text: 26 JULIAN FISCHER AND OLIVIER KNEUSS ∈ From (86), (87) and ((cid:18)88) we deduce that, for (x, y) Q , (cid:19) l − | | − | | ∇ (cid:18)1 M 1/2 + x (1 + x)( 1 + M 1/2) (cid:19) det φ(x, y) = (1 + 2τ) | | + | | ∂ φ (. . .) x M 1/2 x M 1/2 y 2 − | | − | | ≥ (cid:18)1 M 1/2 + x (1 + x)( 1 + M (cid:19)1/2) − | | (1 + 2τ) | | + | | (1 Cτ M 1/2) x M 1/2 x M 1/2 − | | − | | (1 + x)( 1 + M 1/2) ≥ (89) = (1 + 2τ) 1 Cτ M 1/2 | | 1 + τ, x M 1/2 | | ≤ | | where in the last inequality we have used the fact that we have 1 + x M 1/2 | | | | in Q (as well as the smallness condition on M and M 1/2τ). Therefore (9) and l (10) are established. ∞ Step 4.2. From (82a) for p = , (82b), and (83) we deduce (see also the beginning of Step 4) that (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) − | | | − | ≤ − 1 + M 1/2 ≤ | | (90) ∂ φ (. . .) 1 Cτ and φ (. . .) y Cτ M 1/2. y 2 2 x ∈ By (86(cid:12)(cid:12)(cid:12)) and (90) w(cid:18)e h(cid:19)av(cid:12)(cid:12)(cid:12)e for (x,(cid:12)(cid:12)(cid:12)(cid:12)y) Ql (cid:12)(cid:12)(cid:12)(cid:12) − | | − 1 ≤ 1 +(cid:12) M 1/2 (cid:12) − ∂ φ(x, y) Cτ + (cid:12)| | y(1 + x)(∂ φ (.(cid:12). .) 1) x 0 x2(cid:12)(cid:12) M 1/2 y 2 (cid:12)(cid:12) − | | 1 − 1 + M 1/2 + | | φ (. . .) y M 1/2 2 x ≤ 1 | | Cτ + Cτ + | | C M 1/2τ M 1/2 (91) =Cτ, | | ≤ | | where in the last inequality we have used the fact that 1 + x M 1/2 holds in the left quadrilateral Q . l ∈ Similar(cid:12)(cid:12)(cid:12)ly, we have b(cid:18)y (cid:19)(8(cid:12)(cid:12)(cid:12)7) a(cid:12)(cid:12)(cid:12)(cid:12)nd (90) for (x, y) Ql (cid:12)(cid:12)(cid:12)(cid:12) − | | − 0 ≤ (1 + x)(1 M 1/2) − ≤ (92) ∂ φ(x, y) − | | (∂ φ (. . .) 1) Cτ, y 1 x M 1/2 y 2 | | ≤ | | where in the last inequality we have used the fact that 1 + x M 1/2 holds in the left quadrilateral. | | Step 4.3. As the area of the left quadrilateral Q is bounded by M 1/2, (91) l and (92) establish the bound (6). Next from (89), (91) and (92) we directly get (8). It remains to show (7). This however is an easy consequence of the bound (6) ∞ for p = , the fact that on the left edge of our left quadrilateral we have φ = id, as well as the fact that any point in the left quadrilateral has distance of at most | | M 1/2 to the left edge of the quadrilateral. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: JULIAN

text: FISCHER

text: AND

text: OLIVIER

text: KNEUSS

text: From

text: and

text: we

text: deduce

text: that,

text: for

text: (x,

text: y)

text: Q

text: l

text: M

text: x

text: x)(

text: M

text: det

text: φ(x,

text: y)

text: φ

text: x

text: M

text: x

text: M

text: y

text: M

text: x

text: x)(

text: M

text: Cτ

text: M

text: x

text: M

text: x

text: M

text: x)(

text: M

text: Cτ

text: M

text: τ,

text: x

text: M

text: where

text: in

text: the

text: last

text: inequality

text: we

text: have

text: used

text: the

text: fact

text: that

text: we

text: have

text: x

text: M

text: in

text: Q

text: (as

text: well

text: as

text: the

text: smallness

text: condition

text: on

text: M

text: and

text: M

text: Therefore

text: and

text: l

text: are

text: established.

text: Step

text: From

text: for

text: p

text: and

text: we

text: deduce

text: (see

text: also

text: the

text: beginning

text: of

text: Step

text: that

text: M

text: φ

text: Cτ

text: and

text: φ

text: y

text: Cτ

text: M

text: y

text: x

text: By

text: and

text: for

text: Ql

text: M

text: φ(x,

text: y)

text: Cτ

text: x)(∂

text: φ

text: x

text: M

text: y

text: M

text: φ

text: y

text: M

text: x

text: Cτ

text: Cτ

text: C

text: M

text: M

text: =Cτ,

text: where

text: in

text: the

text: last

text: inequality

text: we

text: have

text: used

text: the

text: fact

text: that

text: x

text: M

text: holds

text: in

text: the

text: left

text: quadrilateral

text: Q

text: l

text: we

text: have

text: for

text: (x,

text: y)

text: Ql

text: M

text: φ(x,

text: y)

text: φ

text: Cτ,

text: y

text: x

text: M

text: y

text: where

text: in

text: the

text: last

text: inequality

text: we

text: have

text: used

text: the

text: fact

text: that

text: x

text: M

text: holds

text: in

text: the

text: left

text: quadrilateral.

text: Step

text: As

text: the

text: area

text: of

text: the

text: left

text: quadrilateral

text: Q

text: is

text: bounded

text: by

text: M

text: l

text: and

text: establish

text: the

text: bound

text: Next

text: from

text: and

text: we

text: directly

text: get

text: It

text: remains

text: to

text: show

text: This

text: however

text: is

text: an

text: easy

text: consequence

text: of

text: the

text: bound

text: for

text: p

text: the

text: fact

text: that

text: on

text: the

text: left

text: edge

text: of

text: our

text: left

text: quadrilateral

text: we

text: have

text: φ

text: id,

text: as

text: well

text: as

text: the

text: fact

text: that

text: any

text: point

text: in

text: the

text: left

text: quadrilateral

text: has

text: distance

text: of

text: at

text: most

text: M

text: to

text: the

text: left

text: edge

text: of

text: the

text: quadrilateral.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 30 JULIAN FISCHER AND OLIVIER KNEUSS which directly implies that − ⊂ − ∪ − [f (x) δ, f (x) + δ] [fˆ(x) δ, fˆ(x) + δ] [fˆ (x) δ, fˆ (x) + δ] i i i i i+1 i+1 and we are done. If fˆ (x) > f (x) we deduce, by deﬁnition of fˆ (x) and since ≥ i+1 i i+1 f (x) f (x), that i i+1 − − fˆ (x) = (1 + 2(N i 1))δ. i+1 Hence, proceeding by induction we end up with one of the two following cases: • there exists i < l < N such that − − ≤ ≤ ≤ − (1 + 2(N l 1))δ fˆ (x) f (x) fˆ(x) = (1 + 2(N l))δ, l+1 i l and we are done as before. • If such a l does not exist we have δ = fˆ (x) > f (x) in which case we N i trivially have ∩ − ⊂ − [0, 1] [f δ, f + δ] [fˆ (x) δ, fˆ (x) + δ], i i N N showing the claim. (cid:8) (cid:9) Step 3.3. Recalling that the family f is deﬁned as i − f (x) = min fˆ (x), 1 δ . 1 (cid:8) 1 (cid:9) ≤ ≤ and, for 2 i N, − f (x) = min fˆ(x), f − 2δ , i i i 1 (cid:91) (cid:91) it is enough, in view of Steps 3.1 and 3.2, to prove that ∩ N − ⊂ N − [0, 1] [fˆ(x) δ, fˆ(x) + δ] [f (x) δ, f (x) + δ], i i i i i=1 i=1 in order to show (94) and prove the lemma. First we trivially have ∩ − ⊂ − [0, 1] [fˆ (x) δ, fˆ (x) + δ] [f (x) δ, f (x) + δ] 1 1 1 1 ≥ (cid:54) Finally, suppose that for some i 2 we have f (x) = fˆ(x) (otherwise the claim is i i trivial) and thus, by deﬁnition, − fˆ(x) > f (x) = f − 2δ. i i i 1 ∈ If fˆ(x) [f (x), f − (x)] then the previous equality directly implies that i i i 1 − ⊂ − ∪ − [fˆ(x) δ, fˆ(x) + δ] [f (x) δ, f (x) + δ] [f − (x) δ, f − (x) + δ] i i i i i 1 i 1 { − } and we are done. If fˆ(x) > f − (x) = min fˆ− (x), f − 2δ we deduce that, ≥ i i 1 i 1 i 2 since fˆ− (x) fˆ(x), i 1 i − f − (x) = f − 2δ. i 1 i 2 Hence proceeding by induction we deduce that one of the cases below occur: • ≥ ∈ there exists some i > l 2 such that fˆ(x) [f (x), f − (x)] and f (x) = − i l l 1 l f − (x) 2δ and we done as above. l 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: JULIAN

text: FISCHER

text: AND

text: OLIVIER

text: KNEUSS

text: which

text: directly

text: implies

text: that

text: [f

text: (x)

text: δ,

text: f

text: (x)

text: δ]

text: [fˆ(x)

text: δ,

text: fˆ(x)

text: δ]

text: [fˆ

text: (x)

text: δ,

text: fˆ

text: (x)

text: δ]

text: i

text: i

text: i

text: i

text: and

text: we

text: are

text: done.

text: If

text: fˆ

text: (x)

text: f

text: (x)

text: we

text: deduce,

text: by

text: deﬁnition

text: of

text: fˆ

text: (x)

text: and

text: since

text: i

text: f

text: (x)

text: f

text: (x),

text: that

text: i

text: fˆ

text: (x)

text: i

text: Hence,

text: proceeding

text: by

text: induction

text: we

text: end

text: up

text: with

text: one

text: of

text: the

text: two

text: following

text: cases:

label: list

text: there

text: exists

text: i

text: l

text: N

text: such

text: that

text: l

text: fˆ

text: (x)

text: f

text: (x)

text: fˆ(x)

text: l))δ,

text: i

text: l

text: and

text: we

text: are

text: done

text: as

text: before.

text: If

text: such

text: a

text: l

text: does

text: not

text: exist

text: we

text: have

text: δ

text: fˆ

text: (x)

text: f

text: (x)

text: in

text: which

text: case

text: we

text: N

text: i

text: trivially

text: have

text: [f

text: δ,

text: f

text: δ]

text: [fˆ

text: (x)

text: δ,

text: fˆ

text: (x)

text: δ],

text: i

text: i

text: N

text: N

text: showing

text: the

text: claim.

text: Step

text: Recalling

text: that

text: the

text: family

text: f

text: is

text: deﬁned

text: as

text: i

text: f

text: (x)

text: min

text: fˆ

text: (x),

text: δ

text: and,

text: for

text: i

text: N,

text: f

text: (x)

text: min

text: fˆ(x),

text: f

text: 2δ

text: i

text: i

text: i

text: it

text: is

text: enough,

text: in

text: view

text: of

text: Steps

text: and

text: to

text: prove

text: that

text: N

text: N

text: [fˆ(x)

text: δ,

text: fˆ(x)

text: δ]

text: [f

text: (x)

text: δ,

text: f

text: (x)

text: δ],

text: i

text: i

text: i

text: i

text: in

text: order

text: to

text: show

text: and

text: prove

text: the

text: lemma.

