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Al-Burhan condemns killing of 5 schoolchildren who protested against bread shortage - Daily News Egypt
Politics Al-Burhan condemns killing of 5 schoolchildren who protested against bread shortage
Al-Burhan condemns killing of 5 schoolchildren who protested against bread shortage
Protest leaders called for suspension of talks with the military council
Daily News Egypt July 30, 2019 Comments Off on Al-Burhan condemns killing of 5 schoolchildren who protested against bread shortage
Head of Sudan's ruling military council, Abdel Fattah Al-Burhan, condemned on Tuesday the killing of five schoolchildren in the North Kordofan city of El-Obeid at a rally, describing it as "unacceptable".
On Monday, five Sudanese demonstrators including students were shot dead during a rally attended by school children against a shortage of bread, a day before protest leaders and the ruling generals were expected to hold talks.
"What happened in El-Obeid is a regrettable and upsetting matter and the killing of peaceful citizens is unacceptable and rejected and a crime that requires an immediate and deterrent accountability," he was quoted as saying by the official SUNA news agency.
Residents of Al-Obeid said the rally was against a shortage of bread and fuel in the town, AFP reported. "School children were affected as there is no transport to help them reach their schools. Today, they staged a rally and when it reached downtown there were shots fired."
The protest leaders called on the suspension of talks with the military council and thousands of students took to the streets to denounce the death of students. The authorities announced a night-time curfew in four towns following the deaths in the central town of Al-Obeid.
Sudan's ruling military council and an alliance of opposition groups, signed the first of two documents of a power-sharing deal that activists hope will pave the way for democracy in the country.
The two sides inked the "Political Declaration" in Khartoum in the presence of the African Union and Ethiopian mediators.
Sudan has been in a state of flux since the former president Omar Al-Bashir was ousted in April after large-scale protests against his regime.
Topics: Abdel Fattah al-Burhan protestors Sudan
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https://eklutdvotyzsri.dailynewssegypt.com/2019/07/30/al-burhan-condemns-killing-of-5-schoolchildren-who-protested-against-bread-shortage/
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July 30, 2019 Breaking News


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Tokyo Olympics: Saurabh Chaudhary Qualifies For The Final Of 10m Air Pistol
Kolkata Konnect July 23, 2021 Sports Edit this post
With hope in the Tokyo Olympics, India's ace 10m air pistol shooter Saurabh Chaudhary topped the qualification round at the Tokyo Olympics o...
With hope in the Tokyo Olympics, India's ace 10m air pistol shooter Saurabh Chaudhary topped the qualification round at the Tokyo Olympics on Saturday.
Chaudhury, a Meerut boy, shot 586 to be tied with Chinese Bowen Zhang after qualification, but as the Indian had more 'inner 10s' (27) to his name, he took the first spot foing into the finals.
The other Indian in the fray, Abhishek Verma, was in the reckoning for a place in the finals. But he shot two 9s followed by two 8s in his last four shots, which meant that he finished 17th with a score of 575.
The 19-year-old Chaudhary started off with five shots of 10, leading the competition early. But it was followed by five shots of 9 to end the first series with 95. In the second and third series, he had scores of 98 each.
The fourth series was stellar for him, getting perfect 10s in all the ten shots.
Chaudhary had 43 shots of 10s till shooting a nine on the second-last shot of the fifth series, ending with 98. In the final series, he started off with six 10s followed by three 9s and ending with a 10.
The 31-year-old Verma had an iffy start with 94 in the first series. He was able to overcome it by shooting 96 and 98 in the second and third series respectively. A 97 in the fourth series and 98 in five propelled him to reach inside the top-eight.
Verma looked set to join Chaudhary in the finals till the final series, but the meltdown in the final four shots caused the former's downfall.
The other finalists include Christian Reitz of Germany, Pavlo Korostylow of Ukraine, Javad Foroughi of Iran, Mose Kim of South Korea, Wei Pang of China and Damir Mikec of Serbia.

redpajama_set_name: RedPajamaCommonCrawl


Mary Huff Stevenson, professor emerita of economics at the University of Massachusetts Boston, is an exceptionally astute old friend with whom I used to meet regularly in the 1980s and '90s to discuss economics.
She helped me understand that economic inequality, however unfair and undesirable, might be a sustainable condition in America: the better-off transacting largely with one another and leaving a large swath of Americans out of the picture. She pointed out that more education might not change the situation — it may simply mean that everyone moved ahead a step, not that anyone moved ahead in line. She schooled me on research she had done during a fellowship year on the impetus for the mass library movement in 19th century America: to distract and pacify the restive working classes.
Stevenson has long been a champion of the less fortunate — teaching them, teaching about them. During her decades at UMass, she co-authored two books with noted Northeastern University economist Barry Bluestone: The Boston Renaissance: Race, Space and Economic Change in an American Metropolis, (Russell Sage, 2000) and The Urban Experience: Economics, Society, and Public Policy (Oxford, 2008) with additional co-author Russell Williams.
She also starred with Nobel laureate economist Robert Solow and founding father Benjamin Franklin in a video I presented some years ago to explain the concept of "opportunity cost." In an earlier version, which I can't find online, she played the role of my psychotherapist.
Professor Stevenson, with whom I graduated Brandeis University in 1966, no longer teaches economics, but something called Zumba Gold ®. It is her encore career.
And so, with our most ambitious online effort to date — "New Adventures for Older Workers" — debuting recently, this seemed the right time to offer Stevenson's account of her transition.
Mary Huff Stevenson: I was an economics professor at the University of Massachusetts Boston for 40 years, and I'm now in my encore career as a fitness instructor focusing on older adults and those with physical limitations. It's an unlikely path for someone whose childhood nickname was Supey, short for Superklutz.
Growing up, I was never athletic, never a cheerleader, never had dance lessons. The only evidence that I enjoyed moving around at all was that when I got home from school I'd turn on the TV to watch American Bandstand and I'd try to imitate all the new dance moves.
Around the time I turned 50, I realized that although I was fortunate to be in good health, I should not take this for granted. I joined a gym where I walked on a treadmill and worked out with machines and free weights. However, I injured my shoulder while lifting a suitcase and had to stop the strength training while I was getting physical therapy. Bored with just the treadmill, I wandered into an aerobics class. It was a class in the Nia Technique ® and I was captivated by it.
Nia is an eclectic form of exercise that borrows from martial arts such as tai chi, tae kwon do and aikido; healing arts, including yoga, Alexander Technique and Feldenkrais Method; both modern and jazz dance and the free-spirited moves that Isadora Duncan might have done.
Nia is done "your body's way," so when all the other shoulders in the class were moving, mine wasn't and that was perfectly fine. The shoulder healed, but I never went back to the free weights. Through Nia, I'd reclaimed that energetic teenage girl who enjoyed twirling around her living room.
I knew from my years of teaching economics that a good way to really understand something is to teach it to someone else. That was my motivation for doing the intensive training required of all Nia instructors.
When I began to envision my students, I realized that I would have the greatest impact on other older adults. From there, it became a question of how best to reach them. Knowing that Nia might not appeal to everyone, I got further training to teach Zumba Gold ® (a version of the Latin rhythm exercise class modified for older active adults) and Ageless Grace ® (a seated exercise format that uses 21 unique tools to promote lifelong movement with comfort and ease).
I currently teach classes at senior centers and at a gym for people over 50. I think I am helping older adults to remain physically and mentally agile. As a senior, I know I'm benefiting.
Although this transition can be read as a story of dramatic change, it is also a story of continuity: I've been teaching my entire adult life — the subject matter is just different.


His court appointed lawyer is the person who will review his case and determine what the best thing to do may be. If he has two priors and they have a decent case, he should expect to do some prison time for this charge. Not knowing the judge and district attorney, I cannot comment on what he should do. His attorney will have experience with the judge and the district attorney to know whether what he has been offered is something he should take.
I wad arreated with less than 2 ounces of marijuanna flower.
My 19 year old son was in a large group in a federal park.


