text: 6 Anna Breger, Martin Ehler, and Manuel Gra¨f cf. [10]. In this case, Theorem 1 leads to (cid:12)(cid:12)(cid:12) (cid:90)G f(P)dµk,d(P) − (cid:88)nt ωjtf(Pjt)(cid:12)(cid:12)(cid:12) (cid:46) n−t k(ds−k) (cid:107)f(cid:107)Hps, (5) k,d j=1 so that we have settled that QMC systems do exist, for any s > k(d−k)/p, and can be derived via cubatures. Remark 1. Cubature points {P }n for Π with constant weights ω = 1 are j j=1 t j n called t-designs. For all t = 1, 2, . . ., there exist t-designs, cf. [29]. The results in [5] imply that there are t-designs satisfying (4) provided that k = 1. However, for 2 ≤ k ≤ d, it is still an open problem if the asymptotics (4) can 2 be achieved by t-designs in place of cubatures. 3 Approximation by diﬀusion kernels One example, where integrals over the Grassmannian are replaced with C weighted ﬁnite sums, is the approximation of a function f : G → from k,d ﬁnitely many samples. The approximation scheme developed in [24, 25] works for manifolds and metric measure spaces in general, but we shall restrict the presentation to the Grassmannian. For a function f ∈ L (G ), we denote the (polynomial) best approxima- p k,d tion error by dist(f, Π ) := inf (cid:107)f − g(cid:107) , t Lp g∈Πt Lp where t ≥ 0. It is possible to quantify the best approximation error in depen- dence of the function’s smoothness, see [25, Proposition 5.3] for the following result. Theorem 2. If f ∈ Hs(G ), then p k,d (cid:46) − dist(f, Π ) t s(cid:107)f(cid:107) . t Lp Hps Given f ∈ Hs(G ) we now construct a particular sequence of functions p k,d σ (f) ∈ Π , t = 1, 2, . . ., that realizes this best approximation rate. Note t t ∞ that, since the collection {ϕ } is an orthonormal basis for L (G ), any (cid:96) (cid:96)=0 2 k,d function f ∈ L (G ) can be expanded as a Fourier series by 2 k,d ∞ (cid:88) f = fˆ((cid:96))ϕ . (cid:96) (cid:96)=0 The approach in [24, 25] makes use of a smoothly truncated Fourier expansion of f, ##LTLine## ##LTLine## ##LTLine##

text: Anna

text: Breger,

text: Martin

text: Ehler,

text: and

text: Manuel

text: Gra¨f

text: cf.

text: In

text: this

text: case,

text: Theorem

text: leads

text: to

text: f(P)dµk,d(P)

text: n−t

text: k(ds−k)

text: k,d

text: so

text: that

text: we

text: have

text: settled

text: that

text: QMC

text: systems

text: do

text: exist,

text: for

text: any

text: s

text: k(d−k)/p,

text: and

text: can

text: be

text: derived

text: via

text: cubatures.

text: Remark

text: Cubature

text: points

text: {P

text: }n

text: for

text: Π

text: with

text: constant

text: weights

text: ω

text: are

text: j

text: t

text: j

text: n

text: called

text: t-designs.

text: For

text: all

text: t

text: there

text: exist

text: t-designs,

text: cf.

text: The

text: results

text: in

text: imply

text: that

text: there

text: are

text: t-designs

text: satisfying

text: provided

text: that

text: k

text: However,

text: for

text: k

text: d,

text: it

text: is

text: still

text: an

text: open

text: problem

text: if

text: the

text: asymptotics

text: can

text: be

text: achieved

text: by

text: t-designs

text: in

text: place

text: of

text: cubatures.

text: Approximation

text: by

text: diﬀusion

text: kernels

text: One

text: example,

text: where

text: integrals

text: over

text: the

text: Grassmannian

text: are

text: replaced

text: with

text: C

text: weighted

text: ﬁnite

text: sums,

text: is

text: the

text: approximation

text: of

text: a

text: function

text: f

text: G

text: from

text: k,d

text: ﬁnitely

text: many

text: samples.

text: The

text: approximation

text: scheme

text: developed

text: in

text: works

text: for

text: manifolds

text: and

text: metric

text: measure

text: spaces

text: in

text: general,

text: but

text: we

text: shall

text: restrict

text: the

text: presentation

text: to

text: the

text: Grassmannian.

text: For

text: a

text: function

text: f

text: L

text: (G

text: we

text: denote

text: the

text: (polynomial)

text: best

text: approxima-

text: p

text: k,d

text: tion

text: error

text: by

text: dist(f,

text: Π

text: inf

text: t

text: Lp

text: g∈Πt

text: Lp

text: where

text: t

text: It

text: is

text: possible

text: to

text: quantify

text: the

text: best

text: approximation

text: error

text: in

text: depen-

text: dence

text: of

text: the

text: function’s

text: smoothness,

text: see

text: Proposition

text: for

text: the

text: following

text: result.

text: Theorem

text: If

text: f

text: Hs(G

text: then

text: p

text: k,d

text: dist(f,

text: Π

text: t

text: t

text: Lp

text: Hps

text: Given

text: f

text: Hs(G

text: we

text: now

text: construct

text: a

text: particular

text: sequence

text: of

text: functions

text: p

text: k,d

text: σ

text: (f)

text: Π

text: t

text: that

text: realizes

text: this

text: best

text: approximation

text: rate.

text: Note

text: t

text: t

text: that,

text: since

text: the

text: collection

text: {ϕ

text: is

text: an

text: orthonormal

text: basis

text: for

text: L

text: (G

text: any

text: k,d

text: function

text: f

text: L

text: (G

text: can

text: be

text: expanded

text: as

text: a

text: Fourier

text: series

text: by

text: k,d

text: f

text: The

text: approach

text: in

text: makes

text: use

text: of

text: a

text: smoothly

text: truncated

text: Fourier

text: expansion

text: of

text: f,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: – 15 – the commissioning stage and it could be available very soon. An environmental controlling system can accurately keep the stability of both temperature and humidity for the HRS system. Furthermore, a sub-controlling system of pressure will be installed at this system in the future. The basic parameters of HRS are listed in Table 5. The working wavelength coverage is ≥ 360-1000 nm and the instrumental eﬃciency of the spectrograph is 34% for the peak at a ≥ wavelength coverage of 640-790 nm (RI band), and 10% for λ > 4500 ˚A (the whole working band), based on the tests of November 23, 2010. The spectral resolution is 32,000-106,000 for the spectrograph and it is R = 49800 at a ﬁxed slit width of 0.19 mm (corresponding (cid:48)(cid:48) to 1.8) based on the test of April 12, 2011. The stability of the instrument for the velocity ± measurement is r.m.s. = 6 m/s and the temperature is quite stable even for two weeks. The CCD camera is a back-illuminated ﬁrst-order red-sensitive E2V CCD 203-82. The size × is 4096 4096 pixels with a pixel size of 12 µm. The typical quantum eﬃciency of the CCD − ◦ ∼ − is > 90% under the temperature of 100 C in the wavelength coverage of 500 650 nm, which is shown as the solid line in Figure 15. ∼ The liquid nitrogen (LN) holding time of the system is 20 hours, and the cooling − ◦ − − temperature is 106 C. For Gain = 1.01 e /ADU, the readout noise (RON) is 2.84 e at the − − readout speed of 50k, 4.29 e at readout speed of 100k and 7.88 e at readout speed of 200k. ≥ The distance between the nearby orders of the spectrum is 20 pixels and the two ﬁbers ( (cid:48)(cid:48) (cid:48)(cid:48) 2.4 aperture one and 1.6 aperture one) can work at the same time. The aperture of the ﬁber (cid:48)(cid:48) ∼ (cid:48) × (cid:48) is 100 µm (2.4) and the ﬁber length is 19 m. The FOV for the guiding camera is 3 3 (cid:48)(cid:48) and in the guiding plate the aperture of pupil is 4.0 in front of the ﬁber. The temperature ± ◦ ± ◦ variation of the system is r.m.s. = 0.05 C for a whole night and r.m.s. = 0.34 C for a week. The tip/tilt system is working normally on the Cassegrain focus. The system is currently in its commissioning phase, and its radial velocity precision dose not yet reach its ∼ design goal of a few cm/s. It is expected that this goal will be achieved once commissioning is complete. The guiding CCD camera is a GC1380, and the software AV T UniCamV iewer is used for CCD controlling, exposure time conﬁguration and data collection. The telescope con- trolling software can monitor the guiding uncertainty of the guiding system. An astro-frequency comb calibration system, which is developed by Peking University (PKU), also was installed and is being tested on the 2.16-m telescope. The full spectral wave- ± length range of the astro-frequency comb is 160nm and the central wavelength is 640 20 nm. The observed spacing of the comb teeth is 29.01 GHz. Once working normally, it can greatly improve the measuring accuracy of the stellar radial velocity. The system has signiﬁcant advantages over the I cell (Wilken et al. 2010). It consists of a series of discrete, equally 2

label: paragraph

text: the

text: commissioning

text: stage

text: and

text: it

text: could

text: be

text: available

text: very

text: soon.

text: An

text: environmental

text: controlling

text: system

text: can

text: accurately

text: keep

text: the

text: stability

text: of

text: both

text: temperature

text: and

text: humidity

text: for

text: the

text: HRS

text: system.

text: Furthermore,

text: a

text: sub-controlling

text: system

text: of

text: pressure

text: will

text: be

text: installed

text: at

text: this

text: system

text: in

text: the

text: future.

text: The

text: basic

text: parameters

text: of

text: HRS

text: are

text: listed

text: in

text: Table

text: The

text: working

text: wavelength

text: coverage

text: is

text: nm

text: and

text: the

text: instrumental

text: eﬃciency

text: of

text: the

text: spectrograph

text: is

text: for

text: the

text: peak

text: at

text: a

text: wavelength

text: coverage

text: of

text: nm

text: (RI

text: band),

text: and

text: for

text: λ

text: ˚A

text: (the

text: whole

text: working

text: band),

text: based

text: on

text: the

text: tests

text: of

text: November

text: The

text: spectral

text: resolution

text: is

text: for

text: the

text: spectrograph

text: and

text: it

text: is

text: R

text: at

text: a

text: ﬁxed

text: slit

text: width

text: of

text: mm

text: (corresponding

text: to

text: based

text: on

text: the

text: test

text: of

text: April

text: The

text: stability

text: of

text: the

text: instrument

text: for

text: the

text: velocity

text: measurement

text: is

text: r.m.s.

