text: SHELL et al.: BARE DEMO OF IEEETRAN.CLS FOR JOURNALS 5 0.07 35 0.1 EEM algorithm Average SE Exhaustive search ρ 0.09 Hertz] K = 3, N = 2 ertz] 30 ESEEMM aallggoorriitthhmm 0.08 / H 25 0.07 e 0.06 / Joul /sec 20 0.06 / K = 3, N = 3 s E[bits K = 2, N = 2 SE[bit 15 00..0054 ρ E 0.05 e ge ag 10 0.03 a r er K = 2, N = 3 ve K 0.02 v A A 5 0.01 0.04 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 45 50 55 60 Number of Dual Iterations P (dBm) max Fig. 3: Average EE versus the total number of inner iterations (a) Average SE and ρ versus Pmax for K = 30, 60 and 120. of Dinkelbach’s method required for reaching convergence 0.2 0.1 when using the simulation parameters from Table I with Average EE P = 0dBm, M = 0 and with a cell radius of 1km. 0.18 ρ 0.09 max SEM algorithm rtz] 0.16 EEM algorithm 0.08 e H system shown in Fig. 1, where the RNs are placed halfway / 0.14 0.07 e between the BS and the cell-edge. The channel is modeled by ul o 0.12 0.06 the path-loss [18] and uncorrelated Rayleigh fading obeying /J the complex normal distribution, CN(0, 1). It is assumed bits 0.1 0.05 ρ that the BS-to-RN link has line-of-sight (LOS) propagation, E[ 0.08 0.04 implying that a RN was placed on a tall building. However, the E K e 0.06 0.03 BS-to-UE and RN-to-UE links typically have no LOS, since g a these links are likely to be blocked by buildings and other er 0.04 0.02 v large obstructing objects. An independently-random set of UE A 0.02 0.01 locations as well as fading channel realizations are generated 0 0 for each channel sample. For fair comparisons, the metrics 0 5 10 15 20 25 30 35 40 45 50 55 60 used are the average SE per subcarrier and the average EE per P (dBm) max subcarrier. On the other hand, the sum-rate may be calculated by multiplying the average SE by NW. Additionally, ρ is (b) Average EE and ρ versus Pmax for K = 30, 60 and 120. introduced to denote the average fraction of the total number Fig. 4: Average SE, EE and ρ, and the effect of an increasing of subcarriers that are used for AF transmission. Thus, ρ number of users, K, for a system with simulation parameters quantiﬁes the beneﬁt attained from introducing RNs into the from Table I with N = 128, M = 3 and with a cell radius of system. 1.5km. Fig. 3 illustrates the convergence behavior of Dinkelbach’s method invoked for maximizing the EE for a selection of the maximum EE as well as of the SE attained. Furthermore, small-scale systems, averaged over 104 different channel re- alizations. Since the problem size is small, it is possible to Fig. 4 shows that as Pmax is increased, the SEM algorithm continues to allocate more power in order to achieve a higher generate also the exhaustive-search based solution within a average SE at the cost of EE, while the EEM algorithm reasonable computation time. As seen in Fig. 3, Dinkelbach’s attains the maximum EE and does not continue to increase method converges to the optimal value within forty inner its attainable SE by sacriﬁcing the achieved EE. On the other iterations. This result demonstrates that the EEM algorithm based on Dinkelbach’s method indeed obtains the optimal hand, ρ is inversely proportional to K. This indicates that as the multi-user diversity increases, the subcarriers are less power and subcarrier allocation, even though the relaxed likely to be allocated for AF transmissions, simply because problem is solved and a high receiver’s SNR was assumed. there are more favorable BS-to-UE channels owing to having Additionally, the EEM algorithm may be employed for evaluating the effects of system-level design choices on the more UEs nearer to the cell-center. Moreover, the value of ρ network’s SE and EE. The effect of K on the average EE and decreases as Pmax increases, because there is more power to allocate to the BS-to-UE links for UEs near the cell-center, SE3 is depicted in Fig. 4. As expected, upon increasing K, which beneﬁt from a reduced pathloss as well as from a more the multi-user diversity of the system is increased, since the efﬁcient power ampliﬁer at the BS. scheduler is allowed to choose its subcarrier allocations from a larger pool of channel gains. This results in an increase of both VI. CONCLUSIONS In this paper, the joint power and subcarrier allocation 3N.B. The maximum SE is obtained in the ﬁrst outer iteration of Dinkel- problem was formulated for maximizing the EE in a multi- bcoacnhsu’smmpteitohno.d with q0 = 0, since this equates to zero penalty for any power relay aided multi-user OFDMA cellular network. 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text: SHELL

text: et

text: al.:

text: BARE

text: DEMO

text: OF

text: IEEETRAN.CLS

text: FOR

text: JOURNALS

text: EEM

text: algorithm

text: Average

text: SE

text: Exhaustive

text: search

text: ρ

text: Hertz]

text: K

text: N

text: ertz]

text: ESEEMM

text: aallggoorriitthhmm

text: H

text: e

text: Joul

text: /sec

text: K

text: N

text: s

text: E[bits

text: K

text: N

text: SE[bit

text: ρ

text: E

text: e

text: ge

text: ag

text: a

text: r

text: er

text: K

text: N

text: ve

text: K

text: v

text: A

text: A

text: Number

text: of

text: Dual

text: Iterations

text: P

text: (dBm)

text: max

text: Fig.

text: Average

text: EE

text: versus

text: the

text: total

text: number

text: of

text: inner

text: iterations

text: (a)

text: Average

text: SE

text: and

text: ρ

text: versus

text: Pmax

text: for

text: K

text: and

text: of

text: Dinkelbach’s

text: method

text: required

text: for

text: reaching

text: convergence

text: when

text: using

text: the

text: simulation

text: parameters

text: from

text: Table

text: I

text: with

text: Average

text: EE

text: P

text: M

text: and

text: with

text: a

text: cell

text: radius

text: of

text: ρ

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text: EEM

text: algorithm

text: e

text: H

text: system

text: shown

text: in

text: Fig.

text: where

text: the

text: RNs

text: are

text: placed

text: halfway

text: e

text: between

text: the

text: BS

text: and

text: the

text: cell-edge.

text: The

text: channel

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text: by

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text: o

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text: path-loss

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text: (LOS)

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text: E[

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text: that

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text: RN

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text: a

text: tall

text: building.

text: However,

text: the

text: E

text: K

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text: RN-to-UE

text: links

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text: have

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text: be

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text: An

text: independently-random

text: set

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text: UE

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text: as

text: well

text: as

text: fading

text: channel

text: realizations

text: are

text: generated

text: for

text: each

text: channel

text: sample.

text: For

text: fair

text: comparisons,

text: the

text: metrics

text: used

text: are

text: the

text: average

text: SE

text: per

text: subcarrier

text: and

text: the

text: average

text: EE

text: per

text: P

text: (dBm)

text: max

text: subcarrier.

text: On

text: the

text: other

text: hand,

text: the

text: sum-rate

text: may

text: be

text: calculated

text: by

text: multiplying

text: the

text: average

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text: Additionally,

text: ρ

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text: ρ

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text: the

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text: SE,

text: EE

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text: ρ,

text: and

text: the

text: effect

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text: increasing

text: of

text: subcarriers

text: that

text: are

text: used

text: for

text: AF

text: transmission.

text: Thus,

text: ρ

text: number

text: of

text: users,

text: K,

text: for

text: a

text: system

text: with

text: simulation

text: parameters

text: quantiﬁes

text: the

text: beneﬁt

text: attained

text: from

text: introducing

text: RNs

text: into

text: the

text: from

text: Table

text: I

text: with

text: N

text: M

text: and

text: with

text: a

text: cell

text: radius

text: of

text: system.

text: Fig.

text: illustrates

text: the

text: convergence

text: behavior

text: of

text: Dinkelbach’s

text: method

text: invoked

text: for

text: maximizing

text: the

text: EE

text: for

text: a

text: selection

text: of

text: the

text: maximum

text: EE

text: as

text: well

text: as

text: of

text: the

text: SE

text: attained.

text: Furthermore,

text: small-scale

text: systems,

text: averaged

text: over

text: different

text: channel

text: re-

text: alizations.

text: Since

text: the

text: problem

text: size

text: is

text: small,

text: it

text: is

text: possible

text: to

text: Fig.