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Google Analytics is further explained under the following Link
https://www.google.com/analytics/.
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On this website, the controller has integrated Google Remarketing services. Google Remarketing is a feature of Google AdWords, which allows an enterprise to display advertising to Internet users who have previously resided on the enterprise's Internet site. The integration of Google Remarketing therefore allows an enterprise to create user-based advertising and thus shows relevant advertisements to interested Internet users.
The operating company of the Google Remarketing services is the Google Inc., 1600 Amphitheatre Pkwy, Mountain View, CA 94043-1351, United States.
The purpose of Google Remarketing is the insertion of interest-relevant advertising. Google Remarketing allows us to display ads on the Google network or on other websites, which are based on individual needs and matched to the interests of Internet users.
Google Remarketing sets a cookie on the information technology system of the data subject. The definition of cookies is explained above. With the setting of the cookie, Google enables a recognition of the visitor of our website if he calls up consecutive web pages, which are also a member of the Google advertising network. With each call-up to an Internet site on which the service has been integrated by Google Remarketing, the web browser of the data subject identifies automatically with Google. During the course of this technical procedure, Google receives personal information, such as the IP address or the surfing behaviour of the user, which Google uses, inter alia, for the insertion of interest relevant advertising.
The cookie is used to store personal information, e.g. the Internet pages visited by the data subject. Each time we visit our Internet pages, personal data, including the IP address of the Internet access used by the data subject, is transmitted to Google in the United States of America. These personal data are stored by Google in the United States of America. Google may pass these personal data collected through the technical procedure to third parties.
The data subject may, as stated above, prevent the setting of cookies through our website at any time by means of a corresponding adjustment of the web browser used and thus permanently deny the setting of cookies. Such an adjustment to the Internet browser used would also prevent Google from setting a cookie on the information technology system of the data subject. In addition, cookies already in use by Google may be deleted at any time via a web browser or other software programs.
In addition, the data subject has the possibility of objecting to the interest-based advertising by Google. For this purpose, the data subject must call up the link to
www.google.de/settings/ads
and make the desired settings on each Internet browser used by the data subject.
Further information and the actual data protection provisions of Google may be retrieved under
https://www.google.com/intl/en/policies/privacy/.
With each call-up to one of the individual pages of this website, which is operated by the controller and on which a Google+ button has been integrated, the Internet browser on the information technology system of the data subject automatically downloads a display of the corresponding Google+ button of Google through the respective Google+ button component. During the course of this technical procedure, Google is made aware of what specific sub-page of our website was visited by the data subject. More detailed information about Google+ is available under
https://developers.google.com/+/.
Further information and the data protection provisions of Google may be retrieved under
More references from Google about the Google+ 1 button may be obtained under
https://developers.google.com/+/web/buttons-policy.
On this website, the controller has integrated Google AdWords. Google AdWords is a service for Internet advertising that allows the advertiser to place ads in Google search engine results and the Google advertising network. Google AdWords allows an advertiser to pre-define specific keywords with the help of which an ad on Google's search results only then displayed, when the user utilizes the search engine to retrieve a keyword-relevant search result. In the Google Advertising Network, the ads are distributed on relevant web pages using an automatic algorithm, taking into account the previously defined keywords.
The data subject has a possibility of objecting to the interest based advertisement of Google. Therefore, the data subject must access from each of the browsers in use the link
and set the desired settings.
Further information and the applicable data protection provisions of Instagram may be retrieved under
https://www.instagram.com/about/legal/privacy/.
25. Data protection provisions about the application and use of Jetpack for WordPress
On this website, the controller has integrated Jetpack. Jetpack is a WordPress plug-in, which provides additional features to the operator of a website based on WordPress. Jetpack allows the Internet site operator, inter alia, an overview of the visitors of the site. By displaying related posts and publications, or the ability to share content on the page, it is also possible to increase visitor numbers. In addition, security features are integrated into Jetpack, so a Jetpack-using site is better protected against brute-force attacks. Jetpack also optimizes and accelerates the loading of images on the website.
The operating company of Jetpack Plug-Ins for WordPress is the Automattic Inc., 132 Hawthorne Street, San Francisco, CA 94107, UNITED STATES. The operating enterprise uses the tracking technology created by Quantcast Inc., 201 Third Street, San Francisco, CA 94103, UNITED STATES.
Jetpack sets a cookie on the information technology system used by the data subject. The definition of cookies is explained above. With each call-up to one of the individual pages of this Internet site, which is operated by the controller and on which a Jetpack component was integrated, the Internet browser on the information technology system of the data subject is automatically prompted to submit data through the Jetpack component for analysis purposes to Automattic. During the course of this technical procedure Automattic receives data that is used to create an overview of website visits. The data obtained in this way serves the analysis of the behaviour of the data subject, which has access to the Internet page of the controller and is analyzed with the aim to optimize the website. The data collected through the Jetpack component is not used to identify the data subject without a prior obtaining of a separate express consent of the data subject. The data comes also to the notice of Quantcast. Quantcast uses the data for the same purposes as Automattic.
The data subject can, as stated above, prevent the setting of cookies through our website at any time by means of a corresponding adjustment of the web browser used and thus permanently deny the setting of cookies. Such an adjustment to the Internet browser used would also prevent Automattic/Quantcast from setting a cookie on the information technology system of the data subject. In addition, cookies already in use by Automattic/Quantcast may be deleted at any time via a web browser or other software programs.
In addition, the data subject has the possibility of objecting to a collection of data relating to a use of this Internet site that are generated by the Jetpack cookie as well as the processing of these data by Automattic/Quantcast and the chance to preclude any such. For this purpose, the data subject must press the 'opt-out' button under the link
https://www.quantcast.com/opt-out/
which sets an opt-out cookie. The opt-out cookie set with this purpose is placed on the information technology system used by the data subject. If the cookies are deleted on the system of the data subject, then the data subject must call up the link again and set a new opt-out cookie.
The applicable data protection provisions of Automattic may be accessed under
https://automattic.com/privacy/.
The applicable data protection provisions of Quantcast can be accessed under
https://www.quantcast.com/privacy/.
With each call-up to one of the individual pages of this Internet site, which is operated by the controller and on which a LinkedIn component (LinkedIn plug-in) was integrated, the Internet browser on the information technology system of the data subject is automatically prompted to the download of a display of the corresponding LinkedIn component of LinkedIn. Further information about the LinkedIn plug-in may be accessed under
https://developer.linkedin.com/plugins.
During the course of this technical procedure, LinkedIn gains knowledge of what specific sub-page of our website was visited by the data subject.
LinkedIn provides under
https://www.linkedin.com/psettings/guest-controls
the possibility to unsubscribe from e-mail messages, SMS messages and targeted ads, as well as the ability to manage ad settings. LinkedIn also uses affiliates such as Eire, Google Analytics, BlueKai, DoubleClick, Nielsen, Comscore, Eloqua, and Lotame. The setting of such cookies may be denied under
https://www.linkedin.com/legal/cookie-policy.
The applicable privacy policy for LinkedIn is available under
The LinkedIn Cookie Policy is available under
27. Data protection provisions about the application and use of Pinterest
On this website, the controller has integrated components of Pinterest Inc. Pinterest is a so-called social network. A social network is an Internet social meeting place, an online community that allows users to communicate and interact with each other in a virtual space. A social network may serve as a platform for the exchange of opinions and experiences, or allow the Internet community to provide personal or company-related information. Pinterest enables the users of the social network to publish, inter alia, picture collections and individual pictures as well as descriptions on virtual pinboards (so-called pins), which can then be shared by other user's (so-called re-pins) or commented on.
The operating company of Pinterest is Pinterest Inc., 808 Brannan Street, San Francisco, CA 94103, UNITED STATES.
With each call-up to one of the individual pages of this Internet site, which is operated by the controller and on which a Pinterest component (Pinterest plug-in) was integrated, the Internet browser on the information technology system of the data subject automatically prompted to download through the respective Pinterest component a display of the corresponding Pinterest component. Further information on Pinterest is available under
https://pinterest.com/.
During the course of this technical procedure, Pinterest gains knowledge of what specific sub-page of our website is visited by the data subject.
If the data subject is logged in at the same time on Pinterest, Pinterest detects with every call-up to our website by the data subject—and for the entire duration of their stay on our Internet site—which specific sub-page of our Internet page was visited by the data subject. This information is collected through the Pinterest component and associated with the respective Pinterest account of the data subject. If the data subject clicks on one of the Pinterest buttons, integrated on our website, then Pinterest assigns this information to the personal Pinterest user account of the data subject and stores the personal data.
Pinterest receives information via the Pinterest component that the data subject has visited our website, provided that the data subject is logged in at Pinterest at the time of the call-up to our website. This occurs regardless of whether the person clicks on the Pinterest component or not. If such a transmission of information to Pinterest is not desirable for the data subject, then he or she may prevent this by logging off from their Pinterest account before a call-up to our website is made.
The data protection guideline published by Pinterest, which is available under
https://about.pinterest.com/privacy-policy,
provides information on the collection, processing and use of personal data by Pinterest.
28. Data protection provisions about the application and use of Shariff
On this website, the controller has integrated the component of Shariff. The Shariff component provides social media buttons that are compliant with data protection. Shariff was developed for the German computer magazine c't and is published by GitHub, Inc.
The developers of the component is GitHub, Inc. 88 Colin P. Kelly Junior Street, San Francisco, CA 94107, United States.
Typically, the button solutions provided by the social networks already transmits personal data to the respective social network, when a user visits a website in which a social media button was integrated. By using the Shariff component, personal data is only transferred to social networks, when the visitor actively activates one of the social media buttons. Further information on the Shariff component may be found in the computer magazine c't under
http://www.heise.de/newsticker/meldung/Datenschutz-und-Social-Media-Der-ct-Shariff-ist-im-Usatz-2470103.
Html provided. The use of the Shariff component is intended to protect the personal data of the visitors of our website and to enable us to integrate a button solution for social networks on this website.
Further information and the applicable data protection provisions of GitHub are retrievable under
https://help.github.com/articles/github-privacy-policy/.
29. Data protection provisions about the application and use of Tumblr
