text: a special adaptor for Spark is provided. ABDS Application Manager MapReduce Once the cluster is setup, users can to submit applications Application by using the Pilot-API’s Compute-Unit API to start and 1. Start 2. Submit CU 3. Get CU 4. Stop manage application processes. Compute-Units with type Cluster (ABDS App) Status Cluster Hadoop and Spark are then forwarded to the YARN re- spectively Spark resource manager, which then handles the Pilot-API management of these tasks. With this capability, the Pilot- Pilot-Hadoop Abstraction can be used to manage highly heterogeneous workloads, e. g. bag-of-tasks, coupled tasks, MPI, Hadoop Pilot-Compute Pilot-Data and Spark applications, via a single interface. ABDS Plugin 3.3 Pilot-Data Memory: A Processing Engine Yarn Spark Other for Machine Learning The Pilot-Abstraction provides a low-level mechanisms to HPC / Clouds manage the data across diﬀerent, possible distributed, data stores in conjunction with their task-based computing fo- cusing on the stage-in and out of data related to a set of CUs. Also, it required the developer to either manually Figure 4: Pilot-Hadoop for HPC and Cloud Infras- implement data parallelism or use a higher-level framework, tructures: Pilot-Hadoop provides uniform frame- such as Pilot-MapReduce [56]. Another constraint is the fact work to managing Hadoop and Spark clusters on re- that only persistent storage can be used. The usage of (dis- sources managed by HPC schedulers, such as PBS, tributed) memory for caching of input or intermediate data SGE and SLURM. The Pilot-API provides a uni- (e. g. for iterative machine learning) is not supported. While fying interface for allocating a YARN/Spark cluster this disk-based model is eﬀective for doing many forms of on HPC and for managing Compute-Units that mar- large volume data processing, iterative processing, e. g. for shal a YARN, Spark or other ABDS application. machine learning, requires more sophisticated ways to man- age intermediate data. Pilot-Data Memory adds in-memory capabilities to Pilot- Using the API developers can model complex storage hier- Data and makes it available via the Pilot-API. A particular archies consisting of archival storage, cold data storage for challenge is the integration of the in-memory layer with the raw data, warm and hot storage for pre-processed data used compute layer, which typically requires support for a man- in the model ﬁtting phase and memory for intermediate re- ifold set of tools and programming languages. We focus sults of iterative machine learning algorithms. The runtime on providing a Python-based API and runtime; many scien- system will ensure that data/compute will be co-located if tiﬁc applications are implemented in Python, which makes it possible to improve performance. easy to integrate native code to achieve a good performance. 3.2 Pilot-Hadoop: Supporting Application-Level Further, there is a lack of in-memory frameworks and tools Interoperability for ABDS and HPC for Python – Pilot-Data Memory is an attempt to address these limitations. A particular gap is the ability to manage Having discussed the extensions to the Pilot-Abstraction large amounts of distributed memory - while it is fairly sim- to support Hadoop resources, we explore the usage of ple to manage memory on a single node (using e. g. memory Hadoop and other ABDS frameworks on HPC resources in mapped ﬁles), support for distributed memory typically re- this section (see Figure 1 (b)). Pilot-Hadoop [55] provides quires specialized runtime and processing environments that a framework for executing ABDS applications written for are not compatible with traditional Posix ﬁle APIs. YARN (e. g. MapReduce) or Spark on HPC and cloud re- To process data in an in-memory DU, we extend the DU sources. Pilot-Hadoop has been developed as successor to interface to provide a higher-level MapReduce-based API for SAGA-Hadoop [51] utilizing the handling of multi-node clus- expressing transformations on the data. Using a map and re- ter environments of the Pilot-Agent. duce functions, applications can express abstract operations Figure 4 illustrates the architecture of Pilot-Hadoop. on data without manually creating CUs for partitioning and Pilot-Hadoop uses the Pilot-Abstraction and the BigJob processing the data. The API utilizes a key/value pair tu- implementation to manage Hadoop clusters inside an en- ples as input for the map and reduce function. The runtime vironment managed by an HPC scheduler, such as PBS, system generates the necessary application tasks (Compute- SLURM or SGE, or clouds. The Pilot-Framework is used Units) and run these in parallel considering data locality for dispatching a bootstrap process that generates the nec- and other aspects. Users have the possibility to control this essary conﬁguration ﬁles and for starting the Hadoop pro- placement using a simple, label-based aﬃnity model, which cesses. The speciﬁcs of the Hadoop framework (i. e. YARN allows reasoning about compute and data and provides the and Spark) are encapsulated in an adaptor. The boot- runtime system with hints for optimizing execution. The strap process is then responsible for launching YARN’s re- system is data format agnostic and supports heterogeneous source and node manager processes respectively the Spark data schemes (schema on read). master and worker agents on the nodes allocated by the Figure 5 illustrates the architecture of the Pilot-Data Pilot-Framework. While nearly all ABDS frameworks (e. g. Memory framework. There are multiple levels of coordi- MapReduce, Tez and also Spark) support YARN for re- nation and decision making: The Compute-Data-Manager source management, Spark provides a standalone cluster manages a set of memory and disk-backed Pilot-Data and mode, which is more eﬃcient for dedicated resources. Thus, Pilot-Compute. Applications can submit Compute-Units ##LTFigure##

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text: a

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text: utilizing

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text: of

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text: illustrates

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text: partitioning

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text: read).

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text: ##LTFigure##


text: 2 D. Dimitrov and D. Kjurkchieva other targets from this sample: four of them are newly dis- covered eclipsing stars while two of them are identiﬁed with objects from the lists of ultrashort-period binaries of Norton et al. (2011) and Lohr et al. (2014) based on the SuperWASP photometric survey (Pollacco et al. 2006). Besides photometric data for all six targets in the NSVS database we found also data of four of them in the Super- WASP archive (Butters et al. 2010). The periodogram anal- ysis of these low-precise but numerous data allowed us to determine their ephemeris (Table 1). Figure A1 presents the folded curves of the targets corresponding to their old pho- tometric data. 3 ROZHEN OBSERVATIONS AND DATA REDUCTION The follow-up CCD photometry of the targets in V RI bands was carried out with the three telescopes of Rozhen National Astronomical Observatory (Table 2). The 2-m RCC telescope is equipped with VersArray CCD camera × × (1340 1300 pixels, 20 µm/pixel, ﬁeld of 5.35 5.25 ar- cmin). The 60-cm Cassegrain telescope is equipped with FLI × PL09000 CCD camera (3056 3056 pixels, 12 µm/pixel, × ﬁeld of 17.1 17.1 arcmin). The 50/70-cm Schmidt tele- ◦ scope has FoV around 1 and is equipped with CCD camera Figure 1. Top: the folded light curves of NSVS 4484038 and × FLI PL 16803, 4096 4096 pixels, 9 µm/pixel size. their ﬁts (continuous lines); Bottom: the corresponding residuals. Standard stars of Landolt (1992) and standard ﬁelds of The photometric data in some ﬁlters are shifted vertically by diﬀerent number for a better visibility. Color version of this ﬁgure Stetson (2000) were used for transition from instrumental is available in the online journal. system to standard photometric system. The standard IDL procedures (adapted from DAOPHOT) were used for reduction of the photo- metric data. More than 5 standard stars were chosen in the observed ﬁelds of each target by the requirement to be constant within 0.015 mag during the all observational runs − − and in all ﬁlters. Table A1 presents their V , B V , V R, − and V I determined by our observations (for the targets − they correspond to phase 0.25) as well as their J K colors from the 2MASS catalog. Figures 1-6 present the follow-up Rozhen photometry of our targets. Table A2 presents a sample of the Rozhen photometric data (the full table is available in the online version of the article, see Supporting Information). We calculated times of the observed light minima by the method described in Pribulla et al. (2012). Table A3 − contains their HJD(Min), Epoch, and O C. We obtained low-resolution spectra (Fig. 7) during 5 nights in Feb-May 2011 by the 2-m RCC telescope equipped with focal reducer FoReRo-2. We used grism with 300 lines/mm that allows resolution 5.223 ˚A/pixel in the range 5000-7000 ˚A. These low-resolution spectra do not allow ra- dial velocity measurement but are useful indicators of the temperature and stellar activity. 4 LIGHT CURVE SOLUTIONS Figure 2. Same as Fig. 1 for NSVS 7179685. The photometric data (Figs. 1-6) implied that our tar- gets are contact or overcontact systems that was expected for their ultra-short orbital periods. tures of the targets obtained by comparison of their low- We carried out modeling of the Rozhen photometric resolution spectra (Fig. 7) with those of standard stars with data by the code PHOEBE (Pr´sa & Zwitter 2005) using known temperatures, were almost the same as those of the the following procedure. values of T in Table 3. m We determined in advance the mean temperatures T Firstly we assumed T0 = T and searched for solutions m 1 m of the binaries (Table 3) by their infrared color indices for ﬁxed T0 and q0 = 1 varying the ephemeris, secondary − 1 (J K) from the 2MASS catalog and the calibration color- temperature T , orbital inclination i and potentials Ω . 2 1,2 temperature of Tokunaga (2000). The ”spectral” tempera- We adopted coeﬃcients of gravity brightening 0.32 and (cid:13) c 2014 RAS, MNRAS 000, 1–?? ##LTFigure## ##LTFigure##

label: paragraph

text: D.

text: Dimitrov

text: and

text: D.

text: Kjurkchieva

text: other

text: targets

text: from

text: this

text: sample:

text: four

text: of

text: them

text: are

text: newly

text: dis-

text: covered

text: eclipsing

text: stars

text: while

text: two

text: of

text: them

text: are

text: identiﬁed

text: with

text: objects

text: from

text: the

text: lists

text: of

text: ultrashort-period

text: binaries

text: of

text: Norton

text: et

text: al.

text: and

text: Lohr

text: et

text: al.

text: based

text: on

text: the

text: SuperWASP

text: photometric

text: survey

text: (Pollacco

text: et

text: al.

text: Besides

text: photometric

text: data

text: for

text: all

text: six

text: targets

text: in

text: the

text: NSVS

text: database

text: we

text: found

text: also

text: data

text: of

text: four

text: of

text: them

text: in

text: the

text: Super-

text: WASP

text: archive

text: (Butters

text: et

text: al.

text: The

text: periodogram

text: anal-

text: ysis

text: of

text: these

text: low-precise

text: but

text: numerous

text: data

text: allowed

text: us

text: to

text: determine

text: their

text: ephemeris

text: (Table

text: Figure

text: presents

text: the

text: folded

text: curves

text: of

text: the

text: targets

text: corresponding

text: to

text: their

text: old

text: pho-

text: tometric

text: data.

