnd

text: α

text: in

text: the

text: complement

text: of

text: I

text: inside

text: of

text: I

text: k,c

text: Consider

text: v

text: We

text: consider

text: each

text: preﬁx

text: in

text: turn.

text: Of

text: course

text: u

text: is

text: full-branched.

text: We

text: next

text: compare

text: with

text: by

text: our

text: convention

text: for

text: powers,

text: we

text: certainly

text: ﬁnd

text: that

text: We

text: next

text: compare

text: with

text: certainly

text: Therefore,

text: Compare

text: this

text: with

text: Figure

text: Consider

text: v

text: Θ

text: Arguing

text: just

text: as

text: for

text: the

text: previous

text: case,

text: we

text: ﬁnd

text: that

text: is

text: the

text: minimal

text: element

text: of

text: Since

text: we

text: certainly

text: have

text: that

text: and

text: thus

text: this

text: latter

text: is

text: our

text: current

text: candidate

text: for

text: the

text: maximal

text: length

text: full

text: branched

text: preﬁx

text: of

text: v.

text: We

text: compare

text: it

text: with

text: We

text: ﬁnd

text: that

text: these

text: inﬁnite

text: words

text: agree

text: in

text: their

text: ﬁrst

text: four

text: letters,

text: but

text: in

text: the

text: ﬁfth

text: (a

text: “c

text: ”-position)

text: they

text: ≺j

text: diﬀer.

text: Confer

text: Figure

text: Since

text: we

text: ﬁnd

text: that

text: We

text: thus

text: now

text: compare

text: with

text: Confer

text: Figure

text: we

text: ﬁnd

text: that

text: is

text: the

text: smaller.

text: One

text: easily

text: sees

text: that

text: is

text: it

text: also

text: smaller

text: than

text: We

text: now

text: compare

text: it

text: with

text: See

text: Figure

text: Again,

text: is

text: the

text: smaller.

text: One

text: easily

text: sees

text: that

text: also

text: v

text: Therefore,

text: f(v)

text: The

text: following

text: shows

text: that

text: certain

text: phenomena

text: illustrated

text: in

text: the

text: above

text: examples

text: hold

text: in

text: general.

text: V

text: Lemma

text: If

text: v

text: is

text: of

text: length

text: greater

text: than

text: one,

text: then

text: f(v)

text: has

text: even

text: length.

text: In

text: this

text: case,

text: for

text: N

text: each

text: k

text: a

text: simpliﬁed

text: digit

text: expansion

text: for

text: r

text: (ω

text: is

text: d(k,

text: f(v)

text: k,v

text: Proof.

text: Our

text: convention

text: for

text: powers

text: of

text: words

text: shows

text: that

text: any

text: preﬁx

text: u

text: of

text: odd

text: length

text: greater

text: than

text: one,

text: thus

text: having

text: initial

text: letter

text: c

text: and

text: ﬁnal

text: letter

text: some

text: c

text: has

text: its

text: second

text: power

text: including

text: the

text: ∞i

text: letter

text: c

text: c

text: Already

text: is

text: larger

text: than

text: the

text: preﬁx

text: (c

text: d

text: The

text: ﬁrst

text: statement

text: thus

text: holds.

text: j

text: For

text: any

text: word

text: u

text: c

text: d

text: d

text: we

text: have

text: j

text: d(k,

text: u

text: (k

text: (k

text: d(k,

text: u

text: Thus,

text: since

text: ω

text: is

text: deﬁned

text: to

text: be

text: d(k,

text: f(u)

text: the

text: result

text: holds.

text: k,v

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 28 KARIANE CALTA, COR KRAAIKAMP, AND THOMAS A. SCHMIDT (cid:48)(cid:48) Now, Θ (v) = vv = u(au)2, and hence Θ (v) is admissible on [ζ , ω ). Similarly, each 0 0 k,Θ0(v) k,v Θh(v) is admissible on [ζ , ω ). By considering our ordering on words, it is clear that 0 k,Θh(v) k,v 0 ζ < η < ω . Therefore, the result holds for all Θh(v). k,Θh(v) k,Θh(v) k,v 0 T0he admissi0bility v on all of I and the admissibility of Θ (v) on exactly [ζ , ω ) k,v 0 k,Θ0(v) k,v implies that there must be a shortest extension of v = uau which is admissible for those α immediately to the left of ζ . Lemma 35 shows that f( Θ (v) ) is this extension. The ﬁxed k,Θ0(v) 1 point of R is r (ω ), and we again argue that arbitrarily high powers of this word, k,f(Θ1(v)) 0 k,Θ1(v) f( Θ (v) ), must be admissible just to the left of the corresponding value α, that is of ω . 1 k,Θ1(v) Since any v = uau is always a preﬁx of the square of the corresponding f(v) = ua, we ﬁnd that all of Θ (v) is admissible on an interval ending at ω . By deﬁnition of ζ it 1 k,Θ1(v) k,Θ1(v) follows that this interval is all of [ζ , ω ). We iterate this argument for increasing k,Θ1(v) k,Θ1(v) q, to give that for each q, Θ (v) is admissible on exactly [ζ , ω ). The deﬁnition of q k,Θq(v) k,Θq(v) η shows that it lies strictly between ζ and ω ). k,Θq(v) k,Θq(v) k,Θq(v) Case 3. Suppose that v = Θh(uau) for some uau of Case 2. Thus, v = u(au)h+1. 0 By the proof of Case 2, we can also assume our result for Θh+1(uau) = Θ (v). In partic- 0 0 ular, the left boundary of I does occur at ζ . By Lemma 35, ω = ζ . k,Θ0(v) k,Θ0(v) k,Θ1(v) k,Θ0(v) Arguments as in the previous case yield that all of Θ (v) is admissible on an interval ending at 1 ω , and that this interval is indeed [ζ , ω ). In this case also, induction on q is sukc,Θce1s(vs)ful. That J ⊂ I is here akls,Θo1s(vt)raigkh,Θtf1o(rvw) ard. (cid:3) Θq(v) Θq(v) We are now ready for the following. ∪∞ ∈ Proof of Theorem 19. That (0, γ ) = I follows simply from the fact that for α ∈ I 3,n · k=1 k,1 ∈ V (0, γ ) and x , T (x) = AkC x for some k. Proposition 38 shows that for all v , 3,n α α I = [ζ , ω ) can be partitioned by J = [ζ , η ) and its complement. By Corollary 34 k,v k,v k,v k,v k,v k,v (and the complementary results proven in cases 1 and 3 of the proof of Proposition 38), we have ∈ N have that ω = ω . By Lemma 35, for all q , ω = ζ . Therefore, ∪∞ k,Θ0(v) k,v \ k,Θq(v) k,Θq−1(v) I is a subinterval of I J which has ω as its right endpoint. Finally, the q=0 k,Θq(v) k,v k,v k,v deﬁnition of Θ (v) combined with Lemma 33 shows that lim →∞ ζ = η . Therefore, the left endpoint oqf the union is in fact the right endpoint of Jq . k,Θq(v) k,v (cid:3) k,v We now deﬁne L exactly so that the group identity of Proposition 49 gives that L = − k,v k,v C 1ACR , and thus the main hypothesis of Lemma 12 will be satisﬁed. That synchronization k,v · does occur along J is then only a matter of showing that L A (cid:96) (α) is admissible at all ∈ k,v k,v 0 α J . k,v For further ease, we set − − − − − − w = w = ( 1)n 2, 2, ( 1)n 3, 2. 3,n | | − Note that the length of w is w (cid:2)= 2n 3. One of(cid:3) our ﬁrst goals is to show that as α tends to zero, dα ∞ begins with ever higher powers of w. Recall from (8) that (for any m, n) the element W[1, =) A−2C (A−1C)n−3 A−2C(A−1C)n−2 m−2, equals W = A−1C−1ACA. Just as this is fundamental to understanding the case of α = 0, so is it key to the study of left-orbits for small values of α. For ease of reference, the case of m = 3 is − − − − − − W = A 2C (A 1C)n 3 A 2C(A 1C)n 2 . ##LTLine##

label: paragraph

text: KARIANE

text: CALTA,

text: COR

text: KRAAIKAMP,

text: AND

text: THOMAS

text: A.

text: SCHMIDT

text: Now,

text: Θ

text: (v)

text: vv

text: and

text: hence

text: Θ

text: (v)

text: is

text: admissible

text: on

text: [ζ

text: ω

text: Similarly,

text: each

text: k,v

text: Θh(v)

text: is

text: admissible

text: on

text: [ζ

text: ω

text: By

text: considering

text: our

text: ordering

text: on

text: words,

text: it

text: is

text: clear

text: that

text: k,Θh(v)

text: k,v

text: ζ

text: η

text: ω

text: Therefore,

text: the

text: result

text: holds

text: for

text: all

text: Θh(v).

