text: 10 Publications of the Astronomical Society of Japan, (2014), Vol. 00, No. 0 By deducing M˙ = LR/G(cid:16)M, (cid:17) (cid:18) (cid:19) ≈ − × − αD 0.15 P˙ 4.3 10 5µ 0.285 m R 0.85 30 0.1 6 − × M 0.425 − − I 1P2L 0.85 s yr 1. (A11) M(cid:12) 45 37 It is equivalent to equation (A3). The indices are the same, and the factor is almost the same. The difference is the αD m term instead of the n(ω ) term. s When r is larger than r , magnetic braking takes place. to cr It can work for both spin-up and spin-down, although it mostly works as spin-down. Fig. 7. The functio−n n(ωs) with ωs. The approximation∼by equation (A4) is effective in ωs = 0 0.9. Thus, n(ωs) becomes 0 at ωs 0.349, is positive Appendix 3 Error estimation of period and in ωs <∼ 0.349, and negative in ωs >∼ 0.349. period derivative ωs ≈ 1.35 µ30 67 S(cid:18)2(M) P(cid:19)−1L37−37 . (A6) pulsWe,eoursbuiatlelytcu.seFitrhset fwoeldainsgsummeethaodsettooofbPtaiannPd Pa˙n,dthP˙enoffothlde Here, S (M) is given by equation (18) in GL79 as the light curve with them. If there is no periodicity the resultant 2 −2 folded light curve is ﬂat. If there is a pulsation with P and S2(M) = R6−37 MM(cid:12) 7 . (A7) P˙ it shows a pulse shape. We calculate χ2 which is a sum of squared deviation from the mean, to evaluate the existence of ≤ ≤ Equation (A4) is accurate to within 5% for 0 ωs 0.9. a pulse. We change P and P˙ to ﬁnd the most-likely P and P˙ The behavior of n(ωs) is plotted in ﬁgur∼e 7, where the zero which give the maximum χ2. We can draw distribution of χ2 crossover point is seen to take place at ωs 0.349. in the P and P˙ plane. Figure 3 is an example which we used in this paper to obtain P and P˙ of 4U 1626–67. Thus we can Appendix 2 Lovelace model derive P and P˙ , however, their errors are not obvious. A.3.1 Method 1 - parameter a and the standard LRB95 introduced magnetic outﬂows and magnetic break- estimate - ing to explain the relation between P˙ and M˙ . After many nu- The most primitive and straight-forward method for error- merical integrations they introduced r , where the angular ve- locity ω of accreting matter reaches tthoe maximum (dω /dr = estimation is to assume that the errors ∆P, ∆P˙ corresponds to 0). The amatter tr(cid:16)ansfers(cid:17)the angular mom(cid:18)entum(cid:19)to the paulsar at a difference of pulse number a in the whole time span Ts of the observation. rrt=o ≈rt0o..9r1t×o i1s0g8iveαn0D.b1my eq0u.3aµti03o.0n57(M1˙61−)70in.3LRMMB(cid:12)95 a−s0.15 cm.(A8) ∆P = aTPs2 ∆P˙ = 2TasP22 (A12) a is usually less than 1 pulse. Equation (A12) are led by as In LRB95, α is assumed as 0.01 to 0.1 and Dm is order unity. follows. The number of pulses n(t) since t0 is an integral of rto has basically the same nature as r0 in the GL79 model [equa- the pulse frequency ν(t) from t0 to t. Let us assume that the tion (A2)]. According to rto and rcr, LRB95 demonstrates a frequency ν depends on time with a constant rate ν˙, or we take magne(cid:16)tic out(cid:17)ﬂow case (rto <(cid:18)rcr) a(cid:19)nd magnetic braking of the only the ﬁ(cid:90)rst order of derivatives with time in Taylor expansion disk case (rto > rcr), where rcr is series. Using ν(t(cid:48)) = ν0 + ν˙0(t(cid:48) − t0), rcr ≡ Gω∗M2 13 ≈ 1.5 × 108 MM(cid:12) 13 P 23 cm, (A9) n(t) = t ν(t(cid:48)) dt(cid:48) = ν0(t − t0) + ν2˙0 (t − t0)2. (A13) t0 − and ω∗ = 2π/P. Using the whole time span T = (t t ), s 0 When r is smaller than r , the pulsar shows spin-up with to cr ν˙ magnetic outﬂow. P˙ equation(cid:18)[equa(cid:19)tion(cid:16)(18b) in(cid:17)LRB95] con- n(t) = ν0Ts + 20 Ts2. (A14) sists M˙ and rto. By moving P to the right side, Equation (A14) is also considered as a function of ν0 and ν˙0. 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text: 1 Bounds on the Lambert function and their application to the outage analysis of user cooperation Ioannis Chatzigeorgiou, Member, IEEE Abstract— Problems formulated in terms of logarithmic or exponential equations often use the Lambert W function in their solutions. Expansions, approximations and bounds on W have been derived in an effort to gain a better understanding of the relationship between equation parameters. In this paper, we focus on one of the branches of W, denoted as W− , we derive tractable 1 upper and lower bounds and we illustrate their usefulness in identifying conditions under which user cooperation can yield a lower outage probability than non-cooperative transmission. Index Terms—Lambert function, bounds, cooperation, decode and forward, outage probability, Rayleigh fading. chatz_1-eps-converted-to.pdf I. INTRODUCTION T ∈ C HE Lambert function, denoted as W(z) for z , is deﬁned as the function that satisﬁes W(z) eW(z) = z. (1) Taking logarithms of both sides of (1) and rearranging terms, we obtain the equivalent expression [1] − W(z) = ln(z) ln (W(z)) . (2) Fig. 1. The two real-valued branches of the Lambert W function. Both If z is real, that is z ∈ R, W(z) can take two possible real branches are deﬁned for z ≥ −1/e, wh−ere W−−1(z) ≤ −1 and W0(z) ≥ −1. − ≤ ≥ − The two branch(cid:112)es meet at the point ( 1/e, 1). (cid:112) values for 1/e z < 0. Values satisfying W(z) 1 belong to the principal branch, which is denoted as W (z), while ≤ − 0 · − · values satisfying W(z) 1 belong to the W−1(z) br−anch. where p = 2(e z + 1) for W0 and p = 2(e z + 1) for The two br−anches meet at−the bran−ch point for z = 1/e, W−1. Tractable bounds that provide clearer interpretations of where ≥W0( 1/e) = W−1( 1/e) = 1. All values of W(z) solutions involving the Lambert function were derived in [6], for z 0 belong to the principal branch W0(z). The two [7] but their focus was mainly on the primary branch W0. branches of the W(z) are depicted in Fig. 1. Algorithmic approaches were used in [8] to derive accurate The Lambert W function has been recognised in the solu- approximations to the W−1 branch, for example tion of scientiﬁc and engineering problems and it has been ≈  − (cid:34)− −(cid:18) (cid:19) (cid:35)  introduced as a special function in numerical computation W− (z) ln( z) 2α 1 1 saolsfotwaaprepeparaecdkaignesr,ecseuncth raesseaMrcahplein acnodmmMuAnTicLaAtiBon.s,Itsuhcahs ×  − −1 + ln(−z) 12 −1 (3) 1 1 + α as relaying strategies [2], moment generating functions for 2 modelling signal fading [3] and long-haul cooperative power allocation methods [4]. Numerical evaluation of the Lambert where α = 0.3205. W function makes use of recursive schemes. Non-recursive In this paper, we use functional analysis methods to derive expansions have also been obtained for small and large values upper and lower bounds on the W− branch. Our objective is 1 of z [1], [5]. For instance, the ﬁrst few terms of the series to obtain expressions which might not be as accurate as (3) but expansion of W(z) about the branch point are are simpler, more tractable and can shed light on the essence of solutions that involve the W− function. 1 − − 1 11 The remainder of this paper has been organised as follows. W(z) = 1 + p p2 + p3 + . . . 3 72 Section presents bounds on the natural logarithm which are used as a stepping stone for the derivation of bounds on I. Chatzigeorgiou is with the School of Computing and Communica- the W− function. Section considers a simple cooperative tions, InfoLab21, Lancaster University, LA1 4WA, United Kingdom, (e-mail: 1 i.chatzigeorgiou@lancaster.ac.uk) network and assesses the tightness and usefulness of the ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Bounds

