text: 4 C. Transmission Schemes We give analytical results on the SCDPs of JT, PT-SS and PT-OS cases for a user located at the cluster center. Note that When a random user inside a cluster requests for a ﬁle F taking the cluster-center user as a reference is mainly done for in , since the availability of the requested ﬁle at in-cluster analytical tractability, but it can be seen as an upper bound on SBSs differs with the popularity order of the ﬁle, different the SCDP for randomly located users inside the cluster of transmission schemes, i.e., namely joint transmission (JT) and interest. parallel transmission (PT), are used for delivering the ﬁles in the MPC and LCD ranges, respectively, as described below. A. SCDP Deﬁnition If the requested ﬁle f is in the MPC range, i.e., the m ≤ ≤ (cid:98) (cid:99) popularity order is between 1 m ρM , K SBSs in Assuming that each ﬁle contains S bits, the successful the cluster have the same entire ﬁle. Hence, the requested ﬁle delivery of a ﬁle is deﬁned by the event that S bits are is jointly transmitted to the user as a means to enhance the successfully delivered using bandwidth W and time T. Note content delivery reliability, i.e., increase the received SINR, that in the JT and PT cases, the numbers of information as shown in Fig. 2(a). We denote this case as JT cooperation bits delivered from each SBS are different. In the JT case, scheme. each SBS sends S bits to the user using the same bandwidth. Hence, at the receiver side, the received signals from K SBSs If the requested ﬁle f is in the LCD range, i.e., the are superimposed and thus considered as a single stream. m (cid:98) (cid:99) ≤ (cid:98) (cid:99) − popularity order is between ρM < m ρM + K(M The SCDP is deﬁned as a function of the received signal- (cid:98) (cid:99) ρM ), cooperating SBSs inside the same cluster have disjoint to-interference-plus-noise ratio (SINR), given as partitions of the requested ﬁle. The different partitions need to be transmitted to the user at the same time by parallel pJT = P [WT log (1 + SINR) > S | K] . (4) d,K 2 (multiple) streams, one from each cooperating SBS, as shown in Fig. 2(b). We denote this case as PT cooperation scheme. In the PT-SS case, each SBS sends S bits to the user There are two ways of frequency allocation: i) PT with K orthogonal spectrum assignment (PT-OS) case, each SBS uses employing SIC by sharing the same W bandwidth. The decodability of the received streams depends on the SINR 1 of the overall available spectrum to transmit the stored pKartition of the requested ﬁle to the user; and ii) PT with of each stream and rate requirement. Decoding K streams successive decoding based spectrum sharing (PT-SS) case, K using SIC is theoretically feasible if all K streams achieves higher rate than the target rate for successful transmission [19]. SBSs concurrently transmit K streams containing different Hence, we have (cid:12)  partitions of the requested ﬁle to the user using the same (cid:12) (cid:92) (cid:12) available spectrum resources. In the PT-SS case, at the re-  (cid:12)  (cid:12) ceiver, successive decoding with (multi-stream) interference (cid:12) cancellation (SIC) is used to decode the signal according to pPT-S = P WT log (1 + SINR ) > S K , the received signal power order [19], [20]. More explicitly, d,K ∈{ } 2 i K i 1,...,K the strongest signal is decoded ﬁrst and extracted from the (5) received signal, then proceed to the next decoding layer for the next strongest signal, and so on. where SINR is the received SINR of the stream containing i the i-th partition of the requested ﬁle. If the requested ﬁle is not cached in local cluster, a cache In the PT-OS case, each BS sends S bits by using W K K miss event occurs. In this case, all BSs fetch the requested bandwidth each. The delivery rate is bounded by the stream content from the core network through backhaul links and with the low(cid:20)est achievable rate, so the SCDP is deﬁn(cid:12)(cid:12)ed a(cid:21)s (cid:12) jointly transmit the content to the user in order to reduce (cid:12) the delivery latency. The power consumption consists of the required power for fetching content from the core network pPd,TK-O = P WK T log2(1 + i∈[m1,.i.n.,K]{SINRi}) > KS K . to the cooperative SBSs and the transmission power for (6) delivering content from the cooperative SBSs to the user. The backhauling process increases not only end-to-end delivery We denote R = S (bit/s) as the target rate for successful delay but also energy consumption, compared to the case of content delivery.dIn teTrms of SINR requirement, the SCDP can serving user requests using the local caches [21], [22]. By be rewritten as (cid:104) (cid:12) (cid:105) (cid:12) considering those impacts, the energy efﬁciency of this case (cid:12) is investigated in Section . pJdT,K = P SINR(cid:92)> 2RWd − 1 K (cid:12)(cid:12)(cid:12)(cid:12)  (7) (cid:12) (cid:12) III. SUCCESSFUL(SCCODNTPE)NATNDAELYLISVISERY PROBABILITY pPd,TK-S = P (cid:20)i∈{1,...,K} SINRi > 2KRWd − 1 (cid:12)(cid:12)(cid:12)(cid:12) K(cid:21) (8) evaInluathtiiosnseocftiocnlu,swteer-csetundtryicackaecyhimngetrwicithforSBthSe pceorofpoermraatinocne, pPd,TK-O = P ∈min {SINRi} > 2RWd − 1 K , (9) i [1,...,K] namely the successful content delivery probability (SCDP). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: C.

label: section

text: Transmission

text: Schemes

text: We

text: give

text: analytical

text: results

text: on

text: the

text: SCDPs

text: of

text: JT,

text: PT-SS

text: and

text: PT-OS

text: cases

text: for

text: a

text: user

text: located

text: at

text: the

text: cluster

text: center.

text: Note

text: that

text: When

text: a

text: random

text: user

text: inside

text: a

text: cluster

text: requests

text: for

text: a

text: ﬁle

text: F

text: taking

text: the

text: cluster-center

text: user

text: as

text: a

text: reference

text: is

text: mainly

text: done

text: for

text: in

text: since

text: the

text: availability

text: of

text: the

text: requested

text: ﬁle

text: at

text: in-cluster

text: analytical

text: tractability,

text: but

text: it

text: can

text: be

text: seen

text: as

text: an

text: upper

text: bound

text: on

text: SBSs

text: differs

text: with

text: the

text: popularity

text: order

text: of

text: the

text: ﬁle,

text: different

text: the

text: SCDP

text: for

text: randomly

text: located

text: users

text: inside

text: the

text: cluster

text: of

text: transmission

text: schemes,

text: i.e.,

text: namely

text: joint

text: transmission

text: (JT)

text: and

text: interest.

text: parallel

text: transmission

text: (PT),

text: are

text: used

text: for

text: delivering

text: the

text: ﬁles

text: in

text: the

text: MPC

text: and

text: LCD

text: ranges,

text: respectively,

text: as

text: described

text: below.

text: A.

text: SCDP

text: Deﬁnition

text: If

text: the

text: requested

text: ﬁle

text: f

text: is

text: in

text: the

text: MPC

text: range,

text: i.e.,

text: the

text: m

text: popularity

text: order

text: is

text: between

text: m

text: ρM

text: K

text: SBSs

text: in

text: Assuming

text: that

text: each

text: ﬁle

text: contains

text: S

text: bits,

text: the

text: successful

text: the

text: cluster

text: have

text: the

text: same

text: entire

text: ﬁle.

text: Hence,

text: the

text: requested

text: ﬁle

text: delivery

text: of

text: a

text: ﬁle

text: is

text: deﬁned

text: by

text: the

text: event

text: that

text: S

text: bits

text: are

text: is

text: jointly

text: transmitted

text: to

text: the

text: user

text: as

text: a

text: means

text: to

text: enhance

text: the

text: successfully

text: delivered

text: using

text: bandwidth

text: W

text: and

text: time

text: T.

text: Note

text: content

text: delivery

text: reliability,

text: i.e.,

text: increase

text: the

text: received

text: SINR,

text: that

text: in

text: the

text: JT

text: and

text: PT

text: cases,

text: the

text: numbers

text: of

text: information

text: as

text: shown

text: in

text: Fig.

text: We

text: denote

text: this

text: case

text: as

text: JT

text: cooperation

text: bits

text: delivered

text: from

text: each

text: SBS

text: are

text: different.

text: In

text: the

text: JT

text: case,

text: scheme.

text: each

text: SBS

text: sends

text: S

text: bits

text: to

text: the

text: user

text: using

text: the

text: same

text: bandwidth.

text: Hence,

text: at

text: the

text: receiver

text: side,

text: the

text: received

text: signals

text: from

text: K

text: SBSs

text: If

text: the

text: requested

text: ﬁle

text: f

text: is

text: in

text: the

text: LCD

text: range,

text: i.e.,

text: the

text: are

text: superimposed

text: and

text: thus

text: considered

text: as

text: a

text: single

text: stream.

text: m

text: popularity

text: order

text: is

text: between

text: ρM

text: m

text: ρM

text: K(M

text: The

text: SCDP

text: is

text: deﬁned

text: as

text: a

text: function

text: of

text: the

text: received

text: signal-

text: ρM

text: cooperating

text: SBSs

text: inside

text: the

text: same

text: cluster

text: have

text: disjoint

text: to-interference-plus-noise

text: ratio

text: (SINR),

text: given

text: as

text: partitions

text: of

text: the

text: requested

text: ﬁle.

text: The

text: different

text: partitions

text: need

text: to

text: be

text: transmitted

text: to

text: the

text: user

text: at

text: the

text: same

text: time

text: by

text: parallel

text: pJT

text: P

text: [WT

text: log

text: SINR)

text: S

text: K]

text: d,K

text: (multiple)

text: streams,

text: one

text: from

text: each

text: cooperating

text: SBS,

text: as

text: shown

text: in

text: Fig.

text: We

text: denote

text: this

text: case

text: as

text: PT

text: cooperation

text: scheme.

text: In

text: the

text: PT-SS

text: case,

text: each

text: SBS

text: sends

text: S

text: bits

text: to

text: the

text: user

text: There

text: are

text: two

text: ways

text: of

text: frequency

text: allocation:

text: i)

text: PT

text: with

text: K

text: orthogonal

text: spectrum

text: assignment

text: (PT-OS)

text: case,

text: each

text: SBS

text: uses

text: employing

text: SIC

text: by

text: sharing

text: the

text: same

text: W

text: bandwidth.

text: The

text: decodability

text: of

text: the

text: received

text: streams

text: depends

text: on

text: the

text: SINR

text: of

text: the

text: overall

text: available

text: spectrum

text: to

text: transmit

text: the

text: stored

text: pKartition

text: of

text: the

text: requested

text: ﬁle

text: to

text: the

text: user;

text: and

text: ii)

text: PT

text: with

text: of

text: each

text: stream

text: and

text: rate

text: requirement.

text: Decoding

text: K

text: streams

text: successive

text: decoding

text: based

text: spectrum

text: sharing

text: (PT-SS)

text: case,

text: K

text: using

text: SIC

text: is

text: theoretically

text: feasible

text: if

text: all

text: K

text: streams

text: achieves

text: higher

text: rate

text: than

text: the

text: target

text: rate

text: for

text: successful

text: transmission

text: SBSs

text: concurrently

text: transmit

text: K

text: streams

text: containing

text: different

text: Hence,

text: we

text: have

text: partitions

text: of

text: the

text: requested

text: ﬁle

text: to

text: the

text: user

text: using

text: the

text: same

text: available

text: spectrum

text: resources.

text: In

text: the

text: PT-SS

text: case,

text: at

text: the

text: re-

text: ceiver,

text: successive

text: decoding

text: with

text: (multi-stream)

text: interference

text: cancellation

text: (SIC)

text: is

text: used

text: to

text: decode

text: the

text: signal

text: according

text: to

text: pPT-S

text: P

text: WT

text: log

text: SINR

text: S

text: K

text: the

text: received

text: signal

text: power

text: order

text: More

text: explicitly,

text: d,K

text: i

text: K

text: i

text: the

text: strongest

text: signal

text: is

text: decoded

text: ﬁrst

text: and

text: extracted

text: from

text: the

text: received

text: signal,

text: then

text: proceed

text: to

text: the

text: next

text: decoding

text: layer

text: for

text: the

text: next

text: strongest

text: signal,

text: and

text: so

text: on.

text: where

text: SINR

text: is

text: the

text: received

text: SINR

text: of

text: the

text: stream

text: containing

text: i

text: the

text: i-th

text: partition

text: of

text: the

text: requested

text: ﬁle.

text: If

text: the

text: requested

text: ﬁle

text: is

text: not

text: cached

text: in

text: local

text: cluster,

text: a

text: cache

text: In

text: the

text: PT-OS

text: case,

text: each

text: BS

text: sends

text: S

text: bits

text: by

text: using

text: W

text: K

text: K

text: miss

text: event

text: occurs.

text: In

text: this

text: case,

text: all

text: BSs

text: fetch

text: the

text: requested

text: bandwidth

text: each.