On this website, the controller has integrated components of Tumblr. Tumblr is a platform that allows users to create and run a blog. A blog is a web-based, generally publicly-accessible portal on which one or more people called bloggers or web bloggers may post articles or write down thoughts in so-called blogposts. For example, in a Tumblr blog the user can publish text, images, links, and videos, and spread them in the digital space. Furthermore, Tumblr users may import content from other websites into their own blog.
The operating company of Tumblr is Tumblr, Inc., 35 East 21st St, Ground Floor, New York, NY 10010, UNITED STATES.
Through each call to one of the individual pages of this Internet site, which is operated by the controller and on which a Tumblr component (Tumblr button) has been integrated, the Internet browser on the information technology system of the data subject causes automatically the download of a display of the corresponding Tumblr component of Tumblr. Learn more about the Tumblr-buttons that are available under
https://www.tumblr.com/buttons.
During the course of this technical procedure, Tumblr becomes aware of what concrete sub-page of our website was visited by the data subject. The purpose of the integration of the Tumblr component is a retransmission of the contents of this website to allow our users to introduce this web page to the digital world and to increase our visitor numbers.
If the data subject is logged in at Tumblr, Tumblr detects with every call-up to our website by the data subject—and for the entire duration of their stay on our Internet site—which specific sub-page of our Internet page was visited by the data subject. This information is collected through the Tumblr component and associated with the respective Tumblr account of the data subject. If the data subject clicks on one of the Tumblr buttons, integrated on our website, then Tumblr assigns this information to the personal Tumblr user account of the data subject and stores the personal data.
Tumblr receives information via the Tumblr component that the data subject has visited our website, provided that the data subject is logged in at Tumblr at the time of the call-up to our website. This occurs regardless of whether the person clicks on the Tumblr component or not. If such a transfer of information to Tumblr is not desirable for the data subject, then he or she may prevent this by logging off from their Tumblr account before a call-up to our website is made.
The applicable data protection provisions of Tumblr may be accessed under
https://www.tumblr.com/policy/en/privacy.
On this website, the controller has integrated components of Twitter. Twitter is a multilingual, publicly-accessible microblogging service on which users may publish and spread so-called 'tweets,' e.g. short messages, which are limited to 280 characters. These short messages are available for everyone, including those who are not logged on to Twitter. The tweets are also displayed to so-called followers of the respective user. Followers are other Twitter users who follow a user's tweets. Furthermore, Twitter allows you to address a wide audience via hashtags, links or retweets.
With each call-up to one of the individual pages of this Internet site, which is operated by the controller and on which a Twitter component (Twitter button) was integrated, the Internet browser on the information technology system of the data subject is automatically prompted to download a display of the corresponding Twitter component of Twitter. Further information about the Twitter buttons is available under
https://about.twitter.com/de/resources/buttons.
During the course of this technical procedure, Twitter gains knowledge of what specific sub-page of our website was visited by the data subject. The purpose of the integration of the Twitter component is a retransmission of the contents of this website to allow our users to introduce this web page to the digital world and increase our visitor numbers.
The applicable data protection provisions of Twitter may be accessed under
https://twitter.com/privacy?lang=en.
31. Data protection provisions about the application and use of Xing
On this website, the controller has integrated components of XING. XING is an Internet-based social network that enables users to connect with existing business contacts and to create new business contacts. The individual users can create a personal profile of themselves at XING. Companies may, e.g. create company profiles or publish jobs on XING.
The operating company of XING is XING SE, Dammtorstraße 30, 20354 Hamburg, Germany.
With each call-up to one of the individual pages of this Internet site, which is operated by the controller and on which a XING component (XING plug-in) was integrated, the Internet browser on the information technology system of the data subject is automatically prompted to download a display of the corresponding XING component of XING. Further information about the XING plug-in the may be accessed under
https://dev.xing.com/plugins.
During the course of this technical procedure, XING gains knowledge of what specific sub-page of our website was visited by the data subject.
If the data subject is logged in at the same time on XING, XING detects with every call-up to our website by the data subject—and for the entire duration of their stay on our Internet site—which specific sub-page of our Internet page was visited by the data subject. This information is collected through the XING component and associated with the respective XING account of the data subject. If the data subject clicks on the XING button integrated on our Internet site, e.g. the "Share"-button, then XING assigns this information to the personal XING user account of the data subject and stores the personal data.
XING receives information via the XING component that the data subject has visited our website, provided that the data subject is logged in at XING at the time of the call to our website. This occurs regardless of whether the person clicks on the XING component or not. If such a transmission of information to XING is not desirable for the data subject, then he or she can prevent this by logging off from their XING account before a call-up to our website is made.
The data protection provisions published by XING, which is available under
https://www.xing.com/privacy,
provide information on the collection, processing and use of personal data by XING. In addition, XING has published privacy notices for the XING share button under
https://www.xing.com/app/share?op=data_protection.
With each call-up to one of the individual pages of this Internet site, which is operated by the controller and on which a YouTube component (YouTube video) was integrated, the Internet browser on the information technology system of the data subject is automatically prompted to download a display of the corresponding YouTube component. Further information about YouTube may be obtained under
https://www.youtube.com/yt/about/en/.
During the course of this technical procedure, YouTube and Google gain knowledge of what specific sub-page of our website was visited by the data subject.
YouTube's data protection provisions, available at
https://www.google.com/intl/en/policies/privacy/,
provide information about the collection, processing and use of personal data by YouTube and Google.
33. Payment Method: Data protection provisions about the use of Klarna as a payment processor
On this website, the controller has integrated Klarna components. Klarna is an online payment service provider, which allows purchases on an account or a flexible installment payment. Klarna also offers other services, such as buyer protection and identity or creditworthiness checks.
The operating company of Klarna is Klarna AB, Sveavägen 46, 111 34 Stockholm, Sweden.
If the data subject selects the "purchase on account" or "installment purchase" during the ordering process in our online shop as a payment option, the data of the data subject is automatically transmitted to Klarna. By selecting one of these payment options, the data subject agrees to this transmission of personal data required for the processing of the invoice or installment purchase, or identity and creditworthiness checks.
The personal data transmitted to Klarna is usually first name, surname, address, date of birth, sex, email address, IP address, telephone number, mobile phone number, as well as other data necessary for the processing of an invoice or installment purchase. The processing of the purchase contract also requires such personal data, which are in connection with the respective order. In particular, the exchange of payment information such as bank details, card number, date of validity and CVC code, cumulative number, item number, data on goods and services, prices and taxes, information on the previous purchase behavior or other details of the financial situation of the data subject.
The purpose of the transmission of the data is, in particular, the identification check, payment administration, andfraud prevention. The controller shall provide Klarna with personal data, in particular, if a legitimate interest in the transmission exists. The personal data exchanged between Klarna and the data subject for the data processing shall be transmitted by Klarna to economic agencies. This transmission is intended for identity and creditworthiness checks.
Klarna shall also pass on the personal data to affiliates (Klarna Group) and service providers or subcontractors as far as this is necessary to fulfill contractual obligations or to process the data in the order.
Klarna collects and uses data and information on the previous payment behavior of the data subject as well as probability values for their behavior in the future (so-called scoring) in order to decide on the reasoning, implementation or termination of a contractual relationship. The calculation of scoring is carried out on the basis of scientifically-recognized mathematical-statistical methods.
The data subject is able to revoke the consent to the handling of personal data at any time from Klarna. A revocation shall not have any effect on personal data which must be processed, used or transmitted in accordance with (contractual) payment processing.
The applicable data protection provisions of Klarna may be retrieved under
https://cdn.klarna.com/1.0/shared/content/policy/data/de_de/data_protection.pdf.
If the data subject chooses "PayPal" as the payment option in the online shop during the ordering process, we automatically transmit the data of the data subject to PayPal. By selecting this payment option, the data subject agrees to the transfer of personal data required for payment processing.
The applicable data protection provisions of PayPal may be retrieved under
https://www.paypal.com/us/webapps/mpp/ua/privacy-full.
35. Payment Method: Data protection provisions about the use of Sofortüberweisung as a payment processor
On this website, the controller has integrated components of Sofortüberweisung. Sofortüberweisung is a payment service that allows cashless payment of products and services on the Internet. Sofortüberweisung is a technical procedure by which the online dealer immediately receives a payment confirmation. This enables a trader to deliver goods, services or downloads to the customer immediately after ordering.
The operating company of Sofortüberweisung is SOFORT GmbH, Fußbergstraße 1, 82131 Gauting, Germany.
If the data subject chooses "immediate transfer" as the payment option in our online shop during the ordering process, the data of the data subject will be transmitted to Sofortüberweisung. By selecting this payment option, the data subject agrees to the transmission of personal data required for payment processing.
In the case of purchase processing via direct transfer, the buyer sends the PIN and the TAN to Sofort GmbH. Sofortüberweisung then carries out a transfer to the online merchant after technical verification of the account status and retrieval of additional data to check the account assignment. The online trader is then automatically informed of the execution of the financial transaction.
The personal data exchanged with Sofortüberweisung is the first name, last name, address, email address, IP address, telephone number, mobile phone number, or other data necessary for payment processing. The transmission of the data is aimed at payment processing and fraud prevention. The controller shall immediately transfer other personal data, even if a legitimate interest in the transmission exists. The personal data exchanged between Sofortüberweisung and the controller shall be transmitted by Sofortüberweisung to economic credit agencies. This transmission is intended for identity and creditworthiness checks.
Sofortüberweisung provides personal data to affiliated companies and service providers or subcontractors as far as this is necessary for the fulfillment of contractual obligations or data in order to be processed.
The data subject has the possibility to revoke the consent to the handling of personal data at any time from Sofortüberweisung. A revocation shall not have any effect on personal data which must be processed, used or transmitted in accordance with (contractual) payment processing.
The applicable data protection provisions of Sofortüberweisung may be retrieved under
https://www.sofort.com/eng-DE/datenschutzerklaerung-sofort-gmbh/.
This Privacy Policy has been generated by the Privacy Policy Generator of the External Data Protection Officers that was developed in cooperation with the Media Law Lawyers from WBS-LAW.
© 2018 Julius Frack | Germany | Stuttgart || Impressum || AGB || DATENSCHUTZ || PRIVACY POLICY || Webdesign: MC


