text: 34 L. ANDERSSON AND T.A. OLIYNYK where the constant C is of the form (A.4). With the second term in (A.5) estimated, we use relation ∂2u = aij(u, ∂u)∂ ∂ u + bi(u, ∂u)∂ ∂ u + g(u, ∂u) + h, (A.9) t (cid:2) i j (cid:0) i t (cid:1)(cid:3) to write the third term in (A.5) as − − ∂j 1(D f(u, Du, ∂ u)∂2u) = ∂j 1 D f(u, Du, ∂ u) aij(u, ∂u)∂ ∂ u + bi(u, ∂u)∂ ∂ u + g(u, ∂u) + h . t 3 t t t 3 t i j i t Similar arguments employed above to derive (A.9) show also that (cid:107) − (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) ∂j 1(D f(u, Du, ∂ u)∂2u) − C(1 + u + ∂ u ) (A.10) t 3 t (cid:0)t Ws j,p(Ω) (cid:1) Ws+1,p(Ω) t Ws,p(Ω) for a constant C of the form (A.4). Together, the estimates (A.3), (A.8) and (A.10) show that (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) ∂jf(u, Du, ∂ u) − C u ∞ , ∂ u ∞ (1 + u + ∂ u ). t t Wp,s j(Ω) W1, (Ω) t L (Ω) Wp,s+1(Ω) t Ws,p(Ω) (cid:3) This completes the induction argument and the proof of the proposition. Similar arguments can be used to prove the following variant of Proposition A.9. ∈ Z ≤ ≤ ∞ ∈ R R ∈ ∈ Proposition A.10. Suppose s ≥ , 1 p , f Cs+1( , ), u Wp,s+1(Ω), ∂ u Wp,s(Ω) ≥1 t and the higher time derivatives ∂(cid:96)u (cid:96) 2 are obtained by formally diﬀerentiating t ∂2u = aij(u, ∂u)∂ ∂ u + bi(u, ∂u)∂ ∂ u + g(u, ∂u) + h, t (cid:0) i j i(cid:1) t ∈ ∈ R × R R where h Wp,s+1(Ω), ∂(cid:96)h = 0 and aij, bj, g Cs( n+1, ). Then u satisﬁes the estimate t (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) ∂jf(u) − C u ∞ , ∂ u ∞ (1 + u + ∂ u ) t Wp,s+1 j(Ω) W1, (Ω) t L (Ω) Wp,s+1(Ω) t Wp,s(Ω) ≤ ≤ for 0 j s + 1. ∈ Z ≤ ≤ ∞ ∈ R R ∈ ∈ Proposition A.11. Suppose s ≥ , 1 p , fµν Cs+1( , ), u Wp,s+1(Ω), ∂ u Wp,s(Ω) 1≥ t and the higher time derivatives ∂(cid:96)u (cid:96) 2 are obtained by formally diﬀerentiating t ∂2u = aij(u, ∂u)∂ ∂ u + bi(u, ∂u)∂ ∂ u + g(u, ∂u) + h, t (cid:0) i j i t (cid:1) ∈ ∈ R × R R where h Wp,s+1(Ω), ∂(cid:96)h = 0 and aij, bj, g Cs( n+1, ). Then t (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) ∂ [∂k, fµν(u)∂ ]u − C u ∞ , ∂ u ∞ (1 + u + ∂ u ) µ t ν Wp,s k(Ω) W1, (Ω) t L (Ω) Wp,s+1(Ω) t Wp,s(Ω) ≤ ≤ for 0 k s. (cid:18) (cid:19) (cid:88) (cid:2) (cid:3) Proof. Diﬀerentiating the formula − k 1 k − − [∂k, fνµ(u)∂ ]u = ∂k (cid:96)fνi∂ ∂(cid:96)u + ∂k (cid:96)fν0∂(cid:96)+1u , t µ (cid:96) t i t t t (cid:18)(cid:96)=0(cid:19) (cid:0) (cid:1) (cid:88) (cid:2) we see that − k 1 k − − (cid:3) ∂ [∂k, fjµ(u)∂ ]u = ∂k (cid:96)(Dfji∂ u)∂ ∂(cid:96)u + ∂k (cid:96)fji∂ ∂ ∂(cid:96)u j t µ (cid:96) t j i t t j i t (cid:96)=0 − − + ∂k (cid:96)(Dfj0∂ u)∂(cid:96)+1u + ∂k (cid:96)fj0∂(cid:96)+1∂ u (A.11) t j t t t j

text: L.

text: ANDERSSON

text: AND

text: T.A.

text: OLIYNYK

text: where

text: the

text: constant

text: C

text: is

text: of

text: the

text: form

text: With

text: the

text: second

text: term

text: in

text: estimated,

text: we

text: use

text: relation

text: aij(u,

text: ∂u)∂

text: u

text: bi(u,

text: ∂u)∂

text: u

text: g(u,

text: ∂u)

text: h,

text: t

text: i

text: j

text: i

text: t

text: to

text: write

text: the

text: third

text: term

text: in

text: as

text: ∂j

text: f(u,

text: Du,

text: ∂j

text: D

text: f(u,

text: Du,

text: u)

text: aij(u,

text: ∂u)∂

text: u

text: bi(u,

text: ∂u)∂

text: u

text: g(u,

text: ∂u)

text: h

text: t

text: t

text: t

text: t

text: t

text: i

text: j

text: i

text: t

text: Similar

text: arguments

text: employed

text: above

text: to

text: derive

text: show

text: also

text: that

text: ∂j

text: f(u,

text: Du,

text: u

text: u

text: t

text: t

text: Ws

text: j,p(Ω)

text: t

text: Ws,p(Ω)

text: for

text: a

text: constant

text: C

text: of

text: the

text: form

text: Together,

text: the

text: estimates

text: and

text: show

text: that

text: ∂jf(u,

text: Du,

text: u)

text: C

text: u

text: u

text: u

text: u

text: t

text: t

text: Wp,s

text: j(Ω)

text: (Ω)

text: t

text: L

text: (Ω)

text: t

text: Ws,p(Ω)

text: This

text: completes

text: the

text: induction

text: argument

text: and

text: the

text: proof

text: of

text: the

text: proposition.

text: Similar

text: arguments

text: can

text: be

text: used

text: to

text: prove

text: the

text: following

text: variant

text: of

text: Proposition

text: Z

text: R

text: R

text: Proposition

text: Suppose

text: s

text: p

text: f

text: u

text: u

text: Wp,s(Ω)

text: ≥1

text: t

text: and

text: the

text: higher

text: time

text: derivatives

text: are

text: obtained

text: by

text: formally

text: diﬀerentiating

text: t

text: aij(u,

text: ∂u)∂

text: u

text: bi(u,

text: ∂u)∂

text: u

text: g(u,

text: ∂u)

text: h,

text: t

text: i

text: j

text: t

text: R

text: R

text: R

text: where

text: h

text: and

text: aij,

text: bj,

text: g

text: Cs(

text: Then

text: u

text: satisﬁes

text: the

text: estimate

text: t

text: ∂jf(u)

text: C

text: u

text: u

text: u

text: u

text: t

text: j(Ω)

text: (Ω)

text: t

text: L

text: (Ω)

text: t

text: Wp,s(Ω)

text: for

text: j

text: s

text: Z

text: R

text: R

text: Proposition

text: Suppose

text: s

text: p

text: fµν

text: u

text: u

text: Wp,s(Ω)

text: 1≥

text: t

text: and

text: the

text: higher

text: time

text: derivatives

text: are

text: obtained

text: by

text: formally

text: diﬀerentiating

text: t

text: aij(u,

text: ∂u)∂

text: u

text: bi(u,

text: ∂u)∂

text: u

text: g(u,

text: ∂u)

text: h,

text: t

text: i

text: j

text: i

text: t

text: R

text: R

text: R

text: where

text: h

text: and

text: aij,

text: bj,

text: g

text: Cs(

text: Then

text: t

text: [∂k,

text: fµν(u)∂

text: ]u

text: C

text: u

text: u

text: u

text: u

text: µ

text: t

text: ν

text: Wp,s

text: k(Ω)

text: (Ω)

text: t

text: L

text: (Ω)

text: t

text: Wp,s(Ω)

text: for

text: k

text: s.

text: Proof.

