text: 4 | | ( λ = 0.5). 1 We now analyze some examples in Region 2 where only small ISSs exist, hardly visible in parameter space. We focus on the largest ones recognizable in the bifurcation diagram of Fig.1 c), one centered at k = 6.86 and the other at k = 6.8. They are characterized by a period three and a period ten limit cycles respectively. The re- sults of the diagonalization of the quantum superopera- tor for a) (γ = 0.33, k = 6.86) and b) (γ = 0.33, k = 6.8) are shown in Figs. 4 and 5. For comparison purposes the spectra of the Perron-Frobenius superoperator corre- sponding to the classical map for these parameters are also indicated in Fig. 4 . The Husimi distribution of the invariants displayed in the left panels of Fig. 5 look very similar in both cases and have the structure of strange attractors, thus diﬀer- ing from the asymptotic point-like classical distributions. This is to be expected since, as we have seen in the previ- ous cases, the quantum equilibrium distributions are not aﬀected by the presence of regular regions unless these are really large. For their part, the spectra shown in Fig. 4 also present at ﬁrst sight the characteristic features of chaotic behavior, in the sense that they both have a well deﬁned gap (associated with the decay time towards the invariant) between λ = 1 and the disk where the remain- ing eigenvalues concentrate. However if we look closer at the spectrum of Fig. 4a) we observe that the complex | | ∼ phases of the leading eigenvalues (with λ 0.59) are ∼ multiples of 2π/3, just as the ones in the correspond- ing Perron-Frobenius spectrum. This quantum classical correspondence is conﬁrmed if we look at the structure of the corresponding eigenstate shown in the right panel of Fig. 5 a) which is strongly scarred by the periodic orbit with winding number 2/3 (indicated with crosses) characterizing the island. We now focus on the spectrum of Fig. 4 b) . In this case the limit cycle has period ten, and this periodic- ity manifests itself in the longest-lived eigenvalues of the Perron-Frobenius spectrum, which have phases equal to FIG. 3. (color online) Husimi representation of the invariant the tenth roots of unity. However no traces of this peri- state (left panel) and leading eigenstate (right panel) for a) odic 10-family are present in the quantum spectrum nor (γ = 0.33, k = 4.0), b) (γ = 0.33, k = 5.13), c) (γ = 0.33, k = (cid:126) in the structure of the leading eigenstate of Fig. 5 b). 5.5), d) (γ = 0.33, k = 6.0). = 0.042. In the right panels eﬀ of a) and b) the periodic points of the ISSs are marked with In fact, the distribution of this state is very similar to crosses. the one shown in the panel above, i.e., it is peaked on the period three orbit. Moreover it corresponds to an ∼ eigenvalue with a phase 2π/3 as in the previous case. fairly large, indicating a long transient also in this case. This seems to indicate that limit cycles with high period- The spectrum in Fig.2 c) shows the existence of a second icity that clearly determine the coarse grained classical ∼ real eigenvalue 0.7 considerably larger than the radius dynamics for a given point in parameter space even at of the dense disk. It corresponds to a state with a similar relatively low resolution (in our case N = 300) are not scarred structure. robust enough to manifest in the quantum case. Rather, the quantum behavior is inﬂuenced by regularity regions Finally, the calculation for parameters further away which might not be the closest ones in parameter space, from regular islands gives the typical features of chaotic but correspond to shorter limit cycles. dynamics. Both the equilibrium state and the leading resonance in Fig.3 d) are quantum chaotic attractors. IV. CONCLUSIONS No exceptional long-lived states exist. The spectrum of Fig.2 d) shows the presence of a considerable gap The standard map is a paradigmatic model in the clas- between eigenvalue 1 and the dense disk of eigenvalues sical and the quantum chaos literature, while the dissipa- ##LTFigure##

text: λ

text: We

text: now

text: analyze

text: some

text: examples

text: in

text: Region

text: where

text: only

text: small

text: ISSs

text: exist,

text: hardly

text: visible

text: in

text: parameter

text: space.

text: We

text: focus

text: on

text: the

text: largest

text: ones

text: recognizable

text: in

text: the

text: bifurcation

text: diagram

text: of

text: c),

text: one

text: centered

text: at

text: k

text: and

text: the

text: other

text: at

text: k

text: They

text: are

text: characterized

text: by

text: a

text: period

text: three

text: and

text: a

text: period

text: ten

text: limit

text: cycles

text: respectively.

text: The

text: re-

text: sults

text: of

text: the

text: diagonalization

text: of

text: the

text: quantum

text: superopera-

text: tor

text: for

text: a)

text: (γ

text: k

text: and

text: b)

text: (γ

text: k

text: are

text: shown

text: in

text: Figs.

text: and

text: For

text: comparison

text: purposes

text: the

text: spectra

text: of

text: the

text: Perron-Frobenius

text: superoperator

text: corre-

text: sponding

text: to

text: the

text: classical

text: map

text: for

text: these

text: parameters

text: are

text: also

text: indicated

text: in

text: Fig.

text: The

text: Husimi

text: distribution

text: of

text: the

text: invariants

text: displayed

text: in

text: the

text: left

text: panels

text: of

text: Fig.

text: look

text: very

text: similar

text: in

text: both

text: cases

text: and

text: have

text: the

text: structure

text: of

text: strange

text: attractors,

text: thus

text: diﬀer-

text: ing

text: from

text: the

text: asymptotic

text: point-like

text: classical

text: distributions.

text: This

text: is

text: to

text: be

text: expected

text: since,

text: as

text: we

text: have

text: seen

text: in

text: the

text: previ-

text: ous

text: cases,

text: the

text: quantum

text: equilibrium

text: distributions

text: are

text: not

text: aﬀected

text: by

text: the

text: presence

text: of

text: regular

text: regions

text: unless

text: these

text: are

text: really

text: large.

text: For

text: their

text: part,

text: the

text: spectra

text: shown

text: in

text: Fig.

text: also

text: present

text: at

text: ﬁrst

text: sight

text: the

text: characteristic

text: features

text: of

text: chaotic

text: behavior,

text: in

text: the

text: sense

text: that

text: they

text: both

text: have

text: a

text: well

text: deﬁned

text: gap

text: (associated

text: with

text: the

text: decay

text: time

text: towards

text: the

text: invariant)

text: between

text: λ

text: and

text: the

text: disk

text: where

text: the

text: remain-

text: ing

text: eigenvalues

text: concentrate.

text: However

text: if

text: we

text: look

text: closer

text: at

text: the

text: spectrum

text: of

text: Fig.

text: we

text: observe

text: that

text: the

text: complex

text: phases

text: of

text: the

text: leading

text: eigenvalues

text: (with

text: λ

text: are

text: multiples

text: of

text: just

text: as

text: the

text: ones

text: in

text: the

text: correspond-

text: ing

text: Perron-Frobenius

text: spectrum.

text: This

text: quantum

text: classical

text: correspondence

text: is

text: conﬁrmed

text: if

text: we

text: look

text: at

text: the

text: structure

text: of

text: the

text: corresponding

text: eigenstate

text: shown

text: in

text: the

text: right

text: panel

text: of

text: Fig.

text: a)

text: which

text: is

text: strongly

text: scarred

text: by

text: the

text: periodic

text: orbit

text: with

text: winding

text: number

text: (indicated

text: with

text: crosses)

text: characterizing

text: the

text: island.

text: We

text: now

text: focus

text: on

text: the

text: spectrum

text: of

text: Fig.

text: b)

text: In

text: this

text: case

text: the

text: limit

text: cycle

text: has

text: period

text: ten,

text: and

text: this

text: periodic-

text: ity

text: manifests

text: itself

text: in

text: the

text: longest-lived

text: eigenvalues

text: of

text: the

text: Perron-Frobenius

text: spectrum,

text: which

text: have

text: phases

text: equal

text: to

text: FIG.

text: (color

text: online)

text: Husimi

text: representation

text: of

text: the

text: invariant

text: the

text: tenth

text: roots

text: of

text: unity.

text: However

text: no

text: traces

text: of

text: this

text: peri-

text: state

text: (left

text: panel)

text: and

text: leading

text: eigenstate

text: (right

text: panel)

text: for

text: a)

text: odic

text: are

text: present

text: in

text: the

text: quantum

text: spectrum

text: nor

text: (γ

text: k

text: b)

text: (γ

text: k

text: c)

text: (γ

text: k

text: in

text: the

text: structure

text: of

text: the

text: leading

text: eigenstate

text: of

text: Fig.

text: b).

text: d)

text: (γ

text: k

text: In

text: the

text: right

text: panels

text: eﬀ

text: of

text: a)

text: and

text: b)

text: the

text: periodic

text: points

text: of

text: the

text: ISSs

text: are

text: marked

text: with

text: In

text: fact,

text: the

text: distribution

text: of

text: this

text: state

text: is

text: very

text: similar

text: to

text: crosses.

text: the

text: one

text: shown

text: in

text: the

text: panel

text: above,

text: i.e.,

text: it

text: is

text: peaked

text: on

text: the

text: period

text: three

text: orbit.

text: Moreover

text: it

text: corresponds

text: to

text: an

text: eigenvalue

text: with

text: a

text: phase

text: as

text: in

text: the

text: previous

text: case.

text: fairly

text: large,

text: indicating

text: a

text: long

text: transient

text: also

text: in

text: this

text: case.

text: This

text: seems

text: to

text: indicate

text: that

text: limit

text: cycles

text: with

text: high

text: period-

text: The

text: spectrum

text: in

text: c)

text: shows

text: the

text: existence

text: of

text: a

text: second

text: icity

text: that

text: clearly

text: determine

text: the

text: coarse

text: grained

text: classical

text: real

text: eigenvalue

text: considerably

text: larger

text: than

text: the

text: radius

text: dynamics

text: for

text: a

text: given

text: point

text: in

text: parameter

text: space

text: even

text: at

text: of

text: the

text: dense

text: disk.

text: It

text: corresponds

text: to

text: a

text: state

text: with

text: a

text: similar

text: relatively

text: low

text: resolution

text: (in

text: our

text: case

text: N

text: are

text: not

text: scarred

text: structure.

text: robust

text: enough

text: to

text: manifest

text: in

text: the

text: quantum

text: case.

text: Rather,

text: the

text: quantum

text: behavior

text: is

text: inﬂuenced

text: by

text: regularity

text: regions

text: Finally,

text: the

text: calculation

text: for

text: parameters

text: further

text: away

text: which

text: might

text: not

text: be

text: the

text: closest

text: ones

text: in

text: parameter

text: space,

text: from

text: regular

text: islands

text: gives

text: the

text: typical

text: features

text: of

text: chaotic

text: but

text: correspond

text: to

text: shorter

text: limit

text: cycles.