text: First

text: we

text: trivially

text: have

text: [fˆ

text: (x)

text: δ,

text: fˆ

text: (x)

text: δ]

text: [f

text: (x)

text: δ,

text: f

text: (x)

text: δ]

text: Finally,

text: suppose

text: that

text: for

text: some

text: i

text: we

text: have

text: f

text: (x)

text: fˆ(x)

text: (otherwise

text: the

text: claim

text: is

text: i

text: i

text: trivial)

text: and

text: thus,

text: by

text: deﬁnition,

text: fˆ(x)

text: f

text: (x)

text: f

text: i

text: i

text: i

text: If

text: fˆ(x)

text: [f

text: (x),

text: f

text: (x)]

text: then

text: the

text: previous

text: equality

text: directly

text: implies

text: that

text: i

text: i

text: i

text: [fˆ(x)

text: δ,

text: fˆ(x)

text: δ]

text: [f

text: (x)

text: δ,

text: f

text: (x)

text: δ]

text: [f

text: (x)

text: δ,

text: f

text: (x)

text: δ]

text: i

text: i

text: i

text: i

text: i

text: i

text: and

text: we

text: are

text: done.

text: If

text: fˆ(x)

text: f

text: (x)

text: min

text: fˆ−

text: (x),

text: f

text: 2δ

text: we

text: deduce

text: that,

text: i

text: i

text: i

text: i

text: since

text: fˆ−

text: (x)

text: fˆ(x),

text: i

text: i

text: f

text: (x)

text: f

text: i

text: i

text: Hence

text: proceeding

text: by

text: induction

text: we

text: deduce

text: that

text: one

text: of

text: the

text: cases

text: below

text: occur:

text: there

text: exists

text: some

text: i

text: l

text: such

text: that

text: fˆ(x)

text: [f

text: (x),

text: f

text: (x)]

text: and

text: f

text: (x)

text: i

text: l

text: l

text: l

text: f

text: (x)

text: 2δ

text: and

text: we

text: done

text: as

text: above.

text: l

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: THE PRESCRIBED JACOBIAN INEQUALITY WITH Lp DATA 35 In connection with the analogous estimates for φ , (100) and (103) imply (73). 1 Step 4. In this step we prove the three assertions involving the determinant, ∈ namely (75)-(77). For a. e. (x, y) [0, 1]2 we have (cid:32) (cid:33) (cid:80) (cid:80) (cid:80) (cid:80) − − (cid:48) ∇ 1 4τδM + 2τ M χ{| − | } 2τ M g (y)χ{| − | } φ(x, y) = − (cid:48) j=1 gj(y) x <δ − j=1 j gj(y) x <δ . 2τ N f (x)χ{| − | } 1 4τδN + 2τ N χ{| − | } i=1 i fi(x) y <δ i=1 fi(x) y <δ Denote by S the union of the interior of all the strips, i.e. (cid:0) (cid:1) (cid:0) (cid:1) ∪ {| − | } ∪ ∪ {| − | } S := N f (x) y < δ M g (y) x < δ . i=1 i j=1 j Then, obviously, a. e. outside S we have (cid:18) (cid:19) − ∇ 1 4δMτ 0 φ = − . 0 1 4δNτ (cid:112) ≤ | | Thus, recalling that Mδ, Nδ 2 K (see (95)), (cid:112) ∇ − ≥ − | | \ (104) det φ = 1 4τ(δM +δN)+16δ2MNτ2 1 16τ K a. e. in [0, 1]2 S. | | ≤ Furthermore, assuming τ K 1/2 1/32, we have |∇ | ≤ ∇ \ (105) φ 2 det φ a. e. in [0, 1]2 S. ∈ Trivially for all (x, y) S we have (cid:88) (cid:88) ≤ N M 1 χ{| − | } + χ{| − | }. fi(x) y <δ gj(y) x <δ i=1 j=1 ##LTLine## ##LTLine##

text: THE

text: PRESCRIBED

text: JACOBIAN

text: INEQUALITY

text: WITH

text: Lp

text: DATA

text: In

text: connection

text: with

text: the

text: analogous

text: estimates

text: for

text: φ

text: and

text: imply

text: Step

text: In

text: this

text: step

text: we

text: prove

text: the

text: three

text: assertions

text: involving

text: the

text: determinant,

text: namely

text: For

text: a.

text: e.

text: (x,

text: y)

text: we

text: have

text: 4τδM

text: 2τ

text: M

text: χ{|

text: 2τ

text: M

text: g

text: (y)χ{|

text: φ(x,

text: y)

text: gj(y)

text: x

text: <δ

text: j

text: gj(y)

text: x

text: <δ

text: 2τ

text: N

text: f

text: (x)χ{|

text: 4τδN

text: 2τ

text: N

text: χ{|

text: i

text: fi(x)

text: y

text: <δ

text: fi(x)

text: y

text: <δ

text: Denote

text: by

text: S

text: the

text: union

text: of

text: the

text: interior

text: of

text: all

text: the

text: strips,

text: i.e.

text: S

text: N

text: f

text: (x)

text: y

text: δ

text: M

text: g

text: (y)

text: x

text: δ

text: i

text: j

text: Then,

text: obviously,

text: a.

text: e.

text: outside

text: S

text: we

text: have

text: 4δMτ

text: φ

text: 4δNτ

text: Thus,

text: recalling

text: that

text: Mδ,

text: Nδ

text: K

text: (see

text: det

text: φ

text: K

text: a.

text: e.

text: in

text: S.

text: Furthermore,

text: assuming

text: τ

text: K

text: we

text: have

text: φ

text: det

text: φ

text: a.

text: e.

text: in

text: S.

text: Trivially

text: for

text: all

text: (x,

text: y)

text: S

text: we

text: have

text: N

text: M

text: χ{|

text: χ{|

text: fi(x)

text: y

text: <δ

text: gj(y)

text: x

text: <δ

text: ##LTLine##

text: ##LTLine##


text: THE PRESCRIBED JACOBIAN INEQUALITY WITH Lp DATA 37 Appendix A. Decomposition of C1,1-domains into subdomains that are C1,1-equivalent to the unit square The following lemma has been used in Section . ⊂ R Lemma 16. Let Ω 2 be a bounded domain with boundary of class C1,1. Then ⊂ there exists a ﬁnite number of open sets A Ω such that: • i For every A there exists a C1,1-diﬀeomorphism which maps A to the unit i (cid:83) i square [0, 1]2. • The sets A are pairwise disjoint. • i We have Ω = A . i i Proof. We ﬁrst summarize the proof. • First we reduce the problem to the case of a polygon: We approximate Ω from inside by a polygon whose boundary follows closely the boundary ∂Ω (in a sense to be made explicit below). The outer part may then be decomposed as shown in Figure 3. The outer quadrilaterals (which have one curved side) are easily mapped by a C1,1-diﬀeomorphism to the unit square. It then just remains to treat the case of the (inner) polygon (which possibly contains holes). • Any polygon (including polygons with holes) may be decomposed into a ﬁnite number of triangles, so the problem is reduced to the case of a triangle. Since every triangle may be mapped by an aﬃne map to the reference triangle with corners (0, 0), (1, 0), and (0, 1), we only need to consider the case of the reference triangle. • The reference triangle may be covered by three convex quadrilaterals as shown in Figure 4. Applying Lemma 18, each of these quadrilaterals can ∞ be mapped by a (bi-linear) C -diﬀeomorphism to the unit square. It only remains to make explicit the ﬁrst point. We split its proof into three steps. R → R Step 1. We can ﬁnd (see e. g. Theorem 1.2.6 in [13]) a function F : 2 with the following properties: • F is of class C1,1, • { } { } we have F = 0 = ∂Ω and F > 0 = Ω, • ∇ F is nonzero on ∂Ω. Take δ > 0 small enough and construct a closed polygon P whose boundary is { } { } contained in the set δ/2 < F < 2δ , whose vertices lie on the curve F = δ , and whose edges have length of order δ. Note that for δ small enough, such a polygon exists. Connect every vertex of P to the boundary of ∂Ω with a line segment −∇ pointing in direction F (the green line segments in Figure 3). Let us denote the open regions which are bounded by the polygon P on one side and these line segments and ∂Ω on the other sides as B (see Figure 3). We will show in the i remaining two steps that every B is C1,1-diﬀeomorphic to the unit square [0, 1]2. i ∈ Step 2. Fix some i and let z , z , w , w be the four ”vertices” of B with z , z { } ∈ 1 2 1 2 i 1 2 F = δ and w , w ∂Ω ordered such that 1 2 ∪ ∪ [z , z ] [z , w ] [z , w ] 1 2 1 1 2 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: THE

text: PRESCRIBED

text: JACOBIAN

text: INEQUALITY

text: WITH

text: Lp

text: DATA

text: Appendix

label: section

text: A.

text: Decomposition

text: of

text: into

text: subdomains

text: that

text: are

text: to

text: the

text: unit

text: square

text: The

text: following

text: lemma

text: has

text: been

text: used

text: in

text: Section

text: R

text: Lemma

text: Let

text: Ω

text: be

text: a

text: bounded

text: domain

text: with

text: boundary

text: of

text: class

text: Then

text: there

text: exists

text: a

text: ﬁnite

text: number

text: of

text: open

text: sets

text: A

text: Ω

text: such

text: that:

text: i

text: For

text: every

text: A

text: there

text: exists

text: a

text: which

text: maps

text: A

text: to

text: the

text: unit

text: i

text: i

text: square

text: The

text: sets

text: A

text: are

text: pairwise

text: disjoint.

text: i

text: We

text: have

text: Ω

text: A

text: i

text: i

text: Proof.

text: We

text: ﬁrst

text: summarize

text: the

text: proof.

text: First

text: we

text: reduce

text: the

text: problem

text: to

text: the

text: case

text: of

text: a

text: polygon:

text: We

text: approximate

text: Ω

text: from

text: inside

text: by

text: a

text: polygon

text: whose

text: boundary

text: follows

text: closely

text: the

text: boundary

text: ∂Ω

text: (in

text: a

text: sense

text: to

text: be

text: made

text: explicit

text: below).

text: The

text: outer

text: part

text: may

text: then

text: be

text: decomposed

text: as

text: shown

text: in

text: Figure

text: The

text: outer

text: quadrilaterals

text: (which

text: have

text: one

text: curved

text: side)

text: are

text: easily

text: mapped

text: by

text: a

text: to

text: the

text: unit

text: square.

text: It

text: then

text: just

text: remains

text: to

text: treat

text: the

text: case

text: of

text: the

text: (inner)

text: polygon

text: (which

text: possibly

text: contains

text: holes).

text: Any

text: polygon

text: (including

text: polygons

text: with

text: holes)

text: may

text: be

text: decomposed

text: into

text: a

text: ﬁnite

text: number

text: of

text: triangles,

text: so

text: the

text: problem

text: is

text: reduced

text: to

text: the

text: case

text: of

text: a

text: triangle.

text: Since

text: every

text: triangle

text: may

text: be

text: mapped

text: by

text: an

text: aﬃne

text: map

text: to

text: the

text: reference

text: triangle

text: with

text: corners

text: and

text: we

text: only

text: need

text: to

text: consider

text: the

text: case

text: of

text: the

text: reference

text: triangle.

