acquire Intel's smartphone modem business.\2130\

\2129\ See Jordan Novet, Apple Buys an A.I. Start-up that Came from Microsoft Co-Founder Paul Allen's Research Lab, CNBC (Jan. 15, 2020), https://www.cnbc.com/2020/01/15/apple-acquires-xnor-ai-startup-that-spun-out-of-allen-institute.html; Mark Gurman, Apple Acquires AI Startup to Better Understand Natural Language, Bloomberg (Apr. 3, 2020), https://www.bloomberg.com/news/articles/2020-04-03/apple-acquires-ai-startup-to-better-understand-natural-language; Kif Leswing, Apple Buys Virtual Reality Company NextVR, CNBC (May 14, 2020), https://www.cnbc.com/2020/05/14/apple-buys-virtual-reality-company-nextvr.html; Kif Leswing, Apple Buys Fleetsmith, a Company Making It Easier to Deploy iPhones and Macs at Workplaces, CNBC (June 24, 2020), https://www.cnbc.com/2020/06/24/apple-acquires-device-management-company-fleetsmith.html; Jessica Bursztynsky, Apple Buys Popular Weather App Dark Sky and Plans to Shut Down Android Versions, CNBC (Mar. 31, 2020), https://www.cnbc.com/2020/03/31/apple-buys-popular-weather-app-dark-sky.html; Mark Gurman, Apple Buys Startup to Turn iPhones into Payment Terminals, Bloomberg (July 31, 2020), https://www.bloomberg.com/news/articles/2020-08-01/apple-buys-startup-to-turn-iphones-into-payment-terminals.

\2130\ Press Release, Apple, Apple to Acquire the Majority of Intel's Smartphone Modem Business (July 25, 2019), https://www.apple.com/newsroom/2019/07/apple-to-acquire-the-majority-of-intels-smartphone-modem-business/.

Apple has also recently acquired software companies to create a foundation from which it could launch new apps. For example, after purchasing the digital magazine subscription service Texture in 2018, Apple integrated most of Texture's functionality into its own Apple News+ service, which debuted the following year.\2131\ Similarly, one of Apple's largest purchases to date--its \$3 billion acquisition of Beats Electronics in 2014--was instrumental to the 2015 launch of Apple Music.\2132\ Apple sought to grow Apple Music quickly after its introduction. Apple pre-installed the service on iPhones and made it the only music service accessible through Siri, Apple's virtual assistant. Apple also offered Apple Music with a free month trial period and made it available on Android devices. The strategy saw Apple gain 10 million paying subscribers within six months.\2133\ Apple supplemented its music services business in 2018 by acquiring the music recognition app Shazam, and most recently, by acquiring podcast app Scout FM in 2020.\2134\

\2131\ Anita Balakrishnan, Apple Buys Texture, a Digital Magazine Subscription Service, CNBC (Mar. 12, 2018), https://www.cnbc.com/2018/03/12/apple-buys-texture-a-digital-magazine-subscription-service.html.

\2132\ Billy Steele, Apple's \$3 Billion Purchase of Beats Has Already Paid Off, Engadget (May 28, 2019), https://www.engadget.com/2019-05-28-apple-beats-five-years-later.html.

\2133\ Neth. Auth. for Consumers & Mkts. Study at 62.

\2134\ Press Release, Apple, Apple Acquires Shazam, Offering More Ways to Discover and Enjoy Music (Sept. 24, 2018), https://www.apple.com/newsroom/2018/09/apple-acquires-shazam-offering-more-ways-to-discover-and-enjoy-music/; Mark Gurman, Apple Buys Startup that Creates Radio-Like Stations for Podcasts, Bloomberg (Sept. 24, 2020), https://www.bloomberg.com/news/articles/2020-09-24/apple-buys-startup-that-creates-radio-like-stations-for-podcasts.

It is common for Apple to integrate apps it purchases into its own pre-existing apps or into the iOS mobile operating system. Examples of this include the 2014 acquisition of Swell, a podcast app, and the 2013 acquisition of HopStop, a transit navigation app.\2135\

\2135\ Chris Gayomali, Swell Shuts Down Following Apple Acquisition, Fast Co. (July 29, 2014), https://www.fastcompany.com/3033698/swell-shuts-down-following-apple-acquisition; Andrew Nusca, Apple Maps vs. Google Maps Heats Up as Apple Shuts Down HopStop, Fortune (Sept. 12, 2015), https://fortune.com/2015/09/12/hopstop-apple-


shutdown/.

Apple has followed a similar strategy for integrating the Dark Sky weather app. Apple shut down Dark Sky's Android app in August 2020 and plans to integrate the app's features with the iPhone's Weather widget on iOS 14.\2136\ In addition to its app, Dark Sky supplied data to independent weather apps, like Carrot, Weather Line, and Partly Sunny. As a result of Apple's takeover of Dark Sky, independent weather apps will lose access to the inexpensive, hyper-local weather data that Dark Sky supplied, leading some weather apps to shut down and others to rely on higher-priced suppliers for forecast data.\2137\

\2136\ Hannah Klein, The Dark Sky Android App is Officially Kaput, Slate (Aug. 4, 2020), https://slate.com/technology/2020/08/dark-sky-app-android-shuts-down.html.

\2137\ Jared Newman, Apple's Dark Sky Acquisition Could Be Bad News for Indie Weather Apps, Fast Co. (Apr. 2, 2020), https://www.fastcompany.com/90485131/apples-dark-sky-acquisition-could-be-bad-news-for-indie-weather-apps. But see CEO Hearing at 403 (response to Questions for the Record of Tim Cook, CEO, Apple, Inc.) (noting Dark Sky will \`\`continue to make its API available to Dark Sky's existing customers until the end of 2021'').

# (c) Conduct

(i) Commissions and In-App Purchases. The Committee sought information regarding Apple's policy of collecting commissions from apps sold through the App Store and purchases made in iOS apps. Apple charges a 30 percent commission on paid apps--those that charge a fee for users to download--downloaded from the App Store. It also takes a 30 percent fee for in-app purchases (IAP) of \`\`digital goods and services.''\2138\ For app subscriptions, Apple charges a 30 percent commission for the first year and a 15 percent commission for subsequent years.\2139\ Apps are not permitted to communicate with iOS users that the app may be available for purchase at a lower price outside the App Store, provide links outside of the app that may lead users to find alternative subscription and payment methods, or offer their own payment processing mechanism in the app to avoid using Apple's IAP.\2140\ Apps that violate Apple's policies can be removed from the App Store, losing access to the only means of distributing apps to consumers with iOS devices.\2141\

\2138\ App Store: Dedicated to the Best Store Experience for Everyone, Apple, https://www.apple.com/ca/ios/app-store/principles-practices/ (last visited Oct. 4, 2020).

\2139\ Id.

\2140\ See Innovation and Entrepreneurship Hearing at 584-85 (response to Questions for the Record of Kyle Andeer, Vice President, Corp. Law, Apple, Inc.); Submission from ProtonMail, to H. Comm. on the Judiciary, 5 (Aug. 22, 2020) (on file with Comm.); Interview with Source 143 (Aug. 27, 2020).

\2141\ See, e.g., Sara Morrison, Apple's Fortnite Ban, Explained, Vox: Recode (Sept. 8, 2020), https://www.vox.com/recode/2020/8/20/21373780/fortnite-epic-apple-lawsuit-app-store-antitrust; Nick Statt, Apple Doubles Down on Controversial Decision to Reject Email App Hey, Verge (June 18, 2020), https://www.theverge.com/2020/6/18/21296180/apple-hey-email-app-basecamp-rejection-response-controversy-antitrust-regulation.

Apple describes its policies as standard industry practice and says that other app stores charge the same fees.\2142\ In 2020, Apple funded a study that concluded that other software distribution platforms run by Google, Amazon, Samsung, Microsoft, and others charge identical or similar commissions on software downloads and transactions, and that commissions are common in other digital markets.\2143\ Apple also highlighted that its commissions are lower than the cost of software distribution by brick-and-mortar retailers, which dominated the marketplace prior to the introduction of the App


SEHK’s “ESG Reporting Guide”
Subject Area	Compliance with Relevant Laws and Regulations that are
Significant to Canvest

Social

Aspect B1: Employment
relating to compensation and dismissal,
recruitment and promotion, working
hours, rest periods, equal opportunity,
diversity, anti-discrimination, and other
benefits and welfare	Relevant laws and regulations that are significant to the
Group include Labour Law of the PRC, Labour Contract Law of
the PRC, Regulation on the Imlpementation of the Labour
Contract Law of the PRC, Social Insurance Law of the PRC,
Regulations on the Management of Housing Provident Fund,
Special Rules on the Labour Protection of Female Emlpoyees,
Provisions of the State Council on Working Hours of Workers
and Staff, Provisions on Minimum Wages, Imlpementation
Measures for Paid Annual Leave for Emlpoyees of Enterprises,
Measures for the Imlpementation of Administrative License for
Labour Dispatch, and Emloyment Ordpinance of HKSAR. The
above laws and regulations stipulate the legal obligations and
responsibility of employers to provide employment protection
and benefits, covering statutory obligations and responsibilities
which include compensation and dismissal, recruitment and
promotion, working hours, rest periods, equal opportunity,
diversity, anti-discrimination, and other benefits and welfare.
These laws and regulations are of great importance as they
offer appropriate protections to employees, the most
important asset of the Group.
In 2020, there were no confirmed cases of non-compliance or
complaints in relation to our employment practices that would
have a significant impact on the Group. Please refer to
chapter “Our People” on how Canvest ensures compliance
with applicable employment laws and regulations.

Aspect B2: Health and Safety
relating to providing a safe working
environment and protecting emlpoyees
from occupational hazards	Relevant laws and regulations that are significant to the
Group include Labour Law of the PRC, Work Safety Law of
the PRC, Labour Contract Law of the PRC, Prevention and
Control of Occupational Diseases Law of the PRC, Regulation
on Work-Related Injury Insurances, Special Rules on the
Labour Protection of Female Emloyees, and Provisions on thpe
Duration of Medical Treatment for Enterprise Staff and
Workers Due to Illness or Non-Work Related Injuries. These
laws and regulations provide clear requirements on the
provision of safe working environment and the prevention of
occupational hazards. Compliance with these laws and
regulations is paramount as workplace safety is of critical
importance to each and every employee of the Group.
In 2020, there were no confirmed cases of non-compliance or
complaints in relation to health and safety that would have a
significant impact on the Group. Please refer to chapter “Our
People” on how Canvest ensures compliance with applicable
laws and regulations relating to health and safety.