label: title

text: on

text: the

text: Lambert

text: function

text: and

text: their

text: application

text: to

text: the

text: outage

text: analysis

text: of

text: user

text: cooperation

text: Ioannis

label: author

text: Chatzigeorgiou,

text: Member,

text: IEEE

text: Abstract—

text: Problems

label: abstract

text: formulated

text: in

text: terms

text: of

text: logarithmic

text: or

text: exponential

text: equations

text: often

text: use

text: the

text: Lambert

text: W

text: function

text: in

text: their

text: solutions.

text: Expansions,

text: approximations

text: and

text: bounds

text: on

text: W

text: have

text: been

text: derived

text: in

text: an

text: effort

text: to

text: gain

text: a

text: better

text: understanding

text: of

text: the

text: relationship

text: between

text: equation

text: parameters.

text: In

text: this

text: paper,

text: we

text: focus

text: on

text: one

text: of

text: the

text: branches

text: of

text: W,

text: denoted

text: as

text: W−

text: we

text: derive

text: tractable

text: upper

text: and

text: lower

text: bounds

text: and

text: we

text: illustrate

text: their

text: usefulness

text: in

text: identifying

text: conditions

text: under

text: which

text: user

text: cooperation

text: can

text: yield

text: a

text: lower

text: outage

text: probability

text: than

text: non-cooperative

text: transmission.

text: Index

text: Terms—Lambert

text: function,

text: bounds,

text: cooperation,

text: decode

text: and

text: forward,

text: outage

text: probability,

text: Rayleigh

text: fading.

text: I.

text: INTRODUCTION

text: T

text: C

text: HE

text: Lambert

text: function,

text: denoted

text: as

text: W(z)

text: for

text: z

text: is

text: deﬁned

text: as

text: the

text: function

text: that

text: satisﬁes

text: W(z)

text: eW(z)

text: z.

text: Taking

text: logarithms

text: of

text: both

text: sides

text: of

text: and

text: rearranging

text: terms,

text: we

text: obtain

text: the

text: equivalent

text: expression

text: W(z)

text: ln(z)

text: ln

text: (W(z))

text: Fig.

text: The

text: two

text: real-valued

text: branches

text: of

text: the

text: Lambert

text: W

text: function.

text: Both

text: If

text: z

text: is

text: real,

text: that

text: is

text: z

text: R,

text: W(z)

text: can

text: take

text: two

text: possible

text: real

text: branches

text: are

text: deﬁned

text: for

text: z

text: wh−ere

text: −1

text: and

text: The

text: two

text: meet

text: at

text: the

text: point

text: values

text: for

text: z

text: Values

text: satisfying

text: W(z)

text: belong

text: to

text: the

text: principal

text: branch,

text: which

text: is

text: denoted

text: as

text: W

text: (z),

text: while

text: values

text: satisfying

text: W(z)

text: belong

text: to

text: the

text: br−anch.

text: where

text: p

text: z

text: for

text: and

text: p

text: z

text: for

text: The

text: two

text: br−anches

text: meet

text: at−the

text: bran−ch

text: point

text: for

text: z

text: Tractable

text: bounds

text: that

text: provide

text: clearer

text: interpretations

text: of

text: where

text: All

text: values

text: of

text: W(z)

text: solutions

text: involving

text: the

text: Lambert

text: function

text: were

text: derived

text: in

text: for

text: z

text: belong

text: to

text: the

text: principal

text: branch

text: The

text: two

text: but

text: their

text: focus

text: was

text: mainly

text: on

text: the

text: primary

text: branch

text: branches

text: of

text: the

text: W(z)

text: are

text: depicted

text: in

text: Fig.

text: Algorithmic

text: approaches

text: were

text: used

text: in

text: to

text: derive

text: accurate

text: The

text: Lambert

text: W

text: function

text: has

text: been

text: recognised

text: in

text: the

text: solu-

text: approximations

text: to

text: the

text: branch,

text: for

text: example

text: tion

text: of

text: scientiﬁc

text: and

text: engineering

text: problems

text: and

text: it

text: has

text: been

text: introduced

text: as

text: a

text: special

text: function

text: in

text: numerical

text: computation

text: W−

text: (z)

text: ln(

text: z)

text: 2α

text: raesseaMrcahplein

text: acnodmmMuAnTicLaAtiBon.s,Itsuhcahs

text: −1

text: ln(−z)

text: −1

text: α

text: as

text: relaying

text: strategies

text: moment

text: generating

text: functions

text: for

text: modelling

text: signal

text: fading

text: and

text: long-haul

text: cooperative

text: power

text: allocation

text: methods

text: Numerical

text: evaluation

text: of

text: the

text: Lambert

text: where

text: α

text: W

text: function

text: makes

text: use

text: of

text: recursive

text: schemes.

text: Non-recursive

text: In

text: this

text: paper,

text: we

text: use

text: functional

text: analysis

text: methods

text: to

text: derive

text: expansions

text: have

text: also

text: been

text: obtained

text: for

text: small

text: and

text: large

text: values

text: upper

text: and

text: lower

text: bounds

text: on

text: the

text: W−

text: branch.

text: Our

text: objective

text: is

text: of

text: z

text: For

text: instance,

text: the

text: ﬁrst

text: few

text: terms

text: of

text: the

text: series

text: to

text: obtain

text: expressions

text: which

text: might

text: not

text: be

text: as

text: accurate

text: as

text: but

text: expansion

text: of

text: W(z)

text: about

text: the

text: branch

text: point

text: are

text: are

text: simpler,

text: more

text: tractable

text: and

text: can

text: shed

text: light

text: on

text: the

text: essence

text: of

text: solutions

text: that

text: involve

text: the

text: W−

text: function.

text: The

text: remainder

text: of

text: this

text: paper

text: has

text: been

text: organised

text: as

text: follows.

text: W(z)

text: p

text: Section

text: presents

text: bounds

text: on

text: the

text: natural

text: logarithm

text: which

text: are

text: used

text: as

text: a

text: stepping

text: stone

text: for

text: the

text: derivation

text: of

text: bounds

text: on

text: I.

text: Chatzigeorgiou

text: is

text: with

text: the

text: School

text: of

text: Computing

text: and

text: Communica-

text: the

text: W−

text: function.

