text: 11 (iv) If no bubble precedes bubble i on its branch, then TABLE I the vehicle (i, 1) approaches the intersection at its designated SYSTEM PARAMETERS time τi,1 = τi, with at least a velocity of νnom. Then, by Parameter Symbol Value applying Proposition VII.3 inductively, we see that the last General parameters vehicle (i, m ) of bubble i approaches the intersection with a Car length L 4m vitetlaokceitsyaotfmatoisleta(sLt ν+no∆m)a/nνdnoTmia,ammi o≤unTtia,o1f+tim(meit−o g1o)Tpiaasttatnhde IZMnotaenxre.selsecpnteigoetdnhslliemngitth ∆LvMs,Lm,Le 761002mmkm/h intersection. Thus from (15), we see that claim (iv) is satisﬁed Max. accel. uM 3−m/s2 in this case. Min. accel. um 4m/s2 HD algorithm parameters Now suppose bubble q precedes bubble i on its branch and Nominal speed of crossing νnom 48km/h saNubopowpv,oe,usesTinqca,glmaaiqmrg≤u(mivTe)qan,i1tss+tarnu(ameloqfgo−orubs1u)tbTobitlahetosq=e. iFτnqroP+mro(pomousqrit−iroena1s)VoTnIIiian.3tg,. PNUaoprmpaemirn-eabtleoruinnitnder(o-1vn3e)hinicteler-vaephpircolaecahptpimroeach time σTT0niaomt 1≈≈.211..2538ss we s≤ee that if (cid:64)Fi,1 at any time t ∈ [ts, Tia,1], then we have TMimaxe. p#enrieowd bfourbbelxeescuotniobnraonfcAhlkgorithm 1 TNN¯cks 32.77s Ta Ta + Tiat. However, note from (3) and (15) that Max. # bubbles scheduled ¯ 8 i,1 q,mq Signal-based algorithm parameters ≥ ≥ − ≥ τ τ + τ¯occ τ + (m 1)Tiat + Tiat Ta + Tiat, Green-time 10s i q q q q q,mq where in obtaining the second inequality we have used (15). Recall from the ﬁrst paragraph of the proof of Proposi- The velocity of vehicle j at t , vv(t ), takes a random value ≥ s j s tion VII.3 that for any vehicle (i, j), Ta τ and in uniformly chosen from the interval [0, vM]. Then, the position ≥ i,j∃F i,j pa∈rticular Tia,1 τi. Thus, we conclude that i,1 for all time of vehicle j at ts, xvj(ts) is obtained from (11) as t [ts, Tia,1] = [ts, τi] and that vehicle (i, 1) approaches the min{−(L + L ), xv− (t ) − σ (t )D(vv− (t ), vv(t ))}, intersection at its assigned time τ with a velocity of at least e m j 1 s j s j 1 s j s i νnom. Hence, by using induction over all vehicles in bubble i where xv− (t ) and vv− (t ) are the position and the velocity j 1 s j 1 s and over all bubbles i themselves we conclude that claim (ii) of the last vehicle previously deﬁned on the same branch as holds. that of vehicle j. If there is no previously deﬁned vehicle on ∞ the branch, then xv− (t ) = . Recall from Section that if Theorem VIII.1 does not guarantee the optimal operation j 1 s of the system at the level of individual vehicles under the Tcs < vLMs , then vehicl−es entering the problem domain during the time interval [t T , t ] are within the staging zone proposed hierarchical-distributed coordination approach. How- s cs s − at t . This explains the imposition of the upper bound (L + ever, this result guarantees the optimality at the level of s e L ) on xv(t ). Finally, the number of vehicles spawned on bubbles, on each time t = sT , for the bubbles scheduled m j s s cs a branch at t is determined as follows. The procedure to at t . We believe this is a good compromise in balancing the s s spawn new vehicles described above is repeated as long as the trade-off between optimal vehicle operation and complexity of ∈ − spawned vehicle is in the staging zone, i.e., xv(t ) [ (L + planning and control. − − j s e L + L s), (L + L )]. m e m B. Signal-based trafﬁc coordination IX. SIMULATIONS The vehicle control policy in the simulations with signal- This section presents simulations of our proposed based trafﬁc coordination is given by (14) with g given hierarchical-distributed design and comparisons with a signal- sf by (12) and with g = u . The trafﬁc signaling policy is as based trafﬁc coordination approach under varying trafﬁc con- uc M follows - at any given time only one of the four branches has ditions. Table I speciﬁes the system parameters that we keep the right of way (green or yellow signal). The other branches ﬁxed across all the simulations presented here. The parameters do not have a right of way (red light). When the signal changes Tnom and Tiat are computed parameters, while the remaining to green from red for a branch, it stays green for a period of ones in the table are design choices. time we call green-time and then it turns to yellow. When the signal for a branch is yellow the vehicles on that branch, A. Dynamic trafﬁc generation which are yet to cross the intersection, are divided into two In order to simulate dynamically generated trafﬁc, new groups - those that can come to a stop before the beginning vehicles are spawned every T units of time anywhere in the of the intersection and those that cannot. For the ﬁrst group cs staging zone of each branch as a Poisson arrival process [20]. of vehicles, a virtual vehicle is introduced at position L and In order to ensure safe following at the moment vehicle j is having a velocity of 0 so that the control policy (14) with the spawned, we deﬁne the arrival process on σ rather than on added virtual vehicle ensures that the second group of vehicles j vehicle distances. To be precise, let t = sT be the time at comes to a stop before the intersection. The second group of s cs which vehicle j is spawned. Then, we let σ (t ) = σ, with σ vehicles continue with the usual control policy. The signal j s drawn from the distribution with probability density function changes from yellow to red when the ﬁrst group of vehicles − ≥ 1 + (1/µ)e (1/µ)σ so that σ (t ) 1 and the distribution has all cross the intersection. The virtual vehicle for the branch is j s a mean of 1 + µ. Thus, the smaller the value of µ the greater retained as long as the branch has yellow or red signals. The is the density of the generated trafﬁc. branches get the right of way in a round-robin manner. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (iv)

text: If

text: no

text: bubble

text: precedes

text: bubble

text: i

text: on

text: its

text: branch,

text: then

text: TABLE

text: I

text: the

text: vehicle

text: (i,

text: approaches

text: the

text: intersection

text: at

text: its

text: designated

text: SYSTEM

text: PARAMETERS

text: time

text: τi,

text: with

text: at

text: least

text: a

text: velocity

text: of

text: νnom.

text: Then,

text: by

text: Parameter

text: Symbol

text: Value

text: applying

text: Proposition

text: inductively,

text: we

text: see

text: that

text: the

text: last

text: General

text: parameters

text: vehicle

text: (i,

text: m

text: of

text: bubble

text: i

text: approaches

text: the

text: intersection

text: with

text: a

text: Car

text: length

text: L

text: vitetlaokceitsyaotfmatoisleta(sLt

text: ν+no∆m)a/nνdnoTmia,ammi

text: ∆LvMs,Lm,Le

text: intersection.

text: Thus

text: from

text: we

text: see

text: that

text: claim

text: (iv)

text: is

text: satisﬁed

text: Max.

text: accel.

text: uM

text: in

text: this

text: case.

text: Min.

text: accel.

text: um

text: HD

text: algorithm

text: parameters

text: Now

text: suppose

text: bubble

text: q

text: precedes

text: bubble

text: i

text: on

text: its

text: branch

text: and

text: Nominal

text: speed

text: of

text: crossing

text: νnom

text: time

text: we

text: s≤ee

text: that

text: if

text: at

text: any

text: time

text: t

text: [ts,

text: then

text: we

text: have

text: TMimaxe.

text: p#enrieowd

text: bfourbbelxeescuotniobnraonfcAhlkgorithm

text: TNN¯cks

text: Ta

text: Ta

text: Tiat.

text: However,

text: note

text: from

text: and

text: that

text: Max.

text: bubbles

text: scheduled

text: q,mq

text: Signal-based

text: algorithm

text: parameters

text: τ

text: τ

text: τ¯occ

text: τ

text: (m

text: Tiat

text: Ta

text: Tiat,

text: Green-time

text: i

text: q

text: q

text: q

text: q

text: q,mq

text: where

text: in

text: obtaining

text: the

text: second

text: inequality

text: we

text: have

text: used

text: Recall

text: from

text: the

text: ﬁrst

text: paragraph

text: of

text: the

text: proof

text: of

text: Proposi-

text: The

text: velocity

text: of

text: vehicle

text: j

text: at

text: t

text: vv(t

text: takes

text: a

text: random

text: value

text: s

text: j

text: s

text: tion

text: that

text: for

text: any

text: vehicle

text: (i,

text: j),

text: Ta

text: τ

text: and

text: in

text: uniformly

text: chosen

text: from

text: the

text: interval

text: vM].

text: Then,

text: the

text: position

text: i,j∃F

text: i,j

text: pa∈rticular

text: τi.

text: Thus,

text: we

text: conclude

text: that

text: for

text: all

text: time

text: of

text: vehicle

text: j

text: at

text: ts,

text: xvj(ts)

text: is

text: obtained

text: from

text: as

text: t

text: [ts,

text: [ts,

text: τi]

text: and

text: that

text: vehicle

text: (i,

text: approaches

text: the

text: min{−(L

text: L

text: xv−

text: (t

text: σ

text: (t

text: )D(vv−

text: (t

text: vv(t

text: intersection

text: at

text: its

text: assigned

text: time

text: τ

text: with

text: a

text: velocity

text: of

text: at

text: least

text: e

text: m

text: j

text: s

text: j

text: s

text: j

text: s

text: j

text: s

text: i

text: νnom.

text: Hence,

text: by

text: using

text: induction

text: over

text: all

text: vehicles

text: in

text: bubble

text: i

text: where

text: xv−

text: (t

text: and

text: vv−

text: (t

text: are

text: the

text: position

text: and

text: the

text: velocity

text: j

text: s

text: j

text: s

text: and

text: over

text: all

text: bubbles

text: i

text: themselves

text: we

text: conclude

text: that

text: claim

text: (ii)

text: of

text: the

text: last

text: vehicle

text: previously

text: deﬁned

text: on

text: the

text: same

text: branch

text: as

text: holds.

text: that

text: of

text: vehicle

text: j.

text: If

text: there

text: is

text: no

text: previously

text: deﬁned

text: vehicle

text: on

text: the

text: branch,

text: then

text: xv−

text: (t

text: Recall

text: from

text: Section

text: that

text: if

text: Theorem

text: does

text: not

text: guarantee

text: the

text: optimal

text: operation

text: j

text: s

text: of

text: the

text: system

text: at

text: the

text: level

text: of

text: individual

text: vehicles

text: under

text: the

text: Tcs

text: vLMs

text: then

text: vehicl−es

text: entering

text: the

text: problem

text: domain

text: during

text: the

text: time

text: interval

text: [t

text: T

text: t

text: are

text: within

text: the

text: staging

text: zone

text: proposed

text: hierarchical-distributed

text: coordination

text: approach.