text: k,Θh(v)

text: k,Θh(v)

text: k,v

text: v

text: on

text: all

text: of

text: I

text: and

text: the

text: admissibility

text: of

text: Θ

text: (v)

text: on

text: exactly

text: [ζ

text: ω

text: k,v

text: k,v

text: implies

text: that

text: there

text: must

text: be

text: a

text: shortest

text: extension

text: of

text: v

text: uau

text: which

text: is

text: admissible

text: for

text: those

text: α

text: immediately

text: to

text: the

text: left

text: of

text: ζ

text: Lemma

text: shows

text: that

text: f(

text: Θ

text: (v)

text: is

text: this

text: extension.

text: The

text: ﬁxed

text: point

text: of

text: R

text: is

text: r

text: (ω

text: and

text: we

text: again

text: argue

text: that

text: arbitrarily

text: high

text: powers

text: of

text: this

text: word,

text: f(

text: Θ

text: (v)

text: must

text: be

text: admissible

text: just

text: to

text: the

text: left

text: of

text: the

text: corresponding

text: value

text: α,

text: that

text: is

text: of

text: ω

text: Since

text: any

text: v

text: uau

text: is

text: always

text: a

text: preﬁx

text: of

text: the

text: square

text: of

text: the

text: corresponding

text: f(v)

text: ua,

text: we

text: ﬁnd

text: that

text: all

text: of

text: Θ

text: (v)

text: is

text: admissible

text: on

text: an

text: interval

text: ending

text: at

text: ω

text: By

text: deﬁnition

text: of

text: ζ

text: it

text: follows

text: that

text: this

text: interval

text: is

text: all

text: of

text: [ζ

text: ω

text: We

text: iterate

text: this

text: argument

text: for

text: increasing

text: q,

text: to

text: give

text: that

text: for

text: each

text: q,

text: Θ

text: (v)

text: is

text: admissible

text: on

text: exactly

text: [ζ

text: ω

text: The

text: deﬁnition

text: of

text: q

text: k,Θq(v)

text: k,Θq(v)

text: η

text: shows

text: that

text: it

text: lies

text: strictly

text: between

text: ζ

text: and

text: ω

text: k,Θq(v)

text: k,Θq(v)

text: k,Θq(v)

text: Case

text: Suppose

text: that

text: v

text: Θh(uau)

text: for

text: some

text: uau

text: of

text: Case

text: Thus,

text: v

text: By

text: the

text: proof

text: of

text: Case

text: we

text: can

text: also

text: assume

text: our

text: result

text: for

text: Θ

text: (v).

text: In

text: partic-

text: ular,

text: the

text: left

text: boundary

text: of

text: I

text: does

text: occur

text: at

text: ζ

text: By

text: Lemma

text: ω

text: ζ

text: Arguments

text: as

text: in

text: the

text: previous

text: case

text: yield

text: that

text: all

text: of

text: Θ

text: (v)

text: is

text: admissible

text: on

text: an

text: interval

text: ending

text: at

text: ω

text: and

text: that

text: this

text: interval

text: is

text: indeed

text: [ζ

text: ω

text: In

text: this

text: case

text: also,

text: induction

text: on

text: q

text: is

text: That

text: J

text: I

text: is

text: here

text: ard.

text: Θq(v)

text: Θq(v)

text: We

text: are

text: now

text: ready

text: for

text: the

text: following.

text: Proof

text: of

text: Theorem

text: That

text: γ

text: I

text: follows

text: simply

text: from

text: the

text: fact

text: that

text: for

text: α

text: I

text: V

text: γ

text: and

text: x

text: T

text: (x)

text: AkC

text: x

text: for

text: some

text: k.

text: Proposition

text: shows

text: that

text: for

text: all

text: v

text: α

text: α

text: I

text: [ζ

text: ω

text: can

text: be

text: partitioned

text: by

text: J

text: [ζ

text: η

text: and

text: its

text: complement.

text: By

text: Corollary

text: k,v

text: k,v

text: k,v

text: k,v

text: k,v

text: k,v

text: (and

text: the

text: complementary

text: results

text: proven

text: in

text: cases

text: and

text: of

text: the

text: proof

text: of

text: Proposition

text: we

text: have

text: N

text: have

text: that

text: ω

text: ω

text: By

text: Lemma

text: for

text: all

text: q

text: ω

text: ζ

text: Therefore,

text: k,v

text: k,Θq(v)

text: I

text: is

text: a

text: subinterval

text: of

text: I

text: J

text: which

text: has

text: ω

text: as

text: its

text: right

text: endpoint.

text: Finally,

text: the

text: k,Θq(v)

text: k,v

text: k,v

text: k,v

text: deﬁnition

text: of

text: Θ

text: (v)

text: combined

text: with

text: Lemma

text: shows

text: that

text: lim

text: ζ

text: η

text: Therefore,

text: the

text: left

text: endpoint

text: oqf

text: the

text: union

text: is

text: in

text: fact

text: the

text: right

text: endpoint

text: of

text: Jq

text: k,Θq(v)

text: k,v

text: k,v

text: We

text: now

text: deﬁne

text: L

text: exactly

text: so

text: that

text: the

text: group

text: identity

text: of

text: Proposition

text: gives

text: that

text: L

text: k,v

text: k,v

text: C

text: and

text: thus

text: the

text: main

text: hypothesis

text: of

text: Lemma

text: will

text: be

text: satisﬁed.

text: That

text: synchronization

text: k,v

text: does

text: occur

text: along

text: J

text: is

text: then

text: only

text: a

text: matter

text: of

text: showing

text: that

text: L

text: A

text: (α)

text: is

text: admissible

text: at

text: all

text: k,v

text: k,v

text: α

text: J

text: k,v

text: For

text: further

text: ease,

text: we

text: set

text: w

text: w

text: Note

text: that

text: the

text: length

text: of

text: w

text: is

text: w

text: One

text: our

text: ﬁrst

text: goals

text: is

text: to

text: show

text: that

text: as

text: α

text: tends

text: to

text: zero,

text: dα

text: begins

text: with

text: ever

text: higher

text: powers

text: of

text: w.

text: Recall

text: from

text: that

text: (for

text: any

text: m,

text: n)

text: the

text: element

text: equals

text: W

text: Just

text: as

text: this

text: is

text: fundamental

text: to

text: understanding

text: the

text: case

text: of

text: α

text: so

text: is

text: it

text: key

text: to

text: the

text: study

text: of

text: left-orbits

text: for

text: small

text: values

text: of

text: α.

text: For

text: ease

text: of

text: reference,

text: the

text: case

text: of

text: m

text: is

text: W

text: A

text: (A

text: A

text: ##LTLine##


text: SYNCHRONIZATION FOR AN INFINITE CLASS OF CONTINUED FRACTION TRANSFORMATIONS 39 The following implies that the complement of the union of the J is a Cantor set. This is k,v the main result of this subsection. (cid:91) Theorem 54. We have the following partition ∞ [(cid:15) , 1) = I− . 3,n k,1 ≥ ∈ V k=2(cid:91) Furthermore, for each k 2 and each v , the following is a partition: ∞ ∪ I− = J− I− , k,v k,v k,Θq(v) (cid:48) q=q (cid:48) (cid:48) − where q = 0 unless v = c , in which case q = 1. 1 We defer the proof of this result until page 41. − Note that one extends the deﬁnition of d( k, v) to inﬁnite words in the obvious fashion. Lemma 55. Let k ∈ N and v ∈ V. Assume that η−k←,−v, ζ−k,v ∈ I−k,v. Then dη[1−,∞k,v) = d(−k, v(v(cid:48))∞) and dζ−∞k,v is purely periodic of period d(−k, v(cid:48) ). [1, ) Proof. The result f←o−r η− follows from Lemma 52 (ii). The deﬁnition of ζ− gives the second k,v k,v − (cid:48) − − − · · · − − − − − − (cid:3) result, since d( k, v ) = ( k)c1, ( k 1)d1, , ( k 1)ds−1, ( k)cs 1, ( k 1). The following is an immediate implication of the deﬁnition of the ordering (6). ∈ N Lemma 56. Fix m = 3. For any v, w words and for any k , we have ≺ − (cid:31) − v w if and only if d( k, v) d( k, w) . − Deﬁnition 57. Deﬁne ω− such that dω−∞k,v is purely periodic of period d( k, ( f(v) ), where k,v [1, ) f(v) is the full branched preﬁx of v. ∈ V ≥ Lemma 58. Let v and ﬁx k 2. The α-cylinder set I− is a subset of (ω− , ζ− ]. k,v k,v k,v Proof. The set of α such that dα | | = d(k, v) is contained in the interval (ω, ζ] such that · [1,· v ] L− A (cid:96) (ω) = (cid:96) (ω) and L− A (cid:96) (ζ) = r (ζ). The right endpoint here is exactly ζ = ζ− . k,v 0 0 k,v 0 0 − k,v Due to the order reversing relationship between words and simpliﬁed digits with k < 0, the (cid:3) proof of Lemma 29 shows that ω− is a greatest lower bound for I− . k,v k,v ∈ V ≥ ≥ Lemma 59. Suppose that v = Θ (u) for some u and some p 1. Fix k 2. Then for ∈ N p q , ζ− = ω− . k,Θq−1(v) k,Θq(v) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SYNCHRONIZATION

text: FOR

text: AN

text: INFINITE

text: CLASS

text: OF

text: CONTINUED

text: FRACTION

text: TRANSFORMATIONS

text: The

text: following

text: implies

text: that

text: the

text: complement

text: of

text: the

text: union

text: of

text: the

text: J

text: is

text: a

text: Cantor

text: set.