text: Section

text: considers

text: a

text: simple

text: cooperative

text: tions,

text: Lancaster

text: University,

text: United

text: Kingdom,

text: (e-mail:

text: i.chatzigeorgiou@lancaster.ac.uk)

text: network

text: and

text: assesses

text: the

text: tightness

text: and

text: usefulness

text: of

text: the

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text: 4 reliability compared to non-cooperation. Taking into account [2] T. Cui, T. Ho, and J. Kliewer, “Memoryless relay strategies for two-way ∈ that θ/γ (0, 1), we can use the tight lower bound on W−1 rOeclat.y2c0h0a9n.nels,” IEEE Trans. Commun., vol. 57, no. 10, pp. 3132–3143, shown in (8) and combine it with (12) to state with certainty (cid:114) [3] C. Tellambura and D. Senaratne, “Accurate computation of the MGF that if of the lognormal distribution and its application to sum of lognormals,” IEEE Trans. Commun., vol. 58, no. 5, pp. 1568–1577, May 2010. [4] D. Wu, Y. Cai, L. Zhou, and J. Wang, “A cooperative communication (cid:48) ≥ γ θ 3 θ θ + (14) scheme based on coalition formation game in clustered wireless sensor 2 8 networks,” IEEE Trans. Wireless Commun., vol. 11, no. 3, pp. 1190– cooperation should be avoided. 1200, Mar. 2012. [5] F. Chapeau-Blondeau and A. Monir, “Numerical evaluation of the Lam- Regions of beneﬁcial cooperation as deﬁned by (12) have bert W function and application to generation of generalized Gaussian been plotted in Fig. 3. More speciﬁcally, Fig. 3a shows a noise with exponent 1/2,” IEEE Trans. Signal Process., vol. 50, no. 9, snapshot of θ(cid:48) as a function of γ for θ = 5 dB, while Fig. 3b pp. 2160–2164, Sep. 2002. (cid:48) [6] A. Hoorfar and M. Hassani, “Inequalities on the Lambert W function depicts the dependence of θ on θ for γ = 5 dB. As expected, and hyperpower function,” J. Inequalities in Pure and Applied Math., when the quality of the uplink channel (γ) increases relative to vol. 9, no. 2, 2008. the SNR threshold of non-cooperative transmission (θ), users [7] S. M. Stewart, “On certain inequalities involving the Lambert W function,” J. Inequalities in Pure and Applied Math., vol. 10, no. 4, have the ﬂexibility to choose an SNR threshold for cooperative 2009. (cid:48) transmission (θ ) from an increasingly broader range of values [8] D. A. Barry, L. Li, and D.-S. Jeng, “Comments on “Numerical evaluation (Fig. 3a). This trend is reversed when θ approaches the value of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2”,” IEEE Trans. Signal Process., of γ (Fig. 3b). We observe that the top border of the shaded vol. 52, no. 5, pp. 1456–1458, May 2004. regions is tightly enveloped by the bounds in (13) and (14), [9] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity - which have a much simpler representation than (12). Part I: System description,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927–1938, Nov. 2003. C. Example case [10] Y. Xi, A. Burr, J. Wei, and D. Grace, “A general upper bound to evaluate Consider the case when two users employ rate-1/2 convolu- packet error rate over quasi-static fading channels,” IEEE Trans. Wireless − Commun., vol. 10, no. 5, pp. 1373–1377, May 2011. tional coding with θ = 0.983 dB in non-cooperative mode. [11] Y. Zhao, R. Adve, and T. J. Lim, “Outage probability at arbitrary SNR These two users decide to cooperate and switch to uncoded with cooperative diversity,” IEEE Commun. Lett., vol. 9, no. 8, pp. 700– binary phase shift keying (BPSK) with θ(cid:48) = 5.782 dB [12] in 702, Aug. 2005. [12] I. Chatzigeorgiou, I. J. Wassell, and R. Carrasco, “On the frame error order to maintain a constant transfer rate. As we can deduce rate of transmission schemes on quasi-static fading channels,” in Proc. from Fig. 3b, cooperation will have a negative effect on the Conf. on Information Sciences and Systems, Princeton, USA, Mar. 2008. outage probability if the average uplink SNR is γ = 5 dB. Solving the right-hand side inequality in (13) for γ, we ﬁnd (cid:32) (cid:33) that cooperation will be beneﬁcial only if √ (cid:48) 2 ≥ √θ − θ γ 2 θ 3 ≥ or γ 14.925 dB for this example. Note that γ cannot be (cid:48) determined as easily from (12), where it is both a factor of θ and an input to W− . 1 IV. CONCLUSIONS AND FUTURE WORK In this paper, we focused on the W− branch of the 1 Lambert W function and we derived tractable closed-form upper and lower bounds. We then considered a network of two users and an access point, and we demonstrated that our proposed bounds can identify operational regions of mutually beneﬁcial cooperation based on system characteristics, such as the channel quality and the employed transmission scheme. The derived expressions can be used to make decisions on whether nodes should cooperate or not, if their objective is to improve outage probability. To illustrate the application of the Lambert function and the proposed bounds, we assumed that inter-user channels are perfect and uplink channels are statistically similar. Follow-on work will aim to derive conditions for cooperation in networks where inter-user channels can be reciprocal or independent and uplink channels can be statistically similar or dissimilar. REFERENCES [1] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Computational Math., vol. 5, no. 1, pp. 329–359, Jan. 1996. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: reliability

text: compared

text: to

text: non-cooperation.

text: Taking

text: into

text: account

label: reference

text: T.

text: Cui,

text: T.

text: Ho,

text: and

text: J.

text: Kliewer,

text: “Memoryless

text: relay

text: strategies

text: for

text: two-way

text: that

text: θ/γ

text: we

text: can

text: use

text: the

text: tight

text: lower

text: bound

text: on

text: IEEE

text: Trans.

text: Commun.,

text: vol.

text: no.

text: pp.

text: shown

text: in

text: and

text: combine

text: it

text: with

text: to

text: state

text: with

text: certainty

label: equation

text: C.

text: Tellambura

text: and

text: D.

text: Senaratne,

text: “Accurate

text: computation

text: of

text: the

text: MGF

text: that

text: if

text: of

text: the

text: lognormal

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text: Wu,

text: Y.

text: Cai,

text: L.

text: Zhou,

text: and

text: J.

text: Wang,

text: “A

text: cooperative

text: communication

text: γ

text: θ

text: θ

text: θ

text: scheme

text: based

text: on

text: coalition

text: formation

text: game

text: in

text: clustered

text: wireless

text: sensor

text: networks,”

text: IEEE

text: Trans.

text: Wireless

text: Commun.,

text: vol.

text: no.

text: pp.

text: cooperation

text: should

text: be

text: avoided.

text: Mar.

text: F.

text: Chapeau-Blondeau

text: and

text: A.

text: Monir,

text: “Numerical

text: evaluation

text: of

text: the

text: Lam-

text: Regions

text: of

text: beneﬁcial

text: cooperation

text: as

text: deﬁned

text: by

text: have

text: bert

text: W

text: function

text: and

text: application

text: to

text: generation

text: of

text: generalized

text: Gaussian

text: been

text: plotted

text: in

text: Fig.

text: More

text: speciﬁcally,

text: Fig.

text: shows

text: a

text: noise

text: with

text: exponent

text: IEEE

text: Trans.

text: Signal

text: Process.,

text: vol.

text: no.

text: snapshot

text: of

text: as

text: a

text: function

text: of

text: γ

text: for

text: θ

text: dB,

text: while

text: Fig.