text: shows

text: that

text: as

text: Pmax

text: is

text: increased,

text: the

text: SEM

text: algorithm

text: continues

text: to

text: allocate

text: more

text: power

text: in

text: order

text: to

text: achieve

text: a

text: higher

text: generate

text: also

text: the

text: exhaustive-search

text: based

text: solution

text: within

text: a

text: average

text: SE

text: at

text: the

text: cost

text: of

text: EE,

text: while

text: the

text: EEM

text: algorithm

text: reasonable

text: computation

text: time.

text: As

text: seen

text: in

text: Fig.

text: Dinkelbach’s

text: attains

text: the

text: maximum

text: EE

text: and

text: does

text: not

text: continue

text: to

text: increase

text: method

text: converges

text: to

text: the

text: optimal

text: value

text: within

text: forty

text: inner

text: its

text: attainable

text: SE

text: by

text: sacriﬁcing

text: the

text: achieved

text: EE.

text: On

text: the

text: other

text: iterations.

text: This

text: result

text: demonstrates

text: that

text: the

text: EEM

text: algorithm

text: based

text: on

text: Dinkelbach’s

text: method

text: indeed

text: obtains

text: the

text: optimal

text: hand,

text: ρ

text: is

text: inversely

text: proportional

text: to

text: K.

text: This

text: indicates

text: that

text: as

text: the

text: multi-user

text: diversity

text: increases,

text: the

text: subcarriers

text: are

text: less

text: power

text: and

text: subcarrier

text: allocation,

text: even

text: though

text: the

text: relaxed

text: likely

text: to

text: be

text: allocated

text: for

text: AF

text: transmissions,

text: simply

text: because

text: problem

text: is

text: solved

text: and

text: a

text: high

text: receiver’s

text: SNR

text: was

text: assumed.

text: there

text: are

text: more

text: favorable

text: BS-to-UE

text: channels

text: owing

text: to

text: having

text: Additionally,

text: the

text: EEM

text: algorithm

text: may

text: be

text: employed

text: for

text: evaluating

text: the

text: effects

text: of

text: system-level

text: design

text: choices

text: on

text: the

text: more

text: UEs

text: nearer

text: to

text: the

text: cell-center.

text: Moreover,

text: the

text: value

text: of

text: ρ

text: network’s

text: SE

text: and

text: EE.

text: The

text: effect

text: of

text: K

text: on

text: the

text: average

text: EE

text: and

text: decreases

text: as

text: Pmax

text: increases,

text: because

text: there

text: is

text: more

text: power

text: to

text: allocate

text: to

text: the

text: BS-to-UE

text: links

text: for

text: UEs

text: near

text: the

text: cell-center,

text: is

text: depicted

text: in

text: Fig.

text: As

text: expected,

text: upon

text: increasing

text: K,

text: which

text: beneﬁt

text: from

text: a

text: reduced

text: pathloss

text: as

text: well

text: as

text: from

text: a

text: more

text: the

text: multi-user

text: diversity

text: of

text: the

text: system

text: is

text: increased,

text: since

text: the

text: efﬁcient

text: power

text: ampliﬁer

text: at

text: the

text: BS.

text: scheduler

text: is

text: allowed

text: to

text: choose

text: its

text: subcarrier

text: allocations

text: from

text: a

text: larger

text: pool

text: of

text: channel

text: gains.

text: This

text: results

text: in

text: an

text: increase

text: of

text: both

text: VI.

text: CONCLUSIONS

text: In

text: this

text: paper,

text: the

text: joint

text: power

text: and

text: subcarrier

text: allocation

text: The

text: maximum

text: SE

text: is

text: obtained

text: in

text: the

text: ﬁrst

text: outer

text: iteration

text: of

text: Dinkel-

text: problem

text: was

text: formulated

text: for

text: maximizing

text: the

text: EE

text: in

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text: (cid:90) (cid:90) 14 (cid:98) (cid:100) Malte Henkel and Stoimen S(cid:100)toimenov √ ∞ 2π f(ζ +iλ) = dγ ei(ζ++iλ)γ+F (γ ) = dγ Θ(γ )ei(ζ++iλ)γ+F (γ ) + 0 + + + R + + + + (36) such that by i(cid:90)nverting (31), the two-point functio(cid:90)n F ﬁnally be(cid:100)comes, with x1 = x2 | |− F = πt√22xπ1 (cid:90)R2dζ+dζ− e−(cid:100)i(γ1+γ2)ζ+e−i(γ(cid:90)1−γ2)ζ− Rdγ+Θ(γ+(cid:90))F+(γ+)e−γ+λeiγ+ζ+ | |− = t√2x1 dγ Θ(γ )F (γ )e−γ+λ dζ− e−i(γ1−γ2)ζ− dζ ei(γ+−γ1−γ2)ζ+ π 2π R + + + + R R + − − | |− = δ(γ1 γ2)Θ(γ1)F0,+(γ1)e 2γ1λ t 2x1 (37) where in the last line, two δ-functions were used and F contains the unspeciﬁed 0,+ dependence on the positive constant γ . 1 ∈ − For λ < 0,−we can −use again the sec|on||d|−form of the lemma, with fλ H2 , and ﬁnd F = δ(γ1 γ2)Θ( γ1)F0,−(γ1)e2γ1 λ t 2x1≥. These two forms can be combined into a single one, immediately generalised to d (cid:104)1 dim(cid:12) ensio(cid:12)n(cid:105)s, and assumed con- (cid:12) (cid:12) tinuous in r and rotation-invariant as well (cid:12) (cid:12) · F(t,r) = δx1,x2δ(γ1 −γ2)|t|−2x1 exp −2 γ1t r F0(γ21) (38) 3.3 Discussion Surprisingly, our attempts to establish sufﬁcient criteria that the two-point func- | | → tion F(t,s;r) vanishes in the limit r 0, led to qualitatively different types of results. A) For the Schro¨dinger algebra with the representation (4), the extension to the cor- responding maximal parabolic sub-algebra and the dualisation of the mass M has led to the form (30). It is maximally asymmetric under permutation of its two scal- − ing operators and obeys a causality condition t1 t2 > 0. In applications, it should predict the form of response functions. Indeed, we quoted in section 2 several exam- ples where response functions of non-equilibrium many-body systems undergoing physical ageing are described by (30), or logarithmic extensions thereof. B) For the conformal Galilean algebra with the representation (6), there is no central extension which would produce a Bargman superselection rule for the rapidities γ. An analogous extension to the maximal parabolic sub-algebra and the dualisation of the rapidities rather produced the form (38). It is fully symmetric under the permu- tation of its scaling operators. This is a characteristic of correlation functions. Our result therefore suggests that searches for physical applications of the conformal Galilean algebra should concentrate on studying co-variant correlators, rather than response functions. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: −2

text: r

text: Discussion

text: Surprisingly,

text: our

text: attempts

text: to

text: establish

text: sufﬁcient

text: criteria

text: that

text: the

text: two-point

text: func-

text: tion

text: F(t,s;r)

text: vanishes

text: in

text: the

text: limit

text: r

text: led

text: to

text: qualitatively

text: different

text: types

text: of

text: results.

text: A)

text: For

text: the

text: Schro¨dinger

text: algebra

text: with

text: the

text: representation

text: the

text: extension

text: to

text: the

text: cor-

text: responding

text: maximal

text: parabolic

text: sub-algebra

text: and

text: the

text: dualisation

text: of

text: the

text: mass

text: M

text: has

text: led

text: to

text: the

text: form

text: It

text: is

text: maximally

text: asymmetric

text: under

text: permutation

text: of

text: its

text: two

text: scal-

text: ing

text: operators

text: and

text: obeys

text: a

text: causality

text: condition

text: In

text: applications,

text: it

text: should

text: predict

text: the

text: form

text: of

text: response

text: functions.

text: Indeed,

text: we

text: quoted

text: in

text: section

text: several

text: exam-

text: ples

text: where

text: response

text: functions

text: of

text: non-equilibrium

text: many-body

text: systems

text: undergoing

text: physical

text: ageing

text: are

text: described

text: by

text: or

text: logarithmic

text: extensions

text: thereof.

text: B)

text: For

text: the

text: conformal

text: Galilean

text: algebra

text: with

text: the

text: representation

text: there

text: is

text: no

text: central

text: extension

text: which

text: would

text: produce

text: a

text: Bargman

text: superselection

text: rule

text: for

text: the

text: rapidities

text: γ.