text: dynamics.

text: Both

text: the

text: equilibrium

text: state

text: and

text: the

text: leading

text: resonance

text: in

text: d)

text: are

text: quantum

text: chaotic

text: attractors.

text: IV.

text: CONCLUSIONS

text: No

text: exceptional

text: long-lived

text: states

text: exist.

text: The

text: spectrum

text: of

text: d)

text: shows

text: the

text: presence

text: of

text: a

text: considerable

text: gap

text: The

text: standard

text: map

text: is

text: a

text: paradigmatic

text: model

text: in

text: the

text: clas-

text: between

text: eigenvalue

text: and

text: the

text: dense

text: disk

text: of

text: eigenvalues

text: sical

text: and

text: the

text: quantum

text: chaos

text: literature,

text: while

text: the

text: dissipa-

text: ##LTFigure##


text: gas mixtures as a function of HV is shown in Fig. 7 are shown in the top and bottom panels of Fig. 8, re- in terms of FWHM. The energy resolution shows little spectively. The tail at the lower end of the time spec- variation over the range of voltage studied for both gas trum might be due to the fact that measurements were mixtures and is about 20% (FWHM). performed at high gain. Since the gain is high it might 4.4. Time resolution be possible to have a signal even if the primary ionisa- tion happens to be in the ﬁrst transfer gap. In that case Time resolution of GEM detectors is determined by the signal will be faster than the majority of the signals the spread in signal formation time for diﬀerent events. whose primaries were generated in the drift area. The Time resolution depends on several factors, the most im- results about the time resolution are in agreement with portant being the electron drift velocity. Drift velocity earlier results [2, 18]. depends on the gas type and the electric ﬁeld value. In the present setup, time resolution with the Ar and CO 2 gas mixture in 70:30 proportion was measured using 5. Detector uniformity the 106Ru-Rh β-source. The three-fold scintillator trig- ger described in the previous section was used as start New generation high energy physics experiments re- signal and the signal from the GEM detector processed quire large area detectors. The overall performance of through a fast ampliﬁer was used as the stop signal. The these detectors depends on gain uniformity, energy res- time diﬀerence between the start and the stop signals olution and eﬃciency over the entire active region. Sev- was measured by an ORTEC 567 Time to Amplitude eral factors, like variations in hole diameter, variations Converter (TAC). in gas gap due to inaccurate stretching, etc., can lead to non-uniformity in the detector [19, 20]. Thus it is es- sential to measure the gain uniformity over the entire 70 χCχC22oo //nn nnssddttaaff nntt 6655 ..1100119900 ±±..11 11 //.. 44991111 smuerfaascuereatrheea.gaIninthanisdwenoerkrg, ya rmeseothluotdiohnaisnbloeecnaliusseeddrteo- 60 MSMSiieeggaammnnaa 11 1111..66332255.. 88±± ± ± 00 ..0022..0022 gions and then results obtained across diﬀerent regions 50 have been compared. In the setup shown in Fig. 9, a thick perforated PCB is 40 × placed above the GEM detector. The 10 10 cm2 central 30 × area of the PCB is divided to 7 7 zones of equal area. 20 The 55Fe source was placed on the centre of each zone 10 and the resulting spectrum was recorded. For each zone, sum-up signals from two associated readout connectors 0 0 50 100 150 200 250 300 350 400 have been read out. For the central zones, however, the time (nsec) charge spreads on all four readout connectors. To have similar data taking conditions for all the zones, it was 16 decided to exclude the central zones. In this process, the 14 gain and energy resolution are measured for 36 zones. Fig. 10 shows the eﬀective gain values for each zone 12 of the GEM detector at an applied voltage of 4400 V. The gain has a mean value of 10030 at this voltage with 10 a RMS of 8.8%. As a graphical representation, the rel- 8 ative gain of the detector measured on each zone, has Ar/CO 70:30, 106Ru been plotted in Fig. 11 after normalising to the mean 6 2 value. The gain uniformity result is in good agreement with literature [21]. The energy resolutions obtained for 4 4300 4350 4400 4450 4500 4550 each of the 36 zones at 4400 V is shown in Fig. 12. The HV (V) mean value of the energy resolution (FWHM) is 21% with a RMS of 6.7%. 6. Summary and outlook Figure 8: (top) Time spectrum at 4450 V and (bottom) time resolution as a function of applied HV. × A detailed study of a 10 10 cm2 triple GEM detector ﬁlled with a gas mixture of Ar+CO of 70:30 and 90:10 2 The time spectrum and the plots for time resolution proportions has been performed. Tests were conducted 5 stnuoC )sn( σ t ##LTFigure## ##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## ##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## ##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: gas

label: paragraph

text: mixtures

text: as

text: a

text: function

text: of

text: HV

text: is

text: shown

text: in

text: Fig.

text: are

text: shown

text: in

text: the

text: top

text: and

text: bottom

text: panels

text: of

text: Fig.

text: re-

text: in

text: terms

text: of

text: FWHM.

text: The

text: energy

text: resolution

text: shows

text: little

text: spectively.

text: The

text: tail

text: at

text: the

text: lower

text: end

text: of

text: the

text: time

text: spec-

text: variation

text: over

text: the

text: range

text: of

text: voltage

text: studied

text: for

text: both

text: gas

text: trum

text: might

text: be

text: due

text: to

text: the

text: fact

text: that

text: measurements

text: were

text: mixtures

text: and

text: is

text: about

text: (FWHM).

text: performed

text: at

text: high

text: gain.

text: Since

text: the

text: gain

text: is

text: high

text: it

text: might

label: section

text: Time

text: resolution

text: be

text: possible

text: to

text: have

text: a

text: signal

text: even

text: if

text: the

text: primary

text: ionisa-

text: tion

text: happens

text: to

text: be

text: in

text: the

text: ﬁrst

text: transfer

text: gap.

text: In

text: that

text: case

text: Time

text: resolution

text: of

text: GEM

text: detectors

text: is

text: determined

text: by

text: the

text: signal

text: will

text: be

text: faster

text: than

text: the

text: majority

text: of

text: the

text: signals

text: the

text: spread

text: in

text: signal

text: formation

text: time

text: for

text: diﬀerent

text: events.

text: whose

text: primaries

text: were

text: generated

text: in

text: the

text: drift

text: area.

text: The

text: Time

text: resolution

text: depends

text: on

text: several

text: factors,

text: the

text: most

text: im-

text: results

text: about

text: the

text: time

text: resolution

text: are

text: in

text: agreement

text: with

text: portant

text: being

text: the

text: electron

text: drift

text: velocity.

text: Drift

text: velocity

text: earlier

text: results

text: depends

text: on

text: the

text: gas

text: type

text: and

text: the

text: electric

text: ﬁeld

text: value.

text: In

text: the

text: present

text: setup,

text: time

text: resolution

text: with

text: the

text: Ar

text: and

text: CO

text: gas

text: mixture

text: in

text: proportion

text: was

text: measured

text: using

text: Detector

text: uniformity

text: the

text: β-source.

text: The

text: three-fold

text: scintillator

text: trig-

text: ger

text: described

text: in

text: the

text: previous

text: section

text: was

text: used

text: as

text: start

text: New

text: generation

text: high

text: energy

text: physics

text: experiments

text: re-

text: signal

text: and

text: the

text: signal

text: from

text: the

text: GEM

text: detector

text: processed

text: quire

text: large

text: area

text: detectors.

text: The

text: overall

text: performance

text: of

text: through

text: a

text: fast

text: ampliﬁer

text: was

text: used

text: as

text: the

text: stop

text: signal.

text: The

text: these

text: detectors

text: depends

text: on

text: gain

text: uniformity,

text: energy

text: res-

text: time

text: diﬀerence

text: between

text: the

text: start

text: and

text: the

text: stop

text: signals

text: olution

text: and

text: eﬃciency

text: over

text: the

text: entire

text: active

text: region.

text: Sev-

text: was

text: measured

text: by

text: an

text: ORTEC

text: Time

text: to

text: Amplitude

text: eral

text: factors,

text: like

text: variations

text: in

text: hole

text: diameter,

text: variations

text: Converter

text: (TAC).

text: in

text: gas

text: gap

text: due

text: to

text: inaccurate

text: stretching,

text: etc.,

text: can

text: lead

text: to

text: non-uniformity

text: in

text: the

text: detector

text: Thus

text: it

text: is

text: es-

text: sential

text: to

text: measure

text: the

text: gain

text: uniformity

text: over

text: the

text: entire

text: //nn

text: nnssddttaaff

text: nntt

text: ya

text: MSMSiieeggaammnnaa

text: 88±±

text: gions

text: and

text: then

text: results

text: obtained

text: across

text: diﬀerent

text: regions

text: have

text: been

text: compared.

text: In

text: the

text: setup

text: shown

text: in

text: Fig.

text: a

text: thick

text: perforated

text: PCB

text: is

text: placed

text: above

text: the

text: GEM

text: detector.

text: The

text: central

text: area

text: of

text: the

text: PCB

text: is

text: divided

text: to

text: zones

text: of

text: equal

text: area.

text: The

text: source

text: was

text: placed

text: on

text: the

text: centre

text: of

text: each

text: zone

text: and

text: the

text: resulting

text: spectrum

text: was

text: recorded.

text: For

text: each

text: zone,

text: sum-up

text: signals

text: from

text: two

text: associated

text: readout

text: connectors

text: have

text: been

text: read

text: out.

text: For

text: the

text: central

text: zones,

text: however,

text: the

text: time

text: (nsec)

text: charge

text: spreads

text: on

text: all

text: four

text: readout

text: connectors.

text: To

text: have

text: similar

text: data

text: taking

text: conditions

text: for

text: all

text: the

text: zones,

text: it

text: was

text: decided

text: to

text: exclude

text: the

text: central

text: zones.

text: In

text: this

text: process,

text: the

text: gain

text: and

text: energy

text: resolution

text: are

text: measured

text: for

text: zones.

text: Fig.

text: shows

text: the

text: eﬀective

text: gain

text: values

text: for

text: each

text: zone

text: of

text: the

text: GEM

text: detector

text: at

text: an

text: applied

text: voltage

text: of

text: V.

text: The

text: gain

text: has

text: a

text: mean

text: value

text: of

text: at

text: this

text: voltage

text: with

text: a

text: RMS

text: of

text: As

text: a

text: graphical

text: representation,

text: the

text: rel-

text: ative

text: gain

text: of

text: the

text: detector

text: measured

text: on

text: each

text: zone,

text: has

text: Ar/CO

text: been

text: plotted

text: in

text: Fig.

text: after

text: normalising

text: to

text: the

text: mean

text: value.