text: pp.

text: Sep.

text: A.

text: Hoorfar

text: and

text: M.

text: Hassani,

text: “Inequalities

text: on

text: the

text: Lambert

text: W

text: function

text: depicts

text: the

text: dependence

text: of

text: θ

text: on

text: θ

text: for

text: γ

text: dB.

text: As

text: expected,

text: and

text: hyperpower

text: function,”

text: J.

text: Inequalities

text: in

text: Pure

text: and

text: Applied

text: Math.,

text: when

text: the

text: quality

text: of

text: the

text: uplink

text: channel

text: (γ)

text: increases

text: relative

text: to

text: vol.

text: no.

text: the

text: SNR

text: threshold

text: of

text: non-cooperative

text: transmission

text: (θ),

text: users

text: S.

text: M.

text: Stewart,

text: “On

text: certain

text: inequalities

text: involving

text: the

text: Lambert

text: W

text: function,”

text: J.

text: Inequalities

text: in

text: Pure

text: and

text: Applied

text: Math.,

text: vol.

text: no.

text: have

text: the

text: ﬂexibility

text: to

text: choose

text: an

text: SNR

text: threshold

text: for

text: cooperative

text: transmission

text: (θ

text: from

text: an

text: increasingly

text: broader

text: range

text: of

text: values

text: D.

text: A.

text: Barry,

text: L.

text: Li,

text: and

text: D.-S.

text: Jeng,

text: “Comments

text: on

text: “Numerical

text: evaluation

text: (Fig.

text: This

text: trend

text: is

text: reversed

text: when

text: θ

text: approaches

text: the

text: value

text: of

text: the

text: Lambert

text: W

text: function

text: and

text: application

text: to

text: generation

text: of

text: generalized

text: Gaussian

text: noise

text: with

text: exponent

text: IEEE

text: Trans.

text: Signal

text: Process.,

text: of

text: γ

text: (Fig.

text: We

text: observe

text: that

text: the

text: top

text: border

text: of

text: the

text: shaded

text: vol.

text: no.

text: pp.

text: May

text: regions

text: is

text: tightly

text: enveloped

text: by

text: the

text: bounds

text: in

text: and

text: A.

text: Sendonaris,

text: E.

text: Erkip,

text: and

text: B.

text: Aazhang,

text: “User

text: cooperation

text: diversity

text: which

text: have

text: a

text: much

text: simpler

text: representation

text: than

text: Part

text: I:

text: System

text: description,”

text: IEEE

text: Trans.

text: Commun.,

text: vol.

text: no.

text: pp.

text: Nov.

text: C.

text: Example

text: case

text: Y.

text: Xi,

text: A.

text: Burr,

text: J.

text: Wei,

text: and

text: D.

text: Grace,

text: “A

text: general

text: upper

text: bound

text: to

text: evaluate

text: Consider

text: the

text: case

text: when

text: two

text: users

text: employ

text: convolu-

text: packet

text: error

text: rate

text: over

text: quasi-static

text: fading

text: channels,”

text: IEEE

text: Trans.

text: Wireless

text: Commun.,

text: vol.

text: no.

text: pp.

text: May

text: tional

text: coding

text: with

text: θ

text: dB

text: in

text: non-cooperative

text: mode.

text: Y.

text: Zhao,

text: R.

text: Adve,

text: and

text: T.

text: J.

text: Lim,

text: “Outage

text: probability

text: at

text: arbitrary

text: SNR

text: These

text: two

text: users

text: decide

text: to

text: cooperate

text: and

text: switch

text: to

text: uncoded

text: with

text: cooperative

text: diversity,”

text: IEEE

text: Commun.

text: Lett.,

text: vol.

text: no.

text: pp.

text: binary

text: phase

text: shift

text: keying

text: (BPSK)

text: with

text: dB

text: in

text: Aug.

text: I.

text: Chatzigeorgiou,

text: I.

text: J.

text: Wassell,

text: and

text: R.

text: Carrasco,

text: “On

text: the

text: frame

text: error

text: order

text: to

text: maintain

text: a

text: constant

text: transfer

text: rate.

text: As

text: we

text: can

text: deduce

text: rate

text: of

text: transmission

text: schemes

text: on

text: quasi-static

text: fading

text: channels,”

text: in

text: Proc.

text: from

text: Fig.

text: cooperation

text: will

text: have

text: a

text: negative

text: effect

text: on

text: the

text: Conf.

text: on

text: Information

text: Sciences

text: and

text: Systems,

text: Princeton,

text: USA,

text: Mar.

text: outage

text: probability

text: if

text: the

text: average

text: uplink

text: SNR

text: is

text: γ

text: dB.

text: Solving

text: the

text: right-hand

text: side

text: inequality

text: in

text: for

text: γ,

text: we

text: ﬁnd

text: that

text: cooperation

text: will

text: be

text: beneﬁcial

text: only

text: if

text: √θ

text: θ

text: γ

text: θ

text: or

text: γ

text: dB

text: for

text: this

text: example.

text: Note

text: that

text: γ

text: cannot

text: be

text: determined

text: as

text: easily

text: from

text: where

text: it

text: is

text: both

text: a

text: factor

text: of

text: θ

text: and

text: an

text: input

text: to

text: W−

text: IV.

text: CONCLUSIONS

text: AND

text: FUTURE

text: WORK

text: In

text: this

text: paper,

text: we

text: focused

text: on

text: the

text: W−

text: branch

text: of

text: the

text: Lambert

text: W

text: function

text: and

text: we

text: derived

text: tractable

text: closed-form

text: upper

text: and

text: lower

text: bounds.

text: We

text: then

text: considered

text: a

text: network

text: of

text: two

text: users

text: and

text: an

text: access

text: point,

text: and

text: we

text: demonstrated

text: that

text: our

text: proposed

text: bounds

text: can

text: identify

text: operational

text: regions

text: of

text: mutually

text: beneﬁcial

text: cooperation

text: based

text: on

text: system

text: characteristics,

text: such

text: as

text: the

text: channel

text: quality

text: and

text: the

text: employed

text: transmission

text: scheme.

text: The

text: derived

text: expressions

text: can

text: be

text: used

text: to

text: make

text: decisions

text: on

text: whether

text: nodes

text: should

text: cooperate

text: or

text: not,

text: if

text: their

text: objective

text: is

text: to

text: improve

text: outage

text: probability.

text: To

text: illustrate

text: the

text: application

text: of

text: the

text: Lambert

text: function

text: and

text: the

text: proposed

text: bounds,

text: we

text: assumed

text: that

text: inter-user

text: channels

text: are

text: perfect

text: and

text: uplink

text: channels

text: are

text: statistically

text: similar.

text: Follow-on

text: work

text: will

text: aim

text: to

text: derive

text: conditions

text: for

text: cooperation

text: in

text: networks

text: where

text: inter-user

text: channels

text: can

text: be

text: reciprocal

text: or

text: independent

text: and

text: uplink

text: channels

text: can

text: be

text: statistically

text: similar

text: or

text: dissimilar.

text: REFERENCES

text: R.

text: M.

text: Corless,

text: G.

text: H.

text: Gonnet,

text: D.

text: E.

text: G.

text: Hare,

text: D.

text: J.

text: Jeffrey,

text: and

text: D.

text: E.

text: Knuth,

text: “On

text: the

text: Lambert

text: W

text: function,”

text: Adv.

text: Computational

text: Math.,

text: vol.

text: no.

text: pp.

text: Jan.