text:
\section{Introduction}
A reliable and quantitative prediction of chemically relevant compositions (CRCs) where stable crystals are formed is highly demanded because the synthesis of new inorganic compounds is an important target of physicists, chemists and material scientists.
Density functional theory (DFT) calculations have played a central role in such predictions.
However, exhaustive DFT calculations without prior knowledge of the crystal structures are expensive even for a given composition.
Recent developments and the popularization of machine learning (ML) have facilitated the prediction of currently unknown CRCs.
For example, a ML model with respect to only the composition, which was estimated from a DFT database of the formation energies, predicts currently unknown CRCs\cite{PhysRevB.93.085142}.

Another ML-based approach to search for currently unknown CRCs uses inorganic crystal structure databases such as the Inorganic Crystal Structure Database (ICSD)\cite{bergerhoff1987crystal}.
This approach estimates the CRC probability for a candidate composition using a given compositional similarity.
A procedure to estimate the CRC probability uses the compositional similarity on the basis of the entries in the database itself\cite{hautier2010finding,hautier2010data}.
A different procedure adopts the compositional descriptors defined by a set of well-known elemental representations\cite{PhysRevB.95.144110} as prior knowledge\cite{seko-recommender1}.
Compositional descriptors are useful to avoid the so-called ``cold-start'' problem, which occurs when few known CRCs are available\cite{seko-recommender1}.
The cold-start problem prevents the prediction of CRCs due to the lack of known CRCs.

These approaches based on inorganic crystal structure databases can be referred to as ``recommender systems'' for the discovery of currently unknown CRCs.
Recommender systems\cite{resnick1997recommender,aggarwal2016recommender} developed in the ML community have become increasingly popular in a variety of scientific and non-scientific areas.
In the field of commerce, recommender systems suggest items to a user.
Such a recommender system predicts the rating or preference for an item from an existing dataset comprised of the users' history such as items purchased and numerical ratings given to items to generate a recommendation for the user.

Simple matrix and tensor-based recommender system approaches can be adopted to discover currently unknown CRCs.
These approaches show a robust performance of recommendations for a wide variety of datasets in the ML community\cite{sarwar2000application,frolov2017tensor,symeonidis2016matrix}.
In this study, we build recommender systems to discover currently unknown CRCs using matrix and tensor-based approaches, which are descriptor-free approaches.
We employ different approaches: non-negative matrix factorization (NMF), singular value decomposition (SVD), canonical polyadic (CP) decomposition, and Tucker decomposition.
NMF and SVD are matrix-based recommender systems, while CP and Tucker decompositions are tensor-based recommender systems.
Their performances to discover currently unknown CRCs are compared.
Then using a list of compositions recommended by the best recommender system, we predict the phase stability of compounds using DFT calculations, and examine the performance of discovering currently unknown CRCs using the predicted phase stability.

\section{Methodology}

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.8\linewidth,clip]{Fig-MatrixFactorization.pdf}
\caption{
Schematic illustration of the approximation of the rating matrix by (a) NMF and (b) SVD.
Dimensions of factorized matrices are also shown.
}
\label{recommend:Matrix_decomposition}
\end{center}
\end{figure}

\subsection{Matrix factorization}

A historical dataset used to build a recommender system is simply described as a form of an incomplete matrix or tensor with missing values, called a ``rating matrix'' or a ``rating tensor''.
The underlying assumption of many recommender system approaches is that the complete rating matrix or tensor has a low-ranked structure.
Using this assumption, the ratings of missing elements are predicted as an approximated rating matrix or tensor with a given reduced rank.