text: ROZHEN

text: OBSERVATIONS

text: AND

text: DATA

text: REDUCTION

text: The

text: follow-up

text: CCD

text: photometry

text: of

text: the

text: targets

text: in

text: V

text: RI

text: bands

text: was

text: carried

text: out

text: with

text: the

text: three

text: telescopes

text: of

text: Rozhen

text: National

text: Astronomical

text: Observatory

text: (Table

text: The

text: RCC

text: telescope

text: is

text: equipped

text: with

text: VersArray

text: CCD

text: camera

text: pixels,

text: µm/pixel,

text: ﬁeld

text: of

text: ar-

text: cmin).

text: The

text: Cassegrain

text: telescope

text: is

text: equipped

text: with

text: FLI

text: CCD

text: camera

text: pixels,

text: µm/pixel,

text: ﬁeld

text: of

text: arcmin).

text: The

text: Schmidt

text: tele-

text: scope

text: has

text: FoV

text: around

text: and

text: is

text: equipped

text: with

text: CCD

text: camera

text: Figure

text: Top:

text: the

text: folded

text: light

text: curves

text: of

text: NSVS

text: and

text: FLI

text: PL

text: pixels,

text: µm/pixel

text: size.

text: their

text: ﬁts

text: (continuous

text: lines);

text: Bottom:

text: the

text: corresponding

text: residuals.

text: Standard

text: stars

text: of

text: Landolt

text: and

text: standard

text: ﬁelds

text: of

text: The

text: photometric

text: data

text: in

text: some

text: ﬁlters

text: are

text: shifted

text: vertically

text: by

text: diﬀerent

text: number

text: for

text: a

text: better

text: visibility.

text: Color

text: version

text: of

text: this

text: ﬁgure

text: Stetson

text: were

text: used

text: for

text: transition

text: from

text: instrumental

text: is

text: available

text: in

text: the

text: online

text: journal.

text: system

text: to

text: standard

text: photometric

text: system.

text: The

text: standard

text: IDL

text: procedures

text: (adapted

text: from

text: DAOPHOT)

text: were

text: used

text: for

text: reduction

text: of

text: the

text: photo-

text: metric

text: data.

text: More

text: than

text: standard

text: stars

text: were

text: chosen

text: in

text: the

text: observed

text: ﬁelds

text: of

text: each

text: target

text: by

text: the

text: requirement

text: to

text: be

text: constant

text: within

text: mag

text: during

text: the

text: all

text: observational

text: runs

text: and

text: in

text: all

text: ﬁlters.

text: Table

text: presents

text: their

text: V

text: B

text: V

text: V

text: R,

text: and

text: V

text: I

text: determined

text: by

text: our

text: observations

text: (for

text: the

text: targets

text: they

text: correspond

text: to

text: phase

text: as

text: well

text: as

text: their

text: J

text: K

text: colors

text: from

text: the

text: catalog.

text: Figures

text: present

text: the

text: follow-up

text: Rozhen

text: photometry

text: of

text: our

text: targets.

text: Table

text: presents

text: a

text: sample

text: of

text: the

text: Rozhen

text: photometric

text: data

text: (the

text: full

text: table

text: is

text: available

text: in

text: the

text: online

text: version

text: of

text: the

text: article,

text: see

text: Supporting

text: Information).

text: We

text: calculated

text: times

text: of

text: the

text: observed

text: light

text: minima

text: by

text: the

text: method

text: described

text: in

text: Pribulla

text: et

text: al.

text: Table

text: contains

text: their

text: HJD(Min),

text: Epoch,

text: and

text: O

text: C.

text: We

text: obtained

text: low-resolution

text: spectra

text: (Fig.

text: during

text: nights

text: in

text: Feb-May

text: by

text: the

text: RCC

text: telescope

text: equipped

text: with

text: focal

text: reducer

text: We

text: used

text: grism

text: with

text: lines/mm

text: that

text: allows

text: resolution

text: ˚A/pixel

text: in

text: the

text: range

text: ˚A.

text: These

text: low-resolution

text: spectra

text: do

text: not

text: allow

text: ra-

text: dial

text: velocity

text: measurement

text: but

text: are

text: useful

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text: of

text: the

text: temperature

text: and

text: stellar

text: activity.

text: LIGHT

text: CURVE

text: SOLUTIONS

text: Figure

text: Same

text: as

text: Fig.

text: for

text: NSVS

text: The

text: photometric

text: data

text: (Figs.

text: implied

text: that

text: our

text: tar-

text: gets

text: are

text: contact

text: or

text: overcontact

text: systems

text: that

text: was

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text: for

text: their

text: ultra-short

text: orbital

text: periods.

text: tures

text: of

text: the

text: targets

text: obtained

text: by

text: comparison

text: of

text: their

text: low-

text: We

text: carried

text: out

text: modeling

text: of

text: the

text: Rozhen

text: photometric

text: resolution

text: spectra

text: (Fig.

text: with

text: those

text: of

text: standard

text: stars

text: with

text: data

text: by

text: the

text: code

text: PHOEBE

text: (Pr´sa

text: Zwitter

text: using

text: known

text: temperatures,

text: were

text: almost

text: the

text: same

text: as

text: those

text: of

text: the

text: the

text: following

text: procedure.

text: values

text: of

text: T

text: in

text: Table

text: m

text: We

text: determined

text: in

text: advance

text: the

text: mean

text: temperatures

text: T

text: Firstly

text: we

text: assumed

text: T

text: and

text: searched

text: for

text: solutions

text: m

text: m

text: of

text: the

text: binaries

text: (Table

text: by

text: their

text: infrared

text: color

text: indices

text: for

text: ﬁxed

text: and

text: varying

text: the

text: ephemeris,

text: secondary

text: (J

text: K)

text: from

text: the

text: catalog

text: and

text: the

text: calibration

text: color-

text: temperature

text: T

text: orbital

text: inclination

text: i

text: and

text: potentials

text: Ω

text: temperature

text: of

text: Tokunaga

text: The

text: ”spectral”

text: tempera-

text: We

text: adopted

text: coeﬃcients

text: of

text: gravity

text: brightening

text: and

text: c

text: RAS,

text: MNRAS

text: ##LTFigure##

text: ##LTFigure##


text: Observations and light curve solutions of six NSVS binaries 3 Table 1. List of our ultrashort-period targets Target NSVS ID α δ RNSVS Period References [2000] [2000] [mag] [d] 1 4484038 05 54 16.99 +44 25 34.1 12.50 0.2185 Norton et al. (2011) 2 7179685 06 46 06.47 +36 20 21.1 14.90 0.2097 new 3 4761821 08 01 50.03 +47 14 33.8 13.10 0.2175 Norton et al. (2011) 4 2700153 13 43 51.43 +63 04 22.8 12.48 0.2285 new 5 925605 14 01 46.37 +77 16 38.7 13.73 0.2176 new 6 8626028 21 16 26.81 +25 17 36.2 13.78 0.2174 new Figure 3. Same as Fig. 1 for NSVS 4761821. Figure 5. Same as Fig. 1 for NSVS 925605. Figure 4. Same as Fig. 1 for NSVS 2700153. Figure 6. Same as Fig. 1 for NSVS 8626028. (cid:13) c 2014 RAS, MNRAS 000, 1–?? ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Observations

text: and

text: light

text: curve

text: solutions

text: of

text: six

text: NSVS

text: binaries

text: Table

text: List

label: caption

text: of

text: our

text: ultrashort-period

text: targets

text: Target

label: table

text: NSVS

text: ID

text: α

text: δ

text: RNSVS

text: Period

text: References

text: [mag]

text: [d]

text: Norton

text: et

text: al.

text: new

text: Norton

text: et

text: al.

text: new

text: new

text: new

text: Figure

text: Same

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text: Fig.

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text: for

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text: Fig.

text: for

text: NSVS

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text: for

text: NSVS

text: c

text: RAS,

text: MNRAS

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Observations and light curve solutions of six NSVS binaries 7 Table 5. Semi-major axes of short-period binaries from spectral studies Star Porb M1 + M2 a References CSS J090826.3+123648* 0.1392 0.80 1.049 Drake et al. (2014) CSS J111647.8+294602* 0.1462 0.75 1.061 Drake et al. (2014) CSS J081158.6+311959* 0.1562 0.70 1.083 Drake et al. (2014) CSS J001242.4+130809* 0.1641 0.80 1.171 Drake et al. (2014) CSS J001242.4+130809 0.1641 0.41 0.937 Drake et al. (2014) CSS J090119.2+114254* 0.1867 0.70 1.222 Drake et al. (2014) ± ± BX Tri 0.1926 0.77 0.03 1.284 0.022 Dimitrov & Kjurkchieva (2010) ± ± BW3 V38 0.1984 0.85 0.11 1.356 0.085 Maceroni & Montalban (2004) ± ± SDSS J001641-000925 0.1986 0.88 0.08 1.372 0.057 Davenport et al. (2013) ± ± GSC 1387-475 0.2178 0.94 0.03 1.492 0.019 Rucinski & Pribulla (2008) ± ± CC Com 0.2211 1.09 0.02 1.585 0.011 Pribulla et al. (2007) ± ± 1SWASP J160156.04+202821.6 0.2265 1.43 0.06 1.761 0.033 Lohr et al. (2014) ± ± V523 Cas 0.2337 1.13 0.04 1.663 0.025 Rucinski et al. (2003) ± ± RW Com 0.2373 1.18 0.03 1.704 0.060 Pribulla et al. (2009) BI Vul 0.2518 1.45 1.899 Maceroni & van’t Veer (1996) ± ± 1SWASP J150822.80-054236.9 0.2601 1.62 0.10 2.013 0.065 Lohr et al. (2014) ± ± VZ Psc 0.2612 1.46 0.06 1.950 0.040 Hrivnak et al. (1995) V803 Aql 0.2634 1.58 2.014 Maceroni & van’t Veer (1996) FS Cra 0.2636 1.51 1.984 Maceroni & van’t Veer (1996) 44 Boo=i Boo 0.2678 1.47 1.987 Lu et al. (2001) Note: Stars with WD+dM model are marked with asterisk. Table 6. Global parameters of the targets Name a R1 M M1 L1 Mbol d Jorb R2 M2 L2 MV ± ± ± ± NSVS 4484038 1.55 0.63 0.02 1.01 0.56 0.06 0.23 0.02 5.73 0.11 200 0.151 ± ± ± ± ± ± ± 0.05 0.57 0.02 0.10 0.45 0.04 0.17 0.02 5.928 12 0.034 ± ± ± ± NSVS 7179685 1.461 0.48 0.02 0.95 0.30 0.03 0.06 0.01 6.68 0.13 428 0.118 ± ± ± ± ± ± ± 0.05 0.67 0.02 0.09 0.65 0.06 0.10 0.01 7.284 26 0.027 ± ± ± ± NSVS 4761821 1.541 0.71 0.02 1.04 0.72 0.07 0.23 0.02 6.00 0.10 248 0.137 ± ± ± ± ± ± ± 0.05 0.49 0.02 0.10 0.32 0.03 0.11 0.01 6.397 12 0.030 ± ± ± ± NSVS 2700153 1.650 0.67 0.02 1.15 0.65 0.06 0.22 0.02 5.81 0.11 267 0.190 ± ± ± ± ± ± ± 0.05 0.60 0.02 0.11 0.50 0.05 0.16 0.02 6.060 14 0.042 ± ± ± ± NSVS 925605 1.542 0.73 0.02 1.04 0.62 0.06 0.11 0.01 6.82 0.12 154 0.155 ± ± ± ± ± ± ± 0.05 0.64 0.02 0.10 0.42 0.04 0.024 0.005 8.020 9 0.034 ± ± ± ± NSVS 8626028 1.540 0.63 0.02 1.04 0.57 0.06 0.13 0.01 6.40 0.10 184 0.159 ± ± ± ± ± ± ± 0.05 0.57 0.02 0.10 0.47 0.04 0.09 0.01 7.904 9 0.035 7 CONCLUSIONS ACKNOWLEDGEMENTS The authors gratefully acknowledge observing grant We found well-deﬁned empirical relation “period – semi- support from the Institute of Astronomy and Rozhen Na- major axis” for the short-period binaries and used it for tional Astronomical Observatory, Bulgarian Academy of Sci- estimation of the global parameters of six ultrashort-period ences. The research was supported partly by funds of project targets. RD-08-244 of Shumen University. One of them, NSVS 925605, with huge ﬁll-out factor This research make use of the SIMBAD, Vizier, and Al- and another peculiar characteristics (consisting of M com- adin databases, operated at CDS, Strasbourg, France, and ponents with high activity and non-Planck energy distribu- NASA’s Astrophysics Data System Abstract Service. This tion) probably is at stage of merging. publication makes use of data products from the Two Mi- Our results revealed that all six ultrashort-period tar- cron All Sky Survey, which is a joint project of the Univer- gets with periods 0.20–0.23 days are overcontact systems. sity of Massachusetts and the Infrared Processing and Anal- At the same time the short-period binaries BX Tri and ysis Center/California Institute of Technology, funded by BW3 V38 with periods below 0.20 have detached conﬁgura- the National Aeronautics and Space Administration and the tions. Nefs et al. (2012) spectroscopically conﬁrmed another National Science Foundation. This research also has made detached system with a 0.18 day period containing an M- use of the USNOFS Image and Catalogue Archive operated dwarf. It is worth to study which parameters determine the by the United States Naval Observatory, Flagstaﬀ Station stellar conﬁguration (detached, contact or overcontact) of (http://www.nofs.navy.mil/data/fchpix/). the ultrashort-period binaries. This research was based on data obtained with (cid:13) c 2014 RAS, MNRAS 000, 1–?? ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Observations