text: This

text: is

text: k,v

text: the

text: main

text: result

text: of

text: this

text: subsection.

text: Theorem

text: We

text: have

text: the

text: following

text: partition

text: I−

text: V

text: Furthermore,

text: for

text: each

text: k

text: and

text: each

text: v

text: the

text: following

text: is

text: a

text: partition:

text: I−

text: J−

text: I−

text: k,v

text: k,v

text: k,Θq(v)

text: q=q

text: where

text: q

text: unless

text: v

text: c

text: in

text: which

text: case

text: q

text: We

text: defer

text: the

text: proof

text: of

text: this

text: result

text: until

text: page

text: Note

text: that

text: one

text: extends

text: the

text: deﬁnition

text: of

text: d(

text: k,

text: v)

text: to

text: inﬁnite

text: words

text: in

text: the

text: obvious

text: fashion.

text: Lemma

text: Let

text: k

text: N

text: and

text: v

text: V.

text: Assume

text: that

text: η−k←,−v,

text: ζ−k,v

text: I−k,v.

text: Then

text: d(−k,

text: and

text: dζ−∞k,v

text: is

text: purely

text: periodic

text: of

text: period

text: d(−k,

text: Proof.

text: The

text: result

text: f←o−r

text: η−

text: follows

text: from

text: Lemma

text: (ii).

text: The

text: deﬁnition

text: of

text: ζ−

text: gives

text: the

text: second

text: k,v

text: k,v

text: result,

text: since

text: d(

text: k,

text: v

text: k

text: k

text: k)cs

text: k

text: The

text: following

text: is

text: an

text: immediate

text: implication

text: of

text: the

text: deﬁnition

text: of

text: the

text: ordering

text: N

text: Lemma

text: Fix

text: m

text: For

text: any

text: v,

text: w

text: words

text: and

text: for

text: any

text: k

text: we

text: have

text: v

text: w

text: if

text: and

text: only

text: if

text: d(

text: k,

text: v)

text: d(

text: k,

text: w)

text: Deﬁnition

text: Deﬁne

text: ω−

text: such

text: that

text: dω−∞k,v

text: is

text: purely

text: periodic

text: of

text: period

text: d(

text: k,

text: f(v)

text: where

text: k,v

text: f(v)

text: is

text: the

text: full

text: branched

text: preﬁx

text: of

text: v.

text: V

text: Lemma

text: Let

text: v

text: and

text: ﬁx

text: k

text: The

text: α-cylinder

text: set

text: I−

text: is

text: a

text: subset

text: of

text: (ω−

text: ζ−

text: k,v

text: k,v

text: k,v

text: Proof.

text: The

text: set

text: of

text: α

text: such

text: that

text: dα

text: d(k,

text: v)

text: is

text: contained

text: in

text: the

text: interval

text: (ω,

text: ζ]

text: such

text: that

text: v

text: L−

text: A

text: (ω)

text: (ω)

text: and

text: L−

text: A

text: (ζ)

text: r

text: (ζ).

text: The

text: right

text: endpoint

text: here

text: is

text: exactly

text: ζ

text: ζ−

text: k,v

text: k,v

text: k,v

text: Due

text: to

text: the

text: order

text: reversing

text: relationship

text: between

text: words

text: and

text: simpliﬁed

text: digits

text: with

text: k

text: the

text: proof

text: of

text: Lemma

text: shows

text: that

text: ω−

text: is

text: a

text: greatest

text: lower

text: bound

text: for

text: I−

text: k,v

text: k,v

text: V

text: Lemma

text: Suppose

text: that

text: v

text: Θ

text: (u)

text: for

text: some

text: u

text: and

text: some

text: p

text: Fix

text: k

text: Then

text: for

text: N

text: p

text: q

text: ζ−

text: ω−

text: k,Θq(v)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 40 KARIANE CALTA, COR KRAAIKAMP, AND THOMAS A. SCHMIDT ←−−−−−−− (cid:48) Proof. From Lemma 35, f(Θ (v)) = (Θ − (v) ) . Now Lemma 55 and the deﬁnition of ω− q q 1 k,Θq(v(cid:3)) yield the result. ∈ V ∈ N Lemma 60. For all v and all h , ω− = ω− . k,Θh(v) k,v 0 ∈ V (cid:3) Proof. Proposition 32 shows that f( Θh(v) ) = f(v) for all v . The result thus holds. 0 The following result, and its proof, are completely analogous to Proposition 38 where the case of small α is treated. ∈ V ≥ Proposition 61. For all v , and all k 2, both ⊃ I− = (ω , ζ− ] and I− J− . k,v k,v k,v k,v k,v Proof. Fix k. We argue by induction of the length of the word v. We have already seen that ∈ − (cid:48) ∞ η− I− implies d( k, v(v ) ) gives the sequence of simpliﬁed digits of (cid:96) (η− ). The k,v k,v 0 k,v base cases, given by v = c , are easily veriﬁed. 1 ≥ Case 1. Suppose v = Θh(c) for some h 1 and some c > 1. We have v = c(1 c)h. The 0 argument as for Proposition 38 goes through (compare with the remaining cases). · Case 2. Suppose v = uau in our usual decomposition. Since f(v) = ua, we have L− A k,ua (cid:96) (ω− ) = (cid:96) (ω− ). This equality then implies that given N, there are α suﬃciently close 0 k,v 0 k,v − to, and larger than, ω− such that d( k, u(au)N) is admissible for α. It follows that we k,v (cid:48)(cid:48) can partition I− by subintervals corresponding to the values N. Now, Θ (v) = vv = k,v 0 uauau, and hence Θ (v) is admissible on [ω− , ζ ). Similarly, each Θh(v) is admissible on 0 k,v k,Θ0(v) 0 [ω− , ζ ). By considering our ordering on words, it is clear that ζ > η > ω . k,v k,Θh(v) k,Θh(v) k,Θh(v) k,v 0 0 0 We now proceed inductively for larger values of q. To begin, the admissibility v on all of I− and the admissibility of Θ (v) on exactly [ω− , ζ ) implies that there must be k,v 0 k,v k,Θ0(v) a shortest extension of v = uau admissible for those α immediately to the right of ζ . Lemma 59 shows that f( Θ (v) ) is this extension. Since dω−∞k,Θ1(v) is purely periodic ofkp,Θe0r(iov)d − 1 [1, ] d( k, ( f(Θ (v)) ), arbitrarily high powers of this period give admissible expansions for α just to 1 the right of ω− . Recall that Proposition 32 shows that any v = uau is always a preﬁx of k,Θ1(v) the square of the corresponding f(v) = ua; we hence ﬁnd that all of Θ (v) is admissible on an 1 interval beginning at ω− . We iterate this argument for increasing q, to give that for each k,Θ1(v) q, Θ (v) is admissible on [ζ , ω ). q k,Θq(v) k,Θq(v) From Lemmas 37 and 56 we can conclude that η− > ω− . It follows that J− ⊂ I− . k,Θq(v) k,Θq(v) k,Θq(v) k,Θq(v) Case 3. As in the proof of Proposition 38, the remaining case is that v = Θh(uau). Thus, 0 v = u(au)h+1 in our usual decomposition. The right endpoint of I− is at ζ− . Lemma 55 now yields that the left endpoint of I \ I k,Θh0+i1s(uthaue)point wkit,uh(apuu)hre+l2y k,Θh0(uau)(cid:48) k,Θh0+1(v) periodic lower simpliﬁed digit expansion of period d(k, vu a). In the proof of Proposition 38 we ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: KARIANE

text: CALTA,

text: COR

text: KRAAIKAMP,

text: AND

text: THOMAS

text: A.

text: SCHMIDT

text: Proof.

text: From

text: Lemma

text: f(Θ

text: (v))

text: (Θ

text: (v)

text: Now

text: Lemma

text: and

text: the

text: deﬁnition

text: of

text: ω−

text: q

text: q

text: yield

text: the

text: result.

text: V

text: N

text: Lemma

text: For

text: all

text: v

text: and

text: all

text: h

text: ω−

text: ω−

text: k,Θh(v)

text: k,v

text: V

text: Proof.