The nonnegative matrix factorization (NMF) factorizes nonnegative rating matrix $\bm{X}$ into two nonnegative matrices $\bm{W}$ and $\bm{H}$ with given rank $r$.
$\bm{W}$ and $\bm{H}$ do not have negative elements.
Figure \ref{recommend:Matrix_decomposition} (a) illustrates the behavior of NMF.
The original $(n, m)$-dimensional matrix $\bm{X}$ is approximated as
\begin{equation}
\bm{X} \simeq \bm{W}\bm{H}
\end{equation}
using $(n, r)$-dimensional matrix $\bm{W}$ and $(r, m)$-dimensional matrix $\bm{H}$ with a given rank $r$.
Each of the $r$ dimensions corresponds to a distinctive rating trend.

Another matrix factorization approach is the singular value decomposition (SVD).
Figure \ref{recommend:Matrix_decomposition} (b) illustrates the behavior of SVD.
SVD factorizes original rating matrix $\bm{X}$ into three matrices.
The rating matrix is approximated as
\begin{equation}
\bm{X} \simeq \bm{U}\bm{D}\bm{V}^\top
\end{equation}
using $(n, r)$-dimensional matrix $\bm{U}$, $r$-dimensional diagonal matrix $\bm{D}$ and $(m,r)$-dimensional matrix $\bm{V}$.
The diagonal matrix contains only $r$ largest singular values.
Hence, each of the $r$ dimensions corresponds to a distinctive rating trend as well as the NMF.
The approximated rating matrix is obtained using the algorithms implemented in \textsc {scikit-learn}\cite{scikit-learn}.

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.9\linewidth,clip]{Fig-Tensor.pdf}
\caption{
Schematic illustration of (a) CP and (b) Tucker decomposition.
}
\label{recommend:Tensor_decomposition}
\end{center}
\end{figure}

\subsection{Tensor factorization}
Matrix factorization approaches require that the original data is transformed into a matrix form when the rating depends on three or more factors.
On the other hand, tensor factorization approaches can include information of all factors naturally as a form of the rating tensor.
A tensor factorization technique is the tensor rank decomposition or canonical polyadic (CP) decomposition\cite{hitchcock1927expression}.
Figure \ref{recommend:Tensor_decomposition} (a) illustrates the behavior of CP decomposition.
CP decomposition factorizes rating tensor $\mathcal{X}$ into a set of matrices and a super diagonal core tensor.
Third-order rating tensor $\mathcal{X}$ is approximated using three matrices $\bm{A}^{(1)}$, $\bm{A}^{(2)}$, and $\bm{A}^{(3)}$ as
\begin{equation}
\mathcal{X} \simeq \mathcal{G} \times_1 \bm{A}^{(1)} \times_2 \bm{A}^{(2)} \times_3 \bm{A}^{(3)},
\label{recommend:eqn-tensor}
\end{equation}
where $\mathcal{G}$ denotes the core tensor.

A more flexible tensor factorization is the Tucker decomposition\cite{tucker1966some}, which is also known as higher-order SVD (HOSVD)\cite{de2000multilinear}.
The Tucker decomposition is an extension of SVD, and decomposes rating tensor $\mathcal{X}$ into a set of matrices and a small dense core tensor.
In the Tucker decomposition, $\bm{A}^{(1)}$, $\bm{A}^{(2)}$, and $\bm{A}^{(3)}$ contain the orthonormal vector called mode-1, mode-2 and mode-3 singular vectors, respectively.
Figure \ref{recommend:Tensor_decomposition} (b) illustrates the behavior of Tucker decomposition.
Third-order tensor $\mathcal{X}$ can also be approximated as Eqn. (\ref{recommend:eqn-tensor}), where $\mathcal{G}$ is also the core tensor, but unlike in CP decomposition, the core tensor is not a super diagonal tensor.
In Tucker decomposition, the rank value can be independently given for each mode of the rating tensor.
Herein we adopted \textsc{scikit-tensor}\cite{scikit-tensor} package to use tensor factorization methods.

\section{Datasets}
\label{section:datasets}
We employ three inorganic crystal structure databases of the ICSD, Powder Diffraction File (PDF) by the International Centre for Diffraction Data (ICDD)\cite{ICDD} and SpringerMaterials (SpMat)\cite{SpringerMaterials}.
Only ICSD entries, including entries reported to show a partial occupancy behavior, are regarded as known CRCs.
Both ICDD and SpMat entries, which are not included in the ICSD, are used to validate the recommender systems.

\begin{table}[tbp]
\caption{
Elements included in the datasets of known CRCs.
Rare-earth elements shown in parentheses are excluded in the candidate quaternary and quinary compositions.
}
\label{recommend:elements1}
\begin{ruledtabular}
\begin{tabular}{ll}
Cations & Li,Na,K,Rb,Cs,Be,Mg,Ca,Sr,Ba,Zn,Cd,Hg, \\
        & B,Al,Sc,Y,La,Ga,In,Tl,Si,Ge,Sn,Pb,P,As,Sb,Bi \\
        & Ti,Zr,Hf,V,Nb,Ta,Cr,Mo,W,Mn,Tc,Re, \\
        & Fe,Ru,Os,Co,Rh,Ir,Ni,Pd,Pt,Cu,Ag,Au, \\
        & (Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm,Yb,Lu) \\
Anions  & C,N,O,S,Se,Te,F,Cl,Br,I \\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{table}[tbp]
\caption{
Numbers of ICSD, ICDD and SpMat entries.
Numbers of candidate compositions are also shown.
ICSD entries are regarded as known CRCs.
In the rows of ICDD and SpMat databases, the numbers of entries that are not included in the ICSD are shown.
Numbers in the parentheses indicate the numbers of all entries.
Test dataset is a combined dataset of ICDD and SpMat entries.
Only in Sec. \ref{section:dft}, a combined dataset of ICSD, ICDD, and SpMat entries is used as the training data.
}
\label{recommend:n_compositions}
\begin{ruledtabular}
\begin{tabular}{cccc}
                                & Ternary & Quaternary & Quinary \\
\hline
ICSD                            & 9,313    & 7,742    & 1,321 \\
\multirow{2}{*}{ICDD}           & 2,369    & 2,647    & 639 \\
                                & (9,278)  & (7,864)  & (1,326) \\
\multirow{2}{*}{SpMat}          & 2,708    & 3,066    & 1,169 \\
                                & (10,461) & (8,141)  & (1,893) \\
\multirow{2}{*}{ICDD$+$SpMat}   & 4,134    & 4,961    & 1,616 \\
                                & (12,573) & (11,307) & (2,562) \\
\hline
Candidates   & 7,405,200 & 1,188,038,460 & 23,104,706,560
\end{tabular}
\end{ruledtabular}
\end{table}

To build a recommender system for ternary CRCs, only a set of ternary known CRCs is employed.
Candidate ternary compositions of A$_a$B$_b$X$_x$ are generated, where elements A and B correspond to a cation (66 possible) and element X corresponds to an anion (10 (10 possible).
The possible cations and anions correspond to elements included in the dataset of known CRCs (Table \ref{recommend:elements1}).
One hundred seventy (170) integer sets $(a,b,x)$ that satisfy the condition of $\max (a,b,x) \leq 8$ and are included in the dataset of known CRCs are also considered.
Therefore, there are $66^2 \times 10 \times 170 = 7,405,200$ ternary compositions.
Of these, 9,313 CRCs are known.
The combined database of ICDD and SpMat has 4,134 ternary compositions that are not included in ICSD.
This corresponds to 0.056\% of the candidate compositions.
Table \ref{recommend:n_compositions} summarizes the numbers of database entries and candidate compositions.