text: Diﬀerentiating

text: the

text: formula

text: k

text: k

text: [∂k,

text: fνµ(u)∂

text: ]u

text: ∂k

text: ∂k

text: t

text: µ

text: t

text: i

text: t

text: t

text: t

text: we

text: see

text: that

text: k

text: k

text: [∂k,

text: fjµ(u)∂

text: ]u

text: ∂k

text: u)∂

text: ∂k

text: j

text: t

text: µ

text: t

text: j

text: i

text: t

text: t

text: j

text: i

text: t

text: ∂k

text: ∂k

text: u

text: t

text: j

text: t

text: t

text: t

text: j


text: 38 L. ANDERSSON AND T.A. OLIYNYK Then (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) f H f H , δ 2,s(Q1) 2,s(Q1) (cid:107) (cid:107) (cid:46) 1 (cid:107) (cid:107) g H − g H − δ 0,s (cid:96)(Q1) δ(cid:96) 0,s (cid:96)(Q1) and (cid:107) (cid:107) (cid:46) 1 (cid:107) (cid:107) hδ Hms−(cid:96),s−(cid:96)(Q1) δ(cid:96) h Hms−(cid:96),s−(cid:96)(Q1). Proof. First, a short calculation shows that (cid:107) (cid:107) − − (cid:107) (cid:107) ≤ − − (cid:107) (cid:107) Dsfδ 2L2(Q±) = δ2(s σ) n Dsf 2L2(Q±) δ2(s σ) n Dsf 2L2(Q±), 1 δ 1 and in particular, that (cid:107) (cid:107) ≤ (cid:107) (cid:107) Dsf ± Dsf ± (D.1) δ L2(Q ) L2(Q ) − − ≥ 1 1 since s σ n/2 0 by assumption. Next, we observe that (cid:32) (cid:33) (cid:90) (cid:90) (cid:107) (cid:107) − − (cid:107) (cid:107) D2f 2 = δ2(2 σ) n D2f 2 δ L2(Q1) L2(Qδ) − − | | | | = δ2(2 σ) n D2f dnx + D2f 2 dnx − (cid:0) Q Q+ (cid:1) δ δ which implies via H¨older’s inequality, Theorem A.1, that (cid:0) (cid:1) (cid:107) (cid:107) ≤ − (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) D2fδ 2L2(Q1) ≤ δ2 −n 1 Lp(−Q−δ )− D2f 2L(cid:107)2q(Q−δ (cid:107)) + 1 Lp(Q(cid:107)+δ ) D(cid:107)2f 2L2q(Q+δ ) 2(n 1)/pδ2(2 σ) n+n/p D2f 2 − + D2f 2 (cid:0) L2q(Q ) L2q(Q+)(cid:1) − 1 1 for 1/q + 1/p = 1. Choosing p = n/n 2 and hence q = n/2, the above inequality becomes (cid:107) (cid:107) ≤ − − − (cid:107) (cid:107) (cid:107) (cid:107) D2fδ 2L2(Q1) 2(n 1)(n 2)/nδ2(1 σ) D2f 2Ln(Q−1 ) + D2f 2Ln(Q+1 ) However, by Sobolev’s inequality, Theorem A.2, we have that (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) D2u Ln(Q±(cid:0)) D2f Hn/2−1(Q±), (cid:1) 1 1 and this allows us to conclude that (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) D2f f − + f , (D.2) ≤ δ L2(Q1) Hs(Q1 ) Hs(Q+1 ) since s > n/2 and σ 1. (cid:8) (cid:9) (cid:8) (cid:9) Next, we observe that | − | | | ≤ | | f(δx) f(0) ≤ (cid:107) (cid:107) (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) f (x) x σ | − | max f , f − max f , f − δ δx 0 σ C0,σ(Q+δ ) C0,σ(Qδ ) C0,σ(Q+1 ) C0,σ(Q1 ) ∈ for all x Q . This together with Sobolev’s inequality gives 1 (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) f ∞ f H , δ L (Q1) 0,s(Q1) which, in turn, implies, with the help of H¨older’s inequality, that (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) f f H . (D.3) δ L2(Q1) 0,s(Q1) The inequalities (D.1), (D.2) and (D.3) together with interpolation, see Theorem A.3, then show that (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) f H f H , δ 2,s(Q1) 2,s(Q1) ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: L.

text: ANDERSSON

text: AND

text: T.A.

text: OLIYNYK

text: Then

text: f

text: H

text: f

text: H

text: δ

text: g

text: H

text: g

text: H

text: δ

text: and

text: hδ

text: h

text: Proof.

text: First,

text: a

text: short

text: calculation

text: shows

text: that

text: Dsfδ

text: σ)

text: n

text: Dsf

text: σ)

text: n

text: Dsf

text: δ

text: and

text: in

text: particular,

text: that

label: equation

text: Dsf

text: Dsf

text: δ

text: since

text: s

text: σ

text: by

text: assumption.

text: Next,

text: we

text: observe

text: that

text: σ)

text: n

text: δ

text: σ)

text: n

text: dnx

text: dnx

text: Q

text: Q+

text: δ

text: δ

text: which

text: implies

text: via

text: H¨older’s

text: inequality,

text: Theorem

text: that

text: δ2

text: −n

text: Lp(−Q−δ

text: σ)

text: n+n/p

text: for

text: Choosing

text: p

text: n/n

text: and

text: hence

text: q

text: the

text: above

text: inequality

text: becomes

text: σ)

text: However,

text: by

text: Sobolev’s

text: inequality,

text: Theorem

text: we

text: have

text: that

text: and

text: this

text: allows

text: us

text: to

text: conclude

text: that

text: f

text: f

text: δ

text: since

text: s

text: and

text: σ

text: Next,

text: we

text: observe

text: that

text: f(δx)

text: f

text: (x)

text: x

text: σ

text: max

text: f

text: f

text: max

text: f

text: f

text: δ

text: δx

text: σ

text: for

text: all

text: x

text: Q

text: This

text: together

text: with

text: Sobolev’s

text: inequality

text: gives

text: f

text: f

text: H

text: δ

text: L

text: which,

text: in

text: turn,

text: implies,

text: with

text: the

text: help

text: of

text: H¨older’s

text: inequality,

text: that

text: f

text: f

text: H

text: δ

text: The

text: inequalities

text: and

text: together

text: with

text: interpolation,

text: see

text: Theorem

text: then

text: show

text: that

text: f

text: H

text: f

text: H

text: δ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A TRANSMISSION PROBLEM FOR QUASI-LINEAR WAVE EQUATIONS 41 − − Suppose now that s 2 (cid:96) > n/2. Then (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) D2hδ L∞(Q±) δ2 D2h L∞(Q±) D2h Hs−2−(cid:96)(Q±) h Hs−(cid:96)(Q±) 1 1 1 1 by Sobolev’s inequality. Using H¨older’s inequality, it is not diﬃcult to see that the above inequality implies that (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:46) 1 (cid:107) (cid:107) D2h h − h − . (D.17) δ L2(Q1) H0,s (cid:96)(Q1) δ(cid:96) H2,s (cid:96)(Q1) In either case, (D.16) and (D.17) show that (cid:107) (cid:107) (cid:46) 1 (cid:107) (cid:107) D2h h − (D.18) δ L2(Q1) δ(cid:96) H2,s (cid:96)(Q1) holds. From the inequalities (D.8), (D.10), (D.11), (D.12), (D.18) and interpolation, we conclude that (cid:107) (cid:107) (cid:46) 1 (cid:107) (cid:107) hδ Hms−(cid:96),s−(cid:96)(Q1) δ(cid:96) h Hms−(cid:96),s−(cid:96)(Q1). (cid:3) We will also need the following version of Proposition D.1 for the Hs(Q+) spaces. Since it can be 1 established using similar arguments, we omit the details. ≤ ∈ Z ≥ ≤ ≤ ≤ − − ≥ Proposition D.2. Suppose 0 < δ 1, s, (cid:96) ≥ , n 3, 0 σ 1, 0 σ < s n/2, s (cid:96) 0 ∈ ∈ − 0 f Hs(Q+), g Hs (cid:96)(Q+) and let 1 1 − f(δx) f(0) f (x) = and g (x) = g(δx). δ δσ δ Then (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) f f δ Hs(Q+) Hs(Q+) 1 1 and (cid:107) (cid:107) (cid:46) 1 (cid:107) (cid:107) gδ Hs−(cid:96)(Q+1 ) δ(cid:96) g Hs−(cid:96)(Q+1 ). References 1. R.A. Adams and J. Fournier, Sobolev spaces, 2nd ed., Academic Press, 2003. 2. L. Andersson, B.G. Schmidt, and T.A. Oliynyk, Dynamical compact elastic bodies in general relativity, in preparation. 3. , Dynamical elastic bodies in Newtonian gravity, Class. Quantum Grav. 28 (2011), 235006 (35pp). 4. N. Andersson and G.L. Comer, Relativistic ﬂuid dynamics: Physics for many diﬀerent scales, Living Rev. Relativity 10 (2007), 1. 5. R. Beig and B.G. Schmidt, Relativistic elasticity, Class.Quant.Grav. 20 (2003), 889–904. 6. R. Beig and M. Wernig-Pichler, On the motion of a compact elastic body, Commun. Math. Phys. 271 (2007), 455–465. 7. N. Chamel and P. Haensel, Physics of neutron star crusts, Living Rev. Relativity 11 (2009), 10. 8. Y. Choquet-Bruhat and H. Friedrich, Motion of isolated bodies, Class. Quantum Grav. 23 (2006), 5941–5949. 9. A. Friedman, Partial diﬀerential equations, Krieger Publishing Company, 1976. 10. S. Kind and J. Ehlers, Initial boundary value problem for the spherically symmetric Einstein equations for a perfect ﬂuid, Class. Quantum Grav. 18 (1993), 2123–2136. 11. H. Koch, Hyperbolic equations of second order, Ph.D. thesis, Ruprecht-Karls-Universit¨at, Heidelberg, 1990. 12. , Mixed problems for fully nonlinear hyperbolic equations, Math. Z. 214 (1993), 9–42. 13. H. Lindblad and K.H. Nordgren, A priori estimates for the motion of a selfgravitating incompressible liquid with free surface boundary, JHDE 6 (2009), 407–432. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: TRANSMISSION

text: PROBLEM

text: FOR

text: QUASI-LINEAR

text: WAVE

text: EQUATIONS

text: Suppose

text: now

text: that

text: s

text: Then

text: L∞(Q±)

text: δ2

text: L∞(Q±)

text: h

text: by

text: Sobolev’s

text: inequality.

text: Using

text: H¨older’s

text: inequality,

text: it

text: is

text: not

text: diﬃcult

text: to

text: see

text: that

text: the

text: above

text: inequality

text: implies

text: that

text: h

text: h

text: δ

text: In

text: either

text: case,

text: and

text: show

text: that

text: h

text: δ

text: holds.

text: From

text: the

text: inequalities

text: and

text: interpolation,

text: we

text: conclude

text: that

text: hδ

text: h

text: We

text: will

text: also

text: need

text: the

text: following

text: version

text: of

text: Proposition

text: for

text: the

text: Hs(Q+)

text: spaces.

text: Since

text: it

text: can

text: be

text: established

text: using

text: similar

text: arguments,

text: we

text: omit

text: the

text: details.

text: Z

text: Proposition

text: Suppose

text: δ

text: s,

text: n

text: σ

text: σ

text: s

text: s

text: f

text: Hs(Q+),

text: g

text: Hs

text: and

text: let

text: f(δx)

text: f

text: (x)

text: and

text: g

text: (x)

text: g(δx).

text: δ

text: δσ

text: δ

text: Then

text: f

text: f

text: δ

text: Hs(Q+)

text: Hs(Q+)

text: and

text: gδ

text: g

text: References

text: R.A.

text: Adams

text: and

text: J.

text: Fournier,

text: Sobolev

text: spaces,

text: ed.,

text: Academic

text: Press,

text: L.

text: Andersson,

text: B.G.

text: Schmidt,

text: and

text: T.A.

text: Oliynyk,

text: Dynamical

text: compact

text: elastic

text: bodies

text: in

text: general

text: relativity,

text: in

text: preparation.

text: Dynamical

text: elastic

text: bodies

text: in

text: Newtonian

text: gravity,

text: Class.

text: Quantum

text: Grav.

text: N.

text: Andersson

text: and

text: G.L.

text: Comer,

text: Relativistic

text: ﬂuid

text: dynamics:

text: Physics

text: for

text: many

text: diﬀerent

text: scales,

text: Living

text: Rev.

text: Relativity

text: R.

text: Beig

text: and

text: B.G.