text: How-

text: s

text: cs

text: s

text: at

text: t

text: This

text: explains

text: the

text: imposition

text: of

text: the

text: upper

text: bound

text: (L

text: ever,

text: this

text: result

text: guarantees

text: the

text: optimality

text: at

text: the

text: level

text: of

text: s

text: e

text: L

text: on

text: xv(t

text: Finally,

text: the

text: number

text: of

text: vehicles

text: spawned

text: on

text: bubbles,

text: on

text: each

text: time

text: t

text: sT

text: for

text: the

text: bubbles

text: scheduled

text: m

text: j

text: s

text: s

text: cs

text: a

text: branch

text: at

text: t

text: is

text: determined

text: as

text: follows.

text: The

text: procedure

text: to

text: at

text: t

text: We

text: believe

text: this

text: is

text: a

text: good

text: compromise

text: in

text: balancing

text: the

text: s

text: s

text: spawn

text: new

text: vehicles

text: described

text: above

text: is

text: repeated

text: as

text: long

text: as

text: the

text: trade-off

text: between

text: optimal

text: vehicle

text: operation

text: and

text: complexity

text: of

text: spawned

text: vehicle

text: is

text: in

text: the

text: staging

text: zone,

text: i.e.,

text: xv(t

text: (L

text: planning

text: and

text: control.

text: j

text: s

text: e

text: L

text: L

text: s),

text: (L

text: L

text: m

text: e

text: m

text: B.

text: Signal-based

text: trafﬁc

text: coordination

text: IX.

text: SIMULATIONS

text: The

text: vehicle

text: control

text: policy

text: in

text: the

text: simulations

text: with

text: signal-

text: This

text: section

text: presents

text: simulations

text: of

text: our

text: proposed

text: based

text: trafﬁc

text: coordination

text: is

text: given

text: by

text: with

text: g

text: given

text: hierarchical-distributed

text: design

text: and

text: comparisons

text: with

text: a

text: signal-

text: sf

text: by

text: and

text: with

text: g

text: u

text: The

text: trafﬁc

text: signaling

text: policy

text: is

text: as

text: based

text: trafﬁc

text: coordination

text: approach

text: under

text: varying

text: trafﬁc

text: con-

text: uc

text: M

text: follows

text: at

text: any

text: given

text: time

text: only

text: one

text: of

text: the

text: four

text: branches

text: has

text: ditions.

text: Table

text: I

text: speciﬁes

text: the

text: system

text: parameters

text: that

text: we

text: keep

text: the

text: right

text: of

text: way

text: (green

text: or

text: yellow

text: signal).

text: The

text: other

text: branches

text: ﬁxed

text: across

text: all

text: the

text: simulations

text: presented

text: here.

text: The

text: parameters

text: do

text: not

text: have

text: a

text: right

text: of

text: way

text: (red

text: light).

text: When

text: the

text: signal

text: changes

text: Tnom

text: and

text: Tiat

text: are

text: computed

text: parameters,

text: while

text: the

text: remaining

text: to

text: green

text: from

text: red

text: for

text: a

text: branch,

text: it

text: stays

text: green

text: for

text: a

text: period

text: of

text: ones

text: in

text: the

text: table

text: are

text: design

text: choices.

text: time

text: we

text: call

text: green-time

text: and

text: then

text: it

text: turns

text: to

text: yellow.

text: When

text: the

text: signal

text: for

text: a

text: branch

text: is

text: yellow

text: the

text: vehicles

text: on

text: that

text: branch,

text: A.

text: Dynamic

text: trafﬁc

text: generation

text: which

text: are

text: yet

text: to

text: cross

text: the

text: intersection,

text: are

text: divided

text: into

text: two

text: In

text: order

text: to

text: simulate

text: dynamically

text: generated

text: trafﬁc,

text: new

text: groups

text: those

text: that

text: can

text: come

text: to

text: a

text: stop

text: before

text: the

text: beginning

text: vehicles

text: are

text: spawned

text: every

text: T

text: units

text: of

text: time

text: anywhere

text: in

text: the

text: of

text: the

text: intersection

text: and

text: those

text: that

text: cannot.

text: For

text: the

text: ﬁrst

text: group

text: cs

text: staging

text: zone

text: of

text: each

text: branch

text: as

text: a

text: Poisson

text: arrival

text: process

text: of

text: vehicles,

text: a

text: virtual

text: vehicle

text: is

text: introduced

text: at

text: position

text: L

text: and

text: In

text: order

text: to

text: ensure

text: safe

text: following

text: at

text: the

text: moment

text: vehicle

text: j

text: is

text: having

text: a

text: velocity

text: of

text: so

text: that

text: the

text: control

text: policy

text: with

text: the

text: spawned,

text: we

text: deﬁne

text: the

text: arrival

text: process

text: on

text: σ

text: rather

text: than

text: on

text: added

text: virtual

text: vehicle

text: ensures

text: that

text: the

text: second

text: group

text: of

text: vehicles

text: j

text: vehicle

text: distances.

text: To

text: be

text: precise,

text: let

text: t

text: sT

text: be

text: the

text: time

text: at

text: comes

text: to

text: a

text: stop

text: before

text: the

text: intersection.

text: The

text: second

text: group

text: of

text: s

text: cs

text: which

text: vehicle

text: j

text: is

text: spawned.

text: Then,

text: we

text: let

text: σ

text: (t

text: σ,

text: with

text: σ

text: vehicles

text: continue

text: with

text: the

text: usual

text: control

text: policy.

text: The

text: signal

text: j

text: s

text: drawn

text: from

text: the

text: distribution

text: with

text: probability

text: density

text: function

text: changes

text: from

text: yellow

text: to

text: red

text: when

text: the

text: ﬁrst

text: group

text: of

text: vehicles

text: so

text: that

text: σ

text: (t

text: and

text: the

text: distribution

text: has

text: all

text: cross

text: the

text: intersection.

text: The

text: virtual

text: vehicle

text: for

text: the

text: branch

text: is

text: j

text: s

text: a

text: mean

text: of

text: µ.

text: Thus,

text: the

text: smaller

text: the

text: value

text: of

text: µ

text: the

text: greater

text: retained

text: as

text: long

text: as

text: the

text: branch

text: has

text: yellow

text: or

text: red

text: signals.

text: The

text: is

text: the

text: density

text: of

text: the

text: generated

text: trafﬁc.

text: branches

text: get

text: the

text: right

text: of

text: way

text: in

text: a

text: round-robin

text: manner.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 Jorge Corte´s received the Licenciatura degree in mathematics from Universidad de Zaragoza, Zaragoza, Spain, in 1997, and the Ph.D. degree in engineering mathematics from Universidad Carlos III de Madrid, Madrid, Spain, in 2001. He held post- doctoral positions with the University of Twente, Twente, The Netherlands, and the University of Illinois at Urbana-Champaign, Urbana, IL, USA. He was an Assistant Professor with the Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA, USA, from 2004 to 2007. He is currently a Professor in the Department of Mechanical and Aerospace Engineering, University of California, San Diego, CA, USA. He is the author of Geometric, Control and Numerical Aspects of Nonholonomic Systems (Springer-Verlag, 2002) and co-author (together with F. Bullo and S. Mart´ınez) of Distributed Control of Robotic Networks (Princeton University Press, 2009). He is an IEEE Fellow and an IEEE Control Systems Society Distinguished Lecturer. His current research interests include distributed control, networked games, power networks, distributed optimization, spatial estimation, and geometric mechanics. ##LTFigure## ##LTFigure##

label: paragraph

text: Jorge

text: Corte´s

text: received

text: the

text: Licenciatura

text: degree

text: in

text: mathematics

text: from

text: Universidad

text: de

text: Zaragoza,

text: Zaragoza,

text: Spain,

text: in

text: and

text: the

text: Ph.D.

text: degree

text: in

text: engineering

text: mathematics

text: from

text: Universidad

text: Carlos

text: III

text: de

text: Madrid,

text: Madrid,

text: Spain,

text: in

text: He

text: held

text: post-

text: doctoral

text: positions

text: with

text: the

text: University

text: of

text: Twente,

text: Twente,

text: The

text: Netherlands,

text: and

text: the

text: University

text: of

text: Illinois

text: at

text: Urbana-Champaign,

text: Urbana,

text: IL,

text: USA.

text: He

text: was

text: an

text: Assistant

text: Professor

text: with

text: the

text: Department

text: of

text: Applied

text: Mathematics

text: and

text: Statistics,

text: University

text: of

text: California,

text: Santa

text: Cruz,

text: CA,

text: USA,

text: from

text: to

text: He

text: is

text: currently

text: a

text: Professor

text: in

text: the

text: Department

text: of

text: Mechanical

text: and

text: Aerospace

text: Engineering,

text: University

text: of

text: California,

text: San

text: Diego,

text: CA,

text: USA.

text: He

text: is

text: the

text: author

text: of

text: Geometric,

text: Control

text: and

text: Numerical

text: Aspects

text: of

text: Nonholonomic

text: Systems

text: (Springer-Verlag,

text: and

text: co-author

text: (together

text: with

text: F.

text: Bullo

text: and

text: S.

text: Mart´ınez)

text: of

text: Distributed

text: Control

text: of

text: Robotic

text: Networks

text: (Princeton

text: University

text: Press,

text: He

text: is

text: an

text: IEEE

text: Fellow

text: and

text: an

text: IEEE

text: Control

text: Systems

text: Society

text: Distinguished

text: Lecturer.

text: His

text: current

text: research

text: interests

text: include

text: distributed

text: control,

text: networked

text: games,

text: power

text: networks,

text: distributed

text: optimization,

text: spatial

text: estimation,

text: and

text: geometric

text: mechanics.