text: The

text: reference

text: triangle

text: may

text: be

text: covered

text: by

text: three

text: convex

text: quadrilaterals

text: as

text: shown

text: in

text: Figure

text: Applying

text: Lemma

text: each

text: of

text: these

text: quadrilaterals

text: can

text: be

text: mapped

text: by

text: a

text: (bi-linear)

text: C

text: -diﬀeomorphism

text: to

text: the

text: unit

text: square.

text: It

text: only

text: remains

text: to

text: make

text: explicit

text: the

text: ﬁrst

text: point.

text: We

text: split

text: its

text: proof

text: into

text: three

text: steps.

text: R

text: R

text: Step

text: We

text: can

text: ﬁnd

text: (see

text: e.

text: g.

text: Theorem

text: in

text: a

text: function

text: F

text: with

text: the

text: following

text: properties:

text: F

text: is

text: of

text: class

text: we

text: have

text: F

text: ∂Ω

text: and

text: F

text: Ω,

text: F

text: is

text: nonzero

text: on

text: ∂Ω.

text: Take

text: δ

text: small

text: enough

text: and

text: construct

text: a

text: closed

text: polygon

text: P

text: whose

text: boundary

text: is

text: contained

text: in

text: the

text: set

text: F

text: 2δ

text: whose

text: vertices

text: lie

text: on

text: the

text: curve

text: F

text: δ

text: and

text: whose

text: edges

text: have

text: length

text: of

text: order

text: δ.

text: Note

text: that

text: for

text: δ

text: small

text: enough,

text: such

text: a

text: polygon

text: exists.

text: Connect

text: every

text: vertex

text: of

text: P

text: to

text: the

text: boundary

text: of

text: ∂Ω

text: with

text: a

text: line

text: segment

text: pointing

text: in

text: direction

text: F

text: (the

text: green

text: line

text: segments

text: in

text: Figure

text: Let

text: us

text: denote

text: the

text: open

text: regions

text: which

text: are

text: bounded

text: by

text: the

text: polygon

text: P

text: on

text: one

text: side

text: and

text: these

text: line

text: segments

text: and

text: ∂Ω

text: on

text: the

text: other

text: sides

text: as

text: B

text: (see

text: Figure

text: We

text: will

text: show

text: in

text: the

text: i

text: remaining

text: two

text: steps

text: that

text: every

text: B

text: is

text: to

text: the

text: unit

text: square

text: i

text: Step

text: Fix

text: some

text: i

text: and

text: let

text: z

text: z

text: w

text: w

text: be

text: the

text: four

text: ”vertices”

text: of

text: B

text: with

text: z

text: z

text: i

text: F

text: δ

text: and

text: w

text: w

text: ∂Ω

text: ordered

text: such

text: that

text: [z

text: z

text: [z

text: w

text: [z

text: w

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 40 JULIAN FISCHER AND OLIVIER KNEUSS References [1] Giovanni Alberti, Marianna Cso¨rnyei, and David Preiss. Structure of null sets in the plane and applications. In 4ECM Stockholm 2004, pages 3–22. European Mathematical Society, 2005. [2] Albert Aviny´o, Joan Sol`a-Morales, and Marta Val`encia. A singular initial value problem to construct density-equalizing maps. Journal of Dynamics and Diﬀerential Equations, 24(1):51–59, 2012. [3] John M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63(4):337–403, 1976/77. [4] John M. Ball. Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinburgh Sect. A, 88(3-4):315–328, 1981. [5] Dmitri Burago and Bruce Kleiner. Separated nets in Euclidean space and Jacobians of bi- Lipschitz maps. Geom. Funct. Anal., 8(2):273–282, 1998. [6] Guillaume Carlier and Bernard Dacorogna. R´esolution du probl`eme de Dirichlet pour l’´equation du jacobien prescrit via l’´equation de Monge-Amp`ere. C. R. Math. Acad. Sci. Paris, 350(7-8):371–374, 2012. [7] Gyula Csat´o, Bernard Dacorogna, and Olivier Kneuss. The pullback equation for dif- ferential forms. Progress in Nonlinear Diﬀerential Equations and their Applications, 83. Birkh¨auser/Springer, New York, 2012. [8] Bernard Dacorogna. Direct methods in the calculus of variations, volume 78 of Applied Math- ematical Sciences. Springer, New York, second edition, 2008. [9] Bernard Dacorogna and Ju¨rgen Moser. On a partial diﬀerential equation involving the Jaco- bian determinant. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 7(1):1–26, 1990. [10] Camillo De Lellis. Some ﬁne properties of currents and applications to distributional Jaco- bians. Proc. Roy. Soc. Edinburgh Sect. A, 132(4):815–842, 2002. [11] Guido De Philippis. Weak notions of Jacobian determinant and relaxation. ESAIM Control Optim. Calc. Var., 18(1):181–207, 2012. [12] Konstantinos Koumatos, Filip Rindler, and Emil Wiedemann. Diﬀerential inclusions and Young measures involving prescribed Jacobians. SIAM J. Math. Anal., 47(2):1169–1195, 2015. [13] Steven G. Krantz and Harold R. Parks. The geometry of domains in space. Birkh¨auser Ad- vanced Texts: Basler Lehrbu¨cher. [Birkh¨auser Advanced Texts: Basel Textbooks]. Birkha¨user Boston, Inc., Boston, MA, 1999. [14] Andrea Marchese. Two applications of the theory of currents. PhD Thesis, 2013. [15] Curtis T. McMullen. Lipschitz maps and nets in Euclidean space. Geom. Funct. Anal., 8(2):304–314, 1998. [16] Tristan Rivi`ere and Dong Ye. Resolutions of the prescribed volume form equation. NoDEA Nonlinear Diﬀerential Equations Appl., 3(3):323–369, 1996. [17] Dong Ye. Prescribing the Jacobian determinant in Sobolev spaces. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 11(3):275–296, 1994. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany, E-Mail: julian.fischer@mis.mpg.de Institute of Mathematics, Federal University of Rio de Janeiro, Cidade Universi- taria, 21941909 Rio de Janeiro, Brazil, E-Mail: olivier.kneuss@gmail.com

text: JULIAN

text: FISCHER

text: AND

text: OLIVIER

text: KNEUSS

text: References

label: reference

text: Giovanni

text: Alberti,

text: Marianna

text: Cso¨rnyei,

text: and

text: David

text: Preiss.

text: Structure

text: of

text: sets

text: in

text: the

text: plane

text: and

text: applications.

text: In

text: Stockholm

text: pages

text: European

text: Mathematical

text: Society,

text: Albert

text: Aviny´o,

text: Joan

text: Sol`a-Morales,

text: and

text: Marta

text: Val`encia.

text: A

text: singular

text: initial

text: value

text: problem

text: to

text: construct

text: density-equalizing

text: maps.

text: Journal

text: of

text: Dynamics

text: and

text: Diﬀerential

text: Equations,

text: John

text: M.

text: Ball.

text: Convexity

text: conditions

text: and

text: existence

text: theorems

text: in

text: nonlinear

text: elasticity.

text: Arch.

text: Rational

text: Mech.

text: Anal.,

text: John

text: M.

text: Ball.

text: Global

text: invertibility

text: of

text: Sobolev

text: functions

text: and

text: the

text: interpenetration

text: of

text: matter.

text: Proc.

text: Roy.

text: Soc.

text: Edinburgh

text: Sect.

text: A,

text: Dmitri

text: Burago

text: and

text: Bruce

text: Kleiner.

text: Separated

text: nets

text: in

text: Euclidean

text: space

text: and

text: Jacobians

text: of

text: bi-

text: Lipschitz

text: maps.

text: Geom.

text: Funct.

text: Anal.,

text: Guillaume

text: Carlier

text: and

text: Bernard

text: Dacorogna.

text: R´esolution

text: du

text: probl`eme

text: de

text: Dirichlet

text: pour

text: l’´equation

text: du

text: jacobien

text: prescrit

text: via

text: l’´equation

text: de

text: Monge-Amp`ere.

text: C.

text: R.

text: Math.

text: Acad.

text: Sci.

text: Paris,

text: Gyula

text: Csat´o,

text: Bernard

text: Dacorogna,

text: and

text: Olivier

text: Kneuss.

text: The

text: pullback

text: equation

text: for

text: dif-

text: ferential

text: forms.

text: Progress

text: in

text: Nonlinear

text: Diﬀerential

text: Equations

text: and

text: their

text: Applications,

text: Birkh¨auser/Springer,

text: New

text: York,

text: Bernard

text: Dacorogna.

text: Direct

text: methods

text: in

text: the

text: calculus

text: of

text: variations,

text: volume

text: of

text: Applied

text: Math-

text: ematical

text: Sciences.

text: Springer,

text: New

text: York,

text: second

text: edition,

text: Bernard

text: Dacorogna

text: and

text: Ju¨rgen

text: Moser.

text: On

text: a

text: partial

text: diﬀerential

text: equation

text: involving

text: the

text: Jaco-

text: bian

text: determinant.

text: Ann.

text: Inst.

text: H.

text: Poincar´e

text: Anal.

text: Non

text: Lin´eaire,

text: Camillo

text: De

text: Lellis.

text: Some

text: ﬁne

text: properties

text: of

text: currents

text: and

text: applications

text: to

text: distributional

text: Jaco-

text: bians.

text: Proc.

text: Roy.

text: Soc.

text: Edinburgh

text: Sect.

text: A,

text: Guido

text: De

text: Philippis.

text: Weak

text: notions

text: of

text: Jacobian

text: determinant

text: and

text: relaxation.

text: ESAIM

text: Control

text: Optim.

text: Calc.

text: Var.,

text: Konstantinos

text: Koumatos,

text: Filip

text: Rindler,

text: and

text: Emil

text: Wiedemann.

text: Diﬀerential

text: inclusions

text: and

text: Young

text: measures

text: involving

text: prescribed

text: Jacobians.

text: SIAM

text: J.

text: Math.

text: Anal.,

text: Steven

text: G.

text: Krantz

text: and

text: Harold

text: R.

text: Parks.

text: The

text: geometry

text: of

text: domains

text: in

text: space.

text: Birkh¨auser

text: Ad-

text: vanced

text: Texts:

text: Basler

text: Lehrbu¨cher.

text: [Birkh¨auser

text: Advanced

text: Texts:

text: Basel

text: Textbooks].

text: Birkha¨user

text: Boston,

text: Inc.,

text: Boston,

text: MA,

text: Andrea

text: Marchese.

text: Two

text: applications

text: of

text: the

text: theory

text: of

text: currents.

text: PhD

text: Thesis,

text: Curtis

text: T.

text: McMullen.

text: Lipschitz

text: maps

text: and

text: nets

text: in

text: Euclidean

text: space.

text: Geom.

text: Funct.

text: Anal.,

text: Tristan

text: Rivi`ere

text: and

text: Dong

text: Ye.

text: Resolutions

text: of

text: the

text: prescribed

text: volume

text: form

text: equation.

text: NoDEA

text: Nonlinear

text: Diﬀerential

text: Equations

text: Appl.,

text: Dong

text: Ye.

text: Prescribing

text: the

text: Jacobian

text: determinant

text: in

text: Sobolev

text: spaces.

text: Ann.

text: Inst.

text: H.

text: Poincar´e

text: Anal.