text: m/s

text: and

text: the

text: temperature

text: is

text: quite

text: stable

text: even

text: for

text: two

text: weeks.

text: The

text: CCD

text: camera

text: is

text: a

text: back-illuminated

text: ﬁrst-order

text: red-sensitive

text: CCD

text: The

text: size

text: is

text: pixels

text: with

text: a

text: pixel

text: size

text: of

text: µm.

text: The

text: typical

text: quantum

text: eﬃciency

text: of

text: the

text: CCD

text: is

text: under

text: the

text: temperature

text: of

text: C

text: in

text: the

text: wavelength

text: coverage

text: of

text: nm,

text: which

text: is

text: shown

text: as

text: the

text: solid

text: line

text: in

text: Figure

text: The

text: liquid

text: nitrogen

text: (LN)

text: holding

text: time

text: of

text: the

text: system

text: is

text: hours,

text: and

text: the

text: cooling

text: temperature

text: is

text: C.

text: For

text: Gain

text: e

text: /ADU,

text: the

text: readout

text: noise

text: (RON)

text: is

text: e

text: at

text: the

text: readout

text: speed

text: of

text: e

text: at

text: readout

text: speed

text: of

text: and

text: e

text: at

text: readout

text: speed

text: of

text: The

text: distance

text: between

text: the

text: nearby

text: orders

text: of

text: the

text: spectrum

text: is

text: pixels

text: and

text: the

text: two

text: ﬁbers

text: aperture

text: one

text: and

text: aperture

text: one)

text: can

text: work

text: at

text: the

text: same

text: time.

text: The

text: aperture

text: of

text: the

text: ﬁber

text: is

text: µm

text: and

text: the

text: ﬁber

text: length

text: is

text: m.

text: The

text: FOV

text: for

text: the

text: guiding

text: camera

text: is

text: and

text: in

text: the

text: guiding

text: plate

text: the

text: aperture

text: of

text: pupil

text: is

text: in

text: front

text: of

text: the

text: ﬁber.

text: The

text: temperature

text: variation

text: of

text: the

text: system

text: is

text: r.m.s.

text: C

text: for

text: a

text: whole

text: night

text: and

text: r.m.s.

text: C

text: for

text: a

text: week.

text: The

text: tip/tilt

text: system

text: is

text: working

text: normally

text: on

text: the

text: Cassegrain

text: focus.

text: The

text: system

text: is

text: currently

text: in

text: its

text: commissioning

text: phase,

text: and

text: its

text: radial

text: velocity

text: precision

text: dose

text: not

text: yet

text: reach

text: its

text: design

text: goal

text: of

text: a

text: few

text: cm/s.

text: It

text: is

text: expected

text: that

text: this

text: goal

text: will

text: be

text: achieved

text: once

text: commissioning

text: is

text: complete.

text: The

text: guiding

text: CCD

text: camera

text: is

text: a

text: and

text: the

text: software

text: AV

text: T

text: UniCamV

text: iewer

text: is

text: used

text: for

text: CCD

text: controlling,

text: exposure

text: time

text: conﬁguration

text: and

text: data

text: collection.

text: The

text: telescope

text: con-

text: trolling

text: software

text: can

text: monitor

text: the

text: guiding

text: uncertainty

text: of

text: the

text: guiding

text: system.

text: An

text: astro-frequency

text: comb

text: calibration

text: system,

text: which

text: is

text: developed

text: by

text: Peking

text: University

text: (PKU),

text: also

text: was

text: installed

text: and

text: is

text: being

text: tested

text: on

text: the

text: telescope.

text: The

text: full

text: spectral

text: wave-

text: length

text: range

text: of

text: the

text: astro-frequency

text: comb

text: is

text: and

text: the

text: central

text: wavelength

text: is

text: nm.

text: The

text: observed

text: spacing

text: of

text: the

text: comb

text: teeth

text: is

text: GHz.

text: Once

text: working

text: normally,

text: it

text: can

text: greatly

text: improve

text: the

text: measuring

text: accuracy

text: of

text: the

text: stellar

text: radial

text: velocity.

text: The

text: system

text: has

text: signiﬁcant

text: advantages

text: over

text: the

text: I

text: cell

text: (Wilken

text: et

text: al.

text: It

text: consists

text: of

text: a

text: series

text: of

text: discrete,

text: equally


text: – 7 – ﬁnd a suitable guide star. − − − In 2010 an E2V 55 30 1 348 back illuminated CCD, AIMO was installed on the spectrograph and the CCD controller was made by the Lick Observatory. The size of the × (cid:48)(cid:48) CCD is 1242 1152 pixels with pixel size of 22.5 µm. The pixel scale is 0.457 and the FOV (cid:48) × (cid:48) is 9 .46 8 .77 according to the size of CCD. Figure 5 shows the quantum eﬃciency (Q.E.) of the CCD. It can be seen that the maximum quantum eﬃciency is higher than 90% around 5700 ˚A of the wavelength. Table 3 lists the gains, readout noises and readout speed of the BFOSC CCD at various readout times. Observers can choose diﬀerent options for the speciﬁc observations and the slow readout speed is applied in most of the time. Table 3: Gains, Readout Noises and readout Speed of BFOSC CCD from Huang, Wu & Li (2012). Readout Speed Mode Gain Readout Noise Readout Time − − (e /ADU) (e /pix.) (second) Fast 0 99.58 132.5 5 Fast 1 49.44 103.7 5 Fast 2 24.48 87.35 5 Median 0 2.33 6.80 8 Median 1 1.22 5.93 8 Median 2 0.50 6.06 8 Slow 0 2.43 3.25 28 Slow 1 1.13 2.58 28 Slow 2 0.50 2.46 28 There are six observing modes for the BFOSC instrument: (1) direct imaging; (2) long slit spectroscopy; (3) slitless spectroscopy; (4) echelle grism spectroscopy; (5) coronograph mask; and (6) multiple object Spectroscopy. Figure 6 shows the optical layout of the lens and light path for the direct imaging mode and the spectroscopy observing mode, which could be switched in a few minutes. Since 2012, a multi-object spectroscopy (MOS) observing mode of BFSOC is available for observers by placing a multiple-aperture mask on the aperture wheel (Zhou et al. 2014). Fig. 4.— The transmission curves of the BFOSC ﬁlters, including the Johnson-cousins UBV RI bands as well as the 385LP and Z bands for spectroscopic observations. ##LTLine## ##LTLine## ##LTLine##

text: ﬁnd

text: a

text: suitable

text: guide

text: star.

text: In

text: an

text: back

text: illuminated

text: CCD,

text: AIMO

text: was

text: installed

text: on

text: the

text: spectrograph

text: and

text: the

text: CCD

text: controller

text: was

text: made

text: by

text: the

text: Lick

text: Observatory.

text: The

text: size

text: of

text: the

text: CCD

text: is

text: pixels

text: with

text: pixel

text: size

text: of

text: µm.

text: The

text: pixel

text: scale

text: is

text: and

text: the

text: FOV

text: is

text: according

text: to

text: the

text: size

text: of

text: CCD.

text: Figure

text: shows

text: the

text: quantum

text: eﬃciency

text: (Q.E.)

text: of

text: the

text: CCD.

text: It

text: can

text: be

text: seen

text: that

text: the

text: maximum

text: quantum

text: eﬃciency

text: is

text: higher

text: than

text: around

text: ˚A

text: of

text: the

text: wavelength.

text: Table

text: lists

text: the

text: gains,

text: readout

text: noises

text: and

text: readout

text: speed

text: of

text: the

text: BFOSC

text: CCD

text: at

text: various

text: readout

text: times.

text: Observers

text: can

text: choose

text: diﬀerent

text: options

text: for

text: the

text: speciﬁc

text: observations

text: and

text: the

text: slow

text: readout

text: speed

text: is

text: applied

text: in

text: most

text: of

text: the

text: time.

text: Table

text: Gains,

text: Readout

text: Noises

text: and

text: readout

text: Speed

text: of

text: BFOSC

text: CCD

text: from

text: Huang,

text: Wu

text: Li

text: Readout

text: Speed

text: Mode

text: Gain

text: Readout

text: Noise

text: Readout

text: Time

text: (e

text: /ADU)

text: (e

text: /pix.)

text: (second)

text: Fast

text: Fast

text: Fast

text: Median

text: Median

text: Median

text: Slow

text: Slow

text: Slow

text: There

text: are

text: six

text: observing

text: modes

text: for

text: the

text: BFOSC

text: instrument:

text: direct

text: imaging;

text: long

text: slit

text: spectroscopy;

text: slitless

text: spectroscopy;

text: echelle

text: grism

text: spectroscopy;

text: coronograph

text: mask;

text: and

text: multiple

text: object

text: Spectroscopy.

text: Figure

text: shows

text: the

text: optical

text: layout

text: of

text: the

text: lens

text: and

text: light

text: path

text: for

text: the

text: direct

text: imaging

text: mode

text: and

text: the

text: spectroscopy

text: observing

text: mode,

text: which

text: could

text: be

text: switched

text: in

text: a

text: few

text: minutes.

text: Since

text: a

text: multi-object

text: spectroscopy

text: (MOS)

text: observing

text: mode

text: of

text: BFSOC

text: is

text: available

text: for

text: observers

text: by

text: placing

text: a

text: multiple-aperture

text: mask

text: on

text: the

text: aperture

text: wheel

text: (Zhou

text: et

text: al.

text: Fig.