text: ##LTFigure##

text: ##LTFigure##


text: be seen as predicting the probability of this word given the con- text. This phrasal representation built in (9) and (10) then would be treated as the history for the next step. A standard sigmoidal nonlinearity is used for f and the probability distribution func- tion g is a standard softmax. If we limit the context to sentence levels and move the prob- ability space to the sequence of the words or n-grams, equa- tion (9) and (10) can be seen as composition function building − phrase x(t), of the length n, from sub-phrase x(t 1), of the − length n 1, and the nth word w(t). Equation (11) can be seen as building the unnormalized probability p˜over the phrase x(t). We can rephrase the equations (9), (10) and (11) as (cid:18) (cid:20) (cid:21)(cid:19) − x(t) = f U x(t 1) , and (12) Figure 1: Test contrastive entropy monotonically increasing for w(t) with test set distortion level. y(t) = g(Wx(t)). (13) Here we use the standard sigmoidal non linearity for the function f and the identity function for g. We now deﬁne the score of a(cid:88)length N sentence W as N S(W) = y(t). (14) t=1 The probability of the sentence can now be modeled as an exponential distribution 1 − p(W) = e S(W). (15) Z Figure 2: Training contrastive entropy monotonically increas- where Z is the partition(cid:88)func(cid:16)tion and the contra(cid:17)stive entropy ing with epochs. from (7) can be calculated as − H (T) = 1/N S(Wˆ ) S(W) . (16) c d ∈ W T where Wˆ is the distorted version of W with distortion per- d centage d. Training is done using a contrastive criterion where we try to maximize the distance between the in-domain sentence and its distorted version. This formulation is similar to one followed by Collobert et al. [8] and Okanohara et al. [7] for language modeling and by Smith and Eisner [4] for POS tagging. Mathe- matically, we can deﬁne this pseudo discriminative training ob- jective as (cid:88) (cid:110) (cid:111) − θ(cid:63) = arg min max 0, 1 S(W ) + S(Wˆ ) . (17) Figure 3: Test contrastive entropy increasing with training dis- d d θ ∈ tortion margin. Yellow , red and blue line represent training d D with distortion margin of 50%, 30% and 10% respectively. where Wˆ is the distorted version of sentence W and θ = d d (U, X, W) is the parameter of the model. This simplistic sentence level recurrent neural network respectively. The vocabulary is limited to 10k words with all model is implemented in python using Theano [11] and is avail- words outside this set mapped to a special token < unk >. able at https://github.com/kushalarora/sentenceRNN. We start by examining the distortion generation mecha- 5. Experiments nism. As the evaluation includes the word level models, we need to preserve the word count. To do this, we restrict dis- We use the Pen Treebank dataset with the following splits tortions to only two types: substitution and transpositions. For and preprocessing: Sections 0-20 were used as training data, substitutions, we randomly swap the current word in the sen- sections 21-22 for validation and 23-24 for testing. The train- tence with a random word from the vocabulary. For transpo- ing, validation and testing token sizes are 930k, 74k and 82k sition, we randomly select a word from the same sentence and ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

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text: Observacio´n de lentes gravitatorias con ALMA A. Bonilla1a, O. Tolozaa, I. Fuentesa, V. Mottaa aDepartamento de F´ısica y Astronom´ıa, Facultad de Ciencias, Universidad de Valpara´ıso, Valpara´ıso, Chile. Abstract Gravitational lensing is a fundamental tool for cosmology. A recent instrument which will provide more information for models of these objects is ALMA. Our goal is to select lens candidates to observe with ALMA and then model them using GravLens Software. We had selected 12 quadruple images systems from the CASTLES database, which show a high probability of observ- ing extended sources in the submillimetric range. These new data will allow us to improve existing models. Resumen Las lentes gravitatorias son una herramienta fundamental para la cosmolog´ıa. Un nuevo instru- mento que nos proporcionara´ mayor informacio´n para los modelos de estos objetos, es ALMA. Nuestro objetivo es seleccionar lentes candidatas para observar con ALMA y posteriormente modelarlas mediante el programa GravLens. Seleccionamos de la base de datos de CASTLES 12 sistemas cua´druples, los cuales tienen mayor probabilidad de observar fuentes extendidas en el rango submilime´trico. Estos nuevos datos nos permitira´n mejorar los modelos exitentes para dichos sistemas. Keywords: Lentes gravitatorias, ALMA, submilim´etrico. 2011 MSC: 00-01, 99-00 1. Introduction El efecto lente gravitatoria se produce cuando la luz de un objeto distante (fuente) se desv´ıa, debido a una distribucio´n de masa (lente), produciendo ima´genes de la fuente [2]. Uno de los in- strumentos que nos proporcionara´ mayor informacio´n sobre estos sistemas es el Atacama Large Millimeter/submillimeter Array (ALMA) [5]. Actualmente, en el ciclo cero se encuentran op- erativas 12 antenas para las conﬁguraciones compacta y extendida con receptores cuyas bandas disponibles son las denominadas 3, 6, 7 y 9, que cubren un rango de frecuencia de 84-720 GHz (MANUAL) [6]. 2. Marco teo´rico del efecto lente Utilizando el esquema de la ﬁgura 1 puede deducirse la ecuacio´n de la lente: 1alex.acidjazz@gmail.com Preprint submitted to Asociacio´n Argentina de Astronom´ıa BAAA, Vol. 54, 2011 May 28, 2020 ##LTLine## ##LTLine## ##LTLine##

text: Observacio´n

label: title

text: de

text: lentes

text: gravitatorias

text: con

text: ALMA

text: A.

label: author

text: O.

text: Tolozaa,

text: I.

text: Fuentesa,

text: V.

text: Mottaa

text: aDepartamento

text: de

text: F´ısica

text: y

text: Astronom´ıa,

text: Facultad

text: de

text: Ciencias,

text: Universidad

text: de

text: Valpara´ıso,

text: Valpara´ıso,

text: Chile.

text: Abstract

text: Gravitational

label: abstract

text: lensing

text: is

text: a

text: fundamental

text: tool

text: for

text: cosmology.

text: A

text: recent

text: instrument

text: which

text: will

text: provide

text: more

text: information

text: for

text: models

text: of

text: these

text: objects

text: is

text: ALMA.

text: Our

text: goal

text: is

text: to

text: select

text: lens

text: candidates

text: to

text: observe

text: with

text: ALMA

text: and

text: then

text: model

text: them

text: using

text: GravLens

text: Software.

text: We

text: had

text: selected

text: quadruple

text: images

text: systems

text: from

text: the

text: CASTLES

text: database,

text: which

text: show

text: a

text: high

text: probability

text: of

text: observ-

text: ing

text: extended

text: sources

text: in

text: the

text: submillimetric

text: range.

text: These

text: new

text: data

text: will

text: allow

text: us

text: to

text: improve

text: existing

text: models.

text: Resumen

text: Las

text: lentes

text: gravitatorias

text: son

text: una

text: herramienta

text: fundamental

text: para

text: la

text: cosmolog´ıa.

text: Un

text: nuevo

text: instru-

text: mento

text: que

text: nos

text: proporcionara´

text: mayor

text: informacio´n

text: para

text: los

text: modelos

text: de

text: estos

text: objetos,

text: es

text: ALMA.

text: Nuestro

text: objetivo

text: es

text: seleccionar

text: lentes

text: candidatas

text: para

text: observar

text: con

text: ALMA

text: y

text: posteriormente

text: modelarlas

text: mediante

text: el

text: programa

text: GravLens.

text: Seleccionamos

text: de

text: la

text: base

text: de

text: datos

text: de

text: CASTLES

text: sistemas

text: cua´druples,

text: los

text: cuales

text: tienen

text: mayor

text: probabilidad

text: de

text: observar

text: fuentes

text: extendidas

text: en

text: el

text: rango

text: submilime´trico.

text: Estos

text: nuevos

text: datos

text: nos

text: permitira´n

text: mejorar

text: los

text: modelos

text: exitentes

text: para

text: dichos

text: sistemas.

text: Keywords:

text: Lentes

text: gravitatorias,

text: ALMA,

text: submilim´etrico.

text: MSC:

text: Introduction

text: El

text: efecto

text: lente

text: gravitatoria

text: se

text: produce

text: cuando

text: la

text: luz

text: de

text: un

text: objeto

text: distante

text: (fuente)

text: se

text: desv´ıa,

text: debido

text: a

text: una

text: distribucio´n

text: de

text: masa

text: (lente),

text: produciendo

text: ima´genes

text: de

text: la

text: fuente

text: Uno

text: de

text: los

text: in-

text: strumentos

text: que

text: nos

text: proporcionara´

text: mayor

text: informacio´n

text: sobre

text: estos

text: sistemas

text: es

text: el

text: Atacama

text: Large

text: Millimeter/submillimeter

text: Array

text: (ALMA)

text: Actualmente,

text: en

text: el

text: ciclo

text: cero

text: se

text: encuentran

text: op-

text: erativas

text: antenas

text: para

text: las

text: conﬁguraciones

text: compacta

text: y

text: extendida

text: con

text: receptores

text: cuyas

text: bandas

text: disponibles

text: son

text: las

text: denominadas

text: y

text: que

text: cubren

text: un

text: rango

text: de

text: frecuencia

text: de

text: GHz

text: (MANUAL)

text: Marco

text: teo´rico

text: del

text: efecto

text: lente

text: Utilizando

text: el

text: esquema

text: de

text: la

text: ﬁgura

text: puede

text: deducirse

text: la

text: ecuacio´n

text: de

text: la

text: lente:

text: Preprint

text: submitted

text: to

text: Asociacio´n

text: Argentina

text: de

text: Astronom´ıa

text: BAAA,

text: Vol.

text: May

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Greedy online colouring with buﬀering Wojciech Kordecki Department of Computer Science Faculty of Technical and Economic Science The Witelon State University of Applied Sciences in Legnica e-mail: wojciech.kordecki@pwsz-legnica.eu Anna L(cid:32) yczkowska-Han´ckowiak Faculty of Informatics and Electronic Economy Poznan´ University of Economics e-mail: anna.lyczkowska-hanckowiak@ae.poznan.pl May 28, 2020 Abstract We consider the problem of online graph colouring. Whenever a node is requested, a colour must be assigned to the node, and this colour must be diﬀerent from the colours of any of its neighbours. According to the greedy algorithm the node is coloured by the colour with the smallest possible k. The goal is to use as few colours as possible. We propose an algorithm, where the node is coloured not immediately, but only after the collection of next requests stored in the buﬀer of size j. In other words, the ﬁrst node in the buﬀer is coloured deﬁnitively taking into account all possible colourisations of the remaining nodes in the buﬀer. If there are r possible corrected colourings, then the one with the probability 1/r is chosen. The ﬁrst coloured node is removed from the buﬀer to enable the entrance of the next request. A number of colours in a two examples of graphs: crown graphs and Kneser graphs have been analysed. Keywords: online colouring, greedy algorithm. 2010 Mathematics Subject Classiﬁcation: 05C15, 05C85. 1

text: Greedy

text: online

text: colouring

text: with

text: buﬀering

text: Wojciech

text: Kordecki

text: Department

text: of

text: Computer

text: Science

text: Faculty

text: of

text: Technical

text: and

text: Economic

text: Science

text: The

text: Witelon

text: State

text: University

text: of

text: Applied

text: Sciences

text: in

text: Legnica

text: e-mail:

text: wojciech.kordecki@pwsz-legnica.eu

text: Anna

text: yczkowska-Han´ckowiak

text: Faculty

text: of

text: Informatics

text: and

text: Electronic

text: Economy

text: Poznan´

text: University

text: of

text: Economics

text: e-mail:

text: anna.lyczkowska-hanckowiak@ae.poznan.pl

text: May

text: Abstract

text: We

text: consider

text: the

text: problem

text: of

text: online

text: graph

text: colouring.