text: Non

text: Lin´eaire,

text: Max

text: Planck

text: Institute

text: for

text: Mathematics

text: in

text: the

text: Sciences,

text: Inselstrasse

text: Leipzig,

text: Germany,

text: E-Mail:

text: julian.fischer@mis.mpg.de

text: Institute

text: of

text: Mathematics,

text: Federal

text: University

text: of

text: Rio

text: de

text: Janeiro,

text: Cidade

text: Universi-

text: taria,

text: Rio

text: de

text: Janeiro,

text: Brazil,

text: E-Mail:

text: olivier.kneuss@gmail.com


text: 10 (ω/γ ) H a res (Anti)binding mode Transverse anisotropy 2.5 ξ = 0.2 ξ = 0.1 2 ξ = 0 1.5 1 0.5 0 0 0.5 1 h = k + 3 ξ /2 h = H/H a c Figure 4: Resonance frequency of the binding mode (continuous lines) and the anti-binding mode (dashed lines) as a function of the magnetic ﬁeld and varying dipolar interlayer coupling, with transverse anisotropy. The issue of binding and anti-binding modes and their comparison is rather involved and requires a thorough analysis. For example, a more precise investigation of these modes and their dynamics can be achieved by computing (both analytically and numerically) the time evolution of the two magnetic moments on- and off- resonance upon varying the applied ﬁeld and the coupling. In order not to make the present article too bulky, with the risk of drowning its main message, in the sequel we restrict our discussion to the binding modes and their characteristics, leaving the analysis of anti-binding modes for a separate work. In Fig. 5 we show the results for the resonance ﬁeld for the same setup as in Fig. 4, for a ﬁxed frequency ω˜. The valley and the peak correspond to the easy and hard directions, respectively. We see that as the DI increases the effect of the easy direction is enhanced leading to a wider valley and a lower resonance ﬁeld. The effect of the hard direction is also enhanced but affects a narrower range of ﬁeld directions around θ = π/2. In summary, as the DI increases the easy axis range widens whereas that of the hard axis shrinks. The fact that the amplitude of the resonance ﬁeld globally decreases when the DI increases is simply due to the fact that with a stronger effective ﬁeld, a smaller magnetic ﬁeld is needed to satisfy the resonance condition. 2. Longitudinal anisotropy (LA) The magnetic ﬁeld is still along the x axis (θ = π/2) but now the anisotropy axes are parallel to it but h perpendicular to the MD bond e [see Fig. 3 right]. In this case, the DI and uniaxial anisotropy contributions 12 induce different preferred directions for the two magnetic moments and thereby they are in competition with (cid:0) (cid:1) each other. Therefore, the two magnetic moments may be parallel or anti-parallel to each other and lie in the E − − xz plane. Denoting by θ the polar angle they make to the z axis, the MD energy then reads = 4h sin θ − ± sin2 θ + ξ 1 3 cos2 θ and upon minimizing it with respect to θ we obtain the following minima: θ = π/2 or − sin θ = 2h/ (3ξ 2k). Hence, we have the following three distinct states with their respective energies π E − − θ = = θ , = 2h 2k + ξ, (21a) 1 2 1 2 π − E − − θ = = θ , = 2k ξ, (21b) 1 2 2 2 ##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##

font: VOMPGR+CenturySchL-Roma

text: (ω/γ

font: default

text: H

text: a

text: res

text: (Anti)binding

text: mode

text: Transverse

text: anisotropy

text: ξ

text: ξ

text: ξ

text: h

text: k

text: ξ

text: h

text: H/H

text: a

text: c

text: Figure

text: Resonance

label: caption

text: frequency

text: of

text: the

text: binding

text: mode

text: (continuous

text: lines)

text: and

text: the

text: anti-binding

text: mode

text: (dashed

text: lines)

text: as

text: a

text: function

text: of

text: the

text: magnetic

text: ﬁeld

text: and

text: varying

text: dipolar

text: interlayer

text: coupling,

text: with

text: transverse

text: anisotropy.

text: The

text: issue

text: of

text: binding

text: and

text: anti-binding

text: modes

text: and

text: their

text: comparison

text: is

text: rather

text: involved

text: and

text: requires

text: a

text: thorough

text: analysis.

text: For

text: example,

text: a

text: more

text: precise

text: investigation

text: of

text: these

text: modes

text: and

text: their

text: dynamics

text: can

text: be

text: achieved

text: by

text: computing

text: (both

text: analytically

text: and

text: numerically)

text: the

text: time

text: evolution

text: of

text: the

text: two

text: magnetic

text: moments

text: on-

text: and

text: off-

text: resonance

text: upon

text: varying

text: the

text: applied

text: ﬁeld

text: and

text: the

text: coupling.

text: In

text: order

text: not

text: to

text: make

text: the

text: present

text: article

text: too

text: bulky,

text: with

text: the

text: risk

text: of

text: drowning

text: its

text: main

text: message,

text: in

text: the

text: sequel

text: we

text: restrict

text: our

text: discussion

text: to

text: the

text: binding

text: modes

text: and

text: their

text: characteristics,

text: leaving

text: the

text: analysis

text: of

text: anti-binding

text: modes

text: for

text: a

text: separate

text: work.

text: In

text: Fig.

text: we

text: show

text: the

text: results

text: for

text: the

text: resonance

text: ﬁeld

text: for

text: the

text: same

text: setup

text: as

text: in

text: Fig.

text: for

text: a

text: ﬁxed

text: frequency

text: ω˜.

text: The

text: valley

text: and

text: the

text: peak

text: correspond

text: to

text: the

text: easy

text: and

text: hard

text: directions,

text: respectively.

text: We

text: see

text: that

text: as

text: the

text: DI

text: increases

text: the

text: effect

text: of

text: the

text: easy

text: direction

text: is

text: enhanced

text: leading

text: to

text: a

text: wider

text: valley

text: and

text: a

text: lower

text: resonance

text: ﬁeld.

text: The

text: effect

text: of

text: the

text: hard

text: direction

text: is

text: also

text: enhanced

text: but

text: affects

text: a

text: narrower

text: range

text: of

text: ﬁeld

text: directions

text: around

text: θ

text: In

text: summary,

text: as

text: the

text: DI

text: increases

text: the

text: easy

text: axis

text: range

text: widens

text: whereas

text: that

text: of

text: the

text: hard

text: axis

text: shrinks.

text: The

text: fact

text: that

text: the

text: amplitude

text: of

text: the

text: resonance

text: ﬁeld

text: globally

text: decreases

text: when

text: the

text: DI

text: increases

text: is

text: simply

text: due

text: to

text: the

text: fact

text: that

text: with

text: a

text: stronger

text: effective

text: ﬁeld,

text: a

text: smaller

text: magnetic

text: ﬁeld

text: is

text: needed

text: to

text: satisfy

text: the

text: resonance

text: condition.

text: Longitudinal

text: anisotropy

text: (LA)

text: The

text: magnetic

text: ﬁeld

text: is

text: still

text: along

text: the

text: x

text: axis

text: (θ

text: but

text: now

text: the

text: anisotropy

text: axes

text: are

text: parallel

text: to

text: it

text: but

text: h

text: perpendicular

text: to

text: the

text: MD

text: bond

text: e

text: [see

text: Fig.

text: right].

text: In

text: this

text: case,

text: the

text: DI

text: and

text: uniaxial

text: anisotropy

text: contributions

text: induce

text: different

text: preferred

text: directions

text: for

text: the

text: two

text: magnetic

text: moments

text: and

text: thereby

text: they

text: are

text: in

text: competition

text: with

text: each

text: other.

text: Therefore,

text: the

text: two

text: magnetic

text: moments

text: may

text: be

text: parallel

text: or

text: anti-parallel

text: to

text: each

text: other

text: and

text: lie

text: in

text: the

text: E

text: xz

text: plane.

text: Denoting

text: by

text: θ

text: the

text: polar

text: angle

text: they

text: make

text: to

text: the

text: z

text: axis,

text: the

text: MD

text: energy

text: then

text: reads

text: sin

text: θ

text: θ

text: ξ

text: θ

text: and

text: upon

text: minimizing

text: it

text: with

text: respect

text: to

text: θ

text: we

text: obtain

text: the

text: following

text: minima:

text: θ

text: or

text: sin

text: θ

text: Hence,

text: we

text: have

text: the

text: following

text: three

text: distinct

text: states

text: with

text: their

text: respective

text: energies

text: π

text: E

text: θ

text: θ

text: ξ,

text: π

text: E

text: θ

text: θ

text: ξ,

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text: 15 VI. CONCLUSIONS AND OUTLOOK We have presented in this paper the explicit expressions of several lensing quantities contained in the ampliﬁcation matrix (magniﬁcation, convergence and shear) and the deformation matrix (optical scalars). These results were obtained by the use of the Jacobi map expressed ﬁrst through the zweibeins of the Sachs basis and then within the geodesic light-cone coordinates. We have shown that some particular combinations of the lensing quantities take an elegant expression in the GLC gauge and the explicit GLC-form of the zweibeins is not needed (though it was given for completeness in Appendix ). These expressions are general and can be applied to any inhomogeneous model of Universe as long as no caustic forms on the past light-cone of the observer. We have also seen through these computations the interest of the GLC gauge in dealing with light propagation, emphasizing that the geodesic equation being trivial in this set of coordinates also simpliﬁes the handling of lensing variables. Let us comment brieﬂy on the presence of caustics. In this situation one sees the creation of critical points or lines on a given sphere Σ(w, τ) embedded in the past light-cone. These singularities appear when det JA is equal to zero. B In such a case the Jacobi map is not an invertible matrix and complicated situations with multiple images and inﬁnite ≡ magniﬁcation can be observed. We can also see (from Eqs. (6), (27) and (28)) that this happens when γ det γ = 0 ab in the GLC gauge and we then have a case of strong lensing. The estimation of the angular distance is not anymore valid on these critical lines [44] and a more involved system of coordinates is necessary. The generalisation of the GLC gauge to that case appears as a hard task, despite its simplifying properties in the case of weak lensing. In the second part of this paper we have applied our general formulas to the well-known case of a LTB metric with an oﬀ-center observer. These relations, especially the lensing quantities, were obtained directly after expressing the non-perturbative transformation between LTB and GLC coordinates. They are general and to our knowledge the most explicit expressions presented up to now and they could be used for a broad range of applications, in particular for the exact treatment of light propagation in inhomogeneous universes. We then restricted our study to an LTB over/underdensity (or “bubble”) entirely determined by its local Hubble factor within CDM and ΛCDM backgrounds. We presented the variations of the magniﬁcation µ and the distance modulus diﬀerence ∆m for an over/underdense bubble, pointing at the center of it. We found the correction induced on the magniﬁcation by the bubble to be very ∼ ∼ small at short distances ( 0.015% for d = 10 Mpc) at more signiﬁcant at large distances ( 1.5% for d = 1000 Mpc) in the ΛCDM case as well as CDM (the diﬀerence being negligible for such small under/overdensities). We also showed the consistency of our results by presenting the evolution of the quantities, including the shear and the convergence, when we point at diﬀerent angles from the center of the bubble. We ﬁnally checked that our quantities are well-behaved and that no divergence appears. To ﬁnish, this work improves our understanding of lensing theory and presents some of its most important quantities in an elegant framework. It also illustrates – once again – the general usefulness of the GLC gauge for deriving theoretical expressions as well as working out with more applied situations. We illustrated our results through the LTB model but we can imagine other applications such as, for example, the computation of the lensing quantities in an anisotropic Bianchi model or at second order in perturbations in the Poisson gauge. Lensing beneﬁts from a growing interest in cosmology today, in experimental (e.g. [1, 3–7]) as well as computational aspects (e.g. [45, 46]), these expressions and their wide applicability for diﬀerent models could hence give a nice framework for new lensing predictions and simulations. ACKNOWLEDGMENTS The authors of this paper wish to thank G. Veneziano, B. Metcalf, G. Marozzi, M. Gasperini, P. Fleury and D. Leier for helpful discussions. FN’s research is supported by the project GLENCO, funded under the FP7, Ideas, Grant Agreement n. 259349. GF’s work is supported by the research grant “Theoretical Astroparticle Physics” number 2012CPPYP7 under the program PRIN 2012 funded by the Ministero dell’Istruzione, Universit`a e della Ricerca (MIUR) and by the Italian Istituto Nazionale di Fisica Nucleare (INFN) through the “Theoretical Astroparticle Physics” project. GF also wish to thank the University of Geneva for its hospitality during part of the elaboration of this document.