text: Proposition

text: shows

text: that

text: f(

text: Θh(v)

text: f(v)

text: for

text: all

text: v

text: The

text: result

text: thus

text: holds.

text: The

text: following

text: result,

text: and

text: its

text: proof,

text: are

text: completely

text: analogous

text: to

text: Proposition

text: where

text: the

text: case

text: of

text: small

text: α

text: is

text: treated.

text: V

text: Proposition

text: For

text: all

text: v

text: and

text: all

text: k

text: both

text: I−

text: (ω

text: ζ−

text: and

text: I−

text: J−

text: k,v

text: k,v

text: k,v

text: k,v

text: k,v

text: Proof.

text: Fix

text: k.

text: We

text: argue

text: by

text: induction

text: of

text: the

text: length

text: of

text: the

text: word

text: v.

text: We

text: have

text: already

text: seen

text: that

text: η−

text: I−

text: implies

text: d(

text: k,

text: v(v

text: gives

text: the

text: sequence

text: of

text: simpliﬁed

text: digits

text: of

text: (η−

text: The

text: k,v

text: k,v

text: k,v

text: base

text: cases,

text: given

text: by

text: v

text: c

text: are

text: easily

text: veriﬁed.

text: Case

text: Suppose

text: v

text: Θh(c)

text: for

text: some

text: h

text: and

text: some

text: c

text: We

text: have

text: v

text: c)h.

text: The

text: argument

text: as

text: for

text: Proposition

text: goes

text: through

text: (compare

text: with

text: the

text: remaining

text: cases).

text: Case

text: Suppose

text: v

text: uau

text: in

text: our

text: usual

text: decomposition.

text: Since

text: f(v)

text: ua,

text: we

text: have

text: L−

text: A

text: k,ua

text: (ω−

text: (ω−

text: This

text: equality

text: then

text: implies

text: that

text: given

text: N,

text: there

text: are

text: α

text: suﬃciently

text: close

text: k,v

text: k,v

text: to,

text: and

text: larger

text: than,

text: ω−

text: such

text: that

text: d(

text: k,

text: u(au)N)

text: is

text: admissible

text: for

text: α.

text: It

text: follows

text: that

text: we

text: k,v

text: can

text: partition

text: I−

text: by

text: subintervals

text: corresponding

text: to

text: the

text: values

text: N.

text: Now,

text: Θ

text: (v)

text: vv

text: k,v

text: uauau,

text: and

text: hence

text: Θ

text: (v)

text: is

text: admissible

text: on

text: [ω−

text: ζ

text: Similarly,

text: each

text: Θh(v)

text: is

text: admissible

text: on

text: k,v

text: [ω−

text: ζ

text: By

text: considering

text: our

text: ordering

text: on

text: words,

text: it

text: is

text: clear

text: that

text: ζ

text: η

text: ω

text: k,v

text: k,Θh(v)

text: k,Θh(v)

text: k,Θh(v)

text: k,v

text: We

text: now

text: proceed

text: inductively

text: for

text: larger

text: values

text: of

text: q.

text: To

text: begin,

text: the

text: admissibility

text: v

text: on

text: all

text: of

text: I−

text: and

text: the

text: admissibility

text: of

text: Θ

text: (v)

text: on

text: exactly

text: [ω−

text: ζ

text: implies

text: that

text: there

text: must

text: be

text: k,v

text: k,v

text: a

text: shortest

text: extension

text: of

text: v

text: uau

text: admissible

text: for

text: those

text: α

text: immediately

text: to

text: the

text: right

text: of

text: ζ

text: Lemma

text: shows

text: that

text: f(

text: Θ

text: (v)

text: is

text: this

text: extension.

text: Since

text: is

text: purely

text: periodic

text: d(

text: k,

text: f(Θ

text: (v))

text: arbitrarily

text: high

text: powers

text: of

text: this

text: period

text: give

text: admissible

text: expansions

text: for

text: α

text: just

text: to

text: the

text: right

text: of

text: ω−

text: Recall

text: that

text: Proposition

text: shows

text: that

text: any

text: v

text: uau

text: is

text: always

text: a

text: preﬁx

text: of

text: the

text: square

text: of

text: the

text: corresponding

text: f(v)

text: ua;

text: we

text: hence

text: ﬁnd

text: that

text: all

text: of

text: Θ

text: (v)

text: is

text: admissible

text: on

text: an

text: interval

text: beginning

text: at

text: ω−

text: We

text: iterate

text: this

text: argument

text: for

text: increasing

text: q,

text: to

text: give

text: that

text: for

text: each

text: q,

text: Θ

text: (v)

text: is

text: admissible

text: on

text: [ζ

text: ω

text: q

text: k,Θq(v)

text: k,Θq(v)

text: From

text: Lemmas

text: and

text: we

text: can

text: conclude

text: that

text: η−

text: ω−

text: It

text: follows

text: that

text: J−

text: I−

text: k,Θq(v)

text: k,Θq(v)

text: k,Θq(v)

text: k,Θq(v)

text: Case

text: As

text: in

text: the

text: proof

text: of

text: Proposition

text: the

text: remaining

text: case

text: is

text: that

text: v

text: Θh(uau).

text: Thus,

text: v

text: in

text: our

text: usual

text: decomposition.

text: The

text: right

text: endpoint

text: of

text: I−

text: is

text: at

text: ζ−

text: Lemma

text: now

text: yields

text: that

text: the

text: left

text: endpoint

text: of

text: I

text: I

text: periodic

text: lower

text: simpliﬁed

text: digit

text: expansion

text: of

text: period

text: d(k,

text: vu

text: a).

text: In

text: the

text: proof

text: of

text: Proposition

text: we

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: COMPARATIVE ANALYSIS OF DENGUE VERSUS CHIKUNGUNYA OUTBREAKS IN COSTA RICA 10 by saying that on average zD(t) = 2.230zC(t), i.e., chikungunya cases zC(t) scaled up by a factor of 2.230 to match dengue cases zD(t). 5. Discussion We used a single-outbreak deterministic model to estimate parameters using reported weekly cases for the dengue and chikungunya viruses from the 2015-2016 outbreak in Costa Rica. The reported data shows atypical behaviour where the dengue cases grows exponen- tially fast, (see Figures 2–3), during the ﬁrst 30 weeks of the season. This behaviour is unusual not due to the number of reported cases but rather that the chikungunya virus was present throughout the 2015 outbreak. This has been a challenge for public health oﬃcials mainly because throughout the last 20 years the vector Ae. aegypti was only transmitting dengue in the region. Once another disease transmitted by the same vector is introduced this makes the process of diagnosing patients very diﬃcult. Most cases are clinically di- agnosed since symptoms for both diseases are similar it is a challenge to determine which disease the individual may be infected with. Laboratory diagnosed cases are expensive and are the least of the reported cases. Based on the point estimates obtained from ﬁtting the model to incidence data, we can conclude that it is likely that dengue cases were misdiagnosed and chikungunya cases were under reported in the 2015-2016 outbreak. As the disease was introduced in the country and public health oﬃcials were unaware of the disease and the similarity of symptoms between the two viruses, this can possibly lead to a misdiagnosis. The information criteria scores that we computed in Table 2 conﬁrm that the compart- ments U (infectious and undiagnosed) and D (infectious and diagnosed) are well-suited h h for the dataset ‘Chikungunya’. This is the dataset with poor quality and anomalies, and in such case the need of an undiagnosed compartment pays oﬀ (better score). In a way these scores are reminiscent of the R2 statistic in classical linear regression analysis. On the other hand, the dataset ‘Dengue’ has more quality (no expected anomalies) and in such case one compartment suﬃces to describe the data. In other words, there is no need to increase the complexity of the model. In conclusion, our results show that misdiagnosis of cases can have a major impact on the long term dynamics of dengue and chikungunya. Our model highlights the impor- tance of undiagnosed cases in a dengue/chikungunya outbreak as well as the importance of epidemiological surveillance and correct diagnosis. Routine detection of each virus in both vectors and hosts will be valuable in order to prevent major outbreaks and gauge the severity of the response that is required to combat any potential outbreak. Acknowledgements A. C.-A. thanks the support of the scholarship program Preparation of Data Driven Mathematical Scientists for the Workforce, housed by East Tennessee State University. References [1] Akaike, H. (1974). A new look at the statistical model identiﬁcation. IEEE transactions on automatic control, 19(6), 716–723. [2] Banks, H. T., Hu, S., & Thompson, W. C. (2014). Modeling and inverse problems in the presence of uncertainty. CRC Press.