text: Whenever

text: a

text: node

text: is

text: requested,

text: a

text: colour

text: must

text: be

text: assigned

text: to

text: the

text: node,

text: and

text: this

text: colour

text: must

text: be

text: diﬀerent

text: from

text: the

text: colours

text: of

text: any

text: of

text: its

text: neighbours.

text: According

text: to

text: the

text: greedy

text: algorithm

text: the

text: node

text: is

text: coloured

text: by

text: the

text: colour

text: with

text: the

text: smallest

text: possible

text: k.

text: The

text: goal

text: is

text: to

text: use

text: as

text: few

text: colours

text: as

text: possible.

text: We

text: propose

text: an

text: algorithm,

text: where

text: the

text: node

text: is

text: coloured

text: not

text: immediately,

text: but

text: only

text: after

text: the

text: collection

text: of

text: next

text: requests

text: stored

text: in

text: the

text: buﬀer

text: of

text: size

text: j.

text: In

text: other

text: words,

text: the

text: ﬁrst

text: node

text: in

text: the

text: buﬀer

text: is

text: coloured

text: deﬁnitively

text: taking

text: into

text: account

text: all

text: possible

text: colourisations

text: of

text: the

text: remaining

text: nodes

text: in

text: the

text: buﬀer.

text: If

text: there

text: are

text: r

text: possible

text: corrected

text: colourings,

text: then

text: the

text: one

text: with

text: the

text: probability

text: is

text: chosen.

text: The

text: ﬁrst

text: coloured

text: node

text: is

text: removed

text: from

text: the

text: buﬀer

text: to

text: enable

text: the

text: entrance

text: of

text: the

text: next

text: request.

text: A

text: number

text: of

text: colours

text: in

text: a

text: two

text: examples

text: of

text: graphs:

text: crown

text: graphs

text: and

text: Kneser

text: graphs

text: have

text: been

text: analysed.

text: Keywords:

text: online

text: colouring,

text: greedy

text: algorithm.

text: Mathematics

text: Subject

text: Classiﬁcation:


text: [6] M. Fu¨rer. Improved hardness results for approximating the chromatic number. In Proc. 36th IEEE Symp. Foundations of Computer Science (FOCS ’95), pages 414–421, 1995. [7] M. M. Halld´orsson. Online coloring known graphs. Electron. J. Combin., 7:1–9, 2000. [8] M. M. Halldo´rsson. Online coloring of hypergraphs, 2010. www.ru.is/ faculty/mmh/papers/onhyper-final.pdf. [9] M. M. Halldo´rsson and M. Szegedy. Lower bounds for on-line graph coloring. Theoretical Computer Science, 130:163–174, 1994. [10] S. Irani. Coloring inductive graphs on-line. Algoritmica, 11:53–72, 1994. [11] D. S. Johnson. Worst-case behavior of graph coloring algorithms. In Proc. 5th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Utilitas Mathematicae, pages 513–527, Winnipeg, 1974. [12] H. A. Kierstead and W. T. Trotter. On-line graph coloring. DIMACS, 7:85–92, 1992. [13] L. Lov´asz, M. Saks, and W. T. Trotter. An on-line graph coloring algorithm with sublinear performance ratio. Discrete Math., 75:319– 325, 1989. [14] A. Miller. Online graph colouring, 2004. http://www.cumc.math.ca/ 2005/papers/miller.pdf. [15] H. Zang, J. P. Jue, and B. Mukherjee. A review of routing and wave- length assignment approaches for wavelength-routed optical wdm net- works. SPIE Optical Networks Magazine, 1:47–60, 2000. 12

label: reference

text: M.

text: Fu¨rer.

text: Improved

text: hardness

text: results

text: for

text: approximating

text: the

text: chromatic

text: number.

text: In

text: Proc.

text: IEEE

text: Symp.

text: Foundations

text: of

text: Computer

text: Science

text: (FOCS

text: pages

text: M.

text: M.

text: Halld´orsson.

text: Online

text: coloring

text: known

text: graphs.

text: Electron.

text: J.

text: Combin.,

text: M.

text: M.

text: Halldo´rsson.

text: Online

text: coloring

text: of

text: hypergraphs,

text: M.

text: M.

text: Halldo´rsson

text: and

text: M.

text: Szegedy.

text: Lower

text: bounds

text: for

text: on-line

text: graph

text: coloring.

text: Theoretical

text: Computer

text: Science,

text: S.

text: Irani.

text: Coloring

text: inductive

text: graphs

text: on-line.

text: Algoritmica,

text: D.

text: S.

text: Johnson.

text: Worst-case

text: behavior

text: of

text: graph

text: coloring

text: algorithms.

text: In

text: Proc.

text: Southeastern

text: Conf.

text: on

text: Combinatorics,

text: Graph

text: Theory,

text: and

text: Computing,

text: Utilitas

text: Mathematicae,

text: pages

text: Winnipeg,

text: H.

text: A.

text: Kierstead

text: and

text: W.

text: T.

text: Trotter.

text: On-line

text: graph

text: coloring.

text: DIMACS,

text: L.

text: Lov´asz,

text: M.

text: Saks,

text: and

text: W.

text: T.

text: Trotter.

text: An

text: on-line

text: graph

text: coloring

text: algorithm

text: with

text: sublinear

text: performance

text: ratio.

text: Discrete

text: Math.,

text: A.

text: Miller.

text: Online

text: graph

text: colouring,

text: H.

text: Zang,

text: J.

text: P.

text: Jue,

text: and

text: B.

text: Mukherjee.

text: A

text: review

text: of

text: routing

text: and

text: wave-

text: length

text: assignment

text: approaches

text: for

text: wavelength-routed

text: optical

text: wdm

text: net-

text: works.

text: SPIE

text: Optical

text: Networks

text: Magazine,


text: 6:A 8:B 6:C 8:D 5:A 7:B 5:C 7:D 2:A 4:B 2:B 4:B 1:A 3:B 1:A 3:A C Figure 1: Colouring online for the crown graphs with buﬀering: b = 1 4 (left) and b = 2 (right) • K If k = 1, then = K , the complete graph on n vertices. n,k n • K is the Petersen graph. 5,2 Miller in [14] looks at the class of Kneser graphs as an interesting one and still unconsidered. At ﬁrst let us consider the simplest example, i.e. Petersen graph, using the greedy algorithm with the buﬀer of size b = 1 and b = 2. Let us assume that vertices of Petersen graph are numbered in the lexicographic order: { } { } { } v = 1, 2 , v = 1, 3 , . . . , v = 4, 5 . (5) 1 2 10 In the following example we point out that a buﬀering can both decrease and increase the necessary number of colours. However, as we point out through simulations in Section , the average number colours used by our algorithm with the buﬀer of size b = 2 is a bit smaller than the average number colours given with the buﬀer of size b = 1, i.e. without buﬀering. Therefore we can formulate the following problem. Problem 10. Determine the smallest b that EC(2) − EC(b) − n,k − n,k > δ (6) n 2k + 2 n 2k + 2 8 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

font: default

text: C

label: caption

text: Figure

text: Colouring

text: online

text: for

text: the

text: crown

text: graphs

text: with

text: buﬀering:

text: b

text: (left)

text: and

text: b

text: (right)

label: list

text: K

text: If

text: k

text: then

text: K

text: the

text: complete

text: graph

text: on

text: n

text: vertices.

text: n,k

text: n

text: K

text: is

text: the

text: Petersen

text: graph.

text: Miller

text: in

text: looks

text: at

text: the

text: class

text: of

text: Kneser

text: graphs

text: as

text: an

text: interesting

text: one

text: and

text: still

text: unconsidered.

text: At

text: ﬁrst

text: let

text: us

text: consider

text: the

text: simplest

text: example,

text: i.e.

text: Petersen

text: graph,

text: using

text: the

text: greedy

text: algorithm

text: with

text: the

text: buﬀer

text: of

text: size

text: b

text: and

text: b

text: Let

text: us

text: assume

text: that

text: vertices

text: of

text: Petersen

text: graph

text: are

text: numbered

text: in

text: the

text: lexicographic

text: order:

text: v

text: v

text: v

text: In

text: the

text: following

text: example

text: we

text: point

text: out

text: that

text: a

text: buﬀering

text: can

text: both

text: decrease

text: and

text: increase

text: the

text: necessary

text: number

text: of

text: colours.

text: However,

text: as

text: we

text: point

text: out

text: through

text: simulations

text: in

text: Section

text: the

text: average

text: number

text: colours

text: used

text: by

text: our

text: algorithm

text: with

text: the

text: buﬀer

text: of

text: size

text: b

text: is

text: a

text: bit

text: smaller

text: than

text: the

text: average

text: number

text: colours

text: given

text: with

text: the

text: buﬀer

text: of

text: size

text: b

text: i.e.

text: without

text: buﬀering.

text: Therefore

text: we

text: can

text: formulate

text: the

text: following

text: problem.

text: Problem

text: Determine

text: the

text: smallest

text: b

text: that

text: EC(b)

text: n,k

text: n,k

text: δ

text: n

text: n

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: 2 I. INTRODUCTION Cognitive radio (CR) is a promising spectrum sharing technology that mitigates the spectrum scarcity that resulted from the recent tremendous growth of wireless devices and services over the past decade [1]. As many licensed primary users (PU) block the available spectrum, CR systems exploit the same spectrum resources and allow secondary users (SU) to coexist with PU without degrading the PU quality-of-service (QoS) [2]. CR can operate in three strategies called, interweave, overlay and underlay spectrum sharing. The CR with interweave strategy accesses the PU spectrum when it is not used, i.e., the PU is idle. On the other hand, CR with both overlay and underlay strategies coexists with the PU. In the overlay strategy, the SU uses part of its power to assist the PU transmission in order to compensate its interference impact on the PU. The scenario is more challenging for the underlay strategy, where the SU has to control its power to be within an acceptable level at the PU receiver side. CR systems can be incorporated with other communication techniques, such as full-duplex (FD) and cooperative communications, to improve the spectrum utilization of the communication networks and the SU performance. FD is a spectral efﬁcient paradigm that allows the communication nodes to transmit and receive simultaneously at the same frequency. FD has recently attracted wide attention especially after the progress in self-interference cancellation, which gives a great promise in practical realization [3]. In underlay CR systems, FD is used to compensate the spectral efﬁciency loss of cooperative systems that is used to increase the SU coverage [4], [5]. In addition to that, FD is used to achieve simultaneous sensing and data transmission for SU, or possibly receive data from the other SU node during the transmission according to the channel conditions [6]. The existing research on CR systems avoids sharing the spectrum with in-band FD PU due to the increased interference limitations at the PU side, which can impede the operation of underlay CR systems. Thus, investigating communication systems that can relieve the interference signature on the PU while improving the SU performance become imperative. Improper Gaussian signaling has proven its superiority over proper Gaussian signaling to improve the achievable rate in interference-limited networks [7]–[11]. In CR systems, improper Gaussian signaling is employed in [12], [13], where the PU is assumed to work in half-duplex mode with proper Gaussian signaling. On the other hand, the SU is assumed to use improper Gaussian signaling and have perfect instantaneous channel state information (CSI) of all PU and SU communication links in [12] and average CSI in [13]. Improper Gaussian signaling

text: I.