text: Schmidt,

text: Relativistic

text: elasticity,

text: Class.Quant.Grav.

text: R.

text: Beig

text: and

text: M.

text: Wernig-Pichler,

text: On

text: the

text: motion

text: of

text: a

text: compact

text: elastic

text: body,

text: Commun.

text: Math.

text: Phys.

text: N.

text: Chamel

text: and

text: P.

text: Haensel,

text: Physics

text: of

text: neutron

text: star

text: crusts,

text: Living

text: Rev.

text: Relativity

text: Y.

text: Choquet-Bruhat

text: and

text: H.

text: Friedrich,

text: Motion

text: of

text: isolated

text: bodies,

text: Class.

text: Quantum

text: Grav.

text: A.

text: Friedman,

text: Partial

text: diﬀerential

text: equations,

text: Krieger

text: Publishing

text: Company,

text: S.

text: Kind

text: and

text: J.

text: Ehlers,

text: Initial

text: boundary

text: value

text: problem

text: for

text: the

text: spherically

text: symmetric

text: Einstein

text: equations

text: for

text: a

text: perfect

text: ﬂuid,

text: Class.

text: Quantum

text: Grav.

text: H.

text: Koch,

text: Hyperbolic

text: equations

text: of

text: second

text: order,

text: Ph.D.

text: thesis,

text: Ruprecht-Karls-Universit¨at,

text: Heidelberg,

text: Mixed

text: problems

text: for

text: fully

text: nonlinear

text: hyperbolic

text: equations,

text: Math.

text: Z.

text: H.

text: Lindblad

text: and

text: K.H.

text: Nordgren,

text: A

text: priori

text: estimates

text: for

text: the

text: motion

text: of

text: a

text: selfgravitating

text: incompressible

text: liquid

text: with

text: free

text: surface

text: boundary,

text: JHDE

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: On structural properties of trees with minimal atom-bond connectivity index Darko Dimitrov Institut fu¨r Informatik, Freie Universit¨at Berlin, Takustraße 9, D–14195 Berlin, Germany E-mail: darko@mi.fu-berlin.de Abstract The atom-bond connectivity (ABC) index is a degree-based molecular descriptor, that found chemical applications. It is well known that among all connected graphs, the graphs with minimal ABC index are trees. A complete characterization of trees with minimal ABC index is still an open problem. In this paper, we present new structural properties of trees with minimal ABC index. Our main results reveal that trees with minimal ABC index do not contain so-called B -branches, with k ≥ 5, and that they do not have more k than four B -branches. 4 1 Introduction Description of the structure or shape of molecules is very helpful in predicting activity and properties of molecules in complex experiments. For that purpose, the molecular descrip- tors [40] as mathematical quantities are particularly useful. Among the molecular descriptors, so-called topological indices [17] play a signiﬁcant role. The topological indices can be classi- ﬁed by the structural properties of graphs used for their calculation. For example, the Wiener index [43] and the Balaban J index [4] are based on the distance of vertices in the respective graph, the Estrada index [19] and the energy of a graph [29] are based on the spectrum of the graph, the Zagreb group indices [34] and the Randi´c connectivity index [39] depend on the degrees of vertices, while the Hosoya index [35] is calculated by the counting of non-incident edges in a graph. On the other hand, there is a group of so-called information indices that are based on information functionals [5]. More about the information indices and the discrimi- native power of some established indices, one can ﬁnd in [14–16, 24] and in the works cited therein. Here, we consider a relatively new topological index which attracted a lot of attention in the last few years. Namely, in 1998, Estrada et al. [21] proposed a new vertex-degree-based 1

text: On

label: title

text: structural

text: properties

text: of

text: trees

text: with

text: minimal

text: atom-bond

text: connectivity

text: index

text: Darko

text: Dimitrov

text: Institut

text: fu¨r

text: Informatik,

text: Freie

text: Universit¨at

text: Berlin,

text: Takustraße

text: Berlin,

text: Germany

text: E-mail:

text: darko@mi.fu-berlin.de

text: Abstract

text: The

label: abstract

text: atom-bond

text: connectivity

text: (ABC)

text: index

text: is

text: a

text: degree-based

text: molecular

text: descriptor,

text: that

text: found

text: chemical

text: applications.

text: It

text: is

text: well

text: known

text: that

text: among

text: all

text: connected

text: graphs,

text: the

text: graphs

text: with

text: minimal

text: ABC

text: index

text: are

text: trees.

text: A

text: complete

text: characterization

text: of

text: trees

text: with

text: minimal

text: ABC

text: index

text: is

text: still

text: an

text: open

text: problem.

text: In

text: this

text: paper,

text: we

text: present

text: new

text: structural

text: properties

text: of

text: trees

text: with

text: minimal

text: ABC

text: index.

text: Our

text: main

text: results

text: reveal

text: that

text: trees

text: with

text: minimal

text: ABC

text: index

text: do

text: not

text: contain

text: so-called

text: B

text: -branches,

text: with

text: k

text: and

text: that

text: they

text: do

text: not

text: have

text: more

text: k

text: than

text: four

text: B

text: -branches.

text: Introduction

text: Description

text: of

text: the

text: structure

text: or

text: shape

text: of

text: molecules

text: is

text: very

text: helpful

text: in

text: predicting

text: activity

text: and

text: properties

text: of

text: molecules

text: in

text: complex

text: experiments.

text: For

text: that

text: purpose,

text: the

text: molecular

text: descrip-

text: tors

text: as

text: mathematical

text: quantities

text: are

text: particularly

text: useful.

text: Among

text: the

text: molecular

text: descriptors,

text: so-called

text: topological

text: indices

text: play

text: a

text: signiﬁcant

text: role.

text: The

text: topological

text: indices

text: can

text: be

text: classi-

text: ﬁed

text: by

text: the

text: structural

text: properties

text: of

text: graphs

text: used

text: for

text: their

text: calculation.

text: For

text: example,

text: the

text: Wiener

text: index

text: and

text: the

text: Balaban

text: J

text: index

text: are

text: based

text: on

text: the

text: distance

text: of

text: vertices

text: in

text: the

text: respective

text: graph,

text: the

text: Estrada

text: index

text: and

text: the

text: energy

text: of

text: a

text: graph

text: are

text: based

text: on

text: the

text: spectrum

text: of

text: the

text: graph,

text: the

text: Zagreb

text: group

text: indices

text: and

text: the

text: Randi´c

text: connectivity

text: index

text: depend

text: on

text: the

text: degrees

text: of

text: vertices,

text: while

text: the

text: Hosoya

text: index

text: is

text: calculated

text: by

text: the

text: counting

text: of

text: non-incident

text: edges

text: in

text: a

text: graph.

text: On

text: the

text: other

text: hand,

text: there

text: is

text: a

text: group

text: of

text: so-called

text: information

text: indices

text: that

text: are

text: based

text: on

text: information

text: functionals

text: More

text: about

text: the

text: information

text: indices

text: and

text: the

text: discrimi-

text: native

text: power

text: of

text: some

text: established

text: indices,

text: one

text: can

text: ﬁnd

text: in

text: and

text: in

text: the

text: works

text: cited

text: therein.

text: Here,

text: we

text: consider

text: a

text: relatively

text: new

text: topological

text: index

text: which

text: attracted

text: a

text: lot

text: of

text: attention

text: in

text: the

text: last

text: few

text: years.

text: Namely,

text: in

text: Estrada

text: et

text: al.

text: proposed

text: a

text: new

text: vertex-degree-based


text: 2 graph topological index, the atom-bond connectivity (ABC) index, and showed that it can be a valuable predictive tool in the study of the heat of formation in alikeness. Ten years later Estrada [20] elaborated a novel quantum-theory-like justiﬁcation for this topological index. After that revelation, the interest of ABC-index has grown rapidly. Additionaly, the physico-chemical applicability of the ABC index was conﬁrmed and extended in several studies [3,9,13,28,33,37,47]. Let G = (V, E) be a simple undirected graph of order n = |V | and size m = |E|. For v ∈ V (G), the degree of v, denoted by d(v), is the number of edges incident to v. For an edge (cid:115) uv in G, let d(u) + d(v) − 2 f(d(u), d(v)) = . (1) d(u)d(v) (cid:88) Then the atom-bond connectivity index of G is deﬁned as ABC(G) = f(d(u), d(v)). uv∈E(G) As a new and well motivated graph invariant, the ABC index has attracted a lot of interest in the last several years both in mathematical and chemical research communities and numerous results and structural properties of ABC index were established [6–8, 10–12, 22, 23, 30, 32, 38, 41,44–46]. The fact that adding an edge in a graph strictly increases its ABC index [11] (or equiva- lently that deleting an edge in a graph strictly decreases its ABC index [6]) has the following two immediate consequences. Corollary 1.1. Among all connected graphs with n vertices, the complete graph K has n maximal value of ABC index. Corollary 1.2. Among all connected graphs with n vertices, the graph with minimal ABC index is a tree. Although it is fairly easy to show that the star graph S is a tree with maximal ABC n index [23], despite many attempts in the last years, it is still an open problem the charac- terization of trees with minimal ABC index (also refereed as minimal-ABC trees). The aim of this research is to make a step forward towards the full characterizations of minimal-ABC trees. In Section we give an overview of already known structural properties of the minimal- ABC trees, while in Section we present a few new properties. In the appendix we present some simpler results that are used in the proofs in Section . In the sequel, we present an additional notation that will be used in the rest of the paper. A tree is called a rooted tree if one vertex has been designated the root. In a rooted tree, the parent of a vertex is the vertex connected to it on the path to the root; every vertex except the root has a unique parent. A child of a vertex v is a vertex of which v is the parent. A vertex of degree one is a pendant vertex. The breadth-ﬁrst search is a graph search algorithm that begins at the root vertex and explores all its children vertices, beginning with the most right child and ending with the most left child. Then for each of those children, it explores ##LTLine## ##LTLine##

text: graph

text: topological

text: index,

text: the

text: atom-bond

text: connectivity

text: (ABC)

text: index,

text: and

text: showed

text: that

text: it

text: can

text: be

text: a

text: valuable

text: predictive

text: tool

text: in

text: the

text: study

text: of

text: the

text: heat

text: of

text: formation

text: in

text: alikeness.

text: Ten

text: years

text: later

text: Estrada

text: elaborated

text: a

text: novel

text: quantum-theory-like

text: justiﬁcation

text: for

text: this

text: topological

text: index.

text: After

text: that

text: revelation,

text: the

text: interest

text: of

text: ABC-index

text: has

text: grown

text: rapidly.