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: I. INTRODUCTION Chiral skyrmions are of great interest in fundamental physics and mathematics because of their non-trivial topological properties [1, 2]. Theoretically, the existence of magnetic skyrmions has been predicted in a variety of three- and two-dimensional systems [3–5]. Recently, they have indeed been found in cubic helimagnets [6–8] and in two-dimensional magnetic layers on metal substrates exhibiting strong spin-orbit coupling [9]. In both cases skyrmions easily condense into lattices, while isolated skyrmions are very often unstable. For information storage applications, however, isolated magnetic skyrmions are required, as they have to be created, deleted and manipulated independently of their neighbors [10]. In a few systems only (e.g. Fe/Pd/Ir(111)) isolated skyrmions could be created and stabilized experimentally [11, 12]. While the dependence of the size of these isolated objects on an external magnetic ﬁeld and material parameters have been studied in the framework of continuous model systems [13, 14], the experimentally found nanoscale skyrmions interact with a discrete atomic lat- tice. Hence, in search for stable isolated skyrmions it is important to understand whether their properties diﬀer in continuous and discrete systems [15]. Two aspects appear to be particularly important: First, according to continuum ﬁeld theory skyrmions survive in in- ﬁnitely large ﬁelds [13, 16]. This property is known under the name of topological protection. Experimental results, however, show that at some critical ﬁeld a skyrmion on a discrete lat- tice will collapse [12, 16]. Hence, it is important to study the critical ﬁeld of the collapse and the skyrmion size as a function of ﬁeld and other parameters. The second aspect con- cerns the transition between the regime of skyrmion lattices and that of isolated skyrmions. From ﬁeld theory it is known that the radii of skyrmions in the two phases have a diﬀerent dependence on ﬁeld and Dzyaloshinskii-Moriya (DM) interaction [4, 14, 16]. Particularly, the radius of an isolated skyrmion should increase with the strength of the DM interaction, while the radius of skyrmions in a lattice should, in contrast, decrease [4, 14, 16]. In the published phase diagrams [17–20], however, a diﬀerentiation between those regimes is not addressed. To close this lack of knowledge we investigate the skyrmion radius R and the skyrmion density ρ as a function of the ratio between the strength of the DM interaction D and the exchange interaction J for diﬀerent ﬁelds B by means of Monte-Carlo simulations. We ﬁnd 2

text: I.

label: section

text: INTRODUCTION

text: Chiral

text: skyrmions

text: are

text: of

text: great

text: interest

text: in

text: fundamental

text: physics

text: and

text: mathematics

text: because

text: of

text: their

text: non-trivial

text: topological

text: properties

text: Theoretically,

text: the

text: existence

text: of

text: magnetic

text: skyrmions

text: has

text: been

text: predicted

text: in

text: a

text: variety

text: of

text: three-

text: and

text: two-dimensional

text: systems

text: Recently,

text: they

text: have

text: indeed

text: been

text: found

text: in

text: cubic

text: helimagnets

text: and

text: in

text: two-dimensional

text: magnetic

text: layers

text: on

text: metal

text: substrates

text: exhibiting

text: strong

text: spin-orbit

text: coupling

text: In

text: both

text: cases

text: skyrmions

text: easily

text: condense

text: into

text: lattices,

text: while

text: isolated

text: skyrmions

text: are

text: very

text: often

text: unstable.

text: For

text: information

text: storage

text: applications,

text: however,

text: isolated

text: magnetic

text: skyrmions

text: are

text: required,

text: as

text: they

text: have

text: to

text: be

text: created,

text: deleted

text: and

text: manipulated

text: independently

text: of

text: their

text: neighbors

text: In

text: a

text: few

text: systems

text: only

text: (e.g.

text: isolated

text: skyrmions

text: could

text: be

text: created

text: and

text: stabilized

text: experimentally

text: While

text: the

text: dependence

text: of

text: the

text: size

text: of

text: these

text: isolated

text: objects

text: on

text: an

text: external

text: magnetic

text: ﬁeld

text: and

text: material

text: parameters

text: have

text: been

text: studied

text: in

text: the

text: framework

text: of

text: continuous

text: model

text: systems

text: the

text: experimentally

text: found

text: nanoscale

text: skyrmions

text: interact

text: with

text: a

text: discrete

text: atomic

text: lat-

text: tice.

text: Hence,

text: in

text: search

text: for

text: stable

text: isolated

text: skyrmions

text: it

text: is

text: important

text: to

text: understand

text: whether

text: their

text: properties

text: diﬀer

text: in

text: continuous

text: and

text: discrete

text: systems

text: Two

text: aspects

text: appear

text: to

text: be

text: particularly

text: important:

text: First,

text: according

text: to

text: continuum

text: ﬁeld

text: theory

text: skyrmions

text: survive

text: in

text: in-

text: ﬁnitely

text: large

text: ﬁelds

text: This

text: property

text: is

text: known

text: under

text: the

text: name

text: of

text: topological

text: protection.

text: Experimental

text: results,

text: however,

text: show

text: that

text: at

text: some

text: critical

text: ﬁeld

text: a

text: skyrmion

text: on

text: a

text: discrete

text: lat-

text: tice

text: will

text: collapse

text: Hence,

text: it

text: is

text: important

text: to

text: study

text: the

text: critical

text: ﬁeld

text: of

text: the

text: collapse

text: and

text: the

text: skyrmion

text: size

text: as

text: a

text: function

text: of

text: ﬁeld

text: and

text: other

text: parameters.

text: The

text: second

text: aspect

text: con-

text: cerns

text: the

text: transition

text: between

text: the

text: regime

text: of

text: skyrmion

text: lattices

text: and

text: that

text: of

text: isolated

text: skyrmions.

text: From

text: ﬁeld

text: theory

text: it

text: is

text: known

text: that

text: the

text: radii

text: of

text: skyrmions

text: in

text: the

text: two

text: phases

text: have

text: a

text: diﬀerent

text: dependence

text: on

text: ﬁeld

text: and

text: Dzyaloshinskii-Moriya

text: (DM)

text: interaction

text: Particularly,

text: the

text: radius

text: of

text: an

text: isolated

text: skyrmion

text: should

text: increase

text: with

text: the

text: strength

text: of

text: the

text: DM

text: interaction,

text: while

text: the

text: radius

text: of

text: skyrmions

text: in

text: a

text: lattice

text: should,

text: in

text: contrast,

text: decrease

text: In

text: the

text: published

text: phase

text: diagrams

text: however,

text: a

text: diﬀerentiation

text: between

text: those

text: regimes

text: is

text: not

text: addressed.

text: To

text: close

text: this

text: lack

text: of

text: knowledge

text: we

text: investigate

text: the

text: skyrmion

text: radius

text: R

text: and

text: the

text: skyrmion

text: density

text: ρ

text: as

text: a

text: function

text: of

text: the

text: ratio

text: between

text: the

text: strength

text: of

text: the

text: DM

text: interaction

text: D

text: and

text: the

text: exchange

text: interaction

text: J

text: for

text: diﬀerent

text: ﬁelds

text: B

text: by

text: means

text: of

text: Monte-Carlo

text: simulations.