text: VI.

text: CONCLUSIONS

text: AND

text: OUTLOOK

text: We

text: have

text: presented

text: in

text: this

text: paper

text: the

text: explicit

text: expressions

text: of

text: several

text: lensing

text: quantities

text: contained

text: in

text: the

text: ampliﬁcation

text: matrix

text: (magniﬁcation,

text: convergence

text: and

text: shear)

text: and

text: the

text: deformation

text: matrix

text: (optical

text: scalars).

text: These

text: results

text: were

text: obtained

text: by

text: the

text: use

text: of

text: the

text: Jacobi

text: map

text: expressed

text: ﬁrst

text: through

text: the

text: zweibeins

text: of

text: the

text: Sachs

text: basis

text: and

text: then

text: within

text: the

text: geodesic

text: light-cone

text: coordinates.

text: We

text: have

text: shown

text: that

text: some

text: particular

text: combinations

text: of

text: the

text: lensing

text: quantities

text: take

text: an

text: elegant

text: expression

text: in

text: the

text: GLC

text: gauge

text: and

text: the

text: explicit

text: GLC-form

text: of

text: the

text: zweibeins

text: is

text: not

text: needed

text: (though

text: it

text: was

text: given

text: for

text: completeness

text: in

text: Appendix

text: These

text: expressions

text: are

text: general

text: and

text: can

text: be

text: applied

text: to

text: any

text: inhomogeneous

text: model

text: of

text: Universe

text: as

text: long

text: as

text: no

text: caustic

text: forms

text: on

text: the

text: past

text: light-cone

text: of

text: the

text: observer.

text: We

text: have

text: also

text: seen

text: through

text: these

text: computations

text: the

text: interest

text: of

text: the

text: GLC

text: gauge

text: in

text: dealing

text: with

text: light

text: propagation,

text: emphasizing

text: that

text: the

text: geodesic

text: equation

text: being

text: trivial

text: in

text: this

text: set

text: of

text: coordinates

text: also

text: simpliﬁes

text: the

text: handling

text: of

text: lensing

text: variables.

text: Let

text: us

text: comment

text: brieﬂy

text: on

text: the

text: presence

text: of

text: caustics.

text: In

text: this

text: situation

text: one

text: sees

text: the

text: creation

text: of

text: critical

text: points

text: or

text: lines

text: on

text: a

text: given

text: sphere

text: Σ(w,

text: τ)

text: embedded

text: in

text: the

text: past

text: light-cone.

text: These

text: singularities

text: appear

text: when

text: det

text: JA

text: is

text: equal

text: to

text: zero.

text: B

text: In

text: such

text: a

text: case

text: the

text: Jacobi

text: map

text: is

text: not

text: an

text: invertible

text: matrix

text: and

text: complicated

text: situations

text: with

text: multiple

text: images

text: and

text: inﬁnite

text: magniﬁcation

text: can

text: be

text: observed.

text: We

text: can

text: also

text: see

text: (from

text: Eqs.

text: and

text: that

text: this

text: happens

text: when

text: γ

text: det

text: γ

text: ab

text: in

text: the

text: GLC

text: gauge

text: and

text: we

text: then

text: have

text: a

text: case

text: of

text: strong

text: lensing.

text: The

text: estimation

text: of

text: the

text: angular

text: distance

text: is

text: not

text: anymore

text: valid

text: on

text: these

text: critical

text: lines

text: and

text: a

text: more

text: involved

text: system

text: of

text: coordinates

text: is

text: necessary.

text: The

text: generalisation

text: of

text: the

text: GLC

text: gauge

text: to

text: that

text: case

text: appears

text: as

text: a

text: hard

text: task,

text: despite

text: its

text: simplifying

text: properties

text: in

text: the

text: case

text: of

text: weak

text: lensing.

text: In

text: the

text: second

text: part

text: of

text: this

text: paper

text: we

text: have

text: applied

text: our

text: general

text: formulas

text: to

text: the

text: well-known

text: case

text: of

text: a

text: LTB

text: metric

text: with

text: an

text: oﬀ-center

text: observer.

text: These

text: relations,

text: especially

text: the

text: lensing

text: quantities,

text: were

text: obtained

text: directly

text: after

text: expressing

text: the

text: non-perturbative

text: transformation

text: between

text: LTB

text: and

text: GLC

text: coordinates.

text: They

text: are

text: general

text: and

text: to

text: our

text: knowledge

text: the

text: most

text: explicit

text: expressions

text: presented

text: up

text: to

text: now

text: and

text: they

text: could

text: be

text: used

text: for

text: a

text: broad

text: range

text: of

text: applications,

text: in

text: particular

text: for

text: the

text: exact

text: treatment

text: of

text: light

text: propagation

text: in

text: inhomogeneous

text: universes.

text: We

text: then

text: restricted

text: our

text: study

text: to

text: an

text: LTB

text: over/underdensity

text: (or

text: “bubble”)

text: entirely

text: determined

text: by

text: its

text: local

text: Hubble

text: factor

text: within

text: CDM

text: and

text: ΛCDM

text: backgrounds.

text: We

text: presented

text: the

text: variations

text: of

text: the

text: magniﬁcation

text: µ

text: and

text: the

text: distance

text: modulus

text: diﬀerence

text: ∆m

text: for

text: an

text: over/underdense

text: bubble,

text: pointing

text: at

text: the

text: center

text: of

text: it.

text: We

text: found

text: the

text: correction

text: induced

text: on

text: the

text: magniﬁcation

text: by

text: the

text: bubble

text: to

text: be

text: very

text: small

text: at

text: short

text: distances

text: for

text: d

text: Mpc)

text: at

text: more

text: signiﬁcant

text: at

text: large

text: distances

text: for

text: d

text: Mpc)

text: in

text: the

text: ΛCDM

text: case

text: as

text: well

text: as

text: CDM

text: (the

text: diﬀerence

text: being

text: negligible

text: for

text: such

text: small

text: under/overdensities).

text: We

text: also

text: showed

text: the

text: consistency

text: of

text: our

text: results

text: by

text: presenting

text: the

text: evolution

text: of

text: the

text: quantities,

text: including

text: the

text: shear

text: and

text: the

text: convergence,

text: when

text: we

text: point

text: at

text: diﬀerent

text: angles

text: from

text: the

text: center

text: of

text: the

text: bubble.

text: We

text: ﬁnally

text: checked

text: that

text: our

text: quantities

text: are

text: well-behaved

text: and

text: that

text: no

text: divergence

text: appears.

text: To

text: ﬁnish,

text: this

text: work

text: improves

text: our

text: understanding

text: of

text: lensing

text: theory

text: and

text: presents

text: some

text: of

text: its

text: most

text: important

text: quantities

text: in

text: an

text: elegant

text: framework.

text: It

text: also

text: illustrates

text: once

text: again

text: the

text: general

text: usefulness

text: of

text: the

text: GLC

text: gauge

text: for

text: deriving

text: theoretical

text: expressions

text: as

text: well

text: as

text: working

text: out

text: with

text: more

text: applied

text: situations.

text: We

text: illustrated

text: our

text: results

text: through

text: the

text: LTB

text: model

text: but

text: we

text: can

text: imagine

text: other

text: applications

text: such

text: as,

text: for

text: example,

text: the

text: computation

text: of

text: the

text: lensing

text: quantities

text: in

text: an

text: anisotropic

text: Bianchi

text: model

text: or

text: at

text: second

text: order

text: in

text: perturbations

text: in

text: the

text: Poisson

text: gauge.

text: Lensing

text: beneﬁts

text: from

text: a

text: growing

text: interest

text: in

text: cosmology

text: today,

text: in

text: experimental

text: (e.g.

text: as

text: well

text: as

text: computational

text: aspects

text: (e.g.

text: these

text: expressions

text: and

text: their

text: wide

text: applicability

text: for

text: diﬀerent

text: models

text: could

text: hence

text: give

text: a

text: nice

text: framework

text: for

text: new

text: lensing

text: predictions

text: and

text: simulations.

text: ACKNOWLEDGMENTS

text: The

text: authors

text: of

text: this

text: paper

text: wish

text: to

text: thank

text: G.

text: Veneziano,

text: B.

text: Metcalf,

text: G.

text: Marozzi,

text: M.

text: Gasperini,

text: P.

text: Fleury

text: and

text: D.

text: Leier

text: for

text: helpful

text: discussions.

text: FN’s

text: research

text: is

text: supported

text: by

text: the

text: project

text: GLENCO,

text: funded

text: under

text: the

text: Ideas,

text: Grant

text: Agreement

text: n.

text: GF’s

text: work

text: is

text: supported

text: by

text: the

text: research

text: grant

text: “Theoretical

text: Astroparticle

text: Physics”

text: number

text: under

text: the

text: program

text: PRIN

text: funded

text: by

text: the

text: Ministero

text: dell’Istruzione,

text: Universit`a

text: e

text: della

text: Ricerca

text: (MIUR)

text: and

text: by

text: the

text: Italian

text: Istituto

text: Nazionale

text: di

text: Fisica

text: Nucleare

text: (INFN)

text: through

text: the

text: “Theoretical

text: Astroparticle

text: Physics”

text: project.

text: GF

text: also

text: wish

text: to

text: thank

text: the

text: University

text: of

text: Geneva

text: for

text: its

text: hospitality

text: during

text: part

text: of

text: the

text: elaboration

text: of

text: this

text: document.