text: Additionaly,

text: the

text: physico-chemical

text: applicability

text: of

text: the

text: ABC

text: index

text: was

text: conﬁrmed

text: and

text: extended

text: in

text: several

text: studies

text: Let

text: G

text: (V,

text: E)

text: be

text: a

text: simple

text: undirected

text: graph

text: of

text: order

text: n

text: |V

text: and

text: size

text: m

text: |E|.

text: For

text: v

text: V

text: (G),

text: the

text: degree

text: of

text: v,

text: denoted

text: by

text: d(v),

text: is

text: the

text: number

text: of

text: edges

text: incident

text: to

text: v.

text: For

text: an

text: edge

text: uv

text: in

text: G,

text: let

text: d(u)

text: d(v)

text: f(d(u),

text: d(v))

text: d(u)d(v)

text: Then

text: the

text: atom-bond

text: connectivity

text: index

text: of

text: G

text: is

text: deﬁned

text: as

text: ABC(G)

text: f(d(u),

text: d(v)).

text: uv∈E(G)

text: As

text: a

text: new

text: and

text: well

text: motivated

text: graph

text: invariant,

text: the

text: ABC

text: index

text: has

text: attracted

text: a

text: lot

text: of

text: interest

text: in

text: the

text: last

text: several

text: years

text: both

text: in

text: mathematical

text: and

text: chemical

text: research

text: communities

text: and

text: numerous

text: results

text: and

text: structural

text: properties

text: of

text: ABC

text: index

text: were

text: established

text: The

text: fact

text: that

text: adding

text: an

text: edge

text: in

text: a

text: graph

text: strictly

text: increases

text: its

text: ABC

text: index

text: (or

text: equiva-

text: lently

text: that

text: deleting

text: an

text: edge

text: in

text: a

text: graph

text: strictly

text: decreases

text: its

text: ABC

text: index

text: has

text: the

text: following

text: two

text: immediate

text: consequences.

text: Corollary

text: Among

text: all

text: connected

text: graphs

text: with

text: n

text: vertices,

text: the

text: complete

text: graph

text: K

text: has

text: n

text: maximal

text: value

text: of

text: ABC

text: index.

text: Corollary

text: Among

text: all

text: connected

text: graphs

text: with

text: n

text: vertices,

text: the

text: graph

text: with

text: minimal

text: ABC

text: index

text: is

text: a

text: tree.

text: Although

text: it

text: is

text: fairly

text: easy

text: to

text: show

text: that

text: the

text: star

text: graph

text: S

text: is

text: a

text: tree

text: with

text: maximal

text: ABC

text: n

text: index

text: despite

text: many

text: attempts

text: in

text: the

text: last

text: years,

text: it

text: is

text: still

text: an

text: open

text: problem

text: the

text: charac-

text: terization

text: of

text: trees

text: with

text: minimal

text: ABC

text: index

text: (also

text: refereed

text: as

text: minimal-ABC

text: trees).

text: The

text: aim

text: of

text: this

text: research

text: is

text: to

text: make

text: a

text: step

text: forward

text: towards

text: the

text: full

text: characterizations

text: of

text: minimal-ABC

text: trees.

text: In

text: Section

text: we

text: give

text: an

text: overview

text: of

text: already

text: known

text: structural

text: properties

text: of

text: the

text: minimal-

text: ABC

text: trees,

text: while

text: in

text: Section

text: we

text: present

text: a

text: few

text: new

text: properties.

text: In

text: the

text: appendix

text: we

text: present

text: some

text: simpler

text: results

text: that

text: are

text: used

text: in

text: the

text: proofs

text: in

text: Section

text: In

text: the

text: sequel,

text: we

text: present

text: an

text: additional

text: notation

text: that

text: will

text: be

text: used

text: in

text: the

text: rest

text: of

text: the

text: paper.

text: A

text: tree

text: is

text: called

text: a

text: rooted

text: tree

text: if

text: one

text: vertex

text: has

text: been

text: designated

text: the

text: root.

text: In

text: a

text: rooted

text: tree,

text: the

text: parent

text: of

text: a

text: vertex

text: is

text: the

text: vertex

text: connected

text: to

text: it

text: on

text: the

text: path

text: to

text: the

text: root;

text: every

text: vertex

text: except

text: the

text: root

text: has

text: a

text: unique

text: parent.

text: A

text: child

text: of

text: a

text: vertex

text: v

text: is

text: a

text: vertex

text: of

text: which

text: v

text: is

text: the

text: parent.

text: A

text: vertex

text: of

text: degree

text: one

text: is

text: a

text: pendant

text: vertex.

text: The

text: breadth-ﬁrst

text: search

text: is

text: a

text: graph

text: search

text: algorithm

text: that

text: begins

text: at

text: the

text: root

text: vertex

text: and

text: explores

text: all

text: its

text: children

text: vertices,

text: beginning

text: with

text: the

text: most

text: right

text: child

text: and

text: ending

text: with

text: the

text: most

text: left

text: child.

text: Then

text: for

text: each

text: of

text: those

text: children,

text: it

text: explores

text: ##LTLine##

text: ##LTLine##


text: 21 for every positive u . It follows that −5f(d(u ), 5) + 6f(d(u ) + 1, 4) increases with u . Thus, 1 1 1 1 we obtain ABC(G(cid:48)) − ABC(G) < lim −5f(d(u ), 5) + 6f(d(u ) + 1, 4) − 2f(2, 1) + f(4, 3) 1 1 u1→∞ = −0.00478432. Subcase 2.2. u is not the root of G. 1 Then, d(u ) ≥ 6, and the change of the ABC index is 1 ABC(G(cid:48)) − ABC(G) = −f(d(u ), d(x)) + f(d(u ) + 1, d(x)) 1 1 (cid:88) 5(−f(d(u ), 5) + f(d(u ) + 1, 4)) 1 1 d(u1)−6 + (−f(d(u ), d(y )) + f(d(u ) + 1, d(y ))) 1 i 1 i i=1 −f(d(u ), 2) − f(2, 1) + f(d(u ) + 1, 4) + f(4, 3), 1 1 where x is a parent vertex of u . Applying the same arguments as in Subcase 2.1., we obtain 1 ABC(G(cid:48)) − ABC(G) < −5f(d(u ), 5) + 6f(d(u ) + 1, 4) − 2f(2, 1) + f(4, 3), 1 1 which is identical with (28) from Subcase 2.1. Therefore, the change of the ABC index after applying the transformation T is negative. 2 Applying repeatedly above considered transformations T and T , one can obtain a tree 1 2 with at most 4 B -branches, that has smaller ABC index than the assumed minimal-ABC 4 tree. References [1] M. B. Ahmadi, S. A. Hosseini, P. Salehi Nowbandegani, On trees with minimal atom bond connectivity index, MATCH Commun. Math. Comput. Chem. 69 (2013) 559–563. [2] M. B. Ahmadi, S. A. Hosseini, M. Zarrinderakht, On large trees with minimal atom–bond connectivity index, MATCH Commun. Math. Comput. Chem. 69 (2013) 565–569. [3] M. B. Ahmadi, M. Sadeghimehr, Atom bond connectivity index of an inﬁnite class NS [n] of dendrimer nanostars, Optoelectron. Adv. Mat. 4 (2010) 1040–1042. 1 [4] A. T. Balaban, Highly discriminating distance-based topological index, Chem. Phys. Lett. 89 (1982) 399–404. [5] D. Bonchev, Information Theoretic Indices for Characterization of Chemical Structures, Research Studies Press, Chichester, 1983. [6] J. Chen, X. Guo, Extreme atom-bond connectivity index of graphs, MATCH Commun. Math. Comput. Chem. 65 (2011) 713–722. [7] J. Chen, X. Guo, The atom-bond connectivity index of chemical bicyclic graphs, Appl. Math. J. Chinese Univ. 27 (2012) 243–252. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: for

text: every

text: positive

text: u

text: It

text: follows

text: that

text: increases

text: with

text: u

text: Thus,

text: we

text: obtain

text: ABC(G)

text: lim

text: u1→∞

text: Subcase

text: u

text: is

text: not

text: the

text: root

text: of

text: G.

text: Then,

text: d(u

text: and

text: the

text: change

text: of

text: the

text: ABC

text: index

text: is

text: ABC(G)

text: −f(d(u

text: d(x))

text: f(d(u

text: d(x))

text: f(d(u

text: (−f(d(u

text: d(y

text: f(d(u

text: d(y

text: i

text: i

text: −f(d(u

text: f(d(u

text: where

text: x

text: is

text: a

text: parent

text: vertex

text: of

text: u

text: Applying

text: the

text: same

text: arguments

text: as

text: in

text: Subcase

text: we

text: obtain

text: ABC(G)

text: which

text: is

text: identical

text: with

text: from

text: Subcase

text: Therefore,

text: the

text: change

text: of

text: the

text: ABC

text: index

text: after

text: applying

text: the

text: transformation

text: T

text: is

text: negative.

text: Applying

text: repeatedly

text: above

text: considered

text: transformations

text: T

text: and

text: T

text: one

text: can

text: obtain

text: a

text: tree

text: with

text: at

text: most

text: B

text: -branches,

text: that

text: has

text: smaller

text: ABC

text: index

text: than

text: the

text: assumed

text: minimal-ABC

text: tree.

text: References

text: M.

text: B.

text: Ahmadi,

text: S.

text: A.

text: Hosseini,

text: P.

text: Salehi

text: Nowbandegani,

text: On

text: trees

text: with

text: minimal

text: atom

text: bond

text: connectivity

text: index,

text: MATCH

text: Commun.

text: Math.

text: Comput.

text: Chem.

text: M.

text: B.

text: Ahmadi,

text: S.

text: A.

text: Hosseini,

text: M.

text: Zarrinderakht,

text: On

text: large

text: trees

text: with

text: minimal

text: atom–bond

text: connectivity

text: index,

text: MATCH

text: Commun.

text: Math.

text: Comput.

text: Chem.

text: M.

text: B.

text: Ahmadi,

text: M.

text: Sadeghimehr,

text: Atom

text: bond

text: connectivity

text: index

text: of

text: an

text: inﬁnite

text: class

text: NS

text: [n]

text: of

text: dendrimer

text: nanostars,

text: Optoelectron.

text: Adv.

text: Mat.

text: A.

text: T.

text: Balaban,

text: Highly

text: discriminating

text: distance-based

text: topological

text: index,

text: Chem.

text: Phys.

text: Lett.

text: D.

text: Bonchev,

text: Information

text: Theoretic

text: Indices

text: for

text: Characterization

text: of

text: Chemical

text: Structures,

text: Research

text: Studies

text: Press,

text: Chichester,

text: J.

text: Chen,

text: X.

text: Guo,

text: Extreme

text: atom-bond

text: connectivity

text: index

text: of

text: graphs,

text: MATCH

text: Commun.

text: Math.

text: Comput.

text: Chem.

text: J.

text: Chen,

text: X.

text: Guo,

text: The

text: atom-bond

text: connectivity

text: index

text: of

text: chemical

text: bicyclic

text: graphs,

text: Appl.

text: Math.

text: J.

text: Chinese

text: Univ.