text: The

text: gain

text: uniformity

text: result

text: is

text: in

text: good

text: agreement

text: with

text: literature

text: The

text: energy

text: resolutions

text: obtained

text: for

text: each

text: of

text: the

text: zones

text: at

text: V

text: is

text: shown

text: in

text: Fig.

text: The

text: HV

text: (V)

text: mean

text: value

text: of

text: the

text: energy

text: resolution

text: (FWHM)

text: is

text: with

text: a

text: RMS

text: of

text: Summary

text: and

text: outlook

text: Figure

text: (top)

text: Time

text: spectrum

text: at

text: V

text: and

text: (bottom)

text: time

text: resolution

text: as

text: a

text: function

text: of

text: applied

text: HV.

text: A

text: detailed

text: study

text: of

text: a

text: triple

text: GEM

text: detector

text: ﬁlled

text: with

text: a

text: gas

text: mixture

text: of

text: Ar+CO

text: of

text: and

text: The

text: time

text: spectrum

text: and

text: the

text: plots

text: for

text: time

text: resolution

text: proportions

text: has

text: been

text: performed.

text: Tests

text: were

text: conducted

text: stnuoC

text: )sn(

text: σ

text: t

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text: Rotational subsets of the circle Dr. J. Ramanathan Department of Mathematics Eastern Michigan University Ypsilanti MI 48197 Abstract A rotational subset, relative to a continuous transformation : T → T on T = R Z, is a closed, T / invariant subset of T that is minimal and on which respects the standard orientation of the unit circle. T In the case where is the standard angle doubling map, such subsets were studied by Bullet and Sentenac. T The case where multiplies angles by an integer 2 was studied by Goldberg and Tresser, and Blokh, T d > Malaugh, Mayer, Oversteegen, and Parris. These authors prove that inﬁnite rotational subsets arise as extensions of irrational rotations of the unit circle. This paper studies the extent to which such results hold for general continuous maps of the circle. In particular, we prove the structure theorem mentioned above holds for the wider class of continuous transformations with ﬁnite ﬁbers. Our methods are T more analytic in nature than the works mentioned. The paper concludes with a construction of inﬁnite rotational sets for a class of continuous maps that includes examples that are not equivalent to the model cases treated previously. 2010 MSC Subject Clasiﬁcation: 37B05 Supported in part by a Faculty Research Fellowship from Eastern Michigan University (Fall 2016). Email: jramanath@emich.edu ORCID: 0000-0001-8560-1160

text: Rotational

label: title

text: subsets

text: of

text: the

text: circle

text: Dr.

label: author

text: J.

text: Ramanathan

text: Department

text: of

text: Mathematics

text: Eastern

text: Michigan

text: University

text: Ypsilanti

text: MI

text: Abstract

text: A

label: abstract

text: rotational

text: subset,

text: relative

text: to

text: a

text: continuous

text: transformation

text: T

text: T

text: on

text: T

text: R

text: Z,

text: is

text: a

text: closed,

text: T

text: invariant

text: subset

text: of

text: T

text: that

text: is

text: minimal

text: and

text: on

text: which

text: respects

text: the

text: standard

text: orientation

text: of

text: the

text: unit

text: circle.

text: T

text: In

text: the

text: case

text: where

text: is

text: the

text: standard

text: angle

text: doubling

text: map,

text: such

text: subsets

text: were

text: studied

text: by

text: Bullet

text: and

text: Sentenac.

text: T

text: The

text: case

text: where

text: multiplies

text: angles

text: by

text: an

text: integer

text: was

text: studied

text: by

text: Goldberg

text: and

text: Tresser,

text: and

text: Blokh,

text: T

text: d

text: Malaugh,

text: Mayer,

text: Oversteegen,

text: and

text: Parris.

text: These

text: authors

text: prove

text: that

text: inﬁnite

text: rotational

text: subsets

text: arise

text: as

text: extensions

text: of

text: irrational

text: rotations

text: of

text: the

text: unit

text: circle.

text: This

text: paper

text: studies

text: the

text: extent

text: to

text: which

text: such

text: results

text: hold

text: for

text: general

text: continuous

text: maps

text: of

text: the

text: circle.

text: In

text: particular,

text: we

text: prove

text: the

text: structure

text: theorem

text: mentioned

text: above

text: holds

text: for

text: the

text: wider

text: class

text: of

text: continuous

text: transformations

text: with

text: ﬁnite

text: ﬁbers.

text: Our

text: methods

text: are

text: T

text: more

text: analytic

text: in

text: nature

text: than

text: the

text: works

text: mentioned.

text: The

text: paper

text: concludes

text: with

text: a

text: construction

text: of

text: inﬁnite

text: rotational

text: sets

text: for

text: a

text: class

text: of

text: continuous

text: maps

text: that

text: includes

text: examples

text: that

text: are

text: not

text: equivalent

text: to

text: the

text: model

text: cases

text: treated

text: previously.

text: MSC

text: Subject

text: Clasiﬁcation:

text: Supported

text: in

text: part

text: by

text: a

text: Faculty

text: Research

text: Fellowship

text: from

text: Eastern

text: Michigan

text: University

text: (Fall

text: Email:

text: jramanath@emich.edu

text: ORCID:


text: Remark 17. i. The are half-open intervals that are closed on the left. A I ⊂ ii. If the ﬁnite word is a preﬁx for the word , then . I J A A J I ≥ | | iii. For any ﬁxed and , the collection I n I { | | } : = and is a preﬁx for A J n I J J is a partition of . A I iv. If = and 0 = , then I i1i2...in I i2...in ∈ ⇒ ∈ = x A Tx A 0. I I v. If precedes in lexicographical order, then I J ∈ ∈ ⇒ and = x A y A x < y. I J Each point ∈ T determines a unique inﬁnite word ( ) ∈ AN. Since, by construction, x ι x ◦ ◦ = (3) ι T S ι we have that AN is a Borel measurable factor of (T ). , T Remark 18. The lexicographical ordering on AN and the usual order on [0 1[ are also compatible with the , ∈ factor map . In particular, for any [0 1[, ι x, y , ≤ i. implies ( ) ( ), and x < y ι x ι y ii. ( ) ( ) implies . ι x < ι y x < y Lemma 19. If ∈ AN doesn’t terminate with an inﬁnite sequence of − 1’s, there is an ∈ [0 1[ with the I d x , property that ( ) = . ι x I Proof. Let = and set = for eac\h ∈ N. It is enough to show that I i1i2.... In i1...in n ∞ 6 ∅ = A . In n=1 ⊂ The hypothesis entails that for any there is an with the property that . This in turn I \ m >\n A A n Im In means that ∞ ∞ = A A . In In n=1 n=1 The claim follows since a decreasing sequence of bounded, closed intervals must have a non-empty intersec- tion. Lemma 20. Let ∈ T be a point with the property that ( ) doesn’t terminate with an inﬁnite sequence x0 ι x0 of 0’s. Then is a point of continuity for the map : T → AN. x0 ι 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Remark

text: i.

label: list

text: The

text: are

text: half-open

text: intervals

text: that

text: are

text: closed

text: on

text: the

text: left.

text: A

text: I

text: ii.

text: If

text: the

text: ﬁnite

text: word

text: is

text: a

text: preﬁx

text: for

text: the

text: word

text: then

text: I

text: J

text: A

text: A

text: J

text: I

text: iii.

text: For

text: any

text: ﬁxed

text: and

text: the

text: collection

text: I

text: n

text: I

text: and

text: is

text: a

text: preﬁx

text: for

text: A

text: J

text: n

text: I

text: J

text: J

text: is

text: a

text: partition

text: of

text: A

text: I

text: iv.

text: If

text: and

text: then

text: I

text: I

text: x

text: A

text: Tx

text: A

text: I

text: I

text: v.

text: If

text: precedes

text: in

text: lexicographical

text: order,

text: then

text: I

text: J

text: and

text: x

text: A

text: y

text: A

text: x

text: y.

text: I

text: J

text: Each

text: point

text: T

text: determines

text: a

text: unique

text: inﬁnite

text: word

text: AN.

text: Since,

text: by

text: construction,

text: x

text: ι

text: x

text: ι

text: T

text: S

text: ι

text: we

text: have

text: that

text: AN

text: is

text: a

text: Borel

text: measurable

text: factor

text: of

text: (T

text: T

text: Remark

text: The

text: lexicographical

text: ordering

text: on

text: AN

text: and

text: the

text: usual

text: order

text: on

text: are

text: also

text: compatible

text: with

text: the

text: factor

text: map

text: In

text: particular,

text: for

text: any

text: ι

text: x,

text: y

text: i.

text: implies

text: and

text: x

text: y

text: ι

text: x

text: ι

text: y

text: ii.

text: implies

text: ι

text: x

text: ι

text: y

text: x

text: y

text: Lemma

text: If

text: AN

text: doesn’t

text: terminate

text: with

text: an

text: inﬁnite

text: sequence

text: of

text: there

text: is

text: an

text: with

text: the

text: I

text: d

text: x

text: property

text: that

text: ι

text: x

text: I

text: Proof.

text: Let

text: and

text: set

text: for

text: eac\h

text: N.

text: It

text: is

text: enough

text: to

text: show

text: that

text: I

text: In

text: n

text: A

text: In

text: The

text: hypothesis

text: entails

text: that

text: for

text: any

text: there

text: is

text: an

text: with

text: the

text: property

text: that

text: This

text: in

text: turn

text: I

text: m

text: >\n

text: A

text: A

text: n

text: Im

text: In

text: means

text: that

text: A

text: A

text: In

text: In

text: The

text: claim

text: follows

text: since

text: a

text: decreasing

text: sequence

text: of

text: bounded,

text: closed

text: intervals

text: must

text: have

text: a

text: non-empty

text: intersec-

text: tion.

text: Lemma

text: Let

text: T

text: be

text: a

text: point

text: with

text: the

text: property

text: that

text: doesn’t

text: terminate

text: with

text: an

text: inﬁnite

text: sequence

text: ι

text: of

text: Then

text: is

text: a

text: point

text: of

text: continuity

text: for

text: the

text: map

text: T

text: AN.