In addition to ternary compositions, candidate quaternary A$_a$B$_b$C$_c$X$_x$ compositions are generated, where elements A, B and C denote a cation and element X denotes an anion.
Integer sets $(a,b,c,x)$ satisfy $\max (a,b,c,x) \leq 20$.
The dataset of known quaternary CRCs includes 53 cations, 10 anions, and 798 integer sets.
Therefore the number of quaternary compositions is $53^3 \times 10 \times 798 = 1,188,038,460$, which includes 7,742 known CRCs.
The combined database of ICDD and SpMat includes 4,961 quaternary compositions ($4.2 \times 10^{-4}$\% of the candidate compositions) that are not the entries of ICSD.

Candidate quinary A$_a$B$_b$C$_c$D$_d$X$_x$ compositions are also generated, where elements A, B, C and D denote cations and element X denotes an anion.
Integer sets $(a,b,c,d,x)$ satisfy $\max (a,b,c,d,x) \leq 20$.
The dataset of known quinary CRCs includes 52 cations, 10 anions, 316 integer sets.
Therefore, the number of quaternary compositions is $52^4 \times 10 \times 316 = 23,104,706,560$, where only 1,321 known CRCs are included.
The combined database of ICDD and SpMat contains 1,616 quinary compositions ($7.0 \times 10^{-6}$\% of the candidate compositions) that are not ICSD entries.

\section{Matrix-based recommender system}
\subsection{Rating matrix representations}
The performance of a recommender system strongly depends on the representation of the composition dataset as a rating matrix or tensor.
Experts' knowledge is required to introduce a good representation.
In particular, a composition dataset must be transformed into only two sets of features, which corresponds to users and items in a user-item rating matrix.
The ratings of missing elements are approximately predicted on the basis of feature similarity given by the representation.

We introduce three kinds of matrix representations for the ternary composition dataset.
A composition is decomposed into two feature sets in the following three ways.
\begin{enumerate}
\item \{A\} and \{B, X, $(a,b,x)$\}
\item \{A, X\} and \{B, $(a,b,x)$\}
\item \{A, B\} and \{X, $(a,b,x)$\}
\end{enumerate}
Each of feature sets corresponds to the row or column of the rating matrix.
For example, the first matrix representation is schematically illustrated in Fig. \ref{recommend:MatrixRep}.
In the first representation, the row corresponds to cation A.
The column corresponds to the combination of cation B, anion X, and integer set $(a,b,x)$.
Each composition is expressed by the collection of features included in the row and column.
Only the values of the rating matrix elements corresponding to known CRCs are set to unity.
The other compositions are classified as either currently unknown CRCs or non-existent compositions, and their values are set to zero in the original rating matrix.

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.9\linewidth,clip]{Fig-MatrixRep.pdf}
\caption{
Schematic illustration of a matrix representation for a composition dataset.
}
\label{recommend:MatrixRep}
\end{center}
\end{figure}

\subsection{Discovery performance of unknown CRCs}
The performance of matrix factorization based the recommender systems to discover currently unknown ternary CRCs are evaluated.
Only 4,134 of 7,405,200 compositions (0.056\%) are used as the test dataset of known CRCs (test CRCs).
The test CRCs correspond to ICDD and SpMat entries that are not included in ICSD.
The low percentage obviously confirms that discovering ICDD and SpMat entries by random sampling is not efficient.

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=\linewidth,clip]{Fig-ResultMF.pdf}
\caption{
Numbers of test CRCs included in the top 100 and 3000 compositions of NMF recommender systems for (a) type-1, (b) type-2 and (c) type-3 matrix representations.
Numbers of test CRCs included in the top 100 and 3000 compositions of SVD recommender systems for (d) type-1, (e) type-2 and (f) type-3 matrix representations.
Horizontal axis indicates a given rank for NMF and SVD.
Orange and blue bars denote the top 100 and 300 compositions, respectively.
}
\label{recommend:ResultMF}
\end{center}
\end{figure}

Figure \ref{recommend:ResultMF} shows the rank dependence of the number of test CRCs found in the top 3000 compositions of NMF and SVD recommender systems.
The rank dependences of NMF and SVD are qualitatively similar but weak.
They also exhibit similar matrix representation dependences.
The performance of discovering test CRCs of type-1 and type-2 matrix representations is much better than that of type-3 matrix representations.
This implies that it is important to consider the cation and anion combination as a feature when building a rule to distinguish CRCs and non-existent compositions.

When adopting type-1 and type-2 matrix representations,
NMF recommender systems with type-1 and type-2 matrix representations show the best performance at $r = 38$ and $r = 24$, respectively, indicating that the maximum number of test CRCs are included in the top 3000 compositions of the recommender system.
For type-1 and type-2 representations, 576 and 594 test CRCs (19.2 and 19.8\%) are found in the top 3000 compositions, respectively.
These percentages are much larger than the number of test CRCs discovered by random sampling (0.056\%).
A much larger discovery rate of test CRCs is seen in the sampling of top 100 compositions.
The numbers of test CRCs are 44 and 41 (44.0 and 41.0\%), respectively.
SVD recommender systems with type-1 and type-2 matrix representations show the best performance at $r=12$ and $r=40$, respectively.
The top 3000 compositions include 590 and 640 test CRCs (19.7 and 21.3\%), respectively.
Additionally, 53 and 45 of the top 100 compositions (53.0 and 45.0\%) correspond to test CRCs.
These results indicate that SVD recommender systems perform slightly better than NMF recommender systems.

\section{Tensor decomposition-based recommender system}
\subsection{Rating tensor representation}

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.7\linewidth,clip]{Fig-TensorRep.pdf}
\caption{
Schematic illustration of the tensor representation of a composition dataset.
As a simple example, a tensor representation for binary compositions is illustrated.
In a tensor element highlighted by the orange block, modes for cation type, anion type and integer set indicate Ti, O and $(3,4)$, respectively.
Hence, this tensor element represents the composition of Ti$_3$O$_4$.
}
\label{recommend:TensorRep}
\end{center}
\end{figure}

Figure \ref{recommend:TensorRep} illustrates the tensor representation used in this study.
A mode of the rating tensor means each cation type A, cation type B, anion type X and integer set $\{a, b, x\}$ for ternary compositions A$_a$B$_b$X$_x$.
Therefore, the rating tensor for ternary compositions is a (66, 66, 10, 198)-dimensional tensor.
Similar to the matrix representations, the composition is expressed by the collections of features included in all modes, corresponding to an element of the rating tensor.
Only the values of the rating tensor elements corresponding to known CRCs are set to unity.
The values of tensor elements corresponding to the other compositions are assumed to be zero in the original rating tensor.
By representing quaternary and quinary compositions in tensor forms as well as ternary compositions, the numbers of modes of the rating tensor correspond to five and six for quaternary and quinary compositions, respectively.

\subsection{Discovery performance of unknown CRCs}

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=\linewidth,clip]{Fig-ResultTensor.pdf}
\caption{
Rank dependence of the number of test CRCs included in the top 100 and 3000 compositions of the CP decomposition recommender systems.
Rank dependence of the number of test CRCs included in the top 100 and 3000 compositions of the Tucker decomposition recommender systems.
Here only the performance of Tucker decomposition with a rank value of 30 for the composition integer set is shown.
Orange and blue bars show the top 100 and 3000 compositions, respectively.
}
\label{recommend:ResultTensor}
\end{center}
\end{figure}

Figure \ref{recommend:ResultTensor} shows the number of test CRCs included in the top 3000 ternary compositions of the ranking by CP decomposition.
The horizontal axis indicates a given rank for CP decomposition.
Since only a single value of rank is given in CP decomposition, the maximum value of rank is identical to the number of anion types included in the ternary composition dataset.
The CP recommender system with $r=9$ shows the best performance.
That is, 478 of the top 3000 compositions (15.9\%) correspond to test CRCs.
In the top 100 compositions, 32 test CRCs (32.0\%) are found.
However, CP decompositions detect fewer test CRCs than the best matrix decomposition-based recommender system.
This is attributed to the fact that only a single value of rank is given in CP decomposition.