text: COMPARATIVE

text: ANALYSIS

text: OF

text: DENGUE

text: VERSUS

text: CHIKUNGUNYA

text: OUTBREAKS

text: IN

text: COSTA

text: RICA

text: by

text: saying

text: that

text: on

text: average

text: zD(t)

text: i.e.,

text: chikungunya

text: cases

text: zC(t)

text: scaled

text: up

text: by

text: a

text: factor

text: of

text: to

text: match

text: dengue

text: cases

text: zD(t).

label: section

text: Discussion

text: We

text: used

text: a

text: single-outbreak

text: deterministic

text: model

text: to

text: estimate

text: parameters

text: using

text: reported

text: weekly

text: cases

text: for

text: the

text: dengue

text: and

text: chikungunya

text: viruses

text: from

text: the

text: outbreak

text: in

text: Costa

text: Rica.

text: The

text: reported

text: data

text: shows

text: atypical

text: behaviour

text: where

text: the

text: dengue

text: cases

text: grows

text: exponen-

text: tially

text: fast,

text: (see

text: Figures

text: during

text: the

text: ﬁrst

text: weeks

text: of

text: the

text: season.

text: This

text: behaviour

text: is

text: unusual

text: not

text: due

text: to

text: the

text: number

text: of

text: reported

text: cases

text: but

text: rather

text: that

text: the

text: chikungunya

text: virus

text: was

text: present

text: throughout

text: the

text: outbreak.

text: This

text: has

text: been

text: a

text: challenge

text: for

text: public

text: health

text: oﬃcials

text: mainly

text: because

text: throughout

text: the

text: last

text: years

text: the

text: vector

text: Ae.

text: aegypti

text: was

text: only

text: transmitting

text: dengue

text: in

text: the

text: region.

text: Once

text: another

text: disease

text: transmitted

text: by

text: the

text: same

text: vector

text: is

text: introduced

text: this

text: makes

text: the

text: process

text: of

text: diagnosing

text: patients

text: very

text: diﬃcult.

text: Most

text: cases

text: are

text: clinically

text: di-

text: agnosed

text: since

text: symptoms

text: for

text: both

text: diseases

text: are

text: similar

text: it

text: is

text: a

text: challenge

text: to

text: determine

text: which

text: disease

text: the

text: individual

text: may

text: be

text: infected

text: with.

text: Laboratory

text: diagnosed

text: cases

text: are

text: expensive

text: and

text: are

text: the

text: least

text: of

text: the

text: reported

text: cases.

text: Based

text: on

text: the

text: point

text: estimates

text: obtained

text: from

text: ﬁtting

text: the

text: model

text: to

text: incidence

text: data,

text: we

text: can

text: conclude

text: that

text: it

text: is

text: likely

text: that

text: dengue

text: cases

text: were

text: misdiagnosed

text: and

text: chikungunya

text: cases

text: were

text: under

text: reported

text: in

text: the

text: outbreak.

text: As

text: the

text: disease

text: was

text: introduced

text: in

text: the

text: country

text: and

text: public

text: health

text: oﬃcials

text: were

text: unaware

text: of

text: the

text: disease

text: and

text: the

text: similarity

text: of

text: symptoms

text: between

text: the

text: two

text: viruses,

text: this

text: can

text: possibly

text: lead

text: to

text: a

text: misdiagnosis.

text: The

text: information

text: criteria

text: scores

text: that

text: we

text: computed

text: in

text: Table

text: conﬁrm

text: that

text: the

text: compart-

text: ments

text: U

text: (infectious

text: and

text: undiagnosed)

text: and

text: D

text: (infectious

text: and

text: diagnosed)

text: are

text: well-suited

text: h

text: h

text: for

text: the

text: dataset

text: ‘Chikungunya’.

text: This

text: is

text: the

text: dataset

text: with

text: poor

text: quality

text: and

text: anomalies,

text: and

text: in

text: such

text: case

text: the

text: need

text: of

text: an

text: undiagnosed

text: compartment

text: pays

text: oﬀ

text: (better

text: score).

text: In

text: a

text: way

text: these

text: scores

text: are

text: reminiscent

text: of

text: the

text: statistic

text: in

text: classical

text: linear

text: regression

text: analysis.

text: On

text: the

text: other

text: hand,

text: the

text: dataset

text: ‘Dengue’

text: has

text: more

text: quality

text: (no

text: expected

text: anomalies)

text: and

text: in

text: such

text: case

text: one

text: compartment

text: suﬃces

text: to

text: describe

text: the

text: data.

text: In

text: other

text: words,

text: there

text: is

text: no

text: need

text: to

text: increase

text: the

text: complexity

text: of

text: the

text: model.

text: In

text: conclusion,

text: our

text: results

text: show

text: that

text: misdiagnosis

text: of

text: cases

text: can

text: have

text: a

text: major

text: impact

text: on

text: the

text: long

text: term

text: dynamics

text: of

text: dengue

text: and

text: chikungunya.

text: Our

text: model

text: highlights

text: the

text: impor-

text: tance

text: of

text: undiagnosed

text: cases

text: in

text: a

text: dengue/chikungunya

text: outbreak

text: as

text: well

text: as

text: the

text: importance

text: of

text: epidemiological

text: surveillance

text: and

text: correct

text: diagnosis.

text: Routine

text: detection

text: of

text: each

text: virus

text: in

text: both

text: vectors

text: and

text: hosts

text: will

text: be

text: valuable

text: in

text: order

text: to

text: prevent

text: major

text: outbreaks

text: and

text: gauge

text: the

text: severity

text: of

text: the

text: response

text: that

text: is

text: required

text: to

text: combat

text: any

text: potential

text: outbreak.

text: Acknowledgements

text: A.

text: C.-A.

text: thanks

text: the

text: support

text: of

text: the

text: scholarship

text: program

text: Preparation

text: of

text: Data

text: Driven

text: Mathematical

text: Scientists

text: for

text: the

text: Workforce,

text: housed

text: by

text: East

text: Tennessee

text: State

text: University.

text: References

text: Akaike,

text: H.

text: A

text: new

text: look

text: at

text: the

text: statistical

text: model

text: identiﬁcation.

text: IEEE

text: transactions

text: on

text: automatic

text: control,

text: Banks,

text: H.

text: T.,

text: Hu,

text: S.,

text: Thompson,

text: W.

text: C.

text: Modeling

text: and

text: inverse

text: problems

text: in

text: the

text: presence

text: of

text: uncertainty.

text: CRC

text: Press.


text: Abstract The inﬂuence of pairing correlations on the neutron root mean square (rms) radius of nuclei is investigated in the framework of self-consistent Skyrme Hartree-Fock-Bogoliubov calculations. The continuum is treated appropriately by the Green’s function techniques. As an example the nucleus 124Zr is treated for a varying strength of pairing correlations. We ﬁnd that, as the pairing strength increases, the neutron rms radius ﬁrst shrinks, reaches a minimum and beyond this point it expands again. The shrinkage is due to the the so-called ‘pairing anti-halo eﬀect’, i. e. due to the decreasing of the asymptotic density distribution with increasing pairing. However, in some cases, increasing pairing correlations can also lead to an expansion of the nucleus due to a growing occupation of so-called ‘halo’ orbits, i.e. weakly bound states and resonances in the continuum with low-(cid:96) values. In this case, the neutron radii are extended just by the inﬂuence of pairing correlations, since these ‘halo’ orbits cannot be occupied without pairing. The term ‘anti-halo eﬀect’ is not justiﬁed in such cases. For a full understanding of this complicated interplay self-consistent calculations are necessary. ∗ † yzhangjcnp@tju.edu.cn ‡ § 2 ##LTLine##

text: Abstract

text: The

label: abstract

text: inﬂuence

text: of

text: pairing

text: correlations

text: on

text: the

text: neutron

text: root

text: mean

text: square

text: (rms)

text: radius

text: of

text: nuclei

text: is

text: investigated

text: in

text: the

text: framework

text: of

text: self-consistent

text: Skyrme

text: Hartree-Fock-Bogoliubov

text: calculations.

text: The

text: continuum

text: is

text: treated

text: appropriately

text: by

text: the

text: Green’s

text: function

text: techniques.

text: As

text: an

text: example

text: the

text: nucleus

text: is

text: treated

text: for

text: a

text: varying

text: strength

text: of

text: pairing

text: correlations.

text: We

text: ﬁnd

text: that,

text: as

text: the

text: pairing

text: strength

text: increases,

text: the

text: neutron

text: rms

text: radius

text: ﬁrst

text: shrinks,

text: reaches

text: a

text: minimum

text: and

text: beyond

text: this

text: point

text: it

text: expands

text: again.

text: The

text: shrinkage

text: is

text: due

text: to

text: the

text: the

text: so-called

text: ‘pairing

text: anti-halo

text: eﬀect’,

text: i.

text: e.

text: due

text: to

text: the

text: decreasing

text: of

text: the

text: asymptotic

text: density

text: distribution

text: with

text: increasing

text: pairing.