text: The

text: transmission

text: curves

text: of

text: the

text: BFOSC

text: ﬁlters,

text: including

text: the

text: Johnson-cousins

text: UBV

text: RI

text: bands

text: as

text: well

text: as

text: the

text: and

text: Z

text: bands

text: for

text: spectroscopic

text: observations.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 M. P. SAIKIA (cid:40) 0, if i ≤ k (4.7) Y := . M(AR (a + k + 2 − i), if i ≥ k + 1 a,a+1 Using equation (4.7) in equation (4.6), we can simplify the relation further to the following (4.8) M(ARi ) = M(ARi − ) + Z · M(AR (2, 3, . . . , k + 1)) a,b+1,k a,b,k 1 a,b  where   0, if i ≤ k (4.9) Z := M(AR (a + k + 2 − i) . a,a+1 , if i ≥ k + 1 M(AD(a)) It now remains to show that the expression in the statement satisﬁes equation (4.7). This is now a straightforward application of the induction hypothesis and some algebraic manipulation. (cid:3) Proposition 4.9. Let a, i, j be positive integers such that 1 ≤ i, j ≤ a, then the number of domino tilings of AD(a) with one defect on the southeastern side at the i-th position counted from the south corner and one d(cid:18)efect o(cid:19)n(cid:18)the no(cid:19)rtheast(cid:20)ern side on the j(cid:21)-th position counted from the north corner as shown in Figure 13 is given by − a − 1 a − 1 1, 1 − i, 1 − j 2a(a 1)/2 F ; 2 . i − 1 j − 1 3 2 1 − a, 1 − a Proof. We use induction with respect to a. The base case of induction is a = 2. We would also need to check for i = 1, j = 1, i = a and j = a separately. If a = 2, then the only possibilities are i = 1 or i = a and j = 1 or j = a, so we do not have to consider this case, once we consider the other mentioned cases. We now note that when either i or j is 1 or a, some dominoes are forced in any tiling and hence we are reduced to an Aztec rectangle of size (a − 1) × a. It is easy to see that our formula is correct for this. In the rest of the proof we assume a ≥ 3 and 1 < i, j < a. Let us now denote the region we are interested in this proposition as AD (i, j). Using the dual graph of this region and applying a Theorem 4.4 with the vertices as labelled in Figure 14 we obtain the following identity (see Figure 15 for details), (4.10) M(AD (i, j)) M(AD(a − 1)) = M(AD(a)) M(AD − (i − 1, j − 1)) a a 1 + M(AR − (j)) M(AR − (i)). a 1,a a 1,a (cid:18) (cid:19)(cid:18) (cid:19) Simplifying equation (4.10), we get the following − a − 1 a − 1 (4.11) M(AD (i, j)) = 2a M(AD − (i − 1, j − 1)) + 2a(a 1)/2 a a 1 j − 1 i − 1 where we used Theorem 1.1 and Corollary 4.1. ##LTLine##

text: M.

text: P.

text: SAIKIA

label: equation

text: if

text: i

text: k

text: Y

text: M(AR

text: (a

text: k

text: i),

text: if

text: i

text: k

text: Using

text: equation

text: in

text: equation

text: we

text: can

text: simplify

text: the

text: relation

text: further

text: to

text: the

text: following

text: M(ARi

text: M(ARi

text: Z

text: M(AR

text: k

text: a,b,k

text: a,b

text: where

text: if

text: i

text: k

text: Z

text: M(AR

text: (a

text: k

text: i)

text: if

text: i

text: k

text: M(AD(a))

text: It

text: now

text: remains

text: to

text: show

text: that

text: the

text: expression

text: in

text: the

text: statement

text: satisﬁes

text: equation

text: This

text: is

text: now

text: a

text: straightforward

text: application

text: of

text: the

text: induction

text: hypothesis

text: and

text: some

text: algebraic

text: manipulation.

text: Proposition

text: Let

text: a,

text: i,

text: j

text: be

text: positive

text: integers

text: such

text: that

text: i,

text: j

text: a,

text: then

text: the

text: number

text: of

text: domino

text: tilings

text: of

text: AD(a)

text: with

text: one

text: defect

text: on

text: the

text: southeastern

text: side

text: at

text: the

text: i-th

text: position

text: counted

text: from

text: the

text: south

text: corner

text: and

text: one

text: side

text: on

text: the

text: position

text: counted

text: from

text: the

text: north

text: corner

text: as

text: shown

text: in

text: Figure

text: is

text: given

text: by

text: a

text: a

text: i,

text: j

text: F

text: i

text: j

text: a,

text: a

text: Proof.

text: We

text: use

text: induction

text: with

text: respect

text: to

text: a.

text: The

text: base

text: case

text: of

text: induction

text: is

text: a

text: We

text: would

text: also

text: need

text: to

text: check

text: for

text: i

text: j

text: i

text: a

text: and

text: j

text: a

text: separately.

text: If

text: a

text: then

text: the

text: only

text: possibilities

text: are

text: i

text: or

text: i

text: a

text: and

text: j

text: or

text: j

text: a,

text: so

text: we

text: do

text: not

text: have

text: to

text: consider

text: this

text: case,

text: once

text: we

text: consider

text: the

text: other

text: mentioned

text: cases.

text: We

text: now

text: note

text: that

text: when

text: either

text: i

text: or

text: j

text: is

text: or

text: a,

text: some

text: dominoes

text: are

text: forced

text: in

text: any

text: tiling

text: and

text: hence

text: we

text: are

text: reduced

text: to

text: an

text: Aztec

text: rectangle

text: of

text: size

text: (a

text: a.

text: It

text: is

text: easy

text: to

text: see

text: that

text: our

text: formula

text: is

text: correct

text: for

text: this.

text: In

text: the

text: rest

text: of

text: the

text: proof

text: we

text: assume

text: a

text: and

text: i,

text: j

text: a.

text: Let

text: us

text: now

text: denote

text: the

text: region

text: we

text: are

text: interested

text: in

text: this

text: proposition

text: as

text: AD

text: (i,

text: j).

text: Using

text: the

text: dual

text: graph

text: of

text: this

text: region

text: and

text: applying

text: a

text: Theorem

text: with

text: the

text: vertices

text: as

text: labelled

text: in

text: Figure

text: we

text: obtain

text: the

text: following

text: identity

text: (see

text: Figure

text: for

text: details),

text: M(AD

text: (i,

text: j))

text: M(AD(a

text: M(AD(a))

text: M(AD

text: (i

text: j

text: a

text: a

text: M(AR

text: (j))

text: M(AR

text: (i)).

text: a

text: a

text: Simplifying

text: equation

text: we

text: get

text: the

text: following

text: a

text: a

text: M(AD

text: (i,

text: j))

text: M(AD

text: (i

text: j

text: a

text: a

text: j

text: i

text: where

text: we

text: used

text: Theorem

text: and

text: Corollary

text: ##LTLine##


text: AZTEC RECTANGLES WITH BOUNDARY DEFECTS 5 To state and prove our result, we will need to make some notations and concepts clear. We consider the symmetric diﬀerence on the vertices and edges of a graph. Let H be a planar graph and G be an induced subgraph of H and let W ⊆ V(H). Then we deﬁne G + W as follows: G + W is the induced subgraph of H with vertex set V(G + W) = V(G)∆W, where ∆ denotes the symmetric diﬀerence of sets. Now we are in a position to state our result below. Theorem 3.2. Let H be a planar graph and let G be an induced subgraph of H with the vertices a , a , . . . , a appearing in that cyclic order on a face of H. Consider the skew-symmetric matrix 1 2 2k A = (a ) ≤ ≤ with entries given by ij 1 i,j 2k (3.3) a := M(G + {a , a }), if i < j. ij i j Then we have that Pf(A) (3.4) M(G + {a1, a2, . . . , a2k}) = [M(G)]k−1 . Corollary 3.3. [6, Theorem 2.4] Let G = (V , V , E) be a bipartite planar graph with |V | = 1 2 1 |V | + 1; and let w, x, y and z be vertices of G that appear in cyclic order on a face of G. If 2 w, x, y ∈ V and z ∈ V then 1 2 (3.5) M(G − {w}) M(G − {x, y, z}) + M(G − {y}) M(G − {w, x, z}) = M(G − {x}) M(G − {w, y, z}) + M(G − {z}) M(G − {w, x, y}). (cid:3) Proof. Take n = 2, a = w, a = x, a = y, a = z and G = H \ {a } in Theorem 3.2. 1 2 3 4 1 The proof of Theorem 3.1 follows from the use of some auxillary results. In the vein of those results, we need the following proposition to complete our proof of Theorem 3.2. Proposition 3.4. Let H be a planar graph and G be an induced subgraph of H with the vertices a1, . . . , a2k appearing in that cyclic order a(cid:88)mong the vertices of some face of H. Then k M(G) M(G + {a1, . . . , a2k}) + M(cid:88)(G + {a1, a2l−1}) M(G + {a1, a2l−1}) l=2 k (3.6) = M(G + {a , a }) M(G + {a , a }), 1 2l 1 2l l=1 where {a , a } stands for the complement of {a , a } in the set {a , . . . , a }. i j i j 1 2k Our proof follows closely that of the proof of an analogous proposition given by Ciucu [1]. Proof. We recast equation (3.6) in terms of disjoint unions of cartesian products as follows M(G) × M(G + {a , . . . , a }) ∪ M(G + {a , a }) × M(G + {a , a }) ∪ . . . 1 2k 1 3 1 3 (3.7) ∪M(G + {a , a − }) × M(G + {a , a − }) 1 2k 1 1 2k 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: AZTEC

text: RECTANGLES

text: WITH

text: BOUNDARY

text: DEFECTS

text: To

text: state

text: and

text: prove

text: our

text: result,

text: we

text: will

text: need

text: to

text: make

text: some

text: notations

text: and

text: concepts

text: clear.

text: We

text: consider

text: the

text: symmetric

text: diﬀerence

text: on

text: the

text: vertices

text: and

text: edges

text: of

text: a

text: graph.

text: Let

text: H

text: be

text: a

text: planar

text: graph

text: and

text: G

text: be

text: an

text: induced

text: subgraph

text: of

text: H

text: and

text: let

text: W

text: V(H).

text: Then

text: we

text: deﬁne

text: G

text: W

text: as

text: follows:

text: G

text: W

text: is

text: the

text: induced

text: subgraph

text: of

text: H

text: with

text: vertex

text: set

text: V(G

text: W)

text: V(G)∆W,

text: where

text: denotes

text: the

text: symmetric

text: diﬀerence

text: of

text: sets.

text: Now

text: we

text: are

text: in

text: a

text: position

text: to

text: state

text: our

text: result

text: below.