text: 6 (cid:2) (cid:3) into the new expressions : √ − 1 (1 κ)2 + ωˆ2 = ∆ab∆cd (cid:2)γ (γ ) + 2 γ s1s2(cid:15) (cid:3) , 4d¯ 2 o o ac bd o o a c bd A √ 1 − γˆ2 + γˆ2 = ∆ab∆cd γ (γ ) 2 γ s1s2(cid:15) . (34) 1 2 4d¯ 2 o o ac bd o o a c bd (cid:18) A (cid:19) (cid:0) (cid:1) We can ﬁnally use (cid:16) √ (cid:17) γ γ˙ γbcγ˙ 4u2 γ ∆abγ ∆cd γ = 4u2 ab cd γ γad , s1∆ab(cid:15) ∆cds2 = τo , (35) bc o ad τo (detab γ˙ab)2 o  a o bc o d detab γ˙ab o (cid:18) (cid:19)   to obtain :   (cid:16) (cid:17)  (cid:16) √ (cid:17)   − u 2 γ γ˙ γbcγ˙ γ γ (1 κ)2 + ωˆ2 = τo  ab cd  γ γad + 2 o  , (cid:18) d¯A (cid:19) (cid:16)detab γ˙ab(cid:17)2 o (cid:16)de√tab γ˙ab(cid:17)o    u 2 γ γ˙ γbcγ˙ − γ γ γˆ2 + γˆ2 = τo ab cd γ γad 2 o . (36) (cid:2) 1 2(cid:3) d¯A detab γ˙ab 2 o detab γ˙ab o One can then re-obtain the expression of the magniﬁcation in terms of the GLC quantities, Eq. (29), by using Eqs. ≡ − − | | − A (36) and µ (1 κ)2 + ωˆ2 γˆ 2 1. Let us emphasize that the ratio d¯ /u appears in all the quantities of A τo and because lensing does not depend on the (cid:18)obser(cid:19)ver’s mot(cid:20)ion we ch(cid:21)oose our background distance d¯A with the same observer motion u¯ as the perturbed one, i.e. u¯ = u . In such a way, we can use : τo τo τ0 √ √ d¯ 2 4 γ¯ A = γ¯ , (37) u¯ det γ¯˙ τo ab o where we used (. . .) to label background quantities. Let us now express the deformation matrix and its elements, the optical scalars, in the GLC gauge. By its deﬁnition of Eq. (12) this matrix is given in terms of the zweibeins by : S dsA ω A = a sa = sa sb γ˙ . (38) B dλ B 2Υ A B ab S Let us note that the observer terms of Eq. (27) have disappeared from this expression and that A is symmetric. B S Using Eqs. (38), the property sa sb = γab, and the decomposition of Eq. (13), we get the components of A (the A A B optical scalars) : S tr A γ(cid:18)abγ˙ (cid:19) ω γ˙ (cid:18) (cid:19) θˆ = B = ω ab = , (39) 2 4Υ 4Υ γ S | | tr A 2 − S ω γ˙ 2 − ω2 det γ˙ σˆ 2 = σˆ2 + σˆ2 = B det A = ab . (40) 1 2 2 B 4Υ γ 4Υ2 γ Hence the optical scalars are independent from β as expected (as β describes a U(1)-rotation freedom). These expressions perfectly agree with the general deﬁnitions of Eqs. (16) and (17) with the usual GLC condition kµ = − ωΥ 1δµ. τ Using the relations presented in Eq. (21)(cid:34), or in both equivalent ways from E(cid:35)qs. (14), (15) or Eqs. (11), (38), we can also get the Ricci and Weyl focusing in the GLC gauge. We ﬁnd : ω2 (cid:34) − Υ˙ − 1 (cid:35) Φ00 = 4Υ2 γabγ¨ab Υγabγ˙ab 2γabγ˙a(cid:0)cγcdγ˙db , (cid:1) ω2 (cid:34) − Υ˙ − 1 (cid:35) − ReΨ0 = 4Υ2 γ¨ab Υγ˙ab 2γ˙acγcdγ˙db (cid:0)sa1sb1 sa2sb2(cid:1) , ω2 − Υ˙ − 1 ImΨ = γ¨ γ˙ γ˙ γcdγ˙ sasb + sasb , (41) 0 4Υ2 ab Υ ab 2 ac db 1 2 2 1 ##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: into

text: the

text: new

text: expressions

text: ωˆ2

text: ∆ab∆cd

text: (γ

text: γ

text: o

text: o

text: ac

text: bd

text: o

text: o

text: a

text: c

text: bd

text: A

text: γˆ2

text: γˆ2

text: ∆ab∆cd

text: γ

text: (γ

text: γ

text: o

text: o

text: ac

text: bd

text: o

text: o

text: a

text: c

text: bd

text: A

text: We

text: can

text: ﬁnally

text: use

text: γ

text: γ˙

text: γbcγ˙

text: γ

text: ∆abγ

text: ∆cd

text: γ

text: ab

text: cd

text: γ

text: γad

text: τo

text: bc

text: o

text: ad

text: τo

text: (detab

text: γ˙ab)2

text: o

text: a

text: o

text: bc

text: o

text: d

text: detab

text: γ˙ab

text: o

text: to

text: obtain

text: u

text: γ

text: γ˙

text: γbcγ˙

text: γ

text: γ

text: ωˆ2

text: τo

text: ab

text: cd

text: γ

text: γad

text: o

text: d¯A

text: o

text: u

text: γ

text: γ˙

text: γbcγ˙

text: γ

text: γ

text: γˆ2

text: γˆ2

text: τo

text: ab

text: cd

text: γ

text: γad

text: o

text: d¯A

text: detab

text: γ˙ab

text: o

text: detab

text: γ˙ab

text: o

text: One

text: can

text: then

text: re-obtain

text: the

text: expression

text: of

text: the

text: magniﬁcation

text: in

text: terms

text: of

text: the

text: GLC

text: quantities,

text: Eq.

text: by

text: using

text: Eqs.

text: A

text: and

text: µ

text: ωˆ2

text: γˆ

text: Let

text: us

text: emphasize

text: that

text: the

text: ratio

text: d¯

text: /u

text: appears

text: in

text: all

text: the

text: quantities

text: of

text: A

text: τo

text: and

text: because

text: lensing

text: does

text: not

text: depend

text: on

text: the

text: we

text: our

text: background

text: distance

text: d¯A

text: with

text: the

text: same

text: observer

text: motion

text: u¯

text: as

text: the

text: perturbed

text: one,

text: i.e.

text: u¯

text: u

text: In

text: such

text: a

text: way,

text: we

text: can

text: use

text: τo

text: τo

text: τ0

text: d¯

text: γ¯

text: A

text: γ¯

text: u¯

text: det

text: γ¯˙

text: τo

text: ab

text: o

text: where

text: we

text: used

text: to

text: label

text: background

text: quantities.

text: Let

text: us

text: now

text: express

text: the

text: deformation

text: matrix

text: and

text: its

text: elements,

text: the

text: optical

text: scalars,

text: in

text: the

text: GLC

text: gauge.

text: By

text: its

text: deﬁnition

text: of

text: Eq.

text: this

text: matrix

text: is

text: given

text: in

text: terms

text: of

text: the

text: zweibeins

text: by

text: S

text: dsA

text: ω

text: A

text: a

text: sa

text: sa

text: sb

text: γ˙

text: B

text: dλ

text: B

text: 2Υ

text: A

text: B

text: ab

text: S

text: Let

text: us

text: note

text: that

text: the

text: observer

text: terms

text: of

text: Eq.

text: have

text: disappeared

text: from

text: this

text: expression

text: and

text: that

text: A

text: is

text: symmetric.

text: B

text: S

text: Using

text: Eqs.

text: the

text: property

text: sa

text: sb

text: γab,

text: and

text: the

text: decomposition

text: of

text: Eq.

text: we

text: get

text: the

text: components

text: of

text: A

text: (the

text: A

text: A

text: B

text: optical

text: scalars)

text: S

text: tr

text: A

text: ω

text: γ˙

text: θˆ

text: B

text: ω

text: ab

text: 4Υ

text: 4Υ

text: γ

text: S

text: tr

text: A

text: S

text: ω

text: γ˙

text: ω2

text: det

text: γ˙

text: σˆ

text: σˆ2

text: σˆ2

text: B

text: det

text: A

text: ab

text: B

text: 4Υ

text: γ

text: γ

text: Hence

text: the

text: optical

text: scalars

text: are

text: independent

text: from

text: β

text: as

text: expected

text: (as

text: β

text: describes

text: a

text: freedom).

text: These

text: expressions

text: perfectly

text: agree

text: with

text: the

text: general

text: deﬁnitions

text: of

text: Eqs.

text: and

text: with

text: the

text: usual

text: GLC

text: condition

text: kµ

text: ωΥ

text: 1δµ.

text: τ

text: Using

text: the

text: relations

text: presented

text: in

text: Eq.

text: or

text: in

text: both

text: equivalent

text: ways

text: from

text: or

text: Eqs.

text: we

text: can

text: also

text: get

text: the

text: Ricci

text: and

text: Weyl

text: focusing

text: in

text: the

text: GLC

text: gauge.

text: We

text: ﬁnd

text: ω2

text: Υ˙

text: γabγ¨ab

text: Υγabγ˙ab

text: ω2

text: Υ˙

text: γ¨ab

text: Υγ˙ab

text: ω2

text: Υ˙

text: ImΨ

text: γ¨

text: γ˙

text: γ˙

text: γcdγ˙

text: sasb

text: sasb

text: ab

text: Υ

text: ab

text: ac

text: db

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Figure 3: Frontier expansion times measured visiting an R-MAT graph with scale=21 using a single GPU and varying the number of edges per GPU thread. Since this phase processes the vertices received, instead of their adja- cency lists, fewer computational resources are required compared to those necessary in the frontier expansion phase. As in the expand communication phase, vertices returned by the fold exchange are grouped by sender in diﬀerent blocks and thus, before starting the processing, they are copied to device memory in contiguous locations. After the copy, the frontier update kernel is launched with a thread per vertex. Threads are mapped linearly onto the arrays of vertices and each thread is responsible of updating the level and predecessor of its vertex, if unvisited, and to add it to the output array. As in the frontier expansion kernel, we make use of atomic operations in order to synchronize the accesses to the bitmap and the writes to the output buﬀer. As to the predecessor, since in the fold phase we do not transmit source vertices, the thread, lacking this information, stores the sender processor-column in the predecessor array (the sender saved the predecessor). After the kernel has completed, the output array is copied to host mem- ory and it is appended to the frontier array. 4 Results The performances hereafter reported have been obtained on the Piz Daint system belonging to the Centro Svizzero di Calcolo Scientiﬁco (CSCS). Piz Daint is a hybrid Cray XC30 system with 5272 computing nodes in- 14 ##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##

text: Figure

text: Frontier

text: expansion

text: times

text: measured

text: visiting

text: an

text: R-MAT

text: graph

text: with

text: using

text: a

text: single

text: GPU

text: and

text: varying

text: the

text: number

text: of

text: edges

text: per

text: GPU

text: thread.

text: Since

text: this

text: phase

text: processes

text: the

text: vertices

text: received,

text: instead

text: of

text: their

text: adja-

text: cency

text: lists,

text: fewer

text: computational

text: resources

text: are

text: required

text: compared

text: to

text: those

text: necessary

text: in

text: the

text: frontier

text: expansion

text: phase.

text: As

text: in

text: the

text: expand

text: communication

text: phase,

text: vertices

text: returned

text: by

text: the

text: fold

text: exchange

text: are

text: grouped

text: by

text: sender

text: in

text: diﬀerent

text: blocks

text: and

text: thus,

text: before

text: starting

text: the

text: processing,

text: they

text: are

text: copied

text: to

text: device

text: memory

text: in

text: contiguous

text: locations.

text: After

text: the

text: copy,

text: the

text: frontier

text: update

text: kernel

text: is

text: launched

text: with

text: a

text: thread

text: per

text: vertex.

text: Threads

text: are

text: mapped

text: linearly

text: onto

text: the

text: arrays

text: of

text: vertices

text: and

text: each

text: thread

text: is

text: responsible

text: of

text: updating

text: the

text: level

text: and

text: predecessor

text: of

text: its

text: vertex,

text: if

text: unvisited,

text: and

text: to

text: add

text: it

text: to

text: the

text: output

text: array.

text: As

text: in

text: the

text: frontier

text: expansion

text: kernel,

text: we

text: make

text: use

text: of

text: atomic

text: operations

text: in

text: order

text: to

text: synchronize

text: the

text: accesses

text: to

text: the

text: bitmap

text: and

text: the

text: writes

text: to

text: the

text: output

text: buﬀer.