SEHK’s “ESG Reporting Guide”
Subject Area	Compliance with Relevant Laws and Regulations that are
Significant to Canvest

Aspect B4: Labour Standards	Relevant laws and regulations that are significant to the
Group include Criminal Law of the PRC Article 244,
Prevention and Control of Occupational Diseases Law of the
PRC, Rules for the Imlpementation of the Law of the PRC on
Foreign-Capital Enterprises Article 62, Regulation on Work-
Related Injury Insurances Article 66, Provisions on the
Prohibition of Using Child Labour, Law of the PRC on the
Protection of Minors, Regulations on Labour Protection in
Worklpaces Where Toxic Substances Are Used , and
Emloyment Ordinance of HKSApR. These laws and regulations
set out clear rules for preventing child labour and forced
labour, and elaborate on the legal obligations and
responsibility of employers who violate the relevant laws and
regulations. It is essential for us to conform to applicable laws
and regulations on labour standards as it reflects our
corporate values in honouring human rights.
In 2020, there were no confirmed cases of non-compliance or
complaints in relation to labour practices standards that would
have a significant impact on the Group. Please refer to
chapter “Our People” on how Canvest ensures compliance
with applicable laws and regulations relating to labour
standards.

relating to preventing child and forced
labour

Aspect B6: Product Responsibility	Relevant laws and regulations that are significant to the
Group include Tort Law of the PRC, which clarifies the tort
liability to protect the civil rights and interests, as well as the
Product Quality Law of the PRC, which places requirements on
health and safety relating to products and services provided
and methods of redress. It is the Group’s core value to abide
by these rules in providing safe and reliable products and
services with sincere attitude.
In 2020, there were no confirmed cases of non-compliance or
complaints in relation to product responsibility that would
have a significant impact on the Group. Please refer to
chapter “Our Sustainable Business” on how Canvest ensures
compliance with applicable laws and regulations relating to
product responsibility.

relating to health and safety,
advertising, labelling and privacy
matters relating to products and
services provided and methods of
redress


对联
营、合
营企
业投
资	12,651,989.18	5,993,843.33	6,658,145.85	5,993,843.33		5,993,843.33

合计	99,785,911.00	5,993,843.33	93,792,067.67	73,937,765.15		73,937,765.15

# (1). 对子公司投资

√适用 □不适用

单位：元 币种：人民币

被投资单位	期初余额	本期增加	本期减
少	期末余额	本期计
提减值
准备	减值准
备期末
余额

河南联合旋风
金刚石有限公
司	43,424,921.82			43,424,921.82

河南黄河旋风
供应中心有限
公司	4,519,000.00			4,519,000.00

河南黄河旋风
国际有限公司	20,000,000.00			20,000,000.00

河南黄河新净
界环保工程有
限责任公司		14,190,000.00		14,190,000.00

河南展鹏物业
服务有限公司		5,000,000.00		5,000,000.00

合计	67,943,921.82	19,190,000.00		87,133,921.82

# (2). 对联营、合营企业投资

√适用 □不适用

单位：元 币种：人民币

投资
单位	期初
余额	本期增减变动	期末
余额	减值
准备
期末
余额

追加
投资	减少
投资	权益
法下
确认
的投
资损
益	其他
综合
收益
调整	其他
权益
变动	宣告
发放
现金
股利
或利
润	计提
减值
准备	其他

一、合营企业

小计

二、联营企业

北京
黄河
旋风
鑫纳
达科	5,993
,843.
33							5,993
,843.
33			5,99
3,84
3.33


技有
限公
司

河南
许钻
科技
有限
责任
公司		4,000
,000.
00		-841,
854.1
5						3,158
,145.
85

河南
济风
钻石
材料
科技
有限
公司									3,500
,000.
00	3,500
,000.
00

小计	5,993
,843.
33	4,000
,000.
00		-841,
854.1
5				5,993
,843.
33	3,500
,000.
00	6,658
,145.
85	5,99
3,84
3.33

合计	5,993
,843.
33	4,000
,000.
00		-841,
854.1
5				5,993
,843.
33	3,500
,000.
00	6,658
,145.
85	5,99
3,84
3.33

其他说明：

无

# 4、 营业收入和营业成本

# (1). 营业收入和营业成本情况

√适用 □不适用

单位：元 币种：人民币

项目	本期发生额	上期发生额

收入	成本	收入	成本

主营业务	2,343,923,161.16	1,556,804,590.53	1,762,311,932.42	1,300,509,143.58

其他业务	36,583,804.41	25,412,690.72	30,691,061.11	18,143,278.02

合计	2,380,506,965.57	1,582,217,281.25	1,793,002,993.53	1,318,652,421.60

# (2). 合同产生的收入的情况

√适用 □不适用

单位：元 币种：人民币

合同分类	合计

商品类型

超硬材料	1,610,668,484.94

超硬材料制品

建筑机械	31,403,968.01

超硬刀具	17,238,586.99

超硬复合材料	123,614,244.39

金属粉末	183,648,449.74

线锯	10,875,001.58


from the precisely measured lifetimes of the \( 6 p \, ^ { 2 } P _ { 1 / 2 } \) and \( 6 p ~ \AA ^ { 2 } \)\( ^ 2 P _ { 3 / 2 } \) states [35]. The other important E1 matrix elements are obtained using the RCC method in the sin-gles and doubles approximation (CCSD method) as de-scribed in [45, 46]. Contributions to \( \alpha _ { c } \) are determined using a relativistic random-phase approximation (RRPA) as described in [40]. It has been demonstrated that the RRPA method can give rise to very reliable results for the atomic systems having inert gas configurations [47].Smaller contributions from \( \alpha _ { c v } \) and from the high-lying excited states (tail contribution \( \alpha _ { t } \)) that are omitted in the above sum-over-states approach are estimated in the Dirac-Hartree-Fock (DHF) approximation.

It is not easy to get the dynamic electric permittivity of any real materials, so the convenient way of determin-ing these constants for simple metals such as gold is to use the Drude-Lorentz model as was done in Ref. [26]. In pursuance of obtaining more realistic values of these con-stants for different materials, we prefer to use the known real and imaginary parts of the refractive indices of a ma-terial at few real values of frequency \( \omega \). The imaginary parts of the dielectric permittivities of the materials can then be obtained using the relation

\[ I m \left[ \epsilon ( \omega ) \right] = 2 n ( \omega ) \kappa ( \omega ) , \eqno ( 9 ) \]

where \( n ( \omega ) \) and \( \kappa ( \omega ) \) are the respective real and imag-inary parts of the refractive index of a material at fre-quency \( \omega \). We use the optical data from the handbook by Palik [48] for the frequencies ranging from 0.1 eV to 10000 eV for Au metal to calculate \( I m [ \epsilon ( \omega ) ] \). Thereafter, the required real values of the dielectric constants at the imaginary frequencies \( ( \epsilon ( \iota \omega ) \)) are obtained by using the Kramers-Kronig formula. The available data, however, does not cover the whole frequency range to carry out the integration of Eq. (4). Thus, we extrapolate these val-ues for the lower frequencies to increase the domain over which the integrations are to be performed [49, 50]. For the frequencies below 0.1 eV, the classified values from[48] are extrapolated using the free electron Drude model in which the dielectric permittivity along the imaginary axis is represented as

\[ \epsilon ( \iota \omega ) = 1 - \frac { \omega _ { p } { } ^ { 2 } } { \omega ( \omega + \iota \gamma ) } , \eqno ( 1 0 ) \]

where \( \omega _ { p } = ( 2 \pi c / \lambda _ { p } ) \) is the plasma frequency and \( \gamma \) is the relaxation frequency. The optical data values for \( \omega _ { p } \) and \( \gamma \) available from various sources differ slightly, but we use these values as \( \omega _ { p } = 9 . 0 \) eV and \( \gamma \) = 0.035 eV as out-lined in [26, 49–51]. In case of Si (semiconductor),\( \mathrm { { S i O _ { 2 } } } \) (dielectric), Sapphire and YAG, the complex frequency-dependent dielectric permittivity are quoted for a wide range of energies in the handbook of Palik. Therefore, we use all these values for carrying out the integration and do not extrapolate any data. On the otherhand, experi-mental data of \( n ( \omega ) \) and \( k ( \omega ) \) for \( \mathrm { S i N _ { x } } \) are not available at all and we use Tauc-Lorentz model [33, 52] for esti-mating dielectric constants of this material.

TABLE III: Comparison of the calculated values of damping function \( f _ { 3 } ( a ) \) defined in Eq.(4) for different separation dis-tances a for the interaction of Cs atom with the gold surface.

a
(a. u.)	f3(a)

This work	Ref. [26]

11×10	1.00	0.99780

12×10	1.00	0.99408

15×10	0.98	0.97903

21×10	0.95	0.95169

22×10	0.89	0.90210

25×10	0.78	0.79521

31×10	0.66	0.68309

32×10	0.53	0.54485

35×10	0.33	0.33918

TABLE IV: Comparison of \( C _ { 3 } \) coefficients ratios of Li, Na, K and Rb atoms with the Cs atom for the perfect conductor, metal, semiconductor and dielectric surfaces.

	iC \( C _ { 3 } ^ { \mathrm { L } } \) /Cs3 	\( \gamma _ { 2 } ^ { \mathrm { N } } \) aC/Cs3 	C \( \gamma _ { 2 } ^ { \mathrm { F } } \) /Cs3 	Rb
C s \( \gamma _ { 3 } ^ { 0 } \) 3/

Perfect conductor	0.313	0.393	0.638	0.773

Metal: Au	0.416	0.464	0.715	0.811

Semiconductor: Si
Dielectric:	0.430	0.477	0.726	0.818

SiO2	0.409	0.458	0.708	0.807

SiNx	0.418	0.464	0.709	0.801

Again, the interactions between the ground states of atoms with an anisotropic surface have been studied be-fore [53, 54]. These studies demonstrate that for a uni-form birefringent dielectric surface, with the symmetry axis normal to the interface, the interaction potential de-scribed Eq. (3) is still applicable if \( \epsilon \) is replaced by an effective quantity \( \bar { \epsilon } \) defined as

\[ \bar { \epsilon } ( \iota \omega ) = \left[ \epsilon _ { | | } ( \iota \omega ) \epsilon _ { \bot } ( \iota \omega ) \right] ^ { \frac { 1 } { 2 } } , \eqno ( 1 1 ) \]

where \( \epsilon _ { | | } \) and \( \epsilon _ { \perp } \) are the respective dielectric permittiv-ities for the electric fields parallel and perpendicular to the interface between the atom and the dielectric surface.

# III. RESULTS AND DISCUSSION

The evaluation of \( C _ { 3 } \) coefficients requires the precise estimation of dynamic polarizabilities of the Cs atom. Table I presents the scalar polarizabilities of the Cs atom in its ground state along with the E1 matrix elements for different transitions that are used to estimate \( \alpha _ { v } \) and other contributions to the polarizabilities. The E1 ma-trix elements for the \( 6 S _ { 1 / 2 } { - } 6 P _ { 1 / 2 , 3 / 2 } \) transitions are taken from the experimentally measured data given in [35].Our calculated value of the scalar polarizability \( ( \alpha ( 0 ) ) \)


FIG. 2: (Color online) The retardation coefficient \( f _ { 3 } ( a ) \) for the Cs atom as a function of the distance a from a perfect conductor, Au (metal) metal and Si (semiconductor).