text: We

text: ﬁnd


text: doesn’t provide any speciﬁc information about the skyrmion size or about the density of a skyrmion lattice but rather deﬁnes the very existence of these topological objects. Hence, two skyrmion lattices of diﬀerent density and diﬀerent skyrmion sizes but with identical numbers of skyrmions have identical winding numbers Q and, hence, are indistinguishable in common phase diagrams. It is known, however, that the skyrmion size is strongly parameter- dependent [12, 14]. The aim of this investigation is the derivation of equilibrium states of chiral systems as a function of the eﬀective interaction strength including the information on the skyrmion size and density. For that purpose, similarly to [7], we deﬁne the density order parameter ρ as the ratio between the eﬀective area occupied by skyrmions to the area occupied by a closed packed lattice of skyrmions having the same radius R. The skyrmion radius R has been determined numerically by a two-dimensional surface approximation of the azimuthal angle Θ of the local magnetization texture with a formula which was used originally to describe ◦ 360 N´eel domain walls [26]. In this approximation the angle of the local magnetic moment is given by (cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:18) (cid:19)(cid:21) − r + c r c Θ (r) = arcsin tanh + arcsin tanh (3) 360 w/2 w/2 deﬁning a full rotation of 2π from one to the other side of the domain wall with the ﬁt parameters c and w, and r the distance from the center. According to the previous investigation [12] the magnetization proﬁle of a skyrmion can be well approximated by this formula. The skyrmion diameter can then be determined as the size of the region in which the magnetization has a component in the direction opposite to the magnetic ﬁeld. × To obtain the phase diagrams very large samples (3 105 sites) of rectangular shape with slow D/J gradient along the x axis have been equilibrated at various magnetic ﬁelds B. The magnetocrystalline anisotropy was set to zero for simplicity. The obtained results have been cross-checked via standard calculations without any gradient as explained in ≈ ≈ the previous section. Data presented here correspond to δ(D)/δr 0.005/a or δD/δr J 0.035 meV/a. The method of gradients permits the direct visualization of the magnetic microstructure as a function of the strength of DM interaction. Fig.1a shows the equilibrium micromagnetic structure of a sample subject to a magnetic ﬁeld of strength µB/J = 0.7 at k T/J = 0.1 J in which the transitions from the FM to the Skx, and eventually to the B 6 ##LTLine## ##LTLine## ##LTLine##

text: doesn’t

text: provide

text: any

text: speciﬁc

text: information

text: about

text: the

text: skyrmion

text: size

text: or

text: about

text: the

text: density

text: of

text: a

text: skyrmion

text: lattice

text: but

text: rather

text: deﬁnes

text: the

text: very

text: existence

text: of

text: these

text: topological

text: objects.

text: Hence,

text: two

text: skyrmion

text: lattices

text: of

text: diﬀerent

text: density

text: and

text: diﬀerent

text: skyrmion

text: sizes

text: but

text: with

text: identical

text: numbers

text: of

text: skyrmions

text: have

text: identical

text: winding

text: numbers

text: Q

text: and,

text: hence,

text: are

text: indistinguishable

text: in

text: common

text: phase

text: diagrams.

text: It

text: is

text: known,

text: however,

text: that

text: the

text: skyrmion

text: size

text: is

text: strongly

text: parameter-

text: dependent

text: The

text: aim

text: of

text: this

text: investigation

text: is

text: the

text: derivation

text: of

text: equilibrium

text: states

text: of

text: chiral

text: systems

text: as

text: a

text: function

text: of

text: the

text: eﬀective

text: interaction

text: strength

text: including

text: the

text: information

text: on

text: the

text: skyrmion

text: size

text: and

text: density.

text: For

text: that

text: purpose,

text: similarly

text: to

text: we

text: deﬁne

text: the

text: density

text: order

text: parameter

text: ρ

text: as

text: the

text: ratio

text: between

text: the

text: eﬀective

text: area

text: occupied

text: by

text: skyrmions

text: to

text: the

text: area

text: occupied

text: by

text: a

text: closed

text: packed

text: lattice

text: of

text: skyrmions

text: having

text: the

text: same

text: radius

text: R.

text: The

text: skyrmion

text: radius

text: R

text: has

text: been

text: determined

text: numerically

text: by

text: a

text: two-dimensional

text: surface

text: approximation

text: of

text: the

text: azimuthal

text: angle

text: Θ

text: of

text: the

text: local

text: magnetization

text: texture

text: with

text: a

text: formula

text: which

text: was

text: used

text: originally

text: to

text: describe

text: N´eel

text: domain

text: walls

text: In

text: this

text: approximation

text: the

text: angle

text: of

text: the

text: local

text: magnetic

text: moment

text: is

text: given

text: by

text: r

text: c

text: r

text: c

text: Θ

text: (r)

text: arcsin

text: tanh

text: arcsin

text: tanh

text: deﬁning

text: a

text: full

text: rotation

text: of

text: 2π

text: from

text: one

text: to

text: the

text: other

text: side

text: of

text: the

text: domain

text: wall

text: with

text: the

text: ﬁt

text: parameters

text: c

text: and

text: w,

text: and

text: r

text: the

text: distance

text: from

text: the

text: center.

text: According

text: to

text: the

text: previous

text: investigation

text: the

text: magnetization

text: proﬁle

text: of

text: a

text: skyrmion

text: can

text: be

text: well

text: approximated

text: by

text: this

text: formula.

text: The

text: skyrmion

text: diameter

text: can

text: then

text: be

text: determined

text: as

text: the

text: size

text: of

text: the

text: region

text: in

text: which

text: the

text: magnetization

text: has

text: a

text: component

text: in

text: the

text: direction

text: opposite

text: to

text: the

text: magnetic

text: ﬁeld.

text: To

text: obtain

text: the

text: phase

text: diagrams

text: very

text: large

text: samples

text: sites)

text: of

text: rectangular

text: shape

text: with

text: slow

text: D/J

text: gradient

text: along

text: the

text: x

text: axis

text: have

text: been

text: equilibrated

text: at

text: various

text: magnetic

text: ﬁelds

text: B.

text: The

text: magnetocrystalline

text: anisotropy

text: was

text: set

text: to

text: zero

text: for

text: simplicity.

text: The

text: obtained

text: results

text: have

text: been

text: cross-checked

text: via

text: standard

text: calculations

text: without

text: any

text: gradient

text: as

text: explained

text: in

text: the

text: previous

text: section.

text: Data

text: presented

text: here

text: correspond

text: to

text: δ(D)/δr

text: or

text: δD/δr

text: J

text: meV/a.

text: The

text: method

text: of

text: gradients

text: permits

text: the

text: direct

text: visualization

text: of

text: the

text: magnetic

text: microstructure

text: as

text: a

text: function

text: of

text: the

text: strength

text: of

text: DM

text: interaction.