text: The

text: delivery

text: rate

text: is

text: bounded

text: by

text: the

text: stream

text: content

text: from

text: the

text: core

text: network

text: through

text: backhaul

text: links

text: and

text: with

text: the

text: achievable

text: rate,

text: so

text: the

text: SCDP

text: is

text: jointly

text: transmit

text: the

text: content

text: to

text: the

text: user

text: in

text: order

text: to

text: reduce

text: the

text: delivery

text: latency.

text: The

text: power

text: consumption

text: consists

text: of

text: the

text: required

text: power

text: for

text: fetching

text: content

text: from

text: the

text: core

text: network

text: pPd,TK-O

text: P

text: WK

text: T

text: KS

text: K

text: to

text: the

text: cooperative

text: SBSs

text: and

text: the

text: transmission

text: power

text: for

text: delivering

text: content

text: from

text: the

text: cooperative

text: SBSs

text: to

text: the

text: user.

text: The

text: backhauling

text: process

text: increases

text: not

text: only

text: end-to-end

text: delivery

text: We

text: denote

text: R

text: S

text: (bit/s)

text: as

text: the

text: target

text: rate

text: for

text: successful

text: delay

text: but

text: also

text: energy

text: consumption,

text: compared

text: to

text: the

text: case

text: of

text: content

text: delivery.dIn

text: teTrms

text: of

text: SINR

text: requirement,

text: the

text: SCDP

text: can

text: serving

text: user

text: requests

text: using

text: the

text: local

text: caches

text: By

text: be

text: rewritten

text: as

text: considering

text: those

text: impacts,

text: the

text: energy

text: efﬁciency

text: of

text: this

text: case

text: is

text: investigated

text: in

text: Section

text: pJdT,K

text: P

text: K

text: III.

text: SUCCESSFUL(SCCODNTPE)NATNDAELYLISVISERY

text: PROBABILITY

text: pPd,TK-S

text: P

text: SINRi

text: pceorofpoermraatinocne,

text: pPd,TK-O

text: P

text: ∈min

text: {SINRi}

text: K

text: i

text: namely

text: the

text: successful

text: content

text: delivery

text: probability

text: (SCDP).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 for i = 1, . . . , K. C. SCDP of LCD-PT strategy When the cluster-center user requests for ﬁle f with B. SCDP of MPC-JT strategy (cid:98)ρM(cid:99) < m ≤ (cid:98)ρM(cid:99) + K(M − (cid:98)ρM(cid:99)), which is in thme LCD range, parallel streams containing different partitions of the For the cluster-center user located at y = (0, 0), when ≤ ≤ 0(cid:98) (cid:99) requested ﬁle are simultaneously sent to the user. Considering it requests for ﬁle f with 1 m ρM , which is m different spectrum usages, we study SCDPs for PT-SS and in the MPC range, coordinated joint transmission is used PT-OS cases separately in this section. to combine coherently the received signals from cooperating SBSs. Hence, over each symbol duration time, the cooperating In the PT-SS case, over each symbol duration time, K SBSs SBSs transmit the same symbol s. Assuming equal transmit transmit K different symbols (one symbol in each partition) power P for every SBS and a standard distance-dependent [s1, s2, . . . , sK] to the user at the origin. If all SBSs use the power lawt pathloss attenuation, i.e., r−α, where α > 2 is the same transmit power Pt as in the JT case, the channel output pathloss exponent, the channel output at the user is at the (cid:88)rece(cid:112)iver is (cid:88) (cid:112) (cid:88) (cid:112) (cid:88) (cid:112) − − y = Ptri−α2 his + Ptrj−α2 hjsj + n, (10) y = ∈C Ptri α2 hisi+ ∈ \{C} Ptrj α2 hjsj +n. (14) ∈C ∈ \{C} bj bj Φb bi bj Φb In order to decode multiple streams simultaneously, we use where h denotes the small-scale Rayleigh fading from SIC with respect to a certain order of received signal. The l ∼ CN the l-th SBS to the user, which follows h (0, 1); r detailed analysis of SIC based on power ordering statistics is l l denotes the distance from the l-th SBS to the user; s denotes out of the scope of this paper and has been studied in [20]. l the transmitted symbol of the l-th SBS; and n denotes the For the ease of analysis, we consider here the case where background thermal noise. the user decodes different information streams based on the Considering an interference-limited network and neglecting distance order [23]. After approximating the cluster area by B the background thermal noise, the signal-to-interference ratio the circ(cid:101)le (0(cid:101), R), the(cid:101)decoding order will be from the nearest (SIR) of received signal is g(cid:12)(cid:12)iven by (cid:12)(cid:12) SBS to the K-th neare(cid:101)st SBS to the cluster-center user. We (cid:12)(cid:12)(cid:12) (cid:80) (cid:12)(cid:12)(cid:12)2 ddeisﬁtannecer o=rde[rr,1,w.h.e.r,erKr],tkhe∈d[i1s,taKnc]eisvtehcetodriswtaitnhceinfcroremasitnhge (cid:80) − k ∈C hiri α2 k-th nearest SBS to the cluster-center user. SIRJT = bi | | − . (11) When decoding the information from t{he k-th neares}t SBS, ∈ \{C} hj 2rj α all signals coming from closer SBSs b1, . . . , bk−1 need bj Φb to be successfully decoded and canceled. In this case, the − interference comes from K k remaining SBSs inside the Using (7) and (11), we can obtain the SCDP of JT case as cluster and PPP distributed SBSs outside the cluster. Due to follows. the conditioned number K, the interference distribution is different from the case with PPP-distributed SBSs. For the L2ReWmb −ma1,1t.heFoSrCtDhePcoluf sJteTr-ccaesnetewr iuthseKr wciothoptaerragteitngSISRBθS1s =is twBraitchtak(cid:101)bi∈lit[y1,aKna−ly1si]s,,thweedaisstsruibmuetiothnaotfaitnttheerfekr-itnhgdSeBcoSdsinogutssitdepe given by (0, rk) still follows a homogeneous PPP. The SIR of the k-th stream with SIC is thus given as (cid:101) (cid:32) (cid:33) (cid:80) (cid:90) (cid:90) (cid:89) | | − p(cid:39)JdT,K(0θR1·)· · 0R LI|R (cid:80)Ki=θ11x−i α i=K1 2Rx2i dx1 · · · dxK, SIRk (cid:39) bj∈Φb\Bh(0k,r(cid:101)2k)r|khαj|2rj−α . (15) (12) At the last decoding step, all in-cluster interfering signals are canceled. The remaining interference comes from out-of- where L | (s) is the Laplace transform of the interference cluster SBSs with minimum distance R to the user. Hence, for I x B the last decoded stream, we have coming from SBSs lo(cid:32)cated outsi(cid:90)de of (0, x), giv(cid:33)en by (cid:80) (cid:101) | | − (cid:39) h 2 r α ∞ SIR K |K | − . (16) L − 1 K h 2r α I|x(s) = exp πλbsα2 x2 1 + wα2 dw . (13) bj∈Φb\B(0,R) j j s2/α Using (8), (15), and (16), we now obtain the SCDP of PT-SS case as follows. Proof: See Appendix . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: for

text: i

text: K.

text: C.

text: SCDP

text: of

text: LCD-PT

text: strategy

text: When

text: the

text: cluster-center

text: user

text: requests

text: for

text: ﬁle

text: f

text: with

text: B.

text: SCDP

text: of

text: MPC-JT

text: strategy

text: m

text: K(M

text: which

text: is

text: in

text: thme

text: LCD

text: range,

text: parallel

text: streams

text: containing

text: different

text: partitions

text: of

text: the

text: For

text: the

text: cluster-center

text: user

text: located

text: at

text: y

text: when

text: requested

text: ﬁle

text: are

text: simultaneously

text: sent

text: to

text: the

text: user.

text: Considering

text: it

text: requests

text: for

text: ﬁle

text: f

text: with

text: m

text: ρM

text: which

text: is

text: m

text: different

text: spectrum

text: usages,

text: we

text: study

text: SCDPs

text: for

text: PT-SS

text: and

text: in

text: the

text: MPC

text: range,

text: coordinated

text: joint

text: transmission

text: is

text: used

text: PT-OS

text: cases

text: separately

text: in

text: this

text: section.

text: to

text: combine

text: coherently

text: the

text: received

text: signals

text: from

text: cooperating

text: SBSs.

text: Hence,

text: over

text: each

text: symbol

text: duration

text: time,

text: the

text: cooperating

text: In

text: the

text: PT-SS

text: case,

text: over

text: each

text: symbol

text: duration

text: time,

text: K

text: SBSs

text: SBSs

text: transmit

text: the

text: same

text: symbol

text: s.

text: Assuming

text: equal

text: transmit

text: transmit

text: K

text: different

text: symbols

text: (one

text: symbol

text: in

text: each

text: partition)

text: power

text: P

text: for

text: every

text: SBS

text: and

text: a

text: standard

text: distance-dependent

text: sK]

text: to

text: the

text: user

text: at

text: the

text: origin.

text: If

text: all

text: SBSs

text: use

text: the

text: power

text: lawt

text: pathloss

text: attenuation,

text: i.e.,

text: r−α,

text: where

text: α

text: is

text: the

text: same

text: transmit

text: power

text: Pt

text: as

text: in

text: the

text: JT

text: case,

text: the

text: channel

text: output

text: pathloss

text: exponent,

text: the

text: channel

text: output

text: at

text: the

text: user

text: is

text: at

text: the

text: is

text: y

text: his

text: hjsj

text: n,

text: y

text: ∈C

text: Ptri

text: α2

text: hisi+

text: \{C}

text: Ptrj

text: α2

text: hjsj

text: +n.

text: ∈C

text: \{C}

text: bj

text: bj

text: Φb

text: bi

text: bj

text: Φb

text: In

text: order

text: to

text: decode

text: multiple

text: streams

text: simultaneously,

text: we

text: use

text: where

text: h

text: denotes

text: the

text: small-scale

text: Rayleigh

text: fading

text: from

text: SIC

text: with

text: respect

text: to

text: a

text: certain

text: order

text: of

text: received

text: signal.

text: The

text: l

text: CN

text: the

text: l-th

text: SBS

text: to

text: the

text: user,

text: which

text: follows

text: h

text: r

text: detailed

text: analysis

text: of

text: SIC

text: based

text: on

text: power

text: ordering

text: statistics

text: is

text: l

text: l

text: denotes

text: the

text: distance

text: from

text: the

text: l-th

text: SBS

text: to

text: the

text: user;

text: s

text: denotes

text: out

text: of

text: the

text: scope

text: of

text: this

text: paper

text: and

text: has

text: been

text: studied

text: in

text: l

text: the

text: transmitted

text: symbol

text: of

text: the

text: l-th

text: SBS;

text: and

text: n

text: denotes

text: the

text: For

text: the

text: ease

text: of

text: analysis,

text: we

text: consider

text: here

text: the

text: case

text: where

text: background

text: thermal

text: noise.

text: the

text: user

text: decodes

text: different

text: information

text: streams

text: based

text: on

text: the

text: Considering

text: an

text: interference-limited

text: network

text: and

text: neglecting

text: distance

text: order

text: After

text: approximating

text: the

text: cluster

text: area

text: by

text: B

text: the

text: background

text: thermal

text: noise,

text: the

text: signal-to-interference

text: ratio

text: the

text: R),

text: order

text: will

text: be

text: from

text: the

text: nearest

text: (SIR)

text: of

text: received

text: signal

text: is

text: by

text: SBS

text: to

text: the

text: K-th

text: SBS

text: to

text: the

text: cluster-center

text: user.

text: We

text: ddeisﬁtannecer

text: k

text: ∈C

text: hiri

text: α2

text: k-th

text: nearest

text: SBS

text: to

text: the

text: cluster-center

text: user.

text: SIRJT

text: bi

text: When

text: decoding

text: the

text: information

text: from

text: t{he

text: k-th

text: neares}t

text: SBS,

text: \{C}

text: hj

text: α

text: all

text: signals

text: coming

text: from

text: closer

text: SBSs

text: need

text: bj

text: Φb

text: to

text: be

text: successfully

text: decoded

text: and

text: canceled.

text: In

text: this

text: case,

text: the

text: interference

text: comes

text: from

text: K

text: k

text: remaining

text: SBSs

text: inside

text: the

text: Using

text: and

text: we

text: can

text: obtain

text: the

text: SCDP

text: of

text: JT

text: case

text: as

text: cluster

text: and

text: PPP

text: distributed

text: SBSs

text: outside

text: the

text: cluster.

text: Due

text: to

text: follows.

text: the

text: conditioned

text: number

text: K,

text: the

text: interference

text: distribution

text: is

text: different

text: from

text: the

text: case

text: with

text: PPP-distributed

text: SBSs.

text: For

text: the

text: sJteTr-ccaesnetewr

text: iuthseKr

text: =is

text: given

text: by

text: rk)

text: still

text: follows

text: a

text: homogeneous

text: PPP.

text: The

text: SIR

text: of

text: the

text: k-th

text: stream

text: with

text: SIC

text: is

text: thus

text: given

text: as

text: LI|R

text: α

text: dxK,

text: SIRk

text: At

text: the

text: last

text: decoding

text: step,

text: all

text: in-cluster

text: interfering

text: signals

text: are

text: canceled.