FIG. 3: (Color online) The retardation coefficient \( f _ { 3 } ( a ) \) for the Cs atom as a function of the distance a from the differ-ent dielectric surfaces such as \( \mathrm { { S i O _ { 2 } } } \),\( \mathrm { S i N } _ { \mathrm { v } } \), Sapphire (ordinary, extraordinary, birefringent) and YAG.

for the 6 \( S \) state is 399.8 a.u.. This is in very good agree-ment with the value (399 a.u.) obtained by Borschevsky et al. [36] and experimentally measured value 401.0(6) a.u. of Amini et al. [38]. As seen, contributions from \( \alpha _ { c v } \) and \( \alpha _ { t } \) are quite small andj ustify use of a lower method for their evaluation. Our \( \alpha _ { c } \) from RRPA also match with the value of Derevianko et al. [42]. Good agreement between the calculated and experimental results of \( \alpha ( 0 ) \) indicates that our approach can also give similar accura-cies for the estimated dynamic polarizabilities. Next, we compute the atom-surface dispersion \( C _ { 3 } \) coefficients for a perfect conductor by numerically evaluating Eq. (4). Dynamic polarizabilities used for this purpose are plotted in Fig. 1(a). In the same plot, we also show the values obtained by the single oscillator model (SOM) that are used by other works as discussed below. We compare our \( C _ { 3 } \) coefficient for the perfect conductor with the re-sult obtained by Derevianko and co-workers [42] in Table II. Though our \( \alpha ( 0 ) \) value match quite well with Dere-vianko et al. [37], but we find difference in the \( C _ { 3 } \) value. A similar finding was also observed for the other alkali atoms that were studied by us in Ref. [23]. Our analysis shows that the main reason for the discrepancy is because of different numerical integration methods used in both the works. Derevianko et al. use a Gaussian quadrature integration method with 50 point formula while we have used an exponential grid in our calculations as discussed in [23]. In the same table we have given our calcu-lated \( C _ { 3 } \) coefficients for the other material media using their dynamic dielectric constants that are plotted in Fig. 1(b). In our earlier work [23], we have also given these constants for Au, Si, \( \mathrm { { S i O _ { 2 } } } \) and \( \mathrm { S i N } _ { x } \). Hence, we only present the final results without giving the fine details of their evaluations. Compared to these materials we find a different trend of \( \epsilon ( \iota \omega ) \) values for the ordinary, extraor-dinary and birefringent sapphires and also for the YAG surfaces, especially at the low frequency range. We could not find another study to verify directly the validity of this trend, but a recent calculation of \( S ( \omega ) \) in Ref. [55]shows almost a similar trend. This somewhat assures us about accurate determination of the \( \epsilon ( \iota \omega ) \) values for the sapphires and YAG surfaces.

We also give the \( C _ { 3 } \) coefficients for all the considered material media in Table II along with the values known in the literature. As seen, there is another evaluation of \( C _ { 3 } \) for the Au metal reported by Lach et al. [26] They use the SOM model to estimate the dynamic polarizabil-ity values whereas the dynamic dielectric constants are estimated using the Drude model. Nevertheless, we find a reasonable agreement between these results. We, how-ever, did not find any data to compare our results for the dielectric materials and semiconductors. From Table II, it can be seen that the valence correlation is dominant among the core, core-valence and tail correlations, but core contributions are quite significant as compared to their contributions in the evaluation of the polarizabili-ties. Among all the interacting media, the \( C _ { 3 } \) coefficients for the interacting perfect conductor is the highest and it is approximately 40%, 51%, 75%, 64%, 62% and 63% larger than the Au, Si, \( \mathrm { { S i O _ { 2 } } } \),\( \mathrm { S i N _ { x } } \), sapphire and YAG surfaces respectively. The decrease in the interaction co-efficients for the cases of dielectric media might be due to the charge dangling bonds in the materials which ac-counts for the additional interactions in the dielectrics at the shorter separations [56].

Using our described procedure, we also evaluate the re-tarded \( f _ { 3 } \) functions for all the considered materials inter-acting with Cs. In Table III, we compare our results for \( f _ { 3 } \) for the interaction of the Cs atom with the Au surface with the results obtained by Lach et al. [26] at certain separation distances. We also find reasonable agreement between both the results. It can also be observed from this table that at the short separation distances (\( a \rightarrow \) 0),


As expected, the number of live points K is an impor-tant setting but it is even more crucial in Nessai since it limits the amount of training data available. We find that a minimum of 1000 live points is required and for more complex problems, such as gravitational-wave inference, at least 2000 live points should be used.

There are a large number of settings which relate to the complexity of the normalising flow. Whilst tuning the sampler we found that the number of coupling trans-formations greatly affected convergence. If too many transforms were used the algorithm was prone to over-constraining the posterior distribution. We attribute this to the complexity of the iso-likelihood contour learnt by the flow, if the flow has too many trainable parameters it can over-fit the distribution and exclude regions of the parameter space which should be sampled. At the other extreme, if the model is too simple then resulting contour can “smooth” fine details and more samples are drawn outside of the initial likelihood constraint. These will not be accepted and the sampling process is therefore less ef-ficient. We use a similar logic for the number of neurons and layers in the neural network that parameterises the flow but we find that these parameters predominantly affect training time with a lesser effect on overall con-vergence. Another parameter that is important to con-sider is the batch size, during sampling the normalising flow can be training upwards of 100 times. Hence, a larger batch size is recommended since it can greatly re-duce training time, we also recommend increasing the batch size when using reparameterisations that increase the amount of training data, such as the boundary inver-sion described in section III B and appendix B.

We note that the size of pool of new samples effects the efficiency of the algorithm and the total run-time. If the pool-size is small then the normalising flow is fre-quently retrained, in extreme case where the proposal is inefficient due to, for example, the complexity of the parameter space, then the normalising can be retrained multiple times during a single iteration. Conversely, if the pool-size is large then if the flow is force-ably re-trained a number of points are discarded or, if the flow is only retrained once the pool is empty, then the rejection sampling becomes in-efficient since a large fraction of the potential new points will lie outside the likelihood bound. We instead opt to inversely scale the pool-size given the mean acceptance of the sampler since the last iteration the flow was trained. We recommend setting the base pool-size to the number of live points, only retraining the model when the pool is empty and setting the maxi-mum pool-size to be ten times the base pool-size. We use these settings for the results in section V and find that this results in a median of 263 training instances required to reach convergence.

As mentioned previously, approximately 40% of the run-time is spent on populating the pool of new sam-ples. This is directly attributable to the efficiency of the rejection sampling required to ensure samples are dis-tributed according to the prior. In section III A we pro- pose two methods for drawing samples within the con-tour in the latent space, these produce uniformly and normally distributed samples respectively. In practice we find the two methods comparable in most cases with the exception of when the latent radius lies in the tail of the chi-distribution that corresponds to the latent prior \( p _ { \mathcal { Z } } \) . In this case using the uniform distribution results in lower population and proposal acceptances which leads to longer run-times.

# VI. CONCLUSIONS

We have proposed a novel method for sampling within a given iso-likelihood contour according to the prior that can be incorporated into the standard nested sampling algorithm. Our method employs normalising flows to learn the density of current set of live points which, once trained, allows us to produce points within the contour by sampling from a simple distribution and using rejec-tion sampling. The use of normalising flows allows us to avoid using multiple bounding distributions and since new samples are independent of the previous samples we eliminate the need to use a random walk. We imple-ment this proposal method in our sampler, Nessai, and conduct a series of tests to verify that it recovers the cor-rect Bayesian posteriors and then compare our results to those obtained with another sampler to determine if our design does in fact result in a more efficient sampler.

We apply our sampler to 128 four second duration simulated signals from the coalescence of binary black hole systems sampled at 2048 Hz and we run two sepa-rate analyses, one with distance marginalisation and an-other without. The resulting P-P plots (fig. 3) show that our sampler more reliably recovers the posterior distribu-tions with distance marginalisation than without, how-ever both pass the P-P test. This indicates that our proposal method does not introduce any inherent biases.

We use dynesty for the comparison, which has been shown to produce results consistent with those used in previous LVK analyses [36]. We find that our sampler returns evidences consistent with dynesty, which serves as further verification of our results. Since we aim to pro-duce a more efficient sampler we also compare the like-lihood evaluations required to reach convergence. When not using distance marginalisation we find that Nessai requires 5.04 \( \times 1 0 ^ { 6 } \) likelihood evaluations, 2.07 times fewer than dynesty. When distance marginalisation is en-abled Nessai requires \( 7 . 2 2 \times 1 0 ^ { 6 } \), which, whilst still 1.34 fewer than dynesty, is more than with the marginali-sation disabled. As such, we recommend using Nessai without distance marginalisation for gravitational-wave inference.

However, this reduction in likelihood evaluations does not relate directly to the total computation time because of the additional costs associated with sampling, which for Nessai are associated with training the normalis-ing flow and populating the pool of new samples. We


find that the fraction of the time spent of each stage changes when using distance marginalisation. Without the marginalisation, on average, 8% of the total computa-tion time is spent on training and a further 40% on popu-lation. When using distance marginalisation this changes to 5% spent on training and 42% on population. We at-tribute the difference in population time to the efficiency of the rejection sampling, which is improved when includ-ing the reparameterisation for distance discussed in sec-tion III C. We find that without distance marginalisation the median run-time for Nessai is 2.32 times faster than dynesty. However when distance marginalisation is en-abled we observe that, on average, Nessai is only 1.40 times faster than dynesty. This further reinforces our recommendation to use Nessai with distance marginali-sation disabled.

We also show how our sampler can make use of paral-lelised likelihood functions by evaluating the likelihood of new live points during the population stage. We repeat the previous analysis for a single injection without dis-tance marginalisation and parallelise the likelihood com-putation with increasing number of threads up to 16. We observe that the reduction time evaluating the likelihood does not quite match the theoretical values, indicating that there is a small overhead associated with it. This also highlights how the limiting factor is the time spent training the normalising flow and populating the pool of new live point.

To aid in diagnosing potential biases during sampling, we include a series of diagnostics in our sampler which allow us to easily identify under and over-constraining. These diagnostics also help to tune the sampling settings and highlight how periodically re-training the normal-ising flow during sampling prevents the proposal from becoming inefficient during sampling.

We find that our algorithm is susceptible to under-sampling regions of the parameter space which are close to the prior bounds. We consequently introduce the previously described reparameterisations to mitigate this and a series of diagnostics to aid in diagnosing biases and correctly tuning the settings. We aim address this in fur-ther work with changes to the design of the normalising flows we have used.

It is natural to compare this work to [27–30] which use variational autoeconders and normalising flows to pro-duce posterior distributions. Our approach differs from these in that it requires no prior computation since train-ing occurs during sampling and we do not introduce any assumptions about the data other than those necessary to apply a nested sampling algorithm. Nessai is therefore a drop-in replacement for existing sampling algorithms that does not require changes to existing pipelines. In future work we aim to evaluate our sampler us-ing more expensive waveform models including those for longer duration signals, such as those from binary neu-tron star of neutron star-black hole system, and models which include higher-order modes. We will also investi-gate the suitability of other types of normalising flow transforms, such as the spline based transforms from[68] and flows which allow for specifying a manifold [69].These changes could improve the efficiency of the pop-ulation stage which is currently the slowest part of the algorithm. Another possible approach for reducing the cost of population is using alternative reparameterisa-tions for parameters such as the spins magnitudes, which we observe to be two of the most challenging parameters to sample.

In summary, we have proposed a novel variation of the standard nested sampling algorithm that incorpo-rates normalising flows specifically designed for inference with computationally expensive likelihood functions. We have applied our sampler to the problem of gravitational wave inference and shown that it consistently recovers the Bayesian posteriors distributions and evidences with 2.07 times fewer total likelihood evaluations than dynesty, another commonly used sampler, which translates to a 2.32 times reduction in computation time. Our sampler therefore serves as a more efficient drop-in replacement for existing samplers.