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text: TABLE I other capability to gather, integrate, mine, or fuse data from SYSTEM REQUIREMENTS: THE DESIRABLE CAPABILITIES OF distributed databases. This version of GICoordinator does not GICOORDINATOR AND HUMAN include this functionality. Component Functionalities It is an approach in an organizational structure to make a 1- Specify high-level strategies for Human coordination of agents strategic plan that states that how agents can get from the 2- Revise and reﬁne strategies current state of the world through a sequence of actions to a 3- Organize data (geographic, location-based, desired goal state. Strategic planning in a disaster response and non-geographic) of the problem Spatial Database 4- Integrate the software agent with GIS team includes (1) specify a response objective (a high-level 5- Support information sharing with agents strategy) for the team and decompose it into prioritized sub- and with other information systems, and goals (threads), (2) make a strategic decision by assignment support integration of information systems of agents to threads, and (3) evaluate and adapt a strategic 6- Support human decisions by providing tools for geospatial reasoning, geographic decision to new crisis situations [3]. GIS information management and visualization This capability calculates a set of right choices for making 7- Support the software agent by providing a strategic decision either in execution of the human strategy GIS analysis 8- Interact with human or in adaption of current assignments [5]. These choices 9- Assign agents (ﬁeld units) to human are presented for the IC to select the best choice according high-level strategies in strategic planning to his intuition or to delegate the system to search for the 10- Assign geospatial-temporal macro tasks to agents in centralized scheduling optimal one. Moreover, the system autonomously releases 11- Adapt and revise the strategic decision right agents from a right thread in a right time during tasks Software Agent 12- Adapt and revise the schedule execution in order to revise the strategic decision. 13- Search for an optimal plan 14- Allocate resources (refuges) to damage points under human strategies This capability is required to dynamically assign spatial- 15- Adjust and reﬁne human strategies temporal macro tasks to agents under human strategic deci- 16- Provide an interface to interact with human sions in centralized scheduling in order to minimize the over- 17- Percept and observe the environment via all time of tasks execution. Two main results are achieved the spatial database by running this algorithm: (1) a feasible schedule and (2) an adaption time. A schedule is composed of a number of macro decisions that specify: (1) what task type is going GICoordinator provides geo-technologies and geo- to be done, (2) who (a subset of agents) are assigned to informatics that support human decisions and increase do this assignment, (3) where (a macro geographic object) functionality of the software agent. Three signiﬁcant contains a subset of tasks, (4) when operations start, (5) when purposes are of importance: (1) visualization of information operations ﬁnish, (6) how many tasks are estimated to be via 3D maps or thematic maps e.g. display spatial done, and (7) what task types and how many of them are distribution of tasks, states of macro tasks, action plans, and estimated to be revealed in this location after to ﬁnish this tasks schedule, (2) geospatial reasoning that includes spatial job [6]. relationships analysis, network analysis, proximity analysis, etc, and (3) information management that includes retrieval, This capability calculates an optimal strategic plan, a insert, query, update, and manipulate information of spatial complete schedule, and a overall minimum time of tasks database. It enables the IC to interact with geographic execution. It evolves assignment of agents to threads from information, and it supports human decisions by providing an initial state to a speciﬁed goal state. Results state that a better situational awareness. how and when agents can reach a deﬁned objective. These GICoordinator requires necessary information for compu- results support human decisions and assist the IC to evaluate tation because it is mainly focus on multi-agent coordination the quality of his strategy, reﬁne it, or deﬁne a better strategy. techniques. Different types of information are provided by distributed systems. Therefore GICoordinator requires an- This capability optimally allocates available refuges to res- cued people in medical transportation operations according to human strategic decisions. Results state that which injured persons should be transported to which refuges in order to optimize an objective function. It is difﬁcult for an IC to specify a good strategy and timely revise it during emergency management. An IC may specify a bad or wrong high-level strategy that leads to a very big catastrophe with signiﬁcantly severe consequences. This capability recommends human for adjustment and reﬁnement Fig. 1. The architecture of GICoordinator of his strategy in real-time. ##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##

text: TABLE

text: I

text: other

text: capability

text: to

text: gather,

text: integrate,

text: mine,

text: or

text: fuse

text: data

text: from

text: SYSTEM

label: caption

text: REQUIREMENTS:

text: THE

text: DESIRABLE

text: CAPABILITIES

text: OF

text: distributed

text: databases.

text: This

text: version

text: of

text: GICoordinator

text: does

text: not

text: GICOORDINATOR

text: AND

text: HUMAN

text: include

text: this

text: functionality.

text: Component

text: Functionalities

text: It

text: is

text: an

text: approach

text: in

text: an

text: organizational

text: structure

text: to

text: make

text: a

text: Specify

text: high-level

text: strategies

text: for

text: Human

text: coordination

text: of

text: agents

text: strategic

text: plan

text: that

text: states

text: that

text: how

text: agents

text: can

text: get

text: from

text: the

text: Revise

text: and

text: reﬁne

text: strategies

text: current

text: state

text: of

text: the

text: world

text: through

text: a

text: sequence

text: of

text: actions

text: to

text: a

text: Organize

text: data

text: (geographic,

text: location-based,

text: desired

text: goal

text: state.

text: Strategic

text: planning

text: in

text: a

text: disaster

text: response

text: and

text: non-geographic)

text: of

text: the

text: problem

text: Spatial

text: Database

text: Integrate

text: the

text: software

text: agent

text: with

text: GIS

text: team

text: includes

text: specify

text: a

text: response

text: objective

text: (a

text: high-level

text: Support

text: information

text: sharing

text: with

text: agents

text: strategy)

text: for

text: the

text: team

text: and

text: decompose

text: it

text: into

text: prioritized

text: sub-

text: and

text: with

text: other

text: information

text: systems,

text: and

text: goals

text: (threads),

text: make

text: a

text: strategic

text: decision

text: by

text: assignment

text: support

text: integration

text: of

text: information

text: systems

text: of

text: agents

text: to

text: threads,

text: and

text: evaluate

text: and

text: adapt

text: a

text: strategic

text: Support

text: human

text: decisions

text: by

text: providing

text: tools

text: for

text: geospatial

text: reasoning,

text: geographic

text: decision

text: to

text: new

text: crisis

text: situations

text: GIS

text: information

text: management

text: and

text: visualization

text: This

text: capability

text: calculates

text: a

text: set

text: of

text: right

text: choices

text: for

text: making

text: Support

text: the

text: software

text: agent

text: by

text: providing

text: a

text: strategic

text: decision

text: either

text: in

text: execution

text: of

text: the

text: human

text: strategy

text: GIS

text: analysis

text: Interact

text: with

text: human

text: or

text: in

text: adaption

text: of

text: current

text: assignments

text: These

text: choices

text: Assign

text: agents

text: (ﬁeld

text: units)

text: to

text: human

text: are

text: presented

text: for

text: the

text: IC

text: to

text: select

text: the

text: best

text: choice

text: according

text: high-level

text: strategies

text: in

text: strategic

text: planning

text: to

text: his

text: intuition

text: or

text: to

text: delegate

text: the

text: system

text: to

text: search

text: for

text: the

text: Assign

text: geospatial-temporal

text: macro

text: tasks

text: to

text: agents

text: in

text: centralized

text: scheduling

text: optimal

text: one.

text: Moreover,

text: the

text: system

text: autonomously

text: releases

text: Adapt

text: and

text: revise

text: the

text: strategic

text: decision

text: right

text: agents

text: from

text: a

text: right

text: thread

text: in

text: a

text: right

text: time

text: during

text: tasks

text: Software

text: Agent

text: Adapt

text: and

text: revise

text: the

text: schedule

text: execution

text: in

text: order

text: to

text: revise

text: the

text: strategic

text: decision.

text: Search

text: for

text: an

text: optimal

text: plan

text: Allocate

text: resources

text: (refuges)

text: to

text: damage

text: points

text: under

text: human

text: strategies

text: This

text: capability

text: is

text: required

text: to

text: dynamically

text: assign

text: spatial-

text: Adjust

text: and

text: reﬁne

text: human

text: strategies

text: temporal

text: macro

text: tasks

text: to

text: agents

text: under

text: human

text: strategic

text: deci-

text: Provide

text: an

text: interface

text: to

text: interact

text: with

text: human

text: sions

text: in

text: centralized

text: scheduling

text: in

text: order

text: to

text: minimize

text: the

text: over-

text: Percept

text: and

text: observe

text: the

text: environment

text: via

text: all

text: time

text: of

text: tasks

text: execution.

text: Two

text: main

text: results

text: are

text: achieved

text: the

text: spatial

text: database

text: by

text: running

text: this

text: algorithm:

text: a

text: feasible

text: schedule

text: and

text: an

text: adaption

text: time.

text: A

text: schedule

text: is

text: composed

text: of

text: a

text: number

text: of

text: macro

text: decisions

text: that

text: specify:

text: what

text: task

text: type

text: is

text: going

text: GICoordinator

text: provides

text: geo-technologies

text: and

text: geo-

text: to

text: be

text: done,

text: who

text: (a

text: subset

text: of

text: agents)

text: are

text: assigned

text: to

text: informatics

text: that

text: support

text: human

text: decisions

text: and

text: increase

text: do

text: this

text: assignment,

text: where

text: (a

text: macro

text: geographic

text: object)

text: functionality

text: of

text: the

text: software

text: agent.

text: Three

text: signiﬁcant

text: contains

text: a

text: subset

text: of

text: tasks,

text: when

text: operations

text: start,

text: when

text: purposes

text: are

text: of

text: importance:

text: visualization

text: of

text: information

text: operations

text: ﬁnish,

text: how

text: many

text: tasks

text: are

text: estimated

text: to

text: be

text: via

text: maps

text: or

text: thematic

text: maps

text: e.g.