label: section

text: INTRODUCTION

text: Cognitive

text: radio

text: (CR)

text: is

text: a

text: promising

text: spectrum

text: sharing

text: technology

text: that

text: mitigates

text: the

text: spectrum

text: scarcity

text: that

text: resulted

text: from

text: the

text: recent

text: tremendous

text: growth

text: of

text: wireless

text: devices

text: and

text: services

text: over

text: the

text: past

text: decade

text: As

text: many

text: licensed

text: primary

text: users

text: (PU)

text: block

text: the

text: available

text: spectrum,

text: CR

text: systems

text: exploit

text: the

text: same

text: spectrum

text: resources

text: and

text: allow

text: secondary

text: users

text: (SU)

text: to

text: coexist

text: with

text: PU

text: without

text: degrading

text: the

text: PU

text: quality-of-service

text: (QoS)

text: CR

text: can

text: operate

text: in

text: three

text: strategies

text: called,

text: interweave,

text: overlay

text: and

text: underlay

text: spectrum

text: sharing.

text: The

text: CR

text: with

text: interweave

text: strategy

text: accesses

text: the

text: PU

text: spectrum

text: when

text: it

text: is

text: not

text: used,

text: i.e.,

text: the

text: PU

text: is

text: idle.

text: On

text: the

text: other

text: hand,

text: CR

text: with

text: both

text: overlay

text: and

text: underlay

text: strategies

text: coexists

text: with

text: the

text: PU.

text: In

text: the

text: overlay

text: strategy,

text: the

text: SU

text: uses

text: part

text: of

text: its

text: power

text: to

text: assist

text: the

text: PU

text: transmission

text: in

text: order

text: to

text: compensate

text: its

text: interference

text: impact

text: on

text: the

text: PU.

text: The

text: scenario

text: is

text: more

text: challenging

text: for

text: the

text: underlay

text: strategy,

text: where

text: the

text: SU

text: has

text: to

text: control

text: its

text: power

text: to

text: be

text: within

text: an

text: acceptable

text: level

text: at

text: the

text: PU

text: receiver

text: side.

text: CR

text: systems

text: can

text: be

text: incorporated

text: with

text: other

text: communication

text: techniques,

text: such

text: as

text: full-duplex

text: (FD)

text: and

text: cooperative

text: communications,

text: to

text: improve

text: the

text: spectrum

text: utilization

text: of

text: the

text: communication

text: networks

text: and

text: the

text: SU

text: performance.

text: FD

text: is

text: a

text: spectral

text: efﬁcient

text: paradigm

text: that

text: allows

text: the

text: communication

text: nodes

text: to

text: transmit

text: and

text: receive

text: simultaneously

text: at

text: the

text: same

text: frequency.

text: FD

text: has

text: recently

text: attracted

text: wide

text: attention

text: especially

text: after

text: the

text: progress

text: in

text: self-interference

text: cancellation,

text: which

text: gives

text: a

text: great

text: promise

text: in

text: practical

text: realization

text: In

text: underlay

text: CR

text: systems,

text: FD

text: is

text: used

text: to

text: compensate

text: the

text: spectral

text: efﬁciency

text: loss

text: of

text: cooperative

text: systems

text: that

text: is

text: used

text: to

text: increase

text: the

text: SU

text: coverage

text: In

text: addition

text: to

text: that,

text: FD

text: is

text: used

text: to

text: achieve

text: simultaneous

text: sensing

text: and

text: data

text: transmission

text: for

text: SU,

text: or

text: possibly

text: receive

text: data

text: from

text: the

text: other

text: SU

text: node

text: during

text: the

text: transmission

text: according

text: to

text: the

text: channel

text: conditions

text: The

text: existing

text: research

text: on

text: CR

text: systems

text: avoids

text: sharing

text: the

text: spectrum

text: with

text: in-band

text: FD

text: PU

text: due

text: to

text: the

text: increased

text: interference

text: limitations

text: at

text: the

text: PU

text: side,

text: which

text: can

text: impede

text: the

text: operation

text: of

text: underlay

text: CR

text: systems.

text: Thus,

text: investigating

text: communication

text: systems

text: that

text: can

text: relieve

text: the

text: interference

text: signature

text: on

text: the

text: PU

text: while

text: improving

text: the

text: SU

text: performance

text: become

text: imperative.

text: Improper

text: Gaussian

text: signaling

text: has

text: proven

text: its

text: superiority

text: over

text: proper

text: Gaussian

text: signaling

text: to

text: improve

text: the

text: achievable

text: rate

text: in

text: interference-limited

text: networks

text: In

text: CR

text: systems,

text: improper

text: Gaussian

text: signaling

text: is

text: employed

text: in

text: where

text: the

text: PU

text: is

text: assumed

text: to

text: work

text: in

text: half-duplex

text: mode

text: with

text: proper

text: Gaussian

text: signaling.

text: On

text: the

text: other

text: hand,

text: the

text: SU

text: is

text: assumed

text: to

text: use

text: improper

text: Gaussian

text: signaling

text: and

text: have

text: perfect

text: instantaneous

text: channel

text: state

text: information

text: (CSI)

text: of

text: all

text: PU

text: and

text: SU

text: communication

text: links

text: in

text: and

text: average

text: CSI

text: in

text: Improper

text: Gaussian

text: signaling


text: 28 A-CSI 100 IDL-CSI(simulation) IDL-CSI(analytical) 10-1 γ¯s = 15 dB 10-2 γ¯s = 8 dB γ¯s = 20 dB 10-3 0 0.5 1 1.5 2 2.5 3 R0,s(b/s/Hz) Fig. 8. SU average energy efﬁciency for A-CSI and IDL-CSI improper based designs versus R for γ¯ = 8, 15, 20 dB. 0,s s outage event increases and hence, the SU preferably stays idle and saves its transmit power. Example 7: To investigate the average EE improvement beneﬁt as a result of perfect knowledge of γ , Fig. 8 plots the SU average EE for the SU improper Gaussian signaling design based A- s CSI and IDL-CSI using simulations and derived analytical expression (60). We assume κ = 5 and p = 1 W. First, we observe that the simulation curve has a perfect match with the analytical c expression. At lower SU target rates, the requirements for the SU are ﬂexible and hence, both designs use less power and thus the average EE improves with increasing γ¯ . On the other hand, s as the SU rate increases, the requirements become more stringent, which force the SU to increase its transmit power and therefore, deteriorating the average EE performance. As R increases, 0,s the IDL-CSI based design uses less power to achieve the same target rate. The gap between the two designs increases as γ¯ decreases and hence, the IDL-CSI based design can make use of s the perfect knowledge of the CSI in order to save the transmit power and therefore boost the average EE of the SU system. We observe also from Fig. 8 that as R increases, it reaches 0,s speciﬁc values at which the average EE behavior shows abrupt improvement. As we know from the improper condition (37) that there is a speciﬁc value of R , at which improper signaling 0,s can not be used, thus the solution switches to proper signaling at larger values R . Since the 0,s proper signaling scheme uses less power, it achieves better average EE which interprets this sudden improvement of the average EE performance. Example 8: This example illustrates the beneﬁts that can be reaped at the PU side by the EE egarevA ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A-CSI

text: IDL-CSI(simulation)

text: IDL-CSI(analytical)

text: γ¯s

text: dB

text: γ¯s

text: dB

text: γ¯s

text: dB

text: Fig.

text: SU

text: average

text: energy

text: efﬁciency

text: for

text: A-CSI

text: and

text: IDL-CSI

text: improper

text: based

text: designs

text: versus

text: R

text: for

text: γ¯

text: dB.

text: s

text: outage

text: event

text: increases

text: and

text: hence,

text: the

text: SU

text: preferably

text: stays

text: idle

text: and

text: saves

text: its

text: transmit

text: power.

text: Example

text: To

text: investigate

text: the

text: average

text: EE

text: improvement

text: beneﬁt

text: as

text: a

text: result

text: of

text: perfect

text: knowledge

text: of

text: γ

text: Fig.

text: plots

text: the

text: SU

text: average

text: EE

text: for

text: the

text: SU

text: improper

text: Gaussian

text: signaling

text: design

text: based

text: A-

text: s

text: CSI

text: and

text: IDL-CSI

text: using

text: simulations

text: and

text: derived

text: analytical

text: expression

text: We

text: assume

text: κ

text: and

text: p

text: W.

text: First,

text: we

text: observe

text: that

text: the

text: simulation

text: curve

text: has

text: a

text: perfect

text: match

text: with

text: the

text: analytical

text: c

text: expression.

text: At

text: lower

text: SU

text: target

text: rates,

text: the

text: requirements

text: for

text: the

text: SU

text: are

text: ﬂexible

text: and

text: hence,

text: both

text: designs

text: use

text: less

text: power

text: and

text: thus

text: the

text: average

text: EE

text: improves

text: with

text: increasing

text: γ¯

text: On

text: the

text: other

text: hand,

text: s

text: as

text: the

text: SU

text: rate

text: increases,

text: the

text: requirements

text: become

text: more

text: stringent,

text: which

text: force

text: the

text: SU

text: to

text: increase

text: its

text: transmit

text: power

text: and

text: therefore,

text: deteriorating

text: the

text: average

text: EE

text: performance.

text: As

text: R

text: increases,

text: the

text: IDL-CSI

text: based

text: design

text: uses

text: less

text: power

text: to

text: achieve

text: the

text: same

text: target

text: rate.