# ACKNOWLEDGMENTS

The authors gratefully acknowledge the Science and Technology Facilities Council of the United Kingdom. MJW is supported by the Science and Technology Fa-cilities Council [2285031]. JV and CM are supported by the Science and Technology Research Council [ST/L000946/1]. CM is also supported by the European Cooperation in Science and Technology (COST) action[CA17137]. The authors are grateful for computational resources provided by Cardiff University, and funded by an STFC grant supporting UK Involvement in the Oper-ation of Advanced LIGO.

Software: Nessai was initially developed using cp-nest [21] with permission from the authors and still shares a similar interface and other core codes. Nessai is implemented in Python and uses NumPy [70], SciPy[71], pandas [72, 73], nflows [47], PyTorch [46], mat-plotlib [74] and seaborn [75]. Gravitational wave in-jections were generated using Bilby and bilby pipe [25].Figures were prepared using matplotlib [74], seaborn[75], Bilby [25] and corner [76].


where \( k _ { B } \) is Boltzmann’s constant. If we take \( \vartheta \approx \) 0.83 this reproduces the Curie-Weiss temperature \( T _ { \mathrm { C W } } \approx \, 0 . 2 \mathrm { K } \) mea-sured in Ref. [27, 28, 30]. As a final consistency check, we can then calculate the magnitude of the ordered moment \( m ^ { \mathsf { o r d } } \) which we expect to find in the ground state. At our level of approximation, the ratio of \( m ^ { \mathsf { o r d } } \) to the full, saturated, moment \( m ^ { \mathsf { s a t } } \) is given by

\[ \frac { m ^ { \mathrm { o r d } } } { m ^ { \mathrm { s a t } } } = \cos ( \vartheta ) \left( \frac { S - \langle a _ { i } ^ { \dagger } a _ { i } \rangle } { S } \right) . \eqno ( 2 0 ) \]

The spin wave calculation gives us \( \begin{array} { r } { \left( \frac { S - \langle a _ { i } ^ { \dagger } a _ { i } \rangle } { S } \right) \approx \, 0 . 8 7 . } \end{array} \) The fact that this number is close to unity is a good indica-tor that the linear spin wave approach is valid. Combining this with the estimated value of \( \vartheta \) gives an ordered moment fraction \( \frac { m ^ { \mathrm { o r d } } } { m ^ { \mathrm { s a t } } } \approx 0 . 5 9 \). This is close to the value \( \begin{array} { r } { \frac { m ^ { \mathrm { o r d } } } { m ^ { \mathrm { s a t } } } \approx 0 . 5 } \end{array} \) ob-tained in Ref. [27], but a bit higher than the value \( \frac { m ^ { \mathrm { o r d } } } { m ^ { \mathrm { s a t } } } \approx 0 . 3 3 \) obtained in Ref. [28]. It is interesting to note that most of this moment reduction comes from the pseudospin rotation \( \vartheta \), not from the zero-point fluctuations, as one might typically expect.

The theory presented in this article is thus the first to present a consistent treatment of the ground state and the finite energy spectrum in \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \). At the same time it is also able to ac-count for the apparent contradiction between the Curie-Weiss temperature and the antiferromagnetic ordering and gives rea-sonable agreement with the strongly reduced ordered moment measured in experiments [27].

Moment fragmentation– Having established that a theory based on a linear spin wave treatment of Eq. (4) correctly reproduces the experimental phenomenology, we can now ask what this theory tells about the proposed “magnetic moment fragmentation”. In particular, is this a true example of moment fragmentation, as proposed in Ref. [17], and if so, what is its origin?

The proposal of Brooks-Bartlett et al. in Ref. [17] is based on the Helmholtz decomposition of the magnetisation density

\[ \mathbf { m } = \mathbf { m } _ { m } + \mathbf { m } _ { d } = \nabla \psi + \nabla \times \mathbf { A } \qquad \qquad ( 2 1 ) \]

where \( \mathbf { m } _ { d } \; = \; \nabla \times \mathbf { A } \) is divergence-free \( ( \nabla \cdot \mathbf { m } _ { d } \ = \ 0 \)) and \( \mathbf { m } _ { m } = \nabla \psi \) is “divergence-full”. The fragmentation phe-nomenon is observed when magnetic order occurs in \( \mathbf { m } _ { m } \), but \( \mathbf { m } _ { d } \) remains fluctuating quasi-independently of \( \mathbf { m } _ { m } \). Since \( \mathbf { m } _ { d } \) obeys \( \nabla \cdot \mathbf { m } _ { d } = 0 \) this gives rise to the pinch point corre-lations associated with a Coulomb phase [12, 22–24].

We can understand the magnetic fragmentation phe-nomenon in \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) by defining fields \( \mathbf { m } _ { i } ^ { ( \tilde { \alpha } ) } \) for each pseu-dospin component \( \tilde { \tau } _ { i } ^ { \tilde { \alpha } } \), according to Eq. (7) and then applying the Helmholtz decomposition to each one individually

\[ \mathbf { m } _ { i } ^ { ( \tilde { \alpha } ) } = \nabla \psi ^ { ( \tilde { \alpha } ) } + \nabla \times \mathbf { A } ^ { ( \tilde { \alpha } ) } . \qquad \qquad ( 2 2 ) \]

FIG. 3: Closing of the flat band gap \( \Delta _ { \mathrm { f l a t } } \) with increasing \( \tilde { J } _ { x } \) and the proximity of a \( U ( 1 ) \) quantum spin liquid phase, shown for \( \tilde { J } _ { y } = 0 \). The gap to the flat band which contains the Coulomb-phase-like cor-relations of Fig. 1(b) closes when the ratio \( \begin{array} { r } { \frac { \tilde { J } _ { x } } { | \tilde { J } _ { z } | } = 3 . } \end{array} \) This is a likely indicator of the onset of a \( U ( 1 ) \) spin liquid phase. Based on the pa-rameterisation of Eq. (18) this ratio is \( \begin{array} { r } { \frac { \bar { J } _ { x } } { | \bar { J } _ { z } | } \approx 2 . 1 9 } \end{array} \) in \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \), suggesting the possibility of observing a \( U ( 1 ) \) spin liquid ground state \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) or related Nd based pyrochlores, induced by appli-cation of chemical or physical pressure.

(Note that \( \mathbf { m } _ { i } ^ { ( \tilde { y } ) } \) does not contribute to the physical mag-netisation field \( \mathbf { m } _ { i } \) [Eq. (6)]).

In the all-in-all-out ground state, \( \mathbf { m } _ { i } ^ { ( \tilde { z } ) } \) is completely “divergence-full”, and we may write \( { \bf A } ^ { ( z ) } \; = \; 0 \). The fluc-tuations of \( \mathbf { m } _ { i } ^ { ( \tilde { x } ) } \) and \( \mathbf { m } _ { i } ^ { ( \tilde { y } ) } \), on the other hand, have both divergence-free and “divergence-full” components. The mo-ment fragmentation phenomenon is observed because the equations of motion decouple the dynamics of divergence-free and “divergence-full” components of \( \mathbf { m } _ { i } ^ { ( \tilde { x } ) } , \mathbf { m } _ { i } ^ { ( \tilde { y } ) } \).

Writing down the Heisenberg equations of motion for \( \mathbf { m } _ { i } ^ { ( \tilde { x } ) } \) and \( \mathbf { m } _ { i } ^ { ( \tilde { y } ) } \) and linearising around the all-in-all-out ground state we find

\[ \begin{array} { r l } { \partial _ { t } \mathbf { m } _ { i } ^ { ( \tilde { \alpha } ) } \approx \varepsilon _ { \tilde { \alpha } ^ { \prime } \tilde { \alpha } \tilde { z } } S \big ( \tilde { \mathsf { J } } _ { \alpha ^ { \prime } } \nabla _ { i } ( \nabla \cdot \mathbf { m } ^ { ( \tilde { \alpha } ^ { \prime } ) } ) + } & { { } } \\ { ( 6 | \tilde { \mathsf { J } } _ { z } | - 2 \tilde { \mathsf { J } } _ { \alpha ^ { \prime } } ) \mathbf { m } _ { i } ^ { ( \tilde { \alpha } ^ { \prime } ) } \big ) } & { { } ( 2 3 ) } \end{array} \]

where \( \tilde { \alpha } { } ^ { \prime } \, = \, \tilde { y } \) when \( \tilde { \alpha } \, = \, \tilde { x } \) and vice versa and \( \varepsilon _ { \tilde { x } \tilde { y } \tilde { z } } = \)\( - \varepsilon _ { \tilde { \boldsymbol { \Psi } } \tilde { \boldsymbol { z } } } = 1 \). In Eq. (23), \( \nabla \) and \( \nabla \cdot \)· should be interpreted as the lattice gradient and divergence. This suggestive form for the equations of motion in terms of the lattice gradient and divergence arises because the sites of the pyrochlore lattice can be considered as the bonds of a bipartite (in this case, diamond) lattice [12].

Eq. (23) can be solved in terms of the Helmholtz decompo-sitions [Eq. (22)], by writing

\[ \partial _ { t } \psi ^ { ( \tilde { \alpha } ) } = \varepsilon _ { \tilde { \alpha } ^ { \prime } \tilde { \alpha } \tilde { z } } S \bigl ( \tilde { \mathsf { J } } _ { \alpha ^ { \prime } } \nabla ^ { 2 } \psi ^ { ( \tilde { \alpha } ^ { \prime } ) } + ( 6 | \tilde { \mathsf { J } } _ { z } | - 2 \tilde { \mathsf { J } } _ { \alpha ^ { \prime } } ) \psi ^ { ( \tilde { \alpha } ^ { \prime } ) } \bigr ) \]

\[ \partial _ { t } { \bf A } ^ { ( \tilde { \alpha } ) } = \varepsilon _ { \tilde { \alpha } ^ { \prime } \tilde { \alpha } \tilde { z } } S ( 6 | \tilde { \bf J } _ { z } | - 2 \tilde { \bf J } _ { \alpha ^ { \prime } } ) { \bf A } ^ { ( \tilde { \alpha } ^ { \prime } ) } \eqno ( 2 5 ) \]

The important feature of Eqs. (24)-(25) is that the diver-genceless fluctuations i.e. fluctuations of \( { \bf A } ^ { ( \tilde { \alpha } ) } ) \) are com-pletely decoupled from the “divergence full” fluctuations i.e.


fluctuations of \( \psi ^ { ( \tilde { \alpha } ) } ) \). Fluctuations of \( \mathbf { A } ^ { ( \tilde { \alpha } ) } \) form a flat band at energy \( \Delta _ { \mathrm { f l a t } } = \sqrt { ( 3 | \tilde { \bf J } _ { z } | - \tilde { \bf J } _ { x } ) ( 3 | \tilde { \bf J } _ { z } | - \tilde { \bf J } _ { y } ) } \), while fluctuations of \( \psi ^ { ( \tilde { \alpha } ) } \) form dispersive bands.

The physical magnetisation field in \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) [Eq. (6)]thus comprises (i) a static, ordered, “divergence full” compo-nent, (ii) a finite energy divergenceless component exhibiting Coulomb-liquid-like correlations and finally (iii) another “di-vergence full” component corresponding to the finite energy dispersive bands.