text: The

text: remaining

text: interference

text: comes

text: from

text: out-of-

text: where

text: L

text: (s)

text: is

text: the

text: Laplace

text: transform

text: of

text: the

text: interference

text: cluster

text: SBSs

text: with

text: minimum

text: distance

text: R

text: to

text: the

text: user.

text: Hence,

text: for

text: I

text: x

text: B

text: the

text: last

text: decoded

text: stream,

text: we

text: have

text: coming

text: from

text: SBSs

text: of

text: x),

text: by

text: h

text: r

text: α

text: SIR

text: K

text: |K

text: L

text: K

text: h

text: α

text: I|x(s)

text: exp

text: πλbsα2

text: dw

text: j

text: j

text: Using

text: and

text: we

text: now

text: obtain

text: the

text: SCDP

text: of

text: PT-SS

text: case

text: as

text: follows.

text: Proof:

text: See

text: Appendix

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 objective is to maximize its in-cluster cache service probabil- | 0.45 K=4 ity, that is, to ﬁnd ρ whic(cid:28)h maximizes f(ρ∝K). KK==32 When γ < 1 and M N, since pm 1/mγ, we have [24] (cid:88) 0.4 L ≈ − g(L) = p (L/N)1 γ. (25) 0.35 m m=1 0.3 Using this approximation, pCH, M(ρ) and pCH, L(ρ) can be approximated (cid:18)by tw(cid:19)o continuous functions of ρ, given by (cid:101) 0.25 − (cid:39) (cid:18)M (cid:19)1 γ (cid:110) − (cid:111) pCH, M(ρ) ρ1 γ = pCH, M(ρ), (26) N − 0.20 0.1 0.2 0.M3 PC C0.4ache P0.5ercen0ta.6ge (ρ0).7 0.8 0.9 1 pCH, L(ρ) (cid:39) (cid:101) MN 1 γ [ρ(1 − K) + K]1−γ − ρ1−γ (27) Fig. 4. Cache hit probability vs. MPC cache percentage (ρ) for K = 2,3,4. = pCH, M(ρ). γ = 0.5. Then, th(cid:18)e cac(cid:19)he service probability in (22) is simpliﬁed as hsuecrecetshsefuollpytimsearvl eρdtbhyatlomcaaxlicmaiczheess,thneampeerlcyenthtaegceacohferesqeurveisctes f(ρ|K) (cid:39) (cid:18) M(cid:19)1−γ(cid:104)ρ1−γpJT (θ ) (cid:105) N d,K 1 probability. A user request can be successfully served by local − caches only when: 1) the requested ﬁle is cached inside the M 1 γ − − − − + (ρ(1 K) + K)1 γ ρ1 γ pPT-S(θ ). cluster, 2) the content delivery from the cooperative SBSs N d,K 2 to the user is successful. We then deﬁne the cache service (28) probability as follows. Using (28), we can obtain the optimal ρ as follows. Deﬁnition 1. In the cluster-centric SCN with proposed com- bined caching strategy, the average cache service probability Lemma 4. In a cluster-centric SCN with proposed combined is given as (cid:88) caching strategy, knowing that there are K cooperative SBSs in the cluster, the optimal percentage of cache space assigned ∞ P | for MPC caching is given as psv = (n = K)f(ρ K), (21) K=1 ∗  |    onwhhaevriengf(Kρ|KSB)Siss itnhseidceacthhee scelurvsitceer, pgriovbenabbilyity conditioning ρ = arρg∈[m0,1a]x f(ρ K) − 1/γ −1  (cid:39) K 1 − f(ρ|K) = pCH, M(ρ) pJdT,K(θ1) + pCH, L(ρ) pPd,TK-S(θ2). (22) min K ppJdPdT,,TKK-S ((θθ12)) − 1 + K 1 , 1 , (29) pJT (θ) and pPT-S(θ) are given in (12) and (17), respectively. Herde,,KpCH, M(ρ)d,aKnd pCH, L(ρ) are the probabilities to have where pJdT,K(θ1) and pPd,TK-S(θ2) are given in (12) and (17), the requested ﬁle cached in the MPC and LCD ranges, respectively. respectively, given by Proof: See Appendix . (cid:88) (cid:98) (cid:99) ρM Remark 1. From (29), we can see that the ratio pJdT,K(θ1) is pCH, M(ρ) = pm(cid:88), (23) critical for the optimal cache assignment. When theptPdr,TK-aSn(θs2m) is- (cid:98)m=1(cid:99) −(cid:98) (cid:99) sion reliability of JT scheme is much higher than that of PT ρM +K(M ρM ) (cid:29) ∗ (cid:39) pCH, L(ρ) = pm, (24) sthcahtemmeo,sti.eo.f, tphJdeT,Kc(aθc1h)e spapcPd,eTK-Sw(θo2u)ld, wbee huasevde tρo stor1e, tmheeamnionsgt (cid:98) (cid:99) ρM +1 popular contents. When pJdT,K(θ1) (cid:39) 1, ρ∗ (cid:39) 0, then increasing where p is deﬁned in (2). the content diversity becopPdm,TK-Se(sθ2m)ore beneﬁcial. m Since each cluster performs cooperative caching indepen- Inside each cluster, based on its knowledge about the dent of other clusters, for a random cluster with K SBSs, the number of in-cluster SBSs and out-of-cluster interfering SBS ytilibaborP tiH ehcaC ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: objective

text: is

text: to

text: maximize

text: its

text: in-cluster

text: cache

text: service

text: probabil-

font: default

text: ity,

text: that

text: is,

text: to

text: ﬁnd

text: ρ

text: maximizes

text: f(ρ∝K).

text: When

text: γ

text: and

text: M

text: N,

text: since

text: pm

text: we

text: have

label: equation

text: L

text: g(L)

text: p

text: γ.

text: m

text: Using

text: this

text: approximation,

text: pCH,

text: M(ρ)

text: and

text: pCH,

text: L(ρ)

text: can

text: be

text: approximated

text: continuous

text: functions

text: of

text: ρ,

text: given

text: by

text: γ

text: pCH,

text: M(ρ)

text: ρ1

text: γ

text: pCH,

text: M(ρ),

text: N

text: PC

text: pCH,

text: L(ρ)

text: MN

text: γ

text: K)

text: ρ1−γ

text: Fig.

text: Cache

text: hit

text: probability

text: vs.

text: MPC

text: cache

text: percentage

text: (ρ)

text: for

text: K

text: pCH,

text: M(ρ).

text: γ

text: Then,

text: service

text: probability

text: in

text: is

text: simpliﬁed

text: as

text: hsuecrecetshsefuollpytimsearvl

text: f(ρ|K)

text: (θ

text: N

text: d,K

text: probability.

text: A

text: user

text: request

text: can

text: be

text: successfully

text: served

text: by

text: local

text: caches

text: only

text: when:

text: the

text: requested

text: ﬁle

text: is

text: cached

text: inside

text: the

text: M

text: γ

text: K)

text: γ

text: ρ1

text: γ

text: pPT-S(θ

text: cluster,

text: the

text: content

text: delivery

text: from

text: the

text: cooperative

text: SBSs

text: N

text: d,K

text: to

text: the

text: user

text: is

text: successful.

text: We

text: then

text: deﬁne

text: the

text: cache

text: service

text: probability

text: as

text: follows.

text: Using

text: we

text: can

text: obtain

text: the

text: optimal

text: ρ

text: as

text: follows.

text: Deﬁnition

text: In

text: the

text: cluster-centric

text: SCN

text: with

text: proposed

text: com-

text: bined

text: caching

text: strategy,

text: the

text: average

text: cache

text: service

text: probability

text: Lemma

text: In

text: a

text: cluster-centric

text: SCN

text: with

text: proposed

text: combined

text: is

text: given

text: as

text: caching

text: strategy,

text: knowing

text: that

text: there

text: are

text: K

text: cooperative

text: SBSs

text: in

text: the

text: cluster,

text: the

text: optimal

text: percentage

text: of

text: cache

text: space

text: assigned

text: P

text: for

text: MPC

text: caching

text: is

text: given

text: as

text: psv

text: (n

text: K)f(ρ

text: K),

text: onwhhaevriengf(Kρ|KSB)Siss

text: itnhseidceacthhee

text: scelurvsitceer,

text: pgriovbenabbilyity

text: conditioning

text: ρ

text: f(ρ

text: K)

text: 1/γ

text: −1

text: K

text: f(ρ|K)

text: pCH,

text: M(ρ)

text: pCH,

text: L(ρ)

text: min

text: K

text: ppJdPdT,,TKK-S

text: K

text: pJT

text: (θ)

text: and

text: pPT-S(θ)

text: are

text: given

text: in

text: and

text: respectively.

text: Herde,,KpCH,

text: M(ρ)d,aKnd

text: pCH,

text: L(ρ)

text: are

text: the

text: probabilities

text: to

text: have

text: where

text: and

text: are

text: given

text: in

text: and

text: the

text: requested

text: ﬁle

text: cached

text: in

text: the

text: MPC

text: and

text: LCD

text: ranges,

text: respectively.

text: respectively,

text: given

text: by

text: Proof:

text: See

text: Appendix

text: ρM

text: Remark

text: From

text: we

text: can

text: see

text: that

text: the

text: ratio

text: is

text: pCH,

text: M(ρ)

text: critical

text: for

text: the

text: optimal

text: cache

text: assignment.

text: When

text: is-

text: sion

text: reliability

text: of

text: JT

text: scheme

text: is

text: much

text: higher

text: than

text: that

text: of

text: PT

text: ρM

text: +K(M

text: ρM

text: pCH,

text: L(ρ)

text: pm,

text: sthcahtemmeo,sti.eo.f,

text: wbee

text: huasevde

text: tρo

text: tmheeamnionsgt

text: ρM

text: popular

text: contents.

text: When

text: ρ∗

text: then

text: increasing

text: where

text: p

text: is

text: deﬁned

text: in

text: the

text: content

text: diversity

text: beneﬁcial.

text: m

text: Since

text: each

text: cluster

text: performs

text: cooperative

text: caching

text: indepen-

text: Inside

text: each

text: cluster,

text: based

text: on

text: its

text: knowledge

text: about

text: the

text: dent

text: of

text: other

text: clusters,

text: for

text: a

text: random

text: cluster

text: with

text: K

text: SBSs,

text: the

text: number

text: of

text: in-cluster

text: SBSs

text: and

text: out-of-cluster

text: interfering

text: SBS

text: ytilibaborP

text: tiH

text: ehcaC

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 3: A diagram of the current OSDC clusters, with the solid arrows indicating systems fully operational and accessible with Tukey. The Hadoop clusters are operational and support some of the Tukey services but not all of them. 108 GB Data Set 1.1 TB Data Set mbit/s LLR mbit/s LLR UDR (no encryption) 752 0.66 738 0.64 rsync (no encryption) 401 0.35 405 0.36 UDR (blowﬁsh) 394 0.35 396 0.35 rsync (blowﬁsh) 280 0.25 281 0.25 rsync (3des) 284 0.25 285 0.25 Table 3: Overall transfer speeds in mbit/s and the long distance to local ratio (LLR) between OSDC nodes in Chicago, IL and Livermore, CA, round trip time of 104 ms. metal to a compute or storage cloud in much install. Then the installer runs a script speci- less than a day. Our system starts with one ﬁed at the end of the preseed ﬁle which sets up PXE boot server, a Chef server, and a set of networking on the freshly installed system and servers with IPMI conﬁgured. IPMI is triggered adds another script to be run at boot. Upon re- to boot the servers, which then pull a start- booting, the next script double-checks the IPMI up image and boot options from the PXE boot conﬁguration, ﬁnishes partitioning the disk and server. The options deﬁne the location of a pre- sets up additional RAIDs as necessary, before seed ﬁle which the start-up sequence loads and downloading and installing the Chef client. The uses as a guide to automate the installation of Chef client then checks in with the Chef server the generic Ubuntu Server. The installation uses and runs the “recipes” listed for either a man- the PXE boot server or a preconﬁgured proxy to agement node or a compute node. As a last step, install Ubuntu Server directly from the reposi- a ﬁnal clean up script runs to deliver us a fully tories to avoid any updates being required post- functional OpenStack rack. 9 ##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##

text: Figure

text: A

label: caption

text: diagram

text: of

text: the

text: current

text: OSDC

text: clusters,

text: with

text: the

text: solid

text: arrows

text: indicating

text: systems

text: fully

text: operational

text: and

text: accessible

text: with

text: Tukey.

text: The

text: Hadoop

text: clusters

text: are

text: operational

text: and

text: support

text: some

text: of

text: the

text: Tukey

text: services

text: but

text: not

text: all

text: of

text: them.