text: As

text: to

text: the

text: predecessor,

text: since

text: in

text: the

text: fold

text: phase

text: we

text: do

text: not

text: transmit

text: source

text: vertices,

text: the

text: thread,

text: lacking

text: this

text: information,

text: stores

text: the

text: sender

text: processor-column

text: in

text: the

text: predecessor

text: array

text: (the

text: sender

text: saved

text: the

text: predecessor).

text: After

text: the

text: kernel

text: has

text: completed,

text: the

text: output

text: array

text: is

text: copied

text: to

text: host

text: mem-

text: ory

text: and

text: it

text: is

text: appended

text: to

text: the

text: frontier

text: array.

text: Results

text: The

text: performances

text: hereafter

text: reported

text: have

text: been

text: obtained

text: on

text: the

text: Piz

text: Daint

text: system

text: belonging

text: to

text: the

text: Centro

text: Svizzero

text: di

text: Calcolo

text: Scientiﬁco

text: (CSCS).

text: Piz

text: Daint

text: is

text: a

text: hybrid

text: Cray

text: system

text: with

text: computing

text: nodes

text: in-

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: Computation vs Communication (bitmap transfers) 120 Computation Communication 100 80 s) m me ( 60 Ti 40 20 0 16 64 256 1024 4096 # of processors Figure 11: Computation and transfer times (left and right bar, respec- tively) for BFS operations performed with variable number GPUs on R-MAT graphs taking advantage of the bitmap optimization. 24 ##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##

text: Computation

text: vs

text: Communication

text: (bitmap

text: transfers)

text: Computation

text: Communication

text: s)

text: m

text: me

text: Ti

text: of

text: processors

text: Figure

text: Computation

text: and

text: transfer

text: times

text: (left

text: and

text: right

text: bar,

text: respec-

text: tively)

text: for

text: BFS

text: operations

text: performed

text: with

text: variable

text: number

text: GPUs

text: on

text: R-MAT

text: graphs

text: taking

text: advantage

text: of

text: the

text: bitmap

text: optimization.

text: ##LTFigure##

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text: There is a short GRB in each long one 9 ∼ ness of 60%, hence we do not expect the selection eﬀects Butler N. R., Kocevski D., Bloom J. S., 2009, ApJ, 694, 76 (beyond the limiting ﬂux threshold) to play a dominant role. Butler N. R., Kocevski D., Bloom J. S., Curtis J. L., 2007, The 0.3 s (rest frame) time scale chosen for the ﬁrst ApJ, 671, 656 analysis has a clear interpretation: it is the representative Cabrera J. I., Firmani C., Avila-Reese V., Ghirlanda G., duration of the short GRBs in the D14 catalog. Since the Ghisellini G., Nava L., 2007, MNRAS, 382, 342 main driver for the E evolution is the integration time Cash W., 1976, A&A, 52, 307 iso (Fig. 3, right panel), a longer timescale would result in Cash W., 1979, ApJ, 228, 939 higher values of E . Hence, in order to obtain signif- Cenko S. B., et al., 2009a, GCN 9053 iso,ﬁrst icantly higher (or lower) values of E , we should overcome Cenko S. B., et al., 2009b, GCN 9518 iso the intrinsic scatter of the correlations, namely 0.25–0.3 dex Chornock R., Berger E., Fox D., 2011, GCN 11538 ∼ (cid:46) (cid:38) (a factor 2, i.e. ∆T 0.15 s or ∆T 0.6 s). Chornock R., et al., 2009a, GCN 9028 ﬁrst ﬁrst In summary, we found that the intrinsic spectral prop- Chornock R., et al., 2009b, GCN 9243 erties (peak energy and luminosity) of both the short GRBs Chornock R., Lunnan R., Berger E., 2013, GCN 15307 and the ﬁrst 0.3 s (rest frame) of long ones are actually in- Cucchiara A., et al., 2008, GCN 8713 distinguishable. Hence if the central engine of a long GRB Cucchiara A., et al., 2009a, GCN 9873 ∼ would stop working after 0.3 s, we would have no means Cucchiara A., et al., 2009b, GCN 10031 to distinguish it from a genuine short GRB. Clearly, short Cucchiara A., et al., 2009c, GCN 10065 and long GRBs remains two distinct phenomena, each one Cucchiara A., et al., 2009d, GCN 10202 with its own duration. In particular short GRBs lasts longer Cucchiara A., et al., 2010, GCN 10606 or shorter than 0.3 s. Likewise, we do not expect the short– Cucchiara A., et al., 2012, GCN 12865 like phase at the beginning of long GRBs to always lasts 0.3 Cucchiara A., et al., 2013, GCN 14687 s. Our ﬁndings are in agreement with those in Ghirlanda D’Avanzo P., et al., 2011, GCN 12284 et al. (2009), which found no diﬀerences in the (observed) D’Avanzo P., Salvaterra R., Bernardini M. G., Nava L., spectral properties of short GRBs and the ﬁrst 1–2 s of long Campana S., Covino S., D’Elia V., Ghirlanda G., Ghis- GRBs. We extended this work by comparing the intrinsic ellini G., Melandri A., Sbarufatti B., Vergani S. D., Tagli- (rest frame) properties rather than the observed ones. aferri G., 2014, arXiv:1405.5131 Moreover, we found that the spectral quantities in the de Ugarte Postigo A., et al., 2009, GCN 8766 ﬁrst 0.3 s of long GRBs deﬁne new E –E and E –L de Ugarte Postigo A., et al., 2013, GCN 15470 p iso p iso correlations. These correlations are possibly the manifesta- D’Elia V., et al., 2012, GCN 13494 tion of an underlying physical process common to all GRBs, Firmani C., Cabrera J. I., Avila-Reese V., Ghisellini G., despite the possibly diﬀerent progenitors of short and long Ghirlanda G., Nava L., Bosnjak Z., 2009, MNRAS, 393, GRBs, and the great variety of energetics and spectral prop- 1209 erties involved. Flores H., et al., 2010, GCN 11317 Flores H., et al., 2013, GCN 14491 Frontera F., Amati L., Guidorzi C., Landi R., in’t Zand J., ACKNOWLEDGMENTS 2012, ApJ, 754, 138 We gratefully acknowledge the Referee for the useful Fynbo J. P. U., et al., 2008, GCN 8254 comments and suggestions which greatly improved the pa- Fynbo J. P. U., et al., 2009, GCN 9947 per. The research activity of A. Melandri, M.G. Bernar- Gehrels N., et al., 2004, ApJ, 611, 1005 dini and P. D’Avanzo is supported by ASI grant INAF Gehrels N., et al., 2005, Nature, 437, 851 I/004/11/1. G. Calderone was supported by PRIN INAF Gehrels N., et al., 2008, ApJ, 689, 1161 2011 (1.05.01.09.15). Ghirlanda G., Celotti A., Ghisellini G., 2002, A&A, 393, REFERENCES 409 Amati L., 2006, MNRAS, 372, 233 Ghirlanda G., Ghisellini G., Celotti A., 2004, A&A, 422, Amati L., 2008, in Bianco C. L., Xue S.-S., eds, Rela- L55 tivistic Astrophysics Vol. 966 of American Institute of Ghirlanda G., Ghisellini G., Firmani C., 2006, New Journal Physics Conference Series, Spectrum-Energy Correlations of Physics, 8, 123 and Swift GRBs. pp 3–6 Ghirlanda G., Nava L., Ghisellini G., Celotti A., Firmani Amati L., et al., 2002, A&A, 390, 81 C., 2009, A&A, 496, 585 Arnaud K. A., 1996, in Jacoby G. H., Barnes J., eds, Astro- Ghirlanda G., Nava L., Ghisellini G., 2010, A&A, 511, A43 nomical Data Analysis Software and Systems V Vol. 101 Ghirlanda G., Ghisellini G., Nava L., 2011, MNRAS, 418, of Astronomical Society of the Paciﬁc Conference Series, L109 XSPEC: The First Ten Years. p. 17 Ghirlanda G., Ghisellini G., Nava L., Burlon D., 2011, MN- Avni Y., 1976, ApJ, 210, 642 RAS, 410, L47 Band D., et al., 1993, ApJ, 413, 281 Ghirlanda G., et al., 2012, MNRAS, 422, 2553 Band D. L., Preece R. D., 2005, ApJ, 627, 319 Golenetskii S., et al., 2012, GCN 13736 Berger E., 2013, arXiv:1311.2603 Gruber D., et al., 2011, A&A, 528, A15 Berger E., M. Rauch M., 2008, GCN 8542 Gruber D., et al., 2014, ApJS, 211, 12 Bernardini M. G., Margutti R., Zaninoni E., Chincarini G., Guiriec S., et al., 2010, ApJ, 725, 225 2012, MNRAS, 425, 1199 Guiriec S., et al., 2013, ApJ, 770, 32 Bromberg O., Nakar E., Piran T., Sari R., 2013, ApJ, 764, Hjorth J., Bloom J. S., 2012, The Gamma-Ray Burst - 179 Supernova Connection. pp 169–190 (cid:13) c 2014 RAS, MNRAS 000, 1–10

text: There

text: is

text: a

text: short

text: GRB

text: in

text: each

text: long

text: one

text: ness

text: of

text: hence

text: we

text: do

text: not

text: expect

text: the

text: selection

text: eﬀects

text: Butler

text: N.

text: R.,

text: Kocevski

text: D.,

text: Bloom

text: J.

text: S.,

text: ApJ,

text: (beyond

text: the

text: limiting

text: ﬂux

text: threshold)

text: to

text: play

text: a

text: dominant

text: role.

text: Butler

text: N.

text: R.,

text: Kocevski

text: D.,

text: Bloom

text: J.

text: S.,

text: Curtis

text: J.

text: L.,

text: The

text: s

text: (rest

text: frame)

text: time

text: scale

text: chosen

text: for

text: the

text: ﬁrst

text: ApJ,

text: analysis

text: has

text: a

text: clear

text: interpretation:

text: it

text: is

text: the

text: representative

text: Cabrera

text: J.

text: I.,

text: Firmani

text: C.,

text: Avila-Reese

text: V.,

text: Ghirlanda

text: G.,

text: duration

text: of

text: the

text: short

text: GRBs

text: in

text: the

text: catalog.

text: Since

text: the

text: Ghisellini

text: G.,

text: Nava

text: L.,

text: MNRAS,

text: main

text: driver

text: for

text: the

text: E

text: evolution

text: is

text: the

text: integration

text: time

text: Cash

text: W.,

text: A&A,

text: iso

text: (Fig.

text: right

text: panel),

text: a

text: longer

text: timescale

text: would

text: result

text: in

text: Cash

text: W.,

text: ApJ,

text: higher

text: values

text: of

text: E

text: Hence,

text: in

text: order

text: to

text: obtain

text: signif-

text: Cenko

text: S.

text: B.,

text: et

text: al.,

text: GCN

text: iso,ﬁrst

text: icantly

text: higher

text: (or

text: lower)

text: values

text: of

text: E

text: we

text: should

text: overcome

text: Cenko

text: S.

text: B.,

text: et

text: al.,

text: GCN

text: iso

text: the

text: intrinsic

text: scatter

text: of

text: the

text: correlations,

text: namely

text: dex

text: Chornock

text: R.,

text: Berger

text: E.,

text: Fox

text: D.,

text: GCN

text: (a

text: factor

text: i.e.

text: ∆T

text: s

text: or

text: ∆T

text: s).