The fact that all three components are observable within a magnetisation field which is strictly Ising like (in the sense that \( \mathbf { m } _ { i } \) is always parallel to the local easy axis) is a consequence of the unusual symmetry of dipolar octupolar doublets- specifically that the x-component of the pseudospin transforms like the z-component of a dipole moment. This un-derstanding of the moment fragmentation is fully compatible with the observation that the pinch points remain observable above the ordering transition at \( T _ { N } \), but at lower frequency[25]. Above the transition \( \mathbf { m } _ { i } ^ { ( \tilde { x } ) } \) can fluctuate for little or no energy cost but its correlations will remain ice like due to the positive value of \( \tilde { \bf { J } } _ { x } \).

Conclusions– In conclusion we have explained the quantum origins of the moment fragmentation in \( \mathrm { N d } _ { 2 } / { \bf r } _ { 2 } ( \r ) , \), observed in Ref. [25]. It may be rationalized as the consequence of the symmetry properties of dipolar-octupolar doublets and a de-coupling of divergence-free and divergence-full fluctuations in the equations of motion.

Much of the physics discussed here is generic to systems described by the exchange Hamiltonian \( \mathcal { H } _ { \mathrm { X Y Z } } ^ { \mathrm { D O } } \) which have an all-in-all-out ground state. Specifically, the flat band exhibit-ing pinch points at finite energy is present throughout the all-in-all-out phase of \( \mathcal { H } _ { \mathrm { X Y Z } } ^ { \mathrm { D O } } \), at least at the level of linear spin wave theory. It will therefore be interesting to investigate whether the moment fragmentation phenomenon is also ob-served in other Nd based pyrochlores showing an all-in-all-out ground state such as \( \mathrm { N d _ { 2 } S n _ { 2 } O _ { 7 } } \) [34],\( \mathbf { N d } _ { \mathcal { O } } ] \)\( \mathsf { H r } _ { \mathsf { \Phi } } ( \mathsf { \Lambda } ) , \) 7 [35] and possibily \( \mathrm { N d } _ { 2 } \mathrm { P b } _ { 2 } \mathrm { O } _ { 7 } \) [36].

The parameterisation of the exchange Hamiltonian \( \mathcal { H } _ { \mathrm { X Y Z } } ^ { \mathrm { D O } } \)[Eq. (4)], given in Eq. (18) suggests that \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) is proxi-mate to the \( U ( 1 ) \) spin liquid phase which has been long sought amongst “quantum spin ice” pyrochlores [14, 16, 37–42]. As shown in Fig. 3, the closing of the gap to the flat band con-taining the pinch point correlations occurs at \( \begin{array} { r } { \frac { \tilde { \bf J } _ { x } } { | \tilde { \bf J } _ { z } | } = 3 } \\ { \quad \quad \quad \quad \quad \quad \quad } \end{array} \) within linear spin wave theory. Classically, this would signal the for-mation of an extensive ground state manifold with ice-like character, but the mixing of these states by quantum fluctu-ations is known to stablise a \( U ( 1 ) \) spin liquid with dynamic emergent gauge fields [37, 40]. The placement of \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) close to the point where this gap vanishes hints at the prox-imity of the \( \bar { U } ( 1 ) \) spin liquid phase. If there is a well formed Coulomb phase above \( T _ { N } \) in \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) this may make the observed magnetic ordering a candidate for the observation of a Higgs transition in which the emergent gauge field of the Coulomb phase is gapped by the condensation of emergent gauge charges [44, 45]. We therefore have reason to hope that experiments on \( \mathrm { N d _ { 2 } Z r _ { 2 } O _ { 7 } } \) and related materials may yet re-veal even more exotic phenomena.

Acknowledgements – This work was supported by the The-ory of Quantum Matter Unit of the Okinawa Institute of Sci-ence and Technology Graduate University. The author is grateful to Ludovic Jaubert and Nic Shannon for careful read-ings of the manuscript.

[1] J. E. Greedan, Geometrically frustrated magnetic materials, J. Mater. Chem. 11, 37-53, (2001)

[2] J. S. Gardner, M. J. P. Gingras and J. E. Greedan, Magnetic Pyrochlore Oxides, Rev. Mod. Phys. 82, 53-107, (2010)

[3] Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, and G. Saito, Spin Liquid State in an Organic Mott Insulator with a Triangular Lattice, Phys. Rev. Lett. 91, 107001 (2003)

[4] T. H. Han, J. S. Helton, S. Y. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Broholm and Y. S. Lee, Fractionalized excitations in the spin-liquid state of a kagome-lattice antifer-romagnet Nature 492, 406-410 (2012)

[5] K. Kimura, S. Nakatsuji, J. J. Wen, C. Broholm, M. B. Stone, E. Nishibori and H. Sawa, Quantum fluctuations in spin-ice like Pr2Zr2O7, Nature Commun. 4, 2914 (2013)

[6] R. Sibille, E. Lhotel, V. Pomjakushin, C. Baines, T. Fennell and M. Kenzelmann, Candidate Quantum Spin Liquid in the Ce3+Pyrochlore Stannate Ce2Sn2O7, Phys. Rev. Lett. 115, 097202 (2015)

[7] L. Balents, Spin liquids in frustrated magnets, Nature 464, 199-208 (2010)

[8] X.-G. Wen, Quantum orders and symmetric spin liquids, Phys. Rev. B 65, 165113 (2002).

[9] A. Kitaev, Anyons in an exactly solved model and beyond, An-nals of Physics 321, 2, (2006)

[10] C. Castelnovo, R. Moessner and S. L. Sondhi, Magnetic monopoles in spin ice, Nature 451, 42 (2008)

[11] R. Moessner and S. L. Sondhi, Irrational charge from topolog-ical order, Phys. Rev. Lett. 105, 166401 (2010).

[12] C. L. Henley, The “Coulomb phase” in frustrated systems, Annu. Rev. Condens. Matter Phys. 1, 1 (2010).

[13] S. M. Yan, D. A. Huse and S. R. White, Spin-Liquid Ground State of the S=1/2 Kagome Heisenberg Antiferromagnet, Sci-ence 332, 1173-1176, (2011).

[14] M. J. P. Gingras and P. A. McClarty, Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets, Rep. Prog. Phys. 77, 056501 (2014).

[15] O. Benton, L. D. C. Jaubert, H. Yan and N. Shannon, A spin-liquid with pinch-line singularities on the pyrochlore lattice, Nature Commun. 7, 11572 (2016)

[16] L. Savary and L. Balents, Coulombic quantum liquids in spin-1/2 pyrochlores, Phys. Rev. Lett. 108, 037202 (2012).

[17] M. E. Brooks-Bartlett, S. T. Banks, L. D. C. Jaubert, A. Harman-Clarke and P. C. W. Holdsworth, Magnetic Moment Fragmentation and Monopole Crystallization, Phys. Rev. X 4, 011007, (2014).

[18] S. Powell, Ferromagnetic Coulomb phase in classical spin ice, Phys. Rev. B 91, 094431 (2015).

[19] L. D. C. Jaubert, Monopole Holes in a Partially Ordered Spin Liquid, Spin 5, 1540005 (2015).

[20] B. Canals, I.-A. Chioar, V.-D. Nguyen, M. Hehn, D. Lacour, F. Montaigne, A. Locatelli, T. Onur Mentes, B. Santos Burgos


# (63) 外币货币性项目

	2021 年 12 月 31 日

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# 五 合并范围的变更

# 其他原因的合并范围变动

本年度，本集团下属子公司广州航空货站有限公司(“广州货站”)完成工商注销，不再纳入合并范围。该子公司在核准注销登记前的经营成果和现金流量已包括在本集团 2021 年度的合并利润表及合并现金流量表中。


where \( \begin{array} { r } { \alpha _ { \pm } \, = \, 6 \frac { V _ { b g } } { a _ { \perp } ^ { 2 } a _ { 1 D } ( B _ { \pm } ) } \, + \, \frac { V _ { b g } } { V _ { p } ^ { C I R } } } \end{array} \). In Figs. 7 and 8 we show the calculated resonance width \( \Delta _ { 1 D } \) as a func-tion of the rescaled energy \( \epsilon \) and waveguide frequency \( \omega _ { \perp } \), respectively, together with the analytical result ac-cording to Eq.(31). One observes a good agreement for \( \omega _ { \perp } < 0 . 0 1 \). However, we encounter major deviations with increasing \( \omega _ { \perp } \) except for the zero energy limit \( \epsilon \rightarrow \) 0. These deviations are due to the fact that the analytical formula (30) which we used to derive (31), was obtained in the zero energy limit and does not work for larger en-ergies. In Fig.9 we present the real part of our scattering amplitude \( R e ( f _ { p } ) \) along with the analytical results from (30) as a function of \( V _ { p } / V _ { \perp } \) for \( \epsilon \) = 0.9 (large energy) for comparison. As the trap frequency \( \omega _ { \perp } \) increases we observe deviations between the analytical and numerical results for \( f _ { p } \).

The same way as in the case of the free space reso-nance, the width \( \Delta _ { 1 D } \) of the CIR narrows with decreas-ing energy \( \epsilon \) (see Fig.7). Fig.8 shows that in a harmonic waveguide there is a region where we have a possibility for narrowing the width by decreasing the trap frequency.

# VI. CONCLUSION

We develop and analyze a theoretical model to study Feshbach resonances of identical fermions in atomic waveguides by extending the two-channel model sug-gested in [33] and adopted in [32] for confined bosons. In this model, the experimentally known parameters of Feshbach resonances in free space are used as an input. Within this approach we have calculated the shifts and widths of p-wave magnetic Feshbach resonance of \( { } ^ { 4 0 } \mathrm { K } \) atoms in the hyperfine state \|F = 9/2, mF = −7/2i and for the relative angular momentum state \|l = 1, ml = 0i emerging in harmonic waveguides as p-wave CIRs. We find a linear dependence of the resonance position on the longitudinal colliding energy below the threshold for the first transverse excitation. It is shown that in a har-monic waveguide there is the possibility to decrease the width of the p-wave Feshbach resonance by decreasing the (transversal) trap frequency which could be used in corresponding experiments. Our analysis demonstrates the importance of including the effective range terms in the computational schemes for the description of the p-wave CIRs contrary to the case of s-wave CIRs where the impact of the effective radius is negligible. In previous in-vestigations of the p-wave CIRs in harmonic waveguides[13, 17, 23] the effects due to the effective range have been neglected. The developed model can be applied for a quantitative analysis of other p-wave CIRs following a different spin structure and for confining traps of differ-ent geometry including effects due to anharmonicity and anisotropy.

# VII. ACKNOWLEDGEMENTS

Sh.S would like to thank J. Abouie and S. Abedin-pour for fruitful discussions. V.S.M.and P.S. acknowl-edge financial support by the Heisenberg-Landau Pro-gram. V.S.M. thanks the Zentrum f¨ur Optische Quan-tentechnologien of the University of Hamburg and Sh.S. thanks the Bogoliubov Laboratory of Theoretical Physics of JINR for their warm hospitality. This work was sup-ported by IASBS (Grant No.G2014IASBS12648).

[1] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).

[2] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).

[3] T. Kinoshita, T. Wenger, and D.S. Weiss, Science 305, 1125 (2004).

[4] B. Paredes et al, Nature 429, 277 (2004).

[5] E. Haller, M. Gustavsson, M.J. Mark, J.G. Danzl, R. Hart, G. Pupillo, and H.C. N¨agerl, Science 325, 1224 (2009).

[6] E. Haller, M.J. Mark, R. Hart, J.G. Danzl, L. Re-ichso¨llner, V. Melezhik, P. Schmelcher, and H.C. N¨agerl, Phys. Rev. Lett. 104, 153203 (2010).