text: and

text: light

text: curve

text: solutions

text: of

text: six

text: NSVS

text: binaries

text: Table

text: Semi-major

text: axes

text: of

text: short-period

text: binaries

text: from

text: spectral

text: studies

text: Star

text: Porb

text: a

text: References

text: CSS

text: Drake

text: et

text: al.

text: CSS

text: Drake

text: et

text: al.

text: CSS

text: Drake

text: et

text: al.

text: CSS

text: Drake

text: et

text: al.

text: CSS

text: Drake

text: et

text: al.

text: CSS

text: Drake

text: et

text: al.

text: BX

text: Tri

text: Dimitrov

text: Kjurkchieva

text: Maceroni

text: Montalban

text: SDSS

text: Davenport

text: et

text: al.

text: GSC

text: Rucinski

text: Pribulla

text: CC

text: Com

text: Pribulla

text: et

text: al.

text: Lohr

text: et

text: al.

text: Cas

text: Rucinski

text: et

text: al.

text: RW

text: Com

text: Pribulla

text: et

text: al.

text: BI

text: Vul

text: Maceroni

text: van’t

text: Veer

text: Lohr

text: et

text: al.

text: VZ

text: Psc

text: Hrivnak

text: et

text: al.

text: Aql

text: Maceroni

text: van’t

text: Veer

text: FS

text: Cra

text: Maceroni

text: van’t

text: Veer

text: Boo=i

text: Boo

text: Lu

text: et

text: al.

text: Note:

text: Stars

text: with

text: WD+dM

text: model

text: are

text: marked

text: with

text: asterisk.

text: Table

text: Global

text: parameters

text: of

text: the

text: targets

text: Name

text: a

text: M

text: Mbol

text: d

text: Jorb

text: MV

text: NSVS

text: NSVS

text: NSVS

text: NSVS

text: NSVS

text: NSVS

text: CONCLUSIONS

text: ACKNOWLEDGEMENTS

text: The

text: authors

text: gratefully

text: acknowledge

text: observing

text: grant

text: We

text: found

text: well-deﬁned

text: empirical

text: relation

text: “period

text: semi-

text: support

text: from

text: the

text: Institute

text: of

text: Astronomy

text: and

text: Rozhen

text: Na-

text: major

text: axis”

text: for

text: the

text: short-period

text: binaries

text: and

text: used

text: it

text: for

text: tional

text: Astronomical

text: Observatory,

text: Bulgarian

text: Academy

text: of

text: Sci-

text: estimation

text: of

text: the

text: global

text: parameters

text: of

text: six

text: ultrashort-period

text: ences.

text: The

text: research

text: was

text: supported

text: partly

text: by

text: funds

text: of

text: project

text: targets.

text: of

text: Shumen

text: University.

text: One

text: of

text: them,

text: NSVS

text: with

text: huge

text: ﬁll-out

text: factor

text: This

text: research

text: make

text: use

text: of

text: the

text: SIMBAD,

text: Vizier,

text: and

text: Al-

text: and

text: another

text: peculiar

text: characteristics

text: (consisting

text: of

text: M

text: com-

text: adin

text: databases,

text: operated

text: at

text: CDS,

text: Strasbourg,

text: France,

text: and

text: ponents

text: with

text: high

text: activity

text: and

text: non-Planck

text: energy

text: distribu-

text: NASA’s

text: Astrophysics

text: Data

text: System

text: Abstract

text: Service.

text: This

text: tion)

text: probably

text: is

text: at

text: stage

text: of

text: merging.

text: publication

text: makes

text: use

text: of

text: data

text: products

text: from

text: the

text: Two

text: Mi-

text: Our

text: results

text: revealed

text: that

text: all

text: six

text: ultrashort-period

text: tar-

text: cron

text: All

text: Sky

text: Survey,

text: which

text: is

text: a

text: joint

text: project

text: of

text: the

text: Univer-

text: gets

text: with

text: periods

text: days

text: are

text: overcontact

text: systems.

text: sity

text: of

text: Massachusetts

text: and

text: the

text: Infrared

text: Processing

text: and

text: Anal-

text: At

text: the

text: same

text: time

text: the

text: short-period

text: binaries

text: BX

text: Tri

text: and

text: ysis

text: Center/California

text: Institute

text: of

text: Technology,

text: funded

text: by

text: with

text: periods

text: below

text: have

text: detached

text: conﬁgura-

text: the

text: National

text: Aeronautics

text: and

text: Space

text: Administration

text: and

text: the

text: tions.

text: Nefs

text: et

text: al.

text: spectroscopically

text: conﬁrmed

text: another

text: National

text: Science

text: Foundation.

text: This

text: research

text: also

text: has

text: made

text: detached

text: system

text: with

text: a

text: day

text: period

text: containing

text: an

text: M-

text: use

text: of

text: the

text: USNOFS

text: Image

text: and

text: Catalogue

text: Archive

text: operated

text: dwarf.

text: It

text: is

text: worth

text: to

text: study

text: which

text: parameters

text: determine

text: the

text: by

text: the

text: United

text: States

text: Naval

text: Observatory,

text: Flagstaﬀ

text: Station

text: stellar

text: conﬁguration

text: (detached,

text: contact

text: or

text: overcontact)

text: of

text: the

text: ultrashort-period

text: binaries.

text: This

text: research

text: was

text: based

text: on

text: data

text: obtained

text: with

text: c

text: RAS,

text: MNRAS

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Observations and light curve solutions of six NSVS binaries 9 Table A3. Times of the observed eclipse minima. − HJD(Min) Filter Type of Epoch O C [d] minimum [d] NSVS 4484038 2455539.479830 V Min I 0 0.000563 2455539.479669 I Min I 0 -0.000174 2455539.589299 V Min II 0 0.001572 2455539.589183 I Min II 0 0.001041 2455570.287555 V Min I 141 -0.001048 2455570.287666 I Min I 141 -0.000540 2455570.397084 V Min II 141 0.000236 2455570.397060 I Min II 141 0.000126 NSVS 7179685 2455104.632827 R Min II 0 -0.003361 2455105.472144 R Min II 4 -0.001659 2455105.576676 R Min I 5 -0.003271 2455128.544335 V Min II 114 0.002112 2455128.543839 R Min II 114 -0.000253 2455128.543751 I Min II 114 -0.000672 2455128.648854 V Min I 115 0.000439 2455128.648414 R Min I 115 -0.001659 2455128.648348 I Min I 115 -0.001974 2455262.463012 R Min I 753 0.000620 2455268.335791 I Min I 781 0.000901 2455268.440937 I Min II 781 0.002217 2455563.545264 V Min II 2188 0.002918 2455563.649773 V Min I 2189 0.001197 NSVS 4761821 2455563.547579 V Min II 0 0.000365 2455563.546281 I Min II 0 -0.005603 2455563.656336 V Min I 1 0.000367 2455563.656667 I Min I 1 0.001889 2455568.441653 V Min I 23 0.000509 2455568.441644 I Min I 23 0.000468 2455568.550426 V Min II 23 0.000585 2455568.550388 I Min II 23 0.000411 NSVS 2700153 2455368.313740 V Min I 0 0.006631 Figure A1. Folded light curves based on the NSVS and Super- 2455368.313195 I Min I 0 0.004246 WASP observations of the targets 2455368.425320 V Min II 0 -0.004959 2455368.426162 I Min II 0 -0.001274 2455579.518952 V Min II 924 -0.003699 2455579.518799 I Min II 924 -0.004368 2455580.548178 V Min I 929 0.001440 2455580.660175 V Min II 929 -0.008325 2455581.575046 I Min II 933 -0.003743 2455581.690325 I Min I 934 0.000858 NSVS 925605 2454988.467268 R Min II -1 0.004147 2454989.443816 R Min I 4 -0.008639 2454989.555544 R Min II 4 0.004749 2455040.368512 V Min I 238 -0.010890 2455070.297199 V Min II 375 0.010704 2455070.296609 R Min II 375 0.007993 2455070.296099 I Min II 375 0.005650 2455070.402749 V Min I 376 -0.004296 2455070.402535 R Min I 376 -0.005280 2455070.402487 I Min I 376 -0.005500 2455071.272856 V Min I 380 -0.006176 2455071.272715 R Min I 380 -0.006824 2455071.272511 I Min I 380 -0.007761 2455071.385397 V Min II 380 0.010948 2455071.384643 R Min II 380 0.007483 2455071.383811 I Min II 380 0.003660 NSVS 8626028 2455070.516938 V Min I 0 -0.000789 2455070.516287 R Min I 0 -0.003783 2455070.516218 I Min I 0 -0.004101 2455071.496252 V Min I 5 0.003730 2455071.496096 R Min I 5 0.003013 2455071.496085 I Min I 5 0.002962 2455071.604069 V Min II 5 -0.000347 2455071.603989 R Min II 5 -0.000715 2455071.604153 I Min II 5 0.000039 (cid:13) c 2014 RAS, MNRAS 000, 1–?? ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Observations

text: and

text: light

text: curve

text: solutions

text: of

text: six

text: NSVS

text: binaries

text: Table

text: Times

text: of

text: the

text: observed

text: eclipse

text: minima.