label: table

text: GB

text: Data

text: Set

text: TB

text: Data

text: Set

text: mbit/s

text: LLR

text: mbit/s

text: LLR

text: UDR

text: (no

text: encryption)

text: rsync

text: (no

text: encryption)

text: UDR

text: (blowﬁsh)

text: rsync

text: (blowﬁsh)

text: rsync

text: Table

text: Overall

text: transfer

text: speeds

text: in

text: mbit/s

text: and

text: the

text: long

text: distance

text: to

text: local

text: ratio

text: (LLR)

text: between

text: OSDC

text: nodes

text: in

text: Chicago,

text: IL

text: and

text: Livermore,

text: CA,

text: round

text: trip

text: time

text: of

text: ms.

text: metal

text: to

text: a

text: compute

text: or

text: storage

text: cloud

text: in

text: much

text: install.

text: Then

text: the

text: installer

text: runs

text: a

text: script

text: speci-

text: less

text: than

text: a

text: day.

text: Our

text: system

text: starts

text: with

text: one

text: ﬁed

text: at

text: the

text: end

text: of

text: the

text: preseed

text: ﬁle

text: which

text: sets

text: up

text: PXE

text: boot

text: server,

text: a

text: Chef

text: server,

text: and

text: a

text: set

text: of

text: networking

text: on

text: the

text: freshly

text: installed

text: system

text: and

text: servers

text: with

text: IPMI

text: conﬁgured.

text: IPMI

text: is

text: triggered

text: adds

text: another

text: script

text: to

text: be

text: run

text: at

text: boot.

text: Upon

text: re-

text: to

text: boot

text: the

text: servers,

text: which

text: then

text: pull

text: a

text: start-

text: booting,

text: the

text: next

text: script

text: double-checks

text: the

text: IPMI

text: up

text: image

text: and

text: boot

text: options

text: from

text: the

text: PXE

text: boot

text: conﬁguration,

text: ﬁnishes

text: partitioning

text: the

text: disk

text: and

text: server.

text: The

text: options

text: deﬁne

text: the

text: location

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text: pre-

text: sets

text: up

text: additional

text: RAIDs

text: as

text: necessary,

text: before

text: seed

text: ﬁle

text: which

text: the

text: start-up

text: sequence

text: loads

text: and

text: downloading

text: and

text: installing

text: the

text: Chef

text: client.

text: The

text: uses

text: as

text: a

text: guide

text: to

text: automate

text: the

text: installation

text: of

text: Chef

text: client

text: then

text: checks

text: in

text: with

text: the

text: Chef

text: server

text: the

text: generic

text: Ubuntu

text: Server.

text: The

text: installation

text: uses

text: and

text: runs

text: the

text: “recipes”

text: listed

text: for

text: either

text: a

text: man-

text: the

text: PXE

text: boot

text: server

text: or

text: a

text: preconﬁgured

text: proxy

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text: agement

text: node

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text: compute

text: node.

text: As

text: a

text: last

text: step,

text: install

text: Ubuntu

text: Server

text: directly

text: from

text: the

text: reposi-

text: a

text: ﬁnal

text: clean

text: up

text: script

text: runs

text: to

text: deliver

text: us

text: a

text: fully

text: tories

text: to

text: avoid

text: any

text: updates

text: being

text: required

text: post-

text: functional

text: OpenStack

text: rack.

text: ##LTFigure##

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text: 12 ALIX DERUELLE AND TOBIAS LAMM • L2 estimate : Since (h(t)) ≥ satisﬁes ∂ h = L h, one has, t 0 t t | | ≤ | | − |∇ | | | | | ∂ h 2 ∆ h 2 2 g0(t)h 2 + c(n) Rm(g (t)) h 2 . t g0(t) g0(t) g0(t) g0(t) 0 g0(t) g0(t) × R → R Multiplying this inequality by any smooth cut-oﬀ function ψ2 : M + + gives, after integrating by parts (in space) : ˆ ˆ (cid:68) | | (cid:69) |(cid:0)∇ | ≤ (cid:1) ∂ˆ ψ2 h 2 dµ + 2 ψ2 g0(t)h 2 dµ t g0(t) g0(t) g0(t) g0(t) M M −∇ ∇ | | (cid:0) | | | | (cid:1) ˆ g0(t)ψ2, g0(t) h 2 + ∂ ψ2 + c(n)ψ2 Rm(g (t)) h 2 dµ g0(t) t 0 g0(t) g0(t) g0(t) M ≤ |∇ | | | |∇ | | | | | 4 g0(t)ψ h ψ g0(t)h + ∂ ψ2 + c(n)ψ2 Rm(g (t)) h 2 dµ . g0(t) g0(t) g0(t) t 0 g0(t) g0(t) g0(t) M |∇ | | | |∇ | Hence, bˆy applying Young’s ineˆquality to the term 4 g0(t)ψ g0(t) h g0(t)ψ g0(t)h g0(t), (cid:16)| | |∇ | ≤(cid:17) ∂ ˆψ2 h 2 dµ + ψ2 g0(t)h 2 dµ t g0(t) g0(t) g0(t) g0(t) M M |∇ | | | | | c(n) g0(t)ψ 2 + ∂ ψ2 + ψ2 Rm(g (t)) h 2 dµ . g0(t) t 0 g0(t) g0(t) g0(t) M ∈ × R∗ Now, let (x, R) M and consider the following cutoﬀ function : ψ (y, t) := + R → R x,R ≡ ψ(d (x, y)/R), where ψ : is a smooth function such that ψ| 1, g0(t) ≡ | (cid:48)| ≤ + + [0,1] ψ| ∞ 0 and supR ψ c. ψ is a Lipschitz function satisfying : [2,+ ) x,R + |∇ | ≤ c ≤ ≤ c√ g0(t)ψ , 0 ∂ ψ , x,R g0(t) R t x,R R t almost everywhere, by lemma A.3. Hence, by integrating in time : ˆ ˆ ˆ | | R2 |∇ | ≤ ˆ h 2g0(R2)dµg0(R2) + g0(t)h 2g0(t)dµg0(t)dt Bg0(R2)(x,R) 0 Bg0(t)(x,R) | | h(0) 2 (cid:18)dµ (cid:19) ˆ ˆ g0(0) g0(0) B (x,2R) g0(0) R2 (cid:32) 1 1√ | | | | (cid:33) +c + + Rm(g (t)) h 2 dµ dt R2ˆ Rˆ t 0 g0(t) g0(t) g0(t) 0 B (x,2R) g0(t) ≤ (cid:107) (cid:107) R2 | | c h(0) 2 ∞ Rn + Rm(g (t)) dµ dt, L (M,g0) 0 g0(t) g0(t) 0 B (x,2R) g0(t) ≤ since Vol B (x, s) c(n)sn for any nonnegative s by the Bishop-Gromov g0(t) g0(t) ≥ theorem since Ric(g (t)) 0. Now, as AVR(g (t)) = AVR(g ) > 0 by Appendix 0 0 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ALIX

text: DERUELLE

text: AND

text: TOBIAS

text: LAMM

label: list

text: estimate

text: Since

text: (h(t))

text: satisﬁes

text: h

text: L

text: h,

text: one

text: has,

text: t

text: t

text: t

text: h

text: h

text: c(n)

text: Rm(g

text: (t))

text: h

text: t

text: R

text: R

text: Multiplying

text: this

text: inequality

text: by

text: any

text: smooth

text: cut-oﬀ

text: function

text: ψ2

text: M

text: gives,

text: after

text: integrating

text: by

text: parts

text: (in

text: space)

text: ˆ

text: ˆ

text: ∂ˆ

text: ψ2

text: h

text: dµ

text: ψ2

text: dµ

text: t

text: M

text: M

text: ˆ

text: h

text: ψ2

text: Rm(g

text: (t))

text: h

text: dµ

text: t

text: M

text: h

text: ψ

text: ψ2

text: Rm(g

text: (t))

text: h

text: dµ

text: t

text: M

text: Hence,

text: bˆy

text: applying

text: Young’s

text: ineˆquality

text: to

text: the

text: term

text: h

text: ˆψ2

text: h

text: dµ

text: ψ2

text: dµ

text: t

text: M

text: M

text: c(n)

text: ψ2

text: ψ2

text: Rm(g

text: (t))

text: h

text: dµ

text: t

text: M

text: R∗

text: Now,

text: let

text: (x,

text: R)

text: M

text: and

text: consider

text: the

text: following

text: cutoﬀ

text: function

text: ψ

text: (y,

text: t)

text: R

text: R

text: x,R

text: ψ(d

text: (x,

text: y)/R),

text: where

text: ψ

text: is

text: a

text: smooth

text: function

text: such

text: that

text: ψ|

text: ψ|

text: and

text: supR

text: ψ

text: c.

text: ψ

text: is

text: a

text: Lipschitz

text: function

text: satisfying

text: x,R

text: c

text: c√

text: ψ

text: x,R

text: R

text: t

text: x,R

text: R

text: t

text: almost

text: everywhere,

text: by

text: lemma

text: Hence,

text: by

text: integrating

text: in

text: time

text: ˆ

text: ˆ

text: ˆ

text: ˆ

text: h

text: ˆ

text: ˆ

text: B

text: 1√

text: +c

text: Rm(g

text: (t))

text: h

text: dµ

text: dt

text: Rˆ

text: t

text: B

text: c

text: Rn

text: Rm(g

text: (t))

text: dµ

text: dt,

text: L

text: B

text: since

text: Vol

text: B

text: (x,

text: s)

text: c(n)sn

text: for

text: any

text: nonnegative

text: s

text: by

text: the

text: Bishop-Gromov

text: theorem

text: since

text: Ric(g

text: (t))

text: Now,

text: as

text: AVR(g

text: (t))

text: AVR(g

text: by

text: Appendix

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 ALIX DERUELLE AND TOBIAS LAMM since g (t) has positive curvature operator. Now, by using a barrier function as in the proof ∞0 of the L estimate of theorem 4.1, the result follows by applying the maximum principle for symmetric 2-tensors. (cid:3) Remark 4.6. The proof of theorem 4.5 does not use the expanding structure in an essential way. 5. Heat kernel estimates We follow the same strategy adopted by Grigor’yan and Zhang. Let (Mn, g(t)) ∈ be a complete Ricci ﬂow with bounded curvature for each time slice. t [0,T)  We consider the scalar heat equation with potential coupled with this Ricci ﬂow :  ∂ u = ∆ u + R u, t g(t) g(t) (10) − ∂ g = 2 Ric(g(t)), t ×  on Mn (0, T ). We also consider its conjugate heat equation :  ∂ u = ∆ u, τ g(τ) (11) ∂ g = 2 Ric(g(τ)), τ × − on M (0, t] where τ(s) := t s for some ﬁxed positive time t. ≤ We denote the heat kernel associated to (10) by K(x, t, y, s), for 0 s < t < T , and ∈ x, y M. This heat kernel always exists and is positive : see [Gue02]. It satisﬁes by  ∈ × deﬁnition, for any ﬁxed (y, s) M [0, T ),     · · · · · · ∂ K( , , y, s) = ∆ K( , , y, s) + R K( , , y, s), t g(t) g(t)     − ∂ g = 2 Ric(g(t)), (12) t · lim → K( , t, y, s) = δ . t s y ∈ × · · On the other hand, if (x, t) M (0, T ) is ﬁxed, then K(x, t, , ) is the heat kernel  associated to the conjugatebackward heat equation :     · · − · · ∂ K(x, t, , ) = ∆ K(x, t, , ), s g(s)     − ∂ g = 2 Ric(g(s)), (13) s · lim → K(x, t, , s) = δ . s t x − → Or equivalently, if τ(s) := t s, (y, τ) K(x, t, y, τ) satisﬁes the following forward heat equation :

text: ALIX

text: DERUELLE

text: AND

text: TOBIAS

text: LAMM

text: since

text: g

text: (t)

text: has

text: positive

text: curvature

text: operator.

text: Now,

text: by

text: using

text: a

text: barrier

text: function

text: as

text: in

text: the

text: proof

text: ∞0

text: of

text: the

text: L

text: estimate

text: of

text: theorem

text: the

text: result

text: follows

text: by

text: applying

text: the

text: maximum

text: principle

text: for

text: symmetric

text: Remark

text: The

text: proof

text: of

text: theorem

text: does

text: not

text: use

text: the

text: expanding

text: structure

text: in

text: an

text: essential

text: way.

text: Heat

text: kernel

text: estimates

text: We

text: follow

text: the

text: same

text: strategy

text: adopted

text: by

text: Grigor’yan

text: and

text: Zhang.

text: Let

text: (Mn,

text: g(t))

text: be

text: a

text: complete

text: Ricci

text: ﬂow

text: with

text: bounded

text: curvature

text: for

text: each

text: time

text: slice.

text: t

text: We

text: consider

text: the

text: scalar

text: heat

text: equation

text: with

text: potential

text: coupled

text: with

text: this

text: Ricci

text: ﬂow

text: u

text: u

text: R

text: u,

text: t

text: g(t)

text: g(t)

text: g

text: Ric(g(t)),

text: t

text: on

text: Mn

text: T

text: We

text: also

text: consider

text: its

text: conjugate

text: heat

text: equation

text: u

text: u,

text: τ

text: g(τ)

text: g

text: Ric(g(τ)),

text: τ

text: on

text: M

text: t]

text: where

text: τ(s)

text: t

text: s

text: for

text: some

text: ﬁxed

text: positive

text: time

text: t.