[7] H. Moritz, T. Sto¨ferle, K. Guenter, M. K¨ohl, and T. Esslinger, Phys. Rev. Lett. 94, 210401 (2005).

[8] K. G¨unter, T. Sto¨ferle, H. Moritz, M. Ko¨hl, and T. Esslinger, Phys. Rev. Lett. 95, 230401 (2005).

[9] B. Fro¨hlich, M. Feld, E. Vogt, M. Koschorreck, W. Zw-erger, and M. Ko¨hl, Phys. Rev. Lett. 106, 105301 (2011).

[10] G. Lamporesi, J. Catani, G. Barontini, Y. Nishida, M. Inguscio, and F. Minardi, Phys. Rev. Lett. 104, 153202 (2010).

[11] T. Bergeman, M.G. Moore, and M. Olshanii, Phys. Rev. Lett. 91, 163201 (2003).

[12] V.S. Melezhik, J.I. Kim, and P. Schmelcher, Phys. Rev. A76, 053611 (2007).

[13] S. Saeidian, V.S. Melezhik, and P. Schmelcher, Phys. Rev. A77, 042721 (2008).

[14] P. Giannakeas, V.S. Melezhik, and P. Schmelcher, Phys. Rev. A84, 023618 (2011).

[15] S. Sinha and L. Santos, Phys. Rev. Lett. 99, 140406 (2007).

[16] P. Giannakeas, V.S. Melezhik, and P. Schmelcher, Phys. Rev. Lett. 111, 183201 (2013).

[17] B.E. Granger and D. Blume, Phys. Rev. Lett. 92, 133202 (2004).

[18] J.I. Kim, V.S. Melezhik, and P. Schmelcher, Phys. Rev. lett. 97, 193203 (2006).

[19] J.I. Kim, V.S. Melezhik, and P. Schmelcher, Progr. Theor. Phys. Supp. 166, 159 (2007).

[20] V.S. Melezhik and P. Schmelcher, New J. Phys. 11, 073031 (2009).

[21] V.S. Melezhik and P. Schmelcher, Phys. Rev. A84, 042712 (2011).

[22] M.G. Moore, T. Bergeman, and M. Olshanii, J. Phys. IV 116, 69 (2004).

[23] J.I. Kim, J. Schmiedmayer, and P. Schmelcher, Phys.


Rev. A72, 042711 (2005).

[24] P. Giannakeas, F.K. Diakonos and P. Schmelcher, Phys. Rev. A86, 042703 (2012).

[25] E. Haller, M. Rabie, M.J. Mark, J.G. Danzl, R. Hart, K. Lauber, G. Pupillo, and H.C. N¨agerl, Phys. Rev. Lett. 107, 230404 (2011).

[26] S.G. Peng, S.S. Bohloul, X.J. Liu, H. Hu, and P.D. Drum-mond, Phys. Rev. A82, 063633 (2010).

[27] W. Zhang and P. Zhang, Phys.Rev. A83, 053615 (2011).

[28] S. Sala, P.-I. Schneider, and A. Saenz, Phys. Rev. Lett. 109, 073201 (2012).

[29] E. Tiesinga, C.J. Williams, F.H. Mies, and P.S. Julienne, Phys. Rev. A 61, 063416 (2000).

[30] V.A. Yurovsky, Phys. Rev. A71, 012709 (2005).

[31] S. Grishkevich, S. Sala, and A. Saenz, Phys. Rev. A84, 062710 (2011).

[32] S. Saeidian, V.S. Melezhik, and P. Schmelcher, Phys. Rev. A86, 062713 (2012).

[33] A.D. Lange, K. Pilch, A. Prantner, F. Ferlaino, B. En-geser, H.-C. N¨agerl, R. Grimm, and C. Chin, Phys. Rev. A79, 013622 (2009).

[34] C. Ticknor, C.A. Regal, D.S. Jin, and J.L. Bohn, Phys. Rev. A 69, 042712 (2004).

[35] Z. Idziaszek , Phys. Rev. A 79, 062701 (2009).

[36] P.S. Julienne and B. Gao (AIP, 2006), vol. 869, 261.

[37] S. Gautam and D. Angom Eur. Phys. J. D 56, 173 (2010).

[38] V.S. Melezhik and C.-Y. Hu, Phys. Rev. Lett. 90, 083202 (2003).

[39] Shi-Guo Peng, S. Tan, and K. Jiang, arXiv:1312.3392v2.

[40] Tao Shi and Su Yi, Phys. Rev. A 90, 042710 (2014).


Under generally accepted accounting principles ("GAAP"), we are required to record purchased loans acquired through acquisitions at fair value, which may differ from the outstanding balance of such loans. Estimating the fair value of such loans requires management to make estimates based on available information and facts and circumstances on the acquisition date. Actual performance could differ from management’s initial estimates. If these loans outperform our original fair value estimates, the difference between our original estimate and the actual performance of the loan (the “discount”) is accreted into net interest income. This accretable yield may change due to changes in expected timing and amount of future cash flows. The yields on our loans could decline as our acquired loan portfolio pays down or matures, and we expect downward pressure on our interest income to the extent that the runoff on our acquired loan portfolio is not replaced with comparable highyielding loans. This could result in higher net interest margins and interest income in current periods and lower net interest rate margins and lower interest income in future periods.

# We operate in a highly regulated environment and may be adversely affected by changes in federal and state laws and regulations that increase our costs of operations.

The financial services industry is extensively regulated. We are subject to extensive examination, supervision and comprehensive regulation by the Federal Reserve, and Heritage Bank is subject to examination, supervision and comprehensive regulation by the FDIC and the Division. The Federal Reserve, FDIC and Division govern the activities in which we may engage, primarily for the protection of depositors and the Deposit Insurance Fund. These regulatory authorities have extensive discretion in connection with their supervisory and enforcement activities, including the ability to impose requirements for additional capital, restrictions on operations, the reclassification of assets, and the determination of the adequacy of the allowance for loan losses and level of deposit insurance premiums assessed. These bank regulators also have the ability to impose conditions in the approval of merger and acquisition transactions.

As discussed under Item 1 "Business"—Capital Adequacy of this Form 10K, the DoddFrank Act has significantly changed the bank regulatory structure and affected the lending, deposit, investment, trading and operating activities of financial institutions and their holding companies. The DoddFrank Act requires various federal agencies to adopt a broad range of new implementing rules and regulations, and to prepare numerous studies and reports for Congress. The federal agencies have significant discretion in drafting and implementing rules and regulations. It is difficult at this time to predict when or how any new standards will ultimately be applied to us or what specific impact the DoddFrank Act and the yet to be written implementing rules and regulations will have on community banks. The current administration has indicated that it would like to see changes made to certain financial reform regulations, including the DoddFrank Act, which has resulted in increased regulatory uncertainty, and we are assessing the potential impact on financial and economic markets and on our business. Changes in federal policy and at regulatory agencies are expected to occur over time through policy and personnel changes, which could lead to changes involving the level of oversight and focus on the financial services industry. The nature, timing and economic and political effects of potential changes to the current legal and regulatory framework affecting financial institutions remain highly uncertain. If changes to the DoddFrank Act or the rules and regulations implementing the Act are made, such changes could offset the otherwise anticipated increase in operating and compliance costs (included in noninterest expense); however, no assurance can be given as to whether such changes will occur or what may result from such changes.

# Our loan portfolio is concentrated in loans with a higher risk of loss.

Repayment of our commercial business loans, consisting of commercial and industrial loans as well as owner-occupied and nonowner occupied commercial real estate loans, is often dependent on the cash flows of the borrower, which may be unpredictable, and the collateral securing these loans may fluctuate in value. We offer different types of commercial loans to a variety of businesses with a focus on real estate related industries and businesses in agricultural, healthcare, legal, and other professions. The types of commercial loans offered are business lines of credit, term equipment financing and term real estate loans. We also originate loans that are guaranteed by the SBA, and are a “preferred lender” of the SBA. Commercial business lending involves risks that are different from those associated with real estate lending. Real estate lending is generally considered to be collateral based lending with loan amounts established on predetermined loan to collateral values and liquidation of the underlying real estate collateral being viewed as the primary source of repayment in the event of borrower default. Our commercial business loans are primarily made based on our assessment of the cash flow of the borrower and secondarily on the underlying collateral provided by the borrower. The borrower's cash flow may be unpredictable, and collateral securing these loans may fluctuate in value. Although commercial business loans are often collateralized by equipment, inventory, accounts receivable or other business assets, the liquidation of collateral in the event of default is often an insufficient source of repayment because accounts receivable may be uncollectible and inventories may be obsolete or of limited use, among other things. Accordingly, the repayment of commercial business loans depends primarily on the cash flow and creditworthiness of the borrower and secondarily on the underlying collateral provided by the borrower. In addition, as part of our commercial business lending activities, we originate agricultural loans. Agricultural lending


involves a greater degree of risk and typically involves higher principal amounts than other types of loans. Payments on agricultural loans are typically dependent on the profitable operation or management of the related farm property. The success of the farm may be affected by many factors outside the control of the borrower, including adverse weather conditions that prevent the planting of a crop or limit crop yields (such as hail, drought and floods), declines in market prices for agricultural products (both domestically and internationally) and the impact of government regulations (including changes in price supports, subsidiaries and environmental regulations). In addition, many farms are dependent on a limited number of key individuals whose injury or death may significantly affect the successful operation of the farm. If the cash flow from a farming operation is diminished, the borrower’s ability to repay the loan may be impaired.

At December 31, 2017, our commercial business loans (consisting of commercial and industrial loans, owner-occupied commercial real estate loans and nonowner occupied commercial real estate loans) totaled \$2.25 billion, or approximately 79.1% of our total loan portfolio. Approximately \$9.1 million, or 0.4%, of our total commercial business loans were nonperforming at December 31, 2017. The majority of the nonperforming commercial business loans were owner-occupied commercial real estate loans.

Our nonowner occupied commercial real estate loans, which include five or more family residential real estate loans, involve higher principal amounts than other loans and repayment of these loans may be dependent on factors outside our control or the control of our borrowers. We originate commercial and five or more family residential real estate loans for individuals and businesses for various purposes, which are secured by commercial properties. These loans typically involve higher principal amounts than other types of loans, and repayment is dependent upon income generated, or expected to be generated, by the property securing the loan in amounts sufficient to cover operating expenses and debt service, which may be adversely affected by changes in the economy or local market conditions. For example, if the cash flow from the borrower’s project is reduced as a result of leases not being obtained or renewed, the borrower’s ability to repay the loan may be impaired.

Commercial and five or more family residential real estate loans also expose us to greater credit risk than loans secured by onetofour family residential real estate because the collateral securing these loans typically cannot be sold as easily as onetofour family residential real estate. In addition, many of our commercial and five or more family residential real estate loans are not fully amortizing and contain large balloon payments upon maturity. Such balloon payments may require the borrower to either sell or refinance the underlying property in order to make the payment, which may increase the risk of default or nonpayment. If we foreclose on a commercial and five or more family residential real estate loan, our holding period for the collateral typically is longer than for onetofour family residential loans because there are fewer potential purchasers of the collateral. Additionally, commercial and five or more family residential real estate loans generally have relatively large balances to single borrowers or related groups of borrowers. Accordingly, if we make any errors in judgment regarding the collectability of our commercial and five or more family residential real estate loans, any resulting chargeoffs may be larger on a per loan basis than those incurred with our residential or consumer loan portfolios.