text: An

text: analogous

text: extension

text: to

text: the

text: maximal

text: parabolic

text: sub-algebra

text: and

text: the

text: dualisation

text: of

text: the

text: rapidities

text: rather

text: produced

text: the

text: form

text: It

text: is

text: fully

text: symmetric

text: under

text: the

text: permu-

text: tation

text: of

text: its

text: scaling

text: operators.

text: This

text: is

text: a

text: characteristic

text: of

text: correlation

text: functions.

text: Our

text: result

text: therefore

text: suggests

text: that

text: searches

text: for

text: physical

text: applications

text: of

text: the

text: conformal

text: Galilean

text: algebra

text: should

text: concentrate

text: on

text: studying

text: co-variant

text: correlators,

text: rather

text: than

text: response

text: functions.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Physical ageing and new representations of some Lie algebras of local scale-invariance 5 − − n+1 − n+1 − − n(n+1) − Xn = tn+(cid:0)1∂t 2 (cid:1)tnr∂r 2 xtn n(n+1)ξtn 4 Mtn 1r2 (10) − which also−admits a−more general invariant Schro¨dinger operator S = 2M ∂t ∂2 + 2Mt 1 x+ξ 1 , but without any constraint on neither x nor ξ [48]. This r 2 representation of age(1) is characterised by the scalars (x,ξ,M ). The name of this algebra comes from its use as dynamical symmetry in physical ageing, which can be observed in strongly interacting many-body systems quenched from a disordered initial state to the co-existence regime below the critical temperature Tc > 0 where several equivalent equilibrium states exist. For example, for quenched Ising spins in ≥ d 2 dimensions without disorder, nor frustrations, and with a purely relaxational dynamics without any conservation law, it can be shown that the dynamical expo- nent z = 2 [10]. Assuming age(d) as a dynamical symmetry predicts the form of (cid:104) (cid:105) the two-time linear response function of the average order parameter φ(t,r) with respect to its canonically co(cid:12)(cid:12)njugate(cid:68)magnetic ﬁeld(cid:69)h(s,r(cid:48)) [18, (cid:18)21, 22] (cid:19) (cid:12) (cid:101) (cid:12) (cid:104) (cid:105) δ φ(t,r) − − t (cid:20)r2 (cid:21) R(t,s;r) = δh(s,0) (cid:102) = φ(t,r)φ(s,0) = s 1 aFR s, t −s (11) h=0 − − (cid:48)− − − − (cid:48) −1 FR(y,u) = F0 δ(M M )Θ(y 1)y1+a λR/z(y 1) 1 a exp 2M u (cid:101) where the standard Janssen-d(cid:101)e Dominicis formalism (see e.g. [23]) was used to re- (cid:104) (cid:105) write R = φφ as a correlator of the order parameter scaling o(cid:101)perator φ and the (cid:101) conjugate response operator φ. Both of these are assumed to be quasi-primary under (cid:101) (cid:48) age(d). The ageing exponents a,a ,λR/z are related to x,ξ and x,ξ in a known way, (cid:48) − e.g. a a = 2z (ξ +ξ). F0 is a normalisation constant and the Θ-function expresses the causality condition y = t/s > 1, of which we shall say more in section b(cid:101)elow. Spatial translation-invariance was assumed. The case of a Schro¨dinger-invariance response is obtained if one sets ξ = ξ = 0, (cid:48) hence a = a in (11). Eq. (11) has been conﬁrmed in numerous spin systems (e.g. Ising, Potts, XY, spherical, Hilhorst-van Leeuven, Edwards-Wilkinson,...models) which undergo simple ageing with z = 2, both for the time- and space-dependence; either from a known exact solution or using simulational data. For a(cid:101)detailed review, see [23]. Cur- rent empirical evidence suggests that for quenches to low temperatures T < Tc, one (cid:48) should have for the second scaling dimensions ξ + ξ = 0, hence a = a . However, the full representation (10) of age(1) is needed in the d = 1 Glauber-Ising model, (cid:48) − − where the exact solution rep(cid:48) −rodu(cid:54) ces (11) with a = 0, a a = 21 and λR = 1 [22]. One might anticipate that a a = 0 for quench(cid:54)es to the critical point T = Tc. For critical quenches, one has in general z = 2, such that (11) does no longer apply. However, the form of the auto-response R(t,s) = R(t,s;0) does not contain the precise spatial form so that at least that part of (11) can be used for preliminary tests of dynamical symmetries for generic values of z. In section , various logarithmic representations of these algebras and some of their properties are reviewed. Known applications to physical ageing will be brieﬂy ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Physical

label: title

text: ageing

text: and

text: new

text: representations

text: of

text: some

text: Lie

text: algebras

text: of

text: local

text: scale-invariance

text: Xn

text: xtn

text: Mtn

text: which

text: also−admits

text: a−more

text: general

text: invariant

text: Schro¨dinger

text: operator

text: S

text: ∂t

text: ∂2

text: x+ξ

text: but

text: without

text: any

text: constraint

text: on

text: neither

text: x

text: nor

text: ξ

text: This

text: r

text: representation

text: of

text: is

text: characterised

text: by

text: the

text: scalars

text: (x,ξ,M

text: The

text: name

text: of

text: this

text: algebra

text: comes

text: from

text: its

text: use

text: as

text: dynamical

text: symmetry

text: in

text: physical

text: ageing,

text: which

text: can

text: be

text: observed

text: in

text: strongly

text: interacting

text: many-body

text: systems

text: quenched

text: from

text: a

text: disordered

text: initial

text: state

text: to

text: the

text: co-existence

text: regime

text: below

text: the

text: critical

text: temperature

text: Tc

text: where

text: several

text: equivalent

text: equilibrium

text: states

text: exist.

text: For

text: example,

text: for

text: quenched

text: Ising

text: spins

text: in

text: d

text: dimensions

text: without

text: disorder,

text: nor

text: frustrations,

text: and

text: with

text: a

text: purely

text: relaxational

text: dynamics

text: without

text: any

text: conservation

text: law,

text: it

text: can

text: be

text: shown

text: that

text: the

text: dynamical

text: expo-

text: nent

text: z

text: Assuming

text: age(d)

text: as

text: a

text: dynamical

text: symmetry

text: predicts

text: the

text: form

text: of

text: the

text: two-time

text: linear

text: response

text: function

text: of

text: the

text: average

text: order

text: parameter

text: φ(t,r)

text: with

text: respect

text: to

text: its

text: canonically

text: δ

text: φ(t,r)

text: t

text: R(t,s;r)

text: s

text: aFR

text: s,

text: t

text: −s

text: −1

text: FR(y,u)

text: δ(M

text: M

text: )Θ(y

text: λR/z(y

text: a

text: exp

text: u

text: where

text: the

text: standard

text: Dominicis

text: formalism

text: (see

text: e.g.

text: was

text: used

text: to

text: re-

text: write

text: R

text: φφ

text: as

text: a

text: correlator

text: of

text: the

text: order

text: parameter

text: scaling

text: φ

text: and

text: the

text: conjugate

text: response

text: operator

text: φ.

text: Both

text: of

text: these

text: are

text: assumed

text: to

text: be

text: quasi-primary

text: under

text: age(d).

text: The

text: ageing

text: exponents

text: a,a

text: ,λR/z

text: are

text: related

text: to

text: x,ξ

text: and

text: x,ξ

text: in

text: a

text: known

text: way,

text: e.g.

text: a

text: a

text: (ξ

text: +ξ).

text: is

text: a

text: normalisation

text: constant

text: and

text: the

text: Θ-function

text: expresses

text: the

text: causality

text: condition

text: y

text: t/s

text: of

text: which

text: we

text: shall

text: say

text: more

text: in

text: section

text: Spatial

text: translation-invariance

text: was

text: assumed.

text: The

text: case

text: of

text: a

text: Schro¨dinger-invariance

text: response

text: is

text: obtained

text: if

text: one

text: sets

text: ξ

text: ξ

text: hence

text: a

text: a

text: in

text: Eq.

text: has

text: been

text: conﬁrmed

text: in

text: numerous

text: spin

text: systems

text: (e.g.

text: Ising,

text: Potts,

text: XY,

text: spherical,

text: Hilhorst-van

text: Leeuven,

text: Edwards-Wilkinson,...models)

text: which

text: undergo

text: simple

text: ageing

text: with

text: z

text: both

text: for

text: the

text: time-

text: and

text: space-dependence;

text: either

text: from

text: a

text: known

text: exact

text: solution

text: or

text: using

text: simulational

text: data.