Figure \ref{recommend:ResultTensor} shows the number of test CRCs included in the top 3000 compositions of the ranking by Tucker decomposition.
The candidate values of ranks for the cation mode, anion mode, and integer set mode are given by arithmetic sequences of \{5, 10, 15, 20, 25\}, \{4, 8\} and \{10, 20, 30, $\dots$, 100\}, respectively.
The optimal set of ranks is obtained by a grid search using these candidates.
This means that the performance of discovering test CRCs is examined for 100 combinations of rank values.
The Tucker decomposition recommender system with ranks of (15, 15, 8, 30) shows the best performance; 735 test CRCs (24.5\%) are found in the top 3000 compositions.
In the top 100 compositions, 59 CRCs (59.0\%) are found in the test dataset.
The Tucker recommender system detects more test CRCs than the best matrix-based recommender system.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\linewidth,clip]{Fig-ResultBestRecommend.pdf}
\caption{
Comparison of the dependence of the number of test CRCs as a function of the number of samples obtained from the rankings by the best NMF, best SVD, best CP decomposition, and best Tucker decomposition recommender systems.
Derivatives of these curves correspond to the discovery rate of the test CRCs.
}
\label{recommend:ResultBestRecommend}
\end{center}
\end{figure}

Figure \ref{recommend:ResultBestRecommend} compares the number of test CRCs found by the best NMF, best SVD, best CP decomposition and best Tucker decomposition recommender systems.
The optimal Tucker decomposition recommender system exhibits the best performance among the four systems.
This result is due to the features of Tucker decomposition.
Compared with matrix factorizations, Tucker decompositions naturally consider the low-rank data-structure of the rating tensor related to two or more modes, although only correlations between the two modes of a given representation are in matrix factorizations.
Multiple rank values can be given in a Tucker decomposition, whereas a CP decomposition has a single value for rank.
Therefore, Tucker decompositions allow a hidden low-rank data-structure included in the composition dataset to be extracted more flexibly than matrix factorizations and CP decompositions.

The Tucker decomposition should be the best approach to discover currently unknown CRCs.
Therefore, the performance to discover currently unknown quaternary and quinary CRCs using the Tucker decomposition is examined.
The performances for quaternary and quinary systems are particularly important features required for recommender systems because the numbers of candidate compositions and currently unknown CRCs should be exponentially huge.
As described in Sec. \ref{section:datasets}, only 7,742 and 4,961 of $ 1.2 \times 10^9$ (6.5 $\times 10^{-4}$ and 4.2 $\times 10^{-4}$\%) quaternary compositions are ICSD and ICDD+SpMat entries, respectively.
Even for quinary compositions, only 1,321 and 1,616 of $ 2.3 \times 10^{10}$ (5.7 $\times 10^{-6}$ and 7.0 $\times 10^{-6}$\%) compositions correspond to ICSD and ICDD+SpMat entries, respectively.

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=\linewidth,clip]{Fig-ResultTucker2.pdf}
\caption{
Dependence of the number of test quaternary and quinary CRCs on the number of samples by the Tucker decomposition recommender system up to (a) 100 and (b) 3000 samples.
Diagonal line in (a) indicates the performance of the ideal sampling (i.e. the number of test CRCs is identical to the number of samples).
(c) Dependence of the discovery rate of test CRCs on the predicted rating for ternary, quaternary and quinary compositions.
(d) Relationship between the predicted rating and the discovery rate of the test CRCs.
Horizontal axis indicates the median of bin for the predicted rating.
}
\label{recommend:ResultTucker2}
\end{center}
\end{figure}

Figures \ref{recommend:ResultTucker2} (a) and (b) show the performance to determine test ternary, quaternary and quinary CRCs of the Tucker recommender system.
The discovery rate gradually decreases as the number of samples all for ternary, quaternary and quinary compositions increases.
In the top 20 compositions, 16 and 15 compositions (80 and 75\%) correspond to test CRCs for ternary and quaternary compositions, respectively.
The discovery rate of quaternary test CRCs is higher than expected.
The rate is close to that of ternary test CRCs.
For quaternary compositions, 52 and 524 CRCs are found in the top 100 and 3000 compositions (52.0 and 17.5\%), respectively.
The discovery rates for the top 20, 100, and 3000 compositions are approximately 180,000, 120,000, and 40,000 times larger than the random sampling discovery rate, respectively.
Even for quinary compositions, seven test CRCs (35\%) are found in the top 20 compositions.
In addition, 14 and 82 CRCs are found in the top 100 and 3000 compositions (14.0 and 2.7\%), respectively.
Although the discovery rate of quinary test CRCs is lower than those of ternary and quaternary test CRCs, it is much larger than the random sampling.
In particular, the discovery rate for the top 100 compositions is astonishingly large despite the fact that few known quinary CRCs are included in the quinary composition dataset.

Figures \ref{recommend:ResultTucker2} (c) and (d) show the relationship between the predicted rating and the discovery rate of test CRCs.
Figure \ref{recommend:ResultTucker2} (c) shows the histogram of the discovery rate of the test CRCs obtained with a bin width of predicted rating of 0.1.
Figure \ref{recommend:ResultTucker2} (d) replots the relationship between the median of the bin and discovery rate from the histogram of the discovery rate.
For ternary, quaternary and quinary compositions, the discovery rate of test CRCs is almost proportional to the median of the bin for predicted rating, and slightly smaller than the predicted rating.
This is strong evidence that the predicted rating can be regarded as a figure of merit to identify currently unknown CRCs.
This may also indicate that the predicted rating is almost identical to the discovery rate of currently unknown CRCs because test CRCs can be regarded as a part of currently unknown CRCs.
In addition, quinary compositions with a predicted rating exceeding 0.7 are not observed.
Consequently, the discovery rate of test CRCs is smaller than that of the ternary and quaternary test CRCs.
This is mainly ascribed to the lack of quinary known CRCs.
The number of quinary known CRCs is inadequate to predict currently unknown quinary CRCs.

\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=\linewidth,clip]{Fig-dft1.pdf}
\caption{
Stable compounds on the convex hull of the formation energy computed by the DFT calculations for the 27 pseudo-binary systems containing the top 27 compositions with high predicted ratings.
Compositions with high predicted ratings are shown along with their predicted ratings as the values in parentheses.
Closed triangles and squares denote the stable compounds corresponding ICSD entries and ICDD+SpMat entries, respectively.
}
\label{recommend:dft1}
\end{center}
\end{figure*}