text: Theorem

text: Let

text: H

text: be

text: a

text: planar

text: graph

text: and

text: let

text: G

text: be

text: an

text: induced

text: subgraph

text: of

text: H

text: with

text: the

text: vertices

text: a

text: a

text: a

text: appearing

text: in

text: that

text: cyclic

text: order

text: on

text: a

text: face

text: of

text: H.

text: Consider

text: the

text: skew-symmetric

text: matrix

text: A

text: (a

text: with

text: entries

text: given

text: by

text: ij

text: i,j

text: a

text: M(G

text: {a

text: a

text: if

text: i

text: j.

text: ij

text: i

text: j

text: Then

text: we

text: have

text: that

text: Pf(A)

text: M(G

text: Corollary

text: Theorem

text: Let

text: G

text: (V

text: V

text: E)

text: be

text: a

text: bipartite

text: planar

text: graph

text: with

text: |V

text: |V

text: and

text: let

text: w,

text: x,

text: y

text: and

text: z

text: be

text: vertices

text: of

text: G

text: that

text: appear

text: in

text: cyclic

text: order

text: on

text: a

text: face

text: of

text: G.

text: If

text: w,

text: x,

text: y

text: V

text: and

text: z

text: V

text: then

text: M(G

text: {w})

text: M(G

text: {x,

text: y,

text: z})

text: M(G

text: {y})

text: M(G

text: {w,

text: x,

text: z})

text: M(G

text: {x})

text: M(G

text: {w,

text: y,

text: z})

text: M(G

text: {z})

text: M(G

text: {w,

text: x,

text: y}).

text: Proof.

text: Take

text: n

text: a

text: w,

text: a

text: x,

text: a

text: y,

text: a

text: z

text: and

text: G

text: H

text: {a

text: in

text: Theorem

text: The

text: proof

text: of

text: Theorem

text: follows

text: from

text: the

text: use

text: of

text: some

text: auxillary

text: results.

text: In

text: the

text: vein

text: of

text: those

text: results,

text: we

text: need

text: the

text: following

text: proposition

text: to

text: complete

text: our

text: proof

text: of

text: Theorem

text: Proposition

text: Let

text: H

text: be

text: a

text: planar

text: graph

text: and

text: G

text: be

text: an

text: induced

text: subgraph

text: of

text: H

text: with

text: the

text: vertices

text: appearing

text: in

text: that

text: cyclic

text: order

text: the

text: vertices

text: of

text: some

text: face

text: of

text: H.

text: Then

text: k

text: M(G)

text: M(G

text: M(G

text: k

text: M(G

text: {a

text: a

text: M(G

text: {a

text: a

text: where

text: {a

text: a

text: stands

text: for

text: the

text: complement

text: of

text: {a

text: a

text: in

text: the

text: set

text: {a

text: a

text: i

text: j

text: i

text: j

text: Our

text: proof

text: follows

text: closely

text: that

text: of

text: the

text: proof

text: of

text: an

text: analogous

text: proposition

text: given

text: by

text: Ciucu

text: Proof.

text: We

text: recast

text: equation

text: in

text: terms

text: of

text: disjoint

text: unions

text: of

text: cartesian

text: products

text: as

text: follows

text: M(G)

text: M(G

text: {a

text: a

text: M(G

text: {a

text: a

text: M(G

text: {a

text: a

text: ∪M(G

text: {a

text: a

text: M(G

text: {a

text: a

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 M. P. SAIKIA (cid:18) (cid:19) Corollary 4.2. The number of tilings of AR (2, . . . , b − a + 1) is given by a,b b − 1 2a(a+1)/2 . a − 1 k-1 j Figure 4. Aztec rectangle with k − 1 squares added on the southeastern side and a defect on the j-th position shaded in black; here a = 4, b = 10, k = 6, j = 8 Proposition 4.3. Let 1 ≤ a ≤ b be positive integers with k = b − a > 0, then the number of domino tilings of AR (j) with k − 1 squares added to the southeastern side starting at the a,b (cid:18) (cid:19)(cid:18) (cid:19) (cid:20) (cid:21) second position (and not at the bottom) as shown in the Figure 4 is given by a + k − 1 j − 2 1, 1 − j, 1 − k (4.1) 2a(a+1)/2 F ; 1 . j − 1 k − 1 3 2 2 − j, 1 − a − k − Proof. Let us denote the region in Figure 4 by ARk 1,j and we work with the planar dual graph − a,b of the region ARk 1,j and count the number of matchings of that graph. We ﬁrst notice that a,b the ﬁrst added square in any tiling of the region in Figure 4 by dominoes has two possibilities marked in grey in the Figure 5. This observation allows us to write the number of tilings of − ARk 1,j in terms of the following recursion a,b − − − (4.2) M(ARk 1,j) = M(ARk −2,j 1) + M(AR (2, 3, . . . , k, j)). a,b a,b 1 a,b which can be veriﬁed from Figure 6. Repeatedly using equation (4.2) k − 1 times on succesive iterations, we shall ﬁnally obtain (cid:88) − − k 2 (4.3) M(ARk 1,j) = M(AR − (2, 3, . . . , k − l, j − l)) + M(AR (j − k + 1)). a,b a,b l a,a+1 l=0 ##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##

text: M.

text: P.

text: SAIKIA

text: Corollary

text: The

text: number

text: of

text: tilings

text: of

text: AR

text: b

text: a

text: is

text: given

text: by

text: a,b

text: b

text: a

font: default

text: j

text: Figure

text: Aztec

label: caption

text: rectangle

text: with

text: k

text: squares

text: added

text: on

text: the

text: southeastern

text: side

text: and

text: a

text: defect

text: on

text: the

text: j-th

text: position

text: shaded

text: in

text: black;

text: here

text: a

text: b

text: k

text: j

text: Proposition

text: Let

text: a

text: b

text: be

text: positive

text: integers

text: with

text: k

text: b

text: a

text: then

text: the

text: number

text: of

text: domino

text: tilings

text: of

text: AR

text: (j)

text: with

text: k

text: squares

text: added

text: to

text: the

text: southeastern

text: side

text: starting

text: at

text: the

text: a,b

text: second

text: position

text: (and

text: not

text: at

text: the

text: bottom)

text: as

text: shown

text: in

text: the

text: Figure

text: is

text: given

text: by

text: a

text: k

text: j

text: j,

text: k

text: F

text: j

text: k

text: j,

text: a

text: k

text: Proof.

text: Let

text: us

text: denote

text: the

text: region

text: in

text: Figure

text: by

text: ARk

text: and

text: we

text: work

text: with

text: the

text: planar

text: dual

text: graph

text: a,b

text: of

text: the

text: region

text: ARk

text: and

text: count

text: the

text: number

text: of

text: matchings

text: of

text: that

text: graph.

text: We

text: ﬁrst

text: notice

text: that

text: a,b

text: the

text: ﬁrst

text: added

text: square

text: in

text: any

text: tiling

text: of

text: the

text: region

text: in

text: Figure

text: by

text: dominoes

text: has

text: two

text: possibilities

text: marked

text: in

text: grey

text: in

text: the

text: Figure

text: This

text: observation

text: allows

text: us

text: to

text: write

text: the

text: number

text: of

text: tilings

text: of

text: ARk

text: in

text: terms

text: of

text: the

text: following

text: recursion

text: a,b

text: M(ARk

text: M(ARk

text: M(AR

text: k,

text: j)).

text: a,b

text: a,b

text: a,b

text: which

text: can

text: be

text: veriﬁed

text: from

text: Figure

text: Repeatedly

text: using

text: equation

text: k

text: times

text: on

text: succesive

text: iterations,

text: we

text: shall

text: ﬁnally

text: obtain

text: k

text: M(ARk

text: M(AR

text: k

text: l,

text: j

text: l))

text: M(AR

text: (j

text: k

text: a,b

text: a,b

text: l

text: ##LTFigure##

text: ##LTLine##

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text: Randomization and The Pernicious Eﬀects of Limited Budgets on Auction Experiments † ‡ ∗ Guillaume W. Basse Hossein Azari Souﬁani , Diane Lambert ∗ Guillaume Basse is a graduate student in the Department of Statistics at Harvard University (gbasse@fas.harvard.edu) and his work was supported by a Google Fellowship in Statistics for North Amer- ica. Hossein Azari Souﬁani is a research scientist at Google Research (azari@google.com). Diane Lambert is a retired research scientist (exdlambert@gmail.com). This work was completed at Google Research. The authors thank Google’s Max Lin and George Levitte for their help, as well as Google NYC’s statistics group for early feedback. We also thank three anonymous reviewers for their constructive feedback. ##LTLine##

text: Randomization

label: title

text: and

text: The

text: Pernicious

text: Eﬀects

text: of

text: Limited

text: Budgets

text: on

text: Auction

text: Experiments

label: author

text: Guillaume

text: W.

text: Basse

text: Hossein

text: Azari

text: Souﬁani

text: Diane

text: Lambert

text: Guillaume

label: footer

text: Basse

text: is

text: a

text: graduate

text: student

text: in

text: the

text: Department

text: of

text: Statistics

text: at

text: Harvard

text: University

text: (gbasse@fas.harvard.edu)

text: and

text: his

text: work

text: was

text: supported

text: by

text: a

text: Google

text: Fellowship

text: in

text: Statistics

text: for

text: North

text: Amer-

text: ica.

text: Hossein

text: Azari

text: Souﬁani

text: is

text: a

text: research

text: scientist

text: at

text: Google

text: Research

text: (azari@google.com).

text: Diane

text: Lambert

text: is

text: a

text: retired

text: research

text: scientist

text: (exdlambert@gmail.com).

text: This

text: work

text: was

text: completed

text: at

text: Google

text: Research.

text: The

text: authors

text: thank

text: Google’s

text: Max

text: Lin

text: and

text: George

text: Levitte

text: for

text: their

text: help,

text: as

text: well

text: as

text: Google

text: NYC’s

text: statistics

text: group

text: for

text: early

text: feedback.

text: We

text: also

text: thank

text: three

text: anonymous

text: reviewers

text: for

text: their

text: constructive

text: feedback.