text: The

text: gap

text: between

text: the

text: two

text: designs

text: increases

text: as

text: γ¯

text: decreases

text: and

text: hence,

text: the

text: IDL-CSI

text: based

text: design

text: can

text: make

text: use

text: of

text: s

text: the

text: perfect

text: knowledge

text: of

text: the

text: CSI

text: in

text: order

text: to

text: save

text: the

text: transmit

text: power

text: and

text: therefore

text: boost

text: the

text: average

text: EE

text: of

text: the

text: SU

text: system.

text: We

text: observe

text: also

text: from

text: Fig.

text: that

text: as

text: R

text: increases,

text: it

text: reaches

text: speciﬁc

text: values

text: at

text: which

text: the

text: average

text: EE

text: behavior

text: shows

text: abrupt

text: improvement.

text: As

text: we

text: know

text: from

text: the

text: improper

text: condition

text: that

text: there

text: is

text: a

text: speciﬁc

text: value

text: of

text: R

text: at

text: which

text: improper

text: signaling

text: can

text: not

text: be

text: used,

text: thus

text: the

text: solution

text: switches

text: to

text: proper

text: signaling

text: at

text: larger

text: values

text: R

text: Since

text: the

text: proper

text: signaling

text: scheme

text: uses

text: less

text: power,

text: it

text: achieves

text: better

text: average

text: EE

text: which

text: interprets

text: this

text: sudden

text: improvement

text: of

text: the

text: average

text: EE

text: performance.

text: Example

text: This

text: example

text: illustrates

text: the

text: beneﬁts

text: that

text: can

text: be

text: reaped

text: at

text: the

text: PU

text: side

text: by

text: the

text: EE

text: egarevA

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 4 • Finally, with the aid of numerical results, we ﬁrst examine the accuracy of the PU derived outage probability bound. Then investigate the beneﬁts that can be reaped by employing improper Gaussian signaling for the SU versus different system parameters. And ﬁnally, we explore the availability of the SU direct link beneﬁts on power saving, SU and PU outage probability and SU average EE. The rest of the paper is organized as follows: In Section II, we introduce the system model of the spectrum sharing system with FD system. In Section III, we derive the outage probability expressions for both SU and PU. In Section IV, we design the SU signal parameters to minimize the SU outage probability subject to a given QoS for the PU based on average CSI. In Section V, we study the system performance with perfect CSI knowledge for the SU direct link and look for its advantages on both SU and PU. In Section VI, we present a comprehensive simulation study to illustrate the performance of the proposed improper Gaussian signaling techniques. Finally, Section VI concludes the paper. | | Notation: Throughout the rest of the paper, we use . to denote the absolute value operation ≥ and [x]+ = max(x, 0). 1 ∞ (z) is the indicator function where 1 ∞ (z) := 1 if z x and [x, ) [x, ) { } 1 ∞ (z) := 0 for all z < x. Pr A denotes the probability of occurrence of an event A. The [x, ) E{ } operator . is used to denote the statistical expectation and the mean of a random variable E{ } (RV) X is deﬁned as X¯ = X . II. SYSTEM MODEL Consider an underlay CR system, where a half-duplex SU pair coexists with an in-band full- duplex PU pair as depicted in Fig. 1. In this scenario, we have three simultaneous different transmissions occur in the same network over the same frequency. Before proceeding in describ- ing the system components, we deﬁne the following terms: Deﬁnition 1: The variance and pseudo-variance of a zero mean complex random variable x E{| | } E{ } are deﬁned as σ2 = x 2 and σˆ2 = x2 respectively [16]. x x Deﬁnition 2: The proper signal has a zero σˆ2, while improper signal has a non-zero σˆ2. x x C Deﬁnition 3: The circularity coefﬁcient measures the degree of impropriety of signal x x C | | ≤ C ≤ C C and is deﬁned as = σˆ2 /σ2, where 0 1. = 0 denotes proper signal and = 1 x x x x x x denotes maximally improper signal. ∈ { } The PU nodes are assumed to use zero-mean proper Gaussian signals x , i 1, 2 with a i unit variance, while the SU transmitter uses zero-mean improper Gaussian signals x with a unit s

text: Finally,

text: with

text: the

text: aid

text: of

text: numerical

text: results,

text: we

text: ﬁrst

text: examine

text: the

text: accuracy

text: of

text: the

text: PU

text: derived

text: outage

text: probability

text: bound.

text: Then

text: investigate

text: the

text: beneﬁts

text: that

text: can

text: be

text: reaped

text: by

text: employing

text: improper

text: Gaussian

text: signaling

text: for

text: the

text: SU

text: versus

text: different

text: system

text: parameters.

text: And

text: ﬁnally,

text: we

text: explore

text: the

text: availability

text: of

text: the

text: SU

text: direct

text: link

text: beneﬁts

text: on

text: power

text: saving,

text: SU

text: and

text: PU

text: outage

text: probability

text: and

text: SU

text: average

text: EE.

text: The

text: rest

text: of

text: the

text: paper

text: is

text: organized

text: as

text: follows:

text: In

text: Section

text: II,

text: we

text: introduce

text: the

text: system

text: model

text: of

text: the

text: spectrum

text: sharing

text: system

text: with

text: FD

text: system.

text: In

text: Section

text: III,

text: we

text: derive

text: the

text: outage

text: probability

text: expressions

text: for

text: both

text: SU

text: and

text: PU.

text: In

text: Section

text: IV,

text: we

text: design

text: the

text: SU

text: signal

text: parameters

text: to

text: minimize

text: the

text: SU

text: outage

text: probability

text: subject

text: to

text: a

text: given

text: QoS

text: for

text: the

text: PU

text: based

text: on

text: average

text: CSI.

text: In

text: Section

text: V,

text: we

text: study

text: the

text: system

text: performance

text: with

text: perfect

text: CSI

text: knowledge

text: for

text: the

text: SU

text: direct

text: link

text: and

text: look

text: for

text: its

text: advantages

text: on

text: both

text: SU

text: and

text: PU.

text: In

text: Section

text: VI,

text: we

text: present

text: a

text: comprehensive

text: simulation

text: study

text: to

text: illustrate

text: the

text: performance

text: of

text: the

text: proposed

text: improper

text: Gaussian

text: signaling

text: techniques.

text: Finally,

text: Section

text: VI

text: concludes

text: the

text: paper.

text: Notation:

text: Throughout

text: the

text: rest

text: of

text: the

text: paper,

text: we

text: use

text: to

text: denote

text: the

text: absolute

text: value

text: operation

text: and

text: [x]+

text: max(x,

text: (z)

text: is

text: the

text: indicator

text: function

text: where

text: (z)

text: if

text: z

text: x

text: and

text: [x,

text: [x,

text: (z)

text: for

text: all

text: z

text: x.

text: Pr

text: A

text: denotes

text: the

text: probability

text: of

text: occurrence

text: of

text: an

text: event

text: A.

text: The

text: [x,

text: E{

text: operator

text: is

text: used

text: to

text: denote

text: the

text: statistical

text: expectation

text: and

text: the

text: mean

text: of

text: a

text: random

text: variable

text: E{

text: (RV)

text: X

text: is

text: deﬁned

text: as

text: X¯

text: X

text: II.

text: SYSTEM

text: MODEL

text: Consider

text: an

text: underlay

text: CR

text: system,

text: where

text: a

text: half-duplex

text: SU

text: pair

text: coexists

text: with

text: an

text: in-band

text: full-

text: duplex

text: PU

text: pair

text: as

text: depicted

text: in

text: Fig.

text: In

text: this

text: scenario,

text: we

text: have

text: three

text: simultaneous

text: different

text: transmissions

text: occur

text: in

text: the

text: same

text: network

text: over

text: the

text: same

text: frequency.

text: Before

text: proceeding

text: in

text: describ-

text: ing

text: the

text: system

text: components,

text: we

text: deﬁne

text: the

text: following

text: terms:

text: Deﬁnition

text: The

text: variance

text: and

text: pseudo-variance

text: of

text: a

text: zero

text: mean

text: complex

text: random

text: variable

text: x

text: E{|

text: E{

text: are

text: deﬁned

text: as

text: σ2

text: x

text: and

text: σˆ2

text: respectively

text: x

text: x

text: Deﬁnition

text: The

text: proper

text: signal

text: has

text: a

text: zero

text: σˆ2,

text: while

text: improper

text: signal

text: has

text: a

text: non-zero

text: σˆ2.

text: x

text: x

text: C

text: Deﬁnition

text: The

text: circularity

text: coefﬁcient

text: measures

text: the

text: degree

text: of

text: impropriety

text: of

text: signal

text: x

text: x

text: C

text: C

text: C

text: C

text: and

text: is

text: deﬁned

text: as

text: σˆ2

text: where

text: denotes

text: proper

text: signal

text: and

text: x

text: x

text: x

text: x

text: x

text: x

text: denotes

text: maximally

text: improper

text: signal.