\begingroup
\squeezetable
\begin{table}[htbp]
\caption{
Compositions with a predicted rating of more than 0.2 included in pseudo-binary systems for the top 27 compositions with a high predicted rating.
Stability of the compositions by the DFT calculation is also shown.
}
\label{recommend:dft-table}
\begin{ruledtabular}
\begin{tabular}{ccc}
Composition & Predicted rating & Stability \\
\hline
RbInO$_2$       & 1.01 & $\bigcirc$ \\
Rb$_3$InO$_3$   & 0.64 & $\bigcirc$ \\
RbIn$_5$O$_8$   & 0.20 & $-$ \\
\hline
NaGaS$_2$       & 0.98 & $\bigcirc$ \\
NaGa$_5$S$_8$   & 0.21 & $-$ \\
\hline
KPbCl$_3$       & 0.97 & $-$ \\
\hline
Ca$_2$TiO$_4$   & 0.95 & $\bigcirc$ \\
CaTi$_3$O$_7$   & 0.29 & $-$ \\
Ca$_3$TiO$_5$   & 0.21 & $-$ \\
\hline
BaAs$_2$O$_6$   & 0.93 & $\bigcirc$ \\
\hline
CsZnCl$_3$      & 0.92 & $\bigcirc$ \\
CsZn$_2$Cl$_5$  & 0.25 & $\bigcirc$ \\
\hline
RbInF$_4$       & 0.91 & $\bigcirc$ \\
\hline
Rb$_3$SbO$_3$   & 0.91 & $\bigcirc$ \\
\hline
CsY$_2$F$_7$    & 0.87 & $\bigcirc$ \\
Cs$_2$YF$_5$    & 0.44 & $\bigcirc$ \\
CsYF$_4$        & 0.38 & $-$ \\
\hline
CsYS$_2$        & 0.87 & $\bigcirc$ \\
\hline
Ba$_2$Ga$_2$O$_5$ & 0.85 & $\bigcirc$ \\
BaGa$_4$O$_7$   & 0.55 & $-$ \\
\hline
RbZnCl$_3$      & 0.85 & $\bigcirc$ \\
RbZn$_2$Cl$_5$  & 0.23 & $-$ \\
\hline
AlInO$_3$       & 0.80 & $\bigcirc$ \\
\hline
RbAlS$_2$       & 0.79 & $\bigcirc$ \\
Rb$_3$AlS$_3$   & 0.43 & $\bigcirc$ \\
RbAl$_5$S$_8$   & 0.26 & $-$ \\
\hline
RbScS$_2$       & 0.79 & $\bigcirc$ \\
\hline
TlAlTe$_2$      & 0.79 & $\bigcirc$ \\
TlAl$_5$Te$_8$  & 0.23 & $-$ \\
\hline
K$_2$GeS$_3$    & 0.79 & $\bigcirc$ \\
K$_6$Ge$_2$S$_7$ & 0.25 & $-$ \\
\hline
BaSc$_2$S$_4$   & 0.79 & $\bigcirc$ \\
\hline
KZnCl$_3$       & 0.79 & $-$ \\
KZn$_2$Cl$_5$   & 0.21 & $-$ \\
\hline
CaBi$_2$S$_4$   & 0.78 & $-$ \\
Ca$_2$Bi$_2$S$_5$ & 0.28 & $-$ \\
\hline
CaGa$_2$Te$_4$  & 0.78 & $\bigcirc$ \\
\hline
K$_3$TiF$_7$    & 0.78 & $-$ \\
KTiF$_5$        & 0.26 & $\bigcirc$ \\
\hline
CsInF$_4$       & 0.77 & $\bigcirc$ \\
CsIn$_2$F$_7$   & 0.59 & $\bigcirc$ \\
Cs$_2$InF$_5$   & 0.49 & $\bigcirc$ \\
\hline
Rb$_3$PSe$_4$   & 0.76 & $\bigcirc$ \\
\hline
NaGaSe$_2$      & 0.76 & $\bigcirc$ \\
\hline
LiLaS$_2$       & 0.76 & $\bigcirc$ \\
LiLa$_5$S$_8$   & 0.29 & $-$ \\
\hline
MgAs$_2$O$_6$   & 0.74 & $\bigcirc$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
\endgroup

\section{Discovery of new compounds}
\label{section:dft}
Finally, the discovery of new ternary compounds is demonstrated using a combination of the Tucker recommender system and DFT calculations.
A Tucker decomposition recommender system is constructed using all of the available entries of the ternary compositions included in ICSD, ICDD and SpMat as known CRCs.
The rank values are the same values as those optimized above.
The existence of stable compounds at compositions with high predicted ratings is examined by evaluating the phase stability of the corresponding pseudo-binary systems using DFT calculations.
DFT calculations were performed only for pseudo-binary systems containing the top 27 compositions composed only of elements in Groups 1, 2, 3, 4, 5, 6, 12, 13, 14, 15, 16 and 17 in the periodic table.
For each pseudo-binary system, DFT calculations were performed for all possible prototype structures included in the ICSD.
Totally, the number of DFT calculations was 13,274.

All DFT calculations were performed using the plane-wave basis projector augmented wave (PAW) method\cite{PAW1,PAW2} within the Perdew--Burke--Ernzerhof exchange-correlation functional\cite{GGA:PBE96} as implemented in the \textsc{vasp} code\cite{VASP1,VASP2}.
The cutoff energy was set to 400 eV.
The total energy converged to less than 10$^{-3}$ meV.
The atomic positions and lattice constants were optimized until the residual forces became less than $10^{-2}$ eV/\AA.

Figure \ref{recommend:dft1} shows the stable compounds or CRCs obtained by the DFT calculations for the 27 pseudo-binary systems.
The predicted ratings of the top 27 compositions range from 0.74 to 1.01.
Among the 27 compositions, most (23 compositions) are located on the convex hull, meaning that they are CRCs according to the DFT calculations.
In other words, 85\% of the recommended compositions can be regarded as new CRCs, demonstrating that the predicted rating can be regarded as the discovery rate of new CRCs as described above.
Table \ref{recommend:dft-table} shows the compositions with predicted ratings greater than $0.2$ included in the 27 pseudo-binary systems and their stability obtained by the DFT calculations.
In addition to the top 27 compositions, some other compositions can be CRCs depending on the predicted rating.
In compositions with a predicted rating of 0.2--0.4, only two of the 14 compositions (14\%) correspond to CRCs.
On the other hand, four of the five compositions (80\%), 13 of the 16 compositions (81\%) and 11 of the 12 compositions (92\%) are CRCs with predicted ratings of 0.4--0.6, 0.6--0.8 and 0.8--1.0, respectively.
In other words, compounds with a predicted rating exceeding 0.4 have a high probability of being a CRC.

The remaining 15\% of recommended compositions have a high predicted rating, but do not correspond to the CRCs in the DFT calculations.
Plausible explanations include the following.
(1) A stable or metastable compound does not truly exist at the composition.
(2) A metastable compound can be found at the composition.
(3) A stable compound with a structure other than the prototype structures is observed.

In addition, stable compounds can be found at the other compositions with a high predicted rating.
The compositions are classified into two types.
The first corresponds to compositions omitted in the candidate dataset, which do not satisfy the condition of $\max(a, b, x) \leq 8$.
This type may be predicted with a high rating if a larger maximum value of the integer is considered.
The other corresponds to compositions with a low predicted rating.
They may be the true DFT stable compound when a low rating is predicted due to the lack of known CRCs similar to the corresponding composition.
On the other hand, they may be artificial stable compounds attributed to the use of only the prototype structures for the DFT calculations.

\section{Conclusion}
Herein we apply widely-used recommender system approaches based on NMF, SVD, CP decomposition, and Tucker decomposition to predict currently unknown CRCs.
The present study indicates that the performance of the recommender systems depends on not only the given rank corresponding to a low-rank structure hidden in the rating matrix and tensor but also on the representation of rating matrix and tensor.
Among the considered approaches, a Tucker decomposition recommender system shows the best discovery rate of CRCs.
For ternary and quaternary compositions, at least approximately 60 and 50 of the top 100 recommended compositions are CRCs, respectively.

We also demonstrate the discovery of new ternary compounds using a combination of a Tucker recommender system and DFT calculations.
Among the top 27 compositions whose predicted ratings ranging 0.74 to 1.01, 23 compositions correspond to CRCs.
This suggests that the discovery rate of currently unknown CRCs is surprisingly high (85\%).
Moreover, the predicted ratings reveal additional compositions that may also be CRCs.
The discovery rate of CRCs using DFT calculations is almost identical to the predicted rating of the recommender system.
Consequently, the predicted rating can be regarded as a quantitative figure of merit to determine currently unknown CRCs.
Employing this figure of merit may significantly accelerate the discovery of currently unknown CRCs.

\begin{acknowledgments}
This work was supported by PRESTO, JST, and a Grant-in-Aid for Scientific Research on Innovative Areas ``Nano Informatics" (Grant No. 25106005) from the Japan Society for the Promotion of Science (JSPS).
AS, HH and IT were supported by the ``Materials Research by Information Integration" Initiative (MI$^2$I) from the Japan Science and Technology Agency.
IT was supported by a Grant-in-Aid for Scientific Research (A) from JSPS.
\end{acknowledgments}

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