text: HJD(Min)

text: Filter

text: Type

text: of

text: Epoch

text: O

text: C

text: [d]

text: minimum

text: [d]

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text: I

text: Figure

text: Folded

text: light

text: curves

text: based

text: on

text: the

text: NSVS

text: and

text: Super-

text: I

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text: I

text: WASP

text: observations

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text: c

text: RAS,

text: MNRAS

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: VI. ILLUSTRATIVE EXAMPLE We ﬁrst consider the case of no control as a reference. Here, the system input is driven by vacuum no(cid:20)ise: d(cid:21)u = dv . The (cid:2) (cid:3) 1 To demonstrate the coherent quantum controller design resulting closed loop system is: process of Section , we consider a two mirror optical cavity dTrhivisenexbaymtphleermisalanmoiosdeiﬁocfaitniotennsoitfyaknneaxsamshpolewncoinnsFidiegruerde i3n. dx =(cid:2)Ax d(cid:3)t + Bw1 Bu ddwv11 , [24]. Optically coupled to the second mirror is a controller to where dw = dw1(cid:20) is a quantum Wien(cid:21)er process with be designed via our algorithm. We compare the value of the covarianceCL dv1 cost function obtained with our controller to the no control (1 + 2k )I × 0 case; i.e. when the second mirror is not connected to any other S = n 2 2 . wCL 0 I × system and thus driven by a vacuum noise. We also compare 2 2 our approach with a scheme involving heterodyne detection Equation (22) can then be used to ﬁnd the value of the cost (cid:10) (cid:11) (cid:10) (cid:11) and an optimal classical LQG controller. Our design objective function (20) for a range of values for k . n is to minimize the expected number of photons in the cavity: B. Heterodyne measurement and classical LQG control † − N = a a = 1 x 2 + x 2 0.5, where x and x are Next we consider combining heterodyne measurement with 4 1 2 1 2 the position and momentum operators for the cavity. It will a classical optimal LQG controller. Heterodyne measurement be shown that for all k > 0, it is possible to achieve better introduces an additional vacuum noise input. Similarly, the n performance using a quantum controller designed using our output of the classical controller will contain a vacuum noise method than with the optimal measurement based controller. component when applied to the input of the plant. This is This validates the utility of our method. accounted for by making the following substitutions: du = du˜ + dw ; 3 dy˜ = dy + dw ; 2 Input Controller into (19) to obtain an augmented plant. Here du˜ and dy˜ are Optical Cavity classical signals which represent the input and output of the augmented plant. The resulting equations for the augmented Fig. 3. Quantum Plant: we wish to minimize the expected number of photons plant are as follows: in the cavity. Our plant is of the form (19) with dx = Ax dt + B du˜ + B dw˜; u w˜ dy˜ = Cx dt + D du˜ + D dw˜. u w˜ − γ A = I × ; −√2 2 2 Here A, Bu, C(cid:2), and Du are as before, and (cid:3) B = √ κ I × ; u 2 2 2 (cid:2) √ (cid:3) √ B = √κ I × ; − − w1 1 2 2 Bw˜ =  κ1I2×2 02×2 κ2I2×2 ; C = κ I × ; 2 2 2 Dw˜ = 02×2  I2×2 I2×2 ; D = I × ; u 2 2 dw 1 Dw1 = 02×2; dw˜ =  dw2 ;  Sw1 = (1 + 2kn)I2×2  dw3  γ = 0.2; (1 + 2kn)I2×2 0 0 S = 0 I × 0 . κ = 0.1; w˜ 2 2 1 0 0 I × κ = 0.1. (27) 2 2 2 As with the auxiliary LQG problem in our design algorithm, we treat dw˜ as a standard classical Wiener process with It is sufﬁcient to minimize the cost f−unction (20) with R1 = intensity matrix Sw˜. We now have a standard classical LQG I2×2 and R2 = 02×2. Then N = −14J 0.5. problem. We wish to ﬁnd a controller of the form: Remark 6: We use R = 10 6I × to design both the 2 2 2 heterodyne classical LQG controller and the auxiliary LQG controller. However, the cost function (20) for the resulting dxK = AKxK dt + By dy˜; controller is then evaluated using R = 0 × . du˜ = C x dt. 2 2 2 K K The estimator gain K and the regulator gain F are obtained A. No Control in the usual manner. For this example R = 0 × but for 2 2 2 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: VI.

label: section

text: ILLUSTRATIVE

text: EXAMPLE

text: We

text: ﬁrst

text: consider

text: the

text: case

text: of

text: no

text: control

text: as

text: a

text: reference.

text: Here,

text: the

text: system

text: input

text: is

text: driven

text: by

text: vacuum

text: dv

text: The

text: To

text: demonstrate

text: the

text: coherent

text: quantum

text: controller

text: design

text: resulting

text: closed

text: loop

text: system

text: is:

text: process

text: of

text: Section

text: we

text: consider

text: a

text: two

text: mirror

text: optical

text: cavity

text: dx

text: Bu

text: Optically

text: coupled

text: to

text: the

text: second

text: mirror

text: is

text: a

text: controller

text: to

text: where

text: dw

text: is

text: a

text: quantum

text: process

text: with

text: be

text: designed

text: via

text: our

text: algorithm.

text: We

text: compare

text: the

text: value

text: of

text: the

text: covarianceCL

text: cost

text: function

text: obtained

text: with

text: our

text: controller

text: to

text: the

text: no

text: control

text: )I

text: case;

text: i.e.

text: when

text: the

text: second

text: mirror

text: is

text: not

text: connected

text: to

text: any

text: other

text: S

text: n

text: wCL

text: I

text: system

text: and

text: thus

text: driven

text: by

text: a

text: vacuum

text: noise.

text: We

text: also

text: compare

text: our

text: approach

text: with

text: a

text: scheme

text: involving

text: heterodyne

text: detection

text: Equation

text: can

text: then

text: be

text: used

text: to

text: ﬁnd

text: the

text: value

text: of

text: the

text: cost

text: and

text: an

text: optimal

text: classical

text: LQG

text: controller.

text: Our

text: design

text: objective

text: function

text: for

text: a

text: range

text: of

text: values

text: for

text: k

text: n

text: is

text: to

text: minimize

text: the

text: expected

text: number

text: of

text: photons

text: in

text: the

text: cavity:

text: B.

text: Heterodyne

text: measurement

text: and

text: classical

text: LQG

text: control

text: N

text: a

text: a

text: x

text: x

text: where

text: x

text: and

text: x

text: are

text: Next

text: we

text: consider

text: combining

text: heterodyne

text: measurement

text: with

text: the

text: position

text: and

text: momentum

text: operators

text: for

text: the

text: cavity.

text: It

text: will

text: a

text: classical

text: optimal

text: LQG

text: controller.

text: Heterodyne

text: measurement

text: be

text: shown

text: that

text: for

text: all

text: k

text: it

text: is

text: possible

text: to

text: achieve

text: better

text: introduces

text: an

text: additional

text: vacuum

text: noise

text: input.

text: Similarly,

text: the

text: n

text: performance

text: using

text: a

text: quantum

text: controller

text: designed

text: using

text: our

text: output

text: of

text: the

text: classical

text: controller

text: will

text: contain

text: a

text: vacuum

text: noise

text: method

text: than

text: with

text: the

text: optimal

text: measurement

text: based

text: controller.

text: component

text: when

text: applied

text: to

text: the

text: input

text: of

text: the

text: plant.

text: This

text: is

text: This

text: validates

text: the

text: utility

text: of

text: our

text: method.

text: accounted

text: for

text: by

text: making

text: the

text: following

text: substitutions:

text: du

text: du˜

text: dw

text: dy˜

text: dy

text: dw

text: Input

text: Controller

text: into

text: to

text: obtain

text: an

text: augmented

text: plant.

text: Here

text: du˜

text: and

text: dy˜

text: are

text: Optical

text: Cavity

text: classical

text: signals

text: which

text: represent

text: the

text: input

text: and

text: output

text: of

text: the

text: augmented

text: plant.

text: The

text: resulting

text: equations

text: for

text: the

text: augmented

text: Fig.

text: Quantum

text: Plant:

text: we

text: wish

text: to

text: minimize

text: the

text: expected

text: number

text: of

text: photons

text: plant

text: are

text: as

text: follows:

text: in

text: the

text: cavity.

text: Our

text: plant

text: is

text: of

text: the

text: form

text: with

text: dx

text: Ax

text: dt

text: B

text: du˜

text: B

text: dw˜;

text: u

text: w˜

text: dy˜

text: Cx

text: dt

text: D

text: du˜

text: D

text: dw˜.

text: u

text: w˜

text: γ

text: A

text: I

text: −√2

text: Here

text: A,

text: Bu,

text: and

text: Du

text: are

text: as

text: before,

text: and

text: B

text: κ

text: I

text: u

text: B

text: √κ

text: I

text: Bw˜

text: C

text: κ

text: I

text: Dw˜

text: D

text: I

text: u

text: dw

text: dw˜

text: γ

text: S

text: I

text: κ

text: w˜

text: I

text: κ

text: As

text: with

text: the

text: auxiliary

text: LQG

text: problem

text: in

text: our

text: design

text: algorithm,

text: we

text: treat

text: dw˜

text: as

text: a

text: standard

text: classical

text: Wiener

text: process

text: with

text: It

text: is

text: sufﬁcient

text: to

text: minimize

text: the

text: cost

text: f−unction

text: with

text: intensity

text: matrix

text: Sw˜.

text: We

text: now

text: have

text: a

text: standard

text: classical

text: LQG

text: and

text: Then

text: N

text: problem.