text: ι

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Proof. Put = ([ 0 ]). If ∈ ∩ [ 0], then the invariance of the measure implies that θ0 µ β , β x X α, α T µ Φ( ) = ([ ]) Tx µ α, Tx = ([ ]) + ([ ) µ α, Tβ µ Tα, Tx = ([ 0 ]) + ([ ]) µ Tβ , Tβ µ Tα, Tx = + ([ ]) θ0 µ α, x = + Φ( ) θ0 x . If ∈ ∩ [ 0 ], then x X β , β Φ( ) = ([ ]) Tx µ α, Tx = ([ 0 ]) µ Tβ , Tx = ([ 0 ]) µ β , x = Φ( ) − ([ 0]) x µ α, α = Φ( ) + ([ 0 ]) − 1 x µ β , β = Φ( ) + mod 1 x θ0 . Moreover, is irrational. Otherwise, any ﬁnite (closed) orbit in T under translation by would have a θ0 θ0 non-dense preimage under Φ that is invariant. T The second statement follows directly from the deﬁnition of Φ. If we deﬁne : → T by ( ) = exp(2 Φ( )) and : T → T by ( ) = exp(2 ) , the above φ X φ x πı x τ τ z πıθ0 z considerations show that is a factor map from ( ) to (T ). φ X, T , τ Lemma 12. Let be those points = with the property that for every 0, the intervals ( + ) X0 x X δ > x, x δ − and ( ) contain inﬁnitely many points of . Then: is a dense, uncountable subset of . x, x δ X X0 X Proof. A point is in the complement of in precisely when it is the endpoint of a gap, i.e. a maximal X0 X open interval contained in T\ . Since there are at most countably many such open intervals, we conclude X \ that is countable. As the perfect set is uncountable, is uncountable as well. Furthermore, each X X0 X X0 \ \ point of is nowhere dense. Therefore, is nowhere dense. In other words, is dense. X X0 X X0 X0 Theorem 13. The dynamical system ( ) is uniquely ergodic. X, T ∈ Proof. Let . By proposition 10 and lemma 12, x0 X0 • ∈ ⇒ and Φ( ) Φ( ), and x X x < x0 x < x0 • ∈ ⇒ and Φ( ) Φ( ). x X x > x0 x > x0 ∈ ≥ Therefore, for any and 0, y X n ≤ ⇔ { } ≤ Tny x0 Φ(y) + nθ0 Φ(x0). 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: Put

text: If

text: then

text: the

text: invariance

text: of

text: the

text: measure

text: implies

text: that

text: θ0

text: µ

text: β

text: β

text: x

text: X

text: α,

text: α

text: T

text: µ

text: Φ(

text: Tx

text: µ

text: α,

text: Tx

text: µ

text: α,

text: Tβ

text: µ

text: Tα,

text: Tx

text: µ

text: Tβ

text: Tβ

text: µ

text: Tα,

text: Tx

text: θ0

text: µ

text: α,

text: x

text: Φ(

text: θ0

text: x

text: If

text: then

text: x

text: X

text: β

text: β

text: Φ(

text: Tx

text: µ

text: α,

text: Tx

text: µ

text: Tβ

text: Tx

text: µ

text: β

text: x

text: Φ(

text: x

text: µ

text: α,

text: α

text: Φ(

text: x

text: µ

text: β

text: β

text: Φ(

text: mod

text: x

text: θ0

text: Moreover,

text: is

text: irrational.

text: Otherwise,

text: any

text: ﬁnite

text: (closed)

text: orbit

text: in

text: T

text: under

text: translation

text: by

text: would

text: have

text: a

text: θ0

text: θ0

text: non-dense

text: preimage

text: under

text: Φ

text: that

text: is

text: invariant.

text: T

text: The

text: second

text: statement

text: follows

text: directly

text: from

text: the

text: deﬁnition

text: of

text: Φ.

text: If

text: we

text: deﬁne

text: T

text: by

text: Φ(

text: and

text: T

text: T

text: by

text: the

text: above

text: φ

text: X

text: φ

text: x

text: πı

text: x

text: τ

text: τ

text: z

text: πıθ0

text: z

text: considerations

text: show

text: that

text: is

text: a

text: factor

text: map

text: from

text: to

text: (T

text: φ

text: X,

text: T

text: τ

text: Lemma

text: Let

text: be

text: those

text: points

text: with

text: the

text: property

text: that

text: for

text: every

text: the

text: intervals

text: x

text: X

text: δ

text: x,

text: x

text: δ

text: and

text: contain

text: inﬁnitely

text: many

text: points

text: of

text: Then:

text: is

text: a

text: dense,

text: uncountable

text: subset

text: of

text: x,

text: x

text: δ

text: X

text: X

text: Proof.

text: A

text: point

text: is

text: in

text: the

text: complement

text: of

text: in

text: precisely

text: when

text: it

text: is

text: the

text: endpoint

text: of

text: a

text: gap,

text: i.e.

text: a

text: maximal

text: X

text: open

text: interval

text: contained

text: in

text: T\

text: Since

text: there

text: are

text: at

text: most

text: countably

text: many

text: such

text: open

text: intervals,

text: we

text: conclude

text: X

text: that

text: is

text: countable.

text: As

text: the

text: perfect

text: set

text: is

text: uncountable,

text: is

text: uncountable

text: as

text: well.

text: Furthermore,

text: each

text: X

text: X

text: point

text: of

text: is

text: nowhere

text: dense.

text: Therefore,

text: is

text: nowhere

text: dense.

text: In

text: other

text: words,

text: is

text: dense.

text: X

text: X

text: Theorem

text: The

text: dynamical

text: system

text: is

text: uniquely

text: ergodic.

text: X,

text: T

text: Proof.

text: Let

text: By

text: proposition

text: and

text: lemma

text: and

text: Φ(

text: Φ(

text: and

text: x

text: X

text: x

text: x

text: and

text: Φ(

text: Φ(

text: x

text: X

text: x

text: x

text: Therefore,

text: for

text: any

text: and

text: y

text: X

text: n

text: Tny

text: Φ(y)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 11 µ ν νβ−1 PAM µ=ν −δ (M =N/d) PCD µ=(1−βδ)/(β−1) ν −1 δ 0 1−ν(β−1) 1/β 1 − Fig. 5. The scheme among PCD and PAM that performs better than the other when β > 1 and ν < 1/(β 1), in terms of polynomial scaling in K. Here N = Kν, d = Kδ, and M = Kµ. As with the β ∈ [0, 1) case, we notice that PCD is the better choice when M is small, while PAM is the better choice when M is large. In fact, by comparing the rate expressions in Theorems 6 and 7 using a poly-K analysis, we obtain the following theorem describing the regimes for which either of PCD or PAM is superior to the other. The theorem is illustrated in Fig. 5 and proved at the end of this subsection. Theorem 8. When , and considering only a polynomial scaling of the parameters with β > 1 , PCD outperforms PAM in the regime K µ ≤ min {ν − δ, (1 − βδ)/(β − 1)} , while PAM outperforms PCD in the opposite regime, where N = Kν, d = Kδ, and M = Kµ. When comparing Theorems 3 and 8, we notice that the case β > 1 has the added constraint µ < (1 − βδ)/(β − 1) for the regime where PCD is superior to PAM, indicating that there are values of d for which PAM is better than PCD for a larger memory regime under β > 1 as compared to β ∈ [0, 1). This is represented in Fig. 5 by the additional line segment joining points (1 − ν(β − 1), νβ − 1) and (1/β, 0). As β approaches one from above, this line segment tends toward the segment joining points (1, ν − 1) and (1, 0). With it, the regime in which PCD is better than PAM grows until it becomes exactly the regime shown in Fig. 3 for β ∈ [0, 1). In other words, when β > 1 and as β → 1+, the regimes in which PCD or PAM are respectively the better choice become the same regimes as in the β ∈ [0, 1) case. This seemingly continuous transition suggests that, when β = 1, the system should behave similarly to β ∈ [0, 1), i.e., ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: µ

font: default

text: ν

text: νβ−1

text: PAM

text: µ=ν

text: −δ

text: (M

text: =N/d)

text: PCD

text: ν

text: −1

text: δ

text: 1−ν(β−1)

text: Fig.

text: The

text: scheme

text: among

text: PCD

text: and

text: PAM

text: that

text: performs

text: better

text: than

text: the

text: other

text: when

text: β

text: and

text: ν

text: in

text: terms

text: of

text: polynomial

text: scaling

text: in

text: K.

text: Here

text: N

text: Kν,

text: d

text: Kδ,

text: and

text: M

text: Kµ.

text: As

text: with

text: the

text: β

text: case,

text: we

text: notice

text: that

text: PCD

text: is

text: the

text: better

text: choice

text: when

text: M

text: is

text: small,

text: while

text: PAM

text: is

text: the

text: better

text: choice

text: when

text: M

text: is

text: large.

text: In

text: fact,

text: by

text: comparing

text: the

text: rate

text: expressions

text: in

text: Theorems

text: and

text: using

text: a

text: poly-K

text: analysis,

text: we

text: obtain

text: the

text: following

text: theorem

text: describing

text: the

text: regimes

text: for

text: which

text: either

text: of

text: PCD

text: or

text: PAM

text: is

text: superior

text: to

text: the

text: other.

text: The

text: theorem

text: is

text: illustrated

text: in

text: Fig.

text: and

text: proved

text: at

text: the

text: end

text: of

text: this

text: subsection.

text: Theorem

text: When

text: and

text: considering

text: only

text: a

text: polynomial

text: scaling

text: of

text: the

text: parameters

text: with

text: β

text: PCD

text: outperforms

text: PAM

text: in

text: the

text: regime

text: K

text: µ

text: min

text: {ν

text: δ,

text: βδ)/(β

text: while

text: PAM

text: outperforms

text: PCD

text: in

text: the

text: opposite

text: regime,

text: where

text: N

text: Kν,

text: d

text: Kδ,

text: and

text: M

text: Kµ.

text: When

text: comparing

text: Theorems

text: and

text: we

text: notice

text: that

text: the

text: case

text: β

text: has

text: the

text: added

text: constraint

text: µ

text: βδ)/(β

text: for

text: the

text: regime

text: where

text: PCD

text: is

text: superior

text: to

text: PAM,

text: indicating

text: that

text: there

text: are

text: values

text: of

text: d

text: for

text: which

text: PAM

text: is

text: better

text: than

text: PCD

text: for

text: a

text: larger

text: memory

text: regime

text: under

text: β

text: as

text: compared

text: to

text: β

text: This

text: is

text: represented

text: in

text: Fig.

text: by

text: the

text: additional

text: line

text: segment

text: joining

text: points

text: ν(β

text: νβ

text: and

text: As

text: β

text: approaches

text: one

text: from

text: above,

text: this

text: line

text: segment

text: tends

text: toward

text: the

text: segment

text: joining

text: points

text: ν

text: and

text: With

text: it,

text: the

text: regime

text: in

text: which

text: PCD

text: is

text: better

text: than

text: PAM

text: grows

text: until

text: it

text: becomes

text: exactly

text: the

text: regime

text: shown

text: in

text: Fig.