text: ##LTLine##


text: For each distribution of potential bids and quota limit, we generated 20,000 random query experiments with exactly half the N queries in each experiment assigned to treatment and q half to control. We also generated 20,000 random (query, advertiser) pair experiments, with half the pairs in treatment and half in control. The bid treatment eﬀect or quota treatment eﬀect, depending on the nature of the simulation, was estimated in each experiment. Because the bias depends on the treatment eﬀect for a lognormal, we report simulated bias relative to the true eﬀect. 6.1 Bid treatments Figure 1 describes the simulated relative bias (ratio of the bias to the true eﬀect), where the rows correspond to the diﬀerent quota settings and the columns to the diﬀerent mean √ log treatment bids. The standard errors reported are computed over the 200 draws from the bid distribution and are divided by 200 to reﬂect uncertainty about the simulated mean bias. 1.05 1.1 2 1.5 1.0 l l l l l l 0.5 l l l Randomization 0.0 l l l l l l l l l l (Query, Advertiser) 1.5 l l l l l l l Query 1.0 l l l 0.5 0.0 l l l l l l l l l 100 120 140 100 120 140 100 120 140 Number of auctions in exchange Figure 1: Simulated bias of the eﬀect of a bid treatment relative to the true eﬀect under balanced query randomization and balanced (query, advertiser) randomization. The dots ± show the mean relative bias, and the endpoints show the mean twice its simulated standard error. Columns show the mean log bid under treatment and the rows show the throttling rate. Figure 1 shows that the relative bias is nearly zero for randomized query experiments, while the relative bias has a mean as high as 1.5 for randomized (query, advertiser) pair experiments. That is, the penalty for allowing control and treated advertisers to compete in the same auction is a 50% increase in relative bias. Moreover, the simulated variance of the bid treatment estimate under the random query experiment is no more than its variance under the random (query, advertiser) experiment (see Figure 2.) The ratio of simulated variances for (query, advertiser) randomization versus query randomization is above one for all bid and throttling combinations considered here, and as high as 6 when the treatment 11 tceffe eurt eht ot evitaler saiB 1 3 Nq 2 3 Nq ##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##

text: For

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text: Charles F Manski. Identiﬁcation of treatment response with social interactions. The Econo- metrics Journal, 16(1):S1–S23, 2013. Michael Ostrovsky and Michael Schwarz. Reserve prices in internet advertising auctions: A ﬁeld experiment. In Proceedings of the 12th ACM conference on Electronic commerce, pages 59–60. ACM, 2011. David H Reiley. Field experiments on the eﬀects of reserve prices in auctions: More magic on the internet. RAND Journal of Economics, pages 195–211, 2006. Paul R Rosenbaum. Interference between units in randomized experiments. Journal of the American Statistical Association, 102(477), 2007. D. B. Rubin. Comment: Neyman (1923) and causal inference in experiments and observa- tional studies. Statistical Science, 5(4):472–480, 1990. Donald B Rubin. Comment. Journal of the American Statistical Association, 75(371):591– 593, 1980. Eric J Tchetgen Tchetgen and Tyler J VanderWeele. On causal inference in the presence of interference. Statistical Methods in Medical Research, page 0962280210386779, 2010. David Walker and Lev Muchnik. Design of randomized experiments in networks. Proceedings of the IEEE, 102(12):1940–1951, 2014. 15

text: Charles

label: reference

text: F

text: Manski.

text: Identiﬁcation

text: of

text: treatment

text: response

text: with

text: social

text: interactions.

text: The

text: Econo-

text: metrics

text: Journal,

text: Michael

text: Ostrovsky

text: and

text: Michael

text: Schwarz.

text: Reserve

text: prices

text: in

text: internet

text: advertising

text: auctions:

text: A

text: ﬁeld

text: experiment.

text: In

text: Proceedings

text: of

text: the

text: ACM

text: conference

text: on

text: Electronic

text: commerce,

text: pages

text: ACM,

text: David

text: H

text: Reiley.

text: Field

text: experiments

text: on

text: the

text: eﬀects

text: of

text: reserve

text: prices

text: in

text: auctions:

text: More

text: magic

text: on

text: the

text: internet.

text: RAND

text: Journal

text: of

text: Economics,

text: pages

text: Paul

text: R

text: Rosenbaum.

text: Interference

text: between

text: units

text: in

text: randomized

text: experiments.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: D.

text: B.

text: Rubin.

text: Comment:

text: Neyman

text: and

text: causal

text: inference

text: in

text: experiments

text: and

text: observa-

text: tional

text: studies.

text: Statistical

text: Science,

text: Donald

text: B

text: Rubin.

text: Comment.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: Eric

text: J

text: Tchetgen

text: Tchetgen

text: and

text: Tyler

text: J

text: VanderWeele.

text: On

text: causal

text: inference

text: in

text: the

text: presence

text: of

text: interference.

text: Statistical

text: Methods

text: in

text: Medical

text: Research,

text: page

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text: Walker

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text: Muchnik.

text: Design

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text: networks.

text: Proceedings

text: of

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text: IEEE,


text: 1 INTRODUCTION A search query generates a request for ads to show with search results. A user’s visit to a webpage generates a request to an ad exchange for page ads. In either case, advertisers are chosen to participate in an auction that chooses the ad to be shown. Advertisers often cannot pay for or process all auction requests they are eligible for, so the requests passed on to them are throttled to meet their quota constraints. Auction parameters like reserve prices and throttling schemes can aﬀect an advertiser’s payments and an exchange’s revenue, so A/B experiments are run to test ideas for improving outcomes for advertisers and the ad exchange. Lucking-Reiley (1999) and Einav et al. (2011) experimented with auction formats, Reiley (2006) and Ostrovsky and Schwarz (2011) experi- mented with reserve prices, and Ausubel et al. (2013) experimented with budget constraints. Usually, A/B experiments are analyzed by assuming that the two experiment arms, tra- ditionally called treatment and control, are independent. But, as Blake and Coey (2014) explains, independence fails when the demands of the treatment and control arms aﬀect each other. Such interference is unavoidable when some advertisers in an auction are in treatment and some in control. Kohavi et al. (2009) recognizes that some randomization schemes can give misleading treatment estimates for auction experiments. For more insight into interfer- ence in other applications see Halloran and Struchiner (1995), Hudgens and Halloran (2012), Rosenbaum (2007), Tchetgen and VanderWeele (2010), Aronow and Samii (2013), Eckles et al. (2014), Athey et al. (2015) and Airoldi et al. (2012). Optimal experiment design in the presence of interference has been explored by David and Kempton (1996), Eckles et al. (2014) and Walker and Muchnik (2014). This paper uses the framework of causal models and interference to shed light on auction experiments. Section 2 introduces the main elements of our model: 1. potential outcomes that describe what each advertiser would bid if assigned to treat- ment or if assigned to control, 2. throttling algorithms that determine which of the advertisers eligible to respond to a request for an ad are called to its auction, 3. bid treatments that aﬀect what advertisers bid and throttling treatments that aﬀect when they are called to bid, and 4. eﬀects on the total daily payment of an advertiser or the total daily revenue of the ad exchange. Section 3 deﬁnes two randomization schemes for auction experiments. In query randomiza- tion each request for an ad is randomized to treatment or control, so all participants in an auction are in treatment or all are in control. In (query, advertiser) randomization, each advertiser that is eligible for a query is independently assigned to treatment or control, so treatment bidders and control bidders can compete in the same auction. With either kind of 1

text: INTRODUCTION

text: A

text: search

text: query

text: generates

text: a

text: request

text: for

text: ads

text: to

text: show

text: with

text: search

text: results.

text: A

text: user’s

text: visit

text: to

text: a

text: webpage

text: generates

text: a

text: request

text: to

text: an

text: ad

text: exchange

text: for

text: page

text: ads.

text: In

text: either

text: case,

text: advertisers

text: are

text: chosen

text: to

text: participate

text: in

text: an

text: auction

text: that

text: chooses

text: the

text: ad

text: to

text: be

text: shown.

text: Advertisers

text: often

text: cannot

text: pay

text: for

text: or

text: process

text: all

text: auction

text: requests

text: they

text: are

text: eligible

text: for,

text: so

text: the

text: requests

text: passed

text: on

text: to

text: them

text: are

text: throttled

text: to

text: meet

text: their

text: quota

text: constraints.

text: Auction

text: parameters

text: like

text: reserve

text: prices

text: and

text: throttling

text: schemes

text: can

text: aﬀect

text: an

text: advertiser’s

text: payments

text: and

text: an

text: exchange’s

text: revenue,

text: so

text: A/B

text: experiments

text: are

text: run

text: to

text: test

text: ideas

text: for

text: improving

text: outcomes

text: for

text: advertisers

text: and

text: the

text: ad

text: exchange.

text: Lucking-Reiley

text: and

text: Einav

text: et

text: al.

text: experimented

text: with

text: auction

text: formats,

text: Reiley

text: and

text: Ostrovsky

text: and

text: Schwarz

text: experi-

text: mented

text: with

text: reserve

text: prices,

text: and

text: Ausubel

text: et

text: al.

text: experimented

text: with

text: budget

text: constraints.

text: Usually,

text: A/B

text: experiments

text: are

text: analyzed

text: by

text: assuming

text: that

text: the

text: two

text: experiment

text: arms,

text: tra-

text: ditionally

text: called

text: treatment

text: and

text: control,

text: are

text: independent.

text: But,

text: as

text: Blake

text: and

text: Coey

text: explains,

text: independence

text: fails

text: when

text: the

text: demands

text: of

text: the

text: treatment

text: and

text: control

text: arms

text: aﬀect

text: each

text: other.

text: Such

text: interference

text: is

text: unavoidable

text: when

text: some

text: advertisers

text: in

text: an

text: auction

text: are

text: in

text: treatment

text: and

text: some

text: in

text: control.

text: Kohavi

text: et

text: al.

text: recognizes

text: that

text: some

text: randomization

text: schemes

text: can

text: give

text: misleading

text: treatment

text: estimates

text: for

text: auction

text: experiments.

text: For

text: more

text: insight

text: into

text: interfer-

text: ence

text: in

text: other

text: applications

text: see

text: Halloran

text: and

text: Struchiner

text: Hudgens

text: and

text: Halloran

text: Rosenbaum

text: Tchetgen

text: and

text: VanderWeele

text: Aronow

text: and

text: Samii

text: Eckles

text: et

text: al.

text: Athey

text: et

text: al.