text: For

text: review,

text: see

text: Cur-

text: rent

text: empirical

text: evidence

text: suggests

text: that

text: for

text: quenches

text: to

text: low

text: temperatures

text: T

text: Tc,

text: one

text: should

text: have

text: for

text: the

text: second

text: scaling

text: dimensions

text: ξ

text: ξ

text: hence

text: a

text: a

text: However,

text: the

text: full

text: representation

text: of

text: is

text: needed

text: in

text: the

text: d

text: Glauber-Ising

text: model,

text: where

text: the

text: exact

text: solution

text: ces

text: with

text: a

text: a

text: a

text: and

text: λR

text: One

text: might

text: anticipate

text: that

text: a

text: a

text: for

text: to

text: the

text: critical

text: point

text: T

text: Tc.

text: For

text: critical

text: quenches,

text: one

text: has

text: in

text: general

text: z

text: such

text: that

text: does

text: no

text: longer

text: apply.

text: However,

text: the

text: form

text: of

text: the

text: auto-response

text: R(t,s)

text: does

text: not

text: contain

text: the

text: precise

text: spatial

text: form

text: so

text: that

text: at

text: least

text: that

text: part

text: of

text: can

text: be

text: used

text: for

text: preliminary

text: tests

text: of

text: dynamical

text: symmetries

text: for

text: generic

text: values

text: of

text: z.

text: In

text: section

text: various

text: logarithmic

text: representations

text: of

text: these

text: algebras

text: and

text: some

text: of

text: their

text: properties

text: are

text: reviewed.

text: Known

text: applications

text: to

text: physical

text: ageing

text: will

text: be

text: brieﬂy

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: I. INTRODUCTION Quantum ﬁeld theory in curved spacetimes [1–4] is the natural framework for the study of quantum phenomena in situations where the gravitation itself can be treated classically. Of special interest is quantum ﬁeld theory in de Sitter spacetime. In fact, de Sitter space- time plays a central role in most of inﬂationary models of the early Universe [5–7], where the energy density and pressure of the inﬂaton ﬁeld act approximately as a cosmological constant. Moreover, the ampliﬁcation of quantum ﬂuctuation during an inﬂationary period with an approximately de Sitter background metric, gives a natural mechanism for gener- ating nearly scale-invariant spectrum of primordial inhomogeneities, which can successfully explain the observed CMB anisotropies [8, 9]. De Sitter spacetime is also potentially im- portant for understanding the ﬁnal fate of the Universe if the current accelerated expansion is due to a small cosmological constant, which nowadays is a possibility that is compati- ble with observations [9–12]. On the other hand, previous studies of interacting quantum scalar ﬁelds in de Sitter spacetime have revealed that the standard perturbative expansion gives rise to corrections that secularly grow with time and/or infrared divergences [13–20], signaling a possible deﬁciency of the perturbative approach. This has motivated several authors to consider alternative techniques (see for instance [18–27]) and in particular, to use nonperturbative resummation schemes [28–34]. In the above situations, it is important to study not only test ﬁelds evolving on a ﬁxed background, but also to take into account the backreaction of the quantum ﬁelds on the dynamics of the spacetime geometry. The backreaction problem has been explored by a number of authors in the context of semiclassical gravity (see for instance [35–39]), where the dynamics of the classical metric is governed by the so-called Semiclasical Einstein Equations (SEE). The SEE are a generalization of the Einstein equations that contain as a source (cid:104) (cid:105) the expectation value of the energy-momentum tensor of the quantum matter ﬁelds, T µν [1–4]. Self-consistent de Sitter solutions have been found for the case of free quantum ﬁelds [40–44]. The inﬂuence of the initial state of the quantum ﬁeld on the semiclassical solutions has been studied in Refs. [45, 46]. (cid:104) (cid:105) Since T is formally a divergent quantity, in order to address the backreaction problem µν it is necessary to analyze the renormalization process. For free and interacting quantum ﬁelds in the one-loop approximation, there are well known covariant renormalization meth- 2

text: I.

label: section

text: INTRODUCTION

text: Quantum

text: ﬁeld

text: theory

text: in

text: curved

text: spacetimes

text: is

text: the

text: natural

text: framework

text: for

text: the

text: study

text: of

text: quantum

text: phenomena

text: in

text: situations

text: where

text: the

text: gravitation

text: itself

text: can

text: be

text: treated

text: classically.

text: Of

text: special

text: interest

text: is

text: quantum

text: ﬁeld

text: theory

text: in

text: de

text: Sitter

text: spacetime.

text: In

text: fact,

text: de

text: Sitter

text: space-

text: time

text: plays

text: a

text: central

text: role

text: in

text: most

text: of

text: inﬂationary

text: models

text: of

text: the

text: early

text: Universe

text: where

text: the

text: energy

text: density

text: and

text: pressure

text: of

text: the

text: inﬂaton

text: ﬁeld

text: act

text: approximately

text: as

text: a

text: cosmological

text: constant.

text: Moreover,

text: the

text: ampliﬁcation

text: of

text: quantum

text: ﬂuctuation

text: during

text: an

text: inﬂationary

text: period

text: with

text: an

text: approximately

text: de

text: Sitter

text: background

text: metric,

text: gives

text: a

text: natural

text: mechanism

text: for

text: gener-

text: ating

text: nearly

text: scale-invariant

text: spectrum

text: of

text: primordial

text: inhomogeneities,

text: which

text: can

text: successfully

text: explain

text: the

text: observed

text: CMB

text: anisotropies

text: De

text: Sitter

text: spacetime

text: is

text: also

text: potentially

text: im-

text: portant

text: for

text: understanding

text: the

text: ﬁnal

text: fate

text: of

text: the

text: Universe

text: if

text: the

text: current

text: accelerated

text: expansion

text: is

text: due

text: to

text: a

text: small

text: cosmological

text: constant,

text: which

text: nowadays

text: is

text: a

text: possibility

text: that

text: is

text: compati-

text: ble

text: with

text: observations

text: On

text: the

text: other

text: hand,

text: previous

text: studies

text: of

text: interacting

text: quantum

text: scalar

text: ﬁelds

text: in

text: de

text: Sitter

text: spacetime

text: have

text: revealed

text: that

text: the

text: standard

text: perturbative

text: expansion

text: gives

text: rise

text: to

text: corrections

text: that

text: secularly

text: grow

text: with

text: time

text: and/or

text: infrared

text: divergences

text: signaling

text: a

text: possible

text: deﬁciency

text: of

text: the

text: perturbative

text: approach.

text: This

text: has

text: motivated

text: several

text: authors

text: to

text: consider

text: alternative

text: techniques

text: (see

text: for

text: instance

text: and

text: in

text: particular,

text: to

text: use

text: nonperturbative

text: resummation

text: schemes

text: In

text: the

text: above

text: situations,

text: it

text: is

text: important

text: to

text: study

text: not

text: only

text: test

text: ﬁelds

text: evolving

text: on

text: a

text: ﬁxed

text: background,

text: but

text: also

text: to

text: take

text: into

text: account

text: the

text: backreaction

text: of

text: the

text: quantum

text: ﬁelds

text: on

text: the

text: dynamics

text: of

text: the

text: spacetime

text: geometry.

text: The

text: backreaction

text: problem

text: has

text: been

text: explored

text: by

text: a

text: number

text: of

text: authors

text: in

text: the

text: context

text: of

text: semiclassical

text: gravity

text: (see

text: for

text: instance

text: where

text: the

text: dynamics

text: of

text: the

text: classical

text: metric

text: is

text: governed

text: by

text: the

text: so-called

text: Semiclasical

text: Einstein

text: Equations

text: (SEE).

text: The

text: SEE

text: are

text: a

text: generalization

text: of

text: the

text: Einstein

text: equations

text: that

text: contain

text: as

text: a

text: source

text: the

text: expectation

text: value

text: of

text: the

text: energy-momentum

text: tensor

text: of

text: the

text: quantum

text: matter

text: ﬁelds,

text: T

text: µν

text: Self-consistent

text: de

text: Sitter

text: solutions

text: have

text: been

text: found

text: for

text: the

text: case

text: of

text: free

text: quantum

text: ﬁelds

text: The

text: inﬂuence

text: of

text: the

text: initial

text: state

text: of

text: the

text: quantum

text: ﬁeld

text: on

text: the

text: semiclassical

text: solutions

text: has

text: been

text: studied

text: in

text: Refs.