text: for

text: β

text: In

text: other

text: words,

text: when

text: β

text: and

text: as

text: β

text: the

text: regimes

text: in

text: which

text: PCD

text: or

text: PAM

text: are

text: respectively

text: the

text: better

text: choice

text: become

text: the

text: same

text: regimes

text: as

text: in

text: the

text: β

text: case.

text: This

text: seemingly

text: continuous

text: transition

text: suggests

text: that,

text: when

text: β

text: the

text: system

text: should

text: behave

text: similarly

text: to

text: β

text: i.e.,

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 − Note that the function x (cid:55)→ xe1 x is strictly increasing for x ∈ (0, 1). Since ρ ∈ (0, 1/2), we (cid:0) (cid:1) (cid:0) (cid:1) thus get − − log ρe1 ρ < log 2ρe1 2ρ = −α < log(1) = 0. (cid:8) (cid:0) (cid:1)(cid:9) Applying this to (3), we obtain E 1 − (a) 1 1 − [U0] ≤ √ exp log K + d log 2ρe1 2ρ ≤ √ exp {log K − (1 + t ) log K} = √ K t0, 0 2π 2π 2π where (a) uses (1). This concludes the proof. APPENDIX B DETAILS OF PAM SCHEME FOR β ∈ [0, 1) (PROOF OF THEOREM 2) First, note that it is always possible to unicast from the server to each user the ﬁle that it requested. Since the expected number of users is ρK, we always have R¯PAM ≤ ρK. In what follows, we focus on the regime M = Ω(N/d). Recall that the number of requests for ﬁle n at cluster c is u (c), a Poisson variable with parameter ρdp . n n In the placement phase, we perform a proportional placement. Speciﬁcally, since each cache can store M ﬁles and each cluster consists of d caches, we replicate each ﬁle W on d = p dM n n n caches per cluster. In the matching phase, we ﬁrst construct a fractional matching of users to caches, and then show that this implies the existence of an integral matching. We construct the fractional matching by dividing each ﬁle W into d equal parts, and then mapping each request for ﬁle W to d n n n n requests, one for each of its parts. Each user now connects to the d caches containing ﬁle W n n and retrieves one part from each cache. This leads to a fractional matching where the total data (cid:88) served by a cache k in cluster c is less than one ﬁle if u (c) n ≤ 1, (4) ∈W dn Wn k where W is the set of ﬁles stored on cache k. Let h(x) = x log x + 1 − x be the Crame´r k transform of a unit Poisson random variable. Using the Chernoff bound and the arguments used (cid:40) (cid:41) (cid:88) in the proof of [11, Proposition 1], we have that u (c) − Pr n > 1 ≤ e zdM/N, (5) ∈W dn Wn k where z = (1 − β)ρh(1 + (1 − ρ)/2ρ) > 0. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Note

text: that

text: the

text: function

text: x

text: x

text: is

text: strictly

text: increasing

text: for

text: x

text: Since

text: ρ

text: we

text: thus

text: get

text: log

text: ρ

text: log

text: 2ρ

text: −α

text: Applying

text: this

text: to

text: we

text: obtain

text: E

text: (a)

text: exp

text: log

text: K

text: d

text: log

text: 2ρ

text: exp

text: {log

text: K

text: t

text: log

text: K}

text: K

text: 2π

text: 2π

text: 2π

text: where

text: (a)

text: uses

text: This

text: concludes

text: the

text: proof.

text: APPENDIX

text: B

text: DETAILS

text: OF

text: PAM

text: SCHEME

text: FOR

text: β

text: (PROOF

text: OF

text: THEOREM

text: First,

text: note

text: that

text: it

text: is

text: always

text: possible

text: to

text: unicast

text: from

text: the

text: server

text: to

text: each

text: user

text: the

text: ﬁle

text: that

text: it

text: requested.

text: Since

text: the

text: expected

text: number

text: of

text: users

text: is

text: ρK,

text: we

text: always

text: have

text: R¯PAM

text: ρK.

text: In

text: what

text: follows,

text: we

text: focus

text: on

text: the

text: regime

text: M

text: Ω(N/d).

text: Recall

text: that

text: the

text: number

text: of

text: requests

text: for

text: ﬁle

text: n

text: at

text: cluster

text: c

text: is

text: u

text: (c),

text: a

text: Poisson

text: variable

text: with

text: parameter

text: ρdp

text: n

text: n

text: In

text: the

text: placement

text: phase,

text: we

text: perform

text: a

text: proportional

text: placement.

text: Speciﬁcally,

text: since

text: each

text: cache

text: can

text: store

text: M

text: ﬁles

text: and

text: each

text: cluster

text: consists

text: of

text: d

text: caches,

text: we

text: replicate

text: each

text: ﬁle

text: W

text: on

text: d

text: p

text: dM

text: n

text: n

text: n

text: caches

text: per

text: cluster.

text: In

text: the

text: matching

text: phase,

text: we

text: ﬁrst

text: construct

text: a

text: fractional

text: matching

text: of

text: users

text: to

text: caches,

text: and

text: then

text: show

text: that

text: this

text: implies

text: the

text: existence

text: of

text: an

text: integral

text: matching.

text: We

text: construct

text: the

text: fractional

text: matching

text: by

text: dividing

text: each

text: ﬁle

text: W

text: into

text: d

text: equal

text: parts,

text: and

text: then

text: mapping

text: each

text: request

text: for

text: ﬁle

text: W

text: to

text: d

text: n

text: n

text: n

text: n

text: requests,

text: one

text: for

text: each

text: of

text: its

text: parts.

text: Each

text: user

text: now

text: connects

text: to

text: the

text: d

text: caches

text: containing

text: ﬁle

text: W

text: n

text: n

text: and

text: retrieves

text: one

text: part

text: from

text: each

text: cache.

text: This

text: leads

text: to

text: a

text: fractional

text: matching

text: where

text: the

text: total

text: data

text: served

text: by

text: a

text: cache

text: k

text: in

text: cluster

text: c

text: is

text: less

text: than

text: one

text: ﬁle

text: if

text: u

text: (c)

text: n

text: ∈W

text: dn

text: Wn

text: k

text: where

text: W

text: is

text: the

text: set

text: of

text: ﬁles

text: stored

text: on

text: cache

text: k.

text: Let

text: h(x)

text: x

text: log

text: x

text: x

text: be

text: the

text: Crame´r

text: k

text: transform

text: of

text: a

text: unit

text: Poisson

text: random

text: variable.

text: Using

text: the

text: Chernoff

text: bound

text: and

text: the

text: arguments

text: used

text: in

text: the

text: proof

text: of

text: Proposition

text: we

text: have

text: that

text: u

text: (c)

text: Pr

text: n

text: e

text: zdM/N,

text: ∈W

text: dn

text: Wn

text: k

text: where

text: z

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 15 To ﬁnd a matching between the set of requests and the caches, we serve all requests for ﬁles that are stored on caches for which (4) is violated via the server. For the remaining ﬁles, there exists a fractional matching between the set of requests and the caches such that each request is allocated only to caches in the corresponding cluster, and the total data served by each cache is not more than one unit. By the total unimodularity of adjacency matrix, the existence of a fractional matching implies the existence of an integral matching [14]. We use the Hungarian algorithm to ﬁnd a matching between the remaining requests and the caches in the corresponding cluster. Let τ be the probability that at least one of the caches storing ﬁle W does not satisfy (4). n n (cid:88) − By the union bound, it follows that τ ≤ (KM/N)e zdM/N. By deﬁnition, n N − R¯PAM ≤ τ ≤ KMe zdM/N, (6) n n=1 which concludes the proof of the theorem. While the above was enough to prove the theorem, we will next provide an additional upper bound on the PAM rate, thus obtaining a tighter expression. (cid:84) Let G be the event that the total number of requests at cluster c is less than d, and let c G = G . Using the Chernoff bound, we have c c E K − [G] ≥ 1 − e zd, d where z is as deﬁned above. Conditioned on G , the number of ﬁles that need to be fetched from c the server to serve all requests in the cluster is at most d. The rest of this proof is conditioned on G. Let E be the event that all caches in cluster c satisfy (4). Using (5) and the union bound, we c have that − Pr{E |G} ≥ Pr{E } ≥ 1 − de zdM/N, c c where z is as deﬁned above. Conditioned on E , all the requests in cluster c can be served by c (cid:88) (cid:8) (cid:12) (cid:9) the caches. Therefore, (cid:12) K/d E − [R|G] ≤ d · Pr E G ≤ Kde zdM/N, c c=1 where A denotes the complement of A for any event A. It follows that E E E E − NK − [R] = [R|G] Pr{G} + [R|G] Pr{G} ≤ [R|G] + N Pr{G} ≤ Kde zdM/N + e zd. (7) d ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: To

text: ﬁnd

text: a

text: matching

text: between

text: the

text: set

text: of

text: requests

text: and

text: the

text: caches,

text: we

text: serve

text: all

text: requests

text: for

text: ﬁles

text: that

text: are

text: stored

text: on

text: caches

text: for

text: which

text: is

text: violated

text: via

text: the

text: server.

text: For

text: the

text: remaining

text: ﬁles,

text: there

text: exists

text: a

text: fractional

text: matching

text: between

text: the

text: set

text: of

text: requests

text: and

text: the

text: caches

text: such

text: that

text: each

text: request

text: is

text: allocated

text: only

text: to

text: caches

text: in

text: the

text: corresponding

text: cluster,

text: and

text: the

text: total

text: data

text: served

text: by

text: each

text: cache

text: is

text: not

text: more

text: than

text: one

text: unit.

text: By

text: the

text: total

text: unimodularity

text: of

text: adjacency

text: matrix,

text: the

text: existence

text: of

text: a

text: fractional

text: matching

text: implies

text: the

text: existence

text: of

text: an

text: integral

text: matching

text: We

text: use

text: the

text: Hungarian

text: algorithm

text: to

text: ﬁnd

text: a

text: matching

text: between

text: the

text: remaining

text: requests

text: and

text: the

text: caches

text: in

text: the

text: corresponding

text: cluster.

text: Let

text: τ

text: be

text: the

text: probability

text: that

text: at

text: least

text: one

text: of

text: the

text: caches

text: storing

text: ﬁle

text: W

text: does

text: not

text: satisfy

text: n

text: n

text: By

text: the

text: union

text: bound,

text: it

text: follows

text: that

text: τ

text: (KM/N)e

text: zdM/N.

text: By

text: deﬁnition,

text: n

text: N

text: R¯PAM

text: τ

text: KMe

text: zdM/N,

text: n

text: which

text: concludes

text: the

text: proof

text: of

text: the

text: theorem.