text: and

text: Airoldi

text: et

text: al.

text: Optimal

text: experiment

text: design

text: in

text: the

text: presence

text: of

text: interference

text: has

text: been

text: explored

text: by

text: David

text: and

text: Kempton

text: Eckles

text: et

text: al.

text: and

text: Walker

text: and

text: Muchnik

text: This

text: paper

text: uses

text: the

text: framework

text: of

text: causal

text: models

text: and

text: interference

text: to

text: shed

text: light

text: on

text: auction

text: experiments.

text: Section

text: introduces

text: the

text: main

text: elements

text: of

text: our

text: model:

text: potential

text: outcomes

text: that

text: describe

text: what

text: each

text: advertiser

text: would

text: bid

text: if

text: assigned

text: to

text: treat-

text: ment

text: or

text: if

text: assigned

text: to

text: control,

text: throttling

text: algorithms

text: that

text: determine

text: which

text: of

text: the

text: advertisers

text: eligible

text: to

text: respond

text: to

text: a

text: request

text: for

text: an

text: ad

text: are

text: called

text: to

text: its

text: auction,

text: bid

text: treatments

text: that

text: aﬀect

text: what

text: advertisers

text: bid

text: and

text: throttling

text: treatments

text: that

text: aﬀect

text: when

text: they

text: are

text: called

text: to

text: bid,

text: and

text: eﬀects

text: on

text: the

text: total

text: daily

text: payment

text: of

text: an

text: advertiser

text: or

text: the

text: total

text: daily

text: revenue

text: of

text: the

text: ad

text: exchange.

text: Section

text: deﬁnes

text: two

text: randomization

text: schemes

text: for

text: auction

text: experiments.

text: In

text: query

text: randomiza-

text: tion

text: each

text: request

text: for

text: an

text: ad

text: is

text: randomized

text: to

text: treatment

text: or

text: control,

text: so

text: all

text: participants

text: in

text: an

text: auction

text: are

text: in

text: treatment

text: or

text: all

text: are

text: in

text: control.

text: In

text: (query,

text: advertiser)

text: randomization,

text: each

text: advertiser

text: that

text: is

text: eligible

text: for

text: a

text: query

text: is

text: independently

text: assigned

text: to

text: treatment

text: or

text: control,

text: so

text: treatment

text: bidders

text: and

text: control

text: bidders

text: can

text: compete

text: in

text: the

text: same

text: auction.

text: With

text: either

text: kind

text: of


text: An advertiser does not know which other advertisers are eligible for a query so its bid is independent of all other bids for the query, which implies that the following assumption given in Rubin (1980) holds. Assumption 1 (Stable Unit Treatment Value (SUTVA)). There is only one version of the treatment and there is no interference in the potential bids for eligible advertisers in treatment and control. Because SUTVA holds for bids, the potential bids for all query, eligible advertiser pairs under treatment B((cid:126)1) and under control B((cid:126)0) can be considered separately. 2.2 Potential payments If bidder i’s payment is positive, then any other bidder in the auction pays zero, so a treatment that aﬀects what advertisers bid can aﬀect the payment for both treatment and control advertisers. For example, suppose there are three advertisers in an auction with the potential bids given in the following table and the winner pays what it bid. advertiser B(1) B(0) b 5 2 1 b 4 4 2 b 3 1 3 If all three advertisers participate in the auction and advertiser 2 is the only one assigned to treatment, then it wins and pays Y ((0, 1, 0)) = 4. However, if advertiser 1 is also assigned 2 to treatment and participates in the auction then advertiser 2 loses and pays nothing. This is an example of interference: the payment of an advertiser is aﬀected by the assignment of other advertisers in the auction to treatment and control. Hence, SUTVA does not hold for payments. We make the following assumption about payments. Assumption 2. The potential payment of an advertiser eligible for a query depends only on its assignment to treatment or control and the assignments of the other eligible advertisers for the query. That is Y (Z) = Y (Z ) (1) i i q[i] where Z is the subset of the assignment vector Z on (query, advertiser) pairs for which q[i] the query is q[i]. We extend this assumption to auctions with quota throttling in Section 2.3. 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: An

text: advertiser

text: does

text: not

text: know

text: which

text: other

text: advertisers

text: are

text: eligible

text: for

text: a

text: query

text: so

text: its

text: bid

text: is

text: independent

text: of

text: all

text: other

text: bids

text: for

text: the

text: query,

text: which

text: implies

text: that

text: the

text: following

text: assumption

text: given

text: in

text: Rubin

text: holds.

text: Assumption

text: (Stable

text: Unit

text: Treatment

text: Value

text: (SUTVA)).

text: There

text: is

text: only

text: one

text: version

text: of

text: the

text: treatment

text: and

text: there

text: is

text: no

text: interference

text: in

text: the

text: potential

text: bids

text: for

text: eligible

text: advertisers

text: in

text: treatment

text: and

text: control.

text: Because

text: SUTVA

text: holds

text: for

text: bids,

text: the

text: potential

text: bids

text: for

text: all

text: query,

text: eligible

text: advertiser

text: pairs

text: under

text: treatment

text: and

text: under

text: control

text: can

text: be

text: considered

text: separately.

text: Potential

text: payments

text: If

text: bidder

text: i’s

text: payment

text: is

text: positive,

text: then

text: any

text: other

text: bidder

text: in

text: the

text: auction

text: pays

text: zero,

text: so

text: a

text: treatment

text: that

text: aﬀects

text: what

text: advertisers

text: bid

text: can

text: aﬀect

text: the

text: payment

text: for

text: both

text: treatment

text: and

text: control

text: advertisers.

text: For

text: example,

text: suppose

text: there

text: are

text: three

text: advertisers

text: in

text: an

text: auction

text: with

text: the

text: potential

text: bids

text: given

text: in

text: the

text: following

text: table

text: and

text: the

text: winner

text: pays

text: what

text: it

text: bid.

text: advertiser

text: b

text: b

text: b

text: If

text: all

text: three

text: advertisers

text: participate

text: in

text: the

text: auction

text: and

text: advertiser

text: is

text: the

text: only

text: one

text: assigned

text: to

text: treatment,

text: then

text: it

text: wins

text: and

text: pays

text: Y

text: However,

text: if

text: advertiser

text: is

text: also

text: assigned

text: to

text: treatment

text: and

text: participates

text: in

text: the

text: auction

text: then

text: advertiser

text: loses

text: and

text: pays

text: nothing.

text: This

text: is

text: an

text: example

text: of

text: interference:

text: the

text: payment

text: of

text: an

text: advertiser

text: is

text: aﬀected

text: by

text: the

text: assignment

text: of

text: other

text: advertisers

text: in

text: the

text: auction

text: to

text: treatment

text: and

text: control.

text: Hence,

text: SUTVA

text: does

text: not

text: hold

text: for

text: payments.

text: We

text: make

text: the

text: following

text: assumption

text: about

text: payments.

text: Assumption

text: The

text: potential

text: payment

text: of

text: an

text: advertiser

text: eligible

text: for

text: a

text: query

text: depends

text: only

text: on

text: its

text: assignment

text: to

text: treatment

text: or

text: control

text: and

text: the

text: assignments

text: of

text: the

text: other

text: eligible

text: advertisers

text: for

text: the

text: query.

text: That

text: is

text: Y

text: (Z)

text: Y

text: (Z

text: i

text: i

text: q[i]

text: where

text: Z

text: is

text: the

text: subset

text: of

text: the

text: assignment

text: vector

text: Z

text: on

text: (query,

text: advertiser)

text: pairs

text: for

text: which

text: q[i]

text: the

text: query

text: is

text: q[i].

text: We

text: extend

text: this

text: assumption

text: to

text: auctions

text: with

text: quota

text: throttling

text: in

text: Section

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: bids satisfy B(Z) = B((cid:126)0) = B((cid:126)1). (3) In a throttling experiment, any eligible advertiser would bid the same amount, whether it is throttled or not. This is true if advertisers do not reveal their bids before throttling. 2.5 Eﬀects on revenue If no advertiser is quota constrained, then no advertiser is dropped from a query and the eﬀect τ of the treatment on the total revenue of the ad exchange compares the revenue when every (query, advertiser) pair is treated to the revenue when every (query, advertiser) pair is in control: (cid:88) N − τ = (Y ((cid:126)1) Y ((cid:126)0)) (4) i i i where Y is the payment of (query, advertiser) pair i. If there are quota constraints, then i ∗ τ is aﬀected by random dropping so the eﬀect on total revenue is τ = E(τ), taking the expectation under the dropping scheme p(W ). Note that the sum in this case is taken over the N (query, eligible advertiser) pairs. The eﬀect τ of treatment on the revenue generated by a given advertiser a when there a are no quota constraints is given by (cid:88) − τ = (Y ((cid:126)1) Y ((cid:126)0)). (5) a i i i:a[i]=a ∗ Again, with random throttling the eﬀect of interest is τ = E(τ ), taking the expectation a a under the dropping scheme p(W ). Note that τ considers the eﬀect of treating all eligible a advertisers on the revenue generated only by advertiser a, rather than the eﬀect of treating only the queries for advertiser a. 3 RANDOMIZATION With query randomization, each query is randomized to treatment or control and then all the eligible advertisers for the query are assigned to control if the query is assigned to control or to treatment if the query is assigned to treatment. Formally, an auction experiment is query-randomized if 5

text: bids

text: satisfy

text: B(Z)

text: In

text: a

text: throttling

text: experiment,

text: any

text: eligible

text: advertiser

text: would

text: bid

text: the

text: same

text: amount,

text: whether

text: it

text: is

text: throttled

text: or

text: not.

text: This

text: is

text: true

text: if

text: advertisers

text: do

text: not

text: reveal

text: their

text: bids

text: before

text: throttling.

text: Eﬀects

text: on

text: revenue

text: If

text: no

text: advertiser

text: is

text: quota

text: constrained,

text: then

text: no

text: advertiser

text: is

text: dropped

text: from

text: a

text: query

text: and

text: the

text: eﬀect

text: τ

text: of

text: the

text: treatment

text: on

text: the

text: total

text: revenue

text: of

text: the

text: ad

text: exchange

text: compares

text: the

text: revenue

text: when

text: every

text: (query,

text: advertiser)

text: pair

text: is

text: treated

text: to

text: the

text: revenue

text: when

text: every

text: (query,

text: advertiser)

text: pair

text: is

text: in

text: control:

text: N

text: τ

text: (Y

text: Y

text: i

text: i

text: i

text: where

text: Y

text: is

text: the

text: payment

text: of

text: (query,

text: advertiser)

text: pair

text: i.