text: Since

text: T

text: is

text: formally

text: a

text: divergent

text: quantity,

text: in

text: order

text: to

text: address

text: the

text: backreaction

text: problem

text: µν

text: it

text: is

text: necessary

text: to

text: analyze

text: the

text: renormalization

text: process.

text: For

text: free

text: and

text: interacting

text: quantum

text: ﬁelds

text: in

text: the

text: one-loop

text: approximation,

text: there

text: are

text: well

text: known

text: covariant

text: renormalization

text: meth-


text: function only of R and then substituing into the SEE (71) to obtain an equation of the form Λ = I (R). (80) R s where I depends also on the parameters m2 , ξ , λ and R . The subindex s stands for s R R R 0 symmetric. 2. Broken Phase M In this phase, the solution given in Eq. (78) to the ﬁeld equation already implies 2 > 0. ph (cid:54) ¯ It is important to note that the reason why the φ = 0 solutions are allowed is the presence 0 of the λ term in Eq. (77), which comes as a consequence of imposing the 2PI consistency R (cid:54) ¯ relations. Otherwise, the absence of such term would require that for φ = 0 we had 0 M 2 = 0, and as mentioned before for that case there is no de Sitter invariant vacuum [52]. ph Replacing the non vanishing solution to the ﬁeld Eq. (78) into the gap equation in its quadratic form Eq. (73) (small mass approximation), we obtain a new quadratic equation ¯ 2 for the non symmetric extrema of the potential φ (R), namely 0 (cid:114) (cid:104) (cid:105) −(R B − λRφ¯02 ) ± RB − λRφ¯02 2 − 4R2 A C λ dS 2 dS 2 dS dS R ¯ 2 φ (R) = . (81) 0 3 2A dS M Both branches give a solution with 2 > 0, the smaller being the maximum and the ph larger the minimum of the eﬀective potential. Following the analysis described in paper I, (cid:115) one can show that the condition for the(cid:18)existence(cid:19)of symmetry breaking solutions is − 3 − | | B 2 A C > 0. (82) dS dS dS 2 M Once again, replacing φ (R) and 2 (R) into the SEE gives an equation of the form 0 ph Λ = I (R). (83) R b 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: function

text: only

text: of

text: R

text: and

text: then

text: substituing

text: into

text: the

text: SEE

text: to

text: obtain

text: an

text: equation

text: of

text: the

text: form

text: Λ

label: equation

text: I

text: (R).

text: R

text: s

text: where

text: I

text: depends

text: also

text: on

text: the

text: parameters

text: ξ

text: λ

text: and

text: R

text: The

text: subindex

text: s

text: stands

text: for

text: s

text: R

text: R

text: R

text: symmetric.

text: Broken

text: Phase

text: M

text: In

text: this

text: phase,

text: the

text: solution

text: given

text: in

text: Eq.

text: to

text: the

text: ﬁeld

text: equation

text: already

text: implies

text: ph

font: default

text: It

text: is

text: important

text: to

text: note

text: that

text: the

text: reason

text: why

text: the

text: φ

text: solutions

text: are

text: allowed

text: is

text: the

text: presence

text: of

text: the

text: λ

text: term

text: in

text: Eq.

text: which

text: comes

text: as

text: a

text: consequence

text: of

text: imposing

text: the

text: consistency

text: R

text: relations.

text: Otherwise,

text: the

text: absence

text: of

text: such

text: term

text: would

text: require

text: that

text: for

text: φ

text: we

text: had

text: M

text: and

text: as

text: mentioned

text: before

text: for

text: that

text: case

text: there

text: is

text: no

text: de

text: Sitter

text: invariant

text: vacuum

text: ph

text: Replacing

text: the

text: non

text: vanishing

text: solution

text: to

text: the

text: ﬁeld

text: Eq.

text: into

text: the

text: gap

text: equation

text: in

text: its

text: quadratic

text: form

text: Eq.

text: (small

text: mass

text: approximation),

text: we

text: obtain

text: a

text: new

text: quadratic

text: equation

text: for

text: the

text: non

text: symmetric

text: extrema

text: of

text: the

text: potential

text: φ

text: (R),

text: namely

text: −(R

text: B

text: λRφ¯02

text: RB

text: λRφ¯02

text: A

text: C

text: λ

text: dS

text: dS

text: dS

text: dS

text: R

text: φ

text: (R)

text: dS

text: M

text: Both

text: branches

text: give

text: a

text: solution

text: with

text: the

text: smaller

text: being

text: the

text: maximum

text: and

text: the

text: ph

text: larger

text: the

text: minimum

text: of

text: the

text: eﬀective

text: potential.

text: Following

text: the

text: analysis

text: described

text: in

text: paper

text: I,

text: one

text: can

text: show

text: that

text: the

text: condition

text: for

text: symmetry

text: breaking

text: solutions

text: is

text: B

text: A

text: C

text: dS

text: dS

text: dS

text: M

text: Once

text: again,

text: replacing

text: φ

text: (R)

text: and

text: (R)

text: into

text: the

text: SEE

text: gives

text: an

text: equation

text: of

text: the

text: form

text: ph

text: Λ

text: I

text: (R).

text: R

text: b

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [9] D. N. Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 148, 175 (2003) [arXiv:astro- ph/0302209]; ibidem, Astrophys. J. Suppl. 170, 377 (2007) [arXiv:astro-ph/0603449]. [10] P. A. R. Ade, N. Aghanim et al. [PLANCK Collaboration], arXiv:1303.5076 [astro-ph]. [11] A. G. Riess et al. [Supernova Search Team Collaboration], Astron. J. 116, 1009 (1998) [arXiv:astro-ph/9805201]. [12] S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. 517, 565 (1999) [arXiv:astroph/9812133]. [13] S. Weinberg, Phys. Rev. D 72, 043514 (2005) [arXiv:hep-th/0506236]; ibidem Phys. Rev. D 74, 023508 (2006) [arXiv:hep-th/0605244]. [14] M. van der Meulen and J. Smit, JCAP 0711, 023 (2007) [arXiv:0707.0842 [hep-th]]. [15] D. Seery, JCAP 0711, 025 (2007) [arXiv:0707.3377v3 [astro-ph]]; ibidem JCAP 0802, 006 (2008) [arXiv:0707.3378 [astro-ph]]. [16] D. Seery, Class. Quantum Grav. 27, 124005 (2010) [arXiv:1005.1649 [astro-ph.CO]]. [17] T. Tanaka and Y. Urakawa, Class. Quantum Grav. 30, 233001 (2013) [arXiv:1306.4461 [hep- th]]. [18] A. A. Starobinsky and J. Yokoyama, Phys. Rev. D 50, 6357 (1994). [19] N. C. Tsamis and R. P. Woodard, Nucl. Phys. B724, 295 (2005) [arXiv:gr-qc/0505115]. [20] C.P. Burgess, L. Leblond, R. Holman, and S. Shandera, JCAP 1003, 033 (2010) [arXiv:0912.1608 [hep-th]]; ibidem JCAP 1010, 017 (2010) [arXiv:1005.3551 [hep-th]]. [21] G. Lazzari and T. Prokopec, arXiv:1304.0404 [hep-th]; T. Prokopec, JCAP 1212, 023 (2012)[arXiv:1110.3187v2 [gr-qc]]. [22] A. Rajaraman, Phys. Rev. D 82, 123522 (2010) [arXiv:1008.1271 [hep-th]]. [23] S. Hollands, Ann. H. Poincar´e 13, 1039 (2012) [arXiv:1105.1996 [gr-qc]]. [24] M. Beneke and P. Moch, Phys. Rev. D 87, 064018 (2013) [arXiv:1212.3058 [hep-th]]. [25] A. Riotto and M. S. Sloth, JCAP 0804, 030 (2008) [arXiv:0801.1845 [hep-ph]]. [26] D. Boyanovsky, Phys. Rev. D 85, 123525 (2012) [arXiv:1203.3903 [hep-ph]]. [27] E. T. Akhmedov, JHEP 1201, 066 (2012) [arXiv:1110.2257 [hep-th]]; E. T. Akhmedov amd Ph. Burda, Phys. Rev. D 86, 044031 (2012) [arXiv:1202.1202 [hep-th]]; E. T. Akhmedov, F. K. Popov, and V. M. Slepukhin, Phys. Rev. D 88, 024021 (2013) [ arXiv:1303.1068 [hep-th]]. [28] B. Garbrecht and G. Rigopoulos, Phys. Rev. D 84, 063516 (2011) [arXiv:1105.0418 [hep-th]]. [29] R. Parentani and J. Serreau, Phys. Rev. D 87, 045020 (2013) [arXiv:1212.6077 [hep-th]]; 34

label: reference

text: D.