text: However,

text: in

text: some

text: cases,

text: increasing

text: pairing

text: correlations

text: can

text: also

text: lead

text: to

text: an

text: expansion

text: of

text: the

text: nucleus

text: due

text: to

text: a

text: growing

text: occupation

text: of

text: so-called

text: ‘halo’

text: orbits,

text: i.e.

text: weakly

text: bound

text: states

text: and

text: resonances

text: in

text: the

text: continuum

text: with

text: values.

text: In

text: this

text: case,

text: the

text: neutron

text: radii

text: are

text: extended

text: just

text: by

text: the

text: inﬂuence

text: of

text: pairing

text: correlations,

text: since

text: these

text: ‘halo’

text: orbits

text: cannot

text: be

text: occupied

text: without

text: pairing.

text: The

text: term

text: ‘anti-halo

text: eﬀect’

text: is

text: not

text: justiﬁed

text: in

text: such

text: cases.

text: For

text: a

text: full

text: understanding

text: of

text: this

text: complicated

text: interplay

text: self-consistent

text: calculations

text: are

text: necessary.

text: yzhangjcnp@tju.edu.cn

text: ##LTLine##


text: FIG. 5. (a) Total energy E and the neutron rms radius r obtained by the continuum (ﬁlled tot rms circle) and the discretized (open circle) HFB calculations for 122Zr, and (b) occupation probabilities of the 3p, 2f and 1h orbits around the Fermi energy contributed from the quasiparticle states within ∼ E = 0 7 MeV obtained by the continuum HFB calculation with the SkI4 interaction and the DDDI pairing force with diﬀerent strengths ηV . 0 -935 r 5.04 -940 rms 122Zr 5.00 -945 SkI4 -950 continuum 4.96 discretized E -955 tot (a) 4.92 -960 1.0 0.8 v2 1h 0.6 nlj 11/2 0.2 2f 7/2 3p 0.1 3p3/2 (b) 2f 1/2 5/2 1h 9/2 0.0 = 0.6 0.8 1.0 1.2 1.4 E = 0.0 0.0 0.0 -19.5 -77.1 [MeV] pair = 0.0 0.0 0.0 1.4 2.9 [MeV] uv 2f orbits, which can contribute a larger rms radius rrms as shown in Fig. 3 (c). Therefore nlj in this case, we can only observe an expansion but no shrinkage of the neutron rms radius from the no-pairing to the ﬁnite pairing case. Actually, the diﬀerence between 124Zr and 122Zr is whether the weakly bound 3p orbit is originally occupied or not. The asymptotic 3/2 behavior of this wave function is sensitive to the pairing gap. Finally, we would like to emphasize that the terminology ‘pairing anti-halo eﬀect’ has to be taken with great care. In the original literatures [10], the ‘pairing anti-halo eﬀect’ is actually referred to the fact that the divergence of the HF wave function without pairing at the limit → ε 0 can be avoided in the HFB wave function with ﬁnite pairing gap. It is restricted to the rather academic case of one fully occupied single-particle wave functions with low- (cid:96) value very close to the continuum limit. The realistic case is much more complicated. In particular one has to consider contributions of several partially occupied orbits and the occupation are determined by the pairing correlations. This can lead to a decrease or an 15 ]VeM[ E tot ytilibaborp noitapuccO rrms [fm] ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: (a)

text: Total

text: energy

text: E

text: and

text: the

text: neutron

text: rms

text: radius

text: r

text: obtained

text: by

text: the

text: continuum

text: (ﬁlled

text: tot

text: rms

text: circle)

text: and

text: the

text: discretized

text: (open

text: circle)

text: HFB

text: calculations

text: for

text: and

text: (b)

text: occupation

text: probabilities

text: of

text: the

text: and

text: orbits

text: around

text: the

text: Fermi

text: energy

text: contributed

text: from

text: the

text: quasiparticle

text: states

text: within

text: E

text: MeV

text: obtained

text: by

text: the

text: continuum

text: HFB

text: calculation

text: with

text: the

text: interaction

text: and

text: the

text: DDDI

text: pairing

text: force

text: with

text: diﬀerent

text: strengths

text: ηV

font: default

text: r

text: rms

text: continuum

text: discretized

text: E

text: tot

text: (a)

text: nlj

text: (b)

text: E

text: [MeV]

text: pair

text: [MeV]

text: uv

text: orbits,

text: which

text: can

text: contribute

text: a

text: larger

text: rms

text: radius

text: rrms

text: as

text: shown

text: in

text: Fig.

text: (c).

text: Therefore

text: nlj

text: in

text: this

text: case,

text: we

text: can

text: only

text: observe

text: an

text: expansion

text: but

text: no

text: shrinkage

text: of

text: the

text: neutron

text: rms

text: radius

text: from

text: the

text: no-pairing

text: to

text: the

text: ﬁnite

text: pairing

text: case.

text: Actually,

text: the

text: diﬀerence

text: between

text: and

text: is

text: whether

text: the

text: weakly

text: bound

text: orbit

text: is

text: originally

text: occupied

text: or

text: not.

text: The

text: asymptotic

text: behavior

text: of

text: this

text: wave

text: function

text: is

text: sensitive

text: to

text: the

text: pairing

text: gap.

text: Finally,

text: we

text: would

text: like

text: to

text: emphasize

text: that

text: the

text: terminology

text: ‘pairing

text: anti-halo

text: eﬀect’

text: has

text: to

text: be

text: taken

text: with

text: great

text: care.

text: In

text: the

text: original

text: literatures

text: the

text: ‘pairing

text: anti-halo

text: eﬀect’

text: is

text: actually

text: referred

text: to

text: the

text: fact

text: that

text: the

text: divergence

text: of

text: the

text: HF

text: wave

text: function

text: without

text: pairing

text: at

text: the

text: limit

text: ε

text: can

text: be

text: avoided

text: in

text: the

text: HFB

text: wave

text: function

text: with

text: ﬁnite

text: pairing

text: gap.

text: It

text: is

text: restricted

text: to

text: the

text: rather

text: academic

text: case

text: of

text: one

text: fully

text: occupied

text: single-particle

text: wave

text: functions

text: with

text: low-

text: value

text: very

text: close

text: to

text: the

text: continuum

text: limit.

text: The

text: realistic

text: case

text: is

text: much

text: more

text: complicated.

text: In

text: particular

text: one

text: has

text: to

text: consider

text: contributions

text: of

text: several

text: partially

text: occupied

text: orbits

text: and

text: the

text: occupation

text: are

text: determined

text: by

text: the

text: pairing

text: correlations.

text: This

text: can

text: lead

text: to

text: a

text: decrease

text: or

text: an

text: ]VeM[

text: E

text: tot

text: ytilibaborp

text: noitapuccO

text: rrms

text: [fm]

text: ##LTFigure##

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text: Zhou,

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text: China

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text: Grasso,

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text: Sandulescu,

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text: FIG. 1. Total energy E and neutron root mean square (rms) radius r obtained by the contin- tot rms uum (ﬁlled circle) and the discretized (open circle) Hartree-Fock-Bogoliubov (HFB) calculations for 124Zr with the SkI4 interaction as a function of the strength ηV of the DDDI pairing force, 0 − − where V = 458.4 MeV fm 3. 0 5.10 -930 124Zr continuum r discretized rms 5.08 -940 5.06 -950 5.04 -960 E 5.02 SkI4 tot -970 5.00 = 0.6 0.8 1.0 1.2 1.4 E = -0.7 -2.1 -6.4 -26.0 -83.7 [MeV] pair = 0.2 0.3 0.7 1.5 2.9 [MeV] uv ∼ (0.2 2.9 MeV) are shown under the corresponding pairing strength factor η. One can see that, as the pairing strength increases, the total energy E monotonously tot decreases: the nucleus becomes more bound, due to the attractive pairing interaction. More- over, one can ﬁnd almost no diﬀerence between the total energy obtained by the discretized and by the full continuum HFB calculations. Naively thinking, the corresponding nuclear size is shrinking as the nucleus becomes more bound. However, both the discretized and the continuum calculations show that the neutron rms radius ﬁrst decreases, then reaches a ≈ local minimum at η 1.0 and afterwards increases. The diﬀerence between the discretized and continuum results for the neutron rms radius is more obvious for weak pairing case with a shallow Fermi energy. In order to understand the change of the rms radius as a function of the pairing strength, we ﬁrst plot in Fig. 2 the total neutron density obtained by the continuum HFB calculation for the pairing strength factors η = 0.6, 1.0, and 1.4. A linear scale is used in the main ﬁgure for the inner and the surface region, and a logarithmic scale is used in the insert for the asymptotic region. In the insert we see that as the pairing strength increases, the density in the asymptotic region (r > 10 fm) always decays faster. This corresponds to the ‘anti-halo eﬀect’ discussed in Ref. [10] leading to a shrinkage of the rms radius. However, we also have to consider the 7 ]VeM[ E tot rrms [fm] ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: Total

text: energy

text: E

text: and

text: neutron

text: root

text: mean

text: square

text: (rms)

text: radius

text: r

text: obtained

text: by

text: the

text: contin-

text: tot

text: rms

text: uum

text: (ﬁlled

text: circle)

text: and

text: the

text: discretized

text: (open

text: circle)

text: Hartree-Fock-Bogoliubov

text: (HFB)

text: calculations

text: for

text: with

text: the

text: interaction

text: as

text: a

text: function

text: of

text: the

text: strength

text: ηV

text: of

text: the

text: DDDI

text: pairing

text: force,

text: where

text: V

text: MeV

text: fm

text: continuum

text: r

text: discretized

text: rms

text: E

text: tot

text: E

text: [MeV]

text: pair

text: [MeV]

text: uv

text: MeV)

text: are

text: shown

text: under

text: the

text: corresponding

text: pairing

text: strength

text: factor

text: η.