text: We

text: denote

text: the

text: heat

text: kernel

text: associated

text: to

text: by

text: K(x,

text: t,

text: y,

text: s),

text: for

text: s

text: t

text: T

text: and

text: x,

text: y

text: M.

text: This

text: heat

text: kernel

text: always

text: exists

text: and

text: is

text: positive

text: see

text: It

text: satisﬁes

text: by

text: deﬁnition,

text: for

text: any

text: ﬁxed

text: (y,

text: s)

text: M

text: T

text: K(

text: y,

text: s)

text: K(

text: y,

text: s)

text: R

text: K(

text: y,

text: s),

text: t

text: g(t)

text: g(t)

text: g

text: Ric(g(t)),

text: t

text: lim

text: K(

text: t,

text: y,

text: s)

text: δ

text: t

text: s

text: y

text: On

text: the

text: other

text: hand,

text: if

text: (x,

text: t)

text: M

text: T

text: is

text: ﬁxed,

text: then

text: K(x,

text: t,

text: is

text: the

text: heat

text: kernel

text: associated

text: to

text: the

text: conjugatebackward

text: heat

text: equation

text: K(x,

text: t,

text: K(x,

text: t,

text: s

text: g(s)

text: g

text: Ric(g(s)),

text: s

text: lim

text: K(x,

text: t,

text: s)

text: δ

text: s

text: t

text: x

text: Or

text: equivalently,

text: if

text: τ(s)

text: t

text: s,

text: (y,

text: τ)

text: K(x,

text: t,

text: y,

text: τ)

text: satisﬁes

text: the

text: following

text: forward

text: heat

text: equation


text: 20 ALIX DERUELLE AND TOBIAS LAMM (cid:18) (cid:19) ˆ ≥ 1 + t c K(x, t, y, s)dµ (y) , s < t. g0(s) 1 + s M On the other hand, let (φ ) be any sequence of smooth cut-oﬀ functions approximating k k the constant function 1, then ˆ ˆ − ∂ φ (y)K(x, t, y, s)dµ (y) = ∆ φ (y)K(x, t, y, s)dµ (y) t k g0(t) g0(t) k g0(t) M ≤ M | | C sup ∆ φ , g0(t) k supp(φ ) k ∞ which implies by letting k tending to + , ˆ ≥ ∂ K(x, t, y, s)dµ (y) 0, t g0(t) M i.e. ˆ ∈ K(x, t, y, s)dµ (x) = 1, s < t, y M. M g0(t) ´ Similarly, one can prove the expected lower bound for K(x, t, y, s)dµ (y). M g0(s) (cid:3) Remark 5.2. In particular, proposition 5.1 tells us that the heat semigroup associated to ∆ + R , where (Mn, g (t)) is an expanding gradient Ricci soliton with non negative g0(t) g0(t) 0 ∞ scalar curvature, is L1 contractive but not L contractive unless it is ﬂat. → ∞ Proposition 5.3 (On diagonal upper bound : L2 L bound). Let (Mn, g (t)) ≥ be a 0 t 0 non collapsed Type III solution of the Ricci ﬂow with nonnegative Ricci curvature, i.e. | | ≤ ≥ ≥ ≥ (1 + t) Rm(g (t)) R , Ric(g (t)) 0, AVR(g (t)) V > 0, t 0. 0 g0(t) 0 0 0 0 Then, ≤ c(n, V , R ) ≤ ∈ 0 0 0 < K(x, t, y, s) − , 0 s < t, x, y M. n (t s) 2 ##LTLine## ##LTLine## ##LTLine##

text: ALIX

text: DERUELLE

text: AND

text: TOBIAS

text: LAMM

text: ˆ

text: t

text: c

text: K(x,

text: t,

text: y,

text: s)dµ

text: (y)

text: s

text: t.

text: s

text: M

text: On

text: the

text: other

text: hand,

text: let

text: (φ

text: be

text: any

text: sequence

text: of

text: smooth

text: cut-oﬀ

text: functions

text: approximating

text: k

text: k

text: the

text: constant

text: function

text: then

text: ˆ

text: ˆ

text: φ

text: (y)K(x,

text: t,

text: y,

text: s)dµ

text: (y)

text: φ

text: (y)K(x,

text: t,

text: y,

text: s)dµ

text: (y)

text: t

text: k

text: k

text: M

text: M

text: C

text: sup

text: φ

text: k

text: supp(φ

text: k

text: which

text: implies

text: by

text: letting

text: k

text: tending

text: to

text: ˆ

text: K(x,

text: t,

text: y,

text: s)dµ

text: (y)

text: t

text: M

text: i.e.

text: ˆ

text: K(x,

text: t,

text: y,

text: s)dµ

text: (x)

text: s

text: t,

text: y

text: M.

text: M

text: Similarly,

text: one

text: can

text: prove

text: the

text: expected

text: lower

text: bound

text: for

text: K(x,

text: t,

text: y,

text: s)dµ

text: (y).

text: M

text: Remark

text: In

text: particular,

text: proposition

text: tells

text: us

text: that

text: the

text: heat

text: semigroup

text: associated

text: to

text: R

text: where

text: (Mn,

text: g

text: (t))

text: is

text: an

text: expanding

text: gradient

text: Ricci

text: soliton

text: with

text: non

text: negative

text: scalar

text: curvature,

text: is

text: contractive

text: but

text: not

text: L

text: contractive

text: unless

text: it

text: is

text: ﬂat.

text: Proposition

text: (On

text: diagonal

text: upper

text: bound

text: L

text: bound).

text: Let

text: (Mn,

text: g

text: (t))

text: be

text: a

text: t

text: non

text: collapsed

text: Type

text: III

text: solution

text: of

text: the

text: Ricci

text: ﬂow

text: with

text: nonnegative

text: Ricci

text: curvature,

text: i.e.

text: t)

text: Rm(g

text: (t))

text: R

text: Ric(g

text: (t))

text: AVR(g

text: (t))

text: V

text: t

text: Then,

text: c(n,

text: V

text: R

text: K(x,

text: t,

text: y,

text: s)

text: s

text: t,

text: x,

text: y

text: M.

text: n

text: (t

text: s)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: WEAK STABILITY OF RICCI EXPANDERS WITH POSITIVE CURVATURE OPERATOR 35 0 = ψ∂ (ψu (u + a)) t 1 0 ≤ − ((ψu (u + a))2 c(n, R ) ψ∆ (ψu (u + a)) 1 0 + 0 (ψu (u + a) + a2) g0(t) 1 0 a2t t 1 0 1 − − ∇ ∇ +(∂ ∆ )ψ (ψu (u + a)) 2ψ < g0(t)ψ, g0(t)(u (u + a)) > t g0(t) 1 0 1 0 g0(t) |∇ | ≤ −((ψu (u + a))2 c(n, R ) g0(t1)ψ 2 1 0 + 0 (ψu (u + a) + a2) + 2 g0(t1) (ψu (u + a)) a2t t 1 0 ψ 1 0 − 1 1 +(∂ ∆ )ψ (ψu (u + a)) . t g0(t1) 1 0 − (cid:18) (cid:19) Let us estimate (∂ ∆ )ψ as follows : t g0(t1) (cid:48) (cid:48) · ≤ 1 √ 1 ∂ ψ = η φ + ηφ ∂ d · (x, ) c(θ) + − t t g0( ) r2 t r2r (cid:48) · (cid:48)(cid:48)|∇ | ∆ ψ = ηφ ∆ (d (x, )) + ηφ g0(s)d · 2 , g0(s) g0(s) g0(s) g0( ) g0(s) ≥ − c((cid:18)n) · 1 − c (cid:19) · − − d (x, ) (1 θ)r ((1 θ)r)2 g0(s) ≥ − 1 1 ≥ −c(n, θ) c(n) − + − , θ(1 θ)r2 ((1 θ)r)2 r2 where we used the estimate − · ≤ n 1 ∆ d (x, ) · , g0(s) g0(s) d (x, ) g0(s) ≥ that holds outside the cutlocus of x since Ric(g (s)) 0 together with lemma A.3. There- 0 (cid:18) (cid:19) fore, at (x , t ), the maximum of ψu (u + a) denoted by S satisﬁes : 1 1 1 0 ≤ t t S2 c(n, R , θ)a2 1 + + − S + c(n, R )a4, 0 r2 t r2 0 (cid:18) (cid:19) which implies ≤ ≤ t t a sup u S = sup ψu (u + a) c(n, R , θ)a2 1 + + − , \ 1 × ∞ 1 0 0 r2 t r2 P(x,t,θr) P(x,t,θr/2) M (0,+ ) (cid:18) (cid:19) that is, |∇ | ≤ t t | | sup s g0(s)h 2 c(n, R , θ) 1 + + − sup h 2 , \ g0(s) 0 r2 t r2 g0(s) P(x,t,θr) P(x,t,θr/2) P(x,t,r) (cid:3) which is the desired estimate. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WEAK

text: STABILITY

text: OF

text: RICCI

text: EXPANDERS

text: WITH

text: POSITIVE

text: CURVATURE

text: OPERATOR

text: ψ∂

text: (ψu

text: (u

text: a))

text: t

text: ((ψu

text: (u

text: c(n,

text: R

text: ψ∆

text: (ψu

text: (u

text: a))

text: (ψu

text: (u

text: a)

text: t

text: )ψ

text: (ψu

text: (u

text: a))

text: 2ψ

text: (u

text: a))

text: t

text: −((ψu

text: (u

text: c(n,

text: R

text: (ψu

text: (u

text: a)

text: (ψu

text: (u

text: a))

text: t

text: ψ

text: )ψ

text: (ψu

text: (u

text: a))

text: t

text: Let

text: us

text: estimate

text: )ψ

text: as

text: follows

text: t

text: ψ

text: η

text: φ

text: ηφ

text: d

text: (x,

text: c(θ)

text: t

text: t

text: t

text: ψ

text: ηφ

text: (d

text: (x,

text: ηφ

text: c

text: d

text: (x,

text: θ)r

text: −c(n,

text: θ)

text: c(n)

text: where

text: we

text: used

text: the

text: estimate

text: n

text: d

text: (x,

text: d

text: (x,

text: that

text: holds

text: outside

text: the

text: cutlocus

text: of

text: x

text: since

text: Ric(g

text: (s))

text: together

text: with

text: lemma

text: There-

text: fore,

text: at

text: (x

text: t

text: the

text: maximum

text: of

text: ψu

text: (u

text: a)

text: denoted

text: by

text: S

text: satisﬁes

text: t

text: t

text: c(n,

text: R

text: S

text: c(n,

text: R

text: t

text: which

text: implies

text: t

text: t

text: a

text: sup

text: u

text: S

text: sup

text: ψu

text: (u

text: a)

text: c(n,

text: R

text: t

text: P(x,t,θr)

text: M

text: that

text: is,

text: t

text: t

text: sup

text: s

text: c(n,

text: R

text: θ)

text: sup

text: h

text: t

text: P(x,t,θr)

text: P(x,t,r)

text: which

text: is

text: the

text: desired

text: estimate.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: WEAK STABILITY OF RICCI EXPANDERS WITH POSITIVE CURVATURE OPERATOR 45

text: WEAK

text: STABILITY

text: OF

text: RICCI

text: EXPANDERS

text: WITH

text: POSITIVE

text: CURVATURE

text: OPERATOR


text: WEAK STABILITY OF RICCI EXPANDERS WITH POSITIVE CURVATURE OPERATOR 7 − − − 2 Rm(g ) h gklg Rm(g ) gklg Rm(g ) 0 iklj kl ip 0 jklp jp 0 iklp +g Ric(g ) + g Ric(g ) ik 0 kj jk 0 ki − − − = 2 Rm(g ) h gklh Rm(g ) gklh Rm(g ) 0 iklj kl ip 0 jklp jp 0 iklp − − gkl Rm(g ) gkl Rm(g ) 0 jkli 0 iklj +g Ric(g ) + g Ric(g ) ik 0 kj jk 0 ki − − − = 2 Rm(g ) h h Ric(g ) h Ric(g ) 0 iklj kl ip 0 jp jp 0 ip − − hklh Rm(g ) hklh Rm(g ) ip 0 jklp jp 0 iklp − − − 2 Ric(g ) Rm(g ) hkl Rm(g ) hkl 0 ij 0 jkli 0 iklj +2 Ric(g ) + h Ric(g ) + h Ric(g ) 0 ij ik 0 kj jk 0 ki − = 2 Rm(g ) (h + hkl) 0 iklj kl − − hklh Rm(g ) hklh Rm(g ) ip 0 jklp jp 0 iklp − ∗ ∗ ∗ = (g 1 h h Rm(g )) , 0 ij since, if g = (1 + λ )δ is diagonalized in a orthonormal basis of g at some point, where ∈ R kl k kl 0 λ , for k = 1, ..., n, k (cid:0) (cid:1) − − h + hkl = λ + (1 + λ ) 1 1 δ kl k k kl − = (1 + λ ) 1λ2δ k k kl − ∗ ∗ = (g 1 h h) . kl − ∗ ∗ ∗ − ∗ − ∗ ∇ ∗ ∇ Therefore, R0[h] = g 1 h h Rm(g0) + g 1 g 1 g0h g0h. Now, we introduce the last ﬂow equivalent to (3) that we will study in the next sections : − ∇ (∂ L ) h¯ = R [h¯] + g0(t)R [h¯], (4) t t 0 1 with − ∗ ∗ R0[h¯] := h¯ 1 h¯(cid:16)(cid:16)(cid:0)Rm(g0(t))(cid:1) (cid:17) (cid:17) − ∗ − ∗ ∇ ∗ ∇ +g(t) 1 g(t) 1 g0(t)h¯ g0(t)h¯ ∇ ∇ − ∇ g0(t)R [h¯] := g0(t) g (t) + h¯ ab g (t)ab g0(t)h¯ , 1 ij a 0 0 b ij where,