text: N.

text: Spergel

text: et

text: al.

text: [WMAP

text: Collaboration],

text: Astrophys.

text: J.

text: Suppl.

text: [arXiv:astro-

text: ibidem,

text: Astrophys.

text: J.

text: Suppl.

text: P.

text: A.

text: R.

text: Ade,

text: N.

text: Aghanim

text: et

text: al.

text: [PLANCK

text: Collaboration],

text: [astro-ph].

text: A.

text: G.

text: Riess

text: et

text: al.

text: [Supernova

text: Search

text: Team

text: Collaboration],

text: Astron.

text: J.

text: S.

text: Perlmutter

text: et

text: al.

text: [Supernova

text: Cosmology

text: Project

text: Collaboration],

text: Astrophys.

text: J.

text: S.

text: Weinberg,

text: Phys.

text: Rev.

text: D

text: ibidem

text: Phys.

text: Rev.

text: D

text: M.

text: van

text: der

text: Meulen

text: and

text: J.

text: Smit,

text: JCAP

text: [hep-th]].

text: D.

text: Seery,

text: JCAP

text: [astro-ph]];

text: ibidem

text: JCAP

text: [astro-ph]].

text: D.

text: Seery,

text: Class.

text: Quantum

text: Grav.

text: [astro-ph.CO]].

text: T.

text: Tanaka

text: and

text: Y.

text: Urakawa,

text: Class.

text: Quantum

text: Grav.

text: [hep-

text: th]].

text: A.

text: A.

text: Starobinsky

text: and

text: J.

text: Yokoyama,

text: Phys.

text: Rev.

text: D

text: N.

text: C.

text: Tsamis

text: and

text: R.

text: P.

text: Woodard,

text: Nucl.

text: Phys.

text: C.P.

text: Burgess,

text: L.

text: Leblond,

text: R.

text: Holman,

text: and

text: S.

text: Shandera,

text: JCAP

text: [hep-th]];

text: ibidem

text: JCAP

text: [hep-th]].

text: G.

text: Lazzari

text: and

text: T.

text: Prokopec,

text: [hep-th];

text: T.

text: Prokopec,

text: JCAP

text: [gr-qc]].

text: A.

text: Rajaraman,

text: Phys.

text: Rev.

text: D

text: [hep-th]].

text: S.

text: Hollands,

text: Ann.

text: H.

text: Poincar´e

text: [gr-qc]].

text: M.

text: Beneke

text: and

text: P.

text: Moch,

text: Phys.

text: Rev.

text: D

text: [hep-th]].

text: A.

text: Riotto

text: and

text: M.

text: S.

text: Sloth,

text: JCAP

text: [hep-ph]].

text: D.

text: Boyanovsky,

text: Phys.

text: Rev.

text: D

text: [hep-ph]].

text: E.

text: T.

text: Akhmedov,

text: JHEP

text: [hep-th]];

text: E.

text: T.

text: Akhmedov

text: amd

text: Ph.

text: Burda,

text: Phys.

text: Rev.

text: D

text: [hep-th]];

text: E.

text: T.

text: Akhmedov,

text: F.

text: K.

text: Popov,

text: and

text: V.

text: M.

text: Slepukhin,

text: Phys.

text: Rev.

text: D

text: [hep-th]].

text: B.

text: Garbrecht

text: and

text: G.

text: Rigopoulos,

text: Phys.

text: Rev.

text: D

text: [hep-th]].

text: R.

text: Parentani

text: and

text: J.

text: Serreau,

text: Phys.

text: Rev.

text: D

text: [hep-th]];


text: ibidem, Phys. Rev. D 87, 085012 (2013) [arXiv:1302.3262 [hep-th]]. [30] J. Serreau, Phys. Rev. Lett. 107, 191103 (2011) [arXiv:1105.4539 [hep-th]]. [31] A. Youssef and D. Kreimer, arXiv:1301.3205 [gr-qc]. [32] T. Arai, Phys.Rev. D 86, 104064 (2012). [33] B. Garbrecht, G. Rigopoulos, and Y. Zhu, arXiv:1310.0367 [hep-th]. [34] D. L. L´opez Nacir, F. D. Mazzitelli, and L. G. Trombetta, Phys. Rev. D. 89, 024006 (2014) [arXiv 1309.0864 [hep-th]]. [35] J. B. Hartle and B. L. Hu, Phys. Rev. D 20, 1772 (1979); ibidem, Phys. Rev. D 21, 2756 (1980). [36] P. R. Anderson, C. Molina-Par´ıs, and E. Mottola, Phys. Rev. D 80, 084005 (2009). [37] M. B. Fr¨ob, D. B. Papadopoulos, A. Roura, and E. Verdaguer, Phys. Rev. D. 87, 064019 (2013) [arXiv:1301.5261 [gr-qc]]. [38] J. Z. Simon, Phys. Rev. D 45, 1953 (1992). [39] E. E. Flanagan and R. M. Wald, Phys. Rev. D 54, 6233 (1996) [gr-qc/9602052]. [40] A. A. Starobinsky, Phys. Lett. B 91, 99 (1980). [41] A. Vilenkin, Phys. Rev. D 32, 2511 (1985). [42] M. A. Castagnino, D. D. Harari, and J. P. Paz, Class. Quantum Grav. 3, 569 (1986). [43] D. L. L´opez Nacir and F. D. Mazzitelli, Phys. Rev. D 76, 024013 (2007). [44] S. Wada, T. Azuma, Phys. Lett. B 132, 313 (1983). [45] P. R. Anderson, W. Eaker, S. Habib, C. Molina-Paris and E. Mottola, Int. J. Theor. Phys. 40, 2217 (2001). [46] G. Perez-Nadal, A. Roura and E. Verdaguer, Class. Quant. Grav. 25, 154013 (2008) [arXiv:0806.2634 [gr-qc]]. [47] J. M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D 10, 2428 (1974). [48] G. Amelino-Camelia, S. -Y. Pi, Phys. Rev. D 47, 2356 (1993). [49] P. M. Stevenson, Phys. Rev. D 32, 1389 (1985). [50] P. M. Stevenson and R. Tarrach, Phys. Lett. B176, 436 (1986); P. M. Stevenson, B. Alles, and R. Tarrach, Phys. Rev. D 35, 2407 (1987); P. M. Stevenson, Z. Phys. C 35, 467 (1987). [51] J. Berges, Sz. Borsanyi, U. Reinosa, and J. Serreau, Annals Phys. 320, 344 (2005). [52] F. D. Mazzitelli and J. P. Paz, Phys. Rev. D 39, 2234 (1989). [53] E. Calzetta and B. L. Hu, “Nonequilibrium Quantum Field Theory,” Cambridge University 35

text: ibidem,

text: Phys.

text: Rev.

text: D

text: [hep-th]].

text: J.

text: Serreau,

text: Phys.

text: Rev.

text: Lett.

text: [hep-th]].

text: A.

text: Youssef

text: and

text: D.

text: Kreimer,

text: [gr-qc].

text: T.

text: Arai,

text: Phys.Rev.

text: D

text: B.

text: Garbrecht,

text: G.

text: Rigopoulos,

text: and

text: Y.

text: Zhu,

text: [hep-th].

text: D.

text: L.

text: L´opez

text: Nacir,

text: F.

text: D.

text: Mazzitelli,

text: and

text: L.

text: G.

text: Trombetta,

text: Phys.

text: Rev.

text: D.

text: [arXiv

text: [hep-th]].

text: J.

text: B.

text: Hartle

text: and

text: B.

text: L.

text: Hu,

text: Phys.

text: Rev.

text: D

text: ibidem,

text: Phys.

text: Rev.

text: D

text: P.

text: R.

text: Anderson,

text: C.

text: Molina-Par´ıs,

text: and

text: E.

text: Mottola,

text: Phys.

text: Rev.

text: D

text: M.

text: B.

text: Fr¨ob,

text: D.

text: B.

text: Papadopoulos,

text: A.

text: Roura,

text: and

text: E.

text: Verdaguer,

text: Phys.

text: Rev.

text: D.

text: [gr-qc]].

text: J.

text: Z.

text: Simon,

text: Phys.

text: Rev.

text: D

text: E.

text: E.

text: Flanagan

text: and

text: R.

text: M.