text: One

text: can

text: see

text: that,

text: as

text: the

text: pairing

text: strength

text: increases,

text: the

text: total

text: energy

text: E

text: monotonously

text: tot

text: decreases:

text: the

text: nucleus

text: becomes

text: more

text: bound,

text: due

text: to

text: the

text: attractive

text: pairing

text: interaction.

text: More-

text: over,

text: one

text: can

text: ﬁnd

text: almost

text: no

text: diﬀerence

text: between

text: the

text: total

text: energy

text: obtained

text: by

text: the

text: discretized

text: and

text: by

text: the

text: full

text: continuum

text: HFB

text: calculations.

text: Naively

text: thinking,

text: the

text: corresponding

text: nuclear

text: size

text: is

text: shrinking

text: as

text: the

text: nucleus

text: becomes

text: more

text: bound.

text: However,

text: both

text: the

text: discretized

text: and

text: the

text: continuum

text: calculations

text: show

text: that

text: the

text: neutron

text: rms

text: radius

text: ﬁrst

text: decreases,

text: then

text: reaches

text: a

text: local

text: minimum

text: at

text: η

text: and

text: afterwards

text: increases.

text: The

text: diﬀerence

text: between

text: the

text: discretized

text: and

text: continuum

text: results

text: for

text: the

text: neutron

text: rms

text: radius

text: is

text: more

text: obvious

text: for

text: weak

text: pairing

text: case

text: with

text: a

text: shallow

text: Fermi

text: energy.

text: In

text: order

text: to

text: understand

text: the

text: change

text: of

text: the

text: rms

text: radius

text: as

text: a

text: function

text: of

text: the

text: pairing

text: strength,

text: we

text: ﬁrst

text: plot

text: in

text: Fig.

text: the

text: total

text: neutron

text: density

text: obtained

text: by

text: the

text: continuum

text: HFB

text: calculation

text: for

text: the

text: pairing

text: strength

text: factors

text: η

text: and

text: A

text: linear

text: scale

text: is

text: used

text: in

text: the

text: main

text: ﬁgure

text: for

text: the

text: inner

text: and

text: the

text: surface

text: region,

text: and

text: a

text: logarithmic

text: scale

text: is

text: used

text: in

text: the

text: insert

text: for

text: the

text: asymptotic

text: region.

text: In

text: the

text: insert

text: we

text: see

text: that

text: as

text: the

text: pairing

text: strength

text: increases,

text: the

text: density

text: in

text: the

text: asymptotic

text: region

text: (r

text: fm)

text: always

text: decays

text: faster.

text: This

text: corresponds

text: to

text: the

text: ‘anti-halo

text: eﬀect’

text: discussed

text: in

text: Ref.

text: leading

text: to

text: a

text: shrinkage

text: of

text: the

text: rms

text: radius.

text: However,

text: we

text: also

text: have

text: to

text: consider

text: the

text: ]VeM[

text: E

text: tot

text: rrms

text: [fm]

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


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text: Q.

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text: Perrin,

text: C.

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text: Lalanne,

text: Opt.

text: Exp.

text: F.

text: J.

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text: Kociak,

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text: Hulkko,

text: M.

text: Banik,

text: GO..

text: PMo.tmAkas,elarondd,VM..aC.

text: .AWpkeairdimana,nN,

text: W.

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text: and

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text: ACS

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text: B.

text: de

text: Nijs,

text: F.

text: Benz,

text: S.

text: J.

text: Barrow,

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text: CP.omFopxu,tOer.

text: PHheysss,icsanCdomJ.mJu.nications

text: Baumberg,

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text: F.

text: Alpeggiani,

text: S.

text: D’Agostino,

text: D.

text: Sanvitto,

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text: Fischer,

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text: Becker,

text: N.

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text: aanndd

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text: Optics

text: Express

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text: 10 bility of our technique for use in dielectric systems with exactly the same structure and computational domain is large quality factors, but also tackles the very diﬃcult used for both devices; see Fig. 6. problem of normalizing QNMs for the localized cavities Depicted in Figs. 6(a)-(b), as for the other cavity struc- that are subjected to the Bloch periodic propagation of tures, we ﬁrst conﬁrm a very good agreement between the waveguide, as the dominant outgoing channel37,38; the QNM-calculated Purcell factor and the fully vecto- in this regime, there are additional complexities and rial dipole calculations; here we consider a dipole emit- diﬃculties encountered to regularize an inﬁnite spatial ter placed inside the L3 cavity, at the antinode and integration37,38. aligned along the y-axis, according to the coordinate system shown in the schematic of Fig. 1. As seen, in- 800 800 troducing the waveguide reduces the peak enhancement by close to a factor of 2. The resonance frequencies (cid:126) 400 400 (real part) are computed to be ωL3 = 1.3052 eV and (cid:126) c ωL3W1 = 1.3049 eV, with the corresponding quality fac- c tors of QL3 = 5, 200 and QL3W1 = 3, 300. The associated 0 0 mode-volumes, calculated at the modal antinode at the 1.304 1.305 1.306 1.307 1.304 1.305 1.306 1.307 center of the L3 cavity are also VL3/λ3 = 0.0142 and eﬀ VL3W1/λ3 = 0.0145, which are found to be very similar eﬀ for this particular structure (as is often assumed in the community without proof, but in general this may not be the case, especially for low-Q cavities). In Figs. 6(c) and 6(d), we display the rQNM spatial proﬁle as calculated for both photonic crystal devices, where the in-waveguide section for the coupled device is enhanced for visualization. These two represent xy cuts at the center of the slab. It is evident that the rQNM inside the waveguide behaves diﬀerent than the waveg- uide Bloch mode and somewhat inherits the shape of the L3 mode, with repetitions occurring due to the phase dependence of the cavity QNM58. Therefore, the naive assumption that simply the system behavior inside the waveguide follows the usual W1 propagation, may not be taken. In addition, note that the details of the tran- sition for the system mode to go from a localized mode within the cavity to propagating within the waveguide, that is nontrivial, is also well captured by the calculated rQNM. FIG. 6. (a-b) Comparison between our dipole-normalized rQNM calculation of the generalized Purcell factor, F , P IV. CONCLUSIONS against full-dipole vectorial calculations of Maxwell’s equa- tions, for the isolated L3 cavity and the L3 cavity side-coupled to the waveguide, respectively. The dipole emitter is placed In summary, we have introduced a new normalization at center of the cavity and polarized along the y-axis. (c-d) technique for obtaining regularized QNMs of leaky opti- An xy cut of the calculated rQNM spatial proﬁle at the center cal cavities and plasmonic resonators, and implemented of the slab where in (d) the waveguide portion is nonlinearly it with the widely used FDTD algorithm in real fre- enhanced for better visualization. (e) The propagating Bloch quency space. Our technique requires no spatial integra- mode of a section of the inﬁnite waveguide for comparison tion for post processing normalization of the QNMs, but (see inside text for details). Each colormap is individually rather a set of easy-to-use analytical equations are pro- normalized to one. vided that only require the self-consistent response to a dipole excitation at two diﬀerent frequencies, essentially We use a triangular photonic crystal slab of refrac- exploiting an inverse Green function approach. We ﬁnd tive index of n = 3.5, where the lattice constant is d = this technique to be extremely eﬃcient on both compu- 240 nm, the hole radius is r = 0.28 d and the slab thick- tational memory and run time requirements, and easy × ness is h = 164 nm. The structure is 9 25 d in size, to use. We exempliﬁed this dipole normalization tech- and is ﬁnely meshed with 20 points per lattice period. nique for several diﬀerent arbitrarily-shaped plasmonic In particular, we consider two devices, the isolated L3 resonators, dielectric photonic crystal slabs, and hybrid cavity on its own, and the L3 cavity side-coupled to a devices in order to show its generality and applicability W1 waveguide. To minimize any relevant numerical dis- to a wide range of nanophotonic systems. In particu- crepancies, the L3 cavity is placed to the side such that lar, the spontaneous emission enhancement factor was ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: bility

text: of

text: our

text: technique

text: for

text: use

text: in

text: dielectric

text: systems

text: with

text: exactly

text: the

text: same

text: structure

text: and

text: computational

text: domain

text: is

text: large

text: quality

text: factors,

text: but

text: also

text: tackles

text: the

text: very

text: diﬃcult

text: used

text: for

text: both

text: devices;

text: see

text: Fig.