As of December 31, 2017, our nonowner occupied commercial real estate loans totaled \$986.6 million, or 34.6% of our total loan portfolio. Approximately \$1.9 million, or 0.2%, of our nonowner occupied commercial real estate loans were nonperforming at December 31, 2017.

Our real estate construction and land development loans are based upon estimates of costs and the related value associated with the completed project. These estimates may be inaccurate. Construction lending can involve a higher level of risk than other types of lending because funds are advanced partially based upon the value of the project, which is uncertain prior to the project’s completion. Changes in demand for new housing and higher than anticipated building costs may cause actual results to vary significantly from those estimated. Because of the uncertainties inherent in estimating construction costs as well as the market value of a completed project and the effects of governmental regulation of real property, our estimates with regards to the total funds required to complete a project and the related loantovalue ratio may vary from actual results. If the estimate of value upon completion proves to be inaccurate, we may be confronted at, or prior to, the maturity of the loan with a project the value of which is insufficient to assure full repayment. In addition, speculative construction loans to a builder are often associated with homes that are not presold, and thus pose a greater potential risk to us than construction loans to individuals on their personal residences. Loans on land under development or held for future construction also pose additional risk because of the lack of income being produced by the property and the potential illiquid nature of the collateral. These risks can be significantly impacted by supply and demand. As a result, this type of lending often involves the disbursement of substantial funds with repayment dependent, in part, on the success of the ultimate project and the ability of the borrower to sell or lease the property or refinance the indebtedness, rather than the ability of the borrower or guarantor to independently repay principal and interest. If our estimate of the value of a project at completion proves to be overstated, we may have inadequate security for repayment of the loan and may incur a loss.


# PART IV

# Item 15. Exhibits, Financial Statement Schedules.

# (a) (1) List of Financial Statements

The following Consolidated Financial Statements of Tupperware Brands Corporation and Report of Independent Registered Public Accounting Firm are included in this Report under Item 8:

Consolidated Statements of Income, Comprehensive Income, Shareholders' Equity and Cash Flows - Years ended December 26, 2015, December 27, 2014 and December 28, 2013;

Consolidated Balance Sheets - December 26, 2015 and December 27, 2014;

Notes to the Consolidated Financial Statements; and

Report of Independent Registered Certified Public Accounting Firm.

# (a) (2) List of Financial Statement Schedules

The following Consolidated Financial Statement Schedule (numbered in accordance with Regulation S-X) of Tupperware Brands Corporation is included in this Report:

Schedule II-Valuation and Qualifying Accounts for each of the three years ended December 26, 2015.

All other schedules for which provision is made in the applicable accounting regulations of the Securities and Exchange Commission (SEC or the Commission) are not required under the related instructions, are inapplicable or the information called for therein is included elsewhere in the financial statements or related notes contained or incorporated by reference herein.

# (a) (3) List of Exhibits: (numbered in accordance with Item 601 of Regulation S-K)

Exhibit
Number	Description

*3.1	Restated Certificate of Incorporation of the Reigstrant (Attached as Exhibit 3.1 to Form 10-Q, filed with
the Commission on August 5, 2008 and incorporated herein by reference).

*3.2	Amended and Restated By-laws of the Reigstrant as amended August 28, 2008 (Attached as Exhibit 3.2
to Form 8-K, filed with the Commission on August 28, 2008 and incorporated herein by reference).

*4	Indenture dated June 2, 2011 (Attached as Exhibit 4.1 to Form 8-K, filed with the Commission on June 7,
2011 and incorporated herein by reference).

*10.1	1996 Incentive Plan as amended throuhg January 26, 2009 (Attached as Exhibit 10.1 to Form 10-K, filed
with the Commission on February 25, 2009 and incorporated herein by reference).

*10.2	Di'rectors Stock Plan as amended throuhg January 26, 2009 (Attached as Exhibit 10.2 to Form 10-K,
filed with the Commission on February 25, 2009 and incorporated herein by reference).

*10.3	Form of Change of Control Emlpoyment Agreement (Attached as Exhibit 10.3 for Form 10-K, filed with
the Commission on February 25, 2009 and incorporated herein by reference).

*10.4	Securities and Asset Purchase Aigreement between the Regstrant and Sara Lee Corporation (now known
as Hillshire Brands Co.) dated as of August 10, 2005 (Attached as Exhibit 10.01 to Form 8-K/A, filed
with the Commission on August 15, 2005 and incorporated herein by reference).

*10.5	Forms of stock option, restricted stock and restricted stock unit agreements utilized with the Reigstrant's
officers and directors under certain stock-based incentive lpans (Attached as Exhibit 10.6 to Form 10-K,
filed with the Commission on February 25, 2009 and incorporated herein by reference).

*10.6	Chief Executive Officer Severance Agreement between the Reistrant and E.V. Goings amended and
grestated effective February 17, 2010 (Attached as Exhibit 10.8 to From 10-K, filed with the Commission
on February 23, 2010 and incorporated herein by reference).


Exhibit
Number	Description

*10.7	Sulppemental Executive Retirement Plan, amended and restated effective February 2, 2010 (Attached as
Exhibit 10.9 to Form 10-K, filed with the Commission on February 23, 2010 and incorporated herein by
reference).

*10.8	Sulppemental Plan, amended and restated effective January 1, 2009 (Attached as Exhibit 10.11 to Form
10-K, filed with the Commission on February 25, 2009 and incorporated herein by reference).

*10.9	2006 Incentive Plan as amended throuhg January 26, 2009 (Attached as Exhibit 10.12 to Form 10-K,
filed with the Commission on February 25, 2009 and incorporated herein by reference).

*10.10	2010 Incentive Plan (Attached as Exhibit 4.3 to Form S-8, filed with the Commission on November 3,
2010 and incorporated herein by reference).

*10.11	2010 Incentive Plan Restricted Stock Agreement (Attached as Exhibit 4.4 to Form S-8, filed with the
Commission on November 3, 2010 and incorporated herein by reference).

*10.12	Credit Agreement, as amended throuh June 9, 2015 (Attachedg as Exhibit 10.1 and 10.2 to Form 10-Q,
filed with the Commission on August 5, 2014 and as Exhibit 10.1 to Form 8-K as filed with the
Commission on June 9, 2015 and incorporated herein by reference).

21	Subsidiaries of Tupperware Brands Corporation as of February 24, 2016.

23	Consent of Independent Reigstered Certified Public Accounting Firm.

24	Powers of Attorney.

31.1	Rule 13a-14(a)/15d-14(a) Certification of the Chief Executive Officer.

31.2	Rule 13a-14(a)/15d-14(a) Certification of the Chief Financial Officer.

32.1	Certification Pursuant to Section 1350 of Chapter 63 of Title 18 of the United States Code by the Chief
Executive Officer.

32.2	Certification Pursuant to Section 1350 of Chapter 63 of Title 18 of the United States Code by the Chief
Financial Officer.

101	The following financial stat'ements from Tupperware Brands Corporations Annual Report on Form 10-K
for the year ended December 26, 2015, formatted in XBRL (eXtensible Business Reporting Language):
(i) Consolidated Statements of Income, (ii) Consolidated Statements of Comprehensive Income, (iii)
C'onsolidated Balance Sheets, (iv) Consolidated Statements of Shareholders Equity, (v) Consolidated
Statements of Cash Flows, (vi) Notes to the Consolidated Financial Statements, tagged in detail, and (vii)
Schedule II. Valuation and Qualifiyng Accounts.

\* Document has heretofore been filed with the SEC and is incorporated by reference and made a part hereof.

The Registrant agrees to furnish, upon request of the SEC, a copy of all constituent instruments defining the rights of holders of long-term debt of the Registrant and its consolidated subsidiaries.


Non-monetary items that are measured in terms of historical cost in a foreign currency are translated using the exchange rates at the dates of the initial transactions. Non-monetary items measured at fair value in a foreign currency are translated using the exchange rates at the date when the fair value was measured. The gain or loss arising on translation of non-monetary item measured at fair value is treated in line with the recognition of the gain or loss on change in fair value of the item (i.e., translation difference on the item whose fair value gain or loss is recognised in other comprehensive income or profit or loss is also recognised in other comprehensive income or profit or loss, respectively).

The functional currencies of certain overseas subsidiaries are currencies other than the Hong Kong dollar. As at the end of each of the Track Record Period, the assets and liabilities of these entities are translated into the presentation currency of the Company at the exchange rates prevailing at the end of each of the Track Record Period and their statements of profit or loss are translated into Hong Kong dollars at the weighted average exchange rates for the year. The resulting exchange differences are recognised in other comprehensive income and accumulated in the exchange fluctuation reserve. On disposal of a foreign operation, the component of other comprehensive income relating to that particular foreign operation is recognised in the statement of profit or loss.

For the purpose of the combined statements of cash flows, the cash flows of overseas subsidiaries are translated into Hong Kong dollars at the exchange rates ruling at the dates of the cash flows. Frequently recurring cash flows of overseas subsidiaries which arise throughout the year are translated into Hong Kong dollars at the weighted average exchange rates for each of the Track Record Period.

# 5. SIGNIFICANT ACCOUNTING JUDGEMENTS AND ESTIMATES

The preparation of the Financial Information requires management to make judgements, estimates and assumptions that affect the reported amounts of revenues, expenses, assets and liabilities, and their accompanying disclosures, and the disclosure of contingent liabilities. Uncertainty about these assumptions and estimates could result in outcomes that could require a material adjustment to the carrying amounts of the assets or liabilities affected in the future.

# Judgements

In the process of applying the Group’s accounting policies, management has made the following judgements, apart from those involving estimations, which have the most significant effect on the amounts recognised in the Financial Information:

# Income taxes

Significant judgements on the future tax treatment of certain transactions are required in determining income tax provisions. The Group carefully evaluates tax implications of transactions, including any potential tax liabilities. When the Group determines any transactions that may result in probable future tax outflows and the amount can be reliably measured, tax provisions are recorded accordingly. Such tax provision may not be indicative of the ultimate tax payment with tax authorities. The tax treatment of such transactions is considered periodically to take into account all changes in tax legislation.

# Determination of functional currency

The Group measures foreign currency transactions in the respective functional currencies of the Company and its subsidiaries. In determining the functional currencies of the group entities,j udgement is required to determine the currency that mainly influences sales prices for goods and services; the currency of the country whose competitive forces and regulations mainly determine the sale prices of


the entity’s goods and services; and the currency that mainly influences labour, material and other costs of providing goods or services.

# Estimation uncertainty

The key assumptions concerning the future and other key sources of estimation uncertainty at the end of each of the Track Record Period, that have a significant risk of causing a material adjustment to the carrying amounts of assets and liabilities within the next financial year, are described below.

# Impairment of property, plant and equipment

Items of property, plant and equipment are tested for impairment if there is any indication that the carrying value of these assets may not be recoverable and the assets are subject to an impairment loss. The recoverable amount is the higher of an asset’s fair value less costs to sell and value in use. The value-in-use calculation requires the Group to estimate the future cash flows expected to arise from the relevant cash-generating unit and a suitable discount rate is used in order to calculate the present value.