text: We

text: wish

text: to

text: ﬁnd

text: a

text: controller

text: of

text: the

text: form:

text: Remark

text: We

text: use

text: R

text: to

text: design

text: both

text: the

text: heterodyne

text: classical

text: LQG

text: controller

text: and

text: the

text: auxiliary

text: LQG

text: controller.

text: However,

text: the

text: cost

text: function

text: for

text: the

text: resulting

text: dxK

text: AKxK

text: dt

text: By

text: dy˜;

text: controller

text: is

text: then

text: evaluated

text: using

text: R

text: du˜

text: C

text: x

text: dt.

text: K

text: K

text: The

text: estimator

text: gain

text: K

text: and

text: the

text: regulator

text: gain

text: F

text: are

text: obtained

text: A.

text: No

text: Control

text: in

text: the

text: usual

text: manner.

text: For

text: this

text: example

text: R

text: but

text: for

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Similarly, dw(t) is a column vector of n self-adjoint, non- w commutative operators representing the input to the system and Here, du(t) (a column vector with n components) rep- u dy(t) is a column vector of n self-adjoint, non-commutative resents the inputs to the system and, like dw(t) in (1), is y operators repre(cid:2)senting the outp(cid:3)ut of the system. Their com- assumed to admit the decomposition du(t) = β (t) dt + u mutation relations are as follows: du˜(t). Also, dv (t) and dv (t) (column vectors with n and (cid:2) (cid:3) n components1 respectivel2y) are quantum Wiener provc1esses dwi(t), dwj(t) = 2iΘw,ij dt; covr2responding to the introduced vacuum noise inputs. For con- venience, the vacuum noises are partitioned into two vectors dy (t), dy (t) = 2iΘ dt; i j y,ij dv (t) and dv (t) such that n = n . Then, n = n + n where Θ and Θ are real skew symmetric matrices. is t1he total num2ber of introducve1d vacuuum noise vinputsv.1Subsve2- w y The input signals dw(t) are assumed to admit the decom- quently, we will refer to dv as the direct feedthrough quantum 1 position noises and to dv as the additional quantum noises. Also, F , 2 u˜ dw(t) = β (t) dt + dw˜(t) S , T , F , S , T , F , S , and T are deﬁned for du˜(t), w u˜ u˜ v1 v1 v1 v2 v2 v2 dv (t) and dv (t) respectively as F , S and T were for where the self-adjoint, adapted process βw(t) is the signal dw˜1(t) in (1). F2urthermo(cid:2)re, we(cid:3) assumw˜e thwa˜t F awn˜d F are part of dw(t) and dw˜(t) is the noise part of dw(t). Here, appropriately dimensioned block diagonal matvri1ces withv2each βw(t) is assumed to commute with x(t). The vector dw˜(t) is diagonal block equal to −1 i . This assumption corresponds a quantum Wiener process with Ito products i 1 to the fact that dv and dv represent vacuum noises [3]. 1 2 dw˜(t) dw˜T (t) = F dt The remaining symbols have the same meanings as in (1). We w˜ restrict our attention to the case where n = n . y u where Fw˜ is a non-negative Hermitian matrix. Let Fw˜ = III. PHYSICAL REALIZABILITY S + T , where S is real and T is imaginary. Then w˜ w˜ w˜ w˜ A. Deﬁnitions S describes the intensity of the quantum Wiener process w˜ and is the quantum analog of the intensity matrix for a As in [3], [8], [10], [16]–[18], the concept of physical classical Wiener(cid:2)process. The co(cid:3)mmutation relations for dw˜ realizability means that the system dynamics described by are determined by T : the QSDEs (1) correspond to those of a collection of open w˜ quantum harmonic oscillators. Here, we slightly modify the dw˜(t), dw˜T (t) = 2T dt. deﬁnition of physically realizable given in [3, Deﬁnition 3.1]. w˜ In [3] both fully quantum systems and hybrid systems with Since β (t) is an adapted process, β (t) commutes with w ≥ w quantum and classical degrees of freedom are considered. dw˜(t) for all t 0. Also, dw˜(t) commutes with x(t). However, we restrict our deﬁnition of physically realizable Finally, n, n and n are even (this is because in the quan- w y systems to those that are fully quantum. tum harmonic oscillator, the system variables always occur as Deﬁnition 1: The(cid:2)system varia(cid:3)bles x are said to satisfy the conjugate pairs, see [6]) and A, B, C and D are appropriately canonical commutation relations if dimensioned real matrices describing the dynamics of the system. For further details regarding these models, see [3]. x (t), x (t) = 2iΘ i j ij Remark 1: While it is always possible to describe a   collection of quantum harmonic oscillators by QSDEs of where Θ is of the form  (cid:20) (cid:21) the form (1), not all QSDEs of this form correspond to a  · · ·  J 0 0 collection of quantum harmonic oscillators. The property that  · · ·  0 J 0 tohseciQllaStDorEssi(s1)caclolerrdespphoynsdictaolarecaollilzeacbtiiolintyofanqduainstuamddhreasrmseodniinc Θ = ... ... ... ... ; J = −01 01 . (3) · · · greater detail in Section . 0 0 J We will now further restrict our attention within the class (cid:2) (cid:3) of quantum systems described above. This corresponds to the case where x consists of pairs of position and momentum operators: q (t) p (t) q (t) p (t) . . . T . B. A Class of Quantum System Models 1 1 2 2 Deﬁnition 2: The system described by (1) is physically This paper addresses the problem of implementing an realizable if Θ is of the form (3) and there exists a quadratic H arbitrary, strictly proper, LTI system as a quantum system Hamiltonian operator = 1x(0)T Rx(0), where R is a real, (for example when implementing a coherent controller) by symmetric, n × n matrix, 2and a coupling operator vector L × introducing vacuum noise sources. The resulting quantum = Λx(0), where Λ is a complex-valued 1n n coupling systems are described by the following QSDEs which are a matrix such that the (cid:0)matrices A(cid:0) , B,(cid:1)(cid:1)C and2Dware given by: special case of (1): (cid:2) (cid:3) † A = 2Θ (cid:20)R + Im Λ(cid:21)Λ(cid:20) ; (cid:21) (4a) dx(t) = Ax(t) dt + B du(t) − † u B = 2iΘ Λ ΛT Γ; (4b) + B dv (t) + B dv (t); dy(t) = Cx(tv)1dt +1 dv1(t)v.2 2 (2) C = PT Σ0ny Σ0ny −ΛiΛ++Λi#Λ# ; (4c) ##LTLine## ##LTLine##

text: Similarly,

text: dw(t)

text: is

text: a

text: column

text: vector

text: of

text: n

text: self-adjoint,

text: non-

text: w

text: commutative

text: operators

text: representing

text: the

text: input

text: to

text: the

text: system

text: and

text: Here,

text: du(t)

text: (a

text: column

text: vector

text: with

text: n

text: components)

text: rep-

text: u

text: dy(t)

text: is

text: a

text: column

text: vector

text: of

text: n

text: self-adjoint,

text: non-commutative

text: resents

text: the

text: inputs

text: to

text: the

text: system

text: and,

text: like

text: dw(t)

text: in

text: is

text: y

text: operators

text: the

text: of

text: the

text: system.

text: Their

text: com-

text: assumed

text: to

text: admit

text: the

text: decomposition

text: du(t)

text: β

text: (t)

text: dt

text: u

text: mutation

text: relations

text: are

text: as

text: follows:

text: du˜(t).

text: Also,

text: dv

text: (t)

text: and

text: dv

text: (t)

text: (column

text: vectors

text: with

text: n

text: and

text: n

text: are

text: quantum

text: Wiener

text: dwi(t),

text: dwj(t)

text: dt;

text: to

text: the

text: introduced

text: vacuum

text: noise

text: inputs.

text: For

text: con-

text: venience,

text: the

text: vacuum

text: noises

text: are

text: partitioned

text: into

text: two

text: vectors

text: dy

text: (t),

text: dy

text: (t)

text: dt;

text: i

text: j

text: y,ij

text: dv

text: (t)

text: and

text: dv

text: (t)

text: such

text: that

text: n

text: n

text: Then,

text: n

text: n

text: n

text: where

text: Θ

text: and

text: Θ

text: are

text: real

text: skew

text: symmetric

text: matrices.

text: is

text: total

text: of

text: vacuuum

text: noise

text: w

text: y

text: The

text: input

text: signals

text: dw(t)

text: are

text: assumed

text: to

text: admit

text: the

text: decom-

text: quently,

text: we

text: will

text: refer

text: to

text: dv

text: as

text: the

text: direct

text: feedthrough

text: quantum

text: position

text: noises

text: and

text: to

text: dv

text: as

text: the

text: additional

text: quantum

text: noises.

text: Also,

text: F

text: u˜

text: dw(t)

text: β

text: (t)

text: dt

text: dw˜(t)

text: S

text: T

text: F

text: S

text: T

text: F

text: S

text: and

text: T

text: are

text: deﬁned

text: for

text: du˜(t),

text: w

text: u˜

text: u˜

text: dv

text: (t)

text: and

text: dv

text: (t)

text: respectively

text: as

text: F

text: S

text: and

text: T

text: were

text: for

text: where

text: the

text: self-adjoint,

text: adapted

text: process

text: βw(t)

text: is

text: the

text: signal

text: in

text: assumw˜e

text: thwa˜t

text: F

text: awn˜d

text: F

text: are

text: part

text: of

text: dw(t)

text: and

text: dw˜(t)

text: is

text: the

text: noise

text: part

text: of

text: dw(t).

text: Here,

text: appropriately

text: dimensioned

text: block

text: diagonal

text: βw(t)

text: is

text: assumed

text: to

text: commute

text: with

text: x(t).

text: The

text: vector

text: dw˜(t)

text: is

text: diagonal

text: block

text: equal

text: to

text: −1

text: i

text: This

text: assumption

text: corresponds

text: a

text: quantum

text: Wiener

text: process

text: with

text: Ito

text: products

text: i

text: to

text: the

text: fact

text: that

text: dv

text: and

text: dv

text: represent

text: vacuum

text: noises

text: dw˜(t)

text: dw˜T

text: (t)

text: F

text: dt

text: The

text: remaining

text: symbols

text: have

text: the

text: same

text: meanings

text: as

text: in

text: We

text: w˜

text: restrict

text: our

text: attention

text: to

text: the

text: case

text: where

text: n

text: n

text: y

text: u

text: where

text: Fw˜

text: is

text: a

text: non-negative

text: Hermitian

text: matrix.

text: Let

text: Fw˜

text: III.

text: PHYSICAL

text: REALIZABILITY

text: S

text: T

text: where

text: S

text: is

text: real

text: and

text: T

text: is

text: imaginary.