text: WEAK

text: STABILITY

text: OF

text: RICCI

text: EXPANDERS

text: WITH

text: POSITIVE

text: CURVATURE

text: OPERATOR

text: Rm(g

text: h

text: gklg

text: Rm(g

text: gklg

text: Rm(g

text: iklj

text: kl

text: ip

text: jklp

text: jp

text: iklp

text: +g

text: Ric(g

text: g

text: Ric(g

text: ik

text: kj

text: jk

text: ki

text: Rm(g

text: h

text: gklh

text: Rm(g

text: gklh

text: Rm(g

text: iklj

text: kl

text: ip

text: jklp

text: jp

text: iklp

text: gkl

text: Rm(g

text: gkl

text: Rm(g

text: jkli

text: iklj

text: +g

text: Ric(g

text: g

text: Ric(g

text: ik

text: kj

text: jk

text: ki

text: Rm(g

text: h

text: h

text: Ric(g

text: h

text: Ric(g

text: iklj

text: kl

text: ip

text: jp

text: jp

text: ip

text: hklh

text: Rm(g

text: hklh

text: Rm(g

text: ip

text: jklp

text: jp

text: iklp

text: Ric(g

text: Rm(g

text: hkl

text: Rm(g

text: hkl

text: ij

text: jkli

text: iklj

text: Ric(g

text: h

text: Ric(g

text: h

text: Ric(g

text: ij

text: ik

text: kj

text: jk

text: ki

text: Rm(g

text: (h

text: hkl)

text: iklj

text: kl

text: hklh

text: Rm(g

text: hklh

text: Rm(g

text: ip

text: jklp

text: jp

text: iklp

text: (g

text: h

text: h

text: Rm(g

text: ij

text: since,

text: if

text: g

text: λ

text: )δ

text: is

text: diagonalized

text: in

text: a

text: orthonormal

text: basis

text: of

text: g

text: at

text: some

text: point,

text: where

text: R

text: kl

text: k

text: kl

text: λ

text: for

text: k

text: n,

text: k

text: h

text: hkl

text: λ

text: λ

text: δ

text: kl

text: k

text: k

text: kl

text: λ

text: 1λ2δ

text: k

text: k

text: kl

text: (g

text: h

text: h)

text: kl

text: Therefore,

text: g

text: h

text: h

text: g

text: g

text: Now,

text: we

text: introduce

text: the

text: last

text: ﬂow

text: equivalent

text: to

text: that

text: we

text: will

text: study

text: in

text: the

text: next

text: sections

text: L

text: h¯

text: R

text: [h¯]

text: [h¯],

text: t

text: t

text: with

text: h¯

text: +g(t)

text: g(t)

text: [h¯]

text: g

text: (t)

text: h¯

text: ab

text: g

text: (t)ab

text: ij

text: a

text: b

text: ij

text: where,


text: (cid:102) (cid:101) 4 C. KARIMIANPOUR (cid:102) is an isomorphism. The image of K1 in K1 under the isomorphism (3) is the subgroup K := 0 { − | ∈ } { ∈ | ≡ (k, s(k) 1) k K1 of K1. Consider the compact open congruent subgroups K1 := g K1 g I } ≥ j 2 mod pj , for j 1, of K1. ≥ ∩ Lemma 2. The central extension (1) splits trivially over each of the subgroups K1, j 1, T 1 K1, and ∩ j B1 K1. ⊂ O× n Proof. Using the Hensel’s lemma, it is easy to see that 1 + p . Then, it follows from (3) and ≥ | properties of the n-th Hilbert symbol that, for all i 1, s is trivial. On the other hand, it follows | | Kj1 (cid:3) directly from (3) that s ∩ and s ∩ are trivial. (cid:102) ∼ T1 K1 B1 K1 ∼ ∼ × { } ≥ ∩ ∩ × { } ∩ ∩ × { } We identify Kj1 = Kj1 1 , j 1, B1 K(cid:101)1 (cid:101)= (B1 (cid:101) K1)(cid:101) 1 and T 1 K1 = (T 1 K1) 1 as subgroups of K1. (cid:101) In a similar way, we deﬁne the subg(cid:101)roups T , B and K of G to be the inverse images of the stan(cid:101)dard O torus, Borel, and the maximal compact (cid:101)K = G(cid:101)L( ) subgroups of G respectively(cid:101). Th(cid:101)e central extension G does not split over T . Moreover, T is a Heisenberg group. It is not diﬃ(cid:101)cu(cid:101)lt to se(cid:101)e that Z(T ) = {(dg(s, t), ζ) | s, t ∈ F×n, ζ ∈ µ }, and [T : Z(T )] = n4. Moreover, set A = C (T ∩ K) = {(dg(s, t), ζ) | ∈ F× | | ∈ n } T s, t , n val(s), n val(t), ζ µ . Then, A is a maximal(cid:101)abel(cid:101)ian subgroup of T and [T : A] = n2. In (cid:48)| n × { } addition, β(cid:101) is trivial, where N is the unipotent radical of B. Hence, we can identify N with N 1 . N (cid:110) Under this identiﬁcation, we have the Levi decomposition: B = T N. It is s(cid:101)hown in [11] that the central ≥ extension G splits over K. For j 1, let K denote the family of compact open congruent subgroups { ∈ | ≡ } j ∩ g K g I mod pj of K. Similar to Lemma 2, one can show that G splits over K , T K and ∩ 2 (cid:102) j B K. (cid:102) 4. Branching Rules for G1 (cid:102) First, we present the construction of the principal series representations of G1 following [13]. Fix a → C× faithful character (cid:15)(cid:102): µ . A representation of G1 is g(cid:102)enuine if the central subgroup µ acts by (cid:15). n n Such representations do not factor through representations of G1. The construction of principal series representations of G1 is based on the essential fact that T 1 is a Heisenberg subgroup, and hence its representations are governed by the Stone-von Neumann theorem, which we state here. See [13] for the proof. Theorem 1 (Stone-von Neumann). Let H be a Heisenberg group with center Z(H) such that H/Z(H) ∩ { } is ﬁnite, and let χ be a character of Z(H). Suppose that ker(χ) [H, H] = 1 . Then there is a unique (up to isomorp(cid:102)hism) i(cid:102)rreducible representation π of H with central ch∼aract(cid:102)er χ. Let A be any maximal abelian subgroup of H and let χ be any extension of χ to A. Then π = IndHχ . (cid:102) (cid:102) 0 A 0 (cid:102) ∞ | Note∩that [T 1 : Z(T 1)] = n2 < . Let χ be a(cid:102)genuine character of Z(T(cid:102)1), s(cid:102)o that χ µn = (cid:15). Thus, ker(χ) [T 1, T 1] is trivial. Hence Theorem 1 applies: genuine irreducible smooth representations ρ of T 1 (cid:102) are classiﬁed by genuine smooth characters of Z(T 1). Moreover, dim(ρ) = [T 1 : A1] = n. (cid:102) Let χ0 be a ﬁxe(cid:102)d extension of χ to A1; so that (ρ, IndTA11χ0) is the unique smooth genuine irredu(cid:102)cible r(cid:102)epresentation of T 1 with central character χ. Let us again wr(cid:102)ite ρ for the genuine smooth irreducible (cid:102) representation of T 1, with central character χ, extended trivially over N1 to a representation of B1 = (cid:102) T 1 (cid:110) N1. Then the genuine principal series representation of G1 associated to ρ is IndG1ρ, where I(cid:102)nd B1 (cid:102) (cid:102) denotes the smooth (non-normalized) induction. In the rest of this section, we decompose Res IndG1ρ K1 B1 into irreducible constituents. We drop the adjective “genuine” for simplicity. ##LTLine## ##LTLine##

text: C.

text: KARIMIANPOUR

text: is

text: an

text: isomorphism.

text: The

text: image

text: of

text: in

text: under

text: the

text: isomorphism

text: is

text: the

text: subgroup

text: K

text: (k,

text: s(k)

text: k

text: of

text: Consider

text: the

text: compact

text: open

text: congruent

text: subgroups

text: g

text: g

text: I

text: j

text: mod

text: pj

text: for

text: j

text: of

text: Lemma

text: The

text: central

text: extension

text: splits

text: trivially

text: over

text: each

text: of

text: the

text: subgroups

text: j

text: T

text: and

text: j

text: O×

text: n

text: Proof.

text: Using

text: the

text: Hensel’s

text: lemma,

text: it

text: is

text: easy

text: to

text: see

text: that

text: p

text: Then,

text: it

text: follows

text: from

text: and

text: properties

text: of

text: the

text: n-th

text: Hilbert

text: symbol

text: that,

text: for

text: all

text: i

text: s

text: is

text: trivial.

text: On

text: the

text: other

text: hand,

text: it

text: follows

text: directly

text: from

text: that

text: s

text: and

text: s

text: are

text: trivial.

text: We

text: identify

text: j

text: and

text: T

text: (T

text: as

text: subgroups

text: of

text: In

text: a

text: similar

text: way,

text: we

text: deﬁne

text: the

text: T

text: B

text: and

text: K

text: of

text: G

text: to

text: be

text: the

text: inverse

text: images

text: of

text: the

text: O

text: torus,

text: Borel,

text: and

text: the

text: maximal

text: compact

text: subgroups

text: of

text: G

text: central

text: extension

text: G

text: does

text: not

text: split

text: over

text: T

text: Moreover,

text: T

text: is

text: a

text: Heisenberg

text: group.

text: It

text: is

text: not

text: to

text: that

text: Z(T

text: {(dg(s,

text: t),

text: ζ)

text: s,

text: t

text: F×n,

text: ζ

text: µ

text: and

text: [T

text: Z(T

text: Moreover,

text: set

text: A

text: C

text: (T

text: K)

text: {(dg(s,

text: t),

text: ζ)

text: F×

text: n

text: T

text: s,

text: t

text: n

text: val(s),

text: n

text: val(t),

text: ζ

text: µ

text: Then,

text: A

text: is

text: a

text: subgroup

text: of

text: T

text: and

text: [T

text: A]

text: In

text: n

text: addition,

text: is

text: trivial,

text: where

text: N

text: is

text: the

text: unipotent

text: radical

text: of

text: B.

text: Hence,

text: we

text: can

text: identify

text: N

text: with

text: N

text: N

text: Under

text: this

text: identiﬁcation,

text: we

text: have

text: the

text: Levi

text: decomposition:

text: B

text: T

text: N.

text: It

text: is

text: in

text: that

text: the

text: central

text: extension

text: G

text: splits

text: over

text: K.

text: For

text: j

text: let

text: K

text: denote

text: the

text: family

text: of

text: compact

text: open

text: congruent

text: subgroups

text: j

text: g

text: K

text: g

text: I

text: mod

text: pj

text: of

text: K.

text: Similar

text: to

text: Lemma

text: one

text: can

text: show

text: that

text: G

text: splits

text: over

text: K

text: T

text: K

text: and

text: j

text: B

text: K.

text: Branching

text: Rules

text: for

text: First,

text: we

text: present

text: the

text: construction

text: of

text: the

text: principal

text: series

text: representations

text: of

text: following

text: Fix

text: a

text: C×

text: faithful

text: character

text: µ

text: A

text: representation

text: of

text: is

text: if

text: the

text: central

text: subgroup

text: µ

text: acts

text: by

text: n

text: n

text: Such

text: representations

text: do

text: not

text: factor

text: through

text: representations

text: of

text: The

text: construction

text: of

text: principal

text: series

text: representations

text: of

text: is

text: based

text: on

text: the

text: essential

text: fact

text: that

text: T

text: is

text: a

text: Heisenberg

text: subgroup,

text: and

text: hence

text: its

text: representations

text: are

text: governed

text: by

text: the

text: Stone-von

text: Neumann

text: theorem,

text: which

text: we

text: state

text: here.