text: display

text: spatial

text: done,

text: and

text: what

text: task

text: types

text: and

text: how

text: many

text: of

text: them

text: are

text: distribution

text: of

text: tasks,

text: states

text: of

text: macro

text: tasks,

text: action

text: plans,

text: and

text: estimated

text: to

text: be

text: revealed

text: in

text: this

text: location

text: after

text: to

text: ﬁnish

text: this

text: tasks

text: schedule,

text: geospatial

text: reasoning

text: that

text: includes

text: spatial

text: job

text: relationships

text: analysis,

text: network

text: analysis,

text: proximity

text: analysis,

text: etc,

text: and

text: information

text: management

text: that

text: includes

text: retrieval,

text: This

text: capability

text: calculates

text: an

text: optimal

text: strategic

text: plan,

text: a

text: insert,

text: query,

text: update,

text: and

text: manipulate

text: information

text: of

text: spatial

text: complete

text: schedule,

text: and

text: a

text: overall

text: minimum

text: time

text: of

text: tasks

text: database.

text: It

text: enables

text: the

text: IC

text: to

text: interact

text: with

text: geographic

text: execution.

text: It

text: evolves

text: assignment

text: of

text: agents

text: to

text: threads

text: from

text: information,

text: and

text: it

text: supports

text: human

text: decisions

text: by

text: providing

text: an

text: initial

text: state

text: to

text: a

text: speciﬁed

text: goal

text: state.

text: Results

text: state

text: that

text: a

text: better

text: situational

text: awareness.

text: how

text: and

text: when

text: agents

text: can

text: reach

text: a

text: deﬁned

text: objective.

text: These

text: GICoordinator

text: requires

text: necessary

text: information

text: for

text: compu-

text: results

text: support

text: human

text: decisions

text: and

text: assist

text: the

text: IC

text: to

text: evaluate

text: tation

text: because

text: it

text: is

text: mainly

text: focus

text: on

text: multi-agent

text: coordination

text: the

text: quality

text: of

text: his

text: strategy,

text: reﬁne

text: it,

text: or

text: deﬁne

text: a

text: better

text: strategy.

text: techniques.

text: Different

text: types

text: of

text: information

text: are

text: provided

text: by

text: distributed

text: systems.

text: Therefore

text: GICoordinator

text: requires

text: an-

text: This

text: capability

text: optimally

text: allocates

text: available

text: refuges

text: to

text: res-

text: cued

text: people

text: in

text: medical

text: transportation

text: operations

text: according

text: to

text: human

text: strategic

text: decisions.

text: Results

text: state

text: that

text: which

text: injured

text: persons

text: should

text: be

text: transported

text: to

text: which

text: refuges

text: in

text: order

text: to

text: optimize

text: an

text: objective

text: function.

text: It

text: is

text: difﬁcult

text: for

text: an

text: IC

text: to

text: specify

text: a

text: good

text: strategy

text: and

text: timely

text: revise

text: it

text: during

text: emergency

text: management.

text: An

text: IC

text: may

text: specify

text: a

text: bad

text: or

text: wrong

text: high-level

text: strategy

text: that

text: leads

text: to

text: a

text: very

text: big

text: catastrophe

text: with

text: signiﬁcantly

text: severe

text: consequences.

text: This

text: capability

text: recommends

text: human

text: for

text: adjustment

text: and

text: reﬁnement

text: Fig.

text: The

text: architecture

text: of

text: GICoordinator

text: of

text: his

text: strategy

text: in

text: real-time.

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Meanwhile, we see accumulating evidences of 10-3 10-2 10-1 100 101 failed predictions of the traditional magnetar model (Tong PHsL & Xu 2011). The discovery of a low magnetic ﬁeld mag- Fig.1 Distribution of magnetars on the period period-derivative netar in 2010 challenged the traditional magnetar model di- diagram of pulsars. Blue squares are magnetars, while empty rectly (Rea et al. 2010; Tong & Xu 2012). The low magnetic squares are radio-loud magnetars. Green diamonds are X-ray dim ﬁeld magnetar (SGR 0418+5729) is an old neutron star with × isolated neutron stars, cyan circles are central compact objects, red surface dipole ﬁeld less than 7.5 1012 G. But at the same stars are rotating radio transients, magenta triangles are intermit- time, it can have magnetar-like activities. It will not be too tent pulsars, and black dots are rotation powered pulsars (includ- incorrect to say that none of the predictions of the traditional ing normal pulsars and millisecond pulsars). Figure 1 in Tong & magnetar model is observed. The failed predictions of the Xu (2011), with updates. traditional magnetar model require rethinking the magnetar idea. There are 3+1 things to do concerning magnetars be seen directly form their distribution on the period period- 1. What’s the origin of strong magnetic ﬁeld in magnetars derivative diagram of pulsars (Figure 1). and pulsars? This is relevant to whether AXPs and SGRs The ﬁrst giant ﬂare of magnetars was observed in 1979 are magnetars or not. (Mazets et al. 1979). The magnetar idea (neutron stars with − 2. What’s the emission mechanism of magnetar multiwave magnetic ﬁelds as high as 1014 1015 G) was proposed radiation properties? The multiwave emission mecha- by several authors in 1992 (Duncan & Thompson 1992; nism of magnetars remains illusive. Usov 1992; Paczynski 1992). It was Paczynski (1992) who 3. The birth and environment of magnetars. The environ- pointed that the super-strong magnetic ﬁeld may explain the ment of magnetars possibly includes: fallback disks, super-Eddington luminosity of the 1979 giant ﬂare. Timing pulsar wind nebulae, supernova remnants, and binary observation of the period and period derivative of one SGR companions (if the magnetar is in a binary system). (by RXTE) was thought to be the conﬁrming evidence of 4. The relation between magnetars and other pulsar-like magnetars (Kouveliotou et al. 1998). By assuming that the objects. Magnetars are just a special kind of pulsars. nuetron star is slowed down by emitting magnetic dipole ra- (cid:112) Therefore, we must understand various pulsar-like ob- diation, the neutron star’s surface magnetic ﬁeld is (Tong et jects at the same time. We want to know what’s the rela- al. 2013a) tion between magnetars and X-ray dim isolated neutron × B = 3.2 1019 PP˙ G, (1) stars (XDINSs), central compact objects (CCOs), high magnetic ﬁeld pulsars (HBPSRs), and most importantly where B is the star’s surface magnetic ﬁeld, P is the pul- − normal pulsars. sation period, P˙ is the period derivative. For SGR 1806 20, × − its period and period derivative are 7.47 s and 8.24 10 11, There exist several alternative modelings of magnetars respectively (Kouveliotou et al. 1998). According to the (Tong & Xu 2011). − magnetic dipole braking assumption, SGR 1806 20 is a neutron star with age of 1500 years1 and surface magnetic 1. Twisted magnetosphere model (Thompson, Lyutikov & × ﬁeld as high as 8 1014 G. Therefore, it is a young neutron Kulkarni 2002). The magnetar magnetosphere may be star with super-strong magnetic ﬁeld2(i.e., magnetar). Later, globally twisted. And a partially twisted magnetosphere not only SGRs but also AXPs are thought to be magnetars. In 2008, the traditional magnetar model was (Mereghetti 3 The quantum critical ﬁeld is deﬁned as when the electron cyclotron 1 The characteristic age is deﬁned as P/(2P˙). emnaegrgnyetiecqﬁuaellsdihtsigrheesrt tmhaansstheeneqrugayn,tBumq c=ritimcae2elh¯cv3alu=e,4q.u4an×tu1m01e3leGct.roIdnya- ∼ 2 For normal pulsars, their typical magnetic ﬁeld is 1012 G namics must be employed to treat the microscopic processes. (cid:13)c 2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim www.an-journal.org ##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##