text: Wald,

text: Phys.

text: Rev.

text: D

text: A.

text: A.

text: Starobinsky,

text: Phys.

text: Lett.

text: B

text: A.

text: Vilenkin,

text: Phys.

text: Rev.

text: D

text: M.

text: A.

text: Castagnino,

text: D.

text: D.

text: Harari,

text: and

text: J.

text: P.

text: Paz,

text: Class.

text: Quantum

text: Grav.

text: D.

text: L.

text: L´opez

text: Nacir

text: and

text: F.

text: D.

text: Mazzitelli,

text: Phys.

text: Rev.

text: D

text: S.

text: Wada,

text: T.

text: Azuma,

text: Phys.

text: Lett.

text: B

text: P.

text: R.

text: Anderson,

text: W.

text: Eaker,

text: S.

text: Habib,

text: C.

text: Molina-Paris

text: and

text: E.

text: Mottola,

text: Int.

text: J.

text: Theor.

text: Phys.

text: G.

text: Perez-Nadal,

text: A.

text: Roura

text: and

text: E.

text: Verdaguer,

text: Class.

text: Quant.

text: Grav.

text: [gr-qc]].

text: J.

text: M.

text: Cornwall,

text: R.

text: Jackiw

text: and

text: E.

text: Tomboulis,

text: Phys.

text: Rev.

text: D

text: G.

text: Amelino-Camelia,

text: S.

text: -Y.

text: Pi,

text: Phys.

text: Rev.

text: D

text: P.

text: M.

text: Stevenson,

text: Phys.

text: Rev.

text: D

text: P.

text: M.

text: Stevenson

text: and

text: R.

text: Tarrach,

text: Phys.

text: Lett.

text: P.

text: M.

text: Stevenson,

text: B.

text: Alles,

text: and

text: R.

text: Tarrach,

text: Phys.

text: Rev.

text: D

text: P.

text: M.

text: Stevenson,

text: Z.

text: Phys.

text: C

text: J.

text: Berges,

text: Sz.

text: Borsanyi,

text: U.

text: Reinosa,

text: and

text: J.

text: Serreau,

text: Annals

text: Phys.

text: F.

text: D.

text: Mazzitelli

text: and

text: J.

text: P.

text: Paz,

text: Phys.

text: Rev.

text: D

text: E.

text: Calzetta

text: and

text: B.

text: L.

text: Hu,

text: “Nonequilibrium

text: Quantum

text: Field

text: Theory,”

text: Cambridge

text: University


text: 2 ANAND DEOPURKAR, MAKSYM FEDORCHUK, AND DAVID SWINARSKI Our strategy for proving generic stability of syzygy points follows that of [AFS13] for proving generic stability of ﬁnite Hilbert points. Namely, we prove the semi-stability of the 1st syzygy point of a special singular curve—the balanced ribbon—by a method of [MS11]. In Section , we deﬁne syzygy points of a canonically embedded curve and give a precise statement of our main result. In Section , we recall some preliminary results about bal- anced ribbons. In Sections and , we prove the main theorem. More precisely, Section con- tains the proof assuming the existence of suitable bases for the space of (co)syzygies of the balanced ribbon and the more technical Section is devoted to the construction of these bases. We learned the details of Se´an Keel’s idea to use syzygies as the means to construct the canonical model of M from a talk given by Gavril Farkas at the AIM workshop Log g minimal model program for moduli spaces held in December 2012. This paper grew out of our attempt to implement the roadmap laid out in that talk. We are grateful to AIM for the opportunity to meet. The workshop participants of the working group on syzygies, among them David Jensen, Ian Morrison, Anand Patel, and the present authors, veriﬁed by a computer computation our main result for g = 7. This computation motivated us to search for a proof in the general case. 2. Syzygies of canonical curves In this section, we recall the notions of Koszul cohomology necessary to set-up the GIT problems for all syzygies of canonical curves. We refer to [Gre84] and [AF11b] for a complete treatment of Koszul cohomology and a detailed discussion of Green’s conjecture. Let C be a Gorenstein curve with a very ample dualizing sheaf ω . Associated to C is C (cid:94) (cid:94) (cid:94) the Koszul complex − p+1 p p 1 H0(ω ) ⊗ H0(ωq−1) −f−p+−1−,q−−→1 H0(ω ) ⊗ H0(ωq ) −f−p→,q H0(ω ) ⊗ H0(ωq+1) C C C C C C (cid:88) where the diﬀerentials f are given by p,q (cid:98) − p 1 ∧ ∧ · · · ∧ ⊗ − ∧ · · · ∧ ∧ · · · ∧ ⊗ f (x x x − y) = ( 1)ix x x − x y. p,q 0 1 p 1 0 i p 1 i i=0 (cid:14) The Koszul cohomology groups are K (C) := ker f im f − . p,q p,q p+1,q 1 ≤ ≥ We say that C satisﬁes property (N ) if K (C) = 0 for all (i, q) with i p and q 2. p i,q → Property (N ) means that the natural maps Symm H0(ω ) H0(ωm) are surjective for 0 ≥ C C all m. Property (N ) for p 1 means, in addition, that the ideal of C in the canonical p embedding is generated by quadrics and the syzygies of order up to p are linear. ##LTLine##

text: ANAND

text: DEOPURKAR,

text: MAKSYM

text: FEDORCHUK,

text: AND

text: DAVID

text: SWINARSKI

text: Our

text: strategy

text: for

text: proving

text: generic

text: stability

text: of

text: syzygy

text: points

text: follows

text: that

text: of

text: for

text: proving

text: generic

text: stability

text: of

text: ﬁnite

text: Hilbert

text: points.

text: Namely,

text: we

text: prove

text: the

text: semi-stability

text: of

text: the

text: syzygy

text: point

text: of

text: a

text: special

text: singular

text: curve—the

text: balanced

text: ribbon—by

text: a

text: method

text: of

text: In

text: Section

text: we

text: deﬁne

text: syzygy

text: points

text: of

text: a

text: canonically

text: embedded

text: curve

text: and

text: give

text: a

text: precise

text: statement

text: of

text: our

text: main

text: result.

text: In

text: Section

text: we

text: recall

text: some

text: preliminary

text: results

text: about

text: bal-

text: anced

text: ribbons.

text: In

text: Sections

text: and

text: we

text: prove

text: the

text: main

text: theorem.

text: More

text: precisely,

text: Section

text: con-

text: tains

text: the

text: proof

text: assuming

text: the

text: existence

text: of

text: suitable

text: bases

text: for

text: the

text: space

text: of

text: (co)syzygies

text: of

text: the

text: balanced

text: ribbon

text: and

text: the

text: more

text: technical

text: Section

text: is

text: devoted

text: to

text: the

text: construction

text: of

text: these

text: bases.

text: We

text: learned

text: the

text: details

text: of

text: Se´an

text: Keel’s

text: idea

text: to

text: use

text: syzygies

text: as

text: the

text: means

text: to

text: construct

text: the

text: canonical

text: model

text: of

text: M

text: from

text: a

text: talk

text: given

text: by

text: Gavril

text: Farkas

text: at

text: the

text: AIM

text: workshop

text: Log

text: g

text: minimal

text: model

text: program

text: for

text: moduli

text: spaces

text: held

text: in

text: December

text: This

text: paper

text: grew

text: out

text: of

text: our

text: attempt

text: to

text: implement

text: the

text: roadmap

text: laid

text: out

text: in

text: that

text: talk.

text: We

text: are

text: grateful

text: to

text: AIM

text: for

text: the

text: opportunity

text: to

text: meet.

text: The

text: workshop

text: participants

text: of

text: the

text: working

text: group

text: on

text: syzygies,

text: among

text: them

text: David

text: Jensen,

text: Ian

text: Morrison,

text: Anand

text: Patel,

text: and

text: the

text: present

text: authors,

text: veriﬁed

text: by

text: a

text: computer

text: computation

text: our

text: main

text: result

text: for

text: g

text: This

text: computation

text: motivated

text: us

text: to

text: search

text: for

text: a

text: proof

text: in

text: the

text: general

text: case.