text: See

text: for

text: the

text: proof.

text: Theorem

text: (Stone-von

text: Neumann).

text: Let

text: H

text: be

text: a

text: Heisenberg

text: group

text: with

text: center

text: Z(H)

text: such

text: that

text: H/Z(H)

text: is

text: ﬁnite,

text: and

text: let

text: χ

text: be

text: a

text: character

text: of

text: Z(H).

text: Suppose

text: that

text: ker(χ)

text: [H,

text: H]

text: Then

text: there

text: is

text: a

text: unique

text: (up

text: to

text: representation

text: π

text: of

text: H

text: with

text: central

text: χ.

text: Let

text: A

text: be

text: any

text: maximal

text: abelian

text: subgroup

text: of

text: H

text: and

text: let

text: χ

text: be

text: any

text: extension

text: of

text: χ

text: to

text: A.

text: Then

text: π

text: IndHχ

text: A

text: Note∩that

text: [T

text: Z(T

text: Let

text: χ

text: be

text: character

text: of

text: that

text: χ

text: µn

text: Thus,

text: ker(χ)

text: [T

text: T

text: is

text: trivial.

text: Hence

text: Theorem

text: applies:

text: genuine

text: irreducible

text: smooth

text: representations

text: ρ

text: of

text: T

text: are

text: classiﬁed

text: by

text: genuine

text: smooth

text: characters

text: of

text: Z(T

text: Moreover,

text: dim(ρ)

text: [T

text: n.

text: Let

text: χ0

text: be

text: a

text: extension

text: of

text: χ

text: to

text: so

text: that

text: (ρ,

text: is

text: the

text: unique

text: smooth

text: genuine

text: of

text: T

text: with

text: central

text: character

text: χ.

text: Let

text: us

text: again

text: ρ

text: for

text: the

text: genuine

text: smooth

text: irreducible

text: representation

text: of

text: T

text: with

text: central

text: character

text: χ,

text: extended

text: trivially

text: over

text: to

text: a

text: representation

text: of

text: T

text: Then

text: the

text: genuine

text: principal

text: series

text: representation

text: of

text: associated

text: to

text: ρ

text: is

text: where

text: denotes

text: the

text: smooth

text: (non-normalized)

text: induction.

text: In

text: the

text: rest

text: of

text: this

text: section,

text: we

text: decompose

text: Res

text: into

text: irreducible

text: constituents.

text: We

text: drop

text: the

text: adjective

text: “genuine”

text: for

text: simplicity.

text: ##LTLine##

text: ##LTLine##


text: Keywords: computer simulation/emulation; Busy Beaver Turing ma- chines; intrinsic universality; Turing-completeness; dynamical systems; reprogrammability; behavioural methods 1 Introduction Attempts to answer even the simplest questions about the behaviour of com- puter programs are bedevilled by uncomputability. The concept of asymptotic intrinsic universality introduced here is based upon a Bayesian approach to emu- lation by computer programs of other computer programs. The method provides a means to quantify their reprogramming capabilities, associating them with a deciding procedure that asymptotically recognizes computation with a conﬁ- dence value and sets forth a hierarchy of reprogrammability (see [21]) based upon the likelihood of a system being, in one degree or another, close to (or removed from) Turing universality. In [18], a related conjecture concerning other kinds of simply deﬁned pro- grams was presented, suggesting that all Busy Beaver Turing machines may be capable of universal computation, as they seem to share some of the infor- mational and complex properties of systems known to be capable of universal computational behaviour. We have recently found that most computer programs can be reprogrammed to emulate an increasing number of other (diﬀerent) computer programs of the same size [8] under a similar block emulation transformation or set of compil- ers of increasing size. We also previously advanced a conceptual framework for reprogrammability based upon the display of diﬀerent qualitative output be- haviours [20] and modelled as a type of Turing test to determine computational capabilities [21]. This has been used in connection with an instance of natu- ral computation–in an in-silico simulation of Porphyrin molecules [12] in the context of spatial computing. Here we advance a Bayesian method, namely asymptotic intrinsic univer- sality, that draws everything together and translates the seminal concept of computation universality to degrees of belief and conﬁdence based upon emula- tion and reprogrammability capabilities applicable to natural computation. We test the method with a case study of the set of Turing machines deﬁned by the Busy Beaver functions. 2 Methods (cid:80) 2.1 The classical Turing (cid:83)machine model A Turing machine consists of a ﬁnite alphabet set with symbols = { } { } { } 0, 1, . . . , k and states 1, 2, . . . n 0 , with 0 the “halting state”. The Turing machine “runs” on an one-way unbounded tape and for each pair: (cid:48) the machine’s current “state” n ; and 2

text: Keywords:

label: abstract

text: computer

text: simulation/emulation;

text: Busy

text: Beaver

text: Turing

text: ma-

text: chines;

text: intrinsic

text: universality;

text: Turing-completeness;

text: dynamical

text: systems;

text: reprogrammability;

text: behavioural

text: methods

text: Introduction

text: Attempts

text: to

text: answer

text: even

text: the

text: simplest

text: questions

text: about

text: the

text: behaviour

text: of

text: com-

text: puter

text: programs

text: are

text: bedevilled

text: by

text: uncomputability.

text: The

text: concept

text: of

text: asymptotic

text: intrinsic

text: universality

text: introduced

text: here

text: is

text: based

text: upon

text: a

text: Bayesian

text: approach

text: to

text: emu-

text: lation

text: by

text: computer

text: programs

text: of

text: other

text: computer

text: programs.

text: The

text: method

text: provides

text: a

text: means

text: to

text: quantify

text: their

text: reprogramming

text: capabilities,

text: associating

text: them

text: with

text: a

text: deciding

text: procedure

text: that

text: asymptotically

text: recognizes

text: computation

text: with

text: a

text: conﬁ-

text: dence

text: value

text: and

text: sets

text: forth

text: a

text: hierarchy

text: of

text: reprogrammability

text: (see

text: based

text: upon

text: the

text: likelihood

text: of

text: a

text: system

text: being,

text: in

text: one

text: degree

text: or

text: another,

text: close

text: to

text: (or

text: removed

text: from)

text: Turing

text: universality.

text: In

text: a

text: related

text: conjecture

text: concerning

text: other

text: kinds

text: of

text: simply

text: deﬁned

text: pro-

text: grams

text: was

text: presented,

text: suggesting

text: that

text: all

text: Busy

text: Beaver

text: Turing

text: machines

text: may

text: be

text: capable

text: of

text: universal

text: computation,

text: as

text: they

text: seem

text: to

text: share

text: some

text: of

text: the

text: infor-

text: mational

text: and

text: complex

text: properties

text: of

text: systems

text: known

text: to

text: be

text: capable

text: of

text: universal

text: computational

text: behaviour.

text: We

text: have

text: recently

text: found

text: that

text: most

text: computer

text: programs

text: can

text: be

text: reprogrammed

text: to

text: emulate

text: an

text: increasing

text: number

text: of

text: other

text: (diﬀerent)

text: computer

text: programs

text: of

text: the

text: same

text: size

text: under

text: a

text: similar

text: block

text: emulation

text: transformation

text: or

text: set

text: of

text: compil-

text: ers

text: of

text: increasing

text: size.

text: We

text: also

text: previously

text: advanced

text: a

text: conceptual

text: framework

text: for

text: reprogrammability

text: based

text: upon

text: the

text: display

text: of

text: diﬀerent

text: qualitative

text: output

text: be-

text: haviours

text: and

text: modelled

text: as

text: a

text: type

text: of

text: Turing

text: test

text: to

text: determine

text: computational

text: capabilities

text: This

text: has

text: been

text: used

text: in

text: connection

text: with

text: an

text: instance

text: of

text: natu-

text: ral

text: computation–in

text: an

text: in-silico

text: simulation

text: of

text: Porphyrin

text: molecules

text: in

text: the

text: context

text: of

text: spatial

text: computing.

text: Here

text: we

text: advance

text: a

text: Bayesian

text: method,

text: namely

text: asymptotic

text: intrinsic

text: univer-

text: sality,

text: that

text: draws

text: everything

text: together

text: and

text: translates

text: the

text: seminal

text: concept

text: of

text: computation

text: universality

text: to

text: degrees

text: of

text: belief

text: and

text: conﬁdence

text: based

text: upon

text: emula-

text: tion

text: and

text: reprogrammability

text: capabilities

text: applicable

text: to

text: natural

text: computation.

text: We

text: test

text: the

text: method

text: with

text: a

text: case

text: study

text: of

text: the

text: set

text: of

text: Turing

text: machines

text: deﬁned

text: by

text: the

text: Busy

text: Beaver

text: functions.

text: Methods

text: The

text: classical

text: Turing

text: model

text: A

text: Turing

text: machine

text: consists

text: of

text: a

text: ﬁnite

text: alphabet

text: set

text: with

text: symbols

text: k

text: and

text: states

text: n

text: with

text: the

text: “halting

text: state”.

text: The

text: Turing

text: machine

text: “runs”

text: on

text: an

text: one-way

text: unbounded

text: tape

text: and

text: for

text: each

text: pair:

text: the

text: machine’s

text: current

text: “state”

text: n

text: and


text: (cid:88) (cid:124) (cid:123)†(cid:122) (cid:125) Hˆ (t) = E aˆ aˆ + Hˆ +Hˆ (t) (9) eﬀ i i i ee I i Hˆ0 The many body states are obtained by exact diagonalisation of Hˆ . Fixing the total number 0 of electrons to 4, the 4 particle many-body ground state is where all the levels in the conduction band are empty and all the levels in the valence band are occupied. In Fock state representation it is given by (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) | (cid:105) ≈ | (cid:105) GS 0000 1111 (10) c v | (cid:105) where the occupation numbers n , n , n , n , n , n , n , n relate to the 8 states in Fig. 1(a). | (cid:105) 1 2 3 4 5 6 7 8 The lowest energy excitation ex is due to an electronic transition across the band gap from the highest level in the valence band to the lowest level in the conduction band, which is called exciton. For the system considered, the energy of the lowest singlet excitation √ is E = 1.334 eV above the ground state. The corresponding many body eigenstate is | (cid:105) e≈x | (cid:105) − | (cid:105) ex 1/ 2( 10000111 01001011 ). The energy E is slightly smaller than the diﬀerence ex between the single-particle levels 1 and 5 (1.347 eV from table 1) due to small admixtures of other states. (The experimentally measured gap of 1.07 meV is even smaller. This might be due a diﬀerent shape, but the admixture from lower lying levels (e.g. 1p) neglected in our model √ will also play a role.) Two other many-body states play a very important role in the description | (cid:105) ≈ | (cid:105) − | (cid:105) √ of MEG: The high energy single exciton SX 1/ 2( 00101101 00011110 ) and the double | (cid:105) ≈ | (cid:105) − | (cid:105) exciton DX 1/ 2( 11001001 11000110 ) which are both singlets and close in energy. The detuning between these two states can be controlled by the parameter ∆ (which modiﬁes ≈ − the band gap, see section ) as shown in Fig. 1(b). At ∆ 0.0215 eV the two states show an | − | | | ≈ anticrossing with the diﬀerence E E = 2V , where V 0.6 meV is the DX SX| (cid:105) DX,|SX (cid:105) DX,SX Coulomb matrix element between the states DX and SX . 2.5. Equation of motion for the density matrix The system consists of a classical light ﬁeld and a quantum system with discrete energies. The change to the system induced by the light ﬁeld can be best described by the time evolution (cid:20) (cid:21) of the reduced density operator for the(cid:88)system given by the Lindblad equation [28]. d Njump † − 1 † † ¯h ρˆ (t) = i[ρˆ (t), Hˆ (t)] + Γ Lˆ ρˆ (t)Lˆ (ρˆ (t)Lˆ Lˆ + Lˆ Lˆ ρˆ (t)) (11) dt S S eﬀ j j S j 2 S j j j j S j=1 The time evolution in Eq. (11) was solved in the many particle basis with a 4th order Runge- Kutta method. The jump operators Lˆ describe diﬀerent dissipation processes (with rate Γ /¯h) j j which are restoring the ground state at suﬃciently long time after the excitation. Here we use: Relaxation in the conduction band † † Lˆ = aˆ ↑aˆ ↑ + aˆ ↓aˆ ↓ with strength Γ (12) rel 1 3 2 4 Relaxation Relaxation in the valence band † † Lˆ = aˆ ↑aˆ ↑ + aˆ ↓aˆ ↓ with strength Γ (13) rel 7 5 4 8 Relaxation Recombination across the band gap ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Hˆ

text: (t)

text: E

text: aˆ

text: aˆ

text: Hˆ

text: +Hˆ

text: (t)

text: eﬀ

text: i

text: i

text: i

text: ee

text: I

text: i

text: The

text: many

text: body

text: states

text: are

text: obtained

text: by

text: exact

text: diagonalisation

text: of

text: Hˆ

text: Fixing

text: the

text: total

text: number

text: of

text: electrons

text: to

text: the

text: particle

text: many-body

text: ground

text: state

text: is

text: where

text: all

text: the

text: levels

text: in

text: the

text: conduction

text: band

text: are

text: empty

text: and

text: all

text: the

text: levels

text: in

text: the

text: valence

text: band

text: are

text: occupied.