text: Then

text: w˜

text: w˜

text: w˜

text: w˜

text: A.

text: Deﬁnitions

text: S

text: describes

text: the

text: intensity

text: of

text: the

text: quantum

text: Wiener

text: process

text: w˜

text: and

text: is

text: the

text: quantum

text: analog

text: of

text: the

text: intensity

text: matrix

text: for

text: a

text: As

text: in

text: the

text: concept

text: of

text: physical

text: classical

text: The

text: relations

text: for

text: dw˜

text: realizability

text: means

text: that

text: the

text: system

text: dynamics

text: described

text: by

text: are

text: determined

text: by

text: T

text: the

text: QSDEs

text: correspond

text: to

text: those

text: of

text: a

text: collection

text: of

text: open

text: w˜

text: quantum

text: harmonic

text: oscillators.

text: Here,

text: we

text: slightly

text: modify

text: the

text: dw˜(t),

text: dw˜T

text: (t)

text: dt.

text: deﬁnition

text: of

text: physically

text: realizable

text: given

text: in

text: Deﬁnition

text: w˜

text: In

text: both

text: fully

text: quantum

text: systems

text: and

text: hybrid

text: systems

text: with

text: Since

text: β

text: (t)

text: is

text: an

text: adapted

text: process,

text: β

text: (t)

text: commutes

text: with

text: w

text: w

text: quantum

text: and

text: classical

text: degrees

text: of

text: freedom

text: are

text: considered.

text: dw˜(t)

text: for

text: all

text: t

text: Also,

text: dw˜(t)

text: commutes

text: with

text: x(t).

text: However,

text: we

text: restrict

text: our

text: deﬁnition

text: of

text: physically

text: realizable

text: Finally,

text: n,

text: n

text: and

text: n

text: are

text: even

text: (this

text: is

text: because

text: in

text: the

text: quan-

text: w

text: y

text: systems

text: to

text: those

text: that

text: are

text: fully

text: quantum.

text: tum

text: harmonic

text: oscillator,

text: the

text: system

text: variables

text: always

text: occur

text: as

text: Deﬁnition

text: x

text: are

text: said

text: to

text: satisfy

text: the

text: conjugate

text: pairs,

text: see

text: and

text: A,

text: B,

text: C

text: and

text: D

text: are

text: appropriately

text: canonical

text: commutation

text: relations

text: if

text: dimensioned

text: real

text: matrices

text: describing

text: the

text: dynamics

text: of

text: the

text: system.

text: For

text: further

text: details

text: regarding

text: these

text: models,

text: see

text: x

text: (t),

text: x

text: (t)

text: i

text: j

text: ij

text: Remark

text: While

text: it

text: is

text: always

text: possible

text: to

text: describe

text: a

text: collection

text: of

text: quantum

text: harmonic

text: oscillators

text: by

text: QSDEs

text: of

text: where

text: Θ

text: is

text: of

text: the

text: form

text: the

text: form

text: not

text: all

text: QSDEs

text: of

text: this

text: form

text: correspond

text: to

text: a

text: J

text: 0

text: collection

text: of

text: quantum

text: harmonic

text: oscillators.

text: The

text: property

text: that

text: J

text: Θ

text: J

text: greater

text: detail

text: in

text: Section

text: J

text: We

text: will

text: now

text: further

text: restrict

text: our

text: attention

text: within

text: the

text: class

text: of

text: quantum

text: systems

text: described

text: above.

text: This

text: corresponds

text: to

text: the

text: case

text: where

text: x

text: consists

text: of

text: pairs

text: of

text: position

text: and

text: momentum

text: operators:

text: q

text: (t)

text: p

text: (t)

text: q

text: (t)

text: p

text: (t)

text: T

text: B.

text: A

text: Class

text: of

text: Quantum

text: System

text: Models

text: Deﬁnition

text: The

text: system

text: described

text: by

text: is

text: physically

text: This

text: paper

text: addresses

text: the

text: problem

text: of

text: implementing

text: an

text: realizable

text: if

text: Θ

text: is

text: of

text: the

text: form

text: and

text: there

text: exists

text: a

text: quadratic

text: H

text: arbitrary,

text: strictly

text: proper,

text: LTI

text: system

text: as

text: a

text: quantum

text: system

text: Hamiltonian

text: operator

text: where

text: R

text: is

text: a

text: real,

text: (for

text: example

text: when

text: implementing

text: a

text: coherent

text: controller)

text: by

text: symmetric,

text: n

text: n

text: matrix,

text: a

text: coupling

text: operator

text: vector

text: L

text: introducing

text: vacuum

text: noise

text: sources.

text: The

text: resulting

text: quantum

text: where

text: Λ

text: is

text: a

text: complex-valued

text: n

text: coupling

text: systems

text: are

text: described

text: by

text: the

text: following

text: QSDEs

text: which

text: are

text: a

text: matrix

text: such

text: that

text: the

text: given

text: by:

text: special

text: case

text: of

text: A

text: 2Θ

text: Im

text: dx(t)

text: Ax(t)

text: dt

text: B

text: du(t)

text: u

text: B

text: Λ

text: ΛT

text: Γ;

text: B

text: dv

text: (t)

text: B

text: dv

text: (t);

text: dy(t)

text: C

text: PT

text: −ΛiΛ++Λi#Λ#

text: ##LTLine##

text: ##LTLine##


text: Stephen McCormick 4 ENERGY MINIMISERS IN A FIXED CONFORMAL CLASS Proof. First we write E in terms of φ and g˜, (cid:73) (cid:16) (cid:17) (cid:73) (cid:16) ∇ ∇ − ∇ − ∇ (cid:17) 16πE = ˚gike4φ 4˚ (φ)g˜ + ˚ g˜ 4˚ (φ)g˜ ˚ g˜ dSj ∞ k ij k ij j ik j ik (cid:73) ∇ (cid:16) ∇ − ∇ − ∇ (cid:17) = ˚gik 4˚ (φ)g˜ + ˚ g˜ 4˚ (φ)g˜ ˚ g˜ dSj ∞ k ij k ij j ik j ik − ∇ ∇ − ∇ − ∇ + ˚gik(e4φ 1) 4˚ (φ)g˜ + ˚ g˜ 4˚ (φ)g˜ ˚ g˜ dSj. (4.4) ∞ k ij k ij j ik j ik (cid:12) (cid:73) (cid:16) (cid:17) (cid:12) Lemm(cid:12) a 3.3 can now be used to control the second integrand in Eq. (cid:12)(4.4), (cid:12) (cid:12) − ∇ ∇ − ∇ − ∇ ˚gik(e4φ 1) 4˚ (φ)g˜ + ˚ g˜ 4˚ (φ)g˜ ˚ g˜ dSj k ij k ij j ik j ik S R ≤ (cid:107) − (cid:107) (cid:107) (cid:107) (cid:107)∇ (cid:107) (cid:107)∇ (cid:107) c e4φ 1 ∞ ( g˜ ∞ ˚φ + ˚g˜ ) :S :S 1:S 1:S R R R R ≤ (cid:107) − (cid:107) (cid:107) (cid:107) (cid:107)∇ (cid:107) (cid:107)∇ (cid:107) O(R1/2) e4φ 1 ∞ ( g˜ ∞ ˚φ − + ˚g˜ − ). :S :S 1,2, 3/2 1,2, 3/2 R R Now making use of the continuity and asymptotics of e4φ and g˜, the right-hand-side simply becomes o(1) and therefore vanishes as R tends to inﬁnity. Eq. (4.4) now (cid:73) (cid:16) (cid:17) becomes ∇ ∇ − ∇ − ∇ 16πE = ˚gik 4˚ (φ)g˜ + ˚ g˜ 4˚ (φ)g˜ ˚ g˜ dSj, ∞ k ij k ij j ik j ik (cid:73) (cid:16) (cid:17) which can be expressed in terms of the energy, E˜, of g˜, ∇ − ∇ 16πE = 16πE˜ + 4 ˚gik ˚ (φ)g˜ ˚ (φ)g˜ dSj. ∞ k ij j ik Since (˚g − g˜) ∈ H2− and ∇˚φ = ∂φ(cid:73)= ∇˜ φ,(cid:16)Lemma 3.3 can again(cid:17)be used to conclude 1/2 (cid:73) ∇ − ∇ 16πE = 16πE˜ + 4 g˜ik ˜ (φ)g˜ ˜ (φ)g˜ dSj ∞ k ij j ik − ∇ = 16πE˜ 8 ˜ φdSj. ∞ j It is now simply a matter of applying the divergence theorem and making use of the Hamiltonian constraint (4.1) to complete the proof. Henceforth, when we write E(φ, π), we mean to take (4.3) to be the deﬁnition of the energy, which is well-deﬁned provided both R˜ and the source are integrable. In the vacuum case (ρ = 0), if R˜ = 0 then it is clear from (4.3) that the energy of 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Stephen

text: McCormick

text: ENERGY

text: MINIMISERS

text: IN

text: A

text: FIXED

text: CONFORMAL

text: CLASS

text: Proof.

text: First

text: we

text: write

text: E

text: in

text: terms

text: of

text: φ

text: and

text: g˜,

label: equation

text: 4˚

text: (φ)g˜

text: g˜

text: 4˚

text: (φ)g˜

text: g˜

text: dSj

text: k

text: ij

text: k

text: ij

text: j

text: ik

text: j

text: ik

text: ˚gik

text: 4˚

text: (φ)g˜

text: g˜

text: 4˚

text: (φ)g˜

text: g˜

text: dSj

text: k

text: ij

text: k

text: ij

text: j

text: ik

text: j

text: ik

text: 4˚

text: (φ)g˜

text: g˜

text: 4˚

text: (φ)g˜

text: g˜

text: dSj.

text: k

text: ij

text: k

text: ij

text: j

text: ik

text: j

text: ik

text: a

text: can

text: now

text: be

text: used

text: to

text: control

text: the

text: second

text: integrand

text: in

text: Eq.

text: 4˚

text: (φ)g˜

text: g˜

text: 4˚

text: (φ)g˜

text: g˜

text: dSj

text: k

text: ij

text: k

text: ij

text: j

text: ik

text: j

text: ik

text: S

text: R

text: c

text: g˜

text: ˚φ

text: ˚g˜

text: :S

text: :S

text: R

text: R

text: R

text: R

text: g˜

text: ˚φ

text: ˚g˜

text: :S

text: :S

text: R

text: R

text: Now

text: making

text: use

text: of

text: the

text: continuity

text: and

text: asymptotics

text: of

text: and

text: g˜,

text: the

text: right-hand-side

text: simply

text: becomes

text: and

text: therefore

text: vanishes

text: as

text: R

text: tends

text: to

text: inﬁnity.

text: Eq.