text: If

text: there

text: are

text: quota

text: constraints,

text: then

text: i

text: τ

text: is

text: aﬀected

text: by

text: random

text: dropping

text: so

text: the

text: eﬀect

text: on

text: total

text: revenue

text: is

text: τ

text: E(τ),

text: taking

text: the

text: expectation

text: under

text: the

text: dropping

text: scheme

text: p(W

text: Note

text: that

text: the

text: sum

text: in

text: this

text: case

text: is

text: taken

text: over

text: the

text: N

text: (query,

text: eligible

text: advertiser)

text: pairs.

text: The

text: eﬀect

text: τ

text: of

text: treatment

text: on

text: the

text: revenue

text: generated

text: by

text: a

text: given

text: advertiser

text: a

text: when

text: there

text: a

text: are

text: no

text: quota

text: constraints

text: is

text: given

text: by

text: τ

text: (Y

text: Y

text: a

text: i

text: i

text: i:a[i]=a

text: Again,

text: with

text: random

text: throttling

text: the

text: eﬀect

text: of

text: interest

text: is

text: τ

text: E(τ

text: taking

text: the

text: expectation

text: a

text: a

text: under

text: the

text: dropping

text: scheme

text: p(W

text: Note

text: that

text: τ

text: considers

text: the

text: eﬀect

text: of

text: treating

text: all

text: eligible

text: a

text: advertisers

text: on

text: the

text: revenue

text: generated

text: only

text: by

text: advertiser

text: a,

text: rather

text: than

text: the

text: eﬀect

text: of

text: treating

text: only

text: the

text: queries

text: for

text: advertiser

text: a.

text: RANDOMIZATION

text: With

text: query

text: randomization,

text: each

text: query

text: is

text: randomized

text: to

text: treatment

text: or

text: control

text: and

text: then

text: all

text: the

text: eligible

text: advertisers

text: for

text: the

text: query

text: are

text: assigned

text: to

text: control

text: if

text: the

text: query

text: is

text: assigned

text: to

text: control

text: or

text: to

text: treatment

text: if

text: the

text: query

text: is

text: assigned

text: to

text: treatment.

text: Formally,

text: an

text: auction

text: experiment

text: is

text: query-randomized

text: if


text: (cid:104) (cid:105) (cid:126) m H − H − H − H ∆EVC = 2 Tr⊕4a=1L2a(R2) ( +)12 Tr⊕4a=1L2a(R2) ( 0)12 [TrL2(R2) ( G)12 TrL2(R2) ( 0G)12 ] (37) H − H − where we recall that = I∆+diag (1, 1, 1, 1) and G = ∆+1 are the corresponding second-order 0 0 vacuum ﬂuctuation operators. The zeta function regularization procedure takes proﬁt of the analytical continuation of the divergent quantity ∆EC (37) to the s-complex plane and assigning to the vortex Casimir energy its ﬁnite value V at a regular point. This strategy is justiﬁed from the general theory about the analytical structure of spectral zeta functions of positive operators, in our problem we shall consider the spectral zeta functions H H H H of the PDO +, , G, and G.(cid:16)Thu(cid:17)s,(cid:110)we shall regularize t(cid:16)he vortex Casimir(cid:17)e(cid:111)nergy in the form: 0 0 (cid:126) µ µ2 s − − − ∆EC(s) = ζH (s) ζH (s) ζH (x) ζH (s) , (38) V 2 m2 + 0 G 0G − where µ is a parameter of dimension L 1 needed to keep correct the physical dimensions of energy − away from the physical value s = 1: ∆EC = lim →− ∆EC(s). 2 V s 1 V 2 The spectral heat kernel/zeta function control of divergences in QFT is a procedure that encompasses several diﬀerent but related aspects. 1. Ultraviolet divergences arising in ﬂuctuating topological defects are regularized by using the spectral zeta function of the Hessian operator. In odd dimensional spaces the zeta function giving the − Casimir energy falls in a pole at s = 1 and one must go away from the pole in the s-complex 2 plane to obtain a regularization of ∆EC, but in even dimensions the spectral zeta function is directly − ﬁnite at the value of s = 1. 2 2. The meromorphic structure of the spectral zeta function is clariﬁed when it is obtained via Mellin transform of the heat kernel high temperature expansion. Poles appear in Euler Gamma functions − − Γ(s + n d), i.e., at negative integers or zero values of s + n d. Also, infrared divergences appear 2 2 in the lower Seeley coeﬃcients. Integration of low densities over the whole space gives rise to divergences proportional to the volume, or, the logarithm of the volume, etcetera. Regularization of these divergences requires to restrict the system to a cube of volume V = ld = (mL)d. 3. After these regularizations were performed some renormalizations have to be done. In (1 + 1)- or (2 + 1)-dimensional space-times, where QFT models are usually superrenormalizable, zero point and mass renormalization, taming the divergences due to the tadpoles and self-energy graphs, are enough. 4. It remains to deal with the delicate question of ﬁnite renormalizations. We shall stick to the heat kernel renormalization criterion, tantamount to the vanishing of the tadpole graph. In the limit of inﬁnite particle masses there are no quantum ﬂuctuations, thus there should be no quantum corrections. This means that the contribution of all the coeﬃcients multiplied by non negative powers of mass must be exactly cancelled in the renormalization process. In one and two spatial dimensions only c and c survive when the particles become inﬁnitely heavy and the annihilation 0 1 of their contribution ﬁxes our renormalization criterion. 5. Zero modes, however, respond to rigid motions which survive in the inﬁnite mass regime and the above criterion does not apply to their contributions. 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: m

text: H

text: H

text: H

text: H

text: ∆EVC

text: H

text: H

text: where

text: we

text: recall

text: that

text: I∆+diag

text: and

text: G

text: are

text: the

text: corresponding

text: second-order

text: vacuum

text: ﬂuctuation

text: operators.

text: The

text: zeta

text: function

text: regularization

text: procedure

text: takes

text: proﬁt

text: of

text: the

text: analytical

text: continuation

text: of

text: the

text: divergent

text: quantity

text: ∆EC

text: to

text: the

text: s-complex

text: plane

text: and

text: assigning

text: to

text: the

text: vortex

text: Casimir

text: energy

text: its

text: ﬁnite

text: value

text: V

text: at

text: a

text: regular

text: point.

text: This

text: strategy

text: is

text: justiﬁed

text: from

text: the

text: general

text: theory

text: about

text: the

text: analytical

text: structure

text: of

text: spectral

text: zeta

text: functions

text: of

text: positive

text: operators,

text: in

text: our

text: problem

text: we

text: shall

text: consider

text: the

text: spectral

text: zeta

text: functions

text: H

text: H

text: H

text: H

text: of

text: the

text: PDO

text: G,

text: and

text: shall

text: regularize

text: vortex

text: in

text: the

text: form:

text: µ

text: µ2

text: s

text: ∆EC(s)

text: ζH

text: (s)

text: ζH

text: (s)

text: ζH

text: (x)

text: ζH

text: (s)

text: V

text: G

text: where

text: µ

text: is

text: a

text: parameter

text: of

text: dimension

text: L

text: needed

text: to

text: keep

text: correct

text: the

text: physical

text: dimensions

text: of

text: energy

text: away

text: from

text: the

text: physical

text: value

text: s

text: ∆EC

text: lim

text: ∆EC(s).

text: V

text: s

text: V

text: The

text: spectral

text: heat

text: kernel/zeta

text: function

text: control

text: of

text: divergences

text: in

text: QFT

text: is

text: a

text: procedure

text: that

text: encompasses

text: several

text: diﬀerent

text: but

text: related

text: aspects.

text: Ultraviolet

text: divergences

text: arising

text: in

text: ﬂuctuating

text: topological

text: defects

text: are

text: regularized

text: by

text: using

text: the

text: spectral

text: zeta

text: function

text: of

text: the

text: Hessian

text: operator.

text: In

text: odd

text: dimensional

text: spaces

text: the

text: zeta

text: function

text: giving

text: the

text: Casimir

text: energy

text: falls

text: in

text: a

text: pole

text: at

text: s

text: and

text: one

text: must

text: go

text: away

text: from

text: the

text: pole

text: in

text: the

text: s-complex

text: plane

text: to

text: obtain

text: a

text: regularization

text: of

text: ∆EC,

text: but

text: in

text: even

text: dimensions

text: the

text: spectral

text: zeta

text: function

text: is

text: directly

text: ﬁnite

text: at

text: the

text: value

text: of

text: s

text: The

text: meromorphic

text: structure

text: of

text: the

text: spectral

text: zeta

text: function

text: is

text: clariﬁed

text: when

text: it

text: is

text: obtained

text: via

text: Mellin

text: transform

text: of

text: the

text: heat

text: kernel

text: high

text: temperature

text: expansion.

text: Poles

text: appear

text: in

text: Euler

text: Gamma

text: functions

text: Γ(s

text: n

text: d),

text: i.e.,

text: at

text: negative

text: integers

text: or

text: zero

text: values

text: of

text: s

text: n

text: d.

text: Also,

text: infrared

text: divergences

text: appear

text: in

text: the

text: lower

text: Seeley

text: coeﬃcients.

text: Integration

text: of

text: low

text: densities

text: over

text: the

text: whole

text: space

text: gives

text: rise

text: to

text: divergences

text: proportional

text: to

text: the

text: volume,

text: or,

text: the

text: logarithm

text: of

text: the

text: volume,

text: etcetera.

text: Regularization

text: of

text: these

text: divergences

text: requires

text: to

text: restrict

text: the

text: system

text: to

text: a

text: cube

text: of

text: volume

text: V

text: ld

text: (mL)d.

text: After

text: these

text: regularizations

text: were

text: performed

text: some

text: renormalizations

text: have

text: to

text: be

text: done.