text: While

text: the

text: above

text: was

text: enough

text: to

text: prove

text: the

text: theorem,

text: we

text: will

text: next

text: provide

text: an

text: additional

text: upper

text: bound

text: on

text: the

text: PAM

text: rate,

text: thus

text: obtaining

text: a

text: tighter

text: expression.

text: Let

text: G

text: be

text: the

text: event

text: that

text: the

text: total

text: number

text: of

text: requests

text: at

text: cluster

text: c

text: is

text: less

text: than

text: d,

text: and

text: let

text: c

text: G

text: G

text: Using

text: the

text: Chernoff

text: bound,

text: we

text: have

text: c

text: c

text: E

text: K

text: [G]

text: e

text: zd,

text: d

text: where

text: z

text: is

text: as

text: deﬁned

text: above.

text: Conditioned

text: on

text: G

text: the

text: number

text: of

text: ﬁles

text: that

text: need

text: to

text: be

text: fetched

text: from

text: c

text: the

text: server

text: to

text: serve

text: all

text: requests

text: in

text: the

text: cluster

text: is

text: at

text: most

text: d.

text: The

text: rest

text: of

text: this

text: proof

text: is

text: conditioned

text: on

text: G.

text: Let

text: E

text: be

text: the

text: event

text: that

text: all

text: caches

text: in

text: cluster

text: c

text: satisfy

text: Using

text: and

text: the

text: union

text: bound,

text: we

text: c

text: have

text: that

text: Pr{E

text: |G}

text: Pr{E

text: de

text: zdM/N,

text: c

text: c

text: where

text: z

text: is

text: as

text: deﬁned

text: above.

text: Conditioned

text: on

text: E

text: all

text: the

text: requests

text: in

text: cluster

text: c

text: can

text: be

text: served

text: by

text: c

text: the

text: caches.

text: Therefore,

text: K/d

text: E

text: [R|G]

text: d

text: Pr

text: E

text: G

text: Kde

text: zdM/N,

text: c

text: where

text: A

text: denotes

text: the

text: complement

text: of

text: A

text: for

text: any

text: event

text: A.

text: It

text: follows

text: that

text: E

text: E

text: E

text: E

text: NK

text: [R]

text: [R|G]

text: Pr{G}

text: [R|G]

text: Pr{G}

text: [R|G]

text: N

text: Pr{G}

text: Kde

text: zdM/N

text: e

text: zd.

text: d

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 (cid:26) (cid:27) Using (6) and (7), we obtain − − NK − R¯PAM ≤ min KMe zdM/N , Kde zdM/N + e zd , d which is a tighter bound on the PAM rate and implies Theorem 2. APPENDIX C DETAILS OF PAM SCHEME FOR β > 1 (PROOF OF THEOREM 7) At a high level, the PAM strategy consists in storing complete ﬁles in the caches, replicating the ﬁles across different caches, and then matching the users to the cache that contains their requested ﬁle. Users that cannot be matched to a cache containing their ﬁle are served directly from the server. The above describes PAM strategies very generally; there are many possible schemes for placement and matching within this class of strategies. In this paper, we adopt for β > 1 a strategy that performs a knapsack storage (KS) placement phase that is based on the knapsack problem, and a match least popular (MLP) matching phase in which matching is done for the least popular ﬁles ﬁrst. We refer to this PAM scheme as KS+MLP. A. Excess Users Whatever the strategy, if there are more than d users at a particular cluster, then the excess users must be unmatched. From Lemma 2, we know that the expected total number of these − excess users across all caches is O(K t0), which is o(1). For the proof of Theorem 7, we will generally use asymptotic notation for the expected rate. Thus by always serving these users directly from the server, their contribution to the rate is always o(1). For this reason, we will make the simplifying assumption in what follows that there are no more than d users at each cluster. B. Placement Phase: Knapsack Storage We split the KS policy into two parts. In the ﬁrst part, we determine how many copies of each ﬁle will be stored per cluster. In the second part, we determine which caches in each cluster will store each ﬁle. ##LTLine##

text: Using

text: and

text: we

text: obtain

text: NK

text: R¯PAM

text: min

text: KMe

text: zdM/N

text: Kde

text: zdM/N

text: e

text: zd

text: d

text: which

text: is

text: a

text: tighter

text: bound

text: on

text: the

text: PAM

text: rate

text: and

text: implies

text: Theorem

text: APPENDIX

text: C

text: DETAILS

text: OF

text: PAM

text: SCHEME

text: FOR

text: β

text: (PROOF

text: OF

text: THEOREM

text: At

text: a

text: high

text: level,

text: the

text: PAM

text: strategy

text: consists

text: in

text: storing

text: complete

text: ﬁles

text: in

text: the

text: caches,

text: replicating

text: the

text: ﬁles

text: across

text: different

text: caches,

text: and

text: then

text: matching

text: the

text: users

text: to

text: the

text: cache

text: that

text: contains

text: their

text: requested

text: ﬁle.

text: Users

text: that

text: cannot

text: be

text: matched

text: to

text: a

text: cache

text: containing

text: their

text: ﬁle

text: are

text: served

text: directly

text: from

text: the

text: server.

text: The

text: above

text: describes

text: PAM

text: strategies

text: very

text: generally;

text: there

text: are

text: many

text: possible

text: schemes

text: for

text: placement

text: and

text: matching

text: within

text: this

text: class

text: of

text: strategies.

text: In

text: this

text: paper,

text: we

text: adopt

text: for

text: β

text: a

text: strategy

text: that

text: performs

text: a

text: knapsack

text: storage

text: (KS)

text: placement

text: phase

text: that

text: is

text: based

text: on

text: the

text: knapsack

text: problem,

text: and

text: a

text: match

text: least

text: popular

text: (MLP)

text: matching

text: phase

text: in

text: which

text: matching

text: is

text: done

text: for

text: the

text: least

text: popular

text: ﬁles

text: ﬁrst.

text: We

text: refer

text: to

text: this

text: PAM

text: scheme

text: as

text: KS+MLP.

text: A.

text: Excess

text: Users

text: Whatever

text: the

text: strategy,

text: if

text: there

text: are

text: more

text: than

text: d

text: users

text: at

text: a

text: particular

text: cluster,

text: then

text: the

text: excess

text: users

text: must

text: be

text: unmatched.

text: From

text: Lemma

text: we

text: know

text: that

text: the

text: expected

text: total

text: number

text: of

text: these

text: excess

text: users

text: across

text: all

text: caches

text: is

text: O(K

text: which

text: is

text: For

text: the

text: proof

text: of

text: Theorem

text: we

text: will

text: generally

text: use

text: asymptotic

text: notation

text: for

text: the

text: expected

text: rate.

text: Thus

text: by

text: always

text: serving

text: these

text: users

text: directly

text: from

text: the

text: server,

text: their

text: contribution

text: to

text: the

text: rate

text: is

text: always

text: For

text: this

text: reason,

text: we

text: will

text: make

text: the

text: simplifying

text: assumption

text: in

text: what

text: follows

text: that

text: there

text: are

text: no

text: more

text: than

text: d

text: users

text: at

text: each

text: cluster.

text: B.

text: Placement

text: Phase:

text: Knapsack

text: Storage

text: We

text: split

text: the

text: KS

text: policy

text: into

text: two

text: parts.

text: In

text: the

text: ﬁrst

text: part,

text: we

text: determine

text: how

text: many

text: copies

text: of

text: each

text: ﬁle

text: will

text: be

text: stored

text: per

text: cluster.

text: In

text: the

text: second

text: part,

text: we

text: determine

text: which

text: caches

text: in

text: each

text: cluster

text: will

text: store

text: each

text: ﬁle.

text: ##LTLine##


text: 20 Lemma 5. Let R = {n : x = 1}, where x is the solution to the fractional knapsack problem n n solved in Appendix C-B1. Let denote the event that, in a given cluster, the MLP policy E 2 matches all requests for all ﬁles in R to caches. Then, − Pr{E } = 1 − Ne Ω((log d)2). 2 Proof: Since the MLP policy matches requests to caches starting from the least popular ﬁles, we ﬁrst focus on requests for ﬁles less popular than ﬁle W . Because the ﬁles follow a N2 Zipf distribution, we can write − p p = p N β = 1 . N2 1 2 d(β+1)/2 Every ﬁle less popular than N is stored at most once across all the caches in the cluster. 2 Therefore, under MLP, a request for a ﬁle n > N will remain unmatched only if the cache 2 (cid:48) storing that ﬁle is matched to another request for another ﬁle n > N . Under the KS placement 2 policy, the cumulative popularity of all ﬁles no more popu(cid:16)lar than ﬁ(cid:17)le N stored on a particular 2 cache is less than − β+1 pN2 + pN2+d + pN2+2d + · · · = O p1d 2 . Each unmatched request for ﬁle n > N corresponds to the event that there are at least two 2 requests for the M ﬁles less popular than W stored on a cache. Therefore, by the Chernoff N2 bound (Lemma 3), the probability that a particular request for a ﬁle n > N remains unmatched 2 − is at most e Ω(d). By the union bound, the probability that at least one request for ﬁle n ∈ R − such that n > N is not matched by the MLP policy is at most de Ω(d). 2 Next, we focus on the ﬁles n ∈ {2, . . . , N }. Note that if the KS policy decides to store ﬁle 2 (cid:88) (cid:88) (cid:108)(cid:16) (cid:17) (cid:109) (cid:88) (cid:6) (cid:7) n, it stores it on w caches. Therefore, n N2 N1 p N2 xnwn ≤ (cid:88) (cid:104)(cid:16)1 + 1(cid:17) dpn + (cid:105) (cid:88)4p1(cid:2)(log d)2 (cid:3) 2 n=2 n=2 n=N1+1 ≤ (cid:16)N1 1 (cid:17)+ p1 dp + 1 + N2 4p (log d)2 + 1 n 1 2 (cid:16)n=2 (cid:17) n=N1+1 p (cid:0) (cid:1) ≤ 1 + 1 d(1 − p ) + 4p (log d)2N + N 1 1 2 2 2 p ≤ 1 + 1 d(1 − p ) + Θ d(1+1/β)/2(log d)2 1 2 ≤ d, ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lemma

text: Let

text: R

text: {n

text: x

text: where

text: x

text: is

text: the

text: solution

text: to

text: the

text: fractional

text: knapsack

text: problem

text: n

text: n

text: solved

text: in

text: Appendix

text: Let

text: denote

text: the

text: event

text: that,

text: in

text: a

text: given

text: cluster,

text: the

text: MLP

text: policy

text: E

text: matches

text: all

text: requests

text: for

text: all

text: ﬁles

text: in

text: R

text: to

text: caches.