text: now

text: becomes

text: ˚gik

text: 4˚

text: (φ)g˜

text: g˜

text: 4˚

text: (φ)g˜

text: g˜

text: dSj,

text: k

text: ij

text: k

text: ij

text: j

text: ik

text: j

text: ik

text: which

text: can

text: be

text: expressed

text: in

text: terms

text: of

text: the

text: energy,

text: E˜,

text: of

text: g˜,

text: ˚gik

text: (φ)g˜

text: (φ)g˜

text: dSj.

text: k

text: ij

text: j

text: ik

text: Since

text: (˚g

text: g˜)

text: and

text: ∇˚φ

text: can

text: used

text: to

text: conclude

text: g˜ik

text: (φ)g˜

text: (φ)g˜

text: dSj

text: k

text: ij

text: j

text: ik

text: φdSj.

text: j

text: It

text: is

text: now

text: simply

text: a

text: matter

text: of

text: applying

text: the

text: divergence

text: theorem

text: and

text: making

text: use

text: of

text: the

text: Hamiltonian

text: constraint

text: to

text: complete

text: the

text: proof.

text: Henceforth,

text: when

text: we

text: write

text: E(φ,

text: π),

text: we

text: mean

text: to

text: take

text: to

text: be

text: the

text: deﬁnition

text: of

text: the

text: energy,

text: which

text: is

text: well-deﬁned

text: provided

text: both

text: R˜

text: and

text: the

text: source

text: are

text: integrable.

text: In

text: the

text: vacuum

text: case

text: (ρ

text: if

text: R˜

text: then

text: it

text: is

text: clear

text: from

text: that

text: the

text: energy

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Stephen McCormick 4 ENERGY MINIMISERS IN A FIXED CONFORMAL CLASS → Now let (φ , π ) be a sequence in the constraint set such that E(φ , π ) E . Note n n n n 0 that (4.6) and the Poincar´e inequality imply that there exists a constant K such that for (cid:107) (cid:107) (φ, π) × > K, we have E(φ, π) > E + 1. That is, truncating the beginning H−11/2 L2−3/2 (cid:107) (cid:107) 0 of the sequence if necessary, (φ , π ) × < K. In particular, extracting a sub- n n H−1 L2− 1/2 3/2 × sequence if required, (φ , π ) convergences weakly in H−1 L2− to a limit, (φ∞, π∞). n n ∈ 1/2 3/2 It remains to be shown that (φ∞, π∞) S. − In the following, it will be convenient to let δ = (cid:15)/2, then we assume that 0 (cid:107)∇ (cid:107) max{(cid:107)Φˆ(φn, πn)(cid:107)2,−5/2+2δ0, (cid:107)∇˜˜kππinijj(cid:107)2,−−5/2+2δ0 } < C, and prove below that the inﬁmum j n 2, 5/2+2δ0 is realised in S. Proposition 4.3 and the deﬁnition of Φˆ give 0 (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107)∇ (cid:107) (cid:107) (cid:107) ∆˜ φ − c( R˜ − + ˜ φ 2 − + π 2 − n 2, 5/2+2δ 2, 5/2+2δ n 4, 5/4+δ n 4, 5/4+δ 0 (cid:107) − 0 (cid:107) 0 0 + e 2φnΦ (φ , π ) − ) 0 n n 2, 5/2+2δ ≤ (cid:107) (cid:107) (cid:107)∇ 0(cid:107) c((cid:15))(1 + π 2 − + ˜ φ 2 − ) n 2, 5/4+δ n 2, 5/4+δ (cid:107) (cid:107) 0(cid:107)∇ (cid:107) 0 + (cid:15)( π 2 − + ˜ φ 2 − ), (4.7) n 1,2, 5/4+δ n 1,2, 5/4+δ 0 0 (cid:107) (cid:107) which follows from the assumption Φˆ(φ , π ) − < C and condition (ii) for n n 2, 5/2+2δ 0 the proof of Theorem 4.4, and from Φˆ(φ , π ) = 0 for the proof of Theorem 4.5. n n (cid:107)∇ (cid:107) ˜ πij − Similarly, the assumption (cid:107)∇˜kπinj(cid:107)2,−5/2+2δ0 < C and the deﬁnition of Φi gives j n 2, 5/2+2δ0 (cid:107)∇ (cid:107) ≤ (cid:107)∇ (cid:107) (cid:107) (cid:107) ˜ π − c( ˜ φ − π − n 2, 5/2+2δ n 4, 5/4+δ n 4, 5/4+δ 0 (cid:107) − 0 (cid:107) 0 + e 4φnΦ (φ , π ) − ) i n n 2, 5/2+2δ ≤ (cid:107)∇ (cid:107) (cid:107) (cid:107) 0 c( ˜ φ 2 − + π 2 − + 1) n 4, 5/4+δ n 4, 5/4+δ ≤ (cid:107)∇ (cid:107) 0 (cid:107) (cid:107) 0 c((cid:15))( ˜ φ 2 − + π 2 − + 1) n 2, 5/4+δ n 2, 5/4+δ (cid:107)∇ (cid:107) 0 (cid:107) (cid:107) 0 + (cid:15)( ˜ φ 2 − + π 2 − ). (4.8) n 1,2, 5/4+δ n 1,2, 5/4+δ 0 0 We now recall the scale-broken estimate (Theorem 1.10 of Ref. [1], Proposition 4.13 (cid:16) (cid:17) of Ref. [13]): (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) u C ∆˜ u − + u . (4.9) 2,2,δ 2,δ 2 2,0 (cid:54) Note that the application of the scale-broken estimate here requires (cid:15) = 1. Combining 2 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Stephen

text: McCormick

text: ENERGY

text: MINIMISERS

text: IN

text: A

text: FIXED

text: CONFORMAL

text: CLASS

text: Now

text: let

text: (φ

text: π

text: be

text: a

text: sequence

text: in

text: the

text: constraint

text: set

text: such

text: that

text: E(φ

text: π

text: E

text: Note

text: n

text: n

text: n

text: n

text: that

text: and

text: the

text: Poincar´e

text: inequality

text: imply

text: that

text: there

text: exists

text: a

text: constant

text: K

text: such

text: that

text: for

text: (φ,

text: π)

text: K,

text: we

text: have

text: E(φ,

text: π)

text: E

text: That

text: is,

text: truncating

text: the

text: beginning

text: of

text: the

text: sequence

text: if

text: necessary,

text: (φ

text: π

text: K.

text: In

text: particular,

text: extracting

text: a

text: sub-

text: n

text: n

text: sequence

text: if

text: required,

text: (φ

text: π

text: convergences

text: weakly

text: in

text: to

text: a

text: limit,

text: (φ∞,

text: π∞).

text: n

text: n

text: It

text: remains

text: to

text: be

text: shown

text: that

text: (φ∞,

text: π∞)

text: S.

text: In

text: the

text: following,

text: it

text: will

text: be

text: convenient

text: to

text: let

text: δ

text: then

text: we

text: assume

text: that

text: C,

text: and

text: prove

text: below

text: that

text: the

text: inﬁmum

text: j

text: n

text: is

text: realised

text: in

text: S.

text: Proposition

text: and

text: the

text: deﬁnition

text: of

text: Φˆ

text: give

text: φ

text: c(

text: R˜

text: φ

text: π

text: n

text: n

text: n

text: e

text: 2φnΦ

text: (φ

text: π

text: n

text: n

text: π

text: φ

text: n

text: n

text: π

text: φ

text: n

text: n

text: which

text: follows

text: from

text: the

text: assumption

text: Φˆ(φ

text: π

text: C

text: and

text: condition

text: (ii)

text: for

text: n

text: n

text: the

text: proof

text: of

text: Theorem

text: and

text: from

text: Φˆ(φ

text: π

text: for

text: the

text: proof

text: of

text: Theorem

text: n

text: n

text: πij

text: Similarly,

text: the

text: assumption

text: C

text: and

text: the

text: deﬁnition

text: of

text: Φi

text: gives

text: j

text: n

text: π

text: c(

text: φ

text: π

text: n

text: n

text: n

text: e

text: 4φnΦ

text: (φ

text: π

text: i

text: n

text: n

text: c(

text: φ

text: π

text: n

text: n

text: φ

text: π

text: n

text: n

text: φ

text: π

text: n

text: n

text: We

text: now

text: recall

text: the

text: scale-broken

text: estimate

text: (Theorem

text: of

text: Ref.

text: Proposition

text: of

text: Ref.

text: u

text: C

text: u

text: u

text: Note

text: that

text: the

text: application

text: of

text: the

text: scale-broken

text: estimate

text: here

text: requires

text: Combining

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Stephen McCormick 3 STATIC METRIC EXTENSIONS ∈ G a generic g . However, by writing the energy as the volume integral of a divergence M over (cf. Proposition 4.5 of [3]) the terms combine and the dominant terms cancel out. In particular, the regularised Lagrange function, (cid:90) (cid:90) − (cid:112) L (g; N) = (N∞ N)R(g) reg M ∇ ∇ − ∇ ∇ − + N∞(˚gik˚gjl(˚ ˚ g ˚ ˚ g ) ˚g R(g)), (3.4) M k l ij i k jl G is well deﬁned on all of , and equal to L(g; N) where the latter is deﬁned. The ∈ ﬁrst integral clearly converges since ρ L1 and the second is bound by noting that the ∇ ∇ − ∇ ∇ dominant term in R(g) is ˚gik˚gjl(˚ ˚ g ˚ ˚ g ), when expressed in terms of the k l ij i k jl background connection (see Proposition 4.2 of [3], and the explicit expression for R(g) − ∈ can be found in Appendix A of [15]). Now we show that if (N N∞) W−2,2 then we (cid:90) 1/2 have − · ∗ DL (g; N)[h] = h DR(g) [N], (3.5) reg M ∗ where DR(g) is the formal adjoint of DR(g). This is easily seen by direct calculation, making use of the following Lemma. Lemma 3.3 (Lemma 4.4 of Ref. [3]). Let S be the Euclidean sphere of radius R, E R M \ R be the exterior region to S – the connected component of S containing inﬁnity – R ∈R and A be the annular region between S and S . Suppose u H−1 (E ), then for R ≥ (cid:73) R 2R 3/2 R0 every R R , 0 | | ≤ (cid:107) (cid:107) u dS cR1/2 u − , (3.6) 1,2, 3/2:A R S R where c is independent of R. Note ﬁrst √ · ∗ − ∇ ∇ − ∇ ∇ − ∇ h DR(g) [N] NDR(g)[h] = i(N(˚ hk jh ) + h ˚jN hk˚ N) g, i k ij ij k i M and then taking the integral of this divergence over results in several boundary terms; those on ∂M vanish due to the boundary conditions. The boundary terms at inﬁnity of 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Stephen

text: McCormick

text: STATIC

text: METRIC

text: EXTENSIONS

text: G

text: a

text: generic

text: g

text: However,

text: by

text: writing

text: the

text: energy

text: as

text: the

text: volume

text: integral

text: of

text: a

text: divergence

text: M

text: over

text: (cf.