text: In

text: or

text: space-times,

text: where

text: QFT

text: models

text: are

text: usually

text: superrenormalizable,

text: zero

text: point

text: and

text: mass

text: renormalization,

text: taming

text: the

text: divergences

text: due

text: to

text: the

text: tadpoles

text: and

text: self-energy

text: graphs,

text: are

text: enough.

text: It

text: remains

text: to

text: deal

text: with

text: the

text: delicate

text: question

text: of

text: ﬁnite

text: renormalizations.

text: We

text: shall

text: stick

text: to

text: the

text: heat

text: kernel

text: renormalization

text: criterion,

text: tantamount

text: to

text: the

text: vanishing

text: of

text: the

text: tadpole

text: graph.

text: In

text: the

text: limit

text: of

text: inﬁnite

text: particle

text: masses

text: there

text: are

text: no

text: quantum

text: ﬂuctuations,

text: thus

text: there

text: should

text: be

text: no

text: quantum

text: corrections.

text: This

text: means

text: that

text: the

text: contribution

text: of

text: all

text: the

text: coeﬃcients

text: multiplied

text: by

text: non

text: negative

text: powers

text: of

text: mass

text: must

text: be

text: exactly

text: cancelled

text: in

text: the

text: renormalization

text: process.

text: In

text: one

text: and

text: two

text: spatial

text: dimensions

text: only

text: c

text: and

text: c

text: survive

text: when

text: the

text: particles

text: become

text: inﬁnitely

text: heavy

text: and

text: the

text: annihilation

text: of

text: their

text: contribution

text: ﬁxes

text: our

text: renormalization

text: criterion.

text: Zero

text: modes,

text: however,

text: respond

text: to

text: rigid

text: motions

text: which

text: survive

text: in

text: the

text: inﬁnite

text: mass

text: regime

text: and

text: the

text: above

text: criterion

text: does

text: not

text: apply

text: to

text: their

text: contributions.

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text: (cid:16) (cid:17) (cid:110) (cid:68) (cid:69) (cid:10) (cid:11) (cid:126) ∆EC(s) + ∆E(cid:88)R((cid:88)s) = µ µ2 s 1 5(1 − |ψ|2) − 2V V u−2s −(cid:88)1 1 − |ψ|2 u−2s V V 2 m2 4π k k 4π ∞ H ∞ H + 1 (cid:28) 4 [cn( +)−]a(cid:29)a Γ[s + (cid:111)n − 1] −(cid:16)2Nu(cid:17)−(cid:110)2s(cid:16)− 1 (cid:17) (cid:28) cn( G−) Γ[s + n(cid:29)− 1] + 4πΓ[s] u2n+2s 2 4πΓ[s] u2s+2n 2 n=2 a=1 n=2 (cid:126) 1 1 − | | − 1 · − µ µ2 s 1 1 − | | − 1 · − + 1(cid:88) (cid:16)ψ(cid:88)2 V V u 2s = (cid:17) + 1 (cid:111) ψ 2 V V u 2s + 2π s 2 k k 2 m2 π 2πs 2 k k ∞ H H 1 4 [c ( +)] − c ( G) − − − + n −aa n − Γ[s + n 1] 2Nu 2s , 4πΓ[s] u2s+2n 2 u2s+2n 2 n=2 a=1 where we have used Γ(s + 1) = sΓ(s). The key observation is that, according to the heat kernel renormalization criterion, the contribution of the ﬁrst order Seeley coeﬃcients is exactly cancelled by the minimal subtraction scheme chosen in our mass renormalization prescription. This statement can be − easily checked by looking at the ﬁrst term in the last equality at the physical value s = 1. Therefore, 2 the one-loop BPS vortex mass shift is obtained in this approach by the asymptotic formula: ∆E = lim [∆EC(s) + ∆ER(s)] V →− V V s 1 2 which provides us with the ﬁnal res(cid:88)pon(cid:16)se(cid:88) (cid:17) ∞ (cid:126) 4 − m H − H − − (cid:126) ∆E = [c ( +)] c ( G) Γ[n 3] mN . (40) V 3 n aa n 2 16π 2 n=2 a=1 2.4 One-loop mass shifts of BPS rotationally symmetric vortices: surge of weak quantum forces Use of formula (40) guides us towards the computation of one-loop mass shifts for BPS circularly symmetric vortices, solutions of the PDE system (3) of the form N ψ(x , x ) = f (r) eiNθ ; V (r, θ) = 0 , V (r, θ) = β (r) 1 2 N r θ N r √ where r = x1x1 + x2x2 and θ = arctanx2 are polar coordinates in the plane. In this case the just x1 mentioned PDE system becomes the ODE system: (cid:48) N − (cid:48) r − f (r) = f (r)[1 β (r)] ; β (r) = [1 f2 (r)] . (41) N r N N N 2N N The subindex N in f (r) and β (r) reminds us that the radial proﬁles depend on the vorticity N, i.e., N N they are diﬀerent in diﬀerent topological sectors. The well known procedure for ﬁnding solutions of these (cid:39) ordinary equations proceed in three steps: (1) Solving the (41) near r = 0 one ﬁnds f (r) → D rN (cid:39) N r 0 N and β (r) → E r2, where D and E are integration constants, that are regular solutions near the N r 0 N N N → → → ∞ origin. (2) The asymptotic conditions (4) demand that f (r) 1 and β (r) 1 in the r limit. N N One solves then the (41) system very far from the origin. An smooth sewing between the two regimes requires a precise choice of D and E . (3) This shooting procedure is numerically implemented to N N build interpolating solutions to (41) at intermediate distances. In this way the circularly symmetric BPS 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: ∆EC(s)

text: µ

text: µ2

text: s

text: V

text: V

text: V

text: 4π

text: k

text: k

text: 4π

text: H

text: H

text: [cn(

text: Γ[s

text: cn(

text: G−)

text: Γ[s

text: µ

text: µ2

text: s

text: V

text: V

text: u

text: ψ

text: V

text: V

text: u

text: 2π

text: s

text: k

text: k

text: π

text: k

text: k

text: H

text: H

text: [c

text: c

text: G)

text: n

text: −aa

text: n

text: Γ[s

text: n

text: where

text: we

text: have

text: used

text: Γ(s

text: sΓ(s).

text: The

text: key

text: observation

text: is

text: that,

text: according

text: to

text: the

text: heat

text: kernel

text: renormalization

text: criterion,

text: the

text: contribution

text: of

text: the

text: ﬁrst

text: order

text: Seeley

text: coeﬃcients

text: is

text: exactly

text: cancelled

text: by

text: the

text: minimal

text: subtraction

text: scheme

text: chosen

text: in

text: our

text: mass

text: renormalization

text: prescription.

text: This

text: statement

text: can

text: be

text: easily

text: checked

text: by

text: looking

text: at

text: the

text: ﬁrst

text: term

text: in

text: the

text: last

text: equality

text: at

text: the

text: physical

text: value

text: s

text: Therefore,

text: the

text: one-loop

text: BPS

text: vortex

text: mass

text: shift

text: is

text: obtained

text: in

text: this

text: approach

text: by

text: the

text: asymptotic

text: formula:

text: ∆E

text: lim

text: [∆EC(s)

text: ∆ER(s)]

text: V

text: V

text: V

text: s

text: which

text: provides

text: us

text: with

text: the

text: ﬁnal

text: m

text: H

text: H

text: ∆E

text: [c

text: c

text: G)

text: Γ[n

text: mN

text: V

text: n

text: aa

text: n

text: One-loop

text: mass

text: shifts

text: of

text: BPS

text: rotationally

text: symmetric

text: vortices:

text: surge

text: of

text: weak

text: quantum

text: forces

text: Use

text: of

text: formula

text: guides

text: us

text: towards

text: the

text: computation

text: of

text: one-loop

text: mass

text: shifts

text: for

text: BPS

text: circularly

text: symmetric

text: vortices,

text: solutions

text: of

text: the

text: PDE

text: system

text: of

text: the

text: form

text: N

text: ψ(x

text: x

text: f

text: (r)

text: eiNθ

text: V

text: (r,

text: θ)

text: V

text: (r,

text: θ)

text: β

text: (r)

text: N

text: r

text: θ

text: N

text: r

text: where

text: r

text: and

text: θ

text: are

text: polar

text: coordinates

text: in

text: the

text: plane.

text: In

text: this

text: case

text: the

text: just

text: mentioned

text: PDE

text: system

text: becomes

text: the

text: ODE

text: system:

text: N

text: r

text: f

text: (r)

text: f

text: β

text: (r)]

text: β

text: (r)

text: (r)]

text: N

text: r

text: N

text: N

text: N

text: N

text: The

text: subindex

text: N

text: in

text: f

text: (r)

text: and

text: β

text: (r)

text: reminds

text: us

text: that

text: the

text: radial

text: proﬁles

text: depend

text: on

text: the

text: vorticity

text: N,

text: i.e.,

text: N

text: N

text: they

text: are

text: diﬀerent

text: in

text: diﬀerent

text: topological

text: sectors.

text: The

text: well

text: known

text: procedure

text: for

text: ﬁnding

text: solutions

text: of

text: these

text: ordinary

text: equations

text: proceed

text: in

text: three

text: steps:

text: Solving

text: the

text: near

text: r

text: one

text: ﬁnds

text: f

text: (r)

text: D

text: rN

text: N

text: r

text: N

text: and

text: β

text: (r)

text: E

text: where

text: D

text: and

text: E

text: are

text: integration

text: constants,

text: that

text: are

text: regular

text: solutions

text: near

text: the

text: N

text: r

text: N

text: N

text: N

text: origin.

text: The

text: asymptotic

text: conditions

text: demand

text: that

text: f

text: (r)

text: and

text: β

text: (r)

text: in

text: the

text: r

text: limit.

text: N

text: N

text: One

text: solves

text: then

text: the

text: system

text: very

text: far

text: from

text: the

text: origin.

text: An

text: smooth

text: sewing

text: between

text: the

text: two

text: regimes

text: requires

text: a

text: precise

text: choice

text: of

text: D

text: and

text: E

text: This

text: shooting

text: procedure

text: is

text: numerically

text: implemented

text: to

text: N

text: N

text: build

text: interpolating

text: solutions

text: to

text: at

text: intermediate

text: distances.

text: In

text: this

text: way

text: the

text: circularly

text: symmetric

text: BPS

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