text: In

text: Fock

text: state

text: representation

text: it

text: is

text: given

text: by

text: GS

text: c

text: v

text: where

text: the

text: occupation

text: numbers

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: relate

text: to

text: the

text: states

text: in

text: Fig.

text: The

text: lowest

text: energy

text: excitation

text: ex

text: is

text: due

text: to

text: an

text: electronic

text: transition

text: across

text: the

text: band

text: gap

text: from

text: the

text: highest

text: level

text: in

text: the

text: valence

text: band

text: to

text: the

text: lowest

text: level

text: in

text: the

text: conduction

text: band,

text: which

text: is

text: called

text: exciton.

text: For

text: the

text: system

text: considered,

text: the

text: energy

text: of

text: the

text: lowest

text: singlet

text: excitation

text: is

text: E

text: eV

text: above

text: the

text: ground

text: state.

text: The

text: corresponding

text: many

text: body

text: eigenstate

text: is

text: e≈x

text: ex

text: The

text: energy

text: E

text: is

text: slightly

text: smaller

text: than

text: the

text: diﬀerence

text: ex

text: between

text: the

text: single-particle

text: levels

text: and

text: eV

text: from

text: table

text: due

text: to

text: small

text: admixtures

text: of

text: other

text: states.

text: (The

text: experimentally

text: measured

text: gap

text: of

text: meV

text: is

text: even

text: smaller.

text: This

text: might

text: be

text: due

text: a

text: diﬀerent

text: shape,

text: but

text: the

text: admixture

text: from

text: lower

text: lying

text: levels

text: (e.g.

text: neglected

text: in

text: our

text: model

text: will

text: also

text: play

text: a

text: role.)

text: Two

text: other

text: many-body

text: states

text: play

text: a

text: very

text: important

text: role

text: in

text: the

text: description

text: of

text: MEG:

text: The

text: high

text: energy

text: single

text: exciton

text: SX

text: and

text: the

text: double

text: exciton

text: DX

text: which

text: are

text: both

text: singlets

text: and

text: close

text: in

text: energy.

text: The

text: detuning

text: between

text: these

text: two

text: states

text: can

text: be

text: controlled

text: by

text: the

text: parameter

text: (which

text: modiﬁes

text: the

text: band

text: gap,

text: see

text: section

text: as

text: shown

text: in

text: Fig.

text: At

text: eV

text: the

text: two

text: states

text: show

text: an

text: anticrossing

text: with

text: the

text: diﬀerence

text: E

text: E

text: where

text: V

text: meV

text: is

text: the

text: DX

text: SX|

text: DX,|SX

text: DX,SX

text: Coulomb

text: matrix

text: element

text: between

text: the

text: states

text: DX

text: and

text: SX

text: Equation

text: of

text: motion

text: for

text: the

text: density

text: matrix

text: The

text: system

text: consists

text: of

text: a

text: classical

text: light

text: ﬁeld

text: and

text: a

text: quantum

text: system

text: with

text: discrete

text: energies.

text: The

text: change

text: to

text: the

text: system

text: induced

text: by

text: the

text: light

text: ﬁeld

text: can

text: be

text: best

text: described

text: by

text: the

text: time

text: evolution

text: of

text: the

text: reduced

text: density

text: operator

text: for

text: given

text: by

text: the

text: Lindblad

text: equation

text: d

text: Njump

text: ¯h

text: ρˆ

text: (t)

text: i[ρˆ

text: (t),

text: Hˆ

text: (t)]

text: Γ

text: Lˆ

text: ρˆ

text: (t)Lˆ

text: (ρˆ

text: (t)Lˆ

text: Lˆ

text: Lˆ

text: Lˆ

text: ρˆ

text: (t))

text: dt

text: S

text: S

text: eﬀ

text: j

text: j

text: S

text: j

text: S

text: j

text: j

text: j

text: j

text: S

text: The

text: time

text: evolution

text: in

text: Eq.

text: was

text: solved

text: in

text: the

text: many

text: particle

text: basis

text: with

text: a

text: order

text: Runge-

text: Kutta

text: method.

text: The

text: jump

text: operators

text: Lˆ

text: describe

text: diﬀerent

text: dissipation

text: processes

text: (with

text: rate

text: Γ

text: /¯h)

text: j

text: j

text: which

text: are

text: restoring

text: the

text: ground

text: state

text: at

text: suﬃciently

text: long

text: time

text: after

text: the

text: excitation.

text: Here

text: we

text: use:

text: Relaxation

text: in

text: the

text: conduction

text: band

text: Lˆ

text: aˆ

text: ↑aˆ

text: aˆ

text: ↓aˆ

text: with

text: strength

text: Γ

text: rel

text: Relaxation

text: Relaxation

text: in

text: the

text: valence

text: band

text: Lˆ

text: aˆ

text: ↑aˆ

text: aˆ

text: ↓aˆ

text: with

text: strength

text: Γ

text: rel

text: Relaxation

text: Recombination

text: across

text: the

text: band

text: gap

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: † † Lˆ = aˆ ↑aˆ ↑ + aˆ ↓aˆ ↓ with strength Γ (14) rec 5 1 6 2 Recombination Dephasing in the conduction band † † Lˆ = aˆ ↑aˆ ↑ + aˆ ↓aˆ ↓ with strength Γ deph 1 1 2 2 Dephasing † † (15) Lˆ = aˆ ↑aˆ ↑ + aˆ ↓aˆ ↓ with strength Γ deph 3 3 4 4 Dephasing Dephasing in the valence band † † Lˆ = aˆ ↑aˆ ↑ + aˆ ↓aˆ ↓ with strength Γ deph 5 5 6 6 Dephasing † † (16) Lˆ = aˆ ↑aˆ ↑ + aˆ ↓aˆ ↓ with strength Γ deph 7 7 8 8 Dephasing These jump operators are deﬁned in such a way that they all conserve the total spin indicated by the spin indices. The diﬀerent decoherence mechanism which were assumed phenomenologically in Eq. (11) can be associated to all forms of intrinsic scattering mechanisms other than electron-electron scattering, which has already been included in the eﬀective Hamiltonian. 3. Results We simulate Eq. (11) in order to determine the density matrix as a function of time. In all | (cid:105)(cid:104) | calculations we use τ = 150 fs and t = 200 fs together with the initial condition ρˆ = GS GS 0 at t = 0. Regarding the dissipation mechanisms, dephasing is the fastest scale and we use ≈ Γ = 6 meV, which corresponds to a τ = h¯/Γ 100 fs. Relaxation is Dephasing Dephasing Dephasing typically slightly slower, and we use Γ = 3.3 meV. These are typical values in solids. Relaxation Recombination is typically on the ns time scale. In order to save simulation time, we use a larger ≈ value of Γ = 0.33 meV (corresponding to τ = ¯h/Γ 2 ps), Recombination Recombination Recombination which is still a factor of 10 slower than the other processes. Our main observable is the recombination rate { † } Rec(t) = Γ T r Lˆ ρˆLˆ (17) Recombination rec rec Integrating over time provides the number of recombined electron-hole pairs generated by the pulse. Fig. 2(a) shows the calculated number of recombinations for excitations in the vicinity of the 2p resonance for diﬀerent values of ∆. The average full width at half maximum (FWHM) of ≈ the peaks is about 22 meV which is mainly due to the pulse width τ = 150 fs. It can be seen − that recombination is enhanced when the two excitons are close in energy near ∆ = 0.0215 eV. The other ingredient needed for calculating the yield is the power transferred to the system by the oscillating ﬁeld. It can be deﬁned as the absorbed energy per unit time and can be calculated as d (cid:104) (cid:105) i (cid:104) (cid:105) i (cid:104) (cid:105) P(t) = H = [H (t), H ] = [H (t), H ] (18) 0 eﬀ 0 I 0 dt ¯h ¯h Integrating over time provides the absorbed energy, which is displayed in Fig. 2(b). This has − a similar shape, except for a small dip near ∆ = 0.0215 eV. The yield is then deﬁned as the ratio between the number of recombinations and the absorbed photons. The latter is given by the absorbed energy divided by the photon energy. Due to the ##LTLine## ##LTLine## ##LTLine##

text: Lˆ

text: aˆ

text: ↑aˆ

text: aˆ

text: ↓aˆ

text: with

text: strength

text: Γ

text: rec

text: Recombination

text: Dephasing

text: in

text: the

text: conduction

text: band

text: Lˆ

text: aˆ

text: ↑aˆ

text: aˆ

text: ↓aˆ

text: with

text: strength

text: Γ

text: deph

text: Dephasing

text: Lˆ

text: aˆ

text: ↑aˆ

text: aˆ

text: ↓aˆ

text: with

text: strength

text: Γ

text: deph

text: Dephasing

text: Dephasing

text: in

text: the

text: valence

text: band

text: Lˆ

text: aˆ

text: ↑aˆ

text: aˆ

text: ↓aˆ

text: with

text: strength

text: Γ

text: deph

text: Dephasing

text: Lˆ

text: aˆ

text: ↑aˆ

text: aˆ

text: ↓aˆ

text: with

text: strength

text: Γ

text: deph

text: Dephasing

text: These

text: jump

text: operators

text: are

text: deﬁned

text: in

text: such

text: a

text: way

text: that

text: they

text: all

text: conserve

text: the

text: total

text: spin

text: indicated

text: by

text: the

text: spin

text: indices.

text: The

text: diﬀerent

text: decoherence

text: mechanism

text: which

text: were

text: assumed

text: phenomenologically

text: in

text: Eq.

text: can

text: be

text: associated

text: to

text: all

text: forms

text: of

text: intrinsic

text: scattering

text: mechanisms

text: other

text: than

text: electron-electron

text: scattering,

text: which

text: has

text: already

text: been

text: included

text: in

text: the

text: eﬀective

text: Hamiltonian.

text: Results

text: We

text: simulate

text: Eq.

text: in

text: order

text: to

text: determine

text: the

text: density

text: matrix

text: as

text: a

text: function

text: of

text: time.

text: In

text: all

text: calculations

text: we

text: use

text: τ

text: fs

text: and

text: t

text: fs

text: together

text: with

text: the

text: initial

text: condition

text: ρˆ

text: GS

text: GS

text: at

text: t

text: Regarding

text: the

text: dissipation

text: mechanisms,

text: dephasing

text: is

text: the

text: fastest

text: scale

text: and

text: we

text: use

text: Γ

text: meV,

text: which

text: corresponds

text: to

text: a

text: τ

text: h¯/Γ

text: fs.

text: Relaxation

text: is

text: Dephasing

text: Dephasing

text: Dephasing

text: typically

text: slightly

text: slower,

text: and

text: we

text: use

text: Γ

text: meV.

text: These

text: are

text: typical

text: values

text: in

text: solids.

text: Relaxation

text: Recombination

text: is

text: typically

text: on

text: the

text: ns

text: time

text: scale.

text: In

text: order

text: to

text: save

text: simulation

text: time,

text: we

text: use

text: a

text: larger

text: value

text: of

text: Γ

text: meV

text: (corresponding

text: to

text: τ

text: ¯h/Γ

text: ps),

text: Recombination

text: Recombination

text: Recombination

text: which

text: is

text: still

text: a

text: factor

text: of

text: slower

text: than

text: the

text: other

text: processes.

text: Our

text: main

text: observable

text: is

text: the

text: recombination

text: rate

text: Rec(t)

text: Γ

text: T

text: r

text: Lˆ

text: ρˆLˆ

text: Recombination

text: rec

text: rec

text: Integrating

text: over

text: time

text: provides

text: the

text: number

text: of

text: recombined

text: electron-hole

text: pairs

text: generated

text: by

text: the

text: pulse.

text: Fig.

text: shows

text: the

text: calculated

text: number

text: of

text: recombinations

text: for

text: excitations

text: in

text: the

text: vicinity

text: of

text: the

text: resonance

text: for

text: diﬀerent

text: values

text: of

text: The

text: average

text: full

text: width

text: at

text: half

text: maximum

text: (FWHM)

text: of

text: the

text: peaks

text: is

text: about

text: meV

text: which

text: is

text: mainly

text: due

text: to

text: the

text: pulse

text: width

text: τ

text: fs.

text: It

text: can

text: be

text: seen

text: that

text: recombination

text: is

text: enhanced

text: when

text: the

text: two

text: excitons

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text: 14 Vaddi et al. omy, Manitoba, Canada 3 Harvard University Cambridge, MA 02138, USA 4 University at Buﬀalo, Buﬀalo, NY, 14260, USA

text: Vaddi

text: et

text: al.

text: omy,

text: Manitoba,

text: Canada

text: Harvard

text: University

text: Cambridge,

text: MA

text: USA

text: University

text: at

text: Buﬀalo,

text: Buﬀalo,

text: NY,

text: USA