text: Proposition

text: of

text: the

text: terms

text: combine

text: and

text: the

text: dominant

text: terms

text: cancel

text: out.

text: In

text: particular,

text: the

text: regularised

text: Lagrange

text: function,

text: L

text: (g;

text: N)

text: (N∞

text: N)R(g)

text: reg

text: M

text: N∞(˚gik˚gjl(˚

text: g

text: g

text: ˚g

text: R(g)),

text: M

text: k

text: l

text: ij

text: i

text: k

text: jl

text: G

text: is

text: well

text: deﬁned

text: on

text: all

text: of

text: and

text: equal

text: to

text: L(g;

text: N)

text: where

text: the

text: latter

text: is

text: deﬁned.

text: The

text: ﬁrst

text: integral

text: clearly

text: converges

text: since

text: ρ

text: and

text: the

text: second

text: is

text: bound

text: by

text: noting

text: that

text: the

text: dominant

text: term

text: in

text: R(g)

text: is

text: ˚gik˚gjl(˚

text: g

text: g

text: when

text: expressed

text: in

text: terms

text: of

text: the

text: k

text: l

text: ij

text: i

text: k

text: jl

text: background

text: connection

text: (see

text: Proposition

text: of

text: and

text: the

text: explicit

text: expression

text: for

text: R(g)

text: can

text: be

text: found

text: in

text: Appendix

text: A

text: of

text: Now

text: we

text: show

text: that

text: if

text: (N

text: N∞)

text: then

text: we

text: have

text: DL

text: (g;

text: N)[h]

text: h

text: DR(g)

text: [N],

text: reg

text: M

text: where

text: DR(g)

text: is

text: the

text: formal

text: adjoint

text: of

text: DR(g).

text: This

text: is

text: easily

text: seen

text: by

text: direct

text: calculation,

text: making

text: use

text: of

text: the

text: following

text: Lemma.

text: Lemma

text: (Lemma

text: of

text: Ref.

text: Let

text: S

text: be

text: the

text: Euclidean

text: sphere

text: of

text: radius

text: R,

text: E

text: R

text: M

text: R

text: be

text: the

text: exterior

text: region

text: to

text: S

text: the

text: connected

text: component

text: of

text: S

text: containing

text: inﬁnity

text: R

text: ∈R

text: and

text: A

text: be

text: the

text: annular

text: region

text: between

text: S

text: and

text: S

text: Suppose

text: u

text: (E

text: then

text: for

text: R

text: R

text: every

text: R

text: R

text: u

text: dS

text: u

text: R

text: S

text: R

text: where

text: c

text: is

text: independent

text: of

text: R.

text: Note

text: ﬁrst

text: h

text: DR(g)

text: [N]

text: NDR(g)[h]

text: i(N(˚

text: hk

text: jh

text: h

text: ˚jN

text: hk˚

text: N)

text: g,

text: i

text: k

text: ij

text: ij

text: k

text: i

text: M

text: and

text: then

text: taking

text: the

text: integral

text: of

text: this

text: divergence

text: over

text: results

text: in

text: several

text: boundary

text: terms;

text: those

text: on

text: ∂M

text: vanish

text: due

text: to

text: the

text: boundary

text: conditions.

text: The

text: boundary

text: terms

text: at

text: inﬁnity

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 hedra, where each pair of NiO along the b-direction are 6 edge-shared with both NbO octahedra and BO tetra- 6 4 hedra, as illustrated in Fig. 2. FIG. 2. (Color online) (a) A layer of edge-shared NiO octahe- 6 dra (green) shown in c-projection containing armchair chains along the b-direction, where every BO tetrahedron (blue) in 4 the neighboring layer bridges a pair of NiO and one NbO 6 6 octahedra (yellow) through face-sharing, as shown in (b). B. Magnetic susceptibility FIG. 1. (Color online) (a) The powder XRD pattern from the pulverized as grown Ni NbBO crystal. (b) The XRD pattern 2 6 of Ni NbBO crystal perpendicular to the large surface to 2 6 show (h00) peaks with preferred orientation. Inset is the as grown single crystal. − for the total energy was 10 6 eV. III. RESULTS AND DISCUSSION A. Crystal structure The powder XRD pattern of the polycrystalline sample FIG. 3. (Color online) The temperature dependence of mag- obtained from the pulverized as-grown Ni2NbBO6 single netic suscept−ibility χ(T) and the corresponding inverse sus- crystal sample is shown in Fig. 1(a). All diﬀraction peaks ceptibility χ 1(T) measured at an applied magnetic ﬁeld of − can be indexed to the orthorhombic structure with space 10 kOe in the temperature range 2 300 K for the pulverized group Pnma, without any observable trace of impurity as-grown Ni2NbBO6 crystals. phase. The structural parameters were reﬁned by the Rietveld technique with good quality reﬁnement param- Fig. 3 shows the temperature dependence of magnetic eters (Rwp = 1.67 % and Rp = 1.04 %). The obtained susceptib−ility χ(T) and the corresponding inverse suscep- values of the lattice parameters are a = 10.0690(1) ˚A, tibility χ 1(T) measured at an applied magnetic ﬁeld of b = 8.6266(2) ˚A, and c = 4.4932(3) ˚A, which are in good 10 kOe in the temperature range 2-300 K for pulverized agreement with previously reported values12. Fig. 1(b) powder of the as-grown Ni NbBO crystal. The χ(T) 2 6 illustrates the single crystal XRD pattern with peaks in- curve shows a Curie-Weiss like behavior at high temper- dexed for the preferred orientation perpendicular to the ature and a sharp peak is observed at 24 K, indicating (h00) planes. This compound could also be viewed as lay- the onset of a antiferromagnetic ordering. The ordering ers containing armchair chains of edge-shared NiO octa- temperature T = 23.5 K is deﬁned by the sharp peak 6 N ##LTFigure## ##LTFigure## ##LTFigure##

text: hedra,

text: where

text: each

text: pair

text: of

text: NiO

text: along

text: the

text: b-direction

text: are

text: edge-shared

text: with

text: both

text: NbO

text: octahedra

text: and

text: BO

text: tetra-

text: hedra,

text: as

text: illustrated

text: in

text: Fig.

text: FIG.

text: (Color

text: online)

text: (a)

text: A

text: layer

text: of

text: edge-shared

text: NiO

text: octahe-

text: dra

text: (green)

text: shown

text: in

text: c-projection

text: containing

text: armchair

text: chains

text: along

text: the

text: b-direction,

text: where

text: every

text: BO

text: tetrahedron

text: (blue)

text: in

text: the

text: neighboring

text: layer

text: bridges

text: a

text: pair

text: of

text: NiO

text: and

text: one

text: NbO

text: octahedra

text: (yellow)

text: through

text: face-sharing,

text: as

text: shown

text: in

text: (b).

text: B.

text: Magnetic

text: susceptibility

text: FIG.

text: (Color

text: online)

text: (a)

text: The

text: powder

text: XRD

text: pattern

text: from

text: the

text: pulverized

text: as

text: grown

text: Ni

text: NbBO

text: crystal.

text: (b)

text: The

text: XRD

text: pattern

text: of

text: Ni

text: NbBO

text: crystal

text: perpendicular

text: to

text: the

text: large

text: surface

text: to

text: show

text: peaks

text: with

text: preferred

text: orientation.

text: Inset

text: is

text: the

text: as

text: grown

text: single

text: crystal.

text: for

text: the

text: total

text: energy

text: was

text: eV.

text: III.

text: RESULTS

text: AND

text: DISCUSSION

text: A.

text: Crystal

text: structure

text: The

text: powder

text: XRD

text: pattern

text: of

text: the

text: polycrystalline

text: sample

text: FIG.

text: (Color

text: online)

text: The

text: temperature

text: dependence

text: of

text: mag-

text: obtained

text: from

text: the

text: pulverized

text: as-grown

text: single

text: netic

text: suscept−ibility

text: χ(T)

text: and

text: the

text: corresponding

text: inverse

text: sus-

text: crystal

text: sample

text: is

text: shown

text: in

text: Fig.

text: All

text: diﬀraction

text: peaks

text: ceptibility

text: χ

text: measured

text: at

text: an

text: applied

text: magnetic

text: ﬁeld

text: of

text: can

text: be

text: indexed

text: to

text: the

text: orthorhombic

text: structure

text: with

text: space

text: kOe

text: in

text: the

text: temperature

text: range

text: K

text: for

text: the

text: pulverized

text: group

text: Pnma,

text: without

text: any

text: observable

text: trace

text: of

text: impurity

text: as-grown

text: crystals.

text: phase.

text: The

text: structural

text: parameters

text: were

text: reﬁned

text: by

text: the

text: Rietveld

text: technique

text: with

text: good

text: quality

text: reﬁnement

text: param-

text: Fig.

text: shows

text: the

text: temperature

text: dependence

text: of

text: magnetic

text: eters

text: (Rwp

text: and

text: Rp

text: The

text: obtained

text: susceptib−ility

text: χ(T)

text: and

text: the

text: corresponding

text: inverse

text: suscep-

text: values

text: of

text: the

text: lattice

text: parameters

text: are

text: a

text: ˚A,

text: tibility

text: χ

text: measured

text: at

text: an

text: applied

text: magnetic

text: ﬁeld

text: of

text: b

text: ˚A,

text: and

text: c

text: ˚A,

text: which

text: are

text: in

text: good

text: kOe

text: in

text: the

text: temperature

text: range

text: K

text: for

text: pulverized

text: agreement

text: with

text: previously

text: reported

text: Fig.

text: powder

text: of

text: the

text: as-grown

text: Ni

text: NbBO

text: crystal.

text: The

text: χ(T)

text: illustrates

text: the

text: single

text: crystal

text: XRD

text: pattern

text: with

text: peaks

text: in-

text: curve

text: shows

text: a

text: Curie-Weiss

text: like

text: behavior

text: at

text: high

text: temper-

text: dexed

text: for

text: the

text: preferred

text: orientation

text: perpendicular

text: to

text: the

text: ature

text: and

text: a

text: sharp

text: peak

text: is

text: observed

text: at

text: K,

text: indicating

text: planes.

text: This

text: compound

text: could

text: also

text: be

text: viewed

text: as

text: lay-

text: the

text: onset

text: of

text: a

text: antiferromagnetic

text: ordering.

text: The

text: ordering

text: ers

text: containing

text: armchair

text: chains

text: of

text: edge-shared

text: NiO

text: octa-

text: temperature

text: T

text: K

text: is

text: deﬁned

text: by

text: the

text: sharp

text: peak

text: N

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##