text: Syzygies

text: of

text: canonical

text: curves

text: In

text: this

text: section,

text: we

text: recall

text: the

text: notions

text: of

text: Koszul

text: cohomology

text: necessary

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text: Analysis and Diversion of Duqu’s Driver Guillaume Bonfante Jean-Yves Marion Fabrice Sabatier Aurélien Thierry Université de Lorraine Université de Lorraine Inria Inria LORIA LORIA LORIA LORIA E-mail: firstname.lastname@loria.fr Abstract as we put our hands on Stuxnet and Duqu samples, we wanted to know if our approach, knowing Stuxnet, The propagation techniques and the payload of Duqu could lead to the detection of Duqu. In fact the answer have been closely studied over the past year and it is yes but: we recognize Duqu’s main DLL as related has been said that Duqu shared functionalities with to Stuxnet but only if it is decrypted. The aim of Stuxnet. We focused on the driver used by Duqu dur- this paper is to present an automatic extraction proce- ing the infection, our contribution consists in reverse- dure to bypass stealth techniques such as those used by engineering the driver: we rebuilt its source code and Duqu. The irony of this work is that Duqu itself served analyzed the mechanisms it uses to execute the payload for this purpose. Let us describe in broad terms our while avoiding detection. Then we diverted the driver contribution. into a defensive version capable of detecting injections Infection timeline. The infection detected by in Windows binaries, thus preventing further attacks. CrySys used a malicious Microsoft Word document em- We speciﬁcally show how Duqu’s modiﬁed driver would bedding Duqu. Firstly it exploits a 0-day kernel vulner- have detected Duqu. ability (on TrueType fonts [1]), it installs three com- ponents: a driver (nfrd965.sys), an encrypted DLL (Duqu’s main DLL, NETP191.PNF) and an encrypted conﬁguration ﬁle (netp192.PNF). 1 Introduction The second step takes place at the next reboot. The driver monitors processes loaded by the OS and injects When it was ﬁrst discovered in September 2011 by Duqu’s DLL into a process speciﬁed by the conﬁgura- CrySys [4], it was said that Duqu was close to Stuxnet tion ﬁle, typically services.exe. in that they shared infection and propagation mech- During the third and ﬁnal step, the payload, which anisms. Duqu is an oﬀensive tool used to steal infor- is included in the DLL, is activated. mation: Symantec [7] identiﬁed, amongst other pay- Detecting this malware is challenging because only loads, keyloggers, screen recorders, network monitors the driver is unencrypted on the hard drive. The DLL and service discovery tools. It is kept up to date by is encrypted, packed with UPX, and is injected imme- Command & Control servers and bundled with an auto- diately after decryption so it is only stored decrypted destruction feature typically triggered 36 days after the in RAM. Thus detection could take place when the in- infection. jection is done, which is right after the decryption of The attacks seem to have been successful since the the DLL and before it is activated. malware was not detected during the operations (some Once installed on a ﬁrst target, Duqu receives orders of them lasted a few month) but only post-mortem. for further attacks and propagation from Command Besides, according to Kasperky [5], the latest version & Control servers. Each new controlled machine can of Duqu was found in February 2012, long after the mal- be conﬁgured to connect back to the attacker through ware was ﬁrst discovered and documented. Thus Duqu some routing tunnel, providing access to restricted ar- is a stealthy spyware and the attacks were carefully eas. adapted to each target. At the High Sec Lab, we develop a malware detec- tor based on morphological analysis [3], which is a tech- Defensive version. The defense scenario will be the nique based on control ﬂow graph comparison. As soon following. We modify Duqu’s driver so it monitors

text: Analysis

text: and

text: Diversion

text: of

text: Duqu’s

text: Driver

text: Guillaume

text: Bonfante

text: Jean-Yves

text: Marion

text: Fabrice

text: Sabatier

text: Aurélien

text: Thierry

text: Université

text: de

text: Lorraine

text: Université

text: de

text: Lorraine

text: Inria

text: Inria

text: LORIA

text: LORIA

text: LORIA

text: LORIA

text: E-mail:

text: firstname.lastname@loria.fr

label: footer

text: Abstract

text: as

text: we

text: put

text: our

text: hands

text: on

text: Stuxnet

text: and

text: Duqu

text: samples,

text: we

text: wanted

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text: know

text: if

text: our

text: approach,

text: knowing

text: Stuxnet,

text: The

label: abstract

text: propagation

text: techniques

text: and

text: the

text: payload

text: of

text: Duqu

text: could

text: lead

text: to

text: the

text: detection

text: of

text: Duqu.

text: In

text: fact

text: the

text: answer

text: have

text: been

text: closely

text: studied

text: over

text: the

text: past

text: year

text: and

text: it

text: is

text: yes

text: but:

text: we

text: recognize

text: Duqu’s

text: main

text: DLL

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text: related

text: has

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text: that

text: Duqu

text: shared

text: functionalities

text: with

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text: Stuxnet

text: but

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text: decrypted.

text: The

text: aim

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text: We

text: focused

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text: the

text: driver

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text: Duqu

text: dur-

text: this

text: paper

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text: present

text: an

text: automatic

text: extraction

text: proce-

text: ing

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text: infection,

text: our

text: contribution

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text: dure

text: to

text: bypass

text: stealth

text: techniques

text: such

text: as

text: those

text: used

text: by

text: engineering

text: the

text: driver:

text: we

text: rebuilt

text: its

text: source

text: code

text: and

text: Duqu.

text: The

text: irony

text: of

text: this

text: work

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text: Duqu

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text: analyzed

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text: mechanisms

text: it

text: uses

text: to

text: execute

text: the

text: payload

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text: this

text: purpose.

text: Let

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text: describe

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text: broad

text: terms

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text: while

text: avoiding

text: detection.

text: Then

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text: diverted

text: the

text: driver

text: contribution.

text: into

text: a

text: defensive

text: version

text: capable

text: of

text: detecting

text: injections

text: Infection

text: timeline.

text: The

text: infection

text: detected

text: by

text: in

text: Windows

text: binaries,

text: thus

text: preventing

text: further

text: attacks.

text: CrySys

text: used

text: a

text: malicious

text: Microsoft

text: Word

text: document

text: em-

text: We

text: speciﬁcally

text: show

text: how

text: Duqu’s

text: modiﬁed

text: driver

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text: bedding

text: Duqu.

text: Firstly

text: it

text: exploits

text: a

text: kernel

text: vulner-

text: have

text: detected

text: Duqu.

text: ability

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text: TrueType

text: fonts

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text: installs

text: three

text: com-

text: ponents:

text: a

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text: DLL

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text: main

text: DLL,

text: and

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text: conﬁguration

text: ﬁle

text: Introduction

text: The

text: second

text: step

text: takes

text: place

text: at

text: the

text: next

text: reboot.

text: The

text: driver

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text: the

text: OS

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text: injects

text: When

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text: Duqu’s

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text: Stuxnet

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text: During

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text: anisms.

text: Duqu

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text: oﬀensive

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text: mation:

text: Symantec

text: identiﬁed,

text: amongst

text: other

text: pay-

text: Detecting

text: this

text: malware

text: is

text: challenging

text: because

text: only

text: loads,

text: keyloggers,

text: screen

text: recorders,

text: network

text: monitors

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text: driver

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text: unencrypted

text: on

text: the

text: hard

text: drive.

text: The

text: DLL

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text: service

text: discovery

text: tools.

text: It

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text: kept

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text: date

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text: Command

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text: auto-

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text: decryption

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text: it

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text: stored

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text: destruction

text: feature

text: typically

text: triggered

text: days

text: after

text: the

text: in

text: RAM.

text: Thus

text: detection

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text: infection.

text: jection

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text: which

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text: decryption

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text: the

text: the

text: DLL

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text: before

text: it

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text: activated.

text: malware

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text: not

text: detected

text: during

text: the

text: operations

text: (some

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text: installed

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text: Duqu

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text: orders

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text: lasted

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text: month)

text: but

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text: post-mortem.

text: for

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text: propagation

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text: according

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text: the

text: latest

text: version

text: Control

text: servers.

text: Each

text: new

text: controlled

text: machine

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text: of

text: Duqu

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text: mal-

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text: attacker

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text: ware

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text: documented.

text: Thus

text: Duqu

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text: routing

text: tunnel,

text: providing

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text: is

text: a

text: stealthy

text: spyware

text: and

text: the

text: attacks

text: were

text: carefully

text: eas.

text: adapted

text: to

text: each

text: target.

text: At

text: the

text: High

text: Sec

text: Lab,

text: we

text: develop

text: a

text: malware

text: detec-

text: tor

text: based

text: on

text: morphological

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text: Defensive

text: version.

text: The

text: defense

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text: As

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text: We

text: modify

text: Duqu’s

text: driver

text: so

text: it

text: monitors