text: shows

text: the

text: equilibrium

text: micromagnetic

text: structure

text: of

text: a

text: sample

text: subject

text: to

text: a

text: magnetic

text: ﬁeld

text: of

text: strength

text: µB/J

text: at

text: k

text: T/J

text: J

text: in

text: which

text: the

text: transitions

text: from

text: the

text: FM

text: to

text: the

text: Skx,

text: and

text: eventually

text: to

text: the

text: B

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: p dp s + p s x 0 d ds = ρ dθ, θ 0 x ds = (ρ + x) dθ. ρ θ Let us consider the change in orbit length. The relative elementary path length difference for a particle with a momentum p + dp is − dl ds ds x D dp = 0 = = x , (41) ds ds ρ ρ p 0 0 where D = dx/(dp/p) is the dispersion function from the transverse beam optics. x (cid:90) (cid:90) (cid:90) This leads to a total change in the circumference L of x D dp dL = dl = ds = x ds , (42) 0 0 ρ ρ p C (cid:90) so that 1 D α = x ds . (43) 0 L ρ ∞ Since ρ = in the straight sections, we get < D > α = x m , (44) R where the average < > is considered over the bending magnets only. m Given that the revolution frequency is f = βc/2πR, the relative change is (using the deﬁnition r of the momentum compaction factor) df dβ − dR dβ − dp r = = α , (45) f β R β p r (cid:0) (cid:1) − − (cid:124) (cid:123)(cid:122) (cid:125) E ⇒ dp dβ d(1 β2) 1/2 − − dβ p = mv = βγ 0 = + − − = 1 β2 1 . (46) c p β (1 β2) 1/2 β γ2 (cid:18) (cid:19) So, the relative change in revolution frequency is given by df 1 − dp r = α , (47) f γ2 p r 12 ##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##

text: p

font: default

text: dp

text: s

text: p

text: s

text: x

text: d

text: ds

text: ρ

text: dθ,

text: θ

text: x

text: ds

text: (ρ

text: x)

text: dθ.

text: ρ

text: θ

text: Let

text: us

text: consider

text: the

text: change

text: in

text: orbit

text: length.

text: The

text: relative

text: elementary

text: path

text: length

text: difference

text: for

text: a

text: particle

text: with

text: a

text: momentum

text: p

text: dp

text: is

text: dl

text: ds

text: ds

text: x

text: D

text: dp

text: x

text: ds

text: ds

text: ρ

text: ρ

text: p

text: where

text: D

text: dx/(dp/p)

text: is

text: the

text: dispersion

text: function

text: from

text: the

text: transverse

text: beam

text: optics.

text: x

text: This

text: leads

text: to

text: a

text: total

text: change

text: in

text: the

text: circumference

text: L

text: of

text: x

text: D

text: dp

text: dL

text: dl

text: ds

text: x

text: ds

text: ρ

text: ρ

text: p

text: C

text: so

text: that

text: D

text: α

text: x

text: ds

text: L

text: ρ

text: Since

text: ρ

text: in

text: the

text: straight

text: sections,

text: we

text: get

text: D

text: α

text: x

text: m

text: R

text: where

text: the

text: average

text: is

text: considered

text: over

text: the

text: bending

text: magnets

text: only.

text: m

text: Given

text: that

text: the

text: revolution

text: frequency

text: is

text: f

text: the

text: relative

text: change

text: is

text: (using

text: the

text: deﬁnition

text: r

text: of

text: the

text: momentum

text: compaction

text: factor)

text: df

text: dβ

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text: Fig. 5: Alvarez-type accelerating structure E (z, r) = E = const. (13) 1 0 Fig. 6: Simple uniform ﬁeld model (cid:12) (cid:16) (cid:17)(cid:46) (cid:16) (cid:17) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ω g ω g T = sin RF RF . (14) a 2 v 2 v This simple example shows that the transit-time factor tends towards 1 for smaller gap width g, smaller radio frequency, and higher velocity v of the particle, which is also true in the general case. The reduction in acceleration becomes important for particles with low velocities, like low-energy protons and particularly ions. 2.5 Disc-loaded travelling wave structures Electrons reach a relativistic β close to unity for a kinetic energy of 10 MeV, while protons reach this only at an energy of the order of 10 GeV. Above these energies, the particles have basically the speed of light, and the drift-tube length remains constant. Nevertheless, the drift-tube length would become very long unless the frequency is in the GHz range. So, the idea came up that the ultra-relativistic particles could be accelerated by a travelling wave in a waveguide. In order to to get continuous acceleration, the phase velocity v of the wave has to match ϕ the velocity v of the particle. However, rectangular or cylindrical waveguides have electric ﬁeld modes with an electric ﬁeld in the direction of propagation with phase velocities bigger than c, so that the wave does not remain synchronous with the particle. The phase velocity can be adjusted by inserting irises into the waveguide, and the dimensions of the irises and cells can be tailored to match the phase velocity to the velocity of the particle. Figure 7 shows a sketch of such a disc-loaded travelling wave structure. The electric ﬁeld of an electromagnetic wave travelling in the z direction (see Fig. 8) is given by equation (15). (cid:18) (cid:19) The ﬁeld seen by the particle is − v − E = E cos ω t ω t φ . (16) z 0 RF RF 0 v ϕ When synchronism is satisﬁed with v = v , the particle sees a constant ﬁeld ϕ 5 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: Fig.

text: Alvarez-type

label: caption

text: accelerating

text: structure

text: E

text: (z,

text: r)

text: E

text: const.

text: Fig.

text: Simple

text: uniform

text: ﬁeld

text: model

text: ω

text: g

text: ω

text: g

text: T

text: sin

text: RF

text: RF

text: a

text: v

text: v

text: This

text: simple

text: example

text: shows

text: that

text: the

text: transit-time

text: factor

text: tends

text: towards

text: for

text: smaller

text: gap

text: width

text: g,

text: smaller

text: radio

text: frequency,

text: and

text: higher

text: velocity

text: v

text: of

text: the

text: particle,

text: which

text: is

text: also

text: true

text: in

text: the

text: general

text: case.

text: The

text: reduction

text: in

text: acceleration

text: becomes

text: important

text: for

text: particles

text: with

text: low

text: velocities,

text: like

text: low-energy

text: protons

text: and

text: particularly

text: ions.

text: Disc-loaded

text: travelling

text: wave

text: structures

text: Electrons

text: reach

text: a

text: relativistic

text: β

text: close

text: to

text: unity

text: for

text: a

text: kinetic

text: energy

text: of

text: MeV,

text: while

text: protons

text: reach

text: this

text: only

text: at

text: an

text: energy

text: of

text: the

text: order

text: of

text: GeV.

text: Above

text: these

text: energies,

text: the

text: particles

text: have

text: basically

text: the

text: speed

text: of

text: light,

text: and

text: the

text: drift-tube

text: length

text: remains

text: constant.

text: Nevertheless,

text: the

text: drift-tube

text: length

text: would

text: become

text: very

text: long

text: unless

text: the

text: frequency

text: is

text: in

text: the

text: GHz

text: range.

text: So,

text: the

text: idea

text: came

text: up

text: that

text: the

text: ultra-relativistic

text: particles

text: could

text: be

text: accelerated

text: by

text: a

text: travelling

text: wave

text: in

text: a

text: waveguide.

text: In

text: order

text: to

text: to

text: get

text: continuous

text: acceleration,

text: the

text: phase

text: velocity

text: v

text: of

text: the

text: wave

text: has

text: to

text: match

text: ϕ

text: the

text: velocity

text: v

text: of

text: the

text: particle.

text: However,

text: rectangular

text: or

text: cylindrical

text: waveguides

text: have

text: electric

text: ﬁeld

text: modes

text: with

text: an

text: electric

text: ﬁeld

text: in

text: the

text: direction

text: of

text: propagation

text: with

text: phase

text: velocities

text: bigger

text: than

text: c,

text: so

text: that

text: the

text: wave

text: does

text: not

text: remain

text: synchronous

text: with

text: the

text: particle.