text: Then,

text: Pr{E

text: Ne

text: Ω((log

text: Proof:

text: Since

text: the

text: MLP

text: policy

text: matches

text: requests

text: to

text: caches

text: starting

text: from

text: the

text: least

text: popular

text: ﬁles,

text: we

text: ﬁrst

text: focus

text: on

text: requests

text: for

text: ﬁles

text: less

text: popular

text: than

text: ﬁle

text: W

text: Because

text: the

text: ﬁles

text: follow

text: a

text: Zipf

text: distribution,

text: we

text: can

text: write

text: p

text: p

text: p

text: N

text: β

text: Every

text: ﬁle

text: less

text: popular

text: than

text: N

text: is

text: stored

text: at

text: most

text: once

text: across

text: all

text: the

text: caches

text: in

text: the

text: cluster.

text: Therefore,

text: under

text: MLP,

text: a

text: request

text: for

text: a

text: ﬁle

text: n

text: N

text: will

text: remain

text: unmatched

text: only

text: if

text: the

text: cache

text: storing

text: that

text: ﬁle

text: is

text: matched

text: to

text: another

text: request

text: for

text: another

text: ﬁle

text: n

text: N

text: Under

text: the

text: KS

text: placement

text: policy,

text: the

text: cumulative

text: popularity

text: of

text: all

text: ﬁles

text: no

text: more

text: than

text: N

text: stored

text: on

text: a

text: particular

text: cache

text: is

text: less

text: than

text: O

text: Each

text: unmatched

text: request

text: for

text: ﬁle

text: n

text: N

text: corresponds

text: to

text: the

text: event

text: that

text: there

text: are

text: at

text: least

text: two

text: requests

text: for

text: the

text: M

text: ﬁles

text: less

text: popular

text: than

text: W

text: stored

text: on

text: a

text: cache.

text: Therefore,

text: by

text: the

text: Chernoff

text: bound

text: (Lemma

text: the

text: probability

text: that

text: a

text: particular

text: request

text: for

text: a

text: ﬁle

text: n

text: N

text: remains

text: unmatched

text: is

text: at

text: most

text: e

text: Ω(d).

text: By

text: the

text: union

text: bound,

text: the

text: probability

text: that

text: at

text: least

text: one

text: request

text: for

text: ﬁle

text: n

text: R

text: such

text: that

text: n

text: N

text: is

text: not

text: matched

text: by

text: the

text: MLP

text: policy

text: is

text: at

text: most

text: de

text: Ω(d).

text: Next,

text: we

text: focus

text: on

text: the

text: ﬁles

text: n

text: N

text: Note

text: that

text: if

text: the

text: KS

text: policy

text: decides

text: to

text: store

text: ﬁle

text: n,

text: it

text: stores

text: it

text: on

text: w

text: caches.

text: Therefore,

text: n

text: p

text: xnwn

text: dpn

text: dp

text: (log

text: n

text: p

text: p

text: (log

text: N

text: p

text: p

text: Θ

text: d,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 25 B. The General Shallow Zipf Case (β ∈ [0, 1)) First, notice that the popularity of each ﬁle is − − N β (a) (1 − β)N β 1 − β p ≥ p = ≥ − = , (14) n N A N1 β N N (cid:80) where A is deﬁned in Lemma 8 stated below, and (a) follows from the lemma. N − Lemma 8. Let m ≥ 1 be an integer and let β ∈ [0, 1). Deﬁne A = m n β. Then, m n=1 − − m1 β − 1 ≤ (1 − β)A ≤ m1 β. m Lemma 8 is proved in Appendix D-B. Consider now the following relaxed setup. Suppose that, for every ﬁle n, there are u˜ (c) users (cid:18) (cid:19) n requesting ﬁle n from cluster c, where (1 − β)ρd u˜ (c) ∼ Poisson . n N Since (1−β)ρd/N ≤ p ρd for all n by (14), the optimal expected rate for this relaxed setup can n only be smaller than the rate from the original setup. Indeed, we can retrieve the original setup by simply creating Poisson ((p − (1 − β)/N)ρd) additional requests for ﬁle n at each cluster. n Our relaxed setup is now exactly a uniform-popularities setup except that ρ is replaced by (cid:48) ρ = (1 − β)ρ, which is still a constant. Consequently, the information-theoretic lower bounds obtained in Lemma 6, and inequality (10) that follows it, can be directly applied here, giving the following lemma. (cid:26) (cid:27) ∗ Lemma 9. When N ≥ 10, the optimal expected rate R¯ can be lower-bounded by − − ∗ (1 − β)ρ(1 − e 1/2) (1 − e 1/2)N R¯ ≥ min − d, K . 48 M (cid:26) (cid:27) − When M < (1 − e 1/2)N/2d, the bound in Lemma 9 can be further lower-bounded by − ∗ (1 − β)(1 − e 1/2)2ρ N R¯ ≥ · min , ρK . (15) 96 M (cid:26) (cid:27) (cid:26) (cid:27) Furthermore, the rate achieved by PCD is upper-bounded by − N K t0 N R¯PCD ≤ min ρK, − 1 + √ ≤ min ρK, . (16) M 2π M Consequently, combining (15) with (16) gives us the result of Theorem 4. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: B.

text: The

text: General

text: Shallow

text: Zipf

text: Case

text: (β

text: First,

text: notice

text: that

text: the

text: popularity

text: of

text: each

text: ﬁle

text: is

text: N

text: β

text: (a)

text: β)N

text: β

text: β

text: p

text: p

text: n

text: N

text: A

text: β

text: N

text: N

text: where

text: A

text: is

text: deﬁned

text: in

text: Lemma

text: stated

text: below,

text: and

text: (a)

text: follows

text: from

text: the

text: lemma.

text: N

text: Lemma

text: Let

text: m

text: be

text: an

text: integer

text: and

text: let

text: β

text: Deﬁne

text: A

text: m

text: n

text: β.

text: Then,

text: m

text: β

text: β)A

text: β.

text: m

text: Lemma

text: is

text: proved

text: in

text: Appendix

text: D-B.

text: Consider

text: now

text: the

text: following

text: relaxed

text: setup.

text: Suppose

text: that,

text: for

text: every

text: ﬁle

text: n,

text: there

text: are

text: u˜

text: (c)

text: users

text: n

text: requesting

text: ﬁle

text: n

text: from

text: cluster

text: c,

text: where

text: β)ρd

text: u˜

text: (c)

text: Poisson

text: n

text: N

text: Since

text: p

text: ρd

text: for

text: all

text: n

text: by

text: the

text: optimal

text: expected

text: rate

text: for

text: this

text: relaxed

text: setup

text: can

text: n

text: only

text: be

text: smaller

text: than

text: the

text: rate

text: from

text: the

text: original

text: setup.

text: Indeed,

text: we

text: can

text: retrieve

text: the

text: original

text: setup

text: by

text: simply

text: creating

text: Poisson

text: ((p

text: β)/N)ρd)

text: additional

text: requests

text: for

text: ﬁle

text: n

text: at

text: each

text: cluster.

text: n

text: Our

text: relaxed

text: setup

text: is

text: now

text: exactly

text: a

text: uniform-popularities

text: setup

text: except

text: that

text: ρ

text: is

text: replaced

text: by

text: ρ

text: β)ρ,

text: which

text: is

text: still

text: a

text: constant.

text: Consequently,

text: the

text: information-theoretic

text: lower

text: bounds

text: obtained

text: in

text: Lemma

text: and

text: inequality

text: that

text: follows

text: it,

text: can

text: be

text: directly

text: applied

text: here,

text: giving

text: the

text: following

text: lemma.

text: Lemma

text: When

text: N

text: the

text: optimal

text: expected

text: rate

text: R¯

text: can

text: be

text: lower-bounded

text: by

text: e

text: e

text: R¯

text: min

text: d,

text: K

text: M

text: When

text: M

text: e

text: the

text: bound

text: in

text: Lemma

text: can

text: be

text: further

text: lower-bounded

text: by

text: e

text: N

text: R¯

text: min

text: ρK

text: M

text: Furthermore,

text: the

text: rate

text: achieved

text: by

text: PCD

text: is

text: upper-bounded

text: by

text: N

text: K

text: N

text: R¯PCD

text: min

text: ρK,

text: min

text: ρK,

text: M

text: 2π

text: M

text: Consequently,

text: combining

text: with

text: gives

text: us

text: the

text: result

text: of

text: Theorem

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text: ##LTLine##

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text: POWER-SUM DENOMINATORS 11 ≥ ≥ If n < 2, then M = 1, and we are do(cid:22)ne. Now(cid:23) assume that n 2, so that m 3. n As in the proof of Theorem 3, we set − m 1 L d=ef − . p 1 ≥ Case ε . Since m 3, Theorem 3 implies that ε = 0 if and only if m is a power 2 2 of 2. The latter is equivalent to s (m) = 1, as well as to m having only one digit 2 ≥ equal to 1 in its binary expansion. Thus if ε = 1, then we must have s (m) 2, 2 2 and conversely. This shows (22) for p = 2. ≤ ⇒ Case ε for odd p(cid:18) M . “(cid:19) ”: If ε = 1, t(cid:18)he(cid:18)n we ded(cid:19)uc(cid:19)e from (19) that there p ≤n p exists a positive index j L such that (cid:45) m m p − , that is, v − = 0. j(p 1) p j(p 1) Using (21) we then obtain that − − − s (m) = s (j(p 1)) + s (m j(p 1)). (23) p p p ≥ − ≥ − − As j 1, we conclude by (20) that s (j(p 1)) p 1. Since m > j(p 1) by ≤ − − ≥ p j L, we have s (m j(p 1)) 1. Applying these estimates to (23), we ﬁnally ≥p infer that s (m) p. ⇐ p ≥ ≤ “ ”: We now suppose that s (m) p. The bound p M implies that m > p. p n Therefore, in the p-adic expansion · · · m = α + α p + + α pr, (24) 0 1 r ≥ − ≤ ≤ we have r 1 and α > 0. Since m > j(p 1) when 1 j L, the p-adic r expansion − · · · j(p 1) = β + β p + + β pr (25) j,0 j,1 j,r ≥ ≥ has α β 0 and the digits β , . . . , β cannot all be equal to the digits r j,r j,0 j,r α , . . . , α . By Lucas’s theorem [21, pp. 417–420] (for a modern proof, see [12] or 0 r (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) [24, p. 70]), we obtain m ≡ α α · · · α ω d=ef − (cid:0) (cid:1) 0 1 r (mod p), (26) j j(p 1) β β β j,0 j,1 j,r (cid:54)≡ using the convention that α = 0 if α < β. We will deduce that ω 0 (mod p) β (cid:48) j(cid:48) for some index j. To do so, we construct unique digits β , . . . , β , as follows. ≥ ≥ 0 r (Remember that r 1(cid:32)and α 1.) (cid:33) r (cid:88) • (cid:48) − def Set β = α 1. r r • (cid:48) − − r (cid:48) − − def Set β = min α , (p 1) β iteratively for k = r 1, r 2, . . . , 0. k k (cid:96) (cid:96)=k+1 (cid:48) Roughly speaking, the digits β are “ﬁlled up” by the digits α , until the partial (cid:48) · · · (cid:48) − k (cid:48) k sum β + + β reaches p 1; the remaining β are then set equal to zero. k+1 r k ##LTLine##

text: POWER-SUM

text: DENOMINATORS

text: If

text: n

text: then

text: M

text: and

text: we

text: are

text: assume

text: that

text: n

text: so

text: that

text: m

text: n

text: As

text: in

text: the

text: proof

text: of

text: Theorem

text: we

text: set

text: m

text: L

text: d=ef

text: p

text: Case

text: ε

text: Since

text: m

text: Theorem

text: implies

text: that

text: ε

text: if

text: and

text: only

text: if

text: m

text: is

text: a

text: power

text: of

text: The

text: latter

text: is

text: equivalent

text: to

text: s

text: (m)

text: as

text: well

text: as

text: to

text: m

text: having

text: only

text: one

text: digit

text: equal

text: to

text: in

text: its

text: binary

text: expansion.