text: problem

text: of

text: normalizing

text: QNMs

text: for

text: the

text: localized

text: cavities

text: Depicted

text: in

text: Figs.

text: as

text: for

text: the

text: other

text: cavity

text: struc-

text: that

text: are

text: subjected

text: to

text: the

text: Bloch

text: periodic

text: propagation

text: of

text: tures,

text: we

text: ﬁrst

text: conﬁrm

text: a

text: very

text: good

text: agreement

text: between

text: the

text: waveguide,

text: as

text: the

text: dominant

text: outgoing

text: the

text: QNM-calculated

text: Purcell

text: factor

text: and

text: the

text: fully

text: vecto-

text: in

text: this

text: regime,

text: there

text: are

text: additional

text: complexities

text: and

text: rial

text: dipole

text: calculations;

text: here

text: we

text: consider

text: a

text: dipole

text: emit-

text: diﬃculties

text: encountered

text: to

text: regularize

text: an

text: inﬁnite

text: spatial

text: ter

text: placed

text: inside

text: the

text: cavity,

text: at

text: the

text: antinode

text: and

text: aligned

text: along

text: the

text: y-axis,

text: according

text: to

text: the

text: coordinate

text: system

text: shown

text: in

text: the

text: schematic

text: of

text: Fig.

text: As

text: seen,

text: in-

text: troducing

text: the

text: waveguide

text: reduces

text: the

text: peak

text: enhancement

text: by

text: close

text: to

text: a

text: factor

text: of

text: The

text: resonance

text: frequencies

text: (real

text: part)

text: are

text: computed

text: to

text: be

text: eV

text: and

text: c

text: eV,

text: with

text: the

text: corresponding

text: quality

text: fac-

text: c

text: tors

text: of

text: and

text: The

text: associated

text: mode-volumes,

text: calculated

text: at

text: the

text: modal

text: antinode

text: at

text: the

text: center

text: of

text: the

text: cavity

text: are

text: also

text: and

text: eﬀ

text: which

text: are

text: found

text: to

text: be

text: very

text: similar

text: eﬀ

text: for

text: this

text: particular

text: structure

text: (as

text: is

text: often

text: assumed

text: in

text: the

text: community

text: without

text: proof,

text: but

text: in

text: general

text: this

text: may

text: not

text: be

text: the

text: case,

text: especially

text: for

text: low-Q

text: cavities).

text: In

text: Figs.

text: and

text: we

text: display

text: the

text: rQNM

text: spatial

text: proﬁle

text: as

text: calculated

text: for

text: both

text: photonic

text: crystal

text: devices,

text: where

text: the

text: in-waveguide

text: section

text: for

text: the

text: coupled

text: device

text: is

text: enhanced

text: for

text: visualization.

text: These

text: two

text: represent

text: xy

text: cuts

text: at

text: the

text: center

text: of

text: the

text: slab.

text: It

text: is

text: evident

text: that

text: the

text: rQNM

text: inside

text: the

text: waveguide

text: behaves

text: diﬀerent

text: than

text: the

text: waveg-

text: uide

text: Bloch

text: mode

text: and

text: somewhat

text: inherits

text: the

text: shape

text: of

text: the

text: mode,

text: with

text: repetitions

text: occurring

text: due

text: to

text: the

text: phase

text: dependence

text: of

text: the

text: cavity

text: Therefore,

text: the

text: naive

text: assumption

text: that

text: simply

text: the

text: system

text: behavior

text: inside

text: the

text: waveguide

text: follows

text: the

text: usual

text: propagation,

text: may

text: not

text: be

text: taken.

text: In

text: addition,

text: note

text: that

text: the

text: details

text: of

text: the

text: tran-

text: sition

text: for

text: the

text: system

text: mode

text: to

text: go

text: from

text: a

text: localized

text: mode

text: within

text: the

text: cavity

text: to

text: propagating

text: within

text: the

text: waveguide,

text: that

text: is

text: nontrivial,

text: is

text: also

text: well

text: captured

text: by

text: the

text: calculated

text: rQNM.

text: FIG.

text: (a-b)

text: Comparison

text: between

text: our

text: dipole-normalized

text: rQNM

text: calculation

text: of

text: the

text: generalized

text: Purcell

text: factor,

text: F

text: P

text: IV.

text: CONCLUSIONS

text: against

text: full-dipole

text: vectorial

text: calculations

text: of

text: Maxwell’s

text: equa-

text: tions,

text: for

text: the

text: isolated

text: cavity

text: and

text: the

text: cavity

text: side-coupled

text: to

text: the

text: waveguide,

text: respectively.

text: The

text: dipole

text: emitter

text: is

text: placed

text: In

text: summary,

text: we

text: have

text: introduced

text: a

text: new

text: normalization

text: at

text: center

text: of

text: the

text: cavity

text: and

text: polarized

text: along

text: the

text: y-axis.

text: (c-d)

text: technique

text: for

text: obtaining

text: regularized

text: QNMs

text: of

text: leaky

text: opti-

text: An

text: xy

text: cut

text: of

text: the

text: calculated

text: rQNM

text: spatial

text: proﬁle

text: at

text: the

text: center

text: cal

text: cavities

text: and

text: plasmonic

text: resonators,

text: and

text: implemented

text: of

text: the

text: slab

text: where

text: in

text: (d)

text: the

text: waveguide

text: portion

text: is

text: nonlinearly

text: it

text: with

text: the

text: widely

text: used

text: FDTD

text: algorithm

text: in

text: real

text: fre-

text: enhanced

text: for

text: better

text: visualization.

text: (e)

text: The

text: propagating

text: Bloch

text: quency

text: space.

text: Our

text: technique

text: requires

text: no

text: spatial

text: integra-

text: mode

text: of

text: a

text: section

text: of

text: the

text: inﬁnite

text: waveguide

text: for

text: comparison

text: tion

text: for

text: post

text: processing

text: normalization

text: of

text: the

text: QNMs,

text: but

text: (see

text: inside

text: text

text: for

text: details).

text: Each

text: colormap

text: is

text: individually

text: rather

text: a

text: set

text: of

text: easy-to-use

text: analytical

text: equations

text: are

text: pro-

text: normalized

text: to

text: one.

text: vided

text: that

text: only

text: require

text: the

text: self-consistent

text: response

text: to

text: a

text: dipole

text: excitation

text: at

text: two

text: diﬀerent

text: frequencies,

text: essentially

text: We

text: use

text: a

text: triangular

text: photonic

text: crystal

text: slab

text: of

text: refrac-

text: exploiting

text: an

text: inverse

text: Green

text: function

text: approach.

text: We

text: ﬁnd

text: tive

text: index

text: of

text: n

text: where

text: the

text: lattice

text: constant

text: is

text: d

text: this

text: technique

text: to

text: be

text: extremely

text: eﬃcient

text: on

text: both

text: compu-

text: nm,

text: the

text: hole

text: radius

text: is

text: r

text: d

text: and

text: the

text: slab

text: thick-

text: tational

text: memory

text: and

text: run

text: time

text: requirements,

text: and

text: easy

text: ness

text: is

text: h

text: nm.

text: The

text: structure

text: is

text: d

text: in

text: size,

text: to

text: use.

text: We

text: exempliﬁed

text: this

text: dipole

text: normalization

text: tech-

text: and

text: is

text: ﬁnely

text: meshed

text: with

text: points

text: per

text: lattice

text: period.

text: nique

text: for

text: several

text: diﬀerent

text: arbitrarily-shaped

text: plasmonic

text: In

text: particular,

text: we

text: consider

text: two

text: devices,

text: the

text: isolated

text: resonators,

text: dielectric

text: photonic

text: crystal

text: slabs,

text: and

text: hybrid

text: cavity

text: on

text: its

text: own,

text: and

text: the

text: cavity

text: side-coupled

text: to

text: a

text: devices

text: in

text: order

text: to

text: show

text: its

text: generality

text: and

text: applicability

text: waveguide.

text: To

text: minimize

text: any

text: relevant

text: numerical

text: dis-

text: to

text: a

text: wide

text: range

text: of

text: nanophotonic

text: systems.

text: In

text: particu-

text: crepancies,

text: the

text: cavity

text: is

text: placed

text: to

text: the

text: side

text: such

text: that

text: lar,

text: the

text: spontaneous

text: emission

text: enhancement

text: factor

text: was

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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