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text: Astron. Nachr. / AN (2006) 791 model was also investigated (i.e., corona model of mag- to be high-energy gamma-ray emitters detectable by Fermi- netars, Beloborodov & Thompson 2007; Beleborodov LAT (Cheng & Zhang 2001). However, the X-ray luminous 2009). AXP 4U 0142+61 is not detected in Fermi-LAT observa- 2. Wind braking of magnetars (Tong & Xu 2012; Tong tions (Sasmaz Mus & Gogus 2010). There is also no sig- et al. 2013a). In the wind braking model, magnetars niﬁcant detection in Fermi-LAT observations of all AXPs are neutron stars with strong multipole ﬁeld. A strong and SGRs (Abdo et al. 2010). Then there are conﬂicts be- dipole ﬁeld is no longer needed. A particle outﬂow dom- tween outer gap model in the case of magnetars and Fermi- inates the rotational energy loss rate of magnetars. The LAT observations (Tong, Song & Xu 2010, 2011). AXP multipole ﬁeld is responsible for the braking torque, per- 4U 0142+62 should have been detected by Fermi-LAT. The sistent and burst emissions of magnetars. In the wind present observational upper limits are already below the the- braking model of magnetars, timing events of magne- oretical calculations for some parameter space. There are tars should always be accompanied by radiative events. possibly two solutions for this conﬂict: 3. Magnetothermal evolution model (Vigano et al. 2013). 1. AXPs and SGRs are fallback disk systems. Then most Coupled evolution of magnetic ﬁeld and temperature of of them are not expected to be gamma-ray emitters. neutron stars may explain the surface thermal emission 2. AXPs/SGRs are magnetars braked down by a particle of various kinds of pulsar-like objects. wind. If a particle outﬂow dominates the magnetar’s 4. Fallback disk model (Alpar 2001; Alpar, Ertan & rotational energy loss rate, then the corresponding sur- Kaliskan 2011). A neutron star with a fallback disk may face dipole ﬁeld can be much lower (i.e., 10-100 times explain some aspects of magnetars. And there is already smaller). Meanwhile, in the presence of a particle wind, a disk found in AXP 4U 0142+61 (through optical/IR vacuum gaps can not exist in the magnetosphere. observations, Wang, Chakrabarty & Kaplan 2006). Fermi deeper observations may help us to distinguish be- 5. Accretion induced star-quake model (Xu, Tao & Yang tween the fallback disk model and magnetar model for 2006; Xu 2007). The self-conﬁned quark star surface AXPs and SGRs. can explain the super-Eddington luminosity of magnetar giant ﬂares. Accretion from a fallback disk is responsi- ble for the spindown and persistent emissions. A quark 2.2 Hard X-ray emission cutoff star with a fallback disk may provide another way uni- fying different kinds of pulsar-like objects. The soft X-ray spectral of magnetars are uausally made 6. Quark-nova remnant model (Ouyed, Leahy & Nieber- up of two components: a blackbody component (with tem- ∼ gal 2007, 2011). After the supernova, there may be a perature 0.5 keV) and power law component (with a pho- ∼ − transition from a neutron star to a quark star. This is ton index Γ 3 4). Extrapolating the soft X-ray compo- dubbed as a quark-nova. A quark star with some kind nents, magnetars are not expected to luminous in the hard of quark-nova remnant may explain several pulsar-like X-ray range. However, INTEGRAL observations found that objects (including AXPs and SGRs). many magnetars are detected in hard X-ray (Gotz et al. 7. White dwarf model (Paczynski 1990; Malheiro, Reuda 2006). The hard X-ray can be ﬁtted with a power law with ∼ & Rufﬁni 2012). If the central star of AXP and SGR is photon index Γ 1. And the hard X-ray energy output is a white dwarf, then the rotational energy of the white about half the magnetar’s total electromagnetic energy out- dwarf is enough to power the persistent emissions of put. Therefore, the hard X-ray component of magnetars is AXPs and SGRs. a distinct component compared with the soft X-ray com- ponent. And it is an indispensable part of the magnetar’s The ﬁrst three models are in the magnetar domain (i.e., they energy budget. There are various proposals for the origin all involve neutron stars powered by strong magnetic ﬁeld). of hard X-ray emission both in the magnetar model and the The last four models are more or less beyond the magnetar fallback disk model. And the hard X-ray emission cutoff is model. crucial to distinguish between different models. A possible cutoff in the hard X-ray emission of AXP 4U 0142+61 is reported recently (Wang, Tong & Guo 2013). 2 Toward an understanding of magnetar Using nearly nine years INTEGRAL observations, a possi- multiwave radiation properties ≈ ble cutoff of 130 keV is seen. With a cutoff of 130 keV, we can rule out hard X-ray emission models involving ultra- 2.1 Non-detection in Fermi-LAT observations relativistic electrons. Both the microscope and bulk motion of electrons should be at most mildly relativistic. During the In the traditional model of magnetars, magnetars are nine years interval, the total hard X-ray luminosity is rela- neutron stars with both strong dipole ﬁeld and strong multi- tively stable. Therefore, a persistent source of electrons are pole ﬁeld. Although the magnetic ﬁeld at the magnetar sur- needed rather than transient. Therefore, there must exist a face is very high, the magnetic ﬁeld in the outer magneto- persistent component of particle outﬂow. Hard X-ray Mod- sphere is relatively low. Therefore, particles may be acceler- ulation Telescope (known HXMT, by China) can determine ated in the outer magnetosphere and magnetars are expected the cutoff energy more accurately in the future. www.an-journal.org (cid:13)c 2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ##LTLine##

text: Astron.

text: Nachr.

text: AN

text: model

label: list

text: was

text: also

text: investigated

text: (i.e.,

text: corona

text: model

text: of

text: mag-

text: to

text: be

text: high-energy

text: gamma-ray

text: emitters

text: detectable

text: by

text: Fermi-

text: netars,

text: Beloborodov

text: Thompson

text: Beleborodov

text: LAT

text: (Cheng

text: Zhang

text: However,

text: the

text: X-ray

text: luminous

text: AXP

text: is

text: not

text: detected

text: in

text: Fermi-LAT

text: observa-

text: Wind

text: braking

text: of

text: magnetars

text: (Tong

text: Xu

text: Tong

text: tions

text: (Sasmaz

text: Mus

text: Gogus

text: There

text: is

text: also

text: no

text: sig-

text: et

text: al.

text: In

text: the

text: wind

text: braking

text: model,

text: magnetars

text: niﬁcant

text: detection

text: in

text: Fermi-LAT

text: observations

text: of

text: all

text: AXPs

text: are

text: neutron

text: stars

text: with

text: strong

text: multipole

text: ﬁeld.

text: A

text: strong

text: and

text: SGRs

text: (Abdo

text: et

text: al.

text: Then

text: there

text: are

text: conﬂicts

text: be-

text: dipole

text: ﬁeld

text: is

text: no

text: longer

text: needed.

text: A

text: particle

text: outﬂow

text: dom-

text: tween

text: outer

text: gap

text: model

text: in

text: the

text: case

text: of

text: magnetars

text: and

text: Fermi-

text: inates

text: the

text: rotational

text: energy

text: loss

text: rate

text: of

text: magnetars.

text: The

text: LAT

text: observations

text: (Tong,

text: Song

text: Xu

text: AXP

text: multipole

text: ﬁeld

text: is

text: responsible

text: for

text: the

text: braking

text: torque,

text: per-

text: should

text: have

text: been

text: detected

text: by

text: Fermi-LAT.

text: The

text: sistent

text: and

text: burst

text: emissions

text: of

text: magnetars.

text: In

text: the

text: wind

text: present

text: observational

text: upper

text: limits

text: are

text: already

text: below

text: the

text: the-

text: braking

text: model

text: of

text: magnetars,

text: timing

text: events

text: of

text: magne-

text: oretical

text: calculations

text: for

text: some

text: parameter

text: space.

text: There

text: are

text: tars

text: should

text: always

text: be

text: accompanied

text: by

text: radiative

text: events.

text: possibly

text: two

text: solutions

text: for

text: this

text: conﬂict:

text: Magnetothermal

text: evolution

text: model

text: (Vigano

text: et

text: al.

text: AXPs

text: and

text: SGRs

text: are

text: fallback

text: disk

text: systems.

text: Then

text: most

text: Coupled

text: evolution

text: of

text: magnetic

text: ﬁeld

text: and

text: temperature

text: of

text: of

text: them

text: are

text: not

text: expected

text: to

text: be

text: gamma-ray

text: emitters.

text: neutron

text: stars

text: may

text: explain

text: the

text: surface

text: thermal

text: emission

text: AXPs/SGRs

text: are

text: magnetars

text: braked

text: down

text: by

text: a

text: particle

text: of

text: various

text: kinds

text: of

text: pulsar-like

text: objects.

text: wind.

text: If

text: a

text: particle

text: outﬂow

text: dominates

text: the

text: magnetar’s

text: Fallback

text: disk

text: model

text: (Alpar

text: Alpar,

text: Ertan

text: rotational

text: energy

text: loss

text: rate,

text: then

text: the

text: corresponding

text: sur-

text: Kaliskan

text: A

text: neutron

text: star

text: with

text: a

text: fallback

text: disk

text: may

text: face

text: dipole

text: ﬁeld

text: can

text: be

text: much

text: lower

text: (i.e.,

text: times

text: explain

text: some

text: aspects

text: of

text: magnetars.

text: And

text: there

text: is

text: already

text: smaller).

text: Meanwhile,

text: in

text: the

text: presence

text: of

text: a

text: particle

text: wind,

text: a

text: disk

text: found

text: in

text: AXP

text: (through

text: optical/IR

text: vacuum

text: gaps

text: can

text: not

text: exist

text: in

text: the

text: magnetosphere.

text: observations,

text: Wang,

text: Chakrabarty

text: Kaplan

text: Fermi

text: deeper

text: observations

text: may

text: help

text: us

text: to

text: distinguish

text: be-

text: Accretion

text: induced

text: star-quake

text: model

text: (Xu,

text: Tao

text: Yang

text: tween

text: the

text: fallback

text: disk

text: model

text: and

text: magnetar

text: model

text: for

text: Xu

text: The

text: self-conﬁned

text: quark

text: star

text: surface

text: AXPs

text: and

text: SGRs.

text: can

text: explain

text: the

text: super-Eddington

text: luminosity

text: of

text: magnetar

text: giant

text: ﬂares.

text: Accretion

text: from

text: a

text: fallback

text: disk

text: is

text: responsi-

text: ble

text: for

text: the

text: spindown

text: and

text: persistent

text: emissions.

text: A

text: quark

text: Hard

text: X-ray

text: emission

text: cutoff

text: star

text: with

text: a

text: fallback

text: disk

text: may

text: provide

text: another

text: way

text: uni-

text: fying

text: different

text: kinds

text: of

text: pulsar-like

text: objects.

text: The

text: soft

text: X-ray

text: spectral

text: of

text: magnetars

text: are

text: uausally

text: made

text: Quark-nova

text: remnant

text: model

text: (Ouyed,

text: Leahy

text: Nieber-

text: up

text: of

text: two

text: components:

text: a

text: blackbody

text: component

text: (with

text: tem-

text: gal

text: After

text: the

text: supernova,

text: there

text: may

text: be

text: a

text: perature

text: keV)

text: and

text: power

text: law

text: component

text: (with

text: a

text: pho-

text: transition

text: from

text: a

text: neutron

text: star

text: to

text: a

text: quark

text: star.

text: This

text: is

text: ton

text: index

text: Γ

text: Extrapolating

text: the

text: soft

text: X-ray

text: compo-

text: dubbed

text: as

text: a

text: quark-nova.

text: A

text: quark

text: star

text: with

text: some

text: kind

text: nents,

text: magnetars

text: are

text: not

text: expected

text: to

text: luminous

text: in

text: the

text: hard

text: of

text: quark-nova

text: remnant

text: may

text: explain

text: several

text: pulsar-like

text: X-ray

text: range.

text: However,

text: INTEGRAL

text: observations

text: found

text: that

text: objects

text: (including

text: AXPs

text: and

text: SGRs).

text: many

text: magnetars

text: are

text: detected

text: in

text: hard

text: X-ray

text: (Gotz

text: et

text: al.

text: White

text: dwarf

text: model

text: (Paczynski

text: Malheiro,

text: Reuda

text: The

text: hard

text: X-ray

text: can

text: be

text: ﬁtted

text: with

text: a

text: power

text: law

text: with

text: Rufﬁni

text: If

text: the

text: central

text: star

text: of

text: AXP

text: and

text: SGR

text: is

text: photon

text: index

text: Γ

text: And

text: the

text: hard

text: X-ray

text: energy

text: output

text: is

text: a

text: white

text: dwarf,

text: then

text: the

text: rotational

text: energy

text: of

text: the

text: white

text: about

text: half

text: the

text: magnetar’s

text: total

text: electromagnetic

text: energy

text: out-

text: dwarf

text: is

text: enough

text: to

text: power

text: the

text: persistent

text: emissions

text: of

text: put.