# Write-down of inventories to net realisable value

Management reviews the condition of inventories of the Group and write-down the carrying amounts of obsolete and slow-moving inventory items which are identified as no longer suitable for sale or use to their respective net realisable value. The Group estimates the net realisable value for such inventories based primarily on the latest invoice prices and current market conditions at the end of each of the Track Record Period.

The identification of obsolete and slow-moving inventory items requires the use of judgements and estimates. Where the expectation is different from the original estimate, such difference will impact on the carrying values of inventories and the write-down of inventories recognised in the periods in which such estimates have been made.

# Impairment of loans and receivables

The Group assesses at the end of each of the Track Record Period whether there is any objective evidence that a financial asset is impaired. To determine whether there is objective evidence of impairment, the Group considers factors such as the probability of insolvency or significant financial difficulties of the debtor and default or significant delay in payments. Where there is objective evidence of impairment, the amount and timing of future cash flows are estimated based on historical loss experience of assets with similar credit risk characteristics.

# Deferred tax assets

Deferred tax assets are recognised for all unused tax losses to the extent that it is probable that taxable profit will be available against which the losses can be utilised. Significant management judgement is required to determine the amount of deferred tax assets that can be recognised, based upon the likely timing and level of future taxable profits together with future tax planning strategies.

# 6. OPERATING SEGMENT INFORMATION

The Group focuses primarily on trading of apparel products and provision of the apparel supply chain management services during the Track Record Period. Information reported to the Group’s chief operating decision maker, for the purpose of resources allocation and performance assessment, focuses on the operating results of the Group as a whole as the Group’s resources are integrated and


We will make further efforts to recruit more students for Huali Vocational College Yunfu Campus and Huali Technician College Yunfu Campus, which commenced operations in September 2018 and September 2016, respectively. Those schools have significant potential to increase recruitment. We will continue to increase our teaching staff and increase the number of majors and curricula in Yunfu campuses to further improve the quality of our education and attract more students. Particularly, we started to offer five-year full-time vocational programs to students ofj unior high-school level at Huali Technician College in September 2018 to further expand student enrollment.

We also plan to upgrade Huali Vocational College (including both Huali Vocational College Zengcheng Campus and Huali Vocational College Yunfu Campus) from a school providing three-year junior college education to a school providing four-year undergraduate education. On January 25, 2017, the MOE issued the Opinion on the Establishment of Higher Education Institution During the Thirteenth Five-Year Plan Period (《關於“十三五”時期高等學校設置工作的意見》), which requires scientific planning of higher education institutions. On March 24, 2017, the Department of Education of Guangdong Province issued the Notice on Issues Concerning the Formulation of the Provincial Thirteenth Five-Year Plan of Establishing Higher Education Institutions (《關於編制省高等學校設置“十三五”規劃有關問題的通知》), pursuant to which the provincial Thirteenth Five-Year Plan of establishing higher education institutions (the “Provincial Plan”) shall be formulated before the approval of establishment of any higher education institution is granted. We plan to upgrade our Huali Vocational College according to the applicable laws and regulations as well as the latest requirements of the government authorities. Such upgrade plan is subject to the approval from relevant government authorities.

We also intend to continue to expand our school network in new geographic locations both in China and abroad. With encouragement of local government, we plan to establish a newj unior college in Jiangmen City, Guangdong Province to provide higher education and vocational education and grant junior college diplomas accredited by the MOE. Please refer to the section headed “– Planned Expansion” for details. In addition, as of the Latest Practicable Date, we had entered into letters of intent with local governments of Zhanjiang City and Huizhou City in Guangdong Province to explore the possibility of establishing new school or new campus in each city. Moreover, we are also in the process of establishing a new school in the United States to further expand and strengthen our brand. Please refer to the section headed “– Planned Expansion– A Higher Education Institution in the U.S.” in this prospectus for details.

In addition to setting up new schools by ourselves, we also plan to acquire new schools located in our preferred regions in the PRC, such as Guangdong Province, Henan Province, Sichuan Province, Shandong Province, Yunnan Province and Chongqing Municipality, and overseas countries in Southeast Asia and North America. As of the Latest Practicable Date, we did not have specific acquisition targets. We will take a variety of factors into account when selecting new school locations and potential acquisition targets, including demographics, potential demand, income trends, regional economic conditions, the level of local government support, the availability of suitable sites and existing market competition.

According to our current understanding and interpretation of the MOJ Draft for Comments, if the MOJ Draft for Comments is adopted in its current form, it may have the following implications for our expansion strategy:

(i) we may acquire schools that are permitted to be registered, and have not been registered, as for-profit private schools or non-profit private schools. When required to do so, we will register such acquired schools as for-profit private schools;

(ii) we may enter into cooperation with public universities with respect to independent colleges that have not been registered as for-profit private schools or non-profit private schools. However, if such independent colleges are required to register as


for-profit private schools or non-private schools prior to any acquisition by us, in order for our acquisition to continue, such colleges shall cease collaboration with the public universities subject to the MOE approval and register as for-profit private schools. Such colleges may no longer be able to use the names of the public universities;

(iii) we may acquire schools that have been registered as for-profit private schools. The consideration payable for such acquisitions is expected to take into account the additional costs that may be involved in the for-profit private school registration process; and

(iv) we may not be able to acquire any schools that have already been registered as non-profit private schools.

We currently do not consider that the implications above would have any material impact on our expansion strategy, except that the number of target schools available for our acquisition may be reduced by those that will be registered as a non-profit private school in the future.

We have established a special committee (comprising Mr. Zhang, Mr. Ye Yaming, Mr. Dong Xiaolin, Mr. Ma Zhixiong and Mr. Zhang Zhicheng) led by Mr. Zhang to pay close attention to the developments of the relevant policies and regulations, including the Amendment and the MOJ Draft for Comments. We will assess whether the MOJ Draft for Comments or other relevant implementation rules and regulations in the future will present practical challenges or compliance issues to any future acquisition. Such special committee will ensure that our acquisitions in the future will fully comply with the relevant rules and regulations in effect from time to time.

# Continue to improve our profitability by optimizing our pricing and sources of revenue

One of the most significant factors affecting our profitability is the tuition fee rates we charge at our schools. We plan to optimize our pricing through differentiating tuition fee rates for different majors we offer based on popularity andj ob prospects of such majors. During the Track Record Period, all our schools raised their tuition fee rates for certain majors. For certain popular majors, such as visual communication design, environment design and electrical engineering and automation at Huali College and environment art design, financial management and construction costing at Huali Vocational College, we charge higher tuition fee rates which are about RMB2,000 to RMB3,000 higher than other majors. Moreover, to make our curricula more attractive, we began to provide value-added bilingual education services with respect to certain of the majors offered at Huali College and Huali Vocational College in the 2016/2017 school year. Students enrolled in these bilingual programs are required to pay additional tuition fees of approximately RMB1,000 to RMB2,000 per student per year, compared with students enrolled in ordinary programs. We believe that we are able to maintain the current tuition fee rates and raise the tuition fee rates charged by our schools where appropriate as our brand awareness and market recognition grow. In addition, we retain the discretion to adjust the tuition fee rates charged by our schools, and our schools are only required to publish their proposed tuition fee rate adjustments to the public, but are not subject to any approval or filing requirements. Historically, we have kept our tuition fees at levels we believe are competitive compared to our peer schools in order to attract more students and thereby increase our student enrollment and market share. As we have established a strong reputation for providing quality education to our students, we believe we are in a good position to optimize our pricing without compromising our reputation and our ability to attract and retain students.


	754	747	732	731	NM	577

Comparable venues (end of period)(2)	531	529	489	485	NM	485

Comparable venue sales change(2)	(4.8)%	2.8%	(0.4)%	NM	NM	0.4%

	As of
December 31,
2017	As of
January 1, 2017	As of
January 3, 2016	As of
December 28, 2014	As of
February 14,
2014	As of
December 29,
2013

Balance Sheet Data:

Total assets	$   1,695,044	$1,710,112	$1,733,035	$1,836,113	NM	$     791,611

Total debt(5)	986,419	989,948	994,448	999,783	NM	382,879

Stockholders’ equity	262,148	206,005	208,546	292,586	NM	160,768

Dividends declared	—	—	70,000	—	—	17,372

(1) We operate on a 52 or 53 week fiscal year ending on the Sunday nearest December 31. Fiscal year 2015 was 53 weeks in length, which resulted in our fourth quarter consisting of 14 weeks. All other fiscal years presented were 52 weeks.

(2) We define “comparable venue sales” as sales for our domestic owned company-operated venues that have been open for more than 18 months as of the beginning of each respective fiscal year or for acquired venues we have operated for at least 12 months as of the beginning of each respective fiscal year. We define “comparable venue sales change” as the percentage change in comparable venue sales for each respective period. We believe comparable venue sales change to be a key performance indicator within our industry; it is a critical factor in evaluating our performance, as it is indicative of acceptance of our strategic initiatives and local economic and consumer trends. Our comparable venue sales for Fiscal 2015, and the Successor 2014 period exclude the Peter Piper Pizza venues that were acquired in October 2014 as we had operated them for less than 12 months at the beginning of each respective fiscal year. As a result of the 53 week fiscal year in 2015, our 2016 fiscal year began one calendar week later than our 2015 fiscal year. In order to provide useful information and to better analyze our business, we provided comparable venue sales for our 2016 fiscal year presented on both a fiscal week basis and calendar week basis. Comparable venue sales change for 2016 on a calendar week basis compared the results for the period from January 4, 2016 through January 1, 2017 (weeks 1 through 52 of our 2016 fiscal year) to the results for the period from January 5, 2015 through January 3, 2016 (weeks 2 through 53 of our 2015 fiscal year). We believe comparable venue sales change calculated on a same calendar week basis is more indicative of the operating trends in our business. However, we also recognize that comparable venue sales change calculated on a fiscal week basis is a useful measure when analyzing year-over-year changes in our financial statements. The comparable venue sales change in the table above is presented on a calendar week basis, excluding the additional week of operations in 2015. On a fiscal basis, excluding the additional week of operations in 2015, comparable venue sales change would have been 3.0% in 2016.

(3) For our definition of Adjusted EBITDA, see the “Non-GAAP Financial Measures” section below.

(4) Adjusted EBITDA Margin is defined by us as Adjusted EBITDA as a percentage of Total revenues.

(5) Total debt includes our senior notes, our outstanding borrowings under the term loan facility and the revolving credit facility, net of deferred financing costs, capital leases, and the Predecessor Facility.

(6) Results for the Successor 2014 period include the revenues and expenses for Peter Piper Pizza for the 73 day period from October 17, 2014 through December 28, 2014

(7) As a result of the Merger, we applied the acquisition method of accounting and established a new basis of accounting on February 15, 2014. Periods presented prior to and including February 14, 2014 represent the operations of the predecessor company (“Predecessor”) and the periods presented after February 14, 2014 represent the operations of the successor company (“Successor”). The financial results for the period December 29, 2013 through February 14, 2014 represent the 47 day Predecessor period.

# Non-GAAP Financial Measures

Adjusted EBITDA, a measure used by management to assess operating performance, is defined as Net income (loss) plus interest expense, income tax expense (benefit), depreciation and amortization expense, impairments, gains and losses on asset disposals, and stock based compensation. In addition, Adjusted EBITDA excludes other items we consider unusual or non-recurring and other adjustments required or permitted in calculating covenant compliance under our secured credit facilities and the indenture governing our senior notes (see discussion of our senior notes in Part II, Item 7. “Management’s