text: Chornock

text: R.,

text: et

text: al.,

text: GCN

text: ﬁrst

text: ﬁrst

text: In

text: summary,

text: we

text: found

text: that

text: the

text: intrinsic

text: spectral

text: prop-

text: Chornock

text: R.,

text: et

text: al.,

text: GCN

text: erties

text: (peak

text: energy

text: and

text: luminosity)

text: of

text: both

text: the

text: short

text: GRBs

text: Chornock

text: R.,

text: Lunnan

text: R.,

text: Berger

text: E.,

text: GCN

text: and

text: the

text: ﬁrst

text: s

text: (rest

text: frame)

text: of

text: long

text: ones

text: are

text: actually

text: in-

text: Cucchiara

text: A.,

text: et

text: al.,

text: GCN

text: distinguishable.

text: Hence

text: if

text: the

text: central

text: engine

text: of

text: a

text: long

text: GRB

text: Cucchiara

text: A.,

text: et

text: al.,

text: GCN

text: would

text: stop

text: working

text: after

text: s,

text: we

text: would

text: have

text: no

text: means

text: Cucchiara

text: A.,

text: et

text: al.,

text: GCN

text: to

text: distinguish

text: it

text: from

text: a

text: genuine

text: short

text: GRB.

text: Clearly,

text: short

text: Cucchiara

text: A.,

text: et

text: al.,

text: GCN

text: and

text: long

text: GRBs

text: remains

text: two

text: distinct

text: phenomena,

text: each

text: one

text: Cucchiara

text: A.,

text: et

text: al.,

text: GCN

text: with

text: its

text: own

text: duration.

text: In

text: particular

text: short

text: GRBs

text: lasts

text: longer

text: Cucchiara

text: A.,

text: et

text: al.,

text: GCN

text: or

text: shorter

text: than

text: s.

text: Likewise,

text: we

text: do

text: not

text: expect

text: the

text: short–

text: Cucchiara

text: A.,

text: et

text: al.,

text: GCN

text: like

text: phase

text: at

text: the

text: beginning

text: of

text: long

text: GRBs

text: to

text: always

text: lasts

text: Cucchiara

text: A.,

text: et

text: al.,

text: GCN

text: s.

text: Our

text: ﬁndings

text: are

text: in

text: agreement

text: with

text: those

text: in

text: Ghirlanda

text: D’Avanzo

text: P.,

text: et

text: al.,

text: GCN

text: et

text: al.

text: which

text: found

text: no

text: diﬀerences

text: in

text: the

text: (observed)

text: D’Avanzo

text: P.,

text: Salvaterra

text: R.,

text: Bernardini

text: M.

text: G.,

text: Nava

text: L.,

text: spectral

text: properties

text: of

text: short

text: GRBs

text: and

text: the

text: ﬁrst

text: s

text: of

text: long

text: Campana

text: S.,

text: Covino

text: S.,

text: D’Elia

text: V.,

text: Ghirlanda

text: G.,

text: Ghis-

text: GRBs.

text: We

text: extended

text: this

text: work

text: by

text: comparing

text: the

text: intrinsic

text: ellini

text: G.,

text: Melandri

text: A.,

text: Sbarufatti

text: B.,

text: Vergani

text: S.

text: D.,

text: Tagli-

text: (rest

text: frame)

text: properties

text: rather

text: than

text: the

text: observed

text: ones.

text: aferri

text: G.,

text: Moreover,

text: we

text: found

text: that

text: the

text: spectral

text: quantities

text: in

text: the

text: de

text: Ugarte

text: Postigo

text: A.,

text: et

text: al.,

text: GCN

text: ﬁrst

text: s

text: of

text: long

text: GRBs

text: deﬁne

text: new

text: E

text: –E

text: and

text: E

text: –L

text: de

text: Ugarte

text: Postigo

text: A.,

text: et

text: al.,

text: GCN

text: p

text: iso

text: p

text: iso

text: correlations.

text: These

text: correlations

text: are

text: possibly

text: the

text: manifesta-

text: D’Elia

text: V.,

text: et

text: al.,

text: GCN

text: tion

text: of

text: an

text: underlying

text: physical

text: process

text: common

text: to

text: all

text: GRBs,

text: Firmani

text: C.,

text: Cabrera

text: J.

text: I.,

text: Avila-Reese

text: V.,

text: Ghisellini

text: G.,

text: despite

text: the

text: possibly

text: diﬀerent

text: progenitors

text: of

text: short

text: and

text: long

text: Ghirlanda

text: G.,

text: Nava

text: L.,

text: Bosnjak

text: Z.,

text: MNRAS,

text: GRBs,

text: and

text: the

text: great

text: variety

text: of

text: energetics

text: and

text: spectral

text: prop-

text: erties

text: involved.

text: Flores

text: H.,

text: et

text: al.,

text: GCN

text: Flores

text: H.,

text: et

text: al.,

text: GCN

text: Frontera

text: F.,

text: Amati

text: L.,

text: Guidorzi

text: C.,

text: Landi

text: R.,

text: in’t

text: Zand

text: J.,

text: ACKNOWLEDGMENTS

text: ApJ,

text: We

text: gratefully

text: acknowledge

text: the

text: Referee

text: for

text: the

text: useful

text: Fynbo

text: J.

text: P.

text: U.,

text: et

text: al.,

text: GCN

text: comments

text: and

text: suggestions

text: which

text: greatly

text: improved

text: the

text: pa-

text: Fynbo

text: J.

text: P.

text: U.,

text: et

text: al.,

text: GCN

text: per.

text: The

text: research

text: activity

text: of

text: A.

text: Melandri,

text: M.G.

text: Bernar-

text: Gehrels

text: N.,

text: et

text: al.,

text: ApJ,

text: dini

text: and

text: P.

text: D’Avanzo

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text: supported

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text: ASI

text: grant

text: INAF

text: Gehrels

text: N.,

text: et

text: al.,

text: Nature,

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text: Calderone

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text: PRIN

text: INAF

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text: N.,

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text: al.,

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text: Complexity analysis of quantum walk based search algorithms B L Douglas and J B Wang School of Physics, The University of Western Australia, 6009, Perth, Australia. Abstract We present several families of graphs that allow both eﬃcient quantum walk implementations and eﬃcient quantum walk based search algorithms. For these graphs, we construct quantum circuits that explicitly implement the full quantum walk search algorithm, without reference to a ‘black box’ oracle. These circuits provide a practically implementable method to explore quantum walk based search algorithms with the aim of eventual real-world applications. We also provide a numerical analysis of a quantum walk based search along a twisted toroid family of graphs, which √ requires O( n log(n)) elementary 2-qubit quantum gate operations to ﬁnd a marked node. 1 ##LTLine##

text: Complexity

label: title

text: analysis

text: of

text: quantum

text: walk

text: based

text: search

text: algorithms

text: B

label: author

text: L

text: Douglas

text: and

text: J

text: B

text: Wang

text: School

text: of

text: Physics,

text: The

text: University

text: of

text: Western

text: Australia,

text: Perth,

text: Australia.

text: Abstract

text: We

label: abstract

text: present

text: several

text: families

text: of

text: graphs

text: that

text: allow

text: both

text: eﬃcient

text: quantum

text: walk

text: implementations

text: and

text: eﬃcient

text: quantum

text: walk

text: based

text: search

text: algorithms.

text: For

text: these

text: graphs,

text: we

text: construct

text: quantum

text: circuits

text: that

text: explicitly

text: implement

text: the

text: full

text: quantum

text: walk

text: search

text: algorithm,

text: without

text: reference

text: to

text: a

text: ‘black

text: box’

text: oracle.

text: These

text: circuits

text: provide

text: a

text: practically

text: implementable

text: method

text: to

text: explore

text: quantum

text: walk

text: based

text: search

text: algorithms

text: with

text: the

text: aim

text: of

text: eventual

text: real-world

text: applications.

text: We

text: also

text: provide

text: a

text: numerical

text: analysis

text: of

text: a

text: quantum

text: walk

text: based

text: search

text: along

text: a

text: twisted

text: toroid

text: family

text: of

text: graphs,

text: which

text: requires

text: O(

text: n

text: log(n))

text: elementary

text: quantum

text: gate

text: operations

text: to

text: ﬁnd

text: a

text: marked

text: node.

text: ##LTLine##


text: 50 40 required 30 Steps 20 10 0 0 200 400 600 800 Number of nodes FIG. 8: Number of walking steps required to reach maximum success probability against the size √ of the graph. The solid line represents a N ﬁt to the numerical data. Putting these results together, we see that this family of twisted toroids satisfy both criteria for eﬃcient search via quantum walks, yielding for a graph on N nodes both an eﬃ- cient implementation of the walks themselves (using O(log(N)) 2-qubit gates), and requiring √ O( N) steps of the walk to ﬁnd a marked node with suﬃciently high probability. IV. CONCLUSIONS The examples given in this paper are intended to provide a ‘proof of concept’, presenting graphs for which both eﬃcient quantum search and eﬃcient implementation of quantum walks are possible. By explicitly characterising the internal action of the oracle, these circuits provide a practically implementable method to experimentally test these quantum walk based search algorithms. In some previous studies, quantum speedups have been shown relative to oracles that cannot be eﬃciently implemented. Whilst these studies are very useful from a computational complexity viewpoint, the algorithms involved cannot be practically implemented. 10 ##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## ##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: required

text: Steps

text: Number

text: of

text: nodes

text: FIG.

text: Number

text: of

text: walking

text: steps

text: required

text: to

text: reach

text: maximum

text: success

text: probability

text: against

text: the

text: size

text: of

text: the

text: graph.

text: The

text: solid

text: line

text: represents

text: a

text: N

text: ﬁt

text: to

text: the

text: numerical

text: data.

text: Putting

text: these

text: results

text: together,

text: we

text: see

text: that

text: this

text: family

text: of

text: twisted

text: toroids

text: satisfy

text: both

text: criteria

text: for

text: eﬃcient

text: search

text: via

text: quantum

text: walks,

text: yielding

text: for

text: a

text: graph

text: on

text: N

text: nodes

text: both

text: an

text: eﬃ-

text: cient

text: implementation

text: of

text: the

text: walks

text: themselves

text: (using

text: O(log(N))

text: gates),

text: and

text: requiring

text: O(

text: N)

text: steps

text: of

text: the

text: walk

text: to

text: ﬁnd

text: a

text: marked

text: node

text: with

text: suﬃciently

text: high

text: probability.

text: IV.

text: CONCLUSIONS

text: The

text: examples

text: given

text: in

text: this

text: paper

text: are

text: intended

text: to

text: provide

text: a

text: ‘proof

text: of

text: concept’,

text: presenting

text: graphs

text: for

text: which

text: both

text: eﬃcient

text: quantum

text: search

text: and

text: eﬃcient

text: implementation

text: of

text: quantum

text: walks

text: are

text: possible.

text: By

text: explicitly

text: characterising

text: the

text: internal

text: action

text: of

text: the

text: oracle,

text: these

text: circuits

text: provide

text: a

text: practically

text: implementable

text: method

text: to

text: experimentally

text: test

text: these

text: quantum

text: walk

text: based

text: search

text: algorithms.

text: In

text: some

text: previous

text: studies,

text: quantum

text: speedups

text: have

text: been

text: shown

text: relative

text: to

text: oracles

text: that

text: cannot

text: be

text: eﬃciently

text: implemented.

text: Whilst

text: these

text: studies

text: are

text: very

text: useful

text: from

text: a

text: computational

text: complexity

text: viewpoint,

text: the

text: algorithms

text: involved

text: cannot

text: be

text: practically

text: implemented.

text: ##LTFigure##

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