text: The

text: PU

text: nodes

text: are

text: assumed

text: to

text: use

text: zero-mean

text: proper

text: Gaussian

text: signals

text: x

text: i

text: with

text: a

text: i

text: unit

text: variance,

text: while

text: the

text: SU

text: transmitter

text: uses

text: zero-mean

text: improper

text: Gaussian

text: signals

text: x

text: with

text: a

text: unit

text: s


text: 32 (cid:34) (cid:18) (cid:19) (cid:18) (cid:19) (cid:35) YG − YG − 1 × − (γ ) 1 − − (γ ) 1 M = I − I exp m Is exp m Is . (71) p ¯ p ¯ p ¯ p ¯ 1 p1 2 p2 1 p1 2 p2 Here, we have two cases, either p I¯ > p I¯ or p I¯ < p I¯ . For the ﬁrst, it is clear that M ≥ 0. 1 p1 2 p2 1 p1 2 p2 For the second, the same argument applies by replacing every index 1 by 2 and vice versa in (71). For the other scenario, when p I¯ = p I¯ , we can write M as 1(cid:16)p1 2(cid:17)p2  (cid:16) (cid:17) (cid:0) (cid:1)  (cid:0) (cid:1)  C C ζ p(sm), x −ζ p(sm), x M = I exp I , (72) p ¯ 2 p ¯ 2 1 p1 1 p1 which is found to be non-negative by following (45) and this concludes the proof. APPENDIX D In this appendix, we aim at deriving the equivalent optimization problem in (53). We assume (cid:16) (cid:17) here that the transmission condition in (45) is fulﬁlled, i.e., the SU is allowed to transmit. First, we can write PDL−T p(z), C(z) in (52), when p I¯ (cid:54)= p I¯ , as ou(cid:16)t,s o (cid:17) o (cid:34) 1 p1(cid:18) 2 p2 (cid:19) (cid:16) (cid:17)(cid:35) − − − C 1 I −γ ∆ 1 − I − γ ∆ 1 PDL T p(z), (z) = I − I p ¯ exp s I p ¯ exp s I , (73) out,s o o p ¯ p ¯ 1 p1 p ¯ 2 p2 p ¯ (cid:18) (cid:16) (cid:17) (cid:19) 1 p1 2 p2 1 p1 2 p2 where ∆ = 1(cid:16)− Co(z) 2 (cid:17). Similar to the proof in Appendix C, we assume, ﬁrst, that p I¯ > p I¯ . We calculate Ψthsep(oﬁz)r,Cso(tz)derivative of (73) with respect to as 1 p1 2 p2 ∆ (cid:34) (cid:18) (cid:19) (cid:18) (cid:19) (cid:35) − − − dPDL T γ × −γ ∆ 1 − −γ ∆ 1 out,s = I −s I exp s I exp s I , (74) d∆ p1¯p1 p2¯p2 p2¯p2 p1¯p1 (cid:16) (cid:17) which is always non-positive, independent of th(cid:16)e value (cid:17)of γs, and hence, PoDuLt,−sT p(oz), Co(z) is monotonically decreasing in . Thus, we can replace the original optimization problem by ∆ maximizing ∆ over the optimal local solution pairs p(z), C(z) to obtain the global solution pair o o (p∗, C∗) which yields (53). Also, For the other case in which p I¯ < p I¯ , the same argument s x 1 p1 2 p2 applies as illustrated in Appendix C. On the other hand, when p I¯ = p I¯ , we can write the 1 p1 2 p2 derivative as (cid:18) (cid:19) (cid:18) (cid:19) − − − dPDL T − γ γ ∆ 1 −γ ∆ 1 out,s = Is s I exp s I (75) d∆ p ¯ p ¯ p ¯ 1 p1 1 p1 1 p1 which, from the transmission condition in (45), it follows that the derivative is non-positive hence, same illustrated arguments apply and this concludes the proof. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: YG

text: YG

text: (γ

text: (γ

text: M

text: I

text: I

text: exp

text: m

text: Is

text: exp

text: m

text: Is

text: p

text: p

text: p

text: p

text: Here,

text: we

text: have

text: two

text: cases,

text: either

text: p

text: I¯

text: p

text: I¯

text: or

text: p

text: I¯

text: p

text: I¯

text: For

text: the

text: ﬁrst,

text: it

text: is

text: clear

text: that

text: M

text: For

text: the

text: second,

text: the

text: same

text: argument

text: applies

text: by

text: replacing

text: every

text: index

text: by

text: and

text: vice

text: versa

text: in

text: For

text: the

text: other

text: scenario,

text: when

text: p

text: I¯

text: p

text: I¯

text: we

text: can

text: write

text: M

text: as

text: C

text: C

text: ζ

text: p(sm),

text: x

text: −ζ

text: p(sm),

text: x

text: M

text: I

text: exp

text: I

text: p

text: p

text: which

text: is

text: found

text: to

text: be

text: non-negative

text: by

text: following

text: and

text: this

text: concludes

text: the

text: proof.

text: APPENDIX

text: D

text: In

text: this

text: appendix,

text: we

text: aim

text: at

text: deriving

text: the

text: equivalent

text: optimization

text: problem

text: in

text: We

text: assume

text: here

text: that

text: the

text: transmission

text: condition

text: in

text: is

text: fulﬁlled,

text: i.e.,

text: the

text: SU

text: is

text: allowed

text: to

text: transmit.

text: First,

text: we

text: can

text: write

text: PDL−T

text: p(z),

text: C(z)

text: in

text: when

text: p

text: I¯

text: p

text: I¯

text: as

text: o

text: o

text: C

text: I

text: −γ

text: I

text: γ

text: PDL

text: T

text: p(z),

text: (z)

text: I

text: I

text: p

text: exp

text: s

text: I

text: p

text: exp

text: s

text: I

text: out,s

text: o

text: o

text: p

text: p

text: p

text: p

text: where

text: Co(z)

text: Similar

text: to

text: the

text: proof

text: in

text: Appendix

text: C,

text: we

text: assume,

text: ﬁrst,

text: that

text: p

text: I¯

text: p

text: I¯

text: We

text: calculate

text: Ψthsep(oﬁz)r,Cso(tz)derivative

text: of

text: with

text: respect

text: to

text: as

text: dPDL

text: T

text: γ

text: −γ

text: −γ

text: out,s

text: I

text: −s

text: I

text: exp

text: s

text: I

text: exp

text: s

text: I

text: d∆

text: which

text: is

text: always

text: non-positive,

text: independent

text: of

text: value

text: γs,

text: and

text: hence,

text: PoDuLt,−sT

text: p(oz),

text: Co(z)

text: is

text: monotonically

text: decreasing

text: in

text: Thus,

text: we

text: can

text: replace

text: the

text: original

text: optimization

text: problem

text: by

text: maximizing

text: over

text: the

text: optimal

text: local

text: solution

text: pairs

text: p(z),

text: C(z)

text: to

text: obtain

text: the

text: global

text: solution

text: pair

text: o

text: o

text: (p∗,

text: C∗)

text: which

text: yields

text: Also,

text: For

text: the

text: other

text: case

text: in

text: which

text: p

text: I¯

text: p

text: I¯

text: the

text: same

text: argument

text: s

text: x

text: applies

text: as

text: illustrated

text: in

text: Appendix

text: C.

text: On

text: the

text: other

text: hand,

text: when

text: p

text: I¯

text: p

text: I¯

text: we

text: can

text: write

text: the

text: derivative

text: as

text: dPDL

text: T

text: γ

text: γ

text: −γ

text: out,s

text: Is

text: s

text: I

text: exp

text: s

text: I

text: d∆

text: p

text: p

text: p

text: which,

text: from

text: the

text: transmission

text: condition

text: in

text: it

text: follows

text: that

text: the

text: derivative

text: is

text: non-positive

text: hence,

text: same

text: illustrated

text: arguments

text: apply

text: and

text: this

text: concludes

text: the

text: proof.

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 8 (cid:80) I ◦ 2 p + 1 C γ = i=1 −i Cpi Ψ (p , ) , (10) (cid:16) s (1 (cid:17)2) s s x (cid:112) x C − C − where Ψ (p , ) = 1 + Γ (1 2) 1 /p . By averaging over the exponential statistics s s x s x s I (cid:16) (cid:17) of , we get pi (cid:16) (cid:17) (cid:81) − C C − exp Ψs(p−s,Cx) P (p , ) = 1 γ¯s(1 x2) . (11) out,s s x 2 I C p Ψs(p−s,Cx) + 1 j=1 j pj γ¯s(1 x2) (cid:16) (cid:17) For the proper signaling case, the above expression reduces to (cid:16) (cid:17) (cid:81) − − exp 2R0,s 1 − P (p , 0) = 1 psγ¯s . (12) out,s s 2 I − p 2R0,s 1 + 1 j pj psγ¯s j=1 while for maximally improper case, the expression reduces to (cid:16) (cid:17) (cid:81) − C − exp ( Γ /2p γ¯ ) s s s P (p , 1) = lim P (p , ) = 1 . out,s s Cx→1 out,s s x 2 pjI¯pjΓs + 1 2psγ¯s j=1 B. Primary User Outage Probability The outage probability of PU node i for a given target rate R is deﬁned as 0,pi C { C } P (p , ) = Pr R (p , ) < R . (13) out,pi s x pi s x 0,pi Substituting (5) in (13), we obtain (cid:40) (cid:41) (cid:0) (cid:1) (cid:16)(cid:0) (cid:1) (cid:17) C 2 I − Γ I − I C ≤ P (p , ) = Pr γ2 + p υ + p + 1 γ pi p υ + p + 1 2 p2 2 2 0 , (14) out,pi s x pi p j pj s sj pi p2 j pj s sj s sj x i i − where Γpi = 22R0,pi 1. Similar to the above subsection, the outage probability of PU node i I (conditioned on and υ ) is given by sj pj ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: I

text: p

text: C

text: γ

text: −i

text: Cpi

text: Ψ

text: (p

text: s

text: s

text: s

text: x

text: x

text: C

text: C

text: where

text: Ψ

text: (p

text: Γ

text: /p

text: By

text: averaging

text: over

text: the

text: exponential

text: statistics

text: s

text: s

text: x

text: s

text: x

text: s

text: I

text: of

text: we

text: get

text: pi

text: C

text: C

text: exp

text: Ψs(p−s,Cx)

text: P

text: (p

text: out,s

text: s

text: x

text: I

text: C

text: p

text: Ψs(p−s,Cx)

text: j

text: pj

text: For

text: the

text: proper

text: signaling

text: case,

text: the

text: above

text: expression

text: reduces

text: to

text: exp

text: P

text: (p

text: psγ¯s

text: out,s

text: s

text: I

text: p

text: j

text: pj

text: psγ¯s

text: while

text: for

text: maximally

text: improper

text: case,

text: the

text: expression

text: reduces

text: to

text: C

text: exp

text: Γ

text: γ¯

text: s

text: s

text: s

text: P

text: (p

text: lim

text: P

text: (p

text: out,s

text: s

text: out,s

text: s

text: x

text: pjI¯pjΓs

text: B.

text: Primary

text: User

text: Outage

text: Probability

text: The

text: outage

text: probability

text: of

text: PU

text: node

text: i

text: for

text: a

text: given

text: target

text: rate

text: R

text: is

text: deﬁned

text: as

text: C

text: C

text: P

text: (p

text: Pr

text: R

text: (p

text: R

text: out,pi

text: s

text: x

text: pi

text: s

text: x

text: Substituting

text: in

text: we

text: obtain

text: C

text: I

text: Γ

text: I

text: I

text: C

text: P

text: (p

text: Pr

text: γ2

text: p

text: υ

text: p

text: γ

text: pi

text: p

text: υ

text: p

text: out,pi

text: s

text: x

text: pi

text: p

text: j

text: pj

text: s

text: sj

text: pi

text: j

text: pj

text: s

text: sj

text: s

text: sj

text: x

text: i

text: i

text: where

text: Γpi

text: Similar

text: to

text: the

text: above

text: subsection,

text: the

text: outage

text: probability

text: of

text: PU

text: node

text: i

text: I

text: (conditioned

text: on

text: and

text: υ

text: is

text: given

text: by

text: sj

text: pj

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Abstract With the potential prospects of the Υ(1S) data samples at the running LHC and upcoming → SuperKEKB, the Υ(1S) B ρ weak decay is studied with the pQCD approach. It is found that c (1) the lion’s share of branching ratio comes from the longitudinal polarization helicity amplitudes; → O − (2) branching ratio for the Υ(1S) B ρ decay can reach up to (10 9), which might be hopefully c measurable. 1

text: Abstract

text: With

text: the

text: potential

text: prospects

text: of

text: the

text: data

text: samples

text: at

text: the

text: running

text: LHC

text: and

text: upcoming

text: SuperKEKB,

text: the

text: B

text: ρ

text: weak

text: decay

text: is

text: studied

text: with

text: the

text: pQCD

text: approach.