text: Thus

text: if

text: ε

text: then

text: we

text: must

text: have

text: s

text: (m)

text: and

text: conversely.

text: This

text: shows

text: for

text: p

text: Case

text: ε

text: for

text: odd

text: M

text: If

text: ε

text: we

text: from

text: that

text: there

text: p

text: ≤n

text: p

text: exists

text: a

text: positive

text: index

text: j

text: L

text: such

text: that

text: m

text: m

text: p

text: that

text: is,

text: v

text: j(p

text: p

text: j(p

text: Using

text: we

text: then

text: obtain

text: that

text: s

text: (m)

text: s

text: (j(p

text: s

text: (m

text: j(p

text: p

text: p

text: p

text: As

text: j

text: we

text: conclude

text: by

text: that

text: s

text: (j(p

text: p

text: Since

text: m

text: j(p

text: by

text: p

text: j

text: L,

text: we

text: have

text: s

text: (m

text: j(p

text: Applying

text: these

text: estimates

text: to

text: we

text: ﬁnally

text: ≥p

text: infer

text: that

text: s

text: (m)

text: p.

text: p

text: We

text: now

text: suppose

text: that

text: s

text: (m)

text: p.

text: The

text: bound

text: p

text: M

text: implies

text: that

text: m

text: p.

text: p

text: n

text: Therefore,

text: in

text: the

text: p-adic

text: expansion

text: m

text: α

text: α

text: p

text: α

text: pr,

text: r

text: we

text: have

text: r

text: and

text: α

text: Since

text: m

text: j(p

text: when

text: j

text: L,

text: the

text: p-adic

text: r

text: expansion

text: j(p

text: β

text: β

text: p

text: β

text: pr

text: j,r

text: has

text: α

text: β

text: and

text: the

text: digits

text: β

text: β

text: cannot

text: all

text: be

text: equal

text: to

text: the

text: digits

text: r

text: j,r

text: j,r

text: α

text: α

text: By

text: Lucas’s

text: theorem

text: pp.

text: (for

text: a

text: modern

text: proof,

text: see

text: or

text: r

text: p.

text: we

text: obtain

text: m

text: α

text: α

text: α

text: ω

text: d=ef

text: r

text: (mod

text: p),

text: j

text: j(p

text: β

text: β

text: β

text: j,r

text: using

text: the

text: convention

text: that

text: α

text: if

text: α

text: β.

text: We

text: will

text: deduce

text: that

text: ω

text: (mod

text: p)

text: β

text: for

text: some

text: index

text: j.

text: To

text: do

text: so,

text: we

text: construct

text: unique

text: digits

text: β

text: β

text: as

text: follows.

text: r

text: (Remember

text: that

text: r

text: α

text: r

text: def

text: Set

text: β

text: α

text: r

text: r

text: r

text: def

text: Set

text: β

text: min

text: α

text: (p

text: β

text: iteratively

text: for

text: k

text: r

text: r

text: k

text: k

text: Roughly

text: speaking,

text: the

text: digits

text: β

text: are

text: “ﬁlled

text: up”

text: by

text: the

text: digits

text: α

text: until

text: the

text: partial

text: k

text: k

text: sum

text: β

text: β

text: reaches

text: p

text: the

text: remaining

text: β

text: are

text: then

text: set

text: equal

text: to

text: zero.

text: r

text: k

text: ##LTLine##


text: POWER-SUM DENOMINATORS 3 Example 1. To illustrate Theorems 1 and 2, we compute the table n M q d n n n · · · 19 7 2 3 7 = 42 20 q = 840 · · · · 19 20 11 2 3 5 11 = 330 21 q = 6930 20 which depends on the values of s (n + 1) given by p p 2 3 5 7 p 2 3 5 7 11 and . s (20) 2 4 4 8 s (21) 3 3 5 3 11 p p The rest of the paper is organized as follows. The next section is devoted to preliminaries, including Bernoulli’s formula for power sums and the von Staudt– Clausen theorem on Bernoulli numbers. In Section we consider some properties of the binomial coeﬃcients and we prove ﬁve lemmas. Section contains the proof of Theorem 1. In Section we use a congruence on binomial coeﬃcients due to Hermite and Bachmann to give another formula for the quotients q . In Section we prove n Theorem 2 using p-adic methods, including results of Legendre and Lucas. In the ﬁnal section the theorems are applied to the denominators of the Bernoulli polynomials. 2. Preliminaries In his Ars Conjectandi [4, pp. 96–98] of 1713, Jacob Bernoulli generalized Faul- haber’s formulas, but without giving a rigorous proof. Later a proof followed as a special case of the more general Euler–Maclaurin summation formula, which was independently found (cf. [15, p. 402]) by Euler [9, pp. 17–18] and Maclaurin [22, pp. 676–677] around 1735. In modern terms, Bernoulli’s formula for power sums ≥ (see [7, pp. 106–109]) states that for n 1, − · · · − B (x) B S (x) d=ef 1n + 2n + + (x 1)n = n+1 n+1 , (2) n n + 1 (cid:18) (cid:19) where the nth Bernoulli polynomial B (x(cid:88)) is deﬁned symbolically as n n n − B (x) d=ef (B + x)n d=ef B xn k (3) n k k k=0 and the Bernoulli numbers B , B(cid:88), B , . . . are rational numbers deﬁned by the 0 1 2 generating function ∞ t tk | | − = B ( t < 2π). et 1 k k! k=0 It turns out that B = 0 for odd n > 1 (see, e.g., [17, Section 7.9]). The sequence n of nonzero Bernoulli numbers starts with −1 B = 1, B = , (4) 0 1 2 and continues with 1 − 1 1 − 1 5 − 691 B = , B = , B = , B = , B = , B = , . . . . 2 6 4 30 6 42 8 30 10 66 12 2730 ##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: POWER-SUM

text: DENOMINATORS

text: Example

text: To

text: illustrate

text: Theorems

text: and

text: we

text: compute

text: the

text: table

text: n

text: M

text: q

text: d

text: n

text: n

text: n

text: q

text: q

text: which

text: depends

text: on

text: the

text: values

text: of

text: s

text: (n

text: given

text: by

text: p

text: p

text: p

text: and

text: s

text: s

text: p

text: p

text: The

text: rest

text: of

text: the

text: paper

text: is

text: organized

text: as

text: follows.

text: The

text: next

text: section

text: is

text: devoted

text: to

text: preliminaries,

text: including

text: Bernoulli’s

text: formula

text: for

text: power

text: sums

text: and

text: the

text: von

text: Staudt–

text: Clausen

text: theorem

text: on

text: Bernoulli

text: numbers.

text: In

text: Section

text: we

text: consider

text: some

text: properties

text: of

text: the

text: binomial

text: coeﬃcients

text: and

text: we

text: prove

text: ﬁve

text: lemmas.

text: Section

text: contains

text: the

text: proof

text: of

text: Theorem

text: In

text: Section

text: we

text: use

text: a

text: congruence

text: on

text: binomial

text: coeﬃcients

text: due

text: to

text: Hermite

text: and

text: Bachmann

text: to

text: give

text: another

text: formula

text: for

text: the

text: quotients

text: q

text: In

text: Section

text: we

text: prove

text: n

text: Theorem

text: using

text: p-adic

text: methods,

text: including

text: results

text: of

text: Legendre

text: and

text: Lucas.

text: In

text: the

text: ﬁnal

text: section

text: the

text: theorems

text: are

text: applied

text: to

text: the

text: denominators

text: of

text: the

text: Bernoulli

text: polynomials.

text: Preliminaries

text: In

text: his

text: Ars

text: Conjectandi

text: pp.

text: of

text: Jacob

text: Bernoulli

text: generalized

text: Faul-

text: haber’s

text: formulas,

text: but

text: without

text: giving

text: a

text: rigorous

text: proof.

text: Later

text: a

text: proof

text: followed

text: as

text: a

text: special

text: case

text: of

text: the

text: more

text: general

text: Euler–Maclaurin

text: summation

text: formula,

text: which

text: was

text: independently

text: found

text: (cf.

text: p.

text: by

text: Euler

text: pp.

text: and

text: Maclaurin

text: pp.

text: around

text: In

text: modern

text: terms,

text: Bernoulli’s

text: formula

text: for

text: power

text: sums

text: (see

text: pp.

text: states

text: that

text: for

text: n

text: B

text: (x)

text: B

text: S

text: (x)

text: d=ef

text: (x

text: n

text: n

text: where

text: the

text: nth

text: Bernoulli

text: polynomial

text: B

text: is

text: deﬁned

text: symbolically

text: as

text: n

text: n

text: n

text: B

text: (x)

text: d=ef

text: (B

text: x)n

text: d=ef

text: B

text: xn

text: k

text: n

text: k

text: k

text: and

text: the

text: Bernoulli

text: numbers

text: B

text: B

text: are

text: rational

text: numbers

text: deﬁned

text: by

text: the

text: generating

text: function

text: t

text: tk

text: B

text: t

text: et

text: k

text: k!

text: It

text: turns

text: out

text: that

text: B

text: for

text: odd

text: n

text: (see,

text: e.g.,

text: Section

text: The

text: sequence

text: n

text: of

text: nonzero

text: Bernoulli

text: numbers

text: starts

text: with

text: −1

text: B

text: B

text: and

text: continues

text: with

text: B

text: B

text: B

text: B

text: B

text: B

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