text: Therefore,

text: the

text: hard

text: X-ray

text: component

text: of

text: magnetars

text: is

text: AXPs

text: and

text: SGRs.

text: a

text: distinct

text: component

text: compared

text: with

text: the

text: soft

text: X-ray

text: com-

text: ponent.

text: And

text: it

text: is

text: an

text: indispensable

text: part

text: of

text: the

text: magnetar’s

text: The

text: ﬁrst

text: three

text: models

text: are

text: in

text: the

text: magnetar

text: domain

text: (i.e.,

text: they

text: energy

text: budget.

text: There

text: are

text: various

text: proposals

text: for

text: the

text: origin

text: all

text: involve

text: neutron

text: stars

text: powered

text: by

text: strong

text: magnetic

text: ﬁeld).

text: of

text: hard

text: X-ray

text: emission

text: both

text: in

text: the

text: magnetar

text: model

text: and

text: the

text: The

text: last

text: four

text: models

text: are

text: more

text: or

text: less

text: beyond

text: the

text: magnetar

text: fallback

text: disk

text: model.

text: And

text: the

text: hard

text: X-ray

text: emission

text: cutoff

text: is

text: model.

text: crucial

text: to

text: distinguish

text: between

text: different

text: models.

text: A

text: possible

text: cutoff

text: in

text: the

text: hard

text: X-ray

text: emission

text: of

text: AXP

text: is

text: reported

text: recently

text: (Wang,

text: Tong

text: Guo

text: Toward

text: an

text: understanding

text: of

text: magnetar

text: Using

text: nearly

text: nine

text: years

text: INTEGRAL

text: observations,

text: a

text: possi-

text: multiwave

text: radiation

text: properties

text: ble

text: cutoff

text: of

text: keV

text: is

text: seen.

text: With

text: a

text: cutoff

text: of

text: keV,

text: we

text: can

text: rule

text: out

text: hard

text: X-ray

text: emission

text: models

text: involving

text: ultra-

text: Non-detection

text: in

text: Fermi-LAT

text: observations

text: relativistic

text: electrons.

text: Both

text: the

text: microscope

text: and

text: bulk

text: motion

text: of

text: electrons

text: should

text: be

text: at

text: most

text: mildly

text: relativistic.

text: During

text: the

text: In

text: the

text: traditional

text: model

text: of

text: magnetars,

text: magnetars

text: are

text: nine

text: years

text: interval,

text: the

text: total

text: hard

text: X-ray

text: luminosity

text: is

text: rela-

text: neutron

text: stars

text: with

text: both

text: strong

text: dipole

text: ﬁeld

text: and

text: strong

text: multi-

text: tively

text: stable.

text: Therefore,

text: a

text: persistent

text: source

text: of

text: electrons

text: are

text: pole

text: ﬁeld.

text: Although

text: the

text: magnetic

text: ﬁeld

text: at

text: the

text: magnetar

text: sur-

text: needed

text: rather

text: than

text: transient.

text: Therefore,

text: there

text: must

text: exist

text: a

text: face

text: is

text: very

text: high,

text: the

text: magnetic

text: ﬁeld

text: in

text: the

text: outer

text: magneto-

text: persistent

text: component

text: of

text: particle

text: outﬂow.

text: Hard

text: X-ray

text: Mod-

text: sphere

text: is

text: relatively

text: low.

text: Therefore,

text: particles

text: may

text: be

text: acceler-

text: ulation

text: Telescope

text: (known

text: HXMT,

text: by

text: China)

text: can

text: determine

text: ated

text: in

text: the

text: outer

text: magnetosphere

text: and

text: magnetars

text: are

text: expected

text: the

text: cutoff

text: energy

text: more

text: accurately

text: in

text: the

text: future.

text: ##LTLine##


text: Sabine Hossenfelder Disentangling the Black Hole Vacuum up-modes and dn-modes (following the notation of [8]) ∼ − φ (v) Θ(v)e iω ln(κv)/κ , (2) dn ∼ − − φ (v) Θ( v)eiω ln( κv)/κ , (3) up where Θ is the Heaviside step function, 4M = 1/κ, and normalizations are ommited because we will not need them (we only care about the difference to the standard case). The up-modes cover only the region before the last escaping ray at v = 0, and the dn- modes only cover the space after that ray. Here and in the following we will suppress an index (cid:126)k referring to the momentum of the mode, or ω, l, m referring to its frequency and spherical harmonics composition respectively. One thus has to keep in mind that the following vectors and matrices are a compressed notation that represents inﬁnitely many entries. We will neglect spherical harmonics and backscatterin(cid:0)g on the pote(cid:1)ntial wall. (cid:0) (cid:1) It is common to use the linear combinations φ (v) = c φ + wφ¯ , φ (v) = c φ + wφ¯ (4) d dn up p up dn − − − w = e πω/κ , c = (1 w2) 1/2 , (5) (a bar denotes complex conjugation) for the in-modes, which is sometimes referred to as Wald’s base [11].(cid:90)The usefulness of thes(cid:90)e (orthogonal) modes comes from them being of positive frequency with respect to v, a consequence of the equalities ∞ ∞ − − dve iωvφ (v) = dve iωvφ (v) = 0 . (6) −∞ d −∞ p In these integrals, the prefactor w eats up the branch-cut contribution of the logarithm, resulting in a cancellation of the two parts of the integrals over v > 0 and v < 0. For details, see [8], Appendix H.4. Since φ and φ are of positive frequency, one can d p use them to deﬁne the in-vacuum while at the same time simplifying the Bogoliubov- transformations. The ﬁeld can be expressed in this basis as ΦT = (φ , φ ) , (7) in d p where T denotes transposition and is just used for better display of the vector. The corre- sponding annihilation and creation operators will be denoted as † † † a = (a , a ) , a = (a , a ) (8) in d p in d p and are by default row-vectors. For the out-base one can use ΦT = (φ , φ ) , (9) out out dn ∼ − where φ e iωu. The in-base and out-base are related by the Bogoliubov transfor- out mation − Φ = A¯ TΦ BTΦ¯ . (10) in out out 3 ##LTLine##

text: Sabine

text: Hossenfelder

text: Disentangling

text: the

text: Black

text: Hole

text: Vacuum

text: up-modes

text: and

text: dn-modes

text: (following

text: the

text: notation

text: of

text: φ

text: (v)

text: Θ(v)e

text: iω

text: ln(κv)/κ

text: dn

text: φ

text: (v)

text: Θ(

text: v)eiω

text: ln(

text: κv)/κ

text: up

text: where

text: Θ

text: is

text: the

text: Heaviside

text: step

text: function,

text: and

text: normalizations

text: are

text: ommited

text: because

text: we

text: will

text: not

text: need

text: them

text: (we

text: only

text: care

text: about

text: the

text: difference

text: to

text: the

text: standard

text: case).

text: The

text: up-modes

text: cover

text: only

text: the

text: region

text: before

text: the

text: last

text: escaping

text: ray

text: at

text: v

text: and

text: the

text: dn-

text: modes

text: only

text: cover

text: the

text: space

text: after

text: that

text: ray.

text: Here

text: and

text: in

text: the

text: following

text: we

text: will

text: suppress

text: an

text: index

text: referring

text: to

text: the

text: momentum

text: of

text: the

text: mode,

text: or

text: ω,

text: l,

text: m

text: referring

text: to

text: its

text: frequency

text: and

text: spherical

text: harmonics

text: composition

text: respectively.

text: One

text: thus

text: has

text: to

text: keep

text: in

text: mind

text: that

text: the

text: following

text: vectors

text: and

text: matrices

text: are

text: a

text: compressed

text: notation

text: that

text: represents

text: inﬁnitely

text: many

text: entries.

text: We

text: will

text: neglect

text: spherical

text: harmonics

text: and

text: on

text: the

text: wall.

text: It

text: is

text: common

text: to

text: use

text: the

text: linear

text: combinations

text: φ

text: (v)

text: c

text: φ

text: wφ¯

text: φ

text: (v)

text: c

text: φ

text: wφ¯

text: d

text: dn

text: up

text: p

text: up

text: dn

text: w

text: e

text: πω/κ

text: c

text: (a

text: bar

text: denotes

text: complex

text: conjugation)

text: for

text: the

text: in-modes,

text: which

text: is

text: sometimes

text: referred

text: to

text: as

text: Wald’s

text: base

text: usefulness

text: of

text: (orthogonal)

text: modes

text: comes

text: from

text: them

text: being

text: of

text: positive

text: frequency

text: with

text: respect

text: to

text: v,

text: a

text: consequence

text: of

text: the

text: equalities

text: dve

text: iωvφ

text: (v)

text: dve

text: iωvφ

text: (v)

text: d

text: p

text: In

text: these

text: integrals,

text: the

text: prefactor

text: w

text: eats

text: up

text: the

text: branch-cut

text: contribution

text: of

text: the

text: logarithm,

text: resulting

text: in

text: a

text: cancellation

text: of

text: the

text: two

text: parts

text: of

text: the

text: integrals

text: over

text: v

text: and

text: v

text: For

text: details,

text: see

text: Appendix

text: Since

text: φ

text: and

text: φ

text: are

text: of

text: positive

text: frequency,

text: one

text: can

text: d

text: p

text: use

text: them

text: to

text: deﬁne

text: the

text: in-vacuum

text: while

text: at

text: the

text: same

text: time

text: simplifying

text: the

text: Bogoliubov-

text: transformations.

text: The

text: ﬁeld

text: can

text: be

text: expressed

text: in

text: this

text: basis

text: as

text: ΦT

text: (φ

text: φ

text: in

text: d

text: p

text: where

text: T

text: denotes

text: transposition

text: and

text: is

text: just

text: used

text: for

text: better

text: display

text: of

text: the

text: vector.

text: The

text: corre-

text: sponding

text: annihilation

text: and

text: creation

text: operators

text: will

text: be

text: denoted

text: as

text: a

text: (a

text: a

text: a

text: (a

text: a

text: in

text: d

text: p

text: in

text: d

text: p

text: and

text: are

text: by

text: default

text: row-vectors.

text: For

text: the

text: out-base

text: one

text: can

text: use

text: ΦT

text: (φ

text: φ

text: out

text: out

text: dn

text: where

text: φ

text: e

text: iωu.

text: The

text: in-base

text: and

text: out-base

text: are

text: related

text: by

text: the

text: Bogoliubov

text: transfor-

text: out

text: mation

text: Φ

text: A¯

text: TΦ

text: BTΦ¯

text: in

text: out

text: out

text: ##LTLine##