text: It

text: is

text: found

text: that

text: c

text: the

text: lion’s

text: share

text: of

text: branching

text: ratio

text: comes

text: from

text: the

text: longitudinal

text: polarization

text: helicity

text: amplitudes;

text: O

text: branching

text: ratio

text: for

text: the

text: B

text: ρ

text: decay

text: can

text: reach

text: up

text: to

text: which

text: might

text: be

text: hopefully

text: c

text: measurable.


text: β = x2m2 + x2m2 + x¯2m2 c 1 1 2 2 3 3 − − x x t x x¯ u + x x¯ s, (A23) 1 2 1 3 2 3 β = x2m2 + x2m2 + x2m2 d 1 1 2 2 3 3 − − (cid:113) x x t x x u + x x s, (A24) 1 2 √ 1 3 2 3 − (cid:113)− t = max( α , β , 1/b , 1/b ), (A25) a(b) √ e a(b) 1 2 − | | t = max( α , β , 1/b , 1/b ). (A26) c(d) e c(d) 2 3 [1] K. Olive et al. (Particle Data Group), Chin. Phys. C 38, 090001 (2014). [2] Ed. A. Bevan et al., Eur. Phys. J. C 74, 3026 (2014). [3] H. Li, Phys. Rev. D 52, 3958 (1995). [4] C. Chang, H. Li, Phys. Rev. D 55, 5577 (1997). [5] T. Yeh, H. Li, Phys. Rev. D 56, 1615 (1997). [6] K. K. Sharma and R. C. Verma, Int. J. Mod. Phys. A 14, 937 (1999). [7] R. Dhir, R. C. Verma and A. Sharma, Adv. High Energy Phys. 2013, 706543 (2013). [8] M. Beneke et al., Phys. Rev. Lett. 83, 1914 (1999). [9] M. Beneke et al., Nucl. Phys. B 591, 313 (2000). [10] M. Beneke et al., Nucl. Phys. B 606, 245 (2001). [11] C. Bauer et al., Phys. Rev. D 63, 114020 (2001). [12] C. Bauer, D. Pirjol, I. Stewart, Phys. Rev. D 65, 054022 (2002). [13] C. Bauer et al., Phys. Rev. D 66, 014017 (2002). [14] M. Beneke et al., Nucl. Phys. B 643, 431 (2002). [15] N. Brambilla et al. (QWG group), arXiv:hep-ph/0412158. [16] C. Chang and X. Wu, Eur. Phys. J. C 38, 267 (2004). [17] G. Buchalla, A. Buras, M. Lautenbacher, Rev. Mod. Phys. 68, 1125, (1996). [18] G. Lepage, S. Brodsky, Phys. Rev. D 22, 2157 (1980). [19] T. Kurimoto, H. Li, A. Sanda, Phys. Rev. D 65, 014007 (2001). [20] P. Ball, G. Jones, JHEP, 0703, 069, (2007). [21] G. Lepage et al., Phys. Rev. D 46, 4052 (1992). 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: β

text: c

text: x

text: x

text: t

text: x

text: x¯

text: u

text: x

text: x¯

text: s,

text: β

text: d

text: x

text: x

text: t

text: x

text: x

text: u

text: x

text: x

text: s,

text: t

text: max(

text: α

text: β

text: a(b)

text: e

text: a(b)

text: t

text: max(

text: α

text: β

text: c(d)

text: e

text: c(d)

text: K.

text: Olive

text: et

text: al.

text: (Particle

text: Data

text: Group),

text: Chin.

text: Phys.

text: C

text: Ed.

text: A.

text: Bevan

text: et

text: al.,

text: Eur.

text: Phys.

text: J.

text: C

text: H.

text: Li,

text: Phys.

text: Rev.

text: D

text: C.

text: Chang,

text: H.

text: Li,

text: Phys.

text: Rev.

text: D

text: T.

text: Yeh,

text: H.

text: Li,

text: Phys.

text: Rev.

text: D

text: K.

text: K.

text: Sharma

text: and

text: R.

text: C.

text: Verma,

text: Int.

text: J.

text: Mod.

text: Phys.

text: A

text: R.

text: Dhir,

text: R.

text: C.

text: Verma

text: and

text: A.

text: Sharma,

text: Adv.

text: High

text: Energy

text: Phys.

text: M.

text: Beneke

text: et

text: al.,

text: Phys.

text: Rev.

text: Lett.

text: M.

text: Beneke

text: et

text: al.,

text: Nucl.

text: Phys.

text: B

text: M.

text: Beneke

text: et

text: al.,

text: Nucl.

text: Phys.

text: B

text: C.

text: Bauer

text: et

text: al.,

text: Phys.

text: Rev.

text: D

text: C.

text: Bauer,

text: D.

text: Pirjol,

text: I.

text: Stewart,

text: Phys.

text: Rev.

text: D

text: C.

text: Bauer

text: et

text: al.,

text: Phys.

text: Rev.

text: D

text: M.

text: Beneke

text: et

text: al.,

text: Nucl.

text: Phys.

text: B

text: N.

text: Brambilla

text: et

text: al.

text: (QWG

text: group),

text: C.

text: Chang

text: and

text: X.

text: Wu,

text: Eur.

text: Phys.

text: J.

text: C

text: G.

text: Buchalla,

text: A.

text: Buras,

text: M.

text: Lautenbacher,

text: Rev.

text: Mod.

text: Phys.

text: G.

text: Lepage,

text: S.

text: Brodsky,

text: Phys.

text: Rev.

text: D

text: T.

text: Kurimoto,

text: H.

text: Li,

text: A.

text: Sanda,

text: Phys.

text: Rev.

text: D

text: P.

text: Ball,

text: G.

text: Jones,

text: JHEP,

text: G.

text: Lepage

text: et

text: al.,

text: Phys.

text: Rev.

text: D

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: − − (cid:107) where f is the decay constant; x¯ = 1 x; t = x¯ x; a and C3/2(t) are the Gegenbauer ρ i i (cid:107) moment and polynomial, respectively; a = 0 for odd i due to the G-parity invariance of i the ρ distribution amplitudes. As to the twist-3 distribution amplitudes of the transverse polarization ρ meson, for simplicity, we will take their asymptotic forms [19, 20]: 3 φV (x) = f (1 + t2), (24) ρ ρ 4 3 − φA(x) = f ( t). (25) ρ ρ 2 (cid:39) (cid:39) Because of m 2m and m m + m , both Υ(1S) and B systems are nearly non- Υ(1S) b Bc b c c relativistic. Nonrelativistic quantum chromodynamics (NRQCD) [21–23] and Schro¨dinger equation can be used to describe their spectrum. The eigenfunction of the time-independent Schro¨dinger equation with scalar harmonic oscillator potential corresponding to the quantum numbers nL = 1S is written as ∼ − φ((cid:126)k) e (cid:126)k2/2β2, (26) (cid:104) | | (cid:105) where parameter β determines the average transverse momentum, i.e., 1S (cid:126)k⊥2 1S = β2. Employing the Brodsky-Huang-Lepage ansatz [24, 25] which has been used to structure wave functions for light and heavy mesons [26], (cid:88) → 1 (cid:126)k2⊥ + m2 (cid:126)k2 i qi , (27) 4 x i i (cid:80) (cid:80) where x , (cid:126)k ⊥, m are the longitudinal momentum fraction, transverse momentum, mass i i qi of the valence quarks in hadrons, respectively, with the relations x = 1 and (cid:126)k ⊥ = 0, i i then integrating out (cid:126)k ⊥ and combining with their asymptotic forms, one can obtain [19, 28] i (cid:110) (cid:111) − x¯ m2 + x m2 φ (x) = A xx¯ exp c b , (28) Bc 8 β2 x x¯ 2 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: f

text: is

text: the

text: decay

text: constant;

text: x¯

text: x;

text: t

text: x¯

text: x;

text: a

text: and

text: are

text: the

text: Gegenbauer

text: ρ

text: i

text: i

text: moment

text: and

text: polynomial,

text: respectively;

text: a

text: for

text: odd

text: i

text: due

text: to

text: the

text: G-parity

text: invariance

text: of

text: i

text: the

text: ρ

text: distribution

text: amplitudes.

text: As

text: to

text: the

text: distribution

text: amplitudes

text: of

text: the

text: transverse

text: polarization

text: ρ

text: meson,

text: for

text: simplicity,

text: we

text: will

text: take

text: their

text: asymptotic

text: forms

text: φV

text: (x)

text: f

text: ρ

text: ρ

text: φA(x)

text: f

text: t).

text: ρ

text: ρ

text: Because

text: of

text: m

text: and

text: m

text: m

text: m

text: both

text: and

text: B

text: systems

text: are

text: nearly

text: non-

text: b

text: Bc

text: b

text: c

text: c

text: relativistic.

text: Nonrelativistic

text: quantum

text: chromodynamics

text: (NRQCD)

text: and

text: Schro¨dinger

text: equation

text: can

text: be

text: used

text: to

text: describe

text: their

text: spectrum.

text: The

text: eigenfunction

text: of

text: the

text: time-independent

text: Schro¨dinger

text: equation

text: with

text: scalar

text: harmonic

text: oscillator

text: potential

text: corresponding

text: to

text: the

text: quantum

text: numbers

text: nL

text: is

text: written

text: as

text: e

text: where

text: parameter

text: β

text: determines

text: the

text: average

text: transverse

text: momentum,

text: i.e.,

text: Employing

text: the

text: Brodsky-Huang-Lepage

text: ansatz

text: which

text: has

text: been

text: used

text: to

text: structure

text: wave

text: functions

text: for

text: light

text: and

text: heavy

text: mesons

text: i

text: qi

text: x

text: i

text: i

text: where

text: x

text: m

text: are

text: the

text: longitudinal

text: momentum

text: fraction,

text: transverse

text: momentum,

text: mass

text: i

text: i

text: qi

text: of

text: the

text: valence

text: quarks

text: in

text: hadrons,

text: respectively,

text: with

text: the

text: relations

text: x

text: and

text: i

text: i

text: then

text: integrating

text: out

text: and

text: combining

text: with

text: their

text: asymptotic

text: forms,

text: one

text: can

text: obtain

text: i

text: x¯

text: x

text: φ

text: (x)

text: A

text: xx¯

text: exp

text: c

text: b

text: Bc

text: β2

text: x

text: x¯

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text: ##LTLine##

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