text: The

text: phase

text: velocity

text: can

text: be

text: adjusted

text: by

text: inserting

text: irises

text: into

text: the

text: waveguide,

text: and

text: the

text: dimensions

text: of

text: the

text: irises

text: and

text: cells

text: can

text: be

text: tailored

text: to

text: match

text: the

text: phase

text: velocity

text: to

text: the

text: velocity

text: of

text: the

text: particle.

text: Figure

text: shows

text: a

text: sketch

text: of

text: such

text: a

text: disc-loaded

text: travelling

text: wave

text: structure.

text: The

text: electric

text: ﬁeld

text: of

text: an

text: electromagnetic

text: wave

text: travelling

text: in

text: the

text: z

text: direction

text: (see

text: Fig.

text: is

text: given

text: by

text: equation

text: The

text: ﬁeld

text: seen

text: by

text: the

text: particle

text: is

text: v

text: E

text: E

text: cos

text: ω

text: t

text: ω

text: t

text: φ

text: z

text: RF

text: RF

text: v

text: ϕ

text: When

text: synchronism

text: is

text: satisﬁed

text: with

text: v

text: v

text: the

text: particle

text: sees

text: a

text: constant

text: ﬁeld

text: ϕ

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) Tsihtyeom(cid:74)raetmr(cid:75)ix6.reLperet(cid:74)sne1n,t(cid:75)ant2io,nns3, nn4 b,(cid:74)e nno(cid:75)un, snwit(cid:74)hacno(cid:75)drrenspon, dainndg sduecnh- (cid:74) (cid:75) + 13(cid:74)ψ( (cid:75)young ladies ⊗ own ⊗ animals ) tkh,alt∈n((cid:74)02, 1(cid:75)]=. Th(cid:74)kennw1e(cid:75) h+av(cid:74)De th(cid:75)aan1td: n42 (cid:74)= (cid:75)3l n(cid:74)3 + (cid:75)D4 (cid:48) (cid:74)for (cid:75)some Then s2 − 16 s(cid:74)1 is given by:(cid:75) (cid:74) (cid:75) (cid:74) (cid:75) ϕ ( n1 ⊗ verb ⊗ n3 ) (cid:50)kl ϕ ( n2 ⊗ verb ⊗ n4 ) 112ψ((cid:74)elderly ladies(cid:75) ⊗ (cid:74)own(cid:75) ⊗ (cid:74)dogs ) (cid:75) 1 ⊗ ⊗ Proof. The proof of this result is identical to that of theorem 4, + ψ( elderly ladies own hamsters ) 12 (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) except(cid:74)for(cid:75)the f(cid:74)act th(cid:75)at w(cid:74)hen(cid:75)we consid(cid:74)er (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) 1 ⊗ ⊗ ⊗ ⊗ − ⊗ ⊗ + ψ( young ladies own animals ) ϕ( n verb n ) kl ϕ( n verb n ) 3 2 4 1 3 ⊗ ⊗ which is clearly positive. we(cid:74)get(cid:75)ϕ =(cid:74)µ(cid:75) ι ε app(cid:74)lied(cid:75) to (cid:74) (cid:75) (cid:74) (cid:75) N S N ⊗ ⊗ (cid:48) ⊗ ⊗ ⊗ ⊗ (cid:48) (k n Z D ) + (D Z l n ) + (D Z D ) 1 3 ⊗ ⊗ instead of ϕ = (ε 1 ε ) applied to the same. The result N S N is, however, still a positive quantity by the property of the mor- phisms µ , 1 and ε to map density matrices to density matrices. N S N Thus, w(cid:74) e c(cid:75)an c(cid:74)onclu(cid:75)de a(cid:74)s be(cid:75)fore that: (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) ⊗ ⊗ (cid:50) ⊗ ⊗ ϕ ( n verb n ) ϕ ( n verb n ) . 2 4 kl 1 3 C.2.1 k-hyponymy applied to relative clauses We will consider the containment of the sentence: s = Elderly ladies who own cats 1 in the sentence: s = Women who own animals 2 First of all, let the subject and object space for the vectors corre- R R sponding to the subjects and object of our sentences be 2 and 3 resp(cid:74)ectively. Let: (cid:75) (cid:74) (cid:75) | (cid:105) (cid:104) | | (cid:105) (cid:104) | elderly ladies = e e , young ladies = e e 1 1 2 2 and the density matrix for the hypernym women be: (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) 1 2 women = elderly ladies + young ladies 3 3 Similarly, let: (cid:74) (cid:75) | (cid:105) (cid:104) | (cid:74) cats(cid:75) = n n 1 1 | (cid:105) (cid:104) | (cid:74) dogs(cid:75) = n n 2 2 | (cid:105) (cid:104) | hamsters = n n 3 3 and take the density matrix for animals to be: (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) 1 1 1 animal = cats + dogs + hamsters 2 4 4 The sentence space will not matter in this case, as it gets deleted by the ι morphism,(cid:88)so we just take it to be an unspeciﬁed S. Let the S verb ow(cid:74)n be (cid:75)given by: | (cid:105) (cid:104) | ⊗ | (cid:105) (cid:104) (cid:48)| ⊗ | (cid:105) (cid:104) | own = C e e s s n n . ijkl i k j l ijkl ⊗ ⊗ and the sentence map ψ is given by µ ι ε Then the meaning N S N of sen(cid:74)tenc(cid:75)es s a(cid:74)nd s are given(cid:75) by(cid:74): (cid:75) (cid:74) (cid:75) 1 2 ⊗ ⊗ (cid:74)s (cid:75) = ψ((cid:74)elderly l(cid:75)adie(cid:74)s (cid:75)own(cid:74) cats(cid:75)) 1 ⊗ ⊗ s2 = ψ( w(cid:74)omen ow(cid:75)n (cid:74) an(cid:75)ima(cid:74)ls )(cid:75) 1 ⊗ ⊗ = ψ( elderly ladies own cats ) 6 (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) 1 ⊗ ⊗ + ψ( elderly ladies own dogs 12 (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75) 1 ⊗ ⊗ + ψ( elderly ladies own hamsters )) 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: dainndg

text: sduecnh-

text: ladies

text: own

text: animals

text: Then

text: is

text: given

text: ϕ

text: verb

text: ϕ

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text: Proof.

text: The

text: proof

text: of

text: this

text: result

text: is

text: identical

text: to

text: that

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text: theorem

text: ψ(

text: elderly

text: ladies

text: own

text: hamsters

text: ψ(

text: young

text: ladies

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text: animals

text: ϕ(

text: n

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text: ι

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text: N

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text: sentence:

text: s

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text: elderly

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text: let:

text: n

text: n

text: n

text: n

text: hamsters

text: n

text: n

text: and

text: take

text: the

text: density

text: matrix

text: for

text: animals

text: to

text: be:

text: animal

text: cats

text: dogs

text: hamsters

text: The

text: sentence

text: space

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text: matter

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text: as

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text: unspeciﬁed

text: S.

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text: n

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text: ijkl

text: i

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text: j

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text: and

text: the

text: sentence

text: map

text: ψ

text: is

text: given

text: by

text: µ

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text: the

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text: N

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text: elderly

text: ladies

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text: cats

text: ψ(

text: elderly

text: ladies

text: own

text: dogs

text: ψ(

text: elderly

text: ladies

text: own

text: hamsters

text: ##LTLine##

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